Data post-processing gain resulting from the patchy nature of speckles

TL;DR

The paper addresses how the patchy, azimuthally varying speckle pattern affects exoplanet sensitivity and exposure-time calculations in space-based direct-imaging systems like the Habitable Worlds Observatory. It introduces a weighted-mean framework to combine observations with different speckle-noise levels, deriving a post-processing gain that can exceed the standard $S/N$ improvement of $\sqrt{2}$, with a closed-form gain $g = \left(1 - \left(\frac{\Delta}{p + b_0}\right)^2\right)^{-1/2}$ and a limiting infinite-roll gain $g_\infty = \sqrt{\langle \sigma_\theta^2 \rangle \langle 1/\sigma_\theta^2 \rangle}$. Through both infinite-roll and finite-roll analyses, including a sine-wave speckle toy model, the work shows that a small number of rolls can approach the maximum gain, particularly when starlight dominates the noise budget, while also noting practical overheads and the importance of 2D noise maps for optimal PSF subtraction. The findings imply that ETCs should incorporate non-uniform speckle statistics to more accurately predict detection sensitivity and may influence coronagraph optimization strategies by prioritizing metrics like $\langle 1/I \rangle$ over $\langle I \rangle$. Overall, the study provides a general and potentially impactful enhancement to exposure time planning for exoplanet yields and characterization with ADI-like observing strategies.

Abstract

The data post-processing gain is an important parameter for exposure time calculations used to inform the design of the Habitable Worlds Observatory (HWO). Assuming azimuthally symmetric noise properties is a common simplifying assumption for such simulations, which neglects the patchy nature of the residual diffracted starlight; i.e., speckles. Fortunately, patchiness might prove to be an opportunity that improves the overall sensitivity of observatory assuming photon-noise limited speckle subtraction. We illustrate this effect in the context of angular differential imaging (ADI), which is one of the possible observing strategies being considered for the detection and characterization of exo-Earth with HWO. We show that combining observations of two observatory roll angles leads to a gain in signal-to-noise greater than $\sqrt{2}$ when the patchy starlight dominates other noise sources. The gain can be closer to x2 when the starlight dominates the noise budget by more than an order of magnitude. In other words, combining good and bad observations is better than combining two average ones. This statement is very general as it is a direct consequence of combining data with a weighted mean. It applies more broadly to any combination of observations with varying noise level.

Figures (4)

• Figure 1: Illustration of two observations of a planet with different underlying starlight noise levels. The starlight and other background sources are represented in orange, while the planet signal is shown in blue. In observation 1, the planet sits on top of a bright speckle. In observation 2, the speckle flux is null and the planet only sits on top of the uniform background light. The mathematical notations are defined in the text.
• Figure 2: Average sensitivity gain for a simulated PSF. (Left) Image of a simulated dark hole. The PSF was computed using the CDS pipelineBelikov2024SPIE13092E..66B and convolved with a $1\lambda/D$-radius aperture. (Middle) Azimuthal profile of the speckle field at $5\lambda/D$ from the star compared to a typical exo-Earth flux ratio of $10^{-10}$. (Right) Average gain $g_\infty$ as a function projected separation assuming a negligible planet flux (Starlight only) or comparable planet-starlight flux (Starlight and planet).
• Figure 3: Illustration of a multiple-roll observations, here assuming $N_\theta=3$, with toy model speckle pattern following a sine wave as a function of azimuth $\theta$. The left panel illustrates the position of the planet relative to speckles in an ADI observation. For any number of observations, the roll angles are defined such as to uniformly sample one period of the sine wave. The right panel illustrates the varying speckle intensity as a function of azimuth. The photon noise sources include a uniform background component, the varying speckle, and the planet.
• Figure 4: Average post-processing gain. It is computed as a function of the planet-to-background flux ratio $p/b_0$, the speckle-to-background flux ratio $\Delta_0/b_0$, and the number of roll angles $N_\theta$. The code used to generate this figure is available on Github: https://github.com/jruffio/speckles_snr_gain.