The ISBA model is a LSM defined at global scale on a 0.5∘ × 0.5∘ regular mesh grid that establishes the energy and water budget over continental surfaces. This study operates the ISBA-3L version based on a three-layer soil . The budget equations are solved separately on each grid cell. Still, larger-scale spatial patterns in the radiative and precipitation forcing, the soil composition and the vegetation cover ensure spatial correlations between those cells (for more details, see ). In particular, ISBA gives a diagnostic of the surface runoff (QISBA,sur) and the gravitational drainage (QISBA,sub, i.e., water percolating to the deep layers of the soil) later used as forcing inputs for the RRM denoted CTRIP.

The CTRIP model , is defined on the same mesh grid as ISBA and follows a river network to laterally transfer water from one cell to another, down to the interface with the ocean . The study is based on the CTRIP version from with three reservoirs, as illustrated in Fig. a. The water mass (kg) stored in a groundwater reservoir G and a floodplain reservoir F interacts with the water mass in the surface reservoir S representing the river. Only the surface reservoir S is related to the river network and fills with the surface runoff QISBA,sur, the outflow from upstream cells and the delayed drainage QISBA,sub by means of the groundwater reservoir. Occasionally, when the amount of water in the river exceeds a given threshold (defined by the water level in the reservoir), the river spills into the floodplains.

Within a 0.5∘ × 0.5∘ cell, the surface reservoir is a unique river channel that may gather multiple real river branches. Its rectangular cross section is described by its slope s (–), its width W (m), its bankful depth Hc (m), its length L (m) and finally a Manning or friction coefficient N (sm-1/3) that assesses the reach resistance at the bottom of the river.

Each cell's elevation is deduced from the STN-30p digital elevation model (http://daac.ornl.gov/ISLSCP_II/islscpii.shtml, last access: 20 April 2020). These elevations are then compared to determine the riverbed slope s. Global empirical geomorphologic relationships are used to define the river width W and bankful depth Hc. The arc length between grid cell centers, inflated by a meandering factor μ, results in the river reach length L. More details on these parameters can be found in and .

The Manning coefficient N is generally more complicated to estimate. Following , it should take values between 0.025 and 0.03 for natural streams and values between 0.075 and 0.1 for smaller and mountainous tributaries and also floodplains. Global studies can apply either a constant or a spatially distributed Manning coefficient. However, it is ordinarily accepted that this parameter should vary in space and even in time across the river catchment. Consequently, CTRIP uses a spatially distributed Manning coefficient based on a simple linear relationship between the relative stream size in the current cell, denoted SO, and the size at the river mouth and the source cells, so that 1N=Nmin+(Nmax-Nmin)SOmax-SOSOmax-SOmin. SO is the stream size relative measure at the current cell, SOmax (whose value depends on the network depth) the same measure at the river mouth and SOmin=1 the measure at source cells (namely cells without any upstream cells, according to the river network). The Manning coefficient is then set to be constant in time while its spatial values decrease towards the river outlet (following the river network), with values between Nmin=0.04 and Nmax=0.06.

All these parameters are eventually essential to estimate the spatially and time-varying average cross-sectional flow velocity in the surface reservoir v(t) following the Manning formula : 2v(t)=s12NWhS(t)W+2hS(t)23, where hS is the river water depth estimated from the river storage S by 3hS=SρWL and ρ is the water density. The flow velocity is ultimately used to estimate the discharge leaving the CTRIP cell: 4QoutS(t)=v(t)LS(t).

As the definitions of most of these parameters are based on empirical relationships, we have to be aware that they inevitably have substantial uncertainties.

In this study, we present an OSSE test case over the Amazon River basin, whose hydrology is carefully described in and . This choice was motivated by the present work following and complementing studies over the same domain .

For ISBA-CTRIP, the Amazon basin is composed of a total number of 2028 cells. Based on the basin geomorphology and hydrology , the basin has been split into nine spatial regions. These zones, illustrated in Fig. b, were initially introduced in and will be re-exploited here within the application of data assimilation. For a detailed description of the zones, the reader can refer to .

For the present study, ISBA-CTRIP needs external atmospheric forcing in order to run. Similarly to , such data are provided by Global Soil Wetness Projet 3 (GSWP3, http://hydro.iis.u-tokyo.ac.jp/GSWP3, last access: 20 April 2020) at a 3-hourly time resolution.

