Jean-François Ritz

Owens Valley

Hope Faults

N Stewart

Y Gaudemer

I Manighetti
email: manighetti@geoazur.unice.fr

L Serreau

A Vincendeau

S Dominguez

L Mattéo

J Malavieille

Paléosismologie morphologique à partir de données LiDAR : développement et application d'un code de mesure des déplacements sur les failles, 3D_Fault_Offsets

Keywords: Paléoseismologie, LiDAR, Wairarapa, marqueurs tectono-géomoriques Fault offsets, Paleoseismology, LiDAR, geomorphic markers, Matlab code, Wairarapa fault, strike-slip faults, seismic hazard

The main scope of this PhD thesis is to utilize very high-resolution remote sensing LiDAR data to extract some of the tectono-geomorphic traces imprinted in the morphology from large prehistoric earthquakes. The information that is accessed in these traces is the cumulative slip history of successive large paleoearthquakes along a given fault. The slip history allows the determination of the number of events and the largest slips produced by those respective events. The knowledge of the largest slips produced by historic and prehistoric large earthquakes will enable some inference into the potential magnitude of future events. Characterizing the distribution of surface slip provides important insights into fault mechanics, controls on rupture propagation, and repeatability of rupture at certain points along the fault. However, properly characterizing and measuring the slip distribution from tectonically-displaced geomorphic landforms comes with considerable uncertainties mostly resulting from erosion and depositional processes. These uncertainties could lead to both underestimation and overestimation of the slip, and to conflicting results from different surveys of the same fault. Therefore, we have developed a new MATLAB-based technique, 3D_Fault_Offsets, to mathematically, and hence automatically, characterize the 3D geometry of offset geomorphic markers (defined by 9 geometric features either side of the fault), and then calculate the lateral and vertical components of slip. We believe that the uncertainties obtained from this technique better define the range of potential 'true' offsets compared to more liberal uncertainties offered in other studies, yet they reveal to be fairly large. Upon verification of the code efficacy by successfully re-measuring 3 paleoseismic datasets, we applied it to a strike-slip fault in New Zealand that was historically capable of a large stress drop earthquake (M W ~8.2 in 1855), the Wairarapa fault. We identified and analyzed a total of ~700 displaced geomorphic markers along a 70-km stretch of LiDAR data, making this one of the largest and densest paleoseismic datasets. Measured lateral offsets range from a few meters to about 800 m, but the majority are lower than 80 m, providing the means to examine the most recent lateral fault slips. The vertical offsets range between 0 and ~30 m, and suggest vertical to lateral slip ratios commonly in the range 10-20%. We conducted the statistical analyses of the dense collection of measured offsets separately along the successive major segments that form the investigated fault stretch. In most segments, this analysis revealed 6-7 offset clusters in the range 0-80 m, suggesting the Wairarapa fault ruptured in 6-7 previous large earthquakes. The largest slips we infer for these past earthquakes are large, most in the range 7-15 m. Each earthquake slip seems to vary along the fault length, and be generally greater in its southern part. The Wairarapa fault has thus repeatedly produced large stress drop earthquakes in prehistoric time, which emphasizes the elevated seismic hazard it poses in Southern New Zealand. Therefore, the use of our new code 3D_Fault_Offsets with high resolution topographic data such as LiDAR can lead to better assessments of future behavior of seismogenic faults.

Paléosismologie morphologique à partir de données LiDAR : développement et application d'un code de mesure des déplacements sur les failles, 3D_Fault_Offsets

Résumé L'objectif principal de cette thèse est de tirer de données LiDAR de télédétection à très haute résolution afin d'extraire une partie du traces tectono-géomorphiques imprimées dans la morphologie de grands tremblements de terre préhistoriques. Les informations consultées dans ces traces constituent l'historique des glissements cumulés de grands tremblements paléoséismique successifs le long d'une faille donnée. L'historique des glissements permet de déterminer le nombre d'événements et les glissements les plus importants produits par ces événements. La connaissance des plus grandes glissades produites par des grands séismes historiques et préhistoriques permettra de déduire l'ampleur potentielle des événements futurs. La caractérisation de la distribution du glissement superficiel fournit des informations importantes sur la mécanique des failles, les contrôles de la propagation de la rupture et la répétabilité de la rupture à certains points le long de la faille. Cependant, la caractérisation et la mesure correctes de la distribution des glissements à partir de formes de relief géomorphologiques déplacées par tectonisme sont accompagnées d'incertitudes considérables, résultant principalement de processus d'érosion et de dépôt. Ces incertitudes pourraient entraîner à la fois une sous-estimation et une surestimation du glissement, ainsi que des résultats contradictoires issus d'enquêtes différentes sur le même défaut. Par conséquent, nous avons développé une nouvelle technique basée sur MATLAB, 3D_Fault_Offsets, pour caractériser mathématiquement, et donc automatiquement, la géométrie 3D de marqueurs géomorphiques décalés (définie par 9 entités géométriques situées de part et d'autre de la faille), puis calculer composants latéraux et verticaux du glissement. Nous estimons que les incertitudes générées par cette technique définissent mieux la gamme des "véritables" compensations potentielles par rapport aux incertitudes plus libérales proposées dans d'autres études, pourtant ils se révèlent assez volumineux. Après vérification de l'efficacité du code en mesurant à nouveau 3 ensembles de données paléosismiques, nous avons L'appliqué à une faille de décrochement qui était historiquement capable d'un séisme de chute de contrainte importante (MW ~ 8,2 en 1855), la faille de Wairarapa. Nous avons identifié et analysé un total d'environ 700 marqueurs géomorphiques déplacés le long d'une zone de données LiDAR de 70 km, ce qui en fait l'un des ensembles de données paléosismiques les plus vastes et les plus denses. Les décalages latéraux mesurés vont de quelques mètres à environ 800 m, mais la majorité d'entre eux sont inférieurs à 80 m, ce qui permet d'examiner les plus récents glissements de faille latéraux. Les décalages verticaux varient entre 0 et ~ 30 m et suggèrent des rapports de glissement vertical / latéral généralement compris entre 10 et 20%. Nous avons effectué les analyses statistiques de la collection dense de décalages mesurés séparément le long des principaux segments successifs qui constituent l'étendue de la faille étudiée. Dans la plupart des segments, cette analyse a révélé la présence de 6 à 7 amas décalés dans la plage allant de 0 à 80 m, suggérant la rupture de la faille de Wairarapa lors de 6 à 7 grands séismes précédents. Les plus grandes glissades que nous déduisons pour ces tremblements de terre passés sont importantes, la plupart dans la plage 7-15 m. Chaque glissement sismique semble varier le long de la faille et généralement plus grand dans sa partie sud. La faille de Wairarapa a ainsi provoqué à plusieurs reprises d'importants séismes dus à la chute de contraintes au cours de la période préhistorique, ce qui souligne le risque sismique élevé qu'elle pose dans le sud de la Nouvelle-Zélande. Par conséquent, l'utilisation de notre nouveau code 3D_Fault_Offsets avec des données topographiques à haute résolution telles que LIDAR peut permettre de mieux évaluer le comportement futur des failles sismogènes.

1.a Introduction

One of the fundamental goals of this thesis work is to identify some of the large earthquake recurrence characteristics that might help probabilistic estimates of the potential behavior of forthcoming large earthquakes along a given fault. While of already distinct scientific interest, this type of work can improve assessments of seismic hazard, providing the means to mitigate the impending destruction associated with large earthquakes. In order to anticipate the behavior of these future large earthquakes, in terms of their location, magnitude, and relative timing, an extensive catalogue of the number and magnitude of previous large events is required. Historical records obtained using instrumentation yielding seismological, geodetic, etc. datasets are insufficient to properly characterize the full rupture history of a given fault, since the time period for human observation is severely less than the general recurrence intervals of large earthquakes on most seismogenic faults worldwide. In many countries such as the USA, useful seismicity records extend back only a few centuries (Gutenburg & Richter, 1954;[START_REF] Stover | Seismicity of the United States, 1568-1989[END_REF]) and many active fault zones have no historical record of large earthquakes. For faults that have slipped episodically for many hundreds and thousands of years, even a 3000-year earthquake history such as China's [START_REF] Ambraseys | A history of Persian Earthquakes[END_REF][START_REF] Gu | Catalogue of Chinese earthquakes, 1831 B.C. -1969 A.D[END_REF] covers only a tiny fraction of the history of the faults. An ideal historical earthquake record would a.) include descriptions that enable discerning the earthquake size, b.) associate the earthquake rupture with specific fault zones or segments, c.) give a relative timing per event, and d.) provide the aforementioned information consistently through time and over multiple earthquake cycles (Zielke, Klinger, & Arrowsmith, 2015). In most cases, however, only some of that information can be accessed, distinctly impeding the ability to reconstruct a fault's earthquake history using historical accounts (e.g. [START_REF] Nur | Historical seismicity: Archeoseismology[END_REF].

Paleoseismology and tectonic geomorphology provide some of the datasets that are needed to properly determine earthquake recurrence characteristics from the longest possible earthquake history along a respective fault. Much of the seismic history of a fault is accessible only through paleoseismic studies. Paleoseismology is the study of the location, size, and timing of prehistoric large (i.e., able to break the ground surface, thus with Mw generally ≥ ~6) earthquakes that employ features of the geologic and geomorphic record to deduce the lengths and displacements of ruptures on specific active faults decades, centuries, or millennia after their occurrence [START_REF] Yeats | The Oak Ridge fault system and the 1994 Northridge earthquake[END_REF][START_REF] Yeats | Introduction to special section: Paleoseismology[END_REF]NRC, 2003;[START_REF] Michetti | Future trends in paleoseismology: Integrated study of the seismic landscape as a vital tool in seismic hazard analyses[END_REF][START_REF] Mccalpin | Quaternary happenings in the Overthrust Belt, western Wyoming: 2008 FOP trip--Rocky Mountain Cell[END_REF]. One paleoseismic approach to characterizing a fault's long-term behavior is to analyze that fault's earthquake rupture history, assuming that this history is a reflection of likely future behavior [START_REF] Hutton | The system of the Earth, its duration and stability[END_REF]. Whereas seismologists decipher data recorded concurrent with earthquakes, paleoseismologists interpret geologic evidence created during individual paleoearthquakes in the prehistoric past. Classical paleoseismological studies have dated the stratigraphic record along the fault zone in order to obtain a timing for the individual paleoearthquakes and hence a recurrence interval for a particular fault. However, a major theme to this thesis is to better understand fault behavior during very large magnitude earthquakes, which can be approached with the number and magnitude of previous very large events.

Paleoseismological studies are made possible because large earthquakes generally propagate up through the earth's surface, breaking the surface, and leaving traces and imprints in the topographic geomorphology that are preserved for millennia or more. Each surfacerupturing earthquake event forms a range of coseismic features, usually in the form of laterally and/or vertically displaced geomorphic landforms (termed markers from this point forward) that have been offset, and subsequently record the amount of slip associated with single or multiple rupturing events. Most of the markers along the trace of the fault have alluvial origins, though other types of markers exist (moraines, rock bodies, etc.). Four prominent landforms that are typically used in the reconstruction of surface slip in paleoseismic studies are: fluvial terraces and their corresponding risers, stream channels and their corresponding risers, ridges, and alluvial fans (e.g., Sieh, 1978;[START_REF] Schwartz | Fault behavior and characteristic earthquakes: Examples from the Wasatch and San Andreas Fault zones[END_REF][START_REF] Sieh | Holocene activity of the San Andreas Fault at Wallace Creek, California[END_REF][START_REF] Sieh | The repetition of large-earthquake ruptures[END_REF][START_REF] Field | Uniform California earthquake rupture forecast, Version 2 (UCERF 2)[END_REF]Zielke, Arrowsmith, Ludwig, & Akciz, 2010;Akciz, Ludwig, Arrowsmith, & Zielke, 2010;Zielke Arrowsmith, Ludwig, & Akciz, 2012). Figure 1 shows an example of such features. If the markers are wellpreserved in the geomorphology surrounding the fault, they can be essential in the analysis of the slip distribution along the fault. ) pre-earthquake morphology to B.) the coseismic morphology after one earthquake event to (the MRE in B refers to the 'most recent earthquake') C.) the cumulative slip morphology after several consecutive earthquake events. The coseismic slip records the slip of one event; whereas, the cumulative slip is the record of multiple events. The cumulative slip is generally measured in topography/morphology, but the coseismic slip is assumed to be the smallest locally (i.e. per segment) measured displacement. Another major assumption is that the production rate of the markers is more frequent than the earthquake events that displace them (see text for explanation). MRE is most recent event.

Figure 2 illustrates for the special case of a strike-slip fault the evolution of the tectonogeomorphic marker from its pre-earthquake morphology to its morphology after it recorded the cumulative slip from several rupturing events. Optimal locations for measuring not only the lateral/vertical displacements of offset terraces, but also any geomorphic marker, is where the marker crosses the fault nearly perpendicularly. The geomorphic geometry shown in Fig. 2 is generally created where the strike-slip fault forms the boundary between an area of higher relief and a valley where a large tributary can enter from the area of higher relief. This geometry usually occurs when some oblique slip accompanies the dominant strike-slip motion of the fault. The oblique slip forces the block on one side of the fault to elevate, causing the tributary's direction of flow to be preferentially perpendicular to the fault (McCalpin, 2009). The stream marker depicted in Figure 2 that crosses the fault becomes laterally displaced after one surface rupturing earthquake, offsetting the northern component of the marker from its southern counterpart and redirecting the flow direction along the trace of the fault. The distance (generally in meters) between the northern and southern components of the now displaced stream will equate to the coseismic slip of one earthquake. In paleoseismology, it is assumed that after several subsequent earthquakes, the measurement recorded equates to the cumulative slip of those multiple events since it is impossible to determine if it occurred from other processes. Other process such as interseismic creep (i.e. aseismic activity) can attribute to the slip history from large earthquakes. It can be difficult to prove that the slip came solely from coseismic activity, but geodetic studies can be conducted to assess this at present times (Darby & Beaven, 2001). These geomorphic markers are then offset jointly, to varying degrees, generating groups of offsets along the fault where the smallest local offset is considered to be the coseismic slip from the most recent earthquake (Wallace, 1968;Sieh, 1978;[START_REF] Sieh | Holocene activity of the San Andreas Fault at Wallace Creek, California[END_REF]McGill & Sieh, 1991;Klinger, Etchebes, Tapponnier, & Narteau, 2011;Zielke et al., 2012;Beauprête et al., 2012Beauprête et al., , 2013;;[START_REF] Madden | Compilation of slip-in-the-last-event data and analysis of last event, repeated slip, and average displacement for recent and prehistoric ruptures[END_REF]Manighetti et al., 2015). It is important to note that slip, both coseismic (e.g., [START_REF] Manighetti | Evidence for self-similar, triangular slip distributions on earthquakes: Implications for earthquake and fault mechanics[END_REF] and cumulative (e.g., Manighetti et al., 2001) markedly vary along the fault length. Thus the smallest locally (i.e. per short fault section) measured offset is assumed to be the coseismic displacement from the most recent large earthquake, at the place of the measurement. Because of along-strike slip variation, it might not represent, however, the maximum nor the mean slip produced by the most recent earthquake. Plus, it is a surface measurement, and therefore a slip value generally smaller than the slip originally produced at crustal depth (see [START_REF] Manighetti | Evidence for self-similar, triangular slip distributions on earthquakes: Implications for earthquake and fault mechanics[END_REF]. The cumulative offset values are assumed to be generated solely by coseismic deformation, where inter-seismic and post-seismic deformation cannot be proven [START_REF] Thatcher | The earthquake cycle and its role in the long-term deformation on the continental lithosphere[END_REF]. Again, along-strike cumulative slip variability needs to be considered when interpreting the local slip values.

Utilizing the geomorphic record to constrain the rupture history along a fault is based on a fundamental assumption: the production rate of geomorphic markers that is used to measure displacement is assumed to be greater than the recurrence rate of surface-rupturing earthquakes. As a consequence of this assumption, many conspicuous geomorphic markers are formed jointly in between earthquake events. Therefore, the best fault sites for performing geomorphic paleoseismological studies are regions where (1) slip is localized in a narrow fault zone, (2) many geomorphic features (terraces, streams, ridges, etc.) are constantly being formed to provide ample possibilities for offset markers to be displaced over successive earthquakes, and (3) the fault frequently produces large earthquakes that significantly displace the existing markers. In this framework, strike-slip faults are particularly appropriate.

Measuring the offset distance of many markers along the fault trace will yield the alongfault slip distribution of the respective fault. Clusters of offset geomorphic markers along a certain localized fault section could indicate slip from the same event. However, clusters of offsets with varying mean offset values (identified along a localized section) would represent the cumulative slip amount from an initially indeterminate number of earthquakes. If this fundamental assumption is not realized, and the earthquakes occur more frequently than marker formation, some of the earthquakes will not be resolved in the geomorphic record leading to an overestimation of paleomagnitude of an individual event and an underestimation of the number of events.

Geomorphic evidence is used to measure the lateral and vertical displacements that can be attributed to multi-event and individual paleoearthquakes. Through scaling laws, these displacement measurements can be used to infer earthquake magnitude or seismic moment (see [START_REF] Wells | New empirical relationships among magnitude, rupture length, rupture width, rupture area, and surface displacement[END_REF][START_REF] Manighetti | Earthquake scaling, fault segmentation, and structural maturity[END_REF]. Thus, geomorphic evidence of surface rupture provides valuable information on how an earthquake is characterized including the rupture extent as well as the amount and distribution of fault slip. Geomorphic evidence is also used to ascertain whether these characteristics recur (i.e. whether portions of a fault that exhibited a certain amount of slip in one earthquake, exhibited similar slip in other earthquakes). The individual measurements can then be plotted in a graph like Figure 3 to reconstruct the rupture history. Since paleoseismology relies on the interpretation of geomorphic landforms imprinted in the surface topography surrounding faults, the studies are restricted to analyzing earthquakes capable of displacing these features that can be preserved for long intervals after the event. Therefore, the length of the paleoseismic record is predicated upon the delicate balance between tectonic and alluvial (i.e. erosional) dynamics. Generally, the paleoseismic record is restricted to large (moment magnitude, MW > ~6) or great (MW > 7.8) earthquakes because geologic evidence of small and moderately-sized earthquakes is rarely created or preserved near the surface [START_REF] Wells | New empirical relationships among magnitude, rupture length, rupture width, rupture area, and surface displacement[END_REF].

In this work, the analyses of the along-fault slip distribution will be restricted to strike-slip tectonic settings for several reasons. 1.) Strike-slip faults are often the longest faults on continental landmasses and typically have exposed geomorphic expression. 2.) Many of these faults have historically been acknowledged for experiencing surface ruptures during large and great historical earthquakes. 3.) Since the coseismic deformation along strike-slip faults is horizontal, subsequent earthquakes do not deeply bury, or expose to erosion, traces of earlier events (McCalpin, 2009). Also, the laterally displaced tectono-geomorphic markers are produced with less relief than dip-slip faults leaving them less susceptible to being destroyed through uplift, sedimentation, or erosional processes.

Surface deformation from paleoearthquakes can be assessed using a variety of techniques, most notably field inspection and shallow fault trenching, aerial reconnaissance, and active (digital) remote sensing methods. The first two methods are traditional (i.e. manual) analyses of actual field data, whereas remote sensing involves digital investigation. Field inspection studies (e.g. Wallace, 1968;Sieh, 1978;[START_REF] Sieh | Holocene activity of the San Andreas Fault at Wallace Creek, California[END_REF]Lienkaemper, 2001;[START_REF] Rockwell | Lateral offsets on surveyed cultural features resulting from the 1999 Izmit and Duzce Earthquakes, Turkey[END_REF][START_REF] Ritz | Late Pleistocene to Holocene slip rates for the Gurvan Bulag thrust fault (Gobi-Altay, Mongolia) estimated with10Be dates[END_REF][START_REF] Haeussler | Surface rupture and slip distribution of the Denali and Totschunda Faults in the 3 November 2002 M 7.9 Earthquake, Alaska[END_REF][START_REF] Nazari | Paleoearthquakes determination of magnitude~6.5 on the North Tehran Fault, Iran[END_REF][START_REF] Ritz | Paleoearthquakes of the past 30,000 years along the North Tehran Fault (Iran)[END_REF] are highly relevant, since the natural geomorphic expression can be directly observed. However, it is extremely time-consuming, so most studies only analyze a maximum of 10s of different deformed geomorphic features along any given fault. The small amount of analyzed features does not generally enable proper characterization of the entirety of faults that can stretch for hundreds to over 1000 kilometers. Through aerial reconnaissance (e.g. [START_REF] Rockwell | Lateral offsets on surveyed cultural features resulting from the 1999 Izmit and Duzce Earthquakes, Turkey[END_REF][START_REF] Haeussler | Surface rupture and slip distribution of the Denali and Totschunda Faults in the 3 November 2002 M 7.9 Earthquake, Alaska[END_REF], a more complete image of the fault zone can be obtained through stereoscopic photography. However, there may be zones along the fault covered by dense vegetation, in which the inspection of tectonic features would be impossible through photos.

Therefore, recent years have seen a revolution in the studies of paleoseismology and active tectonics through the incorporation of very-high resolution topographic data by using remote sensing technology such as Light Detection and Ranging (LiDAR) (e.g. McCalpin, 2009;Zielke et al., 2010Zielke et al., , 2012;;Klinger et al., 2011;Meigs, 2013;Manighetti et al., 2015;Haddon, Amos, Zielke, Jayko, & Bürgmann, 2016;Stewart et al., 2018). LiDAR is a type of terrestrial-, or airborne-based active remote sensing technique that transmits pulses of electromagnetic radiation in the visible part of the electromagnetic spectrum to very accurately determine the distance of remote objects, and hence topography. Figure 4 is a schematic representation of the airborne-based technique. During data acquisition, a laser scanner emits pulses of monochromatic light that are dispersed and reflected from an object or surface. Reflected pulses that return to the scanner are detected, recording the precise timing for the pulse to leave and return. The intensity of the returning pulse is recorded along with its time of flight. The latter is converted to a distance between scanner (source) and object (reflector), by taking the speed of light through the medium into consideration. This distance is then further converted to provide absolute geographic coordinates for the reflector using an onboard inertial navigation system (INS) for scanner orientation and GPS for scanner position. Notably, each outgoing laser pulse may generate multiple returning pulses that differ in time and intensity, for example due to the partial pulse reflection by vegetation cover. During post-processing, classification of returning pulses (e.g. by travel time or curvature-based filtering approaches) enables virtual removal of the vegetation coverage [START_REF] Harding | Fault scarp detection beneath dense vegetation cover-Airborne LiDAR mapping of the Seattle fault zone, Bainbridge Island, Washington State[END_REF][START_REF] Haugerud | High-resolution Lidar topography of the Puget Lowland, Washington: A Bonanza for Earth Science[END_REF]. The ability to generate high-resolution "bare-earth" (i.e. last return points) DEMs (Digital Elevation Models, that is, 3D topography), even for densely vegetated, mountainous, and other inaccessible regions, has made LiDAR extremely attractive for earth science studies and revolutionized geomorphic mapping (e.g. [START_REF] Barth | Scale dependence of oblique plate-boundary partitioning: New insights from LiDAR, central Alpine fault, New Zealand[END_REF][START_REF] Haugerud | High-resolution Lidar topography of the Puget Lowland, Washington: A Bonanza for Earth Science[END_REF][START_REF] Howle | Airborne LiDAR analysis and geochronology of faulted glacial moraines in the Tahoe-Sierra frontal fault zone reveal substantial seismic hazards in the Lake Tahoe region, California-Nevada, USA[END_REF]Langridge et al., 2014;Lin, Kaneda, Mukoyama, Asada, & Chiba, 2013;[START_REF] Sherrod | Holocene fault scarps near Tacoma[END_REF]Manighetti et al., 2015). The resolution of a LiDAR scan is determined by the number of pulses (i.e. shot density) and returns per unit area. The shot density is a function of pulse frequency, distance to target, velocity of sensor (for airbornebased sensors), and swath overlap. The return density is more complex, depending on the vegetation cover mainly. LiDAR data have the capacity to provide topography with resolutions generally ~0,2-1m in the horizontal and 10-20 cm in the vertical [START_REF] Carter | Geodetic laser scanning[END_REF]. LiDAR (as seen in Figure 5) helps to address these issues of slip accumulation by providing high-resolution topographic and imagery data for tectono-geomorphic interpretation, enabling:

a.) enormous increase in number of offset measurements to 100s-1000s along entire fault trace (even in zones of vegetation)  Statistical analyses enabled and significant b.) thus, the measurement of slip variability along the entire fault trace c.) analysis of the fine-scale fault geometry to provide an appropriate framework for offset data interpretation d.) approximation of the morphologic response of displaced markers, given the climatic regime they reside in. A philosophical theme of paleoseismological studies is that the basis for rigorous scientific conclusions regarding the paleoseismic history of faults cannot be made through the collection of a few observations only, especially because earthquake slips and cumulative slips are variable along the fault strike (e.g., Manighetti et al., 2001[START_REF] Manighetti | Evidence for self-similar, triangular slip distributions on earthquakes: Implications for earthquake and fault mechanics[END_REF][START_REF] Manighetti | Self-similarity of the largest-scale segmentation of the faults: Implications for earthquake behavior[END_REF]Manighetti et al., , 2015)). A few geomorphic markers will generally be uninterpretable for entire fault or fault region. A large dataset is rather necessary for the implementation of sound contributions to fault behavior through paleoseismic studies. The integration of LiDAR datasets into paleoseismology studies enables the collection of hundreds of offset markers along the faults chosen. Therefore, the statistical basis is encouraging to support the findings.

Given the tremendous amount of slip data along any given fault, statistical approaches must be utilized to extract the most significant and relevant preserved offsets. Many studies (e.g. Wallace, 1968;[START_REF] Sieh | A study of late Holocene displacement history along the South-Central Reach of the San Andreas Fault[END_REF][START_REF] Rockwell | Holocene slip events on the southern Elsinore fault, Coyote Mountains, southern California[END_REF]Rockwell, 1989;McGill & Sieh, 1991;[START_REF] Trifonov | The Talas-Fergana active right-lateral fault[END_REF] have relied on the use of a probability density function (PDFs) to represent an individual fault offset and its uncertainties (Figure 6). Figure 6 is a graph that represents a commonly used Gaussian probability density function (PDF) for one measurement value (in our case, a marker fault offset value). The mean of the measurement lies in the center of the Gaussian curve. One standard deviation, or 1σ, is taken to define the region where there is ~68% of probability that the actual measure is. Whereas, 2σ represents the area where there is 95% probability to find the actual measure. Therefore, PDF functions can be used, both to represent the uncertainty range of individual measures, and the most probable zones of "existence" of a (here offset) value that would result from the combination of many individual measures. In that later case, each of the many individual values has its amplitude and uncertainties represented by an individual PDF. To derive the most probable average value (here offset) that is recorded and represented by the combination of these multiple individual values (here offsets), a summed PDF curve is calculated by summing all the individual PDF curves contributed to the searched value. The most pronounced "peak" in the summed PDF is approximated by a Gaussian PDF function whose peak value provides the searched mean offset and the half or entire widths provide its 1σ or 2σ errors, respectively. The peak offsets or clustered displacements can be inferred to represent the cumulative slip associated with a distinct number of paleoearthquakes. This part will be further discussed in the following.

Paleoseismological surface-slip reconstructions along the great MW 7.8 1857 Fort Tejon, San Andreas earthquake rupture trace (and preceding events) have been instrumental in the formulation of the popular earthquake recurrence (i.e. behavior) models, shaping the understanding of seismic fault behavior along with earthquake forecasting and seismic hazard assessment (e.g., Sieh, 1978;[START_REF] Schwartz | Fault behavior and characteristic earthquakes: Examples from the Wasatch and San Andreas Fault zones[END_REF][START_REF] Sieh | Holocene activity of the San Andreas Fault at Wallace Creek, California[END_REF][START_REF] Sieh | The repetition of large-earthquake ruptures[END_REF]WGCEP, 2008;[START_REF] Field | Uniform California earthquake rupture forecast, Version 2 (UCERF 2)[END_REF]Zielke et al., 2010;[START_REF] Akciz | Century-long average time intervals between earthquake ruptures of the San Andreas Fault in the Carrizo Plain, California[END_REF]Zielke et al., 2012). Fault behavior models, depicted in Figure 7, intend to characterize earthquake magnitude and fault displacement characteristics into a uniformly repeatable model that can be consulted to aid in the prediction of future fault behavior. Fault behavior models are formed from empirical evidence pertaining to the variation of coseismic displacement on a respective fault both at a single point and along the entire trace. When analyzing fault behavior, both spatial and relative temporal aspects must be considered. The assumption surrounding the spatial behavior aspect of these models is that the pattern of surface deformation correlates with the pattern of strain release at depth. Therefore, if certain earthquake rupturing obstructions exist, such as peculiar fault sections, mechanical barriers, etc., they should become evident in the pattern of surface deformation. The temporal aspect of the models assumes that there is a causal link between the timing of major earthquakes and the amount of strain released during each event [START_REF] Scholz | The mechanics of earthquakes and faulting[END_REF]. These fault behavior models can thus add some order to a seemingly chaotic earthquake event interpreted through paleoseismic data.

The objectives of this thesis work are two-fold: 1.) to develop a code to mathematically, and thus objectively, characterize the geomorphic markers (through their 3D geometry), in turn, automating the measurements of their lateral and vertical slip components (in 3 dimensions) along with their uncertainties, and 2.) to apply the code to generate a statistically significant along-fault cumulative slip and rupture history analysis of one major seismogenic, strike-slip fault worldwide. The active strike-slip fault chosen is the Wairarapa fault in northern New Zealand, which produced one of the largest continental earthquakes worldwide with a MW 8.2. This one historically recorded major earthquake event on Wairarapa in 1855 reportedly produced the largest continental coseismic slip ever recorded at 18.7 ± 1 m of horizontal displacement [START_REF] Rodgers | World's largest coseismic strike-slip offset: The 1855 rupture of the Wairarapa Fault, New Zealand, and implications for displacement/length scaling of continental earthquakes[END_REF]. This enormous earthquake poses an extreme seismic hazard for New Zealand and the rupture history should be investigated to ascertain if all previous earthquakes along Wairarapa were as large as the one historic record.

CHAPTER 2: Brief overview of active fault surface traces and geomorphic features

2.a Morphotectonic analysis of the surface: trace of faults

The seismic rupture of a fault usually imprints a clear trace in the ground surface morphology if the earthquake is strong enough (MW ≥ ~ 6) and shallow (≤ ~ 30 km depth) [START_REF] Bonilla | Minimum earthquake magnitude associated with coseismic surface faulting[END_REF][START_REF] Wells | New empirical relationships among magnitude, rupture length, rupture width, rupture area, and surface displacement[END_REF]. Provided that the balance between tectonic processes and alluvial dynamics in shaping the ground morphology favors tectonicallymodified landscapes, the tectonic traces will become more accentuated with successive strong earthquakes. In other words, the recurrence time of the large, surfacing-rupturing earthquakes must be short enough so that the erosional processes do not erase the imprints of previous earthquakes (i.e. slip rate on the fault > erosion rate). Therefore, the imprint of the fault trace becomes successively clearer in the morphology over time, building the traces and the cumulative displacements that shape the tectono-geomorphic landscape (e.g. [START_REF] Tapponnier | Field evidence for active normal faulting in Tibet[END_REF][START_REF] Walsh | Displacement gradients on fault surfaces[END_REF]. Moreover, a strong earthquake occurring on a fault displaces and deforms the morphology and thus all of the landforms (i.e. markers) it contains. For a strong and shallow enough earthquake, as previously mentioned, these displacements (also termed offsets and will be used interchangeably throughout the manuscript) are usually visible on the ground surface and generally range between a few tens of centimeters to about 15 m for continental earthquakes (e.g. [START_REF] Manighetti | Earthquake scaling, fault segmentation, and structural maturity[END_REF]. As strong earthquakes recur, these offsets accumulate and record more slip and become more visible in the topography. The morphology of the surface, therefore, records the rupture history for a very long time period (e.g. Bull, 1991[START_REF] Yeats | The geology of earthquakes (568[END_REF], Burbank & Anderson, 2011) that varies depending upon the climatic regime in which the respective fault resides. Faults are not just one orderly break in the earth's surface and crust, but instead consist of a number of subsidiary faults and fractures. Therefore, any fault is rather a fault zone defined as a group of individual faults being kinematically coupled: there is a major fault, and the latter is generally connected to a dense network of secondary faults of smaller scales. Most of these secondary faults form a more or less wide "damage zone" around or on one side of the master fault, and this damage fault network is especially well expressed at the master fault tip(s) where the secondary faults are commonly referred to as "splay faults". It has been shown that the architecture of the splay faults indicates the direction of the master fault lateral propagation over its long-term evolution (Perrin, Manighetti, & Gaudemer, 2016) (Figure 9). Additionally, faults are segmented laterally at various scales, and this segmentation makes the fault traces punctuated with geometrical complexities [START_REF] Manighetti | Self-similarity of the largest-scale segmentation of the faults: Implications for earthquake behavior[END_REF](Manighetti et al., , 2015)). Given that the main focus of this work is on continental strike-slip faults, it is important to understand the tectono-geomorphic settings that produce the specific markers that are measured. A strike-slip fault is where all displacement occurs predominantly in a direction parallel to the strike of the fault (i.e. slip lineation is predominantly horizontal). By definition, strike-slip faults displace geomorphic markers laterally and approximately parallel to the earth's surface (while some oblique slip will partition some of the slip vertically). Since most young deposits and geomorphic landscapes are planar and also form parallel to the earth's surface, certain problems of discernment between sedimentary or tectonicallydeformed markers can exist in strike-slip settings. This style of deformation differs from that of dip-slip faults, which displace features orthogonally to the earth's surface. However, the geometry of deposits and surfaces laterally displaced offers some distinct advantages. Longer records of seismicity are preserved closer to the surface of the earth than for dip-slip faults where evidence of older paleoearthquakes can be deeply buried and therefore less accessible to investigation (McCalpin, 2009). Also, the laterally displaced tectono-geomorphic markers are produced with less relief than dip-slip faults leaving them less susceptible to being destroyed through uplift, sedimentation, or erosional processes.

In reality, strike-slip faults are generally accompanied by components of transpression (i.e. shortening component that produces uplift) and transtension (i.e. extensional component that causes subsidence along the fault) [START_REF] Weber | Short-term deformation: Geodesy[END_REF]. These dynamic processes, combined with the alluvial dynamics of the respective climatic regime, form the tectono-geomorphic landscape that can be interpreted to extract the slip distribution along the fault. Figure 8 displays a picture of the San Andreas Fault exposing its clearly defined fault trace and corresponding tectono-geomorphic features. The delicate balance of consecutive large earthquakes and a marker formation rate that is more frequent than those earthquakes, but not frequent enough to erase the traces of those earthquakes, allow for the phenomenal preservation of these features in the morphology.

