Satellite altimetry has been designed for sea level monitoring over open ocean areas. However, for some years, this technology has also been used to retrieve water levels from reservoirs, wetlands and in general any inland water body, although the radar altimetry technique has been especially applied to rivers and lakes. In this paper, a new approach for the estimation of inland water level time series is described. It is used for the computation of time series of rivers and lakes available through the web service “Database for Hydrological Time Series over Inland Waters” (DAHITI). The new method is based on an extended outlier rejection and a Kalman filter approach incorporating cross-calibrated multi-mission altimeter data from Envisat, ERS-2, Jason-1, Jason-2, TOPEX/Poseidon, and SARAL/AltiKa, including their uncertainties. The paper presents water level time series for a variety of lakes and rivers in North and South America featuring different characteristics such as shape, lake extent, river width, and data coverage. A comprehensive validation is performed by comparisons with in situ gauge data and results from external inland altimeter databases. The new approach yields rms differences with respect to in situ data between 4 and 36 cm for lakes and 8 and 114 cm for rivers. For most study cases, more accurate height information than from other available altimeter databases can be achieved.

Since the 1990s, monitoring and modelling the water cycle of the Earth system
have become a very important task

Among these remote-sensing techniques is satellite altimetry. Besides its
main design goal of measuring water levels in the ocean, satellite altimetry
can also be used for deriving water levels of inland water bodies, i.e.
lakes, reservoirs, rivers, and wetlands (e.g.

However, because its measurement geometry provides observations along specific ground tracks touching water bodies is by chance. Therefore, big water bodies have a higher probability of being crossed than smaller ones. In addition, because of a repeat orbit configuration, the temporal resolution is limited to 35 (ERS-2, Envisat, SARAL/AltiKa) or 10 (TOPEX/Poseidon, Jason-1, Jason-2) days when only single altimeter missions are used. Thus, the combination of different altimeter systems plays a key role in increasing the temporal and spatial resolution as well as the length of the time series.

Satellite altimetry has to cope with different problems over inland water,
which are mainly caused by the large pulse-limited footprint of radar
altimeters. For altimeter missions using Ku-band such as Envisat, the
resulting footprint varies between 2 km over the ocean and up to 16 km over
the land

The major challenge of inland
altimetry is the handling of different reflections within the large footprint
(water, land, etc.). The shapes of altimeter waveforms vary depending on the
different surface reflections. Waveforms reflected from open ocean or large
lakes show typical Brown-like shapes

The potential of satellite altimetry for the estimation of water level time
series and for understanding the terrestrial water cycle was shown by

The

The

The

The

In contrast to the methods already published in the literature, our approach is based on a rigorous combination of a variety of altimeter missions. In addition, extended outlier detection is applied and optional waveform retracking is implemented. Moreover, the processing contains a full error propagation and provides accuracies for each height measurement. Furthermore, correlations between altimeter measurements are considered in order to achieve more reliable errors for each water level height. The current paper provides detailed information on the estimation of water level time series and performs a comprehensive validation by comparing the results with in situ gauging data and time series from other databases (Hydroweb, River and Lake, and GRLM).

The article is structured as follows: in Sect.

In this paper, altimeter measurements from TOPEX/Poseidon, Jason-1, Jason-2, ERS-2, Envisat, and SARAL/AltiKa are used depending on the data coverage for the inland water bodies under investigation. In principle, data from Geosat, ERS-1, HY-2A, IceSAT, and Cryosat-2 can be used. However, these missions are neglected in the current investigations for a number of reasons, i.e. lack of data over land, non/long-repeat cycle, bad data quality, or missing waveform information. The applied missions can be separated into two groups according to their orbit characteristics. TOPEX/Poseidon was launched in 1992 into an orbit with a repeat cycle of 9.9156 days and a track separation at the Equator of about 300 km. The mission was followed by its successors, Jason-1 and Jason-2. These three altimeter satellites can be used for estimating continuous time series over more than 2 decades. The second group starts with ERS-2 (launched in 1995), followed by Envisat and SARAL/AltiKa. The orbit of these missions is defined by a repeat cycle of 35 days and a track separation of about 80 km at the Equator. The data are available for almost 2 decades with a data gap between October 2010 (end of Envisat core mission) and March 2013 (launch of SARAL/AltiKa). The data for Envisat on its drifting orbit (October 2010–April 2012) are not used. ERS-1 is not yet ready for use in DAHITI but will be integrated in the near future. This will enable extensions of the time series back to 1991.

