Estimation of Water Levels from Surface Areas Using Hypsometry

In the next step, corresponding water levels are estimated for each surface area by using the hypsometric curve. Figure 9 shows the resulting water level time series based on the modified Strahler hypsometric curve (blue). For comparison, water levels based on a linear (dashed red) and polynomial (dashed green) hypsometric curve are shown. Additionally, water levels from satellite altimetry (yellow) and its lower and upper boundaries called confidence range (dashed black) based on the hypsometric curve are shown in Figure 9. For the purpose of quality assessment, all four time series are validated with water levels from the in-situ station (orange). For that purpose, correlation coefficients and RMSE values are estimated twice for each time series: for all water levels and only for water levels within the confidence range, i.e., without extrapolating the hypsometric curve.

**Figure 9.** Comparison of derived water levels from surface areas using different hypsometric curves of Ray Roberts Lake.

Based on the data range used for the computation of the hypsometry curve it can be assumed that the quality of water levels within the confidence range is better than outside due to extrapolation errors. This can be clearly seen in the validation results. The water level time series using the linear hypsometric curve leads to a correlation coefficient of 0.94 and a RMSE of 0.27 m for all points. Within the confidence range, the correlation coefficient decreases slightly to 0.92 while the RMSE is still 0.27 m. For the year 2000, it can be clearly seen that the linear function reconstructs the lower water levels very well. But this linear function has major problems when predicting water levels higher than about 193 m (e.g., 2007, 2015). In contrast, the polynomial function leads to a correlation of 0.93 and a RMSE of 0.31 m for both data sets (points in the confidence range and all points) since all point are in the confidence range. The best result can be achieved by using the modified Strahler approach where the correlation coefficient is 0.96 and the RMSE is 0.24 m using all water levels and 0.22 m within the

confidence range (same correlation). One can conclude that the hypsometry based on the modified Strahler approach performs best. However, extreme events could not be captured by the hypsometric curve because of missing input data. Nevertheless, the result shows that the derived water levels from the hypsometric curve are suited to densify and extend water level time series with good estimates.

#### Estimation of Bathymetry

For the estimation of bathymetry, all surface areas and their corresponding water levels are used. Based on the methodology introduced in Section 4.4, the bathymetry shown in Figure 10 is computed between the minimum and maximum surface area. The resulting bathymetry covers heights between 189.13 m and 194.96 m. Pixels with heights lower than 189.13 m could not be observed and are therefore marked in gray. The minimum of the observed bathymetry will later be used as reference point for the estimation of volume variations. The usage of 543 land-water masks results in a high-resolution bathymetry which has a spatial resolution of 10 m and centimeter resolution in height. The bathymetry clearly shows fine structures in the bays of Ray Roberts Lake.

**Figure 10.** Bathymetry of Ray Roberts Lake derived from land-water masks in combination with water levels derived from the hypsometric model. Additionally, the used altimeter track is highlighted as dotted line.

#### Estimation of Volume Variation Time Series

For the computation of the time series of volume variations, all heights derived from surface areas and satellite altimetry are intersected with the estimated bathymetry. Additionally, volume errors are estimated by considering the given errors of the used surface areas and water levels from altimetry. Figure 11 shows the resulting volume variations time series (orange) based on 874 measurements (518 from surface areas, 331 from water levels, 25 combined).

**Figure 11.** Time series of volume variations of Ray Roberts Lake.

Since direct measuring of ground-truth volumes for lakes and reservoirs is not possible one has to combine water levels from gauging stations and bathymetry for surveys in order to derive in-situ volume for validation purposes. However, for most reservoirs more than one bathymetry survey is available, and consequently, the resulting hypsometry and volumes can differ. This might lead to inconsistencies such as offsets in the volume time series. For Ray Roberts Lake, the volume time series used for validation is based on an older unknown hypsometry and newer hypsometry calculated on 1 October 2008 by the TWDB. Figure 11 shows the first period going until 1 October 2008 (blue) and the second period starting on 1 October 2008 (green). However, no offset exists for Ray Roberts Lake between both periods after updating the hypsometry model. Furthermore, it has to be taken into account that volumes above the maximum observed bathymetry in the survey are extrapolated (dashed black). This means that those volumes can contain errors which result from the extrapolated hypsometry curve.

