Practical Integral Estimators for Gravitational Field Modelling: Basic Formulations and Statistical Characteristics

The mathematical apparatus of integral transformations is often used for gravitational field modelling. A basic assumption of these integrals is the knowledge of relatively accurate data available globally. Practically, however, global data coverage is rarely achieved, and measurement errors always contaminate data. Therefore, integral transformations are appropriately 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 contribution, we systematically formulate practical integral estimators and their respective errors. We present the practical integral estimators in two forms: combined (i.e., combining the restricted integrals for near-zone effects and the truncated spherical harmonic series for far-zone effects) and as a 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. Point-wise errors and global mean square counterparts complement the practical integral estimators. 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 with respect to 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 can be applied to related problems in other potential fields, such as electrostatics or magnetism.