Francesco Salese
email: f.salese@uu.nl

Maarten G Kleinhans

Nicolas Mangold

Veronique Ansan

William Mcmahon

Tjalling De Haas

Gilles Dromart

Estimated Minimum Lifespan of the Jezero Fluvial Delta (Mars)

Keywords:

The paleo-lake floor at the edge of the Jezero delta has been selected as the NASA 2020 rover landing site. In this paper we demonstrate the sequences of lake filling and delta formation, and constrain the minimum lifespan of the Jezero paleo-lake from sedimentological and hydrological analyses. Utilizing imagery provided by the High Resolution Imaging Science Experiment (NASA Mars Reconnaissance Orbiter) and High Resolution Stereo Camera (ESA Mars Express), two main phases of delta evolution can be recognized. 1) Basin infilling before the breaching of the Jezero rim; and 2) the delta formation itself. Our results suggest delta formation occurred over a 90 -550 year period of hydrological activity. Breaching of the Jezero rim occurred in at least three distinct episodes, which spanned a far longer time-period than overall delta formation. This evolutionary history implies the Jezero-lake floor would have been a haven for fine-grained sediment accumulation, and hosted an active environment of significant astrobiological importance.

Introduction

Remnants of ancient deltas provide a critical record of ancient surface flow on Mars. Interpretations of these sedimentary archives are of fundamental importance for quantifying the hydrological history of the planet. A range of delta-types have so far been recognized (e.g., Di Achille and Hynek [2010]; [START_REF] Hauber | Asynchronous formation of Hesperian and Amazonian-aged deltas on Mars and implications for climate[END_REF]; [START_REF] Salese | Geological evidence of planet-wide groundwater system on Mars[END_REF]), including simple stepped deltas [START_REF] Irwin | An intense terminal epoch of widespread fluvial activity on early Mars: 2. Increased runoff and paleolake development[END_REF][START_REF] Di Achille | A steep fan at Coprates Catena, Valles Marineris, Mars, as seen by HRSC data[END_REF][START_REF] Weitz | Formation of a terraced fan deposit in Coprates Catena[END_REF][START_REF] Kraal | Martian stepped-delta formation by rapid water release[END_REF][START_REF] Hauber | Asynchronous formation of Hesperian and Amazonian-aged deltas on Mars and implications for climate[END_REF][START_REF] Salese | Geological evidence of planet-wide groundwater system on Mars[END_REF], complex and lobate deltas [START_REF] Pondrelli | Evolution and depositional environments of the Eberswalde fan delta[END_REF][START_REF] Mangold | The origin and timing of fluvial activity at Eberswalde crater[END_REF][START_REF] Hauber | Asynchronous formation of Hesperian and Amazonian-aged deltas on Mars and implications for climate[END_REF][START_REF] Salese | Hydrological and sedimentary analyses of well-preserved paleofluvial-paleolacustrine systems at Moa Valles, Mars[END_REF] in addition to Gilbert-type deltas [START_REF] Ori | Terraces and Gilbert-type deltas in crater lakes in Ismenius Lacus and Memnonia (Mars)[END_REF][START_REF] Di Achille | Evidence for late Hesperian lacustrine activity in Shalbatana Vallis, Mars[END_REF][START_REF] Hauber | Asynchronous formation of Hesperian and Amazonian-aged deltas on Mars and implications for climate[END_REF][START_REF] Salese | Geological evidence of planet-wide groundwater system on Mars[END_REF]. Whereas stepped deltas are readily explained by a single outflow event [START_REF] Kraal | Martian stepped-delta formation by rapid water release[END_REF][START_REF] De Villiers | Experimental delta formation in crater lakes and implications for interpretation of Martian deltas[END_REF], many studied systems are far more complex because they have much greater catchment areas and often relate to breached craters. The Jezero crater delta, along with Moa Valles-Liberta [START_REF] Salese | Hydrological and sedimentary analyses of well-preserved paleofluvial-paleolacustrine systems at Moa Valles, Mars[END_REF], Eberswalde crater [START_REF] Malin | Mars Global Surveyor Mars Orbiter Camera: Interplanetary cruise through primary mission[END_REF][START_REF] Pondrelli | Evolution and depositional environments of the Eberswalde fan delta[END_REF][START_REF] Mangold | The origin and timing of fluvial activity at Eberswalde crater[END_REF] and Sabrina and Hypanis [START_REF] Adler | Hypotheses for the Origin of the Hypanis Fan-Shaped Deposit at the Edge of the Chryse Escarpment, Mars: Is it a Delta?[END_REF], are among the few Martian deltas with evident stratigraphy, avulsing channels and multilobate depositional patterns. Previous studies have yielded conflicting estimates of the time-scale of evolution within the Jezero delta, with the total duration of hydrological activity ranging from mere decades to upwards of millions of years [START_REF] Fassett | Fluvial sedimentary deposits on Mars: Ancient deltas in a crater lake in the Nili Fossae region[END_REF][START_REF] Ehlmann | Clay minerals in delta deposits and organic preservation potential on Mars[END_REF][START_REF] Schon | An overfilled lacustrine system and progradational delta in Jezero crater, Mars: Implications for Noachian climate[END_REF][START_REF] Goudge | Assessing the mineralogy of the watershed and fan deposits of the Jezero crater paleolake system, Mars[END_REF]. Notably, it remains unclear whether (or not) delta formation occurred during extended epochs of clement climatic conditions (favorable for life), or during punctuated intervals of allogenically forced sedimentary events (e.g., regional impacts, volcanism, or tectonics) (e.g. [START_REF] Brakenridge | Ancient Hot Springs on Mars -Origins and Paleoenvironmental Significance of Small Martian Valleys[END_REF]; [START_REF] Hauber | Asynchronous formation of Hesperian and Amazonian-aged deltas on Mars and implications for climate[END_REF]; [START_REF] Salese | Episodic warming of early Mars by punctuated volcanism[END_REF]). This latter hypothesis would imply that lake conditions may not have been dissimilar from the present day, thus likely prohibitive for life.

Accepted paper in Astrobiology -Vol. 8 (2020) Salese et al., 2020b The aims of this paper are: a) to re-examine the delta-forming discharge from the Neretva Vallis; b) to consider the active lake's water supply and loss mechanics; and c) to determine the geologic timescale of delta formation.

Utilizing high-resolution digital elevation models we estimate the systems hydrological potential, and propose different scenarios of evolution with varied key sediment transport parameters (e.g., grain size). Sediment transport predictions are intricately linked to estimates of grain size, channel depth and gradients, such that minor differences in input conditions may result in sediment budget outputs orders of magnitude apart (perhaps explaining the contradictory estimates of Jezero Crater evolution predicted by previous researchers [START_REF] Fassett | Fluvial sedimentary deposits on Mars: Ancient deltas in a crater lake in the Nili Fossae region[END_REF][START_REF] Ehlmann | Clay minerals in delta deposits and organic preservation potential on Mars[END_REF][START_REF] Schon | An overfilled lacustrine system and progradational delta in Jezero crater, Mars: Implications for Noachian climate[END_REF][START_REF] Goudge | Assessing the mineralogy of the watershed and fan deposits of the Jezero crater paleolake system, Mars[END_REF] (Figure 1). Despite imbedded (and unavoidable) uncertainties, physics-based predictors (used in this work, rather than empirically-based estimates) provide the most robust flow and transport rates available. Empirically-derived hydrological estimates which assume fixed sediment concentrations (e.g. [START_REF] Moore | Martian layered fluvial deposits: implications for Noachian climate scenarios[END_REF]; [START_REF] Irwin | Geomorphology of Ma'adim Vallis, Mars, and associated paleolake basins[END_REF]; [START_REF] Mangold | The origin and timing of fluvial activity at Eberswalde crater[END_REF]; [START_REF] Irwin | Paleohydrology of Eberswalde crater[END_REF]) may yield incorrect estimates of system duration because: 1) considering the power law-dependence of sediment flux on flow shear stress, a simplistic assumption can be wrong by several orders of magnitude, while physics-based methods are always within two orders of magnitude even with a large uncertainty of original grain size; 2) sediment concentration values applied by previous authors are relatively high, and imply unrealistic hyperconcentrated flows (considering the lowgradient slopes and probable hinterland conditions). Morphometry, valley topography, discharge rates of channels and their stratigraphic relationships with lithological units of origins other than the fluvial and lacustrine are investigated in a companion paper (Mangold et al. companion paper).

Jezero crater context

Jezero is a 50-km-diameter crater with a volume of 463 km 3 . It is a shallow impact basin with a major channel system cutting its western rim. The crater contains a large irregular fan deposit with a minor channel entering from the north [START_REF] Fassett | Fluvial sedimentary deposits on Mars: Ancient deltas in a crater lake in the Nili Fossae region[END_REF] (Figure 2a). We model the contribution to the crater infilling and delta formation of the western main valley and we also analyse the breach evolution at the eastern side of the crater. The northern valley does not contain evidence of fan deposition and is overall poorly developed. The crater is breached on the eastern side where there is a connected erosive channel network (Figure 1). The maximum elevation inside the crater basin which water may have reached is -2243m (Figure 2 -Profile R-S). Both the elevation of the delta front and the bottom of the breach is -2410m (Figure 2 -Profiles O-Q and R-S). All parameters (delta surface, volume gradient, eroded sediment valley volume, crater rim diameter, observed depth below rim peak, volume of the breach) were accurately delineated through our Digital Elevation Models (DEMs; see Methods section). The observed depth below the rim peak in this case matches with highest breaching point at -2243m below the Mars datum. Nevertheless, erosion of particular deltaic subenvironments (e.g., offshore delta toe-sets) have been eroded, with this missing geomorphic evidence potentially leading to underestimates of the total active duration for the Jezero system.

