# Assimilation of Synthetic SWOT River Depths in a Regional Hydrometeorological Model

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

^{1/3}s

^{−1}, and the estimation of associated river depths. It was also shown that river depth differences can be assimilated, resulting in a higher root mean square error for roughness coefficients with respect to the true river state. Finally, the last experiment shows how one can take into account more realistic sources of SWOT error measurements, in particular the importance of the estimation of the tropospheric water content in the process.

## 1. Introduction

^{2}drainage area) in 21 days, the duration of a full orbital cycle [9]. Water level measurement errors are expected to be 10 cm aggregating pixels over a 1 km

^{2}water area (e.g., a 10-km reach length for a 100-m-wide river) [10]. This offers a new opportunity for the linking of open water surface elevations, land surface processes, and meteorology more closely on this scale. The SWOT mission will provide relevant information on the temporal evolution of the surface water storage. This information will allow a better understanding of the term, Q, both spatially and temporally, playing an important role in the mass balance equation.

^{2}) [13], but focuses on a smaller basin (Garonne, 56,000 km

^{2}) located in another climatic region. The present study uses a different hydrological model, which has a much higher spatial resolution and a much more ramified river network than other studies (e.g., [13]).

## 2. Catchment and Hydrological Model Description

#### 2.1. The Garonne River Catchment

^{−1}over the Atlantic coastal region to about 600 mm year

^{−1}in the eastern part of the catchment. The upper Garonne and the Ariège rivers regime are characterized by spring snow melt in the Pyrenees [15], while summer flows are very low due to relatively dry summers.

#### 2.2. The ISBA/MODCOU Hydrological Model

_{str}is the Strickler coefficient (m

^{3}s

^{−1}), Q is the discharge (m

^{3}s

^{−1}), W is the width (m), and S

_{o}is the bed slope (-). The formulation is derived from the full Saint-Venant equation system, and supposes that the bed slope, S

_{o}, of the river bed is entirely compensated by the friction slope, S

_{f}. Thus, S

_{o}= S

_{f}means that there is no diffusion, but a simple translation in the time of the flood wave along the river channel. The K

_{str}coefficients are supposed to be constant over time and are spatially distributed for each grid cell of the model. The bed slopes, S

_{o}, are calculated from the Digital Elevation Model SRTM 90 m [26] in the full river network of the Garonne catchment. The discharge regime is permanent and uniform in each grid cell of the model, where the values of the water storage, the discharge, and the river depth simulation are solved with a «Runge-Kutta order 4» method to prevent numerical bias caused by the nonlinearity of the Manning-Strickler formula [27]. This routing method is used in global hydrological models (e.g., in the TRIP model [27,28,29]) or in the regional ISBA/MODCOU model [14]. In the present paper, a time step of 300 s is chosen for the river routing. The geometry of the river channel is assumed to be rectangular. This approximation is based on the assumption that using rectangular or trapezoidal river channel cross-sections leads to close results in term of water depth [14]: in the Garonne river at Tonneins, by imposing discharge values ranging from 0 to 1500 m

^{3}s

^{−1}and considering a low angle (30°) between the river banks and the vertical plane, the river depth difference between the rectangular and trapezoidal channel does not exceed 3 cm for high discharge values. By using a high angle (60°), the river depth difference is about 8 cm for a high discharge regime. This means that simulated river depths in RAPID are slightly overestimated for large flows in comparison with a high angle trapezoidal geometry.

## 3. Assimilation of Synthetic SWOT Data in ISBA/MODCOU: Experimental Design

#### 3.1. The SWOT Mission

#### 3.2. Data Assimilation Algorithm and SWOT Observing System Simulation Experiment

_{str}(noted x). Based on initial values of x, known as the background vector, x

_{b}, the analyzed parameters, x

_{a}, are calculated from Equation (2) [30]. We call “increment” the difference in value between the background, x

_{b}, and the analysis, x

_{a}.

_{a}= x

_{b}+ (

**B**

^{−1}+

**HR**

^{−1}

**H**

^{T})

^{−1}×

**H**

^{T}

**R**

^{−1}× (y

_{o}− H(x

_{b}))

**B**and

**R**are, respectively, the model error matrix and the observation model error matrix, y

_{o}represents the observation vector (river depth or river depth difference as a function of the experiment, see results). H is the observation operator that maps the control vector onto the observation space, it is the composition of the hydrometeorological model integration and the extraction of output values at observation time and space.

**H**is the Jacobian matrix of H, calculated by using a finite difference scheme. It is a linear approximation of the observation operator, H [30].

_{a}and x

_{b}that we decided to impose on the model, and that we call the «limitation of increment».

