Question

If \(g_{FA}\) and \(g_{BA}\) respectively denote the free-air and Bouguer gravity anomalies over a fully compensated mountain, then which one of the following is CORRECT in case of Airy-isostacy?

• A. \(g_{FA} = 0,\ g_{BA} > 0\)
• B. \(g_{FA} > 0,\ g_{BA} = 0\)
• C. \(g_{FA} > 0,\ g_{BA} < 0\)
• D. \(g_{FA} = 0,\ g_{BA} = 0\)

Hint:

Free-air only corrects for elevation (topographic mass still present, so slightly positive); Bouguer removes that mass and exposes the low-density root (a mass deficit, so negative).

Answer 1

The Correct Option is C

Under Airy's model of isostasy, a mountain is supported by a low-density crustal root that displaces denser mantle material, so that the total mass in every vertical column - from the mountain top down through the compensation depth - is the same everywhere ('fully compensated').

Step 1: What the free-air correction does.
The free-air correction only accounts for the change of gravity with elevation (about 0.3086 mGal/m); it removes the effect of being at a different height, but it does NOT remove the attraction of the topographic mass itself sitting between the station and sea level. Because that mass is physically present and pulling on the gravimeter, the free-air anomaly over a mountain retains a small positive signature, \(g_{FA} > 0\), even though the system is isostatically compensated overall.

Step 2: What the Bouguer correction adds.
The Bouguer correction goes a step further and mathematically strips away the gravitational effect of the topographic mass itself (approximated as an infinite slab), as if that rock were not there. Once that mass is removed, all that is left in the anomaly is the effect of the low-density compensating root at depth. A root of density lower than the surrounding mantle is a mass deficit, and a mass deficit produces a negative gravity anomaly, so \(g_{BA} < 0\) over the mountain.

Step 3: Combine.
Hence, for a fully (Airy) compensated mountain, \(g_{FA} > 0\) and \(g_{BA} < 0\) - option \(\boxed{(C)}\), the classic diagnostic used to confirm isostatic compensation from real gravity surveys (near-zero-to-small-positive free-air anomaly alongside a strongly negative Bouguer anomaly that deepens with elevation).