Methods

General approach

We use established hydrological and sediment transport predictors that are well-tested for terrestrial conditions and corrected for Martian gravity [START_REF] Kleinhans | Flow discharge and sediment transport models for estimating a minimum timescale of hydrological activity and channel and delta formation on Mars[END_REF]]. The principle of these formative time-scale is that flows require a certain minimum duration to work (i.e., remove, transport or deposit a known sediment volume), and must involve a certain volume of water that is sometimes constrained in non-overflowing lakes [START_REF] Kleinhans | Flow discharge and sediment transport models for estimating a minimum timescale of hydrological activity and channel and delta formation on Mars[END_REF][START_REF] Roda | Catastrophic ice lake collapse in Aram Chaos[END_REF]. While flow and sediment transport predictors undoubtedly simplify reality, the timescales of morphology development are rather well-constrained by this methodology. We use these predictors to estimate the duration of the Jezero fluvial system and we further test it on well constrained terrestrial (Holocene) cases, including the Wax Lake Delta and Lake Constance, to corroborate general validity (previously assessed by [START_REF] Duller | Reconstructing the timescale of a catastrophic fan-forming event on Earth using a Mars model[END_REF]). In order to obtain a range of possible predictions, we simulated a number of conditions for the most uncertain and sensitive variables (i.e., grain size and channel dimensions). Channel width and slope can be derived from visible images and digital elevation model (DEM) both for terrestrial (Wax Lake Delta and Lake Constance) and Martian (Jezero delta) examples. Channel depth can be directly measured on Earth and estimated on Mars using DEMs (with indications for original water depth ranging between preserved terrace height and bankfull flow conditions). Flow velocity was estimated using a friction law and a measured depth from altimetry data. Channel width, depth and slope were carefully measured using HiRISE DEM's for the Jezero delta (in addition to the Jezero breach because overor under-estimating depth not only affects derived discharge but also, indirectly, the velocity (through the roughness equation, see [START_REF] Kleinhans | Flow discharge and sediment transport models for estimating a minimum timescale of hydrological activity and channel and delta formation on Mars[END_REF]). Sediment flux is calculated using two methods: one assuming a bedload-dominated transport (with mostly rolling and saltating particles and limited energy), and another assuming a suspended load-dominated event (non-cohesive granular material). The sediment mobility, which depends on flow and sediment properties, is used to determine which of the two transport modes is most valid. In such a system, suspended/bedload transport ratio is far larger than one, so classic suspensiondominated sediment transport capacity predictors are employed corrected for Martian conditions [START_REF] Kleinhans | Flow discharge and sediment transport models for estimating a minimum timescale of hydrological activity and channel and delta formation on Mars[END_REF]. Models are based on steady and uniform flows equal to the water surface slope and channel bed surface slope.

In this paper we assume, based on morphological evidence (terraces), a constant (5 metre deep) bankfull discharge. While it is likely that some form of hydrograph is more representative of original flow conditions, the combination of the magnitude-frequency relation of flow and the nonlinearity of sediment transport renders the average sediment transport rate closer to the bankfull discharge than high-peak variance. This is due to the fact that the most extreme discharges (while transporting disproportionally more sediment) are quite rare, while the lowest discharges (while perhaps much more common than the bankfull condition) induce little sediment transport. These assumptions are in agreement with observations that the resulting channel dimensions' scale well to bankfull discharge as a measure for the channel forming discharge of fluvial channels [Leopold et al., 1964].

We assume transport of one sediment size whereas in reality there would have been a mixture. In particular, the Jezero delta is likely composed of a range of grain sizes, suggested by the interior of the delta, which shows variations between the supposed point bar, likely sand facies and the inverted channels above. We modelled scenarios for a range of grain sizes to bracket possible conditions (Table 1). Furthermore, many terrestrial deltas develop floodplains and toesets from fine material transported as washload, which is the part of the suspended load that is composed of particle sizes smaller than those found in appreciable quantities in the bed material. It is in near-permanent suspension and, therefore, is transported through the stream without deposition. This washload cannot be determined from the capacity predictors because it is supply-limited from the hinterland rather than capacity-limited by flow. Furthermore, using the water balance we calculate discharge for basin infilling, but we cannot measure the evaporation or pan-evaporation rate due to the lack of paleoclimate parameters on Mars, or the rate of outflow above -2243m elevation due to the subsequent erosion of the crater rim.

Data

Data used in this study include all types of visible images acquired for this basin, especially High

Case Study

We calculated minimum formation times for the Jezero deltas and basin infilling as a function of channel width, depth, and grain size. Formation times were based on predicted water and sediment discharges in comparison to observed lake volume and erosion and deposition volumes. In particular, for the Jezero delta (Mars) we used channel depth of 5 meters and two different values for channel width: 190m and 50m. The first value is measured using the HiRISE DEM and the second obtained from [START_REF] Fassett | Fluvial sedimentary deposits on Mars: Ancient deltas in a crater lake in the Nili Fossae region[END_REF]. This second width value is merely applied to demonstrate the sensitivity of the model to width parameter (and in doing so allows application of a more conservative estimate of channel width). The only absolute value of Martian grain size was obtained from alluvial fan material inside Gale Crater [START_REF] Williams | Martian Fluvial Conglomerates at Gale Crater[END_REF].

Evidently inappropriate for application at Jezero, in this study we present a range of different D50 values, from fine sand (0.25 mm) to pebble (20 mm). For the D90 value we used the relation 5*D50 [START_REF] Kleinhans | Palaeoflow reconstruction from fan delta morphology on Mars[END_REF]. Sediment and water discharges were derived from the river morphometric parameters using an online supplemental spreadsheet [START_REF] Kleinhans | Flow discharge and sediment transport models for estimating a minimum timescale of hydrological activity and channel and delta formation on Mars[END_REF] based on the equations listed in [START_REF] Kleinhans | Flow discharge and sediment transport models for estimating a minimum timescale of hydrological activity and channel and delta formation on Mars[END_REF]. We then discussed and estimated the longevity of deposition as a function of the inlet (Neretva Vallis) and outlet balance, making comparisons to prior works [START_REF] Fassett | Fluvial sedimentary deposits on Mars: Ancient deltas in a crater lake in the Nili Fossae region[END_REF][START_REF] Ehlmann | Clay minerals in delta deposits and organic preservation potential on Mars[END_REF][START_REF] Schon | An overfilled lacustrine system and progradational delta in Jezero crater, Mars: Implications for Noachian climate[END_REF][START_REF] Goudge | Assessing the mineralogy of the watershed and fan deposits of the Jezero crater paleolake system, Mars[END_REF].

With respect to previous Martian analogue studies applying the same model (e.g. [START_REF] Kleinhans | Flow discharge and sediment transport models for estimating a minimum timescale of hydrological activity and channel and delta formation on Mars[END_REF], [START_REF] Kleinhans | Palaeoflow reconstruction from fan delta morphology on Mars[END_REF]), our approach is more representative of actual deltaic deposition as we used CTX and HiRISE DEMs to measure width, depth, bedload transport, sediment slope failure of the Neretva Vallis and terrestrial analogues unaffected by tidal processes. This allows us to use more accurate input parameters than those applied in previous studies (e.g. [START_REF] Kleinhans | Flow discharge and sediment transport models for estimating a minimum timescale of hydrological activity and channel and delta formation on Mars[END_REF]; [START_REF] Kleinhans | Palaeoflow reconstruction from fan delta morphology on Mars[END_REF]), particularly for river depth, which is one of the most important parameters which affects the sediment transport process and is difficult to constrain on the basis of erosional channel topography alone [START_REF] Salese | Valley formation by groundwater seepage, pressurized groundwater outbursts and crater-lake overflow in flume experiments with implications for Mars[END_REF]. High-resolution topographic data also enable more accurate geological and hydrological analyses, yielding predictions closer to those of typical rivers on Earth.

Model Validation

Model validation was already presented in [START_REF] Kleinhans | Flow discharge and sediment transport models for estimating a minimum timescale of hydrological activity and channel and delta formation on Mars[END_REF] for terrestrial and experimental systems including the Mackenzie and Rhine (NL) [START_REF] Berendsen | Late Weichselian and Holocene palaeogeography of the Rhine-Meuse delta, the Netherlands[END_REF][START_REF] Hill | The Mackenzie Delta: Sedimentary processes and facies of a high-latitude, fine-grained delta[END_REF]. The methodology and the equation introduced by [START_REF] Kleinhans | Flow discharge and sediment transport models for estimating a minimum timescale of hydrological activity and channel and delta formation on Mars[END_REF] was then implemented by [START_REF] Hoke | Formation timescales of large Martian valley networks[END_REF] to assess timescales of formation for seven of the largest ancient Martian valley networks. The model was also used on Mars by [START_REF] Mangold | The origin and timing of fluvial activity at Eberswalde crater[END_REF] and [START_REF] Adeli | Amazonian-aged fluvial system and associated ice-related features in Terra Cimmeria[END_REF], with an aim of understanding the timescale of evolution of the Eberswalde delta and the fluvial system in Terra Cimmeria respectively. de [START_REF] De Villiers | Experimental delta formation in crater lakes and implications for interpretation of Martian deltas[END_REF] tested the method directly for experimental crater lake deltas and Marra et al. [2014] showed it was additionally applicable to erosive experimental systems, and was further supported by a uniquely well-constrained erosional valley case on Mars: the Aram Chaos side valley to Ares Valles [START_REF] Roda | Catastrophic ice lake collapse in Aram Chaos[END_REF]. More recently, the above mentioned model used was applied to reconstruct the timescale of formation of the several Martian delta [START_REF] Kleinhans | Flow discharge and sediment transport models for estimating a minimum timescale of hydrological activity and channel and delta formation on Mars[END_REF][START_REF] Kleinhans | Palaeoflow reconstruction from fan delta morphology on Mars[END_REF] and has already been validated on Earth by [START_REF] Duller | Reconstructing the timescale of a catastrophic fan-forming event on Earth using a Mars model[END_REF], which applied it to reconstruct the timescales of the 1918 catastrophically formed fan in southern Iceland [START_REF] Duller | Architectural analysis of a volcaniclastic jökulhlaup deposit, southern Iceland: sedimentary evidence for supercritical flow[END_REF][START_REF] Duller | Landscape reaction, response, and recovery following the catastrophic 1918 Katla jökulhlaup, southern Iceland[END_REF]. The real known duration of this event was 6-10 hours and the timescale predicted by their modelling is between 2 and 17 hours. Further, to strengthen their test they modelled several hypothetical Martian scenarios applied on the Iceland fan to illustrate potential limitations of using the final topography to estimate flow dimensions and

Accepted paper in Astrobiology -Vol. 8 (2020)

Salese et al., 2020b using a typical value of grain size distribution recorded from Mars missions (as we did in this work). The resulting fan formation timescale was of 0.1 to 40 h (sand system) and 25 to 700 h (gravel system), within two orders of magnitude of the real timescale. If sedimentary information is available and used in conjunction with traditional topographic analyses, then the uncertainty in calculating fan formation and hydrologic timescales both on Earth and Mars can be greatly reduced.

