The potential of river altimetry for model calibration was tested with a process-based hydrological model for the Luangwa River basin. This model relied on distributed forcing allowing for spatially explicit distributed water storage calculations. The model was run on a daily timescale for the time period 2002 to 2016. To reach the objective of this study, the following distinct parameter identification strategies were compared in a step-wise approach: (1) traditional model calibration to observed river flow as a benchmark; (2) identification of parameter sets reproducing the seasonal water storage anomalies based on GRACE data only; (3a) Altimetry Strategy 1: identification of parameter sets directly based on remotely sensed water levels combined with GRACE data; (3b) Altimetry Strategy 2: identification of parameter sets based on remotely sensed water levels by converting modelled discharges into water levels using calibrated rating curves combined with GRACE data; (3c) Altimetry Strategy 3: identification of parameter sets based on remotely sensed water levels by converting modelled discharges into water levels using the Strickler–Manning equation and including river geometry information (cross section and gradient) extracted from Google Earth combined with GRACE data; (4a) Water level Strategy 1: identification of parameter sets based on daily river water level at the catchment outlet only using the Strickler–Manning equation and including river geometry information extracted from Google Earth combined with GRACE data; and (4b) Water level Strategy 2: identification of parameter sets based on daily river water level at the catchment outlet only using the Strickler–Manning equation and including river geometry information obtained from a detailed field survey with an ADCP combined with GRACE data. Note that (1) is completely independent of (2) to (4), where no discharge data were used for the identification of parameter sets.

In this study, a process-based rainfall–runoff model with distributed water accounting and sub-grid process heterogeneity was developed (Ajami et al., 2004; Euser et al., 2015). The river basin was discretized into a grid with a spatial resolution of 10 × 10 km2. Each model grid cell was characterized by the same model structure and parameter sets but forced by spatially distributed, gridded input data (Table 1). Runoff was then calculated in parallel for each cell separately. Subsequently, a routing scheme was applied to estimate the aggregated flow in each grid cell at each time step.

Adopting the FLEX-Topo modelling concept (Savenije, 2010) and extending it to a gridded implementation, each grid cell was further discretized into functionally distinct hydrological response units (HRUs) as demonstrated by Nijzink et al. (2016). Each point within a grid cell was assigned to a response class based on its position in the landscape as defined by its local slope and “Height-above-the-nearest-drainage” (HAND; Rennó et al., 2008; Gharari et al., 2011). Similarly to previous studies (e.g. Gao et al., 2016; Nijzink et al., 2016), the response units plateau, hillslope, terrace, and wetland were distinguished. Reflecting earlier work (e.g. Gharari et al., 2011), all locations with a slope of > 4 % were assumed to be hillslope. Locations with lower slopes were then either defined as wetland (HAND < 11 m), terrace (11 m ≤ HAND < 275 m), or plateau (HAND ≥ 275 m); see Fig. 2. Following this classification, wetlands make up pHRU=8 %, terraces pHRU=41 %, hillslopes pHRU=28 %, and plateaus pHRU=23 % of the total Luangwa River basin area as mapped in Fig. 1b.

Each response class consisted of a series of storage components that were linked by fluxes. The flow generated from each grid cell at any given time step was then computed as the area-weighted flow from the individual response units plus a contribution from the common groundwater component which connects the response units (Fig. 2). Finally, the outflow from each modelling cell was routed to downstream cells to obtain the accumulated flow in each grid cell at any given time step. For this purpose, the mean flow length of each model grid cell to the outlet was derived based on the flow direction extracted from the digital elevation model. The flow velocity, which was assumed to be constant in space and time, was calibrated. With this information on the flow path length and velocity, the accumulated flow in each grid cell was calculated at the end of each time step. The relevant model equations are given in Table 3. This concept was previously successfully applied in a wide range of environments (Gao et al., 2014; Gharari et al., 2014; Fovet et al., 2015; Nijzink et al., 2016; Prenner et al., 2018).

