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All content in this area was uploaded by Rasiah Thayakaran on Jun 13, 2016

Content may be subject to copyright.

Stochastic point process model for ﬁne-scale

rainfall time series

N. I. Ramesh and R. Thayakaran

School of Computing and Mathematical Sciences

University of Greenwich

Old Royal Naval College

Park Row, Greenwich , London SE10 9LS, UK

(e-mail: N.I.Ramesh@greenwich.ac.uk)

Abstract. A stochastic point process model, which is constructed from a class of

doubly stochastic Poisson processes, is proposed to analyse point rainfall time se-

ries observed in ﬁne sub-hourly time scales. Under the framework of this stochastic

model, rain cells arrive according to a Poisson process whose arrival rate is gov-

erned by a ﬁnite-state Markov chain. Each cell of the point process has a random

lifetime during which instantaneous random depths (pulses) of rainfall bursts oc-

cur as another Poisson process. The structure of this model enables us to study

the variability of rainfall characteristics at small time intervals. The covariance

structure of the pulse occurrence process is studied. Second-order properties of the

time series of cumulative rainfall in discrete intervals are derived to model 5-minute

rainfall data, over a period of 48 years, from Germany. The results show that the

proposed model is capable of reproducing rainfall properties well at various sub-

hourly resolutions.

Keywords: Doubly Stochastic Poisson process, Fine-scale rainfall, Point process,

Stochastic models, Rainfall pulse.

1 Introduction

Stochastic point process models for rainfall have been studied extensively

by many authors over the years. Much of the work has focused on models

based on Poisson cluster processes (Rodriguez-Iturbe et al [11], Cowpertwait

1994 [3], Onof 1994 [5], Chandler 2007 [1]) utilizing either the Neyman-Scott

or Bartlett-Lewis processes. Rainfall models based on Markov process have

also considered by some authors (Smith and Karr 1983 [12], Ramesh 1998 [9],

Onof et al 2002 [6], amongst others). However, the majority of the literature

on this topic has concentrated on modelling rainfall data recorded at hourly

or higher aggregation level. In some hydrological applications there is a need

to reproduce rainfall time series at much smaller aggregation level. There

has been some work lately on modelling ﬁne-scale rainfall data using point

process models. Cowpertwait et al 2007 [2] developed a Bartlett-Lewis pluse

model to study ﬁne-scale rainfall structure whereas Ramesh et al, 2011 [10]

considered a class of doubly stochastic Poisson processes to study ﬁne-scale

rainfall intensity using rainfall bucket tip time series.

2 N. I. Ramesh and R. Thayakaran

In this paper, following the approach suggested in Cowpertwait et al

2007 [2], we develop a simple point process model based on a double stochas-

tic Poisson process to analyse rainfall time series collected at sub hourly

ﬁne-scale resolution. Expressions for the second-order properties of the accu-

mulated rainfall in disjoint intervals are derived. The proposed model is ﬁtted

to 48 years of 5-minute rainfall time series from Germany. The results show

that the model is capable of reproducing rainfall properties well at various

sub-hourly resolutions.

2 Model framework

We shall start with a brief description of the doubly stochastic Poisson pro-

cesses (DSPP), as the model we propose is derived from a special class of

this process. A DSPP is a point process where the arrival rate of a Poisson

process itself becomes a stochastic process. A special class of tractable DSPP

emerges when the arrival rate of the point process is governed by a ﬁnite-state

irreducible Markov chain. This process is also called a Markov-modulated

Poisson process (MMPP), see for example, Ramesh 1995 [8] amongst others.

The model we propose in this paper, to study ﬁne-scale rainfall time series,

is based on this class of DSPP.

