In geophysical and geodetic studies, gravity inversion is typically performed such that observed gravity values are first continued downward onto a regular (planar, spherical or spheroidal) surface by solving an inverse integral transform, which originates from a classical solution to the first boundary-value problem in potential theory. A typical example is continuing gravity observed at the topographic surface down to the mean sea level (geoid). Nowadays, gravity-dedicated satellite missions and aerial gravimetry provide gravity data above the topographic surface in addition to classical terrestrial gravity observations. For specific purposes (such as gravity data combination and validation, or quasigeoid determination), satellite and aerial gravity observations have to be continued to the irregular topographic surface. In this study, we address this issue by formulating a functional model for a spectral downward continuation of selected gravitational field quantities to an irregular topographic surface. Moreover, we generalize this functional model to allow for transformation between different types of gravitational field quantities. In particular, we derive spectral weights for estimation of the disturbing potential or disturbing/anomalous gravity at the Earth’s surface by combining the first-, second- and third-order radial gradients of the disturbing potential (disturbing gradients). The correctness of the developed combined spectral estimator is verified in a closed-loop test based on synthetic satellite disturbing gradients. The combined spectral estimator is applied to simulated satellite disturbing gradients polluted by a realistic Gaussian noise. Results of the numerical experiments show that the combined spectral estimator puts the highest importance on the least polluted disturbing gradient, while the contribution of the least accurate disturbing gradient is negligible. An important advantage of this spectral combination method is that no matrix inversion with numerical instabilities requiring regularization is needed.

中文翻译：

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通过光谱加权将引力场量向下延续到不规则表面

在地球物理和大地测量研究中，通常执行重力反演，这样观察到的重力值首先通过求解逆积分变换向下继续向下到规则（平面、球面或椭球体）表面上，该变换源自第一个边界值的经典解势论中的问题。一个典型的例子是在地形表面观察到的持续重力下降到平均海平面（大地水准面）。如今，除了经典的地面重力观测外，重力专用卫星任务和航空重力测量还提供地形表面上方的重力数据。出于特定目的（如重力数据组合和验证，或准地球仪确定），卫星和航空重力观测必须继续到不规则的地形表面。在这项研究中，我们通过为选定的引力场量的光谱向下延续到不规则的地形表面制定功能模型来解决这个问题。此外，我们推广这个功能模型以允许不同类型的引力场量之间的转换。特别是，我们通过组合干扰电位的一阶、二阶和三阶径向梯度（干扰梯度）推导出用于估计地球表面干扰电位或干扰/异常重力的谱权重。在基于合成卫星干扰梯度的闭环测试中验证了开发的组合频谱估计器的正确性。组合频谱估计器应用于模拟卫星干扰梯度，该梯度被真实的高斯噪声污染。数值实验结果表明，组合谱估计器最重视污染最小的干扰梯度，而最不准确的干扰梯度的贡献可以忽略不计。这种谱组合方法的一个重要优点是不需要需要正则化的具有数值不稳定性的矩阵求逆。