Sequential Assimilation of ERS-1 SAR Data into a Coupled Land Surface Hydrological Model Using an Extended Kalman Filter

APRIL 2003 FRANCOIS ET AL. 473 Sequential Assimilation of ERS-1 SAR Data into a Coupled Land Surface–Hydrological Model Using an Extended Kalman Fi...
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Sequential Assimilation of ERS-1 SAR Data into a Coupled Land Surface–Hydrological Model Using an Extended Kalman Filter C. FRANCOIS CETP, Velizy, and ESE, Universite´ Paris Sud, Orsay, France

A. QUESNEY

AND

C. OTTLE´

CETP, Velizy, France (Manuscript received 15 May 2001, in final form 13 September 2002) ABSTRACT A first attempt to sequentially assimilate European Space Agency (ESA) Remote Sensing Satellite (ERS) synthetic aperture radar (SAR) estimations of surface soil moisture in the production scheme of a lumped rainfall– runoff model has been conducted. The methodology developed is based on the use of an extended Kalman filter to assimilate the SAR retrievals in a land surface scheme (a two-layer hydrological model). This study was performed in the Orgeval agricultural river basin (104 km 2 ), a subcatchment of the Marne River, 70 km east of Paris, France. Assimilation was tested over a 2-yr period (1996 and 1997), corresponding to 25 SAR measurements. The improvements observed in simulating flood events demonstrate the potential of sequential assimilation techniques for monitoring surface functioning models with remote sensing data. It was demonstrated that the method could correct for some errors or uncertainties in the input data (precipitation and evapotranspiration), provided that these errors are not greater than 10%. The overall agreement between uncertainties predicted through the extended Kalman filter scheme compared to uncertainties obtained through the ensemble technique reaffirms the validity of the extended Kalman filter scheme but also demonstrates its limits. Questions are raised concerning the determination of sequential model errors.

1. Introduction The difficulty of obtaining a reliable flood forecast in operational hydrology is related to the understanding of initial states and the lack of an accurate model. Regular state measurements (especially through remote sensing) are expected to give additional information to obtain better flood forecasts. Different studies have shown that a better understanding of surface soil moisture improves the estimations of water exchanges at the soil–vegetation–atmosphere interface, the soil water transfers in the unsaturated zone, and, finally, the water flows at the outlets of the watershed (Loumagne et al. 1991; Ottle´ and Vidal-Madjar 1994). At the same time, different authors demonstrated the potential of thermal infrared and microwave remote sensing for estimating surface soil water content (Ulaby et al. 1986; Carlson 1986). The objective of this study is to explain how soil water content estimated from space measurements may be used in a practical way with a hydrological model through sequential assimilation. The most widespread method of sequential correction Corresponding author address: Christophe Francois, ESE, Universite´ Paris Sud, Baˆt 362, 91405 Orsay Cedex, France. E-mail: [email protected]

q 2003 American Meteorological Society

is that proposed by Kalman (Jazwinsky 1970). The value of the adjustment is estimated depending on the model errors and measurement errors at the time of the assimilation, in order to minimize the error variance of the a posteriori estimate. The use of a sequential assimilation process requires the calculation of the model uncertainty and the specification of the measurement errors. This method is commonly used in operational hydrology to assimilate antecedent streamflows in hydrological models (Kitanidis and Bras 1980a,b). In this study we test how soil moisture estimations from the synthetic aperture radar (SAR) on board the European Space Agency (ESA) Remote Sensing Satellite (ERS) can be used to improve flood forecasts. Earlier studies investigated sequential assimilation schemes for soil moisture profile retrieval. Entekhabi et al. (1994) conducted a simulation study using a linearized Kalman filter for assimilating microwave and thermal brightness temperatures into a heat and moisture diffusion equation model (25 soil layers), based on Darcy’s equations. Their article is the first to use a Kalman filter coupled to a radiative transfer model (Njoku and Kong 1977). They demonstrated how surface soil moisture observations can be used to correct deep (1 m) soil water content and temperature simulations. The study,

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however, has limitations. It is a simulation study, for bare and smooth soil conditions, and frequent (hourly) observations are always available. A linearized Kalman filter is used, and the propagation and prediction system (heat and moisture transfer model) as well as the observation system (brightness temperature model) are linearized. Galantowicz et al. (1999) show the possibility of retrieving the measured soil state profile to 1-m depth using real measurements (daily L-band brightness temperature field observations). They also improve the simulation tests within a 4-month time period, including storms and interstorms, with a more realistic observation interval of 3 days. In the same way, Hoeben and Troch (2000) show how to retrieve soil moisture profiles by assimilating radar data in a one-dimensional surface model. They used synthetic radar data to validate their approach. In all of these studies, however, only bare soils are considered. Houser et al. (1998) address the spatial scale of the problem using statistical approaches (statistical corrections, Newtonian nudging, and statistical interpolation) and demonstrate the feasibility of soil moisture data assimilation in a spatially distributed hydrological model. In the same way, Pauwels et al. (2001) show the improvements brought by assimilating the spatial patterns of remotely sensed soil moisture in the TOPMODEL-based Land–Atmosphere Transfer Scheme (TOPLATS) model over small-scale river basins using the nudging to individual observations and the statistical corrections methods. Walker (1999) also addresses the spatial scale by developing a modified Kalman filter assimilation scheme for 3D applications. Finally, Reichle et al. (2000, 2001, 2002) developed and compared two different techniques, the ensemble Kalman filter and the 4D variational methods, to assimilate microwave (L band) synthetic brightness temperatures in a surface model. Their study shows that monitoring the surface scheme with microwave remote sensing data improves the results. In comparison to the previous studies presented above, the approach in our paper is simplified in two ways: two soil layers only are considered in the surface model, and the spatial extent is not treated since a conceptual lumped model is used to simulate the hydrological processes over the whole watershed of size 104 km 2 . The difference with previous studies is threefold: 1) real satellite data (ERS-1 SAR) over a 1-yr period are considered, 2) vegetation cover as well as root water uptake are considered, and 3) an extended Kalman filter (instead of a linearized Kalman filter) is used. This allowed us to address the more general problem of error propagation in (nonlinear) soil–vegetation– atmosphere (SVAT) models. This work concludes studies that were initiated in the context of the Hydrological Atmospheric Pilot Experiment–Mode´ lisation du Bilan Hydrique (HAPEX– MOBILHY) Ottle´ and Vidal-Madjar (1994) show that a better simulation of the water exchanges at the soil–

