Susan J Conway
email: susan.conway@univ-nantes.fr

Matthew R Balme
email: m.r.balme@open.ac.uk

John B Murray
email: j.b.murray@open.ac.uk

Martin C Towner
email: m.towner@imperial.ac.uk

Chris H Okubo
email: cokubo@usgs.gov

Peter M Grindrod
email: p.grindrod@ucl.ac.uk

The indication of martian gully formation processes by slope-area analysis

The formation process of recent gullies on Mars is currently under debate. This study aims to discriminate between the proposed formation processes: pure water flow, debris flow and dry mass wasting, through the application of geomorphological indices commonly used in terrestrial geomorphology. We used high resolution digital elevation models of Earth and Mars to evaluate the drainage characteristics of small slope sections. We have used the data from Earth to validate the hillslope, debris flow and alluvial process domains previously found for large fluvial catchments on Earth, and have applied these domains to gullied and ungullied slopes on Mars. In accordance with other studies our results indicate that debris flow is one of the main processes forming the martian gullies that we studied. The source of the water is predominantly distributed surface melting, not an underground aquifer. We also present evidence that other processes may have shaped martian crater slopes, such as ice assisted creep and solifluction, in agreement with proposed recent martian glacial and periglacial climate. Our results suggest that, within impact craters, different processes are acting on differently oriented slopes, but further work is needed to investigate the potential link between these observations and changes in martian climate.

Martian "gully" landforms were first described by [START_REF] Malin | Evidence for recent groundwater seepage and surface runoff on Mars[END_REF] and defined as features that have an alcove, channel and debris apron with the general appearance of gullies carved by water. Within this definition gullies have a wide range of morphologies (Fig. 1) and they are found in abundance on steep slopes at mid latitudes in both hemispheres on Mars (e.g., [START_REF] Heldmann | Observations of martian gullies and constraints on potential formation mechanisms[END_REF][START_REF] Heldmann | Observations of martian gullies and constraints on potential formation mechanisms II. The northern hemisphere[END_REF]. They are interpreted to be geologically young features because of the pristine appearance and paucity of superposed impact craters. Recent work has suggested that some gullies have been active in the last 3 -1.25 Ma [START_REF] Reiss | Absolute dune ages and implications for the time of formation of gullies in Nirgal Vallis, Mars[END_REF][START_REF] Schon | Unique chronostratigraphic marker in depositional fan stratigraphy on Mars: Evidence for ca. 1.25 Ma gully activity and surficial meltwater origin[END_REF]. [START_REF] Malin | Present-day impact cratering rate and contemporary gully activity on Mars[END_REF] observed new, highalbedo, dendritic deposits (named light-toned deposits) located along the paths of some gullies, and which formed between subsequent images taken by the Mars Orbiter Camera (MOC). These light-toned deposits have been attributed to either dry mass wasting [START_REF] Pelletier | Recent bright gully deposits on Mars: Wet or dry flow?[END_REF][START_REF] Kolb | Modeling the formation of bright slope deposits associated with gullies in Hale Crater, Mars: Implications for recent liquid water[END_REF], or debris flow [START_REF] Heldmann | Possible Liquid Water Origin for Atacama Desert Mudflow and Recent Gully Deposits on Mars[END_REF], involving up to 50 % water [START_REF] Iverson | The physics of debris flows[END_REF]. However, the origins of these deposits are still under debate and it is not clear whether they are related to the formation processes of the gullies, or whether they are formed by a secondary process.

The formation process for martian gullies in general is also still under debate. Three main candidates exist: (1) aquifer outflow, (2) surface melting, or (3) dry granular flow. In the aquifer model the water is either released from a near surface confined aquifer [START_REF] Malin | Evidence for recent groundwater seepage and surface runoff on Mars[END_REF][START_REF] Heldmann | Formation of Martian gullies by the action of liquid water flowing under current Martian environmental conditions[END_REF] or brought up from depth by cryovolcanic processes [START_REF] Gaidos | Cryovolcanism and the recent flow of liquid water on Mars[END_REF]. The main criticism of the aquifer based models is their failure to explain the location of some gullies on isolated hills, impact crater central peaks, mesas and sand dunes.

Melting of near surface ground ice or surface ice has been proposed for the formation of gullies under recent obliquity excursions [START_REF] Costard | Formation of recent Martian debris flows by melting of near-surface ground ice at high obliquity[END_REF]. There is growing support for this model with the most compelling arguments being: (1) the majority of gullies exists at mid-latitudes, (2) the dominance of pole-facing gullies [START_REF] Balme | Orientation and distribution of recent gullies in the southern hemisphere of mars: Observations from High Resolution Stereo Camera/Mars Express (HRSC/MEX) and Mars Orbiter Camera/Mars Global Surveyor (MOC/MGS) data[END_REF][START_REF] Dickson | Martian gullies in the southern midlatitudes of Mars: Evidence for climate-controlled formation of young fluvial features based upon local and global topography[END_REF][START_REF] Kneissl | Distribution and orientation of northern-hemisphere gullies on Mars from the evaluation of HRSC and MOC-NA data[END_REF]) and, (3) observations of coincidence with sites of seasonal surface ice accumulation (Dickson & Head 2009). Granular flow has been suggested as either unassisted [START_REF] Treiman | Geologic settings of Martian gullies: Implications for their origins[END_REF][START_REF] Shinbrot | Dry granular flows can generate surface features resembling those seen in Martian gullies[END_REF], or carbon dioxide assisted flow [START_REF] Musselwhite | Liquid CO 2 breakout and the formation of recent small gullies on Mars[END_REF]. The main criticism of the granular flow model is that it fails to replicate some commonly observed features of gullies, in particular channel sinuosity and complex tributary and distributary systems [START_REF] Mcewen | A closer look at water-related geologic activity on Mars[END_REF].

There is also debate about the type of fluid involved: pure water, or brine. Whilst pure water is not stable under the current surface environment on Mars, it can persist in a metastable form [START_REF] Hecht | Metastability of liquid water on Mars[END_REF], although its flow behaviour may be substantially different to water on Earth (Conway et al. 2010a). Brines are a likely product of water sourced from underground and, moreover, the presence of some common geological compounds can substantially depress the freezing point of water (e.g., [START_REF] Chevrier | Low temperature aqueous ferric sulfate solutions on the surface of Mars[END_REF]. Brines are less likely in a surface melting scenario, because water ice condensed from the atmosphere will have had less opportunity to dissolve salts than an underground water body. Both pure water and brine can support very high concentrations of entrained sediment, and form a flow commonly termed a "debris flow". Debris flow is an attractive candidate process for forming gullies, because large amounts of erosion and deposition can be brought about with only 10 to 50 % water content [START_REF] Iverson | The physics of debris flows[END_REF]. Several authors have proposed debris flow as a potential gully-forming mechanism on Mars due to the supply of loose sediment combined with the steep slopes on which gullies are found (e.g., [START_REF] Malin | Evidence for recent groundwater seepage and surface runoff on Mars[END_REF][START_REF] Balme | Orientation and distribution of recent gullies in the southern hemisphere of mars: Observations from High Resolution Stereo Camera/Mars Express (HRSC/MEX) and Mars Orbiter Camera/Mars Global Surveyor (MOC/MGS) data[END_REF]. The inclusion of debris might also limit evaporation and freezing of the water within the flow.

Debris flows on Earth are commonly triggered by sudden and intense or prolonged rainfall, (e.g., Ben [START_REF] David-Novak | Modern extreme storms and the rainfall thresholds for initiating debris flows on the hyperarid western escarpment of the Dead Sea, Israel[END_REF][START_REF] Decaulne | Spatial and temporal diversity for debris-flow meteorological control in subarctic oceanic periglacial environments in Iceland[END_REF][START_REF] Godt | Alpine debris flows triggered by a 28 July 1999 thunderstorm in the central Front Range[END_REF][START_REF] Crosta | Rainfall-induced landslides and debris flows[END_REF][START_REF] Morton | Contrasting rainfall generated debris flows from adjacent watersheds at Forest Falls, southern California, USA[END_REF] which is not a possible mechanism on Mars under recent climate. However, debris flows can also be triggered by snowmelt, or melting permafrost [START_REF] Harris | Debris flow characteristics in an area of continuous permafrost, St. Elias Range, Yukon Territory[END_REF][START_REF] Decaulne | Debris flow triggered by rapid snowmelt: A case study in the Gleiðarhjalli area, northwestern Iceland[END_REF]. As noted by [START_REF] Lanza | Evidence for debris flow gully formation initiated by shallow subsurface water on Mars[END_REF] infiltration rates on Mars are likely to exceed the low discharge rates produced by a surface melting source. Hence, overland flow is unlikely, unless there is a shallow impermeable barrier, such as near-surface permafrost, or frozen layer formed at the base of the water flow on contact with a cold substrate (Conway et al. 2010a). The dominance of infiltration satisfies the conditions for debris flow triggering, sediment saturation and elevated pore pressures.

The lack of vegetation and the associated lower cohesion of the martian soil, compared to Earth, potentially means that debris flows can be triggered on much lower slope gradients than they are on Earth.

Gullies formed by dilute-water flow and debris flow on Earth can be visually very similar to each other, and the basic structure of gullies can be formed by dry granular flow [START_REF] Mangeney | Numerical modeling of self-channeling granular flows and of their levee-channel deposits[END_REF]. In many geomorphological problems, convergence of visual form means that using images alone can make it very difficult to determine process. The ongoing debate regarding the formation mechanisms of gullies on Mars is a prime example of this. For example some workers have dismissed debris flow as a mechanism for forming martian gullies, because they have not observed the levées that are one of the diagnostic features of debris flow (e.g., [START_REF] Innes | Debris Flows[END_REF]. However, the ability to identify levées depends on viewing geometry and sun angle; metre sized levées are often not visible on 25 cm/pixel air photos of Earth. It is also possible that a combination of the lower gravity and different sediment type on Mars means that the levées might be small compared to those on Earth.

