Assimilation of surface soil moisture into a multilayer soil model: design and evaluation at local scale

. Land surface models (LSM) have improved considerably in the last two decades. In this study, the Interactions between Surface, Biosphere, and Atmosphere (ISBA) LSM soil diffusion scheme is used (with 11 soil layers represented). A simpliﬁed extended Kalman ﬁlter (SEKF) allows ground observations of surface soil moisture (SSM) to be assimilated in the multilayer LSM in order to constrain deep soil moisture. In parallel, the same simulations are performed using the ISBA LSM with 2 soil layers (a thin surface layer and a bulk reservoir). Simulations are performed over a 3 yr period (2003–2005) for a bare soil ﬁeld in south-western France, at the SMOSREX (Surface Monitoring Of the Soil Reservoir Experiment) site. Analyzed soil moisture values correlate better with soil moisture observations when the ISBA LSM soil diffusion scheme is used. The Kalman gain is greater from the surface to 45 cm than below this limit. For dry periods, corrections introduced by the assimilation scheme mainly affect the ﬁrst 15 cm of soil whereas weaker corrections impact the total soil column for wet periods. Such seasonal corrections cannot be described by the two-layer ISBA LSM. Sensitivity studies performed with the multilayer LSM show improved results when SSM (0–6 cm) is assimilated into the second layer (1–5 cm) than into the ﬁrst layer (0–1 cm). The introduction of vertical correlations in the background error covariance matrix is also encouraging. Using a yearly cumulative distribution function (CDF)- matching scheme for bias correction instead of matching over the three years permits the seasonal variability of the soil moisture content to be better transcribed. An assimilation experiment has also been performed by forcing ISBA-DF (diffusion scheme) with a local forcing, setting precipitation to zero. This experiment shows the beneﬁt of the SSM assimilation for correcting inaccurate atmospheric forcing.


