1. Introduction

Knowledge of the rate of rain falling is important for many reasons. It is particularly crucial for modeling groundwater runoff, infiltration, flash flooding events, and so on—all of which are required for effective hydrologic water management. It is also important for the indirect measurement of released latent heat, which is required for the physical understanding of the heat flux process in the atmosphere. Radar meteorologists require information on rain rates to estimate the magnitude of microwave attenuation by high rain rates for radar rainfall algorithm development.

Technically, rain rate is defined as “a measure of the intensity of rainfall by calculating the amount of rain that would fall over a given interval of time if the rainfall intensity were constant over that time period.” (Glickman 2000, p. 522). However, in practice, it is nearly impossible to determine the constancy of rainfall intensity and, thus, most scientists usually assume that rain rate is “rainfall” accumulated over very short time intervals. In this paper, we will define rain rate as the amount of rain falling within time intervals less than one hour. We will also use the terms “rain rate” and “intensity” interchangeably.

Perhaps nowhere else in the world are rainfall measurements more required on a large scale than over tropical open-ocean regions. Accurate measurements of the enormous amounts of latent heat released in the western Pacific (or the central and eastern Pacific during an El Niño event) are essential for the accurate modeling of the earth’s changing climate. An example of the scientific community’s need to better understand tropical rainfall characteristics was shown by the launch of the National Aeronautics and Space Administration’s (NASA) Tropical Rainfall Measuring Mission (TRMM) satellite (Simpson et al. 1988) in 1997. On board the TRMM satellite is an advanced downward-looking precipitation radar (PR). The TRMM PR, along with other onboard instruments, provides input into the estimation of the vertical atmospheric heating profile, which provides vital information for use in weather and climate models. However, the TRMM PR alone is insufficient, as surface rain rate–rainfall measurements are necessary to normalize the heating profiles (Tao et al. 2001) and, perhaps more importantly, to verify the indirect rainfall estimation by the satellite (Amitai et al. 2005). Rainfall accumulated over one hour essentially represents an average over that hour if divided by the time interval. An average provides no knowledge of the variation in rain rates making up the average. The problem is compounded by the fact that rain rates in tropical regions can be extremely high within very short time intervals relative to those in midlatitudes. Rain rates greater than 40 mm h⁻¹ are frequently observed on relatively flat island meteorological stations (Morrissey and Krajewski 1993; Schumacher and House 2000; Kim et al. 2004). Since it is known that hourly rain rates in the tropics are highly skewed, the distribution of shorter-term (e.g., 5 min) rain rates are most likely skewed even more. The question then becomes, how skewed? Since the magnitude of the microwave attenuation problem is extremely sensitive to the skewness of the rain rate distribution, high values of skewness can lead to large uncertainties in the satellite-derived vertical heating profile estimates (L’Ecuyer and Stephens 2002).

Knowledge of the skewness of short-term rainfall accumulations is also very important for the estimation and modeling of the affects of flash floods, rainfall-runoff models, and other hydrological phenomena (Bras and Rodríguez-Iturbe 1985). Unfortunately, most surface rain gauge measurements of rainfall available on tropical islands are taken at resolutions less than one hour (e.g., daily or monthly). There are a few hourly tipping-bucket gauges existing in several Pacific island countries (Morrissey and Krajewski 1993; Greene et al. 2008). However, as mentioned above, even at 1-h resolution, the extremely heavy tropical rain showers that are frequently produced by relatively warm cloud-top systems usually last on the order of only 5–15 min (Leary and Houze 1979; Morrissey and Krajewski 1993).

To better observe tropical high-resolution rainfall, a relatively recent experimental surface instrumentation program has been established under the auspices of the Pacific Islands Global Climate Observation Program (PI-GCOS). The first phase of the project was to install 50 high-quality 12-inch tipping-bucket gauges at several Pacific meteorological services locations. This was accomplished over the last several years. These gauges record 0.254 mm of rainfall per tip and have been calibrated and tested at the University of Oklahoma. A datalogger attached to the gauge, measures and time-stamps each tip. Software is used to filter out double tips. The gauges are cleaned, and the data on the loggers are downloaded once every couple of months, allowing the understaffed local meteorological services to maintain a high-quality rainfall climate network in their island countries. In this study, data from one such gauge located at the Tongan Meteorological Service in Fuaamotu, Tonga, are used to characterize the rain rate and test a point process model developed for the tropics (Morrissey and Krajewski 1993), which is scalable to time resolutions from 5 min to daily. A new analytical expression for the third moment of the rain rate process is derived for the model in this paper. The primary objective of this study is to determine if the model, given the additional expression for the third moment, can successfully reproduce the skewness of the tropical rain rate distribution throughout a range of time scales from 5 min to 24 h.

