Generation of Spatially Heterogeneous Flood Events in an Alpine Region—Adaptation and Application of a Multivariate Modelling Procedure
Abstract
:1. Introduction
2. Study Area and Data Used
3. Data Review and Preparation
3.1. Event Definition
- An analysis of runoff time series, where the consecutive days exceeding a specific threshold (defined by the pth quantile ) are counted, provides a first estimation of . Denote the daily maximum runoff at gauge and time t by , and then the probability of peaks of length L is defined as
- The travel- or concentration time provides additional information about the hydrological response of a watershed and therefore contribute to a thorough event definition. In this context, the concentration time (in hours) is a possible measure to indicate the time interval in which a receptor (e.g., flood-prone area) can be affected by simultaneous flooding from two or multiple tributaries. Grimaldi et al. [38] summarized and presented selected formulas for estimating the concentration time, such as, the formulas of (a) ‘Department of Public Works’, (b) ‘Giandotti’, (c) ‘Kirpich’, and (d) ‘Viparelli’, where only few input parameters are necessary. Full details concerning these formulas of the concentration time and their specified restrictions (e.g., regarding the catchment size) can be found in Grimaldi et al. [38]. Although the empirical formulas to estimate the concentration time are associated with large uncertainties [38], they still provide a valuable instrument to calculate the travel time in investigated watersheds.
- The event definition (i.e., selection of ) may also depend on the intended application of the model, such as reconstruction of flood situations in the past lasting a certain number of days or application in a distinct (re)insurance context [21].
3.2. Event Categorization
3.3. Seasonality of Runoff
3.4. Interpretation of Extremes
4. Heffernan and Tawn Model
4.1. Spatial Dependence Measures
4.2. Statistical Model
4.2.1. Marginal Model
4.2.2. Dependence Model
4.3. Estimation and Simulation
4.3.1. Estimation of Marginal Distribution
4.3.2. Marginal Transformation
4.3.3. Parameter Estimation
4.3.4. Simulation of the Conditioning River Flow X
- Number of simulated extreme events per simulation period: The number of extreme events i.e., where at least one site exceeds ) per simulation period (e.g., all-year-, half-year series) is approximated by a negative binominal distribution.
- Selection of conditioning gauge: The set of extreme events S is partitioned to , where is the set of extreme events where gauge i has the highest non-exceedance probability i.e., is the most extreme one). Then, for an event the probability that gauge i is the most extreme if at least one gauge is extreme has to be estimated using given data. The gauge that is selected as conditioning variable X is drawn from a multinomial distribution with these probabilities. Denote the index of the selected gauge by .
- Drawing the conditioning : According to the transformation to standard Laplace margins and since is close to one and hence set , where e is drawn from an exponential distribution with mean one.
4.3.5. Simulation of
- Randomly choose one from .
- Find the m nearest neighbours of denoted by and compute the vector of componentwise means of .
- Draw a random sample from a uniform distribution with lower bound and upper bound .
- Deliver .
