Testing the LSST Difference Image Analysis Pipeline Using Synthetic Source Injection Analysis

The AL algorithm solves for the matching kernel κ, which uses stars in both images to match the PSF of the template image to the PSF of the science image. Galaxies can be used in addition to stars, although a galaxy provides less information than a star of the same flux level. Thus, this process is not dependent on reliable star–galaxy separation. In the spatially varying (Alard & Lupton 1998; Alard 2000) situation, the matching kernel can be expressed as

$$\begin{eqnarray}&&\kappa (u,v,x,y)={{\rm{\Sigma }}}_{q=1}^{{N}_{k}}{a}_{q}(x,y){k}_{q}(u,v).\end{eqnarray} \tag{ 1 }$$

Here N_κ is the number of kernel bases, a_q is a set of fitted spatially varying coefficients, and k_q is a set of kernel bases. The most commonly used kernel bases are composed with a set of Gaussian functions modulated with a set of polynomials (Alard & Lupton 1998). Another type of kernel basis is composed with delta bases (Bramich 2008; Miller et al. 2008), which are "shape-free" delta functions defined for each kernel pixel coordinate.

Using the matching kernel κ, the difference image of the AL algorithm can be expressed as

$$\begin{eqnarray}&&D(x,y)=S(x,y)-T(x,y)\otimes \kappa (x,y)-B(x,y),\end{eqnarray} \tag{ 2 }$$

where B(x, y) is the fitted sky background.

The ZOGY algorithm was designed for the case where PSFs and flux scaling factors for T and S are properly estimated, and the sky backgrounds of both images have been properly subtracted. ⁸ The difference image from the ZOGY algorithm in Fourier space can be expressed as

$$\begin{eqnarray}&&\hat{D}=\displaystyle \frac{{F}_{t}\hat{{P}_{t}}\hat{S}-{F}_{s}\hat{{P}_{s}}\hat{T}}{\sqrt{{\sigma }_{s}^{2}{F}_{t}^{2}| \hat{{P}_{t}}{| }^{2}+{\sigma }_{t}^{2}{F}_{s}^{2}| \hat{{P}_{s}}{| }^{2}}}.\end{eqnarray} \tag{ 3 }$$

The equivalent PSF of the ZOGY difference image is

$$\begin{eqnarray}&&\hat{{P}_{D}}=\displaystyle \frac{{F}_{t}{F}_{s}\hat{{P}_{t}}\hat{{P}_{s}}}{{F}_{D}\sqrt{{\sigma }_{s}^{2}{F}_{t}^{2}| \hat{{P}_{t}}{| }^{2}+{\sigma }_{t}^{2}{F}_{s}^{2}| \hat{{P}_{s}}{| }^{2}}}.\end{eqnarray} \tag{ 4 }$$

Here the hat symbol represents the Fourier transform of each quantity, P_t and P_s are the PSFs of T and S, F_t and F_s are the flux scaling factors of both images, σ_t and σ_s are background noise levels, and F_D is the normalization constant. The expression in the denominator of $\hat{D}$ plays the role of decorrelating the noise.

However, ZOGY provides no mechanism to include information about the spatial variation of PSFs. One approach to using a spatially varying ZOGY would be to split T and S into subregions within which the spatial variation of PSFs is much smaller (Zackay et al. 2016; Reiss 2017; Hu et al. 2022). Performing the ZOGY algorithm on each subregion is achievable at the expense of more CPU and more complicated bookkeeping for detections near and across the edges of subregions. Such a "cell-based" approach to templates and subtraction could fail if the PSFs are varying rapidly across the images.

To prepare for analysis of LSST data, LSST DESC created the DC2 data set with an end-to-end approach (LSST Dark Energy Science Collaboration (LSST DESC) et al. 2021) to simulate a subset of the full LSST survey. This approach started with a large N-body simulation of galaxies in the Universe, applied LSST-like observations, simulated each exposure, ran the LSST Data Release Processing pipeline, ⁹ and produced data products analogous to those that will be delivered by the Rubin Observatory. The DC2 observations include six optical bands (ugrizy) with coverage of 300 deg² in a wide-fast-deep area and 1 deg² in a deep drilling field. It simulates 5 yr of the LSST survey.