This work builds upon that of Morrissey and Krajewski (1993), who developed a modified version of the cluster-based Poisson point process model to simulate tropical time series of rainfall at different temporal resolutions. The original Morrissey and Krajewski (1993) model hereafter will be referred to as the tropical rainfall (TR) model, whereas the new model incorporating the expression for the third moment will be referred to as the modified tropical rainfall (MTR) model. The development first starts with a description of point process rainfall models in general (section 2). This is followed by short descriptions of the two separate models (section 3), which are combined to constitute the TR model (section 4). The derivation of the third moment expression for the TR model, resulting in the MTR model, will then be given (section 5). The results of an analysis of the capability of the resulting MTR model to characterize the rain rate statistics in Tonga, conducted by fitting both the TR and MTR models to the Tongan tipping-bucket rain gauge data, are then shown (section 8). Finally, a discussion of the results of the analysis of the two models will be given (section 9).

2. Point process rainfall models

During the last 25 yr, it has been hydrologists who have primarily used point process theory (Cox and Isham 1980) to develop models to generate synthetic rainfall time series. For example, a point process model was used to produce synthetic rainfall time series for use in hydrologic risk assessment (Koutsoyiannis 1994). Theoretically, the output from these models should have statistical characteristics similar to the rainfall data used to fit the model. The primary interest in these models comes from their capability to disaggregate or “scale” the time-measuring intervals up to higher resolutions than are possible with a given fixed-resolution rain gauge (e.g., from daily to hourly accumulation periods). Thus, a useful model would be one that preserves the statistical characteristics of rainfall throughout a range of scales. For example, given a model developed and fit using daily rain gauge data, the model can produce synthetic rainfall at the hourly resolution. If hourly data are available for that point, then the statistical characteristics of the hourly and synthetic rainfall should mimic each other, assuming the model provides an adequate fit to the data. Another use of such a model would be to provide input into hydrological models for application in regions where data records are sparse.

There are many variations of these models in hydrologic literature (e.g., Kavvas and Delleur 1981; Waymire and Gupta 1981a,b,c; Smith and Karr 1985; Marien and Vandewiele 1986; Rodríguez-Iturbe 1986; Rodríguez-Iturbe et al. 1987a,b; Rodríguez-Iturbe et al. 1988; Entekhabi et al. 1989; Wheater et al. 2005; Cowpertwait 2004; Cowpertwait et al. 2007), including time–space Poisson process models (e.g., Cox and Isham 1988). Velghe et al. (1994) produced a study comparing the effectiveness of the various classes of the early models. They showed that, by and large, point process models can preserve the rainfall statistics up to the second order at different scale—at least in the regions and time periods studied.

Oddly, there does not appear to be much written on this subject in meteorology literature. The closest seems to be a paper by Smith (1993) using a point process to examine raindrop-size distributions. Further, the only such model developed for tropical rainfall characteristics appears to be that from Morrissey and Krajewski (1993). Given the dearth of literature in meteorological journals on this subject, this paper will provide a summary of the theoretical details of the general point process model, with further details provided in the referenced articles.

3. The Poisson white noise and the Neyman–Scott rectangular pulse models

The basic TR model, and thereby the MTR model, is a superposition of two different point process models. A description of each point process model is given in this section.

a. The PWN model

One of the first model developments using a Poisson counting process for rainfall time series synthesis was by Rodríguez-Iturbe et al. (1984). They developed a model referred to as a Poisson white noise (PWN) model. This model assumes that the rainfall process occurs as a sequence of instantaneous bursts (or “depths”) of rain within a specified time interval. The positioning of these bursts along a time line is governed by a homogeneous Poisson process. Each “burst” of rainfall, and the associated rainfall depth, has a random memory (i.e., statistically independent depths).

