1.1.

SIRS in the Context of Other Work

Most reference correction is done in the time domain. Areas of reference pixels are averaged or a median is computed, and the result is subtracted from photosensitive pixels. The JWST pipeline implementation is typical.⁴ The reference rows and columns are used differently, and the reference rows correction tends to be more widely used.

Because they are sampled only at the beginnings and ends of frames, but appear in all outputs, the reference pixels in rows are used to remove constant offsets between outputs. This is done by robustly computing a spatial average or median of the reference pixels in rows in an output and subtracting it from every pixel in that output. By treating even and odd columns separately, one can also begin to remove some of the alternating column noise (ACN) that is sometimes seen in H2RG detector systems.

In H2RG images (we have seen no ACN in H4RG data), when it is present ACN appears as faint vertical bars in alternate columns. In Fourier space, ACN appears as a feature at the Nyquist frequency with a $1/f$-like shoulder on the low-frequency side. Physically, we believe that the alternating column pattern at the Nyquist frequency likely carries much of the same $1/f$ that appears at low frequency but upconverted to the Nyquist frequency. We refer the interested reader to Ref. 5 for more information on ACN.

When the reference columns are used, JWST’s implementation is typical. A reference pixel signal is calculated using a running median of the reference columns on the left and right. These are optionally smoothed using a kernel having tunable width, multiplied by a tunable gain parameter, and subtracted from the photosensitive pixels. Schlawin et al. (2020)⁶ provided a good recent introduction to this and other advanced time domain techniques in the context of JWST’s Near-Infrared Camera (NIRCam).

SIRS is different. Like JWST’s ${\mathrm{IRS}}^2$, it is built on the observation that the read noise is approximately covariance stationary over a frame readout time. Fourier space, therefore, provides a much less correlated representation than time space. SIRS then seeks the best linear projection of the references onto the data using least squares as the figure of merit.

With careful tuning, it has been learned that the JWST pipeline can approach the cosmetic performance of SIRS in rejecting $1/f$ banding. But SIRS still enjoys quantitative advantages, and moreover, these advantages are realized without the need for any tuning and one knows that the resulting correction is optimal if least squares is accepted as the figure of merit.

1.2.

When Should SIRS be Considered?

SIRS’s main benefit is reducing correlated low-frequency read noise. To quickly determine whether SIRS might help in a particular situation, the key diagnostic is visible banding in slope images (see Fig. 7; we defer detailed discussion of this figure to Sec. 7). If banding is visible, and if the typical band has a height that is more than two lines, then SIRS should help. A “slope image” is the result of least squares fitting straight lines to up-the-ramp sampled IR array data.

More quantitatively, SIRS works when there is temporal correlation in the pixel time series that is longer than the time required to read two lines. The factor of two is needed because the reference columns must critically sample correlated noise to remove it. In Fourier space, this becomes $f_{\text{noise}}<f_{\mathrm{Ny},\text{line}}$, where

Returning to Fig. 7, (1) there is visible banding and (2) the height of a typical band appears to be a few lines. This is a situation where SIRS should help.

1.3.

Paper Outline

The remainder of this paper is structured as follows.

Section 2 presents the scientific rationale for why reducing correlated noise is important. This includes a discussion of why correlated noise is particularly important to Roman.

In Sec. 3, we briefly describe the technical features of HxRG detectors that are most important for this paper. These include the reference pixel layout and a description of the conventional clocking pattern that was used. Teledyne’s user manuals describe several clocking options. This section explains which options were used here.

We then describe the theory in Sec. 5. This includes describing how least squares are used for optimization and presenting the key equations. This section includes a new derivation that can be easily extended to incorporate additional references as they become available.

SIRS’s development was motivated and enabled by the large data sets that the Goddard Detector Characterization Laboratory (DCL) is producing for Roman. In Sec. 7, we use these data to show how SIRS works in practice.

Finally, Sec. 8 describes future work. This includes modifying SIRS to work with the flight clocking pattern that will be very different from what was used here.

