Questions: Gravity Potential Theory and Earth's Gravitational Field

Question 1

Which equation governs the gravitational potential U in a region that contains matter with density ρ?

• A. ∇²U = 0 (Laplace's equation)
• B. ∇²U = −4πGρ (Poisson's equation)
• C. g = −∇²U
• D. ∇U = −4πGρ

Answer: (B)

Poisson's equation ∇²U = −4πGρ applies wherever mass is present (ρ ≠ 0). Laplace's equation ∇²U = 0 is the special case when ρ = 0 — i.e., in mass-free regions such as the air above the ground. The gravity field vector is recovered from the potential as g = −∇U (the negative gradient), not the Laplacian.

Question 2

A gravity anomaly observed at Earth's surface can arise mainly from density variations in the upper crust, not from deeper mantle structure.

• T. True
• F. False

Answer: False

Gravity anomalies integrate the effect of all density contrasts along the entire vertical column beneath the measurement point. Deep density variations — such as thickened oceanic crust, subducting slabs, or mantle plumes — can produce measurable gravity anomalies at the surface. The challenge of gravity interpretation is precisely that signals from different depths superimpose, making it an underdetermined inverse problem.

Question 3

Explain the conceptual difference between the forward problem and the inverse problem in gravity interpretation.

Model answer: The forward problem predicts the surface gravity field produced by a given (assumed) density distribution in the subsurface. The inverse problem works backwards: given observed surface gravity data, it seeks to recover the subsurface density distribution that could explain those observations. The inverse problem is non-unique — many different density models can fit the same gravity data — so additional constraints (e.g., seismic or borehole data) are needed.

Forward modeling is deterministic and always has a unique solution: a specified density model produces one and only one gravity field. The inverse is ill-posed because gravity measurements lose information about depth — a shallow low-density body and a deeper high-density body can produce the same surface anomaly. This non-uniqueness is a fundamental challenge in all potential-field geophysics.