2.1. Study Area
Based on the expected utility theory, the functional relationship between the intensity of disaster-causing factors and the vulnerability of hazard-affected bodies is established, and the loss utility is introduced into the vulnerability analysis of hazard-affected bodies. The expected value of loss utility is used to measure the risk loss caused by possible typhoon disasters, and the typhoon risk level is divided based on the deterministic loss with the same degree of aversion as relative loss aversion. In the new model, it can not only quantify the degree of typhoon risk loss with the loss expectation of loss utility but also reflect the risk attitude of the affected population to typhoon losses.
The degree of dissatisfaction of social groups arising from risky losses can be measured in terms of utility, while in the field of typhoon disaster risk assessment, the main consideration is the degree of dissatisfaction as well as aversion of the affected population due to losses caused by typhoon disasters.
Definition 1. If n typhoons occur in the study area within a certain time scale, and there are m key indicators for typhoon risk loss assessment identification, let their corresponding loss values be lij (i = 1,2, …, n; j = 1,2, …, m), then the first i typhoon has an aversion utility of μ(lj) (j = 1,2, …, m).
The loss utility function is a mathematical measure of the loss utility. It has the following properties: the loss utility
μ increases with the increase in risk loss
l, and the risk loss utility increases with the increase in risk loss. That is to say, the first derivative and the second derivative of the loss utility function
μ with respect to
l are greater than zero. The curve of the loss utility function is shown in
Figure 1 below.
There are three common types of utility functions in general: exponential, logarithmic and binomial [
27,
28]. In view of the research results of Peng et al. [
29] in coal mine safety risk assessment, Gao et al. [
30] in-flight safety evaluation and Jiang and Liu [
31] in the evaluation of dike raising scheme, we determine the expression form of the loss utility function as the following:
where
α > 0 and it is a constant, termed as the risk aversion parameter, which indicates the aversion of the affected group to typhoon risk losses. The larger it is, the higher the degree of aversion to risk losses.
It is reasonable to choose formula (1) as a concrete expression for the loss utility function:
- i.
μ(l)′ > 0 and
μ(l)″ > 0 conform to the property of loss of utility (
Figure 1);
- ii.
When α = 0, .
That is, when the risk aversion attitude is not considered, the loss utility value is equal to the loss value. It can also be interpreted as “risk neutral” as a special case of loss of utility when the utility coefficient α = 0.
The risk aversion parameter in Equation is unknown, and it is generally necessary to determine its value by the “deciding optimal through questioning method”, “curve fitting method” and “utility consistency method”, etc. [
32]. However, due to the difficulty of collecting some disaster relief information, the value of the risk aversion parameter in this paper is taken from the literature, which uses the lower limit of the number of fatalities from different levels of water traffic accidents as the number of accident fatalities l and the amount of fine is used to represent the aversion of the social group
, with the fit was performed to obtain
≈ 0.019 [
33].
Definition 2. If pi represents the probability of Typhoon i, lij (j = 1,2, … m) represents the loss amount of item j caused by typhoon i (such as direct economic loss, casualties, farmland affected area, etc.); Uij represents the loss expectation of item j caused by the impact of typhoon i; then, according to the expected utility theory of von Neumann and Morgenstern, there are:
In the evaluation and analysis of typhoon risk level, Uij can be used to measure the risk level of the typhoon disaster. The larger Uij is, the higher the risk level is; otherwise, the smaller the risk level is.
Since
μ(
lij) =
lij when α = 0, then formula (2) degenerates into the classical quantitative algorithm of loss expectation:
That is, Formula (3) is a special case of Formula (2).
Formula (2) Explanation of the application of the function form used in typhoon risk analysis: Starting from the risk facts, typhoon disasters often cause discontentment and aversion in the affected people, and this dissatisfaction will increase with the aggravation of the degree of loss, and the increasing speed will be faster and faster. Mathematically speaking: For the function describing this dissatisfaction, the first and second derivatives are greater than 0. In this paper, the function describing this phenomenon is taken as the inverse function of the utility function (called loss utility function in this paper), and the specific expression is shown in formula (1): , it can be found that μ(l)′ > 0, μ(l)″ > 0, so it can be considered that the loss utility function is suitable for typhoon disaster risk analysis. In addition, μ(lij) = lij can be understood as a “risk neutral (α = 0)” case. In other words, in the traditional loss expectation formula, the amount of loss is directly substituted into the calculation, and the risk attitude of the affected body is not considered.
Regarding the probability of typhoon occurrence, it is difficult to give a direct probability value because of their random nature. If the probability of a typhoon is calculated based on the frequency of typhoons, such a probability value only characterizes the likelihood of typhoons occurring during the year and does not reflect the intensity of typhoons. In reality, when a typhoon crosses the harbor, the gale force winds not only act directly on the harbor buildings but can also bring huge waves, posing a threat to the structural safety of the harbor structures. Secondly, the violent atmospheric disturbances caused by the approaching typhoon can bring heavy rainfall and trigger secondary flooding. In addition, typhoons moving into shallow offshore waters can also trigger storm surges if they meet with astronomically high tides, causing abnormal sea level rise and triggering the risk of seawater spillover [
31,
34,
35,
36]. Thus, it shows that the hazard of typhoons mainly comes from high winds, huge waves, heavy rainfall and storm surges, which can be characterized by three causative factors, namely wind speed, wave height and water increase. The magnitude of the disaster-causing factors can directly reflect the magnitude of damage when a disaster is likely to occur. Therefore, in this paper, the probability of typhoon occurrence is characterized by the probability of wind speed, wave height and water increase.
