# Application of Sensitivity Analysis for Process Model Calibration of Natural Hazards

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Site Description

^{2}alpine catchment situated between 563 m and 2039 m above sea level. The underlying bedrock on location is Lower Cretaceous limestone. The combination of marl and siliceous limestone found in the vicinity and contributions of weathered schist provided sources of sediment for the torrent processes.

_{max}) was estimated to be between 140 and 160 m

^{3}/s. Available sediment was entrained and approximately 72,000 m

^{3}of total bulk volume was subsequently propagated along the main channel into the settlement, deposited on the alluvial fan, and subsequently flowed into Lake Brienz. Flow velocities between 6 and 10 m/s at the Glyssibrücke was estimated with the superelevation approach. This is based on the forced vortex equation, where the difference in surface elevation of a debris flow is determined as it travels within a known channel curvature [73]. Debris flow materials were described by local experts to be relatively homogeneous, resulting in a viscoplastic mud matrix composed primarily of fine fractions of silt and sand. In particular, the mud matrix of the 2005 event is estimated to have been comprised of 30–70% water; 5–8% of the total sediment volume is comprised of clay. The channelized mud matrix was capable of entraining coarse, unconsolidated materials (e.g., vehicles, large woody debris) and transported boulders between 3 and 5 m in diameter to the bottom of the alluvial fan. For this reason, the 2005 event is referred to as a debris flow.

#### 2.2. Methods

#### 2.3. Data Sources

#### 2.3.1. Initial Conditions and Fixed Parameters

^{−}⅓·s), channels and streets (0.01 m

^{−}⅓·s), and sparsely vegetated settlement areas (0.08 m

^{−}⅓·s) in this study [80].

#### 2.3.2. Input Factors for Calibration

_{v}) input factor is calibrated for, which is the ratio between solid materials and water. Within the solid component, different grain size distributions can be defined. This reflects the importance of grain size on energy dissipation in granular flows, which translates to different effects on geophysical flow mobility [81].

_{v}and discharge data. The amount of precipitation that initiated the debris flow could not be determined from rain or discharge records most proximal to the study site. Consequently, an expert-based estimate of maximum discharge from the main channel was used to produce a series of mud hydrographs based on variations of different volumetric sediment concentrations.

#### 2.3.3. Calibration of Rheological Values

_{v}, where V

_{s}and V

_{w}represent the respective sediment and pore water volumes in a given mixture [82].

_{i}and β

_{i}, are key parameters within the FLO-2D model that define the internal resistance of the mudflow materials. In particular, the study conducted by D’Agostino et al. [69] demonstrated that the simulated results are highly dependent on the correct definition of rheological parameters. While these values are typically determined by lab-based analyses of representative field samples collected shortly post-event (e.g., [67,70,82,83]) to describe process flow behavior, it may not always be possible to obtain such samples. In lieu of post-event field samples, rheological values have been calibrated from existing values reported in the literature (e.g., [67,70]).

_{1–4}) that was compiled from the literature for consideration and subsequent calibration in FLO-2D (Table 3).

_{7}) modifies the defined rheology by introducing a uniform height threshold (in m) to each computational grid cell, so that the volume of materials propagating downslope is retained until the defined height is exceeded.

#### 2.3.4. Observed Reference Data for Model Verification

#### 2.4. Model Description

#### 2.5. Performance Metrics

^{2}is increased by the defined cp value at each step, the regression tree continues to grow. In this study, two regression tree models were generated, based on average gRMSE and F scores, respectively.

## 3. Results

#### 3.1. Model Performance Assessments

_{1–4}and x

_{7}).

^{2}value [76] associated with using all of the data ranged from 0.01 to 0.07, while the adjusted r

^{2}value associated with the subset was relatively higher and ranged from 0.09 to 0.24 (Table 6). For the highest performing model output (i.e., B53), 24.57% of the variance between simulated and reference heights could be explained, an increase of about 17% from using all available observation heights associated with variable confidence levels.

