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Coadd Noise in Pattern-Noise Simulations | |
Below we summarize the impact of pattern noise seen in band 3 and 4
frames on co-adds and temporal outlier detection statistics.
Frames were simulated by Ned Wright with and without pattern noise included. The
pattern noise was embedded using a set of lab darks from the second MIC2 test.
Frames were processed using v2.3x of the WSDS pipeline.
The main steps that concern us here are
Instrumental Calibration and
Frame Co-addition.
The latter included background (offset) matching
(Bmatch),
throughput (or photometric zero-point) matching, and outlier detection
with masking
(AWOD).
Figure 1 shows the simulation layout and co-add geometry. Eight co-adds
were made (using frames with and without pattern noise).
Other WSDC reports shall summarize the impacts on source sensitivity and
detection statistics.
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| Figure 1 - Co-add boundaries (blue) overlayed over simulated region. Red circles are frame centers and black dots are co-add centers. Each co-add is ~1.56° x 1.56° and there is a 3′ linear overlap. The 3 x 3 grid in the bottom co-add shows the regions from which noise statistics were computed. Click to enlarge. |
Here's a summary of the analysis presented below.
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| Figure 2 - Band 3 image products: top row is for no pattern noise and bottom row includes pattern noise. From left to right: Instrumentally calibrated (level-1b) frame (notice the corrugated pattern noise in the bottom frame); co-added region; corresponding depth-of-coverage map (see color bar for range); and outlier mask from the temporal outlier detection module (AWOD). All images measure 6′ on a side. Click to enlarge. |
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| Figure 3 - Band 4 image products: top row is for no pattern noise and bottom row includes pattern noise. From left to right: Instrumentally calibrated (level-1b) frame (notice that the pattern noise is much reduced in band 4, although this could be a one-off frame where the pattern happened to be weak); co-added region; corresponding depth-of-coverage map (see color bar for range); and outlier mask from the temporal outlier detection module (AWOD). All images measure 6′ on a side. Click to enlarge. |
Each pattern and pattern-free co-add was partioned into a 3 x 3 grid and noise statistics were computed in each partition. This enabled us to (i) explore the local variation in sigma over the simulated region and (ii) to sample regions not contaminated by bright extended structure that could bias the noise-sigma estimates.
Since the noise depends on the depth-of-coverage, we computed a noise-sigma using pixels corresponding to an approximately uniform depth-of-coverage centered around the median, i.e, within the range: median coverage ± 0.25. We used the difference between the 50th (median) and 16th percentiles (or more precisely the 15.86 percentile) as a proxy for the RMS in pixel fluctuations. This is because this measure is expected to be robust against outliers and sources in the high tail of the pixel distribution. In the limit of a large number of pixels, this approaches the sample standard-deviation (sigma) of a Gaussian, assuming of course no outliers and that the noise distribution was indeed drawn from a Gaussian population.
As a detail, we note that these sigma measures cannot be used to derive
absolute measures of sensitivity in the co-adds.
In particular, the smallness of the sigmas for band 4 is due to
an erroneous application of a zero-point factor during
co-add generation. This does not affect relative measures of sigma
between the pattern and no-pattern simulations.
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| Figure 4 - Band 3 co-add noise-sigma distribution across all partitions of a 3 x 3 grid on each co-add (giving 36 values in total). This used a robust proxy for the spatial RMS (σ) as described above. The median sigmas for the no pattern/with pattern cases are 0.036645/0.037836 DN/pixel, corresponding to an increase of ~3.3% in the co-add pixel noise due to pattern noise. | Figure 5 - Same as Fig. 4 for band 4. The median sigmas for the no pattern/with pattern cases are 0.0022375/0.0022683 DN/pixel, corresponding to an increase of ~1.4% in the co-add pixel noise due to pattern noise. |
We formally tested whether the co-add sigmas between the pattern and no-pattern noise simulations are significantly different, for each of the 36 partitions. Our null hypothesis is H0: σnopattern = σpattern versus the alternative hypothesis H1: σpattern > σnopattern. This used the F-test which computes the probability of obtaining a ratio F ≤ σ2nopat/σ2pat by chance assuming Gaussian statistics for the pixel-noise distribution.
This test was applied to each partition of the 3 x 3 grid on each of
the 4 coadds, i.e., for the 36 values of
σ2nopat/σ2pat.
Results are tabulated below.
The median depths-of-coverage (Cov) for each region and the number of
pixels (Np) with Cov ± 0.25 used for the sigma computations are also shown.
Most of the chance probabilities are extremely low (Pr < 10-4%),
indicating that σpat > σnopat
to a very high degree of confidence.
The handful of cases where the chance probability is > 20% correspond
to regions contaminated by bright extended structure. Our
proxy for sigma, although robust, is not totally immune to large variations.
These variations will dominate differences in
σpat and σnopat
expected from pattern noise alone.
