Removing Atmospheric Fringes from Zwicky Transient Facility i-band Images using Principal Component Analysis

ZTF's reduction pipeline was designed to process each of the camera's 64 readout-channels separately. We therefore began by gathering images of a common readout-channel together for model generation, with the goal of generating 64 fringe models. Training images were selected by gathering all i-band images between 2019 April 1 and 2020 April 1 and removing images with a limiting magnitude less than 19. This cut on limiting magnitude removed cloudy images from our sample that would fail to have a photometrically accurate measurement of the night sky and thus its atmospheric lines. By including all images within these dates we ensured a representative sample of atmospheric conditions and airmasses which are correlated with the strength of atmospheric fringes. There were 550,365 i-band images included in total, with each fringe model trained on between 8062 and 8898 images.

We began by selecting all fringed images ( I _fringe) for a single readout channel. The location and strength of atmospheric fringes are independent of the astrophysical sources within an image. Therefore we needed to remove sources from each image prior to training in order to reduce the confusion in the PCA variance reduction process. Our preferred method for identifying pixels containing astrophysical sources was to use the quality masks produced by IPAC that labels pixels containing a source in each image's source catalog. In cases where this pixel mask could not be obtained, pixels containing sources were identified by calculating the median absolute deviation for the image and flagging pixels that were 5 standard deviations above or below the image's median value. This process did not clip the atmospheric fringes themselves, which were at the level of the background noise and therefore significantly below the level of these thresholds. All pixels flagged as containing sources were replaced with the value of the image's global median, removing stars from our images without changing the value of the vast majority of pixels. A possible improvement to our method could be to replace the pixel instead with an estimate of the local background. We choose the global median because the local background can be difficult to estimate due to the fringes themselves and in particular in regions where the transverse length scale of the fringes approaches the pixel scale of the image. Each image was then scaled so that they could be used for training together with other images taken at different airmasses and limiting magnitudes. We named each training image a fringe map (${\overrightarrow{F}}_{\mathrm{map}}$), as it traces the relative location and strength of only the atmospheric fringes (Figure 1). Fringe maps were created by subtracting the median from each image, dividing the result by the image's median absolute deviation, and flattening the image into a [1 × N_pixels] array:

$$\begin{eqnarray}&&{\left.{\overrightarrow{F}}_{\mathrm{map}}=\displaystyle \frac{{{\boldsymbol{I}}}_{\mathrm{fringe}}-{\widetilde{{\boldsymbol{I}}}}_{\mathrm{fringe}}}{\mathrm{MAD}({{\boldsymbol{I}}}_{\mathrm{fringe}})}\right|}_{\mathrm{flattened}}.\end{eqnarray} \tag{ 1 }$$

(Because PCA is a linear method that calculates coefficients irrespective of the basis in which the data is represented, performing our analysis in one-dimension has no effect on the results.) Each pixel in the fringe map was treated as a separate variable in our PCA analysis, resulting in 9,461,760 variables (3080 rows by 3072 columns) for each model. The eigenvalues of the feature vector were determined using the randomized Singular Value Decomposition method (Halko et al. 2011; Martinsson et al. 2011) in the scikit-learn Python package (Pedregosa et al. 2011). Each one-dimensional eigenvector was reconstructed into the original image shape to create an eigen-image. The final fringe model was a set of eigen-images that captured orthogonal contributions to the training set variance in each pixel. We also recorded each component's explained variance (${\overrightarrow{R}}_{\mathrm{model}}$), or the amount of training set variance pointed in each eigenvector's direction, for later use in the processing pipeline. We then repeated this process to construct a unique fringe model for each of the camera's 64 readout channels.

Fringes were removed from images by applying the fringe model in these steps:

$$\begin{eqnarray}&&{\overrightarrow{\lambda }}_{\mathrm{map}}=\displaystyle \frac{({\overrightarrow{F}}_{\mathrm{map}}-{\overrightarrow{\mu }}_{\mathrm{model}})\,\cdot \,{{\boldsymbol{v}}}_{\mathrm{model}}^{T}}{\sqrt{{\overrightarrow{R}}_{\mathrm{model}}}},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\overrightarrow{F}}_{\mathrm{bias}} & = & \left[\left[{\overrightarrow{\lambda }}_{\mathrm{map}}\,\cdot \,\left(\sqrt{{\overrightarrow{R}}_{\mathrm{model}}}{{\boldsymbol{v}}}_{\mathrm{model}}\right)\right]\right.\\ & & +\,\left.{\overrightarrow{\mu }}_{\mathrm{model}}\right]\cdot \mathrm{MAD}({\overrightarrow{I}}_{\mathrm{fringe}}),\end{array}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{\overrightarrow{I}}_{\mathrm{clean}}={\overrightarrow{I}}_{\mathrm{fringe}}-{\overrightarrow{F}}_{\mathrm{bias}}.\end{eqnarray} \tag{ 4 }$$