Suppose that the rain cells arrive according to a DSPP on two states

where the arrival rate is switching between the high intensity (φ2) and low

intensity (φ1) states at random times controlled by the underlying Markov

chain that has transition rates λ(for 1 →2) and µ(for 2 →1). Each

rain cell has a random lifetime of length Land a cell originated at time Ti

terminates at time Ti+Li. The cell lifetimes Liare taken to be independent

and exponentially distributed with parameter η. During the lifetime of each

cell, [Ti, Ti+Li), instantaneous random pulses of rainfall at times Tij occur

according to another Poisson process at rate ξ. The process of pulse arrival

terminates with the cell lifetime. Hence each cell of the DSPP generates a

series of pulses during its lifetime and associated with each pulse is a random

rainfall depth, Xij , and therefore the process {Tij , Xij }becomes a marked

point process (Cox & Isham, 1980 [4]). In our derivation of model properties

in Section 3, we treat the pulses in distinct cells as independent but allow

those within a single cell to be dependent. We refer this model as the doubly

stochastic pulse model (DSP).

3 Covariance structure of pulse arrival process

As the properties of the pulse arrival process are functions of those of the cell

arrival process, we shall ﬁrst see the properties of the cell arrival process. The

second-order properties of the two state DSPP can be obtained as functions

of the parameters {λ, µ, φ1, φ2}(Ramesh 1998 [9]). The mean arrival rate

Stochastic point process model for ﬁne-scale rainfall time series 3

of this cell arrival process M(t) is written as E(M(t)) = m=λφ2+µφ1

λ+µ.The

Covariance density of M(t), for t > 0,

cM(t) = λµ

(λ+µ)2(φ1−φ2)2e−(λ+µ)t=Ae−(λ+µ)t(1)

shows that the covariance of the cell arrival process decays exponentially with

time. We shall now study the covariance structure of the pulse arrival process

and focus our attention on deriving an expression for its covariance density

which will then be used in the derivation of the statistical properties of the

aggregated rainfall process in Section 4.

In this DSP model framework, the cell lifetimes Liare assumed to follow

exponential distribution with parameter ηand therefore we have E(Li) = 1

η.

Let N(t) be the counting process of pulse occurrences from all cells. If a cell

is active then it generates a series of instantaneous pulses at Poisson rate

ξduring its lifetime and therefore the mean number of pulses per cell is ξ

η.

Hence the mean arrival rate of pulses is E(N(t)) = mξ

η.

To derive an expression for the covariance density of this DLP process, we

ﬁrst studied the product density of the point process (Cox & Isham, 1980)

at distinct time points. We considered two distinct pulses at time tand t+u

(u > 0), which may come from the same cell or diﬀerent cells, and obtained

an expression for the product density which was then used to obtain the

covariance density of this DLP process for u≥0 as

c(u) = mξ

ηδ(u) + A1e−(λ+µ)u+ [B2−B1]e−ηu (2)

where A1=ξ2A

η2−(λ+µ)2,B1=ξm

η2+ξ2A

η2−(λ+µ)2and B2=ξ2m

η.

Here A1and B1correspond to the contribution from pulses generated by dif-

ferent cells whereas B2corresponds to the contribution from diﬀerent pulses

within the same cell, where the depths of these pulses may be dependent.

4 Properties of the aggregated rainfall

Although our DSP process evolves in continuous time, the rainfall data are

usually available in aggregated form in equally spaced discrete time intervals.

We, therefore, develop expressions for the second-order properties of the ag-

gregated rainfall process which can be used for model ﬁtting and assessment.

Let Y(h)

ibe the total rainfall in disjoint time intervals of ﬁxed length h, for

i= 1,2,..., then it can be expressed as

Y(h)

i=Zih

(i−1)h

X(t)dN(t),

4 N. I. Ramesh and R. Thayakaran

where X(t) is the depth of a pulse at time t. Let E(X(t)) = µxbe the mean

depth of the pulses. The mean of the aggregated rainfall can be written as

EhY(h)

ii=Zih

(i−1)h

E(X(t))dN(t) = mξ

ηµxh. (3)

The variance and autocovariace function of the aggregated rainfall process

can now be worked out using the covariance density of the pulse arrival

process given in (2). In this dervivation, we need to distinguish whether

the pulses at time tand sbelong to the same cell or come from diﬀerent

cells. This will allow us to accommodate some within-cell depth dependence.

However, it is assumed that any two pulses within a cell, regardless of their

location within the cell, have the same expected product moment of depths.

