Draft version April 13, 2017Preprint typeset using LATEX style emulateapj v. 01/23/15

IS FLAT-FIELDING SAFE FOR PRECISION CCD ASTRONOMY?

Michael Baumer, Christopher P. Davis, and Aaron RoodmanKavli Institute for Particle Astrophysics and Cosmology

SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA andPhysics Department, Stanford University, Stanford, CA 94305, USA

Draft version April 13, 2017

ABSTRACT

The ambitious goals of precision cosmology with wide-field optical surveys such as the Dark EnergySurvey (DES) and the Large Synoptic Survey Telescope (LSST) demand, as their foundation, precisionCCD astronomy. This in turn requires an understanding of previously uncharacterized sources ofsystematic error in CCD sensors, many of which manifest themselves as static effective variationsin pixel area. Such variation renders a critical assumption behind the traditional procedure of flatfielding—that a sensor’s pixels comprise a uniform grid—invalid. In this work, we present a methodto infer a curl-free model of a sensor’s underlying pixel grid from flat field images, incorporating thesuperposition of all electrostatic sensor effects—both known and unknown—present in flat field data.We use these pixel grid models to estimate the overall impact of sensor systematics on photometry,astrometry, and PSF shape measurements in a representative sensor from the Dark Energy Camera(DECam) and a prototype LSST sensor. Applying the method to DECam data recovers knownsignificant sensor effects for which corrections are currently being developed within DES. For anLSST prototype CCD with pixel-response non-uniformity (PRNU) of 0.4%, we find the impact of“improper” flat-fielding on these observables is negligible in nominal .7′′ seeing conditions. Theseerrors scale linearly with the PRNU, so for future LSST production sensors, which may have largerPRNU, our method provides a way to assess whether pixel-level calibration beyond flat fielding willbe required.

1. INTRODUCTION

Current and next-generation wide-field optical surveys,including the ongoing Dark Energy Survey (DES) (DarkEnergy Survey Collaboration 2005, 2016) and the up-coming Large Synoptic Survey Telescope (LSST) (LSSTScience Collaboration 2009) will use a variety of cos-mological probes, including gravitational lensing, galaxyclusters, and supernovae, to constrain the nature of darkmatter and dark energy with unprecedented precision.Though each cosmological probe differs in its priorities,all of them demand, at a fundamental level, “precisionCCD astronomy,” placing stringent requirements on in-strumental systematics affecting photometry, astrome-try, and object shape measurement. These requirementsare driving a search for previously unknown sources ofsystematic error within the CCD sensors used in both theDark Energy Camera (DECam) (Flaugher et al. 2015)and LSST.

Several such previously uncharacterized sensor sys-tematics have recently been described in the literature.Plazas et al. (2014) analyzed ‘tree rings’—impurity gra-dients seen as circularly symmetric flux variations in flatfield images—in DECam and found a significant impacton astrometry. Okura et al. (2015) analyzed tree ringsin prototype LSST sensors, where due to improved en-vironmental stability during the growth of the parentsilicon boules, the tree rings are highly suppressed, andfound no significant impact on the two-point shear corre-lation function. Other effects that have been analyzed inisolation include distortions near CCD edges (Bradshawet al. 2015), periodic row/column size variations (Okuraet al. 2016), and stochastic pixel-to-pixel size variation(Baumer & Roodman 2015).

This previous work has taken a “bottom-up” approachto assessing the science impact of sensor systematics—typically using pre-processing to highlight a particulareffect within the (evolving) taxonomy of uncharacter-ized errors and assessing the impact of that individualeffect on particular science observables. Such analyseshave certainly improved our understanding of the waysin which CCDs can lead to systematic errors, but furtherinsight can be gained by considering each of these system-atics to be a different characteristic variation of effectivepixel areas. With this perspective, the astronomer’s pri-mary concern becomes the impact of the superpositionof all these sensor effects (along with effects the remainindividually undescribed) on astronomical observables.This emphasis on superposition is further motivated bythe interpretation of all sensor effects as being causedby spurious lateral electric fields produced by impuritygradients within the silicon bulk of CCD sensors, as sug-gested in Smith & Rahmer (2008), and described fur-ther in Holland et al. (2014), Lupton (2014), and Stubbs(2014).

