The ISBA model is a relatively standard land surface model (LSM) defined over a regular mesh grid at global scale. The model's equations are solved for each grid cell separately from the others. All grid cells are only correlated through the spatial patterns of atmospheric (especially precipitation) and radiative inputs, vegetation cover and soil composition. By taking into account the heterogeneity in precipitation, topography and vegetation within each grid cell and based on the force-restore method , ISBA gives a diagnosis of the water and energy budgets in each grid cell. Especially the ISBA-3L configuration , used in and , has been chosen for the present study. In this version, the soil is divided into three layers: the superficial layer, the root zone and the sub-root zone. Precipitation can either fall directly on the soil surface or be intercepted by the canopy. The soil water content varies with canopy dripping, surface infiltration, soil evaporation, plant evapotranspiration, surface runoff and deep drainage (for more details, see ). Then, ISBA gives a diagnostic of each water budget component, in particular the surface runoff (QISBA,sur) and gravitational drainage (QISBA,sub) which are the main inputs for CTRIP.

The CTRIP RRM is also defined over a regular mesh grid. In this study, it is run at the same resolution as ISBA (0.5∘ × 0.5∘). CTRIP is dedicated to the lateral transfer of water from one cell to the other, up to the continent–ocean interface following a river network . The CTRIP version used in this study is coupled with the ISBA LSM and was subsequently developed by . It consists of a system of three reservoirs (see Fig. b): the surface reservoir S (kg) modelling the river, the groundwater reservoir G (kg) and the floodplain reservoir F (kg).

Only the surface reservoir S sends water from cell to cell based on the TRIP routing network. A cell can receive water from several upstream cells, but sends water into a unique downstream cell based on a space and time-varying flow velocity v(t) estimated with the Manning formula . For any given cell, TRIP inputs are the TRIP outflow from upstream cells Qin,TRIPS(t) and the ISBA surface runoff for that cell QISBA,sur. Moreover, S receives water from the groundwater reservoir G, QoutG(t), and can exchange water mass with the floodplain F, QoutF(t) - QinF(t).

G receives inflows from ISBA gravitational drainage QISBA,sub and outflows to the river reservoir S. This outflow represents more a delayed contribution of the gravitational drainage to the river than a real groundwater dynamic.

The floodplain scheme activates when the water height in the river, hS, exceeds a given critical bankful height Hc. Then, part of the precipitation is intercepted by the floodplain (PF(t)) and the water in the floodplain can either evaporate (EF(t)) or infiltrate into the soil (IF(t)). A detailed description of the floodplain scheme is given in .

ISBA-CTRIP is run in offline mode. This implies that external atmospheric data are needed to force the model. Here, the atmospheric data from the Global Soil Wetness Projet 3 (GSWP3, http://hydro.iis.u-tokyo.ac.jp/GSWP3) are used. The project consists of three global-scale experiments with the objective of investigating long-term changes in the energy–water–carbon cycle components and their interactions. The 3-hourly resolution atmospheric boundary conditions used in the present study were generated by dynamically downscaling the global 2∘-resolution 20th Century Reanalysis . This reanalysis assimilates several atmospheric observations into the Climate Forecast System (CFS) operational model from NCEP (National Centers for Environmental Prediction).

For ISBA-CTRIP, the Amazon basin is composed of a total number of 2028 cells. A sensitivity analysis (SA) of the ISBA-CTRIP has been conducted by . In this analysis, the basin was divided into nine hydro-geomorphological zones which are shown in Fig. . These zones were designed to take into account different components: (1) a hydrological component (the main course is separated from the tributaries which have their own zones); and (2) a geological component (the three major morpho-structural units are distinguishable). The nine zones are the following: (1) the upstream Andean part of the basin until the city of Iquitos, Peru; (2) the mainstream from Iquitos to Óbidos; (3) the mainstream from Óbidos to the river mouth; (4) left-bank tributaries from the Napo River to the Japurá River; (5) left-bank tributaries from the Japurá River to Óbidos, including the Negro River and its drainage area; (6) right-bank tributaries from Iquitos to the Purus River confluence at Anamã; (7) right-bank tributaries from Anamã to Óbidos, including the Madeira River; (8) right-bank tributaries exiting in zone 3, including the Tapajós River and the Xingu River; and (9) left-bank tributaries exiting in zone 3. This subdivision will be used within the DA platform.