_{str}, we obtained the following equation (Equation (4)):

^{2}area (blue box on the bottom, Figure 3). This simple hypothesis will be further investigated in Section 4.4. A perturbed set of forcing and parameters, x

_{b}, is then used to run the background integration (noted SIM_PERT, red box, Figure 3) in which synthetic SWOT data are assimilated to retrieve the unperturbed forcing and parameters. The analysis branch (noted SIM_ANA, green box, Figure 3) for the control vector, x

_{a}, allows then the calculation of the corrected parameters to provide corrected river depths and discharge. The DA analysis is carried out over a sliding time window covering several observation times beyond which a forecast can be issued. The tool used to set up our DA scheme is the Open-PALM (Parallel Assimilation with a Lot of Modularity) software [31,32] developed at CERFACS (Centre Européen de Recherche et Formation Avancée en Calcul Scientifique) and ONERA (Office National d’Etudes et de Recherches Aérospatiales). This software allows the coupling of codes of calculation between them, and in this way, allows the exchange of variables, such as vector or matrix, between these codes. OpenPALM provides a parallel environment based on high performance implementation of the Message Passing Interface standard: This interface is able to perform both data parallelism and task parallelism [33].

#### 3.3. Sensitivity of the River Depth to the Roughness Coefficient, K_{str}

_{str}, is illustrated in Figure 4, for one unique grid cell, with typical conditions of a large rectangular plain section of the lower Garonne river. This test case is mono-dimensional, without the need to analyze the consequences of the effect of the K

_{str}perturbation on the upstream and downstream river depths. Discharge, Q, width, W, and bed slope, S

_{o}, are equal to 500 m

^{3}s

^{−1}, 150 m, and 0.0005 m m

^{−1}, respectively. The K

_{str}values vary between 20 and 40 m

^{1/3}s

^{−1}. Figure 4 shows that for a difference of K

_{str}equal to 5 m

^{1/3}s

^{−1}(between 25 and 30 m

^{1/3}s

^{−1}), the river depth variation is about 32 cm. For a difference of 1 m

^{1/3}s

^{−1}(between 25 and 26 m

^{1/3}s

^{−1}), the river depth variation is about 6.5 cm. It means that an error of one Strickler coefficient equal to x

_{b}− x

_{t}= 1 m

^{1/3}s

^{−1}is related to an error of one river depth, H(x

_{b}) − H(x

_{t}) = 6.5 cm, x

_{t}and H(x

_{t}) being the true Strickler coefficient and the true river depth. This result must be confirmed on the scale of the Garonne river under various hydrological conditions. In this present idealized test case considering one unique grid cell, the impact of a perturbation of K

_{str}on the water level is not shown for downstream river grid cells. The results presented in Section 4 allow the study of how changes in the Strickler coefficient values impact the flood dynamics on the full river network of the Garonne catchment.

_{str}value of 30 m

^{1/3}s

^{−1}. The chosen period experienced strong climate variability with dry and wet periods, allowing the analysis of different ranges of discharge in the selected reach. We imposed four different perturbations to the reference K

_{str}value and quantified the impact on the left balance term of Equation (4) (averaged value over the full period of study, by considering a daily time step). Table 1 shows that by comparing the four different perturbations, the linear approximation is only valid for small perturbations. In addition, it appears that the model is not symmetric. Hence, it is important to limit the perturbations to +5%, and to impose a maximum increment of 1.5 m

^{1/3}s

^{−1}, in order to avoid any instability of the DA system. Table 1 shows that for a perturbation of +/−20%, the system is not at all linear.

#### 3.4. Data Assimilation Experiment Setting

#### 3.4.1. SWOT-Like Observations Generation and Their Processing

^{2}[10]. That is why Strickler coefficients, K

_{str}, and river depths, h, from the reference simulation are aggregated over 10 km reaches (5 to 10 RAPID grid cells) to meet this requirement, before adding the noise on water depths. The aggregation is operated by averaging the values in the considered grid cells, with a weighting proportional to their length (1 or 2 km). The area of each reach varies from 0.5 to 2 km², knowing that the computed width varies from 50 m to 200 m. A total of 165 reaches (Figure 5a) were constituted. Over the study domain in our experimental set-up, reaches were observed between one and four times per repeat period (Figure 5b).