When pixel areas vary across a CCD sensor, a funda-mental assumption underlying the near-ubiquitous prac-tice of flat-fielding astronomical images is violated (tosome degree). Flat fielding assumes that a sensor’s pixelgrid is perfectly uniform, and that the entirety of itspixel response non-uniformity (PRNU), measured as thestandard deviation of a luminosity-corrected coadded flatfield, is caused solely by local variations in the quantumefficiency (QE) of the sensor. In this case, dividing araw image by a flat field image yields a calibrated fluxin each pixel, as is desired. However, if sensor effectscause the pixel grid to deviate from perfect rectilinear-

SLAC-PUB-17051

This material is based upon work supported by the U.S. Department of Energy, Office of Science, under Contract No. DE-AC02-76SF00515.

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ity, dividing raw science image pixel values by flat fieldpixel values yields a calibrated flux per unit (unknown)area—a “pixel surface brightness” rather than a flux—for each pixel in the image. Since astronomical imageanalyses typically assume that each pixel value repre-sents a calibrated flux, it is critical to understand theextent to which deviations from a rectilinear pixel gridbreak this fundamental assumption and potentially ren-der flat-fielding inappropriate.

In this work, we present a novel method for inferringa model of a sensor’s underlying pixel grid from coaddedflat-field images, with results from applying the methodto representative DECam and LSST prototype sensors.In contrast to previous work in this area, we take a “top-down” approach by considering the superposition of allstatic sensor effects simultaneously by inferring a pixelgrid model directly from minimally-processed flat-fielddata. There are several advantages to this method. Thefirst is that rather than requiring a complex sequence ofcalibration exposures, the only data required are severalhundred flat-field exposures, likely already on hand atmost observatories and sensor laboratories. Secondly, byanalyzing flat-field data with minimal pre-processing, weremain sensitive to potential new, as yet uncharacter-ized, sensor effects and unbiased by the pre-processingthat is typically required to highlight a particular effectof interest. Finally, with our top-down approach, we cantest the science impact of all effects at once, allowing usto account for possible constructive and/or destructiveinterference between the different sensor effects charac-terized in previous work.

Since our goal is to place an upper bound on the im-pact of the “improper” flat-fielding described above, wewill assume in this analysis that the assumptions of flat-fielding are maximally wrong—i.e. that all of a sensor’sPRNU is due to the distortion of the underlying pixelgrid rather than variations in sensor QE. For a givensensor, if this upper bound is still within the science re-quirements, then flat-fielding is an acceptable method ofcalibrating pixel fluxes. Plazas et al. (2014) have alreadydemonstrated the need for pixel-level corrections of DE-Cam astrometry to account for tree rings; a central goalof this paper will be to determine if similar pixel-levelcorrections beyond flat-fielding will be necessary for pre-cision astronomy, and therefore cosmology, with LSST.

Having motivated this work and described its funda-mental assumptions, we describe our method for fittinga model of an underlying pixel grid to flat field data andassessing the impact of deviations from grid rectilinear-ity on measurements of simulated PSFs in section 2. Insection 3 we validate the results of our grid model fits onboth simulations and data, and present the results of ourscience impact analysis for a representative DECam andlow-PRNU LSST prototype sensor. Section 4 discussesthe implications of these results, including their general-ization to future production-quality LSST sensors, andsection 5 outlines potential future work informed by theseresults.

2. METHODS

Before fitting an underlying grid model to flat-fielddata, a reasonable parametrization must be chosen forthe underlying pixel grid distortions. Full electrostaticsimulations such as those of Rasmussen (2015) can com-

Figure 1. : A 25× 25 cutout of a fitted pixel grid modelof an LSST sensor, with vertex perturbations magnifiedby a factor of 40 to make grid distortions visible by eye.Pixel color indicates pixel area relative to a unit rectilin-ear grid.

pute pixel boundary shapes with arbitrary precision, butrequire representing the CCD sensor, including defectsin the silicon bulk, as an underlying charge distributionthat cannot be precisely known for each individual de-vice.

Though electrostatic simulations have shown that pixelboundaries can have complex geometries (Rasmussen2015), they remain approximately linear in the limit ofweak pixel grid distortions. This constraint substantiallyreduces the number of parameters required to describethe pixel grid, relative to a full electrostatic simulation.We therefore parametrize pixel grid models by vertex po-sitions, demanding linear pixel boundaries between dis-placed vertices.