2.b Tectono-geomorphic marker analysis: type, formation, and preservation

In tectono-geomorphic environments, the ability to be able to study paleoseismological evidence is dependent upon: the recurrence time of surfacing rupturing events, and the creation and preservation thresholds of the tectono-geomorphic markers. It has been previously discussed in the Chapter 1 that the marker-formation rate must exceed that of the surface rupturing events that displace them. This ensures that the lowest local offset value measured represents the coseismic slip from the most recent earthquake event. Since most of the markers are of alluvial origins, there are many different features that are represented in varying tectonic settings. Despite many alluvial features being formed across the trace of the active strike-slip fault (Figure 10), only several of these common markers are robust for reconstructing the slip history of the respective fault. Markers that contain long, linear aspects on both sides of the fault trace, and strike preferentially perpendicularly to the fault trace create the ideal conditions to measure the offset. The linearity of the marker is important, because there is less obscurity as to where the point of measurement should be. Also, the most accepted method for measuring the offset amount is to take the most representative part of the marker where it intersects (or is projected onto) the fault, termed a piercing point. The piercing points for both the upper and lower sections of the markers are used as end members to the measurement. Condensing the morphology of a marker into one point makes having a linear feature advantageous as to reduce the ambiguity where the piercing point should be. Also, the linear trend of the marker can be projected onto the fault plane if the marker were to be deflected before the intersection with the fault trace. Markers that commonly and consistently possess these characteristics include: stream channels, beheaded streams, terrace risers and treads, ridges, and alluvial fans. The formation and reconstruction of these particular markers will be discussed in the following sections. [START_REF] Suggate | The interpretation of progressive fault displacement of flights[END_REF], Lensen (1964a), andSugimara and[START_REF] Sugimura | Atera Fault and its displacement vectors[END_REF] are recognized as compiling the initial methods used for measuring and interpreting laterally and obliquely offset terraces. A densely studied region of multiple offset river terraces has been documented in New Zealand [START_REF] Lensen | The faulted terrace sequence at the Grey River, Awatere Valley, South Island, New Zealand[END_REF](Lensen, , 1968;;Lensen & Vella, 1971;[START_REF] Suggate | Rate of horizontal fault displacement in New Zealand[END_REF] with the most documented locality being the Saxton River terraces along the Awatere fault [START_REF] Lensen | The faulted terrace sequence at the Grey River, Awatere Valley, South Island, New Zealand[END_REF][START_REF] Knuepfer | Estimating ages of late Quaternary stream terraces from analysis of weathering rinds and soils[END_REF][START_REF] Mccalpin | Tectonic geomorphology and Holocene paleoseismicity of the Molesworth section of the Awatere Fault, South Island, New Zealand[END_REF][START_REF] Mason | Rates of active faulting during late Quaternary fluvial terrace formation at Saxton River, Awatere fault, New Zealand[END_REF]. New Zealand combines both ideal climatic and tectonic conditions for alluvial terrace formation; therefore, there are many studies performed on the slip distribution of displaced terraces along a few different faults in New Zealand including the Waiohine River terraces along Wairarapa fault (e.g. [START_REF] Carne | Using displaced river terraces to determine Late Quaternary slip rate for the central Wairarapa Fault at Waiohine River, New Zealand[END_REF].

2.b.1 Offset Fluvial Terraces

Many terrace risers cross the fault nearly perpendicularly due to the tectonic dynamics. The terraces can be formed either climatically or tectonically. Climatic terraces can be created through base level changes in the valley (inducing aggradation or degradation in the tributary) or from sediment concentration changes upstream of the fault (e.g. [START_REF] Adams | Active tilting of the United States midcontinent: Geodetic and geomorphic evidence[END_REF]Adams, , 1981aAdams, , 1981b;;[START_REF] Dadson | Earthquake-triggered increase in sediment delivery from an active mountain belt[END_REF]. Tectonic terraces can be created by primary or secondary effects of large earthquakes. With an oblique component to slip, tectonic terraces can be created through incision of the up-thrown block. Usually, faulted terraces are separated by more vertical relief than the vertical component of faulting suggests (McCalpin, 2009).

The origin of faulted terraces is dissected into two principal existing models based on whether terrace formation is temporally related to tectonics, or is time independent of faulting. In the time dependent scenario, faulting and terrace formation alternate, so that one or a few earthquakes occur in between successive terrace formations. This "terrace-dependent" model dictates that every paleoearthquake or paleoearthquake cluster is represented by differential offsets of terrace risers and treads. Conversely, climatically-generated terraces indicate no temporal correlation between earthquake timing and terrace formation. This "terraceindependent" model decrees that any number of paleoearthquakes can occur between terrace formations. The two models result in different geometries of faulted terraces.

2.b.1.1 Measuring the offsets of terrace treads

Some linear features of the terrace tread, such as the channel axes, channel margins or risers, and depositional levees (ridges), can be used as a linear geomorphic feature that crosses the fault. It is normally assumed that these features formed contemporaneously with the terrace, and can therefore, be used to measure the offset that exists. However, [START_REF] Knuepfer | Estimating ages of late Quaternary stream terraces from analysis of weathering rinds and soils[END_REF] cites several instances where the geometry indicates a channel that is considerably younger than the terrace tread causing a discrepancy in the offset value. Due to their planar nature, the terrace treads are more difficult to determine the most representative piercing points. Therefore, they have not been used in the analysis of the slip reconstruction in this work. The terrace riser components are much more suitable for these exercises.

2.b.1.2 Measuring the offsets of terrace risers

There exist some guides to measuring offset terrace risers in the field based upon the common erosional processes associated with these features. If the height of the offset riser is constant across the fault, Lensen (1968) suggests that equivalent points of the riser profile (i.e. crests, troughs, or mid-slope) can be used as piercing points on either side of the fault. However, on oblique-slip faults, terraces risers on the up-thrown block are raised, and hence subjected to more erosion making their profiles broader, compared to their down-thrown counterparts. In this scenario, the riser crests have retreated laterally due to erosion, and the riser bases have advanced laterally due to deposition further than equivalent points on the lower riser. Lensen (1968) thus suggests using the riser's midpoint (i.e. mid-slope) for offset measurement in this case due to the assumption of riser erosion by slope decline instead of slope retreat. A conflicting argument comes from [START_REF] Knuepfer | Estimating ages of late Quaternary stream terraces from analysis of weathering rinds and soils[END_REF], who contends that even the midpoint could retreat or advance. Consequently, [START_REF] Knuepfer | Estimating ages of late Quaternary stream terraces from analysis of weathering rinds and soils[END_REF] proposes that averaging the lateral offsets of the crests and troughs would be optimal. My measurements will allow me to come back later to this point. The constantly evolving relationship between fluvial processes and strike-slip faulting often produces the unique landforms that can be analyzed for paleoseismic reconstruction. Consider Figure 11, which displays the common geometry of an active river channel, its corresponding bank (i.e. river terrace), an abandoned channel (T1), and an ancient terrace (R1) all crossed by a strike-slip fault. In Figure 11B, all features are offset by the same amount. After faulting (Figure 11C), the river erodes the lateral offset of its corresponding bank, but the offset is preserved on the abandoned channel (T1) and ancient riser (R1). Over a geological timespan, the river incises laterally to the left abandoning a terrace riser (R2) and an ancient channel (T2). Another earthquake event can occur offsetting all features by a constant amount, but this amount is added to the prior offsets of T1 and R1. This alternation of faulting and terrace formation can continue for many cycles.

This scenario highlights some important aspects of offset terrace interpretation. Firstly, abandoned channels and terrace treads and risers can preserve both the horizontal and vertical components of slip. Secondly, the lateral offsets of riverbanks can be systematically erased through erosion, especially when the respective bank of the river is displaced toward the active channel. Based on these insights, Bull (1991) differentiates between the riverbanks including: the leading edge and the trailing edge. The leading edge defines the riverbanks that are laterally offset into the river's path thus destroying the offset. The trailing edge of the terrace defines the riverbanks that are laterally displaced away from the river's path thus preserving the offset. Thirdly, a terrace riser is abandoned by the active river contemporaneously with the abandonment of the terrace below the riser, not the terrace above the riser [START_REF] Knuepfer | Estimating ages of late Quaternary stream terraces from analysis of weathering rinds and soils[END_REF].

If varying offset measurements are calculated for a channel and its corresponding terrace risers, the difference could be a factor of erosion instead of a difference in the preservation of the true offset. Heavily scoured terrace risers (usually the leading edge) only record the offsets after the abandonment of the surface below them. This model is termed the "lower-terrace" model by [START_REF] Cowgill | Impact of riser reconstructions on estimation of secular variation in rates of strike-slip faulting: Revisiting the Cherchen River site along the Altyn Tagh Fault, NW China[END_REF] and the "strath abandonment" model by Meriaux et al (2005) (depicted in Figure 12A-C). On less-scoured offsets (usually the trailing edge), the true offsets could be preserved after the abandonment of the terrace above the riser. This model is termed the "upper-terrace" model by [START_REF] Cowgill | Impact of riser reconstructions on estimation of secular variation in rates of strike-slip faulting: Revisiting the Cherchen River site along the Altyn Tagh Fault, NW China[END_REF] and the "fill abandonment" model by Meriaux et al (2005) (depicted in Figure 12D-F). These two models describe perfect condition scenarios where all or none of the offset is eroded away. However, in practice, a more intermediate case is likely to prevail in which only part of the offset is eroded. This model is termed the "strath emplacement" model by Meriaux et al (2005). These models are important for understanding and calculating the slip rates of faults; however, they are also beneficial in determining the proper offset value across a single marker.

2.b.2 Stream channels

Laterally offset stream channels have been a prominent feature to document recurrent movement along active strike-slip faults (Wallace, 1968;Sieh, 1978;[START_REF] Sieh | Holocene activity of the San Andreas Fault at Wallace Creek, California[END_REF]Lienkaemper et al., 1989;[START_REF] Grant | Stratigraphic evidence for seven meters of dextral slip on the San Andreas Fault during the 1857 earthquake in the Carrizo Plain[END_REF], 1994;Zielke et al., 2012). Wallace (1968Wallace ( , 1990) ) first acknowledged the complex relationship between fluvial drainage patterns across the fault and tectonically-offset channels. There is a major distinction between drainage patterns affected by tectonic landforms and drainage networks physically offset by tectonic activity. In the first case, stream misalignment, a purely descriptive term for stream segments that do not align across the fault, and stream diversion, streams forced to flow parallel to the fault (by capture or blockage) have no tectonic origins. However, stream offset is a term used for the tectonic displacement of a stream channel. It can sometimes be ambiguous to differentiate between stream misalignment and stream offset.

The thalweg of the stream is usually a robust feature to use as a piercing point on either side of the fault, since it is subjected to persistent stream erosion. Offsets of streams can be attributed to individual paleoearthquakes only if the stream reestablishes its course between each successive earthquake, abandoning the downstream section after each event. However, given the tendency of streams (especially ephemeral: a stream that flows only briefly during and following a period of rainfall in the immediate locality) to be diverted along the fault trace, it is unlikely for streams to record just the coseismic displacement of a single earthquake. Therefore, most streams preserve multi-event along fault slip amounts. This ambiguity in identifying individual paleoearthquakes from the morphology alone is highlighted in Figure 13. Riverbeds often experience deflection due the stream flow trying to resurrect its original path after tectonic misalignment. The offset of a marker implies a previously straight channel that has become curved after displacement. Therefore, the bend in the channel must agree with the sense of slip. A deflection, however, is a bend in the channel resulting from drainage capture (i.e. water will take the easiest path downhill) (Zielke et al., 2015). Therefore, a curve in the channel caused by deflection can be with or against the sense of slip. All offsets are deflections, but not all deflections are offsets.

2.b.2.1 Beheaded streams

When ephemeral streams are offset by amounts that exceed the channel width, the downstream section could become translated enough to completely segregate from the upstream section. With no immediate source for consistent stream erosion, the downstream section becomes a dry riverbed termed a beheaded stream. Figure 14 depicts a stream that has lost its source on the northern part of the fault trace. The beheaded stream can then preserve the offset very well since there is no stream erosion affecting its course. Where large offsets persistently occur, a master stream on the upstream section may have multiple beheaded streams on the downstream side. With the assumption that the master stream would reestablish its linear path after each successive earthquake, the per-event offset amount would be the difference between successive beheaded stream offsets. This has been suggested to be the case in the Waiohine River example on the Wairarapa fault. After one large offset event (depicted in Figure 14B) where the offset is much greater than the channel width, the downstream segment (1d) is transported (to the right on the figure) and beheaded. The throughgoing stream (2u-2'd) later re-establishes a straight course across the fault. Eventually, the beheaded stream will erode headward (1u). After a second, smaller offset event where the offset is about equivalent to the channel width (depicted in Figure 14C), the beheaded channel is further transported (to the right) and is beheaded again from its ponded alluvium (1d & 1u). The downstream segment of the throughgoing channel (2'd) is shifted to the right, but not enough to become beheaded from the master channel. The throughgoing stream again re-established its linear course across the fault by eroding away the offset on the leading edge.

Complications with beheaded streams can arise when there are two or more master streams in close succession. If the amount of offset from successive earthquakes is large enough, a beheaded stream can be recaptured by another master stream (Lienkaemper, 2001). This scenario highlights the potential difficulties in identifying whether individual marker sections were originally connected. Each displaced feature must be studied to indicate causal relationships for an original connection between observed up-fault and down-fault marker.

The geomorphology of perennial (a stream that flows continuously all year round) streams where they cross strike-slip faults might be more complex than for ephemeral streams, since streams are responding to climatic conditions as well as tectonic transport. The landforms that are generated from the relationship of climatic and tectonic activity (i.e. channels, fans, and terraces) are usually unique to each site, based on Quaternary climate changes and lateral slip rate [START_REF] Bull | Adjustments by the Charwell River, New Zealand, to uplift and climatic changes[END_REF].

2.b.3 Ridges and valleys

Through the lateral offset of high-relief terrains such as ridge and valley walls, strikeslip faults generate fault scarps. Generally, the lateral displacement of a ridge will create scarps that face in opposite directions if the net slip vector is closer to purely horizontal than the side-slopes of the ridge. This scenario is depicted in Figure 15, where the net slip vector is nearly horizontal. As can be seen, on the left of the ridges, the light-toned scarps are facing up-valley away from the viewer. On the right side of the ridges, the dark-toned scarps are facing down-valley toward the viewer. On the flat crest of the ridge, there is little to no height difference to the scarp. Contrarily, if there was a large oblique component to the slip vector such that it is more steeply inclined than the ridge side-slopes, then the scarps on opposite sides would face the same direction but there would be a height difference.

2.b.4 Alluvial Fans

Alluvial fans have been integral to classical field studies of paleoseismology, so a brief description is warranted. The geometry of alluvial fans is similar to an offset stream channel, except that an alluvial fan apex exchanges with an incised channel on the downstream section. Historically, matching the alluvial fan (a very broad feature) with its source gully (a narrow feature) on the upper section of the fault was accomplished through the use of topographic maps depicting the alluvial fan morphology. This approach is most successful when the fault zone is coincident with the heads of alluvial fans. Difficulties in measuring offset alluvial fans arise from the discernment between the apex and axis of the feature. Measurements are thus usually difficult (McCalpin, 2009).

2.b.5 Preservation of offset tectono-geomorphic markers

Once paleoseismic evidence is generated, it must be preserved with minimal degradation long enough to be studied. The threshold for preservation at a particular site is subjected to the delicate balance among erosion, deposition, and other tectonic processes. When erosion and deposition are less active, preservation is encouraged. For example, relatively flat surfaces in arid areas (i.e. minimal erosional processes) unobscured by vegetation (i.e. most of San Andreas Fault, USA) can yield identification of even the subtlest fault scarp traces. However, the capability of LiDAR data to create "bare-earth" digital elevation models (DEMs) enables the mapping of fault scarps and evidence hidden beneath even a dense coverage of vegetation on a rugged terrain (e.g. sections of Hope and Wairarapa Faults, NZ).

In many cases, the preservation of paleoseismic evidence is determined by the relative rates of erosion and deposition versus deformation (Wallace, 1986;Bull, 1991). However, these rates cannot be determined generally. Where deformation rates exceed the rates of geomorphic processes, paleoseismic landforms are created. For example, small fault scarps on steep slopes in humid climates (i.e. high erosion potential) will most likely not remain perfectly preserved for multiple earthquake cycles (Zielke et al., 2015) minimizing the length of the paleoseismic record. Whereas, the same fault scarp located in an arid climate could survive for tens of thousands of years and record many more earthquakes along its trace. At present, the number of studies of paleoseismic evidence in arid regions dramatically outweighs those conducted in humid or polar climates (Sieh, 1978;[START_REF] Beauprêtre | Finding the buried record of past earthquakes with GPR-based palaeoseismology: A case study on the Hope fault, New Zealand[END_REF]Beauprêtre et al., , 2013;;Manighetti et al., 2015). Therefore, a goal of this thesis is to contribute rich information that can still be extracted from humid locations, such as New Zealand.

CHAPTER 3: Developing the MATLABbased code, 3D_Fault_Offsets, for cumulative fault offset measurements in topographic data

3.a Introduction

With the advent of very-high resolution topographic data, such as LiDAR, into the studies of active tectonics and paleoseismology, unprecedented opportunities are offered to analyze the fine-scale 3-D geometry of hundreds to thousands of offset tectono-geomorphic markers along many faults worldwide. Having to properly analyze and measure so many offset markers given their three-dimensional geometry and displacements presents new challenges to the paleoseismologist who needs to confidently identify and characterize these multiple offset markers and reconstruct their pre-earthquake morphology in an efficient and systematic manner. Therefore, techniques must be developed in order to create meaningful offset measurements. Indeed, so far, offset measurements have been done "visually" and "manually", either directly on the field through the measurement (by tape, total station, DGPS) of observed displaced features, or using airborne or satellite images (e.g., Gold et al., 2013a;Klinger et al., 2005Klinger et al., , 2006Klinger et al., , 2011;;McGill & Sieh, 1991;Rockwell & Klinger, 2013;[START_REF] Salisbury | LiDAR and field observations of slip distribution for the most recent surface ruptures along the central San Jacinto Fault[END_REF]Scharer et al., 2014a;Sieh, 1978). In the latter case, the offset markers are identified on the images, and their lateral offset is measured by back-slipping half of the image along the fault in the horizontal plane. This classical back-slipping approach thus does not allow measuring the vertical slip, while the lateral slip measure contains a subjective part.

Additionally, many sources of uncertainties affect the fault offset measurements (Figure 16). A first major uncertainty comes from the qualitative assessment of the offset marker (Sieh, 1978;[START_REF] Weldon | Chapter 6 paleoseismology of strikeslip tectonic environments[END_REF]Klinger et al., 2011;Zielke & Arrowsmith, 2012;Scharer, Salisbury, Arrowsmith, & Rockwell, 2014, Manighetti et al., 2015). Because this epistemic uncertainty, which reflects the lack of information regarding the actual shape of the geomorphic feature prior to surface rupturing earthquakes, cannot be quantified, the paleoseismologist will assign a qualitative ranking to each measurement depending on the confidence of the reconstruction and whether the observed separation is of tectonic origin.

A second uncertainty is aleatoric, and related to the range of possible offset values capable of sufficiently restoring the assumed pre-earthquake morphology. This uncertainty, typically recorded like ± 3.6 m, represents the physically plausible offset range and is commonly disposed to be 1 or 2σ (e.g., Klinger et al., 2011;McGill and Sieh, 1991;Scharer et al., 2014a;Zielke et al., 2012). The individual measurements plotted on chart usually take the form of a normal distribution or a bell-shaped Gaussian curve with the mean value being the center.

Another uncertainty is intrinsically linked to the fact that fault planes and marker traces are not perfect geometric lines, but they possess some natural spatial properties. Other uncertainties might contain a component of the type of instruments used in the measurement. For instance, the utilization of topographic data provides uncertainties according to its spatial resolution (e.g. DEM resolution of 0.5 m). Proper indication of the offset range, its quality, and its relation to the uncertainties poses complex problems in the measurement technique as well as the statistical post-processing of the data. Also, erosional processes can severely alter the initial morphology of the offset marker potentially leading to both under-and overestimations in the offset length. Therefore, uncertainty for the amount of erosion that modified the original offset is an issue. Finally, another source of uncertainty arises from the individual interpreter analyzing and recording all of the offset measurements. Offset measurements can be biased, or multiple values can be obtained from different individuals using the same technique.

Therefore, it is important to offer a technique that is semi-automatized in order to reduce user bias or large uncertainties stemming from differing interpretations of the morphology or location of the intersection of a marker with the fault plane. To address the concerns of efficiently analyzing hundreds of offsets, diminishing user bias, properly accounting for the range of potential offset values and uncertainties, and providing a statistical analysis package for post-processing capabilities, we have developed the 3D_Fault_Offsets Matlab code. This work has been published in Journal of Geophysical Research (Stewart et al., 2018), and this paper is presented in the following.

Figure 16. Sketch of several scenarios depicting potential geometric complexities associated with the estimation of the pre-earthquake morphology. Much of the uncertainty in measuring offsets arises from the difficulty in accurately assessing and projecting the marker trace to the fault plane as it was before erosional processes distorted the trace. The actual width of the fault zone coupled with shearing and geomorphic modification could change the initial marker morphology. In A & B, the grey areas indicate the possible range of marker projection onto the fault plane. Therefore, it is important to measure multiple aspects of a marker (i.e. thalweg, terrace riser, etc.) to account for some of these uncertainties. Image taken from Zielke et al., 2015. The objective of 3D_Fault_Offsets was to provide:  unbiased and proper characterization of a marker's complex, both large-and finescale, three-dimensional shape and geometry by eliminating the question of visual subjectivity  three-dimensional reconstruction of the marker, based on its individual geometry, in both its lateral and vertical components of slip  adequate constraint on the uncertainties (i.e. to eliminate the visual subjectivity often attributed to offset measurements) given the complexity of the marker geomorphology that resulted from continued tectonic and erosional deformation 3D_Fault_Offsets mathematically, and hence unbiasedly, characterizes the 3D geometry of an offset marker and solves for the horizontal and vertical components of its slip vector.

Utilizing the variation of apparent offset with topographic attitude and marker geometry allows us to fully exploit high-resolution topographic datasets. Based on the analytical expressions of the fault offsets, we introduced a Monte Carlo approach to the uncertainty analysis of offset measurements by accounting for the 3-D marker geometry and topographic fault intersection. We successfully tested the robustness of this method on both field datasets and remote, high-resolution LiDAR datasets along the 1857 San Andreas Fault (California, USA) rupture trace (e.g. Sieh, 1978;Lienkaemper, 2001;Zielke et al., 2012), the 1872 Owens Valley Fault rupture trace (California, USA) (e.g. Haddon et al., 2016), and the 1888 Hope Fault rupture trace (South Island, New Zealand) (e.g. Manighetti et al., 2015).

3.b 3D_Fault_Offsets JGR paper

Supplementary material is provided in Appendix I.

Introduction

Knowing how much slip a fault has accumulated during one earthquake or over the long-term is important in many aspects including a better understanding of fault kinematics and mechanics (e.g., [START_REF] Armijo | Late Cenozoic right-lateral strike-slip faulting in southern Tibet[END_REF][START_REF] Gaudemer | River offsets across active strike-slip faults[END_REF][START_REF] Gaudemer | Partitioning of crustal slip between linked, active faults in the eastern Qilian Shan, and evidence for a major seismic gap, the 'Tianzhu gap', on the western Haiyuan Fault, Gansu (China)[END_REF][START_REF] Hubert-Ferrari | Morphology, displacement, and slip rates along the north Anatolian fault, Turkey[END_REF]Manighetti et al., 2001;[START_REF] Manighetti | The role of off-fault damage in the evolution of normal faults[END_REF][START_REF] Replumaz | Large river offsets and Plio-Quaternary dextral slip rate on the Red River fault (Yunnan, China)[END_REF][START_REF] Van Der Woerd | Holocene left slip-rate determined by cosmogenic surface dating on the Xidatan segment of the Kunlun fault (Qinghai, China)[END_REF], 2002), relations between earthquake and cumulative slips (e.g., [START_REF] Gaudemer | River offsets across active strike-slip faults[END_REF][START_REF] Li | Co-seismic and cumulative offsets of the recent earthquakes along the Karakax left-lateral strike-slip fault in western Tibet[END_REF][START_REF] Manighetti | Evidence for self-similar, triangular slip distributions on earthquakes: Implications for earthquake and fault mechanics[END_REF][START_REF] Peltzer | Offsets of late Quaternary morphology, rate of slip, and recurrence of large earthquakes on the Chang Ma Fault (Gansu, China)[END_REF][START_REF] Tapponnier | Long-term slip rates and characteristic slip: Keys to active fault behaviour and earthquake hazard[END_REF]Zielke, Klinger, & Arrowsmith, 2015), and earthquake magnitude and stress distribution (e.g., Klinger et al., 2011;[START_REF] Lasserre | Post-glacial left slip-rate and past occurrence of M ≥ 8 earthquakes on the western Haiyuan fault (Gansu, China)[END_REF][START_REF] Manighetti | Generic along-strike segmentation of Afar normal faults, East Africa: Implications on fault growth and stress heterogeneity on seismogenic fault planes[END_REF]Perrin et al., 2016;Zielke et al., 2012). Yet while current earthquake slip can generally be measured or estimated both along-strike and along-dip of the ruptured fault (e.g., [START_REF] Manighetti | Evidence for self-similar, triangular slip distributions on earthquakes: Implications for earthquake and fault mechanics[END_REF][START_REF] Yeats | The geology of earthquakes (568[END_REF], long-term cumulative slip can generally only be measured along the fault strike at the ground surface. Surface slip measurements are thus of critical importance to recover how cumulative displacements, including prehistorical earthquake slips, are distributed along faults.

Fault slips at the surface are generally preserved in the form of fault-offset sublinear geomorphic markers (e.g., Arrowsmith & Zielke, 2009;Burbank & Anderson, 2011;[START_REF] Knuepfer | Changes in Holocene slip rates in strike-slip environments[END_REF]McCalpin, 2009;[START_REF] Sieh | Lateral offset and revised dates of large prehistoric earthquakes at Pallett Creek, southern California[END_REF]Wallace, 1968Wallace, , 1990, Figure 1). These markers are commonly persistent or ephemeral stream channels (Lensen, 1968;Lienkaemper, 2001;McCalpin, 2009;Sieh, 1978;Wallace, 1968Wallace, , 1990;;Zielke et al., 2010; Figure 1) or abandoned alluvial terrace risers (e.g., [START_REF] Gold | Riser diachroneity, lateral erosion, and uncertainty in rates of strike-slip faulting: A case study from Tuzidun along the Altyn Tagh Fault, NW China[END_REF][START_REF] Gold | Faulted terrace risers place new constraints on the late Quaternary slip rate for the central Altyn Tagh Fault, northwest Tibet[END_REF]Lensen, 1964Lensen, , 1968;;[START_REF] Pazzaglia | 9.22 Fluvial terraces[END_REF][START_REF] Van Der Woerd | Uniform postglacial slip-rate along the central 600 km of the Kunlun Fault (Tibet), from 26Al, 10Be, and 14C dating of riser offsets, and climatic origin of the regional morphology[END_REF]. The creation and preservation of these offset markers require an equilibrium between tectonic (fault slip, warping, and tilting), alluvial (sedimentation and erosion), and

• Supporting Information S1 • Data Set S1 • Data Set S2 • Data Set S3 • Data Set S4 • Data Set S5 • Data Set S6
• Table S1 • Table S2 • Table S3 • Table S4 Correspondence to: I. geomorphic (erosion) processes and rates (Bull, 1991;Burbank & Anderson, 2011;Lienkaemper, 2001;Ludwig et al., 2010;McCalpin, 2009;[START_REF] Ouchi | Development of offset channels across the San Andreas Fault[END_REF][START_REF] Salisbury | LiDAR and field observations of slip distribution for the most recent surface ruptures along the central San Jacinto Fault[END_REF]Sieh, 1978;Wallace, 1968;Zielke et al., 2012Zielke et al., , 2015)). These conditions pose a number of challenges (e.g., [START_REF] Gold | Riser diachroneity, lateral erosion, and uncertainty in rates of strike-slip faulting: A case study from Tuzidun along the Altyn Tagh Fault, NW China[END_REF][START_REF] Gold | Faulted terrace risers place new constraints on the late Quaternary slip rate for the central Altyn Tagh Fault, northwest Tibet[END_REF]McCalpin, 2009;[START_REF] Ouchi | Development of offset channels across the San Andreas Fault[END_REF][START_REF] Salisbury | LiDAR and field observations of slip distribution for the most recent surface ruptures along the central San Jacinto Fault[END_REF]Scharer et al., 2014;Zielke et al., 2015): how to evaluate the degree of preservation (or erosion) of an offset marker, that is, its accuracy to represent the actual fault slip? How to measure the three components of the fault displacement across an offset marker? How to perform such complex measurements in an efficient way on a dense population of offset markers?

Until recently, offset markers have been measured "manually," especially along strike-slip faults (e.g., [START_REF] Ansberque | The Longriqu fault zone, eastern Tibetan Plateau: Segmentation and Holocene behavior[END_REF][START_REF] Armijo | Late Cenozoic right-lateral strike-slip faulting in southern Tibet[END_REF][START_REF] Frankel | Spatial variations in slip rate along the Death Valley-Fish Lake Valley fault system determined from LiDAR topographic data and cosmogenic Be-10 geochronology[END_REF][START_REF] Gaudemer | River offsets across active strike-slip faults[END_REF][START_REF] Gold | Riser diachroneity, lateral erosion, and uncertainty in rates of strike-slip faulting: A case study from Tuzidun along the Altyn Tagh Fault, NW China[END_REF][START_REF] Hubert-Ferrari | Morphology, displacement, and slip rates along the north Anatolian fault, Turkey[END_REF]Klinger et al., 2011;[START_REF] Li | Co-seismic and cumulative offsets of the recent earthquakes along the Karakax left-lateral strike-slip fault in western Tibet[END_REF][START_REF] Replumaz | Large river offsets and Plio-Quaternary dextral slip rate on the Red River fault (Yunnan, China)[END_REF][START_REF] Ritz | Slip rates along active faults estimated with cosmic-ray-exposure dates: Application to the Bogd Fault, Gobi-Altaï, Mongolia[END_REF]Rizza et al., 2015;Sieh, 1978;[START_REF] Van Der Woerd | Holocene left slip-rate determined by cosmogenic surface dating on the Xidatan segment of the Kunlun fault (Qinghai, China)[END_REF], 2002). A fault-offset marker is identified visually by the expert in satellite, aerial images, or on the field (Figure 1). Then the expert manually performs a retrodeformation, or back slip, which is the stepwise reconstruction of 2-D horizontal views (more rarely vertical sections) to realign the likely original map-view trace of the marker (e.g., McCalpin, 2009). The preferred reconstruction provides the horizontal offset value along the fault strike or, more rarely, the vertical value along the fault dip. This classical method has shown to be relevant in the many studies that have used it for decades (references above). However, offset measures and their uncertainties are derived solely from visual interpretations that might thus be disputed. Only the horizontal component of the slip is generally estimated. Also, the reconstruction of the marker provides only a map view vision of its original morphology, and the measurement is time consuming.

More recently, the explosion of high-resolution topographic data, especially Lidar that allows the measurement of the bare Earth surface at ≤1 m resolution (e.g., Arrowsmith & Zielke, 2009;[START_REF] Bevis | The B4 project: Scanning the San Andreas and San Jacinto fault zones[END_REF][START_REF] De Pascale | Lidar reveals uniform Alpine fault offsets and bimodal plate boundary rupture behavior[END_REF][START_REF] Frankel | Spatial variations in slip rate along the Death Valley-Fish Lake Valley fault system determined from LiDAR topographic data and cosmogenic Be-10 geochronology[END_REF][START_REF] Haddad | Applications of airborne and terrestrial laser scanning to paleoseismology[END_REF]Lin et al., 2013;Meigs, 2013;Zielke et al., 2012;Zielke et al., 2015), has motivated the development of new, automatized approaches to remotely measure fault slips in the topographic data. These approaches, so far, use an overall measure of the topography [START_REF] Billant | Constraining recent fault offsets with statistical and geometrical methods: Example from the Jasneuf fault (Western Alps, France)[END_REF], planar surfaces [START_REF] Mackenzie | Untangling tectonic slip from the potentially misleading effects of landform geometry[END_REF], or linear geomorphic features (Haddon et al., 2016;Zielke et al., 2012;Zielke & Arrowsmith, 2012) as recorders and markers of the fault displacements.

In particular, Zielke and Arrowsmith (2012) and Zielke et al. (2012) have developed a Matlab code, LaDiCaoz (updated version, LaDiCaoz_v2, released by Haddon et al., 2016), to semiautomatically measure fault offsets across ubiquitous linear geomorphic features such as stream channels and terrace risers. LaDiCaoz correlates two along-fault profiles crossing an offset marker on either side of the fault trace. The best fit correlation provides a measure of the lateral and the vertical offsets of the marker. Errors are estimated from the range of back slip reconstructions that the user estimates as being plausible. This automatized method has proven to be relevant and efficient in the many studies that have used it (Haddon et al., 2016;[START_REF] Ren | Clustering of offsets on the Haiyuan fault and their relationship to paleoearthquakes[END_REF][START_REF] Salisbury | LiDAR and field observations of slip distribution for the most recent surface ruptures along the central San Jacinto Fault[END_REF]Zielke et al., 2010Zielke et al., , 2012)). Its efficiency was especially clear in that it allowed measuring hundreds of offsets along a fault, about 10 times more than ever before. However, LaDiCaoz includes a number of user interactions and primarily analyzes offset markers in 2-D.

Here we have developed a new Matlab code, 3D_Fault_Offsets, which, in topographic data, mathematically identifies and represents in 3-D the most prominent geometric characteristics of ubiquitous geomorphic sublinear markers along faults. Faults can have any dip, but the code is especially appropriate for strike-slip faults. Nine geometric characteristics are identified across each offset marker section on either side of a fault (Figure 2 described below), which are then converted into nine individual horizontal and vertical displacement measurements across each marker. The code also calculates the total 3-D uncertainties on these offsets.

Finally, it provides tools to statistically analyze the collection of measurements, and to reconstruct the likely original marker geometry in both the horizontal and the vertical planes.

To test the robustness of 3D_Fault_Offsets, we have applied it to remeasure the offsets across 88 markers that were analyzed in prior works (Haddon et al., 2016;Lienkaemper, 2001;Manighetti, Perrin, et al., 2015;Sieh, 1978;Zielke et al., 2012). We assume that these previous studies have correctly identified the markers and their respective offsets. The markers extend along three faults, San Andreas and Owens Valley in USA and Hope in New Zealand, that have different slip modes (purely strike slip, strike slip and normal, and strike slip and reverse, respectively) and morphological environments (from moderate to significant erosion conditions). The topographic data that we use have different resolutions (0.25 cm to 1 m). In total, we have performed 303 series of measures (i.e., code runs, each including 9 lateral and 9 vertical offset measures), representing a total of 5,454 lateral and vertical offset measures, and compared these offset measures to previously published field and remote measurements.