The DC2 data set provides two classes of imaging data. For single-frame processing, it provides calibrated exposures (calexps) with sky background subtraction. Each calexp image is a 4000 × 4072 pixel image array with 02 pixel⁻¹ scale. It also has a source catalog (src) that contains measurements of sources in that visit. After producing calexps, multiple visits are coadded together to create deepCoadd exposures. The resampling grid of deepCoadds is defined with tracts and patches. Each tract contains 7 × 7 patches, and each patch contains 4100 × 4100 pixels with a scale of 02 pixel⁻¹. A visual representation of the tract-and-patch structure can be found in Sánchez et al. (2022) or LSST Dark Energy Science Collaboration (LSST DESC) et al. (2021).

We focus on i-band exposures of the DC2 data set for our analysis. We choose "deepCoadd" exposures with seven patches as template exposures. The total area analyzed in this paper is about 0.36 deg².

We undertook this work from the perspective of SN cosmology, and so we are interested in the detection and photometric accuracy of new point sources (SNe) on top of extended sources (galaxies). GCRCatalogs (Mao et al. 2018) is a DESC package that provides users with a consistent interface to access both the input truth catalogs and the output processed catalogs for the DC2 simulations. We use GCRCatalogs to extract a list of potential host galaxies from the input truth table with a range of apparent host magnitudes. Next, we measure the local surface brightness (SB) with 1'' aperture photometry at the center of each galaxy in the simulated images. We sort the galaxy sample into 1 mag arcsec⁻² bins distributed from 20 to 25 mag arcsec⁻² with 1 mag arcsec⁻² bin widths. While in the real Universe, SNe occur throughout galaxies, what we really want to explore here is the dependence of SN detection on underlying SB. Thus, we place the SN at the centers of galaxies with different central SB.

Within each patch and each SB magnitude bin, we select a set of 20 hosts. Galaxies are chosen to satisfy two criteria: (1) each host center is at least 200 pixels away from the image boundaries in both the x- and y-directions, and (2) each host center is at least 100 pixels away from another host center. These criteria help us avoid subtractions near image edges and injecting overlapping synthetic sources. Locations of hosts on a patch are shown in Figure 1.

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After selecting the host set within each patch and each host SB magnitude bin, we match each host set to 10 calexps using the tracts_mapping.sqlite3 database file of the DC2 data set. We inject synthetic sources onto each calexp and then use those as the science exposures for running the image difference algorithm. We define a calexp and its corresponding deepCoadd exposure as an "image pair." In total, we have 70 image pairs. The LSST survey takes images at a range of spatial and rotational dithers; thus, a single calexp often does not fully cover all 20 hosts in each host set. We restrict our sample to calexps that cover at least 15 host galaxies.

Our synthetic source injection procedure is based on the insertFakes module of the pipe_tasks ¹⁰ package of the LSST Science Pipelines. We convert the R.A., decl. of the host to the x, y coordinate on the image. We then simulate point sources based on the calexp PSF model at that location on the image. The synthetic source injection framework converts the magnitude to the instrumental flux and scales the normalized synthetic source model by the instrumental flux. In the single-frame processing, the instrumental flux to magnitude calibration of the images was calculated with respect to a 12 pixel radius aperture. While the PSF model is finite in size, it is always larger than the calibration radius. The PSF model is normalized to sum to 1. We calculate the aperture correction between the full PSF and the smaller calibration radius and scale the model of the PSF by this correction. We then multiply the scaled PSF by the instrumental flux from the calibration to inject a source that has the correct photoelectron contribution for its magnitude. The scaled model is added to the image plane of the calexp without making any changes to its variance plane, which means we ignore the Poisson noise of the synthetic source.

After finishing the injection stage, we run the DIA pipeline on the injected data set using the imageDifferenceDriver.py of the dia_pipe package with v23.0.0 ¹¹ of the LSST Science Pipelines. This stage produces difference images and then runs photometry measurement algorithms on that difference image to produce the DIA source (diaSrc) table, which contains source measurements of the detected source. We obtain positions of variable stars and active galactic nuclei (AGNs) from the truth tables in GCRCatalogs and match them to the diaSrc tables. All variables and AGNs that get matched are removed from the diaSrc tables. As a result, the detections from the diaSrc tables only have synthetic sources and artifacts. ¹² Next, we match the known injection positions to the diaSrc tables to determine whether the injected synthetic sources are detected in the difference exposures. The diaSrc fluxes and the flag information of the detected sources are also extracted from the diaSrc tables for further analysis.