A 25 × 25 cutout from a fitted pixel grid model ofan LSST sensor is shown in Figure 1, with vertex dis-placements scaled up by a factor of 40 to make the griddistortions (with their characteristic pixel-neighbor anti-correlations as observed in Baumer & Roodman (2015))readily visible.

With this model of pixel area distortions, we ana-lytically compute the area of a distorted pixel using astandard cross-product formula for the area of a convexquadrilateral:

Aij =1

2

∣∣(xi+1,j+1 − xi,j)(yi,j+1 − yi+1,j)

−(xi,j+1 − xi+1,j)(yi+1,j+1 − yi,j)∣∣ (1)

When optimizing a pixel grid model, we desire thesecomputed model areas to match the observed areas (fromthe coadded flat fields) as closely as possible. Since equa-tion 1 gives the area of any vertex configuration, this al-lows us to define a goodness-of-fit statistic for a singlepixel as the square of the difference between the com-puted model area and area in the data. By summingover all pixels, we can construct a cost function of the

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form:

C =∑

pixels i,j

(Aij(vertices)−Aij)2 (2)

where Aij is the computed area of the inferred vertexarrangement at pixel (i,j), and Aij is the true area of thepixel, as measured by the flux in a coadded flat field.

To minimize this cost function via gradient descent,we need to calculate the derivative of Aij with respectto the positions of its four neighboring vertices. Thesederivatives can be calculated analytically for each vertexusing equation 1. This gives an overall gradient of C withrespect to the positions of each vertex. The gradientdescent update rule for each vertex position ~xij is thengiven by

~xij → ~xij − λdCd~xij

(3)

where λ is a tuning parameter that sets the step size ofthe parameter update along the gradient direction, andthe negative sign ensures the update is opposite the gra-dient direction, leading to a decrease in the cost functionC. The value of λ must be tuned empirically via a gridsearch to optimize model convergence. Code implement-ing this fitting procedure, built in Python around thenumpy/scipy (van der Walt et al. 2011), matplotlib(Hunter 2007), and pandas (McKinney 2010) libraries,is made available at https://github.com/cpadavis/weak_sauce. Demonstrations of the code are also pro-vided as IPython notebooks (Perez & Granger 2007).

We use the following technique to assess the impact ofthe pixel grid distortions inferred by these fits on crit-ical astronomical observables. We use GalSim (Roweet al. 2015) to construct flux profiles for Moffat PSFswith a nominal seeing value (0.9” for DECam and 0.7”for LSST). Surface integration of these flux profiles overeach pixel of the distorted grid is performed using a two-dimensional adaptation of a code designed for exact vox-elization of phase space in N-body simulations (Powell& Abel 2015). This results in a rendering of the trueflux profile as ‘observed’ by the distorted pixel grid ofthe sensor. This rendered image is then divided by thecoadded flat field, mimicking a standard data reductionpipeline. In this case, however, because the PRNU hasbeen modeled as pixel grid distortions rather than QEvariation, this flat fielding does not properly correct theraw image.

Finally, we use an implementation of the adaptive im-age moments algorithm described in Hirata & Seljak(2003) to compare observed parameters of sources ren-dered onto both our fitted pixel grid and an ideal rec-tilinear grid. This algorithm computes image momentsusing an adaptive elliptical weight matrix. This allowsthe impact of static pixel grid distortions on photometry(zeroth moment), astrometry (first moment), and shapemeasurements (second moments) to be evaluated. Weuse the following (unnormalized) convention for elliptic-ity to allow use of e0 as a proxy for PSF size:

e0 = Ixx + Iyy e1 = Ixx − Iyy e2 = 2Ixy (4)

where Iij are weighted second moments of the renderedsource image.

2.1. Data

A major advantage of this method of assessing sensorsystematics is the simplicity of obtaining the requireddata. While star flats, for example, are a common way ofassessing the impact of sensor imperfections on astrome-try, they require the sensor to be mounted on a telescopewhere time is available for a large number of ditheredexposures. In the lab, dithered arrays of pinholes can beimaged to allow for similar measurements, but require acomplex laboratory setup.