The amount of water required to carve channels and transport and deposit sediment differs substantially between debris flow, water or brine flow (termed "alluvial" throughout the rest of this paper) and granular flow. Determining the amount of water available at the martian surface is important for questions of martian climate, hydrology and the study of potential martian habitats. Hence, an accurate determination of active processes is needed that in turn can constrain the quantity of fluid required to form gullies. Quantitative geomorphological study can provide the tools to discriminate between these three processes.

The recent availability of high resolution digital elevation models (DEMs) of Mars has opened up the possibility of using quantitative geomorphic methods that have, until now, been restricted to analysing landscapes on Earth. By taking well-developed slope-area analyses and other geomorphic process indicators for the Earth and applying them to Mars, this study aims to give insights into both the processes that formed the gullies on Mars and the source of any water involved.

We used three geomorphic tools commonly applied in terrestrial geomorphology to identify active processes forming gullies on Mars: slope-area plots (Fig. 2a), Cumulative Area Distribution (CAD) plots (Fig. 2b) and wetness index maps. These analytical techniques are described in more detail in the following sections. They are usually used to assess active processes within catchment areas and other larger-scale landscape analyses. To test whether they are equally applicable to smaller areas, we first applied them to five study sites on Earth at an equivalent scale to gullies on Mars. Recently deglaciated areas were preferred as these have: (1) a geologically short and well defined slope development history (i.e. since deglaciation) and, (2) a glacial trough valley slope-profile which strongly resembles that of fresh impact craters (compare relationships in [START_REF] Brook | Temporal constraints on glacial valley cross-profile evolution: Two Thumb Range, central Southern Alps, New Zealand[END_REF] and [START_REF] Garvin | Global Geometric Properties of Martian Impact Craters: A Preliminary Assessment Using Mars Orbiter Laser Altimeter (MOLA)[END_REF].

However, suitable quality data could not be found for the alluvial end-member process in glacial environments, so two desert study sites were also included.

When we were satisfied that different geomorphic processes could be discriminated on Earth using slope-area plots, CAD plots and wetness index maps, we applied these analyses to slopes containing gullies on Mars.

Method

Slope-area and Cumulative Area Distribution (CAD) methods

The so-called "stream power law" was first proposed by [START_REF] Hack | Studies of longitudinal stream profiles in Virginia and Maryland[END_REF] and has been widely used to investigate landscape evolution on Earth (e.g., [START_REF] Kirkby | Observed and modelled distributions of channel and gully heads -with examples from SE Spain and Belgium[END_REF][START_REF] Stock | Valley incision by debris flows: Evidence of a topographic signature[END_REF]. It is based on the detachment and transport limited rate of bedrock erosion, otherwise known as the shear-stress incision model, which is stated as follows:

S = kA -θ (1)
where S is local slope, A is upslope drainage area, k is a process related constant, which is different for detachment and transport cases, and θ is the concavity index, which is process dependent. It has also been noted that if the drainage area is plotted against the local slope for drainage basins then process domains can be defined in log-log plots as shown in Fig. 2a (after [START_REF] Montgomery | Channel network source representation using digital elevation models[END_REF]. These process domains were initially schematic, based on few data, but have been supported by later work (e.g., [START_REF] Whipple | Dynamics of the stream-power river incision model: Implications for height limits of mountain ranges, landscape response timescales, and research needs[END_REF][START_REF] Snyder | Landscape response to tectonic forcing: Digital elevation model analysis of stream profiles in the Mendocino triple junction region, Northern California[END_REF][START_REF] Kobor | Systematic variation of bedrock channel gradients in the central Oregon Coast Range: Implications for rock uplift and shallow landsliding[END_REF][START_REF] Marchi | Rocky headwaters in the Dolomites, Italy: Field observations and topographic analysis[END_REF]. [START_REF] Brardinoni | Glacial erosion, evolution of river long profiles, and the organization of process domains in mountain drainage basins of coastal British Columbia[END_REF] added an additional domain in which systems dominated by debris flow deposition, occupy that part of the alluvial domain of [START_REF] Montgomery | Channel network source representation using digital elevation models[END_REF], which is located towards higher drainage areas and steeper slopes (Fig. 2a). This domain was proposed from field observations in glacially modified area and has since been supported by additional observations by [START_REF] Mao | Sediment transfer processes in two Alpine catchments of contrasting morphological settings[END_REF] in a different geomorphic setting. Process information can be obtained both from the position of the data points relative to defined domains on this slope-area plot, and from the trend of the data within these domains, for example, whether the data points plot in a concave, convex, upward trending, or downward trending curve [START_REF] Tucker | Hillslope processes, drainage density, and landscape morphology[END_REF]. The general trend for an alluvial system is shown in Fig. 2a, which passes through several process domains. The data for such plots are generally derived from digital elevation models or topographic maps. The slope and contributing area data are either extracted from the channel only, or the whole drainage basin, depending on the focus of the study. In Fig. 2a these data are taken from every pixel contained within the catchment of the whole fluvial system (encompassing valley hillslopes, tributaries, main channels and estuary system) sampled at a single point in time.

Cumulative Area Distribution (CAD) is the probability distribution of points in the landscape having a drainage area greater than any particular area, A*. The log-log plot of P(A>A*) against A* gives information on the processes acting within a catchment [START_REF] Perera | A physical explanation of the cumulative area distribution curve[END_REF][START_REF] Mcnamara | Channel head locations with respect to geomorphologic thresholds derived from a digital elevation model: A case study in northern Thailand[END_REF]. Interpretation of this index varies, but generally it is split into three areas:

(1) at small drainage areas the plot usually evolves from convex to concave, and represents diffusive erosion, (2) intermediate drainage areas are linear in a loglog plot and this is thought to represent incision, (i.e. channel formation), and (3) at large drainage areas there are small steps where major tributaries join the channel (Fig. 2b). et al. (2006) split domain (1) into three sub-domains (Fig. 2b): (1a) a convex section, representing hillslopes that diverge and do not gather drainage, (1b) linear and steep section in a log-log plot, indicating hillslopes with convergent topography and, (1c) a concave section, which they suggest is a reach dominated by pore pressure triggered landsliding (including debris flows which are triggered by this mechanism).

McNamara

The stream-power law (Eq. 1), and process interpretations in slope-area and CAD plots of [START_REF] Montgomery | Channel network source representation using digital elevation models[END_REF], [START_REF] Brardinoni | Glacial erosion, evolution of river long profiles, and the organization of process domains in mountain drainage basins of coastal British Columbia[END_REF] and [START_REF] Tucker | Hillslope processes, drainage density, and landscape morphology[END_REF] are based on empirical hydraulic geometry functions that are predicated on, and developed for, studies of large fluvial systems with channel morphology well-adjusted to perennial discharge. It could therefore be argued that these systems are unlike the hillslope systems in this study. Hence, we have tested these interpretive analysis techniques on small gully-systems on Earth where we know the active processes in order to demonstrate that they can still be valid. It is, of course, necessary to bear in mind that there is always some uncertainty in inferring process from landscape form, in part due to the intrinsic variability and complexity of natural systems but also due to the effects of vegetation, tectonics, climate and perhaps human interaction with the landscape. However, on Mars the surface processes are likely to be simpler, with little chance of factors, such as rain, vegetation or human action confounding the process domains, so these indices should provide an important addition to the "visual" morphology when inferring process from form.

Application of slope-area method to Mars

The reduced gravitational acceleration of Mars shifts the slope-area boundary of the alluvial slope-area domain vertically (dotted line in Fig. 2a). This means that the unchanneled domain extends to higher slopes for a given drainage area for Mars (extending into the alluvial and debris flow domains for Earth); however the hillslope domain is unaffected. Considering the fact that gullies on Mars do not have large tributary-channel networks it seems unlikely that this domain would be well developed. Appendix 1 gives details of the calculations performed to account for the gravitational acceleration of Mars. The relative gradients and curvatures of the trends described by the alluvial data in slope-area plots are unaffected by the reduced gravity. We have not been able to revise the position of the domain added by [START_REF] Brardinoni | Glacial erosion, evolution of river long profiles, and the organization of process domains in mountain drainage basins of coastal British Columbia[END_REF] as a function of gravitational acceleration because this domain was added empirically, based on field observations.

The slope threshold for dry mass wasting or landsliding in loose material is the same as on Earth [START_REF] Moore | Viking landing sites, remote-sensing observations, and physical properties of Martian surface materials[END_REF][START_REF] Peters | Mojave Mars simulant--Characterization of a new geologic Mars analog[END_REF]. The slope thresholds for pore pressure failure are also unaffected by the difference in gravitational acceleration. Hence there would be no change to these process domains or trends for either dry mass wasting or pore pressure triggered processes such as debris flow.

We note that on Earth, vegetation cover, soil type and geology can have profound impacts on the slope values in a landscape for a given drainage area [START_REF] Yetemen | The implications of geology, soils, and vegetation on landscape morphology: Inferences from semi-arid basins with complex vegetation patterns in Central New Mexico[END_REF]), but we would expect only variations in soil type and geology to affect the data on Mars. Despite these differences in surface properties, basins with similar processes on Earth show a similar pattern or trend of data, but displaced vertically in slope-area plots [START_REF] Yetemen | The implications of geology, soils, and vegetation on landscape morphology: Inferences from semi-arid basins with complex vegetation patterns in Central New Mexico[END_REF].

Datasets and generation of digital elevation models

Slope-area analysis is only possible with high quality elevation data, preferably at a resolution better than 10 m per pixel, or 1:25 000 map scale [START_REF] Montgomery | Channel network source representation using digital elevation models[END_REF][START_REF] Tarolli | Hillslope-to-valley transition morphology: New opportunities from high resolution DTMs[END_REF]. For each of the terrestrial sites 1 m resolution DEMs were derived from airborne laser altimeter (LiDAR) data. These were then resampled to 5 m resolution to match the Mars data, as described below. (2007a, b).

For Mars we used four 1 m resolution DEMs produced using stereo photogrammetry from 25 cm per pixel High Resolution Science Imaging Experiment (HiRISE) images. The DEMs for sites PC, GC, KC and TS were produced by the authors from publically released HiRISE images using methods described by [START_REF] Kirk | Ultrahigh resolution topographic mapping of Mars with MRO HiRISE stereo images: Meter-scale slopes of candidate Phoenix landing sites[END_REF]. Significant metre-scale random noise present in the DEMs of sites GC, KC and TS had a detrimental effect on preliminary slope-area analyses. Hence, all the DEMs were resampled to 5 m per pixel before the reanalysis was performed.