First, a fringe map was regenerated for each fringed image in our sample. The dot product of this fringe map (less the average value of each pixel in the training set ${\overrightarrow{\mu }}_{\mathrm{model}}$) and the model's eigen-images was calculated, and then divided by the square root of that component's explained variance (Equation (2)). This produced the eigenvalues (${\overrightarrow{\lambda }}_{\mathrm{map}}$) for each fringe map. These eigenvalues were used as coefficients to linearly combine re-scaled eigen-images and the average values of the training sets were added back in (Equation (3)). This image, after being re-scaled by the median absolute deviation of the original fringed image, was named a fringe bias (${\overrightarrow{F}}_{\mathrm{bias}}$). When this fringe bias was subtracted from the original fringed image it produced a clean image (${\overrightarrow{I}}_{\mathrm{clean}}$) with significantly less fringing (Equation (4)).

An example of each of the images in this process are shown in Figure 2. This figure visually demonstrates the results of our method. The original image is a 90 s i-band exposure that contains a representative amount of atmospheric fringes. The fringe map is almost entirely devoid of individual sources, although the source pixel identification method does struggle around particularly bright stars. The fringe bias generated from processing the fringe map through the fringe model successfully identifies the location and strength of each atmospheric fringe. The clean image has nearly all of the atmospheric fringing pattern removed while retaining nearly all of the astrophysical sources.

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The arithmetic average of the fringe maps used to train the fringe model for each readout-channel is shown in Figure 3, with each readout-channel placed on a common gray-scale. The pattern that clearly emerges is coherent on the level not just of the readout channel, but also over the full CCD. The etching process for creating a thinned CCD uses a circular buffer that removes layers of the chip to create a uniformly thick device. The average of the training fringe maps identifies the residual thickness variations for each CCD. Work is ongoing to use these psuedo-measurements of the thickness to improve the ZTF data quality pipeline (R. Dekany 2021, private communication). The inner 32 readout channels (16 ≤ rcid < 48) and the outer 32 readout channels (0 ≤ rcid < 16, 48 ≤ rcid < 64) have distinctly different amounts of atmospheric fringing present in their images. The inner readout channels have two layers of AR coating while the top and bottom rows of CCDs have a single layer coating. This causes the outer readout channels to have a higher reflectivity at the longer wavelengths where fringing occurs and thus a larger contrast of the fringing pattern.

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The eigen-images for readout channel 13 are shown in Figure 4 as a representative example of a fringe model. Each pixel in the image was trained as a separate variable and yet the fringe patterns across pixels remains coherent after reconstruction into the original image dimensions. This confirms that the components contain correlated eigenvectors for the different features. In addition to the atmospheric fringes, the PCA training captured large scale variations in the background that remain after flat-fielding. These backgrounds dominate the higher order components. This indicates that some global scale variations in flux remains after the execution of the ZTF flat-fielding pipeline that could possibly be improved by implementing a PCA method (Bernstein et al. 2017). We note that PCA requires setting the number of components, or number of eigenvectors, for the reduction algorithm. The fractional explained variance for all of the components are also shown in Figure 4. On average across the 64 readout channels, the first component captured 64.6% of the pixel variance while the sixth component captured only 5.0% of the pixel variance. In total six PCA components reconstructed 95.0% of the variance seen in our training sample. Adding additional components failed to significantly increase the amount of fractional explained variance as the previous component. We therefore determined that six components was sufficient to capture the variation due to atmospheric fringes.

