Detectors for the James Webb Space Telescope Near‐Infrared Spectrograph. I. Readout Mode, Noise Model, and Calibration Considerations

The James Webb Space Telescope (JWST) was conceived as the scientific successor to NASA's Hubble and Spitzer space telescopes. Of all the JWST "near‐infrared" (NIR; λ = 0.6–5 μm) instruments, the Near‐Infrared Spectrograph (NIRSpec) has the most challenging detector requirements. This paper describes how we plan to operate NIRSpec's two 2048 × 2048 pixel, 5 μm cutoff (λ_co = 5 μm), Teledyne HAWAII‐2RG (H2RG) sensor chip assemblies (SCAs)⁹ for the most sensitive observations, and provides insights into some nonideal behaviors that have been observed in engineering‐grade NIRSpec detectors.

This paper is structured as follows. In § 2, we provide an introduction to JWST, NIRSpec, and NIRSpec's detectors. We have tried to keep this discussion brief, and provide references to more comprehensive discussions in the literature.

In § 3, we present the NIRSpec detector subsystem's baseline MULTIACCUM readout mode. This section includes a detailed discussion of how total noise averages down when multiple nondestructive reads are used sampling up the ramp. MULTIACCUM readout is quite general, and most other common readout modes, including correlated double sampling (CDS), multiple CDS (MCDS; also known as Fowler‐N; Fowler & Gatley 1991), and straight up‐the‐ramp sampling are special cases of MULTIACCUM. The general NIR SCA noise model presented in this section (see eq. [1] and Table 2) is validated using real and simulated test data.

Where practical, our methods and conclusions are anchored by measurement. One advantage of the NIRSpec program is that multiple test SCAs and test facilities are available. These are described in § 4.

Section 5 describes the reset anomaly as it appears in engineering‐grade NIRSpec H2RGs. The reset anomaly is fairly well known in the NIR detector testing community. Here we demonstrate, using real test data, that it is a nearly noiseless artifact for the NIRSpec detectors that have been tested so far. We show that it straightforwardly calibrates out from most science observations and can therefore be safely ignored by most JWST users. However, we show that the reset anomaly can significantly bias dark current measurements if it is not correctly accounted for. In this paper, we describe a method of accounting for the reset anomaly in dark current measurements by fitting a four‐parameter function to up‐the‐ramp sampled pixels.

Finally, in § 6, we describe what is known about random telegraph noise (RTN) within the NIRSpec program. Using real test data, we show that large‐amplitude RTN is a property of only a small and fixed population of pixels for the SCAs that have been studied.¹⁰ Based on these data, we do not expect RTN to significantly impact NIRSpec. While this conclusion may appear to render studies of RTN academic, it actually mitigates the risk that RTN could have a major impact if the affected pixels were to change from integration to integration.

Although our discussion is focused on JWST's NIRSpec, we anticipate that much of what we discuss will be of interest to any astronomer using H2RGs. The noise model is quite general, and we are aware of others having observed both the reset anomaly and RTN. However, one caveat is in order. Integration and testing of the NIRSpec detector subsystem is just beginning now. As such, we anticipate that much remains to be learned about NIRSpec's detectors, and that some of the specifics presented here may change. For this reason, we have tried to focus on general themes, rather than on the measured performance of any particular SCA.

2.1. JWST Mission

2.2. NIRSpec

NIRSpec, which will be the first slit‐based astronomical multiobject spectrograph (MOS) to fly in space, is designed to provide NIR spectra of faint objects at spectral resolutions of R = 100, 1000, and 2700. The instrument's all‐reflective wide‐field optics, together with its novel MEMS (microelectromechanical systems) based programmable microshutter array slit selection device and H2RG detector arrays, combine to allow simultaneous observations of >100 objects within a 3.5^' × 3.4^' field of view with unprecedented sensitivity. A selectable 3^'' × 3^'' integral field unit (IFU) and five fixed slits are also available for detailed spectroscopic studies of single objects. NIRSpec is presently expected to be capable of reaching a continuum flux of 20 nJy (AB > 28) in R = 100 mode, and a line flux of 6 × 10⁻¹⁹ ergs s⁻¹ cm⁻² in R = 1000 mode at S/N > 3 in 10⁴ s.

NIRSpec is being built for the European Space Agency (ESA) by EADS Astrium as part of ESA's contribution to the JWST mission. The NIRSpec microshutter and detector arrays are provided by NASA's Goddard Space Flight Center (GSFC).

2.2.1. NIRSpec Detector Subsystem

All three NIRSpec modes (MOS, IFU, and fixed slits) share the need for large‐format, high detective quantum efficiency (DQE), and ultralow noise detectors covering the λ = 0.6–5 μm spectral range (see Table 1). This need is fulfilled by two λ_co∼5 μm H2RG SCAs. These SCAs, and the two Teledyne SIDECAR¹¹ application‐specific integrated circuits (ASICs) that will control them, represent today's state of the art. This hardware is being delivered to the ESA by the NIRSpec Detector Subsystem (DS) team at GSFC. The DS team will deliver a fully integrated, tested, and characterized DS to ESA for integration into NIRSpec.

The SIDECAR ASIC and NIRSpec SCA, and indeed all JWST SCAs, recently passed a major NASA milestone by achieving Technology Readiness Level 6 (TRL‐6). TRL‐6 is a major milestone in the context of a NASA flight program, because it essentially marks the retirement of invention risk.