In this work, we use the v23.0.0 version of the LSST stack package to provide basic functionalities for synthetic source injection and image difference. We have documented the proper software versions in the GitHub repo LSSTDESC/dia_improvement, ¹³ which includes scripts to reproduce our work.

We evaluate the subtraction quality of the DIA pipeline using the following metrics. These metrics should reflect the detection ability and the photometric measurement accuracy of the pipeline.

• 1.  
  Efficiency ¹⁴ (Eff): the number of detected synthetic sources divided by the number of injected synthetic sources. We expect it to approach unity when synthetic sources are bright and decrease to 0 at the faint magnitude end. The efficiency curve can be fitted using a sigmoid model. In this work, we use the function

  $$\begin{eqnarray}&&\mathrm{eff}(m)=\displaystyle \frac{1}{1+{e}^{-a(m-b)}}\end{eqnarray} \tag{ 5 }$$

  to fit our data, with a and b as fitting parameters and m as the synthetic source magnitude. ¹⁵ Using this equation, we can solve the magnitude depth m_1/2, which corresponds to the synthetic source magnitude, where the efficiency drops to 50%. The magnitude depth m_1/2 is mathematically equivalent to the fitting parameter b.
• 2.  
  Artifact rate (AR): the number of subtraction artifacts per square degree.
• 3.  
  Flux pull (ΔF/σ):

  $$\begin{eqnarray*}&&\mathrm{fluxpull}=(\mathrm{measuredflux}-\mathrm{injectedflux})/(\mathrm{fluxerror}).\end{eqnarray*}$$

  This flux pull should follow a normal distribution with a mean of 0 and a standard deviation of 1. In this analysis, we calculate the pull statistic of the diaSrc PSF flux.

We apply two analysis strategies to sets of DC2 calexps. The first strategy is to focus on measuring efficiency as a function of source magnitude and background SB; we inject synthetic sources from 20–23 mag in 0.5 mag steps and 23–24 mag in 0.1 mag steps. These finer steps focus on the magnitude range where the efficiency drops rapidly. We use this strategy to determine the magnitude depth m_1/2 at which 50% of the sources are recovered.

For the second strategy, we undertook a related but separate campaign of studying efficiency and artifact rate for high signal-to-noise synthetic sources and low signal-to-noise synthetic sources with different subtraction configurations with two consistent sets of 20 mag (MAG20) sources and 23 mag (MAG23) sources.

To get a baseline evaluation of the performance of the DIA pipeline, we run dia_pipe using the default configuration of the pipeline. The default difference imaging algorithm is AL with a set of Gauss-polynomial bases. The AL kernel has three Gaussians with sigma values of 0.7, 1.5, and 3.0 pixels, with respective order 4, 2, and 2 polynomials in kernel (u, v) space. These kernel terms are allowed to spatially vary at first order in x, y across the image. The background level is allowed to vary spatially at second order in x, y.

The upper plot of Figure 2 shows our recovered detection efficiency as a function of synthetic source magnitude. The efficiency follows predicted smooth curve efficiency functions, with a decrease in efficiency as a function of both (1) decreasing injected flux and (2) increasing underlying SB (and thus background noise) at the location the synthetic source is placed. The effect of the underlying host galaxy light is most visible for the brightest hosts, SB = 20–21 mag arcsec⁻², which shows a substantial 0.25 mag m_1/2 shift in the efficiency curve. The DIA pipeline successfully recovers all ¹⁶ synthetic sources brighter than 22 mag. The efficiency starts to drop below 1 around 22 mag, and it drops below 0.5 between 23 and 24 mag. In the lower plot of Figure 2, we focus on the detection efficiency at the faint magnitude (low S/N) limit. The predicted values of the m_1/2 of the fitted function are between 23.40 and 23.71 mag.

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Figure 3 shows the S/N distribution of the detected synthetic sources when injected at 21, 22, 23, and 24 mag. At the synthetic source magnitude of 21, the S/N distribution is far above the detection limit, which means the pipeline should be able to recover all synthetic sources. At the injected magnitude of 23, the S/N distribution overlaps the detection threshold, and the efficiency begins to drop. When the synthetic source magnitude is 24, most injections are below the detection threshold, which explains why the efficiency is low at this magnitude.

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Figure 4 shows the distributions of the detected magnitude and injected magnitude of synthetic sources. At the injected magnitude of (20, 21, 22, 23, 24), more than (99%, 99%, 90%, 57%, 8%) of the flux measurements are within 0.1 mag of the injected magnitudes. At 24 mag, the distribution is significantly biased toward the bright side due to the Malmquist bias (Malmquist 1922, 1925).