The precision of elevation values in the DEMs used here can be estimated based on viewing geometry and pixel scale. For the DEM of site PC, the attendant image pair PSP_004060_1440 (0.255 m/pixel) and PSP_005550_1440 (0.266 m/pixel) have a 12.6º stereoscopic convergence angle. Assuming 1/5 pixel matching error and using a pixel scale of 0.266 m/pixel from the more oblique image, the vertical precision is estimated to be ~ 0.24 m (cf. [START_REF] Kirk | Ultrahigh resolution topographic mapping of Mars with MRO HiRISE stereo images: Meter-scale slopes of candidate Phoenix landing sites[END_REF]. DEMs for sites GC, KC and TS have a similar magnitude of vertical precision. The pixel matching error is influenced by signal-to-noise ratio, scene contrast and differences in illumination between images. Pattern noise can also be introduced by the automatic terrain extraction algorithm, especially in areas of low correlation. Manual editing is necessary to correct spurious topography in areas of poor correlation (e.g., smooth, low contrast slopes and along shadows).

Finally, a synthetic crater was constructed to test whether the results from the Mars study sites in general reflected the process, or instead were a result of the geometry imposed by the impact crater setting (all the Mars study areas were on the inner walls of bowl shaped depressions, but none of the ones on Earth were). A 10 km diameter synthetic crater was created by applying a smooth parabolic radial profile, which was derived by fitting curves through ungullied radial profiles of the craters in sites PC and GC. Metre-scale "pink" (also called "1/f") noise was added to simulate a natural rough surface [START_REF] Jack | Red-shifts and red herrings in geographical ecology[END_REF].

Derivation of drainage area and local slope

Representative slope sections were chosen in each DEM (Figs. 3 and4). For Earth, these were chosen to represent end member and intermediate process domains, including dry mass wasting, debris flow and alluvial processes. On Mars, some areas were chosen that covered the complete slope on which gullies are found, whilst others covered a single gully system, or ungullied slope for comparison. Slope sections always included the drainage divide at the top and extended downslope as far as the visible signs of the distal extent of the gully (or slope) deposits. Where possible lines delineating drainage basins were followed to define the lateral extent of slope sections, but on poorly incised hillslopes this was not always possible and the lateral extent was defined as a straight line. For site KC, on Mars, we chose different configurations of slope sections to test the sensitivity of our analyses to the exact method used to delineate the slope-sections. Careful delineation of slope sections is necessary for two reasons. Firstly, because the larger the sample area, the more processes are included within it, and the more difficult the results will be to interpret. Secondly, if parts of the slope that are integral to the process to be identified are omitted, then the process signal will not be complete.

The slope and the flow directions of each pixel in each DEM were determined using a "Dinf" algorithm. This algorithm gives flow directions in any direction, rather than only towards one of the eight neighbouring pixels [START_REF] Tarboton | On the extraction of channel networks from digital elevation data[END_REF]. This has been shown to produce better results from slope-area analysis because it gives a more accurate approximation of the real path of flow through the landscape [START_REF] Borga | A physically based model of the effects of forest roads on slope stability[END_REF]). For each pixel, the accumulation of flow was calculated from the flow directions by summing the number of pixels located upstream, and multiplying by the pixel area. These analyses were performed using the TauDEM extension for ArcGIS, based on the algorithms developed by [START_REF] Tarboton | A new method for the determination of flow directions and upslope areas in grid digital elevation models[END_REF]. For each DEM the "wetness index" was also calculated. This is the natural logarithm of the ratio of contributing area to slope. It provides information on the potential connectivity of the landscape drainage and the potential ability of the surrounding landscape to route drainage [START_REF] Woods | A connection between topographically driven runoff generation and channel network structure[END_REF]. However, in the case of Earth and particularly in the case of Mars this index should not be interpreted literally as implying that the terrain is "wet". In our study it is used as a visual aid to interpret the spatial variability of the slopearea plot. For example, highly permeable talus slopes on Earth are essentially dry, but they may have moderate to high wetness index. However, we would expect a talus slope on Earth to show a characteristic spatial pattern of wetness index, indicative of dry mass wasting processes. All the DEMs underwent the same processing steps.

We extracted the drainage area and slope for every pixel within the chosen slope sections. To simplify the representation of these data we calculated the mean slope for 0.05 wide logarithmic bins of drainage area, and then constructed the slope-area and CAD plots.

Binning data in this way make the trends in slope-area and CAD plots clearer and is a commonly used display technique (e.g., [START_REF] Snyder | Landscape response to tectonic forcing: Digital elevation model analysis of stream profiles in the Mendocino triple junction region, Northern California[END_REF].

In addition, for one site on Mars (site KC), we visually identified the initiation sites of the gullies on orthorectified HiRISE images. The initiation points for the gullies were defined as the furthest upstream extent defined by a distinct cut, or scarp (Fig. 5a). For each of these locations we extracted the slope and drainage area for the underlying pixel. This analysis was not performed for site PC because edge contamination and noise made it impractical. The analysis was also omitted for site GC because the gullies start at the top of the slope, so would by definition occur at the lowest drainage areas.

Study areas

Earth

All the study sites on Earth are located in the northern hemisphere and most are within the continental USA. Table 1 provides a summary of the sites and Fig. 3 shows the setting of the areas studied.

Site SJ -San Jacinto, California

This site is located in California along a splay of the San Andreas fault, called the San Jacinto fault. This area is a desert with little rainfall (~ 150 mm, annual average recorded by NOAA weather station in nearby Borrego Springs), which has undergone rapid recent uplift caused by the fault system. The landscape has a well developed ephemeral gully network with large alluvial fans. From the study of the 1 m LiDAR data and aerial images we infer the processes forming these fans to be sheet-flow rather than debris flow, based on the lack of levées and lobate terminal deposits. The vegetation is sparse, consisting of small scrub bushes. The underlying geology of the study area is mainly granite, schist and gneiss with minor outcrops of Quaternary older fan deposits [START_REF] Moyle | Water Resources of Borrego Valley and Vicinity, California: Phase 1 -Definition of Geologic and Hydrologic Characteristics of a Basin[END_REF]. For our analyses we used three study areas that contained small complete gully systems, including sources, channels and debris aprons, but avoided large fan systems and debris aprons from neighbouring systems (Figs. 3a and3b:

study areas SJ1, SJ2 and SJ3). Due to the small size of the fans in area SJ1 it is difficult to entirely rule out debris flow as a potential process in forming these alluvial fans.

Site DV -Death Valley, California

This site is located a few kilometres NE of Ubehebe volcano, in Death Valley, California. This is a desert area that has well developed ephemeral gully networks with large alluvial fans. There is little precipitation in this area although the nearby mountains receive as much as 85 mm of rain per year [START_REF] Crippen | Potential hazards from floodflows and debris movement in the Furnace Creek area, Death Valley National Monument[END_REF]) and rare large storms can do much geomorphic work. Debris flows are found on the fans in the area (e.g., [START_REF] Blair | Cause of dominance by sheetflood vs. debris-flow processes on two adjoining alluvial fans, Death Valley, California[END_REF][START_REF] Blair | Sedimentology and progressive tectonic unconformities of the sheetflooddominated Hell's Gate alluvial fan, Death Valley, California[END_REF], but the primary process active in the gullies is alluvial transport [START_REF] Crippen | Potential hazards from floodflows and debris movement in the Furnace Creek area, Death Valley National Monument[END_REF]. We inspected the 1 m LiDAR data for presence of levées and depositional lobes on the fans and found no evidence of these. However, without direct field observations the fact that debris flows do not act on these fans remain an assumption. The bedrock consists of Palaeozoic sedimentary rocks [START_REF] Workman | Geologic map of the Death Valley ground-water model area, Nevada and California, Miscellaneous Field Studies Map 2381-A[END_REF]. We chose two study areas (Fig. 3c: study areas DV1 and DV2) with gully systems that were not affected by neighbouring alluvial fans or gully systems so only receive local rainfall levels.

Site KA -St Elias Mountains, Alaska

This site is located east of the abandoned town of Katalla close to the recently deglaciated mountain range of St Elias, near the coast of Alaska and on the border with Yukon, Canada.

The area has been unglaciated for approximately the last 10 000 years [START_REF] Sirkin | Late Pleistocene and Holocene deglaciation and environments of the southern Chugach Mountains, Alaska[END_REF] and receives very high precipitation, which falls as snow on the upper slopes and rain on the lower. Our study area overlies Tertiary volcanic materials. The slope scarp was generated by the active Ragged Mountain Fault [START_REF] Miller | Geology of the Katalla district, Gulf of Alaska Tertiary province[END_REF]. The area was neither snow covered nor tree covered at the time of survey and the slopes are composed of steep bedrock cliffs that lead directly into large talus aprons. Debris flow tracks are apparent across this talus slope, especially in study areas KA3 and KA4, and might have occurred in study area KA3 as well (Fig. 3d). Study area KA1 has no evidence of debris flow processes (Fig. 3d).

Site FR -Front Range, Colorado

This site is located in the mountainous eastern side of the continental divide. The area was deglaciated around 14 000 to 12 000 years before present [START_REF] Godt | Alpine debris flows triggered by a 28 July 1999 thunderstorm in the central Front Range[END_REF]) and the landscape is dominated by glacially carved valleys. This area has experienced recent debris flows [START_REF] Coe | Debris Flows along the Interstate 70 Corridor, Floyd Hill to the Arapahoe Basin Ski Area, Central Colorado -A Field Trip Guidebook[END_REF][START_REF] Godt | Alpine debris flows triggered by a 28 July 1999 thunderstorm in the central Front Range[END_REF]) and has no permanent snowpack. Our study slopes, located above the tree line, are dominated by Precambrian biotitic gneiss and quartz monzonite, scattered Tertiary intrusions, and by various surface deposits, all of which host debris flows [START_REF] Godt | Alpine debris flows triggered by a 28 July 1999 thunderstorm in the central Front Range[END_REF]. The head and sidewalls of the cirques have large rockfall talus deposits and which have also experienced recent debris flows. These slopes have little or no vegetation. Three of our study areas (Figs. 3e and 3f: study areas FR2 to FR4) include debris flows located on talus. By way of contrast, we also examined a partially vegetated slope (study area FR1) that is unchanneled and which we infer to be dominated by creep processes (Fig. 3e).