The DS (Fig. 1) consists of the following components: focal plane assembly (FPA), two SIDECAR ASICs, focal plane electronics (FPE), thermal and electrical harnesses, and software. The molybdenum FPA is being built by Teledyne and their partner, ITT. The two H2RG SCAs, which are the focus of this paper, are being built by Teledyne.

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The SCA (Fig. 1) was designed by Teledyne and ITT. Starting from the antireflection (AR) coating and going in, SCA components include (1) AR coating, (2) 2K × 2K HgCdTe pixel array, (3) silicon readout integrated circuit (ROIC), (4) balanced composite structure (BCS), (5) molybdenum base, (6) Rigidflex fanout circuit, and (7) a μD‐37 connector. Components 1–4 are built by Teledyne, and components 5–7 are provided by ITT.

Although NIRSpec's DQE requirement is for λ = 0.6–5 μm, the HgCdTe is actually being grown with a somewhat longer cutoff wavelength near to λ_co∼5.3 μm. This is done to ensure the 80% DQE requirement at λ = 5 μm is met, and is accomplished by varying the mole fraction of cadmium in the Hg_{1 - x} Cd_x Te. In practice, proportionally less cadmium is used to achieve longer cutoffs (Brice 1987).

The H2RG ROIC and SIDECAR ASIC are both reconfigurable in software. For example, both can accommodate up to 32 video channels. However, for NIRSpec, we plan to use only four SCA analog outputs. This is driven by power dissipation considerations on‐orbit, and by the need to minimize system complexity. Each NIRSpec detector will return 2048 × 2048 pixels of 16 bit data per frame. These will appear as a contiguous area of 2040 × 2040 photosensitive pixels surrounded by a 4 pixel wide border of non‐photosensitive reference pixels all the way around. Although the reference pixels do not respond to light, they have been designed to electrically mimic regular pixels. Previous testing has shown them to be highly effective at removing low‐frequency drifts, such as the "pedestal effect," which is familiar to HST NICMOS users (Arendt et al. 2002).

In NIRSpec, the four outputs per SCA will appear as thick, 512 × 2048 pixels bands aligned along the dispersion direction. This is done to minimize the possibility of calibration difficulties in spectra that would otherwise span multiple outputs. Raw data will be averaged in the onboard focal plane array processor (FPAP) before being saved to the solid‐state recorder, and ultimately downlinked to the ground. The FPAP is located in the shared integrated command and data‐handling system (ICDH) and is not part of the DS. Averaging is done to conserve bandwidth for the data link to the ground. Following averaging, the data are still sampled up the ramp; however, each up‐the‐ramp data point has lower noise, and the ramp is more sparsely sampled. Detector readout is discussed in detail in § 3.

Before turning to detector readout modes, it is appropriate to comment on the performance of some prototype and engineering‐grade SCAs that have been built so far. In some cases, most notably prototype JWST SCAs H2RG‐015‐5.0μm and H2RG‐006‐5.0μm, the parts met demanding performance requirements, including total noise per pixel σ_total<6 e⁻ rms per 10³ s integration and mean dark current i_dark≤0.010 e⁻ s⁻¹ pixel⁻¹. However, even with such outstanding detectors, getting the most out of NIRSpec will require understanding both the ideal and nonideal detector behaviors.

For most science observations, NIRSpec's detectors will acquire up‐the‐ramp sampled data at a constant cadence of one frame every ≈10.5 s. A frame is the unit of data that results from sequentially clocking through and reading out a rectangular area of pixels. Most often, this will be all of the pixels in the SCA, although smaller subarrays are also possible when faster cadences are needed to observe, e.g., bright targets. Although each of JWST's NIR instruments differs somewhat in the precise details, Figure 2 shows the JWST NIR detector readout scheme.

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Following in the footsteps of NICMOS, we have dubbed this readout pattern MULTIACCUM. We frequently use the abbreviation MULTI‐ n × m, where n is the number of equally spaced groups sampling up the ramp, and m is the number of averaged frames per group. For example, in Figure 2, n = 6 and m = 4. If a NIRSpec user were to see a raw H2RG FITS file, it would have a dimensionality of 2048 × 2048 × n. Each group, in turn, is the result of averaging m 2048 × 2048 pixel frames.

One advantage of up‐the‐ramp sampled data for space platforms is that cosmic rays can potentially be rejected with minimal data loss. Briefly stated, we anticipate that cosmic‐ray hits will appear as discontinuous steps in pixel ramps. These steps can be identified, and samples on either side of the hit can be used to recover the slope. This has previously been done for the HST NICMOS instrument and is being studied for NIRSpec now.

In the JWST usage, the integration time t_int is the time between digitizing pixel [0, 0] in the first frame of the first group and digitizing the same pixel in the first frame of the last group. The small overhead associated with finishing the last group is not included in the integration time.

Other important time intervals include the frame time t_f and the group time t_g. The frame time is the time interval between reading pixel [0, 0] in one frame and reading the same pixel in the next frame within the same group. The group time is the time interval between reading pixel [0, 0] in the first frame of one group and reading the same pixel in the first frame of the next group. For NIRSpec, the integration time is related to the group time as t_int = (n - 1)t_g.

