3.2.1. Estimation of Probability Distribution
The sample data of 19 typhoons in Guangdong Province during 2005–2016 were selected, and the maximum landfall wind speed scale and extreme wave height were noted as
In the above sample data, the minimum value of landing maximum wind speed scale
is 7 and the maximum value is 17, and the minimum value of wave height extreme value
is 1.2 m and the maximum value is 7.0 m. Based on its maximum and minimum values, the discrete domain can be constructed as [5.5, 17.5], [1, 7.6], respectively, at this time, the step of landing maximum wind speed scale is 0.5 and the step of extreme wave height is 0.275. The control points are all 25, then their discrete theoretical domains are as follows
The calculated diffusion coefficients
and
are 1.4919 and 0.8652, respectively, and the probability distributions of landing maximum wind speed scale and extreme wave height are obtained from Equations (1)–(5) of the two-dimensional information diffusion model as shown in
Figure 5 and
Table 2.
From
Table 2 and
Figure 5, it can be seen that the typhoon has the highest probability of occurrence when the maximum wind speed is 13 and the extreme wave height is 4.3 m, followed by typhoons with a maximum wind speed of 12 and 14. The probability of occurrence of different maximum wind speeds and extreme wave height varies, with the overall change pattern increasing and then decreasing, and the overall probability value decreases as the intensity of typhoon risk sources increases later. However, the probability of occurrence of each risk factor of typhoons cannot be used as a measure of risk, for example, the probability values of severe typhoons “Kai Tak” and “Rumbia” are large, but the direct economic loss rate is only 0.0449% and 0.0170%, so it is obviously unreasonable to take the probability value of the disaster factor as the measure of risk.
3.2.2. Estimation of Typhoon Disaster Losses at Different Levels of Risk-Causing Factors
Since the risk of the same direct loss is different for different regions and years, it is obviously unreasonable to compare the risk size by direct economic loss alone. As the output sample and collate the typhoon data to obtain the sample set
.
The domain steps of x, y, z are chosen as 0.5, 0.275 and 0.0175, respectively, and the theoretical domains
,
, and
of x, y, z are constructed as follows:
The diffusion coefficients
,
are calculated as 1.4919, 0.8652, and 0.0553, respectively using Equation (2), which are processed using the three-dimensional normal information diffusion formula of Equation (13) and then superimposed using Equation (13) to generate the original three-dimensional information matrix Q of x, y, z:
After that, fuzzy relation inference is performed using Equation (14) to obtain the fuzzy relation matrix R of x, y, z.
Using the one-dimensional information distribution formula shown in Equations (15) and (16), the maximum wind speed scale x and extreme wave height y of typhoon landfall are fuzzified, and according to the definition of the theoretical domain, the fuzzy sets
= “typhoon maximum wind speed scale is 14” and
= “typhoon extreme wave height is 7.6 m” can be expressed as follows.
Fuzzy approximation reasoning using Equation (17) yields the following affiliation values for the loss set z with respect to the element c in the thesis domain W:
Similarly, the affiliation degrees of all elements in W can be calculated so that the fuzzy output set
of direct economic loss can be obtained by fuzzy approximate inference from the above values
, and R:
The center of gravity method shown in Equation (18) for
is non-fuzzy to obtain its disambiguation value
.
After defuzzifying the output fuzzy set
, the fuzzy value of the output is obtained. The direct economic loss rate is estimated to be 0.2157% for the “maximum wind speed scale of 14 and extreme wave height of 7.6 m”. The direct economic loss values can be obtained for all control points in the same way (due to a large number of control points, it is not appropriate to list them all, the table only indicates the values at typical control points, the same below). The information diffusion calculation results are shown in
Table 3 and
Figure 5.
From
Table 3 and
Figure 6, we can see the estimated values of losses for different wind speed scales and extreme wave heights, and compare with the historical data, the above-inferred values of direct economic loss rate corresponding to different maximum wind speed scale and extreme wave heights of typhoons are basically accurate. For example, the maximum wind speed scale of typhoon “Hagupit” was 15, the extreme wave height was 4.7 m, and the direct economic loss rate was 0.2109%, which was approximated by the three-dimensional information diffusion method to be 0.2167%.