#### 3.2. Parameter Importance and Model Behaviour Assessments

_{2}< 28 and yield stress coefficient α

_{1}< 356 × 10

^{6}. In Figure 7, the cluster of simulation runs (n = 228) indicated by the blue circle produced results with the highest averaged F scores (44% agreement between observed and simulated extents). These simulations were defined by surface detention values < 0.7 m, a volumetric sediment concentration < 32%, and the viscosity coefficient β

_{2}≥ 23.

## 4. Discussion

#### 4.1. Sensitivity Analysis for Model Calibration

#### 4.2. Model Performance Assessment: Potentials and Limitations

_{v}= 45–63%). This variability has been attributed to the local effects of air bubbles or the formation of grain clusters in the mixture, which violate the continuum assumption for mud matrices characterized by a higher C

_{v}. This could provide some explanation for the lower agreement of simulated and observed extents when using literature-based values corresponding to materials from torrent events characterized by a lower C

_{v}to calibrate for higher C

_{v}values (i.e., 45–70%) in this study.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Vectorized post-event orthophotograph [72] capturing the distribution of mud and debris in Brienz.

**Figure 2.**Schematic illustrating how the different sources of data are fed into the dynamic numerical model to generate simulated outputs; summary scalar variables are subsequently produced to support model performance evaluations and provide insight about the potential use of simulated intensity proxies to further develop physical vulnerability curves.

**Figure 3.**Two main sets of reference data are available for the study site including sediment deposition extent and 317 point estimates of sediment deposition heights that are associated with spatial coordinates and confidence levels.

**Figure 4.**Selected examples of post-event photographs of affected buildings in Brienz; sediment deposition heights estimated from the photographs are associated with variable levels of confidence.

**Figure 5.**The two selected maps exemplify the agreement between simulated and observed sediment deposition extents, in addition to the spatial distribution of the simulated deposition heights for two of the best performing simulation runs; simulated maximum velocities provide a quick comparison with the point-based velocity estimate of 6–8 m/s that was determined by local experts at the Glyssibrücke.

Sources of Rheology Values | Model Validation | Reference Data | |||||
---|---|---|---|---|---|---|---|

Cases from Literature | Lab Analyses of Post-Event Field Samples | From Literature | Visual Comparison | Quantification | Sediment Deposition Extent | Sediment Deposition Heights | Number of Sample Points |

Sosio et al. [67] | √ | √ | √ | √ | √ | √ | 18 |

Quan Luna et al. [16] | √ | √ | √ | √ | 13 | ||

Lin et al. [68] | √ | √ | √ | √ | √ | √ | 8 |

D’Agostino et al. [69] | √ | √ | √ | √ | √ | ||

Boniello et al. [70] | √ | √ | √ | √ | √ | ||

Rickenmann et al. [32] | √ | √ | √ |

**Table 2.**Overview of required inputs in FLO-2D and feasible ranges of values specific to the 2005 debris flow event that occurred in Brienz.

Feasible Ranges of Values | ||||||
---|---|---|---|---|---|---|

Initial Conditions | Input Factors for Calibration | Description | Symbol | Units | for FLO-2D Model | for 2005 Debris Flow in Brienz |

√ | computational grid resolution | m | 5 | |||

√ | sediment volume | m^{3} | 70,000–72,000 | |||

√ | (distributed) Manning’s values | n | m^{−⅓}·s | forested areas (0.33), channels and streets (0.01), sparsely vegetated settlement areas (0.08) | ||

√ | resistance parameter for laminar flow | K | 24–50,000 | determined based on floodplain grid element’s Manning’s n value | ||

x_{1} | yield stress, α_{1} | τ_{y} | Pa | 0–∞ | unknown; see Table 3 for potentially applicable values | |

x_{2} | yield stress, β_{2} | 0–∞ | ||||

x_{3} | viscosity, α_{1} | η | Pa·s | 0–∞ | ||

x_{4} | viscosity, β_{2} | 0–∞ | ||||

x_{5} | volumetric sediment concentration (determines mud hydrograph) | C_{v} | 0.03–0.90 | 0.30–0.70 | ||

x_{6} | specific gravity | G_{s} | 2.5–2.8 | 2.5–2.8 | ||

x_{7} | surface detention | m | 0.01–0.50 | model-specific input factor; calibrated with feasible model ranges |

**Table 3.**Summary of selected rheological values from the literature to evaluate with the FLO-2D model with respect to the 2005 debris flow event, in lieu of location-specific post-event field samples; a unique ID is assigned to each group of values.