In OSSEs, we introduce beforehand a reference configuration for the model input parameters that we will consider thereafter to be the truth. From those true parameters, we directly deduce the true run from a ISBA-CTRIP model integration. The synthetic observations used for data assimilation are obtained by perturbing the true observables (variables that are used as observations) using an error model that is representative of the real observation errors. The control variables (the model variables to be corrected with data assimilation) first guess is obtained by directly perturbing the true control variables. Control error also has to be chosen to be representative of the real modeling errors. OSSEs are prerequisite tests to ensure that the implementation of the EnKF algorithm is correct and adapted to the hydrologic problem under consideration (temporal/spatial length scales, sources of uncertainty, observation operator, etc.).

In the EnKF framework, the model M[k-1,k] and observation Hk operators are generally not linear. The main assumption for the EnKF is to use stochastic ensembles to represent first- and second-order moments (namely the means and the covariances) of the control variable errors . Indeed, it is assumed that the distribution of the ensemble is similar to that of the error of the control vector, and it is also assumed that the probability density function (PDF) of the error is Gaussian and thus well described by its first and second moments. The background control variables xkb (the first guess) are therefore represented by an ensemble of ne members: 10Xe,kb=xkb,[1]xkb,[2]…xkb,[ne].

To avoid the collapsing of the ensemble, the observation vector in Eq. () is randomized by adding a supplementary white noise with the same observation error standard deviation σo so that ∀j=1…ny,∀l=1…ne,yk,jo,[l]11=yk,jo+ϵjo,[l],ϵjo≃N(0,σo). An observation ensemble is generated: 12Ye,ko=yko,[1]yko,[2]…yko,[ne]. Note that alternatives exist to the observation randomization chosen here. However, for the present study, we choose to use a full stochastic filter.

Finally, the EnKF analysis step is applied to each member of the ensemble so that 13∀l=1…ne,xka,[l]=xkb,[l]+Ke,kyko,[l]-Hk(xkb,[l]), where Ke,k is the Kalman gain. It is built from the control and observation error covariance matrices P and R and the linearized observation operator H so that (see Appendix  for more details) 14Ke,k=[PHT]e,k[HPHT]e,k+Rk-1.

Figure  summarizes the general OSSE framework used for the present study. The figure reads from top to bottom and from left to right. An assimilation cycle [k-1,k] includes a forecast step where an ensemble of ISBA-CTRIP simulations is integrated, each member having a different spatially distributed Manning coefficient; an analysis step where the ensemble of Manning coefficients is corrected using synthetic observations through the Kalman filter update in Eq. (); and a cycling step where the ISBA-CTRIP model is re-run with these analysis estimates to obtain updated model states.

During one EnKF assimilation cycle, the analysis potentially depends on the following parameters: model spinup, time period (high/low flow), size of the ensemble, and control error. Note that the observation error also has an impact on the analysis, but its value is already fixed for all subsequent experiments (see Sect. ).

A first set of experiments (either model runs or data assimilation runs) will serve as sensitivity tests for the data assimilation platform with respect to the above features. During these sensitivity tests, the different features are tested individually. Table  details the range of variations for each tested feature.

Following the sensitivity tests, a set of three data assimilation experiments will be run and is presented in Table . The data assimilation experiments are divided into two categories: the first one uses water depths as observations, and the second considers water depth anomalies. All experiments are run across a year, corresponding to 17 assimilation cycles of 21 d.

The first experiment, denoted PE1, is configured from the aforementioned sensitivity test outcome. The parameters defining the experiment (spinup, starting date, ensemble size, control error) will be those which provide the best results in the sensitivity tests in Table . The reference level between the observed and simulated water depths is also the same. In other words, there is no bias in the observation. This first idealized experiment serves as a proof-of-concept as the observation type matches exactly the type of the simulated variables. Consequently, with this experiment, we expect to retrieve the true value of the control variables and hence the correct water depths and discharges.