A simple homogeneous Poisson process is given by the probability density function (pdf),

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where N(t) is the number of Poisson-distributed events occurring from time t = 0 to t. The number of events occurring in time interval (t, t + τ) is N(t + τ) − N(t), which follows a Poisson distribution with parameter λ_ɛτ.

Following Rodríguez-Iturbe et al.’s (1984) development, the PWN model is produced from a compound Poisson process (Feller 1968; Cox and Isham 1980). It is assumed that the model consists of a sequence of random bursts (indexed by i) with random magnitudes, X_ɛi i = 0, 1, 2, … , whose position in time t follows a Poisson process. The magnitude of each burst is independent of N(t) and follows an exponentially distribution with parameter α_ɛ [i.e., with pdf, f (X_ɛ) = α_ɛe^−α_ɛX_ɛ]. The rainfall intensity at time t can be written as

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where dN(t) is the shorthand notation for N(t + dt) − N(t), which is equal to 1 if a burst occurs in a time interval (t, t + dt) and is zero otherwise. If all bursts of rain occurring from t = 0 to t are integrated, we obtain the total rainfall accumulated over this time period, that is,

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where X_ɛ0 = 0. To arrive at the representation of a rainfall process consisting of sequential, nonoverlapping time increments, τ (e.g., τ = 1 h) is defined as

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where each sequential time increment is indexed by i.

Expressions for the first two analytical moments for the PWN model were derived by Rodríguez-Iturbe et al. (1984), noting that the covariance of the process is equal to zero (refer to appendix A). Thus, the PWN model has two equations and two unknown parameters: α_ɛ and λ_ɛ. For the third parameter, the temporal accumulation period, τ is set by the user.

b. The RI model

Of all the different variations of Poisson point process rainfall models that have been developed, perhaps the most referenced category of these models is the clustered-based rectangular pulses model first described by Rodríguez-Iturbe et al. (1987a,b; hereafter RI model). There are two main versions of this model: one is based on the Neyman–Scott process and the other is based on the Bartlett–Lewis process. The basic theory behind these two classes of models is the same. Their differences are described below.

The clustered-based rectangular pulses model is based on a “filtered” compound Poisson point process [refer to Cox and Isham (1980) for details on this process]. The RI model assumes that a “storm”–a frontal system, a tropical cloud cluster, or some other large-scale rain-producing event—occurs randomly, and its occurrence is governed by a Poisson process with rate λ, where λ is usually very small (i.e., on the order of 0.001–0.01 storms h⁻¹). Each storm generates a random Poisson-distributed number of “cells,” C, each of which produces rainfall at a constant rate or intensity X (hence the reference to the phrase “rectangular cells”), which is randomly determined from an exponential pdf. Other researchers have selected a geometric distribution from which to generate the number of cells from a storm (e.g., Mondonedo et al. 2006). The number of cells per storm being Poisson-distributed with C ≥ 1 has as its rate parameter ς + 1 (the reason the rate parameter is not simply ς is explained below). The duration of each cell is determined by another random variable, L, which has an exponential pdf with parameter η [i.e., with pdf, f (L) = ηe^−ηL]. The Neyman–Scott process assumes that individual cell displacements, D, from a given storm are determined by a set of random values drawn from an exponential distribution with parameter β. There are no cells belonging to a particular storm that occur at the storm’s origin.

In the Bartlett–Lewis process, the intervals between successive cells are randomly determined by a Poisson process. After a given time, which is exponentially distributed, the Poisson process of generating cells stops. In this paper, we choose to focus on the Neyman–Scott process, primarily since the model we are building upon (i.e., that of Morrissey and Krajewski 1993) successfully used this process for modeling tropical rainfall at several island sites in the Pacific.

Since the parameters for each rectangular-shaped cell are exponentially distributed, the expected values of the cell intensity, X, duration, L, and displacements from the storm origin D are E(X) = 1/α, E(L) = 1/η, and E(D) = 1/β, respectively. Individual cells can be overlapping, so the rain rate over any given time measurement interval is the sum of rainfall from cells active during that interval. Refer to Fig. 1 for a visual representation of the process.

