KiDS+VIKING-450: A new combined optical & near-IR dataset for cosmology and astrophysics

TL;DR

This paper introduces a comprehensive combined optical and near-infrared dataset from KiDS and VIKING surveys, enabling improved cosmological measurements with reduced systematic uncertainties and publicly available data products.

Contribution

The paper presents the creation and validation of a new, large, multi-wavelength dataset that enhances photometric redshift accuracy and stellar mass estimates for cosmology and astrophysics.

Findings

Reduced systematic uncertainties in photo-z compared to optical-only data.

Over 80% of sources have significant detections in all 8 bands.

Photometric stellar mass functions agree with spectroscopic studies.

Abstract

We present the curation and verification of a new combined optical and near infrared dataset for cosmology and astrophysics, derived from the combination of $ugri$-band imaging from the Kilo Degree Survey (KiDS) and $ZY\!J\!H\!K_{\rm s}$-band imaging from the VISTA Kilo degree Infrared Galaxy (VIKING) survey. This dataset is unrivaled in cosmological imaging surveys due to its combination of area ($458$ deg$^2$ before masking), depth ($r\le25$), and wavelength coverage ($ugriZY\!J\!H\!K_{\rm s}$). The combination of survey depth, area, and (most importantly) wavelength coverage allows significant reductions in systematic uncertainties (i.e. reductions of between 10 and 60\% in bias, outlier rate, and scatter) in photometric-to-spectroscopic redshift comparisons, compared to the optical-only case at photo-$z$ above $0.7$. The complementarity between our optical and NIR surveys means that over $80\%$ of our sources, across all photo-$z$, have significant detections (i.e. not upper limits) in our $8$ reddest bands. We derive photometry, photo-$z$, and stellar masses for all sources in the survey, and verify these data products against existing spectroscopic galaxy samples. We demonstrate the fidelity of our higher-level data products by constructing the survey stellar mass functions in 8 volume-complete redshift bins. We find that these photometrically derived mass functions provide excellent agreement with previous mass evolution studies derived using spectroscopic surveys. The primary data products presented in this paper are publicly available at http://kids.strw.leidenuniv.nl/.

Figures

Tables

Table 1. Table 1: Magnitude limits and typical seeing values for each of the KiDS+VIKING photometric bands. Values reproduced from de Jong et al. ( 2017 ); Venemans et al. ( 2015 ) .

Band	${\lambda }_{\mathrm{cen}}$	Exp.time (s)	Mag Limit	PSF FWHM
	(Å)	(s)	$\left(2^{\prime \prime }\mathrm{\hspace{0.17em}5}\sigma \mathrm{AB}\right)$	($\prime \prime$)
$u$	3550	1000	24.2	1.0
$g$	4775	900	25.1	0.9
$r$	6230	1800	25.0	0.7
$i$	7630	1200	23.6	0.8
$Z$	8770	480	22.7	1.0
$Y$	10200	400	22.0	1.0
$J$	12520	400	21.8	0.9
$H$	16450	300	21.1	1.0
$K_s$	21470	480	21.2	0.9

Table 2. Table 2: Measurement statistics for the 13.09 13.09 13.09 million lensing sources that remain after all non-photometry KV450 masks have been applied, per band and as successive bands are added. The columns detail the fraction of sources that have successful GAaP measurements or limits (i.e. where GAaP ran successfully; f good subscript 𝑓 good f_{\rm good} ), and the fraction of sources that returned a significant GAaP flux measurement, and not just an upper limit ( f meas subscript 𝑓 meas f_{\rm meas} ).

Band(s)	$f_{\mathrm{good}}$	$f_{\mathrm{meas}}$
$u$	$0.996$	$0.794$
$g$	$1.000$	$0.990$
$r$	$1.000$	$1.000$
$i$	$1.000$	$0.954$
$Z$	$0.991$	$0.983$
$Y$	$0.990$	$0.965$
$J$	$0.999$	$0.990$
$H$	$0.989$	$0.933$
$Ks$	$0.992$	$0.944$
$ugri$	$0.996$	$0.761$
$ugriZ$	$0.987$	$0.751$
$ugriZY$	$0.977$	$0.732$
$ugriZYJ$	$0.976$	$0.728$
$ugriZYJH$	$0.967$	$0.691$
$ugriZYJH{K}_s$	$0.963$	$0.669$
$griZYJH{K}_s$	$0.967$	$0.820$

Equations

$$I_{\mathrm{new}}=I_{\mathrm{old}}\times {\mathcal{F}}_{\mathrm{r}}$$

$$\begin{array}{rl}{\log }_{10}({\mathcal{F}}_{\mathrm{r}})=-0.4& [Z_{\mathrm{v}}-2.5{\log }_{10}(1/t)\\ & -\tau \left(\sec \chi -1\right)+X_{\mathrm{AB}}-30].\end{array}$$

$$f_{\mathrm{f}}=\frac{{\sum }_{i=1}^n{f}_i{w}_i}{{\sum }_{i=1}^n{w}_i},$$

$${\sigma }_{f_{\mathrm{f}}}={\left[\sum _{i=1}^n{\sigma }_{f_i}^{-2}\right]}^{-\frac{1}{2}}.$$

$${\sigma }_{\Delta i}=\frac{f_{\mathrm{f}}-f_i}{\sqrt{n}{\sigma }_{f_{\mathrm{f}}}},$$