The next step is to head towards more realistic experiments by including new sources of uncertainties in the data assimilation system and seeing how to address them. In this context, two additional experiments denoted PE2 and PE3 will be carried out. As an example of new uncertainties, SWOT will in fact observe water elevations (water surface elevation as referenced to a geoid or an ellipsoid), whilst CTRIP produces water depths (water surface elevation as referenced to the bottom of the river bed). To perform data assimilation, one needs to convert CTRIP water depths (hSCTRIP) into CTRIP water elevations (HaltiCTRIP) or inversely for SWOT. It is highly plausible that this operation induces a bias between the modeled and observed water elevations. A simplified example of such a situation is illustrated in Fig. . In this case, SWOT catches the right water elevation dynamic (as hSSWOT and hSCTRIP are equal), but the direct assimilation of SWOT water elevation HaltiSWOT will induce a bias as the elevations of the river bed (HbedSWOT and HbedCTRIP) are different between CTRIP and SWOT.

A solution for the handling of this issue is to assimilate water depth anomalies instead of water depths. The next data assimilation experiments, denoted PE2 and PE3, will therefore test the feasibility of assimilating anomalies. In these experiments, the water depth anomalies are generated by subtracting a time-averaged reference water depth from the current water depth. For all runs (true, openloop or analysis), this time-averaged reference water depth is computed as the mean (true, openloop or analysis) water depth over the year before the start of the assimilation window. It is therefore different for each member of the ensemble. Firstly, in experiment PE2, there will still be no bias between the observed and simulated river bathymetry to observe how the assimilation of anomalies performs. Similarly to PE1, we expect this experiment to be able to retrieve the true control and state variables. Finally, the last experiment, PE3, which introduces a constant relative bias between CTRIP and SWOT, will be carried out. For this experiment, we anticipate that the assimilation will still be able to retrieve the model state variables. The use of anomalies as observations should limit the impact of the inserted bias. We do not exclude the possibility however that it may be slightly echoed in the control variables.

The objective of the spinup sensitivity tests is to evaluate the minimum spinup period required by the model before applying data assimilation. For this purpose, the model is run several times across 2 years, from 19 December 2006 to 22 December 2008, corresponding to 35 windows of 21 d (735 d).

A first simulation is run using the true Manning spatial distribution (see Eq. ) over the 2-year time period. We then run 18 additional simulations over the same period with a varying length of the spinup period (see Table ). Initially, the simulation setup corresponds to the openloop configuration with the openloop Manning spatial distribution (see Eq. ). At a given time during the first year, the Manning spatial distribution is instantaneously changed to the true distribution (see Eq. ) and the model is run until the end of the 2 years with the true Manning spatial distribution. Table  summarizes, for each run, the date when the Manning coefficients are changed. The spinup period (expressed as a number of windows of 21 d) corresponds to the period between when the Manning distribution is changed and the start of the second year, i.e., 1 January 2008.

To evaluate the spinup impact, the relative difference between the reference run and the test runs is evaluated over the second year of simulation (from 1 January to 22 December 2008) and averaged over every window of 21 d. Figure  presents the results for the spinup sensitivity test. Each test run (on the x axis) is identified by its corresponding spinup period length (expressed as the number of windows of 21 d; see Table ). We then count (on the y axis) the number of 21 d windows during which the relative error between the test run and the reference run is higher than a given threshold. We assume that the spinup period is long enough when this number is equal to 0. This number is evaluated from the basin-averaged relative difference and from the relative difference at the downstream station of Óbidos, both in terms of water depth and discharge. Note that in Fig. , when the number of spinup windows of 21 d is equal to 4 on the x axis, the change from the openloop Manning distribution to the true one is imposed on 9 October 2007. Similarly, when this number is equal to 10, the change is imposed on 5 June 2007. Note also that two thresholds are considered, 0.01 and 0.001. Basin-averaged results are not sensitive to this threshold. There are some differences at Óbidos; however, we retain the basin-averaged results to evaluate the required model spinup period.

From all of the results we conclude that a minimum spinup period of four windows of 21 cycles, i.e., 84 d, is required. This period corresponds to the basin concentration time or, in other terms, the required time for the river network to totally empty. In the following sensitivity tests, the model runs start on 9 October 2007, to be consistent with these results.

