On the computation of area probabilities based on a spatial stochastic model for precipitation cells and precipitation amounts

Abstract

A main task of weather services is the issuing of warnings for potentially harmful weather events. Automated warning guidances can be derived, e.g., from statistical post-processing of numerical weather prediction using meteorological observations. These statistical methods commonly estimate the probability of an event (e.g. precipitation) occurring at a fixed location (a point probability). However, there are no operationally applicable techniques for estimating the probability of precipitation occurring anywhere in a geographical region (an area probability). We present an approach to the estimation of area probabilities for the occurrence of precipitation exceeding given thresholds. This approach is based on a spatial stochastic model for precipitation cells and precipitation amounts. The basic modeling component is a non-stationary germ-grain model with circular grains for the representation of precipitation cells. Then, we assign a randomly scaled response function to each precipitation cell and sum these functions up to obtain precipitation amounts. We derive formulas for expectations and variances of point precipitation amounts and use these formulas to compute further model characteristics based on available sequences of point probabilities. Area probabilities for arbitrary areas and thresholds can be estimated by repeated Monte Carlo simulation of the fitted precipitation model. Finally, we verify the proposed model by comparing the generated area probabilities with independent rain gauge adjusted radar data. The novelty of the presented approach is that, for the first time, a widely applicable estimation of area probabilities is possible, which is based solely on predicted point probabilities (i.e., neither precipitation observations nor further input of the forecaster are necessary). Therefore, this method can be applied for operational weather predictions.

Appendix

We give a brief derivation of Eqs. (6) and (7) for the conditional expectation and variance of the random precipitation amount $\Gamma _t$ at $t \in W$ given $\{E=e\}$. In order to derive (7), we need the following general result for Poisson processes. Let $\{Y_i, i=1,2,\ldots \}$ be a Poisson process in $\mathbb {R}^2$ with locally integrable intensity function $\lambda :\mathbb {R}^2 \rightarrow [0, \infty )$, second-order moment measure $\mu ^{(2)}:\mathcal {B}(\mathbb {R}^2 \times \mathbb {R}^2) \rightarrow [0,\infty ]$ and second-order product density $\varrho ^{(2)}:\mathbb {R}^2 \times \mathbb {R}^2 \rightarrow [0,\infty )$. Furthermore, let $f,g:\mathbb {R}^2 \rightarrow [0,\infty )$ be two nonnegative measurable functions. By using the definition of the second-order moment measure and the result that $\varrho ^{(2)}(x,y)=\lambda (x)\lambda (y)$ for $x,y \in \mathbb {R}^2$, see e.g. Illian (2008), p. 119, we get

$$\begin{aligned}& \mathbb {E}\left( \sum \limits _{i=1}^{\infty } f(Y_i) \sum \limits _{j=1}^{\infty } g(Y_j) \right)= \mathbb {E}\left( \sum \limits _{i,j=1}^{\infty } f(Y_i)\,g(Y_j)\right) \\&\quad= \int \int f(x)\,g(y)\,\mu ^{(2)} (\text{d}(x,y))\\&\quad= \int \int f(x)\,g(y)\,\varrho ^{(2)}(x,y)\,\text{d}(x,y) + \int f(x)\,g(x)\,\lambda (x)\,\text{d}x\\&\quad= \int f(x)\,\lambda (x)\,\text{d}x \int g(y)\,\lambda (y)\,\text{d}y + \int f(x)\,g(x)\,\lambda (x)\,\text{d}x. \end{aligned}$$

We start with Eq. (6) for the conditional expectation $\mathbb {E}(\Gamma _t\,|\,E=e)$ of $\Gamma _t$ given $\{E=e\}$. In the following, we again use the notation introduced in Sect. 4, i.e., let $a_j = \mathbb {E}(A_j\,|\,E=e)$, $c_j = \mathbb {E}(C_j \,|\,E=e)$ and $\tilde{c}_j = \text{ var }(C_j \,|\,E=e)$ for $j=1, \ldots ,n$ and $r=\mathbb {E}(R\,|\,E=e)$. Recall that conditioned on $\{E=e\}$ the point process $\{X_i, i =1, \ldots , Z\}$ is a Poisson process with intensity function $\{\lambda _t, t \in W\}$, where $\lambda _t = \sum _{j=1}^{n} a_j I_{V(s_j)}(t)$ for all $t \in W$. Furthermore, $\{X_i, i =1, \ldots , Z\}$ is conditionally independent of the scaling variables $C_1, \ldots , C_n$ given $\{E=e\}$. By applying the Campbell theorem for point processes (see e.g. Chiu (2013), Theorem 4.1) we get

$$\begin{aligned}& \mathbb {E}(\Gamma _t \,|\, E=e)= \mathbb {E}\left( \sum \limits _{i=1}^Z \sum \limits _{j=1}^n C_j I_{V(s_j)}(X_i) K_p(t,X_i, R)\,\Bigr |\, E=e \right) \\&\quad= \sum \limits _{j=1}^n c_j \,\mathbb {E}\left( \sum \limits _{i=1}^Z I_{V(s_j)}(X_i) K_p(t,X_i, r)\,\Bigr |\, E=e \right) \\&\quad= \sum \limits _{j=1}^n c_j \,\int I_{V(s_j)}(x) \left( 1-\dfrac{|t-x|^2}{r^2} \right) ^p I_{b(t,r)}(x) \sum \limits _{k=1}^{n} a_k I_{V(s_k)}(x)\,\text{d}x\\&\quad= \sum \limits _{j=1}^{n}\, \mathbb {E}(C_j\,|\,E=e) \;a_j\, \int _{V(s_j) \cap b(t,r)} \left( 1-\dfrac{|t-x|^2}{r^2}\right) ^p \text{d}x. \end{aligned}$$