Full text

1 Introduction

Over the last decade observational cosmological estimates have become increasingly restricted by systematic, rather than random, uncertainties (Hildebrandt et al., 2016, 2018; Troxel et al., 2018; Hikage et al., 2018; Planck Collaboration et al., 2018). In particular, estimates made using large photometric samples of galaxies, such as those utilising weak gravitational lensing (e.g. Bacon et al., 2000; Van Waerbeke et al., 2000; Wittman et al., 2000; Rhodes et al., 2001), have moved closer to a regime where increasing sample sizes alone are unlikely to cause a significant improvement in estimate constraints (Becker et al., 2016; Jee et al., 2016; Hildebrandt et al., 2017; Troxel et al., 2018; Hildebrandt et al., 2018). Instead, quantification and reduction of systematic biases is becoming increasingly important, and more frequently is the dominating source of uncertainty in cosmological inference (Mandelbaum, 2018).

One such systematic limitation/bias for many methods of observational cosmological inference (and indeed one that frequently dominates the systematic uncertainty budget; Hildebrandt et al., 2016; Hikage et al., 2018) is also one of the most fundamental: that of estimation of source positions in 3-dimensional space. Specifically, localisation of galaxies along the line-of-sight (i.e. distance) axis is of particular importance. This localisation is primarily achieved through relatively low-precision photometric based methods, referred to as photometric redshift or photo-$z$ (see Hildebrandt et al., 2010, for a summary of photo-$z$ methods).

One method for deriving photo-$z$ estimates involves finding the model galaxy spectrum, from a sample of representative spectrum templates, which best fits the observed galaxy flux in a series of wavelength bands (Ilbert et al., 2006; Benítez, 2000; Brammer et al., 2008; Bolzonella et al., 2000). Such estimates are typically restricted by the quality of the input photometry, the intrinsic redshift distribution of the source galaxy sample, and the degeneracy between various galaxy spectrum models as a function of galaxy redshift (Ilbert et al., 2009; Laigle et al., 2019). In each of these cases, however, additional information can lead to significant benefits in the photo-$z$ estimation process.

For cosmological inference, a weak lensing survey needs to provide reliable galaxy shapes (see, e.g., Massey et al., 2007) and redshift estimates for a statistically representative sample of galaxies over cosmologically significant redshifts (see, e.g., Hildebrandt et al., 2018; Troxel et al., 2018; Hikage et al., 2018). For this purpose there are, therefore, three main properties that determine any weak lensing survey’s cosmological sensitivity: survey area, survey depth, and wavelength coverage. These properties all contribute to both the statistical and systematic uncertainty on cosmological inference. The first two statistics, area and depth, typically govern the raw number of sources (in a given redshift interval) that can be used for inference; a primary contributor to the statistical uncertainty on cosmic shear cosmological inference, and so a key consideration in weak lensing survey design (de Jong et al., 2015; Abbott et al., 2016; Aihara et al., 2018). Similarly, and discussed in more depth below, the wavelength information dictates the limiting photo-$z$ accessible for tomographic binning (Hildebrandt et al., 2018), which is a driving factor in the final signal-to-noise of a cosmic shear estimate. As such wavelength coverage also contributes non-negligibly to the statistical uncertainty of cosmic shear cosmological estimates. On the systematic effects side, area and depth aid in the constraint of many systematics parameters directly, not the least of which is the intrinsic alignment signal (Joachimi et al., 2015). Deeper data allows more galaxy satellites to be observed, and a better constraint on the galaxy-galaxy intrinsic alignment signal to be made. The wavelength baseline is also a primary driving factor in determining the systematic uncertainty on any cosmological inference, primarily because it dictates the signal-to-noise of the shear signal of individual cosmic shear sources.

The reason wavelength information is of significant importance in cosmological inference is many-fold. However we focus on two main reasons, for demonstration: namely reduction of photo-$z$ bias and influence over the redshift baseline. We discuss both of these below.

The importance of wavelength information in the reduction of photo-$z$ bias is driven mostly by the degeneracy between galaxy spectrum models. Even with perfect input photometry, there exist degeneracies between galaxy spectrum models at different redshifts over finite wavelength intervals (see Figure 1 of Buchs et al., 2019, for a nice demonstration). These degeneracies can lead to considerable biases in the source photo-$z$ distribution, when sources are systematically assigned to incorrect parts of redshift-space. Moreover, these effects are increasingly problematic as source photometry becomes noisier and the wavelength baseline becomes shorter. Naturally, the only way to break such degeneracies is by utilising photometry for these sources that extends beyond the wavelength range wherein the degeneracy exists. As such, longer wavelength baselines are fundamental to the breaking of model degeneracies, and therefore to reducing the systematic incorrect assignment of photo-$z$. Such incorrect assignment can lead to considerable bias in estimated redshift distributions (Hildebrandt et al., 2018) which limits cosmological inference, over any given source redshift baseline.