Overview of 3D_Fault_Offsets

Note that 3D_Fault_Offsets is described in greater detail in supporting information ES1 and provided as a zip file in the supporting information named 3D_FAULT_OFFSETS_CODE.zip.

3D_Fault_Offsets is designed to process high-resolution topographic data such as Lidar, but it can be used with any type of topographic data storing surface elevation information on a rasterized grid (digital , McCalpin, 2009;[START_REF] Salisbury | Validation of meter-scale surface faulting offset measurements from high-resolution topographic data[END_REF]Scharer et al., 2014). To operate 3D_Fault_Offsets, the user must first identify the fault trace and the two marker sections offset by the fault. The tectonic relevance of the fault and marker identification is assessed by the user.

The markers must be analyzed on a case-by-case basis, which, in effect, limits the size of the fault zone under analysis. 3D_Fault_Offsets can incorporate any fault dip, and the user is invited to provide this information.

3D_Fault_Offsets functions as scripts written in Matlab. It includes 15 related functions F1 to F7bis and 1 additional routine (F8) made to supplement the analysis. We provide the code as a single wrap-around Matlab file (F0) containing interactive questions and operating instructions that lead the user to easily run the different functions consecutively. Alternatively, the user can choose to use each function individually. The operation of these files requires only very basic knowledge of Matlab procedures. To operate 3D_FAULT_OFFSETS, the user needs to have the matGeom toolbox (it can be freely downloaded from its repository on the Matlab website).

Within a polygon zone defined by the user that roughly includes a marker section on one side of a fault (fault trace drawn by the user; Functions F1 and F2) (Figures 3a and3b), 3D_Fault_Offsets mathematically identifies the topographic data points that characterize nine prominent geometric features prevalent in most geomorphic markers (Figure 2): riverbed, identified as the zone of lowest elevation (Min Z referred to as "river," one point cloud); riser or scarp base and top (referred to as "bot" and "top"), identified through their slope break using the maximum (i.e., maximum convexity of slope) and minimum Laplacian (i.e., maximum concavity of slope) of the topography, respectively (four point clouds on either side of a riverbed); riser or scarp steepest central part or "free face" (referred to as "mid"), identified through the measurement of the maximum gradient of the topography (two point clouds on either side of a riverbed); and the ridge or crest identified as the zone of maximum elevation (referred to as "Max Z"; two point clouds on either side of a riverbed). For every marker, the code searches for the nine specific features systematically along fault parallel topographic profiles that cover the entirety of the polygon zone (two series of profiles on either side of the riverbed, see supporting information ES1). This eventually populates the polygon zone with a great density of points, forming nine individual point clouds.

Using the least squares method, the code then computes a 3-D linear regression through each of the nine point clouds on either side of the fault, in effect creating 18 lines of best fit (Function F3, Figure ES1_B(a) in supporting information ES1). After this first regression, the code removes the artifact points related to the polygon edges and the obvious outliers of the lines through the implementation of the interquartile method (Function F3b, Figure ES1_B(b) in supporting information ES1). It then recalculates the best fit lines from these "cleaned" point clouds (Function F3b, Figure ES1_B(b) in supporting information ES1). The eventual best fit lines characterize the marker geometry in 3-D. The F3e function informs the user of the number of points used in each best fit line calculation, allowing for the verification of its statistical relevance. Each 3-D line of best fit intersects the dipping fault plane creating a piercing point whose x, y, and z coordinates are recorded. The underlying hypothesis is that, as commonly assumed (e.g., McCalpin, 2009), paired piercing points (i.e., characterizing the same geometric feature) on either side of the fault were initially the same "prefaulted" point.

Based upon that assumption, the code computes the three components of the slip vector that joins the paired piercing points. It specifically calculates the horizontal and vertical offsets by subtracting the x and z coordinates of the corresponding piercing points on either side of the fault (Figure 4a; Function F4; if needed, the y component can be derived from the other two slip components). The 18 offset calculations are done systematically regardless of their geomorphic relevance, which can be defined subsequently.

The uncertainties are computed (Function F4) from the various sources of error, which include the resolution of the DEM, the position of each point contributing to a best fit line, the position of the piercing points onto the fault plane, and the errors on the fault position, strike, and dip. The DEM resolution is one of the largest ). This is helpful both to identify the possible outlier points and to judge the geomorphic relevance of the calculated regressions.

The nine lateral and the nine vertical offset measures provide a unique opportunity to examine the variability of the offsets across the entirety of the marker. As commonly performed (e.g., [START_REF] Beauprêtre | Finding the buried record of past earthquakes with GPR-based palaeoseismology: A case study on the Hope fault, New Zealand[END_REF]Beauprêtre et al., , 2013;;[START_REF] Lowell | The application of radiocarbon age estimates to the dating of glacial sequences: An example from the Miami sublobe, Ohio[END_REF]Manighetti, Perrin, et al., 2015;McGill & Sieh, 1991;Scharer et al., 2014;[START_REF] Zechar | Incorporating and reporting uncertainties in fault slip rates[END_REF]Zielke et al., 2012), the code uses probability density functions (PDFs) to derive the most robust offset values and their uncertainties (Figure ES1_F in supporting information ES1). The "best offsets" are estimated by calculating the best fitting Gaussian of the summed PDF function, with their uncertainties being the 1 sigma width of the Gaussian (Figure ES1_F in supporting information ES1). These "best-Gaussian" uncertainties represent the largest possible errors on the offsets for they integrate the full range of offset variability (Figure ES1_F in supporting information ES1). Commonly, a more realistic uncertainty can be derived from the zone of highest and most concentrated peaks in the summed PDF function (Figure ES1_F in supporting information ES1). In the following, we provide both maximum and preferred uncertainties.

The code eventually reconstructs the marker both in a horizontal and a vertical plane (Figure 6 and supporting information ES3, ES4, and ES6). These back slip reconstructions allow the user to verify the geomorphic and tectonic relevance of the offset measures.

Possible Artifacts and Sensitivity of 3D_Fault_Offsets

Possible Artifacts

Different sources of artifacts related to the topographic data or to the calculation methods can affect the point identification and hence the offset calculations. The polygon edges are physical point clouds that disrupt the calculations at the zone edges and produce artifact points. Since these artifact points are systematic, we have included Function F3b to automatically remove them before calculating the final best fit lines (supporting information ES1).

In a similar manner, points at the "internal" tips (i.e., adjacent to riverbed) of the topographic profiles used for calculations may alter some of these calculations. In particular, the Laplacian involves a linear extrapolation of points outside of the domain under analysis, which may produce artifact points misleadingly placed at the tip of the profiles (Figure 5a, ES1_C in supporting information ES1, and ES2_A-B-C in supporting information ES2). Since these artifacts are not systematic, however, we have opted for their manual removal if necessary (Function F3d).

Lidar data, as any other geophysical imaging data, inherently include corduroy artifacts, mainly as a result of the vertical misalignment of the multiple scan lines (e.g., Arrowsmith & Zielke, 2009). The corduroy produces fairly regular troughs and ridges perpendicular to the flight direction with meter-scale wavelengths and decimeter amplitudes, and the effects of these artifacts should be considered when processing the topographic data prior to their use with 3D_Fault_Offsets. In the Lidar data we use here, the corduroy effect was minimized in their prior processing (Arrowsmith & Zielke, 2009;Haddon et al., 2016;Manighetti, Perrin, et al., 2015;Zielke et al., 2012) so that, in the general case, the corduroy height is insignificant compared to the height of the markers' topographic imprint, and therefore, the corduroy artifact does not alter the measurements (Figures 5 and ES1_C in supporting information ES1 and Figure ES2_A in supporting information ES2). More details can be found in supporting information ES1 and ES2.

Some topographic points may display the respective mathematical properties searched by the code while not pertaining to the geometric feature under concern. Those can be points on a neighboring marker, on top of vegetation, on road traces, or simply noise in the data (see example in Figure ES2_D in supporting information ES2). The 3-D views (Function F3c) allow for identifying these artifact points that can then be removed manually.

Overall, the 3-D treatment of topography enables the dense point clouds to generally smooth out these artifacts and reveal the geometric features under concern.

Sensitivity of the Code

We have verified that when the code is used in constant conditions (same DEM, same fault and polygon traces), the repeatability of the measurements is consistent.

We have conducted a series of tests to assess the sensitivity of the calculations to the polygon width, length, shape, and orientation. Those tests are described in detail in supporting information ES2. Basically, the code results are not sensitive to the polygon width provided that the polygon region does not include a significant density of diverting features near the marker under concern. As the following data analysis will better show, the offset results are more sensitive to the polygon length because the geometry of the marker traces commonly varies along their length. This is especially the case when the markers are meandering streams or are subjected to erosion. However, regardless of the polygon length, the code produces stable offset values provided that the marker sections enclosed in the polygons encapsulate the overall map-view geometry of the offset marker. Finally, the point identification, best fit line calculations, and lateral and vertical offsets are not affected by the shape and orientation of the polygons.

The DEM resolution is the most impactful parameter affecting the offset results because the placement of the points and thus the calculation of the best fit lines depends on the density of the elevation points in the polygons. In the following analysis, we compare the results obtained on similar markers using Lidar data at 25 cm and 1 m resolution. We find that, regardless of the DEM resolution, the code's reproducibility is consistent provided that the DEM resolution offers a reasonable point density over a marker, at least 1 point per meter length (more details in the next section).

4. Application of 3D_Fault_Offsets to Measure Lateral and Vertical Offsets on San Andreas Fault

Data and General Approach

We apply 3D_Fault_Offsets to 45 markers (44 stream channels and 1 topographic scarp) along the San Andreas strike-slip fault that were identified in prior works (Lienkaemper, 2001;Sieh, 1978;Zielke et al., 2012). Their lateral offsets were measured both on the field (for 32 of them, Sieh (1978) and Lienkaemper (2001)) and remotely (all 45 markers) using the LaDiCaoz code (Zielke & Arrowsmith, 2012). They vary between ~1 and ~60 m.

We analyze here the northernmost subset of markers reported in Zielke et al. (2012). We picked a few more markers further south that had especially complex traces (low topographic imprint, weathered trace, significant vegetation cover, and trend markedly oblique to the fault; see supporting information ES3) or long traces. The 45 markers had actually been assigned different "qualities" based on the robustness of their identification (Zielke et al., 2012), and the subset analyzed here provides a statistical significance among these qualities (18 markers of high quality, 11 of moderate quality, and 16 of low quality).

We use the same Lidar data as used by Zielke et al. (2012) (data available on www.opentopography.org through the B4 Lidar Project). Their horizontal and vertical resolution is 25 cm (Arrowsmith & Zielke, 2009). However, we have run 3D_Fault_Offsets with a 1 m horizontal and a 0.5 m vertical pixel size as the error inputs, to obtain conservative uncertainties. For each marker, we traced the fault as Zielke et al. (2012) did and considered a constant 90 ± 10°dip at each marker site. We have adopted conservative errors on the fault position (assigned horizontal and vertical errors of 5 and 0.5 m, respectively). Any regression with less than 10 points was discarded from the calculations.

For each marker, we have performed four series of measurements using the 25 cm Lidar data: (i) We have first drawn polygons made to enclose the same marker sections as analyzed by Zielke et al. (2012). These polygons and related calculations are referred to as "AsZ." (ii) We have then used longer and wider polygons relative to the "AsZ," made to enclose a larger portion of the markers and too be less stringent in precisely following the marker traces. These polygons and related calculations are referred to as "Longer." (iii) For each of the AsZ and Longer options, we have made two calculations, the first one entirely automatic referred to as "AUTO," and the second one including a manual user operation to remove the clearest outlying points referred to as "CLICK." Generally, only the "top" and "bot" point clouds were "cleaned," and only a few outliers were removed in each point cloud.

We have then redone the "Longer" measurements (AUTO and CLICK) using the Lidar data at a 1 m resolution.

The four series of measurements in 25 cm Lidar data are reported in ES1_C, and ES2_A to ES2_D) allow for the examination of whether the identified point clouds and the related regressions represent the geomorphic features under concern. At this stage, the user is expected to provide his expertise to decide which regressions and hence which offset measures are geomorphically relevant and should be retained. Reasons for geomorphically irrelevant features include poorly defined regressions, regressions not representing the feature under concern, point clouds containing outliers that could not be removed in any objective manner, etc.; see supporting information ES3). In Table ES-A, we have indicated the offset values that we judge poorly constrained or irrelevant in a geomorphic sense in gray (see table caption) and in black (or red for riverbed) those that we retain as geomorphically meaningful. Later, we refer to "GEOM" as the set of measurements that we consider as geomorphically relevant, and "ALL" as the totality of the measurements with no geomorphic discrimination.

Overall Analysis of the Offset Measures

In the following, we discuss the automatic measures (AUTO) unless it is said differently. When combining the AsZ and Longer approaches, 3D_Fault_Offsets was able to identify 44 out of the 45 markers (43 stream channels and 1 topographic scarp). The marker that could not be identified is the stream channel Sieh5b, which is very short and has a low topographic imprint.

In all 43 stream channels, the code identifies and properly measures (small rms) the riverbed on either side of the fault, in automatic AsZ and Longer measures (supporting information ES3). Outliers rarely exist in Min Z point clouds, and therefore, the automatic identification of the riverbeds is straightforward.

Beyond the riverbeds, 3D_Fault_Offsets identifies additional geomorphic features in most of the markers.

Figure 7 shows the distribution of the identified features that have a geomorphic relevance (values in black in Table ES-A), for the AsZ (Figure 7a) and the Longer approaches (Figure 7b). Overall, considering longer sections of the offset markers allows for better identification and measurement of their geomorphic characteristics. The riverbed is the geomorphic feature most prominently identified in the markers. The riser's steepest slope is the second-most prominent geomorphic feature to be identified well, in ~55% (AsZ) to ~75% (Longer) of the markers. While they are generally not well identified in the AsZ measurements, the riser tops are well-expressed geomorphic features when longer marker sections are considered (identified in ~30% of the cases). The riser bases are more difficult to discriminate, at least in the strictly automatic calculations. It was rare that the point clouds of maximum elevation would highlight any clear geomorphic characteristics of the markers. When the obvious outliers are manually removed ("CLICK" approach, in black in Figure 7), the results are similar but amplified. The major change is that riser bases become better defined.

3D_Fault_Offsets is thus able to identify and mathematically characterize different geomorphic features in a marker. One might argue, however, that the geomorphic relevance of the identified features is not a mathematical output of the code but a subjective ranking by the user. Figure 8a thus examines how the automatic outputs of the code, with no user action, compare with the results derived from user-selected measures. ES2_H(a)). On average, the ALL and GEOM mean lateral offsets are similar within uncertainties in 96% of cases for both the lateral and vertical offsets. Therefore, the offsets derived from the nine systematic and automatic measures of the code are as meaningful as those derived after careful inspection and selection of the geomorphically relevant measures by the expert. This finding should not minimize that the expertise of the user is critical and that the GEOM average offsets are more robustly founded.

Another way to examine the relevance of the automatic approach is to compare the automatic offset measures (AUTO means in 8b for lateral offsets and Figure ES2_H(b) in supporting information ES2 for vertical offsets). Overall, the measures are consistent, similar within uncertainties in 96% of cases for the lateral offsets, and in 100% of cases for the vertical offsets. Therefore, even though a few outlier points exist in the point clouds mathematically defined by the code, it is not necessary to remove them manually. When average offsets are concerned, the dense automatic measures recover the offset values well.

The above findings suggest that for each series of measures with a given polygon set (AsZ or Longer), the four average offsets derived from (i) the nine automatic measures, strict sense (ALL-A, columns AS and AV in Taken together, this analysis suggests that all average offsets discussed above (lateral or vertical) are consistent and equally relevant. We can thus average them to derive a unique global average offset per marker, lateral and vertical, which represents the optimal offset across the marker. Still, we calculate this optimal offset both from the totality of the measures ("ALL," columns BH and BM for lateral and vertical, Table ES-A) and from the geomorphically relevant measures only ("GEOM," columns BI and BN for lateral and vertical, Table ES-A). Figures 9a (lateral) and 9b (vertical) confirm that the two calculations provide similar results, being similar within uncertainties in 98 to 100% of the cases, respectively. We favor the "geomorphic" optimal offsets, however, since they are more robustly founded on the actual marker morphology.

Figure 10 compares our optimal "geomorphic" lateral offsets to prior remote (Figure 10a) and field (Figure 10b) offset measures. The optimal "geomorphic" lateral offsets are similar within uncertainties to prior field and remote measures in ~80% and 90% of cases, respectively. Most differing values are found for low to moderate quality markers. We originally defined the AsZ polygons to represent the marker sections analyzed While our optimal lateral offsets are similar to prior field and remote estimates, their uncertainties are generally larger than the errors previously suggested (Figure ES2_J(a) in supporting information ES2). The latter are generally proposed to be less than 10-20% of the offset. In our study, approximately 45% of the optimal lateral offsets have an uncertainty lower than 20% of the offset. However, this ratio increases regarding the riverbeds (~60%), showing that riverbeds preserve well the lateral offsets. Conversely, a significant population of the optimal lateral offsets have fairly large errors, greater than the offset in ~15% of the cases. While these larger errors partly result from the conservative approach we have followed, they also likely result from the intrinsic variability among the multiple individual offsets we integrated into the offset calculations.

3D_Fault_Offsets also provides the new opportunity to measure vertical offsets on up to nine geomorphic features (optimal vertical offsets in columns BM (ALL) and BN (GEOM) of Table ES-A). There exists no prior measures against which our optimal vertical offsets can be compared. We note that their uncertainties are large, including for riverbeds (Figure ES2_J(b) in supporting information ES2), mainly because vertical slip is insignificant on the San Andreas Fault (mostly <1 m compromising the limits of the data's resolution). Finally, Figure 11 compares the optimal lateral offsets obtained using Lidar data at 25 cm and at 1 m. The offsets are similar within uncertainties in more than 95% of the cases. A few short markers, however, generally less than 10-12 m long (on one or both sides of the fault), could not be resolved well in the 1 m data (uncertainties larger than the offset).

Application of 3D_Fault_Offsets to Measure

Lateral and Vertical Offsets on Owens Valley (USA) and Hope (New Zealand) Faults

We apply 3D_Fault_Offsets to two other marker collections with prior offset measurements to better explore the potential of the code to identify and measure geomorphic markers in different tectonic and geomorphic contexts. We also aim to compare our vertical offsets to prior estimates. Measurements have been done as in section 4. The acronyms AUTO, CLICK, ALL, and GEOM are used similarly.

Owens Valley Fault Markers

The Owens Valley fault (USA) is a NNW-trending, ~80°NE dipping, ~140 km long, right-lateral, seismogenic strike-slip fault with a normal component (e.g., [START_REF] Beanland | The Owens Valley fault zone, eastern California, and surface faulting associated with the 1872 earthquake[END_REF]. Lidar data are available on the fault (www.opentopography.com, EarthScope Southern & Eastern California LiDAR Project), with a possible pixel size of ~25 cm (Haddon et al., 2016). Using the updated version v2 of the LaDiCaoz code, Haddon et al. (2016) have remotely measured the lateral and vertical offsets of 238 geomorphic markers across the fault, most being stream channels. The particularity of these markers is that many of them have low imprints in the topography, which the Lidar data hardly capture (see examples in supporting information ES4a). These subtle traces thus pose an interesting challenge for offset measurements.

We remeasured the lateral and vertical offsets of 20 of those markers (measurements in Table ES-C) chosen to be those having both prior and remote offset measurements, lateral and vertical wherever possible. Eight of them had been qualified as high quality, seven as moderate quality, and five as low quality (Haddon et al., 2016). Their available lateral and vertical offsets range between ~2-35 and ~0-6 m, respectively.

In the following, we have traced the fault as Haddon et al. (2016), considered a constant NE dip of 80 ± 10°at each site, and adopted conservative errors on both the DEM (pixel size of 1 m and vertical error of 0.5 m) and the fault position (assigned horizontal and vertical errors of 5 and 0.5 m, respectively). We discarded regressions with less than 10 points. We have used polygons of similar lengths to the marker sections considered by Haddon et al. (2016) (polygons referred to as "AsH"). Only four markers were long enough to run a second calculation with longer polygons (Table ES-C). In total, we have made 38 series of measures, representing a total of 684 lateral and vertical offset measures.

First, despite their subtle traces, 3D_Fault_Offsets well identifies the 20 markers as they had been identified earlier. In particular, the automatic extraction of their riverbeds is straightforward, with a 100% success rate (Figure 7c, Table ES-C, and examples in supporting information ES4a). The code also well identifies (i.e., with geomorphic relevance) one or both steepest slopes of the stream risers (~55 to 85%, Figure 7c). The top risers are geomorphically well defined in 25-35% of cases, whereas the measured riser bases and zones of maximum elevation more rarely have a geomorphic significance. When the few clearest outliers are manually removed, the above results are enhanced, with a better definition of the riser tops and bases.

As for the San Andreas data set, the mean lateral and vertical offsets obtained from the nine measurements (ALL) and from the geomorphically relevant measurements only (GEOM) are consistent, whether they were measured using AsHad or Longer polygons, and from the AUTO or CLICK approaches (Figure ES2_K in supporting information ES2). Therefore, the ALL and GEOM optimal offsets are also consistent, as shown in Figures 9c and9d. About 60% of our optimal geomorphic lateral offsets are similar within uncertainties to prior field measures (Figure 10d), whereas similarity with prior remote offsets from Haddon et al. (2016) occurs in ~70% of cases (Figure 10c). For vertical offsets, the population of field measures is small (seven measures), hampering any Journal of Geophysical Research: Solid Earth 10.1002/2017JB014863 robust comparison. Currently, similarity between prior vertical field offsets and our optimal vertical offsets is only in 43% of cases (Figure 10f). By contrast, our optimal vertical offsets compare well to the remote vertical offsets of Haddon et al. (2016), with similarity in 75% of the cases (Figure 10f).

The uncertainties on the calculated offsets are generally larger than prior propositions. While ~40 to 70% of the prior field and remote lateral offsets have been suggested to have uncertainties lower than 20% of the offset, about 40% of our optimal lateral offsets record the same ratio (Figure ES2_J(c) in supporting information ES2). With the exception of three poorly constrained lateral offsets (all <4 m), the optimal lateral slips are well constrained, however with uncertainties always less than 40% of the offset. The riverbeds have especially well constrained lateral offsets, with uncertainties lower than 20% of the offset in ~70% of the cases (Figure ES_J(c) in supporting information ES2). Errors previously reported on vertical offsets are also generally low, less than 20% of the offset in most cases (Figure ES2_J(d) in supporting information ES2), which contrasts with the common difficulty to measure small vertical offsets in the field and in topographic data. Uncertainties on a number of prior remote offsets are larger, however. The optimal vertical offsets measured here have uncertainties generally less than 40% of the offset, but this ratio is exceeded for the majority of vertical offsets less than 1 m.

Hope Fault Markers

The Hope fault (New Zealand) is an ENE-trending, NW-dipping (~70°), ~230 km long, seismogenic, strike-slip fault with a reverse component (e.g., [START_REF] Cowan | Late Holocene displacements and characteristic earthquakes on the Hope River segment of the Hope fault[END_REF]Freund, 1971;Langridge & Berryman, 2005;[START_REF] Van Dissen | Hope fault, Jordan thrust, and uplift of the seaward Kaikoura range, New Zealand[END_REF]. Lidar data have been acquired on a 30 km long section of the eastern part of the fault (Manighetti, Perrin, et al., 2015), with a pixel size of 1 m and a vertical resolution of 10-20 cm. Using these remote data, Manighetti, Perrin, et al. (2015) have identified >200 markers and measured their lateral offsets from visual inspection of the back slipped marker traces in map view.

Most analyzed markers are stream channels and alluvial terrace risers. Their greatest particularity is that their trace is significantly weathered due to fast erosion in this region (e.g., Bull, 1991;Manighetti, Perrin, et al., 2015;[START_REF] O'loughlin | Erosion processes in the mountains[END_REF]. Furthermore, many of the markers are covered with vegetation that masks parts of their traces (e.g., Langridge et al., 2014). Therefore, the weathered and vegetated traces of the geomorphic markers across the Hope fault offer new challenging conditions for 3D_Fault_Offsets applications.

We remeasured the lateral offsets of 22 of those markers (82% are stream channels and 18% terrace risers or scarp edges of small reliefs), while measuring for the first time their vertical offsets. We have focused on markers offset by small amounts (most between ~2 and 40 m) to better test the potential of the code in these challenging conditions. Based on the degree of preservation of their geomorphic trace, 12, 8, and 2 of the markers were qualified as high, moderate, and low quality, respectively (Table ES-D; Manighetti, Perrin, et al., 2015).

In the following, we have traced the fault as Manighetti, Perrin, et al. (2015), considered a NW dip of 70 ± 10°a t each site, and adopted conservative errors on both the DEM (pixel size of 1 m and vertical error of 1 m) and the fault position (assigned horizontal and vertical errors of 5 and 1 m, respectively). The acronyms AUTO, CLICK, ALL, and GEOM are used as before. In total, we have made 45 series of measures, representing a total of 810 lateral and vertical offset measures.

In 90% of the markers that are stream channels, the code succeeded to identify the riverbed well in a geomorphic sense (Figure 7d). It also identified well the steepest slopes and tops of the risers and scarps. When the clearest outliers are manually removed (CLICK), the bases of the risers and scarps also represent well the actual geomorphic properties of the markers. As for San Andreas and Owens Valley markers, only the zones of maximum elevation rarely have a geomorphic meaning.

As for the San Andreas and Owens data set, the mean lateral and vertical offsets obtained from the nine measurements (ALL) and from the geomorphically relevant measurements only (GEOM) are consistent, whether they were measured using the AUTO or CLICK approaches (Figure ES2_L in supporting information ES2). Therefore, the ALL and GEOM optimal offsets are also consistent, as shown in Figures 9e and9f.

In 91% of cases, the optimal lateral offsets measured here are similar within uncertainties to the prior remote measures (Figure 10e). The uncertainties on the optimal offsets are also similar to prior estimates, less than 40% of the offset in the majority of cases (Figure ES2_J offsets (<4 m) are poorly constrained with an uncertainty larger than the offset. The riverbed lateral offsets are the best constrained, with ~60% of them having an uncertainty less than 20% of the slip. This shows that riverbeds well record the lateral offsets.

The vertical offsets we measured are low, less than 2 m for the majority of them, with a maximum of ~4 m. As expected from the DEM resolution and the conservative vertical errors we input in the calculations, these low vertical offsets have large uncertainties, and the smallest ones (<1-1.5 m) are not properly resolved (uncertainty similar or greater than the offset, Figures ES2_J(f and f bis ) in supporting information ES2).

Discussion

Of the 88 fault offset geomorphic markers that we analyzed, 3D_Fault_Offsets succeeded to mathematically identify 87 of them, and to measure their lateral and vertical offsets, along with their uncertainties. Overall, the measured offsets compare well to the prior estimates-field and remote. This agreement confirms that 3D_Fault_Offsets may efficiently assist the slip measurements along faults. It also provides new opportunities.

Opportunity to Examine the Preservation of Lateral Offsets Across the Entirety of Markers

Because 3D_F ault_Offsets measures the lateral offset of a marker potentially utilizing nine of its intrinsic geometric features, it offers the opportunity to examine the variability of the lateral offset preservation across the entirety of the marker. If the latter is a stream channel, this makes it possible to examine the preservation of the lateral offset across its "western" and "eastern" (here defined on code-rotated attitudes) risers. We have thus calculated the mean lateral offsets across the western and the eastern risers of the channel markers and analyzed their peak value that represents the most robustly constrained offset. Details can be found in supporting information ES5.

For 70% of the San Andreas stream channels, the west and east riser lateral offsets differ by more than 1 m, that is, more than four pixels, and thus, this difference is significant. About 80% of the channels flowing toward the NE have their western mean lateral offset lower than their eastern mean offset by more than 1 m (Figure 12). About 64% of the channels flowing toward the SW have their eastern mean lateral offset lower than their western offset by more than 1 m (Figure 12). This suggests that the preservation of the lateral offsets across the channels is partly dependent on their flow direction. Across channels flowing toward the NE, the dextral motion of the San Andreas Fault likely leads to the greater trimming of the channel's western trailing edge (trailing edge defined in downstream side of fault), because the channel tends to minimize the lateral translation of its bed (see sketch in supporting information ES5). Across channels flowing toward the SW, a similar tendency of channels to minimize the lateral deviation of their bed leads to the trimming of their eastern trailing edge.

Similarly, about 65% of the Owens Valley channels flowing toward the NE have their western mean lateral offset, hence the offset across their trailing edge, lower than their eastern mean offset by more than 1 m (Figure 12). About 70% of the Hope channels flowing toward the SE have their eastern mean lateral offset, hence the offset across their trailing edge, lower than their western mean offset by more than 1 m (Figure 12).

Trimming of channel trailing edges is thus common along the three faults. While these relations between stream channel flow and sense of fault motion have been discussed for a long time (e.g., McCalpin, 2009;McGill & Sieh, 1991;Wallace, 1990), 3D_Fault_Offsets provides a new opportunity to document them and to quantify the degree of trimming and of preservation of the lateral offsets across the entirety of channel markers.

Opportunity to Examine the Preservation of Vertical Offsets Across the Entirety of Markers

The rich offset record also offers the opportunity to examine the preservation of the vertical offset across the entirety of the marker. First, contrary to lateral offsets (Figure ES2_J in supporting information ES2), errors on riverbed vertical offsets are not systematically lower than those on western and eastern mean vertical offsets (Series A2 of figures in supporting information ES5). This shows that riverbeds are systematically not the geomorphic features to best preserve the vertical offsets. Second, no relation seems to exist between the direction of channel flow, the trimming of either riser, and their vertical offsets. Laterally trimmed risers do not have lower vertical offsets than more preserved edges or beds. In about 60% of the San Andreas stream channels, the riverbed vertical offset differs from that of one or both risers by more than 50 cm (3-5 times the vertical resolution). Generally (~70%), the riverbed has a lower vertical offset than the risers. A reasonable interpretation is that, as a channel goes on flowing after its bed has been vertically offset by the fault motion, it incises the uplifted compartment and hence partly trims the vertical offset, tending to attain a mostly continuous longitudinal profile [START_REF] Ouchi | Flume experiments on the horizontal stream offset by strike-slip faults[END_REF]. By contrast, the two risers are better preserved and hence retain the record of the actual vertical offset better.

The Owens Valley data are too few to examine this issue (Series B2 of figures in supporting information ES5), but 75% of the cases where the riverbed vertical offset differs by more than 50 cm from that of one or both risers are such that the vertical offset preserved by the riverbed is lower than the one preserved by one or both risers. This figure is 76% for Hope channel cases (Series C2 of figures in supporting information ES5). Together these confirm that the riverbeds are not the geomorphic features that best preserve the vertical offsets; those should rather be measured across the flanking risers.

The vertical back slips performed by the code allow the user to examine the plausible vertical reconstructions of the up to nine geomorphic features (see Figure 6 and supporting information ES6, where these reconstructions are shown for the San Andreas markers). The vertical back slip plots confirm that the riverbeds are well identified. Even though they do not generally preserve the entirety of the net vertical offset, they preserve a clear geomorphic signature of the fault vertical motion. The riser steepest slopes are also generally well defined, and their vertical offsets seem to be a fair estimate of the actual net vertical offset across the marker. In many cases, the riser tops are also well defined in the Z dimension and hence provide a relevant record of the vertical offset. The riser bases are generally more poorly constrained. While they are generally not shown on figures of supporting information ES6, because they have no clear geomorphic meaning, the maximum elevation point clouds are expected to provide an accurate record of the net cumulative vertical offset across the markers.

The vertical reconstructions also allow for examining what the original morphology of a marker might have looked like in the vertical dimension. If two marker sections have been incorrectly paired based on the examination of their horizontal traces, the vertical reconstructions should help to detect it (see possible example Sieh145 in supporting information ES3).

Opportunity to Estimate the Vertical to Lateral Slip Ratios

Estimating the ratio between lateral and vertical slip on a strike-slip fault is commonly a challenge because vertical offset measurements are few and possibly small. Yet this information is important to understand

Journal of Geophysical Research: Solid Earth 10.1002/2017JB014863

the kinematics of the fault, the way slip is partitioned on its different segments and branches, and to anticipate the amount of vertical slip that might be expected during a large earthquake on the fault.

3D_Fault_Offsets allows for estimating the ratio of vertical to lateral slip in any series of measurements. For reasons discussed earlier, the steepest slopes are believed to best preserve the net vertical offset, whereas the riverbeds best record the lateral offsets. Therefore, Figure 13 shows the vertical to lateral slip ratios calculated in the three data sets both from the midslope vertical and the riverbed lateral offsets and from the geomorphic optimal vertical and lateral offsets. The San Andreas Fault is the one to have the lowest vertical slip component, in effect showing vertical to lateral slip ratios lower than 10% in almost 70% of the analyzed markers (Figure 13a). The other two faults, Owens Valley and Hope have greater vertical to lateral slip ratios, with ~85% of their total population having a vertical component equal to 10 to 30% of the lateral offset (Figures 13b and13c). This larger fraction of vertical slip is consistent with the two faults having a clear dip-slip component, normal for Owens Valley and reverse for Hope. The vertical to lateral slip ratios found for the Owens Valley fault are consistent with prior estimates (Haddon et al., 2016). By contrast, those found for the Hope fault are larger than previously proposed (~10%, e.g., [START_REF] Cowan | Late Holocene displacements and characteristic earthquakes on the Hope River segment of the Hope fault[END_REF]Berryman et al., 1992;Langridge & Berryman, 2005).

Uncertainties on Lateral and Vertical Offsets

One important issue emphasized from 3D_Fault_Offsets is the significance of the uncertainties on the measured offsets. Several sources of errors are revealed. The first is related to a possible incorrect identification of the two marker sections to be paired (e.g., [START_REF] Mackenzie | Untangling tectonic slip from the potentially misleading effects of landform geometry[END_REF]Scharer et al., 2014;Zielke et al., 2015). The expertise of the user is here fundamental and irreplaceable. However, 3D_Fault_Offsets may assist the user in this identification phase, especially when stream channels are concerned, since it indicates from its very first routine F1 whether it succeeds in identifying the riverbed. A failure to provide this identification may suggest that the proposed marker is not a robust feature. Later on, the success of the code to characterize other geometrical properties of the suggested marker is a further support, or lack of support, of its existence and geomorphic relevance.