As a further independent test of the method we modelled two recent river-dominated, terrestrial fan deltas for which all the input data are available (including precise start dates of delta formation): the Wax Lake delta (Louisiana, USA) and the Lake Constance delta (Austria). Input parameters, such as delta bathymetry, channel width and depth, mouth discharge, are all available from existing literature [START_REF] Muller | The new Rhine delta in Lake Constance[END_REF]Roberts et al., 1997;[START_REF] Roberts | An embryonic major delta lobe: A new generation of delta studies in the Atchafalaya-Wax Lake Delta system[END_REF][START_REF] Wellner | Jet-plume depositional bodies-the primary building blocks of Wax Lake Delta[END_REF][START_REF] Shaw | The importance of erosion in distributary channel network growth[END_REF]Wessels et al., 2015a;Wessels et al., 2015b;[START_REF] Wessels | Tiefenschärfe-Hochauflösende Vermessung Bodensee[END_REF]. The Wax Lake Delta is located in an open but sheltered basin with a microtidal range, whereas the New Rhine Delta is located in a closed basin (lake).

3 Terrestrial delta modelling 3.1 Wax Lake delta

Sedimentological backgorund

The 25-km-long man-made Wax Lake Outlet was completed in October 1941 [START_REF] Roberts | An embryonic major delta lobe: A new generation of delta studies in the Atchafalaya-Wax Lake Delta system[END_REF] and provides the opportunity to model a delta formed by a well-developed river with a precisely known initiation date. Wax Lake Outlet extends south from Six Mile Lake, across the Teche ridge, into Atchafalaya Bay.

Original bottom depth was ~13.7 m below mean sea level and the width was less than 12.2 m [START_REF] Latimer | The Atchafalaya River Study: A Report Based Upon Engineering and Geological Studies of the Enlargement of Old and Atchafalaya Rivers[END_REF]. The Wax Lake Delta (WLD), at its mouth, is a classic river-dominated delta [START_REF] Wellner | Jet-plume depositional bodies-the primary building blocks of Wax Lake Delta[END_REF][START_REF] Falcini | A potential vorticity theory for the formation of elongate channels in river deltas and lakes[END_REF][START_REF] Edmonds | Topset-dominated deltas: A new model for river delta stratigraphy[END_REF] and its form is negligibly affected by the small mean tidal range (0.4 m) and wave climate (0.5 m maximum monthly wave height; [START_REF] Syvitski | The morphodynamics of deltas and their distributary channels[END_REF]). A delta top slope of 0.0085 was measured from detailed bathymetric reports [START_REF] Shaw | Airborne radar imaging of subaqueous channel evolution in Wax Lake Delta, Louisiana, USA[END_REF]).

Researchers have monitored the WLD sedimentation since [START_REF] Morgan | Occurrence and development of mudflats along the western Louisiana coastRep[END_REF] and [START_REF] Morgan | Changes in the Louisiana shoreline[END_REF]. [START_REF] Roberts | Evolution of Sand-Dominant Subaerial Phase, Atchafalaya Delta, Louisiana[END_REF] suggest that by the late 1950s accommodation space within the Atchafalaya Basin was rapidly decreasing. The amount of sand-sized sediment reaching the bay consequently increased.

Prior to 1960, the majority of sediment reaching Atchafalaya Bay was comprised silt-sized (or finer) grains, with most of this material bypassing the bay and depositing seaward of the Point au Fer Shell reef [START_REF] Cratsley | Recent Deltaic Sedimentation, Atchafalaya Bay, Louisiana[END_REF]. After 1960, sand-prone, subaqueous bayhead deltas formed at the mouths of the Lower Atchafalaya River and Wax Lake Outlet [START_REF] Roberts | Evolution of Sand-Dominant Subaerial Phase, Atchafalaya Delta, Louisiana[END_REF].

The WLD, according to [START_REF] Cratsley | Recent Deltaic Sedimentation, Atchafalaya Bay, Louisiana[END_REF], experienced three different phases of sedimentation: (1) deposition of prodelta clays and silty clays from the middle of the 19th century to about 1952; (2) the initiation of coarser-grained deposition around 1952 (mostly silts), and the development of a lobate form between 1952-1962(reported in Shlemon [1975] ] as well); and (3) introduction of bedload sands in the early 1970s. [START_REF] Cratsley | Recent Deltaic Sedimentation, Atchafalaya Bay, Louisiana[END_REF] also showed that thin silt and sand layers were found in the subaqueous delta and that the bulk of the sediments were fine grained and not sand rich.

Deposition between 1962-1972, and the high flood years of 1973-75, which stand out as a period of abnormally high sediment flux, changed bay framework significantly, with the WLD becoming subaerial. Mean discharge and total sediment discharge measured at Simmesport for the years 1951-1989 are summarized in [START_REF] Roberts | An embryonic major delta lobe: A new generation of delta studies in the Atchafalaya-Wax Lake Delta system[END_REF]. [START_REF] Majersky | Facies development in the Wax Lake Outlet delta: Present and future trends[END_REF] shows estimates of delta growth based on a terrain model using both bathymetry and land elevation data collected by the US Army Corps of Engineers.

Between 1989 and 1994, the Wax Lake delta developed at a rate of 3.0 km 2 y -1 [START_REF] Majersky | Facies development in the Wax Lake Outlet delta: Present and future trends[END_REF]. These values clearly show the rapidly expanding nature of this delta and compare favourably with delta growth predicted earlier by [START_REF] Wells | Development of the Atchafalaya River deltas: generic analysis[END_REF] from a genetic growth analysis based on the historic behaviour of subdeltas in the modern Mississippi delta, indicating that conditions of flow and sediment transport are within the expected range for this environment.

Modelling

The sediment timescales and delta formation were calculated for conditions shown within the spreadsheet in the supplementary material. We modelled sediment deposition (delta) between 1952 and 1989 based on variables listed by [START_REF] Roberts | An embryonic major delta lobe: A new generation of delta studies in the Atchafalaya-Wax Lake Delta system[END_REF]. Grain size value (D50 and D90) are taken from [START_REF] Shaw | The importance of erosion in distributary channel network growth[END_REF]. We divided the sedimentary history of this delta in two different periods: silt-dominated before 1961 and sand-dominated post 1961. We found that during the silt-dominated period the delta formed in two years (expected due to the higher mobility of silt). In the sand-dominated period it formed in almost thirteen years.

For this case the calculated timescale is about two factors different from the true timescale (predicted using the spreadsheet: 15 years; reality: 37 years), which is acceptable given all the simplifications and the possible loss of sediment through wave reworking.

New Rhine delta

Sedimentological background

In 1900 (by a treaty between Austria and Switzerland) the position of the mouth of the Alpine Rhine (Alpenrhein), the upstream reach of the river Rhine, was shifted to a new artificial bed that flows into Fussach Bay (close to the town of Fussach in Austria) in the eastern part of Lake Constance [START_REF] Muller | The new Rhine delta in Lake Constance[END_REF]. Since then the Rhine began to build its present delta (the New Rhine delta) into Fussach Bay [START_REF] Muller | The new Rhine delta in Lake Constance[END_REF]. This provides the opportunity to model a delta formed by a well-developed river with an exactly known initiation date.

The Rhine flows into the Lake Constance at an elevation of 396m above mean sea level. The average gradient between the source of the Vorderrhein and the mouth of the Rhine in Lake Constance (over a total length of about 170 km) is 0.0115. Lake Constance, with a volume of 49.4 km 3 and a maximum depth of 252 m is the natural settling basin for the Rhine, which drains a total area of 6122 km 2 . Over 90% of the coarse sediment transported by the Rhine is deposited in Lake Constance and only a very small percentage leaves Lake Constance to be carried downstream to the North Sea [START_REF] Muller | The new Rhine delta in Lake Constance[END_REF] (making the lake an efficient sediment trap). Fine sediments compose the delta while the upstream river is gravel-dominated, and the New Rhine transports 85.5% of the suspended load and 1.2% of the bed load [START_REF] Muller | The new Rhine delta in Lake Constance[END_REF].

The average water flow discharge from 1931 to 1960 was 224 m 3 /s; the average suspended load was 349.5 cm 3 /m 3 , or 454.1 g/m 3 . Thus, the river supplies an average annual suspended load of 2.571 million m 3 , which is deposited into Lake Constance. The amount of bedload (pebbles transported by rolling that constitute less than 2% of the material transported in suspension [START_REF] Muller | The new Rhine delta in Lake Constance[END_REF]) is approximately 40,000 m 3 per year, which is entirely contained within the delta. Seasonal deviations from the average are extreme; during the peak of the thaw period in the Alps, water flow can be ten times greater and the amount of suspended load can increase by more than twenty times. Through extension of the delta out into Fussach Bay, the area of Lake Constance has decreased by approximately 1.2 km 2 in 50 years, and the average depth of Fussach Bay has decreased from 17.2 to 4.06 m. The New Rhine Delta is composed primarily of silty sands; clean sands and pebble deposits are extremely rare. The average grain size decreases from top-set beds (silty sands) through foreset beds (silty sands and silt) to bottom-set beds (silt to clayey silt which grades into silty clay away from the delta).

