# Spatially Consistent Drought Hazard Modeling Approach Applied to West Africa

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data and Study Area

## 3. The Extreme Value Mixture Model

^{2}; $G\left(x|u,\sigma ,\xi \right)$is the conditional GPD cumulative distribution function with location parameter u, scale σ, and shape ξ. The subscripts l and r for the GDP parameters are for the left and right tails of the distribution. In this specification, the tail fraction below the lower threshold, ${\varphi}_{l}=P\left({X<u}_{l}\right),$and the tail fraction above the upper threshold, ${\varphi}_{r}=P\left({X>u}_{l}\right),$ are two implicit parameters defined by the normal bulk model: ${\varphi}_{l}=H\left({u}_{l}\right)$ and ${\varphi}_{r}=1-H\left({u}_{r}\right).$The authors of [45] refer to this model as the “bulk model approach”. To reduce the estimation bias related to a potential misspecification of the central distribution, we adopt the more general parametrized tail fraction approach of [40]. In this specification, ${\varphi}_{l}$ and ${\varphi}_{r}$ are extra parameters to be estimated so that the tail fit is more robust to the bulk model [45]. The cumulative distribution function is given by (2):

^{2}being the mean and variance of the normal bulk distribution; ${u}_{l}$ and ${u}_{r}$ are the location parameters, defining, respectively, the lower and upper tail thresholds; σ

_{l}and σ

_{r}are the scale parameters; ${\xi}_{l}$ and ${\xi}_{r}$ are the shape parameters; ${\varphi}_{r}$ and ${\varphi}_{l}$ are the tail fraction, representing, respectively, the probability of being above the upper threshold and below the lower threshold. We use for estimations the maximum likelihood estimator in the Evmix R package developed by Hu, Y. and Scarrott, C. [45].

## 4. Results

#### 4.1. Main Results from the Mixture Model

#### 4.1.1. Stationarity Tests

#### 4.1.2. Goodness-of-Fit Assessments

#### 4.1.3. The GPD Parameters

_{ul}), is evenly distributed over the entire area (Figure 5B). On average, the tail fraction equals 10.5%, meaning that a drought that can be qualified as extreme has a return period of approximately 10 years. The combination of the threshold value and the tail fraction shows that the intensity of the rainfall deficit characterizing an extreme drought varies greatly within the region, while this type of event has the same return period (9 years). Figure 5C shows the scale parameter (${\sigma}_{l}$) of the left GPD. The scale parameter is the gradient of the return period curve at the return period of the threshold: the higher the scale parameter, the lower the precipitation level of extreme rare events, all things being equal. Figure 5D displays the shape parameter (${\xi}_{l}$). In over 77% of instances, the estimated shape parameter is negative, indicating that the distribution of precipitation has a compact left tail with a defined boundary.

#### 4.2. The Drought Hazard Measurement

#### 4.2.1. The Standard Approach of Drought Hazard

_{i}) is the same whatever the geographical location and statistical characteristics of the raw data, but the empirical frequency (f

_{i}) may differ from one location to another. The DHI is given by:

_{i}) is the same in both approaches, SPI and GNG (Table 1, column 5). According to this drought classification and weighting system, the theoretical value of the DHI, calculated using the theoretical occurrence probability of drought, is 0.24 in both models. If the empirical value of the DHI, calculated using the empirical drought frequency, differs from its theoretical value, it means that the model does not fit the data well.

#### 4.2.2. An Alternative Definition of the Drought Hazard Index

_{A}) that has two advantages over the standard formula. First, we use the inferred percentiles of the GNG distribution instead of a fixed weighting to measure the intensity of each category of droughts. Therefore, the intensity of the different drought categories is specific to each precipitation distribution and varies from one location to another. Second, we measure the frequency of each drought category by its theoretical probability of occurrence, which is the same for all locations. The DHI

_{A}value, being based on the theoretical percentiles and probability of occurrence, is less sensitive to measurement errors related to, for instance, the limited size of the sample. The DHI

_{A}is given by:

_{I}is the theoretical probability of occurrence of a drought of category i; z

_{i}is the standardized value of the theoretical percentile of the GNG distribution, corresponding to the upper bound of the interval defining category i droughts. Therefore, ${z}_{i}$ measures the minimum intensity of category i droughts.

