Wavelet Filter Approach and Z-R Relationship in Meteorological Forecasting

The purpose of this study is to investigate the Z-R relationship for computing rainfall using conventional and wavelet filters technique. Wavelet filter technique was applied to data filtration process.The proposed model was applied to determine the rainfall of five rain gauge meteorological stations in Thailand. The three-hourly rainfall and radar reflectivity data were used in this study. The results indicated that the accumulative rainfall of wavelet filters technique was close to the observed rainfall data more than the results of conventional practice for both calibration and validation processes. Consequently, we are confident that a wavelet filters technique is a useful tool for estimating the rainfall.


Introduction
One of the most important attributes to flood mitigation and water resources management is accurate computation of precipitation over the given areas.Rainfall data is the most important meteorological parameter in hydrological studies and the crop-water requirement, because it constitutes the base of all the computations in water resources management and decision making problems.The obtained rainfall data from rain gauge is the point of rainfall that has been recorded in a particular time scale.
Therefore, the lack of necessary climatological data made it difficult to obtain the reliable rainfall.In addition, the meteorological radar is a very powerful and useful tool for hydrological applications, since it can provide estimates of rainfall fields with a high spatial and temporal resolution over large areas.In the recent past, a few research studies have attempted to forecast the occurrence of precipitation.In measuring rainfall by radar, - relationships are widely used to convert radar measured reflectivity to rainfall intensity; hence the accuracy of the estimation of - relationship is important [1,2].In [3] applied a regression analysis technique to determine the relationship of synchronous datasets between measured rainfall intensity by rain gauge and measured or effective reflectivity by weather surveillance radar (Ze) at the pixel over the rain.However, in reality perfect synchronization between Ze and  is unachievable, except at the closest range and nearest to the ground [4].
Wavelet analysis has been developed in order to provide a performing analysing tool for this kind of signal.It effectively renders possible a time-scale localisation of the process thanks to a projection on a class of functions which in turn makes it possible to extract data within a local neighbourhood.Being a signal's time-scale analysis method, wavelet transform is better than Fourier transform due to its high-resolution features, good localization both in time and scale domains, and its capacity of analyzing signals at multi-time scales [5,6].In addition, wavelet analysis also has the capability of diagnosing the jump features of signals.Therefore, wavelet analysis has been applied to the studies of climate multi-time scales analysis.In the previous studies, wavelet transform is mainly used to analyze the multi-time scales characteristics of annual temperature and rainfall, and many interesting results have been found [7][8][9].Basically, the wavelet decomposition uses a pair of filters to decompose iteratively the original time series such as rainfall 2 and reflectivity.It results in a hierarchy of new time series that are easier to model and predict.These filters must satisfy some constraints such as causality and information lossless [10].

Advances in Meteorology
In this study, the rainfall events of five meteorological stations in Thailand were analyzed for - relationship.The Buffer Probability Technique (BPT) was developed based on GIS basic function and Probability Matching Method.The - pairs can be analyzed for empirical power relationship.The main objective of this study is the comparison between conventional and wavelet filter technique in estimating rainfall using - relationship; the focus was on two different models compared with observed data.