Introduction
It is well known that land surface processes interact strongly with the lower boundary of the atmosphere.In climate and numerical weather prediction (NWP) models, surfaceinteraction processes are represented by land surface models (LSMs).LSMs determine the partitioning of surface energy between sensible and latent heat fluxes, which depend on the quantity of water available in the root zone (Shukla and Mintz, 1982;Koster and Suarez, 1995;Entekhabi et al., 1999).The characterization of soil moisture in deep layers is more important than for surface soil moisture as the superficial reservoir has a small capacity and no memory features.Accurate estimates of root zone soil moisture are also important for many applications in hydrology and agriculture.Therefore, the land dynamics need to be sufficiently accurate.For example, a finer discretization in the vertical soil moisture and temperature profiles allows for a much better description of the nonlinear behavior than two-layer or threelayer models can provide (Reichle, 2000).
Considerable improvements have been made to the initial state of LSMs during the last decade by assimilating remotely sensed near-surface soil moisture data (Houser et al., 1998;Crow and Wood, 2003;Reichle and Koster, 2005;Balsamo et al., 2007).Interest in this was motivated by recent advances in soil moisture remote sensing.Since the 1970s, remote sensing has come to be accepted as a potential tool to access soil moisture at different temporal and spatial scales.Schmugge (1983) shows that the low-frequency microwave range is suitable for measuring the water content of a shallow, near-surface layer.Numerous missions to map surface soil moisture globally have been launched, such as ASCAT (Advanced Scatterometer on board METOP, Wagner et al., 2007), the Advanced Microwave Scanning Radiometer EOS Published by Copernicus Publications on behalf of the European Geosciences Union.
(Earth Observation System)(AMSR-E sensor, Njoku et al., 2003), and the Soil Moisture and Ocean Salinity (SMOS) satellite (Kerr et al., 2001), or are scheduled for launch (Soil Moisture Active Passive (SMAP) satellite, Entekhabi et al., 2004).Moreover, the development of soil moisture retrieval algorithms for preexisting microwave remote sensing missions (Wagner et al., 1999;Owe et al., 2001;Bartalis et al., 2007) offers an opportunity to improve the realism of modeled soil moisture.Several authors (Entekhabi et al., 1994, Houser et al., 1998, Walker and Houser, 2001, Ragab, 1995and Sabater et al., 2007 among others) have shown that combining SSM measured by remote sensing and LSM simulations through a land data assimilation system (LDAS) improves the modeled soil moisture content for deep layers and/or heat fluxes (Pipunic et al., 2013).However, there are considerable uncertainties associated with the use of remotely sensed soil moisture data.In particular, the extent to which the near-surface soil layer represents the underlying soil moisture profile is not well understood.
Satellite data have to be calibrated and validated by observations in situ.The SMOSREX (Surface Monitoring Of the Soil Reservoir Experiment) site located in southwestern France was used for the algorithm validation of SMOS (Saleh et al., 2006(Saleh et al., , 2007)).Over this site, measurements of soil moisture and temperature profiles, meteorological variables and brightness temperatures were obtained for 2003 to 2012 (de Rosnay et al., 2006).Remote sensing and in situ soil measurements are available over fallow and bare soil plots.
A number of studies have been conducted in recent years to investigate the relevance of using variational and Kalman filtering to analyze soil moisture.In 2000, the German Weather Service (Deutscher Wetterdienst) implemented a simplified extended Kalman filter (SEKF) soil moisture analysis using screen-level parameter information (Hess, 2001).Four years later, Balsamo et al. (2004) introduced an online simplified two-dimensional variation (2D-VAR) method instead of a SEKF to retrieve soil moisture.More recently, Météo-France developed an offline SEKF to analyze soil moisture in the SURface EXternalisée (SURFEX) system for research applications (Mahfouf et al., 2009).For the German and French models, the same approach of explicitly computing Jacobians in finite differences based on perturbed simulations is used.In 2010, a new LDAS based on a SEKF was implemented at the European Center for Medium-Range Weather Forecasts (ECMWF) in the global operational Integrated Forecasting System (IFS) (de Rosnay et al., 2013).
In the present work, a SEKF is used to assimilate in situ observations, gathered at SMOSREX over bare soil, into the Interactions between Surface, Biosphere, and Atmosphere (ISBA) LSM.Only the surface soil moisture (SSM) measured from the surface to 6 cm depth at 06:00 LST (Local Standard Time) is assimilated in the LDAS, if the observation is available.This experimental setup is used to represent the daily assimilation of remote sensing data at dawn.For example, SMOS has an ascending Equator crossing time at 06:00 LST.Two versions of ISBA are used: the original force restore scheme (Noilhan and Mahfouf, 1996) and a more complex diffusion scheme using the Richards equation (Decharme et al., 2011).Hereafter, the two ISBA versions are referred to as ISBA-2L and ISBA-DF, respectively.The implementation of SEKF within ISBA-2L has been investigated by several authors, such as Draper et al. (2011), Barbu et al. (2011) and Albergel et al. (2010).In these studies, a twoor three-soil-layer configuration was used.Unlike ISBA-2L, ISBA-DF (diffusion version) allows a fine discretization of the soil to be used (11 soil layers are considered).In this study, a first attempt is made to use the ISBA-DF in a data assimilation experiment.This work focuses on a bare soil site in order to isolate the thermal and water transfers in the soil without any interference from processes related to the vegetation, such as transpiration.Decharme et al. (2011) had already studied differences between ISBA-2L and ISBA-DF using SMOSREX data over the fallow site.The behavior of these two LSMs over bare soil has not yet been compared.The period under investigation extends over three contrasted years from 2003 to 2005, including the 2003 drought.In addition, the role of bias correction in the assimilation is discussed regarding the mass balance in the root zone.Despite the lack of a root zone in the bare soil, this term will be used below to represent the total active soil depth.
The main objectives of this study are (1) to compare the soil moisture simulations from ISBA-2L and ISBA-DF over the bare soil, (2) to evaluate the role of the assimilation of ground observations of SSM for these two LSMs and (3) to investigate different configurations of ISBA-DF LDAS in four sensitivity cases.The SMOSREX data, ISBA-2L and ISBA-DF LSM and the SEKF algorithm are presented in Sect. 2. Assimilation results and sensitivity studies are presented in Sect. 3 and discussed in Sect. 4. Section 5 summarizes the main conclusions of the study.

The SMOSREX site and the soil moisture time series
The SMOSREX long-term experiment (2001-2012) was aimed at improving the modeling of land surface processes in the context of the SMOS mission (de Rosnay, 2006).The SMOSREX site is located in southwestern France (43 • 23 N, 1 • 17 E, at 188 m altitude) and is divided into two parts: a bare soil plot and a fallow plot.Soil moisture was measured at depths of 0-0.06, 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80 and 0.90 m every half hour.Soil probes were calibrated from gravimetric measurements (Schmugge et al., 1980).A weather station measured precipitation, 2 m air temperature and air humidity, 10 m wind speed, atmospheric pressure, and incoming solar and atmospheric radiation every 30 min.An L-band radiometer called LEWIS (Lemaître et al., 2004) Hydrol.Earth Syst.Sci., 18, 673-689, 2014 www.hydrol-earth-syst-sci.net/18/673/2014/ measuring brightness temperature at two polarizations and five different angles is also present at the SMOSREX site.In this study, we focus on the bare soil plot.The percentage of sand and clay observed from 0.1 to 0.90 m depth for bare soil on the SMOSREX site is displayed in Fig. 1a.Only the years 2003Only the years , 2004Only the years and 2005 were considered because, for the bare soil plot, the soil moisture data at depths ranging from 0.20 to 0.90 m are missing from 2006 onwards.The surface soil moisture (SSM) measurements, i.e., from the surface to 6 cm, were used for assimilation and measurements from 10 to 95 cm were used for validation purposes.In order to avoid frozen soil conditions, SSM observations made when the surface temperature was below a threshold of 4 • C were discarded

