Temporal variations of evapotranspiration : reconstruction using instantaneous satellite measurements in the thermal infra red domain

Introduction Conclusions References


Introduction
Evaporation is the largest water loss component of continental surfaces.In semi-arid areas, more than 80 % of the annual available water is lost through evapotranspiration.In most countries, the largest water user is the irrigated agriculture, which represents more than 80 % of all uses, with a low efficiency no greater than 50 % in many cases (PNUE PAM Plan Bleu, 2004).For countries facing water shortage, or likely to suffer from more frequent drought spills under climate change scenarios, there is a great need to rationalize this use, and therefore to monitor more closely the water resources.
Amongst the fluxes that the different actors of the water sector need to assess, evapotranspiration is of major importance.It is also important in the wider context of hydrological prediction and monitoring.
Although the water budget can be fairly easily monitored by the farmer at plot scale, it is much more difficult for regional authorities or national planners to monitor water allocation and use at the relevant scales, i.e. the perimeter and the basin scales.To do so, Remote Sensing (RS) data is increasingly used, because it allows for the description of the surface with a temporal scale lower than a few weeks.This is particularly important to follow the growth of vegetation at most scales ranging from plot to region.
Many methods exist to compute evapotranspiration with the help of RS data (Courault et al., 2005;Kalma et al., 2008).Some of them rely only on the atmospheric demand through different radiation and atmospheric variables derived from remote sensing (Venturini et al., 2008).Since evapotranspiration largely depends on the availability of water, which is often greater in the root zone than at the soil surface, surface losses depend on the intensity of transpiration.Many methods, especially those designed for irrigated agriculture, which is usually not short of water, compute a potential or reference evapotranspiration rate and weigh the latent heat flux by an estimated amount of vegetation present for each pixel, through the use of a vegetation index such as the NDVI (Cleugh et al., 2007).But this does not help when vegetation suffers from water stress, which means that these methods have little applicability in natural lands, for rainfed agriculture areas or for deficit irrigation systems, which are more sensitive to climate fluctuations and drought.
Since evaporation is the most efficient way to dissipate extra energy at the surface, there is a tight coupling between water availability and surface temperature under water Figures stress conditions.Therefore, information in the Thermal InfraRed (TIR) domain is the most appropriate way to assess actual evaporation and soil moisture status at relevant space and time scales (Boulet et al., 2007;Hain et al., 2009).Methods to estimate evapotranspiration from satellite data in the TIR domain are reviewed in Kalma et al. (2008) and Kustas and Anderson (2009).
Geostationary satellite provides information in the TIR domain with a frequency down to 15 min, but for resolutions well above the kilometric scale (Anderson et al., 2011).On the other hand, some sun synchronous satellites (MODIS, AATSR) provide data once or twice a day at kilometric resolution.For shorter space scales, of the same order of magnitude as the average field size in most agricultural systems, data can be available every week or so if data from several platforms (e.g.ASTER, Landsat, ...) are combined.The large temporal gaps between two successive acquisitions with the existing satellites lead to the proposal of the MISTIGRI (MIcro Satellite for Thermal Infrared GRound surface Imaging, Lagouarde et al., 2012) satellite mission by CNES and the scientific community.This mission would provide surface temperature data with a daily revisit and a 50 to 60 m spatial resolution and would therefore be particularly suited to monitor evapotranspiration at field scale.
Most methods using information in the TIR domain rely on once-a-day data, generally acquired around noon, in late morning or early afternoon.As a consequence, the diurnal cycle of the energy budget is not accounted for and most methods compute an instantaneous energy budget at the time of the satellite overpass.They thus provide a single instantaneous evaporation or latent heat flux, whereas a daily average is usually required for hydrological applications.
In order to estimate daily and seasonal evapotranspiration (ET) using remote sensing there is a need to extrapolate daily ET from an instantaneous measurement to reconstruct hourly variations of ET and interpolate ET between two daily ET values to reconstruct sequences of daily cumulated ET.Cloud occurrence is also an issue and no data is acquired under cloudy conditions.Data availability therefore depends on both the overpass frequency and the cloud cover conditions.Figures

