On the convergence of spherical harmonic expansion of topographic and atmospheric biases in gradiometry

Abstract

The  gravity  gradiometric data  are  affected by  the  topographic and  atmospheric  masses.  In order  to fulfill  Laplace-Poisson’s equation and  to simplify  the  downward  continuation process, these effects should be removed from the data. However, if the analytical  downward continuation is considered, the  gravity gradients can  be continued downward disregarding such effects but the result  will be biased.   The topographic and atmospheric biases  can  be expressed in terms  of spherical harmonics and  studying these biases  gives  some ideas  about  analytical downward continuation of these quantities to sea level.  In formulation of harmonic coefficients of the topographic and  atmospheric biases, a  truncated binomial  expansion of topographic height  is used.   In this  paper,  we  show that  the  harmonics are  convergent to  the  third  term  of this  binomial  expansion.  The harmonics of the biases  on Vzz  are  convergent to the first  term  and  they  are  convergent in Vxy  for all the terms.   The harmonics of the other  components of the gravity gradient tensor  are  convergent to the second  terms,  while the third  terms  are  only  asymptotically convergent.  This  means  that  in terrestrial and  airborne gradiometry the  biases  should be  computed just  to the  second  order  term,  while  in satellite gravity gradiometry, e.g. GOCE, the third  term  can  also  be considered.