Land surface models
The ISBA land surface model (Noilhan and Planton, 1989;Noilhan and Mahfouf, 1996) describes the surface processes in weather and climate applications.This model shows the evolution of land surface state variables (surface and soil temperatures, surface and root zone soil moisture content) and the exchanges of heat and water between the low level atmosphere, the vegetation and the soil.In this study, the SUR-FEX (SURFace EXternalisée) modeling platform (Masson et al., 2013) version 7.2 containing the ISBA LSM was used.SURFEX is designed for research and operational applications and includes several options of ISBA.In this study, SURFEX is forced by local atmospheric observations and runs without feedback between the surface and the atmosphere (i.e., SURFEX is used offline).Moreover, to match measurement levels, the total soil depth was set to 0.95 m.The two options of the ISBA LSM (ISBA-2L and ISBA-DF) used in this study are summarized below.

ISBA-2L
The ISBA-2L version of SURFEX is based on the force restore approach, according to Deardorff (1977).This LSM is used in the operational NWP models at Météo-France.The soil is composed of two layers, the first layer being represented by a skin soil top layer 1 cm thick and the second by a bulk reservoir.The first layer is used to compute the SSM and soil evaporation while the second layer is used to compute the total soil moisture and contributes to the evapotranspiration (Boone et al., 1999).Mahfouf and Noilhan (1996) introduced a representation of gravitational drainage.In the force restore equations, the soil moisture dynamics depend on several thresholds -saturation (w sat ), wilting point (w wilt ), and field capacity (w fc ).These parameters are related to the soil textural properties (Mahfouf and Noilhan, 1996).In this work, w fc and w wilt were derived from the clay content observations and set to 0.30 and 0.17 m 3 m −3 , respectively.w sat was derived from the sand content observations and set to 0.45 m 3 m −3 .The modeled surface soil moisture and root zone soil moisture are referred to hereafter as w 1 and w tot , respectively.

ISBA-DF
The ISBA-DF LSM is a new ISBA version including a soil multilayer diffusion scheme (Decharme et al., 2011), which explicitly solves mass and heat-diffusive equations.The ISBA-DF soil hydrology uses the "mixed" form of the Richards equation to describe the water mass transfer within the soil via Darcy's law.Moisture and temperature profiles can be computed according to the vertical soil texture properties.Computations are performed using the SMOSREX soil texture displayed in Fig. 1a.In ISBA-DF, unlike in ISBA-2L, the soil moisture dynamics do not depend on the specification of a volumetric field capacity.In this study, 11 soil layers were considered, corresponding to the locations of observation instruments at the SMOSREX site.Figure 1b shows the soil discretization used in the ISBA-DF LSM and the placement of each probe.The node of each ISBA-DF soil layer corresponds to the depth of each probe: 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80 and 0.90 m.Close to the surface, two layers were considered: 0-0.01 and 0.01-0.05m.Hereafter, soil moisture in the nth layer is denoted by w n and the total soil water w tot is the weighted average of w n .