#### 3.4.2. Description of the Variables Used in the DA Platform

_{b}, contains the 165 Strickler coefficient values for each river reach. The vector, H(x

_{b}), contains the 165 river depths simulated by the model for each river reach. The observation vector, y

_{o}, contains the p SWOT observations by a DA window. If two examples are taken with five river reaches: p = 10 when these five reaches are observed two times during a DA window, or more complicated: p = 20 when the first reach is observed two times, the second reach is observed three times, and the third, fourth, and fifth reaches are observed five times during the window. It presents a sum of 10 SWOT observations in the first example, and 20 observations in the second. The p value is then a function of the number of SWOT passes across the catchment during one DA window, and of the number of rivers reaches observed during one pass of the satellite. The Jacobian matrix,

**H**, contains the sensitivity of the 165 simulated river depths with a 5% perturbation of the p Strickler coefficients located in the p observed reaches. The value of +5% was chosen due to Sub-Section 3.3 (see also Table 1), having a minor impact on the Δh value calculated in Equation (4), for a typical large plain river. The observation error matrix,

**R**, and the background error matrix,

**B**, are diagonal:

**R**contains the errors of the p river depth observations, associated with p diagonal terms.

**B**contains the errors of the 165 background Strickler coefficients, associated with 165 diagonal terms. According to Sub-Section 3.3, in order to avoid problems of non-linearity, we decided to limit the limitation of increment to 1 m

^{1/3}s

^{−1}in each experiment. The different variables of the Extended Kalman Filter equation (Equation (2)) and their dimension are described in Table 2.

#### 3.4.3. Description of the Data Assimilation Experiments

_{b}, of the Strickler coefficient could converge to a stable value and tend to the truth, x

_{t}, through the assimilation windows, and if the simulated river depth, H(x

_{b}), could be well estimated and tend to the true depth, H(x

_{t}). The DA configuration in the first experiment was the same as in previous works related to SWOT DA [13]. In the three following experiments, more realistic experiments were then set up in the DA platform. In the experiments, n° 1, 2 and 4, the chosen period of study was 1995–1998, and 1995–2001 in the experiment, n° 3, three further years were needed to get a full convergence of the system. We will show that the period of study of this third experiment is longer, due to a longer convergence time of the system. During 1995 to 2001, wet and dry periods were observed. The strong climate variability of these years had a direct impact on the discharge: We therefore conclude that it was a good idea to have selected these years for our test case. It allowed us to analyze a wide range of discharge signals. In the following paragraphs, a short description of all four experiments is proposed and synthesized in Table 3.

_{b}, initialized with values equal to 25 m

^{1/3}s

^{−1}. This value corresponded to the averaged reference Strickler coefficient, x

_{t}, in the catchment. Each diagonal term, σ

_{B}

^{2}, of the model error matrix,

**B**, contained values equal to the variance of all the x

_{b}parameters around the truth, x

_{t}, with a minimum value of (1.5 m

^{1/3}s

^{−1})

^{2}. Each diagonal term, σ

_{R}

^{2}, of the observation error matrix,

**R**, contained values equal to (10 cm)

^{2}.

_{b}, initialized with values equal to the reference Strickler parameters on which a Gaussian centered noise, σ

_{xb}, of 5 m

^{1/3}s

^{−1}was added. Each diagonal term, σ

_{B}

^{2}, of the model error matrix,

**B**, contained values equal to the variance of all the x

_{b}parameters around the truth, x

_{t}. A minimum value of (1.5 m

^{1/3}s

^{−1})

^{2}was imposed in the DA platform. Note that σ

_{xb}was always equal to σ

_{B}when σ

_{B}≥ 1.5 m

^{1/3}s

^{−1}, and σ

_{xb}≤ σ

_{B}when σ

_{B}= 1.5 m

^{1/3}s

^{−1}. Each diagonal term, σ

_{R}

^{2}, of the observation error matrix,

**R**, contained values equal to (10 cm)

^{2}.

- In a DA window, there are zero or one river depth observations: It is impossible to assimilate a δh term; and
- In a DA window, there are n observations (n ≥ 2): It is possible to assimilate n − 1 δh terms.

_{b}, initialized with values equal to the reference Strickler parameters on which a Gaussian centered noise, σ

_{xb}, of 5 m

^{1/3}s

^{−1}was added. Each diagonal term, σ

_{B}

^{2}, of the model error matrix,

**B**, contained values equal to the variance of all the x

_{b}parameters around the truth, x

_{t}. A minimum value of (2.12 m

^{1/3}s

^{−1})

^{2}was imposed in the DA platform. Each diagonal term, σ

_{R}

^{2}, of the observation error matrix,

**R**, contained values equal to (14.1 cm)

^{2}.

_{R}

^{2}, of the observation error matrix,

**R**, contained values varying for every reach at each DA window. Data were assimilated over a 48 h duration window. The decision was made to start the experiment with a background vector, x

_{b}, initialized with values equal to the reference Strickler parameters on which a Gaussian centered noise, σ

_{xb}, of 5 m

^{1/3}s

^{−1}was added. Each diagonal term, σ

_{B}

^{2}, of the model error matrix,

**B**, contained values equal to the variance of all the x

_{b}parameters around the truth, x

_{t}. A minimum value of (1.5 m

^{1/3}s

^{−1})

^{2}was imposed in the DA platform.

