Exposure-averaged Gaussian Processes for Combining Overlapping Datasets

Technical Summary: Exposure-Averaged Gaussian Processes for Combining Overlapping Datasets

Problem Statement
Gaussian processes (GPs) are widely used to model stellar variability, employing physically motivated kernels such as damped, driven simple harmonic oscillators (SHOs) to represent granulation and p-mode oscillations. These standard kernels quantify the instantaneous covariance between two points in time. However, this formulation assumes observations are instantaneous samples of a latent signal. In reality, many modern Extreme Precision Radial Velocity (EPRV) spectrographs, particularly those with solar feeds (e.g., HARPS, HARPS-N, EXPRES, NEID, KPF), operate with exposure times that are comparable to the timescales of the stellar variability they aim to measure (minutes to hours).

When exposure times are significant, the observed signal is an exposure-averaged (binned) version of the true latent signal, not an instantaneous point. Naively combining datasets from multiple instruments with differing exposure times using standard instantaneous kernels leads to spurious signals. The model fails to reconcile closely spaced observations that exhibit real radial velocity (RV) differences solely due to their different integration times. Previous attempts to combine such data involved binning all observations to a common, coarse timescale or interpolating between datasets, both of which discard high-frequency information (such as p-mode oscillations) or rely on non-physical interpolation schemes.

Methodology
The authors present a GP framework that explicitly accounts for finite exposure times by computing integrated forms of the instantaneous kernels. The methodology is built on three core components:

1. Kernel Integration Framework: The authors derive analytic expressions for the covariance between:

   • Two binned observations (double integration of the latent kernel).
   • A binned observation and a latent instantaneous point (single integration).
   • Two latent instantaneous points (the standard kernel).
     For stationary kernels, the authors demonstrate that any complex overlap scenario (partial or complete) can be decomposed into sums of "perfectly overlapping" and "completely separate" sub-exposures. This allows the evaluation of the covariance matrix for arbitrary exposure durations ($\delta$) and time separations ($\Delta$).

2. Analytic Solutions for SHO Kernels: The paper provides specific analytic expressions for integrating the stochastically driven, damped SHO kernel (commonly used for granulation and oscillations). This includes helper functions ($I_{SHO}$) to evaluate the integrals for both single and double integration cases, handling the sign flips in the time difference $|t - t'|$ that occur within overlapping exposures.

3. Multi-Component and Multi-Instrument Decomposition:

   • Component Decomposition: The framework extends to multi-component GP models (e.g., sum of granulation, oscillation, and rotation kernels). The authors detail the linear algebra required to condition the full model on data and subsequently extract the predictive mean for individual latent components.
   • Instrumental Drifts: To handle overlapping datasets from multiple instruments, the model treats each instrument's drift as an independent GP component added to a common latent stellar signal. This is implemented via a multivariate GP where the kernel for instrumental drifts is masked by a Kronecker delta, ensuring drift covariances only exist between observations from the same spectrograph. This allows the separation of "stellar" variability from "instrumental" drifts without prescribing a specific functional form for the drift.

Key Results
The framework was validated through two primary applications:

1. Simulated Data: The authors simulated solar RV data mimicking four instruments (KPF, NEID, EXPRES, HARPS-N) with exposure times ranging from 12 to 300 seconds.

   • Exposure Accounting: Models that ignored exposure times produced erroneous predictive means, particularly failing to fit closely spaced observations with different exposure durations. The exposure-aware model correctly predicted the latent signal, allowing the predictive mean to be more extreme than the binned observations (e.g., passing through extrema that were smoothed out by long exposures).
   • Drift Recovery: When synthetic instrumental drifts were injected, the model successfully recovered the relative drifts between instrument pairs with high precision (RMS < 1.5 cm s$^{-1}$). However, the absolute drifts of individual instruments were found to be highly degenerate (often anti-correlated), as the model cannot uniquely distinguish which instrument is drifting without additional constraints.

2. Real Solar Data:

   • KPF & NEID (Single Day): Applied to overlapping data from April 19, 2024, the model successfully separated the stellar signal from instrumental drifts. It identified a smooth drift in KPF relative to NEID, correlating with known operational events (liquid nitrogen fills).
   • Extreme Stellar Signals Project (ESSP): Applied to a 28-day dataset combining HARPS, HARPS-N, EXPRES, and NEID (11,165 observations). The model successfully conditioned on the data to isolate stellar components (oscillations, granulation, supergranulation, and rotation) while correcting for instrument-specific drifts. Due to the computational cost ($O(N^3)$ scaling), hyperparameter fitting was not performed for this large dataset; instead, fixed hyperparameters were used to demonstrate the framework's ability to handle the intricacies of combining disparate datasets.

Significance and Claims
The paper claims that this framework is essential for the analysis of modern EPRV datasets, particularly those involving solar feeds where exposure times vary significantly between instruments. By properly integrating the kernels, the method:

• Preserves high-frequency information (p-modes, granulation) that is lost when data is binned to a common coarse timescale.
• Prevents spurious signals arising from the mismatch of exposure times in overlapping observations.
• Enables the decomposition of complex, multi-instrument datasets into distinct stellar and instrumental components, specifically isolating "pairwise relative drifts" which are robustly constrained.

The authors note that while the current implementation faces scaling limitations for very large datasets ($N > 3000$) due to the non-stationary nature of the integrated kernels (which depend on both time separation and individual exposure times), the framework provides a rigorous statistical foundation. They suggest that future work utilizing state-space models (Kalman filters) will address these scaling issues, enabling robust modeling of stellar variability and instrument drifts over longer timescales. The paper concludes that this approach is most beneficial for high-cadence datasets designed to study stellar variability, rather than standard survey-cadence planet-hunting data where exposure times are negligible relative to the variability timescales.