ArCTIc: CCD CTI Correction Algorithm

• ArCTIc is a physically motivated and empirically calibrated algorithm that corrects CTI trailing in astronomical CCD images by modeling electron capture and release.
• It employs a well-filling function and multi-trap release model to simulate and iteratively invert CTI effects, achieving correction accuracy between 99.5% and 99.9%.
• Empirical calibration using warm pixel trails ensures the algorithm remains effective over decades despite evolving radiation damage.

ArCTIc (Algorithm for Charge Transfer Inefficiency Correction) is a physically motivated and empirically calibrated post-processing software framework developed to correct radiation-induced charge transfer inefficiency (@@@@1@@@@) trailing in CCD images acquired by astronomical telescopes, notably the Hubble Space Telescope’s Advanced Camera for Surveys (ACS). CTI manifests as faint, non-linear, and signal-dependent trailing behind objects due to the trapping and delayed release of electrons in radiation-damaged silicon. The ArCTIc algorithm models both the capture and release processes and uses calibration from hot/warm pixel trails to restore imaging accuracy to better than 99.5%–99.9% across the device lifetime (Massey et al., 5 Sep 2025).

1. Charge Transfer Inefficiency: Physical Origin and Manifestation

The electronic readout of CCDs in a radiation environment is fundamentally limited by CTI, arising from silicon lattice defects that act as charge traps. As charge packets are transferred from pixel to pixel, a fraction of electrons are captured by these traps and later released, typically after a delay governed by trap-specific exponential time constants. The delayed release results in charge trailing—spurious “ghosts” behind bright sources—which perturbs measured photometry, astrometry, and morphology. The severity of CTI trailing increases with accumulated radiation damage, signal level, and the pixel’s distance to the readout register.

In orbit, the density of charge traps evolves with cumulative radiation exposure, being modulated by factors such as solar cycle. Empirically, the damage rate in HST/ACS is modulated by $18.5^{+4.5}_{-0.5}\%$ over an 11-year cycle, peaking approximately $430^{+11}_{-5}$ days after solar minimum (Massey et al., 5 Sep 2025). Traps typically anneal into one of a small set of configurations, each characterized by a distinct release time constant.

2. Mathematical and Algorithmic Model

ArCTIc models the CTI effect by simulating the electron transfer through a population of trap species. Each trap species $i$ is characterized by:

• Trap density per pixel, $\rho_i$
• Characteristic release time, $\tau_i^{(\mathrm{rel})}$ (in units of pixel dwell time)
• A well-filling law for the probability of a trap being exposed, parameterized by $d$ (notch or threshold) and $\beta$ (power-law scaling).

Capture Model

Electrons are instantaneously (or gradually) captured into traps as the charge packet traverses pixel $j$. The number of electrons captured per transfer is: $$n_{\mathrm{capture}} = \sum_{i} \rho_i \cdot V(...),$$ where the well-filling function $V$ models the fractional pixel volume occupied by the charge cloud. By default,

$V() = \left( \frac{S - d}{w - d} \right)^{\beta},$

with $S$ being the signal, $w$ the full well capacity, $d$ the effective notch depth, and $\beta$ the cloud-growth index.

Release Model

Trapped electrons are released exponentially with characteristic times $\tau_i^{(\mathrm{rel})}$: $$f(t_0 + t_{\mathrm{dwell}}) = f(t_0) \cdot \exp\left(-\frac{t_{\mathrm{dwell}}}{\tau_i^{(\mathrm{rel})}}\right).$$ The convolution of multiple trap species yields the observed multi-exponential charge trails behind objects.

Computational Considerations

To achieve both computational efficiency and physical accuracy, ArCTIc employs the “express” method, which groups together multiple transfers using a matrix formalism (denoted $E_n$), approximating the effect of large numbers of pixel shifts while properly updating trap occupancy. This approach significantly reduces computational cost, with runtime on ACS quadrants being $\sim$1 s on commodity hardware.

3. Empirical Calibration Using Trails of Warm Pixels

Achieving high-fidelity correction requires empirical calibration of the trap densities and release times. ArCTIc uses the population of hot/warm pixels as “standard candles”: delta-like sources whose trails directly encode the amplitude and time structure of CTI. The calibration pipeline is as follows:

1. Identification of Warm Pixels: Pixels with consistently high charge excess above the background, stationary in detector coordinates, are identified and checked to exclude astronomical sources.
2. Trail Extraction: For each warm pixel, the net CTI trail is measured by subtracting background and integrating over fixed distances (typically 12 trailing pixels).
3. Parameter Fitting: Observed trail profiles, binned by pixel brightness and register proximity, are fitted to the predicted model trails using a nested sampling approach (e.g., Dynesty). The fitted parameters include trap densities, $\beta$, $d$, and $\tau_i^{(\mathrm{rel})}$ for each trap species.
4. Green’s Function: The calibration step uses a Green’s function (analytic) trail form,

$$^{\mathrm{trail}}(\Delta y) \approx \{ [ ((^{\mathrm{wp}} - d)/(w - d))^{\beta} - ((^{\mathrm{bg}} - d)/(w - d))^{\beta} ] \cdot y \} \sum_{i} [1 - \exp(-1/i)] \exp[(1-\Delta y)/i],$$

where $^{\mathrm{wp}}$ is the warm-pixel signal, $^{\mathrm{bg}}$ the background, $y$ the row, and $\Delta y$ the offset behind the warm pixel.

