Open access peer-reviewed chapter

Analysis of Storm Rainfall in Peninsular Malaysia Using Neyman‐Scott Rectangular Pulse Modeling

By Rado Yendra, Abdul Aziz Jemain and Ibrahim Sulaiman Hanaish

Submitted: March 1st 2017Reviewed: June 7th 2017Published: April 18th 2018

DOI: 10.5772/intechopen.70043

Downloaded: 222

Abstract

This paper aims to estimate Neyman‐Scott rectangular pulse (NSRP) modeling application in representing the storm rainfall that occurred in Peninsular Malaysia. This research utilized hourly rainfall data from 48 rain gauges in Peninsular Malaysia during the period from 1970 to 2008. The raingauge stations are given in four territories, namely northwest, west, southwest, and east. The goodness‐of‐fit test from NSRP model should be done before the other applications of the model. The conclusion of this research revealed that NSRP model is able to show the rainfall data in Peninsular Malaysia.

Keywords

  • storm rainfall
  • NSRP
  • Peninsular Malaysia
  • modelling of storm

1. Introduction

Nowadays, there are many problems regarding climate change and global warming investigated by researchers, especially for the storm rainfall to society. Furthermore, the observation of storm rainfall becomes necessary action in a few sectors, such as agriculture, hydrology, and water resource management. Because of the growth of irrigated agriculture, industrialization, and population, the analyst can be used in forecasting rainfall and making decision. These studies, an intensity extreme rainfall, total rainfall, and heavy rains, have invited much attention of scientists in the world to research, such as research carried out by Lana et al. [1] and Burgueno et al. [2].

There have been a few published works on the behavior of storm rainfall in Peninsular Malaysia. Among them are works on detecting trends in dry and wet spells over the Peninsula during monsoon seasons [3, 4], changes in extreme rainfall events [5], changes in daily rainfall during monsoon seasons [6], and analysis of rainfall variability [7]. In these studies, various objectives and approaches have been highlighted in describing the characteristics of rainfall in this area.

2. Methodology

In this matter, the data used are the hourly rainfall data from 48 rain‐gauge stations from 1970 to 2008. The data can be acquired from the meteorology, drainage, and irrigation department of Malaysia. All stations are divided into four classes, by [8, 9]. Dale [9] has defined five rainfall regions in Peninsular Malaysia, such as west, Port Dickson‐Muar coast, southwest, and east. Nevertheless, a few of stations located on the Port Dickson‐Muar coast were combined with people in the region of the southwest. The lists of stations are given in Table 1, and there are 48 stations that can be delineated in Figure 1.

Figure 1.

Map of Peninsular Malaysia showing geographical regions and the selected 48 stations.

RegionStationsCodeStateLongitudeLatitude
SouthwestKota TinggiS1Johor103.721.76
Batu PahatS2Johor102.931.84
EndauS3Johor103.622.65
LabisS4Johor103.022.38
EastBatu HamparS5Terengganu102.825.45
BertamS6Kelantan102.055.15
BesutS7Terengganu102.625.64
Sg ChanisS8Pahang102.942.81
DabongS9Kelantan102.025.38
DungunS10Terengganu103.424.76
Gua MusangS11Kelantan101.974.88
KemamanS12Terengganu103.424.23
Sg KepasingS13Pahang102.833.02
Kg AringS14Kelantan102.354.94
Kg DuraS15Terengganu102.945.07
MachangS16Kelantan102.225.79
Paya KangsarS17Pahang102.433.90
Kg Sg TongS18Terengganu102.895.36
Ulu TekaiS19Pahang102.734.23
PekanS20Pahang103.363.56
WestAmpangS21Selangor102.003.20
Bkt BenderaS22Pulau Pinang100.275.42
Chin ChinS24Melaka102.492.29
Genting KlangS25W. Persekutuan101.753.24
JasinS26Melaka102.432.31
Kalong TengahS28Selangor101.673.44
KamparS29Perak101.005.71
Kg Saw LebarS30N. Sembilan102.262.76
Ladang BikamS31Perak101.304.05
Kg Kuala SlehS32W. Persekutuan101.773.26
PetalingS33N. Sembilan102.072.94
RompinS34N. Sembilan102.512.72
SerembanS35N. Sembilan101.962.74
Sg BatuS36W. Persekutuan101.703.33
Sg BernamS37Selangor101.353.70
Sg ManggS38Selangor101.542.83
Sg PinangS39Pulau Pinang100.215.39
MerlimauS40Melaka102.432.15
Siti AwanS41Perak100.704.22
Sg Sp AmpatS42Pulau Pinang100.485.29
Teluk IntanS43Perak101.044.02
Tanjung MalimS44Perak101.523.68
NorthwestAlor SetarS45Kedah100.396.11
ArauS46Perlis100.276.43
BalingS47Kedah100.745.58
Kuala NerangS48Kedah100.616.25
Padang KatongS49Perlis100.196.45
Pdg Mat SiratS50Kedah99.676.36

Table 1.