The validation of the volume variations yields a RMSE of 0.025 km<sup>3</sup> and a correlation coefficient of 0.97 using 867 points. The relative error with respect to the volume variations from TWDB is 7.2%. The differences in volume variations are in the range between <sup>−</sup>0.18 km<sup>3</sup> and 0.14 km<sup>3</sup> with zero average (867 points). The larger errors can be explained by the fact that lower water levels (e.g., in 1993/94, 2000) are missing in the calculation step of the used hypsometric curve, which causes a rather large uncertainty/error to an extrapolation. Larger differences for volumes above the extrapolation limit (dashed black) are also visible (e.g., 2015).

#### 5.1.2. Hubbard Creek, Lake

As second study case, Hubbard Creek Lake is presented. It was selected in order to demonstrate the new approach for estimation volume variation having the best input data from satellite altimetry and optical imagery. Hubbard Creek Lake is a freshwater reservoir located about 200 km West of Dallas, Texas which was impounded in 1968 and finally filled in 1970 [36]. In the last four decades, the surface area has varied between 14.74 km<sup>2</sup> and 59.77 km<sup>2</sup> and the water levels has varied between 351.34 m and 361.16 m.

#### Extraction of Input Data

Figure 12 shows the time series of water levels and surface areas derived from satellite altimetry, respectively optical imagery. The water level time series (blue) of Hubbard Creek Lake is based on the altimeter missions of Jason-2 and Jason-3 covering the period between 27 July 2008 and 30 January 2020. The validation of 410 points from altimetry with in-situ data from the gauging station near Breckenridge (ID:08086400) provided by USGS/TWDB results in an RMSE of 0.15 m and a correlation coefficient of 1.00. The water level time series has a temporal resolution of about 10 days. An offset of about 0.15 m occurs because of different vertical datums.

**Figure 12.** Used water level time series from satellite altimetry and surface area time series from optical imagery for Hubbard Creek Lake. Additionally, available validation data provided by TWDB and resulting quality assessment are shown.

The surface area time series (green) is based on 479 data points covering a period of almost four decade between 13 November 1982 and 28 June 2019. It is derived from data measured by Landsat-4/-5/-7/-8 and Sentinel-2A/-2B. For validation, surface area time series provided by TWDB are used, which were derived from two bathymetry surveys on 1 February 1997 (dashed blue) and 1 January 2018 (dashed red). Due to the survey method itself, no surface areas can be provided for areas larger than 60.39 km<sup>2</sup> (1 February 1997–1 January 2018) and 63.48 km<sup>2</sup> (since 1 January 2018) without extrapolation. Comparisons of surface areas in both survey periods lead to offsets of <sup>−</sup>1.70 km<sup>2</sup> and <sup>−</sup>3.26 km<sup>2</sup> which have to be considered. These offsets can be caused by long-term changes of the bathymetry but also improvements in the measurement technique. Then, an overall validation using 479 surface areas from optical imagery with surface areas from survey shows an RMSE of 1.09 km<sup>2</sup> and a correlation coefficient of 0.99.

Both water level time series from satellite altimetry and surface areas from optical imagery are available in a high quality for Hubbard Creek Lake. Furthermore, both data sets cover the full variations of the lake which is essential in order to compute an optimal hypsometric curve.

#### Estimation of Hypsometric Curve

For the estimation of the hypsometric curve, the water levels and surface areas introduced in Figure 12 are used. For Hubbard Creek Lake, 279 data points with a time difference of less than 10 days between both measurements are available. All points covering the full area of data are shown in Figure 13. Additionally, it shows the resulting hypsometric curves based on the modified Strahler approach (blue), linear function (dashed red) and polynomial function (dashed green). The correlation coefficient for all functions is 0.99 but the resulting RMSE of 0.19 m is the best for the modified Strahler approach. Additionally, the two hypsometric curves provided by TWDB (dashed black) are shown for comparison which have similar shapes as the hypsometric curve of the modified Strahler approach. Despite smaller offsets in the height, the agreement for extrapolated values below 14.74 km<sup>2</sup> , respectively 351.38 m are very good. The hypsometric curves of the linear and polynomial functions show their weaknesses for extrapolated values.

**Figure 13.** Hypsometry curves of Hubbard Creek Lake using modified Strahler approach (blue), linear function (dashed red) and polynomial function (dashed green). Additionally, two hypsometric curves (dashed black) from the TWDB are shown as comparison.

Estimation of Water Levels from Surface Areas using Hypsometry

All water levels derived from the hypsometric curves based on surface areas are shown in Figure 14. For Hubbard Creek Lake, all three methods perform similarly well with correlation coefficients of 0.99. However, the best RMSE of 0.21 m can be achieved using the modified Strahler approach. The RMSE of 0.23 m using the polynomial function is only slightly higher. Using the linear function results in an RMSE of only 0.27 m which can be also seen in Figure 14 for low (2014, 2015) and high (1990–1993, 1997–1998) water levels. The data range of the used input data leads to a confidence range which contains all computed water levels.

