Global Gravity Field Modeling from Satellite-to-Satellite Tracking Data

In simple words, parameter estimation is the process of extracting welldefined best ‘guesses’ of geophysical or measurement-system related values from uncertain and erroneous, conflicting and possibly at the same time incomplete data. These data may provide information on the sought-for parameters by way of direct observation, but much more often this relation is very indirect. Anyway, we know that we can never know ‘true’ parameters but we are satisfied with ‘estimates’ as long as these are ‘best’ in some way, like having the least spread for a given breed of data errors. This itself is a challenge common to all natural sciences (and most social sciences where ‘empirical’ studies are relevant). In satellite gravity field determination, the most relevant sought-for quantity is a geophysical field that has an infinite number of degrees of freedom and no preferred degree of truncation—this renders the application of standard and centuries-old methods such as least squares more complicated. This also continues to create confusion among data analysts from different backgrounds. Constructive approximation theory, rooted in functional analysis, provides a modern approach of breaking down the problem to a finite number of parameters without the uneasy feeling of ‘omitting’ something important. But the error bounds provided by approximation theory depend on such abstract ideas as kernel functions that describe the smoothness of a space. Least-squaresmethods are by nomeans the single unchallenged approach in satellite gravity analysis, due to the reason just mentioned. However, they are straightforward to apply, and they lead to error estimates that at least offer a way to address the uncertainty introduced by real measurement systems. J. Kusche (B) · A. Springer Institute for Geodesy and Geoinformation, Bonn University, Bonn, Germany e-mail: kusche@geod.uni-bonn.de A. Springer e-mail: springer@geod.uni-bonn.de © Springer International Publishing AG 2017 M. Naeimi and J. Flury (eds.), Global Gravity Field Modeling from Satellite-to-Satellite Tracking Data, Lecture Notes in Earth System Sciences, DOI 10.1007/978-3-319-49941-3_1 1 2 J. Kusche and A. Springer Least squares methods, in our view, allow to tackle some of today’s challenges in satellite gravity data analysis, like (1) efficiently dealing with large amounts of data and of sought-for parameters, (2) quantifying the effect of data errors and so-called background model errors on spherical harmonics and derived science results more or less comprehensively, and (3) ‘consistently’ combining data sets from different instruments, satellites, and processing chains, in the presence of inconsistencies that are ‘known’ but hard to pinpoint and too costly to remove from a first principles point of view. The development of the least squares method originated in a need for fitting techniques that materialized already in ancient times, with the first astronomical/geodetic observations that relate to the radius of the Earth and the orbital radii of the Earth and the Moon [109]. Later, and well-known in geodesy, meridional arc measurements were carried out in order to determine the flattening of the ellipsoidal Earth. For these measurements, one could not simply derive the parameter of interest from a single measurement and average the data. Laplace (1749–1827) developed several fitting techniques, aiming at e.g. minimizing the maximum error between fitted ellipsoid and data, and later, minimizing the average absolute value of the errors subject to constraining the sum of them to zero. Much of this work was influenced by Boscovitch (1711–1787) who, less known to us today, worked on the same problem and developed first principles of what we would call adjustment theory. Legendre (1752– 1833) published the method of normal equations for least squares to be applied in the analysis of arc measurements, and Gauss (1777–1855) developed the method of weighted least squares as we know it today, albeit without relying to matrix notation at that time. He applied LS to his determinations of planetary and asteroid orbits, and of course in the adjustment of large-scale trigonometric surveys. These lecture notes were compiled on the occasion of the International Autumn School ‘Global Gravity Field Modeling from Satellite-to-Satellite Tracking Data’, organized byDFG’s SFB ‘geo-Q’ atOctober 4–9, 2015 inBadHonnef,Germany.Our aim was to give students with different background an introduction to some concepts from parameter estimation that are common and useful for analyzing satellite gravity data (i.e. typically solving for sets of spherical harmonic coefficients). The focus was on concepts, and technical proofs were avoided.