To evaluate the information content and thus the utility of altimetry data for the selection of feasible model parameter sets, a step-wise procedure as specified in detail below was applied (Table 5). Note that given data scarcity and the related issues of epistemic uncertainties (Beven and Westerberg, 2011; McMillan and Westerberg, 2015) and equifinality (Beven, 2006; Savenije, 2001) we did not aim to identify the “optimal” parameter set in what is frequently considered a traditional calibration approach. In most hydrological applications the available data have limited strength for rigorous model tests (Clark et al., 2015; Gupta et al., 2008; Jakeman and Hornberger, 1993). Thus, to reduce the risk of rejecting good parameters when they should have been accepted (Beven, 2010; Hrachowitz and Clark, 2017), we rather attempted to identify and discard the most implausible parameter sets (Freer et al., 1996) that violate our theoretical understanding of the system or that are inconsistent with the available data (Knutti, 2008). This allowed us to iteratively constrain the feasible parameter space and thus the uncertainty around the modelled hydrograph (Hrachowitz et al., 2014). To do so, a Monte Carlo sampling strategy with uniform prior parameter distributions was applied to generate 5 × 104 model realizations. This random set of solutions was in the following steps used as a baseline and iteratively constrained by identifying parameter sets that do not satisfy pre-specified criteria (see below), depending on the data type and source used.

For each calibration strategy, the performance of all model realizations was evaluated post-calibration with respect to discharge using seven additional hydrological signatures (e.g. Sawicz et al., 2011; Euser et al., 2013) to assess the skill of the model at reproducing the overall response of the system and thus the robustness of the selected parameters (Hrachowitz et al., 2014). The signatures included the logarithm of the daily flow time series (hereafter referred to with the subscript log⁡Q), the flow duration curve (FDC), its logarithm (logFDC), the mean seasonal runoff coefficient during dry periods (April–September; RCdry), the mean seasonal runoff coefficient during the wet periods (October–March; RCwet), the autocorrelation function of daily flow (AC) and the rising limb density of the hydrograph (RLD). An overview of these signatures can be found in Table 6, and more detailed explanations in Euser et al. (2013) and references therein. As performance measures for the model to reproduce the individual observed signatures, the Nash–Sutcliffe efficiency (ENS,log⁡Q, ENS,FDC, ENS,log⁡FDC, ENS,AC; equivalent to Eq. (1) in Table 4) and a metric based on the relative error (ER,RCdry, ER,RCwet, ER,RLD; equivalent to Eq. 3) were used (Euser et al., 2013). The signatures were combined, with equal weights, into one objective function, which was formulated based on the Euclidian distance DE (Eq. 5) so that a value of 1 indicates a “perfect” model (Schoups et al., 2005).

The complete set of all model realizations unsurprisingly resulted in a wide range of model solutions (Fig. 6a), with ENS,Q ranging from -6.4 to 0.78 and with the combined performance metric of all signatures DE ranging from -334 to 0.79 (Fig. 7). With respect to the individual flow signatures, the model performance varied such that the largest range was found in ENS,Q and the smallest in ENS,AC, as visualized in Fig. 7 and tabulated in Table S4. Although containing relatively good solutions, this full set of all realizations clearly also contained many parameter sets that considerably overestimated and/or underestimated flows.

In this study, altimetry data were available at 18 virtual stations. However, would the model performance change if more or less virtual stations were used? To answer this question, n random stations were selected for model calibration, while the remaining stations were used for cross-validation (Klemeš, 1986; Gharari et al., 2013; Garavaglia et al., 2017). This was repeated to cover all combinations of n stations and for n=1,2…17. When applying Altimetry Strategy 3 using altimetry data with the Strickler–Manning equation, this analysis revealed that when increasing the number of calibration stations, the model calibration performance DE,NS,SM gradually decreased, but the ability to meaningfully reproduce the remaining observations which were not used for calibration increased significantly (Fig. 12). Similar results were obtained for Strategies 1 and 2 (compare Fig. 12 with Supplementary Figs. S3 and S4). Also, the model performance with respect to discharge increased when using more virtual stations with an optimum at 7–15 stations depending on the calibration strategy (Fig. S5). This provides evidence that in spite of reduced calibration performance, the simultaneous use of multiple virtual stations can contribute towards more plausible selections of model parameter sets and thus increase the model realism.

