Brighter-fatter effect in near-infrared detectors -- II. Auto-correlation analysis of H4RG-10 flats
BBrighter-fatter effect in near-infrared detectors – II. Auto-correlation analysisof H4RG-10 flats
Ami Choi [email protected]
Christopher M. Hirata
Center for Cosmology and AstroParticle Physics, The Ohio State University, 191 West WoodruffAvenue, Columbus, Ohio 43210, USA
January 27, 2020
ABSTRACT
The Wide Field Infrared Survey Telescope (WFIRST) will investigate the originsof cosmic acceleration using weak gravitational lensing at near infrared wavelengths.Lensing analyses place strict constraints on the precision of size and ellipticity measure-ments of the point spread function. WFIRST will use infrared detector arrays, whichmust be fully characterized to inform data reduction and calibration procedures suchthat unbiased cosmological results can be achieved. Hirata & Choi 2019 introducesformalism to connect the cross-correlation signal of different flat field time samples tonon-linear detector behaviors such as the brighter fatter effect (BFE) and non-linearinter-pixel capacitance (NL-IPC), and this paper applies that framework to a WFIRSTdevelopment detector, SCA 18237. We find a residual correlation signal after account-ing for classical non-linearity. This residual correlation contains a combination of theBFE and NL-IPC; however, further tests suggest that the BFE is the dominant mech-anism. If interpreted as a pure BFE, it suggests that the effective area of a pixel isincreased by (2 . ± . × − (stat.) for every electron in the 4 nearest neighbors,with a rapid ∼ r − . ± . fall-off of the effect for more distant neighbors. We show thatthe IPC inferred from hot pixels contains the same large-scale spatial variations as theIPC inferred from auto-correlations, albeit with an overall offset of ∼ . Subject headings: instrumentation: detectors a r X i v : . [ a s t r o - ph . I M ] J a n
1. Introduction
Weak gravitational lensing (WL) is one of the primary tools the Wide Field Infrared SurveyTelescope (WFIRST) will use to detail the history of cosmic expansion and structure growth. WLrequires high fidelity measurements of galaxy shapes, which for WFIRST will be made on nearinfrared detector arrays consisting of a Teledyne H4RG-10 readout integrated circuit hybridized to2 . µ m cutoff HgCdTe. The WFIRST Science Requirements Document specifies that the pointspread function (PSF) ellipticity must be known with an error of ≤ . ≤ . §
2, we briefly summarize the theoretical predictionsfor the 2-point flat field correlation function contributions of IPC, BFE, and other detector effectsfrom Paper I. In §
3, we describe the data set for a H4RG-10 detector obtained from the Detector See Loose et al. (2007) and Rauscher et al. (2014) for descriptions of this technology. § §
5, we present and compare the main result of this paper: measurementsof the brighter-fatter effect. We conclude and discuss areas of future exploration in §
2. Theoretical background
In this section, we briefly recall the formalism from Paper I used to describe the main detectoreffects of interest: the brighter-fatter effect, (non-linear) inter-pixel capacitance, and classical non-linearity. We describe the main equations and parameters to orient the reader for the measurementsand reference the details from Sections 2 and 3 of Paper I. Table 1 provides a quick referencesummarizing the detector parameters most relevant to this analysis.
The observed signal S in the detector is given in units of data numbers (DN), which are voltagesquantized as 16-bit integers. As the detector is exposed to light, the voltage across the photodiodedecreases, causing S to decrease. In practice, the relation between the accumulated charge, Q , andthe signal drop is non-linear and contains various contributions including IPC. There is evidencethat the IPC increases with increasing signal level, and this non-linear component, NL-IPC, isphenomenologically similar to the BFE in that a greater amount of coupling will also cause a largerchange in FWHM in brighter stars. However, the two effects occur in different stages of the signalmeasurement process (NL-IPC occurs in the conversion of charge to voltage, whereas the BFEoccurs in the collection of charge) and imprint slightly different features on the flat field statistics,as we will investigate later on.In the presence of IPC, the signal drop can be described at a pixel location i, j (column androw indices) by: S initial ( i, j ) − S final ( i, j ) = 1 g (cid:88) ∆ i, ∆ j [ K ∆ i, ∆ j + K (cid:48) ∆ i, ∆ j ¯ Q ] Q i − ∆ i,j − ∆ j , (1)where ¯ Q is the mean accumulated charge ( It in a flat exposure, with current I per pixel given in unitsof e/s and time t in seconds) and g is the gain (units: e/DN). “Initial” is defined here as t = 0, orimmediately following a reset. The kernel matrix K , describing the IPC, satisfies the normalization (cid:80) ∆ i, ∆ j K ∆ i, ∆ j = 1. In the case where the cross-talk is equally distributed to the four nearestneighbors K , = 1 − α , K , ± = K ± , = α , and all others are zero. However, asymmetriesbetween the horizontal and vertical directions ( K , ± (cid:54) = K ± , ) are commonly observed, so weseparately measure α H = K ± , and α V = K , ± ; if these are different then we define α to betheir average ( α H + α V ) /
2. We also allow for diagonal IPC, α D = K ± , ± . In Equation (1), weparameterize the NL-IPC to be dependent on the mean signal level as the kernel matrix K (cid:48) . 4 –We also allow for a classical (total count-dependent) non-linearity in the detectors. This ismodeled by the mapping of charge Q → Q − βQ , where β (units: ppm/e) is the leading-order non-linearity coefficient. We perform this mapping after the IPC convolution. In reality non-linearityand IPC are happening at the same time, but in the case of small fluctuations around a mean signal(as occurs for a flat field) the ordering does not matter. We use the Antilogus et al. (2014) model, in which the pixel areas are modified by existingcharge in accordance with a kernel a ∆ i ∆ j : A i,j = A i,j (cid:88) ∆ i, ∆ j a ∆ i, ∆ j Q ( i + ∆ i, j + ∆ j ) , (2)where A i,j is the effective pixel area, and A i,j is the original pixel area. We will quote a ∆ i ∆ j inunits of 10 − e − , ppm/e, or equivalently %/10 e. Note that while a ∆ i ∆ j is formally dimensionless,the aforementioned choice of units is convenient because a measured value of a ∆ i ∆ j in ppm/e mapsinto the expected order of magnitude of the effect on a star in percent. In this work, we do notstudy the pixel-dependence of the BFE and also assume discrete translation invariance. We alsodefine Σ a = (cid:80) ∆ i, ∆ j a ∆ i, ∆ j ; for a pure BFE (no signal-dependent QE) we should have Σ a = 0. Assome of the tests we conduct later are not sensitive to Σ a , we also define: a (cid:48) ∆ i, ∆ j ≡ a ∆ i, ∆ j − δ ∆ i, δ ∆ j, Σ a . (3)By construction, the a (cid:48) coefficients sum to zero. As these infrared detectors allow multiple samples up the ramp, correlations can be measurednot only between different pixels but also between different frames. The temporal structure of thecorrelations is key to disentangling the BFE and NL-IPC. The correlation function C abcd (∆ i, ∆ j )responds to the BFE and NL-IPC as given by Eq. (51) in Paper I. abcd are indices representingframes, and we assume that a < b and c < d , since these functions contain all the informationbecause of symmetries, but we do not assume anything else about the ordering. The exposureintervals a...b and c...d may be the same, may overlap, or may be disjoint. For the purposes ofPaper II, terms of order α , α , β , a , αβ , and αa are kept, while higher order terms are dropped.First, we consider the non-overlapping correlation function, where a < b < c < d , whichleverages the non-destructive read capability of the infrared detectors to determine whether thereis a correlation between current fluctuations in an earlier part of an exposure and the current 5 –fluctuations in an adjacent pixel at a later part of the exposure and an anti-correlation in the samepixel. This is Eq. (58) in Paper I, through which we can use the ‘observables’ C abcd (∆ i, ∆ j ), I , g , α , α H , α V , and β to solve for the inter-pixel non-linear (IPNL) effects [ K a ] − ∆ i, − ∆ j + [ KK (cid:48) ] ∆ i, ∆ j .The following two special cases of interest can help distinguish between BFE and NL-IPCcontributions to the IPNL effects measured by the non-overlapping correlation function. The firstof these is the equal-interval correlation function, where a = c < b = d – this is the auto-correlationof a single difference image S a − S b , which is most similar to the auto-correlation that one wouldobtain from a CCD. The relevant equations in Paper I are given by Eqs (52-57).The final case of interest involves a = c < b < d in the mean-variance plot, which is a commondiagnostic of the gain of a detector system. The raw gain is more generally written asˆ g raw abcd ≡ M cd − M ab V cd − V ab , (4)where M ab = (cid:104) S a ( i, j ) − S b ( i, j ) (cid:105) and V ab = C abab (0 ,