Yield Stress (T_{y}) | Viscosity (n) | |||||
---|---|---|---|---|---|---|

ID | Sources from Literature | Sample Name | α_{1} | β_{1} | α_{2} | β_{2} |

Dynes/cm^{2} | Poises | |||||

A | O’Brien and Julien [83] | Glenwood 4 | 0.00172 | 29.5 | 0.000602 | 33.1 |

Glenwood 1 | 0.0345 | 20.1 | 0.00283 | 23 | ||

Glenwood 3 | 0.0765 | 16.9 | 0.648 | 6.2 | ||

Glenwood 2 | 0.000707 | 29.8 | 0.00632 | 19.9 | ||

B | O’Brien and Julien [83] | Aspen pit 1 | 0.181 | 25.7 | 0.036 | 22.1 |

Aspen pit 2 | 2.72 | 10.4 | 0.0538 | 14.5 | ||

Aspen natural soil | 0.152 | 18.7 | 0.00136 | 28.4 | ||

Aspen mine fill | 0.0473 | 21.1 | 0.128 | 12 | ||

Aspen natural soil source | 0.0383 | 19.6 | 0.000495 | 27.1 | ||

Aspen mine fill source | 0.291 | 14.3 | 0.000201 | 33.1 | ||

C | Sosio et al. [67] | Scenario A | 0.0013 | 23 | 0.0000283 | 19 |

Scenario B | 0.000093 | 23.5 | 0.000183 | 19 | ||

Scenario C | 0.0011 | 21.8 | 0.0000283 | 18.2 | ||

L1 (deposit sample) | 0.000127 | 22.8 | 0.0000297 | 18.8 | ||

L2 (source area sample) | 0.0004 | 22 | 0.000203 | 18 | ||

D | D’Agostino and Tecca [69] | Rio Dona | 0.05 | 22 | 0.0015 | 22 |

Fiames | 0.152 | 18.7 | 0.0075 | 14.39 | ||

E | Lin et al. [68] | Nan-Ping-Kern | 0.2433 | 4.116 | 0.001386 | 3.372 |

Jun-Kern; Err-Bu | 0.7299 | 12.348 | 0.004158 | 10.116 | ||

Shan-Bu | 0.4055 | 6.86 | 0.00231 | 5.62 | ||

Fong-Chu | 0.85155 | 14.406 | 0.004851 | 11.802 | ||

Tung-Fu Community | 0.811 | 13.72 | 0.00462 | 11.24 | ||

Her-Ser 1 | 1.0543 | 17.836 | 0.006 | 14.612 | ||

Chui-Sue River | 1.2652 | 16.464 | 0.924 | 14.612 | ||

F | Boniello et al. [70] | Fella sx debris flow 1 | 0.000005 | 42.01 | 0.00000002 | 42.23 |

Fella sx debris flow 2 | 0.0383 | 19.6 | 0.000495 | 27.1 |

Input Factors for Model Calibration | Summary Scalar Variables I | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

x_{1}, x_{2}: Yield Stress | x_{3}, x_{4}: Viscosity | x_{5}: Volumetric Sediment Concentration | x_{6}: Specific Gravity | x_{7}: Surface Detention | F | gRMSE1 | gRMSE4 | |||||

Rank | Simulation Run ID | α_{1} | β_{1} | α_{2} | β_{2} | (%) | Very Certain Data Points (m) | All Data Points (m) | ||||