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Successfully training the fringe models required simultaneously holding large numbers of training images in computer memory in order to execute singular value decomposition. The large number of images that trained each of our 64 fringe models ranged in memory from 285 gigabytes to 314 gigabytes. We trained our models on the Cori Haswell nodes at the NERSC Supercomputer located at the Lawrence Berkeley National Laboratory. The Haswell nodes each have 128 gigabytes of RAM, making it impossible to include such a large number of images in our training sets without modifying our method. To adapt to cases where access to computational resources are limited such as this one, our training method was modified by down-sampling the training set. Training images were sorted by limiting magnitude and split into 300 sub-stacks of approximately equal depth. In order to combine each sub-stack into a single training image we calculated the sub-stack's median. While taking the median can distort the correlations between pixels that PCA is attempting to solve for, the median is also more robust than the mean to removing outliers that have contaminated the signal from a different distribution than the one attempting to be measured. Smooth background gradients remain on some ZTF images that can confuse the PCA reduction algorithm if not removed as outliers. We therefore settled on taking the median instead of the mean to combine the sub-stack of fringe maps into a single training image. An alternative approach could be to entirely remove the images with smooth gradients through a different method and then combine the remaining images in the sub-stacks using the mean. These 300 training images totaled approximately 11 GB and were therefore able to be fed simultaneously into our training process. Comparing the fringe models that result from training on down-sampled fringe maps produced comparable results to training on individual fringe maps for smaller sample sizes.

The code used for generating fringe models from ZTF i-band images and cleaning fringed images has been released in the open-source package fringez (Medford 2021). The fringez package includes command line executables and Python functions for downloading the fringe models computed as a result of this work, producing fringe bias images, and cleaning fringed images. This package is available for download and installation at https://github.com/MichaelMedford/fringez, and all fringe models are available for download at https://portal.nersc.gov/project/ptf/iband. In 2019 October, fringez was implemented into the ZTF IPAC data reduction pipeline with fringe models built on 500 training images per readout channel. All i-band images taken previous to this date were also reprocessed by IPAC to remove atmospheric fringes. In 2020 November, fringe models were updated to include the 550,365 i-band images described and investigated in this paper. This implementation of fringez and the current version of fringe models will continue to be a part of the IPAC data reduction pipeline in ZTF-II. The current implementation of fringez can also be generalized to other instruments by simply altering the FITS extensions and header keywords that are currently chosen for ZTF images.

In order to assess the effectiveness of our de-fringing method, we created a procedure for measuring correlated background noise. In this method, aperture photometry is taken at random locations on the background of an image. At the location of each aperture the flux (F_bkg) and its associated measurement error (σ_bkg) are measured, resulting in a set of aperture fluxes ({F_bkg}) and a set of errors ({σ_bkg}). We then compare the standard deviation of background fluxes (std({F_bkg})) to the average error on the background flux measurements (median({σ_bkg})). We call the ratio of these two values the Uniform Background Indicator:

$$\begin{eqnarray}&&{\rm{\Psi }}=\displaystyle \frac{\mathrm{std}(\{{F}_{\mathrm{bkg}}\})}{\mathrm{median}(\{{\sigma }_{\mathrm{bkg}}\})}.\end{eqnarray} \tag{ 5 }$$

For an image of uniform Gaussian noise, Ψ ≈ 1 as the distribution in background fluxes will be equal to the average flux error across all measurements. Here there exists no global variance which is not captured by the local error term. For an image with correlated background noise, Ψ > 1 because the different background values sampled by the apertures will introduce additional variance into the numerator of Equation (5) not captured by the local error terms in the denominator. The flux errors in Equation (5) must be calculated locally and be in the same units as the flux measurement with respect to aperture area. The aperture size should be chosen to be approximately the PSF scale of the instrument so that UBI measurements will indicate what fringe signal (or any other source of correlated background noise) a particular instrument would observe.

We validated using the UBI as a measurement of correlated background noise through a controlled experiment. We generated images containing only Gaussian noise that were the same size as ZTF images. The error on each pixel value was set as the root-mean-square of the median-subtracted pixel values in a 10 pixel by 10 pixel box around that pixel. The size of this square had to be smaller than the spatial scale of the correlated background noise so that it contained minimal variation in the background value. We then laid 50,000 circular apertures onto the image and calculated the aperture flux and aperture flux error of each measurement. The UBI as outlined in Equation (5) was evaluated five times on each image producing an average UBI as well as an error on that measurement.