_{A}considering the three drought categories, from moderate to extreme, defined in Table 1. Therefore, i = 2 to 4, and ${z}_{i}$ is the standardized value of, respectively, the 15.87, 6.68, and 2.27 percentiles of the GNG distribution in Equation (6). Next, to test the sensitivity of the DHI

_{A}to the drought classification, we use a more detailed drought classification, which comes from the USDA (Table 2), and consider four drought categories, from moderate to exceptional. In that case, i = 2 to 5, and${z}_{i}$ is the standardized value of, respectively, the 20th, 10th, 5th, and 2nd percentiles in Equation (6).

_{A}(Figure 7A–C) shows that the area with the highest drought hazard index/DHI is located in northern Nigeria. This area forms a large cluster (hot spot) of high values with ramifications in northern Cameroon and southern Chad (red areas in Figure 7C). The center of Burkina Faso and Cameroon also appear as high-risk areas. By contrast, the drought hazard is relatively low in the coastal area between Guinea and western Ghana, as well as in the Western part of Cote d’Ivoire and South Benin (blue areas). Results for the northernmost part of the study area should be taken with caution due to the poor quality of the data in these arid areas. The spatial pattern that emerges from Figure 7A,B is similar, meaning that results are robust to the number of drought categories used to calculate the DHI

_{A}.

#### 4.2.3. Extreme Drought Hazard Index

_{A}map is shown in Figure 8A. The spatial distribution of extreme hazard is quite similar to that of the drought hazard shown in Figure 7B.

_{A}calculated for one category of droughts only: those with a probability of occurrence less than or equal to 1%. In other words, lower intensity drought categories are given a zero weight in Equation (6). The map of the extreme hazard (Figure 8B) is quite different than that of the DHI

_{A}(Figure 7). The epicenter of the extreme hazard area moves towards the central eastern part of the region. The extreme drought hazard, measured as the severity of the 100-year drought, is higher in the eastern part of the region, with a hot spot spanning three countries, Nigeria, Chad, and Cameroon.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Simulation results. Each of the return plots illustrates the impact of changing one parameter of the left tail GDP, all else being equal. (

**A**) An increase in the left threshold (location parameter μL) shifts the whole distribution upward. (

**B**) A change in the shape parameter (ξ

_{l}) affects the curvature of the plot: the level of extreme precipitation decreases rapidly to the bound for a large negative value of ξ

_{l}; it decreases slowly for a positive value of ξ

_{l}. (

**C**) An increase in the scale parameter σl affects the slope of the distribution: the level of rare extreme precipitation decreases more than that of more frequent events.

## Appendix B

**θ**, results in uncertainty in

**q**= q(

**θ**), the model’s output.

**q**$=\left[{q}_{1},{q}_{2},\dots ,{q}_{n}\right]$ is the vector of the predicted values of precipitation, and

**θ**$=\left[m,s,{{\varphi}_{l},u}_{l},{\sigma}_{l},{\xi}_{l},{\varphi}_{r},{u}_{r},{\sigma}_{r},{\xi}_{rl}\right]$ is the vector of the GNG parameters. To implement the uncertainty and sensitivity analysis, we take advantage of our large sample of estimation results. The mixture model was estimated on 2298 precipitation series, generating 2298 vectors

**θ**of parameter values and 2298 × n output. Therefore, we use the empirical distribution of the parameters instead of using sampling-based methods, which would imply assigning a hypothetical distribution to

**θ**elements. This approach allows us to circumvent the issue of non-independent parameters. As some of the input variables are correlated, parameters cannot be drawn independently.