_{str}, to the truth, x

_{t}. The convergence time of the K

_{str}is irrelevant for the comparison between the four experiments, because the duration of the DA windows, the first guess, x

_{b}, of the K

_{str}at the beginning of the experiment, and the attribution of the diagonal terms in the R and B matrices vary in function to the experiment.

## 4. Results

#### 4.1. Assimilation of River Depths (Experiment 1)

_{B}

^{2}, in the

**B**matrix to (1.5 m

^{1/3}s

^{−1})

^{2}: When the values are too low (<(1 m

^{1/3}s

^{−1})

^{2}), the convergence of the terms (x

_{b}− x

_{t}) to the truth, x

_{t}, is complicated, because the analysis is closer to the background than the observation (for the EKF, there are fewer uncertainties in the background than in the observation). This limitation criterion thus improves the convergence of the terms (x

_{b}− x

_{t}) to the truth, x

_{t}. Concerning the value of the increment (x

_{a}− x

_{b}), when no limitation is imposed, the values of the terms (x

_{b}– x

_{t}) of the DA system are very high in the first windows (between −40 to +70 m

^{1/3}s

^{−1}), and the convergence time is about 800 days. When a limitation of increment of 5 m

^{1/3}s

^{−1}is imposed, the values of the terms (x

_{b}– x

_{t}) of the DA system during the early windows are between -40 and +30m

^{1/3}s

^{−1}until 100 days of assimilation, and the convergence to the truth occurs after 240 days. In the current experiment illustrated in Figure 6, the majority of the K

_{str}values converge to the truth after 700 days of assimilation, considering a limitation of increment fixed to 1 m

^{1/3}s

^{−1}. We consider that the Strickler coefficient convergence to the truth, x

_{t}, is reached when the standard deviation, σ

_{xb}, between all (x

_{b}− x

_{t}) values is equal or lower than 1 m

^{1/3}s

^{−1}.

_{t}.

^{1/3}s

^{−1}, while the true value was 38.7 m

^{1/3}s

^{−1}. The simulated river depth, H(x

_{b}), before SWOT data assimilation was then higher than the true river depth H(x

_{t}), with a positive difference of about 20 cm in normal flow, and more than 80 cm during flood periods. The average true river depth, H(x

_{t}), over the full period of study (365 days) was equal to 2.11 m, with peak values up until 7.5 m. Furthermore, at the beginning of the experiment, the river flow velocity at Lamagistère and upstream was too low (due to low x

_{b}values compared to x

_{t}): The temporal delay between the river depth time series before assimilation and the true time series was negative (with about one day of delay).

_{t}(positive difference of 7.5 m

^{1/3}s

^{−1}), which was equal to 17.5 m

^{1/3}s

^{−1}: The relative difference between the first x

_{b}value and x

_{t}was equal to +42.9%. The simulated river depth, H(x

_{b}), before SWOT data assimilation was then lower than the true river depth, H(x

_{t}), with a negative difference of 20–30 cm in normal flow, and more than 1 m during flood periods. The average true river depth, H(x

_{t}), over the full period of study (365 days) was equal to 2.10 m, with peak values up to 9 m. Furthermore, at the beginning of the experiment, the river flow velocity at Bergerac and upstream was too high (due to high x

_{b}values compared to x

_{t}): The delay between the river depth time series before assimilation and the true time series was positive (with about one day of advance).

_{b}) and H(x

_{t}) were improved: The time difference between the arrival of one H(x

_{b}) peak and one H(x

_{t}) peak was less than 3 h, and the difference between H(x

_{b}) and H(x

_{t}) was inferior to 10 cm for every discharge regime.

#### 4.2. Assimilation of River Depths, in case of Atmospheric Forcing Biases (Experiment 2)

_{str}values to decrease the river depth. The effect of an under-production of water in the river is the opposite: The DA system will decrease the K

_{str}values to increase the river depth. The illustration in Figure 8 shows that after about 1000 days of assimilation, the Strickler coefficient values converge to a stable state, but differ from the true x

_{t}with an average bias of −1.56 m

^{1/3}s

^{−1}in scenario (a), and an average bias of +1.83 m

^{1/3}s

^{−1}in scenario (b). For a water production bias of +10% or −10%, we showed that the impact on the terms (x

_{b}− x

_{t}) at the end of the DA experiment was inferior to 2 m

^{1/3}s

^{−1}, having a minor impact on the flow velocity (phasing delay of 3 h at Tonneins between the “true” hydrograph and the “analyzed” hydrograph).

#### 4.3. Assimilation or River Depth Differences (Experiment 3)

_{t}and h

_{t+1}, follows Equation (5):

_{t}, h

_{t+1}) = 0 and the observation error (Var(h

_{t+1}− h

_{t}))

^{1/2}thus becomes σ

_{R}= 14.1 cm.