Site WF -Westfjords, Iceland

The site is located in NW Iceland and is dominated by fjords and glacially carved valleys.

The last glacial retreat occurred approximately 10 000 years before present [START_REF] Norðdalh | Late Weichselian and early Holocene deglaciation history of Iceland[END_REF].

The valley walls have many active debris flows (Conway et al. 2010b) and on the slopes above Ísafjörður (Fig. 3g: study area WF1) they occur in most years [START_REF] Decaulne | Debris flow triggered by rapid snowmelt: A case study in the Gleiðarhjalli area, northwestern Iceland[END_REF].

The site has a maritime climate, so has high levels of both snow and rainfall, but does not have permanent ice or snow patches. The site is underlain by Miocene basalts, although the debris flows occur most often in glacial till. From this site we chose a study area above the town of Ísafjörður that has very active debris flows (Fig. 3g: study area WF1), two study areas with less active debris flows and more alluvial processes (Figs. 3g and 3h: study areas WF2 and WF3), and one study area dominated by rockfall and rock slide processes, although there are some debris flow tracks visible in the field (Fig. 3h: study area WF4). All these study areas have patchy vegetation, but no trees.

Mars

All the gullies that we studied on Mars were located on the inner walls of craters in the southern hemisphere (Table 2). Slopes both with and without gullies were analysed for comparison. Sites PC, GC and KC were analysed by [START_REF] Lanza | Evidence for debris flow gully formation initiated by shallow subsurface water on Mars[END_REF], because all the sites showed visual evidence of debris flows.

Site PC -Penticton Crater in Eastern Hellas

This site contains the very recent, light-toned deposits observed by [START_REF] Malin | Present-day impact cratering rate and contemporary gully activity on Mars[END_REF] and interpreted by them to be a recent "gully forming" event. These flows were later suggested by [START_REF] Pelletier | Recent bright gully deposits on Mars: Wet or dry flow?[END_REF] to be produced by dry granular flow, or possibly also debris flow. This slope does not have any well defined channels. We used two study areas within the ~ 7.5 km diameter crater for our slope-area analyses, shown in Figs. 4a and4b. Study area PC1 is located over the equator-facing light-toned deposits (Fig. 4a) and study area PC2 on the westfacing crater wall which contains small gullies (Fig. 4b). These gullies appear to be incised into "mantle deposits" [START_REF] Mustard | Evidence for recent climate change on Mars from the identification of youthful near-surface ground ice[END_REF]. The mantle is hypothesised to be the remnants of a previously extensive volatile rich deposit (e.g., [START_REF] Mangold | High latitude patterned grounds on Mars: Classification, distribution and climatic control[END_REF]. This crater is very asymmetric, with the east and north rims being subdued in terms of elevation (the rim is nearly absent on the east side) whilst the southern rim is abrupt and steep.

Site GC -Gasa Crater in Terra Cimmeria

This ~ 7 km wide crater, shown in Figs. 4c and4d, has well developed alcoves or indentations into the rim of the crater. Gully channels are most obvious on the west-facing to pole-facing slopes (Figs. 4c and4d) and the equator-facing slope lacks these well defined alcoves and channels (Fig. 4e). We chose sections on the pole-(study areas GC1 and GC2),

west-(study area GC3) and equator-facing (study area GC4) slopes. This crater is located within a larger crater, which also has gullies on its west-to pole-facing slopes. There is no evidence of mantle deposits being present anywhere within this crater.

Site KC -crater inside Kaiser Crater in Noachis Terra

The study crater, ~ 12 km across is located within the larger Kaiser crater, which not only has gullies down its own rim, but also gullies on the dunes within it [START_REF] Bourke | Alluvial Fans on Dunes in Kaiser Crater Suggest Niveo-Aeolian and Denivation Processes on Mars[END_REF]. Gullies in this crater have alcoves at various positions on the slope, which converge to form well defined tributary networks. Lateral levées bound some of the channels (Figs. 5b and5c). This slope has the subdued appearance often attributed to the presence of volatile rich mantle deposits [START_REF] Mustard | Evidence for recent climate change on Mars from the identification of youthful near-surface ground ice[END_REF]. We chose study areas that encompass the drainage area of two gullies (study area KC2), a single gully (study area KC1) and also the slope section as a whole (study area KC3), all of which are shown in Fig. 4f. We chose study area KC4, an area of the slope not affected by gullies, for comparison (Fig. 4f).

Site TS -crater in Terra Sirenum

This ~ 7 km diameter crater is located to the south of Pickering Crater in Terra Sirenum and contains pole-facing gullies. We analysed an equator-facing slope (Fig. 4g: study area TS1) which has no evidence of channels but contains an apparently well developed talus apron.

There is no evidence of mantle deposits being present on this slope.

Results

Earth

Initially we chose two study areas with talus and with active creep. The slope-area analysis results for these are shown in Fig. 6a. The study areas with well developed talus (WF4 and KA1) show the following pattern on log-log plots: (1) At small drainage area the curves are initially flat.

(2) There is then a linear decrease in slope with increasing drainage area.

(3) The curve then becomes horizontal again at higher drainage area with a lower slope value. Talus slopes that have a mixture of processes (e.g., KA2) show a curve that drops off linearly in log-log plots then flattens at higher drainage areas.

The CAD plot (Fig. 7a) provides additional information: the talus dominated study areas have a very smooth convex shape. The gradient of the curve is low until the drainage area is between approximately 0.001 km 2 after which the curve drops sharply and continues to steepen with increasing drainage area.

The soil creep diffusive process study area (FR1 in Fig. 6a) shows a distinctive signature in slope-area plots: (1) The curve is initially horizontal to gently downwards sloping.

(2) Between drainage areas of 0.0001 to 0.001 km 2 the slope increases linearly with increasing drainage area.

(3) There is then a marked slope turnover at which the curve switches to decreasing slope with increasing drainage area. The soil creep diffusive process study area resembles the talus slopes in CAD plots (FR1, Fig. 7a).

Figs. 6b and 7b show the debris flow study areas that are influenced by talus processes and Figs. 6c and7c show those that are more influenced by alluvial processes.

Generally in slope-area plots debris flow produces a curve that drops off linearly in log-log plots, flattening off before finally dropping away steeply. The difference between the talus study areas (e.g. KA2, Fig. 6a) and the debris flow study areas influenced by talus (Fig. 6b) is subtle in some cases. In a similar way the difference between the debris flow areas influenced by talus processes (Fig. 6b) and those influenced by alluvial processes (Fig. 6c) is also subtle.

Without field information it would be difficult to differentiate talus dominated and debris flow dominated slopes reliably in slope-area plots (e.g., compare Figs. 6a, KA2 and 6b).

However, in CAD plots it is possible to differentiate between the two process types. The Study areas modified by ephemeral water flow have distinct signatures in slope-area plots (Fig. 6d) and in CAD plots (Fig. 7d). In slope-area plots they show a shallow linearly decreasing trend at small drainage areas, which gets steeper at higher drainage areas, and drops into the alluvial domain. The CAD plot drops away from the horizontal slowly and then dips down linearly (or even with a concave profile) until the tail of the curve drops sharply off at the highest drainage areas.

Synthetic Crater

The slope-area and CAD plots for the synthetic crater are easily differentiated from the process study areas that we have examined on Earth. In slope-area plots the synthetic crater produces a hump-backed curve (Fig. 8d): at small drainage areas the curve rises steeply, then levels off and drops at high drainage areas. In appearance the curve is, as expected, nearest to study area FR1, the area dominated by diffusive creep (Fig. 6a). In CAD plots (Fig. 9d) the line follows a smooth convex arc, similar to that shown by talus on Earth, except without a break in gradient.

Mars

The slope-area plots for sites PC and GC (Penticton Crater and Gasa Crater inner slopes)

closely resemble each one another (Fig. 8a andb). The resulting curve can be divided into three zones: (1) A short initial increase in slope with increasing drainage area, followed by a slope turnover at very small drainage areas.

(2) A linear or slightly concave decreasing slope trend with increasing drainage area that continues for most of the plot. ( 3) Finally, at the largest drainage areas, there is a steep decrease in slope with increasing drainage area. For study area PC1 there is a distinct and linear decline in slope with drainage area, whereas for study areas PC2, GC1, GC2 and GC3 this section is slightly concave. The drop-off at the highest drainage areas occurs at lower absolute drainage area values than for site GC. In the CAD plot, study areas PC1 and GC4 have a smooth convex form, whereas study areas PC2,

GC1 and GC2 all have a nearly linear, flattened section at intermediate drainage areas (Figs.

9a and 9b). Study area GC3 lies close to PC1, GC1 and GC2 but without any sign of flattening.

The slope-area plots for gullies in study areas KC1, KC2 and KC3 (Fig. 8c) can be split into three sections as follows:

(1) at small drainage areas the curve is sub-horizontal with a subtle upward trend. This trend is more apparent for the data from individual gullies than the data obtained from the whole slope section and is somewhat variable between gully systems.

(2) At intermediate drainage areas there is a transitional zone, occurring at different drainage areas for each gully system, in which slope drops off markedly with drainage area.

(3) At higher drainage areas there is a gently declining relationship between slope and drainage area, which is the same for all the gully systems.

The ungullied study area (KC4) is also shown in Fig. 8c. This study area has a hump-back shape, resembling that seen for the synthetic crater. The hump occurs across the same slope values as the transition zone (2) for the gullied slopes. In CAD plots (Fig. 9c) study areas KC2 and KC3 have a flattened section at intermediate drainage areas, followed by a steepening decrease at higher drainage areas. The study area without gullies (KC4) has a curve that is convex and initially declines slowly, before dropping off steeply. Study area KC1 has a less flattened profile than study areas KC2, or KC3 and it seems to be a mixture between slope types typified by gullied study areas KC2 or KC3 and ungullied study area KC4.