Furthermore, wavelength information is the primary factor which determines the useful redshift baseline over which cosmological inference can be performed. In particular, photometry that extends redward of the optical bands is essential for the accurate estimation of photo-$z$ beyond a redshift of $z\sim 1$ (for typical ground-based photometric surveys). This intermediate- to high-redshift information is of particular importance to weak lensing cosmological inference, as higher-redshift sources carry considerably more signal-to-noise than their lower-redshift counterparts. This increased signal is critical in the quantification of systematic bias as it allows them to be explored with reduced stacking of sources (e.g. with finer bins containing more homogeneous samples of galaxies), which can alleviate additional biases.

To date, the largest joint optical and near infrared (NIR) dataset for cosmology was a combined Dark Energy Survey (DES) + VISTA Hemisphere Survey (VHS) analysis of the DES Science Verification region, covering ${\sim}150$ deg2 and spanning the ${griZY\!J\!H\!K_{\rm s}}$ bands (Banerji et al., 2015). In this paper we present the integration of two European Southern Observatory (ESO) public surveys; the VISTA Kilo degree INfrared Galaxy (VIKING; Edge et al., 2013; Venemans et al., 2015) survey, probing the NIR wavelengths ($8000-24000$Å), and the Kilo Degree Survey (KiDS; Kuijken et al., 2015; de Jong et al., 2015), probing optical wavelengths ($3000-9000$Å). These combined data represent a significant step forward from the previous state-of-the-art, primarily due to the increase in combined survey area and optimal matching between the two surveys depth (see Sect. 2).

This extension of the wavelength baseline brings with it considerable benefits, particularly for cosmic shear analyses. Hildebrandt et al. (2017) presented cosmological inference from cosmic shear using 450 square degrees of KiDS imaging (referred to as KiDS-450), measuring the matter clustering parameter ($\sigma_{8}$) and matter density parameter ($\Omega_{\rm m}$), which are typically parameterised jointly as $S_{8}=\sigma_{8}\sqrt{\Omega_{\rm m}/0.3}$, to a relative uncertainty of ${\sim}5\%$; an error whose budget was limited essentially equally by systematic and random uncertainties. As a result, we expect that the final KiDS dataset, spanning $1350$ deg2, will in fact be systematics limited in its cosmological estimates (as random uncertainties should downscale by a factor of roughly $\sqrt{3}$). Moreover, constraint and reduction of systematic effects will become of increasing importance in the next few years, and indeed into the next decade with the initiation of large survey programs such as Euclid (see, e.g., Amendola et al., 2018). Using the combined dataset presented here enables us to make considerable progress regarding the challenge of reducing systematics, and that in doing so enables us to perform an updated cosmic shear analysis which better constrains systematic uncertainties and enables the use of higher-redshift sources (Hildebrandt et al., 2018).

Importantly, this dataset is not only useful for cosmological studies. The additional information provided by the near-IR allows better constraint of fundamental galaxy parameters such as stellar mass and star formation rates, which enable the construction of useful samples for galaxy evolution and astrophysics studies. For example, recent use of near-IR data in preselection of ultra-compact massive galaxy candidates (Tortora et al., 2018) has allowed the spectroscopic confirmation of the largest sample of UCMGs to date.

As such, this work focusses on the description and verification of the joint KiDS+VIKING photometric dataset, and on the derivation of higher-level data products which are of interest both for weak-lensing cosmological analyses and non-cosmological science use-cases. The KiDS optical and VIKING NIR data and their reduction are described in Sect. 2. The multi-band photometry and estimation of photo-$z$ are covered in Sect. 3. Model fitting to the broadband galaxy spectral energy distributions is given in Sect. 4, as is the exploration of stellar mass estimates from these fits. We compare the resulting stellar mass function for our dataset to previous works in Sect. 5. The paper is summarised in Sect. 6. The primary data products described in this paper are made publicly available at http://kids.strw.leidenuniv.nl/.

2 Dataset and reduction

In this section we describe the KiDS optical (Sect. 2.1) and VIKING NIR (Sect. 2.2) imaging that is used in this study. KiDS and VIKING are partner surveys that will both observe two contiguous patches of sky in the Galactic North and South, covering a combined area of over 1350 square degrees (Arnaboldi et al., 2007; de Jong et al., 2015, 2017). Observations for KiDS are ongoing, and so joint analysis of KiDS+VIKING is currently limited to the footprint of the third KiDS Data Release (de Jong et al., 2017).

The footprint of the post-masking KiDS-450 dataset presented in Hildebrandt et al. (2017) is shown in Fig. 1 both on-sky and split into each of the KiDS ‘patches’ (where each patch contains one of the $5$ $\sim$contiguous portions of the KiDS-450 footprint). These individual patches divide the KiDS-450 survey area into five sections of (roughly) contiguous data on-sky, centering primarily on fields observed by the Galaxy And Mass Assembly (GAMA, Driver et al., 2011) redshift survey. The geometry of each patch can be seen in the individual panels of Fig. 1, and are named by the GAMA field on which they are focused. The exception is the GS patch, which has no corresponding GAMA field; we nonetheless maintain the naming convention for convenience. Note, though, that future KiDS observations will close the gaps both within and between the patches, and lead to the creation of a contiguous ${\sim}10\leavevmode\nobreak\ {\rm deg}{\times}75\leavevmode\nobreak\ {\rm deg}$ stripe in both the Galactic North and South. Observations of these contiguous stripes in VIKING have already been completed.