To evaluate the impact of the starting date, a set of 17 one-cycle long data assimilation experiments is carried out over the second running year. All experiments have the same general configuration, except for the initial date starting from 1 January 2008 and shifted by 21 d until 22 December 2008. This means that the last experiment starts on 2 December 2008. The performance of each experiment is evaluated by simply evaluating the spatial average difference between the analysis and the true Manning coefficients. Results presented in Fig. a and b indicate that there are no significant differences between the data assimilation experiments (for all 17 experiments, the error of the updated Manning coefficients with respect to the true value of the coefficients is below 5 %).

Note that the results in Fig. a–b show a slight increase in the errors for the experiment starting at the end of the year, presently from 9 September to 2 December 2008. This period corresponds to the low-flow season in the Amazon hydrological cycle. Concerning the sensitivity analysis results , the water depths showed a very low sensitivity to the Manning coefficient during the low-flow season. As a consequence of data assimilation, the EnKF is less effective in low-flow seasons in the correction of the Manning coefficient. The analysis relative error (in Fig. a) and the analysis ensemble dispersion (in Fig. b) are therefore higher in the low-flow season.

For all of the following sensitivity tests, we are only considering therefore one-assimilation-cycle experiments, which will start on 1 January 2008.

The next sensitivity test is dedicated to the ensemble size ne, a critical parameter of any EnKF algorithm. This parameter has to be high enough so as to accurately estimate the Kalman gain matrix but low enough to limit the computational cost (the higher ne, the more model runs are required over each data assimilation window to obtain the analysis estimate of the Manning coefficients).

We consider different ensemble sizes through a one-assimilation-cycle experiment; ne varies between 10 and 200. Figure c compares the analysis Manning coefficient relative error for each ensemble size ne as in Fig. a. Results show that the analysis relative error decreases when the ensemble size ne increases. For an ensemble size ne equal to 20, the analysis error is below 5 %. Also, for an ensemble size ne higher than 50, the analysis relative error has converged to a constant value, while the analysis ensemble dispersion shown in Fig. d stabilizes.

These results indicate that the ensemble size for future data assimilation experiments should be at least equal to 20; we consider ne=25 in the present study to limit computational time.

In this study, we only consider parameter estimation, implying that the background error covariance matrix is associated with the parameter space. We assume that the errors in the Manning coefficients are independent, so that the background error covariance matrix is initially specified as a diagonal matrix, where all diagonal elements correspond to the error variances in the spatially varying Manning coefficients and are equal to the same variance (σb)2, where σb is the background error standard deviation. Similarly to previous sensitivity tests, we study here the sensitivity of the data assimilation results to the value of σb through a one-assimilation-cycle experiment. Figure c shows, in logarithmic scale on the x axis, the relative error of the updated Manning coefficient with respect to the true coefficient. The analysis error curve shows a decreasing behavior until σb is on the order of 0.4. It then increases again.

Note that the actual Manning coefficient error before data assimilation is equal to 0.33 (see the blue curve in Fig. c showing the openloop Manning coefficient error). Consistently, the best data assimilation results are obtained when σb provides a good approximation of the real error standard deviation. Note also that when σb becomes too small, data assimilation is less effective. The EnKF algorithm is known to be under-dispersive. Therefore, for future data assimilation experiments, when updating the error covariance matrix from one cycle to another, we will need to make sure that the ensemble dispersion is high enough to cover possible model behavior over the forecast time window by imposing a minimum value for the error variance. Given the sensitivity test results, the minimum value for σb is set to 0.005. Thus, in the following data assimilation experiments, we use the analysis error covariance matrix as the background error covariance matrix for the next assimilation cycle while applying the minimum threshold value to the matrix diagonal terms.

Figure  gives, for each zone, the time evolution of the mean analysis control variable (red) with its dispersion (even though it is very narrow) compared to the truth (black) and the first guess (blue). Similarly, Fig.  shows, for each zone, the time evolution of the analysis water depth (red) compared to the truth (black) and the openloop (blue). To generate one plot per zone, we use, for each time step, the ensemble of water depths over all cells in the zone and estimate the median value, the first decile and the ninth decile. Furthermore, zone-averaged normalized RMSEn statistics are given in Tables  and  in Appendix .

In general, the PE1 experiment gives very good results as the analysis mean for each zone retrieves the true value with a very low dispersion. However, the data assimilation algorithm features spatially dependent behavior; see Fig. .