The altimetry-based discharge product used in this study is derived from water surface elevations measured by the ENVISAT Radar Altimeter-2 altimeter instrument at Virtual Station (VS). VS is computed where the altimeter track crosses the river. The ENVISAT mission operated from September 2002 to October 2010 on its nominal orbit, which has a 35-day repeat period and an 80 km inter-track distance at the Equator. The water surface elevations measured over the Amazon basin were initially generated by . The final product was referenced to the EGM2008 geoid and the vertical precision ranged from 12 to 30–40 cm for most of the stations (and can reach several metres for the worst stations).

Turning water surface elevation measures into an equivalent discharge requires the use of elevation–discharge rating curves. The rating curves used in this study have been built and validated by , using water surface elevations from ENVISAT and discharges simulated by hydrological–hydrodynamic model MGB-IPH (Model de Grandes Bacias-Instituto de Pesquisas Hidráulicas, ). The model's original version, developed over the Amazon River basin, consists of a large-scale distributed hydrological model coupled with a hydrodynamic module that uses a simple storage scheme for floodplains . The entire basin is divided into 5765 elementary catchments with an area varying between 100 and 5000 km2. A surface scheme is applied for each mini-basin to estimate the main flows and a routing network is used to direct the flows from one elementary catchment to another, down to the outlet. Two approaches are used to estimate the river discharge: 1 – the Muskingum–Cunge method (MC) for basin heads and small tributaries; 2 – the Saint-Venant equations (HD for hydrodynamic) for the main stem and main tributaries. The DEM used is SRTM , and parameters such as the river width and depth are determined using geomorphological relationships calibrated over the sub-basins . Moreover, the model version used to determine the rating curves is the version developed by , where gauge discharges are assimilated into the model via an ensemble Kalman filter (EnKF, ), over the period between 1998 and 2010. The assimilated discharges allow one to correct the simulated discharges over both gauged and ungauged elementary catchments. With a better estimation of discharges, also provide an estimation of discharge uncertainty (modelled as a white noise) for each elementary catchment.

The MGB-IPH discharges were used by as a baseline to estimate the altimetric rating curves such that ∀j,∀i,∀talti,Qmgb,j(i)talti,i=ai×Halti,i-z0,ibi, where

Halti,i is the altimetric water surface elevation at the ith virtual station which is available at time talti,i,

At a national or basin scale, water agencies can share discharge time series, such as the Agencia Nacional de Agua (ANA, hidroweb.ana.gov.br) in Brazil for the Amazon River basin. For the present study, we retrieved discharge time series from 145 ANA in situ stations over the entire basin. These gauge discharges have then been used to evaluate the performances of the DA (but they have not been assimilated into ISBA-CTRIP).

The vector xk ∈ Rnx is called the control vector. It includes the nx uncertain variables to be estimated during the kth DA cycle (within the time interval [k - 1, k]). As stated before, control variables are prognostic or diagnostic variables of the ISBA-CTRIP model. Prognostic variables are the physical unknown in the differential equations' system that describes the model's behaviour. Diagnostic variables are also physical variables, but they are estimated from the prognostic variables. The choice of the control variables determines the observation operator Hk that maps the control variables into the observation space: yk=Hkxk, where yk are the control variables equivalent in the observation space, also called model observations. They are then compared to the measured observations yko (described in Sect. ) during the DA step.

Unlike hydrodynamic models, which directly solve Saint-Venant equations and for which discharge is a model state variable (or prognostic variable), the hydrological model ISBA-CTRIP solve differential equations describing the time evolution of water stock in the river (S), the groundwater (G) and the floodplain (F). Then, water elevation and river discharge are diagnostic variables derived from these prognostic variables. In CTRIP, river discharge QoutS is computed as follows: QoutS=L-1vSkgs-1, with L (m) river section length, v the flow velocity (estimated from the Manning formula) and S the surface water mass.

Therefore, three types of variables can be considered as control variables in the data assimilation scheme: the discharge QoutS (denoted Q in the remaining of the study to simplify notation, which is a diagnostic variable), the river final water stock Send (a prognostic variable) or the river initial water stock 5also a prognostic variable). Definition and complexity of the observation operator Hk, that maps the control space into the observation space, depends on the nature of the control variable. These three options are presented below.

