Further developments of a transient Poisson-cluster model for rainfall

Abstract

Stochastic point processes for rainfall are known to be able to preserve the temporal variability of rainfall on several levels of aggregation (e.g. hourly, daily), especially when the cluster approach is used. One major assumption in most of the applications todate is the stationarity of the rainfall properties in time, which must be reconsidered under a climate change hypothesis. Here, we propose new theoretical developments of a Poisson-based model with cluster, namely the Neyman–Scott Rectangular Pulses Model, which treats storm frequency as a nonstationary function. In this paper, storm frequency is modelled as a linear function of time in order to reproduce an increase (or decrease) of the mean annual precipitation. The basic theory is reconsidered and the moments are derived up to the third order. Then, a calibration method based on the generalized method of moments is proposed and discussed. An application to a rainfall time series from Uccle illustrates how this model can reproduce a trend for the average rainfall. This work opens new avenues for future developments on transient stochastic rainfall models and highlights the major challenges linked to this approach.

Appendix: General expressions of the moments

1.1 First order moment

Random variables ${X_{t-u}(u)}$ and ${\mathrm{d}} {N(t-u)}$ are usually independent. It means that the properties linked to the point generation do not influence the cell characteristics. In particular, the mean rainfall intensity at time ${t}$ is

$$ \begin{aligned} {\mathbb E}\{Y(t)\}=&{\mathbb E}\bigg\{\int\limits_{u=0}^{\infty} X_{t-u}(u) \mathrm{d} {N(t-u)}\bigg\}, \\ =& \int\limits_{u=0}^{\infty} {\mathbb E}\{X_{t-u}(u)\} {\mathbb E}\{{\mathrm{d}} {N(t-u)}\}. \end{aligned} $$

From Cox (1980, p. 75), we obtain ${{\mathbb E}\{N(t,t+\delta)\} = \mu_C \varphi(t) \delta,}$ so that

$$ {\mathbb E}\{Y(t)\}= \mu_C \int\limits_{u=0}^{\infty} {\mathbb E}\{X_{t-u}(u)\} \varphi(t-u) \mathrm{d}u. $$

(6)

Note that, to obtain ${{\mathbb E}\{X_{t-u}(u)\}}$ and then a complete expression for ${{\mathbb E}\{Y(t)\},}$ a bivariate distribution on cell intensity and duration needs to be specified. The mean of the aggregated process ${Y_i^h}$ is

$$ {\mathbb E}(Y_i^h)=\int\limits_{(i-1)h}^{ih} {\mathbb E}\{Y(t)\}\mathrm{d}t. $$

(7)

1.2 Second order moment

To derive the second-order properties of the aggregated process, we need to express the covariance function of the point process ${c(t,u)={\mathbb{C}}{\text{ov}}\big\{N(t,t+\delta_1),N(t+u,t+u+\delta_2)\big\}.}$ According to Cox and Isham (1980, p. 33), the covariance of the counting process between ${t_1}$ and ${t_2}$ is

$$ \begin{aligned} {{\mathbb{C}}{\text{ov}}}&\big\{N(t_1,t_1+\delta_1),N(t_2,t_2+\delta_2)\big\} ={{\text{Pr}}}\big\{N(t_1,t_1+\delta_1){=}N(t_2,t_2+\delta_2){=1}\big\}\\ &-{{\text{Pr}}}\big\{N(t_1,t_1+\delta_1){=1}\big\}{{\text{Pr}}}\big\{N(t_2,t_2+\delta_2) {=1}\big\}+o(\delta_1\delta_2). \end{aligned} $$

Following Cox and Isham (1980, p. 77), two points located at ${t_1}$ and ${t_2 \,>\,_1}$ may

• 1. belong to different clusters, in which case they are independent,

• 2. belong to the same cluster with center ${t_c.}$

The way in which cells are distributed in time can differ depending on whether the Neyman–Scott or Bartlett–Lewis process is studied. In the basic version of the Neyman–Scott process, cell origins are independently separated from storm origins by distances which are exponentially distributed with parameter ${\beta. }$ We have

$$ \begin{aligned} {\rm{Pr}}&\big\{N(t_1,t_1+\delta_1)=N(t_2,t_2+\delta_2)=1\big\} = \mu_C^2 \varphi(t_1) \varphi(t_2) \delta_1 \delta_2 \\ & + {\mathbb E}\{C(C-1)\} \beta^2 \int\limits_{-\infty}^{t_1} \varphi(t_c) {e}^{-\beta(t_1-t_c)} {e}^{-\beta(t_2-t_c)} \delta_1 \delta_2 \mathrm{d}t_c +o(\delta_1\delta_2). \end{aligned} $$

