Math 2.0

Geophysics Gravity field of the Earth Problem

Geodetic boundary value problem

Builds on Gravity anomalyHarmonic functionSpherical harmonicsRobin boundary condition

Problem

The geodetic boundary value problem asks for the gravity potential outside the Earth from gravity data given on its boundary. Its classical linear form, the spherical Stokes problem, is the following. Let σ\sigma be the sphere of radius R>0R > 0 centred at 00 in ℝ3\R^3, and let Δg ⁣:σ→ℝ\Delta g \colon \sigma \to \R be a given continuous function, the gravity anomaly. Find a function TT, harmonic in {x:∥x∥>R}\set{x : \norm{x} > R} and continuously differentiable up to σ\sigma, with T(x)=O(1/∥x∥)T(x) = O(1/\norm{x}) as ∥x∥→∞\norm{x} \to \infty, such that ∂T∂r+2R T=−Δgon σ,\frac{\partial T}{\partial r} + \frac{2}{R}\, T = -\Delta g \quad \text{on } \sigma, where r=∥x∥r = \norm{x}.

Remarks

Expand in spherical harmonics: the degree nn part of the solution has the form (R/r)n+1Tn(R/r)^{n+1} T_n, and the boundary condition reads (n−1)Tn=R Δgn(n-1) T_n = R\,\Delta g_n. So a solution exists only if Δg\Delta g has no degree 11 component, and then the degree 11 part of TT is arbitrary. Once the degree 11 part is fixed, for example by putting the origin at the centre of mass, the solution is unique. It is given by Stokes' formula together with Bruns formula. The boundary condition is of Robin type, and the free degree 11 part shows that, unlike the exterior Dirichlet problem, it is not uniquely solvable without an extra condition. Another form is the fixed gravimetric problem. There the boundary is known from satellite positioning, and the condition becomes ∂T/∂r=−δg\partial T/\partial r = -\delta g with the gravity disturbance δg\delta g. A third form is the free boundary problem of Molodensky.

Reading path

To understand this page from the ground up, read these 16 pages in order. The foundations come first.

Show the path as a graph
Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptGravity anomaly, concept, no evidence levelGravity anomalyconceptRobin boundary condition, concept, no evidence levelRobin boundary condit…conceptSpherical harmonics, concept, no evidence levelSpherical harmonicsconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptGradient, concept, no evidence levelGradientconceptDivergence, concept, no evidence levelDivergenceconceptLaplacian, concept, no evidence levelLaplacianconceptHarmonic function, concept, no evidence levelHarmonic functionconceptGeodetic boundary value problem, problem, no evidence levelGeodetic boundary val…problem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Euclidean space object
  2. Function concept
  3. Gravity anomaly concept
  4. Robin boundary condition concept
  5. Spherical harmonics concept
  6. Total derivative concept
  7. Function of several variables concept
  8. Vector-valued function concept
  9. Differentiable function of several variables concept
  10. Partial derivative concept
  11. Vector field concept
  12. Gradient concept
  13. Divergence concept
  14. Laplacian concept
  15. Harmonic function concept
  16. Geodetic boundary value problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Physical_geodesy
About this page
Tag1F4S3G1R6J
URIm2://1F4S3G1R6J
Kindproblem
WritingW0 stub
MSC86A30, 31B20, 35J25
StatusThis copy of the site has no build data. Run m2 status 1F4S3G1R6J.
AboutGeopotential
UsesGravity anomaly, Harmonic function, Spherical harmonics, Robin boundary condition
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.