In this set up, the variance function turns out to be

Var hY(h)

ii=E(X2)mξ

ηh

+ 2 µ2

xA1ψ1(λ+µ)+2E[XijXik ]B2−B1µ2

xψ1(η) (4)

where ψ1(λ+µ) = [(λ+µ)h−1+e−(λ+µ)h]

(λ+µ)2and ψ1(η) = [ηh−1+e−ηh ]

η2.

Similarly, the autocovariance function for the aggregated rainfall in two

distinct intervals can be derived, by distinguishing the contributions from

pulses within the same cell, and this is given below, for k≥1,

cov hY(h)

i, Y (h)

i+ki=Z(k+1)h

kh Zh

0

cov [X(s)dN(s), X(t)dN (t)]

=µ2

xA1ψ2(λ+µ) + E[Xij Xik]B2−B1µ2

xψ2(η) (5)

where ψ2(λ+µ) = e−(λ+µ)(k−1)h[1−e−(λ+µ)h]2

(λ+µ)2and ψ2(η) = e−η(k−1)h[1−e−ηh]2

η2.

When considering the special case where all pulse depths are independent

E(Xij Xik) can be replaced by µ2

xin equations (4) and (5).

5 Model ﬁtting and assessment

We use our DSP model to analyse 48 years (1960 - 2007) of 5-minute rainfall

data from Dortmund (courtesy of Emschergenossenschaft/Lippeverband) in

the Bochum region around the river Ems in Germany and assess how well

the ﬁtted model reproduces the properties of the rainfall over a range of sub

hourly resolutions. In this work, we shall restrict ourselves to the special

case where the pulse depths X0s

ij are independent random variables with an

exponential distribution. Our model then has 7 parameters but we estimate

the 6 parameters by the method of moment approach using the observed

and theoretical values of the second-order properties. The parameter µxis

Stochastic point process model for ﬁne-scale rainfall time series 5

estimated separately for each month using the sample mean by the following

equation

µx=η

mξ ¯x

where ¯xis the estimated average of hourly rainfall for each month.

The following dimensionless functions, coeﬃcient of variation ν(h) and

the autocorrelation at lag 1 ρ(h) of the aggregated rainfall process, are used

to estimate the remaining 6 parameters of the model.

ν(h) =

EY(h)

i−EhY(h)

ii21/2

E(Y(h)

i), ρ(h) = Corr hY(h)

i, Y (h)

i+1i

The above properties of the aggregated process at 4 diﬀerent agrregation

levels (at h=1/12, 1/3, 1/2 and 1 hour) are used in our estimation. The es-

timates of the functions from the empirical data, denoted by ˆν(h) and ˆρ(h),

are calculated for each month using 48 years of 5-minute rainfall series accu-

mulated at appropriate scales. The estimated values of the model parameters

{ˆ

λ, ˆµ,ˆ

φ1,ˆ

φ2, ˆηand ˆ

ξ}for each month can be obtained by minimizing the

weighted sum of squares of dimensionless functions as given below using stan-

dard routines. Here the weights are taken as the reciprocal of the variance of

the empirical values of the functions calculated separately for the 48 years.

X

h=1

12 ,1

3,1

2,1"1

ˆ

var(ν(h)) (ˆν(h)−ν(h))2+1

ˆ

var(ρ(h)) (ˆρ(h)−ρ(h))2#.

The above objective function is minimized, using the simplex algorithm by

Nelder & Mead, separately for each month to obtain estimates of the model

parameters. Values of ˆµare larger for summer months showing smaller mean

sojourn times (1/µ) in higher rainfall intensity state. The estimates ˆ

φ2and

ˆ

ξare also higher, in general, for the summer months and show that the cell

arrival rates vary from about 55 to 82 per hour whereas the pulse arrival rates

range from 106 to 178 per hour throughout the year. The mean duration of

cell lifetime (1/η) falls between 1.3 to 2 minutes.