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FIG. 1. Description of the GRKAL model (surface scheme and subterranean module). The soil surface is represented by a two-layer system (surface layer and deeper soil layer) with two state variables, w s and w p (soil water contents). The inputs of the surface model are precipitation P, potential evapotranspiration ETP, and vegetation fraction s y (vegetation cover). The model computes the partitioning of precipitation (a), the surface and root zone evapotranspirations (E s and E p ), and the exchange between layers. The output of the surface model is the drainage Drain of the deeper layer. The subterranean model deals with the direct production PR and drainage Drain to simulate the daily streamflows. See text and appendix A for more details and model equations.

vegetation–atmosphere interface with a reinitialization of the surface soil moisture (estimated from thermal infrared remote sensing) leads to better results in the simulation of the water flows at the outlets of the catchment. Therefore, three issues were explored: 1) an adequate modeling of the soil–vegetation–atmosphere interface (hereafter named the surface model) has to be included in the hydrological model, 2) soil moisture data has to be obtained from remote sensing observations, and 3) an assimilation scheme has to be developed. The first issue is explained in Loumagne et al. (1996) and in the present work, the second is developed in Quesney et al. (2000), and the third issue is the subject of our study. This article is presented in four sections: 1) introduction of the hydrological and surface model, 2) data and assimilation methodology, 3) results and further discussions related to the assimilation method, and 4) conclusions. 2. Model description The model used in this study, named GRKAL (Fig. 1), is based on the daily conceptual rainfall–runoff models named GR4 and GRHUM (Edijatno 1991; Loumag-

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ne et al. 1996). The input to the model is the daily precipitation P. The model is then divided into two parts: the surface scheme (itself divided into a surface layer and a deep soil layer) and the subterranean module (see Fig. 1). The input of the surface scheme is the proportion of rain P9 5 (1 2 a)P that enters the soil, and its output is the deep soil layer drainage Drain. The input PR of the subterranean model is the proportion of rain that does not pass through the soil aP plus the drainage coming from the deep soil layer: PR 5 aP 1 Drain (Fig. 1). The subterranean part of the model is described in Loumagne et al. (1996) and has not been modified since then. The transfer function includes a unit hydrograph to split between a routing reservoir and subterranean exchanges. The surface scheme (surface layer and deep soil layer) is described in the following subsections, since some parameterizations have been modified compared to Loumagne et al. (1996) in order to reduce the number of parameters. Surface scheme: General presentation The surface model accounts for the hydrological balance at the soil surface. The soil surface is represented by a two-layer system (surface layer and deep soil layer) with two state variables, which are the soil water contents of the two layers. The inputs of the surface model are precipitation P, potential evapotranspiration ETP, and vegetation fraction s y (vegetation cover). The output of the surface model is the drainage of the deeper layer. The superimposed layers approach requires a hydraulic diffusivity to deal with water exchanges between the two layers (basically, the water transfers from the surface layer to the deeper layer, and, occasionally, water transfers from the deeper layer to the surface layer). The state variables of the surface model are the soil water contents: w s for the surface layer and w p for the deeper layer. The assimilation scheme (where w s is assumed to be observed with a given accuracy) ‘‘corrects’’ w s and w p with the observations. In the next time steps, the model will use better estimates of soil moisture and, consequently, better estimates of rain division and deep layer drainage. The model solves the following two-equation system where k represents the time step, P9 the rain that enters the soil, E s and E p the surface and root zone evapotranspirations respectively, Exc the water exchanges between the two layers, and Drain the drainage from the deeper layer: wsk11 5 wsk 1

P9 2 E s 2 Exc ds

wpk11 5 wpk 1

Exc 2 E p 2 Drain. dp

(1)

More details on the surface model and its different components are given in appendix A.

3. Dataset and methods a. Available data 1) THE ORGEVAL

CATCHMENT

The Orgeval catchment (104 km 2 ), which lies about 70 km east of Paris, France, is an experimental basin managed by the French ‘‘Cemagref’’ institute for 35 years. It is rather flat, its main part being covered with thick tableland loess; the upper layer is a silt–loam soil. The Orgeval watershed is mainly an agricultural area (about 60% of the total land is covered by crops). A more detailed description of the Orgeval basin is given in Quesney et al. (2000). 2) REMOTE

SENSING DATA

Between 1995 and 1997 we assembled a database of in situ surface soil moisture measurements representative of the Orgeval catchment area as well as a remote sensing database derived from both the Landsat thematic mapper (TM) and ERS SAR. It allowed us to develop a methodology to convert radar backscattering signal into surface soil water content at a regional scale. Quesney et al. (2000) showed that the (0–5 cm) surface soil moisture can be estimated from ERS SAR remote sensing data at the basin scale with an accuracy of about 5%. The method and its validation on the Orgeval River basin are fully described in Quesney et al. (2000). These authors have calibrated a semiempirical relationship over a 2-yr period relating the average soil moisture to the ‘‘equivalent bare soil’’ backscattering signal estimated at the regional scale. The equivalent bare soil backscattering signal is derived from the high-resolution radar imagery by averaging the radar signal only over pixels for which the sensibility to surface soil moisture is significant. These pixels correspond to bare soils or low vegetation where the signal can be corrected from the vegetation attenuation effects using a first-order radiative transfer modelization. The area must be large enough to average roughness effects and assume constant mean roughness at the regional scale. This methodology has been validated over different regions, and the results are presented in Le He´garat-Mascle et al. (2002). In the present study, Quesney et al.’s (2000) soil moistures derived from the ERS SAR instrument have been used to control our hydrological model. Because of the repetitiveness of the ERS SAR instrument (35 days), 25 satellite soil moisture estimations were obtained over the 1995–97 period with an accuracy ranging between 4% and 6% (the errors range from 0.005 to 0.02 cm 3 cm 23 in absolute values). 3) SOIL

MOISTURE DATA

Continuous soil moisture measurements have been carried out using different methods [neutron probes, time-domain reflectometers (TDRs), capacitive probes]

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from 1990 to 1997, in the surface and deeper layers (respectively 0–150 and 150–1150 mm). The most reliable data correspond to the 1990–94 period (neutron probes). From 1995 to 1997, TDR measurements were carried out and recalibrated using neutron probe measurements. It is important to point out that these measurements are local measurements. On the contrary, the soil moisture data (‘‘observations’’) derived from the satellite semiempirical algorithm are obtained by comparing a large number of surface soil moisture measurements from all over the Orgeval basin to the radar backscattering signal integrated over the selected pixels in the watershed. 4) DISCHARGE