The full (unmasked) KiDS-450 dataset consists of ${\sim}49$ million non-unique Galactic and extragalactic sources distributed over $454$ overlapping ${\sim}1$ deg2 pointings on sky (see Sect. 2.1). This reduces to ${\sim}33.9$ million unique mostly-extragalactic sources after applying masking of stars and removing duplicated data, distributed over ${\sim}360$ deg2. These unique post-masking sources are shown in Fig. 1, and so the masking around bright stars, for example, can be seen as small circular gaps within the patches. Each source is coloured by its observational coverage statistics: those with full photometric KiDS+VIKING observational coverage are shown in green, those with full KiDS observational coverage but only partial VIKING observational coverage are shown in blue, and those with only KiDS observational coverage are shown in orange. We define the combined KiDS+VIKING-450 sample (hereafter KV450) as those KiDS-450 sources which have overlapping VIKING imaging (i.e. the green sources in Fig. 1). Masking the regions with missing near-IR coverage (i.e. the orange and blue data in Fig. 1), the full KV450 footprint consists of $447$ overlapping pointings111There are a number of pointings, particularly at the survey edges, which have only slight overlap between KiDS and VIKING. This causes the overall loss in area (${\sim}19$ deg2) to be somewhat larger than the loss of only $7$ full pointings would suggest., covering ${\sim}341$ deg2, and consists of ${\sim}31.9$ million unique mostly-extragalactic sources.

2.1 KiDS-450 optical data

The data reduction for the KiDS-450 ${ugri}$-band survey data is described in detail in Hildebrandt et al. (2017) and de Jong et al. (2017), which we briefly summarise here. As stated previously, the full optical dataset consists of $454$ distinct ${\sim}1\leavevmode\nobreak\ {\rm deg}^{2}$ pointings of the OmegaCAM, which is mounted at the Cassegrain focus of ESO’s VLT Survey Telescope (VST) on Cerro Paranal, Chile. Images in the ${ugri}$-bands are available for all of these pointings, with exposure times of $15-30$ minutes and $5\sigma$ limiting magnitudes of $23.8-25.1$; precise values are given in de Jong et al. (2017) and are reproduced here in Table 1. The filter transmission curves for these four optical bands are shown in Fig. 2, along with the atmospheric transmission typical to observations at Paranal.

The optical data for KV450 are reduced using the same reduction pipelines as in KiDS-450. Specifically, the AstroWISE (Valentijn et al., 2007) pipeline is used for reducing the ${ugri}$-band images and measuring multi-band photometry for all sources. Independently, the THELI (Erben et al., 2005; Schirmer, 2013) pipeline performs an additional reduction of the $r$-band data, which is used for cross-validating the AstroWISE reduction and for performing shape measurements for weak lensing analyses.

The only difference between the KiDS-450 and KV450 optical datasets is that the KV450 optical reduction incorporates an updated photometric calibration. KiDS-450 invoked only relative calibration across the ${ugri}$-bands with stellar-locus-regression (SLR, High et al., 2009). The absolute calibration of these data was reliant on nightly standard star observations and the overlap between u- and r-band tiles to homogenize the photometry. Since the publication of Hildebrandt et al. (2017), the first data release from the European Space Agency’s Gaia mission has been made available (Gaia Collaboration et al., 2016). Gaia offers a sufficiently homogeneous, well-calibrated anchor that can be used to greatly improve this absolute calibration. The calibration procedure is described in de Jong et al. (2017) and all optical data used here are absolutely calibrated in this way.

2.2 VIKING infrared data

VIKING is an imaging survey conducted with the Visible and InfraRed CAMera (VIRCAM) on ESO’s 4m VISTA telescope. The KiDS and VIKING surveys were designed together, with the specific purpose of providing well-matched optical and NIR data for ${\sim}1350$ square degrees of sky in the Galactic North and South. As such, the surveys share an almost identical footprint on-sky, with minor differences being introduced due to differences in the camera field of view and observation strategy. VIKING surveys these fields in five NIR bands (${ZY\!J\!H\!K_{\rm s}}$), whose filter transmission curves are shown in Fig. 2, and total exposure times in each band are chosen such that the depths of KiDS and VIKING are complementary.

