Optimization of the Coupled mHM-Land Surface Tempera-

For the calibration of mHM against the satellite retrieved land surface temper-ature the aforementioned T_s module is coupled to mHM. Thus, seven additional parameters are added to mHM and are included in the automated optimization of

the global parameters. For simplicity, the coupled mHM-land surface temperature model is denoted as mHM in the following. Furthermore, all parameters, including the global parameters of mHM (44 parameters) and the T_s module (7 parameters) are herein referred as mHM parameters. Hence, the coupled model has 51global parameters.

The coupled model will be calibrated against land surface temperature or discharge or a combination of both. The performance regarding the two model outputs, i.e., discharge and land surface temperature, is estimated using a weighted objective function. In general the objective function Φ is estimated by

Φ =

Different distance measures φ_i are considered, because discharge Q is only depen-dent on time, whereasT_s is a spatio-temporal variable. Following Duckstein (1984) the exponentpequal to 6 is included to the objective function to assure numerical stability and assure for a compromise solution. The different error measures φ for discharge and land surface temperature are described in the following.

Error Measure for Discharge Q The distance measure which is applied to the discharge estimations is the Nash-Sutcliffe Efficiency (NSE) (Nash and Sutcliffe, 1970). To get satisfying estimations of highflows as well as of lowflows, the NSE is determined for daily discharge (Equation 4.14) and the logarithm of daily discharge (Equation 4.15), respectively. [d] and Q is the mean discharge of all time steps T of the observation.

For the optimization against discharge aloneφ₁andφ₂are considered in the objec-tive function. The weights are chosen to be equal for both criteria (w₁ =w₂ = 0.5).

Error Measure for Land Surface Temperature T_s To assess the model perfor-mance regarding the simulations of T_s an error measure for quantifying the differ-ences between modeled and satellite retrieved T_s has to be found. The satellite

4.4. Methodology retrievals ofT_s have an inherent bias of approximately 2 K to 3 K on the temporal resolution of one day compared to ground measurements (see section 4.3).

A bias correction of the satellite retrievedTs was intended to be avoided, because an additional model accounting for the bias would have to be included into the coupled model. Such a bias model would need to be parameterized. To keep the number of parameters as low as possible the implementation of a bias correction model was avoided.

Hence, the application of common error measures like the mean squared error or any error measure which is sensitive to a bias are not considered. It is assumed that the patterns delivered by the satellite measurements are trustworthy. Thus, an objective comparing the patterns of the satellite retrieved and estimated land surface temperature qualitatively is targeted.

In hydrology common criteria used to determine pattern similarity are usually in-corporating measures accounting for quantitative differences like the mean squared error (Hagen-Zanker, 2006; Wealands et al., 2005; Cloke and Pappenberger, 2008) and thus are inapplicable. A bias resistant, local and non-parametric measure denoted as pattern similarity (P) is developed. Mathematically, the pattern simi-larity criterion can be expressed as

φ₃ = 1

wherei and j are the elements of the spatial domain Ω, which in total consists of N cells, T is the number of time steps, P_ij(t) is the pattern similarity criteria at cell (i, j) at a particular time step t, T_s,ij^(k) is the land surface temperature of the k^th neighbor of the center cell (i, j), and Ts,ij is the land surface temperature of the center cell itself. The Pattern Similarity criterion is normalized with 8, the number of neighbors of the center cell (i, j). The notation without hat (T_s) is used for the satellite derived land surface temperature, while the model simulated temperature is denoted with a hat (Tb_s). The sgn operation determines the sign of the argument a as

sgn(a) =

(1 if a >0

−1 if a≤0. (4.17)

An example for the pattern similarity criterion is depicted in Fig. 4.2.

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Figure 4.2: Schematic description of the pattern similarity criterion according to Equation 4.16b. In the upper left row an example Pattern A with the center pixel having the value 10 is illustrated (e.g., satellite retrieved T_s). Next to it, on its right, the sign of the comparison between the center pixel with its neighboring pixels is shown. If the respective neighboring pixel is larger than the center pixel (green arrow) the value 1 is assigned to this pixel (e.g., 5 pixels in Pattern A), otherwise (red arrow) the value -1 is assigned to them (e.g., 3 pixels in Pattern A).