5) MODEL

X 1 5 X 2 1 K [z 2 h(X 2 )], 2

INPUT DATA

The whole dataset covers the period 1990–97. Daily measurements of precipitation, evapotranspiration (ETP), and vegetation coverage (veg) are available. b. Methodology Our main objective in this study is to test how soil moisture retrievals from ERS SAR measurements may be used to improve flood forecasts. For this purpose the hydrological model described above was used to assimilate the satellite soil moisture data using an extended Kalman filter. The first step is the calibration and validation of the model. The second step is the application of the assimilation scheme. Two different assimilation methods have been applied. The first method, hard updating (Walker 1999), simply consists of replacing the simulated variable (e.g., surface moisture) with the observed one. The second method, the Kalman filter (Jazwinsky 1970), differs from hard updating in two ways: first, the uncertainties of the observations and simulations are taken into account, and second, the correlations between errors in the state variables are taken into account to correct the other state variables (here the deep moisture w p ). 1) THE KALMAN

FILTER

The Kalman filter offers a method to predict the uncertainty of the variables and the strength of the correlation between these variables through the variance– covariance matrix P that characterizes the prognostic model m. The updates for all variables X are then obtained through (Jazwinsky 1970)

(2) 1

where X refers to a priori estimates; X refers to a posteriori (updated) estimates; K is the Kalman gain matrix (size n y 3 nobs ), where n y is the number of variables (2 in our case) and nobs the number of observations at time step k (1 in our case); and z is the vector of observations and h the observation model. The term between brackets in Eq. (2) represents the innovation vector (size nobs ). The Kalman gain K, at one time step k, is computed through the model error variance–covariance matrix P k , the observation error matrix R k , and the Jacobian matrix H k of the observation model h: K k 5 P k H Tk (H k P k H Tk 1 R k ) 21 .

FLOW DATA

Daily streamflow measurements were taken at different places over the basin. In this study, measurements of streamflows at the outlet of the watershed were used to calibrate or estimate the performance of the coupled surface–subterranean model.

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(3)

The model error variance–covariance matrix P k11 is computed given the variance–covariance matrix P k (size n y 3 n y ) at time step k, the Jacobian matrix M k (size n y 3 n y ) of the propagation model m, and the sequential error matrix Q k (size n y 3 n y ): P k11 5 M k P k M Tk 1 Q k .

(4)

In this equation, the first term represents the propagation of the errors already contained in the model at step k (P k ): this error may be increased or decreased by the model M k between steps k and k 1 1. The second term Q k represents all the (unbiased) errors that appear between steps k and k 1 1 because of model inaccuracy (in its equations or parameters), independently of previous errors. In practice, these errors, referred to as ‘‘sequential errors’’ in the following, are frequently assumed to be uncorrelated; that is, Q k is a diagonal matrix, where each element (q s and q p ) represents the variance that is added to the corresponding variable (w s and w p ). These variances (n y variances in general, for the n y state variables) may be seen as additional calibration parameters. They characterize the degree to which the model error increases with time. In the hydrological model we are considering, terms in M are often less than one (negative feedback leading to stable behavior), and therefore the M k P kM Tk term tends to decrease the error P k , while the Q k term is the one that allows the error to increase again, due to model inaccuracy. Commonly, the sequential errors (q s and q p for the two soil moisture variables in our case) are arbitrarily fixed to reasonable values, in order to obtain model errors P comparable to observed errors. In this study we used a sensitivity study on q s and q p to study the corresponding model uncertainties evolution (see the model errors discussion, section 4c). 2) THE

EXTENDED

KALMAN

FILTER

For linear models, M and H are directly the prognostic and observation models. For nonlinear models, on the other hand, M and H are the Jacobian of the surface model m and the observation model h, which are locally

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TABLE 1. Calibration coefficients of the surface hydrological model. These values correspond to a calibration over the 1991–94 period, minimizing Eq. (6). See appendix A and Eqs. (A1)–(A9) for the meaning of the symbols. wsat

wwilt

wmin

a1

a2

a

b

y

ds

dp

pr

h

d1

d2

CLAY

0.40

0.16

0.05

5

9.4

20.6

99.8

0.02

150

2430

0.8

0.50

6.7

8.7

40.5

linearized. The Kalman filter is then named the extended Kalman filter (EKF). After the update process, the whole prognostic model m is used to propagate the state variables X toward the next step, and no linearization is used in the propagation step. In this study, soil moisture is directly derived from remote sensing observations and assimilated; therefore, no observation model is used (h 5 1) and the only difficulty in applying the extended Kalman filter is the error propagation through Eq. (4) and the determination of variances q s and q p to form Q k . 4. Results and discussion a. Model calibration (1990–94) The surface and subterranean models need to be calibrated (15 coefficients for the surface model and 5 coefficients for the subterranean model). The calibration period is 1990–94. Three types of measurements were used to constrain the models during this period: surface moisture measurements, deep soil moisture measurements, and discharge flows (see section 3a). It may be pointed out that this calibration is not optimal since the soil moisture measurements are local measurements and may not be representative of the whole catchment. The deep soil moisture measurements, on the other hand, are expected to be more stable over the catchment, and then better linked to the average hydrolic state of the basin. The objective function used to calibrate the model using these three different sources of data is the Nash criterion (Nash and Sutcliffe 1970) or modeling efficiency (Janssen and Heuberger 1995):

 O (X Na 5 100 1 2  O (X 

n



i51 n



i51

obs i

obs i



2 X isim ) 2 2X

obs

 , )

(5)

2



where X i represents some variable (soil moisture, streamflow), observed or simulated; and Na is expressed as a percentage, 100% meaning a perfect agreement between observations and simulations. A weighted sum of three Nash criteria has been used in the total objective function: J(A) 5 m1Na(w s ) 1 m 2Na(w p ) 1 m 3Na(Q),

