Question 6 of 10: The Fourier Transform in Geophysical Data Processing

Question 6: The Fourier Transform in Geophysical Data Processing (16.66% of paper)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

What it is. The Fourier transform decomposes a signal recorded in space or time into a sum (or continuous integral) of sinusoids of different frequencies (or spatial wavenumbers), each with its own amplitude and phase; it provides an exactly equivalent, invertible representation of the same information in the frequency/wavenumber domain rather than the original space/time domain.

Uses in processing. Filtering (low-pass, high-pass, band-pass) removes unwanted frequency/wavenumber content — e.g. separating a broad regional gravity trend (low wavenumber) from a narrower residual anomaly (higher wavenumber) by simple multiplication in the frequency domain, an operation that is awkward to do directly in the space domain. Deconvolution removes the effect of a known or estimated source wavelet or instrument response by dividing in the frequency domain, sharpening seismic reflections. Potential-field transforms — upward/downward continuation, vertical/horizontal derivatives, reduction to the pole, the analytic signal — are all most efficiently and exactly computed as multiplications by known wavenumber-domain filter functions, then transformed back. Depth estimation from the slope of a field's power spectrum (log power vs. wavenumber) gives a fast, ensemble-average estimate of the mean depth to an assumed statistical population of sources, widely used as a first-pass regional depth-to-basement or depth-to-magnetic-source estimate.

Pitfalls. Aliasing: a signal sampled too coarsely (violating the Nyquist criterion, sample interval greater than half the shortest wavelength present) has its high-wavenumber content folded back and disguised as spurious LOW-wavenumber signal, which can be mistaken for a real, long-wavelength geological feature. Edge effects / spectral leakage: the discrete Fourier transform implicitly assumes the finite record is one period of an infinitely repeating signal, so an abrupt discontinuity between the record's start and end (the true signal rarely being genuinely periodic) generates spurious high-frequency energy (Gibbs-type ringing) unless the data are tapered/windowed and zero-padded before transforming. Non-stationarity: real geological/geophysical signals often have frequency content that changes with position or time (e.g. a survey crossing from a smoothly layered basin into faulted, structurally complex terrain), but a single global Fourier transform gives one frequency description averaged over the whole record and cannot show WHERE a given frequency occurs — a short-window or wavelet transform is needed instead to preserve that spatial/temporal localization. Statistical instability of spectral depth estimates: the power-spectrum depth method assumes an idealized ensemble of many, randomly distributed sources of similar depth; applied to a short window with only a few actual sources, the estimate can be highly unstable and should not be over-interpreted as a precise single-source depth.