Now, we consider the conditional variance $\text{ var }(\Gamma _t \,|\, E=e)$ for a fixed $t \in W$. To simplify the notation we introduce the function $f_j:\mathbb {R}^2 \rightarrow [0,\infty )$ with $f_j(x) = I_{V(s_j)}(x)K_p(t,x,r)$ for all $x \in \mathbb {R}^2$ and $j=1, \ldots ,n$. Obviously, $f_j(x)f_k(x)=0$ for all $x \in \mathbb {R}^2$ if $j\ne k$. Furthermore, $\int f_j(x)\,\text{d}x = I(s_j,t)$ and $\int f_j^2(x)\,\text{d}x = \tilde{I}(s_j,t)$ for $j=1, \ldots ,n$, where $I(s_j,t)$ and $\tilde{I}(s_j,t)$ are defined according to Eqs. (8) and (9). By using the result for Poisson processes shown before and that conditioned on $\{E=e\}$, the scaling variables $C_1, \ldots , C_n$ are independent of each other and of $\{X_i, i=1, \ldots , Z\}$, we get that

$$\begin{aligned}&{\mathbb {E}}\left( \Gamma _t^2\,\left |\,E=e\right) = {\mathbb {E}}\left( \left( \sum \limits _{i=1}^Z \sum \limits _{j=1}^n C_j I_{V(s_j)}(X_i) K_p(t,X_i, R)\right) ^2\left| E=e \right)\right.\right. \\&\quad= {\mathbb {E}}\left( \sum \limits _{j=1}^n \sum \limits _{k=1}^n C_j C_k \sum \limits _{i=1}^Z I_{V(s_j)}(X_i) K_p(t,X_i, R)\right.\\&\left.\left.\qquad \times \sum \limits _{l=1}^Z I_{V(s_k)}(X_l) K_p(t,X_l, R)\,\right |\, E=e \right) \\&\quad= \sum \limits_{j=1}^n \sum \limits_{k=1}^n {\mathbb {E}}(C_j C_k\, | \,E=e) \,{\mathbb{E}}\left( \sum \limits _{i=1}^Z f_j(X_i) \sum \limits_{l=1}^Z f_k(X_l) \,|\, E=e \right) \\&\quad= \sum \limits _{j=1}^n \sum \limits _{k=1}^n \mathbb {E}(C_j C_k\,\left|\,E=e)\right.\\&\qquad \times \left( \int f_j(x)\, a_j\,\text{ d }x \int f_k(x)\, a_k\,\text{d}x + \int f_j(x)\,f_k(x)\, \lambda _x\,\text{d}x \right) \\&\quad= \sum \limits _{j=1}^n {\mathbb {E}}(C_j^2\,|\,E=e)\left( a_j^2\, I^2(s_j,t) + a_j\,\tilde{I}(s_j,t) \right)\\ & \qquad + \sum \limits _{j=1}^n \sum \limits _{\begin{array}{c} k=1\\ k \ne j \end{array}}^n c_j\,c_k\,a_j\,a_k\, I(s_j,t)\,I(s_k,t). \end{aligned}$$

Moreover, according to Eq. (6), we get

$$\begin{aligned} &\bigl (\mathbb {E}\left( \Gamma _t\,|\,E=e\right) \bigr )^2\\&\quad= \left( \sum \limits _{j=1}^{n}\, c_j \,a_j\, \int _{V(s_j) \cap b(t,r)} \left( 1-\dfrac{|t-x|^2}{r^2}\right) ^p \text{d}x\right) ^2\\&\quad= \sum \limits _{j=1}^n \sum \limits _{k=1}^n c_j\,c_k\,a_j\,a_k\, I(s_j,t)\,I(s_k,t)\\&\quad= \sum \limits _{j=1}^n c_j^2\,a_j^2\,I^2(s_j,t) + \sum \limits _{j=1}^n \sum \limits _{\begin{array}{c} k=1\\ k \ne j \end{array}}^n c_j\,c_k\,a_j\,a_k\, I(s_j,t)\,I(s_k,t). \end{aligned}$$

Finally, combining both representation formulas results in

$$\begin{aligned} \text{ var }(\Gamma _t \,|\, E=e)&= \mathbb {E}\left( \Gamma _t^2\,\Bigr |\,E=e\right) - \bigl (\mathbb {E}\left( \Gamma _t\,|\,E=e\right) \bigr )^2\\&= \sum \limits _{j=1}^n \mathbb {E}(C_j^2\,|\,E=e)\left( a_j^2\, I^2(s_j,t) + a_j\,\tilde{I}(s_j,t) \right) - \sum \limits _{j=1}^n c_j^2\,a_j^2\,I^2(s_j,t)\\&= \sum \limits _{j=1}^{n} \tilde{c}_j\left[ a_j \tilde{I}(s_j,t) + a_j^2 I^2(s_j,t) \right] + \sum \limits _{j=1}^{n} c_j^2 a_j \tilde{I}(s_j,t), \end{aligned}$$

which coincides with the representation formula for the conditional variance of $\Gamma _t$ given in (7).