We close with a summary.

3.1.

Reference Pixels

When seen in science data, the HxRG outputs usually appear as thick vertical stripes running the full “height” of the detector. For example, in an H4RG read out using 32 outputs, the pixels from one output would appear in a contiguous 128 pixel wide by 4096 pixel high rectangle. If one looks carefully at an exposure with some light in it, the reference pixels will appear as a four-pixel wide dark frame surrounding the illuminated area (unless the pipeline has cropped them off). The description given here is how the outputs most often appear in science data. Sometimes they appear transposed, but this is a detail of how the instrument builders have chosen to present the data.

Figure 1 shows how these features are physically laid out looking down on a Nancy Grace Roman Space Telescope (“Roman”) H4RG. The important point here is that there are reference rows and reference columns.

Fig. 1

We developed SIRS for use with Roman ground test data. (a) Roman’s Wide Field Instrument (WFI) will fly eighteen $4096\times 4096\text{pixel}$ Teledyne H4RGs. (b) The WFI configuration uses 32 outputs. All HxRGs embed a 4-pixel wide border of reference pixels in the data streams. The Roman fast scan directions use the default Teledyne configuration that alternates. In this figure, the fast scanners run from left to right in the odd numbered white colored outputs and from right to left in the even numbered gray colored outputs. The pixel readout rate is $2\times {10}^5\text{pixels}{\mathrm{s}}^{-1}{\text{output}}^{-1}$. There is a few pixel time overhead at the end of each row before clocking the first pixel of the next row. The prototype H4RG-10 shown in (a) is included for illustrative purposes only. It is not the flight design and some details have been deleted to comply with U.S. legal requirements. This figure should not be relied upon for engineering purposes.

The pixels in the reference rows are usually robustly averaged to apply an additive correction for constant offsets between outputs. Within SIRS, they are used in the same way as in many other pipelines. The reference rows correction is made at $f=0\mathrm{Hz}$, whereas the reference columns are used to correct all other frequencies. In Fourier space, the 0-Hz basis vector is orthogonal to the $f>0\mathrm{Hz}$ basis vectors, so both SIRS and reference rows corrections can be made without greatly affecting one another.

The reference columns and rows do different but complementary things. Only outputs in Figs. 1(a) and 1(b) contain reference columns. Although in practice we find that there is often a useful degree of correlation with the other outputs. For every row that is read out, one gets eight reference samples from the reference columns. At Roman’s 200 kHz pixel rate, the corresponding Nyquist frequency is about $f_c\approx 741\mathrm{Hz}$. SIRS uses the reference columns to correct for frequencies $f>0\mathrm{Hz}$ and legacy reference rows techniques to correct for $f=0\mathrm{Hz}$.

In the remainder of this paper, we focus on the reference columns correction. We refer the reader who is interested in legacy reference rows techniques to the literature. The online JWST user manuals⁴ describe what is arguably the state of the art.

3.2.

SIRS Works with Common HxRG Clocking Patterns

SIRS places no special requirements on how the detector is operated. It works with all of the readout schemes that are described in Teledyne’s user manuals except for those that interleave guide windows. (We are working on enhancements to accommodate guide windows now.) For current Roman testing, we are using Teledyne’s “enhanced clocking mode” and “pixel by pixel reset.” The WFI’s 32 H4RG outputs are read out using a Gen-III Leach Controller from Astronomical Research Cameras, Inc., at $2\times {10}^5\text{pixels}{\mathrm{s}}^{-1}{\text{output}}^{-1}$. There is a little time overhead at the end of each line to allow for clocking to and starting the next line. This overhead is currently about 7 pixel readout times.

Teledyne’s HxRG detectors also offer global and line-by-line reset options. The Goddard DCL’s experience has been that global reset does not work well for low background space astronomy. Early in JWST testing, and subsequently for Roman, the DCL reported that global reset was causing undesirable artifacts (more than one) versus requirements. Because we had no compelling reason to use global reset, we quickly moved on to the other reset options.