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Full Different methods have been developed to scale daily evapotranspiration from a onetime-of-day measurement.These essentially rely on a self preservation or a known diurnal shape of the ratio of the latent heat flux (LE) to a scale factor whose diurnal evolution can in turn be easily resolved.This scale factor is usually either a radiation term (global solar incoming radiation R g , net radiation R n , total incoming radiation ...), the available energy (R n −G where G is the ground heat flux) or a maximum evapotranspiration rate, either a potential evapotranspiration rate or the reference evapotranspiration rate as defined in (Allen et al., 1998).
Often, diurnal self-preservation of the evaporative fraction (EF) is used to solve the problem of scaling ET from a once a day measurement.The evaporative fraction is defined as the ratio between the latent heat flux and the available energy at the land surface (EF = LE/(R n − G)).Shuttleworth et al. (1989), Nichols and Cuenca (1993) and Crago and Brutsaert (1996) observed from in situ measurements on typical few days and in various situations, that EF is nearly constant during daytime under clear sky days.Gentine et al. (2007) investigated the diurnal behavior of EF and its environmental dependencies in details using a Soil-Vegetation-Atmosphere Transfer model applied to a wheat crop in a semi-arid climate.The study showed that EF is almost independent of solar radiation and wind speed, but strongly depends on soil moisture availability and canopy fraction cover.Daytime self-preservation of EF is not always satisfied when fractional vegetation cover is close to 100 %.Indeed, for a fully vegetated surface, EF shows a pronounced rise in the afternoon due to the inversion of sensible heat flux.This effect is stronger with high soil moisture, when EF values exceed unity, and with increasing LAI.Gentine et al. (2007) underlined also that the daytime self-preservation of EF can be revised in order to obtain a concave-up shape of EF more representative of typical diurnal fluctuations.This shape is obtained analytically from a sinusoidal solar radiation forcing by Gentine et al. (2011).Hoedjes et al. (2008)  diurnal shape as a function of relative humidity and incoming solar radiation.The study showed that EF remains fairly constant during daytime under dry conditions and follows a concave-up shape under wet conditions.This work also underlined that using a constant EF value throughout daytime induces significant errors when calculating daily ET.
Other methods using different parameters than EF have also been tested in the past.For instance, Allen et al. (2007) provided an interpretation of the pronounced rise of EF in the afternoon.For theses authors, the assumption of constant EF during the day can underpredict 24 h ET in arid climates where afternoon advection or increased afternoon wind speed may increase ET in proportion to R n −G.They stated that the diurnal self preservation of the stress factor (the ratio of the evaporation rates in actual and potential conditions) during a day appears to be generally valid for agricultural crops that have been developed to maximize photosynthesis and thus stomatal conductance.This ratio may decrease during the afternoon for some native vegetation under water shortage conditions, where plants endeavor to conserve soil water.Under these conditions, the 24 h stress factor must be modeled as some fraction of instantaneous stress factor.This requires local study and measurement to develop the needed functions.Chavez et al. (2008) and Colaizzi et al. (2006) selected and tested several ET extrapolation methods (including those based on EF and the stress factor mentioned previously) to estimate daily ET.
In particular, Chavez et al. (2008) used data on soybean and corn over one summer month.They showed that estimation errors for all methods and both crops vary from −5.7 % (±4.8) to 26.0 % (±15.8).Extrapolated values based on the EF method were closer to observed ET values measured by an eddy covariance system.This method reported an average estimation error of −0.3 mm day −1 for corn.A solar radiationbased ET extrapolation method performed relatively well with an estimation error on daily ET of 2.2 % (±10.1) for both crops.An alfalfa reference ET-based extrapolation fraction method yielded an overall daily ET overestimation of about 4.0 %, (±10.0) for both crops.The results of Colaizzi et al. (2006) also showed that the methods were more efficient when used around noon (12:45 UTC in the study).Each of the five methods tested presented the greater performances at this time of the day (average RMSE of 0.57 mm day −1 ).Crops involved in this study were fully irrigated alfalfa (irrigated to meet the full ET requirement; 304 days), dryland grain sorghum (124 days), partially irrigated cotton (irrigated to meet 50 % of the full ET requirement; 59 days), and bare soil after tilling following a grain sorghum crop (66 days).The climate for this dataset was semiarid.According to their conclusions, scaling with the help of a model based on the grass reference ET is the recommended basis to reconstruct daily ET, but for surfaces having low ET, using a model based on the evaporative fraction may give slightly better estimates with RMSE values of 0.47 mm day −1 (mean observed ET: 1.4 mm day −1 ) for bare soil, 0.47 mm day −1 (mean observed ET: 3.9 mm day −1 ) for cotton and 0.50 mm day −1 (mean observed ET: 4.1 mm day −1 ) for sorghum.
Except for the study by Colaizzi et al. (2006), the different works presented above were generally based on a small range of bio-pedo-climatic conditions and the methods were tested for relatively short time periods.Indeed, the periods of study were often limited to a few days only, and rarely exceeded a few weeks.In some studies, results were obtained for particular and typical situations (stressed, rainy, dry, moist, full cover, bare soil ...) but mostly for isolated days picked from seasonal data sets.Moreover, most studies did not contest the assumption of the self-preservation of the scale factor during the day.
Since the main goal of daily and seasonal ET reconstruction is to estimate daily ET from satellite data operationally (and therefore routinely), it's difficult to implement methods based on biophysical characteristics that are temporally and spatially difficult to infer, such as soil moisture or water stress.For instance, it's not easy to implement, say, a different EF diurnal shape for stressed and unstressed periods, as proposed by Hoedjes et al. (2008).There is moreover no consensus on the general trend of EF diurnal fluctuations, which can exhibit for a given location either a "flat", "tangent-like" or a "concave-up" shape (Van Niel et al., 2011).We thus want to estimate the error Figures associated with the "self preservation" hypothesis which is well suited (and up to now largely used) to reconstruct daily and seasonal ET from instantaneous estimates of ET from satellite data, and then test the impact of using one common shape for the scale factor for all sites and all times on the daily ET reconstruction.One must note that, contrarily to the Evaporative Fraction, there is no documentation, and a fortiori no consensus, on the most common shape of the stress factor during the day, and its self preservation is, to our knowledge, the only tested hypothesis for daily ET reconstruction using methods based on a potential or a reference evapotranspriation rate.Within that context, the objectives of this paper are twofold: 1. To assess the performance of two methods classically used to reconstruct daily (extrapolation) and seasonal (interpolation) ET from sparse instantaneous estimates, as a function of revisit and time of acquisition.

