Whenever the satellite ground track crosses a hydrological object, a so-called virtual station (VS) can be determined. Boundaries of a VS are typically defined based on the type of the water body and the disposition of the altimetry track over the object. All measurements inside a lake or reservoir, for instance, could belong to the same virtual station. This is to assume that the along-track geoid height and altimetric corrections are properly known. In the case of a river, the assumption further implies that the spatial dynamics of the water body is locally negligible. Defining a VS, in effect, allows for reduction of random noise carried by the altimetric measurements.

HydroSat follows a rather flexible approach in defining a VS. The boundaries can be determined using static or dynamic shapefiles, a radial extent around a specific point of interest, or an intersection of the two. Furthermore, HydroSat employs auxiliary sources of information like the water occurrence frequency derived from Landsat imagery by . This allows for removal of measurements over areas with a water occurrence frequency below a certain threshold. Figure S1 shows the block diagram for generating SR water level time series, where the properties of the VS are the basic setup information to be introduced. Defining the VS setup is in fact the only subjective choice made throughout the whole procedure. Nevertheless, flexibility in defining a VS is tolerated as long as reproducibility of the results is guaranteed.

The altimetric water level is initially determined for each sample within the VS. First, range measurements are corrected for geophysical effects (solid Earth tide and pole tide) and path delays caused by the atmosphere (wet tropospheric, dry tropospheric and ionospheric). The water level is then calculated by subtracting the corrected range from the satellite altitude. In the next step, the reference height is changed to geoid according to static gravity field models from XGM2019e , EGM2008 or EIGEN6C3 . To ensure a robust estimation at each overpass, the median of orthometric heights inside the VS is chosen to be the representative height.

It is important to notice that regardless of the choice of the VS, retracker, corrections and statistics by which a typical altimetric water level time series is generated, the result may be affected by many outliers. As already mentioned, missions supporting OLTC and SAR mode are less likely to provide erroneous range measurements. However, neither the OLTC nor the delay Doppler concept is capable of compensating for all unwanted radar processes over inland water bodies. Their usefulness may even be restricted by a number of known factors, e.g., crossing angle for SAR missions. An outlier identification algorithm is, therefore, required to clean the final water level time series from erroneous measurements.

In order to identify the outliers, HydroSat uses an automated, data-driven outlier identification methodology designed within an iterative, non-parametric adjustment scheme. As indicated in Fig. S1, the inputs to the algorithm are the water level time series and stochastic information derived from a number of outlier indicators (e.g., standard deviation of along-track estimates within a VS). The algorithm uses singular spectrum analysis (SSA) for gap-filling, Savitzky–Golay filtering for smoothing and a specially developed outlier identification method. The identification method benefits from applying a local kernel derived based on a local definition of an outlier. The identified outliers are not discarded instantly. The overall scheme allows for a possible correction of the outlying estimation through a retracking method that identifies the leading edge by benefiting from prior information. Such possibility is assessed via comparison of the single water level estimations inside the VS with the water level estimation coming from an ensemble model. The model, which is a by-product of the outlier identification algorithm, is used to bound the search area for identifying the true leading edge of the waveform. After retracking, the newly estimated heights are verified by their similarity to the along-track pattern of outlier-free cycles.

It is important to notice that after rejecting or correcting the outlying measurements, HydroSat does not low-pass-filter the water level time series. It can therefore fully capture the high-frequency behavior within the limitations of satellite sampling. However, this strategy may come at the cost of a higher error level for some stations, e.g., the Mississippi River.

In order to obtain a HR product and cope with the limitation of temporal sampling of single-satellite inland water monitoring, multi-mission altimetry is applied. The multi-mission altimetry for lake monitoring is now a standard approach practiced by various studies and data providers . Assuming that a lake surface is an equipotential surface allows even calibration studies to be performed over lakes. However, for multi-mission altimetry one challenge is posed by the inter-satellite biases, which impede a straightforward combination of water level measurements. Moreover, inaccurate atmospheric corrections (wet tropospheric) may cause large biases of several decimeters, which is even more pronounced for rivers due to inhomogeneous neighboring topography . Unlike lakes, multi-mission studies over rivers are very limited. Only a few studies have been dedicated to water level monitoring of rivers using a multi-mission approach with the focus on improving the temporal resolution . Here the challenge is to combine measurements from different missions at different locations with dissimilar dynamic behavior and hydraulic parameters. In general, HydroSat follows two different approaches for obtaining the HR product over lakes and rivers, outlined in the following sub-sections.

HR lake and reservoir level time series are provided based on the well-known multi-mission concept. As already mentioned, however, the integration of single water level time series is hampered by unknown biases, typically referred to as inter-satellite biases. Studies have been performed to tackle the problem, some targeting the specific case of lakes and reservoirs. provide an overview of research studies aiming at quantifying the absolute altimeter bias of each satellite. They also establish an absolute calibration site over Lake Issyk-Kul in central Asia for estimating the absolute bias of altimetry satellites and retrackers. conduct a global cross-calibration analysis over the oceans to estimate the so-called radial bias – defined as the overall error in an altimeter system – in a relative manner. These results are interpolated by and applied over inland altimetry water level time series. In order to minimize other sources of bias, DAHITI applies identical retracker and geophysical corrections to all measurements. correct for inter-satellite biases based on a maximum likelihood approach either for two missions with overlapping periods or by introducing ICESat-2 as an intermediary for non-overlapping ones.