3.1. Importance of Matching Darks/Skys

3.2. Modeling MULTIACCUM Sampled Data

In this section, we show that a general expression for the total noise variance of an electronically shuttered instrument using MULTIACCUM readout is

In this expression, σ_total is the total noise in units of e⁻ rms, σ_read is the read noise per frame in units of e⁻ rms, and f is flux in units of e⁻ s⁻¹ pixel⁻¹, where f includes photonic current and dark current. The noise model includes read noise and shot noise on integrated flux, which is correlated across the multiple nondestructive reads sampling up the ramp. For the special case of dark integrations, f = i_dark.

Equation (1) can also be used to model CDS and MCDS readout modes, because both are special cases of MULTIACCUM. Table 2 summarizes the parameters used for some common readout schemes. Under ultralow photon flux and ultralow dark current conditions, σ_CDS ≈ 2 σ_read.

An electronically shuttered instrument is one that does not use an opaque shutter to block light from the detectors in normal scientific operations. The main exception to this rule is for taking dark integrations. This readout technique is in widespread use for space‐based astronomical missions and at ground‐based observatories around the world. In an electronically shuttered instrument, the length of an integration is set by the readout pattern, and each pixel sees constant flux during an integration.

JWST testing has demonstrated that dark‐subtracted MULTI‐ n × m sampled data for a pixel (x, y) are usually well modeled by a two‐parameter least‐squares line fit of the form

where s_x,y is the integrating signal in units of e⁻, a_x,y is the y‐intercept, b_x,y is the slope, and t is time.¹² This point is elaborated on in § 5. One widely available implementation is provided by IDL's linfit procedure. However, in practice, we have found that it is much more computationally efficient in IDL to work with full 2048 × 2048 pixel groups of data in parallel, and we compute the standard sums for least‐squares line fitting ourselves. On our Linux and OS X computers, computing the sums directly and in parallel is about 40 times faster than calling linfit sequentially for every pixel in the cube. Moreover, the demands on random‐access memory are greatly reduced, because it is only necessary to read in 2048 × 2048 pixels at any one time. The expressions for the fitted slope b and y‐intercept a are (Press et al. 1992)

In equations (3)–(4), we have dropped the (x, y) subscripts, for the sake of brevity. The terms a and b must be computed for each pixel.

3.3. Derivation of Equation (1)

To correctly model the noise reduction when using multiple nondestructive reads, one must include correlated noise in the integrating charge. Garnett & Forrest (1993) and Vacca et al. (2004) have done this, using slightly different approaches for up‐the‐ramp sampling and MCDS readout modes. However, the JWST readout mode is more general than either of these. Here we extend the previous analysis to cover the more general JWST MULTIACCUM readout mode.

In MULTIACCUM readout, the data are processed in two steps, and both are important for correctly calculating noise correlations. First, the data are averaged into groups of m frames in the onboard FPAP. Subsequently, the n 16 bit unsigned integer‐averaged groups are downlinked to the ground for line fitting using the standard two‐parameter least‐squares fitting in equation (3).

The remainder of this section is necessarily rather mathematical. Readers who are only interested in using equation (1) to model the noise of a detector system may wish to skip to § 3.4. Here we introduce no new material, other than that needed to arrive at equation (1).

Following Garnett & Forrest (1993) and Vacca et al. (2004), the variance in the integrated signal from continuously up‐the‐ramp sampled data can be calculated using propagation of errors, as follows:

where C_i,j is the covariance of the jth data point with respect to the ith data point, and each s_i is the average of m frames. In using equation (5), we have implicitly assumed that each of the partial derivatives is approximately constant within the range of variation of each s_i (Bevington 1969). If this were not true, we would have to include higher‐order partial derivatives. We therefore validate equation (1) for the baseline NIRSpec readout mode in § 3.4.

The covariance terms C_i,j are important because the integrating signal randomly walks away from the best‐fitting line as each successive nondestructive read is acquired. Intuitively, when frame s_i is digitized, the shot noise from frame s_j is already present on the integrating node, and we see that C_i,j = s_j for j<i. Vacca et al. (2004) offer a simple derivation for this relation, as follows. For any two reads i and j with j<i, the associated readout values are s_i and s_j, which are related by

where Δ_{i - j} is the difference in e⁻ between the two reads. One can now write

Because integrating electrons obey Poisson statistics, we see that C_i,j = s_j for j<i.

Using equation (3), the partial derivatives in equation (5) are found to be

Because C_i,j = C_j,i, we can rewrite equation (5) as follows:

Using equation (7) and noting that C_i,i = σ²_i and C_i,j = s_i, where i is the first of the two samples to be acquired, equation (8) can be written

In equation (9), the term 1 / 2(m - 1)t_ff is both important and not obvious at first glance. It comes about because each averaged point sampling up the ramp is, strictly speaking, averaged in both the x‐ and y‐axis directions. The interval over which shot noise is integrated therefore extends from the midpoint of one group to the midpoint of the next. However, σ_g already includes the shot noise from the beginning of the group to its midpoint. For this reason, we must actually subtract the 1 / 2(m - 1)t_ff term in equation (9) to avoid overcompensating for this noise. Although the amount of noise accounted for by this term is small, it shows up clearly in the Monte Carlo simulations that were used to validate the model.

