Journal of Water Resource and Protection, 2012, 4, 866-869 Published Online October 2012 (
A Rainfall Distribution for the Lampao Site in the
Chi River Basin, Thailand
Bungon Kumphon1*, Arun Kaew-Man1, Parakorn Seenoi2
1Department of Mathematics, Mahasarakha m Unive rsity, Mahasarakham, Thailand
2Department of Statistics, Khon Kaen University, Khon Kaen, Thailand
Email: *
Received August 12, 2012; revised September 10, 2012; accepted October 11, 2012
In this study, the four-parameter kappa distribution with L-Moments estimation has been used to fit the distribution of
weekly rainfall data at Lampao in the Chi River Basin, Thailand. The weekly precip itations with probabilit ies 0.75 were
estimated, and the extreme rainfall estimates obtained can be used for water and agriculture management.
Keywords: L-Moments; Four Parameters Kappa Distribution; Goodness of Fit Test
1. Introduction
This paper deals with precipitation frequency analysis in
the central area of Northeast Thailand. Under the Koppen
classification the Northeast climate is called tropical
wet-dry or tropical savannah. The two main mechanisms
that support the rainfall in this region are the southwest
monsoon during May to October, and the tropical cy-
clone depression July to September. Both are more im-
portant for agriculture and in the design and con struction
of various flood control measures, including the
operational management of reservoirs. Reliable estimates
of rainfall quantities at specified recurrence intervals by
operational hydro-meteo-rologists have therefore been
one issue. Ref [1] investigated the distribution of
monthly rainfall in Northeast Thailand, and found the
data could be fitted by either a Gamma distribution or a
log-normal distribution. A more recent study by [2]
showed that the incomplete Gamma distribution fits the
daily rainfall in Thailand, where the rainfall starts from
week 12 up to between week 20 and week 29, and then
increases up to between week 36 and week 39, with very
heavy rainfall in the Northeast, East and Southern
In general, maximum likelihood estimation (MLE) is
too sensitive to investigate extreme values. From our
experience with a quasi-Newton algorithm, we find that a
term in the logarithm function goes negative at some
points xi, and the MLE is computationally inefficient.
The L-Moments approach developed by [3] has been
widely used for parameter estimation because it provides
robust and reliable parameter estimation, particularly
from small samples. It is also computationally more trac-
table than MLE, and less sensitiv e to outliers that may be
present in the sample due to severe drought, flood or
cyclone events. Further, L-Moments are linear combin-
ations of ranked observations and do not involve squares
or cubes, as in the more conventional methods for mom-
ent estimators. The four-parameter kappa distribution
(K4D) also introduced by [4] is a very general distribu-
tion form that includes a variety of distributions—e.g. the
generalized extreme value distribution, the generalized
Pareto distribution, the generalized logistic distribution,
the Gumbel distribution, etc. Several applications and
examples see [5-7]. Ref [8] also used the K4D with L-
Moments estimation (L-ME), in modelling Indian mon-
soon rainfall. Ref [9] found that L-ME and MLE work-
ed equally well for the three parameter kappa distribu-
The K4D with L-Moments procedure is adopted in this
paper, to fit the distribution for the rainfall in a part of the
Chi River Basin—more precisely, on the weekly preci-
pitation data between 1984 to 2010 for the upper North
Eastern region, obtained at the Lampao telemetering
information station and the remote station s of the Hydro-
logy and Water Management Center, Royal Irrigation
department (HWMC) located around the Lampao reser-
voir area. The set of 0.75 probability estimates of the
rainfall then obtained for each week are applicable to
crop water management.
2. Study Area and Data
The Lampao River is a branch of the Chi River, and ex-
tends for 262 km but carries rather little water—viz. ap-
*Corresponding a uthor.
opyright © 2012 SciRes. JWARP
proximately 9300 km3 of water per annum. One head-
wayter in the Nongharn District (Udonthanee Province)
runs south through five districts in Kalasin Province, to
meet the Chi River in the Kammalasai District, Kalasin
Province—cf. Figure 1. In wet seasons, flash floods in
the floodplain of the Chi River basin can be a concern.
However, despite many local drought areas and the low
quality of the soil, these areas produce sticky rice and
other cash crops such as manioc (Cassava) and sugar
cane, and the main income for the population is from
agriculture. Consequently, rainfall and water manage-
ment are of key importance for agriculturists in these
areas given the low rainfall and potential water shortag es
during the growing season. On the other hand, there can
be flooding due to heavy rains towards the end of the
rainy season. The Lampao Reservior, built during 1963
to 1968 and storing 1430 mcm3 of water, assists in agri-
culture over 314,3 00 rais (50,288 hectares) in the h arvest
season and 180,000 rais (28 ,800 hectares) in the dry sea-
son, and in flood prevention. More than 30,000 house-
holds around this river and the reservoir system stand to
gain from effective water management.
The daily rainfall data from the nine telemetering in-
formation stations located around the reservoir (one ma-
ster station; TP1, and eight remote stations; from TP2 to
TP10, where TP7 is nearby TP1, so we decided to study
the master station) was employed in the analysis. The
data were processed according to the water year, begin-
ning on 1st April and ending on 31st March the follow-
ing year, as recorded over the 26 years between 1984 and
2010. The definition of light rain is water in drops of
between 0.1 to 10 mm in diameter [10]. The amount of
rainfall at a station was then obtained by summing up the
Figure 1. Map of Lampao reservoir and the measuring of
telemetering stations in the Lampao River.
daily amounts (>0.1 mm) in that week for the fitted
dis tribu tion . For the rainfall at a probability equal to 0.75,
which is the optimal value for effective rainfall and irri-
gation planning [11,12], were also calculated for TLP3
station because this is a first station at the head of
reservoir, cover the largest area of the irrigation.
3. Backgrond Theory
L-Moments are summary statistics for probability distri-
butions and data samples, analogous to ordinary mom-
ents [3]. They provide measures of location, dispersion,
skewness, kurtosis, and other aspects of the shape of pr o-
bability distributions or data samples. Using the uniform
distribution function as its foundation and based on
shifted Legendre polynomials, each statistical L-Moment
is computed linearly (hence the L reference), giving a
more robust estimate for a given amount of data than
other methods. The sampling properties for L-Moments
statistics are nearly unbiased, even in small samples, and
are near normally distributed. These properties make
them well suited for characterizing environmental data
that commonly exhibit moderate to high skewness.
For the random variables of sample size n drawn from
the distribution of a random variable X with the mean m
and variance s2, the cumulative distribution function of
the K4D for 0, 0kh
 
