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Statistical properties of the rainfall model¶

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In model fitting it is usually necessary to use equations for aggregated properties, because rainfall data are usually sampled over discrete time intervals. Let $Y_{ij}^{h}(x)$ be the aggregated time series of rainfall due to type $i$ storms at point $x = (x, y) ∈ R^2$ in the $j-th$ time interval of width $h$, and let $Y_{j}^h(x)$ be the total rainfall in the $j-th$ interval due to the superposition of the $n$ storm types [1]. Then,

$$Y_{j}^h (x) = \sum_{i=1}^{n} \int_{(j-1)/h}^{jh} Y_i (x, t)dt$$

where $Y_i (x, t)$ is the rainfall intensity at point $x$ and time $t$ due to type $i$ storms $(i = 1, ..., n)$. Since the superposed processes are independent, statistical properties of the aggregated time series follow just by summing the various properties given below [1].

Mean ([2] Eq.12):

$$\mu(h)= \sum_{i=1}^{n} E\lbrace Y_{ij}^{h}(x)\rbrace=h\sum_{i=1}^{n}\frac{\lambda_{i}\nu_{i}}{\chi_{i}\eta_{i}}$$

in the STNSRP model, $\chi_{i}$ must me change by $\chi_{i}^-1$

Covariance ([2] Eq.14):

$$\gamma(x,x, l, h) = \sum_{i=1}^{n} Cov\lbrace Y_{ij}^{h}, Y_{i,j+l}^{h}\rbrace = \frac{\lambda_i(\nu_{i}^{2}-1)[\beta_i^{3}A_i(h, l)-\eta_i^{3}B_i(h,l)]}{\beta_i\xi^{2}\eta_i^{3}(\beta_i^{2}-\eta_i^{2})} - \frac{4\lambda_i \chi_{i}A_i(h,l)}{\xi_i^{2}\eta_i^{3}}$$

when $l =0$,

$$A_i(h,l)=A_i(h)=\eta_i h -1 + e^{-\eta_i h}$$$$B_i(h,l)=B_i(h)=\beta_i h -1 + e^{-\beta_i h}$$

when $l >0$,

$$A_i(h,l)=0.5(1-e^{-\eta_i h})^{2}e^{-\eta_i h(l-1)}$$$$B_i(h,l)=0.5(1-e^{-\beta_i h})^{2}e^{-\beta_i h(l-1)}$$

Probability of not rain in an arbitrary time of length $h$ ([3] Eq.6):

The probability that an arbitrary time interval $[(j − 1)h, j h]$ is dry at a point is obtained by multiplying the probabilities of the independent processes and is given by the following:

$$\phi(h) = \sum_{i=1}^{n} Pr\lbrace Y_{ij}^{(h)}(x)=0 \rbrace = exp(-\lambda_i h + \lambda_i \beta_i^{-1}(\nu_{i} -1)^{-1}\lbrace1-exp[1-\nu_{i} + (\nu_{i}-1) e^{-\beta_i h}]\rbrace -\lambda_i\int_{0}^{\infty}[1-p_{h}(t)]dt)$$

We use the approximation shown in [2] Eq.17 to avoid having to solve the integral:

$$\int_{0}^{\infty}[1-p_{h}(t)]dt=\beta_i^{-1}[\gamma +ln(\alpha \nu_{i}-1)]$$

where $\gamma =0.5771 $ y $\alpha_i = \eta_i/(\eta_i - \beta_i)-e^{-\beta_i h}$

Transition probabilities:

The transition probabilities, $pr \lbrace Y_{i,j+1}^{(h)}(x)>0 | Y_{ij}^{(h)}(x)>0 \rbrace$ and $pr \lbrace Y_{i,j+1}^{(h)}(x)=0 | Y_{i}^{(h)}(x)=0 \rbrace$, denoted as $\phi_{WW}(h)$ y $\phi_{DD}(h)$, respectively, can be expressed in terms of $\phi(h)$ following ([3] Eq.7,8 and 9):

$$\phi_{DD}(h)=\phi(2h)/\phi(h)$$$$\phi(h)=\phi_{DD}(h) + \lbrace 1-\phi_{WW}(h) \rbrace \lbrace(1-\phi(h))\rbrace$$$$\phi_{WW}(h)=\lbrace 1-2 \phi(h) + \phi(2h) \rbrace\lbrace 1 -\phi(h) \rbrace$$

The third moment function ([4] Eq.10):

$$\xi_{h}=E\lbrace Y_{j}^{(h)}(x)-\mu(h)\rbrace^{3}= \sum_{i=1}^{n}[ 6\lambda_i\nu_{i}E(X^{3})(\eta_i h -2 +\eta_i h e^{-\eta_i h}+2e^{-\eta_i h})/\eta_i^{4} + 3\lambda_i\chi_i E(X^{2})\nu_{i}^2 f(\eta_i, \beta_i, h)/{\lbrace2\eta_i^{4}\beta_i (\beta_i^{2}-\eta_i^{2})^{2}}\rbrace + \lambda_i \chi_i^{3}\nu_{i}^{3} g(\eta_i, \beta_i, h)/{\lbrace e\eta_i^{4}\beta_i(e\eta_i^{2}-\beta_i^{2})(\eta_i-\beta_i)(2\beta_i +\eta_i)(\beta_i +2\eta_i)}\rbrace]$$

In the STNSRP model, C is a Poisson random variable, so that $E\lbrace C(C-1)\rbrace = \nu^2$ and $E\lbrace C(C-1)(C-2)\rbrace = \nu^3$ If is a geometric then $E\lbrace C(C-1)\rbrace = 2\nu^2(\nu-1)$ and $E\lbrace C(C-1)(C-2)\rbrace = 6\nu^2(\nu-1)^{2}$

For exponential cell intensities, $E(X_{ijk})$ and $E(X_{ijk}^{k})$ are replaced by $2\chi_i^{2}$ and $6\chi_i^{3}$ respectively.

