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Astronomical Data Analysis Software and Systems X ASP Conference Series, Vol. 238, 2001 F. R. Harnden Jr., F. A. Primini, and H. E. Payne, eds.

Uniform Data Sampling: Noise Reduction & Cosmic Rays
J. D. Offenberg1 , D. J. Fixsen1 , M. A. Nieto-Santisteban2, R. Sengupta1 , J. C. Mather3 , H. S. Stockman2 Abstract. As computers and scientific instruments become more complicated and more powerful (Moore's Law), we can perform astronomical observations never before contemplated. As larger data volumes are acquired, as more complex instruments are designed, and as observatories are placed in distant space locations with constrained downlink capacity, the need for automated, robust image processing tools will increase. We present a robust, optimized algorithm to perform automated processing of array image data obtained with a non-destructive read-out. We present the derivation of the noise effects of this algorithm and compare alternative strategies.

1.

Introduction

The effects of radiation and cosmic rays can be a formidable source of data loss for a space-based observatory. Several solutions to the problem of identifying and removing cosmic rays exist. We evaluate Up-the-Ramp sampling with onthe-fly cosmic ray identification and mitigation, which is described in detail by Fixsen et al. (2000) and compared to Fowler Sampling (Fowler & Gatley 1990). We concentrate on Up-the-Ramp sampling for study because it provides better signal-to-noise in what is probably the most difficult-to-measure regime, the read-noise limit. In the absence of cosmic rays, Up-the-Ramp sampling provides modestly ( 6%) higher signal-to-noise than does Fowler Sampling (Garnett & Forrest 1993). The fact that an Up-the-Ramp sequence can be screened for cosmic rays and other glitches improves this result. Furthermore, on-the-fly cosmic ray rejection allows longer integration times which also improves the signal-to-noise in the faint limit (Offenberg et al. 2001). The following discussion is largely an excerpt from Offenberg et al. 2001. 2. Fowler Sampling

Fowler sampling reduces the effect of read noise to r = r 4/N (for an observation sequence consisting of N samples, N/2 Fowler-pairs). However,
1 2 3

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396 c Copyright 2001 Astronomical Society of the Pacific. All rights reserved.


Uniform Data Sampling: Noise Reduction & Cosmic Rays when a pixel ray essentially that location. the read-noise

397

is impacted by a cosmic ray during an observation, the cosmic injects infinite variance and reduces the signal to noise to zero at If we start with the Fowler sampling signal-to-noise function in limit, from Garnett & Forrest (1993; Eqn. 6), FT SNF = 2 T 1- 2t 2 FT = V0 (1)

r

where F is the flux of the target, T is the observation time, r is the read noise, is the Fowler duty cycle, and t is the time between sample intervals (determined by engineering or scientific constraints on the system). We note that this formula breaks down for relatively small numbers of samples (i.e., t large with respect to T ). FT is the signal, so the remaining terms are the noise, which is the square-root of the variance, V0 . If we consider two cases, "no-cosmic-ray" and "hit-by-cosmic-ray," and combine the variances according to V
comb

=

2 V0 P0 W0 + V1 P1 W (W0 P0 + W1 P1 )2

2 1

(2)

we can rewrite Equation 1 as SN
FC

FT FT (W0 P0 + W1 P1 ) = = 2 2 Vcomb V0 P0 W0 + V1 P1 W1

(3)

As the weight is the inverse of the variance (Wi = 1/Vi), Equation 3 can be rewritten as FT ( P0 + P1 ) P0 P1 V0 V1 = FT + (4) SNFC = V0 V1 P0 /V0 + P1 /V1 V0 is the variance in the no-cosmic-ray case, taken from Equation 1, and P0 is the probability of a pixel surviving without a cosmic ray hit. For simplicity, we define 1 - P to be the probability of a pixel being hit by a cosmic ray per time unit t, so P is the probability of "survival" and P0 = P T/ t. As a cosmic ray hit injects infinite uncertainty, the variance in the cosmic ray case is V1 = . Plugging in to Equation 4, we get the signal-to-noise for Fowler sampling in the read-noise limited case with cosmic rays, Equation 5: SN
FC

= FT

FT P0 +0 = V0 2

r

T 1- P 2t 2

T/(2t)

(5)

SN get

FC

For a given integration time T and minimum read time t, the maximum occurs with duty cycle = 2/3. If we plug this back into Equation 5, we SNF = 2 FT 3 2 T P 3t
T/(2t)

(6)

r

From here, it is possible to find the value of T which gives the best signal-tonoise for a single observation; it occurs at T = - 3t/ ln(P ). If, however, we


398

Offenberg et al.

consider the observation as a series of M equal observations with a specific total observation time, Tobs , the signal-to-noise for the series is SN
FC

FT M = 3 r

2T P 3t

T/(2t)

FT Tobs = 3 r T

2T P 3t

T/2t

(7)