Modelling

We modelled sediment deposition of the New Rhine Delta for the period between 1900 and 2015. We obtained the morphological and sedimentological parameters from previous studies on Lake Constance and the Alpine Rhine such as [START_REF] Muller | The new Rhine delta in Lake Constance[END_REF]; Wessels et al. [2015a]; Wessels et al. [2015b]; [START_REF] Adami | Multidecadal dynamics of alternate bars in the Alpine Rhine River[END_REF]; [START_REF] Wessels | Tiefenschärfe-Hochauflösende Vermessung Bodensee[END_REF]. In particular, the New Rhine delta volume was calculated using the 3m resolution Lake Constance bathymetry constructed by Wessels et al. [2015b], with ArcGIS used for volume calculation. The New Rhine delta is a Gilbert-type [START_REF] Gilbert | Lake Bonneville[END_REF]. Grain size value (D50 and D90) are taken from [START_REF] Tockner | Rivers of Europe[END_REF] and [START_REF] Adami | Multidecadal dynamics of alternate bars in the Alpine Rhine River[END_REF]. Input parameters are shown in paragraph 3.2.1 and in the spreadsheet provided in the supplementary material. For the New Rhine Delta, the calculated timescale are again about two factors different from the true timescale (predicted from modelling: 50 years; reality: 115 years), again within the range of expected uncertainty of sediment transport predictions.

Jezero modeling

The morphological elements suggest several possible scenarios of hydrological activity that are discussed in the Mangold et al. companion paper. Based on that discussion, we split the Neretva Vallis evolution history in two phases: phase 1 concerns the basin infilling just after the breaching of the western rim but before delta formation; phase 2 concerns the development of the delta. Herein we concentrate on the second phase because the first phase is constrained from detailed study by a cover in the crater of presumably volcanic origin. The aim of the modelling is therefore to constrain the period of hydrological activity from: a) observed morphology; and b) flow and sediment transport predictors in the entire active system of inflow, lake and outflow. All calculations are based on observed volumes and on slopes, widths, and lengths of feeder channels, channels on the delta and the eastern crater rim breach, as well as on a range of likely grain sizes.

The results of the simulation using relevant Neretva Vallis morphological and sedimentological parameters are shown in Table 1. Considering the volume detailed in Table 1, initially we find out its suspected filling times (Phase 1) followed a subsequent phase (Phase 2) of delta-formation [START_REF] Kraal | Martian stepped-delta formation by rapid water release[END_REF][START_REF] Kleinhans | Palaeoflow reconstruction from fan delta morphology on Mars[END_REF] Finally, the breach evolution is investigated.

Phase 1 -Basin infilling (Full bank + Low Bank)

We modelled two different scenarios for the Jezero basin infilling phase: one considering Neretva Vallis width and depth calculated in this work and another considering a more conservative (narrower) value of channel width calculated by previous authors [START_REF] Fassett | Fluvial sedimentary deposits on Mars: Ancient deltas in a crater lake in the Nili Fossae region[END_REF][START_REF] Ehlmann | Clay minerals in delta deposits and organic preservation potential on Mars[END_REF]. In this work we estimated Neretva Vallis width of 190m, channel depth of 5m, channel slope of 0.0097 and we considered various grain sizes: from pebble to fine sand (Table 1). The time required to fill the basin, considering a continuous flow within a 190m width and 5m-depth channel, varies between 6,55 and 6,75 years depending on the grainsize considered.

Phase 2 -Delta formation

In this section we show a representative simulation using relevant Jezero morphological and sedimentological parameters listed in Table 1. In particular we consider a 5km 3 delta volume as calculated in this study and as suggested by previous researchers [START_REF] Fassett | Fluvial sedimentary deposits on Mars: Ancient deltas in a crater lake in the Nili Fossae region[END_REF][START_REF] Ehlmann | Clay minerals in delta deposits and organic preservation potential on Mars[END_REF]. The eroded volume of the final and steepened part of the Neretva Vallis is of about 56 km 3 (Mangold et al., companion paper), which is far greater than the sediment stored in the fan. We present two main scenarios that differ in channel width value. We again considered different D50 grain size from 0.25 to 20 mm (from fine sand to pebble). Morphological parameters used for calculations in both scenarios are listed in Table 1. In that phase grain size plays a major role for the delta timing: in fact calculated timescale vary by a factor of fifty depending on whether pebble or fine sands are input. All grain sizes reported hitherto are from distal landing sites (overview in [START_REF] Kleinhans | Flow discharge and sediment transport models for estimating a minimum timescale of hydrological activity and channel and delta formation on Mars[END_REF]) and to date the only genuine measured grainsize in a fluvio-lacustrine environment on Mars are those reported in [START_REF] Williams | Martian Fluvial Conglomerates at Gale Crater[END_REF] measured by the Curiosity rover. We decided to test this model with different grain size in order to show its sensitivity (see spreadsheet in the supplementary material). In the absence of field data and based on the few indications that we have had from the Curiosity rover, in what appears to be a more alluvial environment than fluvial, we have preferred to adopt a conservative approach and to use the D50 values between 8 and 14 mm to calculate the minimum delta formation time. Calculated timescales for the Jezero delta with the above mentioned parameters range between 90 and 145 years. By adopting a more conservative channel width value (50 m), estimated by [START_REF] Fassett | Fluvial sedimentary deposits on Mars: Ancient deltas in a crater lake in the Nili Fossae region[END_REF], and keeping the other parameters unchanged: the delta formation time varies between 330 and 550 years.

A 90-550 years for a fan-delta such as Jezero crater does not seem to be an underestimate when compared to the terrestrial case studies. For instance, A progradation rate of 15m/year has been measured by [START_REF] Muller | The new Rhine delta in Lake Constance[END_REF] in the Rhine delta at Constance lake. For the 5 km long delta at Jezero, a duration of 100-500 years, would correspond to a progradation of 10 to 50 m/year, consistent with the example at Constance Lake.

There is evidence that the Jezero delta underwent intense erosion at the delta front [START_REF] Goudge | Assessing the mineralogy of the watershed and fan deposits of the Jezero crater paleolake system, Mars[END_REF], which could mean a longer formation time if we consider that the delta extended to the areas where today we only see the remnants of a possible ancient delta front. [START_REF] Goudge | Assessing the mineralogy of the watershed and fan deposits of the Jezero crater paleolake system, Mars[END_REF] estimates a possible eroded delta to have originally been 7.8 km 3 in volume. In this study were prefer a more conservative estimate and modelled a delta 15km 3 in volume, which represents the upper limit for the "delta volume" parameter in our modelling. For D50 values between 8 and 14 mm (pebble), keeping unchanged the morphological and sedimentological parameters and modifying only the delta volume (from 5 km3 (real actual delta) to 15km3 (delta hypothesized on the basis of the remnants)), calculated timescale for the Jezero delta then range between 260 and 430 years, which is about three times longer than in the modelling of the deposits present today in the crater. If the calculation is conducted with the more conservative width (50 m), then the delta formation time ranges between 1000 and 1640 years.

Outlet analyses

In this section we establish whether the processes inherent to the Jezero outlet can further constrain the timescale of hydrological activity. It has previously been established that the sedimentary timescale of fan formation is far larger than the minimum hydrological timescale for the filling of the lake to the topmost level of the delta [START_REF] Kleinhans | Flow discharge and sediment transport models for estimating a minimum timescale of hydrological activity and channel and delta formation on Mars[END_REF]; [START_REF] Kleinhans | Palaeoflow reconstruction from fan delta morphology on Mars[END_REF]). The topographic gap between the top of the breach and the delta top makes it clear that at some point the lake overflowed. It is possible that lake overflow processes have the ability to do large amounts of erosional and depositional work in a short period of time, especially during the initial phases of dam breaching [START_REF] Bretz | The Lake Missoula floods and the channeled scabland[END_REF][START_REF] O'connor | Magnitudes and implications of peak discharges from glacial Lake Missoula[END_REF][START_REF] O'connor | Floods from natural rock-material dams[END_REF][START_REF] Roda | Catastrophic ice lake collapse in Aram Chaos[END_REF]. On Mars these overflow processes are thought to be characterized by very high discharges and rapid outlet canyon incision [START_REF] Coleman | Hydrographs of a Martian flood from a breached crater lake, with insights about flow calculations, channel erosion rates, and chasma growth[END_REF][START_REF] Roda | Catastrophic ice lake collapse in Aram Chaos[END_REF][START_REF] Coleman | Hydrographs of a Martian flood from the breach of Galilaei Crater[END_REF], including incision from multiple lake overflow floods (e.g., [START_REF] Salese | Hydrological and sedimentary analyses of well-preserved paleofluvial-paleolacustrine systems at Moa Valles, Mars[END_REF]; [START_REF] Goudge | Characterizing the record of paleolake outlet canyon incision on Mars paper presented at Lunar and Planetary Science Conference[END_REF]) and incision from long-term outflow (e.g., [START_REF] Holo | Incision of the Jezero Crater Outflow Channel by Fluvial Sediment Transport[END_REF]).

The volume of sediment eroded from the rim was estimated using an ArcGIS 10.6 toolbox. This yields an average entire volume of 3 km 3 . From the adjacent rim height, it can be estimated that the pre-breach paleolake level reached at least -2243m in elevation and the post-breach water surface elevation was estimated as the base of the breach at -2410m in elevation, which is the same elevation of the Jezero delta top (Figure 2). The water volume drained from the lake during progressive breach incisions corresponding to the volume contained between -2243 and -2410 meters. This yields a minimum water volume of 238 km 3 that must have been expelled from the crater lake during breaching.

Salese et al., 2020b

The flow flux out of the Jezero crater was likely sediment-poor, ponding water so that the sediment transport capacity of the flow was entirely available for erosion of the channels. This evident water scour is the reverse of the deposition of crater lake deltas from a sediment laden flow that enters a crater lake [START_REF] Kleinhans | Flow discharge and sediment transport models for estimating a minimum timescale of hydrological activity and channel and delta formation on Mars[END_REF][START_REF] Kraal | Martian stepped-delta formation by rapid water release[END_REF] as the gradient of sediment transport integrated over time is the same as the total displaced volume of sediment, allowing the calculation of the timescale [Ts] of formation directly from the volume of displaced sediment [V = 3 km 3 ] and the sediment transport rate [Qs] (corrected for porosity) [START_REF] Kleinhans | Flow discharge and sediment transport models for estimating a minimum timescale of hydrological activity and channel and delta formation on Mars[END_REF].

