*4.3. Uncertainty Analysis*

The storage uncertainty associated with the *A*-*H* relationship is primarily attributed to the use of partial bathymetry information to represent the *A-H* relationship for the entire reservoir. Because the DEM dataset only represents the part of the bathymetry that was above the water surface when the SRTM measurements were collected, it assumes that the unmeasured part below the water surface shares the same *A*-*H* relationship. To quantify the uncertainty associated with this assumption, we compared the storage estimations from three different scenarios (Figure 7). In each case, a simplified cross-sectional view of the reservoir was used—with the water surface area collected by the SRTM (in 2000) indicated as *A*1, and the area of the reservoir bottom indicated as *A*2. Under all scenarios, the storage volume below the DEM water surface was preserved. The first scenario (Figure 7a) follows the algorithm used in this study, which assumes that the *A*-*H* relationship remains the same across the entire profile. The second scenario (Figure 7b) assumes that the area of the reservoir bottom is zero, and thus the *A*-*H* relationship of the unknown part below the water surface has the smallest possible slope of *k*min. The third scenario (Figure 7c) assumes that the minimum area from the MODIS estimations is the area of the reservoir bottom, and thus the *A*-*H* relationship of the unknown part below the water surface has the largest possible slope of *k*max.

**Figure 7.** Illustration of the process for quantifying the uncertainty associated with the extrapolation of the *A*-*H* relationship: (**a**) an example of a simplified reservoir cross section, with a bottom area of *A2* identified by assuming the unmeasured portion shares the same *A-H* relationship, (**b**) the reservoir cross section by assuming the bottom area *A*2 is 0, (**c**) the reservoir cross section by assuming the reservoir bottom area *A*2 equals *A*RS\_min.

Using the slope of the upper portion (i.e., *k*) as estimated from the DEM, the reservoir storage value when the DEM was constructed (i.e., *V*2) can be calculated using Equation (6):

$$V\_2 = V\_\varepsilon - V\_1 = V\_\varepsilon - (A\_\varepsilon + A\_1)(A\_\varepsilon - A\_1)k/2\tag{6}$$

As shown in Figure 7b, the minimum value of *A*2—which is 0—can be used to estimate *k*min via Equation (7):

$$k\_{\rm min} = \frac{2V\_2}{(A\_1 + A\_2)(A\_1 - A\_2)} = \frac{2V\_2}{A\_1^2} \tag{7}$$

Similarly, the maximum value of *A*2—which is equal to the minimum water surface area from MODIS during the research period (Figure 7c)—can be used to estimate *k*max after Equation (8):

$$k\_{\max} = \frac{2V\_2}{(A\_1 + A\_2)(A\_1 - A\_2)} = \frac{2V\_2}{\left(A\_1 + A\_{\text{RS}}^{\min}\right)(A\_1 - A\_{\text{RS}}^{\min})} \tag{8}$$

Thus, for any MODIS remotely sensed area (*ARS*) that is less than *A*1, the storage can range between a minimum possible value of *<sup>V</sup>*RSmin (Equation (9)) and a maximum value of *VRS*max (Equation (10)):

$$V\_{RS}^{\min} = V\_2 - (A\_1 + A\_{RS})(A\_1 - A\_{RS})k\_{\min}/2\tag{9}$$

$$V\_{RS}^{\text{max}} = V\_2 - (A\_1 + A\_{RS})(A\_1 - A\_{RS})k\_{\text{max}}/2\tag{10}$$

Therefore, the uncertainties associated with the constant slope assumption can be represented by the difference between the two storage estimates described below using Equation (11):

$$
\Delta V = (A\_1 + A\_{RS})(A\_1 - A\_{RS})(k\_{\text{max}} - k\_{\text{min}})/2 \tag{11}
$$

The uncertainties associated with this source are illustrated in Figure 8. For all 28 reservoirs, the absolute uncertainty due to the unmeasured *A*-*H* relationship ranged from 0 km<sup>3</sup> to 0.54 km3, with an average of 0.23 km3. Among these reservoirs, the Rihand reservoir had the largest absolute uncertainty (0.54 km3), primarily because this large reservoir was at a relatively high level when the DEM data were collected. The surface area of the Rihand reservoir—as measured by DEM—was 388.96 km2, whereas its surface area at full capacity is 485 km2. Considering all of the reservoirs, we found a significant increasing trend of the absolute uncertainty as the reservoir capacity increased. For every 1 km<sup>3</sup> increase in reservoir capacity, the uncertainty increased by 0.034 km<sup>3</sup> (*p* < 0.01). The averaged relative uncertainty caused by the unmeasured *A*-*H* relationship was 4.68%. However, we observed no significant relationship between the relative uncertainty and the capacity.

The uncertainties from the area estimation algorithm were quantified thoroughly by Zhang et al. [2014]. From this source, the absolute uncertainties were also found to be highly correlated with the storage at capacity, where the absolute uncertainties had an average value of 3.90%. This is a similar uncertainty range but lightly larger than the unmeasured *A-H* relationship.

The vertical error of SRTM DEM could be another source of uncertainty. This was not analyzed in this study because the storage calculation (in this study) was based on the slope of the *A-H* relationship and area estimations, rather than using the absolute elevation value from the SRTM DEM directly. Since the slope of the A-H relationship is determined by the elevation difference of reservoir pixels, the absolute vertical DEM error can be offset during the process, reducing its influence on the storage estimation.

**Figure 8.** Uncertainty analysis results: (**a**) Absolute uncertainty; (**b**) relative uncertainty due to the unmeasured *A*-*H* relationship of SRTM DEM.
