# A timewise kinematic method for satellite gradiometry: GOCE simulations

## Summary (3 min read)

### 1. Introduction

- Gravity field and steady-state Ocean Circulation Explorer (GOCE) is an European Space Agency (ESA) mission soon to be launched, with the c© 2005 Kluwer Academic Publishers.
- The first decomposition of the problem can be obtained from the following considerations.
- 2. Timewise solution A second basic choice is the organization of the GOCE observations in one of three possible ways.
- This method is the most convenient in the mission design phase, because it allows to convert requirements in the error spectrum of the gradiometer into error spectrum of the recovered gravity coefficients.
- The authors show in Section 5 that the polar gaps resulting from the non polar orbit of GOCE introduce an exact symmetry in the functional space of spherical harmonics.

### 2. Gradiometer Calibration

- The observations include two independent components of the gravity gradient.
- Thermal signals are by no means random, they occur with well defined frequencies depending upon the external sources of heat and the control cycles of the thermal stabilization system4.
- The authors are mostly interested in evaluating the effects of the systematic bias components5.
- Third, the authors need to stress once again that the calibration model should not constrain the shape of the calibration as a function of time, and should not use information on this shape illegitimately transfered from the simulation step.
- This selection of base functions does not satisfy the third requirement, because all the base function are periodic of period ∆t and thus have the same value at the two extremes of the arc interval.

### 3. Multi arc solution

- The multi arc approach is a form of the least squares method, in which the list of parameters is split in accordance to the partition of the orbit into arcs (Reigber, 1989; Milani and Melchioni, 1989; Milani et al., 1995).
- This, however, ignores the portion C`g of the normal matrix, forcing Γ`g = 0, and changes the solution ∆g.
- Thus, for the purpose of these simulations, with the main goal of computing a reliable estimate of the error for high degrees, this approximation is sufficient.
- The authors can conclude that, if the arc length ∆t, the number of calibration parameters per arc 2K+2, and the minimum degree l are chosen properly, then it is possible to solve separately for the calibration parameters ` and then to solve for the harmonic coefficients g only.

### 4. Resonant Decomposition

- This makes difficult to perform this computation without using very expensive hardware.
- The above results refer to the geopotential, but in fact apply to whatever harmonic function expanded in spherical harmonics.
- An assessment of the relative size of the off-diagonal blocks was performed in the case of the GPS observables: the off-diagonal block correlations were found to be less than 1% in the most critical cases involving the r = 0 block, less than 0.4% in the other cases .
- The authors can conclude that, even by using h/k = 31/2, not the most appropriate resonance for the value of n of GOCE, the off-diagonal terms can be neglected and a full second iteration is not necessary6.

### 5. Consequences of polar gaps

- It can be checked that the large RMS (and also actual errors) occur in this remainder class only for the zonals m = 0; the other harmonics, with m = 31, 62, . . ., are determined with error/signal ratio well below 1 for all l.
- To understand why the GOCE results leave undetermined the geoid near the poles, the authors have represented in Figure 6 one of the GOCE observables, the radial component of the gravity gradient of the anomalies λ1 V1 and λ2 V2.
- The exact symmetry results in a complete degeneracy of the normal matrix, with the number of zero eigenvalues corresponding to the dimensions of the symmetry group.
- The cap function Φ cannot have a finite spherical harmonics expansion, because it is not an analytic function on the sphere.

### 6. Error sources

- The formal error is of course larger than the one of the naive estimate of Figure 2, but only because of the polar gaps.
- Indeed, for the classes not including low values of m the figures look like the one for r = 1 .
- To assess the origin of the actual errors exceeding the formal ones, it is important to analyze the specific harmonics where the largest actual/formal error ratio takes place.
- To test the relevance of this orbit error on the solution for the gravity coefficients the authors have performed a simulation without the gradiometer bias, with the orbit errors as the only systematic error source: as shown in Figure 10, the actual error in the coefficients becomes statistically consistent with the formal error.
- As a general rule, the formal covariance result is a lower bound for the uncertainty, the actual results can only be worse.

### 7. Conclusions

- As a result the authors have been able to perform a full simulation, including data simulation and differential corrections, with PC class computer resources.
- The authors have improved the theory explaining the remaining uncertainties in the solution for the gravity coefficients, resulting from the polar gaps.
- This is not a surprise: the calibration of accelerometers is one of the main problems of the new generation satellite geodesy missions (Bruinsma et al., 2003).
- How to control the situation of the spurious signals in the frequency domain immediately below fm is going to be the main challenge of the GOCE operations and data processing.
- This work was funded by Agenzia Spaziale Italiana (ASI) under contract I/R/194/02.

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### "A timewise kinematic method for sat..." refers background or methods in this paper

...This is a less natural way of looking at the static part of the gravitational signal: indeed, each spherical harmonic appears as a sum of signals with different frequencies (Kaula, 1966)....

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...This large number 3 This sampling is meant to be the result of a preprocessing stage, in which digital filtering is used to remove the noise at frequencies higher than the upper limit fs ' 1/10 Hz of the measurement band. gocesim2.tex; 13/04/2005; 9:23; p.5 would be a problem, not only for the…...

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...When transformed in the orbital elements of the satellite the contribution of a particular harmonic (l,m) reads (Kaula, 1966) Ulm = GMe R⊕ l ∑ p=0 ∞ ∑ q=−∞ ( R⊕ a )l+1 F̄lmp(I)Glpq(e)Alm cos (vlmpq − ψlm) (6) where Me is the Earth mass, R⊕ its radius, a the semimajor axis of the satellite orbit, e…...

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### "A timewise kinematic method for sat..." refers methods in this paper

...1 There are other methods of data processing to extract gravity field information from the GPS data, as used with CHAMP (Reigber et al., 2002), but they cannot be pushed to the GOCE level of accuracy and resolution. gocesim2.tex; 13/04/2005; 9:23; p.3 These three approaches are equivalent for a perfect distribution of data....

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...1 There are other methods of data processing to extract gravity field information from the GPS data, as used with CHAMP (Reigber et al., 2002), but they cannot be pushed to the GOCE level of accuracy and resolution. gocesim2.tex; 13/04/2005; 9:23; p.3 These three approaches are equivalent for a…...

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...The analysis of the CHAMP and GRACE data should provide more than enough a priori information before the launch of GOCE....

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