The simplified extended Kalman filter
In this section, the classical notations of data assimilation proposed by Ide et al. (1997) are used.The equation for the ith model state forecast and update at time step t i is (1) The equation for the ith state analysis, occurring at time t i is where x indicates the model state and y 0 is the observation vector.The superscripts a, b and o indicate the analysis, the background and the observations, respectively.Analysis increments are the difference between the analysis and the background.M is the nonlinear forecast model and H is the nonlinear observation operator.Both in ISBA-2L and ISBA-DF, the observation vector is y 0 = (SSM).The model state vector is different following the LSM used: x ISBA-DF = [w 1 , w 2 , w 3 , w 4 , w 5 , w 6 , w 7 , w 8 , w 9 , w 10 , w 11 ] .(4) The observations occurred 24 h after the analysis time and H includes a 24 h integration of the forecast model (Draper et al., 2011).The available observations are assimilated at 06:00 LST.The Kalman gain is given by where the Jacobian matrix H is the linearization of H and H T its transpose.H is obtained using a finite difference approach, individually perturbing each component x j of the initial model state x at time t 0 by a small amount δx j .The Jacobian is expressed as the difference of the prognostic states between perturbed and reference runs at the end of the assimilation interval at time t i , divided by the initial perturbation: For all the experiments, the perturbation size is set to 0.001 m 3 m −3 .Mahfouf et al. (2009) have shown that these very small perturbations lead to good approximations of the linear behavior of the observation operator.The Jacobian values are computed with a daily time step (t i − t 0 = 24 h).The examination of the Jacobian matrices is important for understanding the data assimilation performance.B and R are the covariance matrices of the background and observations errors, respectively.The SEKF does not cause B to evolve through a forecast cycle, unlike the traditional EKF.Draper et al. (2009) found that, for assimilating near-surface soil moisture into ISBA LSM, the analyzed soil moisture generated by the EKF and the SEKF were not substantially different.This result is explained by the fact that the increase in the background error during each forward propagation step is balanced by the decrease in the error during the analysis step and the difficulties of specifying the Q matrix (model error).Moreover, Sabater et al. (2007) suggested that, at the SMOSREX location, a fixed background error was more accurate and stable than an evolving background error.The performance of an analysis scheme depends on the use of appropriate statistics for background and observation errors.For all the experiments, the observation standard deviation error for SSM was set to 0.02 m 3 m −3 .This error is consistent with that chosen by Sabater et al. (2007) at the SMOSREX site.
In the ISBA-2L assimilation experiment, B was a diagonal matrix with the standard deviation errors σ w 1 and σ w tot set to 0.02 and 0.005 m 3 m −3 , respectively.The ratio between σ w 1 and σ w tot was equal to one used by Albergel et al. (2010).
Concerning ISBA-DF, the experiments were carried out using a B diagonal matrix with the same background standard deviation error σ w n along the diagonal (0.015 m 3 m −3 ) except that σ w 1 was set to 0.02 m 3 m −3 .The values were adjusted to obtain results not too different from those with ISBA-2L results on average.This representation of the B matrix considers that the soil layer errors are not correlated.However, in ISBA-DF LSM, the layers are linked through diffusive equations.Thus, errors in a given layer should affect the others.Several experiments described below used a tri-diagonal B tri-diag matrix in order to propagated error through the adjacent layers.The B tri-diag matrix is expressed as where the correlation term α is empirically set to 0.5.Numerous studies (Reichle and Koster, 2004;Pellarin et al., 2006;Rüdiger et al., 2007) have shown the necessity to rescale data before assimilating in situ or satellite-derived soil moisture in order to reduce systematic biases between the model and observations.In this study the cumulative distribution function (CDF)-matching technique proposed by Reichle and Koster (2004) is used for all the experiments and described in Sect.3.1.

Design of the experiments
The land surface models were not calibrated.In the case of ISBA-2L, the average of the observed soil texture profile is used (i.e., 20.0 % of clay and 45.3 % of sand) to derive field capacity, wilting point and saturation parameters.In the case of ISBA-DF, the measured profiles of soil texture (Fig. 1a) and soil density are prescribed to the model.The value of the field capacity parameter is not prescribed in ISBA-DF.
Six assimilation experiments were undertaken.The first two were performed using ISBA-2L and ISBA-DF.Hereafter, these experiments are referred to as 2L and DF-REF, respectively.The DF-REF configuration was chosen to be as close as possible to the 2L configuration: the observations concerned the surface soil moisture (0-1 cm) equivalent and the B matrix was assumed diagonal for both experiments.
Then, starting from the DF-REF experiment, several modifications to the LDAS were considered.The experiment DF-H2 assimilated SSM into the second soil layer (1-5 cm) in order to better represent the SSM observations gathered between 0 and 6 cm.Next, the experiment DF-B used a nondiagonal B tri-diag matrix in order to account for the error correlations in adjacent soil layers, as described in Sect.2.3.
In addition, a bias correction technique performed before assimilation was included in the experiment DF-CDF.It   allowed the impact of the temporal window chosen for data rescaling to be studied.The last experiment, DF-NP, evaluated the benefit of assimilation in the case where ISBA-DF LSM was forced with a local forcing, which set the precipitation to zero.The role of the assimilation in LSM in compensating for a less accurate forcing will be shown below.In this experiment, the background state was not as reliable as in previous experiments since there was no precipitation.In this case, the background error σ w n was increased by a factor of three (Albergel et al., 2010) and no bias correction was performed.
The CDF matching was performed for all the experiments (2L, DF-REF, DF-H2, DF-B and DF-CDF) in order to reduce systematic biases between the observations and the model, except for the DF-NP experiment.In the DF-NP experiment, CDF matching could not be performed because the open loop was too distant from the observations.For the other experiments, a CDF matching over the whole three-year period was performed.Additionally, the influence of the CDF matching on the analysis was studied with the DF-CDF experiment.In this experiment, the CDF matching was performed per year rather than over the whole three-year period.
The characteristics of the experiments are listed in Table 1.

Assessment of the day-to-day variability of SSM
In order to better capture the day-to-day variability of w 1 , the seasonal cycle was removed by calculating monthly w 1 anomalies (Albergel et al., 2009).The difference to the mean was calculated for a sliding window of five weeks (if there were at least five measurements during this period), and the difference was scaled to the standard deviation.For each soil moisture w 1 estimate at day (i), a period F was defined, with F = [i, 17 d, i + 17 d] corresponding to a five-week window.
The anomaly is dimensionless and is given by The R ano values were computed for both open-loop and analysis simulations, as described in Sect.2.4.