Once two paired marker sections are properly identified and their section to be matched well defined, a second source of error is analytical. As we discussed it in sections 2 and 3, multiple sources of errors combine.

While the same multiple errors exist in the field measurements of fault offsets, they are rarely accounted for and the uncertainties provided restrict to the range of reconstructions of the marker trace in map view that the user visually estimates as being plausible. Obviously, the actual sources of errors are much more numerous, and the resulting uncertainties are expected to be larger than generally proposed. In most of the measurements we performed, we indeed found errors significantly larger than previously suggested in the field measurements. The LaDiCaoz code was a first attempt to better quantify the errors on the lateral (Zielke et al., 2012) and vertical (Haddon et al., 2016) offsets of sublinear features, but this quantification still included some simplifications that led to smaller errors than those we derive, even though they were greater than field estimates.

As we discussed in sections 2 and 3, 3D_Fault_Offsets may retain certain outlier points of different origins that may introduce an additional error on the offset estimates. However, the density of outliers points is generally low (attested by consistency between AUTO and CLICK measures; see Figure 8b), while the irrelevant offset calculations are smoothed out in the overall statistical analysis of the results (attested by consistency between ALL and GEOM measures; see Figure 9). Therefore, the outlier points and the misleading best fitting regressions that 3D_Fault_Offsets may produce do not alter the final offset measures and hardly increase their uncertainties.

Another source of uncertainty on the offset measures relates to the length of each marker section to be correlated. Because the original geometry of the marker is unknown, the selection of the two marker sections to be paired is disputable in every marker case, especially those with a complex trace (e.g., Zielke et al., 2015). The lateral and vertical offset values measured with 3D_Fault_Offsets vary with the length of the paired marker sections, even if not dramatically in the cases analyzed here (see Figure 8c). Therefore, there exists a variability in the offset measures and hence an additional uncertainty that is related to the user definition of the original prefaulted geometry of the marker. While 3D_Fault_Offsets cannot recognize which marker sections best represent the original trace of a marker, it allows for producing a series of measures made to integrate the full range of plausible lengths, and as such, it allows for estimating the offset uncertainties related to the marker geometry. Generally, it seems beneficial to take a fairly long section of a marker, especially when the latter is sinuous (i.e., meandering), so as to smooth out the marker irregularities and create an average morphology.

We have shown in sections 6.1 and 6.2 that the preservation of the lateral and of the vertical offsets generally differs across the marker. While the lateral offset is generally well preserved across the riverbed, it may be partly trimmed across one or both risers due to the combination of channel flow and fault motion. Conversely, while the vertical offset is generally well preserved across one or both of its risers, it is commonly trimmed in the riverbed. Therefore, an additional uncertainty affects the offsets, which relates to the geomorphic feature(s) across which they are measured. When the offset is averaged across all geomorphic features that characterize a marker, the uncertainties are large (Figure ES2_J in supporting information ES2), partly due to the intrinsic variability of the offset preservation across the marker. When the variability of the offset cannot be confidently related to trimming or to any other well-understood process, the offset averaged across the entirety of the marker should be considered, along with its large uncertainties. When evidence for trimming (or for any other clear process) is clear, the user might thoughtfully select the feature(s) that have best preserved the lateral or vertical offsets. This would refine the offset values and lower their uncertainties. Such a selection might be needed, for instance, when dense collections of offsets are statistically analyzed to recover earthquake slip increments (e.g., Manighetti, Perrin, et al., 2015). Meanwhile, the variability of the offset across the marker may contribute to quantify the trimming.

In the present study, we have analyzed small offsets, less than a few tens of meters. Would the offsets be much larger, the geometry of the fault plane could induce another source of variability in the measured offsets that would need to be taken into account (e.g., [START_REF] Mackenzie | Untangling tectonic slip from the potentially misleading effects of landform geometry[END_REF].

Finally, the proper functioning of 3D_Fault_Offsets relies on a balance between the DEM resolution, the length of the markers to be analyzed, and the height of their vertical imprint. If the latter is too small, at the limit of the DEM vertical resolution, corduroy or any other noise in the Lidar data may prevent the fair identification of the marker. If the marker is very short, the topographic data points might be too few to allow for its identification. From the measurements we have performed here, we suggest that 3D_Fault_Offsets needs a minimum of one topographic data point per meter to identify short markers no longer than ~10 m (on one or both sides of the fault). Topographic data at a resolution greater than 1 m are thus needed to analyze very short markers. Conversely, if the markers under concern are longer than 10-15 m (on each side of the fault), Lidar data at 1 m resolution are appropriate to identify and measure them well. They might even be preferred to higher resolution topographic data for the latter generally introduce noise that may interfere with the point identification (compare resolutions in Figure 5 andES1_C).

Conclusions

We have developed a new Matlab code, 3D_Fault_Offsets, to automate fault slip measurements across offset sublinear geomorphic markers. The code uses topographic data and remotely measures the lateral and vertical offsets of the topographic 3-D traces of the markers, as they express prominent sublinear geomorphic characters such as riverbed, base, middle and top of risers or scarps, and crests or ridges. The code requires only a small amount of user interaction. It thus limits most of the possible bias that are commonly associated Journal of Geophysical Research: Solid Earth 10.1002/2017JB014863 with fault offset measurements (e.g., Scharer et al., 2014), especially with the assigned uncertainties (i.e., offset ranges). We tested the code successfully on three sets of markers (88 at total) offset by different faults, San Andreas, Owens Valley, and Hope. The success of 3D_Fault_Offsets was attested by its ability to reproduce previously made measurements along those faults, providing in the largest number of cases offset estimates that are within the uncertainties of the previous measurements. This successful test indicates that 3D_Fault_Offsets is working well. Considering the quickly growing number of high-resolution topographic data sets (e.g., Opentopography website, e.g., [START_REF] Bemis | Ground-based and UAV-based photogrammetry: A multi-scale, high-resolution mapping tool for structural geology and paleoseismology[END_REF][START_REF] Johnson | Rapid mapping of ultrafine fault zone topography with structure from motion[END_REF] and the anticipated, corresponding growth of the number of studies that perform fault offset measurements, including earthquake slip measures (e.g., Zielke et al., 2015), 3D_Fault_Offsets might be beneficial to a wide range of researchers in the field. Additionally, the largely "hands-off" approach of the code makes the offset measurements not only faster (more than 5,000 offset measurements were performed here) but also more reliable and reproducible.

In addition to lateral offsets, 3D_Fault_Offsets provides a systematic and rich compilation of the vertical offsets. This opens new opportunities to examine the different slip components and their ratios on the faults (see also [START_REF] Billant | Constraining recent fault offsets with statistical and geometrical methods: Example from the Jasneuf fault (Western Alps, France)[END_REF][START_REF] Mackenzie | Untangling tectonic slip from the potentially misleading effects of landform geometry[END_REF], as the relations between alluvial dynamics and fault motion (e.g., [START_REF] Ouchi | Development of offset channels across the San Andreas Fault[END_REF]. 3D_Fault_Offsets also provides a means to quantify the trimming of the channel banks that results from the combination of alluvial dynamics and lateral fault slip. These insights might be useful to estimate the erosion conditions and the relative age of the lateral offsets.

While 3D_Fault_Offsets cannot determine the relevance of matching two marker sections, it allows for the quantitative comparison of these two sections and hence the examination of the geomorphic relevance of their pairing. It might also allow for the measurement of the morphological complexities of the markers that may exist due to their warping, deflection, tilting, distributed faulting, etc. in the fault vicinity (e.g., [START_REF] Ouchi | Development of offset channels across the San Andreas Fault[END_REF]. Therefore, 3D_Fault_Offsets offers an efficient assistance to the geologists willing to analyze fault and marker morphology and to measure a large collection of offsets. As for any remote measurements, the marker identification performed by 3D_Fault_Offsets may gain confidence if cross-checked with field observations. However, the tremendous amount of offset measures the code produces goes much beyond the possibilities of field measurements.

Finally, 3D_Fault_Offsets can be run with any fault dip, and therefore, it should be applicable to dip-slip faults. The only condition is that distinguishable geomorphic markers must exist that are vertically offset by the fault. This might be the case, for instance, for perched streambeds on top of normal fault footwalls (e.g., [START_REF] Monaco | Late Quaternary slip rates on the Acireale-Piedimonte normal faults and tectonic origin of Mt. Etna (Sicily)[END_REF].

CHAPTER 4: Applying 3D_Fault_Offsets to recover past large earthquake slips on the Wairarapa fault, New Zealand 4.a The Wairarapa fault and earthquakes: prior knowledge from literature

4.a.1 General tectonic context of New Zealand

The global active tectonic context of New Zealand is dominated by the complex junction resulting from the tectonic collision of the Australian and Pacific plates (Figure 17). The deformation results from the transitioning of the type of convergent boundary longitudinally across New Zealand. To the northeast of New Zealand and under the North Island, the Pacific Plate converges and is subducted beneath the Australian Plate at ~45 mm/yr (towards the northern component of the Hikurangi Trough) at an azimuth of 261° forming the Hikurangi Subduction Zone (e.g. DeMets, Gordon, Argus, & [START_REF] Demets | Current plate motions[END_REF][START_REF] Demets | Earthquake slip vectors and estimates of present-day plate motions[END_REF]Han, 2003). The Hikurangi trough extends from the Kermadec trench in the north and terminates in the upper South Island. Underneath Fiordland (i.e. the southern tip of New Zealand) and to the south of New Zealand, the Australian Plate converges and is subducted beneath the Pacific Plate forming the Puysegur Subduction Zone. The Puysegur trench is where the Australian plate is subducting beneath the Pacific plate at a rate of ~35 mm/yr (e.g. [START_REF] Demets | Current plate motions[END_REF][START_REF] Demets | Effect of recent revisions to the geomagnetic reversal time scale on estimates of current plate motions[END_REF]. These two subduction zones are connected along almost the entirety of the South Island by a transform fault/boundary known as the Alpine Fault, and some associated faults (see below) [START_REF] Berryman | Age, height, and deformation of Holocene marine terraces at Mahia Peninsula, Hikurangi Subduction Margin, New Zealand[END_REF].

4.a.2 South Island and Marlborough Fault System tectonic context

The main tectonic feature of the South Island is the Alpine fault. The Alpine fault is a more than 650-km long dextral strike-slip fault with a reverse component that forms the boundary between the Pacific and Australian plates. It has had 480 km of total displacement since its origin in the Late Oligocene-Early Miocene (e.g. [START_REF] Berryman | Age, height, and deformation of Holocene marine terraces at Mahia Peninsula, Hikurangi Subduction Margin, New Zealand[END_REF]. Paleoseismic evidence suggests that the Alpine fault has ruptured in major earthquakes (MW > 7.5) with recurrence intervals of ~200 -300 years with the most recent event occurring in 1720 (e.g. [START_REF] Cooper | Estimates for the timing of the last coseismic displacement on the Alpine Fault, northern Fiordland, New Zealand[END_REF]).

In the upper section of the South Island, the Marlborough Fault system, consisting of a set of 4 major, sub-parallel strike-slip dextral faults, is responsible for transferring slip from the dextral-reverse Alpine fault in the southwest to the Hikurangi subduction and North Island dextral faults in the northwest (Figure 18) (e.g. Freund, 1971;Berryman et al., 1992;[START_REF] Barnes | Recognition of active strike-slip faulting from high-resolution marine seismic reflection profiles: Eastern Marlborough fault system, New Zealand[END_REF]Beaupretre et al., 2012)

4.a.3 North Island and North Island Fault System (NIFS) tectonic context

In the lower section of the North Island, the Australian and Pacific plates converge obliquely at ~34 mm yr -1 of shortening and ~26 mm yr -1 of lateral motion (Figure 19) [START_REF] Barnes | Strain partitioning in the transition area between oblique subduction and continental collision, Hikurangi margin[END_REF][START_REF] Nicol | Tectonic evolution of the active Hikurangi subduction margin, New Zealand, since the Oligocene[END_REF]Beaupretre et al., 2013). Part of the shortening is accommodated by thrust faulting and folding nearby the Hikurangi Trench, 150 km east of Wellington [START_REF] Reyners | Plate coupling and the hazard of large subduction thrust earthquakes at the Hikurangi subduction zone, New Zealand[END_REF][START_REF] Darby | Evidence from GPS measurements for contemporary interplate coupling on the southern Hikurangi subduction thrust and for partitioning of strain in the upper plate[END_REF][START_REF] Wallace | Subduction zone coupling and tectonic block rotations in the North Island, New Zealand[END_REF][START_REF] Wallace | Characterizing the seismogenic zone of a major plate boundary subduction thrust: Hikurangi Margin[END_REF]Nicolet al., 2007). The majority of the lateral motion (i.e. margin-parallel motion) is accommodated by a number of NNE-striking dextral strike-slip faults present throughout the North Island comprising the North Island Fault System (NIFS) [START_REF] Van Dissen | Surface rupture earthquakes over the last ∼1000 years in the Wellington region, New Zealand, and implications for ground shaking hazard[END_REF][START_REF] Beanland | The North Island dextral fault belt, Hikurangi subduction margin[END_REF]Mouslopoulou et al., 2007). The major faults composing the NIFS include the Wairarapa fault, Wellington fault and the Ohariu fault to the southwest, the Ruahine and Mohaka faults in the central regions, and the Waimana, Waiotahi, Whakatane, and Waiohau faults to the northeast. These active right-lateral strike-slip faults have average recurrence intervals of meter-scale surface rupture ranging from ~500 -5000 years [START_REF] Van Dissen | Surface rupture earthquakes over the last ∼1000 years in the Wellington region, New Zealand, and implications for ground shaking hazard[END_REF]. In the onshore region, dextral slip of the Shepard's Gully, Ohariu, Wellington, and Wairarapa faults (Figure 19) accounts for up to 18 mm yr -1 [START_REF] Beanland | The North Island dextral fault belt, Hikurangi subduction margin[END_REF] of the lateral motion of the plates (e.g. Berryman, 1990;Van Dissen et al., 1992;[START_REF] Van Dissen | Surface rupture earthquakes over the last ∼1000 years in the Wellington region, New Zealand, and implications for ground shaking hazard[END_REF][START_REF] Rodgers | World's largest coseismic strike-slip offset: The 1855 rupture of the Wairarapa Fault, New Zealand, and implications for displacement/length scaling of continental earthquakes[END_REF][START_REF] Wang | Infrared-stimulated luminescence dating of the late Quaternary aggradation surfaces and their deformation along an active fault, southern North Island of New Zealand[END_REF]. At the surface, these faults typically dip toward the NW and accommodate the convergence across the NIFS with a reversecomponent of slip, that adds to their dominant lateral motion [START_REF] Carne | Using displaced river terraces to determine Late Quaternary slip rate for the central Wairarapa Fault at Waiohine River, New Zealand[END_REF]. The Wairarapa fault is the only one to have ruptured in historic times (i.e. in 1855) (Ristau, 2018).

4.a.4 The Wairarapa fault and earthquakes (existing general and paleoseismic information from literature)

The Wairarapa fault is the easternmost right-lateral, reverse fault in the southern NIFS at the southern end of the North Island of New Zealand (Figure 21). It is about 120 km long from south Palliser Bay in south to Mauriceville in North. It trends N45°E on average. About 90 km of the fault length is exposed in the northern island, while the fault seems to continue further south on ~30-40 km more offshore Palliser Bay [START_REF] Mountjoy | Morphostructure and evolution of submarine canyons across an active margin: Cook Strait sector of the Hikurangi Margin[END_REF][START_REF] Barnes | Recognition of active strike-slip faulting from high-resolution marine seismic reflection profiles: Eastern Marlborough fault system, New Zealand[END_REF].

On its southern land-exposed section, the Wairarapa fault trace is narrow, and the fault creates a topographic step separating the foothills of the greywacke-dominated Rimutaka Range and the alluvial plain of the Plio-Pleistocene Wairarapa Valley to the east (Figure 20). On its northern portions, the Wairarapa fault cuts across the alluvial valley [START_REF] Rodgers | World's largest coseismic strike-slip offset: The 1855 rupture of the Wairarapa Fault, New Zealand, and implications for displacement/length scaling of continental earthquakes[END_REF]. There, it splays into several secondary oblique-trending dextral faults, the principal ones from south to north are called Kaumingi, Masterton-Caterton, Mokonui and Alfredton (Langridge et al., 2005). The splay network draws a fan enlarging towards the NE, what suggests that the Wairarapa fault has been propagating northeastward over its geological history (Perrin et al., 2016). In its southern part, the Wairarapa fault also seems to splay into secondary oblique faults, the principal ones are shown in Figure 22. This suggest that the Wairarapa fault might have also propagated southwestward over its geological history. This history is supposed to be fairly recent, with a fault initiation age of about 2 Ma [START_REF] Villamor | It's our fault -Wairarapa fault slip rate investigations task: Completion report -Is the Wairarapa fault slip rate decreasing to the north[END_REF].

When viewed at a regional scale (i.e. 1:250,000), the central portion of the Wairarapa fault consists of a series of en echelon mostly left-stepping major fault segments [START_REF] Grapes | The Wairarapa Fault[END_REF], in keeping with its dominant dextral motion. The fault also has a reverse vertical component of slip however. It indeed dips by 70-80° towards the NW [START_REF] Villamor | It's our fault -Wairarapa fault slip rate investigations task: Completion report -Is the Wairarapa fault slip rate decreasing to the north[END_REF][START_REF] Carne | Geometry and scale of fault segmentation and deformational bulging along an active oblique-slip fault (Wairarapa fault, New Zealand)[END_REF], and this dip has resulted in a cumulative vertical throw. The total lateral slip accumulated over the entire fault history is suggested to be about 20 km, while the vertical to lateral slip ratio is about 1/10 [START_REF] Carne | Using displaced river terraces to determine Late Quaternary slip rate for the central Wairarapa Fault at Waiohine River, New Zealand[END_REF].

The alluvial dynamics of the Wairarapa fault are favorable for the generation of tectono-geomorphic markers because it behaves as the boundary between the Rimutaka Ranges on the west and the Quaternary alluvial deposits of the Wairarapa plains to the east [START_REF] Begg | Geology of the Wellington area[END_REF][START_REF] Zachriasen | Late Quaternary faulting of the Masterton and Carterton faults[END_REF][START_REF] Lee | Geology of the Wairarapa area[END_REF][START_REF] Townsend | Later Quaternary displacement of the Mokonui Fault, Wairarapa, New Zealand: A preliminary assessment of earthquake generating potential[END_REF]. This situation occurs since some oblique slip accompanies the dominant strike-slip motion of the fault. The oblique slip forces the block on the West side of the fault to elevate, eventually leading to development of tributaries where the direction of flow is preferentially perpendicular to the fault. Alternating aggradation and degradation episodes have occurred repeatedly during the late Quaternary (e.g. Bull, 1991), which has enabled the development of a dense alluvial network and the addition of many alluvial deposit features, such as terrace risers [START_REF] Knuepfer | Estimating ages of late Quaternary stream terraces from analysis of weathering rinds and soils[END_REF]Bull, 1991). The Holocene was a period of heavy precipitation events due to the warmer climate following the glacial stage (Bull, 1991). Therefore, the alluvial dynamics were high for the Wairarapa fault during the Holocene, which enabled alluvial deposits to form repeatedly across the fault trace. This geomorphic setting enables an ideal landscape for the generation of potential tectono-geomorphic markers of alluvial origin. Van Dissen and [START_REF] Van Dissen | Surface rupture earthquakes over the last ∼1000 years in the Wellington region, New Zealand, and implications for ground shaking hazard[END_REF] had excavated a paleoseismic trench across the Wairarapa fault 17 km north of the Waiohine River (southern-most point on Figure 21) to use the 14 C method for dating the observable past earthquake events. They found evidence for 5 past large earthquakes, and estimated their recurrence interval of ~1500 -1600 years. Currently, the only other sources for paleoseismic investigations along the Wairarapa fault include four uplifted beach ridges at Turakirae Head (Hull & McSaveney, 1996), displaced river terraces formed by the Waiohine River [START_REF] Carne | Using displaced river terraces to determine Late Quaternary slip rate for the central Wairarapa Fault at Waiohine River, New Zealand[END_REF], and some other tectonogeomorphically offset features along the Wairarapa fault (Lensen, 1968(Lensen, , 1969;;Grapes et al., 1984;[START_REF] Grapes | The Wairarapa Fault[END_REF][START_REF] Van Dissen | Surface rupture earthquakes over the last ∼1000 years in the Wellington region, New Zealand, and implications for ground shaking hazard[END_REF][START_REF] Little | 1855 Wairarapa Earthquake Symposium[END_REF][START_REF] Rodgers | World's largest coseismic strike-slip offset: The 1855 rupture of the Wairarapa Fault, New Zealand, and implications for displacement/length scaling of continental earthquakes[END_REF][START_REF] Villamor | It's our fault -Wairarapa fault slip rate investigations task: Completion report -Is the Wairarapa fault slip rate decreasing to the north[END_REF]. Of these other paleoseismic studies, Hull and McSaveney (1996) provide the only other estimation of the timing of the past earthquakes. The ages of the ridges that they dated suggest a recurrence interval for large EQs of ~1900 years. This recurrence interval differs from the one estimated by [START_REF] Van Dissen | Surface rupture earthquakes over the last ∼1000 years in the Wellington region, New Zealand, and implications for ground shaking hazard[END_REF]; however, both results indicate a long recurrence interval (1.5 -2kyr) for earthquakes on the Wairarapa fault. Whether this recurrence interval applies to the entire fault will remain uncertain until more trench or other dating studies are conducted. Due to an incongruence among dating results obtained from optically stimulated luminescence (OSL) ages, global slip rate estimates for the Wairarapa fault range from a minimum of 5.9 -6.5 mm yr -1 (Van Dissen & [START_REF] Van Dissen | Surface rupture earthquakes over the last ∼1000 years in the Wellington region, New Zealand, and implications for ground shaking hazard[END_REF] to a maximum of 11.9 ± 2.9 mm yr -1 (Figure 21) (Lensen & Vella, 1971;[START_REF] Grapes | The Wairarapa Fault[END_REF]Wang, 2001;[START_REF] Carne | Using displaced river terraces to determine Late Quaternary slip rate for the central Wairarapa Fault at Waiohine River, New Zealand[END_REF]. The vertical slip rate is estimated to 0.1-2 mm/yr [START_REF] Van Dissen | Surface rupture earthquakes over the last ∼1000 years in the Wellington region, New Zealand, and implications for ground shaking hazard[END_REF][START_REF] Wang | Infrared-stimulated luminescence dating of the late Quaternary aggradation surfaces and their deformation along an active fault, southern North Island of New Zealand[END_REF][START_REF] Carne | Using displaced river terraces to determine Late Quaternary slip rate for the central Wairarapa Fault at Waiohine River, New Zealand[END_REF].

4.a.4.1 The 1855 earthquake Rupture of the Wairarapa Fault

The Wairarapa fault experienced a Mw > 8.1 earthquake in 1855 [START_REF] Rodgers | World's largest coseismic strike-slip offset: The 1855 rupture of the Wairarapa Fault, New Zealand, and implications for displacement/length scaling of continental earthquakes[END_REF]. Historic accounts of the 1855 Wairarapa earthquake indicate intense shaking (maximum of X on the Modified Mercalli Scale), many instances of landslides, and major surface faulting [START_REF] Grapes | The 1855 Wairarapa, New Zealand, earthquake: Analysis of historical data[END_REF] resulting in the deaths of ~9 people. Recent studies have suggested that the surface rupture of the 1855 Wairarapa earthquake produced the largest coseismic displacement associated with a strike-slip fault worldwide. South of the intersection with the Carterton fault, [START_REF] Rodgers | World's largest coseismic strike-slip offset: The 1855 rupture of the Wairarapa Fault, New Zealand, and implications for displacement/length scaling of continental earthquakes[END_REF] inferred from the observation of dextrally offset channels at Pigeon Bush site (Figure 22) that the dextral slip of the 1855 earthquake reached a maximum displacement of ~18.7 m ± 1.0 m. Yet the observations were done long after the earthquake. [START_REF] Rodgers | World's largest coseismic strike-slip offset: The 1855 rupture of the Wairarapa Fault, New Zealand, and implications for displacement/length scaling of continental earthquakes[END_REF] also estimated the average coseismic displacement to be of 15.5 ± 1.4 m (range of 12.9 -18.7 m) and a vertical component of ~2.5 m over the 16 km length that they analyzed south of Featherston (Figure 22). [START_REF] Rodgers | World's largest coseismic strike-slip offset: The 1855 rupture of the Wairarapa Fault, New Zealand, and implications for displacement/length scaling of continental earthquakes[END_REF] infer the very large slip on the southern Wairarapa fault to be attributed to coseismic slip from the most recent 1855 event from several lines of reasoning: The first is that [START_REF] Grapes | The 1855 Wairarapa, New Zealand, earthquake: Analysis of historical data[END_REF] also studied this locality and attributed the slip to be coseismic. A second is the consistently steep and fresh looking expression of the surveyed scarps and preservation of small-relief features offset across them, which is consistent with recent slip. There have been report of increased erosion rates during the 19 th century due to deforestation in this area, so [START_REF] Rodgers | World's largest coseismic strike-slip offset: The 1855 rupture of the Wairarapa Fault, New Zealand, and implications for displacement/length scaling of continental earthquakes[END_REF] argue that the fault scarp should be more highly degraded if it resulted from several earthquakes. A third point is the [START_REF] Rodgers | World's largest coseismic strike-slip offset: The 1855 rupture of the Wairarapa Fault, New Zealand, and implications for displacement/length scaling of continental earthquakes[END_REF] perceived unambiguity of the smallest dextral offsets that were measured at the five single-strand localities. Of these, the site Pigeon Bush 1 is for them the most compelling example of a single-event offset, yet it is also the largest (~18.7 m). Pigeon Bush consists of two beheaded channels that [START_REF] Rodgers | World's largest coseismic strike-slip offset: The 1855 rupture of the Wairarapa Fault, New Zealand, and implications for displacement/length scaling of continental earthquakes[END_REF] argue to have resulted from two large earthquakes; the shorter offset being the one from the most recent 1855 earthquake. Also, the large slips could be attributed to interseimic creep (i.e. aseismic activity) along the fault. However, the New Zealand Department of Scientific and Industrial Research conducted a geodetic survey across the zone in the late 1970s to test for aseismic creep. This data was measured over a 5-year period and reports confirm that no vertical or lateral creep is occurring on Wairarapa (Darby & Beaven, 2001). This conclusion was reinforced by reports from local residents that no displacement of man-made structures like fences has occurred (Darby & Beaven, 2001). The estimates from [START_REF] Rodgers | World's largest coseismic strike-slip offset: The 1855 rupture of the Wairarapa Fault, New Zealand, and implications for displacement/length scaling of continental earthquakes[END_REF] are larger than the displacements values published in previous studies (e.g. [START_REF] Grapes | The Wairarapa Fault[END_REF]Grapes, 1999; see Fig. 24-26 below). Figure 24 depicts all the lateral offset data classified by authors. Figure 25 displays only the best-constrained lateral displacement values (according to [START_REF] Villamor | It's our fault -Wairarapa fault slip rate investigations task: Completion report -Is the Wairarapa fault slip rate decreasing to the north[END_REF], arranged according to their age (note that these ages are highly uncertain since no absolute dates have been calculated). Figure 26 depicts the best-constrained horizontal offsets measured from only streams and channel-type geomorphic markers. As commonly thought, [START_REF] Villamor | It's our fault -Wairarapa fault slip rate investigations task: Completion report -Is the Wairarapa fault slip rate decreasing to the north[END_REF] indeed regard the measurements obtained from streams and channels to be more representative of the actual offset values compared with other types of geomorphic markers such as terrace risers.

Figure 24 (right) shows that, west of the splaying zone, the lowest lateral offsets measured on the Wairarapa fault are of about 12 m, with a significant along-strike clustering of the offset values around this average 12 m offset. This offset amount is also found further east in the main splay zone, up to the Mokanui splay fault. Larger offsets are also found, and they also seem to cluster around a few average values, on the order of 17 m in the westernmost part of the analyzed fault section, then ~24 m over the entire length of the analyzed fault section, then ~33 m also over the entire length of the analyzed fault section.

Larger offsets then become more distributed. While the above "clustered" offset values are also found in the easternmost part of the fault zone, offset amounts are more variable there, spanning a continuum of offset values from about 5 m up to more than 50 m. Best-constrained lateral offsets show the same patterns, with the along-fault existence of lateral offset amounts around 12 m, 17 m, 23 m, and 33 m, and more distributed offset values north-east of the splaying zone. We note that the age discrimination is too imprecise to reveal any clear pattern.

Although the available offset data are fairly numerous, they are insufficient to assess the past earthquake slip offsets. That was my primary motivation to increase the offset dataset through the application of the code 3D_Faults_Offsets to Lidar data that we acquired on the Wairarapa fault. I will thus come back later in this Chapter on this available offset distribution, to discuss the insights from my new measures.

4.c LiDAR data analyzed on the Wairarapa fault

Currently, LiDAR acquisition can be performed via airborne-, or terrestrial-based sensors. Each acquisition method has its advantages, but the method is chosen by the scientist depending upon the desired scale of study from global to highly-localized features and the as well as the cost allotted to the project. Airborne-based methods cover large areas at generally lower resolutions compared to their and terrestrial counterpart, which obtains highly detailed, local elevation data. For example, respective shot densities for terrestrial-based surveys can typically be two or three orders of magnitude larger than those from airborne LiDAR surveys, thus permitting generation of <10 cm DEMs. However, these surveys are extremely localized and not suited to image an entire fault system, as airborne-based techniques can. We have utilized airborne LiDAR data to survey a 70-km stretch of the Wairarapa fault in New Zealand (Figure 27) in order to map the fault trace and offset geomorphic markers along this section. The data were acquired in the framework of the project CENTURISK funded by ANR (PI: I. Manighetti). We also launched a field campaign to 3 different locations along the Wairarapa fault to obtain highly-localized and very high resolution terrestrial LiDAR data to verify the smaller offsets approaching the spatial resolution limits of the airborne LiDAR data. The terrestrial Lidar was provided by Geoscience Montpellier (PI: S. Dominguez and J. Malavieille).

4.c.1 Airborne-based acquisitions

Airborne-based LiDAR data has been integral to the current PhD study and it has been post-processed in a multitude of ways to map and measure offset geomorphic features along the large strike-slip fault in New Zealand, Wairarapa. Large sections of the Wairarapa fault has been scanned with a LiDAR survey and processed by NZ Aerial Mapping (NZAM) Ltd. The information of the acquisition and processing for Wairarapa fault is recorded below. NZAM collected 1-m spatial resolution LiDAR data over a ~70-km long and ~1-km wide stretch of the Wairarapa Fault. The project area for the Wairarapa fault is that shown in Figure 29. LiDAR and digital imagery were collected on April 8-9, 2012 for Wairarapa, respectively, using an Optech ALTM 3100EA LiDAR system and Trimble AIC medium format digital camera. The data was collected flying at an altitude of 900 meters above the ground, and using a field-of-view (FOV) of 20° either side of nadir. The system PRF was set at 70kHZ. The LINZ geodetic reference mark KB79 (for Wairarapa) was used for the collection of GPS receiver station data during the aerial data acquisition. Independent of the aerial acquisition work, Sounds Surveying Limited conducted a field surveyed to verify the accuracy of the processed ground dataset.

The LiDAR sensor positioning and orientation (POS) was determined using the collected GPS/INS datasets and Applanix POSPac software. This work was all undertaken in the NZGD2000 coordinate system using the data collected at the geodetic reference mark for the DGPS processing. The POS data was combined with the LiDAR range files and used to generate LiDAR point clouds in New Zealand Transverse Mercator (NZTM) map projection, but with NZGD2000 ellipsoidal heights. This process was undertaken using Optech DASHMap LiDAR processing software. The point cloud data was then classified into ground, first, and intermediate returns using automated routines tailored to the project land cover and N terrain. These, and subsequent steps were undertaken using TerraSolid LiDAR processing software modules TerraScan, TerraPhoto and TerraModeler.

The Trimble camera images were developed into 8 bit per channel uncompressed TIFF format images. The LiDAR POS data was transformed for use with the camera, and this data was used with the automated classified ground LiDAR point cloud data to produce orthophotos with a ground sample distance of 0.2 m. Comprehensive manual editing of the LiDAR point cloud data was undertaken to increase the quality of the automatically classified ground point dataset. This editing involved visually checking over the data and changing the classification of points into and out of the ground point dataset. Attention was particularly focused on areas of vegetation. The Trimble orthophotos were used as a backdrop when undertaking the manual editing. The height accuracy of the ground classified LiDAR point was checked using the open land-cover survey check site dataset that Sounds Surveying Ltd. surveyed. This was done by calculating height difference statistics between a TIN (Triangulated Irregular Network, to be explained in the next section) of the LiDAR ground points and the check points. The standard deviation is ±0.05m. The positional accuracy of the LiDAR data has been checked by overlaying Sounds Surveying Ltd. surveyed data over the LiDAR data displayed coded by intensity. The data was found to fit well in position.

The supplied products are all in terms of NZTM, Lyttelton 1937, and Wellington 1953 vertical datum. The dataset has been tiled in 1:2000 NZ Topo50 map sheets. The point cloud has been classified into ground, above ground, and unclassified. Figure 28 represents an unclassified (i.e. unprocessed and raw) LiDAR data tile for the Wairarapa dataset. As can be seen in the figure, all of the LiDAR return points are represented including the return points from the vegetation and topographic surfaces. Therefore, these points can be identified and classified according to numerous different methods including intensity and timing of return to the sensor. The longest time for return usually equates to the last return points which are commonly classified to be the topographic surface (i.e. ground) points. Ground LiDAR points are classified as ground returns (i.e. last returns) as well as supplementary points added to the dataset (Figure 28-29). This dataset has been processed in ASCII XYZ (mE mN O) file format. Above ground LiDAR points have been identified as having elevations greater than the points in the ground dataset. This dataset is also processed in ASCII XYZ (mE mN O) file format. Unclassified LiDAR points contain all of the point cloud data collected by the LiDAR sensor. This dataset was processed in LAS 1.1 file format. The orthophotos are 0.2m GSD. The Trimble AIC camera is a semi-metric camera and so the orthophotos are subject to mismatches at photo stitch seam lines. There are also 0.5m interval contours that were processed. The contours were interpolated using the LiDAR ground point cloud dataset. The dataset of the Wairarapa airborne LiDAR was received from NZAM Ltd. in the raw and classified datasets, and split into individual tiles along the fault for easier manipulation of the data. The LiDAR in the raw point cloud format is not visually appeasing for visual interpretation. Therefore, it is necessary to convert this type of dataset into format that can be visually assessed. The most common format for geologic studies is called the digital elevation model (DEM).

