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-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:

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

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)!

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, stronger limits are honoured:

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)

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