Exposure-averaged Gaussian Processes for Combining Overlapping Datasets

TL;DR

This work tackles the problem of combining time series from multiple instruments when exposure times are non-negligible relative to the intrinsic stellar variability timescales. It introduces exposure-integrated Gaussian process kernels by analytically integrating the latent covariance over exposure intervals, deriving $k_{FF}$ and $k_{Ff}$ for overlapping and non-overlapping exposures, with explicit SHO-based kernels for granulation and p-mode oscillations. The framework decomposes multi-component GP models and adds instrument-specific drift GPs to jointly model solar data from several instruments, enabling robust cross-instrument comparisons. Demonstrations on simulated data and real ESSP solar data show improved handling of exposure times and drifts, providing a scalable path for applying GP kernels toany binning scenario in astronomy and beyond.

Abstract

Physically motivated Gaussian process (GP) kernels for stellar variability, like the commonly used damped, driven simple harmonic oscillators that model stellar granulation and p-mode oscillations, quantify the instantaneous covariance between any two points. For kernels whose timescales are significantly longer than the typical exposure times, such GP kernels are sufficient. For time series where the exposure time is comparable to the kernel timescale, the observed signal represents an exposure-averaged version of the true underlying signal. This distinction is important in the context of recent data streams from Extreme Precision Radial Velocity (EPRV) spectrographs like fast readout stellar data of asteroseismology targets and solar data to monitor the Sun's variability during daytime observations. Current solar EPRV facilities have significantly different exposure times per-site, owing to the different design choices made. Consequently, each instrument traces different binned versions of the same "latent" signal. Here we present a GP framework that accounts for exposure times by computing integrated forms of the instantaneous kernels typically used. These functions allow one to predict the true latent oscillation signals and the exposure-binned version expected by each instrument. We extend the framework to work for instruments with significant time overlap (i.e., similar longitude) by including relative instrumental drift components that can be predicted and separated from the stellar variability components. We use Sun-as-a-star EPRV datasets as our primary example, but present these approaches in a generalized way for application to any dataset where exposure times are a relevant factor or combining instruments with significant overlap.

Figures (7)

• Figure 1: When computing the double integral $k_{FF}$, any two exposures can be decomposed into a sum of perfectly overlapping and separate sub-exposures; each sub-integral is evaluated with newly defined exposure times ($\delta$s) and time separations ($\Delta$s). Note that to compute the sum of the three sub-integrals, one must first multiply each sub-integral by the product of the two sub-exposure times before summing, and finally dividing by the original exposure time product to obtain the final result. For example, in Case 3, the total integral $k_{FF}(\delta_1,\delta_2,\Delta) = 1/({\delta_1\delta_2}) \left(k_{FF,sep}(\delta_{a1},\delta_{a2},\Delta_a)*\delta_{a1}\delta_{a2} + k_{FF,over}(\delta_{b})*\delta_{b}^2 + k_{FF,sep}(\delta_{c1},\delta_{c2},\Delta_c)*{\delta_{c1}\delta_{c2}}\right)$.
• Figure 2: Left: PSD for the solar variability kernels used in this work. We use a sum of a granulation kernel and an oscillation kernel. Right: Sample draws for each of the stellar variability kernels (arbitrary offsets added for clarity.
• Figure 3: Simulated time series for our 4 representative instruments, assuming no instrumental noise. The colored horizontal lines indicate each observation's exposure time and lighter points display the observed RV centered within. The "true" stellar variability signal is shown in black in each panel (the sum of the components shown in \ref{['fig:PSD_sim']}). For each instrument, we use the input GP model with fixed hyperparameters to compute the predictive mean for the latent signal (e.g., accounting for the exposure time of each observation), which is shown by the colored line in each panel, with the 1-$\sigma$ confidence region shaded.
• Figure 4: An example of computing the predictive mean on a combined dataset, combining the synthetic time series from Instruments B, C, and D. The top two panels show the predictive mean without exposure-time accounting (teal) as compared to the true signal (black), and the bottom two panels show the same predictive mean after properly accounting for the exposure times. For each case, the lower panel shows the residuals between the prediction and the true synthetic signal. Note the extreme departures when the model is forced to fit two nearby points if it does not account for the exposure times.
• Figure 5: Recovery of instrument drifts from combining noise-free observations from 4 instruments with a model that includes both stellar signals and instrument drifts. Top panel: Recovered, or "predicted" instrument drifts (solid lines with 1-$\sigma$ confidence regions) compared to the true synthetic drifts for each instrument in the drift model case. Recovered drifts are overall well correlated with the true drifts, however, large departures exist, particularly evident between 75 and 100 minutes in this example. Bottom panel: Instead, pairwise relative drifts (e.g., differential drifts between instrument pairs in the top panel) are extremely well constrained, with the recovered pairwise relative drifts (semi-transparent light teal line) only differing from the true relative drifts in the edges of the time series beyond the observations ($t < 0$ min. or $t > 120$ min.); the residuals between the true relative drift and the recovered relative drift for $0 < t< 120$ min has RMS $< 1.5$ cm$\,$s$^{-1}$ for all curves.