Deflection of the Vertical from Vertical Gravity Gradient using Spherical Numerical Integration Expression and Planar Closed-Form Formula

Abstract

The deflection of the vertical (DOV), defined by the angle between the plumb line and the normal to the reference ellipsoid, plays a crucial role in geodesy and geophysics, particularly in geodetic data processing, geoid determination, and gravity field studies. Compared with traditional astro-geodetic methods, the determination of the DOV from gravity observations is more efficient and practically feasible. Moreover, gravity gradients contain richer high-frequency (short-wavelength) information than gravity anomalies. Therefore, they provide a promising data source for high-resolution DOV determination. In this context, this study employs three methods to compute the DOV from the vertical component of gravity gradients: the spherical numerical integration formulation, the planar numerical integration formulation, and the planar closed-form solution. Starting from the spherical formulation, a planar approximation is derived, and a corresponding closed-form solution under this approximation is further developed. Two test regions are selected: one bounded by [0°E, 2°E] in longitude and [0°S, 2°S] in latitude, and another in [150°W, 152°W] [0°, 2°N]. Closed‑loop tests based on the EGM2008 model confirm the theoretical consistency of the three methods. In a separate validation, the results were compared with DOV data from the Scripps Institution of Oceanography (SIO) model. The strong agreement between the computed DOV and the SIO model confirms the accuracy of the algorithms proposed in this study. A comparison of computational times with implementations based on the spherical formulation further demonstrates the high efficiency of the planar approximation. In addition, Graphics Processing Unit (GPU) parallel computing was applied to accelerate the computations. This efficient approach provides a valuable alternative for high-resolution DOV determination.