In the absence of discharge data for hydrological model calibration, as is commonly the case in poorly gauged or ungauged regions, freely and globally available remotely sensed streamwater levels could provide the opportunity to fill this gap, as illustrated in this study as well as in previous studies (e.g. Michailovsky and Bauer-Gottwein, 2014; Pereira-Cardenal et al., 2011; Sun et al., 2012). However, there are several limitations to the approach proposed in this study using altimetry for model calibration.

First, river altimetry data are prone to large uncertainties which increase for smaller river widths as a result of backscatter effects of the surrounding topography (Sulistioadi et al., 2015; Biancamaria et al., 2017; Domeneghetti et al., 2015). Too small rivers could even be missed altogether. In this study, the Luangwa River becomes a small meandering stream in the dry season, resulting in larger altimetry uncertainties. Unfortunately, this uncertainty could not be estimated for the virtual stations used in this study due to data limitations. However, in previous studies in the Zambezi basin, the RMSE relative to in situ stream levels ranged between 0.32 and 0.72 m using Envisat (Michailovsky et al., 2012). Improving altimetry observations such that the uncertainties decrease would improve the identification of feasible parameter sets and simulation of stream levels and flow. However, comparison results between the three altimetry-based calibration strategies are not expected to change since the same altimetry data were used. In other words, Altimetry Strategy 3 is still expected to perform best when decreasing the uncertainties in the altimetry observations.

Second, large uncertainties in the forcing data (precipitation and temperature) with respect to the spatial–temporal variations should not be ignored. This could compromise comparison results between modelled river water levels and altimetry within the basin since it has a low temporal resolution (10 or 35 d). Bias in the precipitation data affects storage calculations and hence the identification of feasible parameter sets based on GRACE (Le Coz and van de Giesen, 2020). This could explain why the flows were frequently overestimated when using GRACE only. In addition, precipitation bias could be compensated through calibration parameters introduced for the discharge–water-level conversion. Therefore, such parameters should be constrained as much as possible. There are also data uncertainties in the cross sections and river gradients extracted from high-resolution terrain data available on Google Earth due to its limited spatial resolution, but more importantly since no information is available below the water surface.

Further, GRACE observations are prone to uncertainties as a result of data (post-)processing, including for example data smoothing (Landerer and Swenson, 2012; Blazquez et al., 2018; Riegger et al., 2012) causing leakage between neighbouring cells of 1∘ (≈ 111 km), which are thus not completely independent of each other. Additionally, GRACE observations are more accurate for large areas. Depending on the applied processing scheme, the error is about 2 cm for basins with an area of around 63 000 km2 (Landerer and Swenson, 2012; Vishwakarma et al., 2018). Also note that due to the coarse temporal resolution, monthly averaged GRACE observations are dominated by slowly changing processes such as the groundwater, soil moisture system, and seasonal variations reflected in all storage components. In addition, open water bodies or wetlands could affect GRACE observations if they are located in or near the basin, for example within a radius of about 300 km, which is the distance often used for data smoothing. In this study, several open water bodies or wetlands were located ≤ 300 km of the Luangwa basin, such as Lake Malawi, Kafue Flats, Cahora Bassa reservoir, Kariba reservoir, Bangweulu, and Tanganyika. These open water bodies and wetlands had a limited impact on the GRACE observations due to limited fluctuations or different temporal variation as illustrated in Fig. 13 for the Cahora Bassa reservoir. These uncertainties in the GRACE observations could influence the identification of plausible parameter sets. For example, feasible parameter sets could be discarded incorrectly, which could distort results obtained by calibrating with respect to altimetry and GRACE simultaneously. However, the comparison between the three altimetry-based calibration strategies is not expected to change since the same GRACE data were used. In other words, Altimetry Strategy 3 is still expected to perform best when considering these uncertainties.