4.3.6. Simulation of the Dependent Event
4.4. HT Model and Seasonal Characteristics of Runoff
5. Results and Discussion
5.1. Event Definition and Seasonality of Runoff
5.2. Spatial Dependence in River Flows
5.3. Estimation Results and Model Checks
5.4. Simulated Extreme Events
6. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
Abbreviations
HT model | Multivariate semi-parametric conditional model introduced by Heffernan and Tawn (2004) [20] |
T | Return period in year |
Annual maximum series | |
AMS derived from daily mean data | |
AMS derived from daily maximum data | |
UoFH | Unit of flood hazard |
All-year (series) | |
General extreme value (distribution) | |
Yearly maximum runoff (of half-year series) | |
May to October (series) | |
November to April (series) |
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i | Name of Gauges | River | ID | Area |
---|---|---|---|---|
1 | Mellau | Bregenzerach | 200261 | 228.6 |
2 | Kennelbach | Bregenzerach | 200329 | 826.3 |
3 | Hopfreben | Bregenzerach | 200246 | 41.7 |
4 | Schönenbach | Subersach | 200287 | 31.1 |
5 | Thal | Rotach | 200311 | 90.1 |
6 | Lingenau | Subersach | 200295 | 111.6 |
7 | Hoher Steg | Dornbirnerach | 200212 | 112.9 |
8 | Enz | Dornbirnerach | 200204 | 51.1 |
9 | Lustenau (H) | Rheintalbinne | 200220 | 77.5 |
10 | Laterns | Frutz | 200154 | 33.4 |
11 | Unterhochsteg | Leiblach | 200394 | 102.4 |
12 | Schruns | Litz | 200048 | 102 |
13 | Garsella | Lutz | 200105 | 95.5 |
14 | Gisingen | Ill | 200147 | 1281 |
15 | Lustenau (E) | Rhein | 200196 | 6110 |
16 | Lech | Lech | 200378 | 84.3 |
17 | Lochau | Ruggbach | 200345 | 7.2 |
p | T (Year) | L (Days) | |||||
---|---|---|---|---|---|---|---|
0 | 1 | 2 | 3 | 4 | |||
R (p,L) | 0.99 | 0.3 | 1 | 0.32 | 0.21 | 0.03 | 0 |
0.997 | 0.9 | 1 | 0.39 | 0.25 | 0.02 | 0 | |
0.999 | 2.7 | 1 | 0.34 | 0.32 | 0.01 | 0 | |
0.9997 | 9.1 | 1 | 0.44 | 0.28 | 0 | 0 |
i | Name of Gauges | (h) | (°) | |
---|---|---|---|---|
1 | Mellau | 1.9–4.8 | 204 | 0.56 |
2 | Kennelbach | 3.3–8.8 | 216 | 0.40 |
3 | Hopfreben | 0.7–2.3 | 205 | 0.52 |
4 | Schönenbach | 0.5–2.0 | 208 | 0.61 |
5 | Thal | 1.3–4.6 | 224 | 0.29 |
6 | Lingenau | 1.4–3.8 | 207 | 0.57 |
7 | Hoher Steg | 1.4–4.3 | 196 | 0.70 |
8 | Enz | 0.8–2.8 | 209 | 0.64 |
9 | Lustenau (H) | 1.4–8.7 | 190 | 0.73 |
10 | Laterns | 0.7–2.1 | 208 | 0.68 |
11 | Unterhochsteg | 1.0–6.3 | 188 | 0.45 |
12 | Schruns | 1.5–3.5 | 188 | 0.80 |
13 | Garsella | 1.2–3.0 | 200 | 0.69 |
14 | Gisingen | 5.3–12 | 192 | 0.76 |
15 | Lustenau (E) | 10–30 | 199 | 0.77 |
16 | Lech | 1.2–3.4 | 191 | 0.63 |
17 | Lochau | 0.4–1.3 | 205 | 0.58 |
Test Criteria | Percentiles (%) | ||
---|---|---|---|
12.5–87.5 | 5–95 | 2.5–97.5 | |
0.87 | 0.95 | 0.98 | |
0.91 | 0.97 | 0.99 | |
0.83 | 0.96 | 0.98 | |
0.87 | 0.98 | 0.99 | |
median () | 0.71 | 0.94 | 0.94 |
median () | 0.65 | 0.82 | 0.94 |
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Schneeberger, K.; Steinberger, T. Generation of Spatially Heterogeneous Flood Events in an Alpine Region—Adaptation and Application of a Multivariate Modelling Procedure. Hydrology 2018, 5, 5. https://0-doi-org.brum.beds.ac.uk/10.3390/hydrology5010005
Schneeberger K, Steinberger T. Generation of Spatially Heterogeneous Flood Events in an Alpine Region—Adaptation and Application of a Multivariate Modelling Procedure. Hydrology. 2018; 5(1):5. https://0-doi-org.brum.beds.ac.uk/10.3390/hydrology5010005
Chicago/Turabian StyleSchneeberger, Klaus, and Thomas Steinberger. 2018. "Generation of Spatially Heterogeneous Flood Events in an Alpine Region—Adaptation and Application of a Multivariate Modelling Procedure" Hydrology 5, no. 1: 5. https://0-doi-org.brum.beds.ac.uk/10.3390/hydrology5010005