Materials and Methods
The selected three-hourly rainfall data of five rain gauge stations from Thai Meteorological Department and the radar reflectivity data from the Department of Royal Rainmaking and Agricultural Aviation at radar Pimai station, Nakhon Ratchasima province, were used in the study.The meteorological stations were sta.387401Mahasarakham, sta.403201Chaiyaphum, sta.405201Roiet, sta.431201Nakhon Ratchasima, and sta.436201Buriram, most of which are located over Chi-Mun basin in the Northeastern region of Thailand.These locations are presented in Table 1 and Figure 1.As the radar reflectivity,  (mm 6 /mm 3 ), commonly varies across many orders of magnitude, therefore  used in this study is the reflectivity expressed in terms of dBz:  (dBz) = 10 log  (mm 6 /mm 3 ) .
(1) Radar reflectivity was obtained by an image analysis process from rainfall events that occurred in Northeastern region of Thailand, from June to August 2009 for long rainfall.The reflectivity was recorded from the Department of Royal Rainmaking and Agricultural Aviation at Pimai station, Nakhon Ratchasima Province, which corresponds to 2.5 km of CAPPI radar products at Pimai site from S-band polarmetric radar that transmits radiation with a wavelength of 10.7 cm and produces a beam width of 1.2 degrees with a maximum range of 480 km, as illustrated in Table 2.The CAPPI products have six minutes and 1 sq.km temporal and spatial resolutions, respectively.This study assumes that there is no bias caused by the bright band effect and different observation altitudes at 1.5 km [11].The CAPPI data lies within 240 km from the radar and the reflectivity values less than 10 dBz and greater than 50 dBz were excluded from the analysis to avoid the effect of noise in the measured radar reflectivity [12].
2.1.- Relationship.The widely accepted relationship of the radar reflectivity factor  and rainfall rate  is given by  the empirical power law relationship [13] which can be defined by  =   , where  and  are coefficients that depend on location and difference in climatology such as season and type of rain.The parameters  and  are related to the intercept and slope of the best-fit line through a plot of  versus  on a log-log plot.Thus, if  value can be measured and thus known, then  value can be found.Many researchers suggested that parameter  does not need to be varied as much as the parameter  [14,15].The most common values for  and  are 200 and 1.6, respectively.Reference [11] found that the existing - relationship that has been used for the Pimai radar site is  = 294  and Window Correlation Matching Method (WCMM).In all processes except TMM, PMM is the process that the sampling volume timing and location problem are not taken into account [1].The matching is done between the Cumulative Distribution Functions (CDFs) of  and , described by the following formula: where () is the probability density function of rainfall from gauge measurement and () is the probability density function of radar reflectivity.The matching of  and  is done at the same probability level.According to the fact that the analysis is done on frequency domain, the timing errors are then eliminated in this method.

Wavelet
where  is an integer index for the finite or infinite sum,   are the real valued expansion coefficients, and   () are the set of real valued functions of  called the expansion set.For the wavelet transform, the two-parameter system is constructed and (3) becomes where  is the set of all integers and the factor 2 /2 maintains a constant norm independent of scale .This parameterization of the time or space location by  and the frequency or the logarithm of scale by  turns out to be extraordinarily effective.

Haar Scaling Function and Wavelets.
The multiresolution formulation needs two closely related basic functions.In addition to the wavelet () that has been discussed and another basic function called the scaling function ().The simplest possible orthogonal wavelet system is generated from the Haar scaling function and wavelet.These are shown in Figure 3.
In order to generate a set of expansion functions, the signal can be represented by the series in or using (5), as where the two-dimensional set of coefficients  , is called the discrete wavelet transform (DWT) of ().
An efficient way to implement DWT is to use a filter process.Normally, the low frequency content of the signal The concept of the wavelet filters technique can be easily applied for signal analysis; this study used the technique to decompose the details (D) from the approximations (A) of rainfall records (y).In wavelet analysis, the approximations are the high-scale, low frequency components of the signal, and the details are low-scale, high frequency components.The decomposition process can be iterated with successive approximation being decomposed in turn, so that one signal is broken down into many lower resolution components.The wavelet decomposition can yield valuable information about signal.The approximations components are processed by the - relationship for rainfall prediction.
In this study the 1D Haar wavelet filters were applied to a rainfall and radar reflectivity series in order to denoise the original signal into the low frequency approximations (A) and the high frequency details (D).The approximations which are the smoothed signal represent the trend of the rainfall and radar reflectivity series whereas the details represent the nuance of them.

Methodology
The methodologies for this research were as follows: (1) The quality control of radar image was performed; the errors due to radial anomality, ANAPROP, and other signals that did not represent reflectivity and errors caused by electronic problems investigation that had been done for both conventional and wavelet filter technique were removed from the measured reflectivity data.
(2) Buffer over rain gauge by Buffer Probability Technique transform process based on the buffer basic function of GIS is constructed to estimate the radar reflectivity over the rain gauge station at the same time.
(3) Average  value in Buffer zone and defined nonzero - pairs are determined.
(4) The radar reflectivity was normalized into threehourly data with the same time interval as rainfall measurement.
(5) From Cumulative Distribution Functions (CDFs), the radar reflectivity and rain rate at the same percentile can be obtained.These matching pairs were then used for - relationship evaluation by conventional and wavelet filter technique.(6) Calibration model by using rainfall event during 1 June-31 August 2009 and verification model by using rainfall event during 1 June-31 October 2011 were obtained.
The - relationship with the studied process is shown in Figure 4.