_{R}and σ

_{B}(see Equation (6)), of the two matrix,

**R**and

**B**(as the experiments detailed in the Section 4.1 and Section 4.2), the minimum value of σ

_{B}is now fixed to 2.12 m

^{1/3}s

^{−1}. This value is calculated from the diagonal terms, σ

_{B,}equal to 1.5 m

^{1/3}s

^{−1}in the

**B**matrix used in the two previous experiments. We show in Figure 9 the temporal evolution of the terms (x

_{b}− x

_{t}) for all reaches in the basin.

_{t}. The convergence of four reaches occur more slowly: One plausible hypothesis is that the observation error is on average larger than for the other reaches, driven by the random Gaussian noise, σ

_{R}, which can be lower or higher than 14.1 cm across the full study period. There are only 50 DA windows in this experiment, and the impact of σ

_{R}on the convergence of x

_{b}is different for one given reach after these 50 windows, when the experiment is remade. After 50 windows or 2100 days of assimilation, the standard deviation, σ

_{xb}, of all (x

_{b}− x

_{t}) terms is about 1.70 m

^{1/3}s

^{−1}: This value is bigger than the σ

_{xb}value presented in the first experiment (Section 4.1), because the diagonal terms, σ

_{R}

^{2}and σ

_{B}

^{2}, imposed in the observation error matrix,

**R**, and the model error matrix,

**B**, are more significant.

#### 4.4. Assimilation of River Depths Considering More Realistic SWOT Errors (Experiment 4)

_{R}, in the observation error matrix,

**R**, vary in space and time (at each DA window). The total error (taking into account the variable and constant errors averaged in time and space) should be equal to 10 cm, and is composed of systematic and non-systematic errors [10]. The non-systematic errors are random and Gaussian, and the systematic errors are constant (see Section 3.1). We are able to approximate by a very simple method the different sources of instrumental error contributing to the changing of the SWOT error measurement values in space and time. Four factors with an impact on the diagonal term values, σ

_{R}, of the observation error matrix,

**R**, are introduced, allowing σ

_{R}to vary at each time step and for every river reach. The first three factors (non-systematic source of errors) are: The surface of the reach, the SWOT look angle, and the radar water surface roughness. The calculation of these simple instrumental-like errors is based on previous works [34]. The last factor having an impact on the σ

_{R}of the

**R**matrix is the wet-troposphere error (systematic source of errors). This error is estimated in this study by the intra-day variability of water content in the troposphere. There are other sources of errors, more difficult to quantify, but they should have a lower impact on the total error budget at the reach scale: They are the ionosphere signal, the dry troposphere signal, the orbit radial component, and the KaRIn random and systematic errors after cross-over corrections. Table 4 summarizes the SWOT error budget [35], after averaging a 1 km² area [10]. It is important to note that all the equations used to calculate these different sources of instrumental errors are greatly simplified, the more realistic and complex full radar equations are not used in our study. The main goal of this fourth experiment is to take into account errors of measurement more realistic than simple Gaussian errors of 10 cm, but it is important to keep in mind that these errors are simplified compared to the detailed error budget calculated by the SWOT simulator used by the JPL and the CNES (see [36]).

#### 4.4.1. SWOT Instrumental Error along the Swath

^{−1}. Wind speed values higher than 8 m s

^{−1}are rare, representing about 1% of all wind speed values over the study period. The wind speeds correspond to SAFRAN reanalysis data at 10 m (3-h time step) averaged over a 10 min period.

#### 4.4.2. The Intra-Day Variability of Water Content in the Troposphere: Impact on the SWOT Error of Measurement

^{−1}) at 10 m (3-h time step) are used to calculate intra-day variances Vi (g kg

^{−1})

^{2}across the study period 1 August 1995–31 July 1998. Considering the 165 river reaches of the DA platform over the full study period, a simple relationship between the intra-day variance of water content, Vi, and the SWOT error measurement, σ

_{R}, is set up: The greater Vi, the greater the measurement error. We would like to build a linear relationship between Vi and σ

_{R}. Once the relationship between Vi and σ

_{R}is established, one can attribute a SWOT error measurement error, σ

_{R}, at each time step for every river reach of the catchment. The slope of the function linking Vi and σ

_{R}is equal to 6.67 cm/(g kg

^{−1})

^{2}. This relationship is based on the assumption that the spatial and temporal averaged wet troposphere error is equal to 4.0 cm (see Table 4), and that the full spatio-temporal SAFRAN reanalysis data distribution of water content is well-known in each river grid cell of the catchment. Eighty percent of the Vi values range between 0 and 1 (g kg

^{−1})

^{2}. Values higher than 3 (g kg

^{−1})

^{2}are rare, representing about 1% of all Vi values over the full study period.