(6)

where Na(w s ), Na(w p ), and Na(Q) respectively represent the Nash criterion on surface moisture, deep moisture, and river flows. Coefficients m1 , m 2 , and m 3 serve to balance the objective function J in order to favor one or the other components. This step in the calibration is

subjective and arbitrary, as it is frequent in multicriteria model calibration [alternative methods may be employed: see Gupta et al. (1999)]. Another alternative is to obtain estimations of the measurement errors and incorporate them into the Nash criterion. It is difficult, however, to quantitatively translate that w s and w p observations are local observations and should not be given the same weight as the river flows in the calibration. This is the reason that a semiempirical objective function was used. Priority was given to the streamflows Q, then to w s and w p , resulting in m1 5 1, m 2 5 1, and m 3 5 5. The minimization method used simulated annealing coupled to the simplex method [see Quesney (1999) for more details]. The obtained coefficients are given in Table 1. The surface thickness d s was forced to 150 mm to correspond to the soil moisture measurements and radar penetration depth. The values obtained after the calibration are satisfactory, especially for the percentage of clay, the wilting point, and maximum soil moisture, for which the calibrated values correspond to the real observed values. Figure 2 shows the evolution of soil moisture in the surface and deeper layer after calibration in 1990–94. The initialization of soil moisture is not an important point for our purpose: a poor initialization is corrected 3 or 5 weeks after the beginning of the simulation (in winter), when precipitation reinitializes both layers to their maximum values. This would not be true in another region with a pronounced dry period. However, even with a good initialization (or reinitialization in rainy periods), the model departs from the measurements (which are here only indicative local measurements). Figure 2 (bottom) also shows the discharge flows simulation. The Nash criterion calculated on the streamflows Na(Q) is 85.1% for this simulation, which is rather satisfactory. It is interesting to note that when w s is lower than 0.20, all the precipitation is directed toward the soil. When w s equals 0.31, precipitation is equally divided between the surface model and runoff. Therefore, the same error in the soil moisture will have different consequences depending on the surface soil moisture: for example, in the Orgeval soil and climatic conditions, if w s 5 0.18, an underestimation of 10% in the soil moisture will have no consequence on the streamflows, but the same underestimation will double the runoff if w s 5 0.30. b. Model validation: Simple simulation (1995–97) The model was validated during the 1995–97 period (see Fig. 3). The first observation is that the Nash cri-

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FIG. 2. Simulation of soil moisture in the surface and the deeper layer, and discharge flows after calibration over the 1990–94 period.

FIG. 3. Validation of the model over the 1995–97 period (surface soil moisture, deeper soil moisture, and streamflows).

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FIG. 4. Ensemble simulations (N 5 10 6 ): model uncertainty (6two standard deviations) compared with measurements over the 1990–94 period for the surface and deeper layers.

terion loses 15.5% compared to the calibration period (69.6% versus 85.1%). This decrease is essentially due to the incorrect simulation of the flood of 26 February 1997. Surface moisture is rather well simulated, except the drying in summer that is underestimated. This has consequences until the next saturation period corresponding to new runoffs (October–November). A lack of data concerning the deep soil moisture does not allow a precise comparison of simulation and observation for the 1995–97 period. However, at the end of the validation period, no bias is observed. c. Remote sensing data assimilation (1995–97) The soil moisture contents derived from the ERS observations are used to correct both the model and the soil moisture simulations, and therefore the discharge flow simulations. As explained in section 3b, two approaches have been tested, hard updating and the extended Kalman filter method. 1) HARD

UPDATING

The soil moisture simulations were replaced by the available observations during the considered period. The results after hard updating show a considerable deterioration of the streamflows simulated during the period. The Nash criterion decreased from 69.6% to 61.9% and the water flows at the outlet were overestimated, with the first flood peak in January 1996 four times its observed value. On the whole, hard updating proves to be a crude method. The poor results show that mea-

surement errors must be taken into account and, consequently, that model errors have to be estimated. 2) MODEL

ERRORS

The purpose of this section is to determine the values of standard deviations q s and q p , to show the evolution of model errors, and to check the linear tangent hypothesis made for the estimation of model errors in Eq. (4). Ensemble simulations were carried out to obtain the probability distribution function (pdf ) of the surface and deep soil moisture during the calibration period (1990– 94): a normal noise (standard deviation q s and q p ) was added at each time step k on the state variables (w s and w p ), and a number N of such simulations was performed. The result obtained with N 5 10 6 is represented in Fig. 4 for the surface soil moisture (top) and the deep soil moisture (bottom), with 6 two standard deviations represented. We have determined q s and q p so that, according to a Gaussian distribution, the percentage of simulations falling within 6 one (two) standard deviation of the measurements are 68% (90%). The obtained percentages are given in Table 2. The obtained values for q s and q p are q s 5 0.016 and q p 5 0.0008 cm 3 cm 23 . Note that covariance q sp is assumed to equal zero (Q assumed to be diagonal), and that q s and q p have been assumed to be constant in time (Q is a constant). These assumptions are subject to discussion. Alternative methods of determining Q in the future are presented in appendix B. The standard deviation of the ensemble simulations has been compared to the standard deviation calculated

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TABLE 2. Percentage of surface (ws) and deep (wp) soil moisture measurements falling within 6one and 6two standard deviations of the ensemble simulations (with qs 5 0.016 cm 3 cm23 and qp 5 0.0008 cm 3 cm23 ). According to a Gaussian distribution, the percentage of simulations falling within 6one and 6two standard deviations of the measurements should be 68% and 90%, respectively. Percentage of measurements falling within 61 s of the ensemble simulations Percentage of measurements falling within 62 s of the ensemble simulations

ws

wp

Gaussian

64.5%

70.3%

68%

86.9%

92.1%

90%

using the Kalman filter [Eq. (4)]. The mean error with respect to the reference simulation (without noise) is always very close to zero (not represented). This is a first indication that the linear tangent hypothesis is fulfilled. The results for the uncertainty prediction are shown in Fig. 5. The first observation is that the introduction of a random noise does not cause the model error to grow unbounded. The negative feedback in the evaporation, exchange, and precipitation terms causes the model error to decrease if the water content is overestimated and to increase if the water content is underestimated. In general, the uncertainty increases in summer during the soil drying period, and decreases each winter when the soil is recharged. The consequence is that the model can recover each year from the previous errors. In our example, after 5 yr of uncontrolled random simulations, the model uncertainty still remains bounded. During these 5 yr, however, the nonlinearity in the model makes the ensemble model error deviate from a

Gaussian distribution (as a simple x 2 test shows, not represented here). This explains the discrepancies observed between the ensemble uncertainty and the extended Kalman filter uncertainty. However, here again, each winter the uncertainty returns to near zero, and becomes Gaussian again. The evaporation globally follows a Gaussian distribution but causes the Kalman filter uncertainties to significantly deviate from their nonlinear values in spring and at the end of the summer, during the transition between soil-limited evaporation and atmosphere-limited evaporation. The exchanges between layers globally distort the Gaussian nature but have only few effects on the uncertainty predictions, so the corresponding errors are low. In fall, however, when both effects coincide, positive feedbacks occur and cause the nonlinearity to become important; therefore, the linear tangent hypothesis used in the Kalman filter scheme fails, and the error is strongly overestimated again. On the other hand, during other periods, marginal effects (such as the threshold wsat that limits the surface mois-

FIG. 5. Model uncertainty predicted for the surface and deeper layers (one standard deviation): comparison between the ensemble simulations and the extended Kalman filter over the 1990–94 period.