2.
To check for possible improvement of the method that performs best.
Within these two main objectives, the interests of the study rely on testing classical methods on a large range of multisite data and to reconstruct ET at daily and seasonal scale.In order to take into account the operational constraints imposed by the existing or future satellite platforms (overpass time, revisit ...), the hypothesis concerning the time and the frequency at which the instantaneous estimates are collected in order to reconstruct ET are also discussed.

Material and methods
In this section, we first present the two main methods generally used to reconstruct daily and seasonal ET which will be tested in this study.The details of the complete dataset used for the test is then described.In order to apply the reconstruction algorithm for clear sky days only, the method to pick these days in the continuous dataset Figures

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Full is presented.Finally, the methods selected to compute the denominator of both evaporative fraction (i.e.available energy) and stress factor (i.e.potential evapotranspiration) are presented and their relative advantages or drawbacks for operational applications are analyzed.

Evaporative fraction (EF) method
The first method is based on the use of the evaporative fraction.The evaporative fraction is defined as the ratio between the instantaneous latent heat flux (LE) and the instantaneous available energy at the land surface As in Chavez et al. (2008), assuming that EF is constant during daytime, the daily cumulative evapotranspiration (ET d ) can be retrieved from the daily available energy (AE d ) and the estimation of EF at the time of satellite overpass: (1) Both R n and G fluxes are determined with relatively good precision from remote sensing data without requiring any additional ground data.This method is therefore particularly suited for mapping daily or seasonal evapotranspiration at large scale.R n is given by: where R g is the global radiation, α the albedo, ε the surface emissivity, R atm the atmospheric longwave radiation, σ the Stefan-Boltzman constant and T s the surface temperature.Also, several empirical functions based on surface temperature and/or NDVI exist to compute the ratio between G and R n (see examples in papers describing the most widely used single source energy balance models: Bastiaanssen et al., 1998;Santanello and Friedl, 2003;Su, 2002 ...).These functions may require to be calibrated for each specific site (Kpemlie, 2009).Introduction

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Full

Stress factor (SF) method
A second method, called SF method, based on the same assumption than the EF method, was tested: where SF is the stress factor computed as the ratio of the instantaneous latent heat flux LE and the instantaneous potential evapotranspiration flux LETp, both estimated at the time of the satellite overpass (SF = LE/LETp).
LETp d is the daily potential evapotranspiration.
LETp is usually derived from a surface energy balance model (Lhomme, 1997) or a reference calculation such as the FAO56 method for grass reference (Allen et al., 1998) or the ASCE Penman-Monteith equation for alfalfa reference (Allen et al., 2007).
Again, the diurnal course of the Stress Factor is neglected, which is consistent with the previous studies.Furthermore, no prior shape of this ratio has been described in the literature.

Experimental datasets
Meteorological and flux data necessary to run and test both methods were obtained over several agricultural fields in different climates.
The first dataset was collected over two cultivated plots, Aurad é (43 were part of the CarboEurope-IP Regional Experiment (Dolman et al., 2006) and the CarboEurope-IP Ecosystem Component.In that context, the data were used for analyzing CO 2 surface-atmosphere exchanges and production of full crop rotation (e.g.Kutsch et al., 2010;Ceschia et al., 2010).For those sites, the Level 3 flux Introduction

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Full products (i.e.non gapfilled) were used.These datasets represent eleven crop cycles to be used for the evaluation of ET extrapolation and interpolation methods on a variety of rainfed and irrigated crops (Table 1).
The experimental setup collected standard meteorological measurements (global incoming radiation, wind speed, air temperature and humidity, rainfall).For each site, the different components (global solar radiation, reflected solar radiation, downward longwave radiation and upward longwave radiation) of the net radiation were measured using a CNR1 radiometer.Soil heat fluxes were measured using heat flux plates close to the surface and a correction to account for the top soil transient heat storage fluctuations.Eddy covariance systems were used to obtain latent heat fluxes.The leaf area index (LAI) was measured using hemispherical photography (Demarez et al., 2008).
For a complete description of the site characteristics and more information on these datasets, see Beziat et al. (2009) for Aurad é and Lamasqu ère, Boulet et al. (2007) for Morocco and Kpemlie (2009) for Avignon.

