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Outline 2-D Fourier transforms The sampling theorem Sampling Aliasing

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1 Outline 2-D Fourier transforms The sampling theorem Sampling Aliasing
Time/Spatial and frequency sampling Nyquist frequency Rules for survey geometry related to target depth Aliasing Effects of finite sampling on frequency and phase Filtering: Removal of the drift (trend) Tapering DC removal Fast Fourier transform (FFT)

2 2-D Fourier transform Maps a function of a 2-D vector x (coordinates) into a function of 2-D wavenumber (spatial frequency) vector k: Inverse transform:

3 Sampling theorem If we know that the Fourier transform of function f(x) equals zero above certain frequency, then f(x) is given by a finite inverse Fourier series: x  [0,L] This is a polynomial function with respect to : Z-transform To determine the 2N coefficients an, we need to specify this function at 2N sampling points zi N = 2N/2 is the Nyquist frequency

4 Summary of sampling If we sample a function f(x) at N points at increments Dx on an interval of length L = NDx, then: The function f(x) is understood as periodic with period L The sampling frequency is The largest recoverable (Nyquist) frequency is The spacing between Fourier frequencies is: This is also the lowest available frequency (apart from f = 0) There exist N Fourier frequencies They can be counted from either j = 0 to N-1 or j = -N/2 to N/2 - 1 NOTE: all of these quantities are properties of the sampling, and not of any particular signal f(x)!

5 Relation of survey geometry to characteristic target depth
Estimated depth to the source of the anomaly determines the design of the survey Dominant wavelength observed on the surface: Therefore, the limits on Dx and L: In practice, the stronger limits are honoured:

6 Edge Effects in Discrete Fourier Transform
In a DFT, the transform time interval is finite, and the signal can be viewed as extrapolated periodically in time or space Note the edge effect of this extrapolation, resulting from using a frequency not equal a multiple of 1/DT in the lecture: This signal has a broad frequency band (although a single sin() function used within the interval)

7 Data Drift removal and tapering helps eliminating the fictitious discontinuities at the edges Drift removed Tapered

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