The effect of downward continuation of gravity anomaly to sea level in Stokes' formula

Stokes' well-known formula integrates gravity anomalies on a sphere to geoidal undulations. Traditionally the effect of continuing the observed gravity anomaly from the Earth's surface to sea level is estimated in a rather rough manner, which significantly degrades the resulting geoidal undulations. In addition, the derived fictitious gravity anomalies at sea level are numerically unstable. This problem is solved by directly deriving a surface integral for the effects on the geoidal undulation and height anomaly. In addition, the solution is stabilized by optimized spectral smoothing by minimizing the mean square error. The final formula is a function of the gravity anomaly, height anomaly and topographic height.