In slope-area plots, study area TS1, an ungullied slope, shows a slope-area turnover at small drainage areas, followed by a decreasing and slightly concave trend in slope with drainage area (Fig. 8d). There is a slight upturn at the highest drainage areas, but this is likely to be an artefact caused by few data-points being used to calculate the mean slope in these bins. In CAD plots (Fig. 9d) study area TS1 has a very smooth convex curve.

The slope and drainage area of the gully head initiation points were recorded for site KC. These data are displayed on Fig. 8c. Interestingly, the locations of the gully heads cluster around the range of drainage areas of the transitional section in the slope-area plot, but are located at higher slope values.

Wetness Index on Earth and Mars

The spatial distribution of the slope-area data is most easily visualised using a wetness index map. Maps of wetness index are presented for Earth (Fig. 10) and for Mars (Fig. 11). The alluvial study areas in Earth sites SJ and DV show very low overall wetness indices -only the channels have significant wetness index (Figs. 10a, 10b, and10c). Debris flow study areas are slightly more complex (Figs. 10d, 10e, 9f, 10g, and 10h): the slopes generally have moderate wetness index, but there are localised paths along which the wetness index is higher. Site WF (Figs. 10g and 10h) is the best example of this pattern, but it is also the area with the highest influence of overland flow. For site KA (Fig. 10d) this signature is poorly developed, but this site has been influenced by talus processes. The creep dominated study area, FR1, has moderate wetness index throughout (Fig. 10e). The talus study areas KA1, KA2 (Fig. 10d) and WF4 (Fig. 10h) show lobe-like areas of low wetness index with widening streaks of higher wetness index in between.

On Mars, study area PC1 (Fig. 11a) and the synthetic crater (Fig. 11h) have similar wetness index maps: the slope generally increases in wetness index going downhill and there are quasi-linear streaks of higher wetness index that increase in value going downslope.

Study area PC2 (Fig. 11b) has overall low wetness index, apart from concentrated lines of high wetness index within the gully alcoves, that spread and become more diffuse in the debris aprons. A similar overall pattern is shown for study areas GC1, GC2 and GC3 (Figs. 11c and11d), but the ridges around the alcoves have very low wetness index. Study area GC2

in particular (Fig. 11c) shows very concentrated slightly sinuous high wetness index lines on its debris apron. However this part of the DEM contains significant noise, making it hard to judge whether this is simply an artefact. Study areas GC4 (Fig. 11e) and TS1 (Fig. 11g) have similar wetness index maps: there is low wetness index at the crest of the slope and where bedrock is exposed and the wetness index generally increases downslope, but this trend is superposed with diffuse linear streaks of higher relative wetness index. Site KC (Fig. 11f) has generally moderate wetness index, with the alcoves and channels of the gullies showing focussed high wetness index flanked by much lower wetness index and the debris aprons having generally high wetness index with diffuse downslope streaking.

Discussion

Comparison of Earth data to previously published slope-area process domains

There are two interlinked methods of determining slope processes from slope-area plots:

(1) The data points fall within domains in the plots which have been found both theoretically and empirically to relate to particular processes, and

(2) The data points exhibit trends and gradients that provide information on active processes.

We compared our data from Earth to the slope-area process domains of [START_REF] Montgomery | Channel network source representation using digital elevation models[END_REF] and the additional domain added by [START_REF] Brardinoni | Glacial erosion, evolution of river long profiles, and the organization of process domains in mountain drainage basins of coastal British Columbia[END_REF], shown as solid lines in Fig. 6. The data from our creep, talus and debris flow analyses fall into the debris flow domain of [START_REF] Montgomery | Channel network source representation using digital elevation models[END_REF]. However, some of our debris flow data drop into the alluvial domain at the highest drainage areas.

Because they are small systems with limited drainage areas, however, only a few points fall within the alluvial domain. Some of our data approach the additional domain added by [START_REF] Brardinoni | Glacial erosion, evolution of river long profiles, and the organization of process domains in mountain drainage basins of coastal British Columbia[END_REF], but do not extend towards sufficiently high drainage areas (or low drainage areas) to enter it (Fig. 6b and6c). Our data from the alluvial systems (Fig. 6d) fall into both the debris flow and alluvial domains. They start to trend downwards in slopearea plots at lower drainage areas than our debris flow systems.

Tucker & Bras (1998) simulated the effects of different dominant processes on slope-area plots and we now compare their model results to the patterns in slope-area plots shown by our data. Our talus systems (Fig. 6a) closely fit their model of a landscape dominated by landsliding (which includes the process of debris flow). In slope-area plots our talus data have an initial flat section at small drainage areas, which represents the slope threshold for the rock wall failure and so differs between localities. At higher drainage areas the curves are again flat, representing the failure threshold of loose talus, which is consistent for all areas at approximately 0.7 gradient, equivalent to a slope of approximately 35°. This is an approximate mean slope angle for talus slopes on Earth [START_REF] Chandler | The Inclination of Talus, Artic Talus terraces, and Other Slopes Composed of Granular Materials[END_REF][START_REF] Selby | Hillslope Materials and Processes[END_REF] and is shown by a dotted horizontal line in Figs. 6 and8. Between these two horizontal sections there is a transition where the dominance shifts from rock wall failure to unconsolidated talus failure.

Within the framework of [START_REF] Tucker | Hillslope processes, drainage density, and landscape morphology[END_REF] the pattern shown by the debris flow slopes on Earth (Figs. 6b and6c) is most consistent with the transition from unsaturated landsliding (dry mass wasting of both talus and rock wall) to pore pressure triggered landsliding (which we interpret to also include debris flow), in a landscape dominated by landsliding. The presence of processes with a slope failure threshold cause data in slope-area plots to fall along horizontal lines. Hence, as the process dominance changes from rock wall failure (highest threshold) to unsaturated landsliding (intermediate threshold) to saturated landsliding (lowest threshold) the curve declines and levels off at the slope value of the saturated landslide threshold in that particular area. As each physical locality has its own saturation threshold this horizontal section occurs at different slope values for different localities but is always located below the dry stability line at 0.7.

In slope-area plots, our data from alluvial systems on Earth (Fig. 6d) show a simple decline of slope with drainage area, possibly steepening at higher drainage areas. The data are scattered at drainage areas > 0.001 km 2 , due to the limitations of the small sizes of the gully systems available. This means a relatively small number of pixels were used to generate each point, leading to random scatter. However, even taking into account the scatter, the data are below the slope threshold for dry slope failure at 0.7 gradient, which suggests a gradual transition from pore pressure dominated landsliding to fluvial processes.

The main feature of our creep dominated hillslope data (FR1, Fig. 6a), is a turnover from increasing slope with drainage area to decreasing slope with drainage area. One of the alluvial systems in site SJ (study area SJ3) shows a weak slope turnover at the lowest drainage areas but none of the other plots show this feature. The slope-area turnover is shown in Fig. 2 and is generally expected to occur in slope-area plots (e.g., [START_REF] Tucker | Hillslope processes, drainage density, and landscape morphology[END_REF]. It usually occurs in, or close to, the "hillslope" domain of [START_REF] Montgomery | Channel network source representation using digital elevation models[END_REF]. The turnover represents a transition from convex slopes dominated by diffusive processes (which include soil creep often modified by plant roots and other biota) to concave slopes dominated by advective, or alluvial processes. Within the diffusive processes domain in slope-area plots, slope increases with drainage area. The most likely reason that most of our data do not show this turnover is that the slopes we studied lack stable vegetation [START_REF] Dietrich | The search for a topographic signature of life[END_REF][START_REF] Marchi | Rocky headwaters in the Dolomites, Italy: Field observations and topographic analysis[END_REF]. Another potential contributing factor is that the bedrock and colluvium in our study areas are not naturally cohesive, for example, clay-rich rocks can exhibit convex creep-dominated slopes in unvegetated badlands on Earth.

The pattern of data in slope-area plots shown by our alluvial systems and some of our debris flow systems (slow decline at small drainage areas followed by a steep decline at higher drainage areas) has been shown from numerous remote sensing and field studies to mark the transition from the colluvial (including debris flow) regime, to that of a fully fluvial regime (e.g., [START_REF] Lague | Constraints on the long-term colluvial erosion law by analyzing slope-area relationships at various uplift rates in the Siwaliks Hills (Nepal)[END_REF][START_REF] Stock | Valley incision by debris flows: Evidence of a topographic signature[END_REF][START_REF] Stock | Erosion of steepland valleys by debris flows[END_REF]. Some have described the transition as a separate linear portion of the plot between the colluvial and the fluvial [START_REF] Lague | Constraints on the long-term colluvial erosion law by analyzing slope-area relationships at various uplift rates in the Siwaliks Hills (Nepal)[END_REF] and some as a gradual curved transition [START_REF] Stock | Valley incision by debris flows: Evidence of a topographic signature[END_REF]. However, both are consistent with Tucker & Bras' (1998) transition from pore pressure triggered landsliding into a fully fluvial system. Our plots do not show a well developed alluvial regime, but this is due to the use of high resolution data of very small areas rather than large, well developed fluvial catchments.

In summary, our terrestrial data are consistent with published slope-area process domains, and provide reassurance that the method is applicable and that the Mars data can be used to infer process in a similar way. The caveat to this is that the environmental differences between Earth and Mars, as detailed in the introduction, must be considered when comparing terrestrial process domains to data from Mars. Furthermore, improved process discrimination can be made by considering CAD profiles in addition to slope-area analysis.

