# Spatial Pattern Oriented Multicriteria Sensitivity Analysis of a Distributed Hydrologic Model

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## Abstract

**:**

## 1. Introduction

## 2. Materials

^{2}covered by mostly sandy soils (Figure 1) and dominated by agriculture (71%) and forest (15%) [21]. The maximum altitude in the basin is 130 m and annually averaged precipitation is around 1000 mm [31]. The monthly mean temperatures vary between 2 and 17 degrees Celsius whereas the annual mean streamflow is around 475 mm [32].

#### 2.1. Satellite-Based Data

#### 2.1.1. Leaf Area Index (LAI)

#### 2.1.2. Actual Evapotranspiration (TSEB)

#### 2.2. Hydrologic Model

_{ref}scaling function introduced in an earlier study [2] has flexibility to change the spatial pattern of AET during the sensitivity analysis as a function of vegetation as compared to the uniform or aspect-driven PET correction factor originally implemented in mHM (Table 2). The study by Samaniego et al. [30] is the key reference describing model formulation and parameter description.

_{ref}, and T

_{avg}, were gridded observational data from the Danish Meteorological Institute resampled from a native resolution of 10 and 20 km, respectively [2]. Readers are referred to Table 1 in Demirel et al. [2] for the complete list of model inputs and data sources.

## 3. Methods

#### 3.1. Objective Functions Focusing on Spatial Patterns

^{2}. The two units are closely related but vary in range; therefore, applying bias insensitive metrics is inevitable.

#### 3.1.1. Goodman and Kruskal’s Lambda

_{ij}is the grid numbers for the class i in first map (A) and to class j in the second map (B); ${c}_{i+}$ is the grid numbers contained in category i in map A; ${c}_{+j}$ is the grid numbers contained in category j in map B; $ma{x}_{i}\left({c}_{i+}\right)$ is the grid numbers in the modal class of map A, i.e., the class with largest number of grids; and $ma{x}_{j}\left({c}_{ij}\right)$ is the number of classes in map B with a given class of map A.

#### 3.1.2. Theil’s Uncertainty Coefficient

#### 3.1.3. Cramér’s V

#### 3.1.4. Mapcurves

#### 3.1.5. Empirical Orthogonal Functions

_{i}represents the covariation contribution of the ith EOF. In our study, we focused on the overall AET pattern performance and thus we averaged S

_{EOF}from the individual months of the growing season into a single overall skill score.

#### 3.1.6. Fractions Skill Score

_{ref}). The MSE is based on all grids (N

_{xy}) that define the catchment area with dimension N

_{x}and N

_{y}. For a certain threshold, the FSS at scale n is given by

#### 3.2. Latin Hypercube Sampling One-Factor-at-a-Time Sensitivity Analysis

## 4. Results

#### 4.1. Exploration of Spatial Metrics Characteristics

^{2}), i.e., all perturbed differently, with a reference LST map using the spatial metrics applied in this study. The details about the applied perturbation strategies can be found in Table 1 of Koch et al. [23]. In that study, Koch et al. [23] conducted an online-based survey with the aim of using the well-trained human perception to rank the 12 synthetic LST maps in terms of their similarity to the reference map. The obtained results were subsequently used to benchmark a set of spatial performance metrics. The same procedure was incorporated in this study to get a better understanding of the metrics selected for this study. We included the survey results in our study and assessed the coefficient of determination R

^{2}between the human perception and the spatial metrics. This helped to differentiate between metrics that contained redundant information and those with unique information content. Figure 2 shows two distinct examples, i.e., one noisy perturbation and one slightly similar map to the reference map, to better explain the results presented in Table 4, which summarizes the spatial scores for the 12 maps sorted based on the survey similarity index (last column).

^{2}values in Table 5, 72.2% of the variance in the human perception is explained by the EOF analysis. The Pearson correlation coefficient (PCC), V, and FSS metrics also performed well in discriminating spatial maps with respect to the human perception as benchmark. However, the U metric explained the lowest variance (40.9%), which indicates that it should not be included in the model calibration because we trust the well-trained human perception as a reference. Moreover, the spatial metrics, i.e., λ, U, V, and MC, are highly correlated (italic fonts in Table 5). All are based on transforming the data into a three category system, which results in redundancy between the four given metrics. This shows that not all of them are required for model calibration. However, we will still evaluate the sensitivity results based on all given spatial metrics in the following sections.