In contrast, the method presented in this paper re-quires only flat field images, which are routinely acquiredin large numbers in both sensor laboratory and observa-tory settings. To suppress the shot noise that dominatesthe variance of individual exposures, we coadd ∼ 500flat field images taken with an LSST prototype CCD ata wavelength of 625 nm and a light level of 75,000e−.Each frame is bias- and overscan-corrected and rescaledto the same average flux level before co-adding to ac-count for shutter timing variability. For DECam, a simi-lar number of r-band dome flats are coadded, with iden-tical pre-processing. In both cases, we consider a singleCCD amplifier block (1/16 of a chip for LSST; 1/2 chipfor DECam) for simplicity, cropping chip edges to focuson effects in the central regions which provide the bulk ofquality wide-field survey data. Bad pixels were maskedboth in the cost function optimization (to avoid warpingtheir neighboring pixels) and the systematics analysis.

We subtract a seventh order polynomial fit from thecoadded flat fields to remove variation in the illuminationpattern. This step has the additional benefit of remov-ing large-scale flux variations that are predominantly dueto QE variations. A primary determinant of the QE ofa pixel is the quality of the anti-reflective (AR) coatingon the pixel surface. Fluctuations in AR coating qualityacross a sensor result in positive pixel-neighbor correla-tions. As is the case with illumination variation, if a pixelis brighter (dimmer) than nominal due to fluctuations inAR coating quality, the relatively large scale of AR coat-ing variations relative to the pixel scale means that itsneighbors will likely also be brighter (dimmer).

On the other hand, local pixel grid distortions arecaused primarily by imperfections in the silicon bulk.These lead to negative pixel-neighbor correlations as pix-els gain and lose area from their neighbors (as shown inBaumer & Roodman (2015)), which are not removablewith a polynomial illumination correction. The fact thatthe large-scale variations removed by this illuminationcorrection are caused primarily by QE variations acrossthe sensor only increases the validity of our worst-case as-sumption that all remaining PRNU is due to distortionsof the underlying pixel grid.

In the following section, we validate the methods pre-sented in the section using both simulations and data,and present results obtained from applying the methodto both the DECam and LSST data described in thissection.

3. RESULTS

3.1. Validation

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One potential concern in fitting this model to data isthat it is apparently underconstrained by a factor of ∼ 2,with N ×M data points from the coadded flat fields and2×(N+1)×(M+1) parameters from allowing each vertexto shift in two dimensions. Expanding this out, we caninterpret the 2NM term as constraining the pixels in thebulk of the sensor, the 2N + 2M terms as constrainingthe boundary axes, and the trailing 2 degrees of freedomas a global translational shift.

Consider the grid of vertex displacements as a vectorfield, which can be represented as the sum of a pure di-vergence and a pure curl term. In this context, a diver-gence of pixel vertices (‘E-mode’) around a given pixelchanges that pixel’s area, while a curl (‘B-mode’) doesnot. Since the optimization of our cost function is an at-tempt to match the pixel areas given by the model withthe pixel areas implied by the flat field, the optimiza-tion process will only introduce curl-free (area-changing)displacements into the pixel grid.

This means that it will not induce an (area-preserving)global translational shift of the sensor grid, leaving these2 parameters fixed. It will also not slide rows andcolumns of vertices past each other (which would leadto a B-mode shear of entire rows and columns of pix-els), which fixes the 2N +2M boundary parameters thatcontrol row and column offsets. This leaves 2NM de-grees of freedom, of which the NM E-mode parametersare well-constrained by the NM data points from theflat field, while the NM B-mode parameters are uncon-strained. Since the fitting procedure begins from a rec-tilinear grid, for which the ’B-mode’ amplitude is zero,no B-mode vertex displacements will be present in theresulting fit. The model will always return a pixel gridmodel with pure E-mode distortions, and therefore willnot be able to constrain B-mode distortions that may bepresent in the pixel grid.

Despite its ability to constrain only E-mode vertex dis-tortions this model is still very useful, since as discussedin Section 1, we expect most sensor systematics to re-sult from spurious transverse electric fields in the siliconbulk. Since Maxwell’s equations ensure that an electro-static field will be curl-free, all electrostatic sensor sys-tematics (even those yet uncharacterized) will cause onlyE-mode distortions in the pixel grid. Such distortions area perfect match for the capability of our model.