The sediment transport rate is calculated from the flow flux through the channel. The following steps were applied to calculate flow flux: a) width, flow depth and gradient of the channel are estimated from HRSC topography (Figure 2a); b) throughout morphological analyses of the breach we identified three main terraces that led us to hypothesize at least three different phases within the same event (Figure 2). These three phases differ in channel depth and width. Furthermore, we model two different breach slope scenarios: 1) considering the slope (0.02) of the actual outlet channel (Figure 2 -Profile E-F); 2) considering a hypothetical pre-breach slope (0.05), estimated through the average of the rim slopes to the north and south of the breach. In both cases are steep breaches and in this situation the flow is usually critical (Froude number = 1). The numerical modelling of crater rim behaviour is subject to great uncertainty due to the lack of knowledge about the grain-size distribution of materials transported and whether the eroding crust was bedrock, unconsolidated or comprised weakly cemented sediment. For this reason we assume a D50 rim grain size of 0.1 meters and D90 of 0.6 m, same values as used for the breach in Aram Chaos from [START_REF] Roda | Catastrophic ice lake collapse in Aram Chaos[END_REF]. Maximum flow depth is estimated from terraces heights, with hydraulic roughness subsequently calculated.

The water depth inferred from terraces (h) is within the expected range based on the resulting width-depth ratio of the flow (about 20 for narrow terrestrial gravel bed rivers of similar slope) and results in reasonable sediment mobility (expressed as the dimensionless Shields number, Table 2) [START_REF] Kleinhans | Flow discharge and sediment transport models for estimating a minimum timescale of hydrological activity and channel and delta formation on Mars[END_REF]. We assumed near-critical flow (Froude number around 0.9) and use this to confine the water depth. In fact, on such steep slopes flow is typically critical, resulting in very efficient sediment transport.

We modelled the outlet with parameter settings and choices for independent variables (depth, slope) and the friction relation for these steep slopes. The resulting erosion timescale was always about three orders of magnitude larger than the time needed for the water to flow out of the crater. Unlike the crater breaching case of [START_REF] Roda | Catastrophic ice lake collapse in Aram Chaos[END_REF], where the authors apply the same method that we use in this work, there was no reasonable combination of slope, grain size and friction for which the timescales of water evacuation and breach erosion overlapped. This means that any outflow with erosive power on the crater rim would have emptied the necessary disc of water faster than that flow could have deepened the breach. In order for the breaching and the delta formation to be coeval, a much larger discharge would be required from the hinterland (in contradiction with outflow calculations).

Outflow evidence's implications

The implication of the timescale calculations is that the breach could not have formed from one overflowing event with the volume of the disc of water, and instead a much longer process of lake activity and spilling is required. This means that the breach formation was not limited by the amount of water within the lake, but only by the flow discharge from the Neretva Vallis. Furthermore, either the discharge must have been very large to form the breach, or flow must have been acting on the crater wall for a sustained period of time.

On the basis of process knowledge for terrestrial breaches certain constraints are possible. The large dimensions of the lake (58 km in diameter) mean it is possible to ignore the effects of both possible floods or constant flow, as these are buffered by incoming floods and cause limited spilling only (with the exception of some seiches during storms and potential small-scale tsunamis in the aftermath of the crater wall collapse). Regardless of the details, this process must have taken a substantial period of time, or such a large discharge as to be incompatible with the late-stage channel and delta. The breach could not have formed in the same event that formed the present Jezero fan deposit, but must have formed in earlier events associated with the entire channel system to the west (before emplacement of the volcanic layer on the crater floor) [Mangold et al., companion paper]. During the time needed for delta formation, erosion of the breach is negligible. We therefore assume constant breach elevation and constant lake level during the delta-forming event.

These findings present certain contradictions to previous work [START_REF] Fassett | Hydrological modeling of the Jezero crater outlet-forming flood[END_REF] suggest that most of the flood-related geomorphic work happened within two weeks while [START_REF] Holo | Incision of the Jezero Crater Outflow Channel by Fluvial Sediment Transport[END_REF] suggest that the observed outlet can be carved in decades to centuries of progressive bed load as the delta forming flows filled the lake. Both estimates are problematic given the combination of outflow and sediment transport capacity of that outflow. The Aram Chaos side-channel case, conversely, did likely form in one outflow event because a much larger volume of water became available and the crater rim was far steeper [START_REF] Roda | Catastrophic ice lake collapse in Aram Chaos[END_REF]. All such calculations depend on the nature of the sediment, unconsolidated rock or impact regolith that is eroded (see page 112 on rock and 114 on breaching in [START_REF] Salese | Valley formation by groundwater seepage, pressurized groundwater outbursts and crater-lake overflow in flume experiments with implications for Mars[END_REF]).

Paleo-evaporation rate on Mars and intermittency

The timescale of lake filling would have likely been additionally affected by evaporation. On Earth 57% of all precipitations on land evaporate and in warm and dry climates up to 96% of the yearly precipitation may evaporate [START_REF] Hendriks | Introduction to physical hydrology[END_REF]. Evaporation is therefore a fundamental parameter in evaluating fluvio-lacustrine lifetimes. On the one hand, high evaporation rates would empty the lake faster following the shutdown of the feeder system. Conversely, given the constraint that the lake must have filled up to the lowest level of the breach in Jezero, evaporation would mean a larger filling timescale with otherwise matching discharge. However, evaporation strongly depends on weather and any estimates here would therefore invoke significant speculation. There are two widely used methods of estimating evaporation rate for terrestrial systems: 1) pan evaporation; and 2) the more complex but realistic Penman-Monteith equation [START_REF] Hendriks | Introduction to physical hydrology[END_REF].

Pan evaporation is common practice in water resource management for dams and reservoirs. A pan coefficient is applied to measured pan evaporation rates in order to derive equivalent evaporation from the water storage of interest. This methodology is widely used due to its simplicity though results in a number of limitations [START_REF] Kohler | Evaporation from pans and lakes: US weather bureau research paper 38[END_REF][START_REF] Shuttleworth | Evaporation[END_REF][START_REF] Lowe | Evaporation from water supply reservoirs: An assessment of uncertainty[END_REF]. As stated by [START_REF] Lowe | Evaporation from water supply reservoirs: An assessment of uncertainty[END_REF], the 95% probability intervals surrounding the estimates of reservoir evaporation on Earth are as large as ±40% of the best estimate using the pan method. Furthermore, one of the main uncertainty of this method is due to pan coefficient itself [START_REF] Lowe | Evaporation from water supply reservoirs: An assessment of uncertainty[END_REF] which is between 0 and 1, 0 in the case of warm climate and low humidity and 1 for cold climate and high humidity [START_REF] Hendriks | Introduction to physical hydrology[END_REF]. The evaporation from a pan is a good indicator of the evaporation from the surrounding environment only when land-surface moisture is in ample supply [START_REF] Brutsaert | The surface roughness parameterization[END_REF][START_REF] Brutsaert | Hydrologic cycle explains the evaporation paradox[END_REF]].

More complex equations, such as for example Penman-Monteith equation, are frequently used to estimate evaporation on Earth's open lakes, taking into account the saturation of air flowing over the lake which act to reduce evaporation. This approach, while more applicable than pan evaporation on a large lake such as Jezero, requires additional knowledge of several environmental parameters. The main and most important input parameters of these equations are: wind speed, air temperature, atmosphere pressure, incoming short wave radiation, relative humidity, and atmosphere density. While estimates in the literature suggest that evaporation modifies the hydrological timescale as much as grainsize changes the sedimentary timescale [START_REF] Wallace | Evaporation of Ice in Planetary-Atmospheres -Ice-Covered Rivers on Mars[END_REF][START_REF] Salese | On laboratory simulation and the temperature dependence of the evaporation rate of brine on Mars[END_REF][START_REF] Sears | On laboratory simulation and the evaporation rate of water on Mars[END_REF], the necessity for variables that are hard to estimate for an early Mars makes it impossible to estimate paleo-evaporation rates with any degree of confidence.

For example, [START_REF] Irwin | Paleohydrology of Eberswalde crater[END_REF] estimated evaporation rate for the Eberswalde Martian case using pan evaporation. They used a pan coefficient of 0.7 and today's parameters (e.g., Solar insolation) to estimate a possible evaporation rate for the Martian paleoenvironment. This resulted in an evaporation rate range of 0.1-1 m/year. While this is the best estimate possible with what is known for Martian conditions, the evaporation rate is based on present meteo-environmental parameters rather than those during the activity of the Eberswalde system, and the obtained value (even for Earth) is large. Note that the greater the temporal duration of a system (i.e., lake), the more significant the value of evaporation rate is for long term predictions.

If the timescale of delta formation was only of the order of years or decades, even such high evaporation would not have significantly modified the order of magnitude of the timescale. We encounter the same issue for the intermittency: if the lake formed in a catastrophic event then intermittency is not as relevant, but if it took centuries then it is likely discharge fluctuated over seasonal or longer time periods.

Interpretation

Based on the evidence found here and in the companion work of Mangold et al., we can deduce that the long Neretva Vallis (200 km) that drained within the Jezero basin was not entirely related with the present delta.

The volume of the missing sediments (56km 3 ) within the entire Neretva Vallis is ten times larger than the present delta volume (5km 3 ), even if the remnants are taken into account. The fact that the volume of the delta is almost the same as the eroded sediment volume from the breach is coincidental because the mechanisms cannot be causally linked. Much more likely is the hypothesis that the current delta is linked to flow event(s) which occurred in the relatively late stage, large valley activity, as well as reliably witnessed by the presence of knickpoints within the terminal part of the Neretva Vallis (Mangold et al., companion paper). Other fluvial sedimentological evidence (e.g., bars) are also evident upstream. We consider it most probable that flow in the Neretva Vallis would have been shallow, wide and relatively weak. This is consistent with the findings of the companion paper (Mangold et al., companion paper): local sediment deposits forming bars but constrained in depth by the rocky valley floor, and a tendency to braid when the valley is wide enough. Furthermore, large discontinuities in channel floor slope suggest that this is not an alluviated equilibrium channel with erodible sediment in the floor, but a bedrock valley formed by an earlier and likely much larger fluvial epoch. The absence of ubiquitous sediment on the valley floor is additional evidence that the sediment stored in the delta was either sourced from the sparse cover of loose sediment in the upstream valley system, or from erosion of the valley floor immediately upstream of the crater lake.

