Practical Integral Estimators for Gravitational Field Modelling and Their Statistical Characteristics I: Theory

Gravitational fields are often modelled by the mathematical apparatus of integral transformations. A basic assumption of these integrals is the knowledge of relatively accurate data available globally. Practically, however, global data coverage is rarely achieved and data are always contaminated by measurement errors. Therefore, integral transformations are properly modified and practical integral estimators are formulated and further employed in numerical experiments. In addition, corresponding statistical characteristics are often desired to indicate the quality of calculated gravitational fields. In this article, we systematically formulate practical integral estimators and their respective errors. We present the practical integral estimators in the combined form (i.e. combining the restricted integrals for the near-zone effects and the truncated spherical harmonic series for the far-zone effects) and in the form of spherical harmonic series. The practical integral estimators form a theoretical basis for an accurate gravitational field modelling, e.g. when solving upward or downward continuation. By employing a unified notation, the mathematical formulas are derived to an unprecedented extent for a broad class of quantities. Namely, the theoretical formulations connect four types of boundary conditions with twenty computed quantities. The practical integral estimators are complemented by point-wise errors and global mean square counterparts. The point errors can be calculated from the errors of the near-zone and far-zone boundary values, the position of the computational point, the size of the integration radius, and the maximum spherical harmonic degree of the far-zone effects. The number of variables is reduced for the global mean square errors, as they are invariant from the horizontal position of computational points. Both statistical characteristics may also be employed in optimisation problems and experimental designs. The basic principles and formulations presented here may be employed in related problems of other potential fields, such as in electrostatics or magnetism.