4.c.1.1 Digital Elevation Models (DEMs)

The DEM represents topography as measured by the heterogeneously distributed 3D LiDAR returns as a 2D grid of elevation values spaced at a constant resolution (Arrowsmith

Unclassified airborne LiDAR tile along Wairarapa fault (BN342224 tile)

Ground-classified airborne LiDAR tile along Wairarapa fault (BN342224 tile)

& Zielke, 2009). A digital elevation model (DEM) can be generated from the LiDAR point clouds by rasterizing (i.e. gridding) the points into an equally-spaced mesh, most commonly displayed as pixels. Each pixel is a digital grid file that records the elevation information interpolated from the LiDAR point cloud. LiDAR-derived DEMs present one type of highresolution topographic map that enables the highly-detailed mapping and surveying of fault geometry and displaced geomorphic markers that are needed to extract invaluable prehistoric earthquake information. A common method for identifying the appropriate DEM spatial resolution (i.e. pixel-size) has been defined by Hu (2003):

s= √(A/n)
where s is the estimated grid size, n is the number of sample points (i.e. total number of points in the point cloud), and A is the area containing the sample points (km2) (Langridge et al., 2014). The A/n relation defines the average shot density (i.e. density of points in the point cloud). However, the points in a point cloud are not evenly spaced, so the density will change throughout the area. The shot density tends to be greatest toward the center of the airborne survey where considerable overlap of different swaths occurs (Figure 30). The shot density is usually lowest towards the edges of the surveys where only one swath is usually performed. Therefore, it is important to find a balance when defining a spatial resolution for the DEM as to not create unrealistic interpolation in areas of low shot density. While the actual spatial resolution depends largely on data acquisition strategy and vegetation coverage, many LiDAR datasets are sufficiently dense to allow for resolutions of ~1m. More recent surveys with shot densities > 10 points/m² enable grid resolutions as high as ~0.25 m. DEMs have the benefit of creating visualizations of the fault area of interest including, hillshade-, slope-, and contour-plots that are not available in field investigations. These visualization approaches have further improved the capabilities of the high-resolution DEMs for tectono-geomorphic studies. The DEMs for this current PhD project have been generated using a variety of rasterizing algorithms at varying resolutions. The generation process for the Wairarapa fault airborne data has been the utilization of the software program LASTools.

Swath Overlap = Higher point density

4.c.1.2 LASTools

The Wairarapa bare-earth DEMs were generated from the raw point cloud data using LASTools. LASTools (Figure 31) is a collection of highly efficient, batch-scriptable, multicore command line tools to classify, tile, covert, filter, raster, triangulate, contour, clip, and polygonise LiDAR data. The software is the fastest and most memory efficient program for batch-scripted LiDAR processing (Isenburg et al., 2006a,b). It can convert billions of LiDAR points into useful products such as DEMs with very little memory requirements. Since, the imported files will be in LAS format, LASTools only requires one tool to convert the point clouds into high-resolution rasterized DEMs. o Kill: This parameter does not generate a raster triangle with edges longer than the specified units (if the points are spaced too far apart -meaning that the interpolation of the point clouds would lead to inaccurate estimates of the elevation data). This parameter was set to 50 units. o Step: the step refers to the pixel resolution (in m). All airborne LiDAR data was set to a step size of 1 (1-m). o Output format: Supported output formats. All DEMs had an extension of ASC to be compatible with 3D_Fault_Offsets. Producing DEMs of individual tile sections (Figure 32) is beneficial for computer processing time and studying highly localized features, but it is necessary to create a mosaic of all tiles together in order to study the entire fault trace at once. In the LasTools' graphical user interface (GUI), there is an option to mosaic all LiDAR tiles together when generating the DEM creating a seamless stitch among all files. Figure 33 displays the mosaic of all of the tiles in the Wairarapa airborne LiDAR dataset. In this format the fault trace and corresponding geomorphic markers can be mapped in high detail using software packages such as ERDAS Ermapper. As explained earlier, during the acquisition of the LiDAR dataset along the Wairarapa fault, optical images (i.e. orthophotos) were taken and processed into the same respective tiles as the LiDAR point clouds. These orthophotos assist in the identification of antropogenic influences on the geomorphology, such as the location of man-made roads, houses, fencelines, etc, and on the presence of vegetation. These orthophotos can thus assist in the determination of deformation due to tectonic or anthropogenic processes. For instance, the presence of irrigation or other farming practices along the fault trace could unnaturally alter the direction of certain alluvial deposits that cross the fault. Figure 34 and 35 show the exact same location along the Wairarapa fault in both bare-earth LiDAR and optical imagery. As can be seen in the optical image (Figure 35), there is farming practices being undertaken that is not evident in the bare-earth DEM.

4.c.2 Terrestrial-based acquisitions during New Zealand field campaign in April 2016

Occasionally, the horizontal or vertical length of the cumulative or coseismic slip of an offset tectono-geomorphic marker can be small enough that it approaches the limits offered by the spatial resolution of the DEM on which it is measured. For instance, a horizontal offset measured at 2 -3 m could result in large errors or inaccurate measurements if the DEM resolution is 1-m. Since the Wairarapa airborne LiDAR has a DEM resolution of 1-m, validation of the offset measurements obtained needed to be verified for smaller offset features along the Wairarapa fault. To validate the efficacy of this airborne LiDAR to produce robust offset measurements even for smaller offsets (i.e. < 10 m), a field campaign was conducted in April 2016 at several locations along the Wairarapa fault in New Zealand to acquire and treat terrestrial LiDAR data . Terrestrial LiDAR is a land-based laser scanner, which combined with a highly accurate differential GPS, behaves the same as airborne or satellite sensors, but at a different scale. The terrestrial scanner measures the radial distance of the surrounding area at different angular resolutions. The LiDAR scanner emits a single beam and uses an interior rotating mirror to distribute the laser covering a large field of view. The main advantage associated with terrestrial LiDAR is that the acquisition yields an extremely high point density that enables the generation of very-high resolution DEMs (e.g. 10 -50 cm). This very-high resolution can be used to test the accuracy of measurements on rougher resolution DEMs. Since the feasibility of obtaining terrestrial LiDAR along these faults was uncertain due to potential dense vegetation or no access to private property, many sites were initially chosen as good probable locations. There were a total of 11 initial sites chosen along the Wairarapa fault named Site A through Site K. After the field campaign, 3 sites were analyzed for the Wairarapa fault, Sites F, I West, and I East. Since Site I was so large, it was divided into 2 sections to allow for easier computer processing of the LiDAR scans. These 3 sites are described below.

Site F on the Wairarapa fault consists of a series of offset river terraces and ancient channels formed by the adjacent Waiohine River (Figure 37). The site is located between Featherston and Woodside, New Zealand on the North Island. These terraces have been studied previously to determine the slip rate of the Wairarapa fault (e.g. [START_REF] Carne | Using displaced river terraces to determine Late Quaternary slip rate for the central Wairarapa Fault at Waiohine River, New Zealand[END_REF]. Site F was the ideal location to acquire terrestrial LiDAR data being that it is free of vegetation, the northern and southern components on either side of the fault are virtually flat, and it was dissected by a well-defined primary fault scarp and a secondary sub-parallel branch. The southern fault trace is the main fault with a fault scarp relief of ~3-4 m. Prior to the field campaign, we had identified and measured ~17 offset geomorphic markers across both the main fault and secondary branch of the fault. Their horizontal back-slip values ranged from 3 ± 1 m to 43 ± 3 m. There was a total of 17 scans for Site F ranging in elevation from 98 m -106 m. Sites I West and I East consist of a series of offset ancient beheaded channels (Figure 38). These sites are located between the towns of Upper Plain and Opaki directly due west of Masterton, New Zealand on the North Island. There are 2 prominent fault traces located at this site with a possible reverse component at the bottom of the slope due to considerable topographic relief on the north side of the fault. The site is located on farm land on a constant slope above a stream channel running parallel to the fault trace. It comprises a large, highslope area with little vegetation. Prior to the field campaign, we had previously identified and measured the horizontal offsets of about 10 geomorphic markers across the entirety of Site I. These offset markers range in horizontal back-slip value from 7 ± 3 m to 66 ± 10m. There was a total of 47 scans for the entire Site I with 18 LiDAR scans attributed to Site I West and 29 LiDAR scans attributed to Site I East.

4.c.2.1 Terrestrial LiDAR data acquisition techniques

A FARO Laser Scanner Focus3D X 330 was the terrestrial LiDAR scanner used during the field campaign in New Zealand. It belongs to the laboratory Geoscience Montpellier (under lead of S. Dominguez). The scanner is a high-speed three-dimensional laser scanner for detailed measurement. It uses laser technology to produce three-dimensional images of complex environments and geometries in only a few minutes per scan. The resulting images are an assembly of millions of 3D measurement points. The scanner is mounted on a tripod and freely revolves 360° horizontally and 300° vertically to obtain the point cloud for a single scan. The FARO Focus3D X 330 can scan objects up to 330 m from the focal point, even in strong sunlit conditions. The scanner has a range from 0.6 -330 m on a reflective surface of 90%. At 307 m, the scanner can acquire 976,000 points/s. This large amount of points translates to a very high point density. The error associated with each point is ± 2 mm, which is defined by a one sigma standard deviation at 10 m and 25 m. The integrated GPS receiver enables seamless combinations of individual scans during postprocessing. The 3D digitized data can be easily imported into post-acquisition software for the reconstruction of topographic surveys. There are several pertinent parameters that need to be toggled to the desired scan location accordingly. For the purpose of this thesis, only the most important parameters to scan quality will be explained here. The two parameters of such importance include the scan range and the scan resolution. The scan range defines the amount of scanner rotation during acquisition and is thus directly related to the area of capture. The scan range was left at maximum for every scan conducted, horizontal from 0° to 360° and vertical from -60° to 90°, in order to maximize the area captured per scan. The scan resolution can be toggled from 1/1 (i.e. full resolution) to 1/32 (i.e. minimum resolution), and it directly affects the quality of the scan and the scanning time. Full resolution (1/1) reduces the noise in the scan data and thus increases the scan quality, which results in an increased scanning time. Every scan was performed at 1/2 resolution resulting in superior quality for the purposes of the project and a scanning time of ~12 minutes per scan. Therefore, each site took about several hours to scan the entire location. With these parameters, the scan recorded ~20,480 points per 360° rotation of the scanner.

Since the sites averaged several hundred meters squared, all of the site locations in New Zealand required multiple scans to capture the entire site with the LiDAR scanner. Therefore, in order to facilitate rapid post-processing stitching of multiple scans, objects of constant size and dimensions were used in each scan so that they could be digitally identified later. These objects enhanced the scan environment with artificial targets. The targets that were used included five spheres of the exact same size. The spheres acted as reference objects in the scans so that there would be no problem during the registration process afterward.

References, or targets, are used to register multiple individual scans that are each on their own coordinate system onto a single, aligned coordinate system. Three of the spheres were kept in the same geographical location for two consecutive scans, while two of the spheres were moved to new locations. Mathematically, three corresponding references in two scans are needed to register each to the other. This process was repeated until the entire site was captured. The scan resolution, diameter of the spheres, and their distance to the scanner and each other had to be considered with their individual placements. Given that the scans are performed in a natural environment, some of the spheres were difficult to locate during registration. All scans for a single location were saved into a "Project" folder to easily discern between locations.

4.c.2.2 Post-processing of Terrestrial LiDAR scans

The purpose of the post-processing of the terrestrial LiDAR scans is to transform the raw individual LiDAR point clouds into very-high resolution gridded DEMs that can be used to analyze and measure offset geomorphic features. The process of this transformation requires the use of a suite of software including FARO SCENE, CloudCompare, and LASTools. The following sections will describe the process through each software package.

FARO SCENE is a comprehensive 3D point cloud processing and managing software that is designed for viewing, administration, and working with extensive 3D scan data obtained from high resolution 3D laser scanners. Since the purpose of FARO SCENE is to process extremely large scan files complete with millions of LiDAR points, the computer's system requirements are very important when considering using the software. For the purposes of generating DEMs, SCENE software is used to register individual scans and potentially join the individual scans into a cluster that incorporates all of the scans of an entire site. Therefore, SCENE software was used to initially stitch together the individual scans of a site.

For, those individual scans that could not be stitched together automatically through SCENE software, manual manipulation was performed using CloudCompare. CloudCompare is a 3D point cloud editing and processing software. It functions to compare and manipulate dense 3D point clouds directly. It relies on a specific octree structure, which is a tree structure where each parent node has exactly eight children. Since most point clouds are acquired by terrestrial LiDAR scanners (i.e. very large point density), CloudCompare is impressive in that it can support huge point clouds on a standard laptop -typically more than 10 million points. The software has the capability to store about 90 million blank points per gigabyte of memory. Therefore, CloudCompare was utilized to manually fit together all of the scans for a site through the recognition of common points in multiple scans.

High-resolution DEMs were then generated from these 3D point clouds using LASTools, and the same techniques described for the airborne data. However, the step-size was reduced to 0.3 enabling the generation of 30-cm resolution DEMs. The final products are depicted in Figures 40414243, and they have been used to measure and verify the horizontal and vertical components of slip for the offset geomorphic markers in these regions.

4.d Mapping the Wairarapa overall fault trace

Figure 44 shows in the color-coded line the extent of the Wairarapa fault section which we mapped in detail in the Lidar data. According to seismic imaging mapping completed by the Holdgate&Grapes (2015), the Wairarapa fault extends about 40 km into the Palliser Bay and the Cook Strait south of the North Island. As any other fault worldwide, the Wairarapa fault is segmented laterally at different scales [START_REF] Manighetti | Self-similarity of the largest-scale segmentation of the faults: Implications for earthquake behavior[END_REF](Manighetti et al., , 2015)). We know that earthquake slip is sensitive to (at least) the largest scales of fault segmentation [START_REF] Manighetti | Earthquake scaling, fault segmentation, and structural maturity[END_REF]. Lateral segmentation is responsible for slip heterogeneity on fault planes, with the zones separating the major segments (commonly referred to as "inter-segments") being zones of slip deficit, while slip is larger on the segments [START_REF] Manighetti | Self-similarity of the largest-scale segmentation of the faults: Implications for earthquake behavior[END_REF](Manighetti et al., , 2015)). We also know that earthquake and fault slips are variable along the fault length, and hence must be measured locally to be relevant (Manighetti et al., 2001[START_REF] Manighetti | Evidence for self-similar, triangular slip distributions on earthquakes: Implications for earthquake and fault mechanics[END_REF]. We thus performed our slip analysis by "fault pieces" along-strike, with these pieces being the major fault segments visible over the 70 km Lidar stretch.

I describe here these major segments that segregate the fault. The Wairarapa fault trace that we mapped is color-coded in Figure 44 to segregate the fault into major segments as defined by various geologic criteria discussed below. The segments are identified as AB, CD, E, FG, H, I, JK, LM, NO, PQ, RS, T, UV, and WXY (this complex labeling results from AB for instance being divided into two smaller segments, A and B; etc.). I remind that, for time reasons, I only analyzed here the segments AB to NO. These AB to NO fault sections actually covers the "core" of the fault, before it splays further north and south into secondary oblique faults (as described in Perrin et al., 2016) (Figure 44). Therefore, we expect to find localized offsets along this investigated section, and larger amounts of displacements than can be found further north and south of our studied section. Altogether, using the fault segmentation as a means to locally investigate slip enables analyzing the offset over fault sections on the order of ~3 (e.g. segment I) -8 km (e.g. segment NO), and hence to consider the earthquake slip along-strike variability.

Although I have measured the offsets along segments PQ -WXY (see Table Wairarapa_Offset Measures

4.d.1 Analysis of fault trace and offset markers per major segments

I provide below the maps of the fault segments which I identified and along which I made the offset marker analysis.

For each segment below, I provide a bare Lidar image of the segment zone (a), a view of the same zone with our fault and marker mapping (b), and a view of the segment fault map where arrows indicate the segment tips (c). Note that, while the fault mapping is accurate, the marker mapping is not, because our main objective was to locate the different markers and discriminate them with colors, not to map their trace precisely. The precise characterization of the marker geometry is done later with the 3D_Fault_Offset code. The colors of the markers are reported in the offset measurement Segment AB is a ~5 km long segment which splays in the south into a secondary oblique fault, and is associated with a small number of secondary faults in its northern compartment. I could make 28 offset measures along segment AB. The southern end of segment AB (i.e. the southern extent of our LiDAR coverage) represents a complex transition into splay faults (Figure 44). Therefore, the southern tip of segment AB marks the bottom of the "core" of the Wairarapa fault where we expect the major slips to have occurred. Segment CD is a ~6 km long segment which is typically made of left-stepping smaller segments in keeping with the dextral motion of the main fault. Therefore, most en echelon segments represented are parts of the main fault, not secondary faults. Segment CD is separated from segment AB by a left step-over. I could make 43 offset measures along segment CD. Segment FG is ~6 km long and has a simpler trace than the other segments. Yet this trace is also segmented at smaller scale (left-stepping arrangement). Segment FG spays at its western tip in its relay zone with segment E. It shows other oblique splays off its trace in its northern compartment and a few small push-ups in its southern compartment. The separation between segments E and FG is a left-step and is marked by the splay tip zone of segment FG. I could make 34 offset measures along segment FG. Segment H is ~4 km long, shorter than the other segments. Yet its limits are clear, marked by clear and large step-overs with both segments FG in south and segment I in north. The stepover with FG is compressive, while normal faults exist in the relay with segment I. As most other segments, segment I splays at its southern tip, while its trace is made of smaller left-stepping segments. I could make 37 offset measures along segment H. Segment I is also short, ~3 km long, and its limits are clear, marked by clear and large step-overs with both segments H in south and segment JK in north. The segment I splays at its southern tip (some of the splay faults have a normal component), while its trace is made of smaller left-stepping segments. I could make 20 offset measures along this segment I. Segment JK is one of the longest, ~7 km long. Its trace is also the simplest, with a fairly narrow and continuous trace. Its separation with segment I in south is a step-over (apparently extensional), while the tip of segment JK in north is mainly marked by the splay network of next segment LM and a push-up. I could make 56 offset measures along this segment JK. Several of these measurements could be done both in airborne and terrestrial Lidar data. Segment LM is also fairly long, ~6 km long. Its trace is also simple, with a fairly narrow and continuous trace, yet showing left-stepping segmentation at smaller scale. The segment splays in the south in the step over with previous segment JK, and also splays in the north in the step-over with the next segment NO. I could make 17 offset measures along segment LM. Segment NO is also among the longest, ~8 km long. Its trace is more complex than that of the other segments, for it is the only segment to show a pronounced bend (20-30°) in its center. It is noteworthy that this bend occurs where the Wairarapa fault starts splaying into large oblique faults in the north (see Figure 44). At the bend, complex secondary faulting has developed. Apart from the bend, the segment trace geometry is fairly simple and continuous. The segment splays at both tips, in opposite compartments. I could make 85 offset measures along segment NO.

4.d.1.1 Segment AB

4.d.1.2 Segment CD

4.d.1.8 Segment LM

4.d.1.10 Overall

As said before, the analysis of the fault has been distributed into major segments to analyze the slips at a more local level. At some places, the slip is distributed among the main fault (MF) and adjacent secondary faults (less pronounced traces thus likely accommodating less total slip than the MF). However, as the fault is segmented in a series of left-stepping en echelon major segments, many of the close, sub-parallel faults appearing to be secondary in nature are actually the overlapping next segment of the MF.

4.e Application of 3D_Fault Offsets

On the Wairarapa fault and its LiDAR data, I used 3D_Fault_Offsets exactly as I did for the JGR paper except we have removed the 'CLICK' analysis to avoid any sense of bias in these measurements. So, all of the measurements have been performed with the 'AUTO' removal of outlying points. Also, since the Wairarapa fault has some oblique slip, a dip angle of 80° ± 10 (towards NW) was considered for every marker. The DEMs that were used were mostly 1m resolution, except for 36 markers that were measured on both the 1-m airborne and 30-cm resolution terrestrial LiDAR. The uncertainty values I used in the code were conservative with a [1m 1m 0.5m] uncertainty for the DEM horizontal and vertical resolution, and a [5m 5m 0.5m] uncertainty for the placement of the fault trace.

4.e.1 Identification of offset markers

I first visually identified the offset markers from their geomorphic imprint in the topography. I used the GIS software ERDAS Ermapper and Imagine to identify and map the fault trace and the geomorphic features that it offset. This software package, as well as other GIS suites, allowed for the illumination of the topography in a very convenient manner for analyzing even very subtle features. Having an adjustable sun angle enables the illumination of features in the topography facing in all cardinal directions. Figures 57a -57c highlight the capabilities of the identification and mapping of geomorphic features using ERDAS Ermapper and Imagine. It is important to remind that the fault trace is precisely mapped in the topography, but the geomorphic features are only 'highlighted'. Certain criteria for identifying the northern and southern components of an offset feature were aided by the mapping software. The sun shading aided in highlighting the slopes for terraces. An offset stream that is still linked is a definitive example of a causal link (orange and red traces in Figure 57c). Also, the abandoned channel (purple trace in Figure 57c) can be confidently reconstructed with the channel to its right (orange trace) given that no other major channels are in the vicinity. The purpose of mapping the geomorphic feature was to identify its location and apparent trace so that it can be uploaded to 3D_Fault_Offsets to measure its displacement. Not every offset geomorphic feature has been mapped, because the purpose was to identify those features whose northern and southern components could have a causal link. Also, since New Zealand is a humid environment, many features were too degraded to be measured confidently. Once all of the geomorphic features were identified in the Ermapper and Imagine software programs, I generated individual DEMs for each individual marker, so that the code could run efficiently given the size of the smaller DEMs.

Then, using 3D_Fault_Offsets, I drew polygons around the identified marker sections in order to encompass the trace of the markers. It was important to focus the polygons on sections of the marker that could have defined its overall trace before a surface rupture. If there was doubt as to the exact location to define the trace of the marker, several polygon lengths and/or widths were used to measure the offset (this will be discussed further in Section 4.e.2 below). One important remark is that the code convention in its current state forces the tracing of the fault according to the direction of its dip (i.e. it uses the right-hand rule). Since the Wairarapa fault dips towards the northwest, we must trace the fault starting from the east. This convention forces the DEM to be rotated clockwise, and thus finally be oriented 180° from true north (i.e. the image appears upside-down). The code thus rotated the LiDAR views of all markers making some cases a bit difficult to well observe the marker and fault. Due to this convention, "west" in the measurement table (i.e. TABLE Wairarapa_Offset Measures-Stewart-PhD Draft 21Oct18) is the actual East, and reciprocally. Also, "north" in the measurement table is the actual South. This is indicated clearly in the measurement table. Also, the DEM figures in the synthesis plates in Appendix II have been rotated 180° to facilitate analysis of the Wairarapa measurements. The rotated DEM (Figure 58b) can cause some disorientation at first, but you are still able to properly identify the pertinent features that should be included in the polygon. First, the causal relationship between the northern and southern counterparts of the marker is evident given the similar sun shading of the slopes and the fact that they are both the oldest terraces in this section of the fault. The polygons were placed to include the possible maximum elevation, the riser top, the riser mid slope, the riser base, and the minimum elevation. Obviously, only the western (i.e. actual eastern) features will be relevant in this example. The polygons' lengths were considered to capture a mostly linear section of the marker, since linear features provide the best estimate to create a piercing point with the fault trace. Figure 59 depicts a situation where the polygons have to be placed according to the marker morphology. The polygon width includes all of pertinent geomorphic features. Stopping the southern polygon (actual northern) before the kink in the channel ensures a linear definition of the marker. Including the portion of the channel after the kink would lead to a smaller offset value and a possible underestimation of the true tectonic offset.

4.e.2 Recognition of the geomorphic features best preserving the lateral or vertical offsets

While analyzing the offset marker sections with the code, I carefully examined the 2D and 3D views of the code-identified geometric features, such as the vertical back-slip figures, the 'calc4' figures showing the linear regressions of all features, and the 3D plots showing the placement of each point on the DEM to identify which geomorphic features were best representing the marker and best preserving its offsets. This generally allowed me to identify the one geomorphic feature among the nine that would best represent the marker. When it exists, this "best-representing feature" is underlined in the measurement table. I will later consider these underlined measures as those possibly the best to represent the lateral or vertical offsets of a given marker. Figure 60 depicts the three-dimensional image of one quadrant of the marker. In this case, it is the northwest (actual southeast) quadrant of marker WF359875, which is a small active channel. This figure is usually utilized as the first step to confirming that the mathematical representation of the geomorphic feature is being realized in the topography of the LiDAR instead of representing an artifact or another irrelevant feature in the polygon. As previously discussed in the JGR paper, airborne LiDAR data can be subjected to artifacts such as the corduroy effect, which produce sudden peaks and valleys (on a cm-scale) that could affect the placement of the points. Therefore, a 3D representation of the specific features aids in determining the true placement of the points. As can be seen in Figure 60, the points for the riser base, riser mid slope, and riser top are well discerned in the topography, especially when considered overall (i.e., locally, some of the points are poorly placed); however, the maximum elevation is likely an artifact of the polygon edge. It becomes more apparent in this figure as to the origin of the points if they are not defining the desired feature. For example, anthropogenic features (such as roads) have sharp edges and can sometimes be picked by points for riser top, riser mid, or riser bottom. Figure 61 will also help to ensure that the geometric feature under concern is more or less linear. The lateral and vertical offset amounts and their corresponding uncertainties are also displayed on this figure, but they should not be considered at this stage, since the user can become biased by offsets with similar amounts. It is also evident now that the maximum elevation from Figure 60 was in fact an artifact of the polygon edge, and it was actually removed during the 'AUTO' removal of outliers. All lateral geomorphically-relevant features will be denoted, and they would be put in bold in the measurement table (red for riverbed). All geomorphically-irrelevant features would be put in pale grey in the measurement table. Based on the well-located points on Figure 60, and the confined 2D spatial variation in both the north and the south on Figure 61, the western mid slope (actual eastern mid slope) would be considered the best lateral geomorphic feature for this marker. It would then be underlined in the measurement table to distinguish this ranking from the rest of the features.

The relevant geomorphic features for the vertical component of slip are regarded to be the same as the lateral features. Since we can only confidently determine the geomorphic relevance of the horizontal offset features, we consider the vertical component to be an extension of this relevance. If the lateral component is well-preserved in the topography, then the vertical component should be as well. Therefore, we then analyze the vertical back-slip figures for the geomorphically-relevant features only (see Appendix II). Figure 62 is a vertical back-slip figure for the eastern (actual western) riser base of marker WF359875. The southern (actual northern) section is back-slipped toward the northern piercing point with the fault trace. Well-constrained vertical offsets are considered when the 2D point dispersion is kept to a minimal (meaning a linear connection of the points as seen in Figure 62). The best vertical component of slip is underlined in the measurement table. During the above examination of the code results (including the lateral and the vertical), I also identified all the features that would appropriately represent the marker: as in the JGR paper, I called them the GEOM features and their offset is in bold in the table. Figures like Figure 62 are provided for every marker measured along the Wairarapa fault in a collection of figures that can be found in Appendix II. Considering the entire population of measurements, the most robustly constrained lateral offset is generally the riverbed (this feature is underlined the most in the measurement table), or the eastern mid slope (discussion further below). For terrace risers, the mid slope is generally the most robustly constrained feature. The maximum elevation is typically never the most robustly constrained feature. For the vertical component of slip, the most robustly constrained feature is generally the mid slope. Largely, the riverbed very nicely shows the vertical slip; however, this preserved slip is usually lower than the original recorded by the morphology (reasoned later in Chapter 4).

4.e.3 Issues and approaches regarding offset measurements

In most marker cases, I made at least two measurements using marker sections of different lengths so as to better capture the actual offsets. These are generally called "Short" and "Long" in the table. Since doubt is prevalent in some cases when determining if the geomorphic representation of the topography is strictly due to tectonic activity, erosion, or both, multiple polygons forms were applied to most of the offset measurements. Figure 63 depicts a marker case that warranted a 'short' and 'long' measurement. The 'short' case captures the linear trend of the terrace riser closest to the fault. The red square in the 'long case' denotes the location of a bend. The bend could have been caused by moderate tectonic activity on the fault that deflected the trend of the marker closest to the fault trace. However, the bend could have also formed from erosional process of the channel that formed the terrace. The uncertainty as to the origin of the shift in the trend of the marker merits the measurement of two different cases that could contribute to the possible range of offset values. Also, in cases with a meandering river, considering multiple lengths of the marker helps to average out the sinuous nature of the channel to obtain an offset amount.

Many of the markers are channels, which are or have been deflected while the fault was displacing them laterally. I have measured the offsets outside of the deflection zone. Wherever possible, I have measured both the minimum and the maximum lateral offsets. An example of this deflection is demonstrated in Figure 64. The deflections of the stream bed are caused by the lateral movement of the fault. The flow regime of the stream is thus redirected close to the fault. Therefore, multiple offsets can be recorded for the same geomorphic marker. The stream can record a surface rupture (or more) after is has already been deflected (i.e. MIN case in Figure 64) and it can also record total slip from the original orientation of the channel (i.e. MAX case in Figure 64).

4.e.4 Offset measures retained

The measurement table, TABLE Wairarapa_Offset Measures-Stewart-PhD Draft 21Oct18, is organized with the titles of each column describing its contents. Columns A-E include descriptive items about each marker including its name, fault segment, whether it is on the main fault (MF) or a secondary fault, its color mapped on the LiDAR data, its type, and its quality rating (next section). As in the code paper, the measurement table shows the lateral offsets in pale violet background (columns N -AE) and the vertical offsets in pale green background (columns AF -AW). When describing the 'mean' offset for any category, I obtained this value by using probability density functions (PDFs) to calculate it (as in the JGR paper), just as I will use it to find my major offsets per segment in the following sections. The mean relevant offsets are at the right of the table, GEOM (column BG+BH for horizontal and column BK+BL for vertical) representing the mean offset calculated per line from the geomorphically-relevant offsets (those values in bold), and the OPTIMAL (or OPT later, column BI+BJ for lateral and column BM+BN for vertical) is the mean offset derived from the different lines of measurements from the GEOM columns (GEOM only). Therefore, if there was only one line of measurement, the GEOM columns will equal the OPT columns.

Since it has been explained in Section 4.e.3 that the riverbed and mid slope measurements are generally the most robust features to represent the horizontal and vertical slip components, respectively, I have created a category that creates the 'mean' of these measurements for each marker. Columns AY+AZ are the 'mean' lateral offset values (obtained using the PDF technique) for the riverbed cases only per marker. For terrace risers, the mid slope value is listed, since it was described to be the most robustly constrained offset value for terraces. If there is only one line of measurement for a particular marker, then these columns simply list the lateral riverbed offset value from columns N+O. Columns BA+BB include the 'mean' vertical offset amounts for the mid slope measurements only. If the western and eastern edges display markedly different estimates, then either the averages for each edge are listed, or only the most geomorphically-relevant is kept. We have of course distinguished the markers offsets across the Main fault (MF, in pale orange background in quality rating column -column F) and across off-fault secondary faults (in pale blue background in quality rating columncolumn F).

4.e.5 Estimating the best preserved offsets and their uncertainties

Given that 3D_Fault_Offsets provides a wealth of data regarding individual markers and their possible range of true offset values, I have the opportunity to discriminate the data and create different significant categories to represent each marker. In order to obtain the most geomorphically relevant results from the statistical analyses, I have divided the lateral dataset in 3 significant categories: Underlined, OPT, and East Mean. The underlined values represent the offset of the one geomorphic feature that we consider as best preserving and representing the component of slip (see above). The OPT values are the overall mean offset values calculated from all geomorphically relevant features per marker. The OPT offset value thus takes into account the full range of potential offset values for the considered marker, and can thus represent the true uncertainty on the actual offset. The East Mean offset value is the mean lateral offset calculated across the actual eastern edge only (western edge in the measurement table). Indeed, as I will show later, the east side of the markers (leading edge) is generally best preserved compared to their western counterparts, likely for similar reasons as discussed in the JGR paper. The East Mean lateral offsets are thus taken to best represent the actual lateral offsets. Since not all markers have a geomorphically relevant eastern edge, there are not as much data as the Underlined and OPT categories.

For the vertical slip, I have divided the dataset into two categories: Mid Slope and Underlined. The OPT vertical value (global mean from all geomorphically-relevant measures) does not generally retain any relevant offset amount for each marker since the vertical degradation processes are generally much quicker than lateral, and thus affect the 9 measured features differently. Therefore, each case must be analyzed individually. The Mid Slope has been chosen for the same reasons as described in the JGR paper in that it best records the vertical offset amount (reasons I will discuss later). The Underlined value has been chosen similar to the lateral Underlined values (i.e., being the geomorphic feature that best preserves the vertical slip), but the vertical Underlined feature can be different than the lateral Underlined feature for the same marker (see the measurement table ).

4.e.6 Rating the offset measures (another source of uncertainty)

We have also rated the quality of the markers mainly depending on the preservation degree and on the robustness of the offset record. Measuring the cumulative lateral offset recorded by a marker requires the recognition of the northern and southern sections of the marker that once were contiguous but are now separated. The reconstruction must also restore properly the original shape of the marker. The recognition and reconstruction of these offsets come along with an epistemic uncertainty that expresses the confidence one can have in a correlation. For instance, there might be cases where the two marker sections are unambiguously recognized (i.e., a channel that is still connected across the fault). However, since their trace is complicated or because erosion has subsequently modified it, there is no way to properly reconstruct the original shape of the offset feature, even if measurements are completed precisely. Therefore, the epistemic uncertainty on the correlations can only be expressed qualitatively. Therefore, we assign each marker according to several characteristics described below.

We rate the offset markers into three classes, based on criteria described in the measurement table. Offset markers rated as 'GOOD' look well preserved and are well resolved in the LiDAR data (resolution ≤1 m), have a simple trace approximately perpendicular to the fault, exhibit clear piercing points onto the fault trace, and still show clearly their original "continuity" across the fault. They are thus unambiguous, whereas their lateral offset can be measured using multiple geometric features. These GOOD quality markers represent ~4% of the total offset population and hence are very few. All qualities for measurements for segments AB-NO are noted in the measurement table (due to time constraints, this analysis has not yet been completed for the remaining segments PQ -WXY). Offset markers rated as MODERATE look well preserved overall, are well resolved in the LiDAR data (resolution ≤1 m), and their correlation is unambiguous. However, their trace might have been slightly eroded, altogether making the restoration of the original marker trace and identification of all geometric features slightly more difficult than for GOOD offset markers. These MODERATE quality offsets represent ~38% of the total offset population. Offset markers rated as POOR are recognized on either side of the fault, and their correlation is likely, but the two marker sections seem to have been partly eroded, have a trace with a complex form or a subtle trace, or strike at a shallow angle to the fault. These POOR quality offsets represent ~57% of the total offset population. The proportion of good quality offsets is thus very low. The predominance of moderate-to low-quality offsets actually enhances some questions regarding the preservation potential of geomorphic markers in humid environments such as New Zealand as discussed in Manighetti et al., 2015 on the Hope fault nearby.

