Question 7 of 8: Gravity Field and Geoid — Stokes’s Formula

Question 7: Gravity Field and Geoid — Stokes’s Formula (20 marks)

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.

Given. Stokes’s integral for the geoid undulation, \(N=\dfrac{R}{4\pi\gamma}\displaystyle\iint_\sigma S(\psi)\,\Delta g\,\mathrm{d}\sigma\) (the standard form is given here).

Find. (a) the derivation approximation and the order of magnitude of its error in \(N\); (b) the meaning of each variable, which are observable and whether directly, and why \(S(\psi)\) is a “kernel”; (c) the additional fundamental data set implied by the formula.

(a) Approximation and its error. Stokes’s formula rests on the spherical approximation: the boundary surface (the geoid) is treated as a sphere of radius \(R\) rather than an ellipsoid, and normal gravity is taken as a constant mean value \(\gamma\). This linearizes the geodetic boundary-value problem and lets the solution be written as a single convolution of gravity anomalies with the Stokes kernel over the whole sphere. It also assumes there are no masses outside the geoid (topographic and atmospheric masses must be removed/reduced beforehand). Because the flattening \(f\approx1/298\) is neglected, the relative error is of order \(f\), which propagates into \(N\) as an error of order \(f\cdot N\approx 3\times10^{-3}N\) — i.e. at the decimetre level (about 0.3 m for the largest undulations of \(\approx100\) m); it is reduced to the centimetre level in modern computations by applying ellipsoidal and higher-order corrections.

(b) Variables, observables and the kernel. In \(N=\dfrac{R}{4\pi\gamma}\iint_\sigma S(\psi)\,\Delta g\,\mathrm{d}\sigma\): \(N\) is the geoid undulation (geoid–ellipsoid separation, the sought quantity); \(R\) is the mean Earth radius; \(\gamma\) is mean normal gravity on the ellipsoid; \(\Delta g\) is the gravity anomaly at the running (integration) point; \(S(\psi)\) is the Stokes function, depending only on the spherical distance \(\psi\) between the computation point and the running point; and \(\mathrm{d}\sigma\) is the surface element of the unit sphere, with the integral taken over the entire sphere \(\sigma\). The single observable is the gravity anomaly \(\Delta g\). It is not observed directly: it is formed as \(\Delta g=g-\gamma\) from measured gravity \(g\) (gravimeter) reduced to the geoid (free-air etc.) minus the computed normal gravity \(\gamma\) — so it needs both a gravity measurement and a known position/height. We call \(S(\psi)\) a kernel function because it is the weighting function of an integral transform: it multiplies the input field \(\Delta g\) inside the integral and distributes each anomaly’s contribution to \(N\) according to its angular distance \(\psi\) — exactly the role of the kernel in a convolution/integral operator.

(c) The implicit additional data set. The integral extends over the whole Earth and requires the gravity anomaly everywhere — so the fundamental implicit requirement is a global set of gravity anomalies \(\Delta g\) (a global gravity/anomaly field), not just local data. In practice this global coverage is supplied by a global geopotential (spherical-harmonic) model derived from satellite gravimetry (e.g. GRACE/GOCE), which furnishes the long-wavelength part of \(\Delta g\) that no local survey can, while terrestrial/airborne gravity fills the short wavelengths (the remove–compute–restore technique). Equivalently, the reference ellipsoid’s normal-gravity field must be known to form \(\Delta g\), and — because Stokes’s formula assumes no masses above the geoid — a digital terrain (elevation and density) model plus the heights of the gravity stations are needed to reduce the observed gravity to the geoid.