The trade between pixel-by-pixel and line-by-line reset is more complicated. Both seem to work well for our applications. For JWST NIRSpec, we eventually selected pixel-by-pixel reset because: (1) each pixel sees the same lag between being reset and the first science sample, (2) each pixel sees the same exposure time independent of position, and (3) better thermal stability on account of the constant clocking cadence. At the time of writing, Roman is using pixel-by-pixel reset for the same reasons, although it is conceivable that this could change before flight.

In any case, we do not believe that the reset mode has any bearing on SIRS assuming that adequate settling time is allowed after resetting pixels. The key requirement is that the pixels always be sampled with the same constant and repeatable cadence apart from the end of line overhead. This requirement can be met with any of Teledyne’s reset options.

In practice, there are many ways to implement Teledyne’s recommended clocking patterns. In our software, the implementation-dependent parameters are defined in the module SIRS.jl. We recommend that users check with their instrument support teams to adjust these settings as necessary.

4.1.

Noise Power Spectrum

Figure 2 shows the average noise power spectrum for two outputs of one flight-grade Roman H4RG tested using a DCL Leach Controller. The two outputs fall essentially on top of one another and we saw no major differences between any of the outputs. Section 7 describes the test configuration in more detail.

Fig. 2

(a) The power spectrum on a log–log plot, in instrumental units, for two different outputs. We saw no qualitative differences between any outputs. Instrumental units are sufficient for making relative comparisons. SIRS is highly effective in rejecting the $1/f$ noise that appears at low frequency. Other notable features include bands at 60 Hz and harmonics due to the U.S. power grid and (b) a “comb” pattern spaced at intervals of the line frequency arising from interpolation over the regularly spaced end-of-line gaps. (b) The same points on a semilog scale. In addition to the features already mentioned, there is a line (of undetermined origin) at 83,688 Hz and its out of band second harmonic appears aliased down at 32,624 Hz. We looked for its third harmonic aliased down to 51,064 Hz, but did not find it. This suggests that either it is not very strong, or it may have been blocked by filters in the readout electronics. We tentatively attribute the enhanced “comb” pattern near 100 kHz to out of band “comb” aliased down.

To make these plots, we started with one 60 frame up-the-ramp sampled dark integration. We least squares fitted straight lines to all pixels and then subtracted the fits leaving residuals, which we identified as read noise. For each frame; we reshaped the outputs as time ordered vectors, interpolated over gaps and statistical outliers, and computed the fast Fourier transform (FFT). The power spectrum is the absolute value of the FFT squared. The plots show the power spectrum averaged over all 60 frames.

4.2.

Covariance Matrix

Figure 3 shows the low-frequency portions of the empirical covariance matrices $\mathbf{V}$ for the medians of lines in output #16. We computed $\mathbf{V}$ in both time and Fourier space using the same input data. We limited the frequency ranges for presentation purposes as it makes the diagonals easier to see. The other outputs are similar.

Fig. 3

(a) The covariance of the row-medians of output #16 computed in the time domain. There is strong off-diagonal covariance, indicating that the different time steps (pixel indices) are correlated in this representation. The diagonal emerges much more clearly in (b) Fourier space. Both (a) and (b) are on the same grayscale. The clouds around the edges in (b) reflect at least in part the uncertainties in the measurement. They become less pronounced as more data is added. The clouds are caused by the $1/f$ noise that SIRS aims to remove. The feature at 120 Hz requires more study, but it may appear so clearly because this line is so much stronger than any others (see Fig. 2).

We computed covariance using the standard relation:

To make $\mathbf{D}$, we started with a set of 100 identical 60 frame sampled up-the-ramp dark integrations. For each one, we fitted and subtracted straight lines to find the residuals, which we identified as read noise. We then extracted output #16 from every frame in the data set yielding 6000 instances of the read noise in output #16. Since we are most interested in low frequencies, we computed the median of each line as a statistically robust estimate of its expectation value. This yielded 6000 vectors, each 4088 elements long (excluding the four reference rows at the top and bottom). To reject constant offsets between frames, we subtracted the mean of each vector from itself. For the Fourier space matrix, we computed the FFT of each vector. In both cases, we put these vectors into a $6000\times 4088$ element matrix $\mathbf{D}$ and computed $\mathbf{V}$ using Eq. (2).