A detailed description of the VIKING survey design and observation strategy can be found in Edge et al. (2013) and Venemans et al. (2015). Briefly, VIRCAM consists of 16 individual HgCdTe detectors, each with a $0.2{\times}0.2$ square degree angular size, but which jointly span a ${\sim}1.2$ square degree field of view, thus leaving considerable gaps between each detector. Observations made by VIRCAM for VIKING therefore implement a complex dither pattern which is able to fill in the detector gaps while also performing jittered observations to enable reliable estimation of complex NIR backgrounds and sampling of data across detector defects. The observation strategy thus involves taking multiple exposures with small (i.e. much less than detector width) jitter steps, taken in quick succession, which are then stacked together to create a ‘paw-print’. The stacked paw-print still has large gaps between the 16 detectors, and so a dither pattern of 6 stacked paw-prints is required in order to create a contiguous ${\sim}1.5$ square degree image, called a tile. The reduction of the data, and the production of these individual data products (reduced exposures, stacked paw-prints, and completed tiles), is carried out by the Cambridge Astronomy Survey Unit (CASU, González-Fernández et al., 2018; Lewis et al., 2010). These reduced data are then transferred to the Edinburgh Royal Observatory Wide Field Astronomy Unit VISTA Science Archive (WFAU VSA, Irwin et al., 2004; Hambly et al., 2008; Cross et al., 2012) where they are benchmarked and stored.

Through the WFAU database, we are able to retrieve any of the 3 levels of data-product described above: exposures, paw-prints, and/or tiles. We opt to work with individual paw-print level data. This is primarily because the tile level data are frequently made up of paw-prints with a range of different point-spread functions (PSFs), and this can lead to complications later in our analysis (specifically regarding flux estimation; see Sect. 3). Therefore, we begin our combination of KiDS and VIKING by first downloading all the available stacked paw-prints from the WFAU. We then perform a recalibration of the individual paw-prints following the methodology of Driver et al. (2016) to correct the images for atmospheric extinction ($\tau$) given the observation airmass ($\sec\chi$), remove the exposure-time ($t$, in seconds) from the image units, and convert the images from various Vega zero-points ($Z_{\rm v}$) to a standard AB zero-point of $30$ (using the documented Vega to AB correction factors, $X_{\rm AB}$; González-Fernández et al., 2018) which roughly translates to an image gain of ${\rm ADU/{e^{-}}}=1$. The recalibration factor used is multiplicative, applied to all pixels in each detector image $I$:

[TABLE]

and is calculated as:

[TABLE]

This preprocessing of each VISTA detector also involves performing an additional background subtraction, which is done using the SWarp software (Bertin, 2010) with a $256{\times}256$ pixel mesh size and $3{\times}3$ mesh filter for the bicubic spline. This allows the removal of small-scale variations in the NIR background with minimal impact on the source fluxes (Driver et al., 2016). Unlike GAMA, however, we do not recombine the individual paw-prints into tiles or large mosaics; we choose instead to work exclusively with the individual recalibrated detectors throughout our analysis.

After this processing, we perform a number of quality control tests to ensure that the imaging is sufficiently high quality for our flux analysis. In particular, we check distributions of background, seeing, recalibration factor (Equation 2), and number counts for anomalies. After these checks, we determined that a straight cut on the recalibration factor was sufficient to exclude outlier detectors, and thus implement the same rejection of detectors as in Driver et al. (2016); namely accepting only detectors with $\mathcal{F_{\rm r}}\leq 5.0$.

After this processing and quality control, we transfer the accepted imaging over to our flux measurement pipeline. Our final sample consists of $301,824$ individual detectors across the 5 VIKING filters, drawn from the WFAU proprietary database v21.3, which are spread throughout the KiDS-450 footprint. This database, however, does not yet contain the full VIKING dataset, as reduction and ingestion of the final VIKING data (taken as recently as February 2018) is ongoing. As such, the final overlap between the KiDS footprint and VIKING is likely to continue to grow with future KiDS+VIKING data releases.

3 Photometry and photometric redshifts

3.1 9-band photometry

Multi-band photometry is extracted from the combined KiDS+VIKING data using the Gaussian Aperture and PSF (GAaP; Kuijken, 2008; Kuijken et al., 2015) algorithm. The algorithm generates PSF-corrected Gaussian-aperture photometry that is particularly well suited for colour-measurements which are used for estimating photometric redshift. The GAaP code differs to other standard photometric codes in that it does not require images to be pixel nor PSF matched in order to extract matched aperture fluxes (such as Source Extractor; Bertin & Arnouts, 1996), and it limits flux estimation to the typically brighter/redder interior parts of galaxies, unlike codes designed for total flux photometry (such as lambdar; Wright et al., 2016). GAaP also utilises purely gaussian photometric apertures and PSFs (hence the name), and therefore performs the required image gaussianisation prior to flux measurement.

The algorithm requires input source positions and aperture parameters, which we define by running Source Extractor (Bertin & Arnouts, 1996) over our theli $r$-band imaging in a so-called hot-mode. This refers specifically to the use of a low deblend threshold, which allows better deblending of small sources. This choice can have the adverse effect, however, of shredding large (often flocculant) galaxies. We choose this mode of extraction as we are primarily interested in sources in the redshift range $0.1\lesssim z\lesssim 1.2$, which are typically small and have smooth surface-brightness profiles. Once we have our extracted aperture parameters, the algorithm then performs a gaussianisation of each measurement image. This removes systematic variation of the PSF over the image and allows for a more consistent estimate of source flux across the detector-plane. This gaussianisation is performed by characterising the PSF over the input image using shapelets (Refregier, 2003), and then fitting a smoothly varying spline to the shapelet distribution. For this reason, it is optimal to provide input images that do not have discrete changes in the shape of the PSF which cannot be captured by this smoothly varying distribution. The smooth function is then used to generate a kernel that, when convolved with the input image, normalises the PSF over the entire input image to a single Gaussian shape with arbitrary standard deviation.