The list of 48 raingauge stations with their respective regions and geographical coordinates.

The Neyman‐Scott rectangular pulse (NSRP) modeling is used to model the rainfall number of each station in Peninsular Malaysia. The single‐site NSRP model is marked by the flexible structure where parameters of model relate to the basically physical features monitored in rainfall. Theoretically, the NSRP model assumes that the sources of storm follow a Poisson process with parameter λ. Additionally, a random figure of E(C)cell origins is displaced from storm provenance by exponentially distributed distance with parameter β. A rectangular pulse, with duration and intensity, showed by other two independent random variables, presumed to be exponentially distributed with parameter ηand E(X)successively, is connected to every original cell. The total intensity on every point of time is given by the number of the active cell intensities in that certain point. Therefore, the NSRP model has a total of five parameters which can be estimated by minimizing an aim function, evaluated as the number of normalized residuals between the characteristic statistics and theoretical expressions that are observed [10, 11]. This model is able to produce statistics estimation values close to the observed values [11].

The main feature of the NSRP model can be summarized as follows:

  1. Every storm arrival, represented by li, i = 1, 2, 3, …, is exponentially distributed in Poisson process with parameter λ.

  2. Every rain cell, cij, i = storm index of i, j = rain cell index of storm i, has Poisson or geometry distribution with a mean of E(C).

  3. every waiting time for cells after the storm origin, bik, i = index storm of i, k = time of rain cell at storm i, will be exponentially distributed with mean β,

  4. Two parameters, intensity xjh, j = jth cell and h = intensity at jth cell which is exponentially distributed with mean E(X), and duration of rain tjs, j = jth cell and s = duration at jth cell which is Exponentially distributed with mean η, form cluster in every rain cell.

These four conditions can be depicted as in Figure 2.

Figure 2.

NSRP modeling, l i storm arrival time, c i j of rain cell, b i k waiting time of rain cell, t j s duration of rain cell, and x j h intensity of rain cell.

Each station’s hourly data are fitted with NSRP and the yielding NSRP parameters (λ,E(x),E,β,η)are noted monthly. To control and make sure that the NSRP model obtained shows the actual rainfall data, the mean of the 1‐h rainfall and probabilities of 1‐ and 24‐h rainfall estimated from the model have been compared with this statistic values calculated from the data which are observed.

3. NSRP modeling

Rodriguez‐Iturbe et al. [11] applied the formula to produce the first and the second statistical moments, whereas the moment is obtained using rainfall data scaling.

There are three statistical moments such as mean, variance and autocorrelation, and the probability of succession rain as Eq. (1)(4) which are used to obtain the NSRP’s parameters.

E(Yi(τ))=ληE(C)E(X)τE1
Var(Yi(τ))=Ω1(λ,E(C),E(X))Ψ1(η,τ)+Ω2(λ,E(C),E(X))Ψ2(β,η,τ)E2
Cov(Yi(τ),Yi+k(τ))=Ω1(λ,E(C),E(X))Ψ3(β,η,τ)+Ω2(λ,E(C),E(X))Ψ4(β,η,τ)E3
1Pr{Yi(τ)=0}E4

This research just have four equations where the others questions explain four equations before

Pr{Yi(τ)=0}=exp(λτ+λβ1(E(C)1)1{1exp[1E(C)+(E(C)1)exp(βτ)]}λ0[1p(t,τ)]dt)E6
p(t,τ)=(exp[β(t+τ)]+1[ηexp(βt)βexp(ηt)]/[ηβ])×exp((E(C)1)β[exp(βt)exp(ηt)]/[ηβ](E(C)1)exp(βt)+(E(C)1)exp[β(t+τ)])E106
Ω1(λ,E(C),E(X))=2λE(C)E(X2)E107
Ω2(λ,E(C),E(X))=λE(C2C)E2(X)E7
Ψ1(η,τ)=1η3(ητ1+exp(ητ))E8
Ψ2(β,η,τ)=Ψ1(η,τ)β2β2η2βτ1+exp(βτ)β(β2η2)E9
Ψ3(β,η,τ)=12η3(1exp(ητ))2exp(η(k1)τ)E10
Ψ4(β,η,τ)=Ψ3(β,η,τ)β2β2η2(1exp(βτ)2exp(β(k1))2β(β2η2)k=autocorrelationoflag1,2,3τ=rainaggregationE11