In Figure 3 we summarize a hypothetical evolution of the Jezero basin. The crater was initially filled by the upstream 200 km-long Neretva Vallis, acting as a closed basin. At some point the eastern rim of the crater breached in at least three episodes from -2243m until -2410m. From then on, the crater floor transformed to an open lake filled for a while with sediment-poor water and then started the growth of the Jezero deposits.

According to the evidence of the sediment at the mouth of the Neretva Vallis, initially and briefly as an alluvial fan and later as a delta. Since the top level of the delta is precisely at the level of the breach, the lake probably overflowed and the delta continued to develop for a while longer. Due to the overflowing condition we cannot constrain an optimum timescale for the delta formation by the maximum water level timescale (as is possible for stepped fans). We are more uncertain about the timescale so we apply a range of grain sizes for a number of channel widths, the main uncertainty in our calculation. We approximate grain size within a logical range [START_REF] Sklar | A mechanistic model for river incision into bedrock by saltating bed load[END_REF][START_REF] Parker | Physical basis for quasi-universal relations describing bankfull hydraulic geometry of single-thread gravel bed rivers[END_REF]). We have estimates of water depth from the present observations of bar heights (that would have been submerged only during flood) and terraces. Our finding, that the sediment timescale is larger than the water timescale is in agreement with our hypothesis so we end up with the above uncertainty as the range. Then, finally, the lake remains wet while the upstream feeder system has shut off until the water has percolated/seeped out and/or entirely evaporated. Even if we

Accepted paper in Astrobiology -Vol. 8 (2020)

Salese et al., 2020b
assume a high evaporation rate of 0.1-1 m per year [Hendricks, 2010], that still adds to the timescale calculation.

Astrobiological implications for rover exploration

The deposits within the MSL 2020 landing site could have preserved biosignatures to investigate on the spot and sample for a return mission. On Earth, microorganisms have pervaded all wet, surface and subsurface environments. In subaqueous environments, microorganisms commonly exist dually in the water column and in sediment pore spaces or as attached biofilms. In addition, sedimentary processing such as hydrodynamic sorting, as evident in terrestrial fluvial and deltaic environments, may concentrate biologically derived carbonaceous particles into fine-grained, organic-rich horizons in sedimentary beds. Among sedimentary systems on Earth, lacustrine (perennial) and deltaic systems have been estimated as high for supporting organic matter (OM) formation, concentration, and preservation [Summons et al., 2011]. This is primarily because of the diverse and prolific microbial life that exist in lakes and the existence of hydraulic gradients across both deltaic upper plains and slopes: organic material sequestered preferentially by the fine-grained sediments of the distal flood plains, and of the bottomsets at the toe of the delta foreslope. By analogy, the deltaic-lacustrine closed basin of western Jezero affords a definite potential for retention of transported and in situ organics and environments that hydrodynamically concentrate organics. Assuming that the lake and delta formed by the processes and on a timescale of decades to millennia (as in our favored interpretation), the following astrobiological implications emerge. In the first place, the gravelly delta itself is an environment poorly favorable to life, because of the high energy in the flow and on the bed. In contrast, it is quite likely that fine-grained sediments were deposited downstream of the delta foreset in bottomsets and on the lake floor. The existence of such fine-grained sediments has been inferred on Mars from lander data, from missing sediment volume in perfectly sediment-trapping crater lakes, and from the ubiquitous fine cover north of the dichotomy [START_REF] Kleinhans | Flow discharge and sediment transport models for estimating a minimum timescale of hydrological activity and channel and delta formation on Mars[END_REF][START_REF] Hauber | Asynchronous formation of Hesperian and Amazonian-aged deltas on Mars and implications for climate[END_REF]. Such fine-grained sediment potentially provided the nutrients, and the low hydrodynamic energy on the lake bed to make this environment astrobiologically significant.

Conclusion

The geological and hydrological analyses performed on the Jezero crater and delta deposits indicate that the minimum lake filling timescale with 463 km 3 of sediment-poor water until -2243m is between 6.55 -27.36 years depending on the channel width (see spreadsheet in the supplementary material). The minimum lake delta formation timescale for 5 km 3 up to -2410m in elevation is 90-150 years with a channel 190 m width and 330-550 years for a channel 50 meters wide. These minimum estimations have been calculated without consideration of intermittency/evaporation/groundwater effects because it is impossible to constrain the impact of these (extinct) processes. The Jezero fan formed as a fluvial delta deposited over a fluvial fan by suspended bed load material, but the timescale uncertainty depends more on unknown grain size than on channel geometry. The discharge of the Neretva Vallis that has been used to form the delta is much smaller than that of the outlet. This confirms that the delta formation and the breach are not coeval because if we compare the two discharges, the basin empties much faster than it fills. Furthermore, the timescale to empty the basin is far smaller than the time scale to carve the breach. This implies that the basin acted as a closed (steady-state) basin for an unknown period with a water table at least at -2243m and then it had at least three catastrophic collapses or it overflowed for a long (unknown time) in order to carve out the breach. The increase in channel size decreases the duration of the system and further corroborates this hypothesis. The time needed to erode the volume of sediment from the breach (3 km 3 ) and to carve the outlet removing its 12 km 3 of sediments, compared to the discharge and the time required to empty the basin, suggest a much longer duration of the inlet (Neretva Vallis) or the breach event. The breach therefore formed over several distinct events sourced by the channel system upstream of the delta, with the delta itself developing far later.

Accepted paper in Astrobiology -Vol. 8 (2020)

Salese et al., 2020b

Our findings about a short duration of formation of the depositional fan of several hundred years do not support the idea of a perennial lake that would have been permissive of the incipience and development of any microbial life. Furthermore, from our analysis, given the high channel discharges, it emerges that the delta toeset dominated by clastic, coarse-deposits (coarse sandstones, conglomerates), is of debatable exobiological significance because of the oxidizing fluids moving through the porous deposits. Conversely, lake floor deposits, paleolake margins and/or delta bottomset are characterised by fine sediments, not trapped in the delta. The latter area is of stronger exobiological significance than the fan: with the suspended sediment serving as food for potential microscopic life, and the lake floor sediment providing a substrate with good preservation potential (e.g. washload sediment settling). However, it should be noted that the Jezero lake may have existed for a long while prior to this ultimate fluvial gasp, and in the short-in-time western Jezero fan deposition may have thus initiated into an already biologically active environments. The short temporal duration of the system hypothesized from this study does not preclude the possibility of discovering in situ organics in the fine-grained deposits, but does reduce the probability of success.

Supplementary Information. is linked to the online version of the paper. Author Information. Correspondence and requests for materials should be addressed to Francesco Salese (f.salese@uu.nl). km 3 up to -2243 m in elevation. The presence of an older and bigger paleodelta than we see at present could be supposed but is not considered as there is no evidence of a paleodelta at this topographic elevation. B) Post-breach phase: the water level drop to -2410 and the water volume within the lake is reduced to 225 km 3 . C) Deposition of the Jezero delta. Note that the top of the Jezero delta has the same elevation of the bottom of the breach. See main text for further description.

Acknowledgements

Author Disclosure Statement. No competing financial interests exist

Table 1 List of parameters and scenarios used to unravel the timing and evolution of the Jezero delta.

Figure 1

 1 Figure 1 Main parameters (grainsize, channel slope, channel depth) used in the hydrological modelling that strongly affect the transport predictors. A) Grain size vs timescale. Based on different grain size, the blue line indicates the amount of time needed to fill the basin whereas the orange line displays the timing to form the delta. The black box indicates the grain-size range that we considered to estimate the Jezero timing (see main text). B) Channel slope vs timescale. The slope is a crucial parameter for estimating delta timescale, as is evident from this plot. Reducing the slope by a factor of 10 implies a timescale reduction by a factor of 100.

Figure 2 A

 2 Figure 2 A) Location map of the topographic profiles within the Jezero crater. Profiles A-B / C-D show the pre-breach rim slope; D-E display outlet channel slope; G-H / I-L / M-N indicate the inlet (Neretva Vallis) depth at three different locations (a zoom of these profiles is in the supplementary material); O-Q shows the delta slope; R-S shows the breach profile with the three main stages in different tones of blue that are described in the main text.

Figure 3

 3 Figure 3 Proposed model of Jezero basin evolution, independent (black) and dependent (red) variables are indicated in the figure. From the bottom to the top: A) Pre-breach stage: the basin was filled at least with 463

  . Francesco Salese is supported by Marie Curie Individual Postdoctoral Fellowship(WET_MARS, Grant Agreement No. 795192). The data (HRSC, CTX, HiRISE, MOLA) that support the findings of this study were obtained freely from the Planetary Data System (PDS) and are publicly available online at https://pds.nasa.gov/index.shtml. The authors wish to thank MRO and HRSC team for these data. Satellite imagery and the Extended Data were generated with ISIS 3 (Integrated Software for Imagers and Spectrometers) available online at https://isis.astrogeology.usgs.gov. All these data were integrated into ArcGIS 10.6 project. French authors are granted by the Centre National d'Etudes Spatiales (CNES). Région Pays de la Loire through the GeoPlaNet project (convention N° 2016-10982) also supported this work. We acknowledge discussions within the rover 2020 team.