Consequently, the covariance between two points at ${t}$ and ${t+u}$ is

$$ \begin{aligned} c &(t,u)={{\mathbb{C}}{\rm{ov}}}\big\{N(t,t+\delta_1),N(t+u,t+u+\delta_2)\big\}\\ &= \left\{\begin{array}{ll} {\mathbb E}\{C(C-1)\} \beta^2 \int\limits_{-\infty}^t \varphi(t_c) {e}^{-\beta(2t+u-2t_c)} \delta_1 \delta_2 \mathrm{d}t_c +o(\delta_1\delta_2)& \hbox{for}\; u>0, \\ {\mathbb E}\{C(C-1)\} \beta^2 \int\limits_{-\infty}^{t+u} \varphi(t_c) {e}^{-\beta(2t+u-2t_c)} \delta_1 \delta_2 \hbox{d}t_c +o(\delta_1\delta_2)& \hbox{for} \;u<0. \end{array} \right. \end{aligned} $$

The Bartlett–Lewis process distributes cell origins according to a Poisson process with rate ${\beta, }$ the storm activity having a random duration which is exponentially distributed with parameter ${\gamma. }$ We obtain in this case

$$ \begin{aligned} c(t,u)&= \left\{\begin{array}{ll} \mu_C \beta \int\limits_{-\infty}^t \varphi(t_c) {e}^{-\gamma(t+u-t_c)} \delta_1 \delta_2 \hbox{d}t_c +o(\delta_1\delta_2) & \hbox{for}\; u>0, \\ \mu_C \beta \int\limits_{-\infty}^{t+u} \varphi(t_c) {e}^{-\gamma(t-t_c)} \delta_1 \delta_2 \hbox{d}t_c +o(\delta_1\delta_2) & \hbox{for} \; u<0. \end{array} \right. \end{aligned} $$

When ${u=0, }$ we have

$$ \begin{aligned} c(t,0)={{\mathbb{V}}{\rm{ar}}}\big\{N(t,t+\delta)\big\}=\mu_C \varphi(t) \delta +o(\delta). \end{aligned} $$

(8)

Note that (8) is valid for both Neyman–Scott and Bartlett–Lewis processes.

The autocovariance of the instantaneous rainfall intensity ${Y(t)}$ with lag ${\tau}$ can be expressed as Rodriguez-Iturbe et al. (1987, Eq.3.2) in terms of the covariance function ${c(t,u){:}}$

$$ {{\mathbb{C}}{\rm{ov}}}\{Y(t),Y(t+\tau)\} = \int\limits_0^{\infty} \int\limits_0^{\infty} {\mathbb E} \{X_{t-u}(u) X_{t+\tau-v}(v)\} c(t,\tau-v+u) {\mathrm{d}}u {\mathrm{d}}v. $$

(9)

Finally, the second-order properties of the aggregated process are obtained as

$$ \begin{aligned} {\mathbb{V}}{\text{ar}}(Y_i^h) &= {{\mathbb{V}}{\text{ar}}}\left\{\displaystyle\int\limits_{(i-1)h}^{ih} Y(t)\mathrm{d} t \right\} \\ &= 2\iint\limits_{\mathop{(i-1)h\leq t \leq i h}\limits_{0\leq \tau \leq i h-t}} {\mathbb{C}}{\text{ov}}\{Y(t),Y(t+\tau)\} \mathrm{d}t \mathrm{d}\tau \end{aligned} $$

and

$$ \begin{aligned} {\mathbb{C}}{\text{ov}}(Y_i^h,Y_{i+k}^h) &={\mathbb{C}}{\text{ov}}\left\{\int\limits_{(i-1)h}^{ih} Y(t){\mathrm{d}} t, \displaystyle\int\limits_{(i+k-1)h}^{(i+k)h} Y(t){\mathrm{d}} t \right\}\\ &= \displaystyle\iint\limits_{\mathop{(i-1)h \leq t_1 \leq ih}\limits_{(i+k-1)h \leq t_2 \leq (i+k)h}} {\mathbb{C}}{\text{ov}}\{Y(t_1),Y(t_2)\} {\mathrm{d}}t_1 {\mathrm{d}}t_2. \end{aligned} $$