CDF-matching technique
The histograms of the distributions of SSM observed at the SMOSREX site, and w 1 modeled by ISBA-2L and ISBA-DF, are displayed in Fig. 2  The CDF matching proposed by Reichle and Koster (2004) was used in order to remove the bias between the observations and the modeled w 1 .This bias correction scheme modifies the observations in order to make their statistical distribution closer to the one from the model.For this, a thirdorder polynomial fit is computed.Following Scipal et al. (2008), a large statistical sample needs to be considered to obtain a robust bias correction scheme.
The results of the CDF matching performed over this period for ISBA-2L and ISBA-DF are shown in Fig. 3. Observations rescaled with ISBA-2L and ISBA-DF do not exceed 0.34 and 0.35 m 3 m −3 , respectively.The CDF matching plays an important role during wet and dry periods and thus reduces the seasonal dynamics by decreasing the SSM values in winter and increasing them in summer.
The rationale for the application of the bias correction is that this study is a first step towards the assimilation of satellite data in ISBA-DF at regional and/or global scales.The methodology described in this paper could be used in future satellite data assimilation studies as the models were not calibrated for this site (a priori parameters are used).In such a context, systematic errors between the observations and the model have to be reduced.In situ observations are assimilated and, as explained above, systematic errors between in situ observations and model values are actually observed.The CDF-matching technique is used to reduce systematic errors.

2L experiment
Simulations performed without data assimilation are called "open-loop" simulations.Figure 4    The annual open-loop and analysis statistical scores for w tot are given in Fig. 5. Assimilating SSM slightly increases the correlation coefficient and decreases the RMSE.
The Jacobian terms ∂w 1 (t) ∂w 1 (t 0 ) and ∂w 1 (t) ∂w tot (t 0 ) in the first thin layer and in the bulk reservoir are illustrated in Fig. 6.Generally, the Jacobian terms have positive values.Zero and small negative values are also found, which represent 39 and 15 % of the Jacobian terms for w 1 and w tot , respectively.Over the 2003-2005 period, 14 and 13 % of the Jacobian terms ∂w 1 (t) ∂w 1 (t 0 ) and ∂w 1 (t) ∂w tot (t 0 ) , respectively, are strictly equal to zero.Null Jacobian values indicate that neither w 1 nor w tot is sensitive to SSM assimilation.During wet periods, the Jacobian term for w 1 is equal or close to zero.For dry soils, SSM assimilation www.hydrol-earth-syst-sci.net/18/673/2014/ Hydrol.Earth Syst.Sci., 18, 673-689, 2014 over a 24 h window does not modify the behavior of w 1 .The mean values of the Jacobian term with respect to w 1 and w tot are 0.15 and 0.51, respectively.Over a 24 h window, the impact of the SSM assimilation is higher in the bulk reservoir than in the first layer.This finding is consistent with previous studies, e.g., by Draper et al. (2009) and Mahfouf (2010).Figure 8 shows the w 1 open-loop and analysis for four different layers i: layer 2 (1-5 cm), layer 4 (15-25 cm), layer 6 (35-45 cm) and layer 8 (35-45 cm).For w 2 and w 4 , the openloop and analysis represent the annual cycle well.After assimilation, w 2 and w 4 are better correlated with observations and the bias is smaller for the w 2 analysis than for the open-loop by 0.004 m 3 m −3 .Concerning w 6 and w 8 , the annual cycle is overestimated compared to the observations.For the whole year 2005, open-loop and analysis w 6 and w 8 are found to be very dry.The same behavior is observed during the summer of 2004.From layer 7 (−45 cm), assimilation decreases the statistical scores slightly.

DF-REF experiment
The Jacobian, the Kalman gain K and analysis increments permit the performance of the data assimilation to be evaluated.Figure 9 shows the time evolution of the Jacobian terms ∂w 1 (t) ∂w n (t 0 ) , the Kalman gain and the increments for each layer n.
The largest Jacobian values are obtained for the layers 2 and 3 (i.e., depths of 1-15 cm).The mean Jacobian values for these layers are 0.19, 0.28 and 0.15, respectively.For deeper layers, the mean Jacobian value is lower than 0.10.The assimilation does not play an important role in the first layer or in layers 5 to 11.During wet periods, the Jacobian in the full soil column is small but not strictly equal to zero.The Kalman gain behaves similarly.During dry periods, the information from the surface does not penetrate very deeply into the soil (less than during wet periods).However, the information from the surface affects the top layers of the soil (1-15 cm) more intensely than during wet periods.The Jacobian values (Fig. 9) show a decoupling of surface layers from deeper layers during dry periods, in relation to lower values of the hydraulic conductivity.From the Kalman gain value, and in contrast to 2L, the LDAS does not perform corrections over the total soil column but in individual layers that vary with the seasonal cycle.Note that the sum of the Jacobian values in each layer is, on average, close to the average of the sum of the Jacobian in the two layers used in 2L.
The analysis increments allow the impact of the data assimilation on the water mass balance to be investigated.whole year 2004, 73 % of the increments are negative.As a result, the assimilation tends to remove water in most instances.For example, from January to May 2004, the increments are essentially negative.In layers 1 to 5 (from the surface to −35 cm), 87 % of the increments are different from zero.For deeper layers, the fraction of non-zero increments decreases, to reach 35 % for the last layer.

