Wavelets and its applications in atmospheric field

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Wavelet Analysis Why? What? And How?
By L. Kiranmayi
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Outline

• Introduction
• History
• Fourier transforms
• Short Time Fourier transforms
• Wavelet transforms
• Wavelets
• Applications

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Introduction

• Data
• Time domain
• Frequency domain

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Fourier Transforms

• The Great Father Fourier - Fourier Transforms

Any Periodic function can be expressed as linear
combination of basic trigonometric
functions (Basis functions used are sine and
cosine)
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Cosine and Sine parts
Real part and imaginary part
t -inf to inf
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How it works?

• Harmonics
• For a series with N points, ith harmonic means
  oscillation with N/i period i.e., fit a sine and
  cosine wave with that period
• Series
• Amplitudes of each sine and cosine wave at each
  frequency/period series of amplitudes as a
  function of frequency/period

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Drawbacks
Stationary
Non Stationary

• Integration from -inf to inf
• Gives frequency content of total time series but
  temporal information is lost!

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Short time Fourier Transforms
Time series
Window function
After multiplication

• Same as usual Fourier transforms, but data is
  modified by multiplication with a window function
• Only part of data at a time is taken and processed

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Drawbacks of STFT

• Frequency and time resolutions are fixed
• (Wider the window width, lesser the time
  resolution and more the frequency resolution and
  vice versa)
• As frequency resolution increases, time
  resolution decreases uncertainty principle

Desired Good time resolution at high frequencies
and good frequency resolution at low
frequencies!
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Wavelets

• Automatic time and frequency resolution
  adjustments
• Flexibility in choosing basic function
• No need to confine to sine and cosines anymore!

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What are Wavelets?

• A small wave
• Extends to finite interval

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Some mathematical expressions
x(t) actual time series ?(t) wavelet function
Integrable and limited to finite region
Total energy finite
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Some typical mother wavelets
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What exactly wavelet transform does?
Scale (expand or contract) translate and the
mother wavelet ?((t-?)/s)
Multiply with the time series
Sum this product for the total time series
Scaled(expanded)
Wavelet coefficient at time ? and scale s
Translated
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Typical picture
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Quantitative information

• Scale and equivalent Fourier period
• pconsts
• Amplitude gives the local power
• Summation of square over all the times gives
  equivalent Fourier power

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Some real life Applications
Time series analysis

• Intraseasonal Oscillations

60E
70E
90E
165E
WT of filtered SST for 4 longitudinal belts from
10 to 12.5 N
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Other applications

• Filtering a non-stationary data
• Through reconstruction of original series from
  coefficients
• Take the coefficients of required scales alone,
  making others zero.

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Two dimensional Wavelet Transform

• Definition
• Useful to obtain time-space variations or spatial
  variations in 2D
• Mainly used in Image processing

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Typical 2D wavelet
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Multiresolution analysis and discrete Wavelet
Transforms

• Varying resolution in time and frequency at
  different levels
• Discrete transforms
• Coefficients at discrete scales and time points

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Data compression

• Image processing
• Main contributions
• FBI finger prints
• JPEG2000
• Audio compression

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Denoising

• Considering all the coefficients with amplitude
  less than a threshold value to be result of noise
• Reconstruct the signal after removing the noise
  coefficients

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Many more

• Edge finding in images - defogging
• Concealed object detection by fusion of images
• Cloud detection and tracking?