0) is the variance of a difference frame. InPaper II we will consider only a = c < b < d , for which the expression for raw gain can be writtenas Eq. (62) from Paper I. In this equation, there are two time-dependent terms within the curlybrackets containing non-linear correction terms; the first involves the start time t a , while the seconddepends on the duration time t ad + t ab . In §
3. Data
Dark and flat illumination frames were acquired for an H4RG-10 detector array labelled asSCA 18237 – a 2.5 µ m cutoff device with 10 µ m pitch pixels – at the Detector CharacterizationLaboratory (DCL) at NASA Goddard Space Flight Center. SCA 18237 was one of the arraysbuilt for the WFIRST infrared detector technology milestone There are some key differences between the detector operation in these tests and the plannedoperation in flight. Most notably, the data here were acquired with a laboratory controller (Gen-IIILeach), rather than the ACADIA flight controller (Loose et al. 2018). Furthermore, the data wereacquired in 64 output channel mode, whereas 32 output channels are planned for flight. Finally,the H4RG-10 has a guide window mode, which was not active during these tests but is planned forflight. Other data have been taken to assess the impact of the guide window on science performance,but are not presented in this study. The technology milestone reports are available at: https://wfirst.gsfc.nasa.gov/science/sdt public/wps/references/WFIRST DTAC4 160922.pdf and https://wfirst.gsfc.nasa.gov/science/sdt public/wps/references/WFIRST DTAC5 nobackup.pdf . ∼
182 s) at the native 16 bit precision of the analog-to-digital converter(ADC) for a file size of 2.2 GB each. The dark and flat exposures were grouped into “sets,” witheach set consisting of a sequence of back-to-back identical exposures, as shown in Figure 1. Theodd-numbered sets contained dark exposures, while the even-numbered sets contained flat fieldexposures, but the number of exposures in each set was varied to provide information on persistenceand hysteresis. Each set has an exposure number, thus we refer to “Set 1, Exposure 1” (S1E1),S1E2, etc. We will often discuss the “first flats” in a sequence, indicating the first flat exposurefollowing a set of darks: S2E1, S4E1, etc. In the presence of persistence and hysteresis effects, thefirst flats show a slightly different signal level and non-linearity curve than the subsequent flats.The ordering of the ADC levels was opposite from the formalism of this paper (i.e., the signal in DNincreases during illumination) so we inverted the ordering, S → − − S , before any processing. The wavelength of illumination was 1.2 µ m, and thus we do not expect to observe quantumyield effects.Some representative dark and flat images from the SCA are shown in Figure 2. The left panelis a dark image (the CDS image S − S for exposure S1E1), with the median taken in 4 × S − S for exposure S1E1, also 4 × S − S were computed for two flats – S2E1 and S4E1 – and the normalized differencewas taken, (S2E1 − S4E1)/ √
2. In each 4 × The sequence was originally designed to study persistence issues. We have received data samples from the DCL in both increasing and decreasing formats, and so we implementedan inversion option in our routines to read the FITS files.
Set 1 10 darks Set 2 5 flats Set 3 8 darks Set 4 11 flats Set 5 11 darks Set 6 10 flats … S3E1 S3E8
Beginning of test
Fig. 1.— The sequence of exposures used in this test, containing interspersed darks and flats. 7 –Quantity Units DescriptionQ ke Charge, current multiplied by time.g e/DN Gain, corrected for IPC and classical non-linearity unlessspecified (e.g. subscript ‘raw’).K IPC kernel matrix, with K , = 1 − α , K , ± = K ± , = α . α % Specifies the IPC kernel, average of horizontal (subscript ‘H’)and vertical (subscript ‘V’) components. Diagonal componentdenoted with subscript ‘D’.K (cid:48) Signal level-dependent NL-IPC kernel matrix (3 × β ppm/e Leading order classical non-linearity coefficient. a ∆ i ∆ j ppm/e BFE kernel coefficients defined in terms of shifts from thecentral pixel (∆ i = ∆ j = 0).Σ a ppm/e Sum of a ∆ i ∆ j over ∆ i ,∆ j .[ K a (cid:48) + KK (cid:48) ] ∆ i, ∆ j ppm/e Inter-pixel non-linearities (IPNL) including linear IPC,non-linear IPC, and BFE.Table 1: Summary of detector parameters. P i x e l Y Dark image: SCA18237 [DN] P i x e l Y Flat image: SCA18237 [DN] P i x e l Y Flat std. dev.: SCA18237 [DN]
Fig. 2.— Dark (left) and flat (middle) images from SCA 18237. The panels show the CDS images S − S , i.e., difference between 1st and 21st frame, and have been median-binned 4 × − S4E1) / √
2. 8 –
4. Characterization based on flat fields
As discussed in Section 5 of Paper I, we want to extract the calibration parameters ( g , α , β , a ∆ i, ∆ j , etc.) from a suite of flat field and dark exposures for SCA 18237. We use solid-waffle ,which is described in Paper I. We summarize the procedure in Figure 3.Figure 3 begins with input of N flat fields and N dark images, where N ≥
2. The SCA isbroken into a grid of N x × N y “super-pixels,” each of size ∆ x × ∆ y physical pixels. Statisticalproperties such as medians, variances and correlation functions are computed in each super-pixel.Note that N x ∆ x = N y ∆ y = 4096 for an H4RG (and 2048 for an H2RG). Super-pixels may be madelarger to improve S/N, but this implies more averaging over the SCA so localized features andpatterns may be washed out. Our default analyses have N x = N y = 32, so there are 32 = 1024super-pixels, each containing 128 ×
128 physical pixels.Each super-pixel is passed through three main steps (grouped with dashed lines in Figure 3):first, “basic” characterization, which measures the gain from the mean-variance plot and correctsit for the IPC (inferred from the CDS autocorrelation function) and the non-linearity β (measuredfrom curvature of the ramp); second, IPNL determination using the non-overlapping correlationfunction, i.e., C abcd (∆ i, ∆ j ) for a < b < c < d ; third, advanced characterization, which iterativelyremoves the biases in gain, IPC, and non-linearity measurements caused by IPNL. We use the“ bfe ” correction scheme for the advanced characterization, since our results show that the BFEdominates over NL-IPC as the main form of IPNL.Figure 4 shows advanced characterization results for SCA 18237 (23 flats and darks). Thefigure shows good pixel percentages, g , α , β , charge per time step, and the central kernel value ofthe inter-pixel non-linearities in each of the 1024 super-pixels. We note that some spatial variationappears in the maps of g , α , and β . Additionally, the IPNL appears to be dominated by noiserather than real fluctuations across super-pixels. The theoretical Poisson noise error on the IPNLfor SCA 18237 is approximately equal to σ ([ K a (cid:48) + KK (cid:48) ] ∆ i, ∆ j ) = 1 (cid:112) N pix ( N flat − It ab )( It cd ) = 1 (cid:112) (23 − × = 1 . × − = 0 .