Using the basic theoretical constructs of the clustered point process model described above, Rodríguez-Iturbe et al. (1987a) defined the rainfall accumulated over time interval τ to be

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where X(t) is the rainfall intensity at time t. Thus, R_i(τ) represents the rainfall accumulated over a time interval τ and is sequentially indexed by i. For example, R₈(1) is the rain rate during the eighth 1-h interval of the time series, and R₈(6) is the rain rate during the eighth 6-h time interval.

Rodríguez-Iturbe et al. (1987a) notes that the number of cells per storm, C, is Poisson distributed with parameter ς. However, a word of caution is due here. It is stated, but not derived, in Rodríguez-Iturbe et al. (1987a) that E(C² − C) = E(C)² + 2E(C). However, Velghe et al. (1994) demonstrated that since the number of cells per storm must be greater than or equal to one (C ≥ 1), it is the variable C − 1 that is Poisson distributed with parameter ς. To show this let C = Y + 1 then Y is Poisson distributed in the interval (0, ∞). Thus, the expression for the expected value of C is

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Since Y has a Poisson rate parameter ς, the following useful relationships are then found:

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noting that E(Y) = ς, E(Y²) = ς² + ς, and E(Y³) = ς³ + 3ς² + ς. Given that E(C) = ς + 1 then ς = E(C) − 1 and substituting this into the second expression in (7) yields E(C² − C) = E(C)² − 1 not E(C² − C) = E(C)² + 2E(C). The expressions for the analytical moments for the RI model are given in appendix B.

The three moment equations, given in appendix B, have five unknown parameters—λ, β, η, α, ς—and two parameters, which can be set by the user, namely, the time resolution τ and the time lag j used in the covariance expression. By selecting different values for τ and j, a system of five equations with five parameters can be constructed. For example, one could select two levels of aggregation, τ = 1 h and τ = 6 h, and a lag of j = 1 h and construct expressions for the mean rainfall at 1 h, the variance at 1 and 6 h, and covariance of 1 and 6 h accumulations with lag 1. This results in five equations. As will be shown, expressions for additional moments and statistics have since been developed for the RI model.

The process of fitting the model to a particular rainfall time series involves equating the analytical moments with moments estimated from a historical rainfall time series. It then becomes a matter of solving a system of Z equations with unknown Z parameter values using the Z moments estimated from the data. Since the number of sets of possible analytical expressions for different moments is theoretically infinite (since τ can be set to any resolution), the optimal set of Z equations for a particular model is unknown. It has been noted by Morrissey and Krajewski (1993) that, given the subjectivity of selecting a system of parameterized equations, logic dictates that one selects the equations that closely represent the moments and time scales of interest. Thus, if one is interested in preserving the variance of the process, then the variance expression should be selected as one of the equations in the system to be solved with perhaps two different values of τ, resulting in two variance equations representing two time scales. The selection process for an optimal equation set remains elusive and is an active area of research.

Another problem presents itself in the parameter estimation process for these models. Given that the system of equations is highly nonlinear, the estimation of the parameters is nontrivial (Foufoula-Georgiou and Guttorp 1987). Even using a sophisticated nonlinear constrained optimization method, the fitting process devolves into a trial and error process of trying different initial parameter values until physical reasonable final parameter values are found that provide a good fit.

Interestingly, given the problems just mentioned, many of the studies referenced above have succeeded in finding very good model fits to rain gauge time series data For example, using the system of equations described above and rain gauge data from Denver, Colorado, Rodríguez-Iturbe et al. (1987a) found that the RI model was able to preserve rainfall statistics up to the second order over the range from 1 to 24 h aggregation periods. However, they did note that the proportion of dry periods was not adequately represented by their model. The fact that the proportion of dry periods is a significant statistic of interest spurred many subsequent efforts to improve the RI model to a point where this statistic is preserved through different scales. For example, Entekhabi et al. (1989) modified the original RI model to include η as a random variable (i.e., not a constant parameter) with the aim of allowing more flexibility in the duration of individual cells. While some improvement was observed, they noted that more research was required for adequate representation of this statistic over a range of scales. Cowpertwait (1991, 1992) developed an analytical expression for the RI model that expresses the probability of a particular time interval being dry. His expression can be added to the present system of equations without the need for any additional parameters. Cowpertwait’s (1991, 1992) expression is

