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Evaluating cosmological biases using photometric redshifts for Type Ia Supernova cosmology with the Dark Energy Survey Supernova Program

Cosmological analyses with Type Ia Supernovae (SNe Ia) have traditionally been reliant on spectroscopy for both classifying the type of supernova and obtaining reliable redshifts to measure the distance–redshift relation. While obtaining a host-galaxy spectroscopic redshift for most SNe is feasible for small-area transient surveys, it will be too resource intensive for upcoming large-area surveys such as the Vera Rubin Observatory Legacy Survey of Space and Time, which will observe on the order of millions of SNe. Here, we use data from the Dark Energy Survey (DES) to address this problem with photometric redshifts (photo-z) inferred directly from the SN light curve in combination with Gaussian and full p(z) priors from host-galaxy photo-z estimates. Using the DES 5-yr photometrically classified SN sample, we consider several photo-z algorithms as host-galaxy photo-z priors, including the Self-Organizing Map redshifts (SOMPZ), Bayesian Photometric Redshifts (BPZ), and Directional-Neighbourhood Fitting (DNF) redshift estimates employed in the DES 3 × 2 point analyses. With detailed catalogue-level simulations of the DES 5-yr sample, we find that the simulated w can be recovered within ±0.02 when using SN+SOMPZ or DNF prior photo-z, smaller than the average statistical uncertainty for these samples of 0.03. With data, we obtain biases in w consistent with simulations within ~1σ for three of the five photo-z variants. We further evaluate how photo-z systematics interplay with photometric classification and find classification introduces a subdominant systematic component. This work lays the foundation for next-generation fully photometric SNe Ia cosmological analyses.

(cosmology:) dark energy↗

Photometric Redshifts for Next-Generation Surveys

Photometric redshifts are essential in studies of both galaxy evolution and cosmology, as they enable analyses of objects too numerous or faint for spectroscopy. The Rubin Observatory, Euclid, and Roman Space Telescope will soon provide a new generation of imaging surveys with unprecedented area coverage, wavelength range, and depth. To take full advantage of these data sets, further progress in photometric redshift methods is needed. In this review, we focus on the greatest common challenges and prospects for improvement in applications of photometric redshifts to the next generation of surveys: ∙ Gains in performance (i.e., the precision of redshift estimates for individual galaxies) could greatly enhance studies of galaxy evolution and some probes of cosmology. ∙ Improvements in characterization (i.e., the accurate recovery of redshift distributions of galaxies in the presence of uncertainty on individual redshifts) are urgently needed for cosmological measurements with next-generation surveys. To achieve both of these goals, improvements in the scope and treatment of the samples of spectroscopic redshifts that make high-fidelity photometric redshifts possible will also be needed. For the full potential of the next generation of surveys to be reached, the characterization of redshift distributions must improve by roughly an order of magnitude compared with the current state of the art, requiring progress on a wide variety of fronts. We conclude by presenting a speculative evaluation of how photometric redshift methods and the collection of the necessary spectroscopic samples may improve by the time near-future surveys are completed.

Astronomy & Astrophysics↗

Dependence of Cosmological Constraints on Gray Photometric Zero-point Uncertainties of Supernova Surveys

Abstract Type Ia supernova (SN) measurements of the Hubble constant, H 0 ; cosmic mass density, Ω M ; and dark energy equation-of-state parameter, w , rely on heterogeneous SN surveys across three decades of observation. These distinct surveys may have undiagnosed, relative photometric zero-point errors. We determine the sensitivities of the SH0ES+Pantheon+ cosmological constraints to unknown gray systematics in the photometric zero-point calibration between the 19 surveys that comprise the Pantheon+ SN compendium. Varying the surveys’ gray zero-points simultaneously with cosmological parameters, we determine that the SH0ES+Pantheon+ measurement of H 0 is robust against gray intersurvey photometric miscalibration. Specifically, uncalibrated intersurvey systematics could represent a source of uncertainty no larger than 0.39 km s −1 Mpc −1 for H 0 . This modest increase in H 0 uncertainty could not account for the 7 km s −1 Mpc −1 “Hubble tension” between the SH0ES measurement of H 0 and the Planck ΛCDM-based inference of H 0 . However, the SH0ES+Pantheon+ best-fit values of Ω M and w are not robust against gray zero-point error, slipping by up to 0.16 and −0.63. Because measurements of Ω M and w depend on intrasurvey cross-band calibration, the hypothetical gray miscalibration underestimates the sensitivity of these measurements to zero-point miscalibration. Because the Pantheon+ compendium contains many surveys that share low- z Hubble flow (HF) and Cepheid-paired SNe, intersurvey photometric calibration errors do not significantly impede the joint use of SH0ES and Pantheon+ to measure H 0 to 1% accuracy. However, H 0 constraints that rely on one HF survey but numerous galactic distance calibration surveys are susceptible to intersurvey photometric miscalibration.