142 ppm / e , (5)where N pix is the number of pixels averaged together and N flat is the number of flat fields used.The range of IPNL values in Figure 4 encompasses about 7 σ . We also verify that the measuredstandard deviation is 0.145 ppm/e, which is consistent with the predicted error.Table 2 shows the difference in the main quantities of interest for SCA 18237 as a function of thenumber of iterations of correction for IPNL. For each quantity, the first row corresponds to meansover N good good super-pixels, and the second row gives statistical uncertainties computed as stan-dard deviations on the mean of N good . The post- αβ -corrected gain variances are also smaller thanthe variances of the raw gain. Choosing ncycle=3 and above yields the same values as presented inTable 2 for ncycle=2 . For the remainder of this work, we will use advanced characterization with 9 – Determine IPNL vianon-overlapping correlation,Section 5.2 of Paper I
Advanced characterization,Section 5.3 of Paper I Basic characterization,Section 5.1 of Paper I
Measure raw C abcd ( Δ i, Δ j)(Paper I, Eq. 73)Measure baseline correctionto filter out low frequencies(Paper I, Eq. 74 & Fig. 2)Use corrected C abcd ( Δ i, Δ j)to solve 5x5 IPNL kernel(Paper I, Eq. 70-72)Is icycle Solve 6 equations for 6 unknowns(Paper I, Eq. 80)Compute error terms assumingthere is BFE and no NL-IPC(Paper I, Eq. 81) yesIs the error mode nlipc ?noCompute error terms assumingthere is NL-IPC and no BFE(Paper I, Eq. 82) Construct S ab (i,j|F k ), S ad (i,j|F k ) andmedian over flats kConstruct pixel mask byrequiring median at (i,j) within 10%Subtract median signal of relevantref pix from super-pix (Paper I,Sec. 5.1)Calculate raw gain via ratio of meanto variance (Paper I, Eq. 59)Calculate horizontal correlation C H and vert, diag correlations C V , C D (Paper I, Eq. 64-65)Calculate slope difference ratio frac_dslope (Paper I, Eq. 66-67)Use raw gain, C H , C V , mean signalM ad , frac_dslope to solvesystem of 5 equations for 5unknowns: g, β , I, α H , α V (Paper I,Eq. 69)Advanced char?yes no Are calculations finishedfor all super-pix(ix=nx, iy=ny)? Read flat & dark cubefor given super-pixnoApply mask (setmasked super-pix to 0)yes Start Input N flats,N darks (N>=2) Input config w/data formats,super-pix geometry(nx=ny=32),adv char ncycle,& other optionsStart from first super-pixel (ix=iy=0)Make plots,output results End Fig. 3.— Flowchart showing overview of analysis procedure including basic characterization, IPNLcalculation via the non-overlapping correlation function, and advanced characterization. Full detailscan be found in the referenced equations and text of Paper I. 10 – ncycle=3 . We have run the characterization for a number of configurations designed to check the stabilityand reproduceability of the g , α , and β parameters. The means and standard deviations of thesevalues are given in Table 3. We also provide a systematic uncertainty for β ramp to account forits stability. We compute this as the sample standard deviation of all of the good super-pixelmeasurements.For every measurement in Table 3, we show three results. The first is “1st,n3,” which is basedexclusively on the first flat illumination in a set (3 flats: S2E1, S4E1, S6E1). These should bethe least affected by persistence/hysteresis effects. The second result is “2nd,n3,” which is basedon the second flat illumination (3 flats: S2E2, S4E2, S6E2). This will be more strongly affectedby persistence and hysteresis from the previous illumination, but less affected by stability issuesassociated with the flat lamp turning on. The third result is “fid,n23,” which contains 23 flat fields(S2E[1-5], S4E[4-11], S6E[1-10]), the subset of the flat field data set to which we had access atthe time of the analysis. This mixes first and subsequent flats, but has the greatest statisticalpower. We see that the non-linearity β ramp changes substantially depending on this test, with the“1st,n3” case giving a result 0.05 ppm/e lower than subsequent flats (i.e., the first ramp is morelinear than the second ramp); moreover the inferred charge in the first ramp is 0 . ± . 15% lower.We are continuing to study how much of this is due to the detector and how much to the testsetup. However there is no detectable change in the gain, IPC, or IPNL coefficients in the 1st vs.subsequent flats.We next vary the quantile level used for estimating the variance in gain. The default is 75(inter-quartile range: ± . σ for a Gaussian), and we compare setting this to 85 (i.e., estimatingthe variance from the difference between 15th and 85th percentiles: ± . σ for a Gaussian). Wewould expect the gains to change if the variance measurement is biased by non-Gaussianity of thesignal (e.g., kurtosis or outliers). The gains change by < . (cid:15) for the IPC correlations. The default is 0.01, whichclips the top 1% and bottom 1% of the pixels before computing a covariance. We set (cid:15) = 0 . 025 tocheck the impact of clipping the top and bottom 2.5%. This is a consistency check for the clippingcorrection factor in Appendix A of Paper I, which changes from f corr = 0 . f corr = 0 . | ∆ f corr | /f corr = 0 . | ∆ α | /α ≈ . β becomes slightlysmaller for all three sets of flats, ranging from 1.6% to 2.8% compared to the fiducial setup. One may notice that Figure 1 shows a total of 26 flats, and that S4E[1-3] were not included in the “fid,n23“ case– this was due to data transfer issues that were only resolved later. 11 –We also check that our choice of the bfe error mode in the advanced characterization schemedoes not affect the output β and BFE+NL-IPC coefficient values by running the same analysisusing the none and nlipc error modes. The resulting values are shown in Table 3 and do not showstrong deviations from the fiducial setup that uses the bfe error mode.Finally, the default calculation uses the first frame as the reference ( t = 0). The first frame inthe data cube is, however, 1 frame after the reset frame, so 2.75 s after the reset. This means thegain computed, in e/DN, is in fact not the slope of the charge vs. signal curve at the reset level,but the charge one frame later. We did one run where the “reference” ( t = 0) is set to frame 0(the reset frame) instead of frame 1. We expect most parameters such as the IPC and IPNL to notchange, but we do expect the gain to change in accordance with∆ g = g | frame 0 − g | frame 1 ≈ − βgIt , . (6)We find that with this change, the changes in IPC are ∆ α = 0 . K a (cid:48) + KK (cid:48) ] , = 0 . 