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