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Two stochastic models of the rainfall process, belonging to different categories, are compared in terms of how well they reproduce certain hyetograph characteristics. The first is the scaling model of storm hyetograph, which belongs to the category of storm-based models. The second is the Bartlett–Lewis rectangular pulse model, the most widespread among the category of point process models. The scaling model is further developed introducing one more parameter to better-fit historical data. The Bartlett–Lewis model is theoretically studied to extract mathematical relationships for the intra-storm structure. The intercomparison is based on the storm hyetographs of a data set from Greece and another one from USA. The different storms are identified in each data set and classified according to their duration. Both models are fitted using the characteristics of storms. The comparison shows that the scaling model of storm hyetograph agrees well with the structure of historical hyetographs whereas the Bartlett–Lewis rectangular pulse model exhibits some discrepancies in either its original version or its random parameter version. However, it is shown that the performance of the Bartlett–Lewis model is significantly improved, and becomes comparable to that of the scaling model, by introducing a power-law dependence of its cell related parameters (duration and rate of arrivals) on the storm duration.
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On the representation of hyetograph characteristics by stochastic rainfall models
Demetris Koutsoyiannis and Nikos Mamassis
Department of Water Resources, Faculty of Civil Engineering,
National Technical University, Athens, Greece
Abstract. Two stochastic models of the rainfall process, belonging to different categories, are
compared in terms of how well they reproduce certain hyetograph characteristics. The first is
the scaling model of storm hyetograph, which belongs to the category of storm-based models.
The second is the Bartlett-Lewis rectangular pulse model, the most widespread among the
category of point process models. The scaling model is further developed introducing one
more parameter to better fit historical data. The Bartlett-Lewis model is theoretically studied
to extract mathematical relationships for the intra-storm structure. The intercomparison is
based on the storm hyetographs of a data set from Greece and another one from USA. The
different storms are identified in each data set and classified according to their duration. Both
models are fitted using the characteristics of storms. The comparison shows that the scaling
model of storm hyetograph agrees well with the structure of historical hyetographs whereas
the Bartlett-Lewis rectangular pulse model exhibits some discrepancies in either its original
version or its random parameter version. However, it is shown that the performance of the
Bartlett-Lewis model is significantly improved, and becomes comparable to that of the scaling
model, by introducing a power-law dependence of its cell related parameters (duration and
rate of arrivals) on the storm duration.
Keywords. Stochastic models; Rainfall; Storm; Hyetograph; Point process; Scaling.
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1. Introduction
Stochastic rainfall models based on point processes, introduced in the 1980s (Waymire and
Gupta, 1981a-b; Smith and Karr, 1983; Rodriguez-Iturbe et al., 1984, Foufoula-Georgiou and
Lettenmaier, 1986; Guttorp, 1986), have been one of the most widespread and useful tools in
analysis and modelling of rainfall. Among them, the cluster-based models (Rodriguez-Iturbe
et al., 1987, 1988) such as the Neyman-Scott rectangular pulse model and the Bartlett-Lewis
rectangular pulse model (especially the latter, see section 3) have offered a more accurate
representation of rainfall and have become the most popular. In these models, storms arrive
following a Poisson process and each storm gives rise to a cluster of rain cells with each cell
having a random time location, duration and intensity. Several applications of these models
have been made and much research resulting in improvements thereof has been conducted
during the 1990s (Cowpertwait, 1991, 1998; Onof and Wheater, 1993, 1994; Onof et al.,
1994; Bo and Islam, 1994; Velghe et al., 1994; Glasbey et al., 1995; Cowpertwait et al.,
1996a, b; Khaliq and Cunnane, 1996; Gyasi-Agyei and Willgoose, 1997, 1999; Verhoest et
al., 1997; Kakou, 1997; Gyasi-Agyei, 1999; Cameron et al., 2000; Koutsoyiannis and Onof,
2001).
It is known that some of the point process models suffer from an inability to describe the
statistical structure of rainfall at a wide range of scales (e.g. Foufoula-Georgiou and Guttorp,
1986; Foufoula-Georgiou and Krajewski, 1995). In addition, they may not represent well the
extreme rainfall events at a range of scales (e.g. Valdes et al., 1985, Verhoest et al., 1997;
Cameron et al., 2000), although recent developments have mitigated this problem either by
explicitly incorporating the process skewness into the parameter estimation procedure
(Cowpertwait, 1998) or by implementing a disaggregation framework using historical daily
rainfall data (Koutsoyiannis and Onof, 2001). These problems may arise from the fact that the
model parameters do not necessarily represent actual physical quantities as they depend on the
time scale that was chosen for their fitting (e.g. Foufoula-Georgiou and Guttorp, 1986; Valdes
et al., 1985). Specifically, the model fitting is usually done in terms of some bulk statistical
properties of the process in the entire time domain such as marginal moments and
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autocorrelations in discrete time and the probability of dry intervals, without focus on the
structure of the individual rainfall events.
More than a decade before the introduction and use of point process based rainfall models,
other types of statistical representations of the rainfall process focusing on the rainfall events
were devised. These ranged from empirical non-dimensionalised mass curves of storms (Huff,
1967) to more theoretically based stochastic models (Grace and Eagleson, 1966; Eagleson,
1970, 1978; Restrepo-Posada and Eagleson, 1982; Woolhiser and Osborn, 1985; Marien and
Vandewiele, 1986; Koutsoyiannis, 1988, 1994; Koutsoyiannis and Xanthopoulos, 1990). A
more sophisticated model of this category, termed scaling model of storm hyetograph, was
proposed by Koutsoyiannis and Foufoula-Georgiou (1993). This is a simple scaling stochastic
model of instantaneous rainfall intensities based on an observed scale-invariance of
dimensionless rainfall with storm duration (see also section 2). The common characteristic of
these approaches was the focusing of the models on the rainy period rather than the
description of the process in the entire time domain. To this aim, they had to partition the time
domain into rainy and dry periods, consider each rainy period as a storm with certain
characteristics such as total duration and depth, and study the internal structure of the storm.
In comparison with point process models, the storm-based models may sometimes have less
elegant mathematical basis, but they may describe better some properties of the rainfall
process, particularly the intra-storm structure (we will discuss this later). Although storm-
based models are dealing with storms only, it is not difficult to combine them with a
stochastic scheme describing storm arrivals so as to compose a model for the entire time
domain. For example, Koutsoyiannis and Pachakis (1996) combined the scaling model of
storm hyetograph with an alternating renewal model to simulate completely the rainfall
process.
A specific difficulty with storm-based models is that they require a definition of a storm in
a manner that a storm can be identifiable from a rainfall record. This is not as easy as it may
seem at first glance, as a storm may contain periods with zero rainfall. Periods of zero rainfall
are contained in ‘storms’ of point process models (e.g., periods not covered by storm cells),
too, but in this case there is no need that the abstract ‘storms’ of the models correspond to
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actual storms of a rainfall time series. (As we discussed above, the fitting of point processes is
done for the entire time domain). The usual convention in storm-based models is to consider
all zero rainfall periods that are shorter than a specified limit (the separation time) as
belonging to a storm, whereas longer zero rainfall periods are assumed to separate different
storms. Huff (1967) assumed a value of 6 hours for this separation time. Restrepo-Posada and
Eagleson (1982) assumed that this time must be defined in a manner that consecutive storms
are statistically independent events and developed a statistical procedure to determine it. In
the same lines, Koutsoyiannis and Xanthopoulos (1990) used a criterion based on the
Kolmogorov-Smirnov test and found the separation time value in the range 5-7 hours using
Greek rainfall data.
Despite of subjectivity in constructing a series of storm hyetographs from a continuous
rainfall record, the different hyetographs, once identified, contain important information on
the structure of the actual rainfall process. Their systematic study may be directly useful to
engineering applications like design storm estimation. In addition, it is useful in rainfall
modelling as it may reveal how well certain characteristics of the hyetographs can be
reproduced by a stochastic model of rainfall. Thus, it can serve as a basis in building an
appropriate stochastic model, improving the structure of an existing model, or choosing the
most suitable among different rainfall models.
Such a study of the structure of rainfall event hyetographs is attempted in this paper. This
study is used to assess how well the structure of hyetographs is described by existing
stochastic models. For comparison, one representative model of each of the two model
categories is chosen. The scaling model of storm hyetograph is chosen for the category of
storm-based models (section 2) and is further developed introducing one more parameter to
better fit historical data. The Bartlett-Lewis rectangular pulse model is chosen for the point
process category (section 3) and is theoretically studied to extract mathematical relationships
for the intra-storm structure. A data set from Greece and another one from the USA (section
4) are used for the study and comparisons. The comparison reveals that the scaling model of
storm hyetograph captures well the structure of historical hyetographs whereas the Bartlett-
Lewis rectangular pulse model may exhibit some discrepancies (section 5). Ways to improve
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the Bartlett-Lewis model performance, by introducing dependence of its cell related
parameters (duration and rate of arrivals) on the storm duration, are also discussed. The paper
concludes by indicating the further research required to implement the studied adaptations
into an operational model (section 6). To increase readability the mathematical derivations
have been excluded from the text and are put separately in appendices (A1-A4).
2. The Scaling Model of Storm Hyetograph
Let D denote the duration of a storm and Ξ(t | D) the instantaneous rainfall intensity at time t
within the storm duration (0 t D). The scaling model of storm hyetograph is based on the
scaling hypothesis, illustrated in the schematic of Figure 1 (Koutsoyiannis and Foufoula-
Georgiou, 1993), i.e.,
{
Ξ(t | D)} =
d {λκ Ξ(λ t | λ D)} (1)
where κ is a scaling exponent, λ is any positive real number and the symbol =
d denotes equality
in distribution. A secondary hypothesis is the weak stationarity (stationarity within the storm),
which results in
E
[Ξ(t | D)] = c1 Dκ (2)
R
Ξ(τ | D) := E[Ξ(t | D) Ξ(t + τ | D)] = ψ(τ / D) D 2κ (3)
where E[ ] denotes expectation, c
1 is a parameter (c1 > 0) and ψ(τ / D) is a function to be
specified. In this paper we specify this function as
ψ(τ / D) = α
τ
D
β
ζ (4)
where α, β, ζ are parameters (α > 0, 0 < β < 1, ζ <1). This expression is slightly modified in
comparison with the original expression by Koutsoyiannis and Foufoula-Georgiou (1993) as
it contains the additional parameter ζ. This additional parameter has been proven to improve
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model fits to real world data. Note that the variance of the instantaneous process is infinite
since ψ(τ / D) tends to infinity as time lag τ tends to zero.