006 ppm/e; and in β ramp are ∆ β ramp = − . 001 ppm/e. The expected changes in gaindo occur: they are ∆ g = − . − . 5. Brighter-fatter effect measurements5.1. IPNL determination via the non-overlapping correlation function We measure the IPNL parameterized by [ K a (cid:48) + KK (cid:48) ] ∆ i, ∆ j via the correlation function fornon-overlapping time slices (see red box in Figure 3 and Section 5.2 in Paper I for details) over pixelseparations (∆ i, ∆ j ) to a maximum separation of 2 pixels in both horizontal and vertical directions.We provide averages of coefficients related by grid symmetries in Table 3 for a “fiducial” choice offrames 3,11,13,21 for each of the three groupings of flats: 1st,n3, 2nd,n3, and fid,n23.We also visualize the coefficients for SCA 18237 in the left panel of Figure 5. The zero-lagcoefficient at (∆ i, ∆ j ) = (0 , 0) has a mean value over all good super-pixels of − . ± . ± . σ statistical uncertainty, and the second error is obtained frompropagating the systematic error in the measurement of β ramp . The four nearest neighbors with(∆ i, ∆ j ) of ( ± , , ± 1) have a mean value of 0 . ± . 002 ppm/e and the four diagonalneighbors have a mean value of 0 . ± . > σ ) that giveevidence for the existence of inter-pixel non-linearities in this detector, although as this part of theanalysis is sensitive to a combination of the BFE and NL-IPC, the exact mechanism cannot yet bedetermined. We note that in Paper I, for a simulation based on parameters similar to SCA 18237,we found biases on the extracted BFE coefficients of 12% in the central component and 2.7% inthe nearest neighbors and determined the cause to be likely related to exclusion of higher orderinteractions in the current formalism. The exact contributions of these higher order terms will berevisited in future work. 12 –The right panels of Figure 5 explore the scenario in which there is no NL-IPC. For this casewhere K (cid:48) = 0, we compute an order- α inverse kernel and convolve it with [ K a (cid:48) ]; the inversekernel K is given by [ K − ] , = 1 + 8 α , [ K − ] ± , = − α H , and [ K − ] , ± = − α V . If theBFE were wholly responsible for the IPNL, the BFE coefficient at zero lag would be given by − . ± . ± . 060 ppm/e.The BFE kernel for CCDs has been found to be long-range: for example, for DECam, a ∆ i, ∆ j ∝ r − ν , where r = (cid:112) ∆ i + ∆ j is the pixel separation and ν ≈ . ν → r ), andthe shortest possible range is ν → ∞ (all missing area from the central pixel appears in the fournearest neighbors). From a fit to the coefficients in the right panel of Figure 5, we find ν = 5 . ± . σ errors based on ∆ χ ). The BFE in this HgCdTe detector is thus much shorter range than fora CCD.CCDs have also shown an asymmetry between the “row” and “column” directions in theBFE. We characterize this quadrupole asymmetry by writing a H = ( a , + a − , ) / a V =( a , + a , − ) / 2. If we interpret the IPNL kernel as BFE, we find a H − a V = 0 . ± . 004 ppm / e or a H − a V a H + a V = 0 . ± . . (7)The BFE kernel is thus much more symmetrical than has been reported for some CCDs (e.g.Coulton et al. 2018). There is a ∼ σ detection of an asymmetry; further investigation will beneeded to establish whether this small asymmetry is in fact due to the BFE, or due to some othersub-dominant effect.We compare the IPNL results for the fiducial 3,11,13,21 frames to two other choices of non-overlapping time slices, 3,7,9,13 and 3,19,21,37. These numbers are also given in Table 3. Fo-cusing on the zero-lag coefficient, we have from Eq. (58) from Paper I that [ K a (cid:48) + KK (cid:48) ] , = g I t ab t cd C abcd (0 , 0) + 2(1 − α ) β . The value of β ramp is slightly smaller for the shorter time interval,and larger for the longer time interval; however the difference in 2 β ramp is much smaller than thecorresponding differences in [ K a (cid:48) + KK (cid:48) ] , for the shorter and longer time intervals. We suggestthat the cause for the differences in [ K a (cid:48) + KK (cid:48) ] , arises from the exclusion of higher order termsin the correlation formalism used in this analysis. Section 5.5 of Paper I describes two tests that can aid our interpretation of the IPNL detectionsof § 5. In particular, Eq. (62) of Paper I gives two time dependencies for the observed raw gain: onepart depends on the start time t a and the classical non-linearity β , while the other part dependson the duration pattern t ab and t ad , β and a , . 13 – In the first test, we measure the mean variance slope ˆ g raw abad (Eq. 62 from Paper I), fix t a , andvary t ab and t ad . We fit an intercept C and a slope C to the equation ln ˆ g raw abad = C + C I ( t ad + t ab ).Eq. (62) enables three different interpretations of C in the cases of no IPNL, IPNL consisting purelyof the BFE, and IPNL consisting purely of NL-IPC: C = 3 β − (1 + 8 α )[ K a ] , + 8(1 + 3 α ) α (cid:48) = β ramp none β ramp − (1 + 8 α )[ K a ] , bfe β ramp − α )[ KK (cid:48) ] , nlipc , (8)Re-arranging the left part of Eq.8 and substituting β ramp = β − Σ a , we can also define a quantityˆ a , , M2 ≡ a , + 8 αa < , > − 32 Σ a − α ) α (cid:48) = 3 β ramp − C (9)= (cid:40) a , + 8 αa < , > bfe − α ) α (cid:48) = 2(1 + 8 α )[ KK (cid:48) ] , nlipc , (10)which is sensitive to the BFE coefficient in the central pixel but also contains a contribution fromNL-IPC.The raw gain is computed for frame triplets from [1,3,5], [1,3,6], ... ,[1,5,18], yielding 14 valuesthat are plotted in the top row of Figure 6 for SCA 18237 as a function of the signal level accumu-lated between the first time slice and the time slice d = 5 ... 