Determination of clear sky days
Cloud occurrence is an issue because no data is acquired under cloudy conditions in the TIR domain which is the most appropriate way to assess actual evaporation and soil moisture status at relevant space and time scales (Boulet et al., 2007;Hain et al., 2009).Data availability depends therefore on both the overpass frequency (also referred to as revisit) and the cloud cover conditions.The extrapolation of ET from an instantaneous measurement to a daily value is computed for all clear sky-days, which correspond to days for which remotely sensed data could be available.
To determine clear sky days for the different datasets, actual incoming solar radiation was compared to outputs of a theoretical clear sky radiation model.The combined Meeus (1999) and Bird and Hulstrom (1981) model (Fig. 1) was selected on the basis of the results obtained during the comparison of five models by Annear and Wells (2007).This empirical model incorporates different atmospheric transmissivity Introduction

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Full coefficients which can be adjusted for calibration.In this intercomparison, it was found that for both years of study the combined Meeus (1999) and Bird and Hulstrom (1981) model performed best based on mean error and RMS error.When the five models were calibrated to all of the clear sky data, the combined Meeus (1999) and Bird and Hulstrom (1981) model had the lowest RMS error for both the application period.A constant ratio between clear sky and extraterrestrial radiances, as proposed by the FAO method, was also calibrated and is shown in Fig. 1 to illustrate the clear sky radiation course during the year according to earth-sun geometry only.
The combined Meeus (1999) and Bird and Hulstrom (1981) model requires air temperature, R atm and relative humidity as inputs and the clear sky radiation is computed as the sum of a direct and a diffuse radiation components.
Clear sky days are selected on the basis of a critical value of the ratio between the incoming solar and the theoretical clear sky radiations.This threshold is not straightforward to define.Based on the comparison of this ratio with a second proxy of cloudiness, the ratio between the diffuse and the total Photosynthetically Active Radiation (PAR), measured in two amongst the three sites, it was established that if the observed radiation was higher than 85 % of the computed clear sky radiation at a specific time corresponding to the choice of the time of the satellite overpass, the day could be defined as clear.
The days classified as clear according to this method were then compared with MODIS (Aqua) cloud mask products obtained at 01:30 p.m.The model applied at 01:30 p.m. produces matching errors with MODIS masks from 6.52 % to 11.72 % (depending on sites).These errors are quite small.Therefore the model and the threshold were kept to select clear sky days for a satellite overpass at midday.One must note that the number of clear sky days does not change significantly when the time of overpass varies from 10:00 a.m. to 02:00 p.m. which were also tested.This is in agreement with the work carried out by Lagouarde et al. (2012) for historical climatic data at five locations in France.Introduction

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Full The performances of the following methods were assessed on a large dataset, whose characteristics are presented in Table 1 and which included 11 years of data on 3 different sites with different climates and different crop types.
In total, both methods to reconstruct daily ET were tested on more than 1600 different days.

Determination of stress periods
The number of stress periods is determined considering evapotranspiration data.A water stress period is identified on the following basis: stress starts when a large deviation between the potential evapotranspiration and the measured actual evapotranspiration rates is observed away from any rain event or any other income of water (i.e.irrigation) and ends with the next income of water.When this deviation lasts more than 4 days in a row, we define arbitrarily the period as stressed.

Evaporative fraction (EF) method: parameterizing the available energy diurnal course (AEd)
The EF method requires the diurnal course of available energy AE, which is not routinely available from meteorological stations or satellite products.proposed parameterization shows RMSE as low as 30 W m −2 between simulated and observed AE for an olive orchad in semi-arid climate.
Since in most of our datasets the diurnal variations of R atm are small, the parameterization used here is a variation of the (Jackson et al., 1983) method, based on the assumption that ET and AE have the same diurnal course as the incoming global solar radiation R g (Fig. 2).
where AE d is the daily available energy and R g d the daily global incoming solar radiation.AE t and R g t are measurements of these components at time t.
A mean fixed value (observation average) of R g is imposed at night to avoid a bias for nocturnal values.
To estimate AE d on cloudy sky days, when RS data are not available, the ratio between daily AE d (computed from Eq. 4) and daily R gd is interpolated linearly between the closest previous and following clear sky days, respectively.representing the extraction of water from the top soil porous medium.Default values typical for herbaceous vegetation are assigned to the plant parameters (0.2 for albedo, 100 s m −1 for the minimum stomatal resistance per LAI, 0.8 m for the maximum vegetation height).Most of the parameters are assigned from a priori averages taken from the literature for crop land use types and not optimized on the datasets.Soil heat flux is modeled as a fixed (0.3) fraction of the net radiation at the ground surface.We ignore here the phase shift between the diurnal fluctuations of the soil heat flux and the net radiation.While important around 10:00 a.m. and 04:00 p.m., the resulting cumulative error is rather small at the daily scale (Gentine et al., 2007).The model allows us to easily compute the evolution of LETp at seasonal scale.

