Hardyʹs multiquadric-biharmonic method for gravity field predictions II

The biharmonic equivalent of Poissonʹs equation is defined here as follows. The biharmonic operator on potential at a point density is double the right-hand side of Poissonʹs equation for the same point density. The mathematical proof of this and other matters are provided in the body of this presentation. This relationship has brought about the replacement of the point mass model with a point density model as the first step of the procedure. The point density model avoids the difficulties associated with point masses. Point masses produce impulse responses at zero distances. Point densities produce null responses at zero distances. Our point density model works extremely well for interpolating scattered geodetic surface data. The unknowns in our system of equations are interpretable as point density anomalies. It is customary to place the point density anomalies at shallow or near zero depths in the vicinity of the surface data. This cannot be done with points masses. We have not been concerned with the possible usefulness of the geodetically estimated point densities for other purposes. They simply provide the basis for smooth interpolation of several geodetic parameters, which fit data values exactly. In any case, we are not competing with seismologists who in a sense are measuring densities, instead of estimating them. In fact, we believe that geodetic interpolation could be improved with the use of seismic density data. That is why we are proposing a joint venture in seismic/geodetic subsurface analysis.