A multi-state Markov chain model to assess drought risks in rainfed agriculture: a case study in the Nineveh Plains of Northern Iraq

Abstract

The occurrence of prolonged dry spells and the shortage of precipitation are two different hazardous factors affecting rainfed agriculture. This study investigates a multi-state Markov chain model with the states of dry spell length coupled with a probability distribution of positive rainfall depths. The Nineveh Plains of Northern Iraq is chosen as the study site, where the rainfed farmers are inevitably exposed to drought risks, for demonstration of applicability to real-time drought risk assessment. The model is operated on historical data of daily rainfall depths observed at the city Mosul bordering the Nineveh Plains during the period 1975–2018. The methodology is developed in the context of contemporary probability theory. Firstly, the Kolmogorov–Smirnov tests are applied to extracting two sub-periods where the positive rainfall depths obey to respective distinct gamma distributions. Then, empirical estimation of transition probabilities determining a multi-state Markov chain results in spurious oscillations, which are regularized in the minimizing total variation flow solving a singular diffusion equation with a degenerating coefficient that controls extreme values of 0 and 1. Finally, the model yields the statistical moments of the dry spell length in the future and the total rainfall depth until a specified terminal day. Those statistical moments, termed hazard futures, can quantify drought risks based on the information of the dry spell length up to the current day. The newly defined hazard futures are utilized to explore measures to avert drought risks intensifying these decades, aiming to establish sustainable rainfed agriculture in the Nineveh Plains.

Appendices

Appendix 1

Scope of the singular diffusion equation with the degenerating coefficient

The purpose of regularization in general is to remove spurious oscillations appearing in a function. Let $u = u\left( {t,x} \right)$ be such a function defined in a domain $\Omega$ included in the t-x-plane. The magnitude of oscillations in $u$ is evaluated with the functional

$$J = \int_{\Omega } {\left| {\nabla u} \right|{\text{d}}\Omega }$$

(29)

which is referred to as the total variation of $u$. The Euler–Lagrange equation in the context of the variational calculus to minimize the functional $J$ in (29) formally becomes

$$\nabla \cdot \left( {\frac{\nabla u}{{\left| {\nabla u} \right|}}} \right) = 0.$$

(30)

The flux ${{\nabla u} \mathord{\left/ {\vphantom {{\nabla u} {\left| {\nabla u} \right|}}} \right. \kern-\nulldelimiterspace} {\left| {\nabla u} \right|}}$ in the left hand side of (30) is a unit vector if $\left| {\nabla u} \right| \ne 0$ and is not well defined if $\left| {\nabla u} \right| = 0$, resulting in the singularity of (30). The proposed approximation of the flux with (16) is a basic method to overcome such singularity. On the other hand, the practical difficulty encountered in the application to the transition probabilities is that there are true abrupt variations in the neighborhoods of the points achieving extreme values of 0 and 1. The idea employed here is to multiply the degenerating coefficient $u\left( {1 - u} \right)$ to both sides of (30) as

$$u\left( {1 - u} \right)\nabla \cdot \left( {\frac{\nabla u}{{\left| {\nabla u} \right|}}} \right) = 0$$

(31)

where the removal of oscillations is inactivated if $u$ is equal to 0 or 1. Using the estimate $u_{\max } \left( {1 - u_{\min } } \right)$ defined with (17), (18), and (19) is to detect the appropriate points of inactivation. However, the singularity of (30) still remains in (31), and its direct solution is difficult to implement. Inspired by the celebrated ROF model, the unsteady term ${{\partial u} \mathord{\left/ {\vphantom {{\partial u} {\partial \tau }}} \right. \kern-\nulldelimiterspace} {\partial \tau }}$ is added to (31) in order to obtain the singular diffusion equation (14), from which the desired MTVF is successfully computed.

Appendix 2

Proofs

Proofs of recursive formulae of (21), (22), (25), and (26) are provided as below. The relations given in (5) should be referred to as well.

