Wavelet Transform Analysis for Nonstationary Rainfall-Runoff-Temperature Processes

1. Wavelet Transform Analysisfor Nonstationary Rainfall-Runoff-Temperature Processes Murat KÜÇÜK, Ekrem TIGLI, and Necati AĞIRALİĞLU Istanbul Technical University ISTANBUL-TURKEY

2. COUNTINOUS WAVELET TRANSFORM AMPLITUDE TIME SCALE main goal • In the present study, main goal is to analysis hydro-meteorological measurements such as gage height, streamflow, water temperature and precipitation measurements of a observation station in a basin; and to compare variations of each measurement with each other in time-scale domain by using continuous wavelet transform. • Transforms • Fourier transform • STFT • Continuous wavelet transform • Applications

3. What is a Transformand Why Do we Need One ? • Transform: A mathematical operation that takes a function or sequence and maps it into another one • Transforms are good things because… • The transform of a function may give additional /hidden information about the original function, which may not be available /obvious otherwise • The transform of an equation may be easier to solve than the original equation (recall Laplace transforms for “Def. eques”) • The transform of a function/sequence may require less storage, hence provide data compression / reduction • An operation may be easier to apply on the transformed function, rather than the original function (recall convolution)

4. What Does a Transform Look Like…? • Complex function representation through two components • Sinusoids as building blocks: Fourier transform • Frequency domain representation of the function discrete countinous

5. Some Transforms • Fourier series • Discrete Fourier transform • Laplace transform • Z-transform

6. Fourier transform Fourier transform decomposes a signal to complex exponential functions of different frequencies

7. Stationary and Non-stationary Signals • Stationary signals’ spectral characteristics do not change with time • Annual mean temperature • Non-stationary signals have time varying spectra 20 Hz 50 Hz 5 Hz Concatenation

8. Stationary and Non-stationary Signals • Stationary signals consist of spectral components that do not change in time • all spectral components exist at all times • FT works well for stationary signals • However, non-stationary signals consists of time varying spectral components • How do we find out which spectral component appears when? • FT only provides what spectral components exist , not where in time they are located. • Need some other ways to determine time localization of spectral components • FT identifies all spectral components present in the signal, however it does not provide any information regarding the temporal (time) localization of these components.

9. Short Time Fourier Transform(STFT) • Choose a window function of finite length • Truncate the signal using this window • Compute the FT of the truncated signal, save. • Slide the window to the right by a small amount • Go to step 3, until window reaches the end of the signal

10. STFT

11. STFT Frequency parameter Time parameter Signal to be analyzed FT Kernel (basis function) STFT of signal x(t): Computed for each window centered at t=t’ Windowing function Windowing function centered at t=t’

12. The Wavelet Transform • Overcomes the resolution problem by using a variable length window • Analysis windows of different lengths are used for different frequencies: • Analysis of high frequencies Use narrower windows for better time resolution • trends, abrupt changes, floods… • Analysis of low frequencies  Use wider windows for better frequency resolution • Seasonal oscillations of hydro climatic series • The function used to window the signal is called the wavelet

13. The Wavelet Transform A normalization constant Translation parameter, measure of time Scale parameter, measure of frequency Signal to be analyzed Continuous wavelet transform of the signal x(t) using the analysis wavelet (.) The mother wavelet. All kernels are obtained by translating (shifting) and/or scaling the mother wavelet Scale = 1/frequency

14. …then scale, and shift through positions High frequency (small scale) Low frequency (large scale)

15. Small scale -Rapidly changing details, -Like high frequency Large scale -Slowly changing details -Like low frequency Scale is (sort of) like frequency

16. “Johns River at Buffalo Bluff Nr Satsuma, in Florida” The station is located 29°35'46" latitude, 81°41'00" longitude in Putnam County, Florida.gage height, streamflow, water temperature and precipitation measurements of a observation station in the basinPeriod of record is between 23 June 2002 and 16 November 2003 for each measurement. Number of records, N is 512 including daily streamflow measurements, gage elevation measurements, water temperature measurements and precipitation measurements, simultaneously. Applications

17. Daily Data Four data sets are used in applications.

18. GAGE HEIGHT-STREAMFLOW 150 days period structures countinous through all time period 1 week and 15 days period structures depend on weather temperature and rainfall

19. 180 days structures of temperature phenomena. Different from streamflow variation Structures with 30 days periods. TEMPERATURE-RINFALL

20. Conclusions • Analysing series can represent hydro-climatic characteristic of region at different time scale by defining periods of the hydro-climatic events by using wavelet transform • In addition, hydro-climatic characteristics of region are found with physical interpretations by using Continuous Wavelet Energy images of each measurement. • Small-scale structures such as 1 to 3 months can be easily seen in CWT image of streamflow measurements.

21. Conclusions • High energy density structure in the image is located about 130-150 days scale through whole time period of the measurement. • Because the gradual incensement in base flow conditions occurring in the basin during winter and early spring is responsible for the high variability at timescales of 4 months, the structure which has 130-150 days period is interpretable for streamflow measurements in this basin. • Application of the method to the precipitation measurements clearly represents small-scale structures which has variation between one week and tree weeks period, easily.

22. Thank you very much