For the estimation of water levels, Sensor Geophysical Data Records (SGDR)
altimeter products are used which provide high-frequency ranges as well as
altimeter waveforms. The altimeter waveforms allow individual retracking in
order to achieve more reliable altimeter ranges, especially for smaller
inland water bodies. Table

List of all altimeter missions used in this study together with their main characteristics.

Perated by

Depending on the investigated inland water body, the original ocean ranges in
the SGDR are very often corrupted. Especially over small lakes and rivers the
altimeter waveforms do not exhibit the typical ocean-like shapes but
quasi-specular shapes. Land-contaminated altimeter waveforms are usually
more peaky and noisy, leading to flat-patched and complex waveforms

In order to convert the range measurements (original or retracked) to water levels serving as input for our Kalman filter approach, numerous preprocessing steps are necessary.

Equation (

First, the range has to be corrected for geophysical effects. For this
purpose, the models and corrections given in
Table

List of applied models and geophysical corrections

For the computation of water level time series within the Kalman filter
approach, normal heights

In order to use altimeter measurements from different tracks and missions a consistent and reliable combination strategy is important. The irregular spaced observations from different locations must be merged into one time series per target, and the optimal combination of measurements with different uncertainties must be ensured. This requirement is fulfilled by our DAHITI approach, which is based on an extended outlier rejection and a Kalman filter for the estimation of water level time series.

The processing strategy for the estimation of water level time series over
inland waters using the DAHITI approach is separated into three steps:
preprocessing, Kalman filtering and postprocessing (cf. Fig.

The preprocessing step includes all necessary tasks for the preparation of the input altimeter heights such as waveform retracking, applying range corrections, calculation of height errors, and rejection of outliers.

In the Kalman filtering step, the computation of the water levels of the investigated water body is performed. In this paper, we apply Kalman filtering in a single location centred on the investigated water body and obtain one computed water level for each epoch. However, there is also an option for performing Kalman filtering on a grid which can be used for investigation of the surface variability of larger lakes.

In the postprocessing step, all water levels from the previous step are merged to form a single water level time series referring to one reference location if the Kalman filtering was performed on a grid. Subsequent outlier detection can be conducted if necessary. The final time series is stored in DAHITI, accessible via the website.

Processing strategy for the computation of water level time series for inland waters in DAHITI in three main steps: preprocessing, Kalman filtering, and postprocessing.

OpenADB holds satellite altimeter data and
derived high-level products. OpenADB provides satellite altimeter data,
geophysical corrections, models, etc., which are also accessible via the
website. The data sets from OpenADB used for this study, and the
methodology used to derive individual water levels are described in
Sect.

In addition to the normal heights of the water levels the Kalman filter requires information on the quality of each measurement. This information is used for the weighting of the individual data sets as well as for the error estimation of water level products. Because of the lack of absolute accuracy, the precision of the heights is computed by analysing the along-track scatter of the measurements.

For this purpose, an “absolute deviation around the median” (ADM) is
estimated by using a sliding box along the altimeter track. The size of the
sliding box varies for large lakes (

Before Kalman filtering is performed, various user-defined outlier rejections
can be applied. Inaccurate water levels must be rejected before Kalman
filtering; precise ones are used for the estimation of the resulting water
levels. The following outlier criteria can be applied in the preprocessing
step:

latitude thresholds,

water height thresholds,

height error (ADM) threshold,

backscatter coefficient (Sigma0) thresholds,

along-track outlier test (SVR).

The backscatter coefficients of altimeter measurements provide information about the reflectance of the surface. This information can be used to reject altimeter measurements affected by ice.

Moreover, outlier detection with support vector regression (SVR)

Example of an outlier detection using error threshold and SVR
along a single satellite track over Lake Erie, which contains an island (between approx. 41

The method of Kalman filtering is applied for
the computation of water level time series in DAHITI. It updates a model by
measurement data of different accuracies and predicts the current state to
the next time epoch

The Kalman filter performs the estimation of water level time series from the
track-wise input heights by combining time-dependent input data available at
irregular intervals and – in the case of larger lakes – at different
locations. Different modified Kalman filter approaches have been used for
geodetic applications (e.g.