Option 1: correcting ISBA-CTRIP discharges: for this option, the control variables, gathered into the vector xk, are the ISBA-CTRIP discharges Qi,k, i = 1 … nx = 2028 (number of TRIP cells in the Amazon basin) estimated for all 2028 cells in the TRIP Amazon basin, at the assimilation cycle k.

The observation operator Hk resumes to a selection operator Sk which selects the observed TRIP cells at the current assimilation cycle:Hk=Sk.This operator is linear. The difficulty with this operator is that, once the assimilation analysis is produced, it is necessary to convert the analysis discharge Qi,ka, i = 1 … nx (i.e. the corrected discharge obtained after assimilation) into the equivalent final water stock Send,i,ka. Indeed, as already stated, in ISBA-CTRIP, discharge is not a prognostic variable. Correction from the assimilation step needs to be propagated to the model prognostic variables, here the river final stock. Moreover, the analysis of the final water stock Send,i,ka will be used as an initial condition for the model run until the next assimilation cycle: ∀i = 1 … nx, ∀k, Send,i,ka = Sini,i,k+1b. However, the exact relationship linking discharge to the final river stock is unknown.

A possible solution consists in inverting Eq. (). Assuming that the discharge estimated by ISBA-CTRIP Qi,ka is the instantaneous flow at the final time of the integration window,Qk,i,kgs-1a=L-1vSend,k,i,[kg]a⟺Qk,m3s-1a=ρ-1L-1vSend,k,i,[kg]a.We obtain that (for more details on this approximation, see Appendix )Send,k,i,[kg]a≈ρLW2/5s-3/10n3/5Qk,i,kgs-1a3/5,with ρ (m3 kg-1) the water density, L (m) the river section length, W (m) the river width, s (–) the riverbed slope and n (–) the Manning coefficient in the riverbed. Then, for experiments with discharges as control variables, the formula in Eq. () will be used to convert corrected discharges into river stock and then propagate the model to the next observation time.

In the framework of the state estimation, the observation variables, at a given day within the Amazon basin, are the discharge estimates derived from ENVISAT water surface elevations at the virtual stations associated with an ISBA-CTRIP cell. The ENVISAT repeatability is 35 days, and therefore a given virtual station will provide an observation every 35 days at best. During the data assimilation experiments, all virtual stations will be used simultaneously. Because of the ENVISAT orbit, the number of available observations at a given day will vary between 0 and 15, and these observations will be assimilated daily via the EnKF. Then, the observation vector yko at the assimilation cycle k (equivalently, at the simulation day k) is composed of the ny,k discharge measures available at day k: yko=yk,1o,yk,2o,…,yk,ny,ko, where yk,jo, j = 1 … ny,k is the jth observation among the ny,k at cycle k.

Measurement errors ϵkm come from errors in ENVISAT water surface elevations, errors in MGB discharges and uncertainties in the rating curve parameters used to turn water surface elevation into discharge. , and noticed that the concavity of the elevation–discharge relationship implies that the higher a water elevation, the more uncertain the corresponding discharge. Therefore, the observation error standard deviation σk,jo, associated with the jth observation at cycle k, is defined with respect to the instantaneous discharge measure yk,jo, i.e.: σk,jo=ηjo×yk,jo,j=1…ny,k, where ηjo ∈ [0, 1] is a constant depending on the virtual station, such that ηjo models the relative error. The observation error standard deviation σk,jo is then a fraction of the instantaneous discharge. For each virtual station, the value of ηjo depends on, first, the deviation from the MGB discharge, noted σjmgb [%] and determined by . As MGB discharges were used to determine ENVISAT discharges from ENVISAT water elevations, σjmgb represents the deviation between the two discharges data. Second, to take into account that MGB discharge is not perfect (in other words, to take into account some deviation from the real discharge), 0.05 is added to σjmgb and the sum is rounding up to the nearest whole number, giving ηjmgb=Eσjmgb+0.05, where the function E is the ceiling function. Finally, ηjo is equal to the first multiple of 5, above ηjmgb. At the end, ηjo varies from 0.10 to 0.35 over the entire basin and is constant in time. Besides, the representativeness error ϵkr induced when a virtual station is associated with cells of the coarse TRIP mesh grid is neglected here.