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FRANCOIS ET AL. TABLE 3. Precipitation (mm) occurring during the days of assimilation.

18 8 3 19 22

Date

P (mm)

Nov 1995 Jan 1995 Mar 1996 Mar 1996 Apr 1996

0.19 2.66 0.39 0.14 0.66

Date 2 6 25 10 29

Jul 1996 Aug 1996 Aug 1996 Sep 1996 Sep 1996

P (mm) 2.89 0.06 0.07 — —

Date 15 3 19 8 16

Oct 1996 Nov 1996 Nov 1996 Dec 1996 Feb 1997

ture w s or the strongly nonlinear exchanges between layers) explain some local underestimation of the uncertainty when the linear tangent hypothesis is used. In both cases, the negative feedback that usually occurs helps things return to normal. Quantitatively, the overall mean error on uncertainty prediction by EKF is 15% for w s and 10% for w p , and the percentage of points with an uncertainty difference EKF-ensemble lower than 25% is 84% for w s and 91% for w p . It is important to note that the errors on uncertainties occurred only after a long elapsed time (several months). If all uncertainties decrease to zero—say every 30 days, as would occur with monthly observation assimilation—then the difference between the linear tangent hypothesis and the full nonlinear ensemble simulations becomes much lower (not represented). The conclusion is that the linear tangent hypothesis used to estimate the model errors in the Kalman filter scheme works well in first approximation when used with a two-layer daily precipitation–evaporation–drainage hydrologic surface model. A better solution, however, is the use of ensemble simulations as shown here. As well as the estimation of model uncertainties, ensemble simulations may be used in the Kalman gain calculation. This improvement of the EKF turns into an ensemble Kalman filter (Evensen 1994, 1997a,b). In our case, however, the application of an ensemble Kalman filter (EnKF) should not be of great importance since the model errors are already well predicted by the EKF scheme. Moreover, the determination of the coefficients q s and q p in the Q matrix still remains the same problem in the EnKF. 3) OBSERVATION

ERRORS

According to section 3a, the conservative value of 6% is chosen for the accuracy of the estimated satellitederived soil moisture. The resulting observation error ranges between 0.005 and 0.02 cm 3 cm 23 in absolute value. 4) IMPLEMENTATION

OF THE EXTENDED

KALMAN

FILTER

Once the covariance matrix of the model errors P is calculated, the Kalman gain may be computed using Eq. (3). In our case, the state variable w s is directly observed, and the observation Jacobian matrix H k is

P (mm) 1.95 4.02 6.38 — 0.04

Date 23 8 27 6 22

P (mm)

Mar 1997 Apr 1997 Apr 1997 Jul 1997 Jul 1997

0.03 — 1.73 — —

Date 10 26 14 30 19

P (mm)

Aug 1997 Aug 1997 Sep 1997 Sep 1997 Oct 1997

— 10.9 0.08 — 2.4

H k 5 (1 0).

(7)

The Kalman gain then reduces to 

Pk2 (1, 1)  P (1, 1) 1 R k

 P

Kk 5  

2 k

2 k

 5 Gain on w . 1Gain on w 2 P (2, 1)  (1, 1) 1 R 2 k



s

(8)

p

k

The Kalman gain on w s (the directly observed variable) is proportional to the uncertainty on its estimation by the model [through the variance P(1, 1) of w s ] and inversely proportional to the observation error, which is the sum of measurement error R k and observation simulation error, which appears here to be once again the variance P(1, 1) of w s . The Kalman gain on w p (the indirectly observed variable) depends on the covariance P(2, 1) 5 P(1, 2). The more the two variables are physically linked, the more the Kalman filter adjusts w p when correcting w s . Each time an observation is available, the Kalman gain is calculated and the variables are corrected using Eq. (2). The variance–covariance matrix of model errors is then updated, and the errors are further propagated until the next observation. 5) INFLUENCE

OF THE DAILY TIME STEP OF THE

MODEL

Problems may arise if a satellite observation occurs during a rainy day before the rain event: in such a case the model would correctly predict higher soil moisture, while the satellite would see a dryer soil. The Kalman filter, incorrectly, would try to correct this apparent discrepancy. To avoid such a problem, observations made on days with precipitation should be removed from the assimilation scheme. Table 3 shows the precipitation occurring during the days of assimilation. Three subsets are defined, A 25 (the original dataset with the 25 observation dates), A19 with 19 dates (where 6 observations with precipitation greater than 2 mm are removed), and A13 with 13 dates (where 12 observations with precipitation greater than 0.1 mm are removed). As described in the following section, the assimilation scheme is tested with these three datasets to assess the importance of the daily time step of the model.