Uncertainties were not only introduced by the data, but also as a result of assumptions and simplifications. First, the reference level h0 was assumed to be equal to the lowest river water level observed to limit the number of calibration parameters (Altimetry Strategies 2 and 3, Water level Strategies 1 and 2). However, uncertainties in the altimetry observations as explained previously influence h0 estimates, which results in a bias between the observed and simulated stream levels in Altimetry Strategies 2 and 3. Second, the roughness was assumed to be constant in time, over the entire cross section, and for all virtual stations throughout the basin (Altimetry Strategy 3). However, this roughness can vary between 15 and 50 m1/3s-1 for natural rivers (Vatanchi and Maghrebi, 2019; Chow, 1959), changing the simulated stream levels between 42 % and 75 % in the Luangwa basin, with the low flows being the most sensitive. Third, all 18 virtual stations were grouped based on their cross-section similarity to limit the number of calibration parameters (Altimetry Strategy 2), but differences within each group remain such that the calibration parameters related to the rating curve vary slightly for each virtual station within a group. Fourth, the assumption of a constant flow velocity in space and time affects the timing of the simulated flow and stream levels, influencing the comparison between model results and altimetry observations (all strategies).

Another limitation is the missing information on absolute flow amounts when directly using (satellite-based) river water levels for model calibration using the Spearman rank correlations as a model performance metric (Altimetry Strategy 1; Seibert and Vis, 2016). This resulted here in an overestimation of intermediate and low flows due to the non-linear relation between stream levels and flows. In contrast, when converting the discharge to streamwater levels, information on absolute flow amounts was included at the cost of introducing additional calibration parameters (Altimetry Strategies 2 and 3), thereby increasing the degrees of freedom and thus the potential for parameter equifinality in the model (Beven, 2006; Sikorska and Renard, 2017; Sun et al., 2012).

Furthermore, it was assumed that the Nash–Sutcliffe efficiency contained sufficient valuable information to describe the model performance with respect to river water level and total water storage when identifying feasible parameter sets. This performance measure is sensitive to the sample size, outliers, bias, and time offset (McCuen Richard et al., 2006). Unfortunately, simulated discharge and stream levels are prone to bias uncertainties as a result of spatio-temporal bias in the rainfall (Le Coz and van de Giesen, 2020). In addition, altimetry observations have a limited sample size for several virtual stations (see Table 2) and are prone to bias due to uncertainties in the reference level h0 as mentioned before. Moreover, a time offset in the simulated flow can occur as a result of rainfall uncertainties. As a comparison, the model performance with respect to altimetry only reached up to DE,NS,SM=-1.3 for Altimetry Strategy 3, while it reached up to ENS,SM,GE=0.61 with respect to daily in situ stream levels for Water level Strategy 1. Therefore, additional study is recommended to confirm this assumption and to assess which performance metric(s) would be most suitable. The model performance with respect to discharge was evaluated with respect to multiple hydrological signatures simultaneously (see Table 6) to assess the model's skill at reproducing the internal dynamics of the system. Even though a few of these signatures have some overlapping information content (McMillan et al., 2017), each signature also contains at least some additional information not included in the other signatures. In general, the ambition is to represent a hydrological system as well as possible in a model, which critically required that the model exhibit sufficient ability to simultaneously reproduce multiple flow signatures (Gupta et al., 2008; Euser et al., 2013; Hrachowitz et al., 2014).

Previous studies have successfully used river altimetry data to calibrate and evaluate rainfall–runoff models using a few virtual stations (Sun et al., 2012; Getirana, 2010; Getirana et al., 2010; Liu et al., 2015). In these studies, the modelled discharge was converted to stream levels by means of a hydraulic model or empirical relations. Our results support several previous findings and added a number of new ones.

Similar to previous studies, the rainfall–runoff model reproduced river flow relatively well when calibrating on remotely sensed streamwater levels, preferably at several virtual stations simultaneously, but discharge-based calibration results performed significantly better (Getirana, 2010). Thus, while river altimetry data cannot fully substitute discharge observations, they at least provide an alternative data source that holds information value where no reliable discharge data are available. In addition, our results suggest that in spite of the typically limited temporal resolution of altimetry observations, these data, when using multiple virtual stations simultaneously, provide enough information to select meaningful model parameter sets (Seibert and Beven, 2009; Getirana, 2010).