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Morrissey and Krajewski (1993) further extended Cowpertwait’s (1991, 1992) work by using the above expression to produce the simple relationship for the expected rainfall given that it is raining, namely,

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With the additions of (8) and (9) to the three moment relations given in appendix B, the RI model now has five expressions and five unknown parameters, λ, β, η, α, ς.

4. The superposition of the RI and PWN models (the TR model)

Morrissey and Krajewski (1993) used 7 yr of hourly rain gauge data (1984–90) from Majuro Atoll in the Republic of the Marshall Islands to test the performance of the RI model and the model developed by Entekhabi et al. (1989) on tropical rainfall. The results indicated that neither of these models adequately preserved the data-estimated moments and statistics spanning an aggregation range from 1 to 48 h. Many different sets of analytical expression sets were tested. A close inspection of the Majuro hourly rainfall time series and work from Ramage (1971) and Houze and Betts (1981) lead Morrissey and Krajewski (1993) to put forth the hypothesis that the rain-producing systems over Majuro came primarily from two sources. One source appeared to be large-scale cloud-cluster systems emanating from the intertropical convergence zone (ITCZ) that tended to produce large amounts of rainfall over 2–3-day periods but occurred only rarely on the order of once or twice a month. The other source was relatively frequent, short “bursts” of light to heavy rain from trade wind showers. Compared to the cloud-clustered systems, these showers occur quite often, sometimes several times per day, and generally last much less than one hour.

From a statistical point of view, the trade wind showers occur in a near-random fashion, while the cloud-cluster systems have a midlatitude-like frontal signature in the sense that their interarrival times and durations are similar to frontal systems. Thus, it was hypothesized that a model developed for midlatitude conditions (i.e., the RI model) might represent the cloud-cluster events, and the PWN model might adequately model the random nature of the trade wind showers. The resulting tropical rainfall model was a superposition of the RI and PWN models, which resulted in the TR model.

Results from the Morrissey and Krajewski (1993) study indicated that the TR model performed much better than either the RI or the Entekhabi et al. (1989) model throughout the range of tested aggregation levels (i.e., 1–24 h). The TR model’s moments, up to the second order, closely matched the empirical Majuro moment estimates, including the probability of zero rain and the conditional rainfall.

Morrissey and Krajewski (1993) assumed, to a first degree of approximation, that the two sources of tropical rainfall were independent. Of course, from a physical point of view, it is very unlikely that both events can randomly occur at the same time and place. There must be some dynamic influence by the large-scale system on the small-scale trade wind showers, which are likely to be overwhelmed by the larger systems. However, from a statistical point of view, assuming independence, the relatively small amount of rainfall from trade wind showers would not add significantly to the total rainfall amount produced by the cloud cluster system. Thus, if the two rainfall-producing systems occurred independently of each other and, thus, happen to occur at the same time, the additional rainfall from trade wind showers compared to the aggregated amount from both systems would probably be negligible. Thus, it was deemed worthwhile to consider the two hypothesized physical sources of Majuro rainfall independently and determine if a point process model using this assumption would adequately fit a tropical rainfall time series. The analytical reasoning behind the use of the “independence” assumption is that it is a relatively simple operation to combine the mathematical representation of two models (i.e., the PWN and RI models). It is well-known that the first two moments of the sum of two independent random variables is simply the sum of the individual moments (Priestley 1981). Therefore, the analytical moments representing the TR model are constructed from the simple addition of the analytical moments from the PWN and the RI models. The probability of zero rainfall is simply the product of the expression for the two models. The resulting TR model moment expressions as given in Morrissey and Krajewski (1993) are

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and the expected value of rainfall conditional on rain is given by

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The addition of the two parameters from the PWN model results in the TR model having five analytical expressions and seven unknown parameters, that is, λ, β, η, α, ς, λ_ɛ, α_ɛ. Selecting different values for τ will result in an equal number of equations and unknowns.