4. NSRP’s parameter estimation and good‐fit test

Rodriguez‐Iturbe et al. [11] and Cowpertwait [10] have used a moment method to estimate NSRP’s parameter. Other methods estimating the same parameters are also conducted by other researchers, who applied the method of log‐likelihood maximum probability. Some researchers, who have provided usual procedure, which is needed to convert hourly rainfall data into aggregate rainfall data, in estimating NSRP’s parameter, are [10, 12, 13]. The application of scaling to obtain the rainfall data of some scales. For example, the 1‐h rainfall scale, 6‐h rainfall scale, and 24‐h rainfall scale used Eqs. (1)(4) and then produced some nonlinear equations. So, the expected parameters of NSRP?s can be obtained numerically by optimizing Eq. (5).

Z(X)=k,τ[1Θk(X,τ)Θk*(τ)]2E5

Θk(τ)is the second moment statistics and rainfall probability from scaled data or generally called as observation statistics, and Θk(x,τ)is the second moment statistics and rainfall probability stated on Eqs. (1)(4) or generally called as theoretical statistics.

The equation solution numerically requires an accurate initial value. Researches on non‐linear numeric model often require it in order to enable them to estimate some required parameters. Some initial values, to estimate the parameters of NSRP, have been presented by Calenda and Napolatino accurately. In fact, it requires testing many of initial values to make zvalue on equation (5) to be optimum. This makes it difficult to perform the numerical solution. Favre et al. [14] has tried the best method on estimating NS parameter easier; The research is conducted by dividing parameters into two sets, which comprise {β,(η)}and {E(C)},λ,(E(X))providing an initial value for parameter {β,(η)}. it can make the estimated numerical solution simpler and easier to handle. The other method of numerical solution of estimating the parameter of NSRP conferred fluctuation scale values linking one parameter with the other four parameters; in addition, based on the four chosen parameters, the value will be optimum. This simplifying numerical solution is also contributed by Calenda and Napolatino [15]. In this paper, the proportion of rainfall cell of each storm will be contributed under Poisson condition; thus E(C2C)=E2(C)1, this result has been well investigated [12].

Good‐fit test, which is used to define the best‐fit distribution in rainfall cell intensity of four given distribution in this research, will be applied by sorting residual value gained from a value of the second moment statistics and observed rainfall probability and from a value of the second moment and theoretical rainfall probability. Velghe et al. [12] used residual value as equation

S=1n[n|1XnXhis,n|]E12

Assume Xnas the second moment and rainfall probability based on theory (NSRP model), Xhis,nas the second moment statistics and rainfall probability based on observation, and n as the number of statistics used in this model. whereas n is 8 representing the average of an hour rainfall; the variance used for the rainfall includes periods of 1, 6, and 24 h, autocorrelation lag 1 for 1‐h rainfall, autocorrelation lag 1 for a 24‐h rainfall scale, a probability of 1‐h rainfall, and a probability of 24‐h rainfall scale.

5. Applications

5.1. Study region

Peninsular Malaysia is located between 1 and 7 north of the equator which is the tropical area. Generally, these areas experience a wet and humid tropical climate throughout the year; this country has characteristic such as high annual rainfall, humidity, and temperature. Peninsular Malaysia has a stable temperature year-round from 25.5 to 32°C. Normally, the annual rainfall is between 2000 and 4000 mm, whereas the annual number of wet days ranges from 150 to 200.

The climate of Peninsular Malaysia describes two monsoons separated by two inter‐monsoons. In May through September, the southwest monsoon (SWM) occurs and the northeast monsoon (NEM) occurs from November to March. The two inter‐monsoons occur in April (FIM) and October (SIM). In Peninsular Malaysia, the main range mountains, widely known at the circumstances as Banjaran Titiwangsa, run southward from the Malaysia‐Thai border in the north, spanning a distance of 483 km and separating the eastern part of the peninsula which receives heavy rainfall. By contrast, regions sheltered by the main range, as shown in Figure 1, are more or less free from its influence.

