## 2 Methodology
### Monte Carlo Simulation  
The Monte Carlo simulation method is a statistical computational technique that simulates probabilistic phenomena in practical problems by randomly generating system inputs, thereby enabling the estimation or prediction of expected outcomes [30]. When the Monte Carlo method is applied to simulate a specific process, the process must be decomposed into random variables describable by probability distributions. Statistical methods are then employed to estimate the numerical characteristics of the model, yielding numerical solutions to the practical problem. The procedure comprises three steps: (1) constructing or describing the probabilistic process, (2) sampling from known probability distributions, and (3) establishing estimators. The law of large numbers and the central limit theorem in probability theory form the theoretical foundation of the Monte Carlo method. The law of large numbers describes the behavior of the sum (or average) of a large number of random variables, stating that the sample mean converges to the expected value of the function. The Central Limit Theorem asserts that, regardless of the distribution of individual random variables, the sum of a sufficiently large number of independent random variables will approximately follow a normal distribution. This theorem provides insight into the distribution of Monte Carlo estimators when the number of samples is large but finite. On the basis of these two theorems, the fundamental principle of the Monte Carlo method can be described as follows: suppose that we are interested in a function \(Y=f(X_{1},X_{2},\ldots,X_{N})\), where the random variables \(X_{1},X_{2},\ldots,X_{N}\) follow known probability distributions. The Monte Carlo method draws samples either directly or indirectly to generate values \(x_{1i},x_{2i},\ldots,x_{ni}\) for each set of random variables and then computes the corresponding output \(y_{i}=f(x_{1i},x_{2i},\ldots,x_{ni})\). Repeating this sampling process independently for \(m\) iterations yields a set of sample outputs \(y_{1},y_{2},\ldots,y_{m}\), which approximately follow a normal distribution. When the number of samples is sufficiently large, the resulting distribution of the function \(Y\) closely approximates its true probabilistic behavior. To formalize this, one first constructs a probability space and defines a statistic \(g(x)\) that depends on the random variable \(x\), with an expected value given by  
\[E(g)=\int g(x)dF(x) \tag{1}\]  
where \(F(x)\) is the cumulative distribution function of \(x\). Then, a simple random sample \(x_{1},x_{2},\ldots,x_{i},\ldots,x_{n}\) is generated from the distribution function. The arithmetic mean of the corresponding statistics \(g(x_{1}),g(x_{2}),\ldots,g(x_{n})\) is used to approximate the desired value \(G\):  
In general, the \(G_{N}\) serves as an unbiased estimator of \(E[g(x)]\). \[\overline{G}_{N}=\frac{1}{N}{\sum_{i=1}^{N}g(x_{i})} \tag{2}\]  
This study uses the Monte Carlo method to generate synthetic typhoons, with the workflow illustrated in Figure 2. On the basis of the prescreened typhoon sample set influencing the target study area, key parameters are extracted and fitted. The annual typhoon count is determined by sampling from the occurrence rate (\(\lambda\)), whereas other key parameters (e.g., \(\theta=[-180^{\circ},180^{\circ}]\), \(D_{min}=[-250\) km, 250 km\(]\)) are stochastically sampled, ensuring physical plausibility. The linear trajectory model assumes uniform climatic conditions within the simulation circle. Nonlandfalling typhoons maintain constant intensity, with their initial generation points derived inversely from the movement direction (\(\theta\)), minimum distance (\(D_{min}\)), and simulation circle radius. These initial points are predefined over the ocean. Typhoon paths are represented as straight lines discretized into sequential points along the directions \(\theta\) and \(D_{min}\).