Sensitivity studies in ISBA-DF
In this section, results of the four experiments presented in Table 1 are examined.

Second soil layer assimilation
The SSM observations are measured between the surface and 6 cm depth.In contrast to the previous experiments, DF-H2 was carried out in order to assimilate SSM in the second layer of ISBA-DF.The depth of this layer (1-5 cm) is closer to the observation depth.CDF matching is performed between w 2 and the observations over the 3 yr period to remove systematic biases.Statistical scores of w 1 analysis for the 3 yr period are displayed in Table 2.Over time, the assimilation in the second layer tends to decrease the RMSE of the w 1 with respect to the DF-REF.However, the correlation coefficient of w 1 computed by DF-H2 is weaker than that simulated by DF-REF.
Yearly statistical scores of w tot are shown in Fig. 5.The RMSE of root zone soil moisture is slightly smaller for the analysis using DF-H2 than DF-REF.Over the full period, assimilating SSM in the second layer decreases the bias by 84 % compared to DF-REF.The non-zero increments increase by 3 % for deeper layers (−35 to −95 cm) with respect to the reference experiment.For layers 1 to 5, no significant impact is observed.
Figure 10 shows the time evolution of w 2 open-loop and the Jacobian term ∂w 2 (t) ∂w 2 (t 0 ) for DF-H2 and ∂w 1 (t) ∂w 2 (t 0 ) for DF-REF.For both experiments, the Jacobian is larger during dry periods than during wet periods.In the wet season (winter) and for rainy events, the Jacobians are very low, indicating that initial soil moisture perturbations are lost during the 24 h model integration.In such situations, the soil moisture in the first two layers is therefore mostly driven by the atmospheric forcing.As seen in Fig. 10, the Jacobian ∂w 1 (t) ∂w 2 (t 0 ) is more sensitive to rainfall events when the SSM is assimilated in the first layer than in the second layer.Since the second layer is less impacted by the atmospheric forcing than the first layer, assimilating the observations in the second layer overcomes the effect of rainfall events and allows more information to be extracted.

The choice of the B matrix
Experiment DF-B was undertaken with the B tri-diag matrix in order to propagate model error in adjacent layers.The statistical scores of w 1 analysis over the 3 yr period are displayed in Table 2.An assimilation experiment with a five-band B matrix was also performed to increase vertical correlations (not shown).The statistical scores of w tot analysis over the 3 yr period were close to those obtained with DF-B.However, the average Kalman gain values increased by 21 % with respect to DF-B.

Impact of the CDF matching on the water mass balance
In this section, the impact of CDF matching on the water mass balance is investigated.CDF matching per year is computed in the experiment DF-CDF but not in DF-REF.A thirdorder polynomial fit is computed for each individual year.
Figure 12  Over the 3 yr period, the correlation coefficient, RMSE, SDD and bias values of the w tot analysis are 0.90, 0.052, 0.052 and −0.006 m 3 m −3 , respectively.A slight increase of the statistical scores is noted with respect to DF-REF (Fig. 5).The main difference between DF-REF and DF-CDF is obtained during 2004 where the RMSE of w tot decreases by 13 %.
The increments provided by the assimilation scheme using CDF matching per year are shown in Fig. 13.For 2005, no significant differences are observed.However, for 2003 and 2004, differences between increments obtained with DF-REF and DF-CDF are clear.Using an annual CDF allows the water mass balance to be conserved in the root zone.For example, in 2003, water from the root zone is removed between January and April in DF-CDF whereas water is still added during the same period in DF-REF.The same feature is observed during the full year of 2004: water in the soil is systematically removed in several consecutive months in DF-REF but in DF-CDF, significant water amounts are added.Moreover, using annual CDF matching improves the simulation of the root zone soil moisture for the years when the SSM distribution is very different from the average SSM distribution.

