Control of the accuracy of the Molodensky's integral equation for the gravity anomalies and disturbances on the Earth's models

The problem of determining the anomalous potential T on the earth's surface can be solved on the basis of various initial available data: gravity anomalies Δg and gravity disturbances δg, their vertical derivatives ∂(Δg)/∂H, ∂(δg)/∂H, gravity gradient anomalies Δ(∂g/∂H) etc. Existing methods of such BVP solution use the integral kernels, elaborated for the sphere and ellipsoid. The attempts to determine the real geoid are closely related to the direct problems of the potential theory, when the mass distribution is assumed to be approximately known (in Molodensky's theory the earth's crust density is used in topographic reductions only for better anomalies interpolation).

Using of the two tipes of related gravity data could be considered as a control, e.g., the anomalous potential T from the gravity anomalies Δg can be used to obtain the gravity disturbances δg, from which we must also get the same anomalous potential T. For the real Earth's surface more flexible is the method of integral equations.

 

(The prime sign indicates a point on the telluroid.)

Molodensky's integral equation for the simple layer density (distributed on the Earth's surface) using the gravity disturbances (1) and gravity anomalies (2) is known, but is usually solved indirectly with an introduction of the small parameter (the Molodensky's parameter k or/and ellipsoid eccentricity e), that lead to the series solution with the well-known integrals. Being the Fredholm equation, the Molodensky's integral equation itself can be solved directly by successive approximations in the ellipsoidal coordinate systems as well as in the spherical one. The integration procedure is probably longer, but any step is of the same type. Then the anomalous potential can be calculated by integration in the form (3).

Figure 1. Simple layers distributed with densities φ on the Earth’s surface S (green). Auxiliary simple layer density χ is distributed on the mean Earth’s sphere Ω  with radius R. In general case, the normal n to the surface, inclination angle α and the radius-vector ρ are slightly different in the two cases. E - reference ellipsoid (blue), the telluroid Σ (red), g - plumb-line.

Some real estimates are possible on the surface and gravity field models. In this study we use the Earth's model in the form of mascons for the surface and gravity field, see Fig. 2. We know all the elements of the anomalous field, the precise coordinates of the points with data and so we can estimate the real theoretical accuracy of the formulas and the number of iterations.

Figure 2. The scheme of the mass forming the anomalous field

In case of gravity anomalies the integration procedure can be considered as an integration over the successively refined  boundary surface. It is enough to find the density distribution of a simple layer on a smoothed surface constructed from the heights of points in the form of the sum of the normal height (from leveling) and the height anomaly from the Stokes approximation.