Laplace’s Equation and Harmonic Functions

Outside any source distribution the gravitational potential $\phi$ satisfies

\[\nabla^2 \phi = \frac{\partial^2\phi}{\partial x^2} + \frac{\partial^2\phi}{\partial y^2} + \frac{\partial^2\phi}{\partial z^2} = 0\]

Solutions are called harmonic functions. Key properties include the maximum principle (no interior extrema) and the mean-value theorem (the value at any point equals the average over a surrounding sphere). The fundamental solution in 3D is the Newton kernel $1/

\mathbf{r} - \mathbf{r}’

$.

Spherical Harmonic Expansion

On a sphere of radius $R$, any harmonic function exterior to the sphere can be written

\[\phi(r, \theta, \lambda) = \frac{GM}{R}\sum_{n=0}^{\infty}\left(\frac{R}{r}\right)^{n+1} \sum_{m=0}^{n} \bigl[C_{nm}\cos m\lambda + S_{nm}\sin m\lambda\bigr] \bar{P}_{nm}(\cos\theta)\]

where $\bar{P}{nm}$ are fully normalised associated Legendre functions and $C{nm}, S_{nm}$ are Stokes coefficients. Global gravity field models (e.g. EGM2008, expanded to degree 2190) are expressed in this form.

Upward and Downward Continuation

In the Fourier domain, upward continuation by height $\Delta z$ multiplies each spatial frequency component by the factor $e^{-

k

\Delta z}$, where $

k

= \sqrt{k_x^2 + k_y^2}$ is the radial wavenumber. This is a stable, low-pass operation. Downward continuation applies $e^{+

k

\Delta z}$ and is inherently unstable — high-frequency noise is amplified exponentially — requiring regularisation or spectral tapering. These operations transform data between different observation levels without re-acquiring surveys.