The impact of missing data on the construction of LISA Time Delay Interferometry Michelson variables

TL;DR

The paper addresses how missing raw phasemeter data propagate through LISA's TDI pipeline and quantifies the resulting augmentation of gaps in second-generation Michelson observables. It develops analytical, FIR-based gap-augmentation formulas for intermediary variables and TDI outputs, and validates them against LISA simulation tools. Key findings show that a single missing sample can yield about 90 s of additional TDI data loss in X_2, with short, frequent gaps posing the greatest risk; planned gaps can lead to several days of data loss over four years, while unplanned gaps contribute modestly. The results provide practical guidance for telemetry management and gap-mitigation strategies, enabling accurate, fast estimates of data loss and informing robust GW inference pipelines in the presence of gaps.

Abstract

We investigate the impact of missing input data on the construction of second-generation Time Delay Interferometry (TDI) variables, which enable data analysis for the Laser Interferometer Space Antenna (LISA). TDI relies on the introduction of precise time delays into the raw interferometric data streams before they are combined to suppress otherwise dominant laser phase noise. We show that a single missing sample, corresponding to 0.25 s of data, will result in an effective data gap of approximately 90 s in the second-generation TDI output if further measures are not taken. This additional gap is largely independent of the initial gap duration, but increases linearly with the order of the fractional-delay filter used for the computations. For a realistic gap scenario, incorporating both planned and unplanned data interruptions consistent with a target duty cycle of ~84%, we find that frequent, short-duration gaps (e.g., a total of 1000 per year, each of which have short durations ~ 100 s) could result in an additional loss in the TDI variables of about one day per year corresponding to a ~0.3% reduction in duty cycle. This amounts to a loss of approximately one day of LISA data suitable for the global-fit per year.

Figures (3)

• Figure 1: LISA's measurement system, adapted from Bayle_2023 with support from O. Hartwig. The Figure shows how each spacecraft uses three main interferometers (IFOs) to detect GWs. The Interspacecraft IFO captures the GW signal, buried within much louder noise, primarily from laser frequency fluctuations. The other two IFOs, the Reference and Test-mass IFOs, help remove noise due to spacecraft motion. Additionally, clock sidebands and pseudorandom noise codes correct for timing and laser noise, while the differential wavefront sensing measures beam angles to minimize angular jitter effects.
• Figure 2: (Top Left/Top right plot): For first/second generation TDI variables( Eq.\ref{['eq:first_gen_factorized_variables']}) we compute the gap augmentation with $\mathcal{K} = 45$ for all Lagrange interpolants as a function of the gap duration. The purple/blue-dashed curves in the top left plot compare the numerical/theoretical gap augmentation in the first generation TDI variables using lisainstrument and pyTDI and Eq. \ref{['eq:gap_augmentation_first_gen_X']} respectively. The second plot is the same but for second generation TDI variables with black/green curves depicting the gap augmentation using pyTDI and Eq. \ref{['eq:gap_augmentation_second_gen_X']}. (Bottom Left/Bottom right plot): Here we compare the Gap Augmentation using lisainstrument and pyTDI against our analytical result of Eq. \ref{['eq:gap_augmentation_first_gen_X']} (left plot) and of Eq. \ref{['eq:gap_augmentation_second_gen_X']} (right plot) as a function of the order of interpolant (set equal for all delay operators). As we can see, there exist three regimes where the gap augmentation changes slope with respect to the order of interpolant (see Eq. \ref{['eq:gap_augmentation_eta_analytical']} for similar behaviour with justification outlined in Appendix \ref{['app:nested_delays_formulas']}). As one can see, our analytical result (dashed blue, yellow and green dots) matches the numerical simulations (solid blue, yelow and green curves) for both plots for various sets of contiguous nans $N^{\text{ifo}}_{\texttt{NaNs}} = \{ 1, 10, 25\}$. A jupyter notebook with code to reproduce these plots is available https://github.com/OllieBurke/gaps_tdi.
• Figure 3: The left/center/right plots show distributions on the resultant duty cycles on the telemetry data when planned/unplanned/both planned and unplanned gap strategies are used via the rates in Tab. \ref{['tab:gap_summary']}.