HydroSat resolves biases in a relative, generic, regional and mission-independent manner (see Fig. S2). First, SR water level time series are categorized into groups of temporally overlapping and non-overlapping time series. For a group of overlapping time series, relative biases are estimated by minimizing a cost function for the merged time series. The cost function represents the difference between the power content of individual SR time series. If stationarity holds, minimizing such a cost function ensures the estimation of a correct relative bias. If stationarity does not hold, or in the case that any unresolved bias remains (scenario 2), remotely sensed surface area time series are used to act as an anchor of biased time series, allowing for estimation of the relevant biases. Here, a 2-D cost function in surface-area–water-level coordinates is minimized within a Gauss–Helmert adjustment scheme. Such a cost function is also used in the case that measurements have no overlap (scenario 1), leading to bias-removed SR lake water level time series. It is important to notice that relative biases are not necessarily estimated between missions but between time series. This allows for consideration of the inaccuracies of geoid or altimetry corrections within one lake and over different tracks.

A relative solution is preferred because absolute estimation of biases requires along-track in situ measurements. The absolute inter-satellite biases over a specific region, on the other hand, are not necessarily applicable elsewhere due to inhomogeneity of correction models at the global scale. It shall also be mentioned that a great portion of lakes and reservoirs are monitored by a few altimetry missions, often times with less-than-sufficient periods of overlap. For instance, the long-term, overlapping, 10 d revisiting Jason series only monitor a small number of lakes given their coarse ground-track pattern. Our proposed method is therefore designed to be least affected by these restricting conditions.

HR water level time series are obtained over rivers based on a method developed by , in which SR time series from individual altimetry missions are merged. This method allows the combination of all VSs from multiple satellite altimeters along a river hydraulically and statistically. As an example, Fig. S4 shows individual SR time series from different satellite missions along the Weser river in Germany. The idea of a HR product is to combine all these measurements for an arbitrary location along the river into a time series with improved temporal resolution.

Figure S3 shows the main processing steps of the densification method. Initially, the time lag due to streamflow between the altimetric virtual stations and the selected location is estimated. The average river width from satellite imagery together with the slope derived from satellite altimetry is used as input to a simple empirical hydraulic equation, which ultimately estimates the average flow velocity and thus the time lag

Using the estimated time lag, the water level hydrographs of all measurements are shifted and stacked at the selected location. The stacked time series is then normalized according to its statistical distribution with the 3rd percentile assigned to 0 and the 85th percentile to 1. Outliers are then identified and removed from the normalized time series by Student's t test for a sliding time window of 1 month. All measurements outside the confidence limit are identified as outliers and removed from the measurements. The outlier-free normalized time series is then rescaled according to the statistical water level distribution of the selected site. Details on the implementation of the method can be found in .

HydroSat provides time series of terrestrial water storage anomaly for major hydrological basins and also over 0.5∘×0.5∘ grid cells. Figure  shows examples of terrestrial water storage anomaly for some selected basins. Figure  (top) shows the long-term trend of the major global hydrological basins. Figure  (bottom) compares the time series of GRACE and GRACE-FO TWSA estimation over the selected basin from HydroSat (black) with two mascon products: CSRv2 (red) and JPLv2 (blue). For visual simplicity and better comparison, only two common and updated mascon products have been selected. In general, in all catchments, TWSA estimation follows the two mascon products well. Minor discrepancies are observed over the Yangtze River basin and Murray–Darling, which can be explained due to the different GIA (glacial isostatic adjustment) models and low signal-to-noise ratio, respectively.

As input data HydroSat uses the ITSG-Grace2018 unconstrained gravity field model from the Institute of Geodesy at the Graz University of Technology .

Figure S6 presents the scheme of the data processing handled in HydroSat to retrieve TWSA from GRACE Level 2 solutions. Each monthly solution contains the full hydrological, cryospheric and GIA signal in the form of fully normalized spherical harmonic (SH) coefficients, after removing the contributions from other phenomena like tides (ocean, solid Earth and atmospheric) and atmospheric and non-tidal oceanic mass changes.

To obtain TWSA, HydroSat applies several corrections on GRACE solutions, known as post-processing steps. The first-degree coefficients are added to the GRACE solutions, accounting for the movement of the Earth's center of mass . Since GRACE estimations of the lowest-degree zonal harmonic coefficient are not accurate, we replace GRACE C2,0 and C3,0 by the coefficients derived from satellite laser ranging (SLR) data . The real shape of the Earth is much closer to an ellipsoid than a sphere. Therefore, each solution is corrected from spherical to ellipsoidal coefficients following the method proposed by . To calculate the geoid anomalies, we remove the long-term (2004–2010) mean of spherical harmonics. Due to imperfect tidal models, GRACE SHs are contaminated by residual tidal aliasing error, a primary and a secondary one . Therefore, HydroSat eliminates the primary and secondary tidal aliasing errors in the main tidal constituents, S1, S2, P1, K1, K2, M2, O2, O1 and Q1, from GRACE monthly solutions using a least-squares Fourier analysis .