Forcing without precipitation
Removing precipitation from the forcing permits the benefit of the SSM assimilation to be evaluated when the forcing is not accurate.No CDF-matching technique is used over the assimilated SSM.The statistical scores of w 1 analysis are shown in Table 2.The statistical scores are improved compared to those of DF-REF.For example, the correlation coefficient is increased by 9 % and the RMSE is decreased by 43 % with respect to the open-loop reference experiment.
The statistical scores computed per year for w tot analysis are displayed in

Discussion
The ISBA-2L LSM is used for operational NWP predictions at Météo-France and has been widely studied.At the moment, unlike ISBA-2L, ISBA-DF is a new LSM that is starting to be tested and is used only for research applications.Soil moisture computed by ISBA-DF is better correlated with observations than that simulated by ISBA-2L.This result is consistent with the finding of Decharme et al. (2011) on the fallow part of the SMOSREX site.Over the 3 yr period, an improvement in terms of statistical scores is found for both w 1 and w tot when simulated by ISBA-DF rather than ISBA-2L.
The SSM assimilation improves the modeling of w 1 and w tot for the two LSMs.In 2L, corrections introduced by the assimilation affect the whole root zone.This feature is not realistic, particularly for a bare soil surface (Draper et al., 2009).The propagation of SSM information to deeper soil layers is consistent with the water transfer physics in ISBA-DF.The assimilation does not have the same impact on the soil moisture content at different soil depths.For example, during dry periods, corrections introduced by the LDAS are located at depths between 1 and 15 cm.Weaker corrections were found to impact the total soil column during wet periods.
As seen in the previous sections, for all the experiments carried out with ISBA-DF LSM, the assimilation of SSM has little impact in the first layer (0-1 cm).For example, the mean Jacobian ∂w 1 (t) ∂w 1 (t 0 ) for DF-REF is equal to 0.034 in the first layer.In contrast, the mean Jacobians ∂w 1 (t) ∂w 2 (t 0 ) , ∂w 1 (t) ∂w 3 (t 0 ) and ∂w 1 (t) ∂w 4 (t 0 ) are larger than 0.1.In the case of ISBA-DF, small Jacobian values are obtained for the top soil layer.Due to the small size of the w 1 reservoir (1 cm), the dynamics of the first layer is driven by the atmospheric forcing to a large extent.Moreover, in the experimental setup, the length of the assimilation window is 24 h.Over this time period, the impact of the initial conditions is reduced by the atmospheric forcing.Similar results were found with DF-H2 with small values of the Jacobians in the first layer.This result is in agreement with a recent study by Medina et al. (2012), also showing a weak impact of the Kalman filter in the first layer.
Accounting for vertical correlations in the background error covariance matrix, B tri-diag , of the SEKF scheme tends to decrease the RMSE of w 1 .Despite an increase in Jacobian and Kalman gain values, no significant improvement to the Hydrol.Earth Syst.Sci., 18, 673-689, 2014 www.hydrol-earth-syst-sci.net/18/673/2014/ analysis of w tot has been noticed.In future studies, it would be desirable to work with an EKF scheme instead of an SEKF to benefit from the evolution of the B matrix.An evolutive B matrix would take the vertical correlation errors into account better.However, more research needs to be done to improve the specification of model errors.Assimilating SSM in the second layer appears to be more physically meaningful.
After assimilation of SSM in the second layer, the bias of w tot analysis is small.Over the 3 yr period, DF-H2 appears to be the best experiment in terms of RMSE and bias.
Numerous studies have proposed to rescale the observations before assimilating them into an LSM.Several authors have performed CDF matching over the entire period of interest (Reichle and Koster, 2004), whereas others have used seasonal-based CDF matching (Barbu et al., 2014;Draper et al., 2009).As shown in Sect.3, using CDF matching per year constrains the mass balance better, essentially for 2004.In southwestern France, the yearly SSM distribution is very different from one year to another and global CDF matching erases the interseasonal variability of the model dynamics.Yearly CDF matching is advocated for reanalysis systems even though it is not suitable for real time monitoring or forecasting.
Other assimilation studies at the SMOSREX site were performed in order to improve the root zone soil moisture (Sabater et al., 2007;Albergel et al., 2010).Unlike this study, they were focused on the fallow part of the SMOSREX site.Over the bare soil, the root zone dynamics and the response to rainfall events are larger than over the fallow area.In this study, similar results to those of Sabater et al. (2007) and Albergel et al. (2010) were found using the 2L model.

