We consider the topographic bias in gravimetric geoid determination when analytically downward continuing the topographic potential from the Earth’s surface or above down to sea level. The total bias is subdivided into those of the Bouguer shell or plate and the terrain. In this process the potential of the Bouguer shell always causes a bias, which increases with the square of the topographic height and typically exceeds 1–2 cm for elevations higher than 1 km. The main understanding today is that the terrain does not provide a potential bias except possibly for masses located inside a dome of height <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\left( {\sqrt 2 - 1} \right)H_{P}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfenced close=")" open="("> <mrow> <msqrt> <mn>2</mn> </msqrt> <mo>-</mo> <mn>1</mn> </mrow> </mfenced> <msub> <mi>H</mi> <mi>P</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, centered along the vertical through the computation point P at height <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$H_{P}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mi>P</mi> </msub> </math></EquationSource> </InlineEquation> with a lateral base radius <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$s_{0} = H_{P}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>s</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>H</mi> <mi>P</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> at sea-level. This result implies that the potential of all terrain masses of arbitrary density located exterior to the dome are unbiasedly downward continued to sea level. In this article a homogenous cylinder and a cone are used to verify the above results. If the computation point P is located at or above the cylinder along its axis and the radius of the cylinder is larger than the height of P, there is no terrain bias. Similar result is obtained for the cone with base radius larger than the height <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$H_{P}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mi>P</mi> </msub> </math></EquationSource> </InlineEquation>, except when P is located at the vertex of the cone, in which case necessary vertical derivatives in the harmonic continuation process formally does not exist, but a solution can still be handled in the numerical/practical application.

Harmonic downward continuation of the gravitational potential along the axes of a homogeneous cylinder and a cone

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2026年5月13日

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L. E. Sjöberg

摘要

We consider the topographic bias in gravimetric geoid determination when analytically downward continuing the topographic potential from the Earth’s surface or above down to sea level. The total bias is subdivided into those of the Bouguer shell or plate and the terrain. In this process the potential of the Bouguer shell always causes a bias, which increases with the square of the topographic height and typically exceeds 1–2 cm for elevations higher than 1 km. The main understanding today is that the terrain does not provide a potential bias except possibly for masses located inside a dome of height $\left( {\sqrt 2 - 1} \right)H_{P}$ $(\sqrt{2} - 1) H_{P}$ , centered along the vertical through the computation point P at height $H_{P}$ $H_{P}$ with a lateral base radius $s_{0} = H_{P}$ $s_{0} = H_{P}$ at sea-level. This result implies that the potential of all terrain masses of arbitrary density located exterior to the dome are unbiasedly downward continued to sea level. In this article a homogenous cylinder and a cone are used to verify the above results. If the computation point P is located at or above the cylinder along its axis and the radius of the cylinder is larger than the height of P, there is no terrain bias. Similar result is obtained for the cone with base radius larger than the height $H_{P}$ $H_{P}$ , except when P is located at the vertex of the cone, in which case necessary vertical derivatives in the harmonic continuation process formally does not exist, but a solution can still be handled in the numerical/practical application.