Testing both methods with in situ data while keeping in mind the future use of remote sensing data
Remotely sensed methods to estimate daily ET are meant to use as little ancillary data as possible (network of meteorological stations or outputs of climate models).They aim at routinely producing instantaneous LE at the time of satellite overpass using energy balance models, and either AE d or LETP d from remotely sensed and meteorological forcing data.In order to restrict our study to the test of the performance of the reconstruction methods of daily (extrapolation) and seasonal (interpolation) ET from sparse instantaneous estimates, we assume that LE and AE are perfectly known for all clear sky days at the time of the satellite overpass (which is taken at midday by default or otherwise stated in this study) and ignore the uncertainties associated with their estimation from energy balance models.This assumption also holds for all inputs to compute LETp since the latter cannot be measured in situ.We therefore use the true in situ LE values measured by eddy covariance and the true AE values measured at ground by the net radiometer and the soil heat flux, both at the time of the IRT data acquisition, as well as the true input data for the LETp estimates.
The EF method uses few input data compared to the SF method (Table 2).Inputs used for EF method could all be derived from remote sensing while inputs used in the Introduction

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Full SF method have to be computed from a model of LETp (an energy balance model).We check below that the LETp model, though not perfect, provides accurate and unbiased estimates at the time of the satellite overpass.

Estimating seasonal ET from instantaneous latent heat flux on clear sky days at satellite overpass
Because data are available only for clear sky days, algorithms to interpolate ET between two successive image acquisition dates are tested.The assumption is to perform a linear interpolation of EF and SF between two successive (clear) days of data and to multiply EF or SF by AE or LETp (respectively) computed during the intermediate cloudy days.
Then, by using and combining both methods to interpolate and to extrapolate, seasonal ET can be simulated.In what follows the same factor (either EF or SF) is used for consistency for the combination of interpolation and extrapolation in order to reconstruct seasonal ET from an instantaneous estimate.We did not test a combination of both methods for interpolation and extrapolation.
Moreover, the study is focused on the reconstruction of ET over an entire growing season, which usually covers several months.However, for some discussions (say, on the optimum revisit frequency), this criteria may not be the most relevant, and another time scale should be considered.For irrigation monitoring or water stress detection for instance, a shorter timescale, typically that of an average interstorm, should be looked at, but this is beyond the scope of this particular study, and would not be feasible with the limited number of water stress events sampled in the various datasets.

Time of day representativeness and revisit frequency
The different instantaneous in situ data (used here as substitutes for the instantaneous estimates that could be later on derived from RS data at the time of satellite overpass) Introduction

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Full This hypothesis is discussed in a later section.The reconstruction performance of daily ET from a one-time day measurement is tested for times of day ranging from 10:00 a.m. to 02:00 p.m.
To deal with the seasonal reconstruction of ET from RS data, the impact of different revisits on the performances of the two methods is also tested from 1 to 16 days.Hoedjes et al. (2008) assessed the validity of EF self-preservation.Assuming EF selfpreservation appears to be valid under dry conditions but no longer valid under wet conditions.

Improvement of the classical method EF
For those conditions, EF shows a concave-up shape.In agreement with the results reported by Lhomme and Elguero (1999), Gentine et al. (2007) and Hoedjes et al. (2008) have shown that assuming a constant EF underestimates actual EF and then ET.Our results corroborate this (Tables 3 and 4).
According to Gentine et al. (2007), EF diurnal course depends on both atmospheric forcing and surface conditions.As presented before, Hoedjes et al. (2008) introduces a more complex parameterization of the EF diurnal cycle: where EF 12obs and EF 12sim are the observed and simulated EF values at noon, respectively.RH is the air relative humidity.Hoedjes et al. (2008) tested the parameterization for an olive tree orchard in Morocco on ten-days wet periods for daytime value only and showed that the errors on ET calculation are reduced to less than 0.5 %, whereas an underestimation of 8 % on average was observed when assuming EF self preservation.Introduction

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Full In what follows, the parameterization is tested on the entire dataset i.e. the eleven agricultural sites and multiple crop types, which amounts to 1683 clear sky days of data.

Results
Since each method is primarily dependent on the model accuracy when computing the the available energy (AE) and potential evapotranspiration (LET P ), we first show the model performance in estimating AE and LETp.Then the results of both EF and SF methods in reconstructing daily ET are presented, together with the impact of the time of satellite overpass and the time of revisit on this performance, because once again these methods are meant to be applied operationally.Eventually, a similar analysis is carried out for the seasonal ET reconstruction, and the interest of a proposed improvement of the classical EF method is shown for the daily ET reconstruction.

Available energy diurnal course
An overestimation of about 10 % is found between the estimated (Eq. 4) and the measured daily components of the available energy (not shown).An overestimation of the same order of magnitude has been also reported by Anderson et al. (1997) on a different dataset.Subsequently, the following corrected parameterization of AE is used: Further research is required to investigate what are the physical reasons behind this overestimation, but this rather simple parameterization (Eq.6) seems to be systematic Figures

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Full enough to be routinely used in the modeling framework presented in Anderson et al. (1997).With Eq. ( 6), biases between simulated and observed ET are reduced by 25 % up to 40 % (depending on sites) compared to Eq. ( 4).RMSE values are also reduced by 18 % on average.Nash efficiency fluctuates between 0.4 and 0.6 when using Eq. ( 6) instead of 0.28 and 0.4 if Eq. ( 4) is used.The corrected parameterization (Eq.6) leads to a RMSE as low as 38 W m −2 on average for every site.
It must be pointed out that only diurnal values are used to compute all statistics, defined by a threshold on incoming solar radiation values (R g > 10 W m −2 ).