Moreover, for a given assimilation cycle and also between different cycles, the observations are considered uncorrelated in space and time. The observation error covariance matrix at cycle k Rk is then a diagonal definite positive square matrix.

In the EnKF framework, the model and observation operator are not linear. Therefore, the main idea is to use stochastic ensembles to represent the control variables PDFs along with the error models . First, the background control variables xkb are stochastically represented by an ensemble of ne members: Xe,kb=xkb,[1]xkb,[2]…xkb,ne. Each member is estimated separately through the EnKF prediction step. For each control variable case (see Sect. ), each member of the control ensemble Xe,kb is estimated by integrating the model operator from the corresponding analysis member at the previous assimilation cycle, while adding external uncertainties (see Sect. ): ∀l=1…ne,xkb,[l]=M[k-1,k]xk-1a,[l]. Then, the background control ensemble must be compared to the observations. Depending on the control variables nature, the model operator is already included (option 3) or not (option 1 and 2) within the observation operator. Besides, following , an additional noise ϵko is added to the observation vector yko to avoid ensemble collapse. The observation ensemble thus obtained is noted: Ye,ko=yko,[1]yko,[2]…yko,ne.

Finally, the EnKF analysis step is applied to each member of the ensemble such that ∀l=1…ne,xka,[l]=xkb,[l]+Kk,eyko,[l]-Hkxkb,[l], where the direct non-linear observation operator is applied to convert the control variables into equivalent model observations.

The particularity of the EnKF is that the Kalman gain (Ke,k) is stochastically estimated from the different control and model observation ensemble, as follows: Ke,k=PHTe,kHPHTe,k+Rk-1.

In the framework of state estimation, the sampling error can introduce artificial correlations into the background/analysis error covariance matrices, and generate spurious correlations between two distant grid cells in the mesh . The ensemble size ne is limited by computational constraints. Therefore, before the EnKF analysis step, a numerical processing of the matrices [PHT]e,k and [HPHT]e,k is necessary to suppress spurious correlations that can potentially degrade the analysis. Localization methods are designed to reduce these problems.

There exist two types of localization techniques . The first one is called B-localization. It is based on explicitly modifying the background error covariance matrix Pe,kb. It consists in multiplying the matrix Pe,kb by a correlation matrix generated from a radial function, namely a function of the two/three spatial dimensions which monotonously decrease with the distance between control variables . This modified matrix replaces Pe,kb in the calculation of the Kalman gain matrix Ke,k. The other common localization technique is called R-localization or local analysis. This one consists in performing the analysis step in characteristic sub-spaces of the overall problem space.

However, all these localization techniques described above have been developed for atmospheric modelling where problems are in two or three dimensions. The use of localization in hydrology is more limited. Several studies exist to improve subsurface flow modelling , but these approaches have a dimensionality close to atmospheric approaches, as they take place in a continuous medium in two or three dimensions. Other studies using localization allow estimation of better model parameters, still continuously defined in two or three dimensions .

The localization method used with the CTRIP river routing model is of the B-localization type. However, it can not be simply defined on a two-dimensional radial function. Indeed, the river flow is along several one-dimensional flow directions, modelled by the routing network. The localization technique must consider the routing network to decorrelate adjacent cells on the mesh grid but located in two different sub-catchments. Nevertheless, along a same flow direction, the correlation between two distinct cells depends on the distance between the two cells. Then, for each assimilation cycle, the localization consists in a localization mask delimiting an influence area for each observation. These influence areas gather a limited number of neighbouring downstream and upstream cells around the observed cell with respect to the river routing network. We chose a fixed localization scale for simplicity and as a first step in the feasibility study of the development of a localization method for a hydrology application.

To determine the number of cells defining the influence area, the basin subdivision into nine hydro-geomorphological zones is used with a mean flow velocity for each zone. The influence area, for a given observed cell, is given by the criteria below. For an influence area of size p cells, the area is composed of the following.

The observed cell.

