NUMERICAL STUDIES ON THE HARMONIC DOWNWARD CONTINUATION OF BAND-LIMITED AIRBORNE GRAVITY

Abstract

In this paper, the numerical stability and efficiency of methods of harmonic downward continuation from flying altitudes are treated for sampled gravity field data. The problem is first formulated in its continuous form, i.e. as the inverse solution of the spherical Dirichlet problem, and is then approximated by Gaussian quadrature to yield a finite system of linear equations. The numerical stability of this system is investigated for both error-free gravity data and for the noisy and band-limited gravity measurements usually obtained from airborne gravity surveys. It can be shown that the system becomes ill-conditioned, once the ratio between flying altitude and data sampling rate exceeds a certain limit. It can also be shown that noisy measurements tend to generate a solution that is practically useless, long before the system becomes ill-conditioned. Therefore, instead of treating the general solution of the discrete downward continuation problem, the more modest question is studied, for which range of flying altitudes and sampling rates, the numerical solution of the discrete linear system can be considered as practically useful. 'Practically useful' will be defined heuristically as of sufficient accuracy and stability to satisfy the requirements of the user. The question will be investigated for the specific application of geoid computation from gravity data sampled at flying altitudes. In this case, a stable solution with a standard deviation of a few centimeters is required. Typical flight parameters are heights of 2-6 km, a minimum half-wavelength resolution of 2 km, and data noise between 0.5 and 1.5 mGal. Different methods of geoid determination, different solution techniques for the resulting systems of linear equations, and different minimization principles will be compared. As a result operational parameters will be defined which, for a given noise level, will result in a geoid accuracy of a few centimeters for the estimated band-limited gravity field spectrum.