Table 1 - Summary of F-test statistics for band 3 (see text for details):
F = σ2nopat/σ2pat Cov(nopat) Cov(pat) Np(nopat) Np(pat) Pr(≤ F)
0.956415 11.5360 11.6317 87900 73521 1.403e-10
0.861750 11.4549 11.5391 157343 88402 2.243e-140
0.996390 11.5004 11.4885 111581 120586 2.691e-01
0.942883 12.0076 12.0084 390491 384987 3.419e-75
0.908003 11.9857 11.9972 374487 380092 3.143e-193
0.981811 11.9682 11.9808 358295 373111 1.438e-08
0.941375 11.4501 11.4786 169648 135454 4.149e-32
0.956332 11.5560 11.6559 85246 72789 1.930e-10
0.876726 11.4764 11.4145 136285 227675 6.622e-161
1.013889 11.4254 11.5444 200688 83836 9.911e-01
0.938251 11.4267 11.4271 192779 201057 1.164e-45
0.922125 11.5124 11.3847 103667 277112 1.728e-55
0.973010 11.9789 12.0009 365637 379650 3.449e-17
0.933515 11.9675 11.9591 368966 353555 2.986e-95
0.975400 11.9813 11.9766 372842 375252 1.305e-14
0.966112 11.3639 11.3839 321142 295555 5.571e-22
0.962362 11.3496 11.3653 371601 326881 5.395e-30
0.915909 11.4599 11.3860 157764 283003 1.513e-86
0.948483 11.6248 11.4949 70350 116973 2.628e-15
0.883140 11.5750 11.5324 75085 90660 6.302e-71
0.962829 11.5282 11.4578 93385 160973 4.018e-11
0.900046 11.9852 11.9895 370221 379586 4.195e-228
0.923526 11.9858 11.9888 376130 378344 4.402e-132
0.920393 11.9632 11.9453 356526 335305 1.093e-131
0.953288 11.4321 11.4090 208752 240339 6.610e-30
0.922871 11.4018 11.4165 259531 226738 3.337e-87
0.942644 11.3678 11.4213 326301 219562 3.275e-52
0.933309 11.7007 11.6852 84284 80119 2.236e-23
0.928756 11.6334 11.5585 69275 77701 8.181e-24
0.995205 11.6174 11.6739 68965 74212 2.603e-01
0.892084 11.9942 12.0021 378841 383527 2.661e-272
0.903997 11.9792 11.9947 364242 382757 9.004e-209
0.953375 12.0106 11.9866 388772 363794 7.850e-49
0.904083 11.4662 11.4087 154378 244095 4.962e-106
0.926477 11.3763 11.3950 316686 275747 6.444e-96
0.934977 11.3410 11.3900 398273 289104 6.594e-85
Table 2 - Summary of F-test statistics for band 4:
F = σ2no-pat/σ2pat Cov(nopat) Cov(pat) Np(nopat) Np(pat) Pr(≤ F)
1.042155 11.3995 11.4517 209614 146489 1.000e+00
0.984679 11.3529 11.4091 285590 195545 9.905e-05
1.004168 11.3895 11.3806 226700 230722 8.400e-01
0.998341 11.9087 11.9130 327656 325778 3.176e-01
0.982694 11.8681 11.8961 289171 319081 7.654e-07
0.918030 11.8521 11.8461 274810 277756 3.920e-112
0.904612 11.3718 11.4012 272146 227257 5.275e-138
0.947239 11.4383 11.4994 173074 115637 2.515e-24
0.975253 11.4024 11.3519 217565 302528 1.499e-10
0.981703 11.3411 11.4055 325727 210000 1.511e-06
1.000420 11.3526 11.3408 296312 325350 5.465e-01
1.011503 11.3894 11.3246 233659 350379 9.988e-01
0.998937 11.8649 11.9084 278093 320429 3.859e-01
1.026373 11.8459 11.8234 265976 231532 1.000e+00
1.008121 11.8625 11.8651 281151 288166 9.845e-01
1.045813 11.3061 11.3328 401410 365202 1.000e+00
0.958239 11.3058 11.3136 419284 399461 1.086e-42
1.005068 11.3813 11.3281 260589 360577 9.178e-01
0.925845 11.4546 11.3848 151559 247085 1.655e-62
0.988485 11.4316 11.4049 176948 216981 5.292e-03
0.944725 11.4042 11.3724 206747 273427 1.768e-43
0.941939 11.8724 11.8837 278320 296214 4.987e-58
0.910337 11.8788 11.8843 296344 296790 1.492e-144
0.855448 11.8279 11.7991 241034 217071 1.697e-305
0.974494 11.3701 11.3531 298241 324273 3.024e-13
0.952174 11.3440 11.3570 353691 318959 4.788e-46
0.971612 11.3199 11.3556 396190 317646 5.907e-18
0.885055 11.5432 11.5184 103197 109446 2.993e-88
0.971738 11.4885 11.4418 123851 161147 4.142e-08
1.029524 11.4652 11.4972 138482 120847 1.000e+00
1.005769 11.9077 11.9266 316870 336160 9.498e-01
0.955244 11.8818 11.9098 286933 322197 9.633e-37
0.976678 11.9369 11.8945 343028 300878 1.176e-11
0.960404 11.4081 11.3689 233483 305209 1.520e-25
0.970651 11.3256 11.3423 397906 363441 2.113e-20
1.010201 11.3068 11.3400 443191 370548 9.994e-01