Figure 3 reveals the key insight behind both SIRS and JWST’s ${\mathrm{IRS}}^2$.³ The covariance matrix is more nearly diagonal in Fourier space. The clouds along the bottom and left edges of Fig. 3(b) are due at least in part to uncertainties in the measurement. They become less pronounced when more data are used to make the measurement. Because the covariance matrix is roughly diagonal in Fourier space, we pay a much smaller penalty for ignoring the off diagonal covariance than we would if working in the time domain as most pipelines do today.

5.1.

Derivation of Key Equations

This derivation leads to almost the same equations as appear by Rauscher et al. (2017).³ However, we eliminate a filter that is no longer needed and introduce a more compact notation that better reflects the references that are used here. Compared to our previous derivation, this derivation has the advantage that it can be more easily generalized to handle more reference streams. It also lends itself to purely numerical implementation, which may be advantageous when many reference streams are used.

Each frame of data contains normal pixels (grouped by output), reference columns on the left and right sides, and reference rows on the top and bottom. The data from the normal pixels and reference columns can be formatted into time series vectors; ($\mathbf{n}$) normal pixels, ($\mathbf{l}$) left reference columns, and ($\mathbf{r}$) right reference columns. Projected into Fourier space these become $\mathcal{n}$, $\mathcal{\ell }$, and $\mathcal{r}$. In this paper, boldface type indicates vectors and calligraphy fonts generally indicate Fourier space. Vectors that appear in calligraphy fonts are therefore complex.

There are gaps in the time series that complicate projection into Fourier space. The gaps arise because of the time overheads (typically a few pixel readout times) required to start clocking new lines and because the normal pixels are sampled much more regularly than the reference columns. To accommodate the gaps, we use the incomplete real Fourier transform (IRFT) rather than the more common FFT. Our IRFT is a specific type of non-uniform discrete Fourier transform⁸ that avoids interpolation. We refer the interested reader to Appendix A for more information. It is not necessary to understand the IRFT’s implementation to understand the SIRS algorithm.

Once in Fourier space, we model the read noise in the normal pixels as a linear combination of the read noise in the references:

Because the Fourier components are uncorrelated, we can solve for them independently. Let $\mathcal{n}$, $\mathcal{\ell }$, and $\mathcal{r}$ represent one element each of their corresponding vectors. Using Eq. (3), we can write a set of $m\gg 2$ equations in the two unknowns $\alpha$ and $\beta$:

The least squares solution of Eq. (4) is

To further simplify Eq. (7), it is helpful to define some sums:

The first two sums are real and the last three are complex. With these substitutions, Eq. (7) simplifies to:

In Eq. (3), we defined $\alpha$ and $\beta$ as vectors in Fourier space, but Eqs. (13) and (14) are for single Fourier components. Equations (8)–(14) must therefore be evaluated for all frequencies that will be used to make the reference correction.

There are other ways to find the least squares solution to Eq. (5). The least squares itself are just a generalization of the familiar solution for fitting a two-parameter straight line to data. We could, therefore, have subtracted the right side of Eq. (5) from the left side, summed the squares of the deviates (remembering that the deviates are complex numbers), and minimized this sum over $\alpha$ and $\beta$. The result is the same, although the geometric approach presented here scales more gracefully as additional reference streams become available.

5.2.

Example $\alpha$ and $\beta$

We developed SIRS using data from 18 Roman flight candidate H4RGs. Some of these tests are described in more detail in Sec. 7. Here we briefly discuss the measured values of $\alpha$ and $\beta$ for one Roman H4RG flight candidate: SCA 20663. These results are broadly representative.