#### 4.2. Latin Hypercube Sampling One-Factor-at-a-Time Sensitivity Analysis

_{ref}-a parameter seems to be the second most important parameter affecting streamflow dynamics of the Skjern River model (see PB at Figure 3 and Figure 4), whereas another PTF parameter is ranked as the second most important parameter affecting KGE (Table A1).

_{ref}-a). Second, we can recognize different patterns on the maps such as land cover patterns (see Figure 1) from root fraction maps (especially the one for pervious areas), LAI patterns from ET

_{ref}-c map, and soil patterns from pedo-transfer function maps (e.g., ptfksconst). The geoparam 2, 3, and 4 parameters are also identified as sensitive influencing streamflow dynamics (see KGE at Figure 3). However, their maps are not shown in Figure 4 as they are similar to the map for geoparam 1 (uniform effect). Further, the map for the ptflowdb parameter is completely dark blue, showing that it has no effect on the simulated spatial pattern of AET.

#### 4.3. Random Parameter Sets Based on the 17 Sensitive Parameters Evaluated against NSE and FSS

## 5. Discussion

#### Utility of the Multicriteria Spatial Sensitivity Analysis

## 6. Conclusions

- Based on the detailed analysis of spatial metrics, the EOF, FSS, and Cramér’s V are found to be relevant (nonredundant) pairs for spatial comparison of categorical maps. Further, the PCC metric can provide an easy understanding of map association, although it can be very sensitive to extreme values.
- Based on the results from sensitivity analysis, vegetation and soil parametrization mainly control the spatial pattern of the actual evapotranspiration in the mHM model for this study area.
- Besides, the interception, recharge, and geological parameters are also important for changing streamflow dynamics. Their effect on spatial actual evapotranspiration pattern is substantial but uniform over the basin. For interception, the lacking effect on the spatial pattern of AET is due to the exclusion of rainy days in the spatial pattern evaluation.
- More than half of the 47 parameters included in this study have either little or no effect on simulated spatial patterns, i.e., noninformative parameters, in the Skjern Basin with the chosen setup. In total, only 17 of 47 mHM parameters were selected for a subsequent spatial calibration study.
- The sensitivity maps are consistent with parameter types, as they reflect land cover, LAI, and soil maps of the Skjern Basin.
- Combining NSE with a spatial metric strengthens the physical meaningfulness and robustness of selecting behavioral models.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

ID | Parameter | Description | Normalized Sensitivity | |||
---|---|---|---|---|---|---|

LHS-OAT Random | LHS-OAT Behavioral | |||||

KGE | FSS | KGE | FSS | |||

1 | ptfhigconst | Constant in pedo-transfer function (ptf) for soils with sand content higher than 66.5% | 0.394 | 0.207 | 0.367 | 0.19 |

2 | ptfhigdb | Coefficient for bulk density in pedo-transfer function for soils with sand content higher than 66.5% | 0.261 | 0.17 | 0.243 | 0.151 |

3 | ptfksconst | Constant in pedo-transfer function for hydraulic conductivity of soils with sand content higher than 66.5% | 0.366 | 0.003 | 0.765 | 0.005 |

4 | ptfkssand | Coefficient for sand content in pedo-transfer function for hydraulic conductivity | 0.469 | 0.005 | 1 | 0.006 |

5 | ptfkscurvslp | Exponent in pedo-transfer function for hydraulic conductivity to adjust slope of curve | 0.005 | 0.002 | 0.007 | 0.004 |

6 | rotfrcoffore | Root fraction for forested areas | 1 | 1 | 0.746 | 1 |

7 | rotfrcofperv | Root fraction for pervious areas | 0.03 | 0.008 | 0.024 | 0.01 |