18. The value of each data point is givenby the mean over all super-pixels, with errors on the mean. The IPNL at zero-lag measured fromMethod 1 is used to compute the slopes for the pure BFE and pure NL-IPC interpretations, withthe central value passing through the center of the measured data points (i.e. the intercept for theseslopes is unimportant). A systematic error related to the modeling of the non-linearity (“sys nl”)is also indicated in each panel of Figure 6. This is based on fitting a 5th order polynomial to themedian signal levels in the detector. For both this curve and the quadratic ( β ) model, we computedthe expected raw logarithmic gain ln g raw a,b,d for Poisson statistics , compute the difference, and plotan error bar showing the peak − valley range. Note that the absolute gain does not enter becausewe are using ln g raw a,b,d . This procedure is intended only to give an indication of the magnitude ofsystematic errors due to deviation of the classical non-linearity from the β model, and in this paperwe have not attempted any corrections.SCA 18237 appears consistent with a pure BFE interpretation within systematic error. We canquantitatively compare the various estimates for ˆ a , ,M using results from Method 1 as describedin Eq. 10. We have ˆ a , ,M = − . ± . ± . − . ± . ± . − . ± . ± . a , ,M is quantitatively consistent with the pure BFE interpretation. The formula can be derived following the procedure in Paper I for any signal S ( t ). It is: 14 – In the second test, we measure the mean variance slope, fix t ab and t ad , and vary t a , fittingan intercept C (cid:48) and slope C (cid:48) to ln ˆ g raw abad = C (cid:48) + C (cid:48) It a . As for the previous test, we can use thedetected IPNL from Method 1 to inform different interpretations of the slope where C (cid:48) = 2 β − α ) α (cid:48) = β ramp none β ramp bfe β ramp − α )[ KK (cid:48) ] , nlipc . (11)For this test, C (cid:48) is only sensitive to NL-IPC, as the none and bfe cases give identical predictions.We can re-write the above to isolate interesting quantities. First, we have β − α ) α (cid:48) = C (cid:48) ,which is an alternate way of determining the non-linearity with no leading-order sensitivity to Σ a albeit a dependence on NL-IPC. We can also isolate a combination of Σ a and α (cid:48) :Σ a − α ) α (cid:48) = C (cid:48) − β ramp . (12)The raw gain is computed for frame triplets from [1,3,5], [2,4,6], ... ,[14,16,18], yielding 14 valuesthat are plotted in the middle row of Figure 6 for SCA 18237 as a function of the signal levelaccumulated between the first time slice and the time slice a = 1 , ..., 14. As before, the none and bfe slopes are plotted. While the data points seem to prefer the bfe slope, there is clearly a changein slope at both low and high signal levels, which warrants further investigation in future studies.For the 1st flats with SCA 18237, we measure β − α ) α (cid:48) = 0 . ± . β ramp value from basic characterization of 0 . ± . ± . 032 given ahypothesis that the NL-IPC contribution from α (cid:48) is 0. Then Σ a − α ) α (cid:48) = − . ± . ± . α ) α (cid:48) = − . ± . ± . ± ± 2% of the IPNL signal measured in Method 1. However, given that higher-order termsappear to be biasing the Method 1 measurement by ∼ 10% (see Paper I), we urge caution ininterpreting this result. We believe the hot pixel test ( § This method uses the equal-interval correlation function in adjacent pixels as given by Eq. (52)and Eq. (55) for a time-translation-averaged version in Paper I. As summarized in Sections 3.8.1 g raw a,b,d = [ S ( t d ) − S ( t b )] / {− t a S (cid:48) ( t a ) S (cid:48) ( t d )[ S (cid:48) ( t d ) − S (cid:48) ( t b )] + t d [ S (cid:48) ( t d )] − t b [ S (cid:48) ( t b )] } , where S is the signal curve and S (cid:48) is its derivative. 15 –and 5.4.3 of Paper I, we can fix the starting time t a and fit the combination g C abab ( ± , / ( It ab )as a function of t ab , fitting g It ab C abab ( (cid:104)± , (cid:105) ) = C (cid:48)(cid:48) + C (cid:48)(cid:48) It ab . The slope is given by C (cid:48)(cid:48) = − αβ + α Σ a + [ K a ] (cid:104) , (cid:105) + 2[ KK (cid:48) ] (cid:104) , (cid:105) = − αβ ramp none − αβ ramp + [ K a (cid:48) ] (cid:104) , (cid:105) bfe − αβ ramp + 2[ KK (cid:48) ] (cid:104) , (cid:105) nlipc . (13)We can add 8 α H β ramp to the left hand part of Eq. 13 to obtain C (cid:48)(cid:48) + 8 αβ ramp = [ K a ] (cid:104) , (cid:105) + 2[ KK (cid:48) ] (cid:104) , (cid:105) − α Σ a = [ K a (cid:48) + 2 KK (cid:48) ] (cid:104) , (cid:105) − α Σ a (14)IPC is measured via basic characterization of frame triplets from [1,2,3], [1,2,4],..., [1,2,18],and CDS auto-correlations are computed for [frame 3 - frame 1], [frame 4 - frame 1],..., [frame 18 -frame 1]. The bottom panel of Figure 6 shows the measurements from Method 3 with predictionsfrom Method 1 over plotted. SCA 18237 agrees well with the pure BFE interpretation.We can compare the estimate of [ K a (cid:48) + 2 KK (cid:48) ] (cid:104) , (cid:105) − α Σ a with the Method 1 result of [ K a (cid:48) + KK (cid:48) ] (cid:104) , (cid:105) . For a pure BFE interpretation, these quantities would be equivalent; for a pure NL-IPC interpretation, the two would differ by a factor of 2. For SCA 18237, this test measures0 . ± . . ± . . ± . An alternative method to assess the IPC is to use hot pixels observed during dark exposures.The method relies on the fact that if the pixel ( i, j ) is hot (i.e., the photodiode leaks significantcurrent even in the absence of illumination), then a signal (in DN) will appear in the neighboringpixels due to capacitive coupling. This method is in principle more direct than the flat field method;it does not involve the BFE or other sources of correlations between pixels. It also enables one toexplore a wide range of signal levels, including very low signal levels where control of systematicsis difficult with flats. The main drawback is that it only probes the specific pixels that are hot,and one must beware of issues involving hot pixel selection and the possibility that hot pixelsmay behave differently from the science-grade pixels in ways other than being hot. The proceduredescribed here is qualitatively similar to the one used in Hilbert & McCullough (2011) to makeon-orbit measurements of the IPC for the infrared channel of the Wide Field Camera 3.The solid-waffle system selects hot pixels as follows. First, for each dark D k , we make theCDS image S , ( i, j | D k ) from the 1st and 65th time frames. We provisionally select pixels to be“hot” if the average M , ( i, j ) = N dark (cid:80) N dark k =1 S , ( i, j | D k ) of these images is in a given signalrange (specified as a minimum and maximum DN, e.g., 1000–2500). We next impose additionalcuts on the pixels to ensure that they are isolated (so that our IPC measurements are not affected 16 –by other nearby hot pixels) and repeatable (so that we are not selecting cosmic rays or pixels affectedby random telegraph noise). We impose the isolation cut first since we also want to be isolatedfrom unstable pixels. This cut requires the 5 × (cid:15) i times the pixel itself, i.e., M , ( i + ∆ i, j + ∆ j ) < (cid:15) i M , ( i, j ) for | ∆ i | ≤ , | ∆ j | ≤ , (∆ i, ∆ j ) (cid:54) = (0 , . (15)The default is (cid:15) i = 0 . 