Indeed, when we fit our model to flat fields synthe-sized from pixel grids with known E-mode distortions(pixel displacements that are the gradient of a scalarfield), our fitted model captures the input vertex posi-tions with high fidelity. This is demonstrated in Figure2, which shows a tight correlation between the knownand recovered locations of pixel vertices for sensors withpure E-mode vertex distortions.

If B-mode distortions are present in a pixel grid, theirimpact on the results of this fitting procedure dependson their degree of spatial correlation. For example, whenfitting to flat fields from known grids with both E- andB-mode power (constructed by perturbing a rectilineargrid with independently-drawn Gaussian vertex displace-ments), the recovery of individual vertex positions is de-graded (see the left panel of Figure 3), since does notcapture the B-mode distortions present in the input grid.

However, since the B-modes are spatially uncorrelatedin this case, the residuals that result from failing to cap-

Figure 2. : When fitting to flat fields from pixel grids withpure E-mode distortions, the model returns the inputvertex offsets with high fidelity. This will be the case forall electrostatic (curl-free) sensor systematics.

ture them in the model average out under a PSF profile.The right panel of Figure 3, which plots the astromet-ric shifts measured on the recovered grid against thosemeasured on the input grid, shows that these spatially-uncorrelated uncertainties in individual vertex positionsdo not affect the analysis of the impact of pixel grid dis-tortions on PSF observables.

So as long as there are no B-mode distortions that arespatially correlated at large scales, our model will capturethe systematic impact of all sensor effects properly. All ofthe individual effects mentioned in section 1 (tree rings,edge effects, and periodic step-and-repeat errors from thelithographic mask, and intrinsic pixel-to-pixel size varia-tion) all result in pure E-mode distortions, which meansthey will all be captured by this fitting method.

One possible source of B-mode distortions is “litho-graphic jitter”—pixel-to-pixel imperfections in the litho-graphic mask. While this effect is not constrained toinduce pure E-mode distortions, any B-modes inducedby errors in the mask are likely to be spatially uncorre-lated, and therefore unlikely to affect PSF observables,as shown in Figure 3. We are not aware of any sourcesof spatially-correlated B-mode distortions of CCD pixelgrids. However, should any exist, they would not beconstrained by this fitting procedure, and would requireseparate characterization. In this work, we will proceedunder the assumption that all sensor systematics affect-ing astronomical observables are fundamentally electro-static (i.e. curl-free), and present further validations ofthis method on real data.

To demonstrate that our fitted grid is not only a the-oretically robust solution, but a physically relevant one,we illustrate the pixel shapes (widths and heights) froma model fit to an LSST sensor with known anisotropy inpixel boundary strengths and DECam sensor with treerings.

Figure 4 illustrates how a pixel grid fit to an LSSTcoadded flat field is very close to rectilinear—the RMSdeviation of pixels from their nominal position is ∼1/1000 of a pixel side length. Given that the vertex dis-placements remain small in the absence of an explicit reg-

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Figure 3. : For flat fields from pixel grids containing spa-tially unstructured B-modes, the model’s reconstructionof the pixel grid is very noisy. However, since these errorsinduced by the uncorrelated B-modes average out underPSF profile (FWHM ∼ 3.5 pixels), the systematic errorsinduced by distortions in the input and recovered gridson PSF observables show excellent correlation.

ularizing constraint, this figure illustrates directly thatexplicit regularization is unnecessary. We observe thatthe scale of the vertex displacements in the parallel di-rection (vertical) is slightly larger than in the serial di-rection (horizontal). This is expected behavior fromboth electrostatic simulations (Rasmussen 2015) anddata (Baumer & Roodman 2015), where the anisotropyis explained via the relative electrostatic weakness ofthe pixel clock boundaries (parallel) with respect to thestronger, static channel stop implants (serial).

Figure 4. : A histogram of vertex displacements from thenominal pixel grid, illustrating the relative susceptibilityof the weaker clock leaves which serve as the pixel bound-aries in the parallel direction to the stronger channel stopimplants bounding the serial direction, as predicted inRasmussen (2015).

Figure 5 illustrates our correct inference of pixel shapesdue to a dominant tree-ring distortion in a DECam sen-sor. Since tree rings are caused by radial transverse elec-tric fields, pixel widths (heights) are expected to be mostaffected when the tree ring is fluctuating in the horizontal(vertical) direction. This pattern is observed in Figure5, showing that the fitted pixel geometries reproduce theexpected pixel shape anisotropy of tree ring features inDECam. This expected behavior is realized in the pixel

grid fits despite no explicit model pressure (the modelknows only about pixel fluxes, not shapes), demonstrat-ing that this method infers physically reasonable pixelgeometries from real data.