79 ASTRONOMY AND ASTROPHYSICS↗

Considerations for Optimizing the Photometric Classification of Supernovae from the Rubin Observatory

The Vera C. Rubin Observatory will increase the number of observed supernovae (SNe) by an order of magnitude; however, it is impossible to spectroscopically confirm the class for all SNe discovered. Thus, photometric classification is crucial, but its accuracy depends on the not-yet-finalized observing strategy of Rubin Observatory's Legacy Survey of Space and Time (LSST). We quantitatively analyze the impact of the LSST observing strategy on SNe classification using simulated multiband light curves from the Photometric LSST Astronomical Time-Series Classification Challenge (PLAsTiCC). First, we augment the simulated training set to be representative of the photometric redshift distribution per SNe class, the cadence of observations, and the flux uncertainty distribution of the test set. Then we build a classifier using the photometric transient classification library snmachine, based on wavelet features obtained from Gaussian process fits, yielding a similar performance to the winning PLAsTiCC entry. We study the classification performance for SNe with different properties within a single simulated observing strategy. We find that season length is important, with light curves of 150 days yielding the highest performance. Cadence also has an important impact on SNe classification; events with median inter-night gap <3.5 days yield higher classification performance. Interestingly, we find that large gaps (>10 days) in light-curve observations do not impact performance if sufficient observations are available on either side, due to the effectiveness of the Gaussian process interpolation. This analysis is the first exploration of the impact of observing strategy on photometric SN classification with LSST.

79 ASTRONOMY AND ASTROPHYSICS↗

A Southern Photometric Quasar Catalog from the Dark Energy Survey Data Release 2

We present a catalog of 1.4 million photometrically selected quasar candidates in the southern hemisphere over the ~5000 deg 2 Dark Energy Survey (DES) wide survey area. We combine optical photometry from the DES second data release (DR2) with available near-infrared (NIR) and the all-sky unWISE mid-infrared photometry in the selection. We build models of quasars, galaxies, and stars with multivariate skew-t distributions in the multidimensional space of relative fluxes as functions of redshift (or color for stars) and magnitude. Our selection algorithm assigns probabilities for quasars, galaxies, and stars and simultaneously calculates photometric redshifts (photo-z) for quasar and galaxy candidates. Benchmarking on spectroscopically confirmed objects, we successfully classify (with photometry) 94.7% of quasars, 99.3% of galaxies, and 96.3% of stars when all IR bands (NIR YJHK and WISE W1W2) are available. The classification and photo-z regression success rates decrease when fewer bands are available. Our quasar (galaxy) photo-z quality, defined as the fraction of objects with the difference between the photo-z z p and the spectroscopic redshift z s , |Δz| ≡ |z s - z p |/(1 + z s ) ≤ 0.1, is 92.2% (98.1%) when all IR bands are available, decreasing to 72.2% (90.0%) using optical DES data only. Our photometric quasar catalog achieves an estimated completeness of 89% and purity of 79% at r < 21.5 (0.68 million quasar candidates), with reduced completeness and purity at 21.5 < r ≲ 24. Among the 1.4 million quasar candidates, 87,857 have existing spectra, and 84,978 (96.7%) of them are spectroscopically confirmed quasars. Finally, we provide quasar, galaxy, and star probabilities for all (0.69 billion) photometric sources in the DES DR2 coadded photometric catalog.