Proof of (21)

$$\begin{gathered} {\text{E}}\left[ {T - t\left| {X_{t} = i} \right.} \right] \\ = P_{i0} \left( {1 + {\text{E}}\left[ {T - \left( {t + 1} \right)\left| {X_{t + 1} = 0} \right.} \right]} \right) \\ + P_{i1} \left( {1 + {\text{E}}\left[ {T - \left( {t + 1} \right)\left| {X_{t + 1} = i + 1} \right.} \right]} \right) \\ = P_{i0} + \left( {1 - P_{i0} } \right)\left( {1 + {\text{E}}\left[ {T - \left( {t + 1} \right)\left| {X_{t + 1} = i + 1} \right.} \right]} \right) \\ = 1 + \left( {1 - P_{i0} } \right){\text{E}}\left[ {T - \left( {t + 1} \right)\left| {X_{t + 1} = i + 1} \right.} \right] \\ = 1 + P_{i1} {\text{E}}\left[ {T - \left( {t + 1} \right)\left| {X_{t + 1} = i + 1} \right.} \right] \\ \end{gathered}$$

(32)

Proof of (22)

$$\begin{gathered} {\text{E}}\left[ {\left( {T - t} \right)^{2} \left| {X_{t} = i} \right.} \right] \\ = {\text{E}}\left[ {\left( {1 + T - \left( {t + 1} \right)} \right)^{2} \left| {X_{t} = i} \right.} \right] \\ = {\text{E}}\left[ {1^{2} + 2\left( {T - \left( {t + 1} \right)} \right) + \left( {T - \left( {t + 1} \right)} \right)^{2} \left| {X_{t} = i} \right.} \right] \\ = 1 + P_{i1} \left( {2{\text{E}}\left[ {T - \left( {t + 1} \right)\left| {X_{t + 1} = i + 1} \right.} \right] + {\text{E}}\left[ {\left( {T - \left( {t + 1} \right)} \right)^{2} \left| {X_{t + 1} = i + 1} \right.} \right]} \right) \\ \end{gathered}$$

(33)

Proof of (25)

$$\begin{gathered} {\text{E}}\left[ {\sum\limits_{k = t}^{k < N} {r_{k + 1} } \left| {X_{t} = i} \right.} \right] \\ = P_{i0} \left( {{\text{E}}\left[ {r_{t + 1} \left| {X_{t + 1} = 0} \right.} \right] + {\text{E}}\left[ {\sum\limits_{k = t + 1}^{k < N} {r_{k + 1} } \left| {X_{t + 1} = 0} \right.} \right]} \right) + P_{i1} \left( {0 + {\text{E}}\left[ {\sum\limits_{k = t + 1}^{k < N} {r_{k + 1} } \left| {X_{t + 1} = i + 1} \right.} \right]} \right) \\ = P_{i0} \left( {{\text{E}}\left[ {r_{t + 1} \left| {r_{t + 1} > 0} \right.} \right] + {\text{E}}\left[ {\sum\limits_{k = t + 1}^{k < N} {r_{k + 1} } \left| {X_{t + 1} = 0} \right.} \right]} \right) + P_{i1} {\text{E}}\left[ {\sum\limits_{k = t + 1}^{k < N} {r_{k + 1} } \left| {X_{t + 1} = i} \right. + 1} \right] \\ = P_{i0} \left( {\frac{\alpha }{\beta } + {\text{E}}\left[ {\sum\limits_{k = t + 1}^{k < N} {r_{k + 1} } \left| {X_{t + 1} = 0} \right.} \right]} \right) + P_{i1} {\text{E}}\left[ {\sum\limits_{k = t + 1}^{k < N} {r_{k + 1} } \left| {X_{t + 1} = i} \right. + 1} \right] \\ \end{gathered}$$

(34)

with

$$\begin{gathered} {\text{E}}\left[ {\sum\limits_{k = t}^{k < t + 1} {r_{k} } \left| {X_{t} = i} \right.} \right] \\ = {\text{E}}\left[ {r_{t} \left| {X_{t} = i} \right.} \right] = \left\{ {\begin{array}{*{20}c} 0 & {{\text{if }}i > 0} \\ {{\text{E}}\left[ {r_{t} \left| {X_{t} = 0} \right.} \right]} & {{\text{if }}i = 0} \\ \end{array} } \right. \\ = \left\{ {\begin{array}{*{20}c} 0 & {{\text{if }}i > 0} \\ {{\text{E}}\left[ {r_{t} \left| {r_{t} > 0} \right.} \right]} & {{\text{if }}i = 0} \\ \end{array} } \right. \\ = \left\{ {\begin{array}{*{20}c} 0 & {{\text{if }}i > 0} \\ {\frac{\alpha }{\beta }} & {{\text{if }}i = 0} \\ \end{array} } \right. \\ \end{gathered}$$