To complete the derivation, we need an expression for σ_g. For the ith group, the FPAP performs straight 16 bit integer averaging of the m frames:

For simplicity, we do not attempt to model truncation errors associated with integer arithmetic. As before, we use propagation of uncertainty to write an expression for σ_g:

Because the signal within each averaged group is referenced to the first read in that group, the reads in one group are not correlated with those in any other. As such, all groups have the same value of σ_g. Moreover, in this case, the partial derivatives in equation (11) are both equal to 1/m, and using equation (10), we can write the following:

Substituting equation (12) into equation (9) and simplifying, we arrive at equation (1).

3.4. Validation of Equation (1)

We have validated equation (1) using Monte Carlo simulations, by comparing our results to others in the literature, and by modeling real data (see § 5.2).

3.4.1. Monte Carlo Simulations

To validate equation (1), we simulated JWST NIRSpec MULTI‐22 × 4 integrations for a range of fluxes. The simulation parameters were as follows: t_int = 890.4 s, σ_read = 14 e⁻ rms, and 0.001≤f<64 e⁻ s⁻¹ pixel⁻¹. Because f includes dark current, the lowest flux simulations indicate the ultimate noise floor of the system, while higher flux pixels indicate what might be seen when observing bright stars.

We simulated 2048 × 2048 pixel data cubes by incrementally adding integrated flux, one frame at a time. The integrated flux during any one frame time was distributed according to the Poisson distribution. Once all flux had been accumulated, normally distributed read noise was added to all pixels in all frames. Following plans for JWST operation, the data were then rebinned into n groups of m averaged frames. Finally, equation (3) was used to compute pixel slopes; these were converted into an integrated signal by multiplying by the integration time; and the standard deviation of each two‐dimensional 2048 × 2048 pixel image was calculated.

The results (see Fig. 3) are in excellent agreement with equation (1), with all deviations being within the statistical uncertainty of the Monte Carlo simulation.

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3.4.2. Comparison to Other Authors

It is helpful to consider a few limiting cases for comparison to previous results in the literature. For the case m = 1, straight up‐the‐ramp sampling, both Garnett & Forrest (1993) and Vacca et al. (2004) contain results that can be compared to our equation (1). In particular, Vacca et al.'s equation (53) is in complete agreement with our result.

In a similar manner, Garnett & Forrest (1993) computed the total noise in read‐noise– and shot‐noise–dominated regimes for continuous up‐the‐ramp sampling. For read‐noise–dominated observations, the noise computed using equation (1) is

For the shot‐noise–dominated regime, equation (1) becomes

Equations (13) and (14) should compare to Garnett & Forrest's equations (19) and (23), multiplied by T²_int. However, they do not, and the difference lies in differing definitions of the integration time. In Garnett & Forrest (1993), the integration time T_int is defined as the entire integration time on the detector node, beginning when the reset switch is opened and ending when the final signal level is sampled. For most astronomical instruments, this is not correct, and the integration time should be defined as shown in Figure 2.

Expressing t_int, the correct integration time in terms of the integration time in Garnett & Forrest's notation T_int, we find

where δ_t is the time between successive pedestal or signal samples. With this correction to Garnett & Forrest's equations (19) and (23), our equations (13)–(14) are in complete agreement with theirs. For completeness, we note that a similar error exists in Garnett & Forrest's results for Fowler sampling. A correction of the form

should be made to their results for Fowler sampling.

3.5. Effect of Neglecting Covariance Terms

If covariance terms in equation (5) are neglected, equation (1) simplifies as

where we have introduced the new symbol σ˜_total to unambiguously represent the approximate noise. The first term represents read noise being averaged down, and the second term accounts for shot noise on integrated flux under the incorrect assumption that noise in the multiple nondestructive reads is uncorrelated.

In the following, we consider two limiting cases: (1) the read‐noise–dominated regime and (2) the shot‐noise–dominated regime. In both cases, we compare the total noise per pixel computed using equation (1) to that computed using the approximate relation, equation (17).

3.5.1. Read‐Noise–Dominated Regime

We first consider the read‐noise–dominated regime. This applies, for example, when measuring the total noise of an SCA having little or no dark current under ultralow photon flux conditions. JWST SCA H2RG‐015‐5.0μm was a good example, having dark current ≤0.006 e⁻ s⁻¹ pixel⁻¹ when tested at the University of Hawaii and at the Space Telescope Science Institute/Johns Hopkins University (STScI/JHU; Rauscher et al. 2004; Figer et al. 2004). We adopt as our metric the ratio ξ = σ_total/ σ˜_total. For the read‐noise–dominated case, this simplifies to

and we see that neglecting the covariance terms does not cause significant errors in this case.

3.5.2. Shot‐Noise–Dominated Regime

In the shot‐noise–dominated regime, the situation is very different. Making the simplifying assumption m = 1, we compute ξ for straight up‐the‐ramp sampling:

From equation (19), we see that for large n and in the shot‐noise–dominated regime, equation (17) underestimates the total noise by 9.5%. As a cross‐check, we note that this result is consistent with Garnett & Forrest's equation (24). Because of this significant error using equation (17), it is particularly important to use equation (1) for modeling up‐the‐ramp sampled data when shot noise is important. For completeness, in the baseline NIRSpec MULTI‐22 × 4 readout mode and in the shot‐noise–dominated regime, ξ = 1.071 and we see that equation (17) underestimates the noise by 7.1%. Equation (1) should clearly be used in this case.