The Kalman filter uses input observations to update the current state of the
system and predict the model of the following time epoch. This is performed
in a continuous loop consisting of two steps (an update and a prediction
step) running consecutively for every period of time

All computations can be referred to one location (centre of the target) or performed on a computation grid. The latter is optional and can be applied for special investigations on surface variability of larger lakes. The standard solution – also used for all computations within this study – assumes uniform lake surfaces in balance with gravity and merges all water heights of one update step to one location. Surface differences owing to systematic height, geoid errors or hydrodynamic effects from wind and waves are neglected. In practice, our approach automatically creates a grid by means of a recursive algorithm used on an initial grid node as a reference point. A land water mask provides information on the extent of the water body and the grid. The grid node separation can be chosen manually depending on the extent of the investigated inland water. Thus, normally we define only one grid node over the target. However, in cases where surface differences are expected, a smaller grid node distance can be chosen. The computations will then be performed for all grid nodes and different water levels for the whole lake surface.

Procedure of Kalman filtering starting with an initialization step followed by a progressive loop containing one update and one prediction step.

In the following, the basic equations of the Kalman filter are introduced. The algorithm consists of an observation model and a dynamic model.

The observations for each step

The vector length of

The uncertainties of the water levels are described in

The dynamic model of the Kalman filter approach describes the transition of the system state from epoch

The applied Kalman filter procedure as used in the DAHITI approach is described in detail below.

The Kalman filter approach begins with an initialization step which is
necessary before starting the recursive loop. The initial state vector

In the update step, new altimeter water levels are introduced in order to
update the parameters of the current state

The parameter update of vector

In parallel, the corresponding covariance matrix

After the parameter vector and the covariance matrix of the current epoch

The Kalman filter provides water heights

If Kalman filtering is performed on a single grid node, the final water level and error are immediately available. If it is computed on a grid, a “mean” one-dimensional time series is computed. Instead of simply averaging all grid node heights, we select only the best water levels per epoch. Only water levels are selected that fulfill certain error criteria of Kalman filtering errors. In general, the limit for the maximum height error is set to values between 5 and 10 cm. The selected limit depends on the resulting height errors. Therefore, the limit is selected manually in such a manner that only reliable heights are used for the final time series. The remaining water levels are averaged for each epoch by using the formal errors for the weighting factors. Finally, a time series of water levels and their formal errors over the entire period of time are obtained.

In a last step, an outlier rejection is performed. The water level time
series can still contain outliers because of bad quality of data, ice
coverage, orbit manoeuvres, etc. For the detection of those outliers, SVR can
be applied again – now on the full time series. Complete tracks showing
significant differences with respect to the other points of the water level
time series can be rejected. This time, radial base functions instead of a
linear kernel are used to perform the regression since a constant water level
over time cannot be assumed. The radial base function kernel of
the SVR allows us to fit the time series including seasonal variations and
trends. Figure

Example of applied SVR using radial base functions for outlier rejection on a resulting water level time series (Lake Erie) of the Kalman filtering step. The estimated regression function (cyan) and its confidence intervals (dotted cyan) are plotted. The result of the regression shows valid (blue) and rejected (red) altimeter heights. Each rejected water level height represents one complete satellite overflight.

In this chapter, water level time series
resulting from the Kalman approach are presented and validated. Since it is
not possible to show results for all inland water bodies, we focus on the
selected study areas introduced in Sect.

For altimetry-derived water level time series, in situ measurements from gauging stations are the most important validation data sets. In order to perform reliable comparisons, only those inland water bodies are selected as study areas for which in situ data are available. Since we have access to many gauging stations in North and South America, we focus our study on these two continents.

Another criterion for the selection of inland water bodies is the
availability of external altimetry-derived time series to demonstrate the
performance of our Kalman filter method compared with other approaches. Each
study case is observed by at least one other group (i.e. Hydroweb, River &
Lake, or GRLM). Thus, those targets in North and South America are selected
which are best represented by other inland altimetry databases for as long a time
period as possible. We end up with the 16 lakes and 20 river
crossings illustrated in Fig.

