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-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.

Appendices

Appendix A: Equator crossings

This appendix gives the general description for the dependency on the parity. In Sect. 2.1, the relation between one descending and all ascending arcs was used. This general description relates all descending to all ascending arcs. Consider again a circular orbit and an arbitrarily located equator crossing of an ascending arc at longitude $\lambda ^\mathrm{a}_0$. The equator crossing of the $p$th ascending arc can then be found by:

$$\begin{aligned} \lambda ^\mathrm{a}_p = \lambda ^\mathrm{a}_0 + \frac{2\pi }{\beta }p, \,\text{ with}\,p = 0 \ldots \beta -1. \end{aligned}$$

(20)

Starting from $\lambda ^\mathrm{a}_0$, the location of the first (in the timely sense) equator crossing of a descending arc is shifted by $180^{\circ }$ minus the angle passed by the Earth during the traveling of the satellite from equator crossing of the ascending to one of the descending arc. The equator crossings of the $q$th descending arc is then given by:

$$\begin{aligned} \varLambda ^\mathrm{d}_q = \varLambda ^\mathrm{a}_0 + \pi - \pi \frac{\alpha }{\beta } + \frac{2\pi }{\beta }q,\, \text{ with} \,q = 0 \ldots \beta -1. \end{aligned}$$

(21)

Similar to $p$, $q$ refers also to the East–West numbering of the equator crossing of the descending arcs and not to their timely occurrence.

The equator crossings of ascending and descending arcs will coincide if a set of $(p,q)$ can be found for which the difference of ascending and descending arcs becomes zero:

$$\begin{aligned} \varLambda ^\mathrm{a}_p - \varLambda ^\mathrm{d}_q = 0. \end{aligned}$$

(22)

Inserting Eqs. (20) and (21) yields

$$\begin{aligned}&\varLambda ^\mathrm{a}_0 + \frac{2\pi }{\beta }p - \varLambda ^\mathrm{a}_0 - \pi + \pi \frac{\alpha }{\beta } - \frac{2\pi }{\beta }q = 0 \nonumber \\&\quad \Leftrightarrow \frac{2p}{\beta } - 1 + \frac{\alpha }{\beta } - \frac{2q}{\beta } = 0 \nonumber \\&\quad \Leftrightarrow 2p - \beta + \alpha - 2q = 0 \nonumber \\&\quad \Leftrightarrow | 2p - 2q | = \beta - \alpha \end{aligned}$$

(23)

The absolute bracket is added because the difference $\left(\beta - \alpha \right)$ will always be positive. Since all quantities are integer values, Eq. (23) has only a solution in case of even parity of $\left(\beta - \alpha \right)$.

Appendix B: Condition number

Table 4 $log_{10}$ of the condition number for two repeat conditions each with and without order-limiting and different maximum degree for the case of noise-free data. Condition numbers beyond the computational accuracy are marked italics