Вычисление первой и второй вертикальных производных силы тяжести

Summary

Following Molodensky's suggestions anomalies of the vertical gradient of gravity were used to achieve a greater accuracy in the determination of the figure of the Earth by gravimetrical methods. The existing methods of computing this quantity do not take into account inclinations of the physical surface of the Earth. Using the Laplace equation, the second derivative ∂² T/∂v ² (1) of the disturbing potentialT is expressed by the second derivatives ofT along the tangentsτ ₁ andτ ₂ to the physical surface of the Earth in mutually perpendicular planes and by the derivatives of gravity anomalies (2). The derivatives ∂² T/∂τ ²₁ and ∂² T/∂τ ²₂ have been determined using the Molodensky method [4] of solving his integral equation for the single layer density. In the zero approximation, the Noumerov formula [2] was obtained; however, the results obtained using this formula should be referred to the physical surface of the Earth, not to the Listing geoid. The correction of the first approximation is given by formula (16). The second vertical derivative of gravity anomalies can be determined using the expression (20).