LETp simulation model
This uncalibrated model performs well for unstressed periods corresponding to the interval between an irrigation or a rainfall event and the occurrence of water stress or the next income of water, whichever comes first.For those days there is a bias at noon ranging between 44.5 W m −2 and 87.5 W m −2 depending on the dataset.
Figure 3 shows a scatterplot of computed vs. observed LETp at noon during unstressed periods for Aurad é dataset in 2006.One can note that with the chosen default values of the parameters, the model performs effectively well but tends to overestimate LETp.

EF and SF methods of extrapolation
Both methods show similar performances for the reconstruction of daily ET from an instantaneous measurement at midday on sites exhibiting very few stress events (Table 3).On these sites, the EF method shows global RMSE of 0.78 mm day −1 while global RMSE is about 0.73 mm day −1 for the SF method.The mean bias on those unstressed sites calculated for EF method is about −0.39 mm day −1 and about −0.31 mm day −1 for the SF method.However, the method based on EF tends to outperform the method based on SF for most sites that exhibit a significant water stress Figures

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Full ( Aurad é, 2007;Lamasqu ère, 2006Lamasqu ère, , 2007;;Morocco, 2004).On Aurad é 2007, a site presenting 5 stress periods, RMSE is about 0.60 mm day −1 when ET is reconstructed with the EF method, and about 0.81 mm day −1 with the SF method.On this site, the EF method produced an efficiency of 0.70 whereas the SF method leads to an efficiency of 0.30.
For those datasets, the EF method shows a very small bias (absolute value less than 0.11 mm day −1 ) whereas the bias is commonly greater on other sites.A similar observation can be made for RMSE values (0.45 mm day −1 average for sites with water stress, 0.65 mm day −1 elsewhere).For Aurad é, in 2007, where we detect a significant number of 5 water stress periods, RMSE values between observed and simulated ET are 0.6 mm day −1 when calculated with the EF method and 0.81 mm day −1 when using the SF method.

Time of day representativeness
In the previous paragraphs, reference time is noon.In what follows we want to assess the impact of overpass time on the reconstruction of the diurnal cycle.
Both methods were tested for different time of overpass in order to estimate the most relevant hour to scale diurnal ET.
We observed that the number of available data is similar from 10:00 a.m. to 02:00 p.m. for each site.
Figure 4 shows the influence of the time of overpass on the estimation of the water lost through evaporation (computed with EF method) at seasonal scale.The Nash criteria testifying the efficiency of the evaporation reconstitution, presented on Fig. 4, is higher when the reference instantaneous measurement is taken at noon.Before and after, this efficiency is deteriorated: the furthest the time of day representativeness is from noon, the weakest the efficiency.Note that, in that case, seasonal scale means that only the evaporation occurring during clear sky days is taken into account.
The evaporation of cloudy days is not evaluated and does not amount to the total cumulative quantity.It is shown that the actual water lost through evapotranspiration is 1718 Figures

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Full underestimated by 5 % to 18 % (depending on the sites) at noon by our model.This underestimation increases with an earlier or later time of overpass (from 14 % to 37 % at 10:00 a.m.).This underestimation remains relatively small within the 11:00 a.m. to 01:00 p.m. time frame.
Those results are in agreement with the theoretical study by Gentine et al. (2011) which is based on an analytical estimation of peak latent heat flux as a response to a sinusoidal radiation forcing.

Reconstructing seasonal ET for one-day revisit frequency
According to Fig. 5, with a revisit frequency of one day, the water stress is often overestimated when performing a linear interpolation between two successive days of available data, which implies that ET is underestimated.Results show indeed a significant underestimation of ET, with cumulative differences of 50 mm for some sites as Aurad é in 2007, Avignon in 2004 or Lamasqu ère in 2006 (Table 4).
Actually, the assumption of performing a linear interpolation of SF or EF for cloudy days can be discussed.For those days indeed, SF and EF are often bigger than what would happen if full radiation was available: during cloudy or overcast days, the evaporation process can be limited by the low available energy while during the previous and the following clear sky days the available energy (and thus the atmospheric evaporation demand) is sufficient to produce water stress.Moreover, the ratio which defines SF or EF does not have a real typical course during cloudy sky conditions and it sometimes makes a great difference for the daily ET reconstruction algorithm for cloudy days.
During cloudy sky periods, AE has a similar diurnal course as R g and the evolution of the EF ratio is closer to reality than the evolution of SF ratio when performing a linear interpolation, meaning that the EF method is more relevant.Introduction

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Full What can be observed is that this negative bias between computed SF and actual SF is greater than the bias between computed and actual EF.This can be partly explained by the fact that AE is similar in stressed and unstressed conditions whereas LET drops significantly while LETp often increases significantly at the same time during stressed conditions.Inaccurate prediction of stress through temporal interpolation leads therefore to higher discrepancies for the SF method than for the EF method.
SF method displays the largest underestimation in ET estimations (Fig. 5).The results obtained to compute water lost through evapotranspiration (i.e. a seasonal accumulation in mm, Table 4) on sites exhibiting several periods of water stress are improved when using the EF method and in particular when extrapolation and interpolation are combined to compute seasonal evapotranspiration.Indeed, for sites where the underestimation of ET is the most important, we can note that the gap is greater when ET is modeled with the SF method.On Aurad é 2007, EF method underestimates actual ET by 57 mm while the SF method produces an underestimation of 95 mm.On Lamasqu ère 2006, actual ET is underestimated by 55 mm with the EF method and by 77 mm with the SF method.