Moving from continuous time to discrete time, we divide the time domain into k intervals
of length Δ = δ D assuming for simplicity that k = 1 / δ = D / Δ is integer. Let Yi denote the
incremental rainfall depth at time interval i (1 i k). The first- and second-order moments
of the incremental depths are
E
[Yi] = c1 D 1 + κ δ (5)
Var[Yi] = D 2(1 + κ) δ2 (c1
2 + c2)(δβε) – c1
2 (1 – ε)
1 – ε (6)
Cov[Yi, Yi + m] = D 2(1 + κ) δ2 (c1
2 + c2)[δβ f(m, β) – ε] – c1
2 (1 – ε)
1 – ε (7)
so that
Corr[Yi, Yi + m] = (c1
2 + c2)[δβ f(m, β) – ε] – c1
2 (1 – ε)
(c1
2 + c2)(δβε) – c1
2 (1 – ε) (8)
where
c
2 = α (1 – ε)
(1 – β) (1 – β/2)c1
2 (9)
ε = ζ (1 – β) (1 – β / 2) (10)
and
f
(m, β) =
1
2[(m – 1)2 – β + (m + 1)2 – β] – m 2 – βm > 0
1‚ m = 0
(11)
These expressions are slightly different from the original ones due to the additional parameter
ζ. (To get the original expressions it suffices to set ζ = 0). Their derivation is given in
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Appendix A1. The relevant statistics of the total depth, H, of the storm, directly obtained from
(5) and (6) by setting δ = 1, are
E
[H] = c1 D 1 + κ (12)
Var[H] = c2 D 2(1 + κ) (13)
From the above equations we may observe the following:
1. The mean
E[H] and the standard deviation Std[H], if plotted on a log-log paper versus
duration D, will be parallel straight lines with slope 1 + κ. Usually κ is negative so that
this slope is less than 1. Consequently, the coefficient of variation CV[H] := Std[H] / E[H]
is constant, independent of D.
2. The same is true for the statistics of the incremental depths, i.e., E[Yi] and Std[Yi] if the
number of intervals k (and, consequently the dimensionless interval length δ) is constant.
However, if we assume a constant interval length Δ, the mean incremental depth becomes
E[Yi] = c1 D κ Δ, which for negative κ is a decreasing function of D. The behaviour of
Std[Yi] is similar.
3. For large lag
m the autocorrelation structure is long-memory structure due to the presence
of the term f(m, β). (This term characterises also the fractional Gaussian noise model.)
4. The lag-one autocorrelation coefficient is generally an increasing function of storm
duration D.
The model includes five parameters in total: the scaling exponent κ, the mean value
parameter c
1, the variance parameter c
2 (or, equivalently, α) and the correlation decay
parameters β and ζ. The first two can be estimated by least squares from E[H] = c1 D 1 + κ
using storm data classified according to duration. The third parameter can be estimated from
c2 = Var[H] / D 2(1 + κ). The remaining two parameters can be estimated by least squares from
δβε
1 – ε = E[Y2
i]
E2[Yi] E2[H]
E[H2] (14)
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in combination with (10). Equation (14) is derived by combining (5), (6), (12) and (13).
Alternatively, the three parameters c2, β and ζ can be estimated simultaneously by minimising
the fitting error of the model in Var[H], Var[Y] and Corr[Yi, Yi + 1].
There have been a few applications of the model with different rainfall data sets. These
include modelling of (a) point rainfall data in Northern Greece (Koutsoyiannis and Foufoula-
Georgiou, 1993); (b) areal rainfall in Italy (Mamassis et al., 1994) in a rainfall forecast
framework; (c) intense rainfall at a point in Greece (Mamassis, 1997); (d) point rainfall in
Florida, USA, in order to construct a continuous rainfall simulation model (Koutsoyiannis and
Pachakis (1996); and (e) intense point rainfall based on intensity-duration-frequency curves at
Athens, Greece, in an urban storm design framework (Koutsoyiannis and Zarris, 1999).
Notably, in case (d) comparisons of simulated to historical data were made using not only
typical statistical descriptors, but also descriptors used in chaos literature such as correlation
dimensions and correlation integrals. The results showed a very satisfactory agreement
between simulated and historical statistical descriptors.
3. The Bartlett-Lewis (BL) rectangular pulse point process
As mentioned above, the Bartlett-Lewis rectangular pulse model (Rodriguez-Iturbe et al.,
1987, 1988; Onof and Wheater, 1993, 1994) is a stochastic model for the entire time domain
(as opposed to the scaling model, which applies to rainy periods only). The model has been
applied widely and there is much experience in calibrating it to several climates, which
accumulated evidence on its ability to reproduce important features of the rainfall field from
the hourly to the daily scale and above. The general assumptions of the Bartlett-Lewis
rectangular pulse model are (see Figure 2): (1) storm origins ti occur following a Poisson
process with rate λ; (2) origins tij of cells of each storm i arrive following a Poisson process
with rate β; (3) cell arrivals of each storm i terminate after a time Vi exponentially distributed
with parameter γ; (4) each cell has a duration Wij exponentially distributed with parameter η;
and (5) each cell has a uniform intensity Xij with a specified distribution.
In the original version of the model, all parameters are assumed constant. In the modified
version, the parameter η is randomly varied from storm to storm with a gamma distribution
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with shape parameter α and scale parameter ν. Subsequently, parameters β and γ also vary so
that the ratios κ := β / η and φ := γ / η are constant.
The distribution of the uniform intensity Xij of cells is typically assumed exponential with
parameter 1 /
μX, where μX := E[X] (indices i and j are omitted due to stationarity).
Alternatively, it can be chosen as two-parameter gamma with mean μX and standard deviation
σX. Thus, in its most simplified version the model uses five parameters, namely λ, β, γ, η, and
μX (or equivalently, λ, κ, φ, η, and μX) and in its most enriched version seven parameters,
namely λ, κ, φ, α, ν, μX and σX.
The equations of the BL model, in its original or the modified (random parameter)
configuration, may be found in the appropriate references (Rodriguez-Iturbe et al., 1987,
1988; Onof and Wheater, 1993, 1994). These equations relate the statistical properties of the
rainfall process in discrete time in the entire time domain, to the model parameters and serve
as the basis for model fitting using these statistical properties. However, the focus of the
present study is not on the entire time domain, but rather on the storm event only. Therefore,
we need equations of the same statistical properties for the storm event. If we neglect the
possibility of overlapping of storms (assuming that λ << β, γ, η) then it is easy to derive such
equations, as the process of interest in rainy periods becomes similar to the Poisson
rectangular pulses model (Rodriguez-Iturbe et al., 1984). In Appendix A2 we have derived
expressions for the rainfall intensity in continuous time in a storm. It may be seen that the
process in a storm is not strictly stationary because of the boundary (origin) effect. If for
simplification we neglect this effect, we find that the instant intensity Ξ(t | D) at time t in a
storm of duration D has mean
E
[Ξ(t | D)] = κ E[X] (15)
and covariance
Cov[Ξ(t | D), Ξ(t + τ | D)] = κ E[X 2] eη τ (16)
for the original BL model, and
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Cov[Ξ(t | D), Ξ(t + τ | D)] = κ E[X 2]
ν + φ D
ν + φ D + τ
α + 1
(17)
for the random parameter BL model.
It is easy then to derive similar expressions in discrete time (see Appendix A3). For both
model versions the mean in discrete time with step Δ is
E
[Yi] = κ E[X] Δ (18)
where Yi is the incremental rainfall depth at time interval i of length Δ. The second-order
moments for the original model version are
Var[Yi] = 2 κ E[X2]
η2 (η Δ – 1 + eη Δ) (19)
Cov[Yi, Yi + m] = κ E[X2]
η2 (1 – eη Δ )2 eη (m – 1) Δ (20)
Corr[Yi, Yi + m] = (1 – eη Δ )2 eη (m – 1) Δ / 2 (η Δ – 1 + eη Δ) (21)
where m > 1. The corresponding equations for the random parameter model version are
Var[Yi] = 2 κ E[X2] (ν + φ D)2
α (α – 1) (θ1
α – 1 + α – 1
θ1α) (22)
Cov[Yi, Yi + m] = 2 κ E[X2] (ν + φ D)2
α (α – 1) []
(1/2) (θm + 1
α – 1 + θm – 1
α – 1 ) – θm
α – 1 (23)
Corr[Yi, Yi + m] = []
(1/2) (θm + 1
α – 1 + θm – 1
α – 1 ) – θm
α – 1 / (θ1
α – 1 + α – 1
θ1α) (24)
where
θm = ν + φ D
ν + φ D + m Δ (25)
The mean and variance of the total storm depth can be determined from (18) and (19) (or (22)
for the random parameter model version), respectively, setting Δ = D. The above equations
may serve as a basis for parameter estimation using information from the rainy periods only.
11
We can notice in the above equations that (under the assumptions made) the statistics of
the rainfall process within a storm depend on the parameters κ and η (or the parameters of the
latter’s distribution), and the parameters of the distribution of Xij (namely, E[X], E[X2]). In the
random parameter model version they are also affected by the parameter φ whereas in neither
version is affected by the parameters λ. In fact, the three parameters κ, E[X] and E[X2] are
combined in two, namely κ E[X], κ E[X2]. Therefore, it is no loss of generality to assume that
Xij is exponentially distributed, in which case E[X2] = 2
E2[X], so that we have two
independent parameters, namely κ and μΧ E[X]. In addition, we have the parameter η for the
original model version and parameters α, ν and φ for the random parameter model version.
Contrary to the scaling model, the mean E[Yi] and the standard deviation Std[Yi] of the
incremental depths do not depend on duration D and are constant for all storms in the original
model version. Also constant, independent of duration, is the autocorrelation coefficient of
any lag for this version. In the random parameter version the standard deviation and the
autocorrelation coefficient depend on duration D but the mean is still constant. Another
difference among the two model versions is that the decay of the autocorrelation function is
milder in the random parameter model version.
4. Data sets
Two data sets were used for the exploration and comparison of models. The first data set is
point rainfall at the Zographou meteorological station, Athens, Greece (latitude 37o58΄26΄΄N,
longitude 23ο47΄16΄΄E, altitude 219 m), for 6 years (1993-99). The available time resolution
of measurements is ten minutes and the depth resolution is 0.1 mm. The mean annual rainfall
in the area is around 450 mm. The separation time to identify different events was assumed to
be 6 h. Only intense storms were chosen from the available record, i.e., those with hourly
depth exceeding 5 mm or daily depth exceeding 15 mm. Thus, a total of 81 storms were
extracted, which were classified in five classes (marked as 1 to 5) according to their durations,
as shown in Table 1. The available temporal resolution Δ = 10 min was used to assemble
Table 1. The basis for selecting these classes was to have approximately the same number of
storms within each class. For each class a mean duration E[D] was estimated from data, equal
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to the mean of durations of all storms in this class, and assigned as the representative duration
for the class. Standard deviations of storm durations, as well as means and standard deviations
of the total and incremental depths, and lag-one autocorrelation coefficients of incremental
depths were also estimated from data for each class and are also shown in Table 1. The storms
were further grouped into two larger classes (A and B, also shown in Table 1), which were
necessary to achieve reliable estimates of autocorrelation coefficients for larger lags. In
addition, to assess the effect of the temporal resolution, the set of storms was processed using
a larger temporal resolution, Δ = 1 h, and the resulting characteristics of the classes are
tabulated in Table 2 in a similar manner as in Table 1.