1 | B53 | Aspen pit 1 | 0.181 | 25.7 | 0.036 | 22.1 | 0.45 | 2.8 | 0.80 | 53.42 | 0.71 | 0.92 |

2 | B54 | Aspen pit 1 | 0.181 | 25.7 | 0.036 | 22.1 | 0.45 | 2.8 | 1.00 | 53.34 | 0.72 | 0.92 |

3 | B52 | Aspen pit 1 | 0.181 | 25.7 | 0.036 | 22.1 | 0.45 | 2.8 | 0.60 | 53.22 | 0.71 | 0.92 |

4 | E48 | Chui-Sue River | 1.2652 | 16.464 | 0.924 | 14.612 | 0.50 | 2.5 | 1.40 | 53.15 | 0.87 | 0.94 |

5 | B34 | Aspen pit 1 | 0.181 | 25.7 | 0.036 | 22.1 | 0.45 | 2.65 | 1.00 | 53.02 | 0.72 | 0.93 |

6 | E103 | Chui-Sue River | 1.2652 | 16.464 | 0.924 | 14.612 | 0.50 | 2.65 | 1.40 | 52.99 | 0.87 | 0.94 |

7 | B41 | Aspen pit 1 | 0.181 | 25.7 | 0.036 | 22.1 | 0.45 | 2.8 | 0.50 | 52.98 | 0.71 | 0.92 |

8 | E158 | Chui-Sue River | 1.2652 | 16.464 | 0.924 | 14.612 | 0.50 | 2.8 | 1.40 | 52.97 | 0.87 | 0.93 |

9 | B33 | Aspen pit 1 | 0.181 | 25.7 | 0.036 | 22.1 | 0.45 | 2.65 | 0.80 | 52.96 | 0.71 | 0.93 |

10 | E49 | Chui-Sue River | 1.2652 | 16.464 | 0.924 | 14.612 | 0.50 | 2.5 | 1.50 | 52.95 | 0.84 | 0.92 |

**Table 5.**Overview of the top 10 performing simulations based on lowest gRMSE1 values calculated with very certain reference height data only.

Input Factors for Model Calibration | Summary Scalar Variables I | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

x_{1}, x_{2}: Yield Stress | x_{3}, x_{4}: Viscosity | x_{5}: Volumetric Sediment Concentration | x_{6}: Specific Gravity | x_{7}: Surface Detention | F | gRMSE1 | gRMSE4 | |||||

Rank | Simulation Run ID | α_{1} | β_{1} | α_{2} | β_{2} | (%) | Very Certain Data Points (m) | All Data Points (m) | ||||

1 | E112 | Chui-Sue River | 1.2652 | 16.464 | 0.924 | 14.612 | 0.30 | 2.80 | 1.00 | 44.66 | 0.60 | 0.88 |

2 | A6 | Glenwood 4 | 0.00172 | 29.5 | 0.000602 | 33.1 | 0.35 | 2.50 | 1.00 | 42.71 | 0.60 | 0.90 |

3 | E2 | Chui-Sue River | 1.2652 | 16.464 | 0.924 | 14.612 | 0.30 | 2.50 | 1.00 | 44.72 | 0.60 | 0.87 |

4 | E57 | Chui-Sue River | 1.2652 | 16.464 | 0.924 | 14.612 | 0.30 | 2.65 | 1.00 | 44.81 | 0.60 | 0.87 |

5 | B1 | Aspen pit 1 | 0.181 | 25.7 | 0.036 | 22.1 | 0.30 | 2.50 | 0.80 | 42.84 | 0.61 | 0.89 |

6 | B21 | Aspen pit 1 | 0.181 | 25.7 | 0.036 | 22.1 | 0.30 | 2.65 | 0.80 | 42.69 | 0.61 | 0.89 |

7 | B41 | Aspen pit 1 | 0.181 | 25.7 | 0.036 | 22.1 | 0.30 | 2.80 | 0.80 | 42.58 | 0.61 | 0.89 |

