Presentation on theme: "Outline 2-D Fourier transforms The sampling theorem Sampling Aliasing"— Presentation transcript:
1 Outline 2-D Fourier transforms The sampling theorem Sampling Aliasing Time/Spatial and frequency samplingNyquist frequencyRules for survey geometry related to target depthAliasingEffects of finite sampling on frequency and phaseFiltering:Removal of the drift (trend)TaperingDC removalFast Fourier transform (FFT)
2 2-D Fourier transformMaps a function of a 2-D vector x (coordinates) into a function of 2-D wavenumber (spatial frequency) vector k:Inverse transform:
3 Sampling theoremIf 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-transformTo determine the 2N coefficients an, we need to specify this function at 2N sampling points ziN = 2N/2 is the Nyquist frequency
4 Summary of samplingIf 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 LThe sampling frequency isThe largest recoverable (Nyquist) frequency isThe spacing between Fourier frequencies is:This is also the lowest available frequency (apart from f = 0)There exist N Fourier frequenciesThey can be counted from either j = 0 to N-1 or j = -N/2 to N/2 - 1NOTE: 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 surveyDominant 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 spaceNote 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 DataDrift removal and tapering helps eliminating the fictitious discontinuities at the edgesDrift removedTapered