Firstly, the control variable for zones 1, 2, 3, 4, 5 and 9 converges instantaneously (in only one assimilation cycle) toward the true values and remains at these true values for all following cycles.

These observations can be explained with the global sensitivity analysis results for water depths in and using Fig. .

Firstly, zones 1, 2 and 3 correspond to the river main stem, whilst zones 4, 5 and 9 correspond to the main left-bank tributaries, namely the Caquetá/Japurá River (zone 4) and the Negro River (zone 5). In these zones, as the Manning coefficient is directly corrected in the first assimilation cycle, the analysis water depths (red) overlap the true water depth (black). Furthermore, Fig.  for these zones shows that the openloop (blue) and true (black) water depths have a very similar variability in time but differ by a constant bias. The global sensitivity analysis results in these zones showed a constant first-order sensitivity in time to the Manning coefficient all year long. This first-order sensitivity means that the contribution of the Manning coefficient to the water depth is linear. Correcting the Manning coefficient in these zones equates therefore to correcting the bias between the openloop and the true water depths.

The assimilation of anomalies has been tested over two experiments denoted PE2 and PE3 (see Table ). Note that the observation error standard deviation σo remains equal to 10 cm as, with these experiments, we only aim at testing the feasibility of assimilating water depth anomalies. In the PE2 experiment, there is no difference of bathymetry between the simulated and observed water anomalies, whilst the river bankful depth is different in the PE3 experiment. Figure  gives, for each zone, the time evolution of the mean analysis control variable for PE2 (orange) and PE3 (purple) with their dispersion compared to the truth (black) and the first guess (blue). Again, zone-averaged normalized RMSEn statistics for these experiments are given in Tables  and  in Appendix .

The general configuration of experiments PE1 and PE2 is the same. The only difference between the two experiments is the nature of the observations: water depths for PE1 and water anomalies for PE2. Like experiment PE1, experiment PE2 (the orange line in Fig. ) gives very good results. All control variables converge toward the true values more or less rapidly. The control variable for zones 4 to 8 instantaneously (in only one assimilation cycle) converges toward the true value, while the convergence is slower for the remaining zones as around five assimilation cycles are needed to retrieve the true value. This slower convergence for these zones can be explained by the fact that the magnitude of the observed water anomalies is generally smaller compared to the water depths assimilated in PE1. The ratio between the observation error and the observations themselves is also then smaller, resulting in a smaller EnKF gain. The control variable correcting increment is smaller for the anomalies therefore than for the water depth, and more cycles are needed to converge.

As for the PE3 experiment results – the purple line in Fig.  – the assimilation still gives good results, but not as good as in previous experiments. The control variables still instantaneously converge toward the truth in zones 4, 5 and 6. Concerning the other zones, there is no clear convergence towards the true value. Instead, the analysis control variables either get closer to but remain distinct from the true value (zones 2 and 3) or temporarily deviate from the truth during the experiment (zones 1, 7, 8 and 9). Still, despite the control variables not clearly converging towards the truth, the simulated water depths using the analysis control variables, presented in Fig. , display a very low deviation from the truth, confirming the general good performance of the data assimilation procedure.

Comparing the control variables and water depth time variations, it appears that the control variables are deviating from the truth mainly when water depths are decreasing, in between the high-flow and low-flow seasons. During this period, the model goes from a state where floods occur to a state where there is no flood, particularly in zones 2–3 and 7–8 with a clear seasonal cycle. On the other hand, no flood event was spotted in zones 4, 5 and 6, where the best results were obtained. In ISBA-CTRIP, the activation/deactivation of the flooding scheme is triggered by the simulated water depth exceeding/becoming lower than the river bankful depth. However, in experiment PE3, this river bankful depth differs between the model and the observation because we artificially inserted a bias between the simulated and observed water depths. More specifically, the river bankful depth is lower in the model than in the observations. Therefore, the control variables deviating from the true value when the water depth is decreasing are indicative of the simulated water depths presenting floods when there is no flood in the observations. The activation of the flood scheme changes the dynamics of the water depth in the river. As part of the water in the river is spilled into the floodplains, water-level variations in the river are slower. The flooded model needs then a stronger variation of the Manning coefficient in order to catch the non-flooded observed water level. Ultimately, the stronger variations of the estimated Manning coefficient allow the retrieval of the true water depths.