This empirical calibration is repeated across epochs (178 ACS epochs from 2002–2025 in the reference implementation), ensuring the model tracks temporal evolution of radiation damage and trap densities.

4. Iterative CTI Correction in Image Post-processing

Correction proceeds by iteratively “inverting” the forward CTI model:

• Forward Model: Given an estimate of the true scene, the CTI model is applied to simulate expected trailing.
• Iterative Residual Minimization: The corrected image is updated as $A^\prime = A + (A - B)$, where $A$ is the current corrected image and $B$ is the forward model output. This process is repeated (typically $\sim$5 times) until the forward simulation of the corrected scene matches the observed image.
• Handling Read Noise: As read noise is not subject to CTI during readout but is untrailed during correction, each iteration slightly increases noise in the corrected image. This sets a practical limit to correction performance for faint sources.

The algorithm’s design allows for batch processing of large data sets with minimal performance penalty, making routine post-processing of HST survey data feasible.

5. Correction Accuracy and Impact on Observational Data

ArCTIc achieves superior correction efficiency, removing 99.5% of all CTI trailing averaged over the ACS operational lifetime and 99.9% in the most recent, most highly damaged data (Massey et al., 5 Sep 2025). The corrected data show:

• Elimination of CTI-Induced Biases: Systematic photometric underestimation and morphological distortion are essentially removed, restoring reliable measurement of source brightness, position, and shape.
• Consistency Across Background Levels: Empirical calibration ensures the algorithm self-consistently accounts for background dependence (“shadowing” of traps), with no significant systematic residuals remaining in dark or science field images.
• Stability Across Temporal Evolution: Calibration at the hot pixel level ensures ArCTIc tracks changes in trap population due to ongoing radiation damage and annealing, maintaining effectiveness over decades.

For science drivers such as weak gravitational lensing or astrometric missions, the correction performance is sufficient to prevent CTI from exceeding the allowable systematic error budgets even for the most stringent requirements.

6. Physical Foundations and Broader Applicability

ArCTIc’s underlying model structure—capture per well-filling law, multi-species exponential release, iterative correction—is directly transferable to other CTI-dominated devices, provided appropriate empirical calibration data are available. The well-filling law and multi-trap formalism parallel models adopted in Gaia, Euclid, LSST, and XRISM correction frameworks (Prod'homme et al., 2011, Short et al., 2013, Israel et al., 2015, Snyder et al., 2020, Kanemaru et al., 2020), confirming the universality of ArCTIc’s core approach.

The modularity of the model, parameterized entirely by empirically measurable quantities, also enables adaptation to device-specific readout schemes (e.g., varying dwell times, device geometry) and operational conditions (temperature, background, clocking speed).

7. Limitations and Future Directions

While ArCTIc achieves near-complete removal of CTI trailing, a fundamental limitation is set by the amplification of read noise during the correction process (Massey et al., 2014, Israel et al., 2015). As noise is untrailed in hardware but untrailed in software correction, this can introduce small, irreducible residual errors, especially in very low-signal regimes. Additionally, potential improvements include:

• Refinement of Express Approximations: Further reduction of residuals at extremely low electron counts may be possible by refining the express grouping scheme.
• Upgrade for Multi-Phase Clocking: Enhanced accuracy might be realized by explicit simulation of multi-phase clocking schemes or 3D trap mapping.
• Application to Extended Devices: Broader adoption in future and current space missions will require appropriate in situ calibration campaigns, as device-specific trap populations, dwell times, and well-filling parameters must be empirically constrained for each detector.

Summary Table: Key ArCTIc Components and Parameters

Component	Parameter(s)	Purpose
Trap densities	$\rho_i$	Amplitude of CTI per trap species
Well-filling function	$d$, $\beta$, $w$	Volume of pixel exposed to trapping
Trap release times	$\tau_i^{(\mathrm{rel})}$	Governs decay profile of trails (multi-exponent)
Empirical calibration	Hot/warm pixel trails	Fits model parameters epoch-by-epoch
Express approximation	$E_n$	Efficient simulation of many transfers
Iterative inversion	$A^\prime = A + (A-B)$	Correction of observed images

The ArCTIc algorithm represents a comprehensive, empirically validated approach to restoring image fidelity in radiation-damaged CCDs, combining physically motivated capture/release modeling, efficient computational techniques, and robust per-epoch empirical calibration. Its deployment on ACS/HST data has demonstrably enabled science requiring sub-percent level systematic error budgets over two decades of operation (Massey et al., 5 Sep 2025).