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

Question 10: Non-Uniqueness in Potential Field Methods (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 of non-uniqueness

A potential field observed only on a bounding surface (the Earth's surface, or a survey plane above it) does NOT uniquely determine the 3D distribution of the property (mass or magnetization) generating it: infinitely many different internal distributions can reproduce EXACTLY the same external field. This is a fundamental result of potential theory (related to Gauss's/Skeels' non-uniqueness theorem), not a limitation of any particular survey or algorithm — more data of the SAME kind, collected more densely or for longer, cannot by itself remove this ambiguity.

Mathematical demonstration: the spherical-layer equivalent-source theorem

The clearest demonstration is Newton's classical equivalence theorem for a spherically symmetric mass distribution. For an exterior observation point at distance r from the centre of ANY spherically symmetric distribution of total mass M (a point mass, a uniform sphere, or a thin concentric spherical layer), the gravitational potential and field are identical:

$$U(r) = \frac{GM}{r}, \qquad g(r) = \frac{GM}{r^{2}}$$

depending ONLY on the total enclosed mass M and the distance r — not on how that mass is distributed inside the sphere. A uniform sphere of mass M, a thin concentric spherical layer of the same mass M and any radius smaller than r, and a point mass M at the centre are therefore completely indistinguishable from any exterior gravity measurement.

Example of non-uniqueness arising in interpretation

A compact, dense, SHALLOW ore body and a larger, less dense, DEEPER body can be constructed to produce an identical surface gravity anomaly, because the anomaly's shape and amplitude trade off between the body's density contrast, volume and depth in a way that a single gravity profile alone cannot separate; a geologist forward-modelling only the gravity data has no mathematical way to prefer one geometry over the other from the gravity data alone. The same issue arises for magnetics: an inversion given only the surface magnetic anomaly, without further constraint, will typically converge to a smooth, minimum-structure model that fits the data but need not resemble the true, more localized geological source.

Ways to address non-uniqueness

Geophysical: combine potential-field data with an INDEPENDENT geophysical method governed by different physics (seismic reflection/refraction for depth to a boundary, electrical/EM methods for conductivity contrasts, radiometrics for near-surface lithology) — because each method has a different, largely uncorrelated non-uniqueness "blind spot," combining methods constrains the solution space far more than adding more of the same type of potential-field data ever could. Constrained/regularized inversion (imposing smoothness or bound constraints, or anchoring to a reference model) selects the most PLAUSIBLE member of the non-unique family rather than an arbitrary one, without ever claiming the result is the unique true answer. Geological: drill-hole intersections and downhole logs provide direct, local ground truth on depth, geometry and physical properties at specific points, which can be locked into the model as hard constraints; surface geological mapping (contacts, structural trends, known stratigraphy) constrains plausible geometries; and independently known physical-property ranges for the rock types present (from hand-sample or downhole measurements, Question 1) bound the density or susceptibility contrast an acceptable model may use, narrowing the family of mathematically valid models to those that are also geologically credible.