04-Geol-B10 · May 2016
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
EGBC National Exam — Geological Engineering, 04-Geol-B10-1 Gravity and Magnetic Fields, 2016-May. Closed book; no calculator permitted. All ten questions require an answer in essay format, with diagrams used wherever appropriate. The exam instructs "choose six (6) of the following ten (10) questions, the first six as they appear in the answer book will be marked, each of equal value".
Reference texts: Telford, Geldart & Sheriff, Applied Geophysics, 2nd ed. (physical properties ch.2 & 5; gravimeters and gravity reduction ch.2; magnetometers and magnetic surveying ch.4–5; forward/inverse modelling throughout); Kearey, Brooks & Hill, An Introduction to Geophysical Exploration, 3rd ed. (survey design, data processing and interpretation workflow ch.6 & 7); Blakely, Potential Theory in Gravity and Magnetic Applications (potential-field theory, Fourier-domain filters, reduction-to-pole ch.2, 9 & 12).
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.
The Fourier transform decomposes a spatial signal — here, a gridded gravity or magnetic anomaly f(x,y) — into a sum (integral) of sinusoids of different spatial wavenumbers k, each with its own amplitude and phase:
$$F(k_x,k_y) = \iint f(x,y)\, e^{-i2\pi(k_x x + k_y y)}\, dx\, dy$$
Every wavenumber component of F corresponds to a particular spatial wavelength (λ = 1/|k|) of variation in the original grid; short wavelengths (high k) represent sharp, local, shallow-source detail while long wavelengths (low k) represent smooth, broad, deep or regional structure.
Once the data are in the wavenumber domain, an enormous range of otherwise difficult spatial operations become simple multiplications: differentiation (vertical/horizontal derivatives, Question 7) becomes multiplying by a power of ik; upward/downward continuation becomes multiplying by e∓2π|k|Δz; reduction-to-pole (Question 6) becomes multiplying by a direction-dependent filter; and general low-pass, high-pass or band-pass noise filtering becomes simple selective attenuation of chosen wavenumber ranges. After filtering in the wavenumber domain, an inverse Fourier transform returns the processed grid to the spatial domain for mapping. Because gridded data are processed via the Fast Fourier Transform (FFT) algorithm, these operations are also computationally efficient even on large regional grids.
The Fourier transform mathematically assumes the input data are periodic and of infinite extent; a finite survey grid is neither, so the implicit wrap-around at the grid edges introduces spurious high-wavenumber content and ringing artefacts (Gibbs phenomenon), which is why practitioners pad and taper (window) a grid's edges before transforming. Real station spacing imposes a Nyquist wavenumber limit (the shortest wavelength that can be represented is twice the station spacing); any true signal or noise with a shorter wavelength than this aliases — it is misrepresented as a longer, spurious wavelength — so survey design (Question 3) must anticipate the shortest wavelength of interest. Gridding itself, performed before the transform, is an interpolation that smooths and can distort the true wavenumber content, especially where station spacing is irregular. The 2D Fourier transform also implicitly assumes the data are statistically uniform (stationary) across the whole grid, which is rarely exactly true of real geology, and any filter that amplifies high wavenumbers (derivatives, downward continuation) amplifies noise right along with genuine signal, which is the single most common practical pitfall in Fourier-domain processing of field data.