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TABLE 4. Nash criterion (river flows) results with a simple simulation (S) and after assimilation (A) (1995–97 period), where A 25 corresponds to the assimilation with 25 observation dates, A19 is the assimilation with 19 dates (6 observations with precipitation greater than 2 mm are removed), and A13 is the assimilation with 13 dates (12 observations with precipitation greater than 0.1 mm are removed). These dates are removed because of the daily time step of the model (when precipitation occurs the day of assimilation, the observation may not take this into account, depending on the time of precipitation and observation). Na (%) Simple simulation S Assimilation A 25 Assimilation A19 Assimilation A13

6) RESULTS

69.6 80.8 85.1 84.2

OF THE ASSIMILATION

The results with the three observation datasets are given in Table 4. In all cases, from November 1995 to October 1997, the SAR estimations of surface soil moisture used in the assimilation with the EKF improve the simulation of the flows. With the whole dataset (A 25 ) the Nash criterion increases from 69.6% to 80.8% after the assimilation. With A19 , when 6 observation dates with precipitation greater than 2 mm are removed, the result reaches 85.1% (see Fig. 6), and 84.2% with A13 when 12 observation dates with precipitation greater than 0.1 mm are removed. Therefore, A19 is the best compromise. These results show that it is important to take into account and remove observation dates with

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precipitation. A threshold of 2 mm appears to be a convenient one to use. 7) DISCUSSION It is satisfactory to note that the result on the validation period after assimilation (Nash 5 85.1%) is 15 points higher than before assimilation, and it reaches the same value than during the calibration period. Compared with the results obtained by the hard-updating method, the use of a finer-assimilation method that takes into account the model and measurement errors is therefore fully justified. The indirect corrections on w p using measurements of w s are weak in our case, but visible when comparing Figs. 3 and 6. Adjustment of surface and deep soil moisture has a variable influence on the simulated flows. As explained previously, when soil moisture is high, the sensitivity is at its highest. During strong rain events, the underground flows increase, thus resulting in a significant rise in the flows to the discharge system. For the period under study, this case occurred once, the flood of 26 February 1997. The weak correction of soil moisture improves to a significant degree the peak value. The difference between real and estimated peak maximum value is approximately divided by 3, passing from 4.5 to 1.6 m 3 s 21 , the observation being around 11 m 3 s 21 . On the other hand, when the rates of precipitation are weak, the assimilation of soil moisture does not have any influence on the flows regardless of the level of saturation of the surface. If sur-

FIG. 6. Results after the extended Kalman assimilation over the 1995–97 period with dataset A19 .

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FIG. 7. Kalman gains Ks and Kp for the surface and deeper soil moisture during the extended Kalman filter assimilation over the 1995–97 period with dataset A19 .

face soil moisture is low (,28%), the assimilation has only few consequences on the streamflow simulations (as explained at the end of section 4a). In order to understand how the root zone moisture may be updated through surface moisture observations, the Kalman gains for w s and w p are represented in Fig. 7. As expected, the Kalman gain for w s (K s ) is more important than the w p gain (K p ). In the dry down periods, when the model error is greatest, K s reaches its maximum, but it decreases in summer, when observations are less accurate because of vegetation coverage: the K s maxima therefore occur in spring (March–April) and fall (August–November). The gain K p on w p is much lower than K s . There are two periods when the root zone moisture may be corrected. The first one is winter (November–March), when the surface moisture reaches large values and causes important transfers down to the root zone. The correlation between surface and root zone moisture is therefore important through the exchange term, and the root zone moisture may be corrected if the observation appears to differ from the surface moisture simulation. The secondary period (much more reduced) occurs in summer, when the vegetation uptake by the root in the surface zone, controlled by the moisture in the deep zone, increases the correlation between the surface and deep moisture; the gain is very limited, however, because in the summer period the observations are correlatively TABLE 5. Comparison of the Nash criterion (river flows) when the daily input potential evapotranspiration data (ETP) are modified from their initial values. Results with a simple simulation (S) and after assimilation (A) are presented (1995–97 period), where A 25 , A19 , and A13 correspond to assimilations with 25, 19, and 13 observation dates, as explained in Table 4. Boldface type indicates best assimilation compromise. 0.90*ETP 0.95*ETP S A 25 A19 A13

Na Na Na Na

(%) (%) (%) (%)

71.7 79.7 84.2 82.2

71.5 80.5 84.8 83.7

ETP 69.6 80.8 85.1 84.2

affected by important errors, due to the same presence of vegetation. Generally, K p appears to be quite low and causes few modifications in the root zone moisture. Our feeling is that the scheme, as we applied it, underestimates K p . We suspect that the main reason for this is the assumption that Q is diagonal (i.e., that there is no correlation between the errors on w s and w p in a single time step), which is certainly wrong. This assumption causes the covariance terms in P to be underestimated and, in turn, the gain K p on the deep moisture to be underestimated. The solution would be to have a less arbitrary estimate of Q and the covariance error matrix P as proposed in appendix B. A parallel method of investigation is to improve the determination of the hydrological index to obtain better accuracies in summer, when the correlation between w s and w p through root water uptake would allow a more precise correction of w p through surface observations. d. Assimilation with errors on input data The Kalman assimilation method may also be used to correct erroneous input data. In order to assess this, the evapotranspiration and precipitation data during the November 1995 to October 1997 period were artificially modified in the range 610% from their initial values. Table 5 (evapotranspiration) and Table 6 (precipitation) gather the Nash criterion Na obtained from these difTABLE 6. Comparison of the Nash criterion (river flows) when the daily input precipitation data (P) are modified from their initial values. Results with a simple simulation (S) and after assimilation (A) are presented 1995–97 period), where A 25 , A19 , and A13 correspond to assimilations with 25, 19, and 13 observation dates, as explained in Table 4. Boldface type indicates best assimilation compromise.

1.05*ETP 1.10*ETP 66.2 80.9 85.2 84.7

63.6 80.9 85.1 84.9

S A 25 A19 A13

Na (%) NA (%) Na (%) Na (%)