Comparison of Earth data to published CAD process domains

Comparison of all our CAD plots for Earth (Fig. 7) to the published process domains for CAD (Fig. 2) reveals that our data do not generally follow the cited trends. This is possibly because we are studying small areas, rather than large catchments. However, the shape of the curve outlined by our data in CAD plots does allow process discrimination and does follow some of the framework outlined by [START_REF] Mcnamara | Channel head locations with respect to geomorphologic thresholds derived from a digital elevation model: A case study in northern Thailand[END_REF]. Specifically region 1 on Fig. 2 has three sub-regions whose shapes can be recognised in our datasets. The talus data (Fig. 7a) and synthetic crater (Fig. 9d) are both convex in their CAD plots, resembling most closely region 1a of [START_REF] Mcnamara | Channel head locations with respect to geomorphologic thresholds derived from a digital elevation model: A case study in northern Thailand[END_REF]. They describe this region as representing "hillslopes that diverge and do not gather drainage." Our alluvial data and some of our debris flow data show a flattening of the CAD plot curve in the middle region, giving a steep linear section, corresponding to either region 1b or region 2 (Fig. 2b) which [START_REF] Mcnamara | Channel head locations with respect to geomorphologic thresholds derived from a digital elevation model: A case study in northern Thailand[END_REF] describe as slopes that are convergent (1b), or channel forming (2). Two debris flows (WF2 and WF3 in Fig. 7c) show a concave section, which would correspond to region 1c of [START_REF] Mcnamara | Channel head locations with respect to geomorphologic thresholds derived from a digital elevation model: A case study in northern Thailand[END_REF] and which they attribute to pore pressure triggered landsliding or debris flow.

The similarity of talus and debris flow in slope-area plots can be attributed to their similarly linear long profiles. However, the two processes produce different patterns in CAD plots because talus slopes tend to disperse drainage but debris flow slopes tend to have convergent drainage. This can also be seen in the wetness index plots (Fig. 10).

This difference of behaviour in CAD and wetness index plots, in addition to the information from the slope-area plots, shows that we can detect slopes dominated by alluvial, debris flow and dry mass wasting on the basis of these parameters, even for small catchments such as individual gullies or debris flow tracks. However, it should be noted here that these analyses have been performed on relatively few sample sites on Earth and some of the differences are subtle. Future work has to include extending this analysis to a greater number of test sites on Earth to verify that this kind of process discrimination is robust. Using these initial results we continue and apply these methods of process discrimination to Mars.

Process domains for gullies on Mars

In slope-area plots all the Mars slope sections, except study area TS1, fall below the slope threshold for dry mass wasting (dotted line in the plots in Fig. 8). This means that talus-like dry mass wasting is not a dominant process in these areas. However, study area TS1, visually similar to talus on Earth, is not only above the slope threshold for dry mass wasting, but also bears a signature similar to talus on Earth in the combination of its slope-area plot, CAD plot and wetness index map Within the process domains of [START_REF] Montgomery | Channel network source representation using digital elevation models[END_REF] the majority of the Mars data lie within the debris flow domain, with some data located in the debris flow deposition domain added by [START_REF] Brardinoni | Glacial erosion, evolution of river long profiles, and the organization of process domains in mountain drainage basins of coastal British Columbia[END_REF] and a few in the alluvial domain. The difference in gravity between Earth and Mars requires an upwards slope adjustment to the alluvial channels domain boundary (see Fig. 2a) in slope-area plots (Appendix 1), but does not change the gradient of the line. This is marked by the dash-dot line on the plots in Fig. 8. This shift places more data in the unchanneled domain, but does not place any additional data into the alluvial or debris flow domains. This in itself distribution does not provide very detailed information on the formation mechanisms for gullies. However, by combining slope-area trends, CAD plots and wetness index maps we can make more detailed assessments. We examine each of the study areas on Mars in turn and then discuss the overall implications for the gully formation processes.

Synthetic Crater

The pattern in slope-area plots of the interior of impact craters is, in part, a result of the inherent shape of the crater slope which in turn is due to the impact process and the modification that occurs immediately afterwards. The slope of a fresh impact crater is concave and exponentially shaped in profile [START_REF] Garvin | Global Geometric Properties of Martian Impact Craters: A Preliminary Assessment Using Mars Orbiter Laser Altimeter (MOLA)[END_REF]. Thus in slope-area plots it resembles a well developed alluvial system on Earth (e.g., [START_REF] Hack | Studies of longitudinal stream profiles in Virginia and Maryland[END_REF]. This reinforces the uncertainty in inferring a unique process from slope form. In CAD plots, however, the synthetic crater data show a similar pattern to that of talus slopes on Earth, indicating that at short length-scales this type of slope cannot channelise flow on its own. This interpretation is supported by the wetness index plot (Fig. 11), which shows a slowly coalescing flow, rather than discrete areas of fluid concentration.

Site PC -Penticton Crater in Eastern Hellas

In slope-area plots the slope turnover is well expressed for both study areas in site PC (Fig. 8a). This suggests a strong diffusive or creep influence on both slopes. Study areas PC1 and PC2 both resemble either poorly developed talus or debris flow in slope-area plots.

In the CAD plot (Fig. 9a); however, study area PC2 has the distinctive profile associated with debris flow, whereas study area PC1 more closely resembles talus. Talus processes can only be active in study area PC1 at small drainage areas, where it lies on the dry mass wasting threshold in slope-area plots. Hence the shape of the CAD curve must be explained by another process, which has a slope threshold but does not concentrate drainage. This unknown process must be pore pressure triggered as it is below the slope for dry mass wasting. In addition, the wetness index plot reveals that study areas PC1 and PC2 are very different: study area PC1 has a similar wetness index map to the synthetic crater (Fig. 11h), whereas study area PC2 resembles debris flow areas on Earth (e.g., Fig. 10f) with strongly concentrated high wetness index within alcoves and channels, becoming more diffuse down slope on the debris aprons.

The combined evidence suggests that the west-facing slope, which contains small gullies, has been modified by debris flow, whereas the equator-facing slope is more similar to dry mass wasting deposits. This agrees with the interpretation of [START_REF] Pelletier | Recent bright gully deposits on Mars: Wet or dry flow?[END_REF], who, using numerical modelling, concluded that the new bright toned deposits on this slope were more similar in form to deposits of dry granular flows than debris flows.

Site GC -Gasa Crater in Terra Cimmeria

In the slope-area plot for site GC (Fig. 8b), the slope-turnover occurs at very small drainage areas (one or two pixels) and is thus partly abbreviated. This suggests that creep has not strongly influenced this site. This interpretation is supported by the observation that the gully heads originate at the very top of the slope. Study areas GC1, GC2 and GC3 resemble either poorly developed talus on Earth (study area KA2, Fig. 6a) or debris flows on Earth (Figs. 6b and6c) in slope-area plots. However, in CAD plots (Fig. 9b) they have a flattened mid-section, resembling debris flow systems on Earth. Their wetness index plots (Figs. 11c and11d) have strong similarities with debris flow systems on Earth (e.g., Fig. 10g): showing flow concentration in the alcove and channel with more diffuse flow on the debris apron.

Study area GC2 (Fig. 11c) shows a similar pattern of wetness index to the alluvial systems on Earth, with focussed flow throughout.

In slope-area plots (Fig. 8b) study area GC4 has a flatter profile than study areas GC1, GC2 and GC3. The drop in slope at high drainage areas in GC4 is probably an artefact of the low number of pixels included in the slope calculations in the last 5 to 10 points. In the CAD plot (Fig. 9b), study area GC4 has a similar shape to talus systems on Earth (Fig. 7a).

The talus interpretation for GC4 is supported by additional evidence: (1) there is no evidence for channels (Fig. 4e), ( 2) the wetness index plot (Fig. 11e) is similar to talus slopes on Earth and (3) part of the slope-area curve lies on the threshold for dry mass wasting (Fig. 8b). The dip of the slope-area curve away from the threshold for dry mass wasting suggests that another process with a lower slope threshold is acting, either without having an effect on the CAD plot, or with the same CAD plot as talus. We hypothesise that this may be the same unknown process as noted in study area PC1.

The combined evidence suggests that the pole and east facing slopes of the crater have been affected by debris flow processes and the equator-facing slope by mass wasting and an unknown process.

Site KC -crater inside Kaiser Crater in Noachis Terra

Our ungullied study area (KC4) shows patterns in slope-area (Fig. 8c) and CAD plots (Fig. 9c) very similar to the synthetic crater and creep slopes on Earth. The difference between this study area and the gullied study areas (KC1 to KC3) is presumably a result of the process of gully formation. Study areas KC1 to KC3 do not have slope-area plots (Fig. 8c) that fit easily within the framework established so far. However, if we refer to the modelling work of [START_REF] Tucker | Hillslope processes, drainage density, and landscape morphology[END_REF] then the patterns in slope-area plots can be explained. At small drainage areas our curves for study areas with gullies have a horizontal or slightly positive trend compared to our ungullied study area, which has a definite positive trend. This suggests the weak influence of diffusive processes (which generate a positive relationship in slopearea plots) combined with slope threshold processes (which tend to produce horizontal trends). As all the data are below the dry mass wasting threshold, this threshold process is likely to be a pore pressure triggered process, such as debris flow. At intermediate drainage areas there is a transitional region which occurs at a similar drainage area to the slopeturnover in the ungullied section. At high drainage areas the gullied study areas show a slightly decreasing sub-horizontal trend, as opposed to the ungullied study area which has a well defined decrease in slope with drainage area. This also can be attributed to a pore pressure triggered threshold process but at a lower slope threshold than the previous process.

In CAD plots (Fig. 9c) study areas KC1 to KC3 are consistent with debris flow processes.

The wetness index plots for these study areas (Fig. 11f) are similar to terrestrial debris flow study areas which have been influenced by alluvial processes (e.g., site WF, Figs. 10g and10h). This suggests that the first pore pressure threshold in slope-area plots is due to debris flow and the second lower one due to an unknown process, which again could be the same process affecting sites PC and GC.

In slope-area plots, the gully heads on this slope (Fig. 8c) coincide with the drainage area of the slope turnover in study area KC4 and the transitional study areas of KC1 to KC3.