The second data set is point rainfall at the Parrish raingauge Florida, USA (altitude 40 m),
for 19 years (1971-89). The available time resolution of measurements is 15 minutes and the
depth resolution is 0.1 in 3 mm. The mean annual rainfall in the area is 1290 mm. The
separation time to identify different events was assumed to be 6 h. All storms (1643 in total)
of the 19-year period were assembled from the available record. The 1643 storms were
classified in 10 classes (marked as 1 to 10) according to their durations, as shown in Table 3
for temporal resolution Δ = 15 min and in Table 4 for a larger temporal resolution, Δ = 2 h.
Here we intentionally chose the 2 h (rather than the ‘standard’ 1 h) resolution to assess the
performance of models in describing rainfall events at a temporal resolution as coarser as
possible. This is almost one order of magnitude coarser than the original of 15 min; an even
coarser resolution would obscure the event structure too much (the mean storm duration is
2.28 h). As in the Zographou data set, a second grouping using two classes (A and B) was
made, which is also shown in Table 3 and Table 4.
The standard deviations of the total depths (Std[H] =: σ
H) shown in Table 1 through Table
4 are corrected according to
σ2
H =
σ΄2
Hμ2
H (1 + κ)2 σ2
D / μ2
D
1 + (1 + κ)2 σ2
D / μ2
D
(26)
13
where σ
H and σ΄
H are the corrected and the raw estimate of standard deviation of the total depth
of each class, respectively, μ
H is the average total depth of the same class, μD and σ
D are the
mean and standard deviation of the duration of each class, and κ is the scaling exponent of (1)
estimated from (12). This correction was necessary because the raw estimate of variance
using the data values of H of each class is affected by the variability of duration in each class
(expressed by σ2
D), which in some cases (the classes with the larger durations) is very
significant. The justification of (26) is given in Appendix A4, where it is also shown that an
analogous correction for the mean of H is not necessary.
Another correction may be needed for the mean of D in the class with the lowest durations
(class 1). This problem is apparent in class 1 of Table 3, where all 608 events seemingly have
duration equal to 0.25 h. However, this is due to the available temporal resolution of Δ = 0.25
h. All storms with durations smaller than Δ = 0.25 h are erroneously assigned a duration of
0.25 h. Therefore, the actual mean of class 1 of Table 3 must be less than 0.25 h. Assuming
that durations are exponentially distributed, it can be shown that the correct mean duration of
this class is about 0.05 h. However, we preferred not to apply this correction and exclude this
class from further processing.
5. Comparisons and further developments
The scaling model of storm hyetograph was fitted to both Zographou and Parrish data sets,
using the finest available temporal resolution (10 and 15 min, respectively) and the
methodology already described in section 2. The estimated parameters are shown in Table 5.
In a similar manner the parameters of the BL model were estimated for both data sets.
Specifically, the product κ μX, which has a common value for both model versions, was
estimated from (18) as the average of E[Y] of all classes divided by Δ. Then the product
κ E[X2] and the parameter η of the original version or the parameters α, ν and φ of the random
parameter version were estimated simultaneously by minimising the fitting error of the model
in Var[H], Var[Y] and Corr[Yi, Yi + 1] for all classes. Parameters κ and μX were then determined
from the known products κ μX and κ E[X2] assuming an exponential distribution for X, i.e.,
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E[X2] = 2 μ2
X. The estimated parameters are shown in Table 6 for the original BL model
version and in Table 7 for the random parameter BL model version.
Now we are able to intercompare the scaling model and the BL model based on their
agreement with the historical storm data. This is done graphically in Figure 3 through Figure
9. Each of these figures contains a sequence of points corresponding to the historical data and
four curves corresponding to the scaling model, the two versions of the BL model, and one
additional version of the latter which will be discussed later in this section. Furthermore, each
of these figures includes four cases, of which cases (a) and (c) refer to the Zographou and
Parrish data sets, respectively, for the finer available temporal resolutions (Δ = 10 and 15 min,
respectively). Similar are cases (b) and (d) but they refer to the coarser temporal resolutions
(Δ = 1 and 2 h for Zographou and Parrish, respectively). We emphasise that the model
parameters were not fitted separately for the coarser temporal resolutions, but rather those
estimated from the finest resolutions were used in cases (b) and (d), too. In this way we can
test whether a parameter set estimated from one resolution is appropriate for another
resolution, too.
Figure 3 refers to the mean of total storm depth as a function of storm duration
(logarithmic plot). To plot the empirical means, the mean duration of each duration class was
used. One theoretical curve is plotted for both versions of the BL model, which was
determined by (18) for Δ = D. This is a straight line with slope equal to one for all cases. On
the other hand, the scaling model yields another straight line with slope equal to 0.46 in the
Zographou case and 0.40 in the Parrish case. Clearly, the scaling model outperforms the BL
model in all cases. The latter exhibits large departures from data, the most significant being
the one for Parish with Δ = 2 h (Figure 3(d)). This indicates the inappropriateness of a specific
parameter set for a temporal resolution different from the one that was used for the model
fitting. A more careful observation reveals that the appearing discrepancies are not due to
inappropriate parameters but rather are related to the model structure, which implies a straight
line with slope equal to one, regardless of the parameter values.
Figure 4 depicts the models’ behaviour regarding the standard deviation of total storm
depth as a function of storm duration. Here all models performed equally well for Zographou,
15
but there are some slight departures of the BL model for Parrish, especially in the case with Δ
= 2 h (Figure 4(d)).
Figure 5 refers to the mean of incremental storm depth. As in Figure 3, one theoretical
curve is plotted for both versions of the BL model, which again was determined by (18) for
the appropriate time interval Δ of each case. According to the BL model, the mean of
incremental depth is constant, independent of the duration D. However, the data show a clear
decreasing trend of the mean incremental depth with the increase of storm duration. This trend
is very well captured by the scaling model in all cases. Thus, the scaling model outperforms
the BL model, especially in case of Figure 5(d) (Parrish, Δ = 2 h) where the BL model
predicts values far higher than the actual ones. Again the problem here is due to the model
structure rather than the parameter values.
Figure 6 shows the empirical and theoretical standard deviation of incremental storm depth
as a function of storm duration. Again here the empirical data indicate a decreasing trend of
the standard deviation of incremental depth with the increase of storm duration. The scaling
model captures this trend in all cases, whereas the BL model suggests a constant standard
deviation of the incremental storm depth, independent of storm duration, or even an opposite
(increasing) trend in the case of the random parameter model version (more apparent in
Figure 6 (b)).
Figure 7 depicts the variation of the lag-one autocorrelation coefficient of incremental
storm depth with storm duration. Empirical data suggest an increasing trend of the
autocorrelation coefficient with duration, which is captured well by the scaling model and
much less by the random parameter model version, whereas the original BL model implies
that this coefficient should be constant. The departures of BL model from data is not very
significant in this case.
Figure 8 and Figure 9 depict the empirical and theoretical autocorrelation functions for lags
up to 10 of incremental storm depth for small and large storm durations, respectively (classes
A and B, respectively). In the case of small durations (Figure 8) all models perform almost
equally well. However, Figure 9, in comparison with Figure 8, suggests that empirical
autocorrelation functions tend to decay more slowly for large durations than for small ones.
16
This again is captured by the scaling model, whereas in both BL model versions the
theoretical curves are identical in both Figure 8 and Figure 9.
Additional comparisons were made to assess the effect of seasonality to model parameters.
Thus, the entire analysis for the Parrish data set was repeated using two seasons: from June to
September (790 mm total depth; rainy season) and from October to May (dry season). Plots
similar with those of Figure 3 through Figure 9 were constructed using the single parameter
set estimated from data of the whole year. These plots, which have not been included in the
paper, indicated that the single parameter set was almost equally good for both seasons.
The entire analysis indicates a better performance of the scaling model in comparison with
the BL model in terms of capturing the empirical characteristics of storms. The worse
performance of the BL model is mainly due to the fact that it does not capture the empirical
relationship of the means of total and incremental storm depths with duration (Figure 3 and
Figure 5). On the contrary, the fit of the scaling model is perfect in this case, owing to the
scaling exponent κ whose value is –0.54 for Zographou and –0.60 for Parrish (Table 5). Had
this value been closer to zero (as indeed was in most of the case studies mentioned in section
2), the departure of the BL model (which in fact assumes κ = 0) would be lower.
The examined model BL versions, i.e., the original and random parameter versions, both
imply a common expression of the mean incremental depth, i.e., E[Yi] = κ E[X] Δ (equation
(18)). We note that the random parameter version varies randomly the parameter η, and
consequently β, so that both the mean cell duration 1 /
η and the mean interarrival time of
cells 1 /
β become increasing functions of the storm durations. Specifically, these functions
can be expressed by
E
[1 / η | D] = ν + φ D
α , E[1 / β | D] = ν + φ D
κ α (27)
as it follows from the results of Appendix A2 (equation (A.32)). This means that in a storm of
a large duration, cells tend to last longer on average (higher 1 / η), and at the same time are
more spaced out (higher 1 / β). These two tendencies are mutually counterbalanced, as far as
average rainfall intensity is concerned, because the two relations in (27) are proportional to
17
each other. In other words, this is the consequence of the assumption of a constant κ, which
must be considered as the main reason why both BL model versions have practically the same
performance in this study.
Thus, the question arises whether a modified version of the Bartlett-Lewis model would be
in better agreement with the empirical data. To assess this, we attempted to introduce
dependence of the parameter κ on duration, described by a power relationship, i.e.,
κ = κ0 D κ1 (28)
We assume a similar relationship for η, i.e.,
η = η0 D η1 (29)
in which case the parameter β varies accordingly as
β = β0 D β1 (30)
where β0 = κ0 η0 and β1 = κ1 + η1. We expect that both exponents κ1 and η1 (and consequently
β1, too) are smaller than zero. In this manner, the mean cell duration 1 /
η and the mean
interarrival time of cells 1 / β will be increasing functions of the storm durations, which agrees
with the remark of Rodriguez-Iturbe et al. (1988) that storms made up of cells with longer
durations tend to last longer and to have longer interarrival times between cells.