1. We also require this 5 × N dark darks: N dark max k =1 S ,b ( i, j | D k ) − N dark min k =1 S ,b ( i, j | D k ) ≤ (cid:15) r N dark N dark (cid:88) k =1 S , ( i, j | D k ) for 2 ≤ b ≤ , (16)where (cid:15) r is a repeatability parameter (default: 0.1). Note that we impose this criterion for inter-mediate frames b : we want pixels that exhibit the same time history in every dark exposure.A hot pixel can be used to give an estimate of the IPC using CDS images from any final frame b – that is, from S ,b ( i, j | D k ). This is useful because by varying b , we may determine how the IPCvaries with signal level. The steps are as follows: • [ Optional, default = on ]: Perform a lowest-order non-linearity correction on the CDS frames, S ,b ( i, j | D k ) → S ,b ( i, j | D k )[1 + βgS ,b ( i, j | D k )], where βg is the product of non-linearity andgain (units: DN − ). • We construct the median of the dark frames: M ,b ( i, j ). • If the pixel ( i, j ) is hot, then we extract the 3 × M ,b ( i + ∆ i, j + ∆ j ) for | ∆ i | , | ∆ j | ≤ 1. We also extract a “background” estimate B from the mean of the surrounding5 × − × • We construct the IPC kernel for that hot pixel from:ˆ K (∆ i, ∆ j ) = M ,b ( i + ∆ i, j + ∆ j ) − B (cid:80) i (cid:48) = − (cid:80) j (cid:48) = − [ M ,b ( i + ∆ i (cid:48) , j + ∆ j (cid:48) ) − B ] . (17)Averages of 4 pixels such as α (average of K in 4 nearest neighbors) and α D (4 diagonalneighbors) can also be reported.The aforementioned procedure gives an estimate of α for each selected hot pixel and each timesample b .In Figure 7, we show plots from solid-waffle for SCA 18237. The results here are for hotpixels in the 1000–2000 DN range. We used only the first 5 dark frames for this analysis. Wedivided the SCA into 16 1024 × α hot pix − α autocorr =0 . α as a function of signal level, which can be seen all the way from tens of DN (comparable to thereset noise) all the way up to 2 . × DN (roughly half full well).If NL-IPC were responsible for the entirety of the inter-pixel non-linearity signal observed inMethod 1, then we would have to have [ KK (cid:48) ] (cid:104) , (cid:105) = 0 . 22 ppm/e. Expanding the kernel, we see that[ KK (cid:48) ] (cid:104) , (cid:105) = (1 − α ) α (cid:48) , so this suggests that the signal dependence of α in DN units would be gα (cid:48) = g − α [ KK (cid:48) ] (cid:104) , (cid:105) = 5 . × − DN − . (18)The relevant range of signal levels is up to 15 kDN (the signal level at frame 21), so we wouldexpect α (as inferred from the hot pixels) to change by 5 . × − DN − × 15 kDN = 0 . 81% fromlow values up to 15 kDN. Instead, the differences in Figure 8 over this range are 0.1–0.2%. Adetailed quantitative comparison is not possible since NL-IPC can depend (in principle) on bothsignal level and contrast: the flat field measures the dependence on signal level at low contrast,whereas the hot pixel test measures the dependence on signal level in one pixel with a backgroundof near zero. The two measurements need not be exactly the same (and in the formalism of Donlonet al. 2018, they are not). Nevertheless, the very low NL-IPC that we observe in the hot pixel testsuggests that Method 1 is seeing primarily BFE rather than NL-IPC. 6. Discussion and Future Work We have analyzed the flat field statistics of a prototype WFIRST H4RG-10 detector array(SCA 18237) following the procedures introduced in Paper I. In summary, we started with a basiccharacterization of the detector, where we constructed CDS images per flat per super pixel, com-puted a median over the flats per super pixel and a mask, performed a reference pixel subtraction,computed the raw gain, horizontal and vertical correlations, and an estimate of ramp curvature.We then solved for the IPC and non-linearity corrected gain, horizontal and vertical α components,current, and classical non-linearity β . The following step was to measure the non-overlapping cor-relation function, which is almost a direct test for the presence of inter-pixel non-linearities. We 18 –then used the non-overlapping correlations, interpreted as either BFE or NL-IPC, to de-bias theoriginal “basic characterization” parameters of g , α , etc. with an iterative process. We performeda range of robustness checks to ensure the stability of our results when various analysis choiceswere modified. We conducted four complementary investigations to help build the interpretationof the mechanism(s) behind the inter-pixel non-linearities: (1) raw gain vs interval duration, (2)raw gain vs interval center, (3) equal-interval correlation function in adjacent pixels, and (4) IPCmeasurement on hot pixels. The main results can be recapitulated as follows. • There is large-scale spatial variation of the IPC at the ∼ . 3% level. The same spatial variationis observed in both the autocorrelation and hot pixel tests for IPC. There is a ∼ . 06% overalloffset between the two methods that is under investigation. IPC and its spatial variation willbe further investigated with single pixel reset tests during WFIRST flight detector acceptancetesting. • We have used the formalism built up in Paper I to detect residual correlations between dif-ference frames of flat fields where the time intervals do not overlap. SCA 18237 shows adecrement in the central kernel value at high S/N. While this non-overlapping correlationmethod provides the highest S/N measurements, the underlying mechanism includes contri-butions from both the BFE and NL-IPC. If interpreted as pure BFE, this measurement wouldindicate an Antilogus coupling coefficient to the 4 nearest neighbors of a (cid:104) , (cid:105) = 0 . ± . a (cid:104) , (cid:105) = 0 . ± . 003 ppm/e. This effect is of thesame order of magnitude compared to Plazas et al. (2018), who use spot illumination on anH2RG device (with different geometry: 18 µ m pitch pixels). The differences between theiranalysis and ours complicate a more quantitative comparison. • The main effect of the BFE on weak lensing analyses is generally through the stars used toestimate the PSF. The WFIRST Science Requirements Document uses a reference star witha total fluence of 8 . × collected electrons per exposure. If we use an obstructed Airy diskcentered on a pixel center and with no extra spreading due to aberrations, charge diffusion, orimage motion (the most extreme case), and the BFE kernel for SCA 18237, then averaged overthe duration of the exposure the area of the central pixel is modified by − . 