Due to the requirement that the input imaging not have discrete changes in the PSF parameters, we require that the GAaP algorithm be run independently on subsets of the data that were taken roughly co-temporally. In the optical this is trivial; the $1.2$ square degree stacks of jittered observations, called ‘pointings’, are always comprised of individual exposures with small offsets that were taken essentially cotemporally, due to the design of the detector array and survey observation strategy. This, combined with the stability of the PSF pattern across the field of view that is inherent to observations made at the Cassegrain focus of a Ritchey-Chrétien telescope, means that the stacked pointings are optimised for use in GAaP. Using the KiDS pointings for optical flux measurements with GAaP results in at most four flux estimates for any one KiDS source in the limited corner-overlap regions between adjacent pointings, or two flux estimates at the pointing edges. However, as in KiDS-450, we mask these overlap regions such that the final dataset contains only 1 measurement of all sources within the footprint, rather than performing a combination of these individual flux estimates. As such, our final flux and uncertainty estimates in the optical are simply those output directly by GAaP.

Conversely, the VISTA tiles are particularly sub-optimal for use in GAaP, due to the large dithers between successive paw-prints which are necessary to fill a contiguous area on-sky. Stacking such exposures with large dithering offsets can lead to significant discrete changes in the PSF of the stacked image, and this problem is exacerbated by the strong PSF variations over the focal plane inherent to observations made with such a fast telescope. Therefore, in order to streamline the data handling and avoid non-contiguous PSF patterns we decided to extract the VISTA NIR photometry from single VISTA detector images of individual paw-prints, as recommended in González-Fernández et al. (2018). In practice, the paw-print level data are provided as individual detector stacks, rather than as a mosaic of the telescope footprint.

Accordingly, we gaussianise the PSF of each paw-print detector in the VIKING survey separately, and run GAaP on these units. As there is no one-to-one mapping between KiDS pointings and VIKING paw-prints, we are required to associate individual VIKING detectors with overlapping KiDS pointings on-the-fly. Furthermore, the VISTA dither pattern results in anywhere between 1 and 6 independent observations of a given source within the tile. This typically results in multiple flux measurements per source and band as most sky positions within the tile are covered by at least 2 paw-prints in the $ZYHK_{\rm s}$-bands and at least 4 paw-prints in the $J$-band. Therefore, for each source we calculate a final flux estimate, $f_{\rm f}$, that is the weighted average of the $n$ individual flux measurements, $f_{\rm i}$:

[TABLE]

where the weight for each source is the individual GAaP measurement inverse variance $w_{i}=\sigma_{f_{i}}^{-2}$. The final flux uncertainty is the uncertainty on this weighted mean flux:

[TABLE]

To test whether the GAaP flux uncertainties are suitable for use in estimating the final flux this way, we examine the distribution of sigma deviations between the final (weighted mean) flux and the individual estimates:

[TABLE]

where $n$ is the number of flux measurements that went into the computation of $f_{\rm f}$ and $\sigma_{f_{\rm f}}$. In the limit where the individual flux uncertainties $\sigma_{f_{i}}$ are perfectly representative of the scatter between the individual measurements, the distribution of $\sigma_{\Delta{\rm i}}$ values should be a Gaussian with 0-mean and a standard deviation of 1. When the flux uncertainties are not representative of the scatter in the individual measurements, the distribution may deviate in mean, standard deviation, or both. In particular, systematic bias in the flux uncertainties as a function of flux will shift the mean of the distribution away from 0 (and/or give the distribution an obvious skewness), while over- or under-estimation of the uncertainties as a whole will cause the distribution standard deviation to decrease or increase, respectively. Figure 3 shows the distributions of $\sigma_{\Delta{i}}$ for each of the five VIKING bands. The figure shows that our flux uncertainties in the $ZYHK_{\rm s}$-bands are appropriate and (for the vast majority of estimates) Gaussian; roughly 20% of our individual flux estimates have a scatter that is not well described by the simple final Gaussian uncertainty on our flux estimate, however this is not surprising given that the individual GAaP flux estimates are purely shot noise; they do not capture the full uncertainty in cases where there is considerable zero-point uncertainty, sky background, correlated noise, or other systematic effects which contribute to the flux uncertainty. Figure 3 also demonstrates that our flux uncertainties tend to be under-estimated in the $J$-band by roughly $30\%$. Encouragingly, however, the distributions show no sign of systematic bias in the flux uncertainties, which would be indicated by a significant skewness of these distributions.