In the testing framework of this paper it is easy to implement this adaptation of the BL
model, because the storm duration D is known. All equations of the original model version
are still valid, if we simply substitute κ and η with their expressions from (28) and (29),
respectively. In the parameter sets obtained in this case, five parameters can be estimated
from the data, namely μΧ, κ0, κ1, η0 and η1, while the two additional parameters λ and γ are
beyond the scope of this work. The parameter κ1 was set equal to κ of the scaling model and
all other parameters were estimated in the manner described above. All fitted parameters of
this model version are given in Table 8. The resulting modelled characteristics of total and
incremental depths are depicted in Figure 3 through Figure 9, marked as ‘BL/additional’
among with the curves of the other model, which were described above. We observe that in
18
Figure 3 and Figure 5, where the two known versions of the BL model had the worst
performance, the additional BL version becomes identical to the scaling model and, thus, it
agrees very well with empirical data. In all other figures the additional BL model version
outperforms the known two versions and fits the empirical data equally well with (in some
instances slightly better than) the scaling model.
6. Conclusions and discussion
An intercomparison of two stochastic models of the rainfall process, belonging to different
categories, has been attempted in this paper. The first is the scaling model of storm
hyetograph, which belongs to the category of storm-based models. The second is the Bartlett-
Lewis rectangular pulse model, the most widespread among the category of point process
models. The scaling model is further developed introducing one more parameter to better fit
historical data. The Bartlett-Lewis model is theoretically studied to extract mathematical
relationships for the intra-storm structure. The intercomparison is based on the storm
hyetographs of a data set from Greece and another one from the USA. The different storms
are identified in each data set and classified according to their duration. A systematic study of
the structure of storm hyetographs is used to assess how well certain characteristics of the
hyetographs can be reproduced by the two stochastic models. Both models are fitted using the
characteristics of storms. The comparison shows that the scaling model of storm hyetograph
agrees well with the structure of historical hyetographs whereas the Bartlett-Lewis rectangular
pulse model exhibits some discrepancies in either its original version or its random parameter
version. However, it is shown that the performance of the Bartlett-Lewis model is
significantly improved, and becomes comparable to that of the scaling model, by introducing
a power-law dependence of its cell related parameters (duration and rate of arrivals) on the
storm duration.
It must be emphasised that the fitting of the Bartlett-Lewis model used in this paper serves
the purposes of this intercomparison study only and it is not appropriate for an operational
modelling application. Specifically, the fitting procedure used refers to a part of the model’s
parameter set and it is not free of simplifying assumptions (e.g. ignorance of storm
19
overlapping) and subjectivities (e.g., choice of storm classes). Despite of the assumptions and
subjectivities, the testing framework is able to locate structural weak points of the Bartlett-
Lewis model, which may be responsible for the known difficulties to describe the statistical
structure of rainfall at a wide range of scales (particularly at the sub-storm timescales) using
the same parameter values.
The proposed adaptation of the Bartlett-Lewis model by introducing a power-law
dependence of its cell related parameters on the storm duration, which is a sort of merging of
the Bartlett-Lewis point process model to the scaling model, seems to be a promising
improvement. Further research is required, however, to implement this adaptation into an
operational model to be used for simulation. Specifically, the power laws may be expressed in
terms of the Bartlett-Lewis model ‘storm’ duration V rather than the measurable duration of
each storm event D that was used in this study. In addition, expressions for the statistics of the
incremental depth over the entire time domain are needed, in addition to those derived in this
study that are focused on the interior of a storm. Such expressions will make feasible a more
robust parameter technique for the adapted model.
Acknowledgments. We thank Christian Onof and an anonymous reviewer for their detailed, positive
and constructive reviews which resulted in significant improvements of the paper.
20
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24
Appendix: Derivation of equations
A1. Expressions of variances and covariances in discrete time for the modified
version of the scaling model
The expected value of the product Yi Yi + m is
E[Yi Yi + m] = E[Y1 Y1 + m] =
0
Δ
mΔ
(m + 1)Δ
E[Ξ(t1) Ξ(t2)] dt1 dt2 =
0
Δ
mΔ
(m + 1)Δ
RΞ(t2t1) dt1 dt2 (A.1)
The double integral can be simplified by transforming its integration area, that is,
E[Yi Yi + m] =
(m – 1)Δ
mΔ
RΞ(τ) [τ – (m – 1)Δ] dτ +
mΔ
(m + 1)Δ
RΞ(τ) [(m + 1)Δ τ] dτ (A.2)
A translation of the integration variable results in
E[Yi Yi + m] =
Δ
Δ
RΞ(m Δ + τ) (Δ |τ|) dτ (A.3)
Substituting
RΞ(τ) from (3) and also using (4) we get
E[Yi Yi + m] = α D 2κ
Δ
Δ
m Δ + τ
D
β
ζ (Δ |τ|) dτ (A.4)
which setting Δ = δ D and τ = χ Δ = χ δ D becomes
E[Yi Yi + m] = α δ2D 2(1 + κ)
–1
1
[δβ |m + χ|βζ] (1 |χ|) dχ (A.5)
After algebraic manipulations this becomes (for m 1)
E[Yi Yi + k] = α δ2D 2(1 + κ) {(1/2)[(m – 1)2 – β + (m + 1)2 – β] – m 2 – β} δβζ(β – 1)(β – 1/2)
(β – 1)(β – 1/2) (A.6)
which by virtue of (9), (10) and (11) writes
25
E[Yi Yi + k] = D 2(1 + κ) δ2 (c1
2 + c2)[δβ f(m, β) – ε]
1 – ε (A.7)
Then using (5), the derivation of (7) is straightforward.
For
m = 0, (A.5) writes (due to symmetry)
E[Yi
2] = 2 α δ2D 2(1 + κ)
0
1
(δβ χβζ) (1 χ) dχ (A.8)
or
E[Yi
2] = 2 α δ2D 2(1 + κ) δβζ(β – 1)(β – 1/2)
(β – 1)(β – 1/2) (A.9)
which again can be written in the form of (A.7) with f(0, β) = 1 (by virtue of (10)). The
derivation of (6) is then straightforward. Furthermore, (13) is a direct consequence of (6)
obtained for δ = 1.
A2. Expressions for the rainfall intensity in continuous time in a storm for the
rectangular pulse Bartlett-Lewis model
We assume that the rate of storm arrivals λ is much less than the rate of cell arrivals β
within a storm (λ << β) so that we can ignore the possibility of storm overlapping. The
duration W of each cell is exponentially distributed with parameter η (FW(w) = 1 – exp(η w))
and its intensity X is uniform through duration W with a specified distribution FX(x).
The rainfall intensity Ξ(t) at time t is the sum of the contributions of several cells Xj that
arrived some time tj before t and lasted more than ttj. We divide the time before t into small
intervals Ii of length Δa setting a0 t, a1 = tΔa, a2 = t – 2Δa, …, and Ii := (ai, ai – 1]. We
denote by Δni the number of cell arrivals within each Ii. The probability that Δni = 1 is β Δa
whereas the probability that Δni = 0 is 1 – β Δa (we can neglect the probability Δni > 1
because Δa is small). We also denote by Xi the intensity of the rain cell arrived at the interval
Ii and ΔΞ(t, ai) the rainfall intensity at time t due to that cell. Thus,
26
Ξ(t) =
i
ΔΞ(t, ai) (A.10)
Now, if
Δni = 0 then obviously ΔΞ(t, ai) = 0 whereas if Δni = 1 then
ΔΞ(t, ai) =
Xiif Wi > ai
0if
Wi < ai
(A.11)
Consequently,
E[ΔΞ(t, ai) | Δni = 1] = E[Xi] P(Wi > ai) (A.12)
where P( ) denotes probability, or,
E[ΔΞ(t, ai) | Δni = 1] = E[X] [1 – FW(ai)] (A.13)
and
E[ΔΞ(t, ai)] = E[ΔΞ(t, ai) | Δni = 1] P(Δni = 1) (A.14)
or
E[ΔΞ(t, ai)] = E[X] [1 – FW(ai)] β Δa (A.15)
Taking expected values in (A.10), combining (A.15) and also assuming Δa 0, we get
E[Ξ(t)] =
0
t
E[X] [1 – FW(a)] β da + E[X] [1 – FW(t)] (A.16)
where we have assumed that the storm have started at time t = 0. The last term in the right-
hand side of (A.16) is the contribution of the first cell of the storm, which in the Bartlett-
Lewis model is assumed to arrive at t = 0. Substituting the exponential expression of FW(a) in
(A.16) we get
E[Ξ(t)] = β E[X]
0
t
exp(–η a) da + E[X] exp(–η t) (A.17)
27
and after algebraic manipulations
E[Ξ(t)] = κ E[X] + (1 – κ) E[X] exp(–η t) (A.18)
where we have substituted β / η = κ. The second term in the right-hand side of (A.18)
represents the boundary effect. For large t (A.18) simplifies to (15).
Furthermore, we consider the product
Ξ(t) Ξ(t + τ) =
i
ΔΞ(t, ai)
j
ΔΞ(t + τ, aj) (A.19)
for τ > 0. When ai aj, ΔΞ(t, ai) is independent to ΔΞ(t + τ, aj), because they correspond to
different rain cells, and therefore
E[ΔΞ(t, ai) ΔΞ(t + τ, aj)] = E[ΔΞ(t, ai)] E[ΔΞ(t + τ, aj)] (A.20)
When ai = aj, ΔΞ(t, ai) and ΔΞ(t + τ, aj) correspond to the same cell. Therefore, if Δni = 1 then
ΔΞ(t, ai) ΔΞ(t + τ, ai) =
X 2
iif Wi > ai + τ
0if
Wi < ai + τ
(A.21)
whereas if Δni = 0 then obviously ΔΞ(t, ai) ΔΞ(t + τ, ai) = 0. Consequently,
E[ΔΞ(t, ai) ΔΞ(t + τ, ai) | Δni = 1] = E[X 2
i][1 – FW(ai + τ)] (A.22)
and
E[ΔΞ(t, ai) ΔΞ(t + τ, ai)] = E[X 2
i][1 – FW(ai + τ)] β Δa (A.23)
Taking expected values in (A.19), combining (A.20) and (A.23) and also assuming Δa
0, we get
E[Ξ(t) Ξ(t + τ)] =
0
t
E[X 2
i][1 – FW(a + τ)] β da + E[X 2
i][1 – FW(t + τ)]
+ E[Ξ(t)] E[Ξ(t + τ)] (A.24)
28
and
Cov[Ξ(t), Ξ(t + τ)] =
0
t
E[X 2
i] [1 – FW(a + τ)] β da + E[X 2
i] [1 – FW(t + τ)] (A.25)
Again we have assumed that the storm have started at time t = 0 and the last term in the right-
hand side of (A.25) is the contribution of the first cell of the storm. Substituting the
expression of FW(a) in (A.25) we get
Cov[Ξ(t), Ξ(t + τ)] = β E[X 2
i]
0
t
exp[–η (a + τ)] da + E[X 2
i] exp[–η (t + τ)] (A.26)
and after algebraic manipulations
Cov[Ξ(t), Ξ(t + τ)] = κ E[X 2
i] exp(–η τ) + (1 – κ) E[X 2
i] exp[–η (t + τ)] (A.27)
where we have substituted β / η = κ. The second term in the right-hand side of (A.27)
represents the boundary effect. For large t (A.27) simplifies to (16).