Figure 5. : Maps of relative pixel widths (a) and heights(b) from a fit to a coadded DECam flat field, showingpreferential variation in pixel shape along the radial di-rection of the tree rings, as expected. This demonstratesthat the pixel grid models inferred by our method arephysically relevant.

An even more direct test of this method is illustrated inFigure 6, which shows a comparison of the radial profileof radial astrometric distortions as measured using DE-Cam star flats and the prediction from our pixel gridmodel. The radial profiles of tree rings as measuredby our model are well-aligned with those measured byPlazas et al. (2014). The small differences are likelythe results of either noise (since our model, derived fromcoadded flat fields with negligible photometric noise, canbe sampled much more densely than star flats allow onthe sky) or tree ring structure being washed out in ourmodel by local QE defects present in DECam sensors(scratches, smudges, etc.) that our method incorrectlymodels as pixel size variation. This latter concern is notan issue in analyzing the LSST sensor, where the cos-metic quality is extremely high, but should be consideredwhen applying this method to sensors with significantvariations in both QE and pixel size.

Having demonstrated the robustness of this methodfor inferring sensor systematics from flat field data viapixel grid models, we present results from applying ourmethods to data from a representative DECam CCD andan LSST prototype sensor in the following sections.

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Figure 6. : Radial profiles of radial astrometric distortionas measured using DECam star flats from Plazas et al.(2014), compared with the prediction from our pixel gridmodel, showing good agreement. The radial profile pre-sented in this work is smoother than the one derived fromstar flats, as our full grid model is derived from coaddedflat fields with negligible photometric (shot) noise.

3.2. Results from DECam

Figure 7. : At left, a one-amplifier cutout of a coaddedDECam flat field, which the pixel grid aims to modelduring the fit. At right, a map of the flux residuals forthe fitted grid model. The lack of similarity betweenthe data and residuals demonstrates that the model ispicking up the relevant spatial structure in the flat field,in particular the prominent tree rings.

We apply our method to a coadded flat field from aDECam sensor (shown at left in Figure 7) and obtainthe residuals illustrated in the right panel in the same

figure. The spatial structure of the fitting residuals isdistinct from that of the data, indicating that the fittedmodel contains nearly all the relevant spatial structurepresent in the data.

Figure 8 presents the impact of sensor effects on 5 keyastronomical observables as modeled across a DECamchip using our method. Included are photometry andastrometry, PSF shape, parametrized by PSF size e0,and unnormalized ellipticities e1 and e2 as defined insection 2. A flat field (smoothed with a nominal PSFkernel to highlight medium-scale structure) is includedat left so structure in the systematics maps can be visu-ally compared to structure in the raw data. While thereare indeed grid distortions at the single pixel scale in areal sensor, as we showed in section 3.1, the systematicimpact of these is negligible for any source that is well-sampled, thus justifying the smoothing of the flat fieldfor visual comparison.

Tree rings provide the dominant spatial structure in all5 systematics maps. The impact of the tree rings is mostevident in the astrometric offset, where their imprint isobserved at the ∼ 10 mas level. Given this significantimpact, DES is currently implementing a catalog-levelfix for this using star flats as described in Plazas et al.(2014).

The variations in photometry and PSF size correlatewith tree rings since the effective area of pixels near treering peaks is larger than those near troughs. This consti-tutes an approximately affine stretching of the local pixelgrid near tree ring extrema which is unaccounted for intraditional PSF estimation. The effect on PSF shape pa-rameters agrees with expectation based on Figure 5, ase1 is primarily affected in areas where tree ring varia-tions run vertically and horizontally, while e2 is affectedonly when the tree rings run diagonally. The issue ofDECam tree rings impacting PSF behavior is currentlybeing addressed within the DES collaboration.

3.3. Results from LSST

Applying our method to the LSST data describedin section 2.1 yields a fitted grid model that captures99.85% of the flux deviations present in the data. Figure9 shows the flat field the model is trying to match, alongwith the map of flux residuals, whose RMS of 0.0006%is negligible compared with the flat field PRNU of 0.4%.Again, these residuals are uncorrelated spatially with thetarget flat field, indicating that the model was successfulin picking up spatial structure in the data.

