Question 7 of 10: High-Pass vs. Low-Pass Filtering of Geophysical Data

Question 7: High-Pass vs. Low-Pass Filtering of Geophysical Data (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.

Low-pass filtering passes long-wavelength (low spatial-frequency) signal and attenuates short-wavelength (high spatial-frequency) signal, so it isolates broad, regional trends and smooths out local, high-frequency noise or shallow near-surface effects. Examples: a simple moving-average or Gaussian spatial smoothing filter; upward continuation, a physically-based low-pass filter that mathematically projects a potential-field (gravity/magnetic) dataset upward to a higher observation level, $\hat f(z+\Delta z)=\hat f(z)\,e^{-k\Delta z}$ in the wavenumber domain, which attenuates short-wavelength (shallow-source) anomalies much more strongly than long-wavelength (deep-source) ones, effectively isolating the regional field; a Butterworth low-pass filter with a chosen cutoff wavenumber. Used for: isolating a REGIONAL background trend (e.g. a deep crustal density/susceptibility source) for subtraction, or simply de-noising a data grid before interpretation.

High-pass filtering passes short-wavelength signal and attenuates long-wavelength signal, so it isolates local, near-surface anomalies by removing the broad regional background. Examples: the first or second vertical derivative, which acts as a high-pass filter because differentiation in the wavenumber domain multiplies the spectrum by $|k|$ (first derivative) or $k^2$ (second), strongly boosting short-wavelength (shallow) content relative to long-wavelength (deep) content — the standard "residual" or edge-sharpening filter in potential-field interpretation; downward continuation (mathematically the inverse of upward continuation, $\hat f(z-\Delta z)=\hat f(z)\,e^{+k\Delta z}$), which sharpens shallow sources even further but is numerically unstable at high wavenumbers; and regional-residual separation by fitting and subtracting a low-order polynomial trend surface (the subtracted residual is functionally a high-pass result). Used for: isolating a shallow, local exploration target (an ore body, a cavity, a buried structure) from the deep regional field that would otherwise mask it, and for sharpening edges/contacts for structural (fault, lithological boundary) interpretation.

Artifacts and pitfalls. Vertical-derivative (high-pass) filters strongly amplify high-wavenumber noise (since the multiplier $|k|$ or $k^2$ grows without bound at short wavelength), so raw, un-smoothed data must be low-pass filtered first or the derivative result is dominated by amplified noise rather than genuine shallow signal. Sharp-cutoff filters (an idealized "brick-wall" low- or high-pass in the wavenumber domain) introduce ringing/Gibbs-phenomenon oscillations in the spatial domain around any strong anomaly edge, which can be mistaken for a real adjacent feature; a smoother taper (e.g. cosine-tapered cutoff) avoids this. Upward continuation loses genuinely useful near-surface information if continued too far, and downward continuation amplifies noise catastrophically if pushed too close to the source depth. Polynomial-trend regional-residual separation is inherently subjective — too low a polynomial order leaves real regional signal in the "residual," while too high an order removes genuine local target signal along with the regional trend, so the choice of order must be justified against independent geological knowledge of the expected regional wavelength.