Wavelet for Climate Data Analysis

Though there are a lots of factors affecting climate time series, non-linear interaction between different scales or interference of frequency component from its sidebands is associated with amplitude modulation which can be seen with frequency components in Fourier Spectrum without time information.

If some fundamental physical properties of the climate system undergoes secular changes which is followed by frequency modulation. Some of physical properties are increase in moisture content due to global warming. If the there is not finite time of occurrence then Fourier Spectrum is not sufficient to get dominant frequencies.

Another properties associated, abrupt change in frequency of climate time series is due to occurrence of catastrophic event with long term impact, which can be analysed with spikes or peaks with corresponding frequencies in Fourier Spectrum but contains no time information at which such abrupt changes occurred.

Similarly, another short term effect associated with sudden finite amplitude perturbation. This is specially due to volcanic eruption which causes global temperature variations, which can be detected in Fourier Spectrum with large number of component without time information.

For this all Wavelet Transform could be better solution which can give time information as well as frequency information though both time and frequency information could not be determined at the same time according to Heigenburg Principal, small frequency band can be localized at the small time interval.

Multi Resolutional Analysis

When a signal is analysed using Wavelet Transform as analytical tool, wavelet function and scaling functions are used as high pass filter and complementary low pass filter respectively. If we go through multi resolutional analysis in depth, we can note that at first decomposition, original signal is decomposed into low band signal and high band signal. As the high band signal is not further decomposed rather low band signal is further decomposed, we will have high scale value at high frequency band resulting poor frequency resolution.

As the low band signal is further decomposed, with smaller scaling value resulting better frequency resolution. Smaller the scaling value better the frequency resolution is. As further decomposition is done at higher level, the signal will be down sampled resulting poorer time resolution at lower frequency band.

Thus it can be stated; wavelet transform has poor frequency resolution at higher frequency and better time resolution where as it has better frequency resolution at low frequency band and poorer time resolution.