The JWST Project began working with Teledyne¹³ on the H2RG SCA for space astronomy in 1998. Two pathfinder SCAs were produced during the development program. These were the 1024 × 1024 pixel HAWAII‐1R, the first Teledyne SCA to incorporate reference pixels in the imaging area, and the 1024 × 1024 pixel HAWAII‐1RG, which added a programmable guide window. Although the guide window will be used to some extent by all JWST NIR instruments, it will be most heavily used by the TFI.

Beginning in late 2002, the first science‐grade H2RGs began to be produced. For purposes of this article, a science‐grade SCA is one that has excellent performance but is nonetheless non–flight grade. Reasons why a part might be science grade, instead of flight grade, include differences in packaging and changes in the fabrication process. Table 3 summarizes the properties of all of the SCAs that we discuss in this article. The two science‐grade parts had serial numbers H2RG‐006‐5.0μm and H2RG‐015‐5.0μm. H2RG‐006‐5.0μm was a fully substrate‐removed part, whereas the substrate‐on H2RG‐015‐5.0μm was only thinned. Although these two detectors were tested extensively at Teledyne, the University of Hawaii, and STScI/JHU, these early tests did not include the extensive sets of darks that are needed for the statistical analysis presented in §§ 5 and 6.

Beginning in 2006, the NIRSpec DS team at GSFC began to receive engineering‐grade NIRSpec SCAs. Because the packaging was somewhat different from that used earlier, Teledyne hybridized the lowest graded HgCdTe layers first. These lower grade layers have yielded engineering‐grade detectors with dark current and total noise exceeding NIRSpec requirements. However, these engineering‐grade detectors were also the first to be used in a fully flight representative MULTI‐22 × 4 readout mode, and with 50 ramps used for each dark current and total noise test. Where possible, we have cross‐checked our conclusions based on the large data sets by comparison to available data from the earlier science‐grade SCAs. For this reason, although the specific performance parameters of these engineering‐grade SCAs are not fully flight representative vis à vis dark current and total noise, we believe that the general conclusions regarding the reset anomaly and RTN are valid. As new and better SCAs arrive, we plan to continue testing these parameters and others to enable the best possible ranking for flight selection.

4.1. Test Facilities

It is not uncommon to observe a reset anomaly in MULTIACCUM sampled data from JWST H2RGs (Fig. 4). The anomaly is characterized by nonlinearity in the early frames following pixel reset. Although the reset anomaly appears to be unrelated to response linearity,¹⁵ these early frames nonetheless fall below below a line projected through the later, asymptotic portion of the ramp. Fortunately, the reset anomaly is nearly noiseless for the JWST SCAs that have been tested so far, and it usually subtracts out during dark or sky subtraction. Nevertheless, its potentially detrimental side effects must be considered for the most accurate measurement of dark current.

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Depending on the part, we have found that the fraction of affected pixels can range from just a few percent to a significant fraction of the SCA. Tests of the engineering‐grade λ_co = 5 μm NIRSpec SCA H2RG‐S016 revealed that over 15% of the pixels could not be satisfactorily modeled by a straight line (Q_line<0.1, where Q is the integrated χ² probability density giving the probability that the fit's χ² could have been obtained by chance fluctuation within the error bars; Press et al. 1992, eq. [6.2.3]). On the other hand, the reset anomaly was barely noticeable in at least one outstanding prototype SCA, the H2RG‐015‐5.0μm. This detector is one of four JWST SCAs in regular use at the University of Hawaii 2.2 m telescope (Hall et al. 2004).

The reset anomaly can introduce systematic errors into dark current measurements if it is not correctly accounted for. As illustrated in Figure 4, if a two‐parameter line is fitted through all points, the early frames cause the fitted line to overestimate the asymptotic slope, and thereby the dark current.

One common solution is to discard the first few frames of each integration. Clearly, this is an inefficient use of time. Furthermore, complete and unbiased removal of the reset anomaly is nontrivial. For JWST SCAs, the reset anomaly has been observed to have time constants ranging from seconds to hours before the pixels reach the asymptotic portion of the ramp. Moreover, different pixels in the same SCA have different time constants. Even by discarding the first few frames, it is difficult to consistently identify the asymptotic portion of the ramp, and a systematic bias tending to overestimate the dark current remains.

One solution that does not require discarding data is to extract the asymptotic slope, using a function that allows for the reset anomaly early in the ramp. Recent JWST testing has demonstrated that MULTIACCUM sampled data from pixels showing the reset anomaly can be well modeled by a four‐parameter function that includes linear and exponential components. We speculate that the exponential term may be related to RC charging effects in the ROIC/detector components of the hybrid. The equation is of the form

where s_x,y is the integrating signal, t is time, and a_x,y, b_x,y, c_x,y, and d_x,y are the four fitting parameters. The parameters c_x,y and d_x,y are negative quantities. Bacon et al. (2004) used the same equation for modeling the dark current of pixels in a λ_co = 9.1 μm detector array made by Teledyne (known then as Rockwell Scientific). Of the nonlinear pixels (Q_line<0.1), more than 70% are well fitted by the four‐parameter model (Q_4‐param>0.1). Of the remaining nonlinear pixels, many were hot pixels or were corrupted by RTN (see § 6).