Table 2

 2 List of scenarios for the three different phases of the Jezero crater breach.

	Suspe	Nondim.	-		0,9	1,3	2,0	3,5	0,9	1,3		2,0	3,5	0,9	1,3	2,0	3,5
	nsion	Transport	km	3	0,00	0,00	0,00	0,00	0,000	0,000	0,000	0,0000	0,000	0,000	0,000	0,000
	domi	rate	/da	0095	0109	0128	0156	0251	0288	0338	413	0251	0288	0338	0413
	nated	Volum.	y		3	3		6	8	1907	1649	1388	1124	1907	1649	1388	1124
		Transport	-		2037	1760	1482	1200
		rate
		Water/sedi
		ment ratio
	Form	Time scale	yea	6,55	`6,6	6,65 6,75 26,55 26,76 27,03	27,36	26,55 26,76 27,03 27,36
	ative	to filling	rs
	time	basin until -
	scale	2243 (H 2 O)
	Form	Time scale	yea	143,	125,	106,	87,3 545,8 475,8 404,5	331,7	1637,	1427,	1213,	995,0
	ative	to form	rs		6	2		5							3	4	6
	time	delta
	scale
	Calculation	Parameters		Units					Scenario Phase 1
									1	2		3		4	5	6	7	8
	Input		Channel width		m		4000	4000	4000	4000	4000	4000	4000	4000
			Channel depth		m		5	10		15		20	25	30	35	37
			Channel slope		m/m	0,032	0,038	0,042	0,045	0,048	0,05	0,052	0,053
	Discharge	Friction factor		-		0,31	0,37		0,41	0,44	0,47	0,49	0,51	0,52
	calculation	Velocity			m/s	3,9	5,5		6,7		7,8	8,7	9,5	10,3	10,6
			Froude number		-		0,9	0,9		0,9		0,9	0,9	0,9	0,9	0,9
			Discharge			km	3 /day	6,7	19,0		34,9	53,7	75,1	98,6	124,1	135,0
	Bedload		Shields parameter	-		0,12	0,19		0,24	0,29	0,34	0,38	0,42	0,44
	transport
			Nondim. Transport	-		0,23	0,49		0,75	1,05	1,36	1,66	1,96	2,10
			rate
			Volum. Transport		km	3 /day 0,0094 0,0204 0,0320 0,0440 0,0567 0,0692 0,0821 0,0876
			rate
			Water/sediment		-		716	932		1090	1220	1325	1425	1512	1540
			ratio
	Total load	Shields parameter	-		0,6	1,5		2,5		3,6	4,8	6,0	7,3	7,8
	transport
	Suspension	Nondim. Transport	-		0,1	0,8		2,4		5,6	10,8	18,0	28,1	33,3
	dominated	rate
			Volum. Transport		km	3 /day 0,004	0,032	0,102	0,232	0,449	0,752	1,173	1,391
			rate
			Water/sediment		-		1547	597		342	231	167	131	106	97
			ratio
	Erosion		Sediment breach		km	3	3	3		3		3	3	3	3	3
			volume
	Discharge	Inlet				km	3 /day 0,2	0,2		0,2		0,2	0,2	0,2	0,2	0,2
	Formative time	Time scale to empty	months 0,29	0,10		0,05	0,035	0,025	0,019	0,015	0,014
	scale		the basin until -
			2410m
			Time-scale to carve	months 436	60		19		8	4	3	2	1,5
			the outlet
			(suspension
			dominated)


Marie Le Guilly
email: marie.le-guilly@insa-lyon.fr

Nassia Daouayry
email: nassia.daouayry@airbus.com

Pierre-Loic Maisonneuve
email: pierre-loic.maisonneuve@airbus.com

Ammar Mechouche
email: ammar.mechouche@airbus.com

Jean-Marc Petit
email: jean-marc.petit@insa-lyon.fr

Vasile-Marian Scuturici
email: marian.scuturici@insa-lyon.fr

Contextualisation of Datasets for better classification models: Application to Airbus Helicopters Flight Data

Keywords: data contextualization, failure anticipation, classification

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In the helicopters industry, predictive maintenance is crucial and Airbus Helicopters (AH) seeks to anticipate failure as soon as possible. One solution is to analyze flight data, as most helicopters are equipped with flight recorders for hundreds of parameters. Such an amount of data makes it possible to analyse "low-level signals"' over longer periods of time, and to detect failures earlier.

In this context,AH has gathered data on hundreds of thousands flight hours: to face such a huge amount of data, a Big Data platform has been deployed at AH to enable the storing and processing of large quantities of data [START_REF] Mechouche | Helicopter big data processing and predictive analytics: Feedback and perspectives[END_REF].

Using this platform, digital twins have been devised to identify as soon as possible small variations on core physical sensors. They are mainly based on physical models and expert knowledge, but AH combines these with machine learning techniques to build predictive models from the data. To build such models, AH faces generic and recurrent issues that are well-known in machine learning, such as data cleaning, accuracy , or explainability. But in addition to these classic issues, AH also seeks to build models corresponding to the normal behavior of the system, and has to use data fitting the behavior algorithms have to model. Indeed, an important filtering step is performed to identify the data that is adequate to deal with the considered problem: complex systems such as helicopters go through many different phases, and only a subset of the data is relevant for a given model, as they are the only one for which the laws of normal behavior of the system apply. It is therefore necessary to identify the correct context for the considered task, which is the subset of data corresponding to the desired phases on which the model is applied. We define this problem as contextualization, according to the term used by AH experts. Thus, it consists in determining the flight phases where considered parameters have lesser variability and are less subject to pilot maneuvers and external parameters not recorded by the system. At AH, this crucial step is dealt with by relying on experts knowledge who specify how to filter flight data.

The contextualization problem can seem as a simple problem at first hand (mainly data selection), but turns out to be a nightmare in practice. Identifying the appropriate data is clearly not an easy task, and depends on the final objective for the classification model. In addition, contexts are tightly linked with the application they concern, so solutions are often specific to one given situation. For systems such as helicopters, contextualization is also important as they are systems governed by physical laws, that apply only in specific contexts: the purpose of classification models is therefore to produce outputs coherent with these laws. To this end, these models have to be trained on data consistent with the physical model they represent.

In this paper, we propose our ongoing work to address the contextualization challenge. We seek to identify the appropriate context for a classification task, by identifying the subset of data more likely to capture the normal behavior. To do so, we seek the data favoring the existence of a function between the features and the class to predict. As the correct context should follow some underlying function the model seeks to define, we propose to remove the regions of the data preventing the existence of that function, and to only keep the data more likely to correspond to a normal behavior. We then show how this approach can be applied to AH classification datasets.

Based on these considerations, we made the following contributions: (1) Proposing a generic solution for contextualization, in order to define filters that can be used to reduce the dataset to a given context; (2) Experiments on AH data showing how identifying context elements can improve the accuracy of classifiers; (3) Confronting a contextualization proposed by AH experts to additional context elements proposed by our method.

Section 2 introduces the preliminaries. In section 3, we propose our approach to better contextualize datasets, and in section 4, we focus on AH's data, to show how we built a context for the considered dataset, and develop the lessons drawn from this collaboration based on the experimentations that have been conducted. Finally section 5 presents the related work before concluding in section 6.

Preliminaries

Functional dependencies

We first recall basic notations and definitions (see [START_REF] Levene | A guided tour of relational databases and beyond[END_REF]). Let U be a set of attributes. A relation schema R is a name associated with attributes of U , i.e. R ⊆ U . A database schema R is a set of relation schemas. Let D be a set of constants, A ∈ U and R a relation schema. The domain of A is denoted by dom(A) ⊆ D. A tuple t over R is a function from R to D. A relation r over R is a set of tuples over R. If X ⊆ U , and if t is a tuple over U , then we denote the restriction of t to X by t

[X]. If r is a relation over U , then r[X] = {t[X], t ∈ R}. Definition 1. Let R be a relation schema, X ⊆ R and C ⊆ R\X. A FD on R is an expression of the form R : X → C (or simply X → C when R is clear from context) Definition 2. Let r be a relation over R and X → C a functional dependency on R. X → C is satisfied in r, denoted by r |= X → C, if and only if for all t 1 , t 2 ∈ r, if t 1 [X] = t 2 [X] then t 1 [C] = t 2 [C].

Supervised classification in machine learning

Let's consider a set of N training samples {(x 1 , y 1 ), ..., (x N , y N )} where x i is the feature vector of the i-th example and y i its label (or class). The number of different labels K, is limited and much smaller than the number of samples.Given this, classification is the task of learning a target function g (a classifier) that maps each example x to one of the k classes, with the lowest error rate. It is possible to express a classification problem using relational databases notations. In the sequel, we will therefore consider a relation r 0 (A 1 , . . . , A n , C) with N tuples, where for any tuple t i , t i [A 1 . . . A n ] = x i and t i [C] = y i . In addition, we consider that traditional feature selection methods (see [START_REF] Arauzo-Azofra | Empirical study of feature selection methods based on individual feature evaluation for classification problems[END_REF]) have been applied and consider the subset X ⊆ {A 1 . . . A n } of selected features.

To evaluate the performances of an algorithm, we use accuracy, which is the proportion of samples that are correctly classified by a model. This score lies between 0 and 1, and ideally should get as close as possible to 1. Given a model M over a relation r, accuracy is defined as follows:

accuracy(M, r) = # of correct predictions |r| 2.

Existence versus determination of a function

We use the link between FDs and classification, developed in [START_REF] Guilly | Evaluating classification feasability over datasets using functional dependencies[END_REF]. We only underline here it relies on the notion of function, as classifier seeks to define a function from the features to the class, while the FD X → C can say whether or not such a function exists or not: the FD X → C is satisfied if and only if there exists a function from X to C. If the FD is not satisfied, it means some pairs of tuples have the same value on X, but different classes. Such tuples are called counterexamples:

Definition 3. Let r be a relation over R and X → C a FD f on R. The set of counterexamples of f over r is denoted by CE(X → C) and defined as follows:

CE(X → C, r) = {(t 1 , t 2 )|t 1 , t 2 ∈ r, t 1 [X] = t 2 [X] and t 1 [C] = t 2 [C]}
Counterexamples are important as they identify pairs of tuples for which the classifier cannot perform correctly, as for the same input, it always predicts the same output. The proportion of counterexamples therefore directly impacts the quality of the classification: it can be evaluated using measure G 3 , and contrary to [START_REF] Kivinen | Approximate inference of functional dependencies from relations[END_REF] that presents this measure as an error, we propose it as follows:

G 3 (X → C, r) = max({|s||s ⊆ r, s |= X → C}) |r|
Measure G 3 is of crucial importance for the classification problem, as in the subset s defined for G 3 , there exists a function between the left and right hand side of the dependency. For classification, measure G 3 is therefore a way to bound the accuracy a classifier can reach on the considered dataset, as it is necessary limited by the existence of counterexamples. As a result, the following result holds, for which the details and proof are given in [START_REF] Guilly | Evaluating classification feasability over datasets using functional dependencies[END_REF]: Proposition 1. Let X ⊆ R be a set of features, C ∈ R the class to be predicted, r a relation over R, and M a classifier from X to C. Then:

accuracy(M, r) ≤ G 3 (X → C, r)
In the setting of contextualization, G 3 can be seen as a way to identify whether or not a dataset follows a function, and to identify zones that are therefore more likely to correspond to a normal behavior of the system.