5. The derivation of the third moment expression (the MTR model)

Modeling rainfall at high temporal resolution (i.e., rain rate) requires knowledge of the third moment of the rainfall process. One prominent feature of high-resolution rainfall accumulations is the highly skewed nature of its distribution (Hosking and Stow 1987; Wilks 1989; Del Genio and Kovari 2002). Given the relatively intense, short-lived nature of tropical rainfall, the rain rate pdf is skewed even more than midlatitude rainfall intensity. Thus, a point process model without an expression for the skewness is unlikely to capture these common rain rate extremes. This makes it unlikely that model estimates of the probability of rain rates exceeding a given threshold would be useful for various applications.

Therefore, a modification is made to the TR model’s set of analytical expressions to include an equation for the third central moment of the rainfall process. The MTR model is actually identical to the TR model but with different parameter values resulting from the use of the third moment expression. With the analytical expression for the third moment built into the MTR model, it is hoped that the probability of extreme rain rate values can be adequately estimated by the model. To derive the required third central moment expression, it will be necessary to initially produce two individual expressions for this moment: one for the PWN model and one for the RI model. It will then be shown that the simple addition of these expressions results in a single analytical expression for the third central moment for the MTR model.

While it was mentioned above that the first two moments of the sum of two independent processes is simply the sum of those moments, it is not as well known nor immediately clear that this applies to the third central moment as well. In fact, for two independent random variables, X and Y, with means m_X and m_Y, respectively, the following relationship, for example, holds for the third central moment μ₃:

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This is shown by a simple expansion of the third central moment expression for these two variables:¹

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Note that the independence assumption makes the products of the expected values of the zero mean variables within the last two terms equal to zero. It is this fact that allows the simple construction of the third central moment expression for the MTR model.

a. The third central moment for the RI model

A rather long expression for the third central moment for the RI model was developed by Cowpertwait (1998) as

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where

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Note that the parametric expression for E[C (C − 1)(C − 2)] is given in the third equation in (7). No new parameters have been introduced in this expression. Also, since X is exponentially distributed, its third raw moment is E(X³) = 6/α³ (see below for derivation).

b. Development of the third central moment for the PWN model

To the author’s knowledge, the third central moment for the PWN model is not given in the relevant literature and must be derived. The moment generating function (MGF) for a compound Poisson process was selected as the method of choice for the derivation which follows.

The MGF is defined as M_A(s) = E(e^sA) with s ∈ R for random variable A. The nth raw moment of A can be found from

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For example, the MGF for an exponentially distributed random variable A with parameter ϕ is

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The second and third raw moments of A are, therefore,

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The moment generating function for the PWN model is the same as that for a compound Poisson process (Feller 1968) and is given as

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where M_Xɛ (s) is the MGF for X_ɛ, the exponentially distributed rain intensity. For the PWN model, M_Xɛ (s) is given by

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Substituting Eq. (24) into (23) reduces the MGF function for the PWN model to

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Setting s equal to zero and taking the first three derivatives with respect to s results in the first three raw moments for the PWN model:

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Using E(R_ɛⁿ) = μ′_n yields an expression for the third central moment for the PWN model:

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Substituting the expressions in (26) into (27) yields the parametric expression for the third central moment for the PWN model:

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where the right-hand side of (28), E(X_ɛ³) = 6/α_ɛ³ is determined from (22).

Now that the third central moment of the PWN model has been derived, according to the relationship shown in (15), the third central moment for the TR model (i.e., the MTR model) is the simple addition of expressions (17) and (28). With this derivation, the MTR model is complete. A summary of the analytical expressions for the relevant moments follows.