In Figure 5, we show the results of this experiment for varying aperture sizes. For all Gaussian images and aperture sizes we found Ψ ≈ 1, confirming our interpretation that this value indicates no correlated background noise on an image. The location and scale of the Gaussian noise was also found to have no effect on this measurement. Figure 5 shows three example images with correlated background noise and the UBI values that they produce, covering the range of Ψ that we detected throughout this analysis. These example images show the correlation between a quantitatively larger Ψ and a qualitative increase in the appearance of atmospheric fringes.

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Having verified that the UBI is a valid indicator of correlated background noise, we next measured the effect of removing atmospheric fringes on the UBI. A sample of g-band, r-band and i-band images was created by downloading one random image for each filter, readout-channel and field observed in the ZUDS survey from the week of 2020 February 1 to 2020 February 8 with a limiting magnitude greater than 19. This sampling method ensured a representative sample of airmasses and limiting magnitude for ZTF observations, while removing images of extremely low quality from the sample. Our final sample contained 1408 g-band, 1407 r-band, and 1472 i-band images across a range of stellar densities. Each of the i-band images was cleaned using our method described in Section 3.2. The UBI was then calculated for each of the fringed images, as well as the cleaned images.

The distribution of the UBI for all of the images in our sample is shown in Figure 6, with Ψ ≥ 1 as expected. The g-band and r-band images each have a nearly normal UBI distribution with medians of Ψ = 1.13 and Ψ = 1.16, indicating a small but measurable amount of correlated background noise remaining after flat-fielding. The i-band images contain far more correlated background noise and are notably bimodal, with one population averaging Ψ = 1.33 and a second wider distribution averaging a larger Ψ = 1.91. This split is caused by the location of the readout channel on the image plane. The inner 32 readout channels (16 ≤ rcid < 48) with their additional layer of AR coating are less susceptible to atmospheric fringing. The outer 32 readout channels (0 ≤ rcid < 16, 48 ≤ rcid < 64) experience significantly more fringing due to a lack of an additional later of AR coating. Less than 20% of the images have a Ψ < 1.72 and the 20% most fringed images have a Ψ ≥ 2.20. The cleaned i-band images show indistinguishable behavior between the inner and outer readout channels forming a single distribution averaging Ψ = 1.15. This population appears similar to the g-band and r-band populations, indicating successful removal of atmospheric fringes. However the g-band and r-band images have significantly longer tails with an 80th percentile of Ψ = 1.28 and Ψ = 1.32 respectively, compared to an 80th percentile for cleaned i-band images of Ψ = 1.20. This indicates that there is correlated background noise occurring in g-band and r-band images that PCA analysis could potentially model and remove. It is clear that the process of removing atmospheric fringes significantly reduces correlated background noise from i-band images.

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It is reasonable to predict that the UBI will increase with airmass, as the presence of atmospheric fringes is caused from the column of atmosphere through which images are taken. This is found to be the case in Figure 7, where collecting the images into airmass bins and calculating the median UBI shows a trend toward larger UBI for larger airmass in i-band images. There is a slight increase in the g-band and r-band images as well, although the effect is relatively weak. The cleaned i-band images have comparable UBI values to the g-band and r-band images, again confirming the effectiveness of our method.

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While the UBI is a useful measurement of the presence of correlated background noise, we sought to quantitatively measure the improvement in photometric precision that resulted from removing atmospheric fringes. First we measured the photometric error caused by atmospheric fringes relative to a reference catalog. Next we measured how de-fringing removes this error by injecting faint artificial sources into single images. Last we measured how de-fringing recovers lost signals by injecting extremely faint sources into images before combining them through co-addition. For each of these experiments we will demonstrate how our method significantly improves the photometric precision of faint source measurements affected by atmospheric fringes.

First we measured the photometric error caused by atmospheric fringes. For each of the i-band images in our sample, we used SExtractor (Bertin & Arnouts 1996) to generate an instrumental photometric aperture catalog of astrophysical sources. We then cross-matched the original and cleaned images with Pan-STARRS1 (PS1) (Chambers et al. 2016) i-band catalogs downloaded from the Vizier database using astroquery (Ginsburg et al. 2019) to create cross-matched catalogs. A zero-point was calculated for each image using matching sources with PS1 i-band magnitudes less than 17. This zero-point was used to transform the instrumental photometric catalogs into ZTF i-band magnitudes comparable with PS1 i-band magnitudes, as well as to calculate a 5σ limiting magnitude for each image. After removing images with a limiting magnitude less than 21 to ensure reasonable measurement of faint sources, our final sample size was 738 images. For each pair of fringed and cleaned images, the cross-matched catalogs were used to find the variance in the difference between ZTF and PS1 magnitudes for stars within a magnitude bin. The photometric error caused by fringes was then calculated as:

$$\begin{eqnarray}&&{\sigma }_{\mathrm{mag}}=\mathrm{std}({m}_{\mathrm{PS}1}-{m}_{\mathrm{ZTF}})\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{\sigma }_{\mathrm{fringe}}=\sqrt{{\sigma }_{\mathrm{mag},\mathrm{fringed}}^{2}-{\sigma }_{\mathrm{mag},\mathrm{cleaned}}^{2}}.\end{eqnarray} \tag{ 7 }$$

The photometric error we measured due to fringing is shown in Figure 8, plotted separately for different faint magnitude bins against the UBI of the fringed images before cleaning. Images with larger UBI values have larger amounts of additional magnitude scatter relative to the PanSTARRS catalog than those images with smaller UBI values. Those images with Ψ ≤ 1.3 appear to have marginal improvement to their photometric precision due to removing fringes. However, images with Ψ > 1.3 show significant photometric errors due to fringing that our method removes. Images with Ψ = 1.75 have a 0.21 mag error on 19.5 mag stars, 0.35 mag error on 20.5 mag stars and a 0.43 mag error on 21.5 mag stars. Images with Ψ = 2.0 have a 0.28 mag error on 19.5 mag stars, 0.46 mag error on 20.5 mag stars and a 0.57 mag error on 21.5 mag stars. The worst 10% of i-band images in the outer readout channels have Ψ ≥ 2.4, resulting in errors as large as 0.21 mag for 18.5 mag stars and up to 0.39 mag for 19.5 mag stars. Images with Ψ ≥ 2.3 have very few sources fainter than 19.5 mag in the figure, indicating that these images fail to pass the limiting magnitude cut. Our method enables the recovery of the faint end of the luminosity function that would otherwise be unobservable in ZTF i-band images.

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Another way to measure the effect of atmospheric fringes on photometric precision of faint sources is through the injection and recovery of fake sources. We selected an i-band image with Ψ = 1.78 as a representative image. The image's zero-point and 5σ limiting magnitude were calculated using a cross-match to PanSTARRS1 sources as described above. A PSF for the image was derived using psfex (Bertin 2011) and executed using wrappers written in the galsim (Rowe et al. 2015) Python package. 100 sources with magnitudes equal to the 5σ limiting magnitude were injected into the original image using this PSF model at random locations, including Poisson noise. The image was then cleaned using the fringez package. Photometric catalogs using both aperture and PSF photometry were calculated on the original and cleaned images using SExtractor with a 3 pixel 1σ detection threshold. Measurements in the catalogs at the location of the injected fake sources were then recovered for comparison to the true modeled flux. We also calculated the ideal aperture corrected flux of the injected fake sources by calculating the median ratio of aperture to PSF fluxes of high signal-to-noise astrophysical sources from the catalogs, for both fringed and cleaned images. We then repeated this process 50 times for a total of 5000 fake sources injected to form our sample. We also duplicated this experiment with the SExtractor BACKGROUND parameter set to LOCAL and GLOBAL to test the effect of forcing the source identification algorithm to attempt to characterize local variations in the background noise. Figure 9 shows the results of this experiment. The distributions show the fractional offset of the measured flux to the injected flux scaled by the theoretical error on a 5σ source. Plots are drawn in log-scale to highlight the long tail of overestimated measurements for images that have not been cleaned, with a Gaussian distribution drawn in gray as a reference.

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We first note some observations about the quality of recovered fake sources on the images which have not been cleaned. Setting the SExtractor BACKGROUND parameter to LOCAL improves photometric measurements on these fringed images. GLOBAL computes the background flux across the entire image and underestimates the amount of background underneath a source sitting on a bright fringe. These sources will have overestimated brightness and form the long tail of measurements with larger than expected fluxes. LOCAL computes the background flux with a rectangular annulus around the source that prevents attributing the additional brightness to the source but instead to the background. Switching to a LOCAL background therefore reduces this tail. Aperture photometry performs better than the PSF photometry on these fringed images with a reduced tail for both SExtractor settings. This prominent tail may be due to a bias in the PSF model caused by including stars that fall on fringes, inflating the wings of the PSF model. When that model is applied to stars that fall on particularly bright fringes the increased background may be included in the measured flux, resulting in an overestimate of the flux. Aperture photometry does not suffer from this potential model bias and therefore has a less prominent tail of overestimated fluxes.