Contextualization of a classification dataset

The objective is to propose a methodology for the contextualization of classification datasets. The proposed solution considers there should be a function between the features and the class to predict. The idea is to identify the regions in the initial dataset in which a function is likely to exist, and therefore in which the FD f eatures → class is likely to be satisfied. On the opposite, regions with a high proportion of counterexamples should be removed, as they are likely regions where the model hypothesis are not verified.

To contextualize a dataset, we propose an iterative approach, that is summarized on figure 1. The process starts with an initial classification dataset. It is then discretized, to smooth the data variability and to better identify counterexamples. Then, G 3 is computed, and a classifier is trained and tested, to obtain an accuracy measure. Measure G 3 allows to evaluate the existence of a function, while the accuracy guaranties the performances of the model. These two measures are taken into account to determine the next step in the process.

If the domain experts are not satisfied with the measures, the counterexamples are enumerated, to identify filters to remove the tuples that cause too many counterexamples. The key is to find balance between removing regions of the data while keeping as many tuples as possible. The filters can take different forms: here, we propose to define filters in the form of conjunctions of conditions allowing to remove groups of tuples. To identify such groups, visualizations are proposed, to observe what tuples are the most involved in counterexamples. Once the filters are determined, based on these visualizations and in collaboration with domain experts, tuples are removed, providing a new dataset. This process is repeated until satisfaction. When the proposed contextualization is not satisfying, solutions have to be proposed to refine it, and to therefore remove tuples from the dataset. The challenge is to determine what are the tuples to remove and why. We therefore propose to determine filters that can be applied to the dataset, to remove tuples and lower the number of counterexamples in the dataset. Such filter should ideally remove as few tuples as possible, while removing as many counterexamples as possible. Indeed, one tuple might be involved in many counterexamples: in this case, it should be removed.

Many solutions can be considered for the filters: one solution from example is too order the tuples by the number of counterexamples they are involved in, and to set a threshold to remove all the tuples involved in more counterexamples than this threshold. But it does not explain what are the characteristics of the removed tuples: if a domain expert wishes to understand why a tuples is removed, she has to manually check each counterexample. In this paper, we propose to define filters in the form of conjunction of conditions applied to the dataset, making the overall process explainable. These filters define, in simple terms, regions of the dataset containing more counterexamples than others, while concerning only a few tuples. This can be performed using visualizations proposing, for each feature, histograms showing the distribution of values among counterexamples, and the number of tuples taking a given value. The histograms can then be used to identify values having, on a given feature, few tuples involved in many counterexamples. The filters then integrate a condition removing such values from the dataset. Such filters are interpretable by domain experts, who can analyze whether or not these filters make sense with the desired context. By comparing these two visualizations, it appears there is a zone that does not contain many tuples, but many counterexamples. As a result, one condition for a contextualization filter could be to remove all tuples for which A ≥ 15 and A ≤ 25. This gives an interpretable filter, removing a few tuples and improving measure G 3 . Similar work can be performed for each feature of the dataset, creating a filter that is a conjunction of conditions over all features.

4 Application to AH flight data

AH classification datasets

Using helicopters flight data, AH is developing tools such as virtual sensors, that aim at monitoring the aircraft health and usage. They use the historical flight data to learn a predictive model for a given parameter. The predicted value is compared to the one given by the physical sensor: an alert is raised if the difference between the two values is too high. An example of such a virtual sensor has been proposed by AH for the oil pressure of the helicopter Main Gear Box (MGB) [START_REF] Daouayry | Datacentric helicopter failure anticipation: The mgb oil pressure virtual sensor case[END_REF]. We reuse the data from this study to perform the experiments of this paper. As a first contextualization had been done by AH domain experts, we used and compared two datasets, with 10 attributes selected and discretized by AH experts: the raw dataset corresponds to the flight data without any contextualisation, for a given period of time, randomly mixing tuples from several flights; the expert-Contextualized dataset is a subset of the raw one containing tuples filtered by AH experts (around 50% of the raw data). The impact of contextualization was analyzed, by comparing accuracy for a random forest algorithm (baseline column of table 1). The accuracy for the expert-contextualized dataset is much higher than for the raw one,confirming the expert contextualization pertinence. Moreover, G 3 = 95.53% for raw dataset and G 3 = 95.51% for expert-contextualized one. The proportion of counterexamples is therefore reasonable and the two datasets have similar G 3 values. The contextualization seems to have preserved the proportion of counterexamples: they have decreased in absolute number, but not with respect to the size of the dataset. New contextualization might therefore increase the model's accuracy.

Comparison of AH datasets

Additional contextualization using G 3

We applied our methodology from figure 1 to the two datasets, but first verified that the counterexamples were evenly distributed among the flights. Figure 3 shows a histogram of the percentage of counterexamples among flights: most flights have a very low rate of counterexamples, so any removal of counterexamples affects a large number of flights, avoiding the model to overfit on a subpart We then analyzed the two plots made for each feature such as figures 4a and 4b for the pressure. Low pressure values have more counterexamples, while containing an important number of tuples. It can also be noted that the domain contextualization removes a significant part of counterexamples, but other regions could be cleaned further from counterexamples with additional contextualization, for example for pressure values over 5.6 bar.

A first filter was designed (Filter 1 in table 1). For the pressure, this filter removes all the data for which it is below 3.2 bar and above 6.4 bar, as these regions have few tuples but many counterexample (see figure 4). Similar rules were applied for the other features of the dataset. The results in table 1 show the positive effect of this filter on classifier's accuracy. It was decided to improve again the contextualization, so we obtained filter 2 by adding additional rules to the ones from filter 1. Table 1 shows that accuracy is improved by filter 2. After this second iteration, the obtained contextualization was considered satisfying.

Finally, it should be noted from table 1 that there is a significant gap between the highest accuracy on the raw dataset and the lowest accuracy for the expertcontextualized one. Even with the best filter, it is not possible to reach the result obtained using expert knowledge: the best approach consists in taking the valuable domain expert knowledge into account, before refining it using tools such as counterexamples and G 3 .

Take away lessons

These experimentation showed how contextualization can be used to improve the accuracy of classifiers for AH virtual sensors. Contextualization is an important problem, but it is not easy to address because the proposed solutions are often domain-specific, or included in the "data preparation" steps that our left to data scientists judgment: our solution could in comparison be applied for other types of application and involves domain experts in the loop. There is also a qualitative aspect to this approach, that aims at taking a step back from the model, to understand what is being done, and understand the limitations. This is directly related to the explicability of the model, a crucial notion in aeronautics: the prediction of what can be seen as a simple classification algorithm output can put into question human lives getting back into an aircraft or not.

Related work

We applied our contextualization technique in the context of predictive maintenance for helicopters, a growing topic in the industry. Virtual sensors such as the ones used for the experiments of this paper [START_REF] Daouayry | Datacentric helicopter failure anticipation: The mgb oil pressure virtual sensor case[END_REF] are interesting solutions in this context. Similarly, [START_REF] De Benedetti | Anomaly detection and predictive maintenance for photovoltaic systems[END_REF] proposes a virtual sensor to anticipate failures on photo-voltaic systems. Additionally, [START_REF] Sundareswara | Interpretable unsupervised feature extraction and learning of abnormal system state transitions in aircraft sensor data[END_REF] presents a failure anticipation approach for aircraft systems. In this case, the learning is done only on flight phases predefined by experts. More generally, in most works developed in the industry, data is always combined with domain knowledge in order to speed-up accurate predictive models development. However, this combination still is often not optimal, and we believe this is a lever for improving accuracy of predictive models developed in the industry.

Functional dependencies are of high interest for data cleaning, a necessary prerequisite for data contextualization. The authors from [START_REF] Kwon | Effects of data set features on the performances of classification algorithms[END_REF] showed that if there is a functional dependency between features, it is likely to affect the classifier negatively. Specific dependencies have been proposed to identify inconsistencies in a dataset, and eventually repair it. Matching dependencies [START_REF] Fan | Dependencies revisited for improving data quality[END_REF] for data repairing uses matching rules to relax the equality on functional dependencies and assign values for data repairing. In Holoclean [START_REF] Rekatsinas | Holoclean: Holistic data repairs with probabilistic inference[END_REF], dependencies are used to clean automatically a dataset.

Conclusion

In this paper, we addressed the problem of contextualization of classification datasets, applied to the flight data of AH. This problem is crucial, and appears in many data science industrial applications, but has yet not been addressed as massively as other traditional machine learning problems. We proposed a methodology, and conducted experiments on data from a virtual sensor developed by AH, and showed how our method could improve the contextualization and, as a consequence, the accuracy of the datasets.

Fig. 1 :

 1 Fig. 1: Overview of the solution proposed to contextualize a classification dataset

Fig. 2 :

 2 Fig. 2: Toy example for filter design

Figure 2

 2 presents visualizations used to define filters. For a given feature A, figure2ashows for each value taken by this feature, the proportion of tuples involved in counterexamples, and therefore how much they contribute to the value of G 3 . Figure2bis an histogram of values for the considered feature.

Fig. 3 :

 3 Fig. 3: Distribution of flights for each proportion of counterexamples

Fig. 4 :

 4 Fig. 4: Counterexamples and distribution for pressure values

Table 1 :

 1 Accuracy of random forest models on the oil pressure datasets

		Baseline	Filter 1	Filter 2
	Dataset	# tuples accuracy # tuples accuracy # tuples accuracy
	Raw	1969533 53.97% 607248 57.28% 468630 61.71%
	Expert-contextualized 541342 73.94% 281947 76.02% 100165 78.61%