In the case of the modified Bartlett-Lewis rectangular pulse model (A.18) represents the
conditional mean for given η. The probability density function of η is
fH(η) = να ηα – 1 eν η
Γ(α) (A.28)
We assume that the distribution function of the storm duration D is practically equal to that of
V (see section 3 and Figure 2), that is
fD|H (d| η) = γ eγ d = φ η eφ η d (A.29)
From the Bayes’ theorem (e.g., Papoulis, 1965, p. 177) it follows that
fH|D (η| d) = fD|H (d| η) fH(η) /
0
fD|H (d| η) fH(η) dη (A.30)
or
29
fH|D (η| d) = φ η eφ η d να ηα – 1 eν η
Γ(α) /
0
φ η eφ η d να ηα – 1 eν η
Γ(α) dη (A.31)
which after algebraic manipulations becomes
fH|D (η| d) = (ν + φ d)α + 1 ηα e–(ν + φ d) η
Γ(α + 1) (A.32)
This is the gamma probability density with shape parameter α + 1 and scale parameter ν + φ d.
The mean of Ξ(t) conditional on D will be
E[Ξ(t)|D = d] =
0
{κ E[X] + (1 – κ) E[X] exp(–η t)} fH|D (η| d) dη (A.33)
which reduces to
E[Ξ(t)|D = d] = κ E[X] + (1 – κ) E[X]
ν + φ d
ν + φ d + t
α + 1
(A.34)
If we neglect the last term, which represents the boundary effect, we arrive at the same
expression as in the original Bartlett-Lewis model.
The covariance function of Ξ(t) conditional on D will be
Cov[Ξ(t), Ξ(t + τ)|D = d] =
0
{κ E[X 2
i] exp(–η τ) + (1 – κ) E[X 2
i] exp[–η (t + τ)]} fH|D (η| d) dη (A.35)
which reduces to
Cov[Ξ(t), Ξ(t + τ)|D = d] = κ E[X 2
i]
ν + φ d
ν + φ d + τ
α + 1
+ (1 – κ) E[X 2
i]
ν + φ d
ν + φ d + t + τ
α + 1
(A.36)
If we neglect the last term, which represents the boundary effect, we get (17).
30
A3. Expressions of variances and covariances in discrete time for the Bartlett-
Lewis rectangular pulse model
It may be easily shown assuming a stationary process Ξ(t) that (A.3) results in
Cov[Yi, Yi + m] =
Δ
Δ
Cov[Ξ(t), Ξ(t + m Δ + τ) (Δ |τ|) dτ (A.37)
Substituting the covariance term of the right-hand side of (A.37) from (16) we get for the
original Bartlett-Lewis model
Cov[Yi, Yi + m] = κ E[X 2]
Δ
Δ
eη |m Δ + τ| (Δ |τ|) dτ (A.38)
which after algebraic calculations results in (20) when m > 0 or in (19) when m = 0.
Similarly, substituting the covariance term of the right-hand side of (A.37) from (17) we
get for the modified Bartlett-Lewis model
Cov[Yi, Yi + m] = κ E[X 2]
Δ
Δ
ν + φ d
ν + φ d + |m Δ + τ|
α + 1
(Δ |τ|) dτ (A.39)
which after algebraic calculations results in (23) when m > 0 or in (22) when m = 0.
A4. Derivation of equation for the correction of rain depth variance at each class
Let Ω denote the event that a storm’s duration belongs to a certain class and μD := E[D | Ω],
σ2
D := Var[D | Ω]. Let also μ΄
H := E[H | Ω] (the mean of the total storm depth when the storm’s
duration belong to the certain class), μ
H := E[H | D = μD] (the mean of the total storm depth
when the storm’s duration equals μD), σ΄2
H := Var[H | Ω] and σ2
H := Var[H | D = μD]. We need
to estimate μ
H and σ2
H given μ΄
H and σ΄2
H, which can be directly estimated from the available
data.
We assume that each of the conditional moments E[H | D] and Var[H | D] is a power
function of D, at least for durations in the certain class, i.e.,
31
E
[H | D] = c1 D 1 + κ1 (A.40)
Var[H | D] = c2 D 2(1 + κ2) (A.41)
where for generality we have assumed different κ in each case (κ1 and κ2). This assumption is
true for the scaling model, with κ1 = κ2 =
κ, and approximately valid for the BL model, in
which case κ1 = 0 whereas κ2 < 0. (As shown in Figure 4, the power relation for the variance
of H approximately holds for the BL model as well). (A.40) and (A.41) can be written as
E
[H | D] = μ
H (D / μD)1 + κ1 (A.42)
Var[H | D] = σ2
H (D / μD) 2(1 + κ2) (A.43)
The mean
μ΄
H is given by
μ΄
H = E[H | Ω] =
ξ Ω
E[H | ξ] fD|Ω(ξ) dξ =
ξ Ω
μ
H (D / μD)1 + κ1 fD|Ω (ξ) dξ (A.44)
where fD|Ω( ) denotes the conditional probability density function of D in Ω. To proceed with
an approximation of μ΄
H we take the Taylor series of E[H | D] around D = μD:
E[H | D] = μ
H + (1 + κ1)(DμD) (μ
H / μD) + (1/2) κ1 (1 + κ1) (DμD)2 (μ
H / μ2
D) + ... (A.45)
Assuming that κ1 is small, the third term and all terms beyond can be neglected, so that
E[H | D] μ
H + (1 + κ1)(DμD) (μ
H / μD) (A.46)
Substituting (A.46) into (A.44) and since
ξ Ω
fD|Ω(ξ) dξ = 1,
ξ Ω
(DμD) fD|Ω (ξ) dξ = 0 (A.47)
we get
μ΄
H μ
H (A.48)
32
which indicates that no correction is needed to obtain μ
H from μ΄
H.
Similarly, we have
E[H2 | D] = σ2
H (D / μD) 2(1 + κ2) + μ2
H (D / μD) 2(1 + κ1) (A.49)
whose Taylor expansion around D = μD is
E[H2 | D] = (σ2
H + μ2
H) + (2 / μD) [(1 + κ2) σ2
H + (1 + κ1) μ2
H] (DμD)
+ (1 / μ2
D) [(1 + κ2) (1 + 2 κ2) σ2
H + (1 + κ1) (1 + 2 κ1) μ2
H] (DμD)2 +
+ (2/3) (1 / μ3
D) [κ2 (1 + κ2) (1 + 2 κ2) σ2
H + κ1 (1 + κ1) (1 + 2 κ1) μ2
H] (D μD)3 + ... (A.50)
Assuming that κ1 and κ2 are small, the fourth term and all terms beyond can be neglected, so
that
E[H2 | D] (σ2
H + μ2
H) + (2 / μD) [(1 + κ2) σ2
H + (1 + κ1) μ2
H] (DμD)
+ (1 / μ2
D) [(1 + κ2) (1 + 2 κ2) σ2
H + (1 + κ1) (1 + 2 κ1) μ2
H] (DμD)2 (A.51)
We note however that if σ2
H = 0, (A.51) results in
E[H2 | D] μ2
H + (2 / μD) (1 + κ1) μ2
H (DμD)
+ (1 / μ2
D) (1 + κ1) (1 + 2 κ1) μ2
H (DμD)2 (A.52)
which is not consistent with (A.46). Indeed, in this special case Var[H | D] = 0 and E[H2 | D]
= E2[H | D], which from (A.46) is
E[H2 | D] μ2
H + (2 / μD) (1 + κ1) μ2
H (DμD) + (1 / μ2
D) (1 + κ1)2 μ2
H (DμD)2 (A.53)
Thus, to make (A.51) consistent with (A.46), we replace (1 + 2 κ2) and (1 + 2 κ1) by (1 + κ2)
and (1 + κ1), respectively, so that it writes
E[H2 | D] (σ2
H + μ2
H) + (2 / μD) [(1 + κ2) σ2
H + (1 + κ1) μ2
H] (DμD)
+ (1 / μ2
D) [(1 + κ2)2 σ2
H + (1 + κ1)2 μ2
H] (DμD)2 (A.54)
Besides
33
E[H2 | Ω] =
ξ Ω
E[H2 | ξ] fD|Ω(ξ) dξ (A.55)
which after substituting E[H2 | ξ] from (A.51) and also noticing that, in addition to (A.47),
ξ Ω
(DμD)2 fD|Ω (ξ) dξ = σ2
D (A.56)
yields
E[H2 | Ω] (σ2
H + μ2
H) + (σ2
D / μ2
D) [(1 + κ2)2 σ2
H + (1 + κ1)2 μ2
H] (A.57)
Now, considering that σ΄2
H = Var[H | Ω] = E[H2 | Ω] – E2[H | Ω] and μ΄
H μ
H, we find that
σ΄2
H σ2
H + (σ2
D / μ2
D) [(1 + κ2)2 σ2
H + (1 + κ1)2 μ2
H] (A.58)
and solving for σ2
H
σ2
H
σ΄2
Hμ2
H (1 + κ1)2 σ2
D / μ2
D
1 + (1 + κ2)2 σ2
D / μ2
D
(A.59)
If we set κ1 = κ2 = κ in (A.59) we find (26).
It may be shown that (A.59) is exact in the special case where H can be expressed as H =
Φ D, where Φ is a random variable independent of D. In this case κ1 = κ2 = 0. In another
special case where H can be expressed as the sum of D independent variables Φi (assuming
that D is integer), in which κ1 = 0 and κ2 = –0.5, (A.59) would become exact if we omit the
factor (1 + κ2)2 in the denominator. Thus (A.59) becomes exact for both special cases if we
slightly modify it to become
σ2
H
σ΄2
Hμ2
H (1 + κ1)2 σ2
D / μ2
D
1 + [max(0, 1 + 2 κ2)]2 σ2
D / μ2
D
(A.60)
34
To verify (A.59) and (A.60) two series of stochastic simulations were performed. In the
first one, for a given κ, H was generated as H = Φ D1 + κ, where Φ is a random variable
independent of D, so that both (A.40) and (A.41) are valid for κ1 = κ2 = κ. In the second
series, H was generated as the sum of D correlated variables Φi (where D is integer),
assuming that the sequence of Φi is Markovian with mean μ standard deviation σ and lag-one
autocorrelation ρ > 0. It can be shown that in the second case,
E[Η | D] = μ D (A.61)
Var[H | D] = σ2 D (1 – ρ2) – 2ρ (1 – ρD)
(1 – ρ)2 (A.62)
Consequently,
κ1 = d(ln(E[Η | D]))
d(ln D) – 1 = 0 (A.63)
κ2 = 1
2 d(ln(Var[Η | D]))
d(ln D) – 1 = 1
2 + ρ 1 – ρD (1 – D ln ρ)
D (1 – ρ2) – 2 ρ (1 – ρD) (A.64)
Both (A.59) and (A.60) yielded good approximations in both series of simulations, even with
large values of σD / μD, which verifies their appropriateness.