Results
Three-hourly rainfall data and radar reflectivity at five stations in the upper Chi-Mun basin of Thailand during the period from June to August 2009 are filtered by Haar wavelet function as shown in Figures 5 and 6. 3 and Figure 7 that the coefficients of - relationship from the conventional technique were in the range from 218.50 to 376.00, being larger at sta.436201 Buriram and sta.403201Chaiyaphum.The  coefficient did not vary significantly, remaining in the range from 1.025 to 1.491.The highest values occurred at sta.405201 Roiet and sta.431201Nakhon Ratchasima.The coefficients of determination were all above 0.834.For the data filtration by wavelet technique, the values of the coefficient varied widely from 132.82 to 156.02 whereas the  coefficient values were high, ranging from 1.183 to 1.558.The coefficients of determination were all above 0.951 suggesting a good - relationship.However, this accepted model is necessary for verification with other data for evaluating the performance of the model.

Model Verification.
The calibrated model parameter from the previous section was further validated using another climatological data set that occurred from 1 June 2011 to 31 October 2011, which were not considered in calibrating process.These data of five stations were used to calculate the rain rate; if  value can be measured and thus known, then rain rate or  value can be found.The verification results and performance statistics for - relationship at each station were presented in Table 4.The correlation coefficients for conventional and wavelet filters technique were all above 0.892 and 0.926, respectively, whereas the standard error of       This clearly indicates that the  WF - WF relationship by wavelet filter technique developed in the present study can be used with confidence in estimating rainfall for this study area.The results show that the overall correlation coefficients between the estimated and observed rainfall for all station were high range, which are acceptable.

Discussion
This paper applied a wavelet filter technique to signify the data preprocessing as an important step in the construction of - relationship for rainfall estimation.The wavelet filters can extract the chaotic signal components from the original     rainfall and dBz data for the calibration.Therefore, this study demonstrates that the importance of wavelet filters technique is a practical tool for the prediction of three-hourly rainfall in Upper Chi-Mun basin of Thailand.

Figure 1 :
Figure 1: Locations of the five meteorological stations.

Figure 2 :
Figure 2: Relationship between (a) radar reflectivity and ground surface rainfall (b) cumulative density function of - matching method.
where both  and  are integer indices and  , () are the wavelet expansion functions that usually form an orthogonal basis.The set of expansion coefficients  , are called the discrete wavelet transform (DWT) of () and it is the inverse transform.The wavelet system is a two-dimensional expansion set (basis) for some class of one-dimensional signal.The wavelet expansion provides a time-frequency localization of the signal.First generation wavelet systems are generated from a single scaling function (wavelet) by simple scaling and translation.The two-dimensional parameterization is achieved from the function (mother wave) () by  , () = 2 /2  (2   − ) ,  ∈ ,

Figure 4 :
Figure 4: Studies process for rainfall estimating by - relationship comparison between the conventional and wavelet filter technique.

Figure 5 :
Figure 5: Rainfall data filtration by Haar wavelet filter type for each station.

Figure 7 :
Figure 7: Scatter plot of - pairs for conventional practice and  WF - WF pairs for wavelet filters technique for each station. .

Figure 8 :
Figure 8: Time series plot of mean observed rainfall at sta.387401 Mahasarakham compared with conventional and wavelet filters technique.

Figure 14 :Figure 15 :
Figure 14: Time series plot of mean observed rainfall at sta.431201 Nakhon Ratchasima compared with conventional and wavelet filters technique.

Figure 16 :Figure 17 :
Figure 16: Time series plot of mean observed rainfall at sta.436201 Buriram compared with conventional and wavelet filters technique.

Table 1 :
Locations of the five meteorological stations.

Table 2 :
Characteristics of radar at Pimai station, Nakhon Ratchasima Province.

Table 3 :
Result of - relationship for each station during 1 June-31 August 2009 period calibration.

Table 4 :
Result of statistical - relationship for each station during June-October 2011 period verification.