Since the Copula functional model is not restricted by the form of marginal distribution, it is able to organically combine different degrees of correlation and correlation patterns among random variables while building a joint structure [
37,
38]. Therefore, this paper adopts the Copula function for the probability of typhoon occurrence in structure function construction.
If the continuous random variables
x,
y,
z represent the extreme wave height, water increase, and wind speed in the study area under the influence of a typhoon, respectively, assume that they obey marginal distributions of
Q(x),
G(y) and
H(z). The joint probability distribution of the occurrence of each risk factor is denoted as
F(x,y,z), According to Sklar’s theorem [
11], there exists a unique Copula function C such that:
Common three-dimensional asymmetric Copula functions mainly have the following three types (Q(x), G(y) and H(z) are abbreviated as u, v, w to make the formula more concise):
- (1)
Gumbel Copula function:
- (2)
Frank Copula function:
M6 Copula function: It is qualitatively a combination of two two-dimensional Gumbel copulas with different parameters, so it is also called Asymmetric Gumbel nested copula, with M6
1 Copula, M6
2 Copula, and M6
3 Copula. The expressions of its function structure are as follows:
where,
u,
v and
w are edge distribution functions, respectively;
θ1,
θ2,
θ11,
θ12,
θ13,
θ21,
θ22, and
θ23 are all parameters of the Copula function and can be estimated by maximum likelihood method.
If the
M62 Copula function is used in this paper to construct a three-dimensional joint probability distribution, and if these three variables obey the Gumbell distribution, Pearson distribution and Pearson distribution, respectively, then
where,
b and
a represent the reciprocal of position coefficient and scale coefficient of the Gumbell distribution, respectively; Γ(
α1) and Γ(
α2) are gamma functions of α1 and α
2, respectively;
α1,
β1,
α2,
β2 are the shape scale parameters of the Pearson-III distribution, and their values are all greater than 0.
a01 and
a02 are positional unknown parameters of Pearson’s three-type distribution.
Direct economic losses and casualties are chosen as indicators of the vulnerability of the disaster-bearing population, such as
L1 and
L2. Let the vulnerability loss to a study area due to the combined effect of the contributing factors be
Lj (x,y,z), then the expected loss due to the typhoon in the study area
Uj can be expressed as
where
Lj (x,y,z) is the vulnerability function characterizing the risk-bearing body. It can be calculated by identifying the constituent sample from the hazard intensity-loss data recorded in the disaster. Because it is calculated based on direct economic losses or affected populations, it is the absolute loss generated by the occurrence of a typhoon in a given study area. If the value of
URj represents the aversion utility of the absolute losses caused by the occurrence of a particular typhoon, then
Absolute loss aversion reflects the degree of aversion caused by a certain absolute loss to the affected population in a fixed study area. The larger the value of absolute loss aversion, the higher the degree of aversion, and vice versa. For typhoon hazards, it is applied to measure the amount of typhoon risk for a particular city.
Since the same loss amount poses different risk sizes for cities of different scales. At this point, it is obviously not reasonable to compare the amount of typhoon risk losses in two different regions by comparing their loss utilities of absolute loss. In contrast, relative loss aversion utility needs to be introduced. The relative loss effect is the calculation of utility for relative losses, which requires calculating the proportion of losses in total assets and then calculating their utility. If the gross product of a study region is denoted as GrossV, the relative loss aversion effect for a typhoon disaster is set to
U′Rj, then we can have
In the previous part, we used L(x,y,z) to characterize the potential loss scenario for the hazard-bearing region under some combination of extreme wave height, water increase, and wind speed occurring simultaneously, which is a stochastic loss. For decision-making on typhoon risk, compared to stochastic losses L(x,y,z) we are more interested in the magnitude of the fixed loss l. Theoretically, the dissatisfaction that can be brought about by the losses faced by the disaster-bearing body is the same as the dissatisfaction that can be brought about by the direct losses l. Therefore, we can solve the Equation (6) to get l. At this point, l denotes the definitive loss value with the same degree of aversion as the absolute loss expectation for each typhoon hazard. Similarly, l′ can be inversely calculated according to formula (8). l′ represents the definitive loss value with the same degree of aversion as the total expected relative loss for each typhoon hazard. Something should be noted as follows:
- i.
Since Equation (8) is calculated under the condition of relative loss, it is independent of the city scale. It means that the same value brings the same degree of aversion for different cities. l′ clearly has a range of [0,1].
- ii.
The transformation of Equations (6) and (8) actually converts the random loss into a fixed loss. For instance, l′ = 30%, then it can be said that the typhoon risk faced by the city is equal to 30% of the total output loss of the city in the sense of relative loss aversion.
When mapping typhoon risk levels map, risk partitioning can be performed by drawing expected loss aversion contours. The area between the abscissa and the ordinate and the first contour is marked as a first-level risk area, and the area between the first contour and the second contour is marked as a second-level risk area so that other level areas can be similarly marked outward. The higher the number, the higher the risk level in the region. Finally, the risk level of each typhoon is judged by its fallout point (F, L′).