Rainfall Distribution Pattern Investigation withUnit Disaggregation Curve (UDC) Technique

S.Tantanee1, S.Patamatumkul2 and T.Oki3
1:Graduate School, Khon Kaen University, 2:Assoc.Prof., Khon Kaen University,
3:Assoc.Prof.,University of Tokyo
Engineering Faculty, Naresuan University, Thailand. e-mail: sarintipt@nu.ac.thAbstract
The technique of Unit Disaggregation Curve (UDC) has been developed in this study in order to investigate the patternof rainfall distribution within a year. The proposed technique is based on the concept of Disaggregation model. UDC isconstructed from the historical rainfall records, which have been identified as signal. The data used in the study are therecords from 4 stations distributed over Northeastern of Thailand, of which the length of records is at least 50 years.
Applying wavelet to those signals, approximations or trend, and details or noise, are separated. The analysis emphasizeson the approximations that apparently demonstrate similar pattern of distribution in each year. The rainfall distribution
pattern graphs of each year are recognized from these approximations. The obtained UDC is generated from therepresentative pattern of rainfall distribution graph within a year. The UDC is expressed in the unit curve that is able toapply for any level of annual rainfall. By coupling the signal of details, the further study can be undertaken to develop
the new technique of rainfall prediction process.
Keyword: wavelet, rainfall, disaggregation

1. Introduction
Rainfall is the most significant parameter forwater resources planning; especially in plantation
planning, not only amount of rainfall but also thedistribution of rainfall within a year is need. Several
stochastic models have been proposed for modellinghydrologic time series for long time such as
autoregressive model (AR) ( Yevjevich,1963),
fractional Guassian noise model (Matalas and Willis,1971) and autoregressive moving-average models(ARMA)(Carlson et. al.,1970) etc. The objective of thisstudy is to develop a technique that can investigate therainfall distribution within a year, which is a part of
rainfall prediction model. The analysis was undertakenwith rainfall records of 4 stations distributed over
Northeastern part of Thailand by applying wavelet
analysis with rainfall records and constructing a Unit
Disaggregation Curve (UDC). Figure 1 shows thelocation of the studied stations.
The proposed UDC can demonstrate thecharacteristics of rainfall distribution within a year interm of trend that noises already eliminated and thecurve also is convenient to utilize as disaggregation
tool for rainfall prediction.

2. Wavelet andDisaggregation model
application for Unit Disaggregation Curve(UDC)
As wavelet can be easily applied for signal
analysis, this study used wavelet to separate the details(noises) from the approximations (trend) of rainfall
records. In wavelet analysis, the approximations are thehigh-scale, low-frequency components of the signal,and the details are the low-scale, high-frequencycomponents. The process of 3-layer decomposition
with wavelet is demonstrated in Fig.2 as wavelet
decomposition tree. The decomposition process can beiterated, with successive approximations being
decomposed in turn, so that one signal is broken down
into many lower resolution components. The wavelet
decomposition can yield valuable information ofsignal.
In order to eliminate the details from theapproximations in this study, the decomposition
process of wavelet was applied to rainfall records,which was identified as a signal process. The obtained
approximation curve will apparently demonstratecharacteristics of rainfall distribution within a studiedyear. UDC technique was developed from the basic
concept of linear disaggreagtion model that subseriesof monthly data sum exactly to key series of annualdata.

Fig.1: The 4 Studied stations(Khon Kaen, Nakhon
Ratchasima, Ubon Ratchathani and Udon Thani.)
Fig.2: Wavelet decomposition tree

The basic linear dependence model (Valencia andShaake, 1973) that has been applied to this study is
Y= AX+Bε(1)Where Y is the current observation of the series beinggenerated (subseries or dependence series which ismonthly series in this case); Y is generated dependent
on X series ( key series or independent series which isannual series in this case).; ε represent the current valueof a completely random series (stochastic term which isseparated as details).
The process of UDC construction is shown in
Fig. 3.

illustrate pattern of distribution as Fig.4. Considering
the curve obtained from 3-level filter, some
information of approximations is separated andincluded into the details as shown in the curve ofdetails in Fig.5(c).
252015da10yMonthly rainfall series

5Wavelet filter
01Details

Approximations

23456789101112(4a) 1 level-wavelet filter

Polynomial or mean Fitting

2520

Approximation distribution curve
15

Unit Disaggregation Curveda10yFig.3: The UDC construction process
53. Studied Data
The selected 4 rainfall stations distributed
over Northeastern part of Thailand were tested in thisstudy. The records range of monthly and annual rainfallof these stations is at least 50 years. The information ofrecords is shown in Table 1. The studied process wastested with 2 variables series of the number of rainydays (day) and the amount of rainfall (mm).