The sources from the cleaned catalogs are more accurately and precisely measured under all conditions. There is a slight tail of overestimated fluxes when performing PSF photometry but it is far closer to a Gaussian distribution than without cleaning. For both LOCAL and GLOBAL backgrounds, the distribution of recovered sources in the cleaned catalogs very nearly resembles a Gaussian distribution. There exists only a few sources with measured fluxes exceeding the injected flux causing a deficit of fainter sources. There still remains a noticeable deficit of sources at the faint end of the brightness distribution, indicating that our method is not removing all additional flux in the image background. Aperture photometry remains the best way to evaluate sources even where atmospheric fringes have been removed. Failure to clean images containing atmospheric fringes results in a systematic overestimation of the flux of faint sources. Applying our method enables near-ideal recovery of 5σ sources for a variety of measurement methods.

It is particularly difficult to overcome the effects of fringes when combining multiple images of the same field to recover sources fainter than a single image's limiting magnitude. Changing atmospheric conditions and various observational airmasses will alter the strength of the fringes, while dithering and sky motion will place astrophysical sources onto slightly different pixels for each exposures. Failure to remove fringes will contribute significant excess flux to a co-addition of multiple images. We demonstrate here this effect quantitatively by injecting extremely faint sources into individual images and attempting to measure them in a co-added image.

100 i-band images of the same field were zero-pointed using the previously outlined method. Sources in each image were measured and the signal-to-noise versus magnitude was fit to extrapolate the 0.5σ magnitude of all images. Each image was injected with 100 sources at this 0.5σ magnitude at a common list of coordinates in R.A. and decl. These sources would be expected to appear as 5σ sources after combining the 100 i-band images because signal-to-noise increases as the square root of the total exposure time. All images were then cleaned with the fringez package. SCAMP (Bertin 2006) was run to find astrometric projection parameters for each of the images such that they could be transformed onto a common reference frame. All fringed and cleaned images were then combined into two separate co-additions using SWarp (Bertin et al. 2002), produced with a median combination filter and background subtraction. Aperture photometry catalogs were then generated on the final co-additions using SExtractor with the LOCAL background setting. Sources at the locations of the injected signals were recovered and their signal-to-noise measured as the ratio of their aperture flux to their locally determined aperture flux error. We repeated this process 50 times for a total of 5000 fake sources distributed over 50 fringed and 50 cleaned co-additions.

Figure 10 shows the quantitative and qualitative photometric improvements to these recovered sources. Sources observed on fringed co-additions peak at a signal to noise of 3, pushing them below the typical 5σ observable threshold. These sources also have a long tail of excessive flux stretching as high as 10σ due to the additional flux caused by atmospheric fringes. Sources observed on cleaned co-additions are much closer to their theoretical distribution. These sources peak at exactly 5σ and the large majority of sources are within the normal signal-to-noise range of 4–6. There still exists a tail of excessive fluxes, demonstrating that our method is failing to remove all excessive correlated background from the individual exposures. However our method clearly produces an observed flux distribution that is much closer to a Gaussian distribution.

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The largest difference between the two populations appears in their yields. Of the 5000 sources injected into each of the fringed and cleaned images, 1161 sources were recovered from cleaned co-additions and only 156 sources were recovered from fringed co-additions. Over 95% of the 0.5 sources injected into the fringed images failed to be recovered after co-addition. The order of magnitude increase in the number of sources recovered by our method enables ZTF i-band surveys to recover faint sources that would otherwise have been extremely unlikely to observe. We note that forced photometry at the known locations of the fake sources may have increased the recovered yields for both populations but would not be an accurate representation of the observation process undertaken for unknown sources.

Our analysis demonstrates the power of our method to remove photometric error due to atmospheric fringes and enable the recovery of faint sources in both single images and co-additions that would otherwise have been undetectable. Failure to implement a method for removing atmospheric fringes greatly reduces the effectiveness of the i-band filter for observing any sources fainter than 18th magnitude. Our method increases the photometric precision of the i-band to that of g-band and r-band images that do not suffer from atmospheric fringes.