Furthermore, the monthly solutions are contaminated by noise from different sources including the high-frequency noise in the spherical harmonic coefficients due to the orbit geometry and sensor noise. In order to reduce the high-frequency noise and retrieve mass changes, Gaussian filtering with a radius of 400 km is applied . Moreover, to correct for leakage due to filtering, existing methodologies are employed, e.g., a data-driven approach proposed by and forward-modeling . Finally, HydroSat corrects the GIA signal following . The uncertainty for the GRACE-based TWSA at each month is estimated by propagating the calibrated error in the GRACE and GRACE-FO Level 2 solutions.

HydroSat estimates the water volume anomaly for lakes and reservoirs by combining water level and surface water area variation obtained from satellite altimetry and imaging measurements. Figure  shows selected time series of water volume anomaly for some small, medium and large lakes with different climate characteristics. A statistically representative joint time period for altimetry and imagery data is crucial to obtain a reliable estimate of water volume. For the Tana, Sobradinho and Dale Hollow lakes, for example, simultaneous surface water area and water level measurements are available for more than 10 years, resulting in a reliable area–volume model. Over Arkabutla Lake and Barren Lake, the joint time period is rather short but representative as it covers the entire statistical distribution of both variables. On the other hand, the time series of lake volume anomaly of the McConaughy, Harlan County, Rathbun and Mark Twain lakes carry mismodeling error as their joint time periods are short and non-representative of the entire distribution.

HydroSat follows a straightforward approach relying on a monotonic relationship of water level and surface area (Fig. S7). The algorithm starts by acquiring the water area from satellite imagery (explained in Sect. ) and the time series of the water level (explained in Sect. ). To derive the time series of the water volume anomaly and also the water-level–area–volume model, the following steps are performed:

generating monthly water level time series with the same temporal sampling as the surface area time series;

Standard-rate river discharge Q at selected gauges can be estimated from space-based water level H (or river width W) measurements using an empirical relationship between river height (or width) and discharge . The most common form of this relationship is the so-called rating curve Q=F(H) or Q=F(W). Rating curves are conventionally generated for simultaneous measurements of space-based water level (or river width) and in situ river discharge. Once the model is developed, the discharge can be determined from water level or width measurements. The restriction however remains that for deriving a rating curve, simultaneous measurements are required, meaning the availability of in situ measurements during the satellite era.

Globally a great portion of existing gauges are not active during the satellite era, although they provide a wealth of legacy data. For such gauges, suggest a statistical approach based on quantile mapping of in situ discharge and altimetry water level measurements. Since the quantile functions of discharge and river water level (width) have the same x axis (cumulative probability), it is possible to connect their y axis directly and obtain F(H) or F(W). As this approach does not involve the time coordinate explicitly, the requirement for synchronous datasets is obsolete. This is to say that the pre-satellite river discharge data records can be salvaged and turned into usable data for the satellite altimetry or imagery time frame.

The method is further improved by to infer a non-parametric model for estimating the river discharge and its uncertainty. The algorithm employs a stochastic quantile mapping function scheme by iteratively (1) generating realizations of river discharge and height (width) time series using a Monte Carlo simulation, (2) obtaining a collection of quantile mapping functions by matching all possible permutations of simulated river discharge and height (width) quantile functions, and (3) adjusting the measurement uncertainties according to the point cloud scatter. The flowchart in Fig. S8 describes the procedure of SR discharge estimation using spaceborne river height or width.

HydroSat provides high-rate (HR) river discharge time series based on the method developed by that goes beyond the conventional one-on-one relationship between virtual station (or reach) and (legacy) in situ station explored in SR discharge products. A multitude of altimetric discharge time series over a river network are used in this approach to estimate time series of daily river discharge. This is fundamentally done via assimilation of multiple altimetric discharge – the SR time series – and in situ measurements using a linear dynamic system. The dynamic system consists of a stochastic process model that benefits from the cyclostationarity of discharge. This model is informed by the covariance and cross-covariance generated out of old in situ data. The process model is then combined with observation equations fed by several altimetric and in situ discharge time series to form a linear dynamic system. Ultimately, the system is solved using the Kalman filter, followed by smoothing of the solutions using the Rauch–Tung–Striebel (RTS) scheme . In fact, the Kalman filter produces an a posteriori discharge estimate with a likelihood function of discharge based on the available observations and the prior information derived from the stochastic process model. In the case of an observation gap, the posterior estimates rely on the stochastic process model and its cyclostationary mean discharge. Figure S9 represents the flowchart for estimating the HR river discharge. The implementation details of the method can be found in .