11 h
Fxh kx
 
The probability density function is
 
11 1
fx kxFx
 
 and 0
where ξ is a location para-
meter, α is a scale parameter and h, k are the shape
parameters (skewness and kurtosis), respectively. The
quantile function (inverse cumulative distribution func-
tion) is
xF kh
 
Special cases of Equation (1) take the form of different
distribution functions, such as the generalized Pareto
distribution, the generalized ex treme value distribution or
the generalized logistic d istribution wh en and h =
1, 0, –1 respectively. In the same way, for k = 0 and h = 1,
0, –1, we have the exponential distribution, the Gumbel
distribution and the logistic distribution, respectively;
and for k = 1 and h = 1, 0, the distributions are the
uniform distribution and reverse exponential distribution,
Let 1:1 1:
X, be the order statistics such that the
L-Moments of X are defined by
Copyright © 2012 SciRes. JWARP
 
where r is the rth L-Moment of a distribution and E(Xi:r)
is the expected value of the ith smallest observation in a
sample of size r. The first four L-Moments of a random
variable X can be written as
33:32:3 1:3
44:43:42:4 1:4
133 .
[3] demonstrated the utility of estimators based on the
L-Moment ratios in hydrological extreme analysis. The
second moment is often scaled by the mean, so that a
coefficient of variability is determined—viz,
 