35
Tables
Table 1 Characteristics of storms in Zographou meteorological station according to their
duration (Time interval Δ = 10 min; durations in h, depths in mm).
Class range Duration Storm depth Incremental depth
Class Dmin Dmax
Number
of storms E[D] Std[D] E[H] Std[H] E[Y] Std[Y] Corr[Yi,Yi+1]
1 0.17 2.00 19 1.09 0.65 12.99 5.09 1.95 3.18 0.39
2 2.17 4.00 14 2.95 0.53 18.27 13.07 1.03 1.87 0.43
3 4.17 12.00 18 7.67 2.64 23.49 13.60 0.51 1.28 0.63
4 12.17 18.00 17 14.90 1.75 36.34 17.84 0.41 0.88 0.58
5 18.17 40.00 13 24.78 6.91 60.55 38.68 0.41 1.11 0.60
A 0.50 5.00 38 2.28 1.35 16.63 12.26 1.24 2.39 0.51
B 5.17 40.00 43 16.08 1.47 39.50 29.02 0.41 1.01 0.58
All 0.17 40.00 81 9.57 8.85 28.77 25.37 0.50 1.26 0.57
Table 2 Characteristics of storms in Zographou meteorological station according to their
duration (Time interval Δ = 1 h; durations in h, depths in mm).
Class range Duration Storm depth Incremental depth
Class Dmin Dmax
Number
of storms E[D] Std[D] E[H] Std[H] E[Y] Std[Y] Corr[Yi,Yi+1]
1 1 2 19 1.95 0.78 12.99 5.80 6.67 6.29 -0.30
2 3 4 14 3.86 0.77 18.27 13.04 4.74 6.03 0.02
3 5 12 18 8.50 2.60 23.49 13.74 2.76 4.93 0.25
4 13 18 17 15.88 1.80 36.34 17.85 2.29 3.45 0.42
5 19 40 13 25.46 6.79 60.55 38.78 2.38 4.61 0.55
A 1 5 38 3.11 1.47 16.63 12.26 5.36 6.65 0.10
B 6 40 43 16.91 7.40 39.50 28.02 2.34 4.07 0.45
All 1 40 81 10.43 8.82 28.77 25.37 2.76 4.64 0.38
36
Table 3 Characteristics of storms in Parrish raingauge according to their duration (Time
interval Δ = 15 min; durations in h, depths in mm).
Class range Duration Storm depth Incremental depth
Class Dmin Dmax
Number
of storms E[D] Std[D] E[H] Std[H] E[Y] Std[Y] Corr[Yi,Yi+1]
1 0.25 0.25 608 0.25 0.00 3.40 1.30 3.40 1.30
2 0.50 0.50 139 0.50 0.00 10.10 5.60 5.10 3.70 0.23
3 0.75 0.75 113 0.75 0.00 15.60 9.70 5.20 4.70 0.32
4 1.00 1.00 78 1.00 0.00 20.10 13.00 5.00 5.20 0.32
5 1.25 1.50 112 1.38 0.13 18.40 12.77 3.40 4.00 0.39
6 1.75 2.25 109 1.98 0.23 21.20 14.75 2.70 3.80 0.40
7 2.50 3.25 107 2.83 0.28 25.10 17.76 2.20 3.90 0.50
8 3.50 4.50 137 3.98 0.38 26.90 19.76 1.70 3.40 0.49
9 4.75 7.00 119 5.73 0.73 28.80 20.52 1.30 3.00 0.49
10 7.25 26.75 121 12.08 4.83 47.00 27.44 1.00 2.10 0.35
A 1.00 4.00 491 2.17 0.96 22.40 15.79 2.58 3.99 0.45
B 4.25 26.75 292 8.12 4.60 35.70 26.25 1.10 2.51 0.44
All 0.25 26.75 1643 2.28 3.47 16.21 18.68 1.78 3.22 0.48
37
Table 4 Characteristics of storms in Parrish raingauge according to their duration (Time
interval Δ = 2 h; durations in h, depths in mm).
Class range Duration Storm depth Incremental depth
Class Dmin Dmax
Number
of storms E[D] Std[D] E[H] Std[H] E[Y] Std[Y] Corr[Yi,Yi+1]
1 2 2 1124 2.00 0.00 9.30 10.30 9.30 10.30
2 4 4 227 4.00 0.00 25.50 17.70 12.70 13.70 0.02
3 6 6 132 6.00 0.00 26.70 21.20 8.90 13.00 0.08
4 8 8 64 8.00 0.00 31.60 19.30 7.90 10.60 0.07
5 10 10 35 10.00 0.00 35.10 19.70 7.00 9.00 0.27
6 12 14 27 12.80 1.00 47.70 26.04 7.50 9.10 0.09
7 16 28 34 19.60 4.00 69.10 30.98 7.10 8.80 0.26
A 8 12 52 10.60 0.95 37.40 19.79 7.02 8.57 0.12
B 14 28 44 18.20 4.17 66.45 32.02 7.27 9.12 0.23
All 2 28 1643 3.54 3.28 16.21 18.68 9.16 11.16 0.11
Table 5 Fitted parameters of the scaling model of storm hyetograph.
Parameter Zographou Parrish
κ –0.54 –0.60
c1 (mm) 11.3 16.2
c2 (mm2) 45.3 116.6
ζ 0.92 0.00
β 0.21 0.34
Table 6 Fitted parameters of the Bartlett-Lewis rectangular pulse model – original version.
Parameter Zographou Parrish
μΧ (mm h–1) 9.8 14.6
κ 0.53 0.97
η (h–1) 4.83 6.36
38
Table 7 Fitted parameters of the Bartlett-Lewis rectangular pulse model – random parameter
version.
Parameter Zographou Parrish
μΧ (mm h–1) 9.2 15.3
κ 0.56 0.80
ν (h) 0.34 1.17
α 2.02 7.86
φ 0.0157 0.0001
Table 8 Fitted parameters of the Bartlett-Lewis rectangular pulse model – additional version.
Parameter Zographou Parrish
μΧ (mm h–1) 13.1 16.5
κ0 0.86 0.98
κ1 -0.54 -0.60
η0 (h–1) 7.92 8.24
η1 -0.33 -0.34
39
List of Figures
Figure 1 Explanatory sketch for the scaling model of storm hyetograph.
Figure 2 Explanatory sketch for the Bartlett-Lewis rectangular pulse model.
Figure 3 Empirical and theoretical mean of total storm depth as a function of storm duration:
(a) Zographou, Δ = 10 min; (b) Zographou, Δ = 1 h; (c) Parrish, Δ = 15 min; (d) Parrish, Δ = 2
h.
Figure 4 Empirical and theoretical standard deviation of total storm depth as a function of
storm duration: (a) Zographou, Δ = 10 min; (b) Zographou, Δ = 1 h; (c) Parrish, Δ = 15 min;
(d) Parrish, Δ = 2 h.
Figure 5 Empirical and theoretical mean of incremental storm depth as a function of storm
duration: (a) Zographou, Δ = 10 min; (b) Zographou, Δ = 1 h; (c) Parrish, Δ = 15 min; (d)
Parrish, Δ = 2 h.
Figure 6 Empirical and theoretical standard deviation of incremental storm depth as a
function of storm duration: (a) Zographou, Δ = 10 min; (b) Zographou, Δ = 1 h; (c) Parrish, Δ
= 15 min; (d) Parrish, Δ = 2 h.
Figure 7 Empirical and theoretical lag-one autocorrelation coefficient of incremental storm
depth as a function of storm duration: (a) Zographou, Δ = 10 min; (b) Zographou, Δ = 1 h; (c)
Parrish, Δ = 15 min; (d) Parrish, Δ = 2 h.
Figure 8 Empirical and theoretical autocorrelation function of incremental storm depth for
small storm durations: (a) Zographou, Δ = 10 min; (b) Zographou, Δ = 1 h; (c) Parrish, Δ = 15
min; (d) Parrish, Δ = 2 h.
40
Figure 9 Empirical and theoretical autocorrelation function of incremental storm depth for
large storm durations: (a) Zographou, Δ = 10 min; (b) Zographou, Δ = 1 h; (c) Parrish, Δ = 15
min; (d) Parrish, Δ = 2 h.
41
Figures
Ξ(t,D)
1
D1
D2
1
D1
D2
D
t = λ D
t t = D
Figure 1 Explanatory sketch for the scaling model of storm hyetograph.
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X22
X21 X23
X24
t21 t2t22 t23 t24
t21 t2t22 t23 t24
v2
w21 w22
w23 w24
t1t2t3time
time
Figure 2 Explanatory sketch for the Bartlett-Lewis rectangular pulse model.
42
10
100
110100
D (h)
E[H] (mm)
(b)
10
100
110100
D (h)
E[H] (mm)
(a)
Empirical Scaling & BL/additional BL/original & BL/random parameter
10
100
110100
D (h)
E[H] (mm)
(d)
10
100
0.1 1 10 100
D (h)
E[H] (mm)
(c)
Figure 3 Empirical and theoretical mean of total storm depth as a function of storm duration:
(a) Zographou, Δ = 10 min; (b) Zographou, Δ = 1 h; (c) Parrish, Δ = 15 min; (d) Parrish, Δ = 2
h.
43
1
10
100
1000
1 10 100
D (h)
Std[H (mm)
(b)
1
10
100
1000
1 10 100
D (h)
Std[H] (mm)
(a)
Empirical Scalin g BL/original BL/random parameter BL/additional
1
10
100
1 10 100
D (h)
Std[H (mm)
(d)
1
10
100
0.1 1 10 100
D (h)
Std[H] (mm)
(c)
Figure 4 Empirical and theoretical standard deviation of total storm depth as a function of
storm duration: (a) Zographou, Δ = 10 min; (b) Zographou, Δ = 1 h; (c) Parrish, Δ = 15 min;
(d) Parrish, Δ = 2 h.
