Stochastic simulation of precipitation data for preserving key statistics in their original domain and application to climate change analysis

Abstract

We propose a new method to estimate autoregressive model parameters of the precipitation amount process using the relationship between original and transformed moments derived through a moment generating function. We compare the proposed method with the traditional parameter estimation method, which uses transformed data, by modeling precipitation data from Denver International Airport (DIA), CO. We test the applicability of the proposed method (M2) to climate change analysis using the RCP 8.5 scenario. The modeling results for the observed data and future climate scenario indicate that M2 reproduces key historical and targeted future climate statistics fairly well, while M1 presents significant bias in the original domain and cannot be applied to climate change analysis.

Appendix 1: Example of M2 parameter estimation procedure

As an example, the parameter estimation procedure of M2 for the exponent c = 2 is presented below. The mean, variance, and lag-1 correlation for the original variable Z can be expressed as:

$$ {\mu}_Z=E\left({Z}_t\right)=E\left({N}_t^2\right)=\gamma \left(2,0\right)={\mu}_N\gamma \left(1,0\right)+\left(2-1\right){\sigma}_N^2\gamma \left(0,0\right)={\mu}_N^2+{\sigma}_N^2 $$

(A1)

$$ \begin{array}{l}{\sigma}_Z^2=E{Z}^2-{(EZ)}^2=E\left({N}^4\right)-{\left(E\left({N}^2\right)\right)}^2=\gamma \left(4,0\right)-{\left(\gamma \left(2,0\right)\right)}^2={\mu}_N\gamma \left(3,0\right)+\hfill \\ {}3{\sigma}_N^2\gamma \left(2,0\right)-{\left({\mu}_N^2+{\sigma}_N^2\right)}^2\hfill \end{array} $$

(A2)

$$ {\rho}_{1,z}=\frac{\gamma \left(2,2\right)-{\left(\gamma \left(2,0\right)\right)}^2}{\gamma \left(4,0\right)-{\left(\gamma \left(2,0\right)\right)}^2} $$

(A3)

$$ \gamma \left(2,2\right)={\mu}_N\gamma \left(1,2\right)+2{\sigma}_N^2{\rho}_1{{}_{,}}_N\gamma \left(1,1\right)+\gamma \left(0,2\right) $$

(A4)

Here, μ _z , σ ² _Z , and ρ _1,z are known values from the original amount data employing Eqs. (4)–(6), but for the original data before transformation. ${\widehat{\mu}}_N$ and ${\widehat{\sigma}}_N^2$ are estimated with two known values of μ _Z and σ ² _Z by numerically solving Eqs. (A1) and (A2). Note that Eq. (A2) is a function of μ _Z and σ ² _Z .

Furthermore, ρ _1,N is numerically solved with Eqs. (A3) and (A4) after estimating ${\widehat{\mu}}_N$ and ${\widehat{\sigma}}_N^2$. Additionally, ${\widehat{\sigma}}_{\varepsilon}^2$ is estimated from Eq. (7).