Question 2 of 8: Gravity Terrain Corrections, Nettleton's Method, Magnetic Anomaly Comparison

Question 2: Gravity Terrain Corrections, Nettleton's Method, Magnetic Anomaly Comparison (equal value)

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.

(i) When is a terrain correction required? A terrain (topographic) correction is required whenever the ground surface near a gravity station is NOT flat — i.e. whenever hills rise above the station or valleys cut below it within a radius of a few kilometres (conventionally out to Hammer zone M, about 21.9 km, though the correction is dominated by the nearest zones). The Bouguer correction, $\Delta g_B=0.0419\rho$ mGal/m, models the mass between the station and the reference datum as an INFINITE HORIZONTAL SLAB of thickness equal to the station's elevation; it removes the attraction of that idealized slab but makes no allowance for real topography. Two distinct real-world departures from the infinite-slab model both act to REDUCE observed gravity below what the flat-slab correction accounts for, and both are corrected in the SAME (positive) sense: a hill standing above the station has mass located above the station elevation that pulls UP on the gravimeter (opposing gravity, since gravity is measured downward) — mass the slab model never counted, so it must be added back; a valley cut below the surrounding terrain represents a mass DEFICIT that the slab model assumed was present (rock that would have pulled down is missing), so the missing downward pull must also be compensated by an addition. The terrain correction $\Delta g_T$ is therefore always ADDED (never subtracted), $g_B=g_{obs}-g_t+(\Delta g_L+\Delta g_{FA}-\Delta g_B+\Delta g_T)$ per the formula sheet, and free-air plus Bouguer corrections alone are insufficient precisely because neither one has any information about the actual 3-D shape of the terrain — they only know the station's elevation, not what is beside it.

(ii) Nettleton's method for near-surface density. Nettleton's method estimates the correct Bouguer reduction density $\rho$ directly from the survey's own data, without a rock-sample density measurement, by exploiting the fact that an INCORRECT reduction density leaves a spurious correlation between the Bouguer anomaly and topography. The procedure: survey a gravity profile across an isolated topographic feature (classically a hill or ridge with no significant subsurface density anomaly of its own); reduce the raw readings to a Bouguer anomaly profile using a RANGE of trial densities (e.g. 1.8, 2.0, 2.2, ... 2.8 g/cm³); plot each resulting profile against the topographic elevation profile. If the trial density is too LOW, the Bouguer correction under-corrects for the topographic mass and the reduced profile still curves UPWARD in sympathy with the hill (positive correlation with elevation); if the trial density is too HIGH, the correction over-corrects and the profile curves DOWNWARD under the hill (anti-correlated with elevation, an over-corrected "trough" appearing where the hill is highest). The CORRECT near-surface density is the one for which the reduced Bouguer profile is FLATTEST — shows no residual correlation, positive or negative, with the topography — since only the true rock density removes the topographic mass exactly.

(iii) Comparing two magnetic anomalies of the same source shape/size/direction of magnetization.

(a) Shallower source. The narrow, sharp anomaly on the LEFT of the printed profile (A in the sketch) comes from the shallower source. For sources of the same shape, size and magnetization direction, the horizontal scale of an anomaly (its half-width or peak-to-trough distance) is proportional to the depth of the source: a compact source at depth $z$ produces a field that changes over lateral distances of order $z$. A shallow body therefore gives a short-wavelength anomaly and a deeper body a broad one (B, on the right). On the printed profile B's peak-to-trough distance is several times A's, so B's source is several times deeper.

(b) Stronger magnetization. The broad anomaly B comes from the more strongly magnetized source. On the printed profile the two anomalies have about the SAME amplitude. For a compact, dipole-like body the anomaly amplitude scales as $M/z^3$ (the formula sheet's $B_r,B_\theta\propto M/r^3$). If both bodies had the same magnetization, the deeper body's anomaly would be smaller by a factor of about $(z_B/z_A)^3$; since it is not smaller, B's magnetization must be larger by roughly that factor. With a depth ratio of about 6 (the ratio used for the sketch), $M_B/M_A\approx6^3\approx216$; even for an elongate (2-D) body, whose amplitude falls as $1/z^2$, the ratio would be about 36. So equal amplitude at greater depth means much stronger magnetization, and the sharp shallow anomaly A comes from the WEAKER source.

(c) Estimating depth from the profile alone. The half-width rule from the formula sheet, $z\le1.3\,x_{1/2}$ (where $x_{1/2}$ is the horizontal distance from the anomaly's peak to the point where it has fallen to half its peak amplitude), gives a quick upper-bound depth estimate for each anomaly directly from the profile with no other information required: measure $x_{1/2}$ for Anomaly A and for Anomaly B separately (B's larger half-width gives the larger depth estimate, consistent with part (a); for a dipolar anomaly with a positive and a negative lobe, half the peak-to-trough distance can be used in place of $x_{1/2}$), and multiply each by 1.3. A more rigorous alternative is Peters' half-slope method (locate the two points of maximum slope on each flank of the anomaly and measure their horizontal separation, which is proportional to depth for a given source geometry) or a full curve-matching inversion against master curves for the assumed source shape (sphere, cylinder, dyke); the half-width rule is the fastest and is what the printed formula sheet supports directly.