2% in J -band and − . 5% in H -band. Since WFIRST aims for PSF size calibration at the 7 . × − level, theBFE will have to be accurately measured and corrected. The WFIRST Wide Field InstrumentCalibration Plan presented at the WFIRST System Requirements Review in 2018 estimatedthat if 5% (in an RSS sense) of the PSF size error budget is allocated to BFE, then BFEcoefficient a < , > needs to be measured to an accuracy of σ a,< , > (req (cid:48) t) = 0 . / e.Here we have achieved a statistical error on a < , > of 0.003 ppm/e. This is encouraging but isby no means the end of the story since we measured a single IR detector and used low contrastdata (fluctuations around a flat); further laboratory studies will be needed to validate theaccuracy of the BFE model for the high contrasts expected when observing PSF stars. • In order to determine whether the BFE or NL-IPC is the dominant mechanism behind the 19 –measured IPNL, we can run other tests such as measuring the dependence of the raw gainon either the start time or the interval duration. A third test involves the scaling of theadjacent-pixel covariance as a function of signal level. All of these tests favor the BFE ratherthan NL-IPC as the dominant mechanism. • The hot pixel analysis is a more direct way of assessing the IPC and can be used to investigatea wide range of signal levels. We find evidence for NL-IPC, with the IPC coefficient increasingfor greater signal in the hot pixel. However this NL-IPC is at least a factor of ∼ µ m pitch pixels compared to 18 µ m pitch pixels for the H2RG devices analyzed in Donlonet al. (2016). In future work, we will compare the hot pixel results to those obtained frommeasurements of single pixel reset data. • The most significant limitation of the present BFE measurement is that the current model forthe correlation function C abcd (∆ i, ∆ j ) keeps only the leading-order non-linear terms, i.e., weuse the quadratic β -model for non-linearity, and drop terms of order a or aβ . Simulationsshow that this induces a bias of ∼ 12% for SCA 18237-like parameters. This limitation is notfundamental, and will be remedied in a future paper. • The BFE kernel is shorter range than observed in thick CCDs, with an observed fall-off of ∝ r − . ± . . This makes physical sense given the thin geometry of the HgCdTe detectors andagrees qualitatively with the model described in Plazas et al. (2017, 2018) wherein higher-signal pixels have smaller depletion regions, which is more of a local effect compared to the caseof CCDs. There is also only a small horizontal vs. vertical asymmetry in the non-overlappingcorrelation function; if ascribed to the BFE, this asymmetry suggests ( a H − a V ) / ( a H + a V ) =0 . ± . solid-waffle tools on a larger sample of SCAs, including on flat fields of all of the WFIRST flight 20 –candidate SCAs. The lessons learned will feed back into calibration planning for the WFIRSTmission. Acknowledgements We thank the Detector Characterization Laboratory personnel, Yiting Wen, Bob Hill, andBernie Rauscher at NASA Goddard Space Flight Center for their efforts enabling the existenceand access to the data analyzed in this series of papers, and we thank Chaz Shapiro, Andr´esPlazas, and Eric Huff for helpful discussions. We thank Jay Anderson and Arielle Bertrou-Cantoufor useful presentations to the Detector Working Group on their analyses of non-linearities inthe HST/WFC3-IR and Euclid H2RG detectors. We thank the anonymous referee for helpfulsuggestions that improved the clarity of this paper. We are also grateful for the use of OhioSupercomputer Center (1987) for computing the results in this work. AC and CMH acknowledgesupport from NASA grant 15-WFIRST15-0008. During the preparation of this work, CMH has alsobeen supported by the Simons Foundation and the US Department of Energy. Software: Astropy(Astropy Collaboration et al. 2013, 2018), fitsio (Sheldon 2019), Matplotlib (Hunter 2007), NumPy(Oliphant 2006–), SciPy (Jones et al. 2001–) REFERENCES Antilogus, P., Astier, P., Doherty, P., Guyonnet, A., & Regnault, N. 2014, Journal of Instrumenta-tion, 9, C03048, doi: Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: Astropy Collaboration, Price-Whelan, A. M., Sip˝ocz, B. M., et al. 2018, AJ, 156, 123, doi: Baumer, M. A., & Roodman, A. 2015, Journal of Instrumentation, 10, C05024, doi: Bernstein, G. M., & Jarvis, M. 2002, AJ, 123, 583, doi: Cheng, L. 2009, Master’s thesis, Rochester Institute of TechnologyCoulton, W. R., Armstrong, R., Smith, K. M., Lupton, R. H., & Spergel, D. N. 2018, AJ, 155, 258,doi: Donlon, K., Ninkov, Z., & Baum, S. 2016, in Proc. 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J., Boehm, N., Cagiano, S., et al. 2014, PASP, 126, 739, doi: Sheldon, E. 2019, A python package for FITS input/output wrapping cfitsio This preprint was prepared with the AAS L A TEX macros v5.2. 22 – S u p e r p i x e l Y / Good pixel map (%) S u p e r p i x e l Y / Gain map g (e/DN) S u p e r p i x e l Y / IPC map α (%) S u p e r p i x e l Y / Non-linearity map β (ppm/e) S u p e r p i x e l Y / Charge It n, n + 1 (e): S u p e r p i x e l Y / IPNL [K a + KK ′ ] 0, 0 (ppm/e): −1.6−1.4−1.2−1.0−0.8−0.6 Fig. 4.— Advanced characterization for SCA 18237. 23 –Quantity Units ncycle0 1 2 3 4Gain g e/DN 2.1574 2.0636 2.0643 2.0643 2.06430.0022 0.0016 0.0016 0.0016 0.0016IPC α H % 2.0761 1.6301 1.6330 1.6330 1.63300.0138 0.0070 0.0046 0.0046 0.0046IPC α V % 2.2023 1.7410 1.7439 1.7439 1.74390.0100 0.0053 0.0053 0.0053 0.0053IPC α D % - 0.1881 0.1881 0.1881 0.1881- 0.0029 0.0029 0.0029 0.0029Non-linearity β ramp ppm/e 0.5597 0.5832 0.5830 0.5830 0.58300.0010 0.0010 0.0010 0.0010 0.0010Table 2: Statistics of gain, IPC, and ramp curvature as a function of ncycle iteration for a stack of 3SCA 18237 flats. Two rows of values are provided for each quantity with the first row correspondingto the mean value over all good super-pixels and the second row corresponding to the standarderror on the mean. The measurements converge by three iterations. 24 – Quantity Units Flat type, number Uncert Notes1st,n3 2nd,n3 fid,n23 stat.(3) stat.(23) sys.