5.2. Goodness of fit of NSRP

Table 2 provides information about the parameters of the NSRP model for rainfall occurring in November dan December for 48 terminals in Peninsular Malaysia. The NSRP model with parameters, which is identified for every terminal, and various statistic values are the initial foundation for contraction of the rainfall data. In particular, the mean and probability values of the 1‐ and 24‐h rainfall amount are then calculated. To describe the condition of a data set, these statistics are chosen.

RegionStationNovemberDecember
λE (X)E (C)βηλE (X)E (C)βη
SouthwestS10.02593.702.560.1162.230.01256.827.880.0781.48
S20.028221.461.460.0202.080.02170.674.580.1092.56
S30.03315.581.390.2212.220.0125.965.700.0812.16
S40.00374.4016.280.0011.480.01585.305.410.1121.96
EastS50.01212.694.340.0683.030.02111.943.220.1973.08
S60.0225.285.770.0982.230.0124.3115.230.0812.13
S70.0128.6513.680.0341.500.0094.9322.140.0261.03
S80.02789.302.800.1932.310.01294.9910.560.0972.22
S90.0176.247.690.0621.640.0115.3916.080.0611.60
S100.0145.1711.080.0541.080.0103.8614.210.0450.71
S110.0267.873.810.0882.120.0104.7510.910.0551.56
S120.01356.9514.200.0711.420.01035.4136.410.0601.32
S130.02995.272.670.1322.310.01269.417.450.0541.67
S140.0227.245.490.0501.920.0126.2317.260.0661.83
S150.0218.168.950.0471.910.0134.9920.430.0401.24
S160.0154.7114.230.0951.720.0083.6594.520.0973.23
S170.0207.413.750.0661.980.0145.276.690.0531.68
S180.0208.069.270.0541.770.0137.8713.760.0341.01
S190.02313.265.630.1263.310.0126.2644.970.1114.95
S200.0258.363.540.0831.540.0116.0910.710.0550.96
WestS210.0075.6710.950.0372.300.0087.625.650.0962.82
S220.0278.642.830.0741.740.0128.832.900.0671.97
S240.0282.7111.380.4783.020.0125.965.700.0812.16
S250.03782.692.570.1212.080.00985.796.550.0422.03
S260.0316.287.300.7864.920.0095.1510.320.0762.62
S280.0516.692.440.1771.950.0165.905.050.0591.86
S290.03810.652.510.0892.130.02611.142.480.0761.94
S300.01579.974.670.0442.150.01444.105.340.0661.80
S310.0328.783.260.0852.180.03811.582.800.7983.23
S320.048110.721.950.2632.520.015107.193.470.0512.63
S330.02661.913.220.1392.160.01660.174.110.0802.02
S340.02360.324.920.0802.170.01444.126.910.0611.69
S350.0347.093.150.1692.140.0146.564.530.0692.07
S360.038129.972.350.0512.480.03397.962.420.1702.46
S370.0368.513.460.1862.740.02011.083.780.0912.83
S380.03763.644.470.6583.490.02497.533.780.1983.71
S390.0307.273.570.2812.070.0106.873.980.1031.82
S400.0248.454.980.1414.310.0114.976.610.1022.36
S410.0316.562.550.2362.050.0178.573.860.0922.31
S420.0286.783.200.1001.650.0166.773.190.1221.79
S430.0337.643.380.1862.190.0248.673.100.1812.01
S440.0377.943.310.1102.220.0228.863.120.0862.43
NorthwestS450.0072.0916.270.1132.050.0075.467.400.0862.58
S460.0105.7016.490.0542.610.0058.2324.890.0673.98
S470.0125.045.970.1272.390.0069.605.570.0973.34
S480.0224.115.160.1471.890.0083.626.920.0801.51
S490.07867.531.010.1792.570.00316.5022.200.0481.09
S500.0116.898.100.0612.430.0087.684.090.0982.86

Table 2.

List of NSRP parameter for the 48 raingauge stations.

To control how well the representation of the rainfall data is made by the NSRP model obtained, the mean of the 1‐h rainfall and the probabilities of the 1‐ and 24‐h rainfall estimated from the model are compared with these statistics values calculated from the observed data. Part of the results, focusing on the month of November and December only, is displayed in Table 3. It can be seen that there are no major differences between the estimated and the observed values of the statistics of interest.