8 | infshapef | Infiltration (inf) shape factor | 0.051 | 0.008 | 0.06 | 0.011 |

9 | ET_{ref}-a | Intercept | 0.383 | 0.052 | 0.388 | 0.056 |

10 | ET_{ref}-b | Base coefficient | 0.165 | 0.021 | 0.176 | 0.022 |

11 | ET_{ref}-c | Exponent coefficient | 0.046 | 0.008 | 0.047 | 0.011 |

12 | slwintreceks | Slow (slw) interception | 0.113 | 0 | 0.236 | 0 |

13 | rechargcoef | Recharge coefficient (coef) | 0.14 | 0 | 0.309 | 0 |

14 | geoparam1 | Parameter for first geological formation | 0.13 | 0 | 0.081 | 0 |

15 | geoparam2 | Parameter for second geological formation | 0.045 | 0 | 0.032 | 0 |

16 | geoparam3 | Parameter for third geological formation | 0.175 | 0 | 0.105 | 0 |

17 | geoparam4 | Parameter for fourth geological formation | 0.038 | 0 | 0.025 | 0 |

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**Figure 2.**Two synthetic land surface temperature (LST) maps for the Ahlergaarde sub-basin of Skjern to compare the spatial metrics. Map 9 (

**left**) was generated using a noisy perturbation, while map 6 (

**right**) was more similar to the reference map (

**middle**).

**Figure 3.**Average accumulated relative parameter sensitivities over 100 behavioral (NSE > 0.5) initial parameter sets and one-at-a-time parameter perturbations. The different colors indicate different model processes. The red coded parameters refer to soil moisture, yellow to evapotranspiration, ice blue to interflow, light blue to percolation, and dark blue to geology.

**Figure 4.**Average of 100 sensitivity maps (entire Skjern Basin) based on the difference between 100 behavioral (NSE > 0.5) initial parameter sets and one-at-a-time parameter perturbations (100 runs per parameter).

**Figure 5.**Scatter plot of 329 (NSE > 0.0) model runs from a total of 1700 random parameter sets as a function of spatial fit (

**a**) FSS (

**b**) EOF between remote sensing actual evapotranspiration and simulated actual evapotranspiration from mHM. Green box in both figures shows the new behavioral regions when spatial and temporal thresholds are applied together.

Variable | Description | Period | Spatial Resolution | Remark | Source |
---|---|---|---|---|---|

LAI | Fully distributed 8-day time varying LAI dataset | 1990–2014 | 1 km | 8 day to daily | MODIS and Mendiguren et al. [33] |

AET | Actual evapotranspiration | 1990–2014 | 1 km | daily | MODIS, TSEB |

**Table 2.**Setup of newly introduced parameters for dynamic ET

_{ref}scaling function for sensitivity analysis: parameter types, initial values, and range.

Parameter | Unit | Description | Initial Value ** | Lower Bound | Upper Bound |
---|---|---|---|---|---|

ET_{ref}-a | - | Intercept | 0.95 | 0.5 | 1.2 |

ET_{ref}-b | - | Base Coefficient | 0.2 | 0 | 1 |

ET_{ref}-c | - | Exponent Coefficient | −0.7 | −2 | 0 |

**Table 3.**Overview of the 10 metrics which were used in the sensitivity analysis. The first three metrics were regarding time series of streamflow, while the latter seven were used to evaluate spatial patterns of AET.

Description | Best Value | Abbreviation | Group | Reference |
---|---|---|---|---|

Nash–Sutcliffe Efficiency | 1.0 | NSE | Streamflow | [47] |

Kling–Gupta Efficiency | 1.0 | KGE | Streamflow | [48] |

Percent Bias | 0.0 | PB | Streamflow | |

Goodman and Kruskal’s Lambda | 1.0 | λ | Spatial pattern | [49] |

Theil’s Uncertainty coefficient | 1.0 | U | Spatial pattern | [50] |

Cramér’s V | 1.0 | V | Spatial pattern | [51] |

Map Curves | 1.0 | MC | Spatial pattern | [52] |

Empirical Orthogonal Function | 0.0 | EOF | Spatial pattern | [23] |

Fraction Skill Score | 1.0 | FSS | Spatial pattern | [27] |

Pearson Correlation Coefficient | 1.0 | PCC | Spatial pattern | [53] |

**Table 4.**Comparison of 12 perturbed maps [23] based on spatial metrics. The first seven columns present the metrics used in this study, while the last column gives the survey similarity reported in Koch et al. [23]. Map 1 has the highest similarity, which means that it is the most similar map to the reference, while map 7 is the least similar.

