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Stochastic point process model for fine-scale rainfall time series

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Abstract

A stochastic point process model, which is constructed from a class of doubly stochastic Poisson processes, is proposed to analyse point rainfall time series 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 governed 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 occur 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.
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(φ1φ2)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)) =
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 u0 as
c(u) =
ηδ(u) + A1e(λ+µ)u+ [B2B1]eηu (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
(i1)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
(i1)h
E(X(t))dN(t) =
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)
ηh
+ 2 µ2
xA1ψ1(λ+µ)+2E[XijXik ]B2B1µ2
xψ1(η) (4)
where ψ1(λ+µ) = [(λ+µ)h1+e(λ+µ)h]
(λ+µ)2and ψ1(η) = [ηh1+eηh ]
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 k1,
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]B2B1µ2
xψ2(η) (5)
where ψ2(λ+µ) = e(λ+µ)(k1)h[1e(λ+µ)h]2
(λ+µ)2and ψ2(η) = eη(k1)h[1eηh]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
¯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)
iEhY(h)
ii21/2
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)) ν(h)ν(h))2+1
ˆ
var(ρ(h)) ρ(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) 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) 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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... The aim of this paper is to develop a class of doubly stochastic Poisson process (DSPP) models with exponentially decaying pulses to describe the probabilistic structure of the rainfall at a single rain-gauge. The model we consider here is similar in structure to those described in Thayakaran and Ramesh (2017), Ramesh and Thayakaran (2012) and Ramesh (1998) but is different in its form due to exponential pulses. The proposed model is applied to a set of sub-hourly rainfall data from Bracknell in England, obtained from the U.K. Meteorological Office, to illustrate its application in the modelling of temporal rainfall. ...
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 Over a decade ago, point rainfall models based upon Poisson cluster processes were developed by Rodriguez-Iturbe, Cox and Isham. Two types of point process models were envisaged: the Bartlett–Lewis and the Neyman–Scott rectangular pulse models. Recent developments are reviewed here, including a number of empirical studies. The parameter estimation problem is addressed for both types of Poisson-cluster based models. The multiplicity of parameters which can be obtained for a given data set using the method of moments is illustrated and two approaches to finding a best set of parameters are presented. The use of a proper fitting method will allow for the problems encountered in regionalisation to be adequately dealt with. Applications of the point process model to flood design are discussed and finally, results for a model with dependent cell depth and duration are given. Taking into account the spatial features of rainfall, three multi-site models are presented and compared. They are all governed by a master Poisson process of storm origins and have a number of cell origins associated with each storm origin. The three models differ as to the type of dependence structure between the cell characteristics at different sites. Analytical properties are presented for these models and their ability to represent the spatial structure of a set of raingauge data in the South-West of England is examined. Continuous spatial-temporal models are currently being developed and results are presented for a model in which storm centres arrive in a homogeneous Poisson process in space-time, and cells follow them in time according to a Bartlett–Lewis type cluster. Examples of simulations using this model are shown and compared with radar data from the South-West of England. The paper concludes with a summary of the main areas in which further research is required.
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One of the most difficult problems in rainfall modelling is often the fitting of theoretical models to data. In this paper a spectral approach is proposed for fitting single-site models, by considering the approximate likelihood functions of collections of sample Fourier coefficients. An objective function is derived, which is the same as that used in Whittle's method for time series parameter estimation, and is shown to have an interpretation as a quasi-likelihood when the time series is non-Gaussian. The method requires knowledge of the theoretical spectral density of models to be fitted; the form of this spectral density is given for a wide class of point-process-based rainfall models. A variant of the method is also developed for when a number of independent replications of a rainfall process are available. Large-sample properties of the estimators are derived, and the method is illustrated with some data from the south-west of England.