$f(\eta_i, \beta_i, h)$ and $g(\eta_i, \beta_i, h)$ are derived bellow:

$$f(\eta_i, \beta_i, h) = -2\eta^{3}\beta^{2}\exp(-\eta h)-2\eta^{3}\eta^{2}\exp(-\beta h)+\eta^{2}\beta^{3}\exp(-2\eta h)+2\eta^{4}\beta \exp(-\eta h)+2\eta^{4}\beta \exp(-\beta h)+2\eta^{3}\beta^{2}\exp(-(\eta+\beta) h)-2\eta^{4}\beta\exp(-(\eta+\beta)h)-8\eta^{3}\beta^{3} h+11 \eta^{2}\beta^{3}-2\eta^{4}\beta+2\eta^{3}\eta^{2}+4\eta\beta^{5} h+4\eta^{5}\beta h-7 \beta^{5}-4\eta^{5}+8\beta^{5} \exp(-\eta h)-\beta^{5}\exp(-2\eta h)\ -2 h \eta^{3}\beta^{3}\exp(-\eta h)-12\eta^{2}\beta^{3}\exp(-\eta h)+2 h \eta \beta^{5}\exp(-\eta h)+4 \eta^{5}\exp(-\beta h)$$

$$g(\eta_i, \beta_i,h)=12\eta^{5}\beta\exp(-\beta h)+9\eta^{4}\beta^{2}+12\eta \beta^{5}\exp(-\eta h)+9 \eta^{2}\beta^{4}+12\eta^{3}\beta^{3}\exp(-(\eta+\beta)h)-\eta^{2}\beta^{4}\exp(-2\eta h)-12 \eta^{3}\beta^{3}\exp(-\beta h)-9\eta^{5}\beta-9\eta \beta^{5}-3\eta\beta^{5}\exp(-2\eta h)-\eta^{4}\beta^{2}\exp(-2\beta h)-12\eta^{3}\beta^{3}\exp(-\eta h)+6\eta^{5}\beta^{2} h-10\beta^{4}\eta^{3}h+6\beta^{5}\eta^{2}h-10\beta^{3}\eta^{4} h+4\beta^{6}\eta h-8\beta^{2}\eta^{4}\exp(-\beta h)+4\beta\eta^{6} h+12\beta^{3}\eta^{3}-8\beta^{4}\eta^{2}\exp(-\eta h)-6\eta^{6}-6\beta^{6}-2\eta^{6}\exp(-2\beta h)-2\beta^{6}\exp(-2\eta h)+8\eta^{6}\exp(-\beta h)+8\beta^{6}\exp(-\eta h)-3\beta\eta^{5}\exp(-2\beta h)$$

The following aggregated statistics only affect to the STNSRM¶

For each storm, the number of cells $\nu$ that overlap a point in $R^2$ is a Poisson random variable with mean ([5] Eq.3):

$$\nu_C = \nu \phi_c^2/(2\pi) $$

where $\nu_C$ denotes the two-dimensional Poisson process (cells per $km^2$)

The probability that a cell overlaps a point $x$ given that it overlapped a point $y$ a distance $d$ from $x$ ([5], Eq.9)

$$ P(\phi, d)\approx \frac{1}{30} \sum_{i=1}^{4} \lbrace 2f(\frac{2 \pi i}{20}) + 4f(\frac{2\pi i + \pi}{20})\rbrace - \frac{1}{30} f(0) $$

where

$$f(y)=(\frac{\phi d}{2 cosy}+1)exp(\frac{-\phi d}{2cosy}), 0 \le y < \pi/2, f(\pi/2)$$

Cross-correlation ([5], Eq.6):

$$\gamma(x,y,l,h)=\sum_{i=1}^{n} Cov\lbrace Y_{ij}^{h}(x), Y_{i,j+l}^{h}(y)\rbrace = \sum_{i=1}^{n} [\gamma_i(x,x,l,h)-2\lambda_i\lbrace1-P(\phi_i, d)\rbrace\nu_{Ci}E(X^2)A_i(h,l)/\eta_i^3]$$

References¶

[1]: Cowpertwait, P., Ocio, D., Collazos, G., De Cos, O., Stocker, C. Regionalised spatiotemporal rainfall and temperature models for flood studies in the Basque Country, Spain (2013) Hydrology and Earth System Sciences, 17 (2), pp. 479-494.

[2]: Cowpertwait, P.S.P. Further developments of the neyman‐scott clustered point process for modeling rainfall (1991) Water Resources Research, 27 (7), pp. 1431-1438.

[3]: Cowpertwait, P.S.P., O'Connell, P.E., Metcalfe, A.V., Mawdsley, J.A. Stochastic point process modelling of rainfall. I. Single-site fitting and validation (1996) Journal of Hydrology, 175 (1-4), pp. 17-46.

[4]: Cowpertwait, P.S.P. A Poisson-cluster model of rainfall: High-order moments and extreme values (1998) Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 454 (1971), pp. 885-898.

[5]: Cowpertwait, P.S.P., Kilsby, C.G., O'Connell, P.E. A space-time Neyman-Scott model of rainfall: Empirical analysis of extremes (2002) Water Resources Research, 38 (8), pp. 6-1-6-14.