Results from applying the same analysis to an LSSTprototype sensor are shown in Figure 10. None of thesystematics residual maps are correlated with the fluxresiduals from the model fit (cf. Figure 9), so the ob-served structure is not due to a fitting artifact.

Spatial structure is observed in each of the LSST sys-tematics maps, albeit at a much smaller scale than thosefrom DECam. The scatter induced in photometric mea-surements is well below 1 mMag, while in the astrometricmap, a median deviation of about 1 milliarcsecond is ob-served. The spatial structure visible in the systematicsmaps is generally observed at locations where the gra-dient of the coadded flat field is highest. This is wherethe assumption of a locally-rectilinear (affine) pixel gridis maximally violated.

While photometric and astrometric precision will be

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Figure 8. : A summary of systematic errors induced by sensor effects across a DECam chip, as modeled using ourmethod. A flat field (a) (smoothed to highlight medium-scale structure) for comparison with systematics maps froma representative DECam amplifier, showing significant impact on photometry (b), astrometry (c), PSF size (d), andPSF ellipticity components e1 (e) and e2 (f). Tree rings constitute the dominant source of systematic error, affectingastrometry most significantly.

Figure 9. : A single amplifier segment from a coadd of∼ 500 flat field images (a) from an LSST prototype sen-sor, and the residual map (b) resulting from fitting apixel grid model as described in section 2, showing thatthe model reproduces 99.85% of the RMS flux variationspresent in the coadded flat field.

of general concern to nearly all users of LSST data, thePSF shape parameters, also affected by the modeled grid

distortions, are of primary importance to weak gravita-tional lensing analyses. As Figure 10 shows, the fittedpixel grid distortions do cause some spatial structure toappear in the maps of e0, e1, and e2. However, sincecosmology from weak lensing relies on correlation func-tions of shape rather than individual measurements, therequirement on PSF shape measurements is expressed asa correlation function, ξPSF (e, e) as elucidated in Jarviset al. (2016). For a fixed level of PSF leakage (contamina-tion of galaxy shapes by unremoved PSF), a requirementon PSF shear auto-correlations can be derived; we usethe requirements given in LSST Science Council (2011).Computing the correlation functions of our derived PSFshape errors yields structure-free correlations orders ofmagnitude below LSST requirements.

ObservableFlux Astrometry

ξ(e1, e1) ξ(e2, e2)(mMag) (mas)

Shift µ 0.08 1.2< 10−8 < 5 × 10−10

Scatter σ 0.31 0.7LSST Req. 5 10 2 × 10−5 10−7

Table 1: This table shows the photometric and astro-metric mean shifts µ and induced scatter σ due to in-duced by the pixel grid distortions fit by our model. PSFshape errors are quoted as upper bounds on correlationfunctions. For a nominal sensor with a PRNU of 0.4%, allthree fall well below LSST requirements (LSST ScienceCouncil 2011).

A summary of results from this analysis is presented inTable 1. The mean shift µ refers to the average offset in-duced by the fitted pixel grid distortions, and the inducedscatter σ is the scatter in an observable quantity across asensor chip due to these effects. Overall, this shows thatthe impact of all static sensor effects taken together, in-cluding unknown effects that impact flat fields, are neg-ligible compared to LSST single-exposure requirements

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Figure 10. : Similar to Figure 8, systematics maps from an LSST prototype sensor showing the impact of pixel griddistortions on photometry (b), astrometry (c), PSF size (d), and PSF ellipticity components e1 (e) and e2 (f), witha lightly smoothed flat field (a) for comparison. The observed structure in the systematics maps is genuine, as it isuncorrelated with the residuals shown in Figure 9.

(these systematics should be even further suppressed byLSST’s many exposures and dithering strategy). This in-dicates that flat-fielding will remain an appropriate strat-egy for calibrating LSST images. However, this partic-ular sensor has a PRNU of only 0.4%, much lower thanthe LSST CCD requirement of 5%. We consider the im-plications of this in the following section.

4. DISCUSSION

Although we have shown that the impact of static sen-sor effects on LSST science observables is modest for asensor with very low PRNU, it is possible that some ofthe sensors included in the LSST focal plane could havea PRNU approaching 5%. In this section, we considerthe impact of such high PRNU if it is due to pixel griddistortions rather than local QE variation (which wouldbe correctable with typical flat-fielding).