Although $\alpha$ and $\beta$ clearly differ from detector to detector, the general characteristic of $1/f$ noise with spectral lines at 60 Hz and harmonics is always seen. In practice, SIRS must be trained for every detector. For any one detector, each test lasted about 8 h. We detected no time-dependent changes in $\alpha$ and $\beta$ for any of these detectors.

Figure 4 shows the low-frequency behavior on a semilog scale as a function of output number ($\alpha$ and $\beta$ are calculated separately for each output). The H4RG’s outputs are numbered 1 to 32, with output #1 being closest to the left reference columns that $\alpha$ operates on and output #32 being closest to the right reference columns that $\beta$ operates on. Output #16 is just to the left of the array’s center line.

Fig. 4

$\alpha$ and $\beta$ for 3 of H4RG 20663’s 32 outputs: (a) output #1, (b) output #16, and (c) output #32. As expected, the relative weights taken by $\alpha$ and $\beta$ shift depending on which reference columns are closest. Broadly speaking, outputs 2 to 15 look intermediate between outputs 1 and 16 and similarly outputs 18 to 31 look intermediate between outputs 17 and 32.

The HxRG’s reference pixels are physically located in the ROIC where they appear to be in images. As one would expect from the physical proximity, the relative importance of $\alpha$ and $\beta$ shifts depending on which reference columns are closest. Although the noise is dominated by $1/f$ at low frequencies, many spectral bands and lines are present. We have not tried to track down all of their causes, but we find that cryocoolers often appear at about 8 Hz, as seems to be the case here.

Figure 5 shows the same data on a linear scale and includes the highest frequencies near $f_{\mathrm{Ny}}=200\mathrm{kHz}$. Although we only show one output, these plots are typical if allowance is made for the shifting relative importance of $\alpha$ and $\beta$.

Fig. 5

(a) This figure replots Fig. 4 (center) on a linear scale. Visible features include $1/f$ noise and lines and bands associated with the US power grid at 60, 120, 180, and 240 Hz. (c) Because the reference pixels are the first pixels read in every line, the absolute value of the phase increases from zero at 0 Hz to $\frac{\pi }{2}$ at $f_{\mathrm{Ny},\text{line}}$. (b) There is very little amplitude near the Nyquist frequency, $f_{\mathrm{Ny}}=200\mathrm{kHz}$, and (d) phase is almost entirely noise at these frequencies. We zero-out frequencies $f>f_{\mathrm{Ny},\text{line}}$ when working with the existing Roman data.

Figure 5(a) again shows the strong low-frequency correlation. It also highlights some features that one often sees with US power: lines at 60 Hz (and harmonics) with a strong band at 180 Hz on account of the three phase power and fluorescent lights in the lab. Figure 5(c) shows that at 0 Hz, everything is in phase. But because the reference pixels are the first pixels read on every line, the absolute value of the phase increases to $\frac{\pi }{2}$ at $f_{\mathrm{Ny},\text{line}}$. The phase noise increases as the amplitude decreases.

Figures 5(b) and 5(d) show the high-frequency behavior near $f_{\mathrm{Ny}}=200\mathrm{kHz}$. Our conclusion is that there is no useful reference information at these frequencies for these detectors. We are, therefore, zeroing-out $\alpha$ and $\beta$ for $f>f_{\mathrm{Ny},\text{line}}$ in the current Roman test data.

This was a pleasant surprise. In JWST NIRSpec, there was significant ACN at $f_{\mathrm{Ny}}$. This mixed with low frequency $1/f$ to produce a high-frequency peak [see Fig. 8(a) of Ref. 3]. The ACN was caused by a Teledyne-proprietary design choice in the H2RG column buses.^3,5 Roman’s H4RG-10s do not seem to have this spectral feature. Although we do not have the same insight into the relevant details of the H4RG’s ROIC design, Rauscher did discuss ACN with Teledyne while they were designing the H4RG and what could be done to improve it.