The empirical and ﬁtted values of the mean, standard deviation, coef-

ﬁcient of variation and lag 1 autocorrelation of the aggregated rainfall are

displayed in Figures 1 and 2. In almost all cases a near perfect ﬁt, exact

ﬁt in some cases, was obtained for all properties. An exception is the lag1

autocorrelation at 1 hour aggregation level where there appears to be a slight

underestimation. Nevertheless the diﬀerences in the correlations are less than

0.1 and the model does well at small time-scales. One point to note here is

that h=10 minutes aggregation was not used in the ﬁtting but the model has

certainly reproduced all the properties well for this time-scale. This reveals

that the model is capable of producing estimates of the quantities not used

in the ﬁtting which adds strength to this DSP modelling framework.

6 N. I. Ramesh and R. Thayakaran

●●●●●●●●●●●●

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

0.04 0.10

Mean plot for 60 minute levels of aggregation

mm / hour

●●●●●●●●●●●●

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

0.02 0.05

Mean plot for 30 minute levels of aggregation

mm / hour

●●●●●●●●●●

●●

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

0.020 0.035

Mean plot for 20 minute levels of aggregation

mm / hour

●●●●●●●●●●●●

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

0.005 0.020

Mean plot for 10 minute levels of aggregation

mm / hour

●●●●●●●●●●

●●

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

0.004 0.009

Mean plot for 5 minute levels of aggregation

mm / hour

FITTED OBSERVED

●●●●●●●●●●●●

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

0.0 0.3 0.6

SD plot for 60 minute levels of aggregation

mm / hour

●●●●●●●●●●●●

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

0.00 0.20

SD plot for 30 minute levels of aggregation

mm / hour

●●●●●●●●●●●●

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

0.00 0.15

SD plot for 20 minute levels of aggregation

mm / hour

●●●●●●●●●●●●

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

0.00 0.08

SD plot for 10 minute levels of aggregation

mm / hour

●●●●●●●●●●●●

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

0.00 0.04

SD plot for 5 minute levels of aggregation

mm / hour

FITTED OBSERVED

Fig. 1. Observed ﬁtted values of the mean and Standard deviation of the aggre-

gated rainfall for DSP model at h=5,10,20,30,60 minutes aggregations.

Stochastic point process model for ﬁne-scale rainfall time series 7

●●●●●●●●●●●●

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

0 4 8

CV plot for 60 minute levels of aggregation

cv

●●●●●●●●●●●●

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

0 4 8

CV plot for 30 minute levels of aggregation

cv

●●●●●●●●●●●●

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

0 4 8

CV plot for 20 minute levels of aggregation

cv

●●●●●●●●●●●●

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

048

CV plot for 10 minute levels of aggregation

cv

●●●●●●●●●●●●

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

048

CV plot for 5 minute levels of aggregation

cv

FITTED OBSERVED

●●●●●●●●●●●●

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

0.0 0.6

AC1 plot for 60 minute levels of aggregation

correlation

●●●●●●●●●●●●

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

0.0 0.6

AC1 plot for 30 minute levels of aggregation

correlation

●●●●●●●●●●●●

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

0.0 0.6

AC1 plot for 20 minute levels of aggregation

correlation

●●●●●●●●●●●●

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

0.0 0.6

AC1 plot for 10 minute levels of aggregation

correlation

●●●●●●●●●●●●

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

0.0 0.6

AC1 plot for 5 minute levels of aggregation

correlation

FITTED OBSERVED

Fig. 2. Observed ﬁtted values of the coeﬃcient of variation and autocorrelation of

the aggregated rainfall for DSP model at h=5,10,20,30,60 minutes aggregations.

8 N. I. Ramesh and R. Thayakaran

6 Conclusions and future work

A DSP model has been developed to study the properties of ﬁne-scale rainfall

time series. Second-order moment properties of the aggregated rainfall have

been derived and used to the model 5-minute rainfall data. The empirical

properties of the rainfall are in very good agreement with the ﬁtted theoretical

values over a range of sub hourly time scales, including those that are not

used in ﬁtting. This suggests that the model is capable of reproducing the

ﬁne scale structure of the rainfall process well and has potential application

in many areas. Despite this, there is potential to develop the model further

to accommodate third order moments and also to explore its capability to

handle aggregations at higher levels. Further developments to study other

hydrological properties of interest are also envisaged.

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