Gravitational Potential
The gravitational potential due to a density distribution $\rho(\mathbf{r}’)$ is
\[U(\mathbf{r}) = -G \int_V \frac{\rho(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|}\, dV'\]The vertical gravity component measured at the surface is $g_z = -\partial U/\partial z$. In the absence of mass, $U$ satisfies Laplace’s equation $\nabla^2 U = 0$; in regions containing mass it satisfies Poisson’s equation $\nabla^2 U = 4\pi G \rho$.
Reductions and Anomalies
Raw measurements are corrected through a sequence of reductions:
| Correction | Formula | Purpose |
|---|---|---|
| Free-air | $+0.3086\,\Delta h$ mGal/m | Elevation above datum |
| Bouguer slab | $-2\pi G \rho_0 h$ | Mass of rock between datum and station |
| Terrain | Numerical integration | Irregular topography |
| Latitude | $g_0(\phi)$ from GRS80 | Reference spheroid |
The Bouguer anomaly $\Delta g_B$ isolates density contrasts relative to the assumed background density $\rho_0 \approx 2670$ kg m$^{-3}$.
Density Inversion
Given $N$ surface observations $\mathbf{d}$ and $M$ sub-surface cells with unknown density contrasts $\mathbf{m}$, the forward problem is linear: $\mathbf{d} = \mathbf{G}\mathbf{m}$. Because $N \ll M$ the system is underdetermined, and a regularised least-squares solution is sought:
\[\hat{\mathbf{m}} = \arg\min_{\mathbf{m}} \|\mathbf{G}\mathbf{m} - \mathbf{d}\|^2 + \lambda \|\mathbf{W}\mathbf{m}\|^2\]where $\mathbf{W}$ encodes smoothness or depth-weighting constraints and $\lambda$ controls regularisation strength.