Gauss-Markov Model (GMM) of an Integrated Approach for Regional Gravity Field Determination and De­sign Optimization of a Hybrid Gravimetric and Vertical Directions Network Using Adjusted Spherical Cap Harmonics (ASCH) Modelling

This study presents a Gauss–Markov model (GMM) for gravity field determination, in which the disturbing potential is parameterized using adjusted spherical cap harmonics (ASCH) in local spherical coordinates (r, θ, α). The ASCH para­me­tri­zation within W = V+ Z (Z = cent­ri­fu­gal potential) means a gravity field representation over a spherical area placed round the local cap pole (φ₀, λ₀) with an opening angle θ₀, and the scaling s=π/2·θ₀, using the gene­ra­lized Le­gendre polynomial P_n(k),m. With the relation k_SCHA ≈ (1/s)·n_SH much less ASCH parameters p are needed for the same resolution as for SH.

The GMM related to the above ASCH parametrization, developed in recent years at the Laboratory for GNSS and Navigation at Karlsruhe University of Applied Sciences and implemented in the   is presented. The GMM includes direct observations or a priori information on the ASCH parameters (C'_n(k),m, S'_n(k),m),  which can be derived from global spherical harmonic (SH) models. In addition, geometric fitting points (H B, L, h) are used to reduce long-wavelength components of geoid or quasi-geoid models N = h − H resulting from the ASCH parametrization. Furthermore, surface gravity observations g(x) are incorporated as observation equations. Topographic and isostatic reductions are not applied within the ASCH GMM.

Latvia is one of the few countries where at present high-quality vertical direction (VD) observations for the astronomical latitude and longitude (Φ, Λ)_x, observed by digital zenith ca­me­ra, developed at the Institute of Geodesy and Geoinformatics of the University of Latvia, are available (~450 points). This enables the design of an integrated hybrid gravimetric and vertical direction (VD) network. The implementation of the corresponding highly nonlinear Gauss–Markov observation equations for astronomical latitude and longitude, formulated in terms of the ASCH parameters (C'_n(k),m, S'_n(k),m), is discussed. In this way reduction free surface obser­va­tions (Φ, Λ)_x can be used, while vertical deflections (ξ, η)_x require, depending on the type of modeling reductions, and do not provide any additional information. 

In the second part, this study introduces a comprehensive methodological framework for the optimization of a hybrid gravity and vertical direction (VD) network within the above GMM parametrized by ASCH. As a first approach, a network reduction method is applied for the 1^st order design, enabling the optimization of the number and spatial distribution of combined gravity g(x) and vertical direction (φ, Λ)_x observations within a given area. As a second approach, a spectral optimization method based on eigenvalue analysis is employed to solve the first-order design problem. The proposed framework provides a robust basis for optimizing future measurement campaigns and for improving regional gravity field and quasi-geoid modelling.