Computing the Electronic Gain for Detectors Read Out Up-The-Ramp

TL;DR

This work presents a likelihood-based framework to estimate detector electronic gain from up-the-ramp, nondestructive reads, addressing limitations of the traditional photon transfer curve by exploiting the full ramp covariance and marginalizing over the per-ramp count rate. The method remains accurate under photon-noise and Gaussian read-noise assumptions and accommodates mild nonlinearity via a second-order correction. Validation with synthetic data shows Gaussianity of the gain posterior and unbiasedness with sufficient ramps, and application to Roman Space Telescope WFI data yields per-pixel gain maps with ~3.5% precision, revealing that gain variations account for much of the flatfield structure. The approach provides a practical, scalable path to pixelwise gain calibration across large detector arrays, with public code for implementation and replication of results.

Abstract

The electronic gain -- the conversion between photoelectrons on a pixel and the digital number recorded to disk -- gives physical units to an astronomical image and sets the relation between pixel value and photon noise. This paper presents a new, likelihood-based approach to derive the gain from images taken up-the-ramp, where the detector is read out nondestructively many times before being reset. Our method makes full use of the individual reads assuming an ideal detector subject to photon noise and Gaussian read noise. We extend the method to account for slight nonlinearities in the relation between photoelectrons and measured counts. We demonstrate that our likelihood-based approach provides a consistent (i.e. asymptotically correct) and nearly unbiased estimator of the gain both with and without fitting for nonlinearity. Finally, we apply this approach to a single detector from the Wide-Field Instrument on the Roman Space Telescope, and show how pixel-to-pixel gain variations describe much of the variations in pixel response seen in flatfield images. Code to compute gain and regenerate figures in this paper is available at https://github.com/RomanSpaceTelescope/SOCReferenceFileCode.

Figures (6)

• Figure 1: Log-likelihood as a function of gain and read noise, marginalized separately over 100 ramps and then combined over all of those ramps. The red contours show $1\sigma$, $2\sigma$, and $3\sigma$ thresholds from $\Delta \chi^2$ thresholds. The left panel interprets the likelihood as a probability density in gain and read noise; the right panel interprets the likelihood as a probability density in the logarithm of the gain and read noise. The blue point and green contours are from fitting a quadratic form to the log likelihood at nine points near its peak. When interpreting the likelihood as a probability density in log gain and log read noise, its logarithm is very accurately described by a quadratic form (equivalently, the likelihood is very nearly Gaussian). In this case, shown in the right panel, the green contours---the $1\sigma$, $2\sigma$, and $3\sigma$ thresholds of a Gaussian likelihood described by the best-fit quadratic form---provide an excellent description of the likelihood.
• Figure 2: Probability distributions in $\log g$ for the same 100 ramps shown in the right panel of Figure \ref{['fig:likelihood_2d_test']} computed two ways: by a full numerical integral over the joint distribution shown in Figure \ref{['fig:likelihood_2d_test']} (solid blue line), and from the covariance matrix computed from the inverse of the quadratic form coefficients (orange dashed line). The two probability distributions are almost indistinguishable. The probability distribution of the gain for each pixel can thus be accurately computed using the approach outlined in this paper, at a cost of $\lesssim$20 times the cost of fitting the ramps used to determine the gain.
• Figure 3: Left: same as the right panel of Figure \ref{['fig:likelihood_2d_test']}, but for just 10 ramps of 30 reads each. The 10 ramps have random count rates ranging from 0 to 100 electrons/read. The joint probability distribution remains accurately Gaussian, though not to the same degree as with 100 ramps. Right: the marginalized probability distribution, computed as in Figure \ref{['fig:marginalized_pdists']}, likewise remains well-approximated by a Gaussian.
• Figure 4: Distribution of $z$-scores (difference between true log(gain) and best-fit log(gain) divided by the standard error of the fit) compared to a unit Gaussian. If the posterior probability distribution of the gain is reliable, the histogram and the unit Gaussian should match. The two distributions are indeed an excellent match in both cases. This strongly suggests that if the noise model of Gaussian read noise plus Poisson photon noise is accurate, the method presented in this paper will produce reliable gain values and reliable uncertainties on those gain values.
• Figure 5: Distributions of $z$-scores of log(gain) using two approaches to fitting for nonlinearity. The blue curves fit a three-dimensional likelihood in $\log g$, $\log \sigma_{\rm r}$, and nonlinearity coefficient $\alpha$, while the orange curves use a two-dimensional likelihood in $\log g$ and $\log \sigma_{\rm r}$ at the $\alpha$ that produces the largest gain. In the latter case the derived gain is adjusted to account for the small bias expected from using the nonlinearity coefficient that maximizes the gain. The left panel uses 50 ramps, each of 50 reads; the right panel uses ten times as much synthetic data.