If we hold Tobs constant and find the optimum T , we find it at -2t/ ln(P ). In either case, it is important to note that there is an optimal value for T , and extending the observation beyond that time will ruin the data. It is worth noting that the result assumes that all cosmic ray events can be identified a posteriori. This is not necessarily the case, particularly when it is considered that, in the one-image case, the fraction of pixels surviving without a cosmic ray impact is P -3/ ln(P ) = e-3 0.05; for the multi-image case, the fraction of survivors is P -2/ ln(P ) = e-2 0.14. In both cases, the number of "good" pixels is so low that separating themfromthe impacted pixels will not be a trivial task. For example, the median operation would not be able to identify a good samples, as more than half of the samples would be impacted by cosmic rays. In practice, the detector will often saturate before this limit is reached, but this shorter integration time means that less-than-optimal signal-to-noise will be obtained. 3. Up-the-Ramp Sampling

Up-the-Ramp sampling reduces the effect of read noise to r = r 12/N , for N uniformly-spaced samples with equal weighting (which is the optimal weighting for the read-noise limited case). When a pixel is impacted by a cosmic ray, the Up-the-Ramp algorithm preserves the "good" data for that pixel. The exact quality of the preserved data depends on the number of cosmic ray hits and their timing within the observation. For example, a cosmic ray hit which just trims off the last sample in the sequence has minimal impact compared to a cosmic ray hit that occurs in the middle of the observation sequence. The variance of a Uniformly-sampled sequence with Ni samples is proportional to 1/Ni (Ni +1)(Ni - 1). If an Up-the-Ramp sequence is broken into i chunks by a cosmic ray, the variance becomes Vi = V
U i j =0

N (N +1)(N - 1) (Nj )(Nj +1)(Nj - 1)

(8)

When there are zero cosmic ray events, of course, V0 = VU . If there is one cosmic ray event during the sequence, the variance becomes V
1

=V

U

N (N +1)(N - 1) Ni(Ni +1)(Ni - 1) + (N - Ni )(N - Ni +1)(N - Ni - 1)

(9) dis, we will ray

If we assume (as is reasonable) that the cosmic ray events are randomly tributed over time and find the expectation value for all values of 0...Ni...N find that the typical V1 VU 2 (plus a small term in N -1 , which we ignore for simplicity). If we perform a similar computation for two cosmic


Uniform Data Sampling: Noise Reduction & Cosmic Rays

399

events, we find that V2 VU 10/3 (again, plus lower-order terms which we ignore). In general, we find that it is possible to find a valid result with a finite variance for any sequence broken up by cosmic ray events provided we have at least two consecutive "good" samples (for all practical purposes, we can ignore the situation where this is not the case). To simplify the following, we consider only three cases: The no-cosmic-ray case V0 = VU , the one-cosmic-ray case V1 = 2 VU and all multiple-cosmic-ray cases combined as one, V2+ = VU / 2 , where 2 is a small but non-zero number, roughly 0.3. The Up-the-Ramp signal-to-noise function for the read-noise limited case (Garnett & Forrest 1993; Eqn. 20) is FT FT N2 - 1 SNU = = (10) 6N VU 2r We combine the variances in the three possible cases with the three-case equivalent to Equation 2, and thus arrive at
1/2 FT P1 + P2+ = P0 + (11) UC 2 VU where Pi is the probability of a pixel being impacted by i cosmic rays during the integration. We note, as did Garnett & Forrest, that there would be no reason to limit the number of samples to anything less than the maximum possible number, so we can set N = T/ t. Using the definition of P described earlier, P0 = P T/ t, P1 = (T/ t)(1 - P )P T/ t-1 and P2+ = 1 - (P0 + P1 ). Putting these values back into Equation 11, we get:

SN

= FT

P2+ P0 P1 + + V0 V1 V2+

1/2

SN

UC

(12) If we seek the maximum value of SNUC with respect to T , we find that SNUC is strictly increasing if T t (otherwise we would have an integration shorter than one sample time, which would be useless), P = 0 and 0 < < 1 (both of which are true by construction). This result applies whether we are considering one independent integration or a series of observations to be combined later. As the derivative is strictly positive, the signal-to-noise continues to increase with the sample time, although as T , the gain in signal-to-noise asymptotically approaches zero. So, extending the observing time while using Up-the-Ramp sampling with cosmic ray rejection does not damage the data (although we might be spending time with little or no gain). As noted earlier for the Fowler-sampling case, there is an optimal observing time, beyond which further observation reduces the overall signal-to-noise. References Fixsen, D. J., et Fowler, A. M. & Garnett, J. D. & Offenberg, J. D., al. 2000, PASP, 112, 1350 Gatley, I. 1990, ApJ, 353, L33 Forrest, W. J. 1993, Proc. SPIE, 1946, 395 et al. 2001, PASP, 113, in press

FT = 2

r

T 2 - t2 (1 - )P 6T t

T/ t

+(1 - )

T (1 - P )P t

T/ t-1

1/2

+