Risk evaluation of ice-jam disasters using gray systems theory: the case of Ningxia-Inner Mongolia reaches of the Yellow River

Abstract

Risk evaluation for natural disasters is an important part of the emergency management, disaster prevention and mitigation. Because of the complexity and uncertainty of practical evaluation problems, the evaluation information available generally needs depiction of interval gray numbers instead of real numbers. This paper presents an evaluation method with three-parameter interval gray numbers which can deal with dynamic multiple indicators in order to evaluate efficiently the ice-jam disaster risk of Ningxia-Inner Mongolia reaches of the Yellow River in China. The gray range transformation is introduced into the process of model building to eliminate the incomparability of different dimensions. Moreover, model GM(1,1) is used to simulate and predict the development trend of risk vector. As the results show, while the ice-jam disaster risk of Ningxia-Inner Mongolia reaches of the Yellow River reveals certain wave characteristics, the overall trend remains smooth. The risk degree of ice-jam disaster with Bayangol and Toudaoguai is expected to decrease in the years between 2013 and 2015, while that with Sanhu River tends to increase.

Attachment: modeling process of gray forecasting model GM(1, 1)

Define the risk measure sequence as $X^{ ( 0 )} { = }\left( {x^{(0)} ( 1 ) , { }x^{(0)} ( 2 ) ,\ldots , { }x^{(0)} (n )} \right)$, where $x^{(0)} (k) \ge 0,$ $k = 1,2, \ldots n$; $X^{(1)}$ is the first-order accumulating generator sequence (1-AGO) of $X^{(0)}$:$X^{(1)} = \left( {x^{(1)} (1),x^{(1)} (2), \ldots ,x^{(1)} (n)} \right)$.

Here $x^{(1)} (k) = \sum_{i = 1}^{k} {x^{(0)} (i)}$, $k = 1,2, \ldots n$; $Z^{(1)}$ is the sequence close to the mean generation of $X^{(1)}$:$Z^{(1)} = \left( {z^{(1)} (1),z^{(1)} (2), \ldots ,z^{(1)} (n)} \right)$,In which $z^{(1)} (k) = 0.5\left( {x^{(1)} (k) + x^{(1)} (k - 1)} \right),k = 2,3 \ldots ,n$.

Define

$$x^{(0)} (k) + az^{(1)} (k) = b$$

as the base form of GM(1,1).

If $\hat{a} = (a,b)^{T}$ is the parameter vector of GM(1,1), then the order least square (OLS) of GM(1,1) is

$$\hat{a} = (B^{T} B)^{ - 1} B^{T} Y$$

Here $Y = \left[ {\begin{array}{*{20}c} {x^{(0)} (2)} \\ {x^{(0)} (3)} \\ \vdots \\ {x^{(0)} (n)} \\ \end{array} } \right]$ $B = \left[ {\begin{array}{*{20}c} { - z^{(1)} (2)} & 1 \\ { - z^{(1)} (3)} & 1 \\ \vdots & \vdots \\ { - z^{(1)} (n)} & 1 \\ \end{array} } \right]$.

Set $X^{\left( 0 \right)}$ as nonnegative quasi-smooth sequence, and $X^{ ( 0 )} { = }\left( {x^{(0)} ( 1 ) , { }x^{(0)} ( 2 ) ,\ldots , { }x^{(0)} (n )} \right)$. $X^{(1)}$ is the 1-AGO sequence of $X^{(0)}$, and $Z^{(1)}$ is the sequence close to the mean generation of $X^{(1)}$, so $[a,b]^{T} = (B^{T} B)^{ - 1} B^{T} Y$, then

$$\frac{{{\text{d}}x^{(1)} }}{{{\text{d}}t}} + ax^{(1)} = b$$

is the whitenization equation of GM(1,1).

Set $\hat{a} = [a,b]^{T} = (B^{T} B)^{ - 1} B^{T} Y$, then the time response sequence of the model GM(1,1) is

$$\hat{x}^{(1)} (k) = \frac{b}{a} + \left( {x^{(0)} (1) - \frac{b}{a}} \right)e^{ - a(k - 1)} ,\quad k = 2,3, \cdots n.$$

Its reduction value is

$$\hat{x}^{(0)} (k) = \hat{x}^{(1)} (k) - \hat{x}^{(1)} (k - 1).$$