*4.2. Estimation of Hypsometric Curve*

Each lake and reservoir has a fixed area-height relationship that depends on its bathymetry and can be described by a hypsometric curve. In hydrological applications, hypsometric curves are used to assign water levels and surface areas of inland water bodies. For this purpose, mathematical functions are adapted to describe the relationship between water levels and surface areas. Due to the bathymetry and the surrounding topography, the adjusted function must always be monotonically increasing. Former studies used linear or polynomial functions to fit the hypsometric curve [27,28,30,31]. However, these applied functions do not capture all variations of the area-height relationship caused by the bathymetry.

In 1952, Strahler developed a percentage hypsometric curve that relates the horizontal cross-sectional area of a drainage basin to the relative elevation above basin mouth, allowing direct comparisons between different basins [33]. In his study, hypsometric curves are used to analyze the erosion of topography within basins. To fit the most natural hypsometry curves, a function with three variables (*a*,*d*,*z*) is used, which is shown in the Equation (1). In this function, *x* gives the normalized basin area and *y* the normalized basin height.

$$y = \left[\frac{d-x}{x} \cdot \frac{a}{d-a}\right]^z \tag{1}$$

The original formula of Strahler (Equation (1)) is based on two constants *a* and *d* which fulfill the condition *d* > *a*. The general form of the function is defined by the exponent *z* greater than 0. All functions intersect the points *A* and *B*. Between the intersection points *A*(*a*, 1) and *B*(*d*, 0), the values on the *y*-axis are always limited between 0 and 1, while the values on the *x*-axis are limited between *a* and *d*. Figure 5(left) shows examples of hypsometric curves based on the original Strahler approach for different *z* and constant *a* (0.05) and *d* (1.00).

**Figure 5.** Original Strahler approach [33] (left) and modified Strahler approach based on water levels and surface areas (right).

In general, it can be assumed that the bathymetry of lakes or reservoirs has similar characteristics as a drainage basin. Therefore, we have modified the original Strahler approach shown in Equation (1) to estimate hypsometic curves for lakes and reservoirs.

$$y = \left[\frac{(\mathbf{x}\_{\min} - \mathbf{x})}{(\mathbf{x}\_{\min} - \mathbf{x}\_{ip})} \cdot \frac{(\mathbf{x}\_{\max} - \mathbf{x}\_{ip})}{(\mathbf{x}\_{\max} - \mathbf{x})}\right]^2 \cdot y\_{scale} + y\_{min} \tag{2}$$

The original Strahler approach can easily be used to analyze entire drainage basins where the minima and maxima of area and height are known [33]. Then, input data is normalized to fulfill Equation (1). However, in our study, the minima and maxima of water levels from satellite altimetry and surface areas from optical imagery of a lake or reservoir are unknown. The main reason for this is that the water body under investigation was never empty during the measurement period. Also extreme events like floods or droughts can be missed due to the lower temporal resolution of both data sets. This leads to an unknown bathymetry below lowest observed water level or the smallest surface area. Therefore, only a section of the hypsometric curve is known for the estimation, which requires a modification of the original Strahler approach.

In Equation (2), six parameters of the resulting hypsometry curve are adjusted. *xmin* defines the minimum surface area and *xmax* the maximum surface area of the hypsometry curve. The minimum water level is defined as *ymin* and the variations of water level is defined as *yscale*. The exponent *z* describes the shape of the hypsometric curve. The intersection points *A*(*xip*, *ymin* + *yscale*) and *B*(*xmin*, *ymin*) can be expressed by five parameters of the modified Strahler approach. Between *A* and *B*, the function is always monotonically increasing, which is directly related to the bathymetry. In general, the surface areas can vary between 0 km<sup>2</sup> if the lake is empty and the maximum surface area if the lake is filled. However, water levels refer to a specific reference (e.g., sea level) that does not reflect the bottom of the lake or reservoir. Since the depths of inland waters are not known, an assumption must be applied for an as accurate as possible *ymin*. In [34], the area-volume-depth relationship of lakes was investigated on the basis of the Hurst coefficients. In this study, we use the area-depth relationship to define rough limits for the minimum water level *ymin*. In Figure 5(right), examples of hypsometric curves based on the new modified Strahler approach are shown for different *z* and constant *ymin*, *yscale*, *xmin*, *xip* and *xmax*.

Since water levels and surface areas are usually not captured on the same day, both data must be linked in a suitable way. Within a few days, the water level or surface areas can change significantly (e.g., flood events). In order to maximize the number of pairs and to minimize possible errors, only water levels and surface areas whose difference between the two measured data is less than 10 days are assigned. Finally, the resulting pairs are used to fit the function of the modified Strahler approach in Equation (2). The hypsometric curve can be used to calculate water levels from surface areas or vice versa. Detailed examples and discussion of hypsometric curves for selected inland water bodies are shown in Section 5.1.

#### *4.3. Estimation of Water Levels from Surface Areas using Hypsometry*

Estimating lake bathymetry requires water levels for each surface area and land-water masks. This information is necessary to estimate bathymetry between the smallest and largest surface area. Therefore, we use the calculated hypsometric curve to derive the water levels of all observed surface areas. Based on satellite altimetry, water level time series for larger lakes can be derived with an accuracy of less than 10 cm beginning in 1992. For smaller lakes and reservoirs the accuracy normally yields between 10 cm and 40 cm. The temporal resolution of the altimetry-derived time series is limited to 10–35 days for smaller lakes with only one satellite overflight track. The water levels derived from surface areas can also be used to densify the altimetry-based water level time series and extend them to preceding years since 1984. Moreover, for smaller lakes and reservoirs, surface areas can be derived more accurately than water levels due to the measurement technique. The advantage of using the hypsometric curve to derive water levels from surface areas is that water level errors from altimetry are minimized. However, additional errors due to extrapolation or time-dependent changes of bathymetry may occur.

#### *4.4. Estimation of Bathymetry*

To estimate volume variations, the next step is to calculate a bathymetry of the lakes or reservoirs. The calculation of the bathymetry can be done in different ways. Simple assumptions such as regular or pyramidal shapes of the bathymetry can be used. This might be accurate enough for lakes with more or less regular shapes, however, for reservoirs with large volume changes, this assumption may lead to large errors in volume change estimation. Therefore, we estimate a bathymetry between the minimum and maximum observed surface area based on all available high-resolution land-water masks and corresponding water levels derived from the hypsometric curve. The resulting bathymetry finally allows us to compute volume variations referring to the minimum observed surface area, since the underlying bathymetry is unknown.

The schematic processing strategy for the bathymetry is shown in Figure 6. In the first step, all land-water masks are stacked according to water levels which are derived from the hypsometric curve in descending order. Then, each pixel column is analyzed separately to calculate the resulting height of the bathymetry. Therefore, a median filter of size five is applied to each pixel column in the direction of decreasing water levels. This is shown as an example for two pixels. In the first example (green), all pixels in column are correctly classified. In the second example (blue), not all pixels in the

column are correctly classified because there are several changes between land and water. Using the median filter reduces the influence of corrupted land-water pixels on the final height. As long as the result of the median filter is water, the current height is set for the pixel. If the result of the median filter is land, the filtering is stopped for the current pixel column. Finally, this processing step leads to a bathymetry between the minimum and maximum observed surface area with a spatial resolution of 10 m.

**Figure 6.** Processing strategy for the estimation of the bathymetry.
