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

Abstract

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-ofthumb, 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 e.g. 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.

Combination of GRACE monthly gravity field solutions from different processing strategies

“…1 (bottom). The density of the ground tracks of the satellites directly determines the achievable spatial and spectral resolution of the derived monthly gravity field (Weigelt et al 2013;Klokočník et al 2015 Fig. 1 GRACE monthly gravity fields available for this study (top; dark blue color indicates missing monthly gravity fields) and the maximum longitudinal spacings in the ground track of the GRACE-A satellite (bottom)…”

Ground track density considerations on the resolvability of gravity field harmonics in a repeat orbit

“…The practical experience of gravity field recovery from various authors, however, does not support this rule (Weigelt et al, 2009(Weigelt et al, , 2013Visser et al, 2012 and many experiments performed in the Astronomical Institute, Ondřejov, at NOAA, Silver Spring and in Faculté des Sciences, de la Technologie et de la Communication). The rule looks too stringent.…”

“…While these rules are solely based on the pattern of equator crossings (Wagner et al, 2006;Klokočník et al, 2008;Weigelt et al, 2009Weigelt et al, , 2013, the present work takes into account a possible latitude (and inclination) dependence. The authors here analyze the actual ground track patterns of CHAMP, GRACE and GOCE over all their available latitudes and then derive a more general rule for M max .…”

“…The problem of spatial sampling has been studied repeatedly (Wagner et al, 2006;Klokočník et al, 2008;Weigelt et al, 2009) and simple rules have been derived to limit the maximum order for unconstrained solutions (inversions) for the gravity field parameters or their variations from observations of a single satellite. Here we work with the latest rule from Weigelt et al (2013) which distinguishes the maximum attainable order according to the parity of the two parameters defining the repeat orbit or orbital resonance, b the number of nodal satellite's revolutions in a nodal days (a, b co-prime integers, the ratio b/a irreducible). This rule, that the resolvable order (in a repeat near polar orbit) should be b for odd parity (b À a) and b/2 for even parity (b À a) orbits, arose from the discovery that the number of distinct and equally spaced equatorial crossings (ascending and descending passes) for odd parity (b À a) is 2b while for even parity orbits it is only b.…”

The impact of common versus separate estimation of orbit parameters on GRACE gravity field solutions

“…The quality of the gravity field models is not necessarily characterized by their spatial resolution, which is limited by the density of the ground track pattern during the corresponding time intervals (Weigelt et al, 2013), but by the error and signal content of the spherical harmonic coefficients (in the spectral domain), or equivalently, the grid values of geoid heights or quantities derived therefrom (in the spatial domain).…”