79 ASTRONOMY AND ASTROPHYSICS↗

The Young Supernova Experiment Data Release 1 (YSE DR1): Light Curves and Photometric Classification of 1975 Supernovae

We present the Young Supernova Experiment Data Release 1 (YSE DR1), comprised of processed multicolor PanSTARRS1 griz and Zwicky Transient Facility (ZTF) gr photometry of 1975 transients with host–galaxy associations, redshifts, spectroscopic and/or photometric classifications, and additional data products from 2019 November 24 to 2021 December 20. YSE DR1 spans discoveries and observations from young and fast-rising supernovae (SNe) to transients that persist for over a year, with a redshift distribution reaching z ≈ 0.5. We present relative SN rates from YSE's magnitude- and volume-limited surveys, which are consistent with previously published values within estimated uncertainties for untargeted surveys. We combine YSE and ZTF data, and create multisurvey SN simulations to train the ParSNIP and SuperRAENN photometric classification algorithms; when validating our ParSNIP classifier on 472 spectroscopically classified YSE DR1 SNe, we achieve 82% accuracy across three SN classes (SNe Ia, II, Ib/Ic) and 90% accuracy across two SN classes (SNe Ia, core-collapse SNe). Our classifier performs particularly well on SNe Ia, with high (>90%) individual completeness and purity, which will help build an anchor photometric SNe Ia sample for cosmology. We then use our photometric classifier to characterize our photometric sample of 1483 SNe, labeling 1048 (~71%) SNe Ia, 339 (~23%) SNe II, and 96 (~6%) SNe Ib/Ic. YSE DR1 provides a training ground for building discovery, anomaly detection, and classification algorithms, performing cosmological analyses, understanding the nature of red and rare transients, exploring tidal disruption events and nuclear variability, and preparing for the forthcoming Vera C. Rubin Observatory Legacy Survey of Space and Time.

79 ASTRONOMY AND ASTROPHYSICS↗

Photometric redshifts for the S-PLUS Survey: Is machine learning up to the task?

The Southern Photometric Local Universe Survey (S-PLUS) is a novel project that aims to map the Southern Hemisphere using a twelve filter system, comprising five broad-band SDSS-like filters and seven narrow-band filters optimized for important stellar features in the local universe. In this paper we use the photometry and morphological information from the first S-PLUS data release (S-PLUS DR1) cross-matched to unWISE data and spectroscopic redshifts from Sloan Digital Sky Survey DR15. We explore three different machine learning methods (Gaussian Processes with GPz and two Deep Learning models made with TensorFlow) and compare them with the currently used template-fitting method in the S-PLUS DR1 to address whether machine learning methods can take advantage of the twelve filter system for photometric redshift prediction. Using tests for accuracy for both single-point estimates such as the calculation of the scatter, bias, and outlier fraction, and probability distribution functions (PDFs) such as the Probability Integral Transform (PIT), the Continuous Ranked Probability Score (CRPS) and the Odds distribution, we conclude that a deep-learning method using a combination of a Bayesian Neural Network and a Mixture Density Network offers the most accurate photometric redshifts for the current test sample. In conclusion, it achieves single-point photometric redshifts with scatter (σ NMAD ) of 0.023, normalized bias of -0.001, and outlier fraction of 0.64% for galaxies with r_auto magnitudes between 16 and 21.

79 ASTRONOMY AND ASTROPHYSICS↗

CIRCLEZ : Reliable photometric redshifts for active galactic nuclei computed solely using photometry from Legacy Survey Imaging for DESI

Photometric redshifts for galaxies hosting an accreting supermassive black hole in their center, known as active galactic nuclei (AGNs), are notoriously challenging. At present, they are most optimally computed via spectral energy distribution (SED) fittings, assuming that deep photometry for many wavelengths is available. However, for AGNs detected from all-sky surveys, the photometry is limited and provided by a range of instruments and studies. This makes the task of homogenizing the data challenging, presenting a dramatic drawback for the millions of AGNs that wide surveys such as SRG/eROSITA are poised to detect. This work aims to compute reliable photometric redshifts for X-ray-detected AGNs using only one dataset that covers a large area: the tenth data release of the Imaging Legacy Survey (LS10) for DESI. LS10 provides deep grizW1-W4 forced photometry within various apertures over the footprint of the eROSITA-DE survey, which avoids issues related to the cross-calibration of surveys. We present the results from CIRCLEZ, a machine-learning algorithm based on a fully connected neural network. CIRCLEZ is built on a training sample of 14 000 X-ray-detected AGNs and utilizes multi-aperture photometry, mapping the light distribution of the sources. The accuracy (σNMAD) and the fraction of outliers (η) reached in a test sample of 2913 AGNs are equal to 0.067 and 11.6%, respectively. The results are comparable to (or even better than) what was previously obtained for the same field, but with much less effort in this instance. We further tested the stability of the results by computing the photometric redshifts for the sources detected in CSC2 and Chandra-COSMOS Legacy, reaching a comparable accuracy as in eFEDS when limiting the magnitude of the counterparts to the depth of LS10. The method can be applied to fainter samples of AGNs using deeper optical data from future surveys (for example, LSST, Euclid), granting LS10-like information on the light distribution beyond the morphological type. Along with this paper, we have released an updated version of the photometric redshifts (including errors and probability distribution functions) for eROSITA/eFEDS.

