Two-dimensional geoid modelling: some remarks on Chapman's algorithm

C Ayala,J Pous,M Fernández,M Torné

doi:10.1111/j.1365-246x.1996.tb04739.x

C Ayala, J Pous + Show 2 more

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https://doi.org/10.1111/j.1365-246x.1996.tb04739.x

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Abstract

SUMMARY An analytical solution and its application to compute 2-D geoid anomalies, based on Brun’s formula (Heiskanen & Moritz 1967), was developed by Chapman (1979). In this paper we present the derived solution from a thorough review of Chapman’s algorithm. Apart from some typographical errors in the mathematical expressions, we have found that some of the precautions given by the author are not necessary when the algorithm is applied, in particular those regarding the unit system, topography and density contrast. Recent studies have shown that geoid anomalies, when used with other geophysical anomalies, are able to constrain plausible models of lateral density variations at depth and therefore to determine the Earth’s internal structure. Chapman’s algorithm (1979), based on Stokes’ theorem, is a tool used to calculate 2-D and 3-D geoid anomalies produced by a given body. However, due to the restrictions imposed, the algorithm is limited to the modelling of marine tectonic structures. The goal of this research note is to extend the application of Chapman’s algorithm and to fix some typographical errors found in its formulation. The geoid anomaly for any arbitrary 2-D body is

Highlights

Chapman’s algorithm (1979), based on Stokes’ theorem, is a tool used to calculate 2 - D and 3-D geoid anomalies produced by a given body
While the first is an implicit condition in the calculation of the geoid anomaly, the second refers to a particular case where topography is not considered
After integration we find that the contribution to the geoid anomaly due to the ith line segment for one polygon is

Summary

Introduction

Chapman’s algorithm (1979), based on Stokes’ theorem, is a tool used to calculate 2 - D and 3-D geoid anomalies produced by a given body. The geoid anomaly for any arbitrary 2-D body is (Chapman 1979), where p is the density, G is the Newtonian constant 6.67 x lo-” m3 kg-ls-’, y is normal gravity, 9.8 m s-’, primed coordinates indicate the observation point, unprimed coordinates are the integration variable, y is the horizontal distance and z is the depth, positive downward.

Results

Conclusion