6. Summary of the MTR model equations

In the original description of the RI model given in Rodríguez-Iturbe et al. (1987a) and Entekhabi et al. (1989), the expected values of the cell parameters were left in the equations rather than in their equivalent parameter representation. This was done to leave some flexibility in the equations in the event researchers wanted to select different pdfs for the cell characteristics. Following that practice, a summary of the MTR model moment equations are given as

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where f (η, β, τ) and g(η, β, τ) are given in Eqs. (18) and (19), respectively. Two dimensionless expressions can be substituted for the covariance and the third moment, that is, 1) the lagged correlation ρ(l) by simply dividing Eq. (31) by the variance Eq. (30) and 2) the skewness using

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The following are the parameter substitutions for the cell characteristic distributions:

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The MTR model is a six-analytical expression model (i.e., the mean, variance, probability of zero rain, conditional rain rate, correlation and the skewness) with seven parameters, λ(h⁻¹), ς(cells per storm), β(h⁻¹), α(h mm⁻¹), η(h⁻¹), λ_ɛ(h⁻¹), α_ɛ(h mm⁻¹), whereas the TR model has five analytical expressions and seven parameters. In addition, two parameters can be set by the user, that is, τ and l. Given seven unknown parameters, at least seven analytical expressions are needed. Such a system can be easily set up using different values of the τ and l user-set parameters in the expressions. The difficulty here lies in the optimal choice of a system of equation, which provides the optimal fit.

7. Parameter estimation

The method of moments is commonly used to estimate the parameters of the point process models using available rain gauge data. This method is based on equating the observed or “estimated” moments from the data with the analytical moments given by the above equations [Eqs. (29)–(35)]. The resultant system of nonlinear equations and their corresponding estimated moments is solved for the parameters by formulating the problem as an optimization task. An appropriate minimization routine is applied with the objective of obtaining a set of parameters that provides an optimum fit to the data. Entekhabi et al. (1989) developed a setup scheme for the minimization routine, which not only improves the convergence to a solution but also normalizes the system of equations so that the minimization is not biased toward the larger magnitude components of the system. They defined a parameter vector, v = (ν₁, ν₂, … , ν_n), containing the n × 1 parameters (e.g. ν₁ = E(X) = 1/α, ν₂ = E(C) = ς + 1, ν₃ = β, and so on), where n is the number of parameters. The vector m(v) = [m₁(v), m₂(v), … ., m_n(v)] contains the analytical moments corresponding to the observed moments as functions of the parameter vector v. A moment vector is then constructed, d = (d₁, d₂, … , d_n), containing the n × 1 moments estimated from the data that correspond to the m(v) vector. Thus,

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which defines a system of n nonlinear equations with n unknown parameters, v. Entekhabi et al. (1989) suggested that Eq. (37) be solved by minimizing the sum of the squared differences and normalizing the components. They defined two diagonal matrices: 𝗗 = diag(d) and 𝗠(v) = diag[m(v)]. The resulting equation to minimize becomes

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The minimization routine used is the sequential quadratic programming method, which is a constrained nonlinear algorithm (Gill et al. 1981, p. 180).

8. Comparison of the TR and MTR models with Tongan rain gauge data

a. Objective

Our primary objective in this study is to determine, given the addition of the skewness expression, how the MTR model compares to the TR model given the same initial conditions. It is expected that the addition of the skewness expression in the fitting process will result in MTR model output with an improved skewness representation of the data. However, what is not known a priori is how the skewness expression will affect the overall representation of other data statistics by the model.

To examine this question, two experiments were conducted. Experiment 1 was set up to examine the fit of the MTR and TR models to data statistics at the high time resolutions of τ = 5, 15, 30, and 60 min to determine if any improvement occurs in the model’s ability to simulate rainfall intensity and accurately characterize its statistics. Experiment 2 was designed to test the fits of these models throughout a range of scales from 5 min to 24 h (i.e., τ = 5 min, and 1, 6 and 24 h; refer to Table 1 for the experimental setup).

Parameters were estimated for the TR and MTR models using the nonlinear optimization method with the fitness function described in the section 7. The parameters for both the TR and MTR models were constrained to realistic values as follows:

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The initial parameter values for both experiments were

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It should be noted that, since this is really a trial and error process, these initial parameter values were those that provided consistent results. In other words, small variations in the values of this parameter set tended to produce the same solution. As in Cowpertwait et al. (2007), since the mean (i.e., E[R(τ)]) is only a function of τ, it is combined with the variance to produce the dimensionless variable the coefficient of variation (CV; i.e., σ/x) for plotting the results. It is important to note that, since the TR and MTR models are of the same construct (i.e., a superposition of the RI and PWN models), the only difference between them is that the skewness is included in the fitting process for the MTR model and not the TR model. However, the skewness expression can still be used to assess the skewness of the TR model with a given set of parameters. In this manner a comparison of the differences in the skewness for each model at different resolutions can be conducted.