0.90*P

0.95*P

P

1.05*P

1.10*P

51.6 76.9 80.2 81.3

60.8 80.4 84.5 85.1

69.6 80.8 85.1 84.2

74.1 76.6 80.9 78.6

70.6 67.0 71.4 67.4

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ferent databases when a simple simulation (S) or an EKF assimilation (A) is used. In this test, the three datasets A 25 , A19 , and A13 have been tested again to assess the stability of the results. The results show that the assimilation is almost always efficient, especially when the water system loss (water system input) is overestimated (underestimated), especially with A19 , as expected, where the system can recover from an error as high as 10% for evapotranspiration (with a Nash criterion about 85% after assimilation), and between 210% and 15% for precipitation (with a Nash criterion higher than 80% after assimilation). In fact, the greater the water inputs, the more the estimation error on the surface soil moisture is limited by the negative feedback effects, and the lower the soil moisture adjustment values: the assimilation scheme is then unable to account for corrections. This effect is particularly marked when the modified input data is precipitation. Precipitation (when it is greater than zero) essentially determines the Jacobian matrix component M(1, 1), which controls the negative feedback. When the precipitation is underestimated by 10% or 5%, or when the water demand (ETP) is overestimated, the river flows simulations are greatly improved when observations are assimilated. The Nash criterion increases by between 20% and 25%. On the other hand, when precipitation is overestimated by 10%, the assimilation may be unable to correct this error and may not perform as well as the original simulation. This effect does not occur when assimilation dates with precipitation are removed (see Table 6, line 3). 5. Conclusions and discussion This study demonstrates that it is possible to adapt the correction of the model when measurements are available according to the quality of the model parameterization and data errors. Even in the ideal case tested here, wherein the model has been calibrated on large series of measurements, the improvement is important during the validation period, as high as 15% on the Nash criterion (from 69.6% to 85.1%). This shows the advantage of assimilation compared to hard updating, which degrades the model simulations by assuming that all of the measurements are accurate. We did not observe local degradation, such as false detections or nondetection of floods, because we took into account the soil moisture measurements. This study also demonstrates that the assimilation of SAR data can improve flood forecasting. However, as a consequence of the daily time step of the model and input data, it is important to remove the observations with precipitation greater than 2 mm. This has an important influence on the performance of the filter (4 points on the Nash criterion, from 80.8% to 85.1%). Finally, it has been shown that the assimilation could correct for some errors or uncertainties in the input data, provided that the error is not higher than 10% (especially for the precipitation). The perfor-

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mances after the assimilation with erroneous input data are always greater than Nash 5 80%. Concerning the applied methodology, the extended Kalman filter proved to be efficient in our case. Some points, however, should be improved in the application of the method. The values we obtained for q s and q p should be seen as calibration parameters. The forecast of model errors appears to be fairly reproduced through the linear tangent hypothesis but may be improved using ensemble methods (the ensemble method, however, does not solve the problem of the estimation of sequential model error Q). The proper estimation of Q appears to be the most important factor to be addressed in all assimilation studies, since this governs the way the corrections are performed on the model, regardless of the assimilation method used (Kalman, variational, or other). The diagonal assumption for Q, especially, is critical and leads to the underestimation of deep soil moisture adjustments. A method is proposed in appendix B to obtain a reliable estimate of Q (or a direct estimate of variance–covariance matrix P) using Monte Carlo simulations, provided uncertainties are known on each parameter and elementary processes of the model. For future works this method will be implemented to test its feasibility and consider more rigorous procedures of determining the sequential errors. The adjustment of deep soil moisture would also be improved with reliable surface moisture observations in summer [this is currently difficult due to vegetation covering of the surface; see Quesney et al. (2000)]. Acknowledgments. The authors thank anonymous reviewers for their suggestions and the care they had in reading the original manuscript. The authors are grateful to O. Talagrand for fruitful discussions about the method developed in this study. We also gratefully acknowledge T. Tokedira for the English corrections of the manuscript. This study has been carried out in the framework of a scientific cooperation with M. Normand and C. Loumagne from Cemagref, which provided us with the meteorological and hydrological data and the GRHUM model. This work was funded by the French National program for Hydrological Research (INSU–PNRH). APPENDIX A Description of the Surface Model GRKAL The GRKAL model solves the following two-equation system, where k represents the time step; E s and E p , respectively, the surface and root zone evapotranspirations; Exc the water exchanges between the two layers; and Drain the drainage from the deeper layer: wsk11 5 wsk 1

P9 2 E s 2 Exc ds

wpk11 5 wpk 1

Exc 2 E p 2 Drain. dp

(A1)

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For each time step, the terms (P 2 P9) and Drain are routed to the subterranean model (see Loumagne et al. 1996). The proportion P9 of the rain that enters the surface model is closely linked to the surface soil moisture relative to its saturation level (wsat ): P9 5 P(1 2 a),

(A2)

d. Water exchanges between layers The water exchanges between the two layers are parameterized following the approach of Bernard et al. (1986) based on the Darcy’s equations. The exchange term between both layers is named Exc and is calculated through a diffusivity coefficient d wspwp:

where P is the total precipitation and a is given by

1 1 2 2,

w a 5 max 1, a1 s wsat

1

Exc 5 d wspwp ws 1

a2

(A3)

where wsat , a1 , and a 2 are calibration coefficients. One may note that parameter a also influences the subterranean model by determining its input (P 2 P9).

E soil 5

E

esoil dt,

k

where esoil 5 min(ETP, elim ),

(A4)

where t is time and elim is given by (Cognard 1996): elim 5 a exp(bw s2 ),

1 1 2 ,d

d wspwp 5 min d1 wmoy 5

wmoy wp max

d2

1

p

2

dp 1 ds

2

1 P9 w 1 1 wp , 2 s ds

(A9)

where d1 and d 2 are calibration coefficients and w pmax is the maximum water content in the deeper layer. The diffusivity is limited by d p /(d p 1 d s ) in order to avoid excessive and unrealistic exchanges between the layers (if both layers had the same thickness, the diffusivity would be limited to 0.5). The water fluxes between the two layers depend on the mean water content—through the hydraulic diffusivity, Eq. (A9)—and on the soil moisture gradient.

(A5)

where a and b are calibration coefficients (linked to the soil type). b. Vegetation transpiration The transpiration is calculated using a limiting factor depending only on the deep water content w p : Eveg 5 ETP(1 2 e 2y (wp2wwilt) ),

(A8)

We assume here that the proportion of the rain P9 that enters the surface model is first absorbed by the surface layer, and then the gradient (w s 1 P9/d s 2 w p ) creates a potential for water transfer, generally down to the deeper layer. The diffusivity coefficient is calculated through the following formulation:

a. Soil evaporation It is assumed that the surface layer is the only one to contribute to the soil evaporation. The evaporation esoil is then calculated as the minimum of the potential evaporation etp and the limit evaporation elim that the soil may supply [a function of the surface soil moisture content w s , as proposed by Bernard et al. (1986)]:

2

P9 2 wp . ds

(A6)

where wwilt is the wilting point and y an aridity coefficient.

e. Drainage The drainage from the deeper layer is the output of the surface model and the input of the subterranean model (together with the runoff P 2 P9 from the surface). A gravitational drainage from the deeper layer is calculated using the percentage of clay (CLAY) in the soil, following the formulation proposed by Mahfouf and Noilhan (1996). APPENDIX B Determination of Sequential Model Error Q

c. Evapotranspiration The total evapotranspiration in each layer is finally given through the root proportion in the surface layer p r , the fraction of vegetation s y , and the remaining proportion of soil evaporation (1 2 h) for a fully vegetated area (s y 5 1) (the effective proportion factor for the vegetation contribution is thus hs y ): E s 5 (1 2 hsy )E soil 1 pr hsy E veg E p 5 (1 2 pr )hsy E veg .

