Comment on Sjöberg (2006) “The topographic bias by analytical continuation in physical geodesy” J Geod 81(5):345–350

The analytical, or harmonic, downward continuation of the external gravity potential into the topographic masses gives rise to a bias, which is called the analytical (downward) continuation (ADC) bias (Ågren in J Geod 78:314–332, 2004a) or the topographic bias (Sjöberg in J Geod, 2006). In Sjöberg (J Geod, 2006), a proof is presented that this bias is exactly equal to a simple two-term expression, which depends only on the topographic height and density in the evaluation point P. The expression is simple and inexpensive to evaluate. In this paper, we wish to question the validity of the expression given in Sjöberg (J Geod, 2006) for realistic terrains. The topographic bias is commonly defined as the difference between the true (internal) and the analytically downward continued external geopotential, evaluated at sea level. Typically both are evaluated as external or internal spherical harmonic (SH) expansions, which may however not always converge. If they do converge, they have been well known in the literature (e.g., Ågren (J Geod 78:314–332, 2004a), Wang (J Geod 71:70–82, 1997)) to produce a bias that contains additional terms over and beyond the simple expression. Below we analyze the additional terms that arise when applying the method to realistic terrains. Also, for realistic terrains, analytical downward continuation may not even be strictly possible. In practice, for discrete data sets, it is always possible, but then, an implicit smoothing of the terrain, or terrain potential, always takes place.