4.f Overview of the offset collection

We have applied 3D_Fault_Offsets to 671 total markers (on all segments from AB to WXY) along the 70-km stretch of airborne LiDAR data covering the Wairarapa fault. The average density of measured markers along the fault trace is ~10 markers/km, making this one of the largest and densest datasets for paleoseismic studies. Provided that some of these markers were analyzed using 'SHORT' and 'LONG' cases, I performed over 1000 offset measurements for both the lateral and vertical components of slip. The naming convention for the markers follow: 'WFxxxxxx'. 'WF' refers to Wairarapa Fault, which is followed by the Easting coordinate (in UTM) of the marker location. This naming convention allowed for me to add and remove markers without disrupting the names along the fault trace. Figure 65 displays the location of the measured offsets along the fault. Of the 671 geomorphic markers measured, 530 are channels (perennial, ephemeral, or abandoned), 130 are terrace risers, 11 are ridges, and 1 is the edge of an alluvial fan. Of the 530 channels, 80 are beheaded. Abandoned or beheaded channels allowed for very reliable and confident offset reconstructions, which will be discussed later. The lateral offsets range between a few meters to about 800 m, but 83% are lower than 80 m. Also, the population of offsets with a lateral slip of less than 80m are the youngest offsets inferring that their preservation is the best since it has been exposed to erosional processes for less time. Therefore, I will focus my analyses on this range. The vertical offsets range between 0 and ~30 m: ~71% of all vertical offsets are less than 5 m (absolute value).

The uncertainties on the lateral offsets are generally ~0-20% of the slip. For lateral cases regarding the best geomorphic features (i.e. underlined), 72% of the uncertainties are within 0-20% of the lateral offset value (Figure 66). For cases where only the offset values along the eastern edge of the marker were considered (i.e. Mean East), 64% of the uncertainties are within 0-20% of the lateral offset value. For the optimal mean values (OPT), only 52% of the uncertainties fall into the range of 0-20% of the lateral offset value. Uncertainties are especially large on the Optimal mean offsets for those integrate the variable preservation of the lateral offset across the west, bed and east sides of the channels. The east means approach considers the mean of a maximum of 4 features, which leads to smaller uncertainties. Whereas, the best geomorphic feature cases only consider one of the 9 features which is the best preserved one, and thus have the lowest possible uncertainties. The entire data collection allows for the examination of whether the (GEOM) lateral offset is similar or not across the west and east sides of a marker. The global analysis of the mean lateral offsets across the west and east sides of the marker population reveals that, in about 60% of the population, the mean lateral offset is greater across the eastern side. Since the eastern side of the markers is also their leading edge, this is compatible with the eastern leading edge being more preserved from erosion than the western trailing edge, as found and discussed in our JGR paper in Chapter 3. Figure 67 shows how the eastern edge can retain a better preservation of the actual offset value. Therefore, I infer that, in many cases, the mean East lateral offsets best preserve the actual lateral offsets, and I will thus use the Mean East offsets as reference offsets in the following.

Since the vertical offsets on the oblique-slipping Wairarapa fault are generally small compared to the lateral components of slip (normally <~20% of the lateral slip to be discussed later), their uncertainties are commonly a larger percentage of their offset value. The uncertainties for the vertical offsets usually fall between ~0 -30% of the vertical slip. For the cases where only the single, best geomorphic feature is considered to constrain the vertical slip (i.e. underlined), 55% of the uncertainties fall between 0 -30% of the vertical offset. Another important note is that ~15% of the uncertainties for underlined cases are greater than the offset value themselves, meaning that these vertical offsets are not constrained. For the cases where only the maximum gradient of a riser edge (i.e. mid-slope) is considered, 41% of the uncertainties are within 0-30% of the vertical offset value. In the mid-slope cases, ~23% of the uncertainties are larger than the actual offset value, meaning that these vertical offsets are not constrained. The percentage of uncertainties that fall in a lower percentile range of the actual vertical offset differs for the two discriminatory categories. The underlined cases record the lowest uncertainty because only the single, best preserved feature is used to constrain the vertical offset value. The mid-slope approach can incorporate up to a maximum of 4 midslope offset values per marker. Therefore, the uncertainties are larger for this approach. Below I will analyze the offsets for each major segment. Each segment contains between about 20 and 80 measurements.

4.g Statistical analysis of offsets along individual segments

The statistical analysis is as described in the JGR paper: each offset value and its uncertainties are represented by an individual Gaussian PDF. Summing all the individual Gaussian pdf results in a Summed PDF curve whose peaks indicate the possible offset clusters and thus the offset amounts most represented and hence likely best preserved in the data collection. In prior works (see Zielke et al., 2010), these peaks have been called COPD for cumulative offset probability distribution, and I will thus use that term throughout the manuscript.

It is very important to note that since the uncertainties we find on most offsets are large: 1) the summed PDF curves generally do not show markedly pronounced peaks; 2) some of the global peaks might be biased by local offsets having lower uncertainties than the others. Because of that, I will search for the most relevant compromise between such "local peaks" (which are somehow "artificial") and an average offset that would best represent the entire subset.

The figures I present below, basically present the same features for each segment: -Along each segment (AB -NO only; PQ-WXY have been omitted), I have analyzed the offsets along the main fault only (MF) and along the Secondary faults (where applicable and provided in Appendix III), and I have performed the analysis by discriminating the "good and moderate" quality measures on the one hand (red colors; thicker when Good), and all quality measures on the other hand (in blue); The Secondary faults are shown in grey, and they have not been considered when finding the peaks for the COPD plots; The Secondary faults were analyzed to see how their slip is recorded compared to the main fault offsets; the plots for the Secondary fault analysis are included in Appendix III -To better see the peaks in the lateral summed PDF curves, I show both the original curve (in black) and the curve stretched vertically by 3 (grey pale curve) -I have measured numerically the offset peak values and estimate their uncertainties approximating the peaks as Gaussian curves. I considered large conservative uncertainties, generally between 1 and 2σ. -I have performed these analyses on the range 0-80m for the lateral slips. Greater slips will be shown in Appendix III. -All lateral offset peak values or COPD are reported in a dedicated Table ( In the following, I present the lateral best preserved offsets for each segment in a series of figures completed as follows:

The first line of plots (a-b-c) shows the statistical analysis for the Good and Moderate quality lateral offsets, in the "best geomorphic feature" case ( COPD plots were also applied to the vertical components of slip for the two categories described in Section 4.e.5, Figure a representing the statistical analysis for the vertical Underlined category and Figure b representing the Mid Slope category. The vertical offsets were only considered for the same measurements as the lateral components (lateral range to 80m offsets). Thus, using the same markers as the 0-80m lateral offset range allows for evaluating the evolution of both lateral and vertical components of slip with increasing offset values. Understanding both the lateral and vertical component of slip is useful in assessing the total slip vector for a rupture event. Therefore, it is necessary to try and link the vertical slip that accrued with a specified lateral slip amount. Since we have conducted COPD analyses for both lateral and vertical slip on the same sets of measurements (markers recording <80 m of lateral slip along the main fault only), we can highlight features in the vertical COPDs to try and link them with a lateral COPD slip amount. The following figures with the Underlined (i.e. "best" geomorphic feature) vertical COPD plots are formatted so the individual markers have been highlighted according to the lateral COPD group in which they were mostly associated. The 'GOOD&MODERATE' quality data is depicted with a thicker line than the 'POOR' data. If the marker had clearly no affiliation with any lateral COPD group, it was omitted from the plot. Since the Underlined cases generally agree with the Mid-Slope cases, we will focus the linking of the vertical and lateral slip components on the Underlined (i.e. "best" geomorphic cases).

4.g.1 Segment AB

In segment AB, four major peak offsets, or COPDs, are well-resolved in the offset collection (Figure 69) that comprises 28 offset measures. In the majority of markers (~46%), the vertical to lateral slip ratio is below 10%, and at most up to 30% in some cases (Figure 71). The majority of cases falling below 10% are indicative of the dominantly strike-slip nature of the Wairarapa fault. In segment CD (43 offset measures), three lateral offsets are well preserved (in all graphs -Figure 72), averaging 4.0 ± 2.2 m (4.5 ± 2.5 for all qualities), 13.9 ± 3.0 (13.9 ± 3.1 for all qualities), and 20.3 ± 2.7 m (20.7 ± 2.9 for all qualities). Additional COPDs are suggested at 30.2 ± 3.2 m (29.9 ± 3.2 for all qualities), 39.3 ± 4.6 m (39.4 ± 4.6 for all qualities), 48.0 ±1.1 m (48.1 ± 2.4 for all qualities), 67.0 ± 4.0 m (64.4 +-5.2 for all qualities), and 73.6 ± 5.0 m (73.2 ± 4.9 for all qualities). In 75% of the markers, the vertical to lateral slip ratio is below 20% and at the most up to 30% in a few cases (larger values are estimated to be irrelevant, Figure 74). There is a greater balance between the population that falls in between 0-30% of the V/H ratio. Only 40% of the cases fall below 10% of the V/H ratio, but this result is still consistent with the dominant lateral slip with an oblique component. The trend of larger amounts of oblique slip in segment CD could be attributed to the complex nature of its trace and the small restraining bend (bend that accommodates local contraction) (<5°) in the middle of the segment. In segment E (26 offset measures), three lateral offsets are well preserved (in all graphs -Figure 75), averaging 12.1 ± 3.4 m (12.1 ± 3.1 for all qualities), 29.1 ± 4.5 m (28.9 ± 4.7 for all qualities), and 53.6 ± 2.0 m (53.6 ± 2.0 for all qualities). Additional COPDs are About 60% of the markers show a vertical to lateral slip ratio below 10%, but about 30% of the markers show a higher vertical to lateral slip ratio between 20 and 40% (Figure 77). The trend is again consistent with a dominant strike-slip fault with an oblique component. The 60% of cases under 10% of V/H ratio could be consistent with the mostly linear trace of the segment. More than 60% of the markers show a vertical to lateral slip ratio below 10%, with about 27% of the cases to 30% (Figure 80). Similar to segment E, segment FG has 60% of the cases with a V/H ratio less than 10%. Segment FG also has a fairly linear trace with the exception of a small step-over in the middle of the segment. In segment H (37 offset measures), the lateral offset record is less robust than in the prior segments in that less well-constrained measures exist (Figure 81). However, when considering the entire data set, several recurrent and hence well-constrained COPD peak offsets are revealed, on average at 8.6 ± 2.0 m (8.9 ± 2.1 for all qualities), 21.9 ± 4.4 m (21.1 ± 4.0 for all qualities), and 30.9 ± 3.8 m (30.8 ± 3.9 for all qualities). Additional smaller COPDs are found at (4.6 ± 2.5 for all qualities), 39.2 ± 11.9 m (39.8 ± 3.7 for all qualities), (50,7 ± 3.1 for all qualities), 60.3 ± 2.4 m (60 ± 3.2 for all qualities) and (71.5 ± 4.6 for all qualities). About 60% of the markers show a vertical to lateral slip ratio below 10%, with about 25% of the cases up to 30% (Figure 83). The V/H ratio is consistent with the other segments preceding it. The vertical to lateral slip ratios are more distributed for segment I compared to prior segments with larger values up to 40% (Figure 86). Segment I is also short and bounded by large left-stepping step-overs, and its trace is more complex than segment H. The fault trace indicates many contractional step-overs existing along segment I. It is also bounded by contractional bulges. It seems that segment I is a product of localized uplift, which could account for the measured increase in oblique slip. Almost 50% of the markers show a vertical to lateral slip ratio below 10%, but 30% of them have this ratio up to 30% (Figure 89). Segment JK transitions again into longer segments (~7 km) with a very simple and fairly narrow, linear trace. The trace helps to explain the dominant strike-slip motion. In segment LM (17 offset measures), the number of measures is too few to robustly constrain the lateral offsets (Figure 90). However, the entire collection suggests wellpreserved lateral offsets at 12.6 ± 2.8 m (10.2 ± 3.8 for all qualities), 18.1 ± 0.9 m (same -for all qualities), 29.7 ± 2.9 m (27.8 ± 4.1 for all qualities), 37.0 ± 1.3 (same -for all qualities), 49.7 ± 3.1 m (50.0 ± 3.4 for all qualities), 68.5 ± 4.6 m (68.0 ± 5.5 for all qualities), and (76.7 ± 1.6 for all qualities). About 90% of the markers show a vertical to lateral slip ratio below 20% (Figure 92). With the exception of the segment transition zones, LM also possesses a fairly narrow and linear fault trace. The dominant strike-slip motion is indicative of this fairly simple trace. About 85% of the markers show a vertical to lateral slip ratio below 20% (Figure 95). Segment NO possesses the most pronounced bend (20-30°) of the Wairarapa fault. However, either side of the bend consists of a fairly linear trace. The V/H ratio is consistent with the other segments and the dominant strike-slip motion of the Wairarapa fault.

4.g.2 Segment CD

4.g.3 Segment E

4.g.4 Segment FG

4.g.5 Segment H

4.g.6 Segment I

4.g.7 Segment JK

4.g.8 Segment LM

4.g.9 Segment NO

4.g.9 Segments AB -NO notes on the vertical offsets

Even though the vertical offsets are variable, likely due to differential erosion along the fault, the vertical linking plots suggest that most lateral COPD groups correspond to distinctive and increasing vertical offsets. The red lateral group (lateral offset in range ~10-14m, see later) generally coincides with vertical peaks of ~0-1m. The green lateral group (lateral offset in range (~17-21m) generally coincides with vertical slips ranging from ~3-4 m. The yellow group (lateral offset in range ~27-32m) generally coincides with vertical slips ranging from ~5-6 m. The violet group (lateral offset in range ~38-41m) generally coincides with vertical slips ranging from ~7-9 m. The pale blue group (lateral offset in range ~48-53m) generally coincides with vertical slips ranging from ~9-10 m. And the dark blue group (lateral offset in range ~62-67m) generally coincides with vertical slips ranging from ~10-14 m. Therefore, even though vertical slips are much more variable than lateral slips, likely due to differential erosion and local vertical motion enhancements, there seems to be a fairly linear relation between lateral and vertical peak offsets (see Figure 96). Lensen (1968Lensen ( , 1969) ) range from 4.5 -140m, and present no clear clusters. Measurements from [START_REF] Villamor | It's our fault -Wairarapa fault slip rate investigations task: Completion report -Is the Wairarapa fault slip rate decreasing to the north[END_REF] are similar to those completed by Lensen (1968); however, there remains discrepancy between these datasets when they are compared to the [START_REF] Grapes | The Wairarapa Fault[END_REF] clusters.

Figure 97b shows the best lateral displacement values (best defined by [START_REF] Villamor | It's our fault -Wairarapa fault slip rate investigations task: Completion report -Is the Wairarapa fault slip rate decreasing to the north[END_REF] for all types of geomorphic features in the database with respect to their age (not absolute). Figure 97c represents data from offset streams and channels, which is regarded by the authors to be most representative of the true offset value since other geomorphic features (e.g. terrace risers) can be subjected to post-displacement erosion. Of course, the thalwegs of streams and channels can be subjected to their own factors of degradation.

I have reported in black dots the average 'mean' (calculated using COPDs) offsets discussed above (those for all qualities offsets). The segments I analyzed are located on the Xaxis. Our measures agree well with prior estimates, and altogether, the offset data suggest clusters on the AB-NO fault section, around 10-13m, 17-22m, 28-32m, ~40m, ~50m, 60-63m (see especially the data regarded as the author's most robust-offset Figure 97c). However, our dataset seems to provide a clearer separation among the clusters of points that could possibly infer separate large earthquake events. The datasets from Lensen (1968) and [START_REF] Villamor | It's our fault -Wairarapa fault slip rate investigations task: Completion report -Is the Wairarapa fault slip rate decreasing to the north[END_REF] are not numerous enough to establish a trend of along-fault slip on the Wairarapa fault. Our trends seem to be in agreement with those of [START_REF] Grapes | The Wairarapa Fault[END_REF]; however, we are able to establish a more robust history of cumulative slip on the fault as evidenced by our dataset above ~50 m of offset.

Taken at first glance, the clusters of data points would suggest that past large earthquakes repeatedly produced similar amounts of lateral slips on the order of 10 meters. However, we will analyze these trends by distinguishing the quality of data points to establish a more robust interpretation. It is also important to note that our data provides insight into offset measurements with a much larger lateral slip, and hence an older record. Globally, about 70% of the markers show a vertical to lateral slip ratio below 20%, with an additional 14% of the population up to 30% (Figure 98). This agrees with the dominant motion of horizontal slip with a smaller, yet significant degree of oblique reverse slip on the Wairarapa fault.

4.h.2 Global lateral to vertical slip ratio

4.h.3 Insights on marker erosion

It has been well established that the robust characterization of tectono-geomorphic markers and their corresponding measurements of displacement are frequently hindered by erosional processes. Generally, the evolution of geomorphic markers involves a range of physical, chemical, biological, and anthropological mechanisms that are individually difficult to quantify. Attempting to define the preservation of landforms by their source parameters would be an unrealistic undertaking. However, the erosion that resulted from these various properties may contain some pertinent information regarding the degree of erosion. Through the proper characterization and identification of different features of a single marker exposed to varying degrees of erosion, some quantifiable aspects of erosion can start to be provided. I will present certain quantifiable aspects of erosion in this section. Figure 99 depicts the preservation of the eastern (leading edge) versus western (trailing edge) edges per marker of the entire dataset. Only cases where both edges were geomorphically relevant were considered. This phenomenon occurs for reasons explained in Section 4.f. Since the eastern edge is larger in ~60% of the cases, I deemed the eastern edge to preserve the actual offset more than the western edge (which is the reason for having an East Mean category in the COPD analyses). Also, quantitatively comparing the eastern, riverbed, and western edges of a respective marker enables to partial quantification of the amount of erosion. Figure 100 shows that the vertical slip across riverbeds is generally smaller than the vertical slip across the risers. This measures the incision the river has produced subsequent to the vertical slip. Fluvial erosional processes are extremely efficient as degrading the thalweg of a river. If the cohesive strength of the channel bed is lower than the shear exerted on it by the flowing water, the sediment on the bed can be mobilized incising the channel. The bed of the channel will be incised faster than the ridges if they are not in direct contact with the water. Therefore, the vertical slip of the riverbed was compared against the vertical slip of the best geomorphic feature (i.e. underlined cases) for the same respective marker. When the riverbed was the underlined feature, it was omitted from the analysis. As can be seen in Figure 100, 87% of these cases saw a greater incision in the riverbed compared to another feature on the same marker, inferring that the riverbed is generally retaining an underestimate of the true vertical slip. Quantifying the amount of slip that has been incised (i.e. the difference between the "best" preserved vertical offset and that across the riverbed), enables the analysis of the degree at which the channels have been incised. Figure 101 Here we take the lateral offset amount as a proxy of time; the greater the lateral slip, the longer the time of fault activity. The figure thus compares the lateral slip amount (underlined cases) with the amount of vertical incision (difference between the underlined vertical slip and the vertical slip across the riverbed, for a respective marker (cases where underlined feature is the riverbed have been ignored). Even though the data are distributed, we see that, for most lateral slips greater than ~40m, the vertical incision increases overall with the amount of lateral displacement, and hence with time. For the smallest lateral offsets, there is no clear correlation between lateral slip amount and incision degree. More work needs to be done to analyze these preliminary results. Figure 103 is a first proposition for the potential earthquake events that caused the cumulative slip measured at the surface. We have plotted the 3 categories of COPD lateral offsets (see figure caption), as a function of the total fault length. Figure 103 is only composed of the most robustly-constrained 'GOOD&MODERATE' quality data. The colors for the dots represent our tentative interpretation of the data based on the similarity "at first glance" of the COPD values. These COPD values are directly derived from Table 1 (also available externally as TABLE Wairarapa_COPD versus Length-Stewart-PHD 11Nov18). Hence, it is likely that similar cumulative slip values along different fault segments were produced by the same event (or number of events). The size of the points represents how well the COPD value is constrained (see caption of figure). On the top figure of Figure 103 22), where the fault branches into the Pa Valley and Alfredton fault systems. However, [START_REF] Grapes | The Wairarapa Fault[END_REF] recorded lateral slip components of ~10m near Mauriceville (an observation that is represented in this study as well -see Figure 97a-bc). Shermer et al., 2004 recorded a maximum lateral slip of 4-7m (average ~5 m) along the Alfredton fault. Trenching results show that 2 earthquakes occurred on the Alfredton fault, the most recent event coinciding with the timing of the 1855 Wairarapa earthquake. Therefore, it is likely that the average displacements on the Alfredton fault are a continuation of the rupture of the 1855 Wairarapa earthquake and can be used to constrain the along-fault slip distribution of our dataset.

4.h.4 Past large earthquake slips

The grey dots depict a subset of the population that is not well-represented across the length of the fault. The COPDs for the grey dot collection are only clearly present in segments AB, CD, H and JK. The lateral slips are between ~3 and 5 m (see TABLE Wairarapa_COPD versus Length-Stewart-PHD 11Nov18). These offsets are much smaller than those reported for the most recent 1855 earthquake, while no more recent earthquake is known on the Wairarapa fault. Furthermore, if these 3-5m slip would have been produced by an earthquake, this earthquake would have been strong [START_REF] Manighetti | Earthquake scaling, fault segmentation, and structural maturity[END_REF], likely with a magnitude greater than 6.5. Yet no earthquake of this size has been reported on the Wairarapa fault since the large 1855 event. Together these suggest that the markers that have preserved these small offsets have been dramatically eroded. As can be seen in Figure 102, even the smallest offset can be exposed to large degrees of erosion along the Wairarapa fault. Also, Darby&Beaven (2001) have demonstrated through 5-year geodetic study that there is no aseismic creep along the Wairarapa fault. Therefore, it is likely that the grey dots represent markers that record an underestimation of the true offset amount due to the erosion which they were subjected. I will thus ignore these small lateral offsets in the following (note however that "scenario1" in Table 3&4 (TABLE Wairarapa_COPD and Slip Increments-Stewart-PHD 11Nov18 found externally) refers to an attempt to consider them; they then produce earthquakes slips locally much lower than anywhere else on the fault, and much lower than the 1855 slip, which confirms they should be ignored).

The red dots thus represent the smallest and hence most recent lateral slips preserved along the fault. It is a COPD population that is well-represented throughout every segment (AB -NO). The lateral slips range from 8.6± 2 (segment H) -13.9±3m (segment CD) with an average of 10.5 ± 3.5m. Since the red dot population is the best-resolved in the morphology, and it consists of the smallest offsets, it is inferred that they represent the slips from the most recent earthquake, the 1855 Mw 8.2 on Wairarapa. The average dextral slip obtained in this scenario is smaller than that reported by Rodgers&Little (2006) of 15.5 ± 1.4m. The maximum slip is also smaller than their proposed slip of 18.7±1.0m. However, their study consisted of measurements from only 16 localities along the section of the fault we refer to as segment FG. Provided that we have a larger population with average measurements 2/3 the value of those reported by Rodgers&Little (2006) we propose that the maximum offset value of 18.7±1.0m is the result of two large earthquake events.

The green dot population is also well-represented in all of the fault segments except for segment I. Segment I is a very short (~3km) segment with a small population of data (20 measurements). Therefore, it is unlikely that the amount of measurements is sufficient to represent all prehistoric earthquakes that have occurred. The lateral slips for the green dot group range from 15.4±3.1 (segment JK) -21.9±4.4m (segment H) with an average slip of 19.0±3.7m. Since the data is well-represented in the morphology, and the average slip is essentially twice of the data representing the most recent event (i.e. red dot group), we infer that the green dot group represents the cumulative slip from the 2 past large earthquake events that occurred on the Wairarapa fault.

The purple dot group is well-represented in all of the segments along the study region. The lateral slips revealed by the COPD analysis range from 27.4±2.2 (segment JK) -32±3.3m (segment FG) with an average slip of 29.8±3.9m. We infer that the purple dot group likely resulted (at least at most segment sites) from the cumulative effect of the 3 past large earthquake events.

The cyan dot group is well-represented in the morphology except for segments FG, I and NO. The lateral slips range from 37±1.3 (segment LM) -41.2±4.9m (segment JK) with an average slip of 38.7±4.3m. We infer that the cyan dot group contains the cumulative slip from at least the past 4 large earthquake events.

The yellow dot group is well-represented in the morphology except for segment AB. The lateral slips range from 47.5±3 (segment FG) -53.6±2m (segment E) with an average slip of 49.6±3.6m. We infer that the yellow dot group cumulated the slip from at least the past 5 large earthquake events.

The blue dot group is resolved in every segment except segment I; however, they are not as well constrained as the previous groups. The lateral slips range from 57.1±7 (segment FG) -68.5±4.6 (segment LM) with an average slip of 62.3±5m. The blue dot group is certainly not well-constrained throughout the length of the study area, but the average slip still falls into a dominant pattern with the last 5 groups. Therefore, the blue dot group may contain the cumulative slip from the past 6 large events.

The maroon dot group is poorly constrained in the morphology and is only represented in segments AB, CD, I, JK, and NO. The lateral slips range from 72.5±9.1 (segment I) -79.8±11.7m (segment JK) with an average slip of 75.0±8.4m. Since the maroon dot group is so poorly constrained, and the slip values do not follow the same pattern, we infer that the cumulative slip could have resulted from at least 7-8 large earthquakes.

Therefore, the 'GOOD&MODERATE' COPD data suggests 6 past large earthquake events that can be well-resolved in the morphology. Figure 104 is the same interpretation of the COPD data as Figure 103 using the 'ALL' quality data. Therefore, there is more data (albeit less well-constrained). These COPD values are directly derived from Table 2 (TABLE Wairarapa_COPD versus Length-Stewart-PHD 11Nov18). They show the same distribution as those discussed above. I thus don't repeat the above inferences, and report the reader to the table. 3), empty symbols are for moderately constrained (regular font in the table), and X-symbols are for poorly constrained (italics in the table). The segments that we have defined are labeled along the x-axis.

From the difference between each successive COPD that is presented in Figures 103&104, we have also estimated the slip increments that add to form the measured COPDs. Figure 105 presents these slip increments that propose a scenario for the slips attributed the individual paleoearthquakes proposed in Figures 103&104. These interpretations are defined and organized in Table 3&4. Figure 105 represents solely "Scenario 2" in the Table 3&4 (TABLE Wairarapa_COPD and Slip Increments-Stewart-PHD 11Nov18 found externally). Figure 105 is the same scenario comparing the results from the 'GOOD&MODERATE' (top) and 'ALL' (bottom) qualities. The increment analysis is conducted per segment (AB -NO) along the fault.

If we follow the scenario from Figures 103&104, the slip increments for the most recent earthquake (for 'GOOD &MODERATE') range from 7.2±4.6 (segment NO) -13.9±3m (segment CD) with an average slip increment of 10.5±3.5m. For 'ALL' quality data, the slip increments range from 7.9±4.7 (segment NO) -13.9±1m (segment CD) with an average slip increment of 10.1±3.6m.

For the 'GOOD&MODERATE' quality data, the slip increments for the penultimate earthquake range from 4.3±4.8 (segment JK) -13.3±4.8m (segment H) with an average slip increment of 8.5±5.6m. For 'ALL' quality data, the slip increments range from 5.9±3.9 (segment JK) -12.2±4.5m (segment H) with an average slip increment of 8.7±5.1m.

For the 'GOOD&MODERATE' quality data, the slip increments for the 3 rd -to-last earthquake range from 9.0±5.8 (segment H) -14.1±4.8m (segment FG) with an average slip increment of 11.1±5.1m. For 'ALL' quality data, the slip increments range from 6.9±6.3 (segment E) -14.6±4.4m (segment FG) with an average slip increment of 11.0±5.1m.

For the 'GOOD&MODERATE' quality data, the slip increments for the 4 th -to-last earthquake range from 7.1±6.2 (segment AB) -13.8±5.4m (segment JK) with an average slip increment of 9.5±6.6m. For 'ALL' quality data, the slip increments range from 5.1±4.6 (segment AB) -16.5±4.0m (segment JK) with an average slip increment of 9.4±6.5m.

For the 'GOOD&MODERATE' quality data, the slip increments for the 5 th -to-last earthquake range from 8.7±4.7 (segment CD) -12.7±3.4m (segment LM) with an average slip increment of 11.0±5.0m. For 'ALL' quality data, the slip increments range from 6.5±4.0 (segment JK) -13.0±3.6m (segment LM) with an average slip increment of 10.8±5.4m.

For the 'GOOD&MODERATE' quality data, the slip increments for the 6 th -to-last earthquake range from 7.5±2.2 (segment E) -12.5±7.6m (segment NO) with an average slip increment of 9.7±5.1m. For 'ALL' quality data, the slip increments range from 7.6±2.7 (segment E) -12.9±7.6m (segment NO) with an average slip increment of 10.4±5.7m.

For the 'GOOD&MODERATE' quality data, the slip increments for the 7 th -to-last earthquake range from 6.6±6.4 (segment CD) -14.1±7.4m (segment NO) with an average slip increment of 10.1±10.2m. For 'ALL' quality data, the slip increments range from 4.8±4.8 (segment E) -13.4±4.5m (segment JK) with an average slip increment of 9.9±6.9m. However, the slip increments for this event are very poorly constrained. Therefore, we do not infer to provide robust values for a 7 th event.

Figure 105 shows that the slip increments were large for most past earthquakes, in the range of 7-15m. There is variability in the increment amounts both along strike of a single rupture, and from event to event on a single segment. Thus, there does not appear to be characteristic behavior for the slips increments in a single rupture or at the same location over multiple earthquakes. However, the average slip increments per event all hovered around 10 m of slip per event (the actual average for all events is 10.1±5.1m). Therefore, there does seem to be some repeatability in the total amount of slip per event. Certain segments may account for more of the deformation from event to event. The slip increments between 7-15m are hence very large lateral displacements. Prehistoric earthquakes on the Wairarapa fault were thus all or mostly of very high stress drops, as the 1855 earthquake, which is an important finding for seismic hazard assessments.

5.a 3D-Fault-Offsets Code capabilities

Many remote offset measuring techniques rely on biased approaches from the user making the measurements. Whether the user is forced to visually manipulate the offset, or the user manually traces the marker itself, too much interaction with the measurement will lead to discrepancies among interpreters. 3D_Fault_Offsets attempts to remove user bias by enabling minimal interaction with the characterization or measurement itself. A polygon is placed roughly around the marker under concern, but the code automatically characterizes the marker 3D topography and geometry using mathematical, and hence unbiased approaches.

A major contribution of 3D_Fault_Offsets relies on its ability to simultaneously measure 9 pertinent geometric features across a single marker. These specific geometric features are prominent in most geomorphic markers analyzed in tectono-geomorphic studies. Previous studies (e.g. Lensen 1968;Meriaux et al., 2005;[START_REF] Cowgill | Impact of riser reconstructions on estimation of secular variation in rates of strike-slip faulting: Revisiting the Cherchen River site along the Altyn Tagh Fault, NW China[END_REF]) have questioned the most appropriate feature of a terrace riser to record both the lateral and vertical components of slip. For instance, the riser crest could retreat from erosion, while the riser base could advance laterally from deposition of the aforementioned erosion causing an overestimation and an underestimation, respectively, to the true offset amount. Whereas, the steepest gradient (i.e. mid-slope) of the riser could remain well-preserved. The code is able to measure all 3 features, and the interpreter can compare the offsets and their characterization to conclude on the most-representative feature. We have persistently used the Underlined (i.e. "best" geomorphic case) throughout our analyses. The code enables the opportunity to identify the geomorphic feature that has the greatest preservation potential, so that the truest offset value can be obtained. If there is not one equivocally expressed "best" feature, then the code enables the averaging of the geomorphic features that are relevant.

Since the code can characterize the full marker simultaneously, it enables the ability to study the preservation potential of the entire marker. For instance, the eastern versus western preservation study is highly applicable in a tectonic setting with high alluvial activity. The code provides a step in the direction of partially quantifying some of the erosion of the markers by comparing the western, riverbed, and eastern extents of the marker. Especially in the case of dextral slip, the erosion to the western side of the marker can be partially quantified. Also, as displayed earlier, the river incision can be quantified as well. This will allow for the vertical component of slip to be addressed quantitatively. This has rarely if ever been done so far. Therefore, the preservation potential of both the lateral and vertical components of slip can be estimated in more than just a qualitative manner.

5.b Large prehistorical earthquake record on Wairarapa fault

We were able to resolve 6 -7 major prior large earthquake events in the ground morphology along Wairarapa. Previous studies had resolved only 5 prior events, yet with smaller and hence less robust datasets. The slip increments that we derived range between ~7 -15m. Therefore, prehistorical earthquakes had a very large stress drop [START_REF] Manighetti | Earthquake scaling, fault segmentation, and structural maturity[END_REF]. Therefore, the earthquake record obtained from the ground morphology can contribute to seismic hazard estimates for the Wairarapa fault. The bottom figure represents only the individual lateral offset data for segments AB -NO using the Underlined cases. Even though the grey data has not been verified, the trends it depicts are highly consistent with the lateral data for segments AB -NO, and therefore can contain significant information. The amount of measurements decreases exponentially with increasing lateral offset. This finding is consistent with the most recent earthquake events being the best-resolved in the morphology (see Manighetti et al., 2015;Zielke et al. 2015).

These individual measurements are the source for the COPD analyses and the extraction of the most-representative cumulative earthquake slips. Despite the seemingly scattered data, clusters of data points are arranged semi-linearly with increasing lateral slip. Also, the PQ -WXY segments generally contain the same number of measurements with generally the same amount of lateral slip as the measurements along segments AB -NO. Therefore, we are able to infer that the "core" of the Wairarapa fault (i.e. the 70-km stretch of LiDAR data) consistently produces similar amounts of slip along this section of the fault. have not been identified yet for segments PQ -WXY. Hence, the data for PQ -WXY is formatted in grey dots, because their offset values have not been double-checked or analyzed in detail yet. The grey data points are still significant because they were measured in the same fashion as the measurements in AB -NO.

The amount of cumulative slip for the vertical component appears to decrease significantly at the intersection of segments NO and PQ. The cumulative vertical slip for segments AB -NO can increase to ~20m (absolute value); whereas, the cumulative vertical slip along segments PQ -WXY terminate ~10m of slip. That is a significant decrease in slip. As such a corresponding decrease is not clearly observed for the lateral slip, more work needs to be done to understand the reasons of such a vertical decrease in the splay zone (Figure 107). It is possible that the splay faults are more strike-slip than the "core" of the Wairarapa fault. This would indicate that the largest lateral slips occur along the southern core section of the Wairarapa fault. Figure 108 is a histogram showing the vertical to lateral slip ratio along segment UV. I should remind the reader that the data on segments PQ -WXY has not been completely verified yet; however, it contains significant information because the measurements were completed in the same way as segments AB -NO. The OPT dataset for the lateral component was compared against the Mid-Slope dataset for the vertical component. There is 65% of cases where the vertical to lateral slip ratio is less than 10% of the lateral slip. The 65% of cases is larger than any of the segments in the southern section (AB -NO) and is an increase the average of ~40% (for segments AB-NO) by ~25%. This increase of cases less than the 10% threshold is in keeping with a transition to a purer strike-slip motion, and hence a smaller vertical component of slip in the northern section of the "core" of the Wairarapa fault.