To verify the calibration of our imaging and flux estimates, we compare our estimates for a sample of KV450 stars to those measured by SDSS and/or 2MASS. Stars are particularly useful for this purpose as GAaP yields not only reliable colours but also total magnitudes for these sources (Kuijken et al., 2015), and therefore we need not be concerned with aperture effects in the flux comparisons. As the CASU pre-reduction assigns a photometric zeropoint to each VISTA paw-print based on a calibration with 2MASS, residuals in our multi-band photometry with 2MASS (particularly in the $JHK_{\rm s}$-bands) would indicate problems with our pipeline. Similar offsets with respect to SDSS in the $Z$-band would also be cause for concern. Hence these comparisons are used as quality control tests, typically on the level of a KiDS pointing. The distributions of the pointing-by-pointing offsets between our GAaP photometry and SDSS/2MASS are shown in Fig. 4, per band. The figure shows the PDFs of these residuals, as well as Gaussian fits to the distributions. In the $Z$-band, we have two lines: the solid line is a direct comparison to SDSS, while the dashed line is an extrapolation of 2MASS $J-H$ colours to the $Z$-band. A similar extrapolation is shown in the $Y$-band. Both of these extrapolations have significant colour-corrections, and so should be taken somewhat cautiously. Encouragingly, however, in all the cases where we have fluxes that can be directly compared to one-another (i.e. in all but the $Y$-band), the direct comparison residuals are centred precisely on 0. Furthermore, in all cases the fluctuations between pointings are all within $\left|\Delta m\right|<0.02$.

As a final test of the fidelity of our fluxes, we compare colours of KV450 stars with the same measured in 2MASS, to demonstrate that our observed colours are consistent with, but less noisy than, those from 2MASS. The distributions of KV450 and 2MASS $J-H$ and $H-K_{\rm s}$ colours can be seen in Fig. 5. As expected, the KV450 colours show considerably less scatter, suggesting that they are a better representation of the underlying, intrinsic stellar colour distribution (Wright et al., 2016), and are therefore superior to the colours of 2MASS.

Now confident that our fluxes are appropriate, we can further verify the appropriateness of our sample definition and effective-area calculations by comparing our measured galaxy number counts (in our extraction band, $r$) with previous works from the literature. Figure 6 shows the $r$-band number counts for the KV450 dataset compared to the compendium of survey number counts presented in Driver et al. (2016). We show the KV450 dataset both with and without the removal of stellar sources described in Sect. 4.2. Furthermore, we show the number counts for the sample of lensing sources used for cosmological inference (Hildebrandt et al., 2018). The lensing subset is constructed of sources which are suitable for shape measurement as described in detail in Hildebrandt et al. (2017). This lensing sample consists of $13.1$ million sources, all of which fall within the $r$-band magnitude range $20\lesssim m_{r}\lesssim 25$, are unblended, and are resolved.

We see that the all-galaxy sample is lacking in number counts at the brightest magnitudes; we attribute this to our hot-mode source extraction biasing against the extraction of the largest, brightest galaxies, as has been noted previously in earlier KiDS datasets (see, e.g., Tortora et al., 2018). Otherwise, the observed counts of both the all-galaxy- and lensing-only-samples are in excellent agreement with the literature compendium of $r$-band counts from Driver et al. (2016), suggesting that our sample definitions and area calculations are appropriate.

Unlike KiDS-450, we also require the final lensing sample to have full 9-band photometric coverage; i.e. successful photometric measurements are required for every source in all 9 bands. Table 2 provides the photometric measurement statistics for the lensing sample in KV450, as a function of individual band and for combinations of bands. The statistics shown are the fraction of sources with successful GAaP measurements ($f_{\rm good}$) in all 9 bands, for all sources that fall both within the area of mutual KiDS+VIKING coverage.

The table demonstrates that GAaP returns a successful flux measurement for greater than 98% of all lensing sources in all bands. However, as the failures are different in each band, the full sample ends up with successful estimates in all 9-bands for greater than 96% of would-be lensing sources. Therefore, the requirement of a successful GAaP measurement acts to trim our final lensing sample down by less than $4\%$. Furthermore, the lensing sample has significant detections in all 9-bands for over 66% of the full sample, and for over 69% of sources that have no GAaP failures (i.e. where there is data in all 9-bands). We note, however, that the $u$-band has the lowest number of significant detections within the dataset, by a considerable margin, but that this is primarily a reflection of physics rather than of the imaging depth. The rapidly declining nature of galaxy SEDs in this wavelength range at all redshifts, conspiring with the lower sensitivity of the band compared to, say, the $g$-band, means that the $u$-band experiences significantly more non-detected sources than any other band over our redshift window. Explained differently: the $i$-band, for example, sees no such dearth in detections despite being shallower than the $u$-band, courtesy of its probing a typically more luminous part of the galaxy SED (at $z\lesssim 1$ this is primarily because of the flux increase associated with the $4000\AA$ break). Removing the $u$-band from our considerations of detection statistics, we find that we have significant detections in the ${griZY\!J\!H\!K_{\rm s}}$-bands for $82\%$ of lensing sources in the dataset. This is a vindication of the combined KiDS+VIKING survey design, whereby limiting magnitudes were designed specifically with the goal of sampling the 9-band SEDs of the $r$-band selected KiDS sample.