Estimation of Bathymetry

Figure 15 shows the resulting bathymetry of Hubbard Creek Lake calculated by using 636 land-water masks and their corresponding water levels. In 2015, the Hubbard Creek Lake was half empty which enables us to estimate nearly the majority of the bathymetry except remaining lower water levels (gray). The bathymetry varies between 351.71 m and 361.23 m. The absolute minimum water level derived from the TWDB hypsometric curve is 351.71 m. The fine structures at the lake inlets can be clearly seen in the bathymetry, but also the historic rivers and their valleys before Hubbard Creek Lake was dammed.

**Figure 14.** Comparison of derived water levels from surface areas using different hypsometric curves of Hubbard Creek Lake

**Figure 15.** Bathymetry of Hubbard Creek Lake derived from land-water masks in combination with water levels derived from the hypsometric model. Additionally, the used altimeter track is highlighted as dotted line.

Estimation of Volume Variation Time Series

The combination of the observed bathymetry with water levels from satellite altimetry and water levels derived from the hypsometric curve results in the time series of volume variations shown in Figure 16. Additionally, volume errors resulting from satellite altimetry and optical imagery are

provided. The volume variations are validated again with absolute volume time series provided by USGS/TWDB. Since the volume time series relies on the three different hypsometric curves, we split the time series into three phases. The first phase (green) is based on a hypsometric curve from 1 January 1962 by USGS. The second (blue) and third (red) phases rely on hypsometric curves calculated by the TWDB on 1 February 1997 and 1 January 2018. However, for Hubbard Creek Lake only insignificant offsets remain for the different three time periods which are not corrected.

**Figure 16.** Volume time series of Hubbard Creek Lake

For validating the time series of volume variations there are 1021 data points. They show a RMSE of 0.008 km<sup>3</sup> and a correlation coefficient of 0.99 for Hubbard Creek Lake. The volume variations reach a maximum of 0.34 km<sup>3</sup> , and the errors with respect to the in-situ values vary between <sup>−</sup>0.04 km<sup>3</sup> and <sup>−</sup>0.02 km<sup>3</sup> . When adding the constant volume below the lowest observed water level, which is 0.071 km<sup>3</sup> derived from validation data, absolute volumes can be computed. The relative errors with respect to the full volume is only 2.0%. The relative error with respect to the volume variations is 2.8%. This example clearly shows the potential of the new approach when input data are evenly distributed and of good quality.

#### 5.1.3. Palestine, Lake

Palestine Lake is the third water body presented in detail which is located about 150 km southeast of Dallas, Texas. Palestine Lake is a reservoir built in the 1960s impounding the Neches River [37]. It has an average surface area of 87.01 km<sup>2</sup> and an average water level of 104.73 m. It shows only smaller seasonal variations in the surface area (18.61 km<sup>2</sup> ) and water level (1.27 m). Palestine Lake was chosen to demonstrate and discuss the challenges when estimating volume variations based on less optimal input data.

#### Extraction of Input Data

Figure 17 shows the used water levels (top) from satellite altimetry and surface areas (bottom) from optical imagery of Palestine Lake. Palestine Lake is crossed by the two latest altimeter missions Sentinel-3A and Sentinel-3B that have been launched on 16 February 2016, respectively 25 April 2018. Thus, no information is available before 2016. Moreover, the repeat cycle of those satellites is 27 days, and therefore, the temporal resolution of the water level time series is almost three times sparser than for Ray Roberts Lake and Hubbard Creek Lake, where Jason data is available. The resulting water level

time series covers only the time span between 14 March 2016 and 27 February 2019. The validation with in-situ water levels measured by the gauging station near Frankston (ID: 08031400, USGS) shows a RMSE of 0.13 m and a low correlation coefficient of 0.84 by using only 36 measurements. An offset of −0.37 m between both time series occurs caused by different vertical datums. However, the water levels (104.11 m–105.38 m) measured over the last three years can only capture 37% of the full range of water level changes (103.17 m–106.58 m) shown in the in-situ time series (black) in Figure 17.

**Figure 17.** Used water level time series from satellite altimetry and surface area time series from optical imagery for Palestine Lake. Additionally, available validation data provided by TWDB and resulting quality assessment are shown.