In order to understand and interpret the slip distribution of paleoseismic displacements on strike-slip faults, the pattern and amount of variability must be disseminated, both along strike of the rupture in each event (i.e. spatial variation) and between successive events (temporal variation). Figure 109 depicts a refined proposition for along-fault slip profiles using the 'ALL' quality lateral COPD data. This is the same figure as Figure 104, but the colors have been removed to allow for a more objective view. The colors that were shown in Figure 104 were a first-step proposition based on the rough similarity of the COPD values. The lateral offsets are expected to vary along the length of the fault. Therefore, this scenario is likely more plausible than the first. It is reasonable because it is constrained for the 1855 rupture by the northern measure on the Alfredton fault by Schermer et al. (2004). The slip profiles, as well as the slip increment data, suggest an asymmetric cumulative slip distribution, which is commonly observed on faults (Manighetti et al., 2001) and earthquakes [START_REF] Manighetti | Evidence for self-similar, triangular slip distributions on earthquakes: Implications for earthquake and fault mechanics[END_REF]. The slip profiles also suggest an asymmetric earthquake slip distribution for the 1855 earthquake, which is commonly observed on earthquakes [START_REF] Manighetti | Evidence for self-similar, triangular slip distributions on earthquakes: Implications for earthquake and fault mechanics[END_REF].

In terms of hazard assessment, it is crucial to understand whether the Wairarapa fault always ruptures with lengths similar to the 1855 earthquake, or if shorter segments of the fault rupture independently. Trenching results on the Alfredton fault completed by Schermer et al. (2004) suggest that 2 surface rupturing events occurred on the Alfredton fault since 2900 BP. Lateral offset values compiled by Lensen (1969[START_REF] Beanland | The North Island dextral fault belt, Hikurangi subduction margin[END_REF]), and Schermet et al. (2004) on the Alfredton fault not only recorded clusters of 4-7m for a single-event displacement, but the collection of measurements also revealed clusters at 8-12m and 18-19m. The second cluster of 8-12m is twice as much as the single-event range, suggesting that the penultimate Wairarapa fault event is also constrained by a northern rupture extent on the Alfredton fault. The lateral slip tapering on the Alfredton fault as well as the data we present for the varying slip increments (i.e. 7-15m per event with an average of ~10m of slip) on the "core" (i.e. segments AB -NO) of the Wairarapa fault all suggests that each recurring earthquake event has a very large stress drop.

5.c Discussion on past earthquake slip preservation in ground morphology: new insights on actual uncertainties

One of the great questions posed in tectono-geomorphic studies is the preservation potential of the geomorphology outside of semi-arid climatic regions such as the location of the San Andreas fault. Given the humid conditions of New Zealand, there should be little uncertainty as to the sufficient marker production rate, but the preservation potential could be a concern. Despite that, the 'GOOD' and 'MODERATE' quality measurements accounted for 42% of the entire range of measurements, they still represent a statistically significant population. Most of the comparisons among the 'GOOD+MOD' analyses with the 'ALL' quality analyses provided very similar results, indicating that even the 'POOR' quality measurements contain valuable preserved offset data. It is possible to give some acknowledgement to 3D_Fault Offsets when addressing the 'POOR' quality measurements. Since the code identifies 9 features, these 'POOR' quality markers could retain at least one feature that can be well-characterized by the code.

The lateral cumulative slip measurements documented indicate that there is a long slip history recorded in the morphology. This observation provides some insight into the preservation potential of tectonic environments in humid climates. The erosion rate in this setting may be too fast to obtain paleosesimic records of >10 earthquake events. However, we were still able to infer 6-7 well-resolved large earthquake events in the morphology, which is still a significant and rich history of earthquakes, especially when considering the potential magnitude of earthquakes on Wairarapa. Given the large magnitude of the 1855 earthquake on Wairarapa, power laws dictate that the frequency of those events are very low. In fact, the recurrence interval on Wairarapa is estimated to be ~1200 years (Nicol et al. 2012). Therefore, the 6-7 well-resolved large earthquakes are a significant history considering it potentially accounts for the past ~7200 years.

Given that the epistemic uncertainties related to marker identification are so large, the aleatoric uncertainties do not seem to resemble the qualitative doubt. For instance, Klinger et al. (2011) asserted that 85% of their uncertainties were less than 20% of the offset value with 37% of the population under 10%. In this study, we show similar results when we only consider one geomorphic feature to represent the marker (72% of 'best geomorphic' cases under 20% of uncertainty). However, when we consider the full range of potential geomorphic features (i.e. OPT cases), which we believe to represent the actual offset range of the marker, only 52% of these cases are lower than 20% of the offset. Therefore, the uncertainties reported in prior literature may be too liberal.

CHAPTER 6: Conclusions and perspectives for future work

6.a Conclusions

3D_Fault_Offsets has provided the ability to produce an extremely rich dataset of individual offset values in both the lateral and vertical components using tectono-geomorphic markers. The robustness of these measurements is ascertained through the comprehensive mathematical characterization of individual geomorphic markers. The ability to discern among the lateral and vertical offset values for individual features on a respective marker allows for the opportunity to assess the true offset value. The efficacy of 3D_Fault_Offsets has been verified using 3 distinct datasets composed of both field and remote measurements of different techniques. The code allows for better interpretations regarding the degrees of erosion that may affect the preservation potential of markers. The code was then applied to a prominent strike-slip fault in the humid climate of North Island, New Zealand, the Wairarapa fault, that has ruptured one time historically with a Mw ~8.2 in 1855. A collection of 671 individual lateral and vertical measurements were performed along the fault length of the 70km stretch of LiDAR acquired along the "core" of the Wairarpa fault, representing more than 1000 measures. This collection is one of the largest and densest paleoseismic datasets. The code offered a statistical method, the use of probability density functions, for analyzing the individual measurements that represent the cumulative slip history of the fault. The alongfault slip distribution revealed a large lateral slip throughout the "core" of the Wairarapa fault. However, the cumulative vertical slip significantly decreased at the junction of the southern and northern sections of the "core", which is defined geologically by the fault splaying into a network of secondary oblique faults. Therefore, it seems that the southern portion of the Wairarapa fault accommodates most of the slip during earthquakes. The analysis of the remaining offset measures along the PQ to WXY segments will provide the missing information to more properly describe the slip distribution along the fault. So far, through my analysis of the cumulative slip history on the AB to NO fault section, a total of 6 wellresolved individual large earthquakes could be inferred to have occurred on the Wairarapa fault. The lateral cumulative slip measurements recorded provided information that could have suggested a longer paleoseismic history. However, the COPD analysis of these measurements were too poorly resolved to provide any robust conclusions. This observation provides some insight into the preservation potential of tectonic environments in humid climates. The erosion rate in this setting may be too fast to obtain paleosesimic records of >10 large earthquake events. However, through the analysis of the 6 well-resolved large earthquakes, the slip increments for each event were inferred to range from ~7-15m. This slip per event suggests that all prior large earthquakes on the Wairarapa fault produced very large stress drop earthquakes, which is an important factor for seismic hazard assessments. Also, the use of very high resolution topographic data is invaluable for contributing to tectonogeomorphic studies.

6.b Future work

The type of rich data set that has been produced on the Wairarapa fault using 3D_Fault_Offsets offers many opportunities for future work. First, the segments for PQ -WXY must be analyzed in the same fashion to develop potential slip profiles and slip increment history for the entire length of the onland Wairarapa fault. The analyses of the rest of these segments will provide greater detail into the possible rupture extents of the prehistoric earthquakes that occurred on Wairarapa. This study suggests that the earthquakes on the Wairarapa fault possess a very large displacement to rupture length ratio, suggesting that the Wairarapa fault is immature (as confirmed by its young age, see [START_REF] Manighetti | Earthquake scaling, fault segmentation, and structural maturity[END_REF]. Therefore, the analyses of the rest of the segments will provide further insight into this topic.

3D_Fault_Offsets can then be applied to a variety of different faults in a variety of different settings. The code can be used for case studies on faults with varying degrees of structural maturity to see how displacement values evolve with increasing fault maturity. The code should also be tested on normal and reverse faults (possibly with some evolution of the code) to ascertain its efficacy on markers that have undergone dip-slip fault motion. Given that the vertical component of slip is more susceptible to erosion than the lateral component, dip-slip motion faults will have to be chosen carefully. However, since 3D_Fault_Offsets can start to quantify this erosion (through vertical incision of the riverbed for instance), it could be useful in this realm. The code should also be applied to fault zones located in more arid climatic conditions to see if a longer slip history can be extracted. Finally, some analyses regarding the behavioral aspects of ruptures on individual faults and potentially faults in general can be inferred from this type of dataset, which could potentially be used to improve existing fault behavior models.

  2.a Morphotectonic analysis of the surface: trace of faults 2.b Tectono-geomorphic marker analysis : type, formation, and preservation 3.a Introduction to 3D_Fault_Offsets 3.b 3D_Fault_Offsets JGR paper 4.a The Wairarapa fault and earthquakes: prioer knowledge from literature 4.b Geomorphic markers and the cumulative offsets (prior knowledge from literature) 4.c LiDAR data analyzed on the Wairarapa fault 4.d Mapping the overall Wairarapa fault trace 4.e Application of 3D_Fault_Offsets 4.f Overview of the offset collection 4.g Statistical analysis of offsets along individual segments 4.h Statistical analysis of best preserved offsets 4.h.1 Comparison of our measured offsets with prior offset measurements 4.h.2 Global lateral to vertical slip ratio 4.h.3 Insights on marker erosion 4.h.4 Past large earthquake slips 5.a Code capabilities 5.b Large prehistorical earthquake record on Wairarapa fault 5.c Discussion of past slip preservation: new insights on actual uncertainties 6.a Conclusions 6.b Future Work

Figure 1 .

 1 Figure 1. Image taken along the southern San Andreas Fault at a famous geomorphic marker, Wallace Creek. The image shows how a strike-slip tectonic fault can laterally offset alluvial features. The red arrows indicate that it is a right-lateral strike-slip fault. The white arrows represent the offset amount of these streams to be the cumulative displacement from multiple earthquakes. Image modified from Wallace, 1990.

Figure 2 .

 2 Figure 2. Schematic representation along the trace of a strike-slip fault of the evolution of a single tectono-geomorphic ephemeral stream marker from A.) pre-earthquake morphology to B.) the coseismic morphology after one earthquake event to (the MRE in B refers to the 'most recent earthquake') C.) the cumulative slip morphology after several consecutive earthquake events. The coseismic slip records the slip of one event; whereas, the cumulative slip is the record of multiple events. The cumulative slip is generally measured in topography/morphology, but the coseismic slip is assumed to be the smallest locally (i.e. per segment) measured displacement. Another major assumption is that the production rate of the markers is more frequent than the earthquake events that displace them (see text for explanation). MRE is most recent event.

Figure 3 .

 3 Figure 3. Schematic representation of an along-fault slip reconstruction. A.) Individual offset measurements for individual markers are plotted versus the length of the fault. The vertical lines represent the uncertainty range on the offsets and the colors indicate quality of the marker (darker is more reliable). B.) Then, a plausible earthquake history can be reconstructed inferring the number of paleoearthquakes responsible for the different slips. Given the density of the measurements, some reconstructions will be more reliable than others (Zielke et al., 2015).

Figure 4 .

 4 Figure 4. LiDAR acquisition through airborne-based methods. It is an active form of remote sensing sending out laser pulses in the visible portion of the electromagnetic spectrum, reflecting off objects or the ground surface, returning to the sensor, and recording the timing and intensity when those pulses return to the sensor. The pulses are converted to absolute geographic coordinates enabling the generation of very-high spatial resolution topographic base maps. The pulses can be partially returned, distinguishing between the types of the reflectors. Post-processing techniques can eliminate all objects above the Earth's surface, exposing only the raw earth, creating images called 'Bare-Earth Digitial Elevation Models'. Figure modified from Carter et al., 2007.

Figure 5 .

 5 Figure 5. A image of a section of the Wairarapa fault on the southern part of the North Island in New Zealand displaying optical imagery (left) at 1 m resolution and a 'bare-earth' DEM (right) at 1 m resolution. The 'bare-earth' DEM easily highlights the unprecendented ability to discern hundreds of subtle offset markers along the fault trace that is inaccessible using the optical image (Wairarapa Lidar data from the present study).

Figure 6 .

 6 Figure 6. Plot showing a normal-distribution bell-shaped curve (i.e. probability density function) with the different standard deviations representing the uncertainties (Chandler, 2012).

Figure 7 .

 7 Figure 7. Conceptual models attempting to define the recurring behavior along seismically-active faults. Each model was formed through surface-slip reconstructions of the MW 7.8 1857 Fort Tejon rupture trace. A.) Variable slip model enables variable displacement per event at a point and randomly sized earthquakes. B.) Uniform slip model (Sieh, 1981) defines constant slip per event both at a point and along the entire trace resulting in constant large events with more frequent smaller events. C.) Characteristic earthquake model (Wells & Coppersmith, 1984) defines constant slip at a point with a variable slip rate from constant large earthquakes and infrequent moderate earthquakes.

Figure 8 .

 8 Figure 8. Air photo of the well-defined fault trace with associated pressure ridges along the San Andreas Fault. Successive large earthquakes have imprinted the highly visible fault trace into the ground morphology. The trend of

Figure 9 .

 9 Figure 9. Sketch depicting slip accommodation on various sections of the fault. The left side of the sketch represents the main fault trace; whereas, the right side represents splay networks branching out from the main fault. The graph plots the horizontal displacement values comparing the amount of slip on accommodated on the main fault compared to the splay network. The slip decreases in the direction of splay network enlargement because it is the direction of long-term fault propagation and hence age and slip decrease. While the splay network accommodates less slip than the master fault core, the amount is not negligible and should be addressed in a paleoseismological analysis of the full rupture history. Modified from Nicol et al., 2017.

Figure 10 .

 10 Figure 10. A sketch depicting common features formed from strike-slip tectonics, alluvial dynamics, or a combination of the two. Only several of these potential markers are robust for reconstructing the slip history along the fault (see text) (Costa & Baker, 1981).

Figure 11 .

 11 Figure 11. Sketch of hypothetical development of a terrace sequence offset by an upthrown right-lateral strike-slip fault. Hashes point to lower surface. T represents the channels and R represents the risers (Knuepfer, 1988).

Figure 12 .

 12 Figure 12. Block diagrams exhibiting two terrace models for measuring the offset between offset terrace risers. Lower (A-C) and upper (D-F) terrace reconstructions are depicted. See text for details (Cowgill, 2007).

Figure 13 .

 13 Figure 13. Sketch of commonly observed relations between horizontally displaced stream channels to highlight the potential difficulties in reconstructing slip. The question marks represent questionable realignments. Each marker must be studied intensely to create confidence that the northern and southern components of the marker were once a linear feature across the fault (Zielke et al., 2015).

Figure 14 .

 14 Figure 14. Schematic diagrams showing development of beheaded streams in a left-lateral strike-slip tectonic setting. See text for detailed discussion (Klinger et al., 2000).

Figure 15 .

 15 Figure 15. Oblique aerial photograph of laterally displaced ridges and valleys along the Awatere Fault, New Zealand (McCalpin, 2009).

Figure 1 .

 1 Figure 1. Field view of a dextrally offset channel, Hope fault, New Zealand (picture is courtesy of J. Malavieille). The channel is the 16/17-no marker originally measured by Manighetti, Perrin et al. (2015) and that was measured here (see Table ES-D).

Figure 2 .

 2 Figure 2. Idealized representation of the nine geometric features identified mathematically by the 3D_Fault_Offsets code.

Figure 3 .

 3 Figure 3. (a) Hillshade representation of a DEM region (here at 1 m resolution) including two offset channels (southern San Andreas Lidar data; Zielke et al., 2012) and fault trace (blue) drawn by the user. The white arrow indicates the ZA6808a marker that is illustrated in following figures. (b) Once the user has traced the fault, the code rotates the DEM and fault to an E-W attitude. When the user has traced the two polygons around the marker sections of interest, the code identifies the points of minimum elevation within the polygons and populates the DEM with those points (white). They underline the riverbed. In this figure and following (Figures 3-6 and additional ones in supporting information ES1), the example has been processed using 1 m resolution Lidar data to enhance the clarity of the figures.

Figure 4 .

 4 Figure 4. (a) Using the "cleaned" best fit lines, the code calculates the lateral and vertical offsets by subtracting the x and z coordinates of the piercing points of their respective best fit lines on either side of the fault. Uncertainties on the offsets are also calculated. The calculations are here strictly automatic (AUTO approach). (b) Similar offset calculations but performed after the user have manually removed a few outlier points in the point clouds (CLICK approach).

Figure 5 .

 5 Figure 5. (a-d) 3-D rotated views of the NE side of the marker ZA6808a shown in Figures3 and 4. The distribution of the identified points and corresponding best fit lines can be seen onto the DEMs and compared between the two DEM resolutions (25 cm and 1 m). The views show that the "click" calculations we did in the 25 cm data (Figure5c) could have been more optimal (a few outlier points obviously remain to be removed). The corresponding measurements are in Tables ES-A and ES-B.

Figure 6 .

 6 Figure 6. (a) Horizontal back slip of the marker performed by the code, using the offset value provided by the user (here, optimal lateral offset). (b) Vertical back slip of the riverbed of the marker. The back slip is performed using the "direct" vertical offset (see details in supporting information ES1), but the mean vertical offset is shown along with its uncertainty. Here the vertical offset is about null, what explains the large relative uncertainties. Data are in Table ES-B.

  or the marker length was too short to permit any longer polygon. The lack of a CLICK series indicates that no or too few obvious outliers were to be removed. The 18 best fit line calculations and their visualization in map (supporting information ES3) and rotating 3-D views (examples in Figures 5,

Figure 7 .

 7 Figure 7. Identified geomorphic features (i.e., with geomorphic relevance) in the three data sets. (a) San Andreas data set (Table ES-A) with measures made using AsZ polygons. (b) San Andreas data set (Table ES-A) with measures made using longer polygons. (c) Owens Valley data set (Table ES-C). (d) Hope data set (Table ES-D). See discussion in text.

Figure 8 .

 8 Figure 8. (a) Comparison of the mean lateral offsets measured across the 44 San Andreas markers derived from the totality of the offset measures per marker (column AS in Table ES-A, means noted ALL) and from the geomorphically relevant measures only (column AY in Table ES-A, means noted GEOM). (b) Comparison of the mean lateral offsets measured across the 44San Andreas markers, with no user action (i.e., strictly automatic approach, "AUTO"), and with a user action that consists of removing the few clearest outlier points from some of the best fit lines, generally the tops and bases ("CLICK") (data in Table ES-A). (c) Comparison of the mean lateral offsets measured across the 44 San Andreas markers, using short (AsZ) and longer (longer) polygons (data in Table ES-A). In all plots, the empty symbols indicate poorly constrained lateral offsets with uncertainties ≥10 m.

Figure 9 .

 9 Figure 9. Comparison of the optimal lateral and vertical offsets obtained for the three data sets from the totality of the offset measures per marker (ALL) and from the geomorphically relevant measures only (GEOM) (data in Tables ES-A, ES-C, and ES-D). (a-f) The empty symbols indicate poorly constrained offsets with large uncertainties (≥10 m for lateral and ≥2.5 m for vertical).

  Figure ES2_I in supporting information ES2 confirms that the mean lateral offsets estimated with AsZ polygons are similar overall to the lateral offsets measured by Zielke et al. (2012) (~90% similar within uncertainties).

Figure 10 .

 10 Figure 10. Comparison of geomorphic optimal (GEOM) lateral and vertical offsets obtained in present study with prior field and remote measures. (a) San Andreas data set, lateral offsets compared to prior remote measures. (b) San Andreas data set, lateral offsets compared to prior field measures. (c) Owens Valley data set, lateral offsets compared to prior remote measures. (d) Owens Valley data set, lateral offsets compared to prior field measures. (e) Hope data set, lateral offsets compared to prior remote measures. (f) Owens Valley data set, vertical offsets compared to prior remote measures. The empty symbols indicate poorly constrained lateral offsets with uncertainties ≥10 m.

Figure 11 .

 11 Figure 11. Comparison of the optimal lateral offsets measured across the 44 San Andreas markers using Lidar data at 25 cm and at 1 m resolution. Data are in Tables ES-A (25 cm resolution) and ES-B (1 m resolution).

  (e) in supporting information ES2). Two small lateral Journal of Geophysical Research: Solid Earth 10.1002/2017JB014863 STEWART ET AL. 3D_FAULT_OFFSETS

Figure 12 .

 12 Figure 12. Histogram showing the percentage of channel markers in the three data sets that have a lower mean lateral offset across their trailing or their leading edge. See text.

Figure 13 .

 13 Figure13. Vertical to lateral slip ratios measured in the three data sets. The ratios are calculated from both the optimal geomorphic offsets (GEOM) and the mean vertical and lateral offsets recorded by the steepest slopes and the riverbeds, respectively.

Figure 17 .

 17 Figure 17. Overview of the tectonic setting of New Zealand, which is dominated by the interaction of the Pacific plate and Australian plate. In the North Island, the Pacific plate is subducting beneath the Australian plate at the Hikurangi trough and Kermadec trench. South of the South Island, the Australian plate is subducting beneath the

  . The four main structures associated with the Marlborough Fault System include the Hope Fault, Clarence Fault, Awatere Fault, and Wairau Fault. The Marlborough Fault System accommodates most of the ~43 mm/yr N40°W convergent motion between the Australian and Pacific plates. A small amount of obliquity (i.e. on the order of a few mm/yr) in the convergence across the Marlborough Fault System results in a reverse component of slip in addition to the fault's dominant right-lateral strikeslip motion (Beaupretre et al., 2012).

Figure 18 .

 18 Figure 18. Tectonic setting of the upper component of the South Island, New Zealand. The four major active strikeslip faults composing the Marlborough Fault System are depicted as well as the Alpine fault to the SW. The inset displays the Australian-Pacific plate convergence with arrows and values for velocity directions and rates in mm/yr, respectively (Beavan et al. 2002). Figure from Beaupretre et al., 2012.

Figure 19 .

 19 Figure 19. Tectonic setting of the lower part of the North Island, New Zealand. Major active strike-slip faults have been highlighted. Small open circles are main towns along the Wairarapa fault trace. The inset displays the Australian-Pacific convergence setting globally across New Zealand with the box showing the location of the zoomed figure. Figure from Carne et al., 2013.

Figure 20 .

 20 Figure 20. Simplified geology map depicting the relationship between the Wairarapa fault (red line) and the greywacke-dominated basement rock overlaying Tertiary sediments and Quaternary alluvial terraces and fans. Mapping and ages of terraces and fans by Begg and Johnston (2000) and Lee and Begg (2002). Faults from GNS database (http://data.gns.cri.nz/af/). Figure from Villamor et al., 2008.

Figure 21 .

 21 Figure 21. Active faults from GNS active database (http://data.gns.cri.nz/af/). Stars are the main towns. Values are previously published geological slip rate estimates from Van Dissen and Berryman (1996; Tea Creek Site), Schermer et al., 2004 (Alfredton trench site), Wellman (1972; Waiohine site), Wang and Grapes (2007; Waiohine site), Little et al., 2009 (Waiohine site), and Carne et al., 2011(Waiohine site). Figure from Villamor et al., 2008.

Figure 22 .

 22 Figure 22. Figure A: Locations of the fault offset measurements conducted by Rodgers and Little (2006) to the north and south of the town of Featherston (16 sites). Figure B: Labeled sites where slip measurements were undertaken by Rodgers and Little (2006). Dextral slip values recorded are the total offset at single-strand localities, and inferred minimums (one strand only) are at two-strand localities. Both figures from Rodgers and Little (2006).

Figure 23 .

 23 Figure 23. Map of cumulative and coseismic displacement value locations along the Wairarapa fault from previously published studies. From Villamor et al., 2008.

Figure 24 .

 24 Figure 24. Plots of horizontal displacement measures available along the Wairarapa fault. Splay fault junctions are indicated along the x-axis. The types of geomorphic markers are not discriminated. Both A and B plots show the same data over different lateral offset ranges. From Villamor et al., 2008.

Figure 26 .

 26 Figure 26. Best-constrained horizontal displacements measured across streams and channels, hence best preserved features, along the Wairarapa fault. Offsets are classified according to age of offset terrain. From Villamor et al., 2008.

Figure 27 .

 27 Figure 27. Figure showing the project area and data coverage (outlined in the orange polygon) of airborne LiDAR collected by New Zealand Aerial Mapping Ltd. along the Wairarapa fault trace. The total stech of data is ~70 kmlong.

Figure 28 .

 28 Figure 28. An unclassified, raw LiDAR point cloud taken from the Wairarapa dataset (BN342224 tile). All of the points collected during the airborne acquisition are present in this figure including points returned from vegetation and the topographic surface (i.e. ground). Figure created using CloudCompare software.

Figure 29 .

 29 Figure 29. The same LiDAR point cloud data as Figure 28 (BN342224 tile). This point cloud represents the points that have been classified as the ground points by NZAM. These points represent only the topographic surface of the section of the fault that has been scanned. From here, the points can be rasterized (i.e. set to be equi-distant) to make a bareearth digital elevation model (DEM). Figure created using CloudCompare software.

Figure 30 .

 30 Figure 30. Figure depicting multiple paths by the aircraft along the same section creating areas of overlap and hence higher point densities. Figure modified from http://azgeology.azgs.arizona.edu.

Figure 31 .

 31 Figure 31. LASTools' graphical user interface (GUI) for generating DEMs from LiDAR point cloud data (LAS files). Step 1 -Blast2dem  This tool can read billions of LiDAR points from the LAS format, triangulate them using a seamless TIN (Triangulated Irregular Network) algorithm, and raster the TIN onto a DEM that can optionally be tiled. The output supports multiple formats, but ASC is relevant for the purposes of using it in 3D_Fault_Offsets.  For the ASC format, the actual elevation, slope, and intensity values are stored.  The tool utilizes the TIN (Triangulated Irregular Network) interpolation algorithm, a high-performance code for generating raster DEMs from mass points via stream Delaunay triangulation (Isenburg et al. 2006a,b). A TIN model represents the landscape as a surface composed of continuous and non-overlapping triangles constructed between the cloud of LiDAR returns. Within each triangle, the surface is represented by a plane. A raster DEM is then constructed by sampling the triangulated surface elevation at the DEM nodes. Greater weight is given to areas of higher shot densities, and the TIN algorithm can successfully interpolate for areas of no point returns. Areas void of any points in close vicinity produce artifacts of triangulatedappearing pixels and become illegible, making it easier to discern places of highly accurate data. Isenburg et al. (2006a,b) computed a billion-triangle terrain

Figure 32 .

 32 Figure 32. A 1-m hillshaded DEM of the Wairarapa fault generated using the LASTools software according to the process described above. This DEM is the same data tile as the raw point clouds depicted in Figures 28 and 29 (BN342224).

Figure 33 .

 33 Figure 33. Mosaic of the entire coverage of teh Wairarapa airborne LiDAR dataset. The hilshaded DEM covers all 70 km of the Wairarapa fault that was scanned. The figure was created using ERDAS Ermapper software.

Figure 34 .

 34 Figure 34. Bare-earth DEM along the Wairarapa fault displaying an active channel with right-laterally offset river terraces and risers. An unnatural feature is also positioned to the west of the channel. To the east of the channel and south of the fault, there are perfectly straight lines that seem to be of anthropogenic origin.

Figure 35 .

 35 Figure 35. Optical image (i.e. orthophoto) to the same extents as the DEM in Figure 34. It is now clear that the straight lines visible in the DEM are due to farming practices, and this must be considered during the interpretation of offset features in this location.

Figure 36 .

 36 Figure 36. Google Earth image of the locations along the Wairarapa fault where terrestrial LiDAR scans were conducted. There were many other sites under consideration, but these locations offered the best land coverage, free from dense vegetation to collect LiDAR scans. Inset shows the more general location of the sites.

Figure 37 .

 37 Figure 37. Close-up view of Site F where terrestrial LiDAR scans were collected. This site was ideal for its minimal elevation change and it being virtually free of vegetation. Image courtesy of Google Earth.

Figure 38 .

 38 Figure 38. Close-up view of Site I where terrestrial LiDAR scans were obtained. This site had some considerable topographic relief on the north of the fault with some beheaded stream channels. Image courtesy of Google Earth.

Figure 39 .

 39 Figure 39. The terrestrial LiDAR scanner (TLS) used for the acquisiton of the data along Wairarapa fault. The LiDAR scanner is coupled with high precision kinematic GPS data of the same location to create the DEMs. Figure modified from https://www.bgs.ac.uk.

Figure 40 .

 40 Figure 40. Terrestrial LIDAR DEM of Site F overlain onto the 1m airborne LiDAR data (with our fault and marker intrepretations, see further below).

Figure 41 .

 41 Figure 41. Terrestrial LiDAR DEM of Site F along the Wairarapa fault at 30-cm resolution.

Figure 42 .

 42 Figure 42. Terrestrial LiDAR DEM of Site I-West along the Wairarapa fault at 30-cm resolution.

Figure 43 .

 43 Figure 43. Terrestrial LiDAR DEM of Site I-West along the Wairarapa fault at 30-cm resolution, with our fault and markers interpretation (see further below).

  Figure44shows in the color-coded line the extent of the Wairarapa fault section which we mapped in detail in the Lidar data. According to seismic imaging mapping completed by the Holdgate&Grapes (2015), the Wairarapa fault extends about 40 km into the Palliser Bay and the Cook Strait south of the North Island. As any other fault worldwide, the Wairarapa fault is segmented laterally at different scales[START_REF] Manighetti | Self-similarity of the largest-scale segmentation of the faults: Implications for earthquake behavior[END_REF](Manighetti et al., , 2015)). We know that earthquake slip is sensitive to (at least) the largest scales of fault segmentation[START_REF] Manighetti | Earthquake scaling, fault segmentation, and structural maturity[END_REF]. Lateral segmentation is responsible for slip heterogeneity on fault planes, with the zones separating the major segments (commonly referred to as "inter-segments") being zones of slip deficit, while slip is larger on the segments[START_REF] Manighetti | Self-similarity of the largest-scale segmentation of the faults: Implications for earthquake behavior[END_REF](Manighetti et al., , 2015)). We also know that earthquake and fault slips are variable along the fault length, and hence must be measured locally to be relevant(Manighetti et al., 2001[START_REF] Manighetti | Evidence for self-similar, triangular slip distributions on earthquakes: Implications for earthquake and fault mechanics[END_REF]. We thus performed our slip analysis by "fault pieces" along-strike, with these pieces being the major fault segments visible over the 70 km Lidar stretch.I describe here these major segments that segregate the fault. The Wairarapa fault trace that we mapped is color-coded in Figure44to segregate the fault into major segments as defined by various geologic criteria discussed below. The segments are identified as AB, CD, E, FG, H, I, JK, LM, NO, PQ, RS, T, UV, and WXY (this complex labeling results from AB for instance being divided into two smaller segments, A and B; etc.). I remind that, for time reasons, I only analyzed here the segments AB to NO. These AB to NO fault sections actually covers the "core" of the fault, before it splays further north and south into secondary oblique faults (as described inPerrin et al., 2016) (Figure44). Therefore, we expect to find localized offsets along this investigated section, and larger amounts of displacements than can be found further north and south of our studied section. Altogether, using the fault segmentation as a means to locally investigate slip enables analyzing the offset over fault sections on the order of ~3 (e.g. segment I) -8 km (e.g. segment NO), and hence to consider the earthquake slip along-strike variability.Although I have measured the offsets along segments PQ -WXY (seeTable Wairarapa_Offset Measures-Stewart-PhD 11Nov18), I have not analyzed their results; therefore, segments PQ -WXY will be omitted from the analyses (except on Figures 98, 106, and 107 towards the end of the manuscript).

Figure 44 .

 44 Figure 44. SRTM of the southern section of the North Island, New Zealand complete with fault mapping of the Wairarapa fault, and other major dextral-slip faults in this region composing some of the North Island Fault System (NIFS). The upper figure also shows the bathymetric fault mapping in the Cook Strait by the National Institute of Water and

Figure 45b .

 45b Figure 45b. Same as 47a with our fault and marker mapping.

Figure 46a .

 46a Figure 46a. Bare earth DEM displaying the junction between segments FG and H. Two parallel stands are evident in the morphology that are separated by a considerable distance, while a push-up relief has formed in the segment relay zone.

Figure 46b .

 46b Figure 46b. Same as 48a with our fault and marker mapping.

Figure 47a .

 47a Figure 47a. Bare earth DEM of 'Pigeon Bush' site (inside the red square). 'Pigeon Bush' is the site where Rodgers and Little (2006) measured the 1855 earthquake offset they claim to be the largest ever recorded for any strike-slip fault worldwide. It is located within segment FG. It consists of two beheaded channels.

Figure 47b .

 47b Figure 47b. Same as 49a with our fault and marker mapping.

Figure 48a .

 48a Figure 48a. Bare earth DEM of segment AB

Figure 48c .

 48c Figure 48c. The red arrows delineate the extents of segment AB. The green arrows mark the commencement of segment CD.

Figure 49a .

 49a Figure 49a. Bare-earth DEM of segment CD.

Figure 49b .

 49b Figure 49b. The green arrows delineate the extents of segment CD. The red arrow marks the termination of segment AB while the blue arrows represents the beginning of segment E.

Figure 49c .

 49c Figure 49c. The green arrows delineate the extents of segment CD. The red arrow marks the termination of segment AB while the blue arrows represents the beginning of segment E.

Figure 50b .Figure 51a .

 50b51a Figure 50b. Mapped fault trace and geomorphic markers along segment E. The blue arrows delineate the extents of segment E. The green arrow marks the termination of segment CD while the red arrow signifies the beginning of segment FG.

Figure 51b .

 51b Figure 51b. Mapped fault trace and geomorphic markers along segment FG overlain onto the Bare earth DEM. Segment FG is longer (~6 km) and with a more overall singular trace than the previous segments.

Figure 51c .

 51c Figure 51c. Mapped fault trace of segment FG.

Figure 52b .

 52b Figure 52b. Mapped fault trace and geomorphic markers along segment H. The yellow arrow delineate the extents of segment H. The red arrow marks the termination of segment FG and the green arrow represents the start of segment I.

Figure 53b .

 53b Figure 53b. Mapped fault trace and geomorphic markers along segment I. The green arrows delineate the extents of segment I. The yellow arrow marks the termination of segment H and the orange arrow show the beginning of segment JK.

Figure 53c .