(3)Charge, It n,n +1 ke 1.4536 1.4617 1.4632 0.0016 0.0015Gain g e/DN 2.0643 2.0652 2.0639 0.0016 0.0014IPC α % 1.6884 1.6830 1.6877 0.0070 0.0042IPC α H % 1.6330 1.6243 1.6342 0.0046 0.0021IPC α V % 1.7439 1.7417 1.7411 0.0053 0.0036IPC α D % 0.1881 0.1835 0.1846 0.0029 0.0009Non-linearity β ramp ppm/e 0.5830 0.6332 0.6351 0.0010 0.0010 0.0313Alternative setupsGain g e/DN 2.0645 2.0647 2.0637 0.0016 0.0014 IQR 85IPC α % 1.6810 1.6769 1.6806 0.0073 0.0042 (cid:15) = 0 . β ramp ppm/e 0.5665 0.6209 0.6251 0.0010 0.0010 Not ref. pix. corr.Non-linearity β ramp ppm/e 0.5816 0.6321 0.6342 0.0010 0.0010 Error mode ‘none’Non-linearity β ramp ppm/e 0.5726 0.6200 0.6211 0.0015 0.0010 Error mode ‘nlipc’Alternative intervalsNon-linearity β ramp ppm/e 0.5701 0.6488 0.6551 0.0014 0.0014 0.0455 Frames 3,7,9Non-linearity β ramp ppm/e 0.6377 0.6611 0.6650 0.0010 0.0010 0.0330 Frames 3,19,21Non-overlapping correlation function (Method 1)BFE+NL-IPC Coefficients - frames 3,11,13,21, baseline-corrected, error mode ‘bfe’[ K a (cid:48) + KK (cid:48) ] , ppm/e -1.2004 -1.0936 -1.0772 0.0154 0.0045 0.0541 Central pixel[ K a (cid:48) + KK (cid:48) ] < , > ppm/e 0.2142 0.2232 0.2241 0.0074 0.0020 Nearest neighbor[ K a (cid:48) + KK (cid:48) ] < , > ppm/e 0.0468 0.0449 0.0489 0.0074 0.0020 Diagonal[ K a (cid:48) + KK (cid:48) ] < , > ppm/e 0.0103 0.0186 0.0166 0.0073 0.0020[ K a (cid:48) + KK (cid:48) ] < , > ppm/e 0.0018 0.0102 0.0042 0.0052 0.0015[ K a (cid:48) + KK (cid:48) ] < , > ppm/e -0.0076 0.0013 0.0014 0.0073 0.0020BFE+NL-IPC Coefficients - frames 3,7,9,13 baseline-corrected, error mode ‘bfe’[ K a (cid:48) + KK (cid:48) ] , ppm/e -1.0446 -0.9054 -0.9128 0.0303 0.0088 0.0786 Central pixel[ K a (cid:48) + KK (cid:48) ] < , > ppm/e 0.2548 0.2845 0.2375 0.0155 0.0042 Nearest neighbor[ K a (cid:48) + KK (cid:48) ] < , > ppm/e 0.0713 0.0584 0.0522 0.0159 0.0042 DiagonalBFE+NL-IPC Coefficients - frames 3,19,21,37 baseline-corrected, error mode ‘bfe’[ K a (cid:48) + KK (cid:48) ] , ppm/e -1.2276 -1.1731 -1.1637 0.0077 0.0049 0.0572 Central pixel[ K a (cid:48) + KK (cid:48) ] < , > ppm/e 0.2078 0.2100 0.2101 0.0034 0.0009 Nearest neighbor[ K a (cid:48) + KK (cid:48) ] < , > ppm/e 0.0496 0.0485 0.0530 0.0034 0.0010 DiagonalBFE+NL-IPC Coefficients - frames 3,11,13,21, baseline-corrected, error mode ‘none’[ K a (cid:48) + KK (cid:48) ] , ppm/e -1.2086 -1.1018 -1.0853 0.0155 0.0046 Central pixel[ K a (cid:48) + KK (cid:48) ] < , > ppm/e 0.2155 0.2248 0.2257 0.0074 0.0020 Nearest neighbor[ K a (cid:48) + KK (cid:48) ] < , > ppm/e 0.0468 0.0449 0.0489 0.0074 0.0020 DiagonalBFE+NL-IPC Coefficients - frames 3,11,13,21, baseline-corrected, error mode ‘nlipc’[ K a (cid:48) + KK (cid:48) ] , ppm/e -1.1827 -1.0760 -1.0600 0.0152 0.0045 Central pixel[ K a (cid:48) + KK (cid:48) ] < , > ppm/e 0.2045 0.2123 0.2128 0.0072 0.0020 Nearest neighbor[ K a (cid:48) + KK (cid:48) ] < , > ppm/e 0.0468 0.0449 0.0489 0.0074 0.0020 DiagonalMean-variance relation (Method 2)ˆ a , ,M ppm/e -1.2843 -1.0765 -1.0791 0.0398 0.0089 0.0939 β − α ) α (cid:48) ppm/e 0.3957 0.3769 0.3790 0.0234 0.0050 (cid:80) a − α ) α (cid:48) ppm/e -0.3747 -0.5125 -0.5122 0.0469 0.0100 0.0626Adjacent pixel correlations (Method 3)[ K a (cid:48) + 2 KK (cid:48) ] < , > − α (cid:80) a ppm/e 0.2194 0.2123 0.2067 0.0073 0.0020 Table 3: Averaged results for SCA 18237, based on stacks of flat ramps. Advanced characterizationwith ncycle=3. 25 – -0.0009 0.0066 0.0127 0.0035 0.00170.0081 0.0469 0.2034 0.0464 0.00200.0143 0.2325 -1.0772 0.2276 0.01850.0049 0.0539 0.2328 0.0483 -0.00010.0013 0.0058 0.0209 0.0027 0.0034 ∆ i = − j = − j = +2 ∆ i = +2 BFE+NL-IPC Coefficients - no IPC correction [ K a + KK ϕ ] ∆ i ∆ j (ppm/e) -0.0014 0.0053 0.0070 0.0019 0.00170.0068 0.0380 0.2625 0.0380 -0.00010.0072 0.2976 -1.2529 0.2922 0.01240.0030 0.0451 0.2952 0.0391 -0.00250.0010 0.0039 0.0151 0.0005 0.0035 ∆ i = − j = − j = +2 ∆ i = +2 BFE Coefficients - with linear IPC correction a ∆ i ∆ j (ppm/e), assumes K =0 Fig. 5.— The Method 1 BFE+NL-IPC coefficients (left panel) and IPC-corrected coefficients(right panel) for SCA 18237. These coefficients were measured on a stack of 23 flats, with thefull characteristics given in the third column on Table 3. Note that the IPC-corrected coefficientsassume that the IPC is linear, i.e. the non-overlapping correlations are ascribable entirely to theBFE and not NL-IPC. The 1 σ statistical uncertainty for each coefficient is 0.0045 ppm/e (leftpanel) and 0.0051 ppm/e (right panel). The central value at zero lag (∆ i, ∆ j ) = (0 , 0) carriesan additional systematic uncertainty of 0.0531 ppm/e (left panel) and 0.0603 ppm/e (right panel)propagated from a standard deviation in β ramp of 0.0307 ppm/e. 26 – [ke]0.900.920.940.960.98 l n g r a w , , d sys nl Raw gain vs. interval duration pure BFEpure NL-IPC0 5 10 15Signal level It [ke]0.9100.9150.9200.9250.9300.935 l n g r a w a , a + , a + sys nl Raw gain vs. interval center pure BFEpure NL-IPC5 10 15 20 25Signal level It [ke]0.0300.0310.0320.0330.0340.0350.0360.037 g C d d (⟨ , ⟩) / [ I t d ] sys nl CDS ACF vs. signal pure BFEpure NL-IPCbeta only Fig. 6.— Visual comparison of BFE predictions from Method 1 vs measurements from Methods 2and 3 for SCA 18237. We only use first flats of a set, with 3 available for SCA 18237. 27 – hot pixel locations: H4RG-18237 α [%]1.51.61.71.81.9 h o t p i x e l α [ % ] hot pixels vs. autocorr. IPC α Fig. 7.— Hot pixel IPC analysis for SCA 18237 (5 dark exposures, 6604 hot pixels). Parametersare: hot pixel brightness 1000–2000 DN; (cid:15) i = 0 . 1; and (cid:15) r = 0 . Left panel : the locations of selectedhot pixels. Right panel: The comparison of median IPC from hot pixels (vertical axis) versus theadvanced auto-correlation analysis (horizontal axis), binned into 1024 × σ errors on the mean. 28 – -0.30-0.25-0.20-0.15-0.10-0.050.000.050.100.150.200.250.30 50 100 200 500 1k 2k 5k 10k 20k α ( ho t p i x ) - α ( au t o c o rr) [ % ] Signal level [DN]IPC offsets: SCA 18237 16k-32k DN [2081 pix]8k-16k DN [2719 pix]4k-8k DN [2955 pix]2k-4k DN [3748 pix]1k-2k DN [6604 pix]500-1k DN [4453 pix] Fig. 8.— The hot pixel IPC as a function of signal level, for SCA 18237. We subtracted theauto-correlation α from the vertical axis, however exactly the same auto-correlation α map wasused as a reference for every point (i.e., by construction it has no time or signal dependence). Eachpoint style reflects a selection of pixels, from S , = 16000 − S ,b (varying b ; b = 2 is the first point shown, b = 65 is the last, and in between we have shown onlysome of the points – b = 1 , , , × n for integer nn