#### 4.4.3. SWOT DA Experiment Using Realistic Errors of Measurement

**R**, is built considering the two sources of errors described in Section 4.4.1 and Section 4.4.2: Each term, σ

_{R}, varies at each time step of the experiment for every river reach, and the spatio-temporal average of σ

_{R}over the full period of the study and domain is equal to 10 cm. After 365 days of assimilation, the majority of the parameters converges to the truth, x

_{t}, with a standard deviation of ~1 m

^{1/3}s

^{−1}. During the DA windows n° 259 and 448, and 518 and 896 days of assimilation, the standard deviation, σ

_{xb}, between all (x

_{b}− x

_{t}) values decreases, respectively, to 0.76 and 0.79 m

^{1/3}s

^{−1}(Figure 11). To understand these low values, we analyzed the temporal evolution of the Vi variable over the full study period: It showed that during the winter months the values of Vi are equal to 0.5 (g kg

^{−1})

^{2}on average (the days 518 and 896 of this experiment are in the winter season), and 1.5 (g kg

^{−1})

^{2}on average during the summer months. The low values of Vi during the winter season lead to low SWOT error measurements, σ

_{R}, and then to a better convergence of the K

_{str}values to the truth, x

_{t}. The convergence time of this DA experiment is around 600 days when the x

_{b}values of each reaches convergence at the truth, x

_{t}.

## 5. Conclusion and Perspectives

_{str}, of the river bed.

_{str}parameter around the known truth. In the first experiment, we showed that the parameters converge to the truth after ~2 years, with an average error of ±1 m

^{1/3}s

^{−1}, and with an associated average error on river depths of ±5 cm. The choice of the maximum increment value (the difference between the analysis, x

_{a}, and the a priori value of the Strickler coefficient, x

_{b}) has an impact on the time of convergence. In the second experiment, we showed the importance of the atmospheric forcing and its impact on the convergence of the K

_{str}parameters. An error of ±10% water produced by ISBA led to an error of ±1.5 m

^{1/3}s

^{−1}, but had a minor impact on the river flows. The impact of greater errors on the discharge phasing should be taken in consideration. The third experiment illustrated the interest of assimilating the difference of river depths instead of absolute open surface water elevations that are equivalent to river depths in our study. The bathymetry elevation is unknown in RAPID, and the satellite will not provide measurements of river depths. The duration of a DA window should be equal to 42 days, in order to be sure that every reach is observed at least two times during one DA window. The quality of the convergence (±1.69 m

^{1/3}s

^{−1}) was slightly lower in comparison with the first experiment, but the time of convergence was equivalent (~2 years). In the last experiment, we proposed the introduction of more realistic SWOT error measurements varying in time and space, which depended on the surface of the observed reach, the look angle of the satellite, the surface water roughness, and the intra-day variability of the water content in the troposphere. The impact of the last factor seems to play an important role: The convergence was on average better during the winter months than during the summer months, because of a low temporal variability of the troposphere water content.

#### 5.1. The Choice of the Data Assimilation Method and the Experimental Design

_{b}, and the analysis, x

_{a}, calculated by the DA system). Other methods, such as the Ensemble Kalman Filter (EnKF) [40], could be tested to see whether it is possible to reduce the convergence time of the parameters, or to decrease the standard deviation, σ

_{xb}, around the truth, x

_{t}, at the end of one given DA experiment. In our study, we assumed that the model is linear: We should use it in the domain of linearity by imposing small perturbations to the parameter to correct. The EnKF can be used in the domain of non-linearity and could be tested. It has gained popularity because of its simple conceptual formulation and relatively easy implementation, requiring no derivation of a tangent linear operator or adjoint equations, and no integrations backward in time [40]. Concerning the description of the different variables and data assimilation experiments described in Section 3.4, another way in which to build the experimental design could be proposed: One important one is the testing of what the impact is on different lengths of river reaches on the convergence of the Strickler coefficients in all four DA experiments, the SWOT error measurement being directly linked to the length of the reaches. Another point is the attribution of the different parameters used in the DA platform: Choice of the DA window duration, threshold of the increment (x

_{a}− x

_{b}) limitation, and attribution of the model error matrix. The choice of the attribution of the first a priori value, x

_{b}, at the beginning of each experiment also plays an important role, having a direct impact on the time of convergence of the Strickler coefficients. The study period can present a more or less spatial and temporal variability of precipitation reaching the surface, impacting directly the flood dynamics in the river network, and thus the increment term (x

_{a}− x

_{b}), which depends on the discharge regime. All these factors shortly described could have different impacts on the convergence of the system.

#### 5.2. Choice of the Observation to Assimilate and the Parameter to Correct

#### 5.3. Assimilation of River Depths Considering More Realistic Meteorological and SWOT Errors

^{−1})

^{2}could therefore be more difficult to estimate in case A than the same Vi value during case B.

#### 5.4. Extension of SWOT DA in Other Catchments of the World

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The Garonne catchment with the names of main rivers (blue) and tributaries (black), and the names of main cities (dark black).