79 ASTRONOMY AND ASTROPHYSICS↗

The clustering of DESI-like luminous red galaxies using photometric redshifts

ABSTRACT We present measurements of the redshift-dependent clustering of a DESI-like luminous red galaxy (LRG) sample selected from the Legacy Survey imaging data set, and use the halo occupation distribution (HOD) framework to fit the clustering signal. The photometric LRG sample in this study contains 2.7 million objects over the redshift range of 0.4 < z < 0.9 over 5655 deg2. We have developed new photometric redshift (photo-z) estimates using the Legacy Survey DECam and WISE photometry, with σNMAD = 0.02 precision for LRGs. We compute the projected correlation function using new methods that maximize signal-to-noise ratio while incorporating redshift uncertainties. We present a novel algorithm for dividing irregular survey geometries into equal-area patches for jackknife resampling. For a five-parameter HOD model fit using the MultiDark halo catalogue, we find that there is little evolution in HOD parameters except at the highest redshifts. The inferred large-scale structure bias is largely consistent with constant clustering amplitude over time. In an appendix, we explore limitations of Markov chain Monte Carlo fitting using stochastic likelihood estimates resulting from applying HOD methods to N-body catalogues, and present a new technique for finding best-fitting parameters in this situation. Accompanying this paper, we have released the Photometric Redshifts for the Legacy Surveys catalogue of photo-z’s obtained by applying the methods used in this work to the full Legacy Survey Data Release 8 data set. This catalogue provides accurate photometric redshifts for objects with z < 21 over more than 16 000 deg2 of sky.

79 ASTRONOMY AND ASTROPHYSICS↗

The dark energy survey 5-yr photometrically identified type Ia supernovae

ABSTRACT As part of the cosmology analysis using Type Ia Supernovae (SN Ia) in the Dark Energy Survey (DES), we present photometrically identified SN Ia samples using multiband light curves and host galaxy redshifts. For this analysis, we use the photometric classification framework SuperNNovatrained on realistic DES-like simulations. For reliable classification, we process the DES SN programme (DES-SN) data and introduce improvements to the classifier architecture, obtaining classification accuracies of more than 98 per cent on simulations. This is the first SN classification to make use of ensemble methods, resulting in more robust samples. Using photometry, host galaxy redshifts, and a classification probability requirement, we identify 1863 SNe Ia from which we select 1484 cosmology-grade SNe Ia spanning the redshift range of 0.07 < z < 1.14. We find good agreement between the light-curve properties of the photometrically selected sample and simulations. Additionally, we create similar SN Ia samples using two types of Bayesian Neural Network classifiers that provide uncertainties on the classification probabilities. We test the feasibility of using these uncertainties as indicators for out-of-distribution candidates and model confidence. Finally, we discuss the implications of photometric samples and classification methods for future surveys such as Vera C. Rubin Observatory Legacy Survey of Space and Time.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

Amalgame: cosmological constraints from the first combined photometric supernova sample

ABSTRACT Future constraints of cosmological parameters from Type Ia supernovae (SNe Ia) will depend on the use of photometric samples, those samples without spectroscopic measurements of the SNe Ia. There is a growing number of analyses that show that photometric samples can be utilized for precision cosmological studies with minimal systematic uncertainties. To investigate this claim, we perform the first analysis that combines two separate photometric samples, SDSS and Pan-STARRS, without including a low-redshift anchor. We evaluate the consistency of the cosmological parameters from these two samples and find they are consistent with each other to under 1σ. From the combined sample, named Amalgame, we measure ΩM = 0.328 ± 0.024 with SN alone in a flat ΛCDM model, and ΩM = 0.330 ± 0.018 and w = $-1.016^{+0.055}_{-0.058}$ when combining with a Planck data prior and a flat wCDM model. These results are consistent with constraints from the Pantheon+ analysis of only spectroscopically confirmed SNe Ia, and show that there are no significant impediments to analyses of purely photometric samples of SNe Ia. The data and results are made available at https://github.com/bap37/AmalgameDR.