MAP_ID | λ | U | V | MC | EOF * | FSS | PCC | Survey Similarity |
---|---|---|---|---|---|---|---|---|

1 | 0.68 | 0.59 | 0.81 | 0.76 | 0.02 | 0.96 | 0.95 | 0.86 |

6 | 0.49 | 0.50 | 0.73 | 0.72 | 0.05 | 0.97 | 0.86 | 0.75 |

8 | 0.28 | 0.30 | 0.57 | 0.53 | 0.06 | 0.96 | 0.86 | 0.64 |

12 | 0.39 | 0.37 | 0.63 | 0.59 | 0.06 | 0.89 | 0.86 | 0.61 |

5 | 0.50 | 0.44 | 0.70 | 0.65 | 0.05 | 0.91 | 0.87 | 0.59 |

2 | 0.20 | 0.26 | 0.52 | 0.50 | 0.08 | 0.89 | 0.79 | 0.59 |

10 | 1.0 | 1.0 | 1.0 | 1.0 | 0.0 | 1.0 | 1.0 | 0.57 |

11 | 0.00 | 0.04 | 0.20 | 0.32 | 0.21 | 0.87 | 0.37 | 0.42 |

4 | 0.00 | 0.07 | 0.26 | 0.35 | 0.25 | 0.77 | 0.27 | 0.36 |

3 | 0.00 | 0.17 | 0.39 | 0.40 | 0.15 | 0.93 | 0.70 | 0.29 |

9 | 0.20 | 0.21 | 0.48 | 0.47 | 0.16 | 0.87 | 0.48 | 0.23 |

7 | 0.00 | 0.00 | 0.04 | 0.29 | 0.28 | 0.73 | −0.01 | 0.10 |

**Table 5.**Coefficient of determination (R

^{2}) between spatial metrics and survey similarity. Bold values mark metrics with highest ability to reproduce survey similarity. Italic values highlight spatial metrics which are highly correlated.

R^{2} Score | λ | U | V | MC | EOF | FSS | PCC | Survey Similarity |
---|---|---|---|---|---|---|---|---|

λ | 1 | 0.97 | 0.88 | 0.97 | 0.71 | 0.51 | 0.59 | 0.46 |

U | 1 | 0.90 | 0.99 | 0.72 | 0.59 | 0.63 | 0.41 | |

V | 1 | 0.93 | 0.90 | 0.72 | 0.85 | 0.59 | ||

MC | 1 | 0.77 | 0.61 | 0.67 | 0.49 | |||

EOF | 1 | 0.79 | 0.96 | 0.72 | ||||

FSS | 1 | 0.84 | 0.52 | |||||

PCC | 1 | 0.69 | ||||||

Survey Similarity | 1 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Demirel, M.C.; Koch, J.; Mendiguren, G.; Stisen, S.
Spatial Pattern Oriented Multicriteria Sensitivity Analysis of a Distributed Hydrologic Model. *Water* **2018**, *10*, 1188.
https://doi.org/10.3390/w10091188

**AMA Style**

Demirel MC, Koch J, Mendiguren G, Stisen S.
Spatial Pattern Oriented Multicriteria Sensitivity Analysis of a Distributed Hydrologic Model. *Water*. 2018; 10(9):1188.
https://doi.org/10.3390/w10091188

**Chicago/Turabian Style**

Demirel, Mehmet Cüneyd, Julian Koch, Gorka Mendiguren, and Simon Stisen.
2018. "Spatial Pattern Oriented Multicriteria Sensitivity Analysis of a Distributed Hydrologic Model" *Water* 10, no. 9: 1188.
https://doi.org/10.3390/w10091188