9. Summary

This paper develops a new expression for the third moment to be included in a clustered point process model designed to produce synthetic tropical rain rate. The original TR model designed for tropical conditions was a superposition of the standard RI Neyman–Scott point process model and a Poisson white noise process model, that is, the PWN model. Noting that one of the hallmarks of tropical rainfall is its highly skewed distribution, it was realized that this model needed to include an expression for the skewness to adequately simulate the extreme rain events that occur in the tropics.

Using an assumption of two independent rainfall producing processes, the expression for the skewness of the RI model derived by Cowpertwait (1998) was combined with one derived here for the PWN model. The expressions shown in (15) and derived in (16) then allowed the relatively simple construction of an analytical expression for the skewness of the original TR model, which resulted in the new MTR model.

Using high-resolution tipping-bucket rain gauge data from Tonga, the TR and MTR models were fit to the data at resolutions at 1 h and higher (experiment 1) and 24 h and higher (experiment 2). The results indicated that the MTR did indeed significantly improve the model’s representation of the rainfall skewness throughout all the time resolutions tested. However, since the constructs of the original TR model are the same as the MTR model, with the only difference being a new analytical expression for the third moment, it is not surprising that the improvement in skewness representation would come at a cost of poorer representation of other statistics. This is indeed what was found. Overall, the error statistics indicated that both models performed about equally well within the boundaries of statistical uncertainty. Comparing all the statistics and τ values tested, both models performed with an average percent error between about 15% and 20%.

From an observation of the resulting parameter values, it appears that to model the skewness accurately, as opposed to the first and second order moments, the PWN model gets priority in the superposition and vice versa. This was shown with the MTR model by the higher frequency of random bursts of rainfall from the PWN model occurrence parameter. For the user of the MTR model, the results suggest that if the user needs to reproduce the higher-order statistics well, then the representation of the lower-order statistics will be compromised somewhat. Perhaps the best use of this model is in simulating the probability of rain at different resolutions, given its relatively low percent error at most resolutions. The results also indicate if simulated rainfall at a given τ value is required, then the parameters for the MTR model should be obtained from the model fit to data at or near this resolution.

Given the relatively short record length of the tipping-bucket data used in this study, it is probably safe to assume that these results err on the high side. Given a longer data record, parameters for different months or seasons could be determined and the associated percent errors will, in all likelihood, be smaller to some degree. Fortunately, high-resolution tipping-bucket data are currently being collected at many topical Pacific island sites, and the data record continues to grow.

It should be noted that the MTR model is not the first point process model used to simulate finescale rainfall. Recently, Cowpertwait et al. (2007) developed a similar model based upon three layers of rainfall “pulses,” which also occur in a Poisson process. Their model consists of “storms,” which generate random “cells” in which random “bursts” of rainfall emanate. The model, fit to a 60-yr record of rain gauge data from the temperate climate of New Zealand, provided an excellent fit to the data by reproducing the data time series statistics at scales ranging from 5 min to 24 h. It will be very interesting to compare the MTR model with that of Cowpertwait et al.’s (2007) fit to tropical rain gauge data.

As far as the author is aware, this effort represents a first attempt at using point process theory to model tropical rain rate, as opposed to “rainfall” (i.e., over a period of one hour or greater). While the results indicate that the MTR model reproduces the rain rate characteristics in Tonga reasonably well for all the statistics tested, there is certainly room for improvement in the model. Cowpertwait et al.’s (2007) model may indeed provide that improvement when fit to tropical data.

Lastly, this effort illustrated one area that is in need of additional research with point process rainfall models. This is the development of criteria for selecting the optimal set of analytical expressions used in fitting the model to data. Currently, the selection process for this and the initial parameter values is quite subjective. For example, it is possible that another set of seven equations fit with different τ fit to the same dataset would produce a model with altogether different results.