**θ**, we examine the distribution of a selected number of precipitation quantiles using boxplots. To assess the importance of each individual element of

**θ**with respect to the uncertainty in q(

**θ**), we draw on regression analysis [55,56].

**θ**

_{k}the vector of the parameters of the distribution k, and q

_{i}(

**θ**

_{k}) the theoretical quantile i of the distribution k, i = 1%, 5%, 10%, …, 90%, 95%, 99%, and k = 1 to 2298. Figure A2(B)shows the uncertainty in the forecast errors with respect to the value of the model’s parameters by precipitation quantile. Uncertainty tends to be higher in extreme quantiles, with more outliers in the lowest and highest quantiles.

**Figure A2.**Representation of uncertainty in the model’s results. (

**A**) Uncertainty in the predicted precipitation values by quantile. (

**B**) Uncertainty in the forecast error by precipitation quantile.

**q**, and ${\theta}_{j}$ the element j of vector

**θ**. The PCC measures the strength of the relationship between the two variables with the effect of other parameters ${\theta}_{p\ne j}$removed. Whatever the correlation degree between the GNG parameters, the PCC allows us to assess the importance of each element of

**θ**, holding the other one constant. The PCC between q and ${\theta}_{j}$ is given by the correlation coefficient between $\left(q-\widehat{q}\right)$and ${(\theta}_{j}-\widehat{{\theta}_{j}})$ with:

^{2}of the full model and the R

^{2}of the reduced model given by Equation (A2). The full model includes all elements of

**θ**so that the associated coefficient of determination is close to 1. The CPD represents the fraction of the variance of q explained by θ

_{j}, controlling for all other model’s variables.

_{i}with i = 1 p, 5 p, 10 p, …, 90 p, 95 p, 99 p. The absolute value of PCCs are provided in Figure A3(A). As expected, the figure shows that precipitation values in the lowest and highest quantiles are strongly correlated with the left and right GPD parameters, respectively. The most important parameters in the extremes of the distribution are, by importance order, location, scale, and shape parameters of the corresponding GPD. In the center of the distribution, the most important parameter is the Gaussian mean. We note that the PCCs are symmetrically distributed around the median axis. For instance, the PCC between u

_{r}and the q

_{i}is equal to the PCC between u

_{l}and q

_{1−i}, with q

_{i}the i-th quantile. We note also that precipitation values from the 10th to the 90th quantiles are correlated, but in a decreasing (increasing) way, with the location parameters of the left (right) GPD and all other parameters held constant.

**Figure A3.**Sensitivity analysis. (

**A**) Partial correlation coefficients in absolute value. (

**B**) Coefficients of partial determination.

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**Figure 2.**Results of simulations: probability density of a GPD–Normal–GPD mixture model (orange) and of Gaussian distribution (blue). The two distributions have the same mean.

**Figure 3.**Stationarity and independence test results. (

**A**) ADF unit root test, with intercept and trend in the test equation. Red cells: the test does not reject the null hypothesis of unit root. (

**B**) Pettit test: red cells: rejection of the no change point null hypothesis. (

**C**) Autocorrelation test on the POT series: Box-Pierce/Ljung-Box Q-statistics for serial correlation up to the order 1. Red cells: rejection of no serial correlation at lag 1.

**Figure 4.**Goodness-of-fit test results. (

**A**) Kolmogorov–Smirnov test; Red cells: bootstrap p-value < 0.05, the GNG does not fit the data. (

**B**) Diebolt and Mariano test; Loss function: absolute error. H

_{0}: Both forecasts have the same accuracy. Green cells: H

_{0}is rejected against the alternative hypothesis (H

_{1a}) that the GNG is more accurate than the gamma model. Red cells: H

_{0}is rejected against the alternative hypothesis (H

_{1b}) that the GNG is less accurate than the gamma model. Yellow cells: H

_{0}is not rejected against both alternatives H

_{1a}and H

_{1b}.