This coincident relationship matches the observations made by many authors who have studied gullies on Earth (e.g., [START_REF] Hancock | Channel head location and characteristics using digital elevation models[END_REF]. Our channel heads lie mainly in the domain attributed to "pore pressure landsliding channel initiation" processes, but some also lie in the "unchanneled" domain [START_REF] Mcnamara | Channel head locations with respect to geomorphologic thresholds derived from a digital elevation model: A case study in northern Thailand[END_REF]. Notably the gully heads occur below the dry mass wasting threshold, again suggesting that these martian gullies are initiated by a pore pressure threshold process. The gully heads occur on slope gradients of 0.55 similar to those described by [START_REF] Lanza | Evidence for debris flow gully formation initiated by shallow subsurface water on Mars[END_REF], but at drainage areas an order of magnitude lower. This is possibly due to the different approach used by [START_REF] Lanza | Evidence for debris flow gully formation initiated by shallow subsurface water on Mars[END_REF] to measure the contributing area, and possible differences in their interpretation of the location of channel initiation. The co-occurrence of the gully heads with the slope-turnover in slope-area plots suggests that the gullies are a result of whole-slope drainage, as previously found by [START_REF] Lanza | Evidence for debris flow gully formation initiated by shallow subsurface water on Mars[END_REF], either at the surface or shallow subsurface. Our work provides additional evidence to support the conclusions of [START_REF] Lanza | Evidence for debris flow gully formation initiated by shallow subsurface water on Mars[END_REF] that these gullies originate from a distributed source and hence supports the surface melting model for martian gully formation, rather than an aquifer source model. Further, this observation provides additional evidence that a threshold process, probably debris flow, is forming these gullies, as previously suggested by [START_REF] Lanza | Evidence for debris flow gully formation initiated by shallow subsurface water on Mars[END_REF].

From the combination of the slope-area, CAD and wetness index plots we infer that the gullies in this crater are produced by debris flow and were initiated by surface, or near subsurface, flow of water. Creep and an unknown process were likely to have been the dominant processes on the ungullied crater slopes. This is consistent with the setting of these gullies within the ice-rich mantle deposits which is likely to be susceptible to melting, providing a distributed source of water for the gullies.

Site TS -crater in Terra Sirenum

Unlike the other areas we have studied on Mars, parts of the slope-area data for study area TS1 at lower drainage areas (Fig. 8d) are above the threshold slope for dry mass wasting.

This is an indication that rock strength limited dry mass wasting is occurring in the upper parts of the slope. In CAD plots (Fig. 9d) this study area has the classic shape of a talus or creep slope. However, the slope-area trend shown by study area TS1 is very different from that of the synthetic crater (Fig. 8d), which we assume to have been similar to the starting point for study area TS1. This assumption carries the implication that the slope in study area TS1 has evolved over time from concave to linear in profile. Study area TS1 shows a very similar trend in slope-area plots as study area GC4 (Fig. 8b), but originates above the 0.7 slope threshold. As discussed previously for study area GC4, in the framework of Tucker & Bras (1998) such a pattern is likely to reflect a gradual transition from the dominance of a dry mass wasting threshold at lower drainage areas to the dominance of a pore pressure triggered slope threshold due to an unknown process at higher drainage areas. However, in the case of TS1 this signal not only includes dry mass wasting of non-cohesive material, but rockwall mass wasting as well. The wetness index map shows that the slope does not concentrate drainage, except for some diffuse linear areas, again resembling talus slopes on Earth. The combination of the slope-area plot, CAD plot and wetness index map suggests a dominantly dry mass wasting evolution of this slope, which fits well with the visual observations.

Solifluction on slopes on Mars

In many of the Mars study sites we have inferred an unknown process that is responsible for a second, lower-slope pore pressure triggered threshold in the slope-area plots. However, this process seems to produce slopes that yield a CAD plot that is similar to talus on Earth, i.e. it does not concentrate drainage. As suggested by [START_REF] Tucker | Hillslope processes, drainage density, and landscape morphology[END_REF] another threshold process which would produce a similar response in slope-area plots to pore pressure induced landsliding is solifluction. Solifluction in frozen landscapes comprises the combined action of gelifluction and frost creep, and describes the slow, down slope movement of water saturated debris or soils. Solifluction requires freezing and thawing to generate elevated pore pressures and occurs at lower slope angles than pore pressure induced failure, which can trigger landslides and debris flow [START_REF] Harris | An investigation of periglacial slope stability in relation to soil properties based on physical modelling in the geotechnical centrifuge[END_REF]. This process is consistent with the recent observations of periglacial landform assemblages on Mars (Balme & Gallagher 2009;Balme et al. 2009;[START_REF] Soare | Stratigraphical evidence of late Amazonian periglaciation and glaciation in the Astapus Colles region of Mars[END_REF]. [START_REF] Dietrich | The search for a topographic signature of life[END_REF] suggested that the lack of biotic processes on Mars would promote erosion by rilling and gullying and stripping of the fine surface materials, given a suitable water source. This would lead to a slope-area plot that lacked a distinct slope turnover, similar to the slope-area plots seen in the Death Valley data (our site DV -Fig. 6d).

Implications for the formation process of martian gullies

However, inspecting the trends in the slope-area plots for the Mars systems in Fig. 8, one of the most apparent differences from Earth is the presence of this slope turnover. This indicates that creep is a more dominant process on martian hillslopes than on those we studied on Earth; contradictory to the predictions made by [START_REF] Dietrich | The search for a topographic signature of life[END_REF]. The creep signal in most published slope-area plots on Earth is induced predominantly by biota, hence on Mars the creep must be facilitated using a different mechanism. [START_REF] Perron | Icedriven creep on Martian debris slopes[END_REF] Virtually every gully that we have studied on Mars has the distinct signal of debris flow as the dominant gully forming process. [START_REF] Lanza | Evidence for debris flow gully formation initiated by shallow subsurface water on Mars[END_REF] also found visual and morphometric evidence of debris flows in these areas. The notable exception is area PC1, the slope containing the new light-toned deposits. However, this area does not include gullies of a normal form (Fig. 1) as they lack well defined alcoves and channels. Examination of a far greater number of DEMs containing gullies would be needed to confirm debris flow as the main gully forming process on Mars. However, if this is the principal mechanism, brings up the following hypotheses and predictions for the formation of gullies on Mars:

(1)

The high sediment concentrations and low infiltration rates could protect the water from evaporation.

pole-facing slopes during periods of high obliquity, similar to the model proposed by [START_REF] Costard | Formation of recent Martian debris flows by melting of near-surface ground ice at high obliquity[END_REF] for gully formation on pole-facing slopes. Our study sites also show this asymmetry: pole-facing slopes are longer and have a greater variety of slope angles and are more concave, whereas the equator-facing slopes are shorter and have a more uniform distribution of slopes and are more linear. There is a marked difference in geomorphological process between crater walls with different aspects in the two craters that we studied (sites PC and GC). The observed asymmetry of process and form supports the model of a climatic influence on gully formation and general slope development of the craters. However, many more sites would have to be studied to verify this for gullies in general.

Conclusions

We have shown the potential of applying quantitative geomorphological analysis techniques commonly used on Earth for discriminating between different active processes on Mars.

Specifically we have validated the use of slope-area plots, cumulative area distribution (CAD) plots, and wetness index maps on small slope sections of less than one square kilometre. We have shown that pure water (alluvial) flow, debris flow and dry mass wasting dominated slopes can be satisfactory discriminated on Earth. By applying these techniques to four areas of Mars containing recent gullies we have inferred that debris flow is the dominant gully forming process. However, we have also inferred that, as on Earth, gully formation on Mars is a complex process: slopes on Mars are likely to have been affected by a variety of processes that lead to a mixture of signals from our geomorphological analyses. Despite this, we have not found the distinctive geomorphological fingerprint of pure water flow on slopes that host gullies. Its absence, however, does not prove the absence of the process. Our results are consistent with the possibility that ice driven creep and solifluction are, or have recently been, active in modifying crater slopes on Mars.

From the location of gully heads within the landscape, and by studying the form of alcoves, it is apparent that at least two of the sites examined contain gullies which have been formed from a widely distributed source of water. This is most easily explained by a surface melting source for the water. The model of [START_REF] Costard | Formation of recent Martian debris flows by melting of near-surface ground ice at high obliquity[END_REF] provides a mechanism by which the cause of this melting was increased insolation during past high obliquity excursions. Our preliminary observations of an asymmetry in process and form around the impact craters provides additional support for this model, but we cannot rule out surface melting at present day, or during other epochs.

Our geomorphological evidence for debris flow as an active process in forming gullies is reinforced by visual observations. Debris flow as a process leaves distinct geomorphological features, such as failure scars and lobate deposits, which have been observed both here and in previous studies (Dickson & Head 2009;[START_REF] Levy | Identification of gully debris flow deposits in Protonilus Mensae, Mars: Characterization of a water-bearing, energetic gully-forming process[END_REF][START_REF] Lanza | Evidence for debris flow gully formation initiated by shallow subsurface water on Mars[END_REF]. Unfortunately the topographic data on Mars are not yet sufficient for the discrimination of these features and flanking levées in DEMs, which would allow accurate estimation of individual flow volumes and thus estimation of the volumes of water needed to form the gullies (Conway et al. 2010b).

where k and a are positive constants. Following [START_REF] Snyder | Landscape response to tectonic forcing: Digital elevation model analysis of stream profiles in the Mendocino triple junction region, Northern California[END_REF], [START_REF] Whipple | Dynamics of the stream-power river incision model: Implications for height limits of mountain ranges, landscape response timescales, and research needs[END_REF] we use the assumptions of conservation of mass (water) and steady uniform flow to obtain the following expression of basal shear stress:

3 / 2 3 / 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = W gSQ C f b ρ τ (3)
where ρ is the density of water, C f is a dimensionless friction factor, g is the acceleration due to gravity, S is the local channel slope, Q is the stream discharge and W is the stream width.

We then include a relationship for basin hydrology and hydraulic geometry given by:

Q = k q A c (4) W = k w Q b (5)
where k q and k w are constants, A is the drainage area and b and c are positive dimensionless constants. Combining 2-5, leads to:

E = k e A m S n (6)
where:

k e = k b k w -2a/3 k q 2a(1-b)/3 ρ a g 2a/3 (7) m = (2ac/3)(1-b) (8) n = 2a/3 (9)
Given this, if we now define a constant k m for Mars, based on the assumption that gravitation acceleration is approximately one third that on Earth:

k m = (1/3) 2a/3 k e = (1/3) n k e (7)
To derive (1), in the main text, we have to include the expectation that over long timescales, uplift rate (U) and erosion rate compete to change the landscape elevation (z):

n m e S A k U E U t z - = - = ∂ ∂ ( 8
)
where t is a given time-step. Now if we assume that the system is in equilibrium in which erosion is balanced by uplift rate, ∂ z/ ∂ t = 0, then:

S = (U/k e ) 1/n A -m/n (9)
Comparing this to equation (1), in the main text, we have:

k = (U/k e ) 1/n (10) θ = -m/n (11)
If we then include our k m constant (7) for Mars in (9) we get: .