 53c Figure 53c. Fault trace of segment I. Segment I has again a short length (~3.1 km) and is bounded by large step-overs.

Figure 54b .

 54b Figure 54b. Mapped fault trace and geomorphic markers along segment JK. The orange arrows delineate the extents of segment JK. The green arrow represents the termination of segment I while the blue arrow indicates the commencement of segment LM.

Figure 54c .

 54c Figure 54c. Mapped fault trace of segment JK. It is fairly long (~7.1 km) and linear with only one major strand.

Figure 55a .

 55a Figure 55a. Bare-earth DEM of segment LM.

Figure 55b .

 55b Figure 55b. Mapped fault trace and geomorphic markers overlain onto the bare-earth DEM along segment LM. The blue arrows delineate the extents of segment LM. The orange arrow indicates the termination of segment JK while the red arrow demarcates the start of segment NO. The junction of segments JK and LM is marked by a pressure ridge.

Figure 55c .

 55c Figure 55c. Fault trace of segment LM.

Figure 56b .

 56b Figure 56b. Mapped fault trace and geomorphic markers along segment NO. The red arrows delineate the extents of segment NO. The orange arrows display the termination of segment LM and the green arrow shows the start of segment PQ.

Figure 56c .

 56c Figure 56c. Mapped fault trace of segment NO.

Figure 57a .

 57a Figure 57a. A DEM of the section of the Wairarapa fault. In this case the sun angle is not ideal. Major features can be discerned, but it is difficult to ascertain the all possible offset geomorphic features confidently.

Figure 57b .

 57b Figure 57b. The same DEM with a better sun illumination angle and aspect ratio. We are able to better identify offset features.

Figure 58a .

 58a Figure 58a. The original DEM of marker WF365779 of segment LM. Notice the original orientation of the DEM and fault trace.

Figure 58b .

 58b Figure 58b. The rotated DEM after the code processes the dip direction according the direction of the fault trace. Also, notice how the polygons encompass the terrace and include all of its pertinent geomorphic features. The image on the right exhibits the faulted terrace in its natural orientation to allow for easier discrimination of the features.

Figure 59 .

 59 Figure 59. An example of polygon placement for marker WF355941 in segment JK. The blue arrow indicates the location of a kink in the channel that might alter the offset amount.

Figure 60 .

 60 Figure 60. A 3D plot of the placement of each geometric feature in the northwest quadrant of the DEM of marker WF359875. This figure helps to see if the placement of the points corresponding to a specific geometric feature is actually representing that feature instead of an artifact.

Figure 61 .

 61 Figure 61. The 'Calc4' figures represent the linear regressions and piercing points of all the geometric features on a 2D plot. This allows for the observation of the 2D spatial variation along the marker, and to verify if the feature is more or less linear. The lateral and vertical offset amounts are also visible, but they should not be considered at this stage, for the user can be biased by like amounts.

Figure 61

 61 Figure 61 also assists in the identification of the geomorphic features actually representing the marker. A version of Figure 61 can be found for each respective marker in a collection of synthesis figures per marker in Appendix II. It shows the two-dimensional spatial variation of the points on the DEM. It becomes more apparent in this figure as to the origin of the points if they are not defining the desired feature. For example, anthropogenic features (such as roads) have sharp edges and can sometimes be picked by points for riser top,

Figure 62 .

 62 Figure 62. A vertical back-slip figure of marker WF359875. The figure presents the geomorphic feature under concern, here the marker's east riser base, in a 2D representation of the points that populated the DEM. It back-slips the southern component to the northern. It allows me to identify the best-constrained vertical offset by considering the spatial width of the points.

Figure 63 .

 63 Figure 63. Example of the short and long case measurements for the terrace riser of marker WF365779 (rotated, thus upside down view from the code). Both polygons capture the overall linear trend of the marker, but the small bend in the middle warrants two separate measurements. The red square in the 'long' case denotes the location of the bend.

Figure 64 .

 64 Figure 64. Deflection of an active channel due to its repeated lateral displacements.

Figure 65 .

 65 Figure 65. A Google Earth image showing the location of the measured geomorphic markers by 3D_Fault_Offsets along the available LiDAR dataset. The yellow pins represent the markers. The figure shows well the density of offsets along the entire trace.

Figure 66 .

 66 Figure 66. Plots showing the ratio of the uncertainty to the lateral offset value for the three discriminatory categories: 'underlined' (i.e. best geomorphic feature), 'east mean' (i.e. mean of offset across eastern edge of marker) and 'OPT' (i.e. mean of all geomorphically relevant features) cases.

Figure 67 .

 67 Figure 67. Sketch detailing how the trailing edge of an offset channel can be incised and lose the preservation of the true offset amount (from Stewart et al., 2018).

Figure 68 .

 68 Figure 68. Plots showing the ratio of the uncertainty to the vertical offset value for the two discriminatory categories: 'underlined' (i.e. best geomorphic feature) and 'Mid-slope' (i.e. mean of all geomorphically relevant maximum slope gradient) cases.

  Figure a, underlined value in TABLE Wairarapa_Offset Measures-Stewart-PhD 11Nov18), the Mean East offset (Figure b), and the overall Optimal offset (Figure c). Then the second line of figures shows the same for the "All quality offsets" (thus Figures d-e-f in correspondence with Figures a-bc, respectively).

Figure 69 .

 69 Figure 69. COPD plots for the 3 lateral offset categories 'Best geomorphic' (i.e. Underlined), 'mean East', and 'OPT', for segment AB. The rows are discriminated according to the inclusion of only the GOOD+MOD and ALL quality ratings. The peak values are clear in the 3 "categories" of the lateral offsets, "best geomorphic feature" (Figure 69-a&d), mean East (Fig 69-b&e), and optimal (Fig 69-c&f), and in both the "Good and Moderate quality" offsets (Figure 69-a-b-c) and the entire offset collection (Figure

Figure 70a .

 70a Figure 70a. COPD plot (with a range to 80 m on the lateral COPD plots) for the vertical offset 'Underlined' (i.e. best geomorphic) category for segment AB.

Figure 70b .

 70b Figure 70b. COPD plot (with a range to 80 m on the lateral COPD plots) for the vertical offset 'Mid-Slope' (i.e. maximum gradient) category for segment AB. The 2 categories of Underlined and Mid Slope generally agree with the positions of most individual measures. The peaks in the summed PDF curves are not very meaningful however, except possibly those around +0.6, +2.5, and -2 --4 m (Figures 70a&70b).According to Figure 70a, the lateral COPD group ~4 (grey) and ~9.5 m (red) congregate mostly around the 0.6 m of vertical slip. Even though vertical offsets are fairly distributed, the

  According to Figure 70a, the lateral COPD group ~4 (grey) and ~9.5 m (red) congregate mostly around the 0.6 m of vertical slip. Even though vertical offsets are fairly distributed, the vertical slips seem to increase from red (~0m) to green (~-4m) to violet (~-9m) as do their corresponding lateral COPD offsets.

Figure 71 .

 71 Figure 71. The vertical to lateral slip ratio of offset measurements for segment AB.

Figure 72 .

 72 Figure 72. COPD plots for the 3 lateral offset categories 'Best geomorphic' (i.e. Underlined), 'mean East', and 'OPT', for segment CD. The rows are discriminated according to the inclusion of only the GOOD+MOD and ALL quality ratings.

Figure 73a .

 73a Figure 73a. COPD plot (with a range to 80 m on the lateral COPD plots) for the vertical offset 'Underlined' (i.e. best geomorphic) category for segment CD.

Figure 73b .

 73b Figure 73b. COPD plot (with a range to 80 m on the lateral COPD plots) for the vertical offset 'Mid-Slope' (i.e. maximum gradient) category for segment CD.The peaks in the vertical summed PDF curves are not very meaningful, except those around +1, and +5 m (Figures 73a&73b). Even though vertical offsets are fairly distributed, the positive side of Figure73ashows a discrete increase of the vertical slips from the red (~1-2m), to green-yellow (~5m), to violet (~8m), then to dark blue (~10m), in accordance with the discrete increase of their corresponding lateral COPD offsets.

  Even though vertical offsets are fairly distributed, the positive side of Figure73ashows a discrete increase of the vertical slips from the red (~1-2m), to green-yellow (~5m), to violet (~8m), then to dark blue (~10m), in accordance with the discrete increase of their corresponding lateral COPD offsets.

Figure 74 .

 74 Figure 74. The vertical to lateral slip ratio of offset measurements for segment CD.

Figure 75 .

 75 Figure 75. COPD plots for the 3 lateral offset categories 'Best geomorphic' (i.e. Underlined), 'mean East', and 'OPT', for segment E. The rows are discriminated according to the inclusion of only the GOOD+MOD and ALL quality ratings.

Figure 77 .

 77 Figure 77. The vertical to lateral slip ratio of offset measurements for segment E.

Figure 78 .

 78 Figure 78. COPD plots for the 3 lateral offset categories 'Best geomorphic' (i.e. Underlined), 'mean East', and 'OPT', for segment FG. The rows are discriminated according to the inclusion of only the GOOD+MOD and ALL quality ratings. In segment FG (34 offset measures), four lateral offsets are well preserved (in all graphs -Figure 78), averaging 9.3 ± 2.7 m (10.0 ± 3.3 m for all qualities), 17.9 ± 3.5 m (17.2 ± 3.2 for all qualities), 32.0 ± 3.3 m (31.8 ± 3.0 for all qualities), and 47.5 ± 3.0 m (47.5 ± 3.6

Figure 79a .

 79a Figure 79a. COPD plot (with a range to 80 m on the lateral COPD plots) for the vertical offset 'Underlined' (i.e. best geomorphic) category for segment FG.

Figure 79b .

 79b Figure 79b. COPD plot (with a range to 80 m on the lateral COPD plots) for the vertical offset 'Mid-Slope' (i.e. maximum gradient) category for segment FG. The corresponding vertical offset values form clusters around ~0, ~2, and 4 -5 m (Figures 79a&79b).Figure 79a suggests (on its positive side) a discrete increase of the vertical slips from the red (~0m), to green-yellow (~2-5m), to pale blue (~7m), in accordance with the discrete increase of their corresponding lateral COPD offsets.

  Figure79asuggests (on its positive side) a discrete increase of the vertical slips from the red (~0m), to green-yellow (~2-5m), to pale blue (~7m), in accordance with the discrete increase of their corresponding lateral COPD offsets.

Figure 80 .

 80 Figure 80. The vertical to lateral slip ratio of offset measurements for segment FG.

Figure 81 .

 81 Figure 81. COPD plots for the 3 lateral offset categories 'Best geomorphic' (i.e. Underlined), 'mean East', and 'OPT', for segment H. The rows are discriminated according to the inclusion of only the GOOD+MOD and ALL quality ratings.

Figure 82a .

 82a Figure 82a. COPD plot (with a range to 80 m on the lateral COPD plots) for the vertical offset 'Underlined' (i.e. best geomorphic) category for segment H.

Figure 82b .

 82b Figure 82b. COPD plot (with a range to 80 m on the lateral COPD plots) for the vertical offset 'Mid-Slope' (i.e. maximum gradient) category for segment H. The corresponding vertical offset values are very distributed (in both positive and negative sides) and hence difficult to interpret. The majority of them form a cluster around 0 vertical slip, with additional offsets suggested around ~+4 and ~ -3/-5m (Figures 82a&82b).The correspondence with the lateral COPD cannot be done.

  The correspondence with the lateral COPD cannot be done.

Figure 83 .

 83 Figure 83. The vertical to lateral slip ratio of offset measurements for segment H.

Figure 84 .

 84 Figure 84. COPD plots for the 3 lateral offset categories 'Best geomorphic' (i.e. Underlined), 'mean East', and 'OPT', for segment I. The rows are discriminated according to the inclusion of only the GOOD+MOD and ALL quality ratings. In segment I (20 offsets measured), because the data are few, only the smallest offset is very well constrained (Figure 84) at, on average, 10.8 ± 3.4 m (10.3 ± 3.8 for all qualities). Yet other COPDs are revealed fairly clearly, on average at (18.3 ± 1.6 for all qualities), 29.1 ±

Figure 87 .

 87 Figure 87. COPD plots for the 3 lateral offset categories 'Best geomorphic' (i.e. Underlined), 'mean East', and 'OPT', for segment JK. The rows are discriminated according to the inclusion of only the GOOD+MOD and ALL quality ratings. In segment JK (56 offset measures), the four smallest COPDs are well-constrained (Figure 87) and they are, on average, at 3.4 ± 1.8 m (5.1 ± 2.7 for all qualities), 11.1 ± 3.7 m (9.3 ± 2.3 for all qualities), 15.4 ± 3.1 m (15.2 ± 3.1 for all qualities), and 27.4 ± 2.2 m (27.6

Figure 88a .

 88a Figure 88a. COPD plot (with a range to 80 m on the lateral COPD plots) for the vertical offset 'Underlined' (i.e. best geomorphic) category for segment JK.

Figure 88b .

 88b Figure 88b. COPD plot (with a range to 80 m on the lateral COPD plots) for the vertical offset 'Mid-Slope' (i.e. maximum gradient) category for segment JK. The corresponding vertical offset values form clusters around ~0, ~2, ~4-5, and ~ 10 m (Figures 88a&88b).Figures 88a suggests (on its positive side) a discrete increase of the vertical slips from the grey-red (~0m), to green (~4-5m), to violet (~7m) to pale blue (~10-11m), in accordance with the discrete increase of their corresponding lateral COPD offsets.

  Figures 88a suggests (on its positive side) a discrete increase of the vertical slips from the grey-red (~0m), to green (~4-5m), to violet (~7m) to pale blue (~10-11m), in accordance with the discrete increase of their corresponding lateral COPD offsets.

Figure 89 .

 89 Figure 89. The vertical to lateral slip ratio of offset measurements for segment JK.

Figure 90 .

 90 Figure 90. COPD plots for the 3 lateral offset categories 'Best geomorphic' (i.e. Underlined), 'mean East', and 'OPT', for segment LM. The rows are discriminated according to the inclusion of only the GOOD+MOD and ALL quality ratings.

Figure 91a .

 91a Figure 91a. COPD plot (with a range to 80 m on the lateral COPD plots) for the vertical offset 'Underlined' (i.e. best geomorphic) category for segment LM.

Figure 91b .

 91b Figure 91b. COPD plot (with a range to 80 m on the lateral COPD plots) for the vertical offset 'Mid-Slope' (i.e. maximum gradient) category for segment LM. The vertical offsets are very distributed, yet suggest dominant clusters around ~0-2, and ~6-7m. Figures 91a suggests a discrete increase of the vertical slips from the red-green (~0-2m), to yellow (~6m), to pale blue (~7m), and to dark blue (~14m), in accordance with the discrete increase of their corresponding lateral COPD offsets.

Figure 92 .

 92 Figure 92. The vertical to lateral slip ratio of offset measurements for segment LM.

Figure 93 .

 93 Figure 93. COPD plots for the 3 lateral offset categories 'Best geomorphic' (i.e. Underlined), 'mean East', and 'OPT', for segment NO. The rows are discriminated according to the inclusion of only the GOOD+MOD and ALL quality ratings.In segment NO (85 offsets measures, the largest density of measurements on a segment), five lateral offsets are well preserved (Figure93) at, on average, 7.2 ± 4.6 m (7.9 ± 4.7 for all qualities), 19.3 ± 3.2 m (19.2 ± 3.9 for all qualities), 30.6 ± 4.4 m (31.8 ± 4.4 for all

Figure 94a .

 94a Figure 94a. COPD plot (with a range to 80 m on the lateral COPD plots) for the vertical offset 'Underlined' (i.e. best geomorphic) category for segment NO.

Figure 94b .

 94b Figure 94b. COPD plot (with a range to 80 m on the lateral COPD plots) for the vertical offset 'Mid-Slope' (i.e. maximum gradient) category for segment NO.

  Figures 94a suggests (on its positive side) a discrete increase of the vertical slips from the red (~0m), to green-yellow (~3-4m), to pale blue (~4-7m), and to dark blue (~9-11m), in accordance with the discrete increase of their corresponding lateral COPD offsets.

Figure 95 .

 95 Figure 95. The vertical to lateral slip ratio of offset measurements for segment NO.

Figure 96 .

 96 Figure 96. Scatter plots showing the relation between the lateral and vertical COPD peak values.

Figure 97c .

 97c Figure 97c. Channel lateral offset measurements plotted along the length of the Wairarapa fault. The data represents only the geomorphic markers of streams and channels, which previous authors found to be the most robustly constrained. In black are my measurements.

Figure

  Figure97arepresents the lateral offset dataset classified by author. For the sector of the Wairarapa fault beginning with our segment AB, the measurements from[START_REF] Grapes | The Wairarapa Fault[END_REF] range from 11.5 to 380 m and present 2 well-defined clusters at ~12 and 24m. Lateral offset values fromLensen (1968Lensen ( , 1969) ) range from 4.5 -140m, and present no clear clusters. Measurements from[START_REF] Villamor | It's our fault -Wairarapa fault slip rate investigations task: Completion report -Is the Wairarapa fault slip rate decreasing to the north[END_REF] are similar to those completed byLensen (1968); however, there remains discrepancy between these datasets when they are compared to the[START_REF] Grapes | The Wairarapa Fault[END_REF] clusters.Figure97bshows the best lateral displacement values (best defined by[START_REF] Villamor | It's our fault -Wairarapa fault slip rate investigations task: Completion report -Is the Wairarapa fault slip rate decreasing to the north[END_REF] for all types of geomorphic features in the database with respect to their age (not absolute). Figure97crepresents data from offset streams and channels, which is regarded by the authors to be most representative of the true offset value since other geomorphic features (e.g. terrace risers) can be subjected to post-displacement erosion. Of course, the thalwegs of streams and channels can be subjected to their own factors of degradation.

Figure 98 .

 98 Figure 98. The vertical to lateral slip ratio for the entire Wairarapa fault (segments AB -WXY) covered by LiDAR data.

Figure 99 .

 99 Figure 99. Histogram showing the preservation of the western and eastern edges of the entire dataset.

Figure 100 .

 100 Figure 100. The vertical offset amounts of the riverbed versus the 'best geomorphic' representation on the riser edge. This graph provides insight into the incision that the river has committed on the thalweg (i.e. bed) compared to the riser edges.

Figure 101 .

 101 Figure 101. A plot showing the degrees of erosion, specifically for the bed of river, compared to the "best" preserved vertical offset. It is a histogram of the percentage of incision of the riverbed.

  is a histogram of the degrees of incision of the riverbed. The histogram shows that all degrees of incision are well represented in the data, from 0 to greater than 100 % of the vertical riverbed slip. The varying percentages indicate that the riverbed can slightly or severely underestimate the true vertical offset amount. More work needs to be done on these degrees of incision to determine their distribution along the fault and their possible major causes.

Figure 102 .

 102 Figure 102. Plot showing the lateral slip (underlined cases) versus the amount of riverbed incision (difference between "best" preserved vertical offset and that across the riverbed). It shows how the incision evolves with increasing cumulative lateral slip, that is time.

Figure 102

 102 Figure 102 is an attempt to understand how the vertical incision evolves with time.Here we take the lateral offset amount as a proxy of time; the greater the lateral slip, the longer the time of fault activity. The figure thus compares the lateral slip amount (underlined cases) with the amount of vertical incision (difference between the underlined vertical slip and the vertical slip across the riverbed, for a respective marker (cases where underlined feature is the riverbed have been ignored). Even though the data are distributed, we see that, for most lateral slips greater than ~40m, the vertical incision increases overall with the amount of lateral displacement, and hence with time. For the smallest lateral offsets, there is no clear correlation between lateral slip amount and incision degree. More work needs to be done to analyze these preliminary results.

Figure 103 .

 103 Figure 103. Plots of our 'GOOD+MOD' quality COPD horizontal offsets plotted along the length of the fault. The segments are illustrated in the x-axis. The 3 mean offset cases (underlined, East Mean and OPT) are represented (from Table 1 (TABLE Wairarapa_COPD versus Length-Stewart-PHD 11Nov18 found externally). The COPD in bold in the table are represented in the largest symbols, those in standard font in medium size symbols, and those in italics in the smallest symbols. The color groups represent our first interpretation of the group of cumulative slip values that we grouped based on a rough similarity of their lateral slip amount. The groups might have thus been created during successive individual events. The red square at ~121 km in the top figures represents the average 1855

  Figure103is a first proposition for the potential earthquake events that caused the cumulative slip measured at the surface. We have plotted the 3 categories of COPD lateral offsets (see figure caption), as a function of the total fault length. Figure103is only composed of the most robustly-constrained 'GOOD&MODERATE' quality data. The colors for the dots represent our tentative interpretation of the data based on the similarity "at first glance" of the COPD values. These COPD values are directly derived from Table1(also available externally as TABLE Wairarapa_COPD versus Length-Stewart-PHD 11Nov18). Hence, it is likely that similar cumulative slip values along different fault segments were produced by the same event (or number of events). The size of the points represents how well the COPD value is constrained (see caption of figure). On the top figure of Figure103, the red square at ~121 km represents the average displacement measured bySchermer et al., 2004 at several locations on the Alfredton fault, which branches north from the Wairarapa fault. Schermer et al., 2004 postulated that the southern part of the Alfredton fault ruptured in 1855 resulting from the historic Wairarapa fault earthquake. Prior to the work ofSchermer et al. 2004, the 1885 Wairarapa rupture was conventionally regarded to have terminated in Mauriceville (Figure22), where the fault branches into the Pa Valley and Alfredton fault systems. However,[START_REF] Grapes | The Wairarapa Fault[END_REF] recorded lateral slip components of ~10m near Mauriceville (an observation that is represented in this study as well -see Figure97a-bc). Shermer et al., 2004 recorded a maximum lateral slip of 4-7m (average ~5 m) along the Alfredton fault. Trenching results show that 2 earthquakes occurred on the Alfredton fault, the most recent event coinciding with the timing of the 1855 Wairarapa earthquake. Therefore, it is likely that the average displacements on the Alfredton fault are a continuation of the rupture of the 1855 Wairarapa earthquake and can be used to constrain the along-fault slip distribution of our dataset.The grey dots depict a subset of the population that is not well-represented across the length of the fault. The COPDs for the grey dot collection are only clearly present in segments AB, CD, H and JK. The lateral slips are between ~3 and 5 m (see TABLE Wairarapa_COPD versus Length-Stewart-PHD 11Nov18). These offsets are much smaller than those reported for the most recent 1855 earthquake, while no more recent earthquake is known on the Wairarapa fault. Furthermore, if these 3-5m slip would have been produced by an earthquake, this earthquake would have been strong[START_REF] Manighetti | Earthquake scaling, fault segmentation, and structural maturity[END_REF], likely with a magnitude greater than 6.5. Yet no earthquake of this size has been reported on the Wairarapa fault since the large 1855 event. Together these suggest that the markers that have preserved these small offsets have been dramatically eroded. As can be seen in Figure102, even the smallest offset can be exposed to large degrees of erosion along the Wairarapa fault. Also, Darby&Beaven (2001) have demonstrated through 5-year geodetic study that there is no aseismic creep along the Wairarapa fault. Therefore, it is likely that the grey dots represent markers that record an underestimation of the true offset amount due to the erosion which they were subjected. I will thus ignore these small lateral offsets in the following (note however that "scenario1" in Table3&4(TABLE Wairarapa_COPD and Slip Increments-Stewart-PHD 11Nov18 found externally) refers to an attempt to consider them; they then produce earthquakes slips locally much lower than anywhere else on the fault, and much lower than the 1855 slip, which confirms they should be ignored).The red dots thus represent the smallest and hence most recent lateral slips preserved along the fault. It is a COPD population that is well-represented throughout every segment (AB -NO). The lateral slips range from 8.6± 2 (segment H) -13.9±3m (segment CD) with an average of 10.5 ± 3.5m. Since the red dot population is the best-resolved in the morphology, and it consists of the smallest offsets, it is inferred that they represent the slips from the most recent earthquake, the 1855 Mw 8.2 on Wairarapa. The average dextral slip

Figure 104 .

 104 Figure 104. Plots of our 'ALL' qualities COPD horizontal offset data plotted along the length of the fault. Same caption as in prior figure. The color groups represent our first interpretation of the group of cumulative slip values that could be grouped together based on a "first glance" similarity of their values.

Figure 105 .

 105 Figure 105. A scenario for the slip increments for each large earthquake that we can resolve in Wairarapa fault rupture history. The top figure is data from the 'GOOD+MOD' quality ratings only, while the lower figure is data from 'ALL' qualities. The different symbols indicate the successive increments. Solid symbols are for well-constrained (bold in Table3), empty symbols are for moderately constrained (regular font in the table), and X-symbols are for poorly constrained (italics in the table). The segments that we have defined are labeled along the x-axis.

Figure 106 .

 106 Figure 106. Plot depicting all the lateral slip measures along-fault per marker. The top figure displays data using the OPT category for the entire length of the LiDAR data (70 km) (segments AB -WXY). The bottom figure represents the data for the "best" geomorphic cases (i.e. underlined) (segments AB -NO).

Figure 106

 106 Figure 106 consists of scatter plots depicting all the lateral slip measures for each individual marker along the length of the fault. The top figure represents data for the entire 70-km length of the LiDAR data. It includes the individual measurements for every segment (AB -WXY). I used the OPT dataset for the top figure because the underlined cases have not

Figure 107 .

 107 Figure 107. Plot depicting all the vertical slip measures along-fault per marker. The top figure displays data using the OPT category for the entire length of the LiDAR data (70 km) (segments AB -WXY). The bottom figure represents the data for the "best" geomorphic cases (i.e. underlined) (segments AB -NO).

Figure 107

 107 Figure 107 consists of scatter plots depicting all the vertical slip measures for each individual marker along the length of the fault. The top figure represents data for the entire 70-km length of the LiDAR data. It includes the individual measurements for every segment (AB -WXY). I used the Mid-Slope dataset for the top figure because the underlined cases

Figure 108 .

 108 Figure 108. Vertical to lateral slip ratio along segment UV (northern section of the "core" of the Wairarapa fault).

Figure 109 .

 109 Figure 109. Plot of the lateral COPD peaks per segment (AB -NO). The data used is 'ALL' qualities. It is the same representation as Figure 95, but the colors have been removed to allow for a more objective view. The suggested slip profiles have been drawn (thick grey lines) as another, more likely proposition for cumulative slip per large earthquake event.

Table ES

 ES

-A for lateral and vertical, respectively), (ii) the nine automatic measures with manual removal of the few clearest outlier points (ALL-C, columns AS and AV in Table ES-A for lateral and vertical, respectively); (iii) the most geomorphically relevant automatic measures only (GEOM-A, columns AY and BB in Table ES-A for lateral and vertical, respectively), and (iv) the most geomorphically relevant measures with manual removal of the clearest outlier points (GEOM-C, columns AY and BB in Table ES-A for lateral and vertical, respectively) are all consistent. They can thus be combined into a mean global offset, per polygon set (columns BG and BL in Table ES-A for lateral and vertical, respectively). This allows for further

  Table provided in Supplementary Material (excel file named

TABLE Wairarapa_Offset Measures-Stewart-PhD 11Nov18).

 Wairarapa_Offset

Table 1 -

 1 GOOD+MOD &

Table 2 -

 2 ALL Qualities). The COPD peak offsets constrained from multiple well-constrained individual offsets are presented in bold in the table. Those derived from only a few well-constrained individual offsets are in standard font. Additional offsets are suggested in italics: they are poorly constrained. Finally, the few values in pale grey in Table1&2(also available externally as

TABLE Wairarapa_COPD versus Length-Stewart-PHD 11Nov2018) indicate

  the existence of individual offsets and whose value is close to a neighboring COPD, yet they do not form any peak in the summed pdf. They are listed to indicate that offsets with this amount exist in the collection.

Table 1 . Table showing all of the COPD results for segments AB -NO for the 'GOOD+MOD' quality data.Table 2 . Table showing all of the COPD results for segments AB -NO for the 'ALL' quality data.

 12

Table 3 . Table showing the COPD and slip increment analysis for segments AB -NO for the 'GOOD+MOD' quality data.Table 4 . Table showing the COPD and slip increment analysis for segments AB -NO for the 'ALL' quality data.

 34

Acknowledgements

The work of completing a thesis is a long and arduous endeavor that would not be possible without support of many. I would like to take this opportunity to acknowledge and

Acknowledgments

The Lidar data used in the present study can be found on www.opentopography. org (B4 LiDAR Project and EarthScope Southern & Eastern California LiDAR Project) and in the prior papers that published them (Zielke et al., 2012; Haddon et al., 2016; Manighetti, Perrin, et al., 2015). The measurement data sets produced in the present study are available in the supporting information. This work has been funded by Université Côte d'Azur, Nice, France. We greatly thank the two reviewers (O. Zielke and O. Bellier) and the Associate Editor for their thorough constructive comments that greatly helped us to improve our manuscript.

The preservation of fresh fault scarps was identified by Grapes (1999) from Palliser Bay (southern extent on land of the Wairarapa fault) to Mauriceville (Figures 21 & 23). Schermer et al. (2004) observed that the Alfredton fault, a northward continuation of the Wairarapa fault, appeared to become reactivated about 250 -330 years B.P., indicating that the 1855 rupture may have extended about 30 km north of Mauriceville. This observation coupled with other onshore evidence infers that the 1855 rupture could have been ~120 km long onshore alone. After this event, there have been no large earthquakes on the Wairarapa fault. Recent field investigations of offset terraces suggest 2.7 m of relative vertical slip on the fault at the coast in 1855 decreasing to ~1.8 m near Featherston [START_REF] Grapes | The 1855 Wairarapa, New Zealand, earthquake: Analysis of historical data[END_REF]. The 1855 rupture is inferred to have originated offshore into Palliser Bay which is the likely location of the epicenter [START_REF] Darby | Possible source models for the 1855 Wairarapa Earthquake, New Zealand[END_REF][START_REF] Grapes | The 1855 Wairarapa, New Zealand, earthquake: Analysis of historical data[END_REF]. Seismic reflection data imaged the southwestern termination of the offshore continuation of the Wairarapa fault into the Cook Strait over ~35 -40 km [START_REF] Barnes | Recognition of active strike-slip faulting from high-resolution marine seismic reflection profiles: Eastern Marlborough fault system, New Zealand[END_REF]. The surface rupturing evidence from both on-and offshore indicate a maximum rupture length of about 145 -160 km. Even with this "increased" length, the 1855 earthquake was a very large stress drop earthquake, with among the greatest slip to length ratio ever observed in continental earthquakes [START_REF] Rodgers | World's largest coseismic strike-slip offset: The 1855 rupture of the Wairarapa Fault, New Zealand, and implications for displacement/length scaling of continental earthquakes[END_REF][START_REF] Manighetti | Earthquake scaling, fault segmentation, and structural maturity[END_REF]. One key question of this work is to know if forthcoming large earthquakes could be as strong and energetic as the 1855 rupture based on the slip history of Wairarapa.

Along the Alfredton fault, the northern extent of the Wairarapa fault, the 1855 rupture produced a horizontal coseismic displacement of 4 -7 m (Schermer et al., 2004). This proposed offset amount along the northern component of the rupture is markedly smaller than the displacement recorded for the central section of the Wairarapa fault (i.e. 15.5 ± 1.4 m; [START_REF] Rodgers | World's largest coseismic strike-slip offset: The 1855 rupture of the Wairarapa Fault, New Zealand, and implications for displacement/length scaling of continental earthquakes[END_REF]. Although this data suggests a decreasing trend northward for the 1855 rupture slip, it remains unclear as to how this trend is attained. On the Wairarapa fault to the north of the intersection with the Carterton fault, there are no reported displacement measurements associated with the 1855 rupture. There have only been some recorded estimates of the cumulative offset in this region of the fault. At the Tea Creek site (Figure 21), Van Dissen and [START_REF] Van Dissen | Surface rupture earthquakes over the last ∼1000 years in the Wellington region, New Zealand, and implications for ground shaking hazard[END_REF] recorded a cumulative displacement of a stream crossing the Wairarapa fault to be ~40 m that they propose accumulated from 4 surface-rupturing events yielding a mean single-event displacement amount of ~10 m at this location.

4.b Geomorphic markers and cumulative offsets along the Wairarapa fault (prior knowledge from literature)

Both single-event displacement studies from the most recent earthquake and studies investigating the cumulative slip along Wairarapa have been previously conducted (e.g. Lensen, 1968Lensen, , 1969;;Grapes et al., 1984;[START_REF] Grapes | The Wairarapa Fault[END_REF][START_REF] Van Dissen | Surface rupture earthquakes over the last ∼1000 years in the Wellington region, New Zealand, and implications for ground shaking hazard[END_REF][START_REF] Little | 1855 Wairarapa Earthquake Symposium[END_REF][START_REF] Rodgers | World's largest coseismic strike-slip offset: The 1855 rupture of the Wairarapa Fault, New Zealand, and implications for displacement/length scaling of continental earthquakes[END_REF][START_REF] Villamor | It's our fault -Wairarapa fault slip rate investigations task: Completion report -Is the Wairarapa fault slip rate decreasing to the north[END_REF]. All offset measures performed before 2008 have been compiled by [START_REF] Villamor | It's our fault -Wairarapa fault slip rate investigations task: Completion report -Is the Wairarapa fault slip rate decreasing to the north[END_REF], yet in an unpublished GNS report. Therefore, this data compilation is not available. Only the figures produced by [START_REF] Villamor | It's our fault -Wairarapa fault slip rate investigations task: Completion report -Is the Wairarapa fault slip rate decreasing to the north[END_REF] can be found, and I present them below. The location of the published offsets is displayed in Figure 23. In Figures 242526, the measured horizontal displacement values are plotted against the length of the Wairarapa fault. The major splay faults in the north are indicated. The basically single trace "core" of the Wairarapa fault is thus west of the Carterton intersection, while faulting becomes splaying and hence more distributed east of the Carterton intersection. The total range of lateral offsets is shown to the left while a focus on the smallest lateral offsets, in the range 0-50m, is made to the right. We examined the LiDAR data with the Ermapper software allowing for variable sunshading. The sun-shading allowed for the illumination of the fault trace beautifully, as well as most of the geomorphic markers the fault offsets. As most active strike-slip faults, the Wairarapa fault has a fairly linear and continuous trace at ground surface that dissects all the markers and features that stand on the ground. Observed in more detail, the fault is seen to form a small vertical scarp whose dip oscillates along the fault about the vertical. The fault is also segmented laterally into discrete sections arranged in a left-stepping geometry and separated by step-overs or contractional relays. The inter-segment zones consist of wide areas of distributed, more disorganized, secondary fissuring. Examples are shown in Figures 4546. We have used this natural segmentation of the fault to conduct our offset analysis on discrete separate fault sections, which are described below. The LiDAR data also well reveal most of the geomorphic markers offset along the fault. Examples are shown in Figures 454647below, and in the next section.

4.d.1.3 Segment E

4.h.1 Comparison of our measured offsets with prior offset measurements