**Figure 5.**GNG parameters. (

**A**) Location parameter (u

_{l}) after standardization; (

**B**) probability of being below the left threshold (ϕ

_{ul}); (

**C**) scale parameter (σ

_{l}); (

**D**) shape parameter (ξ

_{l}).

**Figure 7.**Alternative drought hazard index (DHI

_{A}) mapping. (

**A**) DHI

_{A}calculated using Equation (6) and three drought categories, from moderate to extreme, given in Table 1. (

**B**) DHI

_{A}calculated using Equation (6) and four drought categories, from moderate to exceptional, given in Table 2. (

**C**). Hot and cold spots of the DHI

_{A}given in Figure 8A.

**Figure 8.**Extreme drought hazard index. (

**A**) DHI

_{A}calculated for severe to exceptional droughts. (

**B**) standardized value of the 100-year drought return level.

**Table 1.**Drought categories and thresholds of [18].

Drought Category | Weight W _{i} | Threshold Values for the SPI | Thresholds in Percentiles | Prob. of Occurrence (ϕ _{i}) | Empirical Frequency (f_{i}): SPI | Empirical Frequency (f_{i}): GNG |
---|---|---|---|---|---|---|

1. Mild drought | 0 | $[-1;0]$ | $[15.87\%;50\%]$ | 0.341 | 0.354 | 0.356 |

2. Moderate drought | 1 | $[-1.5;-1]$ | [6.68%; 15.87%] | 0.92 | 0.079 | 0.096 |

3. Severe drought | 2 | $[-2;-1.5]$ | $[2.27\%;6.68\%]$ | 0.04 | 0.040 | 0.041 |

4. Extreme drought | 3 | ≤−2 | $2.27\%$ | 0.0227 | 0.029 | 0.0227 |

DHI (Equation (5)) | 0.24 | 0.243 | 0.246 |

Drought Category (USDA) | Percentiles | Theoretical Prob. of Occurrence (ϕ _{i}) | Empirical Frequency: GNG (f _{i}) |
---|---|---|---|

1. Abnormally dry | $[20\%;30\%]$ | 10% | 0.096 |

2. Moderate drought | $[10\%;20\%]$ | 10% | 0.099 |

3. Severe drought | $[5\%;10\%]$ | 5% | 0.053 |

4. Extreme drought | $[2\%;5\%]$ | 3% | 0.027 |

5. Exceptional drought | $\le 2\%$ | 2% | 0.021 |

Total | 30% | 0.295 |

SPI | GNG | |||
---|---|---|---|---|

Value | Count | Percent | Count | Percent |

0 | 286 | 12.45 | 6 | 0.26 |

1 | 758 | 32.99 | 1406 | 61.18 |

2 | 715 | 31.11 | 791 | 34.42 |

3 | 424 | 18.45 | 89 | 3.87 |

4 | 95 | 4.13 | 6 | 0.26 |

5 | 19 | 0.83 | 0 | 0 |

6 | 1 | 0.04 | 0 | 0 |

Total | 2298 | 100 | 2298 | 100 |

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Araujo Bonjean, C.; Sy, A.; Dury, M.-E.
Spatially Consistent Drought Hazard Modeling Approach Applied to West Africa. *Water* **2023**, *15*, 2935.
https://doi.org/10.3390/w15162935

**AMA Style**

Araujo Bonjean C, Sy A, Dury M-E.
Spatially Consistent Drought Hazard Modeling Approach Applied to West Africa. *Water*. 2023; 15(16):2935.
https://doi.org/10.3390/w15162935

**Chicago/Turabian Style**

Araujo Bonjean, Catherine, Abdoulaye Sy, and Marie-Eliette Dury.
2023. "Spatially Consistent Drought Hazard Modeling Approach Applied to West Africa" *Water* 15, no. 16: 2935.
https://doi.org/10.3390/w15162935