(35)

Proof of (26)

$$\begin{gathered} {\text{E}}\left[ {\left. {\left( {\sum\limits_{k = t}^{k < N} {r_{k + 1} } } \right)^{2} } \right|X_{t} = i} \right] \\ = {\text{E}}\left[ {\left. {\left( {r_{t + 1} + \sum\limits_{k = t + 1}^{k < N} {r_{k + 1} } } \right)^{2} } \right|X_{t} = i} \right] \\ = {\text{E}}\left[ {\left. {r_{t + 1}^{2} + 2r_{t + 1} \sum\limits_{k = t + 1}^{k < N} {r_{k + 1} } + \left( {\sum\limits_{k = t + 1}^{k < N} {r_{k + 1} } } \right)^{2} } \right|X_{t} = i} \right] \\ = {\text{E}}\left[ {\left. {r_{t + 1}^{2} } \right|X_{t} = i} \right] + 2{\text{E}}\left[ {\left. {r_{t + 1} \sum\limits_{k = t + 1}^{k < N} {r_{k + 1} } } \right|X_{t} = i} \right] + {\text{E}}\left[ {\left. {\left( {\sum\limits_{k = t + 1}^{k < N} {r_{k + 1} } } \right)^{2} } \right|X_{t} = i} \right] \\ = P_{i0} {\text{E}}\left[ {\left. {r_{t + 1}^{2} } \right|X_{t + 1} = 0} \right] + 2P_{i0} {\text{E}}\left[ {\left. {r_{t + 1} \sum\limits_{k = t + 1}^{k < N} {r_{k + 1} } } \right|X_{t + 1} = 0} \right] \\ + P_{i0} {\text{E}}\left[ {\left. {\left( {\sum\limits_{k = t + 1}^{k < N} {r_{k + 1} } } \right)^{2} } \right|X_{t + 1} = 0} \right] + P_{i1} {\text{E}}\left[ {\left. {\left( {\sum\limits_{k = t + 1}^{k < N} {r_{k + 1} } } \right)^{2} } \right|X_{t + 1} = i + 1} \right] \\ = P_{i0} {\text{E}}\left[ {\left. {r_{t + 1}^{2} } \right|r_{t + 1} > 0} \right] + 2P_{i0} \left( {{\text{E}}\left[ {\left. {r_{t + 1} } \right|r_{t + 1} > 0} \right]{\text{E}}\left[ {\left. {\sum\limits_{k = t + 1}^{k < N} {r_{k + 1} } } \right|X_{t + 1} = 0} \right]} \right) \\ + P_{i0} {\text{E}}\left[ {\left. {\left( {\sum\limits_{k = t + 1}^{k < N} {r_{k + 1} } } \right)^{2} } \right|X_{t + 1} = 0} \right] + P_{i1} {\text{E}}\left[ {\left. {\left( {\sum\limits_{k = t + 1}^{k < N} {r_{k + 1} } } \right)^{2} } \right|X_{t + 1} = i + 1} \right] \\ = \frac{{\alpha \left( {\alpha + 1} \right)}}{{\beta^{2} }}P_{i0} + \frac{2\alpha }{\beta }P_{i0} {\text{E}}\left[ {\left. {\sum\limits_{k = t + 1}^{k < N} {r_{k + 1} } } \right|X_{t + 1} = 0} \right] \\ + P_{i0} {\text{E}}\left[ {\left. {\left( {\sum\limits_{k = t + 1}^{k < N} {r_{k + 1} } } \right)^{2} } \right|X_{t + 1} = 0} \right] + P_{i1} {\text{E}}\left[ {\left. {\left( {\sum\limits_{k = t + 1}^{k < N} {r_{k + 1} } } \right)^{2} } \right|X_{t + 1} = i + 1} \right] \\ \end{gathered}$$

(36)