The surface area time series (green) derived from optical imagery satellites Landsat-4/-5/-7/-8 and Sentinel-2A/-2B is shown in Figure 17. The surface area has varied between 73.77 km<sup>2</sup> and 92.38 km<sup>2</sup> since the 1980s. However, it can be clearly seen that the average seasonal variations is much smaller, between 85 km<sup>2</sup> and 90 km<sup>2</sup> only. Similar variations can also be seen in the period where water levels from satellite altimetry are available. For validation, the surface area time series from TWDB is used. However, the surface area time series is based on two bathymetric surveys. The first ship survey was performed in June 2003 [38]. The second bathymetric survey was undertaken in July/August 2012 using a multi-frequency, sub-bottom profiling depth sounder [37]. This results in an inconsistent jump of 0.591 km<sup>3</sup> on 1 August 2012 when the bathymetry respectively rating curve was changed. Since the optical imagery provides a homogeneous surface area time series, we applied an offset of 0.591 km<sup>3</sup> to the in-situ surface areas between 13 May 1999 and 1 August 2012 to achieve a consistent time series for validation. Additionally, for each rating curve an extrapolation limit is shown (dashed black). After homogenizing the in-situ surface area time, an RMSE of 2.14 km<sup>2</sup> and a correlation coefficient of 0.80 using 301 data points can be achieved. The offset between both time series is <sup>−</sup>3.145 km<sup>2</sup> .

Despite the long surface area time series, only data between February 2016 and April 2018 where water levels from satellite altimetry are available can be used for the estimation of the hypsometric curve that is shown in Figure 18.

#### Estimation of Hypsometric Curve

Only 32 data points are available for calculation having a time difference of less than 10 days between the measurement of water level and surface area. The hypsometric curves based on the modified Strahler approach, the linear function and the polynomial function are almost identical, which is a significant indication of the linear dependence between water level and surface area in the fitted range of data (orange rectangle). However, the correlation coefficient is only 0.64 which

shows that the used input data is noisy. This can have a strong impact on the quality of resulting water levels from surface areas in the next step. Especially, for the reconstruction of water levels based on surface areas less than about 84 km<sup>2</sup> a rapid decrease in quality is expected due to extrapolation issues. Additionally, both hypsometic curves (dashed black) provided by TWDB are shown in Figure 18. One can clearly see the discrepancies between both curves and with respect to the curves fitted to the remote sensing data sets. The TWDB curves are already shifted by the datum offsets between altimetry and in-situ heights, which improves the consistency but is not able to align all curves. Especially, the different gradients are not impacted by shifts or offsets. The possible reasons for the different hypsometric curves is unknown. It can be related to the input data but also to the validation data. A significant change in bathymetry is also possible. A new bathymetry survey may help to clarify this issue.

**Figure 18.** Hypsometry curves of Palestine Lake using modified Strahler approach (blue), linear function (dashed red) and polynomial function (dashed green). Additionally, two hypsometric curves (dashed black) from the TWDB are shown as comparison.

#### Estimation of Water Levels from Surface Areas using Hypsometry

The reconstructed water levels based on the three hypsometric curves are shown in Figure 19. The performances of the three approaches within the confidence range (dashed black) are identical with a good RMSE of 0.21 m, but a poor correlation coefficient of only 0.52. Considering all 344 surface areas for the validation with in-situ water levels leads to an increase of the correlation coefficient from 0.52 to 0.81, respectively 0.82. However, the RMSE values decreased from 0.21 m to 0.25 m–0.27 m which is twice as high as from satellite altimetry only (RMSE: 0.13 m). This result clearly shows the problem if input data for hypsometric curves are noisy and not accurate enough. The extrapolation is incorrect for lower water levels (e.g., 2006, 2011), but also within the confidence range, differences between reconstructed water levels and in-situ are visible.

#### Estimation of Bathymetry

Figure 20 shows the resulting bathymetry of Palestine Lake based on 520 surface areas and corresponding water levels. The bathymetry based on all land-water masks covers heights between 102.40 m and 105.19 m. This example clearly shows that the bathymetry can only be estimated reliably for flatter areas in the North and West of Palestine lake. Everywhere else at the shores, the minimum bathymetry (gray) is reached very fast which indicates steep lake shores. This clearly shows the

lack of input data for lower water levels, respectively surface areas for a more accurate estimation of the bathymetry.

**Figure 19.** Comparison of derived water levels from surface areas of three hypsometric curves calculated for Palestine Lake.

**Figure 20.** Bathymetry of Palestine Lake derived from land-water masks in combination with water levels derived from the hypsometric model. Additionally, the used altimeter tracks are highlighted as dotted line.