Because we thought that similar ACN might still exist in the Roman H4RGs, SIRS fits for both low and high frequencies as shown here. If we had known in advance that only the low frequencies were important, it would have simplified the implementation somewhat. For now, we have left this feature in as things may change when the detectors are mated to the real flight electronics.

In any event, we do not expect SIRS to be particularly powerful at suppressing ACN. Unlike JWST’s ${\mathrm{IRS}}^2$, which uses a specialized clocking pattern to fully sample ACN, conventional clocking patterns do not sample reference often enough to be very effective at suppressing ACN.

7.1.

Noise Comparison

We can qualitatively compare the amount of correlated noise in the SIRS corrected set of data and the standard corrected set of data by inspecting the slope images from a representative exposure. Figure 7 shows the slope image from fitting the 60 samples up-the-ramp of a single exposure for SIRS corrected up-the-ramp data and up-the-ramp data using the standard correction. In Fig. 7, the horizontal banding from correlated noise in the slow readout direction is evident and stronger in the data corrected using only the last four rows of reference pixels.

Both the SIRS and standard reference corrections were done frame-by-frame, prior to fitting for slope. Although reference correction can be done before or after slope fitting and will often give about the same result, the two are not mathematically equivalent. We view reference subtraction as part of differentially sampling the detector and therefore do it before slope fitting.

Fig. 7

Representative comparison of slope images derived from (a) SIRS corrected ramp and (b) a standard reference subtraction corrected ramp. Beside each image the output median for the 32 outputs is plotted to highlight the horizontal banding. The slope image derived from the SIRS corrected data show substantial reduction in the horizontal banding. The grayscale in the images is up-the-ramp group slope in units ${\mathrm{e}}^{-}{\mathrm{s}}^{-1}{\text{pixel}}^{-1}$. Since submitting this paper, a JWST colleague has pointed out that with careful tuning, one can approach SIRS’ low-frequency noise suppression using a reference columns option that is built into the JWST pipeline. We agree but believe that that SIRS still offers less visible statistical advantages without any need for tuning.

For both SIRS and JWST’s ${\mathrm{IRS}}^2$, we are often asked if it is acceptable for the up-the-ramp samples to be averaged into “groups” before reference correction. To a first approximation, the answer is yes. Since the operations are linear (apart from outlier rejection), the result will be about the same to within the roundoff errors.

To quantify the excess noise in data using only the standard correction, we can examine the residual variation in each output along the slow readout direction after subtracting off the output mean slope. Figure 8 shows the residuals of a central output (#15) for both sets of data for a representative exposure in Fig. 8(a). In Fig. 8(b), the ratio of the standard deviation of the SIRS corrected data residuals to data residuals using the standard correction is plotted for each output of the same exposure. For the exposure shown in Fig. 8, the SIRS corrected slope image data has $\sim 33\%$ less variation. Overall, when looking at a total of 105 exposures, the residual variation in the SIRS corrected data is about 25% less than the residual variation in data corrected using the standard reference pixel correction.

Fig. 8

Representative comparison of residuals in the median slope in output #15 as a function of pixel row. (a) The residual between the output median and the average value of the output median over all active pixel rows for both SIRS and standard reference pixel corrected data. (b) The ratio of the standard deviation of the residuals in (a) (SIRS divided by the standard correction) for all output channels. The residuals of the SIRS corrected slope image data are more strongly peaked at zero and show less variation than the data using the standard correction.