Reconstructing seasonal ET for different revisit frequency
Up to now it was assumed that an instantaneous estimate is available for each clear sky day, which corresponds to an everyday revisit frequency.In this section we analyze the evolution of the performances of both methods when selecting different revisit frequencies, from 1 day (typical of many low resolution satellites such as MODIS) to Introduction

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Full perhaps not the best one to investigate the optimum revisit strategy, since errors tend to be smoothed out at the monthly scale.It therefore represents "climatological errors".Moreover, for most datasets exhibiting water stress, the performance criteria vary rather chaotically from one frequency to the other (Fig. 6a).Indeed, when the revisit frequency is greater than one day, some clear sky days are not observed and therefore some water stress periods are not detected.But this lack of detection does not occur for each combination of observed clear sky days.
On sites evaporating mostly at a potential rate (Fig. 6b and c), results do not vary significantly with the revisit frequency, even if we can point out that after 10 days of revisit, the performances of the interpolation algorithm drop significantly.The EF method outperforms the SF method at any revisit frequency, and the deterioration with increasing revisit frequency is more pronounced for the SF method.

About the time of overpass
Impacts of the time of overpass on seasonal reconstruction performances are in agreement with those presented in Sect.3.1.4.For each revisit frequency, it appears that noon is the most representative hour to reconstruct seasonal ET.Again, the criterion used to assess the performance is the difference between observed and simulated seasonal cumulative evapotranspiration.
One can note that with an earlier (or later) time of overpass, results are more significantly and quickly deteriorated, but the general trend remains the same than when noon is used.

An alternative efficient operational method to reconstruct seasonal ET
In our study, we did not find a consistent pattern for SF diurnal fluctuations.This might be due to the variable discrepancies between stomatal functioning in actual and potential conditions, respectively.Again, in order to select one operational method for daily Introduction

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Full and seasonal ET reconstruction, we decided to improve the best performing method (the EF method) with the known diurnal shape of EF instead of self preservation.
According to the work of Hoedjes et al. (2008) which introduces a bivariate linear relationship to parameterize the EF diurnal cycle (Eq.6), an improvement of the classical method to reconstruct daily ET (ET d ) from EF is tested.This parameterization depends on two routinely available atmospheric forcing parameters, the incoming solar radiation and the relative humidity of the air.In order to be consistent with the previous parameterizations, a variation is used here to account for the bias observed in the reconstruction of daily AE: The coefficient 1.1 corrects for the overestimation of diurnal AE when instantaneous AE at midday is used (see Sect. 3.1.1 and Anderson et al., 1997).With this parameterization (called "variable method" on Fig. 7) of the concave-up shape of EF during the day, an important improvement can also be observed for all but one datasets.This is particularly true when looking at the water lost through ET during the season (Table 5).
It is shown that ET is underestimated by an average of 15.8 % using constant EF.The error is reduced to an average of 1.9 % using the variable EF parameterization (Eq.7).
The parameterization (Eq.7), tested here on crops, allows a great improvement in the reconstruction of daily ET for every site and the large number of climatological situations sampled in our datasets.But, this parameterization allows great improvement because the climatic years of data tested are relatively little stressed.Indeed, on little stressed days, the shape proposed with Eq. ( 7) fits well with the real shape of EF, whereas for stressed days, the shape of EF is more constant during daytime.To improve the method, a criterion based on a rough indicator of the presence or not of water stress should be determined Introduction

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Full in order to choose the better parameterization of EF (constant or concave-up shape) to use to reconstruct ET.