This analysis is repeated for a pattern B (e.g., simulation of Tb_s), as depicted in the lower row. The results of both comparisons are multi-plied and increased by 1. Thus, the dissimilar pixels between pattern A and pattern B become 0, while similar become 2. The elements of the resulting matrix are summed up and divided by twice of the number of neighbors (e.g., 8). For the given example, the pattern similarity criterion is 0.75, meaning that three quarter of the neighbors showed the same relation to its center value.

The criterion is based on a 3×3 pixel search raster. Its center cell is subtracted from the eight neighboring cells. The difference becomes negative and the sgn = −1, if the value of the center cell is greater than the neighbor. In the opposite case the sign becomes positive (sgn = 1). This procedure is applied to both fields under comparison, i.e., the satellite retrieved T_s and the modeled land surface temperature Tb_s. The two resulting 3×3 signum matrices are multiplied with each other. The resulting matrix has a negative entry (-1) where the elements of both factors had different signs and a positive entry (+1) where the factors had the same sign.

Thus, a negative entry appears when the modeled grid cell shows a different ten-dency compared to the measured land surface temperature. The entry is positive, when the grid cell tendencies are in correspondence. In order to avoid the results to be canceled out when summed up, the eight single results are increased by one.

4.4. Methodology Hence, for full correspondence the sum of the elements of the search raster yields 16 while it is zero for full disagreement. Finally, the sum is scaled between zero and one. Hence, a P_ij(t) of 1 means full agreement of patterns, i.e., no dissimilarity.

The scaling assures comparability to other error measures like the Nash Sutcliffe Efficiency or the correlation coefficient. The pattern similarity of 0 does not only correspond to full dissimilarity, but means that the two patterns are inverse to each other. A P_ij(t) of 0.5 marks randomly diverging patterns.

The 3×3 local search window is applied to every cell (i, j) within the domain Ω and all time steps t of the patterns under comparison. The overall pattern similarity is then calculated as the mean of the single values (see Equation (4.16a)).

Numerical tests showed that a combination of the pattern similarity criteria with another bias resistant criteria, i.e., the Pearson correlation coefficient, result in the best model performances regarding discharge and land surface temperature. Thus, a forth error measure is considered in the objective function

φ₄ =ρ= Cov(T_s,Tb_s) σ_Tsσ_Tb

s

(4.18) whereρdenotes the Pearson correlation coefficient, Cov(T_s,Tb_s) the covariance be-tween the spatio-temporal fields ofT_sandTb_s, andσ_Ts andσ

cTs denotes the standard deviation of the satellite retrieved Ts and modeled land surface temperature Tbs, respectively.

The criteria for pattern similarity φ3 and φ4 are equally weighted (w3 = w4 = 0.5) in the objective function if applied for calibration against the land surface temperature (only φ₃ and φ₄ are considered in the objective function).

The calibration with respect to a combination of land surface temperature and discharge data are conducted using all four error measures φ₁, φ₂, φ₃, and φ₄ as objective. The weights are defined asw₁ =w₂ = ¹₃ and w₃ =w₄ = ¹₆. The higher weighting of discharge error measures is chosen to ensure the partitioning of water to the different fluxes and states of the hydrologic cycle. In comparison with other weighting schemes this setup has proven to perform best.

To avoid the dominance of any objective φ_i (i= 1, ..4) the objectives are normal-ized by their potential ranges:

φ_i = φi−φ^min_i

φ^max_i −φ^min_i (4.19)

where min and max denote the upper and the lower bound of the particular objectivei, respectively. φ^min_i andφ^max_i are determined based on 55 000 simulations in two of the catchments under investigation, i.e., Ems and Neckar, using random

parameters. To ensure sampling over the entire parameter domain a stratified sampling strategy was applied to generate 55 000 parameter sets (Morris, 1991).