Astronomy & Astrophysics↗

Joint cosmic density reconstruction from photometric and spectroscopic samples

ABSTRACT We reconstruct the dark matter density field from spatially overlapping spectroscopic and photometric redshift catalogues through a field-level forward modelling approach. Instead of directly inferring the underlying density field, we find the best-fitting initial Gaussian fluctuations that will evolve into the observed cosmic volume. To account for the substantial uncertainty of photometric redshifts we employ a differentiable continuous Poisson process. As an initial test, we construct a mock based on the upcoming Prime Focus Spectrograph combined with photometric sample modelled on the Subaru Hyper Suprime-Cam. Depending on the statistic of interest, we find improvements in cosmic structure classification equivalent to 50–100 per cent more spectroscopic targets by combining relatively sparse spectroscopic with dense photometric samples.

Horowitz, B.↗

Photometric Objects around Cosmic Webs (PAC) Delineated in a Spectroscopic Survey. I. Methods

We provide a method for estimating the projected density distribution ${\bar{n}}_{2}{w}_{p}({r}_{p})$ of photometric objects around spectroscopic objects in a spectroscopic survey. This quantity describes the distribution of photometric sources with certain physical properties (e.g., luminosity, mass, and color) around cosmic webs (PAC) traced by the spectroscopic objects. The method can make full use of current and future deep and wide photometric surveys to explore the formation of galaxies up to medium redshift (z s < 2) 3 with the aid of cosmological spectroscopic surveys that sample only a fairly limited species of objects (e.g., emission line galaxies). As an example, we apply the PAC method to the CMASS spectroscopic and HSC-SSP PDR2 photometric samples to explore the distribution of galaxies for a wide range of stellar masses from 10 9.0 to 10 12.0 M ⊙ around massive galaxies at z s ≈ 0.6. Using the abundance-matching method, we model ${\bar{n}}_{2}{w}_{p}({r}_{p})$ in N-body simulation using Markov chain Monte Carlo sampling, and we accurately measure the stellar–halo mass relation and stellar mass function for the whole mass range. We can also measure the conditional stellar mass function of satellites for central galaxies of different mass. The PAC method has many potential applications for studying the evolution of galaxies.

79 ASTRONOMY AND ASTROPHYSICS↗

Measuring He i Stark Line Shapes in the Laboratory to Examine Differences in Photometric and Spectroscopic DB White Dwarf Masses

Abstract Accurate helium White Dwarf (DB) masses are critical for understanding the star’s evolution. DB masses derived from the spectroscopic and photometric methods are inconsistent. Photometric masses agree better with currently accepted DB evolutionary theories and are mostly consistent across a large range of surface temperatures. Spectroscopic masses rely on untested He i Stark line-shape and Van der Waals broadening predictions, show unexpected surface temperature trends, and are thus viewed as less reliable. To test this conclusion, we present in this paper detailed He i Stark line-shape measurements at conditions relevant to DB atmospheres ( T electron ≈12,000–17,000 K, n electron ≈ 10 17 cm −3 ). We use X-rays from Sandia National Laboratories’ Z -machine to create a uniform ≈120 mm long hydrogen–helium mixture plasma. Van der Waals broadening is negligible at our experimental conditions, allowing us to measure He i Stark profiles only. H β , which has been well-studied in our platform and elsewhere, serves as the n e diagnostic. We find that He i Stark broadening models used in DB analyses are accurate within errors at tested conditions. It therefore seems unlikely that line-shape models are solely responsible for the observed spectroscopic mass trends. Our results should motivate the WD community to further scrutinize the validity of other spectroscopic and photometric input parameters, like atmospheric structure assumptions and convection corrections. These parameters can significantly change the derived DB mass. Identifying potential weaknesses in any input parameters could further our understanding of DBs, help elucidate their evolutionary origins, and strengthen confidence in both spectroscopic and photometric masses.