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Here we give a comparison of the i-band m_1/2 to the previous DESC DC2 DIA (Sánchez et al. 2022). That work evaluated the m_1/2 of the LSST DIA pipeline using about 15 deg² of the DC2 data set with artificial SN Ia light curves overlaid onto exposures. The estimated m_1/2 in the i band is 23.50 mag for all sources.

In this work, we estimate the m_1/2 in the i band using about 0.36 deg² of the DC2 data set with carefully controlled synthetic point sources distributed across different host brightness. The fitted m_1/2 is between 23.40 mag and 23.71 mag, depending on the underlying SB.

Alard & Lupton (1998) and Alard (2000) suggest expanding the AL kernel coefficients a_q (x, y) (Section 2.1) as a function of a power series of positions. The highest-order d_s of the power series defines the kernel spatial degree of freedom. The implementation of the AL algorithm in the DIA pipeline can be configured to run with different values of d_s . In this section, we explore the pipeline performance with d_s varying from 1 to 4. The pipeline default value of d_s is 1. Our analysis was constrained to a maximum spatial order of d_s = 4 due to limitations in v23.0.0 of the LSST Science Pipelines, which have since been lifted. Raising d_s beyond 4 offers increased kernel flexibility, thereby potentially reducing the artifact rate. However, this approach carries the risk of overfitting, since the number of spatial polynomials scales proportionally with ${d}_{s}^{2}$ (Alard 2000; Bramich et al. 2013). Additionally, it would result in an increase in computational time.

The primary goal of exploring the spatial degrees of freedom is to reduce artifacts while maintaining high efficiency and flux accuracy. We focus here on high-S/N sources at 20 mag (MAG20), which is far above the detection limit, and low-S/N sources at 23 mag (MAG23), which is close to the m_1/2. Since the comparison focuses on the region that is away from the image edge, ¹⁷ it is possible that the high d_s s are performing worse near the image edge. More evaluations near the image boundary are still needed in future work.

In Table 1, we show the efficiency, artifact rate, ¹⁸ mean, and standard deviation of the flux pull distribution for each spatial degree of freedom at MAG20 and MAG23. We find that the results from high S/N and low S/N are consistent. Up to d_s = 4, the artifact rate is a monotonically decreasing function of d_s . The fluctuation of the recovered mean pull of the flux of the injected sources is less than 0.020. The fluctuation of the standard deviation is less than 0.010. The fluctuation of the efficiency is less than 0.013.

In Figure 5, we show the postage stamps of difference images from different spatial degrees of freedom. Each column of stamps is made at the same position, while each row corresponds to a specific d_s . By comparing the stamps, we find that high d_s s, especially d_s = 4, can produce cleaner difference images. As a result, the artifact rate decreases as the d_s increases.

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The LSST Science Pipelines are configured to detect any possible source and characterize it with flags to indicate detections that are not real astrophysical sources. These 109 flags identify pixel-level features such as saturation, bad pixels, and cosmic rays as well as characteristics of the source in the subtraction such as moment and dipole fits. We use these flags to reject artifacts, which we here define as detections that are not in circular regions with 08 radii of known synthetic sources. The data are from MAG20 and MAG23, with host SB between 20 and 21 mag arcsec⁻². In this synthetic source injection analysis, we split the detected sources from each diaSrc table into two tables: one for the detected synthetic sources (MAG20: 1184 entries; MAG23: 1003 entries) and the other for detections that are artifacts (MAG20: 2446 entries; MAG23: 2464 entries).

We use the efficiency and artifact rate to examine the power of each flag to classify sources into real and artifact. We calculate these metrics for all flags and show the full table in Table 2. From this information in this table, we can select a small set of flags where (1) the efficiency of each flag is close to 1 at MAG20 and (2) each flag has a clear physical interpretation related to artifact classification. We document this "selected flags" set in Table 3. To evaluate the efficacy of this flag set, we compare the efficiency and artifact rate before and after applying the union of these flags to our detection. At MAG20, we find that the efficiency drops from 1.000 to 0.999, and the artifact rate drops from 695 deg⁻² to 69 deg⁻². At MAG23, the efficiency drops from 0.847 to 0.791, and the artifact rate drops from 700 deg⁻² to 68 deg⁻². These results suggest that using the "selected flags" to sift artifacts significantly improves the artifact rate with only a small decline in efficiency. We strongly recommend the use of these flags when looking to identify a set of real astrophysical detections in diaSrc tables.