Meanwhile, the conditional probability distribution of the extreme wave height is calculated based on the existing probability distribution of the maximum wind speed scale-extreme wave height (
Table 2). For example, when the maximum wind speed scale is 12, the probability values of different extreme wave heights are summed by
. Then the conditional distribution columns of different extreme wave heights are obtained by dividing each probability value by S, respectively (see
Table 4). Additionally, the direct economic loss rates under different wind speed scales are obtained by using the risk synthesis rule (risk = probability × loss) according to
Table 3 and
Table 4 (see
Table 5).
Comparing the three-dimensional information diffusion model based on conditional probability with the inference results of direct economic loss rate based on the two-dimensional information diffusion model (
Figure 7), when the wind speed scale is greater than 10, the slope of the vulnerability curve increases continuously. The direct economic loss rate increases significantly, showing an overall upward trend, which aligns with the cognitive law that the loss increases with the disaster level, reflecting the rationality of the information diffusion method. When the maximum wind speed scale is greater than 12, the projected value based on the 3D information diffusion model is slightly larger than that of the 2D information diffusion model, while when the maximum wind speed scale is greater than 12, the projected value is slightly smaller than that of the latter, analyzing this reason, because the model in this paper takes into account the influence of the extreme wave height, the extreme wave height corresponding to the sample data below level 12 is small. In contrast, the extreme wave height corresponding to the high wind speed scale typhoon is high, so the projected result shows the accuracy and stability of the 3-D information diffusion model and the necessity of considering other factors such as extreme wave height.
The advantages of the 3D information diffusion model are further analyzed, and the results of the 2D and 3D information diffusion approximations are compared in
Table 6 (the 2D information diffusion model only considers the maximum wind speed scale).
As can be seen from
Table 6, the two-dimensional information diffusion model only considers the maximum wind speed scale, so the inferred values of direct economic loss rates for the same wind speed scale are the same, while the three-dimensional information diffusion model has different results due to the consideration of the extreme wave height factor. The actual typhoon data show that the maximum wind speed at landfall is the same, but the direct economic loss rate is not the same, such as Typhoon 0601 “Chanchu” and Typhoon 0814 “Hagupit”, which maximum wind speed scale at landfall is 15, and the direct economic loss rate is 0.1677% and 0.2109%, respectively, and the two-dimensional information diffusion approximation is 0.2314%, while the three-dimensional information diffusion approximation is 0.2119% and 0.2167%, which is closer to the real value. Therefore, the estimation results of the 3D information diffusion model are more accurate and closer to the true values than the 2D information diffusion model when estimating the losses of typhoon disasters, which reflects the necessity of considering multiple contributing factors for typhoon disaster risk assessment.
3.2.3. Typhoon Risk Coupling
Since the probability distributions of typhoon maximum wind speed scale and extreme wave height and the vulnerability function of typhoon maximum wind speed scale-extreme wave height-direct economic loss rate obtained by 3D information diffusion are discrete distributions, the integral calculation form of the risk model is applied:
The average annual frequency of typhoons from 2005–2016 = 1.58, then the expected value of losses from the typhoon risk assessment model based on the update time factor under the conditions of maximum wind speed scale and extreme wave height is = 0.2542%.
The arithmetic mean of the calculated direct economic loss rate is 0.1273%, which is lower than the expected value calculated using the information diffusion technique because the calculation of the mean value considers the probability of occurrence of each loss value to be equal. While the probability density histogram shows that the probability of occurrence of each class of typhoon is different, and the nonlinearity of wind speed scale and extreme wave height sway the degree of loss, so the risk value of the disaster is not equal to the arithmetic mean.
The mean value of annual risk for typhoons is R′ = 0.2011%.
The comparison of the annual disaster loss expectation and mean value of the vulnerability indicator (direct economic loss rate) using the information diffusion method is shown in
Table 7.
From the above table, it is easy to see that the inferred values of expected losses of annual typhoon disasters based on the information diffusion method are all greater than the arithmetic mean. The inferred values of the direct economic loss rate using the two-dimensional information diffusion model are 12.53% higher than the arithmetic mean. The inferred values of the three-dimensional information diffusion model are 26.40% higher than the arithmetic mean and 12.33% higher than the inferred values of the two-dimensional information diffusion model, which shows that using the mean value as the risk degree of the typhoon will lead to low evaluation results. In addition, it is similar to the expected 0.26% of direct economic loss rate of the coastal cities of south-central Guangdong calculated in the paper [
35]. Additionally, it also shows that it is more reasonable to use the expected loss as the quantitative index of risk.