The number of cells within the influence area depends on the mean flow velocities (averaged over a year of simulation) in the zone in which the considered cell is situated. Those mean velocities are calculated from the free run simulation, namely the ISBA-CTRIP simulation realized without any assimilation step. The ISBA-CTRIP resolution is 0.5∘ × 0.5∘, or approximately 50 km × 50 km. Given the river meanderings, the minimal covered distance through a cell is of 50 km. Furthermore, by comparing the free run discharge to in situ and ENVISAT discharges, it seems that the free run underestimate discharge (and so the flow velocity). Consequently, to fix the number p of cells defining the influence area in each hydro-geomorphological zone, the following steps have been performed:

the mean velocity for the cells into a given zone is converted into an equivalent distance in km,

The background error covariance matrices [PHT]e,k et [HPHT]e,k are estimated from the control variable ensemble using the definition suggested by , and . To get a large ensemble, while maintaining a reasonable computational time, the ensemble size ne has been set to a hundred members. Details on how they are exactly calculated are given in Appendix . These matrices have a nx × ny,k and ny,k × ny,k size, respectively. The elements in the error covariance matrices, depend only on the definition of the background ensemble stored in the control matrix Xe,kb and the parameter uncertainties taken into consideration to generate H(Xe,kb). In the framework of state estimation, we choose to consider uncertainty into the initial condition and uncertainty into the precipitation forcing .

Perturbation of the initial condition: the vector containing initial surface reservoir storage for each nx = 2028 CTRIP cell at the assimilation cycle k is called ck. To ease the notations, we will omit, as much as possible, the assimilation cycle k subscript, knowing that, for all randomly perturbed variables and constants, a new ensemble is generated at each cycle.

We used the Amazon basin division into ns = 9 hydrogeomorphological zones (see Fig. ). Initial conditions are perturbed by applying a multiplying factor over each zone ηsc,[l] such that ∀s=1…ns,∀l=1…ne,cs,k[l]=ηic,[l]⋅cs,k,where cs,k is the reduction of the initial condition ck to the only zone s. For the perturbation to vary from one member to another, the value ηsc,[l] is the realization of a Gaussian distribution, different for each member [l] = 1 … ne and for each hydrogeomorphological zone s. The Gaussian distributions used have a mean value of 1 and a standard deviation of σηs,kc whose values are detailed in Table .

The ηic,[l] values depend on the assimilation cycle k and on the hydrogeomorphological zone in which the ith cell is.

Firstly, a more important perturbation is applied to cells situated on the river mainstream (zones 2 and 3), as we assume that the uncertainties are more important in those zones. Indeed, discharges in these zones are the highest of the entire basin. Besides, several cells are confluence cells and are subject to backwater effects. As ISBA-CTRIP does not model the backwater effects, the water stock uncertainties in these cells are increased.

Figure  displays the mean analysis RMSEn (RMSEnia,†‾ defined in Eq. ) for each ENVISAT virtual station (Fig. a) and for each in situ station (Fig. b). First of all, results between the ENVISAT RMSEn and the in situ RMSEn are similar, due to the similarity between ENVISAT and gauge discharge time series at most stations. According to Fig. a, the assimilation worked quite well along the mainstream and the main left-bank tributaries, namely the Negro River, the Japurá and the Icá, with several stations where RMSEn is below 20 %. The assimilation performances are more moderate over right-bank tributaries, where RMSEn is mostly between 20 and 60 %. Over the entire basin, RMSEn remains high in all basin heads, along small tributaries and also at most confluences; see for example the Jutaí–Solimões confluence (RMSEn above 60 %), the Purus–Solimões/Madeira–Solimões/Tapajós–Amazon confluences (RMSEn above 40 %) or the Xingu–Amazon confluence (RMSEn above 80 %).

Figure  compares the mean analysis discharge in red line at the 12 stations previously introduced in Sect. . For most stations, we can see that the mean analysis discharges recovers a seasonal cycle closer to the observations than the free run. It is especially true for stations along the mainstream, namely Sao Paulo de Olivenca, Manacapuru and Óbidos (Fig. , panels 1–3). Also, for stations along right-bank tributaries (Fig. , panels 7–12), the analysis seasonal discharge peak is lowered compared to the free run seasonal discharge peak and fits better the observations. This shows the good functioning of the assimilation platform.