0123456789101112101112(4b) 2 level-wavelet filter

1412Table 1 Details of rainfall records of the 4 stations
StationKhon
Kaen
NakhonRatchasima
Ubon
Ratchathani
Udon
ThaniYear

1951
2002
1951
2002
1951
2002
1951
2002

Latitude –longitude

16.26N
102.50E

No.
Yearofrecord52Avg.
Rainfall(mm)1212.5Meannumberof wetdays
87.21

108yad6414.58N
102.05E

52

1092.686.73

15.15N104.52E

52

1580.8100.8817.23N
102.48E

52

20123456789

period1460.3101.42(4c) 3 level-wavelet filter
Fig 4: Approximation Distribution Curve of Number ofrainy days of Khon Kaen station. (red line: Arithmetic
mean and Green line: Polynomial fitting)

4. Results
By analyzing with 1-3 level of wavelet filter,
it is found that the obtained approximation distribution
curve of 2-level filter is the most appropriate to

From the approximation curve of 2-levelfilter, the UDC was constructed by cumulating eachmonth variables into annual value and made it into aunit curve. The example of unit curve of UDC for
number of rainfall are shown in Fig.6. Similarly, 2level filter is also appropriate for investigate theapproximation distribution curve of monthly rainfall(mm). Fig.7 and Fig. 8 show the curves of rainfall
distribution of Udon Thani station and the UDC ofrainfall of Ubon Ratchathani station, respectively.

15105yad0-55.Conclusion
Studying with 4 rainfall stations data records,
the obtained UDC from analyzing of rainfall andnumber of rainy days can illustrate the pattern ofdistribution within a year very well. The UDC isexpressed in the unit curve that is able to apply for any
level of annual rainfall. The UDC technique is the firststep of wavelet-disaggregation model for rainfallprediction. Further study is undertaken to construct therainfall prediction model that can predict annual
rainfall and monthly rainfall within a predicted year.

-10-15

1

2

34

5

678

910

1112

period(5c) 3 level-wavelet filter

Fig 5: Details curve of Number of rainy days ofKhon Kaen station.(red line: AR generated sequences)
1.

0.9
80.8
0.7

60.6
40.50.4

2
yad0.3
00.2
0.1

-2

0-4-6

1234

56789101112period1234

567period

891011121112Fig 6: The UDC of number of rainy days ofNakhon Ratchasima station(5a) 1 level-wavelet filter
400
8350

6300

4250

2

yad200
mm

0-2

100-4

50-6-8150

0123456789period(5b) 2 level-wavelet filter

10111234567891012Fig 7: The 2-level approximation distribution curve ofrainfall (mm) of Udon Thani station. (red line:Arithmetic mean and Green line: Polynomial fitting)

10.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
01234567
period89101112Fig 8: The UDC of number of rainfall (mm) ofUbon Ratchathani station

References
Box, G.E.P. and Jenkins, G., Time series analysis,
Forecasting and Control. Holden-Day Inc., SanFrancisco, 1970.
Burrus, C.S., Gopinath, R.A., and Guo H., Introduction towavelets and Wavelet transforms. Prentice-HallInternational Inc., Texas, 1998
Carlson, R.F., MacCormick, A.J.A., and Watts, D.G.,
Application of linear model to four annual stream
flow series. J. Water Resources Research, 6,4,1070-1078, 1970.
Matalas, N.C. and Willis, J.R., Statistical properties ofmultivariate fractional noise processes. J. Water
resources Research, 7,6, 1460-1468, 1971.
Salas, J.D., Delleur, J.W., Yevjevich, V., and Lane, W.L.,
Applied modelling of hydrological time series.
Water Resources Publications, Colorado, 1985.
Valencia, D., and Shaake, J.C., Disaggregation process instochastic hydrology. J. Water Resources Research,9, 3, 580-585, 1973.
Yevjevich, V. M., Fluctuation of wet and dry years. Part 1.
Hydrology paper 1, Colorado State University, Fort
Collins, Colorado, 1963