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

The methodology to generate SWOT-like observations has only been very briefly shown in

Figure 3 (blue boxes). It is described in more detail in the following paragraphs.

SWOT data are generated using the simple methodology used in previous studies [

12,

13]. First, SWOT orbit ground tracks crossing the study domain are selected. For each orbit, polygons of SWOT swaths are created. These polygons are used to select all model pixels that lie in these swaths’ polygons, for each observation time during a 21-day repeat period. Then, this pixel’s selection is replicated for all repeat cycles (i.e., 21-day time spans) during the whole study time period. Then, for all observation times and all selected pixels, a noise on water depth is simply added to the reference simulation water depths (SIM_TRUE) to generate SWOT-like data. For the three first experiments, the instrument noise is a white noise with a 10 cm standard deviation, for all selected pixels. A less simplistic additive noise on water depth is computed for the last experiment (see

Section 4.4).

With this methodology, all pixels within the river network that lies in the swath polygons could be potentially observed. Yet, SWOT is conceived to observe rivers wider than 100 m (requirement) and could even observe rivers wider than 50 m (goal). To take this limitation into account, only river portions that were wider than 50 m were considered in this study. Furthermore, the 10 cm accuracy on water elevation is only met after aggregating SWOT data over 1 km

^{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).

Many sources of errors (orbit errors, SAR interferometry processing errors, impact of the surrounding topography and vegetation, delays induced by the crossing of the atmosphere by the electromagnetic wave) are therefore not considered, as the corresponding errors are not Gaussian.

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

In the DA platform, windows with a duration of 2 to 42 days were tested, as a function of the conditions imposed in the chosen DA experiment (see

Section 4). The background vector, x

_{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 Δ

h 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

We set-up four different DA experiments (

Table 3): The goal was to analyze whether the a priori values, x

_{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.

In the first experiment, we assimilated river depths over 48 h duration windows, across a period of three years. The choice of this duration was based on one main argument: The duration corresponds to the time taken by the water to flow from the upstream to the downstream section of the Garonne river observed by SWOT, i.e., from Saint-Gaudens to Bordeaux (see

Figure 1). This means that by using a window duration of 48 h, a roughness coefficient perturbation at Saint-Gaudens impacts all river reaches located downstream (until Bordeaux) during this window duration. It was decided that the experiment will commence with a background vector, x

_{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}.

In the second experiment, we assimilated data over 48 h duration windows by perturbing the amount of runoff and drainage produced by ISBA (increasing or decreasing the production with a bias of ±10%), and tested the impact on the convergence through the assimilation windows. It is important to represent well the atmospheric forcing: In DA, an error of quantification or repartition of precipitation will have a direct impact on the convergence of the parameters to correct, and thus on the quality of the river flow simulation. In this way, it is important to quantify the impact of a forcing error in ISBA on the convergence of the Strickler coefficients. It was decided that the experiment should start 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. 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 the third experiment, river depth differences were assimilated over a 42-day duration window in the model, considering that the satellite would not observe river depths, but, instead, open water surface elevations. A solution for the use of water surface elevation in our system was to assimilate the water surface elevation difference, δh, between two consecutive observations, which was equal to the river depth difference. In order to assimilate δh in the model, two or more river depth observations in one DA window were required. One can imagine two different scenarios:

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.

To respect the case, n° 2, for all reaches of the Garonne basin, we decided to extend the assimilation window to 42 days, to be sure that at least two observations were present for each reach in the assimilation window. It was decided 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 (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}.

In the fourth experiment, we increased the realism of the SWOT error measurement, by implementing time-and-space variable errors of observation: Each diagonal term, σ_{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.

Note that the common criterion for the comparison of these four different DA experiments is the quality of the convergence of the Strickler coefficients, K_{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.