A numerical test of the topographic bias

In 1962 A. Bjerhammar introduced the methodof analytical continuation in physical geodesy, implyingthat surface gravity anomalies are downward continuedinto the topographic masses down to an internal sphere(the Bjerhammar sphere). The method also includes ana-lytical upward continuation of the potential to the surfaceof the Earth to obtain the quasigeoid. One can show thatalso the common remove-compute-restore technique forgeoid determination includes an analytical continuationas long as the complete density distribution of the topog-raphy is not known. The analytical continuation impliesthat the downward continued gravity anomaly and/or po-tential are/is in error by the so-called topographic bias,which was postulated by a simple formula of L E Sjöbergin 2007. Here we will numerically test the postulated for-mula by comparing it with the bias obtained by analyticaldownward continuation of the external potential of a ho-mogeneous ellipsoid to an inner sphere.The result shows that the postulated formula holds: At theequator of the ellipsoid, where the external potential isdownward continued 21 km, the computed and postulatedtopographic biases agree to less than a millimetre (whenthe potential is scaled to the unit of metre).Keywords:analytical continuation, homogeneous ellip-soid, topographic bias