Map of selected study areas of lakes (blue) and rivers (red) in North America (left) and South America (right).

The first study areas are the Great Lakes of North America, comprising Lake
Superior (82 000 km

In addition to the Great Lakes, the Great Slave Lake (27 200 km

In addition to the lakes in North America, two lakes in the very south
of South America are selected for validating our approach. Lake Argentino
(1466 km

For the analysis of rivers, the Amazon Basin is selected as the study area;
it is the largest basin in the world and covers about 7 000 000 km

Water level time series of Lake Superior from DAHITI (1992–2014), Hydroweb (1992–2011), River & Lake (2002–2010) and GRLM (1992–2014) compared with in situ data (Ontonagon, 1992–2014) and shifted to the water level height of the in situ data. Additionally, differences between heights from altimetry and in situ data are plotted for periods in which both data sets are available.

Water level time series of Lake Athabasca from DAHITI (1992–2014), Hydroweb (1992–2011), River & Lake (2002–2010) and GRLM (2002–2014) compared with in situ data (Lake Athabasca, 1992–2013) and shifted to the water level height of the in situ data. Additionally, differences between heights from altimetry and in situ data are plotted for periods in which both data sets are available.

Water level time series from gauges have a high relative accuracy, but some points must be kept in mind in the use of in situ data. The absolute comparison of heights from gauges and satellite altimetry is often very difficult since location, reference height and vertical datum of gauges are not always precisely known or may even be unknown. This leads to height offsets between water level time series from gauge and altimetry, which must be considered in the validation step. In particular, the comparison between water levels from altimetry and in situ data over rivers shows in most cases remaining offsets. In general, almost no altimeter satellite track crosses the river at the location of a gauging station, which leads to additional offsets because of the river slope. To avoid handling the uncertainties of in situ data, only relative comparisons with water level time series from altimetry are performed.

In order to rank our results with respect to other time series derived from altimeter data, we download water levels from three external inland altimeter databases, namely Hydroweb, River & Lake, and GRLM. These results are based on various altimeter missions, and diverse approaches were performed to compute the water level time series. As a consequence, these external time series cover different time periods with temporal resolutions between 10 and 35 days. This has to be kept in mind when the different time series of the four databases are compared.

We choose three of the aforementioned water bodies
in order to present detailed results of our DAHITI approach. The targets are
selected to represent three diverse inland water body types featuring
different characteristics. Lake Superior (Fig.

Water level time series of the Madeira River from DAHITI (2002–2014), Hydroweb (2002–2010), and River & Lake (2002–2010) compared with in situ data (Humaitá, 2007–2014) and shifted to the water level height of the in situ data. Additionally, differences between heights from altimetry and in situ data are plotted for periods in which both data sets are available.

Figure

Relevant parameters for the estimation of the water level time series for DAHITI. This table shows information about the altimeter missions used for selected lakes and rivers, applied retracking (Retr.), ice coverage (Ice), and applied outlier criteria which are used for the processing of water level time series. Finally, the number of data points and percentage of outliers of the final water level time series are given.

Missions: TOPEX/Poseidon (TP), Jason-1 (J1), Jason-2 (J2), ERS-2 (E2), Envisat (EV), SARAL/AltiKa (SA).

The DAHITI water levels coincide very well with the daily in situ data of
Ontonagon. The correlation coefficient

Figure

The DAHITI water level shows a very good agreement with in situ data in
summer, and almost no outliers owing to ice coverage are visible in winter
compared with time series from Hydroweb and River & Lake. The overall
consistency with the gauge data yields a correlation coefficient of 0.90 and
an rms difference of 15.1 cm using 1279 points in the period between 1992
and 2014. The usage of a median filter leads to slightly worse rms
differences of 15.3 cm for Lake Athabasca. The differences between in situ
data and Hydroweb (rms

Figure

The DAHITI time series show good consistency with in situ observations and
clear advances over established approaches. However, some problems remain,
especially for smaller lakes and rivers. For larger lakes, the assumption of
a uniform surface level may no longer be justified. In addition to height
differences due to systematic errors in geophysical corrections or the
geoid, hydrodynamic effects caused by wind and waves can cause horizontal
lake level differences. Currently, these are neglected when combining
observations from diverse parts of the lake. Moreover, measurements
(altimetry as well as in situ) feature non-uniform accuracies observed over
areas with different surface conditions. This effect can be seen when we
compare the DAHITI water level time series of Lake Superior with additional
gauging stations. The five possible comparisons lead to rms differences
varying by 2 cm (between 4.4 and 6.6 cm;
Table