The histograms of the residuals per output channel in Fig. 8 appear coarsely normal, but to assess this quantitatively, we can look at statistical properties for both sets of residuals to test for normality. Normally distributed residuals would indicate random variation with pixel row and that the optimal model for the slope image data in an output is indeed a constant value independent of pixel row. Non-normality would suggest additional parameters are needed to describe the data. Figure 9 shows the calculated skewness, kurtosis, and the Durbin–Watson statistic for the residuals in all 32 outputs as 3 histograms. Skewness and kurtosis are the commonly defined standardized third and fourth moments of a probability distribution, respectively. The Durbin–Watson statistic is a measure of the serial autocorrelation in the residuals.⁹ For a normal distribution, skewness would be 0, kurtosis would be 3, and the Durbin–Watson statistic would be 2 assuming no autocorrelation. For the representative exposure in Fig. 9, the SIRS corrected data residuals have less skewness, data residuals from both corrections have excess kurtosis, and the SIRS corrected data have less autocorrelation. These results in conjunction with the results of Fig. 8 suggest that the residuals from both types of data are not completely normally distributed. But they suggest that the residuals from the SIRS corrected data could be represented by a Gaussian distribution with narrower standard deviation with enhanced tails.

Fig. 9

Statistical properties for output residuals from Fig. 8: distribution of (a) skewness calculated for each output’s residuals, (b) kurtosis calculated for each output’s residuals, (c) Durbin–Watson statistic calculated for each output’s residuals. For a normal distribution, skewness would be zero, kurtosis would be 3, and the Durbin–Watson statistic would be 2. The SIRS corrected data residuals have less skewness, data residuals from both corrections have excess kurtosis, and the SIRS corrected data have less autocorrelation.

Normality is a key feature to look for in this data because it implies the pixel noise properties could permit averaging over multiple pixels to increase precision and accuracy by the reduction of uncorrelated noise. We compare how the pixel noise averages down for both types of reference pixel correction using square apertures of increasing size in the central output (#15) of their respective slope images. The square aperture sizes are $3\times 3$, $5\times 5$, $9\times 9$, $21\times 21$, $51\times 51$, and $101\times 101\text{pixels}$. The mean signal in an aperture is calculated for each exposure. The standard deviation of the mean as a function of aperture size is then calculated. The variation in the mean should decrease with increasing aperture size for data containing few correlations. Fig 10 shows the results for computing aperture means for 105 exposures and estimating the standard deviation of that mean for each aperture across those exposures. Figure 10(b) shows that the variation (noise) of the aperture mean in the SIRS corrected slope data decreases more rapidly with aperture size and to nearly half the value of the variation of the aperture mean in the slope data corrected with the standard correction. The SIRS corrected data, however, does still plateau at noise levels above the theoretical limit. This is likely due to residual $1/f$ noise not completely removed in this simple algorithm. Figure 10(c) shows the relative standard deviation (standard deviation divided by absolute value of mean). We can see that the relative standard deviation is smaller for the SIRS corrected data, and that it decreases below the mean value in contrast to the relative standard deviation for the data corrected using the standard correction. This is consistent with the larger amount of correlation in the data using the standard correction.

Fig. 10

(a) Image of the region ($101\times 101$) where the aperture averaging is done with red rectangles overlaying the smaller apertures. (b) Standard deviation of the mean as a function of aperture, calculated using 105 exposures. (c) Relative standard deviation (standard deviation divided by absolute value of mean) as a function of aperture, calculated using 105 exposures. The variation (noise) of the aperture mean in the SIRS corrected slope data decreases more rapidly with aperture size and to nearly half the value of the variation of the aperture mean in the slope data. However, both corrections bottom out after the aperture is larger than $9\times 9$ indicating there is still room for improvement; this is likely due to residual $1/f$ noise that is not removed by the current algorithm. Nevertheless, the relative standard deviation is smaller for the SIRS corrected data, and it decreases below the mean value in contrast to the relative standard deviation for the data corrected using the standard correction.

7.2.

Comparison to Subtracting Reference Columns

Since submitting this paper, a JWST colleague has compared SIRS to one of the JWST pipeline reference correction methods using column 4. Using the side reference columns and smoothing them over 11 rows, he was able to produce images that were visibly closer to Fig. 7(a) with regard to horizontal banding. Cosmetically, the result was much cleaner than just using the reference rows, and the visible difference was much less dramatic compared to Fig. 7.