**Figure 2.**(

**a**) Illustration of SWOT (© NASA-http://swot.jpl.nasa.gov/): The widths of the two rectangles represent two swaths of 50 km wide, observed by the two antennas of the satellite. They can observe either ocean or continental open water surfaces. (

**b**) Illustration of the left and right swaths of SWOT, with the range direction perpendicular to the satellite track (which is the azimuth direction).

**Figure 3.**Schematic illustration of the twin experiment. The red boxes describe the perturbed parameters of the model (SIM_PERT). The blue boxes represent the truth SIM_TRUE (reference simulation) to which white noise is added to generate SWOT-like measurement error. The green boxes represent the corrected parameters of the model after a DA window: We obtained corrected parameters of the model (SIM_ANA) being used for the next DA window (black line), allowing the calculation of new corrected parameters at the next window’s termination.

**Figure 4.**River depth, h, as a function of the roughness coefficient, K

_{str}, for a large rectangular river channel, with geomorphological parameters and a discharge value typical of large plain rivers.

**Figure 5.**(

**a**) River widths computed in the DA platform over the Garonne catchment, considering three river width classes where virtual SWOT observations are assimilated; (

**b**) SWOT observation number for a 21-day cycle for every 10-km reaches of the Garonne catchment.

**Figure 6.**Temporal evolution (1 August 1995 to 31 July 1998) of the difference between the a priori control vector (x

_{b}) and the true control vector (x

_{t}) of the Strickler coefficients in the DA system. The 165 black curves represent the value (x

_{b}− x

_{t}) at all 165 reaches defined over the Garonne catchment. The red horizontal line is equivalent to a zero difference between x

_{b}and x

_{t}. After 3 years (i.e., 548 48 h-assimilation windows), we represent the standard deviation, σ

_{xb}, between the 165 values (x

_{b}− x

_{t}) in the basin. The X terms represent, from top to bottom, the minimum value, the 1st quartile, the 2nd quartile, the median, the 3rd quartile, and the maximum value of (x

_{b}− x

_{t}) at the end of the experiment (day n° 1096).

**Figure 7.**Temporal evolution of the river depth at Lamagistère (

**a**) and Bergerac (

**b**), over the period of 1 August 1995 to 31 July 1996. Black curves correspond to the truth and red curves are the result from the assimilation of the river depth.

**Figure 8.**Temporal evolution (1 August 1995 to 31 July 1998) of the difference between the a priori control vector (x

_{b}) and the true control vector (x

_{t}) of the Strickler coefficients in the DA system. Scenario (

**a**) represents a −10% perturbation of water produced by ISBA, and scenario (

**b**) represents a +10% perturbation. The 165 black curves represent the value (x

_{b}− x

_{t}) of all 165 reaches defined over the Garonne catchment. The red horizontal line is equivalent to a zero difference between x

_{b}and x

_{t}. After 3 years (i.e., 548 48 h-assimilation windows), we represent the standard deviation, σ

_{xb}, between the 165 values (x

_{b}− x

_{t}) in the basin. The X terms represent from top to bottom the minimum value, the 1st quartile, the 2nd quartile, the median, the 3rd quartile, and the maximum value of (x

_{b}− x

_{t}) at the end of the experiment (day n° 1096).

**Figure 9.**Temporal evolution (1 August 1995 to 15 April 2001) of the difference between the a priori control vector (x

_{b}) and the true control vector (x

_{t}) of the Strickler coefficients in the DA system. The 165 black curves represent the value (x

_{b}− x

_{t}) at all 165 reaches defined over the Garonne catchment. The red horizontal line is equivalent to a zero difference between x

_{b}and x

_{t}. After 5 years (i.e., 50 42 day-assimilation windows), we represent the standard deviation, σ

_{xb}, between the 165 values (x

_{b}− x

_{t}) in the basin. The X terms represent from top to bottom the minimum value, the 1st quartile, the 2nd quartile, the median, the 3rd quartile, and the maximum value of (x

_{b}− x

_{t}) at the end of the experiment (day n° 2100).

**Figure 10.**SWOT error measurement as a function of the look angle for a wind speed of 0 m s

^{−1}, 4 m s

^{−1}, or 8 m s

^{−1.}

**Figure 11.**Temporal evolution of the standard deviation, σ, between the 165 (x

_{b}− x

_{t}) terms, over the 1 August 1995–31 July 1998 period.

**Table 1.**Impact of the relative and absolute perturbation of the K

_{str}reference coefficient on the river depths, h (cm).