Nevertheless, mean analysis discharge for all displayed stations presents a chaotic behaviour with numerous local minima and maxima. We can assume that this behaviour is present for all CTRIP cells in the basin. Moreover, for a given cell, most of these sudden variations are asynchronous with ENVISAT observation dates for this cell. For example, at Serrinha in the left panel in Fig. , an ENVISAT observation is available on the 4th day of the 35-day repeat period when big off-peaks appear on the 25th day of the same 35-day repeat period. The right panel in Fig.  displays the Serrinha station (red circle) with all ENVISAT observations available during the 25th day (yellow circles). On inspection of the contribution of all these observations to the analysis control variable at Serrinha (not shown here), we find that it is observation number 4 that has the highest impact on the analysis (and not observation number 5, as could be expected). This observation 4, located on a very small Negro tributary, has a low discharge value and is responsible for the low corrected discharge at Serrinha after the assimilation step.

These abrupt variations are completely artificial and directly result from the assimilation processing. Indeed, for days with unrealistic peaks/off-peaks, there are multiple ENVISAT observations available on the basin, which impact many cells all over the basin, even if they are located on other sub-catchments or tributaries. This is due to the construction of the error covariance matrices [PHT]e,k and [HPHT]e,k. As these matrices are generated from the ensemble with a limited number of members, some spurious elements may appear in the matrices and link two cells that are very distant in the basin or even situated on different sub-basin. This first experiment highlights the necessity to treat the error covariance matrices to limit such spurious elements.

In the SE1-diag experiment, the error covariance matrices are forced to be diagonal. The objective of such processing on the error covariance matrices is to limit the impact of a given observation only to the observed cell. According to Table , the assimilation experiment allowed one to reduce the global ENVISAT, an in situ RMSE, when compared to the free run. However, among all three experiments, it is the one which gives the worst global performances. In this experiment, the chaotic behaviour of the mean analysis discharge is not present any more (not shown here). Nevertheless, the mean analysis discharge remains close to the free run discharge, except for regular peaks/off-peaks at an observation time when it is closer to the observed discharge. Therefore, the information brought by only one local observation is not enough. With the localization (see next section), whose results are presented in the following section, the information of several neighbouring VS is used and should constrain the analysis discharge more.

SE1-local uses the localization treatment presented in Sect. . Figure  displays the RMSEn evolution from the SE1-direct to SE1-local experiments for both ENVISAT and gauge discharge. Green colours indicate that the SE1-local experiment reduced the RMSEn compared to the SE1-direct experiment, while yellow to red colours indicate that the SE1-local experiment increased them. The RMSEn is mostly improved over the entire basin and more particularly along major right-bank tributaries. However, the RMSEn is generally degraded along the mainstream. These maps show the good performances of the localization method over tributaries. Now, it appears that, compared to gauge discharges (see Fig. b), the SE1-direct experiment gives better results, especially along the mainstream. Indeed, Table  details the local RMSEnia,† at ENVISAT/in situ stations located along the mainstream and confirms that SE1-direct gives better results. As the global RMSEn is defined as the mean of the RMSEn weighted by the maximum discharge at the station (see Eq. ), this explains why the SE1-direct experiment gives a better global RMSEn compared to gauge discharge.

Then, Fig.  displays the mean analysis discharge for SE1-local compared to the free run discharge, with corresponding ENVISAT discharge and gauge discharge, at the 12 in situ stations already used in Figs.  and . Except for stations along the mainstream (Fig. , panels 1–3) and also at Boa Sorte (Fig. , panel 12), the analysis discharge shows less sharp variations. From these results, we can say that the localization scheme is necessary and improves the assimilation.

The localization mask has been built to avoid the effect of spurious correlations between distant cells or ones situated on different sub-basins. The current localization scheme meets this constraint. Indeed, results for the SE1-local experiment are globally improved compared to the previous experiments.

Nevertheless, along the mainstream, the initial experiment without localization gives better results. We can interpret that by the fact that discharge along the mainstream integrates hydrological processes from all the upstream basins. So when, in the SE1-local experiment, we limit the impact of the observation to only close cells, we suppress part of the information brought by distant cells to mainstream cells.

Therefore, the current localization mask should be improved. The main difficulty here is to determine the size of the influence area for each observation. Currently, this size is predetermined and is constant in time according to averaged flow velocity. A potential development is to consider an influence area size that can vary in time, according to the hydrological season (high-flow/low-flow season). For example, during high-flow season, the flow velocity is higher so is the size of the influence area. Thus, the error covariance matrices would depend on the river time and space dynamic (as if there were defined from a well-sampled and significant ensemble).