Matthias Weigelt

An improved sampling rule for mapping geopotential functions of a planet from a near polar orbit

One of the limiting factors in the determination of gravity field solutions is the spatial sampling. Especially during phases, when the satellite repeats its own track after a short time, the spatial resolution will be limited. The Nyquist rule-of-thumb for mapping geopotential functions of a planet, also referred to as the Colombo–Nyquist rule-of-thumb, provides a limit for the maximum achievable degree of a spherical harmonic development for repeat orbits. We show in this paper that this rule is too conservative, and solutions with better spatial resolutions are possible. A new rule is introduced which limits the maximum achievable order (not degree!) to be smaller than the number of revolutions if the difference between the number of revolutions and the number of nodal days is of odd parity and to be smaller than half the number of revolutions if the difference is of even parity. The dependence on the parity is reflected in the eigenvalue spectrum of the normal matrix and becomes especially important in the presence of noise. The rule is based on applying the Nyquist sampling theorem separately in North–South and East–West direction. This is only possible for satellites in highly inclined orbits like champ and grace. Tables for these two satellite missions are also provided which indicate the passed and (in case of grace) expected repeat cycles and possible degradations in the quality of the gravity field solutions.

Dependency of Resolvable Gravitational Spatial Resolution on Space-Borne Observation Techniques

The so-called Colombo-Nyquist (Colombo, The global mapping of gravity with two satellites, 1984) rule in satellite geodesy has been revisited. This rule predicts that for a gravimetric satellite flying in a (near-)polar circular repeat orbit, the maximum resolvable geopotential spherical harmonic degree (lmax) is equal to half the number of orbital revolutions (nr) the satellite completes in one repeat period. This rule has been tested for different observation types, including geoid values at sea level along the satellite ground track, orbit perturbations (radial,along-track, cross-track), low-low satellite-to-satellite tracking, and satellite gravity gradiometry observations (all three diagonal components). Results show that the Colombo–Nyquist must be reformulated. Simulations indicate that the maximum resolvable degree is in fact equal to knr + 1, where k can be equal to 1, 2, or even 3 depending on the combination of observation types. However, the original rule is correct to some extent, considering that the quality of recovered gravity field models is homogeneous as a function of geographical longitude as long as l max < nr/2.

Regional Gravity Field Recovery from GRACE Using Position Optimized Radial Base Functions

Global gravity solutions are generally influenced by degenerating effects such as insufficient spatial sampling and background models among others. Local irregularities in data supply can only be overcome by splitting the solution in a global reference and a local residual part. This research aims at the creation of a framework for the derivation of a local and regional gravity field solution utilizing the so-called line-of-sight gradiometry in a GRACE-scenario connected to a set of rapidly decaying base functions. In the usual approach, the latter are centered on a regular grid and only the scale parameter is estimated. The resulting poor condition of the normal matrix is counteracted by regularization. By contrast, here the positions as well as the shape of the base functions are additionally subject to the estimation process. As a consequence, the number of base functions can be minimized. The analysis of the residual observations by local base functions enables the resolution of details in the gravity field which are not contained in the global spherical harmonic solution. The methodology is tested using simulated as well as real GRACE data.

On the Comparison of Radial Base Functions and Single Layer Density Representations in Local Gravity Field Modelling from Simulated Satellite Observations

The recovery of local (time-variable) gravity features from satellite-to-satellite tracking missions is one of the current challenges in Geodesy. Often, a global spherical harmonic analysis is used and the area of interest is selected later on. However, this approach has deficiencies since leakage and incomplete recovery of signal are common side effects. In order to make better use of the signal content, a gravity recovery using localizing base functions can be employed. In this paper, two different techniques are compared in a case study using simulated potential observations at satellite level – namely position-optimized radial base functions and a single layer representation using a piecewise continuous density. The first one is the more common approach. Several variants exist which mainly differ in the choice of the position of the base function and the regularization method. Here, the position of each base is subject to an adjustment process. On the other hand, the chosen radial base functions are developed as a series of Legendre functions which still have a global support although they decay rapidly. The more rigorous approach is to use base functions with a strictly finite support. One possible choice is a single layer representation whereas the density is discretized by basic shapes like triangles, rectangles, or higher order elements. Each type of shape has its own number of nodes. The higher the number of nodes of a particular element, the more complicated becomes the solution strategy but at the same time the regularity of the solution increases. Here, triangles are used for the comparison. As a result, the radial base functions in the employed variant allow a modeling with a minimum number of parameters but do not achieve the same level of approximation as the discretized single layer representation. The latter do so at the cost of a higher number of parameters and regularization. This case study offers an interesting comparison of a near localizing with a strictly localizing base function. However, results can currently not be generalized as other variants of the radial base functions might perform better. Also, the extension to a GRACE-type observable is desirable.

Representation of Regional Gravity Fields by Radial Base Functions

The research aims at an investigation of the optimal choice of local base functions, to derive a regional solution of the gravity field. Therefore, the representation of the gravity field is separated into a global and a residual signal, which includes the regional details. To detect these details, a superposition of localizing radial base functions is used. The base functions are developed from one mother function, and modified by four parameters. These arguments can be separated into two coordinates, one scale factor and a shape parameter. The observations of a few residual gravity fields are simulated by orbit integration and the energy-balance technique, in order to test the current approach. After selecting a region of interest, the parameters of the base functions are estimated. In order to get the optimal positions, two searching algorithms are compared. In the first algorithm the scale factors are estimated, while the positions and shape parameters are fixed. This method requires no initial values, because of the linear, but ill-posed and maybe ill-conditioned problem, but usually a regularization is necessary. The second algorithm searches possible positions for one base function in each step, until a termination condition is fulfilled, and improves the positions and scale factors in one adjustment. The results in the second case are better and faster for the test fields, but they depend on the initial values, the number of iterations and an assumption of an approximate constant orbit height

Evaluation of EGM2008 by comparison with global and local gravity solutions from CHAMP

New gravity field models incorporate GRACE data for the long wavelengths since it is one of the best available data sources. However, considering e.g. the degree difference RMS between EGM2008 and GGM02s, also discrepancies between these models occur which cannot solely be explained by numerical inaccuracies. Their validation is difficult since comparisons with existing GRACE models will always be biased. Maybe the best independent data set on a global scale is the CHAMP data. One the other hand, it is known that the accuracy of these solutions is approximately one order of magnitude worse than GRACE-only solutions. In this research, it has been investigated if CHAMP can serve as an indicator in the comparison of EGM2008 and GGM02s. The primary data source of CHAMP is its position and velocity measurements derived from GPS. They are based on purely kinematic orbits, which are independent of any a priori information and are provided by the Institute for Astronomical and Physical Geodesy at the Technical University Munich for the period of April 2002 to February 2004. The comparison is based on a global solution and a local refinement with Slepian functions in order to further improve the quality of the CHAMP solution. The latter is adopted because it can make better use of information in high-latitude areas where the data density is higher. However, the solutions are solely based on GPS measurements which have a poorer quality compared to the K-band ranging system of GRACE and thus prevents a real statement about the quality of the EGM2008.