(A7)

The determination of Q is one of the key points in the application of the Kalman filter, since it determines the variance–covariance matrix of errors P and, finally, the Kalman gain K. However, no satisfactory method yet exists for precisely determining it. Two different ways of determining Q are given here: an empirical method and a method extensively based on Monte Carlo simulations. Let us recall the definition of Q: X k11 5 m(X k ) 1 N(0, Q),

(B1)

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where N(0, Q) is a normal distribution with zero mean and variance–covariance matrix Q, X is the state variable encompassing variables w s and w p , and m is the model given in Eq. (A1). a. The empirical method If the true values T k of X k were known, the problem would be solved: it would be easy to compute the statistics of T k11 2 m(T k ), to check that one obtains a Gaussian distribution with a zero mean (this is one hypothesis in the Kalman filter), to determine the standard deviations of this distribution, and to identify it as the sequential variances q s and q p involved in Eq. (4) (diagonal elements of Q). It would also be possible to compute the covariance of T k11 2 m(T k ) for all variables, and thus suppress the hypothesis that Q is diagonal. Since the true T k is not known, one could use sufficiently good measurements (say a wide spatial distribution of gravimetric soil moisture measurements G k over the whole catchment in our case) and consider them as ground truth. The obtained distribution would be G k11 2 m(G k ), the variances and covariances of which could be determined, so that an approximated value of Q is obtained. b. The Monte Carlo method The previous method for determining Q is based on actual measurements. Another possible method is based on ensemble methods: each calculus line of model code may be associated with a process error (either uniform or Gaussian), assuming that all parameters and quantities in this line are known and exact. The scheme begins with the estimation of parameter uncertainties and continues with the intermediate and prognostic equations. Such an idea is an extension of previous works by Spear and Hornberger (1980), further developed in Franks and Beven (1997), and applied in Franks et al. (1999). However, only the parameters were affected with errors in these works (not the equations), and uniform errors were always retained (instead of Gaussian). The final step is to run the model with all errors to create an ensemble of simulations (say 10 000 simulations). For each simulation, at each calculus code line, an error will be added according to its statistical properties. Finally, one obtains an ensemble of simulations, and thus the whole pdf of the model state, which represents all the available information. One can see whether the obtained pdf is Gaussian or not. Generally this will not be the case because of uniform noises that have been chosen for poorly modeled processes and/or poorly known parameters, and because of the model nonlinearity. This method allows us to obtain all of the information available from the model, together with the variance– covariance matrix of model errors (if relevant, i.e., for

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Gaussian pdf’s), provided that the errors were estimated correctly for each code line. Note that this method not only provides an estimation of Q but also a direct estimation of the variance–covariance matrix P of the model error at each time step. In this case there is no need to assume that Q is diagonal or constant. The assumption of unbiased errors is no longer necessary; the prognostic variables for which the obtained pdf is biased may be corrected (should the obtained bias value be trusted), or the bias may be added as an additional state variable and determined through the filter. Such a method does not avoid arbitrary decisions (the choice of noises at each calculus line) but forces these decisions to be explicit and, therefore, to be subject to discussion and reconsideration. REFERENCES Bernard, R., J. V. Soare`s, and D. Vidal-Madjar, 1986: Differential bare field drainage properties from airborne microwave observation. Water Resour. Res., 22, 869–875. Carlson, T. N., 1986: Regional scale estimates of surface moisture availability and thermal inertia. Remote Sens. Rev., 1, 197–247. Cognard, A. L., 1996: Suivi de l’e´tat hydrique des sols par te´le´de´tection spatiale (radar et thermographie infrarouge) et mode´ lisation hydrologique a` l’e´chelle du bassin versant. Ph.D. thesis, Universite´ Paris XI Orsay, 135 pp. Edijatno, 1991: Mise au point d’un mode`le e´le´mentaire pluie-de´bit au pas de temps journalier. Ph.D. thesis, Universite´ de Strasbourg, 242 pp. Entekhabi, D., H. Nakamura, and E. G. Njoku, 1994: Solving the inverse problem for soil moisture and temperature profiles by sequential assimilation of multifrequency remotely sensed observations. IEEE Trans. Geosci. Remote Sens., 32, 438–448. Evensen, G., 1994: Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res., 99 (C5), 10 143–10 162. ——, 1997a: Advanced data assimilation for strongly nonlinear dynamics. Mon. Wea. Rev., 125, 1342–1354. ——, 1997b: Application of ensemble integrations for predictability studies and data assimilation. Monte Carlo Simulations in Oceanography: Proc. ‘Aha Huliko’ a Hawaiian Winter Workshop, Honolulu, HI, University of Hawaii at Manoa, 11–22. [Available online at http://fram.nrsc.no/;geir/.] Franks, S. W., and K. J. Beven, 1997: Bayesian estimation of uncertainty in land surface–atmosphere flux predictions. J. Geophys. Res., 102 (D20), 23 991–23 999. ——, ——, and J. H. C. Gash, 1999: Multi-objective conditioning of a simple SVAT model. Hydrol. Earth Syst. Sci., 3, 477–489. Galantowicz, J. F., D. Entekhabi, and E. G. Njoku, 1999: Tests of sequential data assimilation for retrieving profile soil moisture and temperature from observed L-band radiobrightness. IEEE Trans. Geosci. Remote Sens., 37, 1860–1870. Gupta, H. V., L. A. Bastidas, S. Sorooshian, W. J. Shuttleworth, and Z. L. Yang, 1999: Parameter estimation of a land surface scheme using multi-criteria methods. J. Geophys. Res., 104 (D16), 19 491–19 504. Hoeben, R., and P. A. Troch, 2000: Assimilation of active microwave observation data for soil moisture profile estimation. Water Resour. Res., 36, 2805–2819. Houser, P. R., W. J. Shuttleworth, J. S. Famiglietti, H. V. Gupta, K. Syed, and D. C. Goodrich, 1998: Integration of soil moisture remote sensing and hydrologic modeling using data assimilation. Water Resour. Res., 34, 3405–3420. Janssen, P. H. M., and P. S. C. Heuberger, 1995: Calibration of process-oriented models. Ecol. Modell., 83, 55–66.

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