Mathematically, this makes sense. Smoothing the columns with a kernel can be implemented with a filter in Fourier space. SIRS can also be viewed as applying (more complex) filters in Fourier space. Viewed this way, SIRS still enjoys compelling statistical advantages, even if the difference is less obvious to the eye.

The first is that SIRS is statistically optimal using least squares as the figure of merit. To the extent that the the deviates follow an approximately normal distribution, it is also optimal with regard to maximum likelihood and bayesian figures of merit. The second is that SIRS does the tuning with superhuman precision. It is impossible to reproduce the complex structure shown in Fig. 5 using only a small handful of tuning parameters such as window height and gain. Finally, SIRS uses frequency-dependent gain and phase information that is discarded in the implementations that we are aware of.

7.3.

Response Comparison

To look for any potential differences in the measured response when using the SIRS correction versus the standard correction, we can examine the mean signal as a function of frames up-the-ramp using the second data set with flatfield illumination. The array mean is estimated in each frame after subtracting the baseline signal in the first frame and plotted as a function of time in Fig. 11. No significant differences are apparent in the response implied from the two distinct reference pixel correction schemes.

Fig. 11

(a) Mean signal in illuminated frames up-the-ramp for a representative illuminated exposure. The array mean is estimated in each frame after subtracting the baseline signal in the first frame and plotted as a function of time. The means for the SIRS corrected data lie on top of the means for the data corrected using the standard reference subtraction. The dashed black line shows the linear trend expected for constant signal. (b) The fits of the slopes and corresponding residuals for the mean signals of the SIRS corrected data and the data corrected using the standard reference pixel correction. (b) The residuals to the fits. The response rolls off as the detector system nears saturation around $80{\mathrm{ke}}^{-}$, but there are no significant differences in response due to the SIRS correction. The best fit slopes, ${\beta }_{1,\mathrm{SIRS}}:381.73\pm 0.501$ and ${\beta }_{1,\mathrm{std}}:381.67\pm 0.486$, for both data sets are similar and indistinguishable within the error bars of the fit.

8.1.

Planned Roman Upgrades

The DCL acquired the data that underlie this paper using flight candidate H4RGs and non-flight Leach controllers. For flight, most of the same detectors will be built into “triplets.” A triplet consists of (1) H4RG detector, (2) interconnect cable, and (3) ACADIA application specific integrated circuit (ASIC). The differences, such as the ACADIA replacing the Leach controller, a different clocking pattern, and additional references, will be available.

The ACADIA is a new IR array controller ASIC that aims to improve on the Teledyne SIDECAR¹⁰ ASICs that were used by JWST, Euclid, and many other observatories. Because it is new, we cannot say anything about the performance of flight ACADIAs at this time other than we understand that they are meeting their requirements. For more information about ACADIA, we refer the interested reader to Ref. 11.

Roman plans to use the H4RG’s “guide window” for observatory guiding. Although the Roman project has tested guide windows extensively, they were not used for the data that are the basis of this paper. Using the guide window requires a more complex clocking pattern that interleaves controlling the guide window with reading out normal pixels. The new timing pattern will have different gaps and overheads than the one that was used for this paper. This will in turn require modifications to the SIRS software. Depending on how extensive the modification are, SIRS may be superseded by something that uses the same mathematical techniques, but that is adapted to use all of the references available at triplet level optimally including the H4RG’s reference output.

8.2.

Planned JWST Upgrades

With JWST launch now imminent, we plan further work optimizing SIRS for JWST NIRCam and NIRISS. NIRSpec already provides ${\mathrm{IRS}}^2$ readout which will be superior to SIRS.

The primary limitation that we are aware of concerns the ACN that we have only briefly touched on here in this paper because it falls outside the scope of the Fourier domain reference corrections that are our main focus. At the time of writing, SIRS implemented a very simple version of reference rows correction to remove constant drifts between outputs. We believe that with a little more work on the reference rows correction, we should be able to produce a version of SIRS that is superior to JWST’s pipelined reference correction in all respects.