Applying multi-link signal-processing to noise models

In Section 3.4, the signal processing required to recover the line-of-sight displacement from 8 interspacecraft measurements was presented. In a multi-link GRACE with 3 optical heads there will be 18 link measurements. The steps required to transform these 18 one-way link measurements into a round-trip line-of-sight displacement can be found in Appendix A. Round-trip combinations are formed along each link, suppressing the distant spacecraft laser frequency displacement noise and fibre fluctuations on the two spacecraft. The round-trip measurements are then added to two weighted averages to cancel the rotation of both spacecraft. While laser frequency displacement noise, fibre fluctuations and rotation coupled error are all suppressed, shot noise, DEHI noise and cyclic error are uncorrelated in the different link beatnotes and therefore sum incoherently in the data combinations. In this section a multi-link transfer function is derived for each noise source in the one-way link measurements.

In the first stage of signal processing, simple TDI combinations are formed along each link, subtracting a delayed copy of the spacecraft 2 measurement from the spacecraft 1 measurement, to obtain 9 round-trip displacement measurements. This requires an estimate of the interspacecraft range4 and interpolation to synchronise the data [109].

4_{The pseudo random codes used for DEHI could be used to estimate the delay although another}

Chapter 7 Evaluating the multi-link GRACE concept architecture

The TDI combination is used to cancel laser frequency displacement noise from spacecraft 2 and the fibre fluctuations along the link. If there is an error in the range estimation then the noise will not cancel completely. There are a number of reasons this could be the case, some of the reasons are discussed in Section 7.2.4.

An error, ∆τ, in the estimate of the interspacecraft delay will result in residual laser frequency displacement noise with a root-power spectral density given in Eq.5.1. In order for the residual pre-stabilised, 30 Hz/√Hz, laser 2 noise to be below the 30 nm/√Hz residual laser noise requirement for GRACE-FO [1] the interspacecraft delay would need to be known to better than 1 ms. This is considerably higher than the required 6 ns in the GRACE-FO LEGOP TDI experiment. This is a benefit of using pre-stabilised lasers on both spacecraft.

In addition to the spacecraft 2 laser frequency displacement noise there will also be residual pathlength noise due to errors in the estimation ofτ. The fibre fluctuations are suppressed by the transfer function:

HResFibre(f) = 2 sin(πf∆τ /2) (7.3) The large interspacecraft time delays used in the TDI combinations will also affect the frequency response of the other noise terms. The round-trip measurement of spacecraft 1 laser frequency displacement noise will be shaped by the following transfer function:

HRTLas(f) = 2 sin(2πf τ) (7.4) where 2τ represents the round-trip interspacecraft delay equivalent to two passes along the interspacecraft link.

The rotation-to-pathlength error is indistinguishable from interspacecraft displacement in the round-trip measurement. It is assumed however that over the round-trip measurement time (∼2 ms) the rotation does not change significantly and therefore a transfer function is not applied to the rotation coupled error.

The other noise sources - shot noise, DEHI noise, and cyclic error - are assumed to be un- correlated over the 18 link measurements. Therefore, in the round-trip TDI combination, they sum incoherently. These noise terms have a gain:

HRTNoise(f) =

√

2 (7.5)

In the next two stages of signal processing successive weighted averages are performed to cancel the rotation-to-pathlength error due to rotation of both spacecraft. The spacecraft weights are normalised to ensure that the recovered centre of mass displacement has the correct scaling for the displacement signal. Therefore the laser frequency displacement noise will not be affected by the weighted average. The uncorrelated noise in each of the round-trip measurements will however be affected by the weighted average. Assuming the symmetric triangle optical head arrangement shown in Figure 7.1 the gain from each weighted average will be:

HAvgNoise(f) =

√

3w= √1

3 (7.6)

7.2 Predicting the sensitivity of a multi-link GRACE

The noise is suppressed assuming the optical heads surround the centre of mass. In Sec- tion 3.5.4 there was a discussion about SNR penalties for using optical head configurations that did not enclose the centre of mass. If this is the case then the gain of this transfer function would be greater.

Therefore the uncorrelated noise - shot noise, DEHI noise, and cyclic error - will have a total multi-link gain of:

HMLNoise(f) =

√

2

3 (7.7)

The weighted averages are used to cancel the rotation-to-pathlength error. In Section 3.5.3 the effect of an uncertainty in the optical head positions on the suppression of rotation-to- pathlength error was considered. In the symmetric 3 optical head system, weights equal to w = 1/3 are needed to cancel the error due to pitch and yaw. If the optical head A, which only has error from yaw, moves by ∆z then the error due to unsuppressed yaw will be:

δ˜xResYaw(f) =w∆zδθ˜yaw(f) (7.8)