For small lakes and rivers, land contamination of waveforms is the largest problem because nearly all altimeter measurements are affected. For rivers, almost no nadir measurements may occur, and even these can originate from river branches and distort the water level time series from the investigated target. Moreover, the river slope can influence the time series, as well as the comparison with in situ data. The crossings between river and altimeter track can vary slightly (up to 1 km) because of orbit instabilities so that the reflections originate from different areas which do not exhibit the same water level. The most important challenge remaining is the handling of inter-mission biases and retracker biases. The usage of radial errors from a global crossover analysis and the restriction to one common retracker works reasonably well; however, small discrepancies remain in the time series. Moreover, the quality of the single altimeter measurements could surely be further improved by combining different retracking algorithms depending on the waveform shapes. This remains a major challenge and offers enormous potential for future work.

The validation of water level time series of DAHITI for Lake Superior, Lake Athabasca, and the Madeira River compared with in situ data and time series from Hydroweb, River & Lake, and GRML showed clear improvements. To evaluate the impact of the outlier rejection and Kalman filtering on the improvements of the DAHITI time series, an alternative approach using a simple median filter instead of a Kalman filter was applied.

The resulting rms differences for three inland waters decreased slightly by 0.1 to 0.2 cm, which indicates that the combination strategy has only a moderate effect on the overall accuracy. The strongest improvements are currently due to rigorous outlier detection and data retracking. However, the Kalman filter has a considerable potential when upgraded by dynamic modelling and used for real-time applications.

Water level time series of selected lakes and reservoirs from DAHITI, Hydroweb, River & Lake, and GRLM compared with in situ data. For each comparison of water level time series from altimetry with in situ data an rms difference and squared correlation is computed. The number of points from the final water level time series which were used for the validation is given in the third column (No.) of each altimeter data set. The smallest rms difference for each target is highlighted in bold, the largest one in italic.

Source of in situ data:

Water level time series of selected rivers of the Amazon Basin from DAHITI, Hydroweb and River & Lake compared with in situ data.
For each comparison of water level time series from altimetry with in situ data an rms difference and squared correlation is computed.
The number of points from the final water level time series which were used for the validation is given in the third column (No.) of each altimeter data set.
Additionally, the distance to the nearest in situ station (upstream (

Source of in situ data:

The results for Lake Superior, Lake Athabasca, and the Madeira River
presented in Sect.

Table

Table

DAHITI results show rms differences with respect to the gauge data between 4 and 36 cm. It is obvious that accuracy declines with lake extent and ice coverage. For some lakes, the differences between DAHITI and in situ data vary by more than a factor of 2 with different lake gauges. Especially for Lake Erie the difference between the rms values can reach up to 8.1 cm. For most lakes, the relations between the different rms values are similar for the different altimeter products.

For most lakes the DAHITI water levels are more consistent with in situ data than the results from external altimeter databases. In addition, the temporal resolutions of the time series are significantly higher, as indicated by the number of points used for validation. Of course, the different time periods of the other altimeter data sets have to be taken into account, too. The most notable improvements through the DAHITI approach with respect to the existing databases can be seen for smaller lakes. For example, for the Lake of the Woods, the DAHITI consistency with in situ data is more than twice as good as the other altimeter products, improving the rms differences from about 36 cm to approximately 16 cm.

The validation results for different rivers in the Amazon Basin are
summarized in Table

The rms differences between altimeter time series and in situ data vary between 8 and 114 cm in the case of DAHITI. For most virtual stations, the consistency with the gauge is considerably lower than for lakes. It is not possible to prove a dependence between river width and distance to the gauge, not only because of the altimeter time series but also because of the accuracies of the in situ data which also contain measurement errors. Also, the angle at which the satellite track crosses the river has a strong impact on the quality of the water level time series. Furthermore, distances of tenths of kilometres between the in situ station and the nearest crossing altimeter track make it more difficult to prove dependences due to unpredictable river flow effects.

