Question 10 of 10: Non-Uniqueness in Potential-Field Interpretation

Question 10: Non-Uniqueness in Potential-Field Interpretation (Choose 6 of 10 – 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.

The concept

Recovering a subsurface density or susceptibility distribution from surface potential-field measurements is an inverse problem, and it is fundamentally non-unique: infinitely many different subsurface source distributions can produce EXACTLY the same field measured at the surface. This is a mathematical property of potential theory itself — the forward operator (integrating a 3-D source distribution via Green's function to a field on a 2-D external surface) is information-losing, so the mapping from sources to surface field cannot be inverted one-to-one, no matter how good or dense the data are.

A formula showing how it arises — the buried sphere

The vertical gravity anomaly of a buried sphere of radius R, density contrast Δρ, centred at depth z, measured at horizontal offset x, is:

$$g_z(x) = \dfrac{4}{3}\pi G\,\Delta\rho\, R^{3}\,\dfrac{z}{\left(x^{2}+z^{2}\right)^{3/2}}$$

Crucially, Δρ and R appear in this formula only through the single product ΔρR³ (the sphere's equivalent point mass), never separately. Consequently, for a fixed depth z, any two spheres whose ΔρR³ products are equal produce an IDENTICAL surface anomaly g_z(x) at every offset x — a large, low-density-contrast sphere and a small, high-density-contrast sphere at the same depth are indistinguishable from gravity data alone, however precisely that data is measured. (The same logic applies to a magnetic dipole's anomaly, which depends on susceptibility and volume only through their product, the dipole moment.)

Example in geophysical interpretation

In mineral resource estimation, a compact, high-grade (dense) ore body and a larger, lower-grade (less dense) body at the same depth can produce statistically indistinguishable surface gravity anomalies, so a residual anomaly alone cannot, by itself, determine whether a target is small-and-rich or large-and-lean — a genuine, practically important ambiguity that gravity data alone cannot resolve.

Geophysical and geological ways to address non-uniqueness

Because the ambiguity is intrinsic to potential-field data, resolving it requires information OUTSIDE the gravity (or magnetic) data set itself: independent geophysical methods governed by different physics (magnetics, electromagnetics/resistivity, seismic reflection/refraction for geometric constraints) over-determine the problem and break trade-offs that gravity alone cannot; direct borehole or drilling control provides ground truth on depth, size and density/susceptibility at discrete points, anchoring the otherwise-unconstrained model; geological constraints — known outcrop geometry, a regional geological/structural model, or realistic bounds on density/susceptibility contrast from rock-property databases — restrict the plausible model space to geologically sensible solutions; and formal inversion with regularization (minimum-structure or "Occam's" inversion) selects the smoothest/simplest model consistent with the data among the non-unique family, with the explicit understanding that this yields A plausible model, not THE unique answer, and that acquiring more gravity stations alone does not, by itself, resolve this class of ambiguity.