Conclusions
Two methods to reconstruct daily and seasonal evapotranspiration from an instantaneous estimate at the time of satellite overpass during clear sky days were compared.
Both methods were tested on a large range of sites and vegetation types under contrasted climatic conditions.One uses as a proxy the evaporative fraction (EF) to extrapolate instantaneous ET to daily values by assuming a self preservation of EF during the day and interpolate between two successive clear sky days, the second does the same but with the stress factor (SF) which needs more input data and cannot be derived from RS data only.We found that for sites with no more than two periods of water stress longer than four days, EF and SF reconstruction methods show similar results.
However, for sites with a larger number of water stress periods, the EF method tends to outperform the SF method both for daily and seasonal reconstruction.Furthermore, the extrapolation results are significantly improved by modifying the parameterization of EF in order to take into account the diurnal fluctuations of EF as an empirical bilinear function of solar radiation and relative humidity of the air.An improved parameterization of SF could also be used, but a particular diurnal shape of SF is difficult to find and then to parameterized.Some developments about the SF diurnal shape needs to be led.Both methods could be improved in reducing the bias due to errors in the LETp simulation model (SF method) or AE simulation model (EF method).For the second (SF) method, the energy balance model used to compute LETp could be improved, for example by tuning some of the unknown parameters (minimum resistance etc) in order to minimize the difference between the surface temperature in potential conditions and the observed remotely sensed radiative temperature in unstressed conditions.For the first method, the universality of the empirical correction factor of about 10 % calculated Introduction Full when modeling AE should be tested on a wider range of surface and climatic conditions.
Finally, the EF method to reconstruct daily and seasonal ET has been tested here with in situ data.In order to evaluate the method when using real remote sensing data, a study including errors on instantaneous EF when the later is derived from remote sensing model needs to be carried out.Full  Full    Full Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | also revised the assumption of EF daytime self preservation in order to obtain a better estimate of evapotranspiration.They parameterized EF Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | 54 97 N, 01• 10 61 E) and Lamasqu ére (43 • 49 65 N, 01 • 23 79 E), separated by 12 km and located near Toulouse (South West France).The second is situated near Sidi Rahal in the Haouz plain in Morocco (31.67250 • N, 7.59597 • W).The third one in Avignon in South Easter France (43.92 • N; 4.88 • E).Aurad é, Lamasqu ère and Avignon Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Various formulations were proposed for estimating AE from an instantaneous estimate at a given time of the day (see Chavez et al., 2008).Sobrino et al. (2007) used parametric equations which allow to derive daily net radiation from instantaneous net radiation as a function of the day in the year and the acquisition time of satellite data (surface temperature and albedo).Hoedjes et al. (2008) used a parameterization of AE based on a function of global incoming radiation (R g ) and atmospheric thermal irradiance (R atm ).In their study, the Discussion Paper | Discussion Paper | Discussion Paper | d are computed using an energy balance model described byGentine et al. (2007).It is a dual-source energy budget model which requires various input data related to the atmosphere, such as air temperature, wind speed, relative humidity and global radiation, the vegetaion development and physiology, such as LAI, vegetation height, minimum stomatal resistance, and the soil.Some of these data can be taken from nearby meteorological stations and from remote-sensing informations.But others, like the minimum surface resistance to transpiration, as well as the various parameters of the aerodynamic resistances, are more difficult to infer without a proper in situ measurement.LETp is computed by specifying minimum values for the stomatal closure due to water stress and zero-value for the soil resistance to evaporation Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | are taken at midday by default.According to Gentine et al. (2007), midday is the most representative hour to reconstruct ET from the EF.
Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |

Acknowledgements.
Fundings from the French space agency (Centre National d'Etudes Spaciales, CNES) for the MiSTIGRI (MicroSatellite for Thermal Infrared Ground surface Imaging) phase A study, the MISTRALS (Mediterranean Integrated STudies at Regional And Local Scales) SICMed (Continental Surfaces and Interfaces in the Mediterranean area) program and the European FP7 projetc Sirrimed (FP7-KBBE-2009-3 Grant Agreement Number 245159) are gratefully acknowledged.Avignon and SudOuest data were acquired and processed in the frame of the CARBOEUROPE-IP and the CARBOFRANCE project funded by the European FP7 Program (GOCECT-2003-505572) and the French Ministry in charge of Environment (GICC programme).For the R3 dataset, in addition to IRD, financial support was provided by EC in the frame of the WATERMED project (contract ICA3-CT-1999-00015) and IRRIMED project (contact ICA3-2002-10080) and by the French Programme National de T él éd étection Spatiale (PNTS) and the French space agency (CNES).The publication of this article is financed by CNRS-INSU.Discussion Paper | Discussion Paper | Discussion Paper | Shuttleworth, W. J., Gurney, R. J., Hsu, A. Y., and Ormsby, J. P.: FIFE: the variation in energy partition at surface flux sites, IAHS Publ., 186, 67-74, 1989.Sobrino, J. A., G ómez, M., Jim énez-Mu ñoz, J. C., and Olioso, A.: Application of a simple algorithm to estimate daily evapotranspiration from NOAA-AVHRR images for the Iberian Peninsula, Remote Sens. Environ., 110, 139-148, 2007Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |

Figure 1 :
Figure 1: Observed clear sky radiations vs computed clear sky radiations at noon with FAO model and Meeus, Bird and Hulstrom model , on Lamasquère 2007

Figure 4 :
Figure 4 : Deterioration due to time of acquisition

Figure 7 :Fig. 7 .
Figure 7 : Simulation of EF with the parameterization using Rg and RH -Morocco 2004, example for day of year 73

Table 2 .
Inputs data used for the two methods and in the SVAT model.

Table 3 .
Statistic comparison of method EF and method SF.Reconstruction of ET on clear sky days from a one-time-day measurement.

Table 4 .
Seasonal ET (mm) simuled with EF and SF methods for one-day revisit frequency.