Figure 4 shows a direct comparison of all three fitting methods. The data are taken from a single pixel in a dark integration. A linear fit of the entire ramp clearly overestimates the dark current. The linear fit of the asymptotic portion of the ramp and the four‐parameter fit provide much better results. Although both of these methods are comparable in their quality of fit, the four‐parameter fit does not require any data to be discarded. Furthermore, the asymptotic portion of the ramp does not have to be identified for each pixel in the array.

5.1. Noiseless Calibration of the Reset Anomaly

NIRSpec testing has shown that the reset anomaly is highly repeatable for a given pixel. A direct comparison between populations of pixels that are affected by the reset anomaly and those that are not indicates that the reset anomaly contributes almost no additional noise (Fig. 5). Although the dark current properties of these engineering‐grade SCAs are unacceptable for NIRSpec, the noise properties of the two populations are essentially identical.

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We cross‐checked these conclusions against the science‐grade SCA H2RG‐006‐5.0μm. Although the available data sets do not allow us to make the same statistical comparison that we make above for more recent parts, we have compared the measured total noise using 88 samples taken at the beginning of MULTI‐145 × 1 sampled integrations to 88 samples taken at the very end. In this case, we find that using the first 88 frames degrades the total noise by only a few percent compared to using the last 88 frames. We used 88 frames as the basis of this comparison because the NIRSpec baseline MULTI‐22 × 4 readout mode allows 88 frames per 1008 s integration.

The reset anomaly calibrates out during matching dark or sky subtraction. Figure 6 shows the subtraction of a median dark integration from an individual dark integration. The subtraction is performed using a matching MULTI‐88 × 1 median dark cube that was created from a median combination of 50 individual dark integrations, pixel by pixel, within the 2048 × 2048 × 88 pixel cube. The subtracted images have offsets and residual slopes that are the equivalent to a_x,y and b_x,y, respectively, in equation (2). The distribution of offsets is centered at zero, which indicates that the reset anomaly has an identical shape from one integration to the next. The scatter in the offset a_x,y is completely dominated by kTC noise associated with resetting the pixel at the beginning of the integration. In § 5.2, we show that the small residual slope is consistent with shot noise on integrating dark current, as predicted by equation (1) with f = i_dark.

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5.2. Unbiased Dark Current Measurements

We tested the success of the four‐parameter model for measuring dark current using real data from NIRSpec H2RGs. In particular, we (1) tested whether the dark current inferred from the four‐parameter fit could account for the observed noise of the test SCAs, and (2) compared the success of the four‐parameter fit to the more traditional methods discussed above. These tests included a statistical analysis of the noise properties of pixels in the engineering‐grade NIRSpec SCAs H2RG‐S015 and H2RG‐S016. We also performed less extensive spot checks on the engineering‐grade NIRSpec SCA H2RG‐S002.

We expect the measured total noise to be about equal to the noise predicted by equation (1). The observed noise per pixel is given by the standard deviation in the pixel's integrated signal over many integrations. We analyzed 50 individual integrations taken in the DCL, as described in § 4.1.2. To remove the instrumental signature of the reset anomaly, we subtracted a median dark integration from each individual integration. As described in § 5.1, the reset anomaly is highly repeatable. A nearly noiseless subtraction was obtained, as illustrated in Figure 6. The subtraction for each pixel generally results in a small residual slope b_x,y, with an offset a_x,y.

To calculate the noise for each pixel (x,y), we fitted a two‐parameter line to the residual slope in each of the 50 dark‐subtracted integrations, using equation (2). The a_x,y term, which is completely dominated by kTC noise, was discarded. The b_x,y term was used to calculate the integrated signal, as follows:

The analysis produced 50 two‐dimensional images of the residual signal. As expected, the mean value of each pixel is zero e⁻ to well within the uncertainties. The noise of each pixel was computed as

Ideally, we expect the measured noise (eq. [22]) to equal the modeled total noise (eq. [1]). In other words, the ratio of measured to model noise values should be 1.0. In equation (1), the variable f is the dark current of each pixel measured using the four‐parameter fit. The read noise per frame, σ_read, is approximated using the spatial averaging technique. In spatial averaging, two CDS integrations, INT0 and INT1, are used to infer the average noise. Each CDS integration is represented by a data cube. The first two dimensions are the (x,y) pixel position, and the third dimension gives the sample number, which can have the value 0 or 1. The read noise σ_read was calculated as follows:

Because statistical outliers can corrupt spatial averaging noise measurements, iterative σ‐clipping with a 3 σ threshold was used to reject outliers.

We analyzed the noise characteristics of pixels with the reset anomaly in SCAs H2RG‐S015 and H2RG‐S016. The dark current used in equation (1) was obtained from the four‐parameter fit. For each pixel, the measured noise was compared to the mean predicted noise. The results are shown in Figure 7. The success of the four‐parameter fit is highlighted by the agreement between the measured and modeled noise values. The ratio of the two noise terms for SCAs H2RG‐S015 and H2RG‐S016 are 0.97 and 1.02, respectively. These ratios are for the modes of the distributions.

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For comparison purposes, the dark current was also measured using the other fitting techniques described above: (1) linearly fitting the entire ramp and (2) linearly fitting the asymptotic portion at the end of the ramp. For consistency, the asymptotic portion of the ramp was designated to be sample numbers greater than 50. The results in Figure 7 indicate that a linear fit of the entire ramp is a poor estimate of the dark current. The measured and modeled noise values do not agree within an acceptable uncertainty. The linear fit of the asymptotic portion at the end of the ramp does much better. The results are comparable to the four‐parameter fit. The ratio of the two noise terms for SCAs H2RG‐S015 and H2RG‐S016 are 1.01 and 1.00, respectively. While this method provides adequate results, it requires data to be discarded and does not provide consistent results, due to varying time constants.