where 1
is the measure of location. As with the defi-
nitions and the meaning of the ratios between ordinary
moments, the coefficients of L-kurtosis and L-skew-
ness are defined as
for , where3r3
is the
measure of skewness (L-Cs) and 4
is the measure of
kurtosis (L-Ck).
4. Application to Weekly Rainfall
The exact distribution of parameter estimators obtained
by this method is difficult to derive in general, so we ex-
plored the fit between the theoretical distribution (K4D)
and the real data set, as shown in Figure 2. Parameters
and a goodness of fit test via the Kolmogorov-Smirnov-
test (KS) and the Anderson-Darling test (AD) with a 0.05
significance level computed for the data at all nine
stations are tabulated in Table 1 and shown that K4D fit
well with all the rainfall for nine stations. A close inspec-
tion of the parameters shows values of the respective
parameters h > 0 and , suggesting that the underly-
ing distribution tends towards the generalized Pareto
distribution rather than the generalized extreme value
distribution fo r all stations.
As an application of this methodology to the estima-
tion of the maximum amount of rainfall at the 0.75 prob-
ability each week, the example observations were shown
at one station above the reservoir within the Chi Basin
using the parameters in Equations (1) and (2) estimated
at TLP3 station. The estimated values have been
computed as presented in Table 2. The highlighted area
Figure 2. The exploration for goodness of fit for K4D at
TLP3 station.
Table 1. Parameters estimation and goodness of fit test for
the K4D distribution.
site locationscaleh k KS (p-value) AD (p-value)
TLP1–23.355169.4270.2731.165 0.280 (0.953 ) 0.0402 (0.887)
TLP2–14.444176.8330.3211.088 0.240 (0.975 ) 0.0360 (0.943)
TLP3–37.121237.3760.4551.127 0.061 (0.26 7) 1.370 (0.211)
TLP4–18.790178.1920.2921.101 0.320 (0.923 ) 0.031 (0.988)
TLP5–18.872166.9700.3781.121 0.243 (0.974 ) 0.031 (0.990)
TLP626.010146.2380.2830.787 0.535 (0.711) 0.037 (0.933)
TLP826.051112.9080.1570.731 0.3880 (0.861) 0.040 (0.943)
TLP920.012152.2460.2940.819 0.717 (0.544) 0.043 (0.812)
TLP 1048.534117.7650.1360.538 1.101 (0.308) 0.051 (0.649)
Table 2. The estimation of the maximum amount of rainfall
for each week at the 0.75 probability, TLP3 station.
Week AmountWeek Amount Week Amount
1 0.000 19 61.487 37 73.535
2 0.000 20 61.835 38 62.461
3 0.000 21 76.256 39 37.234
4 0.000 22 69.715 40 49.580
5 0.305 23 65.838 41 39.601
6 0.000 24 72.143 42 22.817
7 1.566 25 74.050 43 4.943
8 0.000 26 60.660 44 4.855
9 5.483 27 64.666 45 0.000
10 0.000 28 56.508 46 0.039
11 1.659 29 65.155 47 0.000
12 16.165 30 41.593 48 0.000
13 23.785 31 79.242 49 0.000
14 10.443 32 82.184 50 0.000
15 16.638 33 103.833 51 0.000
16 19.828 34 103.358 52 0.000
17 43.136 35 84.370
18 45.875 36 110.496
shows the rainy season from week 19 to week 40. The
K4D realdata
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Copyright © 2012 SciRes. JWARP
awareness weeks are weeks 33 to 36, with very heavy
rain (84 to 110 mm.). This is useful information for the
hydro-meteorologists, and for reservoir planners who
manage water release and storage before and during the
awareness weeks 33 to 36. The best weeks for agri-
culturists to plan drying processes for crops are 1 to 6, 8,
10 and 45 to 52 because there is then no rainfall.
5. Conclusion
For the planning and design of crop scheduling and the
design of water management in Northeast Thailand, the
distribution of weekly rainfall was investigated from data
on weekly rainfall for a part of Chi River Basin sur-
rounding the Lampao Reservoir. The data is fitted well
by K4D with L-Moments estimation, and there is some
evidence for a generalized Pareto distribution. There is
usually one distribution that passes the goodness-of-fit
test. Although, ther e might be more than one distribu tion
for this relatively small region (from the family of such
distributions). For the estimated rainfall at the specific
probability 0.75, there is low to no rainfall in the dry
season, which is the best time for drying crops or any
associated activity that has no water requirement. On the
other hand, there is very high value of rainfall in the
rainy season.
6. Acknowledgements
Financ ial support was provided by the Faculty of Scienc e ,
and Mahasarakham University Development Fund, Maha-
sarakham University. The data was provided by the Hy-
drology and Water Management Center for the upper
north eastern Royal Irrigation department, Thailand. The
author thanks Professor Roger Hosking for his support
and helpful suggestions.
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