Conclusions
This study investigated the assimilation performances of two versions of the land surface model ISBA: (1) ISBA-2L with a soil composed of two layers and (2) ISBA-DF with a soil divided into 11 layers.This work was carried out on the SMOSREX bare soil site during the period 2003-2005.This was a first attempt to explore the potential of assimilating SSM using an SEKF over France with ISBA-DF.Regarding open-loop simulations, w 1 and w tot simulated by ISBA-DF outperformed those computed by ISBA-2L.
The largest impact of the SSM assimilation with ISBA-DF concerns the first 45 cm of soil.For dry periods, the SSM assimilation plays an important role from the surface to 15 cm whereas, during wet periods, weak corrections are applied for the entire soil column.These seasonal variations cannot be represented with a single bulk layer in the root zone.For the first time, the evolution of the Jacobian terms, the Kalman gain and the increments with depth have been examined.
The best improvement of w tot analysis was obtained by assimilating SSM in the second layer.This result shows the benefits of working with a multilayer LSM.Assimilating the SSM in the second layer and adding vertical correlation errors are recommended for future research.Yearly CDF matching leads to a better conservation of the water mass balance in the root zone and is encouraged for data reanalyses.
This work shows the potential to improve the soil moisture in the deep layers by assimilating SSM in ISBA-DF.This study will be extended to the whole of France by assimilating satellite-derived SSM such as ASCAT and SMOS.In addition, the assimilation of brightness temperature from the L band radiometer LEWIS in ISBA-DF would be interesting.The impact on the other terms of water balance (surface evaporation and runoff) was not examined because of the lack of suitable observations over the bare soil plot of SMOSREX.
Fig. 1. a) Sand and clay soil profile (in %) measured at the SMOSREX bare-soil site.b) Soil profile model used with ISBA-DF.The depth of each layer is given in meters.Red dots are places where soil moisture observations were made.
shows the time series of w 1 , w tot and observations for the open-loop and the analysis simulations over the period 2003-2005.The annual w 1 is generally well reproduced by the model.In winter, the modeled w 1 are smaller than the observed values, except for the winter of 2005.The daily cycles and the rainfall responses are overestimated.The greatest differences between the open-loop and the analysis simulations concern the winter and spring of 2003 and 2004.For 2003, the analysis values are higher than the open-loop values.The contrary is seen for 2004.No marked differences between open-loop and analysis simulations are observed for 2005.The temporal behavior of the w tot is well represented except for the drought Hydrol.Earth Syst.Sci., 18, 673-689, 2014 www.hydrol-earth-syst-sci.net/18/673/2014/

Fig. 4 .
Fig. 4. 2L time series of open-loop simulation (red line), analysis simulation (blue line) and SSM observations at the SMOSREX site (black dots) for 2003-2005: in the surface layer (a) and in the total reservoir (b).The units are in m 3 m −3 .

Fig. 5 .Fig. 6 .Fig. 6 .
Fig.5.Yearly and 3 yr period statistical scores for w tot analysis for experiments presented in Table1 and 2Lexperiment.O.L. means open-loop simulations.Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Fig. 6.Time evolution of the two Jacobian terms of ISBA-2L in (top) the first layer and (bottom) the bulk reservoir.

Figure 7 Fig. 7 .
Figure 7 shows w 1 open-loop and analysis time series for DF-REF together with observations for 2003-2005.The statistical scores for w 1 open-loop and analysis simulations over the 3 yr period are listed in Table2.The w 1 simulations performed with and without assimilation are better correlated with observations for ISBA-DF than ISBA-2L.Moreover,

Fig. 9 .Fig. 9 .
Fig. 9. Representation of the Jacobian (top), the Kalman gain (middle) and increments (bottom) provided of the SSM assimilation for DF-REF.Soil layer depths (in cm) are represented in y-axis and the time is represented in x-axis.
Compared to DF-REF, a slight increase in the correlation coefficient and a decrease in RMSE are observed.The SDD falls by 42 % relative to the reference experiment.Yearly statistical scores of w tot analysis are shown in Fig. 5. Statistical scores of DF-REF and DF-B are close.The B tri-diag matrix has an important impact on the Kalman gain.Figure11shows the difference between the Kalman gain computed by the assimilation in DF-B and DF-REF.Adding correlation terms to the B matrix increases the Kalman gain values from 1 to 35 cm depth and for deeper layers during wet periods.For these layers during dry periods, the Kalman gain is strictly zero in DF-REF.Over the 3 yr period, the mean Kalman gain values increase by 27 % in the DF-B relative to DF-REF.

Fig. 11 .Fig. 12 .
Fig. 11.Difference of the Kalman Gain provided by the SSM assimilation in DF-B and DF-REF.Soil layer depths (in cm) are represented in y-axis and the time is represented in x-axis.

Fig. 5 .Fig. 13 .Fig. 14 .
Fig. 13.Representation of daily increments (m 3 .m−3 ) provided by the SSM assimilation in DF-CDF with a year per year CDF matching.Soil layer depths (in cm) are represented on the y-axis and time on the x-axis .

Table 1 .
Description of the ISBA-DF experiments.Bold face indicates changes in the assimilation experiments with respect to ISBA-DF reference experiment (DF-REF).

Table 2 .
w 1 open-loop and analysis scores (r, RMSE, SDD and bias) computed over the 3 yr period.Statistical scores are performed with rescaled observations.