The Relationships between Wind Speed and Temperature Time Series in Bangkok, Thailand.

In this research we investigate the relationships between wind speed and temperature time series data in Bangkok, Thailand, from the time interval of January 2009 to December 2011 using wavelet transform (WT), cross wavelet transform (XWT) and wavelet coherence (WTC). The results from all three wavelet analysis show the strong periodicity around period 1 day (hourly data) and period band 256-450 days (daily data) variations that are exhibited in both wind speed and temperature data across the entire power spectrum from 2009 to 2011. These two oscillations are connected with the natural day time effects and the annual natural season cycle. Although the daily periodic for the temperature is appeared nearly uniform all year but it is not the case for wind speed. In 2009 this wind speed oscillations appear only from mid-February to mid-April in summer and from the fourth week of May to the third week of August in rainy season. XWT also detects strong high common power between the wind speed and temperature at a period band of 14-25 days in summer 2009, a period band of 4-8 days in summer 2009, July 2009, summer 2010 and summer 2011. WTC shows the coherence period band around 10-30 days appeared in summer and rainy season and 32-50 days in summer 2009 and rainy season in 2010. From these three wavelet analysis, the wind speed and temperature time series data show the strong correlation especially at 1 day and 256-450 days period band and also at several different scales. This studied will be helpful in predicting the wind speed and temperature for the future used.


Introduction
During the last decade, wavelet analysis has been successfully used as a time-frequency tool to study several fields such as intra-seasonal oscillations in wind speed and oceanic wave, tropical convection, intra-decadal changes in ENSO Monsoon system [1]. It is able to detect variations of power within time series data. The wavelet transform (WT) is a very adequate tool to analyze the time series that contain information at different time scales such as diurnal, seasonal and annual time's scales. From the timefrequency representation provided by the WT [2,3], one can easily determine significant time series oscillations and how these oscillations vary with respect to time [4,5,6].

The wavelet transform
A wavelet is a function with zero mean that is localized in both frequency and time. We can characterize a wavelet by how localized it is in time (t) and frequency ( or the bandwidth). One particular wavelet used in this research, the Morlet, is defined as [4,5,6,7] i ( ) e e

Cross-wavelet transforms
The XWT is a useful tool to study the relationships between two time-series. The wavelet coherence is the square of the XWT over the individual power WT [1,3,4,7]: where S in eq.4, is the smoothing operator which is essential in coherence analysis.

Experiment
First, the wavelet transform is applied to wind speed and temperature data in order to investigate their spectral behaviours and how they vary with respect to time. From there wavelet power spectrum and with an appropriate choice of the analysis frequency band, we can determine the significant oscilla-tions frequencies as well as their durations and time of occurrences. After that, the cross wavelet transform is used in order to study coherency and the phase difference between the two time series. The study is carried out using hourly and daily average data extending over a period of 3 years (January 2009 to December 2011).

Results and Discussion
The daily wind speed and temperature data at Bangkok from the year 2009 to 2011 is shown in figure  1. The average wind speed and temperature over these 3 years period are 3.11 m/s and 28.5 o C, respectively. It is obvious from this figure that the wind speed and temperature time series show the periodicity in one year period. Wind speed and temperature of Thailand is under the influence of monsoon wind system of seasonal character. The southwest monsoon which starts in May and ends in mid-October brings a stream of warm moist air from the Indian Ocean towards Thailand. While the northeast monsoon normally starts in mid-October and ends in mid-February brings the dry and cool air to Thailand. Observing figure 1 does not give much information about the relationships between wind speeds and temperature data.

Wavelet Power Spectrum
The wavelet power spectrum for daily wind speed fluctuation at Bangkok, Thailand over a period of 3 year (January 2009 to December 2011) has been presented in figure 2. The horizontal axis is time scale (in day) and the vertical axis is the period (1/frequency). The color in the figure stands for the structure of wing speed variety (the power ranges from weak (white shades) to strong (black shades)). With this technique, the difference in time-series data will be mapped into wavelet region and into various different scales. In this research, we use Morlet wavelet as a basis due to the fact that it is polyvalent in analyzing non-stationary time-series. The WT power spectra of wind speed in figure 2 is evaluated using 3 years data and over range of period from 2 to 512 days (i.e., frequency from 1/512 = 0.00195 to 1/2 =0.5 cycles/day). As we can see from this figure, there is one dominant oscillation equals to period of 365 days. This oscillation is obviously connected with natural annual periodicities. Other significant oscillations of period from 32 to 64 days occur in summer and less power in rainy season in the period between January to October for the year 2009. However in 2010 this oscillation has moved to the period from January to May with less power and relatively short period from January to May 2011. Another oscillation of period from 10 to 30 days occurs in summer 2009 between February to May and extended up to mid-July with less power. In 2010, these oscillations are characterized by relatively short durations from one to a few weeks in mid-March. These oscillations moved to summer and rainy season in 2011.

Cross wavelet analysis
The XWT between wind speed and temperature time series is shown in figure 4. One significant peak appears near the edge effect (at the bottom) correspond to the period of 256-450 days. One peak appears at a period of

Wavelet coherence
Wavelet coherence between wind speed and temperature time series is shown in Fig 5. The colour black shading represents the wavelet squared coherence. The thick black line represents the 95% significance level. The vectors, only plotted for the squared coherence quarter than 0.5, denote the phase relationship between the time series. The vector pointing right is for in-phase relation; left for anti-phase; up for wind

Conclusion
The relationships between wind speed and temperature time series data show the strong correlation at 1 day and 256-450 days period band. The WXT and WTC are useful to study the interaction and the relative phase between the two time series.