While we are encouraged by the excellent agreement between measured and modeled noise for these SCAs, this agreement depends in part on the conversion gain g. As explained in § 4.1.1, conversion gain was measured using the photon transfer method (Janesick et al. 1987), and for consistency in this argument, we used the mode of the distribution of g values for each SCA. Ideally, g would be individually measured for each pixel, and an IPC correction would be applied. Doing this accurately requires larger data sets than are available for these engineering‐grade parts, and also a better knowledge of the IPC than is available at the present time. We therefore plan to revisit the agreement between measured and modeled noise as more complete data sets, including good measurements of IPC, become available for NIRSpec's flight and flight spare SCAs in late 2007 and 2008.

5.3. Note on Obtaining Convergence in 4‐Parameter Fitting

In this section, we show that large‐amplitude RTN affects a small and fixed population of pixels. This confirms a previous finding by C. McMurtry (2004, private communication). We believe that small‐amplitude RTN, close to the noise floor of the SCA, can probably be tolerated so long as it does not cause pixels to exceed their stringent total noise budgets. If substantiated by future testing of NIRSpec flight SCAs, we plan to monitor and track RTN using standard pixel operability maps.

RTN has been observed in several JWST H2RG SCAs, and also in four H1RGs at the University of Rochester (Bacon et al. 2005). RTN is characterized by a digital‐like toggle between two (or more) levels. For this reason, RTN has also been referred to as "popcorn mesa noise" (Rauscher et al. 2004) and "burst noise" (Bacon et al. 2005). Because RTN has been observed in both regular and reference pixels, the noise is thought to originate in the ROIC. One likely explanation points to single‐charge defects in the unit cell MOSFET, which is the first amplifier seen by a detector diode.

Figure 8 illustrates a few manifestations of RTN in JWST H2RG pixels. In each case, the data are distributed between two (or more) distinct states. However, the distribution characteristics of these states vary from pixel to pixel. In particular, the states can vary in size and in the frequency and magnitude of the scatter.

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These variations make the detection of RTN difficult and time consuming. We have developed a simple algorithm to detect RTN pixels in MULTIACCUM sampled data. The algorithm consists of a two‐step process designed to identify pixels that share the following two characteristics: (1) unusually noisy sample ramps and (2) sharp rises and falls associated with the digital toggle between the two states.

The first step identifies noisy ramps. Consider a typical pixel with RTN (e.g., Fig. 9 a). To remove any offsets and correlated noise effects, a median dark integration is subtracted from the individual integration (Fig. 9 b). The noise in this ramp is revealed by the large degree of scatter. Two distinct readout states are revealed. While these two states are apparent in Figure 9 b by inspection, they are more clearly illustrated by the histogram in Figure 9 c. The scatter in these pixels tends to be larger than the average scatter σ_avg. We flag all pixel ramps with a sample scatter beyond ±5 σ_avg as potential RTN pixels. Although this high threshold has the advantage that it results in few false detections, it also means that we miss smaller amplitude RTN pixels.

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This first step, however, cannot distinguish between RTN pixels and those that are naturally noisy. The algorithm tends to return false detections due to "hot" pixels that do not necessarily exhibit the two (or more) distinct states that are associated with RTN. These pixels have a high degree of scatter, because they typically have high dark current and poor median dark subtraction. For future detector operation, we expect to have pixel masks that will allow us to identify and avoid these "hot" pixels. At the time of this analysis, however, we implemented a second step to isolate RTN pixels.

This second step identifies pixel ramps that exhibit sharp, distinct rises and falls. This characteristic is typical of RTN, which is identified by the toggling between two (or more) levels. In comparison, the noise in "hot" pixels is due to large dark current and does not tend to toggle up and down. Instead, the charge increases steadily, just as it does in well‐behaved pixels. The only difference is that the increase tends to be larger. Differencing successive data points provides an easy analysis of the pixel behavior. The toggle in an RTN pixel will produce a differential plot similar to the one shown in Figure 9 d. Again, the pixel differentials will have an average scatter σ_avg. Of these pixels flagged in step one, all ramp differentials with scatter beyond ±5 σ_avg are flagged as RTN pixels.

The success of this algorithm is highlighted by its false‐detection rate of less than 1%. Nonetheless, we note that the algorithm's success is limited by the chosen threshold. For the present purpose of studying RTN characteristics, we choose a ±5 σ_avg threshold to best isolate pixels with RTN from those that may be affected by other noise sources. Therefore, our sample of RTN pixels represents a lower limit on the actual number of RTN pixels within the array. A ramp could potentially have two states confined within the 5 σ_avg threshold, and would thereby go undetected. Setting the threshold lower would increase the number of detections, but it would also increase the chance of a false detection due to the other sources of scatter. A possible solution utilizes multiple‐Gaussian fitting to identify the two unique populations apparent in Figure 9 c (Bacon et al. 2005).