S = (U/k m ) 1/n A -m/n (12) S = 3(U/k e ) 1/n A -m/n (13)
Average elevation is given relative to the Mars datum, as defined from the MOLA dataset.

The average elevation has been estimated from the MOLA dataset and relief from the HiRISE DEMs.

  debris flow dominated study areas(Figs. 7b and 7c) show the following pattern: (1) The curve drops away from the horizontal slowly (but faster than the talus slopes) at small drainage areas. (2) The curve then either dips down linearly, or follows a flattened convex path, and (3) at high drainage areas the curve drops away sharply with increasing drainage area.

  observed using Mars Orbiter Laser Altimeter (MOLA) data that slopes on Mars have average gradients well below 35° and suggested that ice driven creep is the cause. Other potential creep mechanisms include frost heave and shrink-swell in clays and hydrated salts, both of which produce creep on un-vegetated and un-bioturbated slopes on Earth. These mechanisms however would require widespread and relatively large amounts of liquid water, which is considered unlikely under current or geologically recent martian climate. Hence, we believe that ice driven creep provides the best explanation for the signals seen in our slope-area data from Mars. In accordance with their results, most of the slopes we studied on Mars also have average gradients well below 35°, with the exception of slope TS1, whose average gradients are partially above 35°.

Fig. 1 .

 1 Fig. 1. HiRISE images of a variety of gullies on Mars. Image credits: NASA/JPL/UofA. (a)

Fig. 3 .

 3 Fig. 3. Hillshade representations made from digital elevation models of the study site

Fig. 9 .

 9 Fig. 9. Cumulative Area Distribution plots for the study areas on Mars. (a) Plots for Site PC,

Fig. 11 .

 11 Fig. 11. Wetness index maps made from digital elevation models of the study site locations

Table 1

 1

	lists the

Table 1 .

 1 Summary table for the study sites on EarthAverage elevation is given relative to datum, for A-D this is NAD 1983 and for Site E this isWGS 1984, in both cases the difference between the datum and sea level is approximately 60 m. Abbreviations: NCALM -National Center for Airborne Laser Mapping supported by the USA's National Science Foundation, ARSF -Airborne Research and Survey Facility supported by the UK Natural Environment Research Council.

	Site	Location	Date Flown	Data Source	(mm/year)	precipitation	Approx.	Landscape-type	Latitude	Longitude	elevation (m)	Average	Relief (m)
		San Jacinto Fault
	A	(SJF Segment 3) -Santa Rosa	mid 2005	NCALM B4 Project	150		desert	33° 25' 58.55" N	116° 28' 57.55" W	597
		Mountains
	B	Death Valley California	28/02/2005	NCALM	<85		desert	39° 38' 01.77" N	105° 49' 13.88" W	3664	5
	C	St. Elias, Alaska	02-15/9/2005 NCALM	2000		periglacial	60° 18' 18.59" N	144° 32' 14.98" W	490
	D	Front Range, Colorado	30/09/2005	NCALM	600		periglacial	37° 04' 28.50" N	117° 26' 37.60" W	258
	E	Westfjords, Iceland	05/08/2007	ARSF	700		periglacial	66° 04' 13.20" N	023° 07' 14.19" W	271

Table 2 .

 2 Summary table for the study sites on Mars

					Average
	Site	HiRISE image pair	Latitide Longitude	elevation (m)	Relief (m)
	F	PSP_001714_1415 PSP_001846_1415	-38.4°	96.8°	-2648	1124
	G	PSP_004060_1440 PSP_005550_1440	-35.7°	129.4°	300	1205
	H	PSP_003418_1335 PSP_003708_1335	-46.1°	18.8°	595	687
	J	PSP_003674_1425 PSP_005942_1425	-37.4°	229.0°	1904	961

Acknowledgements

Thanks go to Grant Meyer and one anonymous reviewer for their constructive comments which greatly improved this manuscript. This work would not have been possible without a postgraduate studentship grant from the U.K. Natural Environment Research Council (NERC). We thank the NERC ARSF for obtaining the LiDAR data on which part of this paper relies. We thank the UK NASA RPIF-3D Facility at UCL for enabling the production of one of the HiRISE DEMs. Additional funding was awarded to S.J.C. by Earth and Space Awards, the Geological Society's W.G. Fearnsides Award, The Dudley Stamp Fund and the British Society for Geomorphology's postgraduate funds. P.M.G. is funded by an STFC Aurora Fellowship (ST/F011830/1).Thanks to Jon Yearsley for creation of spatialPattern script to create pink noise in MatLab.

Thus for a given drainage area on Earth we would expect the slope on Mars to be three times smaller. Or, in log-log terms: (14)

(2)

The energy released by grain interactions within the flow could retard freezing.

(3) Basal freezing (Conway et al. 2010a) or a permafrost layer could facilitate the saturation of the sediment that is required to generate the high pore water pressures to trigger debris flow.

(4) Expected depositional features include levées and lobes.

(5) Expected erosional features include discrete slip scars.

Points 1-3 of are hard to observe or test, but the erosional and depositional features can be detected in the high resolution HiRISE images. Failure scars have been noted by other authors (Dickson & Head 2009) from HiRISE images and are present within our study areas.

Depositional lobes have also been noted by other authors [START_REF] Levy | Identification of gully debris flow deposits in Protonilus Mensae, Mars: Characterization of a water-bearing, energetic gully-forming process[END_REF][START_REF] Lanza | Evidence for debris flow gully formation initiated by shallow subsurface water on Mars[END_REF]). Visual observations have been made of debris flow levées [START_REF] Lanza | Evidence for debris flow gully formation initiated by shallow subsurface water on Mars[END_REF]), but DEMs from HiRISE are not yet of sufficient quality to reliably resolve debris flow levées.

High quality DEMs would allow the estimation of individual flow volumes (Conway et al. 2010b), which could be used to constrain models of gully formation. This should be a priority for future work, as it would allow more accurate estimates of the amounts of water associated with formation of gully landforms.

A debris flow, once triggered, results in more erosion and deposition with less water than pure water flow. This means that high discharges, invoked by other workers [START_REF] Heldmann | Formation of Martian gullies by the action of liquid water flowing under current Martian environmental conditions[END_REF][START_REF] Hart | Gully Slopes and Discharges on Lyot Crater's Central Peak[END_REF], are not required to form martian gullies. Modelling has shown that surface melting produces only small amounts of liquid water [START_REF] Williams | Ancient melting of mid-latitude snowpacks on Mars as a water source for gullies[END_REF]. This has been one of the major criticisms of the surface melting model. However, if gullies are formed mainly by debris flow, points (1) and ( 2) above indicate that relatively small amounts of water are needed.

Implications for the water source of martian gullies

The observed relationship in slope-area plots between the slope turnover and the location of gully heads in site KC on Mars is an important observation and indicates that the transition from concave to convex topography is closely linked to gully formation. This would not be expected in an aquifer system, as channel formation would be controlled predominantly by the location of aquifer bodies rather than the shape of the landscape [START_REF] Fetter | Applied Hydrogeology[END_REF]. Our work indicates that a widely distributed source of surface or shallow subsurface flow in site KC would be the most satisfactory explanation, in support of the conclusions of [START_REF] Lanza | Evidence for debris flow gully formation initiated by shallow subsurface water on Mars[END_REF]. Because our data do not show a definite trend in slope-area plots this indicates that the channels originate from shallow sub-surface flow [START_REF] Hattanji | Thresholds for bed load transport and channel initiation in a chert area in Ashio Mountains, Japan: An empirical approach from hydrogeomorphic observations[END_REF][START_REF] Jaeger | Channel and perennial flow initiation in headwater streams: Management implications of variability in source-area size[END_REF][START_REF] Imaizumi | Channel initiation by surface and subsurface flows in a steep catchment of the Akaishi Mountains, Japan[END_REF], or more likely surface flow in a soil poor landscape [START_REF] Larsen | Geologic versus wildfire controls on hillslope processes and debris flow initiation in the Green River canyons of Dinosaur National Monument[END_REF]. A potential source for this near surface water is the mantle deposits, which have been observed on both this slope and in site PC2 and has been linked to gully formation by other authors [START_REF] Christensen | Formation of recent martian gullies through melting of extensive water-rich snow deposits[END_REF][START_REF] Aston | Identifying Martian gully evolution[END_REF][START_REF] Van Gasselt | Periglaical Geomorphology of Tempe Terra, Mars: role of mantling deposits in lobate debris apron formation and evolution[END_REF].

The development of equally spaced incised alcoves in site GC can either be attributed to geological controls (e.g., faulting), or landscape self organisation from an interlinked debris flow-alluvial system [START_REF] Perron | Formation of evenly spaced ridges and valleys[END_REF]. We argue against a structural control, because there is a lack of these organised alcoves on the equator-facing slope. Hence, considering that we conclude debris flow to be the dominant gully forming process on this crater slope, it would seem most likely that these self-organised alcoves are a result of this process. This kind of self organisation requires a landscape that responds to a distributed water source as on Earth rather than an aquifer source. [START_REF] Kreslavsky | North-south topographic slope asymmetry on Mars: Evidence for insolation-related erosion at high obliquity[END_REF] and [START_REF] Kreslavsky | Periods of active permafrost layer formation during the geological history of Mars: Implications for circum-polar and mid-latitude surface processes[END_REF] found that pole-facing slopes between 40-50° latitude in both hemispheres were systematically gentler than equatorfacing slopes. They suggest that this is due to insolation asymmetry and melting of ice on

Appendix 1

The derivation of the shear stress erosion model relies on the assumption that erosion rate (E) is a power law of bed shear stress (τ b ):

(2)