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Spatial modeling of drought events using max-stable processes

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Abstract

With their severe environmental and socioeconomic impact, drought events belong to the most far-reaching natural disasters. Effects are tremendous in rain-fed agricultural areas as in Africa. We analyzed and modeled the spatio-temporal statistical behavior of the Normalized Difference Vegetation Index as a risk indicator for drought, reflecting its stochastic effects on vegetation. The study used a data set for Rwanda obtained from multitemporal satellite remote sensor measurements during a 14-year period and divided into season-specific spatial random fields. Maximal deviations from average conditions were modeled with max-stable Brown–Resnick processes taking methodological and computational challenges into account. Those challenges are caused by the large spatial extent and the relatively short time span covered by the data. Extensive simulations enabled us to go beyond the observations and, thus, to estimate several important characteristics of extreme drought events, such as their expected return period.

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Acknowledgements

This work was mainly undertaken while M. Oesting was at ITC, University of Twente, The Netherlands. The authors are grateful to Professor Michael L. Stein and two anonymous referees for valuable comments and suggestions.

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Authors and Affiliations

  1. Department Mathematik, University of Siegen, 57068, Siegen, Germany

    Marco Oesting

  2. Faculty for Geo-Information Science and Earth Observation (ITC), University of Twente, PO Box 217, 7500 AE, Enschede, The Netherlands

    Alfred Stein

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Appendix: Bivariate density of Brown–Resnick processes

Appendix: Bivariate density of Brown–Resnick processes

Analogously to Genton et al. (2011), the bivariate density \(f_{\zeta (x_1;S),\zeta (x_2;S)}(\cdot ,\cdot ; \vartheta _\gamma (S))\) for the random vector \((\zeta (x_1;S),\zeta (x_2;S))\), \(x_1,x_2 \in {\mathscr {X}}\), is given by

$$\begin{aligned}&f_{\zeta (x_1;S), \zeta (x_2;S)}(z_1,z_2; \vartheta _\gamma (S)) \nonumber \\&\quad = \exp \left[ - \sum _{k=1}^2 \frac{1}{z_k} {\varPhi }\left( \frac{\sqrt{\gamma (x_1-x_2; \vartheta _\gamma (S))}}{2} + (-1)^k \frac{\log (z_1) - \log (z_2)}{\sqrt{\gamma (x_1-x_2; \vartheta _\gamma (S))}}\right) \right] \nonumber \\&\qquad \cdot \left[ \prod _{k=1}^2 \frac{1}{z_k^2}{\varPhi }\left( \frac{\sqrt{\gamma (x_1-x_2; \vartheta _\gamma (S))}}{2} + (-1)^k \frac{\log (z_1) - \log (z_2)}{\sqrt{\gamma (x_1-x_2; \vartheta _\gamma (S))}}\right) \right] \nonumber \\&\qquad \cdot \frac{\exp \left( -\frac{1}{8} \gamma (x_1-x_2; \vartheta _\gamma (S))\right) }{z_1^{3/2} z_2^{3/2} \sqrt{\gamma (x_1-x_2; \vartheta _\gamma (S))}} \cdot \exp \left( -\frac{(\log (z_1)-\log (z_2))^2}{4\gamma (x_1-x_2; \vartheta _\gamma (S))}\right) , \quad z_1,z_2>0, \end{aligned}$$
(30)

where \({\varPhi }\) and \(\varphi\) denote the cumulative distribution function and the probability density function of the standard normal distribution, respectively. In order to calculate the bivariate density \(f_{Z(x_1;S),Z(x_2;S)}(\cdot ; \theta (S))\) for the vector of temporal maxima \((Z(x_1;S),Z(x_2;S))\) of transformed NDVI values, we apply the density transformation formula to (30) and obtain

$$\begin{aligned}&f_{Z(x_1;S),Z(x_2;S)}(z_1,z_2; \vartheta (S)) \nonumber \\&\quad = \left[ \prod _{k=1}^2 \frac{1}{\sigma (x_k;S)} \left( 1+\xi (S)\frac{z_k - \mu (x_k;S)}{\sigma (x_k;S)}\right) ^{1/\xi (S)-1}\right] \nonumber \\&\qquad \cdot f_{\zeta (x_1;S),\zeta (x_2;S)}\left( \left( \left( 1 + \xi (S) \frac{z_k - \mu (x_k;S)}{\sigma (x_k;S)}\right) ^{\frac{1}{\xi (S)}}\right) _{k=1,2}; \vartheta _\gamma (S)\right) \end{aligned}$$
(31)

where \(\vartheta (S) = (\vartheta _{GEV}(S), \vartheta _{\gamma }(S))^\top\).

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Oesting, M., Stein, A. Spatial modeling of drought events using max-stable processes. Stoch Environ Res Risk Assess 32, 63–81 (2018). https://doi.org/10.1007/s00477-017-1406-z

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