Using our two‐pass algorithm, we have observed large‐amplitude RTN to occur in a fixed, small subset of pixels. For SCA H2RG‐S16, 99 integrations were tested. Figure 10 shows a histogram that illustrates the repeatability of RTN detections per pixel from integration to integration. A vast majority of pixels have zero detectable RTN features at the ±5 σ_avg threshold in any of the 99 integrations sampled, as indicated by the peak at bin 0, which reaches beyond the extent of the plot to just under 100%. Less than 1% of pixels exhibited RTN characteristics at the ±5 σ_avg threshold. For a majority of those that did, RTN was subsequently detected in that pixel for 99% of the integrations, as indicated by the peak at bin 99. The noticeable rise in bin 1 and falloff in bin 100 is a result of the statistical nature of the magnitude of the scatter. These features can also be partly attributed to the algorithm's <1% false‐detection rate.

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For the engineering‐grade JWST SCAs that have been studied to date, these results for H2RG‐S16 are typical, and only a small percentage of pixels appear to show large‐amplitude RTN at T = 37 K. Using a more sensitive detection algorithm, Bacon et al. (2005) found that 11% of the pixels in the SCA they tested manifested RTN at T = 37 K; and moreover, that there were significant temperature dependencies. These included the size of the largest transition decreasing with increasing temperature (Bacon et al. 2005). The difference in the percentage of RTN pixels reflects differences in detection algorithms, and possibly device‐to‐device variation.

As science‐ and flight‐grade SCAs become available for JWST, we plan to continue and extend these studies of RTN. One interesting conjecture is that there may be a continuum of pixels affected by RTN (blending into the read noise), and that the lower one sets the threshold, the more RTN pixels one finds. However, even if this conjecture were substantiated, it is not clear to us that a pixel should be disqualified from use if it meets all operability requirements while manifesting low‐level RTN. At some level, RTN becomes one of many components that contribute to the overall noise of a pixel. Viewed in this light, RTN is a noise component that has the advantage that it is easily identified and can therefore be fixed in future SCA designs.

The repeatability of large‐amplitude RTN is good news. The feature is typically one of the noise components that can cause a pixel to fail to meet operability requirements. Locating and handling RTN pixels in real‐time pipeline processing is costly and inefficient. Because large‐amplitude RTN is confined to a fixed, small subset of pixels, it is a feature that can be tracked using a pixel operability mask. Because tracking operable pixels is a standard part of calibration for flight instruments, we expect large‐amplitude RTN to have a negligible impact on JWST calibration pipelines.

Additional study is needed to understand how repeatable small‐amplitude RTN is. Although we hypothesize that small‐amplitude RTN is also a property of a fixed population of pixels, it would be good to confirm this by testing. Doing this correctly requires a better RTN detection algorithm than we have at the current time, and we plan to test this hypothesis as better detection algorithms are developed.

Likewise, it would be helpful to know exactly where in the signal chain RTN arises. We know that a significant fraction of the RTN, perhaps all of it, originates in the ROIC. We know this because we see RTN in both reference pixels, which are not connected to the HgCdTe detectors, and in regular pixels. Others have also used specialized readout software to show that RTN originates in the ROIC (Bacon et al. 2004). Simple physical arguments suggest that the origin lies in the first MOSFET in the signal chain, although it would clearly be better to experimentally pinpoint the origin. Doing this could facilitate design improvements to eliminate the RTN.

For similar reasons, it would be helpful to identify the physical mechanism that is the underlying cause of the reset anomaly. As with RTN, additional study would be helpful. One area that we plan to explore more fully is whether the reset anomaly alters a pixel's response to light. Although there has been no clear evidence of this in the JWST program so far, it will be tested when we characterize the linearity and photometric stability of the DS.

In this paper, we describe the JWST NIRSpec baseline MULTIACCUM readout mode, present a general noise model for NIR detector data acquired using multiple nondestructive reads, and discuss recent NIRSpec SCA test results. We believe that the noise model is applicable to most astronomical NIR instruments. Our major findings and recommendations are as follows:

• 1.  
  The total noise in common NIR detector operating modes, including CDS, MCDS (Fowler‐N), and MULTIACCUM, can be modeled using equation (1) and the parameters listed in Table 2. This noise model includes read noise, shot noise on integrated charges, and covariance terms between multiple nondestructive reads. If these covariance terms are neglected, and read noise and shot noise are simply added in quadrature, we show that errors of ≈9.5% in the predicted noise for bright sources are possible. The sense of the error is to underpredict noise when covariance terms are neglected.
• 2.  
  Many NIRSpec H2RG SCAs have shown a reset anomaly. This appears as nonlinearity in the early reads following reset. Although the reset anomaly does not appear to be related to response linearity, we plan to verify this by test for NIRSpec. If the reset anomaly is not correctly accounted for during calibration, it can lead to systematic overestimation of the dark current. We show how the reset anomaly can be noiselessly calibrated out using matching darks, and how dark current can be accurately measured in the presence of the reset anomaly using four‐parameter fits.
• 3.  
  As has previously been reported, NIRSpec H2RGs are often affected by RTN. Using new test data, we show that large‐amplitude RTN is often a property of only a small and fixed population of pixels. For flight operations, we plan to monitor and track RTN using pixel operability maps.

These conclusions, particularly with regard to the reset anomaly and RTN, are largely based on testing engineering‐grade SCAs. This was done because the required large data sets are only available from engineering‐grade parts at this time. We therefore plan to confirm these findings using better SCAs as they become available.