44
0
2
4
6
8
10
0102030
D (h)
E[Y] (mm)
(b)
0
1
2
0102030
D (h)
E[Y] (mm)
(a)
Empirical Scaling & BL/additional BL/original & BL/random parameter
0
5
10
15
20
25
30
0 5 10 15 20
D (h)
E[Y] (mm)
(d)
0
1
2
3
4
5
6
0 5 10 15 20
D (h)
E[Y] (mm)
(c)
Figure 5 Empirical and theoretical mean of incremental storm depth as a function of storm
duration: (a) Zographou, Δ = 10 min; (b) Zographou, Δ = 1 h; (c) Parrish, Δ = 15 min; (d)
Parrish, Δ = 2 h.
45
0
2
4
6
8
10
0102030
D (h)
Std[Y] (mm)
(b)
0
1
2
3
0102030
D (h)
Std[Y] (mm)
(a)
Empirical Scalin g BL/original BL/random parameter BL/additional
0
4
8
12
16
20
0 5 10 15 20
D (h)
Std[Y] (mm)
(d)
0
1
2
3
4
5
6
0 5 10 15 20
D (h)
Std[Y] (mm)
(c)
Figure 6 Empirical and theoretical standard deviation of incremental storm depth as a
function of storm duration: (a) Zographou, Δ = 10 min; (b) Zographou, Δ = 1 h; (c) Parrish, Δ
= 15 min; (d) Parrish, Δ = 2 h.
46
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30
D (h)
Corr[Yi, Yi+1]
(
b
)
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30
D (h)
Corr[Yi, Yi+1 ]
(a)
Empirical Scaling BL/original BL/random parameter BL/additional
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20
D (h)
Corr[Yi, Yi+1]
(
d
)
-0.4
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Figure 7 Empirical and theoretical lag-one autocorrelation coefficient of incremental storm
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Figure 8 Empirical and theoretical autocorrelation function of incremental storm depth for
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Figure 9 Empirical and theoretical autocorrelation function of incremental storm depth for
large storm durations: (a) Zographou, Δ = 10 min; (b) Zographou, Δ = 1 h; (c) Parrish, Δ = 15
min; (d) Parrish, Δ = 2 h.
... 2.3.2 Rectangular pulse cluster based stochastic rainfall models .Stochastic rainfall models based on point processes have been one of the most widespread and useful tools in the analysis and modelling of rainfall (Koutsoyiannis and Mamassis, 2001). ...
... In cluster based models, storms are modelled as clusters of rain cells and each cell is a pulse with a random duration and random intensity, which is constant for the duration of the pulse (Smithers and Schulze, 2000a;Frost et al., 2004). In these models, both storms and the origin of cells for each storm arrive follow a Poisson process (Koutsoyiannis and Mamassis, 2001;Koutsoyiannis, 2003). The two abovementioned cluster based models have slight differences (Entekhabi et al., 1989). ...
... These methods treat a rainfall event as a cluster of many small rainy cells with a random period of rain (Rodríguez-Iturbe et al. 1987;Rodriquez-Iturbe et al. 1988;Onof and Wheater 1994). However, point process models require a large number of parameters (5-7), and have shown some disagreements (Koutsoyiannis and Mamassis 2001). Scalebased methods use the inherent scale-invariant structure of rainfall (Schertzer and Lovejoy 1987;Gupta et al. 1993) with the assumption that the statistical properties of rainfall occurrences aggregated at different time scales are related through a simple scaling function. ...
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The time distribution of extreme rainfall events is a significant property that governs the design of urban stormwater management structures. Accuracy in characterizing this behavior can significantly influence the design of hydraulic structures. Current methods used for this purpose either tend to be generic and hence sacrifice on accuracy or need a lot of model parameters and input data. In this study, a computationally efficient multistate first-order Markov model is proposed for use in characterizing the inherently stochastic nature of the dimensionless time distribution of extreme rainfall. The model was applied to bivariate extremes at 10 stations in India and 205 stations in the United States (US). A comprehensive performance evaluation was carried out with one-hundred stochastically generated extremes for each historically observed extreme rainfall event. The comparisons included: 1-h (15-min); 2-h (30-min); and, 3-h (45-min) peak rainfall intensities for India and (US) stations, respectively; number of first, second, third, and fourth-quartile storms; the dependence of peak rainfall intensity on total depth and duration; and, return levels and return periods of peak discharge when these extremes were applied on a hypothetical urban catchment. Results show that the model efficiently characterizes the time distribution of extremes with: Nash–Sutcliffe-Efficiency > 0.85 for peak rainfall intensity and peak discharge; < 20% error in reproducing different quartile storms; and, < 0.15 error in correlation analysis at all study locations. Hence the model can be used to effectively reproduce the time distribution of extreme rainfall events, thus increasing the confidence of design of urban stormwater management structures.
... The other, less well known problem was identified by Marani (2003). While one of the strengths of models based upon Poisson cluster processes is their ability to capture rainfall variability over a range of scales (hence its use in disaggregation; see Koutsoyiannis and Mamassis, 2001), they underestimate this variability for scales equal to or larger than a few days. ...
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The use of Poisson cluster processes to model rainfall time series at a range of scales now has a history of more than 30 years. Among them, the randomised (also called modified) Bartlett-Lewis model (RBL1) is particularly popular, while a refinement of this model was proposed recently (RBL2; Kaczmarska et al., 2014). Fitting such models essentially relies upon minimising the difference between theoretical statistics of the rainfall signal and their observed estimates. The first statistics are obtained using closed form analytical expressions for statistics of the orders 1 to 3 of the rainfall depths, as well as useful approximations of the wet-dry structure properties. The second are standard estimates of these statistics for each month of the data. This paper discusses two issues that are important for the optimal model fitting of RBL1 and RBL2. The first issue is that, when revisiting the derivation of the analytical expressions for the rainfall depth moments, it appears that the space of possible parameters is wider than has been assumed in past papers. The second issue is that care must be exerted in the way monthly statistics are estimated from the data. The impact of these two issues upon both models, in particular upon the estimation of extreme rainfall depths at hourly and sub-hourly timescales, is examined using 69 years of 5 min and 105 years of 10 min rainfall data from Bochum (Germany) and Uccle (Belgium), respectively.
... The other less well-known problem was identified by Marani (2003). While one of the strengths of models based upon Poisson-cluster processes is their ability to capture rainfall variability over a range of scales (hence its use in disaggregation -see Koutsoyiannis and Mamassis (2001); Koutsoyiannis et al. (2003); Kossieris et al. (2016)), they underestimate this variability for scales equal to or larger than a few days. ...
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Full-text available
The use of Poisson-cluster processes to model rainfall time series at a range of scales now has a history of more than 30 years. Among them, the Randomised (also called modified) Bartlett–Lewis model (RBL1) is particularly popular, while a refinement of this model was proposed recently (RBL2) (Kaczmarska et al., 2014). Fitting such models essentially relies upon minimising the difference between theoretical statistics of the rainfall signal and their observed estimates. The first are obtained using closed form analytical expressions for statistics of order 1 to 3 of the rainfall depths, as well as useful approximations of the wet-dry structure properties. The second are standard estimates of these statistics for each month of the data. This paper discusses two issues that are important for optimal model fitting of the RBL1 and RBL2. The first is that, when revisiting the derivation of the analytical expressions for the rainfall depth moments, it appears that the space of possible parameters is wider than has been assumed in the past papers. The second is that care must be exerted in the way monthly statistics are estimated from the data. The impact of these two issues upon both models, in particular upon the estimation of extreme rainfall depths at hourly and sub-hourly timescales is examined using 69 years of 5-min and 105 years of 10-min rainfall data from Bochum (Germany) and Uccle (Belgium), respectively.
... Usually, if long series of rainfall data are available, their temporal resolution is often unfortunately lower than that required for the application of a statistical model; in contrast, if high-resolution rainfall data are available, their sample is often not sufficiently long to perform reliable statistical analyses. In these cases, the synthetic generation of long series of high-resolution rainfall events using stochastic point process representations of rainfall (e.g., through Monte Carlo approach) [5][6][7][8][9][10] can be a very practical alternative. ...
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The aim of this paper is to present a stochastic model to generate sub-hourly rainfall events at a given point. Historical events used as the input have been extracted by the sub-hourly rainfall series available for a defined rain gauge station based on a fixed inter-event time and selected if their average intensity was larger than a critical fixed one. The sub-hourly events generated by applying the proposed methodology are completely stochastic and their main characteristics, i.e., shape, duration and average intensity, have been derived as a function of the statistics of the historical events analyzed. In order to characterize the shape, dimensionless hyetographs have been derived. They have been statistically modelled by using the Beta cumulative distribution. Average intensity and duration of the historical events were first modelled separately by fitting several probability distributions and selecting the best one using the more common statistical criteria. Then, their correlation was modelled using the Frank’s copula. In order to test the methodology, two sites in Sicily, Italy, where 10 min’ recorded rainfall data were available, were analyzed. Finally, comparison between the statistics of the simulated events and those of the measured data demonstrates the good performance of the model.
... Five categories for duration as D1, D2… D5 can be selected in the RDP model. The intervals [0-3], [3-6], [6][7][8][9][10][11][12], [12][13][14][15][16][17][18][19][20][21][22][23][24] and more than 24 hours are selected for duration of storm patterns. For example if duration of one observed storm event is 7 hours, this event belongs to the third category of Group 1(D3). ...
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Lack of storm patterns (storm hyetograph) in many catchments is an important issue in hydrological analysis. So, in many studies various methods are developed to generate storm pattern. There are uncertainties in generated storm patterns due to uncertainty of generating method (model uncertainty) and uncertainty of the variables affecting the storm patterns such as the total depth of rainfall, rainfall duration and dimensionless hyetograph (inherent uncertainty). This study developed the Rain Data Processor (RDP) and the Rain Pattern Generator (RPG) models to generate storm patterns based on Mass curve method with considering inherent and model uncertainty in ungauge catchments by using the Monte Carlo simulation and Bootstrap resampling. Methodology of this study is applied in Iran (Seymareh catchment).According to the statistics of generated peak intensity by the RPG model; there is an acceptable agreement between observed and generated hyetographs. Also, the RPG model is more accurate than triangular hyetograph model in generation of storm pattern.