Compared with time series from Hydroweb and River & Lake, the new DAHITI
approach can improve the gauge consistency for most of the targets. The
improvement can reach several decimetres. Many correlation coefficients in
Table

This paper presents a new method for estimating water level time series over inland waters using multi-mission satellite altimetry data. It is based on careful data preprocessing (including waveform retracking), a Kalman filter approach, and a rigorous outlier detection. The introduced method is the basis of DAHITI, an online database for inland water level time series from satellite altimetry observations operated by the Deutsches Geodätisches Forschungsinstitut der Technischen Universität München (DGFI-TUM).

The study demonstrates the performance of the new method for numerous lakes and rivers in North and South America. A comprehensive validation is performed by comparison with time series of water level variations from in situ gauging stations. Moreover, a comparison with external altimetry-derived water level variations is presented based on data from Hydroweb (LEGOS), the River & Lake database (ESA-DMU), and the Global Reservoir and Lake Monitor (USDA).

The lake level data sets computed with the presented approach yield accuracies between 4 and 36 cm depending on the surface extent of the lake and climate conditions (i.e. ice coverage). For rivers, the performance is considerably lower, with rms differences varying between 8 and 114 cm. Here the accuracy mainly depends on the crossing angle of the altimeter track and the surrounding conditions. Also, other surrounding conditions – such as topography, quality of waveforms and their retracked water heights – can influence the resulting water level time series. Especially in the Amazon Basin the river meander can also change over the years because of strong seasonal variations.

For most study cases, the new approach yields significant accuracy improvements compared with water level variations provided by established inland altimeter databases, especially for smaller lakes and rivers. In addition, the temporal resolution of the DAHITI lake time series is significantly improved compared with other data sets, allowing for the detection of sub-monthly temporal changes.

The reasons for the improved performance of the presented approach are multiple: first, a larger observation data set is used as input as a multi-mission concept is realized. All available altimeter missions are cross-calibrated and incorporated into the computations. Second, the applied preprocessing consists of a robust outlier elimination and optional retracking. This ensures that only highly accurate data will be used. Moreover, the Kalman filter approach permits the optimal combination of all data sets and also includes the accuracies of the input data for weighting. This also enables rigorous error propagation and the computation of formal errors for each water level height. Further comparisons for the three selected areas show that using the Kalman filter approach instead of a median approach leads to slightly decreased rms differences. This indicates that the major improvements in the water level times of DAHITI are due to the extended outlier rejection. In future, the Kalman filter approach will also be used for (near-)real-time analysis and integration of altimeter data (with the so-called Operational Geophysical Data Record, OGDR). This enables daily actualization of the water level time series and may also be used for short-time predictions. Furthermore, the introduction of a dynamic model in the Kalman filter will cause an increase in the temporal resolution of the water level time series. For the development of the dynamic model, external data sets such as GRACE, precipitation, etc. can be used.

In spite of the improved water level time series of DAHITI compared with results from Hydroweb, River & Lake and GRLM, there are still some challenging tasks which have to be taken into account to make further improvements. Retracking is the most challenging task in using altimeter data for smaller water bodies. The mixture of different waveform shapes – such as ocean-like, specular, and other ones – makes it difficult to choose a suitable retracking algorithm. Each retracker is optimized for special waveform shapes, but switching the retracking algorithm to achieve the best ranges will lead to retracker biases which have to be taken into account. Furthermore, inter-mission offsets can also arise because of the different characteristics of the measurement systems (e.g. Ku-band (Envisat) and Ka-band (SARAL/AltiKa)).

All presented water level time series as well as results for many additional
targets are freely available in the “Database for Hydrological Time Series
over Inland Waters” (DAHITI) at

The altimeter observations and geophysical corrections are taken from OpenADB
(

We thank the Agência Nacional de Águas (ANA); National Oceanic and Atmospheric Administration (NOAA); government of Canada; US Geological Survey (USGS); and Ministerio de Planificación Federal, República Argentina, for providing the in situ data.

We also thank the reviewers for helpful comments and suggestions which helped us to improve the quality of this paper. This work was supported by the German Research Foundation (DFG) and the Technische Universität München within the funding programme Open Access Publishing. Edited by: B. Schaefli