8 | A26 | Glenwood 4 | 0.00172 | 29.5 | 0.000602 | 33.1 | 0.35 | 2.65 | 1.00 | 42.57 | 0.61 | 0.90 |

9 | B26 | Aspen pit 1 | 0.181 | 25.7 | 0.036 | 22.1 | 0.35 | 2.65 | 1.00 | 43.18 | 0.61 | 0.92 |

10 | B46 | Aspen pit 1 | 0.181 | 25.7 | 0.036 | 22.1 | 0.35 | 2.80 | 1.00 | 43.29 | 0.61 | 0.92 |

**Table 6.**Comparison of parameter combinations and summary scale variables for the top performing simulations runs for each of the six sets of rheologies obtained from published literature to back-calculate the 2005 debris flow in Brienz.

Model Inputs for Calibration | |||||||||

x_{1}, x_{2}: Yield Stress | x_{3}, x_{4}: Viscosity | x_{5}: Volumetric Sediment Concentration | x_{6}: Specific Gravity | x_{7}: Surface Detention | |||||

Rank | Simulation Run ID | α_{1} | β_{1} | α_{2} | β_{2} | ||||

1 | B53 | Aspen pit 1 | 0.181 | 25.7 | 0.036 | 22.1 | 0.45 | 2.80 | 0.80 |

2 | E48 | Chui-Sue River | 1.2652 | 16.464 | 0.924 | 14.612 | 0.50 | 2.50 | 1.40 |

3 | A55 | Glenwood 4 | 0.00172 | 29.5 | 0.000602 | 33.1 | 0.45 | 2.80 | 1.50 |

4 | D50 | Rio Dona | 0.05 | 22 | 0.0015 | 22 | 0.30 | 2.80 | 1.30 |

5 | F300 | Fella sx debris flow 2 | 0.0383 | 19.6 | 0.000495 | 27.1 | 0.30 | 2.65 | 1.30 |

6 | C60 | Scenario A | 0.0013 | 23 | 2.83 × 10^{−5} | 19 | 0.30 | 2.80 | 1.30 |

Summary Scalar Variables I | |||||||||

F | gRMSE1 | gRMSE4 | Adjusted r^{2} | ||||||

Rank | Simulation Run ID | (%) | Very Certain Data Points (m) | All Data Points (m) | Very Certain Data Points | All Data | Simulated Maximum Velocity (m/s) | ||

1 | B53 | Aspen pit 1 | 53.42 | 0.71 | 0.92 | 0.24 | 0.07 | 2–4 | |

2 | E48 | Chui-Sue River | 53.15 | 0.87 | 0.94 | 0.08 | 0.04 | 0.5–1 | |

3 | A55 | Glenwood 4 | 52.03 | 0.86 | 0.93 | 0.11 | 0.06 | 1–2 | |

4 | D50 | Rio Dona | 46.27 | 0.72 | 0.96 | 0.10 | 0.06 | 2–8 | |

5 | F300 | Fella sx debris flow 2 | 46.11 | 0.72 | 0.95 | 0.11 | 0.06 | 6–8 | |

6 | C60 | Scenario A | 45.73 | 0.72 | 0.97 | 0.09 | 0.01 | 6–8 |

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**MDPI and ACS Style**

Chow, C.; Ramirez, J.; Keiler, M. Application of Sensitivity Analysis for Process Model Calibration of Natural Hazards. *Geosciences* **2018**, *8*, 218.
https://doi.org/10.3390/geosciences8060218

**AMA Style**

Chow C, Ramirez J, Keiler M. Application of Sensitivity Analysis for Process Model Calibration of Natural Hazards. *Geosciences*. 2018; 8(6):218.
https://doi.org/10.3390/geosciences8060218

**Chicago/Turabian Style**

Chow, Candace, Jorge Ramirez, and Margreth Keiler. 2018. "Application of Sensitivity Analysis for Process Model Calibration of Natural Hazards" *Geosciences* 8, no. 6: 218.
https://doi.org/10.3390/geosciences8060218