RegionStationNovemberDecember
AOAEPOPEPO2PE2AOAEPOPEPO2PE2
SouthwestS12.742.720.080.080.570.573.313.540.110.120.510.47
S23.454.380.110.060.640.612.732.670.110.110.560.58
S30.290.320.100.060.580.580.190.190.080.080.440.45
S42.612.580.090.080.550.692.803.520.110.100.490.47
EastS50.230.220.060.060.420.440.270.260.080.070.470.49
S60.290.300.140.140.660.630.320.360.160.170.600.52
S70.890.930.200.200.700.690.960.980.270.270.720.72
S82.912.960.090.090.580.574.235.620.130.130.580.48
S90.470.500.160.160.690.640.540.610.180.190.660.57
S100.700.760.210.210.730.650.710.760.210.220.640.58
S110.360.370.120.120.680.660.320.340.140.140.560.52
S126.687.660.200.200.700.598.319.420.260.270.680.58
S133.183.250.100.100.650.623.433.600.110.110.560.53
S140.450.450.150.150.730.730.580.690.190.200.690.57
S150.780.800.220.220.790.781.001.040.290.290.730.73
S160.530.580.190.200.650.570.740.880.220.230.570.46
S170.280.280.100.100.600.600.300.290.120.120.570.58
S180.800.830.210.210.750.740.971.450.250.260.750.74
S190.400.520.140.130.710.610.480.690.200.230.640.53
S200.470.480.120.120.640.640.700.760.170.170.610.55
WestS210.180.180.090.090.460.450.120.130.050.050.330.32
S220.380.380.110.110.650.660.160.160.050.050.390.39
S240.290.290.100.150.580.570.190.190.080.080.440.45
S253.803.740.120.120.680.712.372.450.080.080.480.46
S260.280.290.120.110.560.570.170.180.090.090.410.40
S280.430.430.160.150.750.790.270.260.110.110.550.58
S290.490.480.130.120.700.740.370.370.090.090.600.61
S302.702.630.090.090.560.581.811.790.100.100.510.51
S310.440.430.130.130.680.730.360.380.120.090.630.63
S324.144.130.110.110.730.732.162.130.070.070.510.52
S332.422.400.100.100.590.591.971.940.080.080.490.50
S343.213.170.140.140.650.662.432.450.120.120.560.55
S350.360.350.130.130.640.670.210.210.080.080.480.50
S364.674.640.110.110.770.783.153.150.100.100.630.63
S370.400.390.140.130.670.690.300.290.090.090.540.56
S382.992.980.110.110.630.632.472.420.090.090.530.55
S390.370.370.110.110.600.590.150.160.050.050.350.34
S400.240.230.120.110.560.590.160.160.080.080.400.41
S410.260.260.100.100.590.600.240.240.080.080.500.49
S420.370.370.120.120.650.660.200.200.070.070.430.44
S430.400.390.130.130.630.650.320.320.090.090.530.53
S440.450.440.150.150.710.740.260.250.090.090.570.58
NorthwestS450.110.110.090.090.280.290.120.110.060.060.290.30
S460.370.380.170.170.570.570.230.280.100.100.400.34
S470.150.150.080.080.380.380.100.100.040.040.250.25
S480.250.250.130.130.580.570.130.130.070.070.360.35
S492.452.080.100.100.510.850.960.900.070.070.210.22
S500.250.250.100.100.500.500.090.090.040.040.290.29

Table 3.

The representation of the statistics which are estimated from the NSRP model compared with the statistic obtained from the analyzed data for the 48 raingauge terminal.

AO = mean of 1-h rainfall (observed data), AE = mean of 1-h rainfall (NSRP model), PO = probability of 1-h rainfall (observed data), PE = probability of 1-h rainfall (NSRP model), and PO2 = probability of 24-h rainfall (observed data, PE2 = probability of 24-h rainfall (NSRP model).


6. Conclusions

The results of this study proved that the Neyman‐Scott rectangular pulse model is able to imitate the pattern of actual rainfall in Peninsular Malaysia by comparing the parameters as well as the spatial distribution of the means and probabilities of 1‐ and 24‐h rain. Thus, the results from the NSRP model fitting for each station are valid to be used for further analysis, that is, to evaluate the behavior of storm rainfall.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Rado Yendra, Abdul Aziz Jemain and Ibrahim Sulaiman Hanaish (April 18th 2018). Analysis of Storm Rainfall in Peninsular Malaysia Using Neyman‐Scott Rectangular Pulse Modeling, Engineering and Mathematical Topics in Rainfall, Theodore V Hromadka II and Prasada Rao, IntechOpen, DOI: 10.5772/intechopen.70043. Available from:

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