Schaeuble, M. -A. (ORCID:0000000188695203)↗

Photometric Recalibration of the SDSS Stripe 82 to a Few Millimagnitude Precision with the Stellar Color Regression Method and Gaia EDR3

By combining spectroscopic data from the LAMOST DR7, Sloan Digital Sky Survey (SDSS) DR12, and corrected photometric data from the Gaia EDR3, we apply the stellar color regression (SCR) method to recalibrate the SDSS Stripe 82 standard stars catalog of Ivezic' et al. With a total number of about 30,000 spectroscopically targeted stars, we have mapped out the relatively large and strongly correlated photometric zero-point errors present in the catalog, ~2.5% in the u band and ~1% in the griz bands. Our study also confirms some small but significant magnitude dependence errors in the z band for some charge-coupled devices. Various tests show that we have achieved an internal precision of about 5 mmag in the u band and about 2 mmag in the griz bands, which is about five times better than previous results. We also apply the method to the latest version of the catalog (v4.2), and find modest systematic calibration errors of up to ~1% along the R.A. direction and smaller errors along the decl. direction. The results demonstrate the power of the SCR method when combining spectroscopic data and Gaia photometry in breaking the 1% precision barrier of ground-based photometric surveys. Our work paves the way for the recalibration of the whole SDSS photometric survey and has important implications for the calibration of future surveys. Future implementations and improvements of the SCR method under different situations are also discussed.

79 ASTRONOMY AND ASTROPHYSICS↗

Dark Energy Survey Year 6 Results: Photometric Dataset for Cosmology

We describe the photometric dataset assembled from the full 6 yr of observations by the Dark Energy Survey (DES) in support of static-sky cosmology analyses. DES Y6 Gold is a curated dataset derived from DES Data Release 2 (DR2) that incorporates improved measurement, photometric calibration, object classification and value-added information. Y6 Gold comprises nearly 5000 deg$^{2}$ of grizY imaging in the south Galactic cap and includes 669 million objects with a depth of i$_{AB}$ ∼ 23.4 mag at a signal-to-noise ratio ∼ 10 for extended objects and a top-of-the-atmosphere photometric uniformity <2 mmag. Y6 Gold augments DES DR2 with simultaneous fits to multiepoch photometry for more robust galaxy shapes, colors, and photometric redshift estimates. Y6 Gold features improved morphological star–galaxy classification with an efficiency of 98.6% and a contamination of 0.8% for galaxies with 17.5 < i$_{AB}$ < 22.5. Additionally, it includes per-object quality information, and accompanying maps of the footprint coverage, masked regions, imaging depth, survey conditions, and astrophysical foregrounds that are used for cosmology analyses. After quality selections, benchmark samples contain 448 million galaxies and 120 million stars. This publication is complemented by data access and documentation.

79 ASTRONOMY AND ASTROPHYSICS↗

The photometric functions of Phobos and Deimos. III - Surface photometry of Phobos

Mariner 9 television pictures of Phobos are used to study the uniformity of a certain photometric scattering parameter over the surface of the satellite, assuming that the scattering law for Phobos can be represented by the Hapke-Irvine equation. Ways of avoiding topographically irregular areas are discussed, phase functions for Phobos are obtained from scans along the photometric equator, and the photometric behavior of selected areas not along this equator is examined. Searches are conducted for anomalously bright areas and evidence of solid rock on the surface. The results indicate that: (1) at least three large regions on the surface of Phobos are covered by a dark material of complex texture which scatters light according to the Hapke-Irvine law; (2) the average intrinsic phase coefficient of this material is 0.020 mag/deg over the phase-angle range from 20 to 80 deg; (3) even the topographically irregular areas have photometric properties similar to those of the three smooth regions; (4) the surface of Phobos is rougher than that of Deimos on the largest scale; and (5) apparent bright patches are most likely caused by local tilts of the surface.

Noland, M.↗

Determination of planetary photometric functions

The determination and removal of the photometric properties of the atmosphere and surfaces of pale planetary objects are of signal importance in the production of planetary mosaics and are themselves important scientific endeavors. The work done at the JPL Image Processing Laboratory in this area in support of the Voyager project is reviewed. Topics discussed include data-base generation, parameter determination, and photometric function removal. The importance of determining the photometric function for a planet in mosaicking together sets of images is stressed. It is noted that techniques have been developed and software has been written to allow scientists to easily determine the parameters to several photometric functions.

Mosher, J.↗