Now that we can identify artifacts using the "selected flags" discussed in Section 4.6, we examine their physical origins using artifacts from AL-based subtractions on the MAG20 data set. We define four types of artifacts that are commonly seen in difference images. The morphology of these artifacts is shown in Figure 6.

• 1.  
  Saturation artifact: caused by saturated pixels, where the PSF shape is distorted. As a result, an optimal kernel solution does not exist.
• 2.  
  Clustering-pixels artifact: a clustering-pixels artifact appears as clusters of positive pixels and negative pixels. It comes from subtraction residuals of background sources or background noise.
• 3.  
  Dipole artifact: offset positive and negative lobes. Inaccurate image registration or nonoptimal image resampling can cause dipole artifacts. It is a subclass of the clustering-pixels artifact.
• 4.  
  Deconvolution artifact: a ring pattern. It is found in AL difference images when the full width at half-maximum (FWHM) of the science image is sharper than the FWHM of the template image. In this scenario, the matching kernel has to "deconvolve" the PSF of the template but is not able to do so because the necessary information is not available.

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To help us understand the relationship between flags and artifact morphology, we split the "selected flags" defined in Section 4.6 into three groups: saturation flags, dipole flags, and shape flags. As their names suggest, saturation flags are set to True if the detected source contains saturated pixels. We name detected artifacts with the saturation flag set to True saturation-flag artifacts and name the rest non-saturation-flag artifacts. Dipole flags are set to True for dipole artifacts. It is possible that a detected source visually appears as a dipole but has no dipole flags set to True. This could be related to the trigger mechanism and flux separation of dipole measurements. When the DIA pipeline is making a dipole classification for a detected source, it fits the fluxes of the negative and positive lobes and their corresponding centroids simultaneously. A detected source is classified as a dipole if the quadrature sum of the fitted fluxes is above the corresponding minimum S/N threshold and the fitted fluxes weighted by the total flux are both smaller than the maximum flux ratio threshold. Any detection that visually appears as a dipole but does not satisfy these criteria is not classified as a dipole artifact. Also, the fitting algorithm itself may have degeneracy in estimating dipole fluxes. Dipoles from faint and highly separated sources are almost indistinguishable from dipoles from bright and closely separated sources (Reiss 2016). The inaccuracy of dipole flux estimation could further affect dipole classification. Possible improvements such as using presubtraction images in dipole fitting are discussed in Reiss (2016). Shape flags are set to True if the detected source has second moments that are inconsistent with a Gaussian shape. More detailed definitions of these flag sets are given in Table 4.

Artifact morphology can be affected by the relative seeing conditions of template images and science images. We split our image pairs defined in Section 3.3 into three classes based on the FWHM of their coadd (template)/calexp (science) image PSFs. These classes are defined as

• 1.  
  broad: calexp FWHM ≥ coadd FWHM;
• 2.  
  near: calexp FWHM < coadd FWHM and (calexp FWHM − coadd FWHM) / (coadd FWHM) > −0.05; and
• 3.  
  sharp: (calexp FWHM − coadd FWHM) / (coadd FWHM) ≤ −0.05.

In the broad class, calexps have larger PSFs than coadd exposures, which is the condition that the AL algorithm is capable of matching the coadd PSF to the calexp PSF. In the near class, calexp PSFs are smaller than coadd PSFs. However, the relative difference of FWHMs is still close. In the sharp class, calexp PSFs are much smaller than coadd PSFs. The AL matching kernel has to "deconvolve" the coadd PSF to match the calexp PSF.

Using the above definitions, we can study artifact morphology in each FWHM class. We start with calculating the proportions of artifacts in each flag group. The results are shown in Table 5. The fraction of artifacts in each class with saturation flags set to True is nearly 50%. Since they are subtraction residuals of bright background sources (starts, galaxies, etc.), it is important to understand at what background source S/N this issue occurs most frequently. We match background sources from the calexp src (Section 3.1) to artifacts using a one-direction-matching ¹⁹ method with a 2'' search radius. Figure 7 shows the artifact fraction with respect to the calexp source S/N. ²⁰ There is an increasing trend between the artifact fraction and the source S/N. This comparison shows that using saturation flags to identify pixel issues can help us to remove saturation artifacts, especially when the source S/N is greater than 600.