ΔK_{str} | Averaged Δh |
---|---|

−5% (−1.5 m^{1/3} s^{−1}) | +1.5 cm |

+5% (+1.5 m^{1/3} s^{−1}) | +1.0 cm |

−20% (−6 m^{1/3} s^{−1}) | +7.5 cm |

+20% (+6 m^{1/3} s^{−1}) | +3.5 cm |

**Table 2.**Synthetic description of the variables used in the assimilation equations (symbol, description, and dimension). The p value corresponds to the number of SWOT observations during a DA window.

Symbol | Variable | Dimension |
---|---|---|

x_{b} | A priori vector of the Strickler coefficients | 165 |

x_{t} | Truth vector of the Strickler coefficients | 165 |

x_{a} | Analysis vector of the Strickler coefficients | 165 |

y_{o} | Observation vector of the synthetic SWOT river depths during a DA window | p |

H(x_{b}) | Model equivalent of the y_{o} vector, with the river depths simulated by RAPID during a DA window | p |

R | Observation error matrix of river depths | p × p |

B | Model error matrix of Strickler coefficients | 165 × 165 |

H | Jacobian matrix: sensitivity of H(x_{b}) to a perturbation of the Strickler coefficient in the model | 165 × p |

Criterion | Experiment 1: River Depth Assimilation | Experiment 2: River Depth Assimilation by Perturbing ISBA Outputs | Experiment 3: Difference of River Depth Assimilation | Experiment 4: River Depth Assimilation Considering More Realistic SWOT Errors |
---|---|---|---|---|

Assimilation type | River depths | River depths; ISBA runoff and drainage perturbed ±10% | river depth differences | River depths |

Window duration | 48 h | 48 h | 42 days | 48 h |

Beginning of the experiment | 1 August 1995 | 1 August 1995 | 1 August 1995 | 1 August 1995 |

End of the experiment | 31 July 1998 | 31 July 1998 | 15 April 2001 | 31 July 1998 |

First a priori value x_{b} | Each value of Strickler coefficient are equal to 25 m^{1/3} s^{−1} | “Truth” of the 165 Strickler coefficient values perturbed with a centered Gaussian noise, σ_{xb}, equal to 5 m^{1/3} s^{−1} | “Truth” of the 165 Strickler coefficient values perturbed with a centered Gaussian noise, σ_{xb}, equal to 5 m^{1/3} s^{−1} | “Truth” of the 165 Strickler coefficient values perturbed with a centered Gaussian noise, σ_{xb}, equal to 5 m^{1/3} s^{−1} |

Definition of the matrix B | Diagonal, each value σ_{B}^{2} is equal to the variance of all the x_{b} values around the truth. Minimum imposed value = (1.5 m^{1/3} s^{−1})² | Diagonal, each value σ_{B}^{2} is equal to the variance of all the x_{b} values around the truth. Minimum imposed value = (1.5 m^{1/3} s^{−1})² | Diagonal, each value σ_{B}^{2} is equal to the variance of all the x_{b} values around the truth. Minimum imposed value = (2.12 m^{1/3} s^{−1})² | Diagonal, each value σ_{B}^{2} is equal to the variance of all the x_{b} values around the truth. Minimum imposed value = (1.5 m^{1/3} s^{−1})² |

Definition of the matrix R | Diagonal, each value σ_{R}^{2} is equal to (10 cm)² | Diagonal, each value σ_{R}^{2} is equal to (10 cm)² | Diagonal, each value σ_{R}^{2} is equal to (10 cm)² | Diagonal, each value σ_{R}^{2} varies in the time and the space |

Hydrology Error Component | Height Error (cm) |
---|---|

Ionosphere signal (constant) | 0.80 |

Dry troposphere signal (constant) | 0.70 |

Wet troposphere signal (variable) | 4.00 |

Orbit radial component (constant) | 1.62 |

KaRIn Random and systematic errors after cross-over correction (constant) | 7.74 |

Surface of the reach + look angle + wind speed (variable) | 4.40 |

Total allocation | 9.95 |

Unallocated margin (constant) | 1.04 |

Total error | 10.0 |

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**MDPI and ACS Style**

Häfliger, V.; Martin, E.; Boone, A.; Ricci, S.; Biancamaria, S.
Assimilation of Synthetic SWOT River Depths in a Regional Hydrometeorological Model. *Water* **2019**, *11*, 78.
https://doi.org/10.3390/w11010078

**AMA Style**

Häfliger V, Martin E, Boone A, Ricci S, Biancamaria S.
Assimilation of Synthetic SWOT River Depths in a Regional Hydrometeorological Model. *Water*. 2019; 11(1):78.
https://doi.org/10.3390/w11010078

**Chicago/Turabian Style**

Häfliger, Vincent, Eric Martin, Aaron Boone, Sophie Ricci, and Sylvain Biancamaria.
2019. "Assimilation of Synthetic SWOT River Depths in a Regional Hydrometeorological Model" *Water* 11, no. 1: 78.
https://doi.org/10.3390/w11010078