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Precipitation data for long period with short time scales extending from limited minute to daily time steps is the principal requirement in water resources engineering for planning, design, and operation of water resources projects. However, the availability of measurements of these data faced several obstacles in various places in the world. In this study, the Intensity Duration Frequency (IDF) relationships of precipitation amount at Najaf city region in Iraq are generated based on a stochastic technique for disaggregation of daily time scale precipitation into fine-scale durations, i.e. the modified Bartlett-Lewis rectangular pulses of the seven parameters (BLRP) model (λ, κ, φ, α, υ, μχ, and σχ). The model applied for daily precipitation over a period of 35 years by employing the Climate Forecast System Reanalysis (CFSR) grid station data after implementing bias correction with respect to the monthly available data at Najaf city meteorological station. Eight sub-daily precipitation data sets with storm durations of 5, 10, 20, 30, 60, 120, 360, and 720 min were generated by using Hyetos-Minute R-package. Three statistical continuous distributions used to fit the estimation of extreme values data sets which included the Generalised Extreme Value (GEV), Gumbel (EV-1), and Log-Pearson type III (LP-3). Finally, the IDF curves were generated based on GEV and LP-3 for six return periods (2, 5, 10, 25, 50, and 100 years) using Sherman equation for each return period. The goodness of fit between the theoretical and developed distributions was tested by applying the Kolmogorov–Smirnov, Anderson–Darling and chi-squared test at 5% significance level. Results showed affirmative agreements between the both datasets from the three distributions at 5% significance level. Furtherly, the probabilistic model selection by two numerical criteria AIC and BIC nominated that GEV and LP-3 as appropriate models for simulation extreme precipitation values at Najaf city for different return periods.
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The rainfall intensity–duration–frequency (IDF) curves play an important role in water resources engineering and management. The applications of IDF curves range from assessing rainfall events, classifying climatic regimes, to deriving design storms and assisting in designing urban drainage systems, etc. The deriving procedure of IDF curves, however, requires long-term historical rainfall observations, whereas lack of fine-timescale rainfall records (e.g. sub-daily) often results in less reliable IDF curves. This paper presents the utilization of remote sensing sub-daily rainfall, i.e. Global Satellite Mapping of Precipitation (GSMaP), integrated with the Bartlett-Lewis rectangular pulses (BLRP) model, to disaggregate the daily in situ rainfall, which is then further used to derive more reliable IDF curves. Application of the proposed method in Singapore indicates that the disaggregated hourly rainfall, preserving both the hourly and daily statistic characteristics, produces IDF curves with significantly improved accuracy; on average over 70% of RMSE is reduced as compared to the IDF curves derived from daily rainfall observations.
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A design storm definition procedure for the sizing of channels within mixed urban and lowland rice drainage systems is developed, with reference to the Northern Delta in Vietnam. By analyzing a 20-years-long time series of rainfall observations recorded at Hanoi monitoring station, typical storm events were identified and a set of suitable design storms was determined and recommended for the optimal design storm selection. A procedure to select an optimal design hyetograph is then defined, based on a combination of the design storm method and continuous simulation. A set of simulation experiments on a fictitious basin was finally carried out to support the choice of the storm temporal pattern providing the best performance in terms of reproduction of runoff peaks.
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A design storm definition procedure for the sizing of channels within mixed urban and lowland rice drainage systems is developed, with reference to the Northern Delta in Vietnam. By analyzing a 20-years-long time series of rainfall observations recorded at Hanoi monitoring station, typical storm events were identified and a set of suitable design storms was determined and recommended for the optimal design storm selection. A procedure to select an optimal design hyetograph is then defined, based on a combination of the design storm method and continuous simulation. A set of simulation experiments on a fictitious basin was finally carried out to support the choice of the storm temporal pattern providing the best performance in terms of reproduction of runoff peaks
Conference Paper
Full-text available
Based on the recently developed scaling model of storm hyetograph, a conditional simulation scheme is presented, which can be used for stochastic forecasting of the temporal evolution of rainfall. The scaling model is fitted to hourly rainfall data of Greece and Italy. In addition, the model is tested for capturing statistical properties that are not explicitly used for the fitting. The scheme is formulated so as to use any information known for the rainfall event, as a condition for the simulation. The conditional simulation scheme is applied in two steps: first we generate the duration and total depth of the event and then we disaggregate the total depth into sequential hourly depths. Two different types of conditions are examined. The first one concerns the incorporation of preceding hourly rainfall depths. The second is related to information given by meteorological forecasts from which we can approximately estimate the duration and total depth of the event.
Thesis
In this thesis, we study the evolution of rainfall at a single site and over a network of sites by generalising existing point process based models. Stochastic models based on clustered point processes, such as the Neyman-Scott and the Bartlett-Lewis processes, have been used recently in the description of the behaviour of rainfall at a fixed point in space. In such models, storms are idealised as cluster origins that arrive in a Poisson process and are followed by a number of rain cell origins, the cluster members. A rectangular pulse is associated with each rain cell origin, having independent random duration and intensity. In this thesis, a class of models with rain cell duration and intensity being dependent random variables has been developed and the main properties have been derived. For the description of the evolution of a rainfall event at many distinct spatial locations, we consider a master clustered point process which is decomposed into subprocesses according to a marking mechanism, depending on the location(s) that are affected by a storm and its rain cells. Each cluster of the sub-processes is randomly translated in time, in order to allow different sites to experience the same event at different times. Some of the model's parameters remain the same at all generated sub-processes, while others vary in a stochastic or deterministic way. We follow two approaches in modelling the probability that a storm or a rain cell affects a particular subset of sites. One is to describe the spatial structure of a rainfall event by assigning to each of its elements a band of random width, location and orientation, that intersects the catchment area. Alternatively, the probability that two sites experience the same event can be expressed as a deterministic function of the distance between the sites. The models are fitted to rain gauge data from the South-West of England.
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A concise stochastic model for the nondimensional thunderstorm rainfall process at a point is proposed. The accumulated precipitation process for individual thunderstorms is nondimensionalized by dividing the precipitation at any time by the total precipitation and the elapsed time by the total duration. The dimensionless process is divided into 100 equal time increments, and the depth increments are rescaled to range between 0 and 1. The sequence of rescaled increments Z1, Z2,…, Z9 are assumed to represent a nonhomogeneous Markov process in discrete time with continuous state space. The expected value of the kth rescaled increment, given the k-1st increment, is assumed to be a linear function of that increment, and the marginal distribution of the first increment and the conditional distributions are assumed to be described by the beta distribution. An analyses of data for 275 thunderstorms observed at the Walnut Gulch Experimental Watershed in southeastern Arizona showed that the proposed model structure is a good approximation for this region. The number of model parameters can be reduced from 26 to a minimum of 10 by approximating the 2 parameters in the conditional expectation function and the conditional beta parameter as polynomial functions of the dimensionless time. Likelihood ratio tests and the Akaike information criterion suggest that the dependence parameters are independent of storm amount and duration, but the conditional beta parameter αk is larger for short-duration storms than for long-duration storms. A 13-parameter model is recommended for disaggregating thunderstorm rainfall in southeastern Arizona.
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This article presents a review of recent advances in rainfall modeling, estimation, and forecasting geared towards hydrologic science and applications. Only peer reviewed articles in U.S journals or by U.S investigators in other journals during the period of 1991 to 1994 are presented, in conformance with the requirements of IUGG. Whenever necessary, reference is made to research articles published during the period of 1987 to 1990, as review of rainfall research during that period was not covered in the previous IUGG volume. Georgakakos and Kavvas's (1987) article presents an insightful review of rainfall research for the period of 1983 to 1986.
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This technical note presents a comparison of cluster-based point rainfall models using the historical hourly rainfall data observed between 1949 and 1976 at Denver, Colorado. The Denver data are used to analyze the performance of three classes of models, namely, the Bartlett-Lewis model, the geometric Neyman-Scott model and the Poisson Neyman-Scott model. The original formulation of the structure of each model, as well as the modified description developed in order to improve the zero depth probability, is considered in this study Rodriguez-Iturbe et al. (1987a) concluded that it is unlikely that empirical analysis of rainfall data can be used to choose between the Bartlett-Lewis model and the Neyman-Scott model. In a subsequent paper, Rodriguez-Iturbe et al. (1987b) argued that the choice of the distribution of the number of cells per storm for the Neyman-Scott model, either geometric or Poisson, has no general bias effect on the stochastic structure. Some investigators (e.g., Burlando and Rosso, 1991), however, reported results contradictory to those of the previous authors. In light of these observations this note investigates the performance of the cluster-based models. For the Denver data the geometric Neyman-Scott model yields better results compared to the Poisson Neyman-Scott model. Moreover, the Bartlett-Lewis model is shown to be very sensitive to the sets of moment equations used in the parameter estimation. This sensitivity is not observed in the Neyman-Scott scheme and is believed to be a drawback for applying the Bartlett-Lewis model in hydrologic simulation studies.
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The partial differential momentum and continuity equations governing two-dimensional overland flow on a plane surface with vertical inflow are developed and normalized.Using extensive data from the literature on such subjects as rainfall, topographic characteristics, and unsteady overland flow, the individual terms in these normalized equations are compared on an order-of-magnitude basis. This comparison results in a systematic and justifiable simplification in the equations and thus in the similarity criteria given by heir coefficients. The criteria obtained are realizable over a limited, but significant, range of prototype rainfalls and land forms.
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Point precipitation is represented by Poisson arrivals of rectangular intensity pulses that have random depth and duration. By assuming the storm depths to be independent and identically gamma distributed, the cumulative distribution function for normalized annual precipitation is derived in terms of two parameters of the storm sequence, the mean number of storms per year and the order of the gamma distribution. In comparison with long-term observations in a subhumid and an arid climate it is demonstrated that when working with only 5 years of storm observations this method tends to improve the estimate of the variance of the distribution of the normalized annual values over that obtained by conventional hydrologic methods which utilize only the observed annual totals.
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Part 3 constitutes the final part of three parts devoted to the mathematical structure of rainfall. The objective is to illustrate the scope of the tools developed in part 2 in the mathematical description of rainfall and rainfall-driven processes. A general overview of the three-part series is given in part 1.
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Recent work on modeling rainfall occurrence has used continuous time point process models. In many cases the available data is not sufficiently precise to properly fit such a model. For example, if only occurrence or nonoccurrence of precipitation is given daily, a more appropriate model would be a binary time series. In this work we relate some parameters of the time series thus obtained to parameters of the underlying (but not observed) continuous time process. We carry out the details for several models proposed in the literature, such as the Neyman-Scott Poisson cluster process and the Smith-Karr renewal Cox process. An application of the theory is illustrated on some Washington State rainfall data.
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This paper provides some useful results for modeling rainfall. It extends work on the Neyman-Scott cluster model for simulating rainfall time series. Several important properties have previously been found for the model, for example, the expectation and variance of the amount of rain captured in an arbitrary time interval (Rodriguez-Iturbe et al., 1987a), In this paper additional properties are derived, such as the probability of an arbitrary interval of any chosen length being dry. In applications this is a desirable property to have, and is often used for fitting stochastic rainfall models to field data. The model is currently being used in rainfall time series research directed toward improving sewage systems in the United Kingdom. To illustrate the model's performance an example is given, where the model is fitted to 10 years of hourly data taken from Blackpool, England.