Strikingly, only limited studies combined altimetry with GRACE observations in the calibration procedure (Kittel et al., 2018). As altimetry observations only describe water-level variations with no information on the flow amounts, GRACE provides additional valuable information to constrain the river discharge by improving the rainfall–runoff partitioning as demonstrated in previous studies (Rakovec et al., 2016; Bai et al., 2018; Dembélé et al., 2020). Combining both data sources in the calibration procedure allowed for a more accurate identification of feasible parameter sets. The model performance range with respect to discharge improved from DE,5/95=-8.4–0.77 when using only altimetry to DE,5/95=0.19–0.75 when combining GRACE and altimetry for Altimetry Strategy 3 (see Fig. S8).

In contrast to previous studies, altimetry data originated from five different satellite missions rather than a single one. As a result, altimetry data were available at 18 locations for the time period 2002 to 2016. This gave the opportunity to analyse the effect of combining different numbers of stations for calibration and evaluation. This study illustrated that better predictions can be achieved when using more virtual stations for calibration. Furthermore, this study demonstrated that in particular the combination of altimetry with information on river geometry (cross section, gradient) proved beneficial for the selection of feasible parameter sets within relatively narrow bounds, comparable to the benchmark using discharge. Using more accurate cross-section information obtained from a detailed field survey rather than Google Earth based estimates improved the water-level simulations, modelled rating curve, and discharge simulations during intermediate and low flows significantly for which on-site cross-section data were available. That is why it is recommended to acquire accurate cross-section information on locations concurring with altimetry overpasses (not done in this study).

For future studies, it will be interesting to improve Altimetry Strategy 3 using additional data sources. For instance, the combination of altimetry observations with river width estimates derived from Landsat or Sentinel-1/2 (Pekel et al., 2016; Hou et al., 2018) may bear some potential as the combination of the two different hydraulic variables complements each other and increases the temporal sampling (Huang et al., 2018; Tarpanelli et al., 2017; Sichangi et al., 2016). During high flows for example, river width estimates can be more accurate than altimetry observations, especially when floodplains are inundated and small water-level changes cause large river width changes. Alternatively, the altimetry observations used here could be combined with river surface water-level slope estimates based on CryoSat observations which provide water-level information at lower temporal resolution (every 369 d) but higher spatial resolution (equatorial inter-track distance of 7.5 km) (Schneider et al., 2017; Jiang et al., 2017). This allows for the estimation of the energy gradient based on stream levels as required in the Strickler–Manning equation, instead of the bed slope based on topography, which proved to be a good first estimate in the absence of more reliable data. In addition, CryoSat observations are available annually such that there can be more overlap with altimetry observations in contrast to topography data. With the upcoming SWOT (Surface Water Ocean Topography) mission, more accurate altimetry observations should be available as well as river slope observations and width. The repeat cycle will be 21 d and across-track resolution between 10 and 60 m, increasing the number of observation points available within a specific area (Biancamaria et al., 2016; Langhorst et al., 2019; Oubanas et al., 2018). As a result, hydrological models can be calibrated with respect to river altimetry and width simultaneously at multiple locations even for small river basins, improving the identification of plausible parameter sets and hence the model realism as illustrated in Sect. 4.2. It will also be very valuable to improve cross-section estimates with respect to the submerged part of the cross section as already explored in previous studies (Domeneghetti, 2016) or to use drone observations to obtain more accurate cross-section information and estimates of the river slope and roughness (Entwistle and Heritage, 2019). By improving the river profile description, the simulated stream levels become more accurate, which is crucial when using this time series for model calibration. As illustrated with Water level Strategies 1 and 2, improving the cross section resulted in a more accurate rating curve (Fig. 11), stream-level simulation (see Fig. 10), and discharge simulation (Fig. 8). Clearly, it will be interesting to analyse and disentangle different individual sources of uncertainty related to the discharge–water-level conversion from the hydrological model in a more data-rich region (Renard et al., 2010). Unfortunately, this was not possible in this study due to the scarcely available in situ observations in Luangwa. As concluded by Renard et al. (2010), reliable estimates of the data uncertainty are required to disaggregate multiple sources of uncertainty in rainfall–runoff modelling successfully.