In Figure 11, we investigate this by multiplicativelyscaling up the vertex displacements from our fitted modelby a factor of 10, resulting in a flat field with ∼ 4%PRNU. The photometric and astrometric shifts obtainedfrom this analysis are shown as black points shown inFigure 11.

Interestingly, directly scaling up the photometric dis-tortions observed with the original fitted pixel grid(blue), yields the distribution illustrated by the red his-togram in Figure 11. The agreement between these twodistributions shows that even for pixel grids near theworst-case LSST PRNU, the systematic errors on scienceobservables respond linearly to increased grid distortions.This linearity allows us to predict the scale of the system-atics that might arise from any sensor of moderately-lowPRNU. Specifically, the values quoted in Table 1 canbe scaled linearly (quadratically for the correlation func-tions) to other PRNU values as a way to estimate sys-tematic impact of static effects in lower quality sensors.However, given the generality of the framework proposedin this work, it would also make sense to apply it to in-

coming production sensors to test whether their uniquepixel grid structure led to different systematic effects.

5. CONCLUSIONS

We have presented a method to infer a model of anunderlying pixel grid that agnostically models the super-position of all electrostatic (curl-free) static sensor ef-fects present in flat fields, reproducing observed flat fieldfluxes with very high (∼ 99%) fidelity. We have usedthese pixel grid models to analyze the impact of sensorsystematics on photometry, astrometry, and PSF shapemeasurements for both DECam and LSST prototype sen-sors.

For the DECam sensor, our findings recover behaviorof tree rings known from both electrostatic simulationand star flat data, and agree with previous work in find-ing significant impact of static pixel grid distortions onastrometry and PSF observables, for which independentcorrections are currently being developed within the DEScollaboration.

For the LSST prototype, we find that for a sensor withlow PRNU (0.4%), the impact all static sensor effectstaken together—both known and unknown—on these ob-servables is negligible, but that the impact scales linearlywith the PRNU. Therefore, for sensors with PRNU val-ues closer to the specification of 5%, additional work toseparate and model the effects of local QE variation andpixel grid distortions may be required.

In cases where a fitted pixel grid model uncovers an un-acceptably high level of systematics, or in the case of im-ages with exceptional seeing, where the impact of staticpixel grid distortions would be magnified, our methodcould potentially be used as the basis of a correction toameliorate them. The primary advantage of such a cor-rection would be that the sampling of systematics mapscan be arbitrarily dense, rather than being limited by thefinite density of real star flats, for example. A proper cor-rection could require extending this framework to incor-

9

Figure 11. : The correspondence between photometric(a) and astrometric (b) shifts from the LSST sensor withPRNU=0.4% scaled up by a factor of 10 and the shiftsobtained by depositing PSF profiles on a grid with vertexperturbations scaled by the same factor shows that weremain in the linear distortion regime even for relativelyhigh PRNU.

porate additional data (for example, flat fields with non-uniform illumination patterns) to disentangle local QEvariation from true pixel grid distortions. We leave con-sideration of such correction algorithms as future work.

Overall, we conclude that for sensors with low PRNU,calibrating LSST images using traditional flat-fielding,despite the presence of varying pixel size, would havenegligible impact on LSST science, even in the worst casethat all of the PRNU is due to pixel size variation. How-ever, for sensors with PRNU closer to the nominal worstcase, more characterization may be required to disentan-gle the individual contributions of pixel grid distortions

and local QE variation to the PRNU to form the basis ofa more detailed correction. As long as LSST productionsensors have sufficiently low PRNU, we anticipate thatthis will not be necessary.

We’d like to thank Devon Powell for advice in adapt-ing his phase-space integration code, Andres Plazas forproviding DECam star flat measurements, and Peter Do-herty for providing LSST flat field data. Many thanksto Robert Lupton and Gary Bernstein for their insight-ful comments on the manuscript. We are grateful forthe extraordinary contributions of our CTIO and DE-Cam colleagues in achieving the excellent instrument andtelescope conditions that have made this work possible.This work was performed in part under DOE ContractDE-AC02-76SF00515. This material is based upon worksupported by the National Science Foundation GraduateResearch Fellowship under Grant No. DGE-114747.

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