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Next, we study the artifacts that have no saturation flags set to True. These non-saturation-flag artifacts are detected under different seeing conditions with different artifact rates. For our analysis, we find that the rates of non-saturation-flag artifacts per square degree are 286 (broad), 312 (near), and 414 (sharp), which suggests that as the calexp PSFs become sharper, the artifact rate increases (Table 5).

The morphologies of non-saturation-flag artifacts are shown in Figures 8, 9, and 10. By inspecting postage stamps of these artifacts, we find that the majority of artifacts in the broad class are dipole artifacts and clustering-pixels artifacts. In the near class, the artifact morphology is similar to the broad class. However, in the sharp class, there are more ring artifacts, which indicates the PSF matching kernel is deconvolving the seeing condition of the template image.

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To understand the source properties of the non-saturation-flag artifacts, we match them to the src using the one-direction-matching method with a 2'' matching radius. ²¹ Of the artifacts, 80.2% have matches in the src, which means most of these artifacts are also subtraction residuals of background sources. In Figure 11, S/N distributions of matched sources in each FWHM class are plotted. Most sources have S/Ns close to or higher than 200, which means bright sources are the main contribution to detection artifacts. This is reasonable because we can only observe artifacts from bright sources where subtraction residuals can exceed sky background fluctuations. It is hard for faint sources to produce visible artifacts because their subtraction residuals are overwhelmed by sky background noise. Other than bright sources, we also find clusters of faint sources in the S/N distributions of each class. In Figure 12, we plot a few stamps of faint sources from coadd exposures, calexps, and their corresponding artifacts from difference exposures. These plots show that peak pixels are the main causes of these artifacts. As to the artifacts that have no matches from the src, we show their morphology in Figure 13. Clearly, background noise is the origin of these artifacts. The artifact rate of this type is 62 deg^–2 in the broad class, 57 deg^–2 in the near class, and 79 deg^–2 in the sharp class.

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Because dipole artifacts are prevalent in all classes with the AL algorithm, we investigate whether the ZOGY algorithm also produces dipole artifacts at the same locations. Figure 14 shows postage stamps of AL dipoles and postage stamps of the ZOGY algorithm at the same positions. Both the kernel matching method (AL) and the cross-filtering method (ZOGY) have dipole artifacts with the same orientations at the same locations.

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Here, we give a brief discussion of several different possible causes of dipole artifacts.

(1) Image registration. It is possible that dipoles are caused by small errors in the relative astrometric registration of the images. Applying astrometric calibration may reduce the registration error. However, even subpixel errors could be magnified by bright sources, thereby producing visible dipole artifacts that have positive–negative separations of multiple pixels. (2) Sampling. Another possible origin of dipoles is the sampling process used for observing or warping exposures. Errors introduced in this process can be magnified by the brightness of background stars, which would result in dipole artifacts. (3) Moving object. An object with a small motion between the images can create a dipole that represents real astrophysical change. However, there are no stars with proper motion in the DC2 simulation, so this particular effect is not applicable to our testing here. (4) PSF matching. It is also possible that the AL algorithm finds an approximate kernel solution. An asymmetric kernel convolving a symmetric coadd source and subtracted by a symmetric calexp source could produce a dipole artifact. However, we also find dipole artifacts using the ZOGY cross-filtering method, which has no kernel fitting involved (Figure 14). We do not entirely exclude the kernel fitting explanation because the PSF models used for cross-filtering are measured from images. It is thus also possible that the approximate PSF used for cross-filtering caused the dipole artifact, which is mathematically similar to a nonoptimal AL matching kernel. (5) Poisson fluctuation. In the bright region, the Poisson fluctuation of background sources can exceed the background sky fluctuation, which leads to local bright/faint residuals. These residuals can spread to nearby pixels after convolution operation, which may appear as dipole artifacts in difference images. More accurate estimations of variance and covariance may increase the detection thresholds and thereby remove these artifacts from the diaSrc table. (6) Differential chromatic refraction (DCR). The atmosphere of the Earth refracts incident light, thereby deflecting the observed positions of sources (FILIPPENKO 1982). This effect could result in dipole artifacts in difference images. Since the DC2 images do not include this effect, we exclude DCR as the cause of the artifacts in our analysis.

From this discussion, we conclude that the origin of the dipole artifacts is still undetermined. To thoroughly understand the cause will require control of the image resampling in more detail and carefully considering the resulting PSFs of the calexps and coadd templates.