For completeness, we investigate the cause of the GAaP failures in our dataset. These typically occur when either there are data missing, or when the algorithm is unable to compute the measurement aperture given the image PSF. The latter can occur when the PSF full-widths at half-maximum (FWHM) of the measurement image is considerably larger than the input (detection) aperture (Kuijken et al., 2015). As such, the input aperture size can be a source of systematic bias in the GAaP flux measurement procedure, as smaller input apertures are more likely to hit the aperture-PSF limit in one of our non-detection bands. We conclude, however, that this is unlikely to introduce significant biases into our subsequent analyses as less than 1.2% of sources per-band are affected by the GAaP measurement failure. Nonetheless, in future releases of KiDS+VIKING data, a recursive flux measurement method will be invoked, whereby sources that fail in any band due to this effect are subsequently re-measured with an artificially expanded GAaP input aperture.

After applying the requirement of successful (i.e. $f_{\rm good}$) 9-band photometric estimation, we finish with a final lensing sample of ${\sim}12.6$ million sources, which are drawn from an effective area of $341.3$ deg2 (see Sect. 2). This is a slight reduction in the effective area from KiDS-450 ($360.3$ deg2), however this area will recover somewhat in future KiDS+VIKING releases, as the final (full) VIKING area is processed and released by CASU (see Sect. 2.2).

3.2 Photometric redshifts

Photometric redshifts are estimated from the 9-band photometry using the public Bayesian Photometric Redshift (bpz; Benítez, 2000) code. We use the re-calibrated template set of Capak (2004) in combination with the Bayesian redshift prior from Raichoor et al. (2014); hereafter R14. We utilise the maximum amount of photometric information per source, providing BPZ with both flux estimates and limits (where available). Finally, input fluxes are extinction corrected before use within the BPZ code, using Schlegel et al. (1998) dust maps and per-band absorption coefficients.

We test the accuracy of our KiDS+VIKING photo-$z$ estimates using a large sample of spectroscopic redshifts collected from a number of different surveys:

• •

zCOSMOS (Lilly et al., 2009);

• •

DEEP2 Redshift Survey (Newman et al., 2013);

• •

VIMOS VLT Deep Survey (Le Fèvre et al., 2013);

• •

GAMA-G15Deep (Kafle et al., 2018);

• •

ESO-GOODS (Popesso et al., 2009; Balestra et al., 2010; Vanzella et al., 2008).

This combined spectroscopic calibration sample, matched to KV450, includes $>33,000$ sources extending over a 95% $r$-band magnitude quantile range of $r\in\left[19.76,24.75\right]$. Within the sample, $96\%$ of sources have full 9-band photometric information returned by GAaP, and $77\%$ have significant detections in all 9-bands. This sample is therefore a reasonable match to the full KV450 dataset, which extends slightly deeper $(r\in\left[20.82,25.18\right])$ and has $96\%$ and $67\%$ coverage and detection fractions, respectively (see Table 2). Detailed information on the collation of this spectroscopic calibration sample can be found in Hildebrandt et al. (2018).

We note here that, importantly, our testing and quality verification of the photo-$z$ extend only to the maximum likelihood point-estimate values returned from the redshift fitting code: the $\rm Z\_B$ values. This is because for the analyses performed with the photo-$z$ within KiDS, only the point-estimates are ever used; the full photo-$z$ PDFs are never considered. Therefore, we note here for clarity that the statistics presented here all extend to the Z_B values only, and no quality testing of the full photo-$z$ PDFs is presented.

Figure 7 shows a comparison of our photo-$z$ with the spectroscopic calibration sample. The figure shows the standard photo-$z$ vs. spec-$z$ distributions for 3 separate photo-$z$ realisations, as well as annotated statistics for each distribution as a function of photo-$z$. These statistics are calculated using the distribution of $(z_{\rm B}-z_{\rm spec})/(1+z_{\rm spec})\equiv\Delta z/(1+z)$ values, and are:

• •

$\sigma_{\rm m}$: the normalised median-absolute-deviation of $\Delta z/(1+z)$;

• •

$\eta_{3}$: the fraction of sources with $|\Delta z/(1+z)|>3\sigma_{\rm m}$; and

• •

$\zeta_{0.15}$: the fraction of sources with $|\Delta z/(1+z)|>0.15$.

The three photo-$z$ realisations include the initial KiDS-450 4-band photo-$z$ as presented in Hildebrandt et al. (2017), an updated version of the 4-band photo-$z$ using the R14 prior, and KiDS+VIKING 9-band photo-$z$ (also with the R14 prior). Comparing the two 4-band photo-$z$ setups, we see that the R14 prior is effective in suppressing outliers in the low photo-$z$ portion of the distribution by over $30\%$, but shows worse performance at the highest redshifts, where the outlier rate and scatter increase by factors of $1.14$ and $1.13$ respectively. The 9-band photo-$z$, however, shows significant improvement over both 4-band setups. In particular, the inclusion of the NIR data allows us to constrain photo-$z$ in the $z_{\rm B}>0.9$ range ($\sigma_{\rm m}=0.096$) to almost the same level of precision as for the $z_{\rm B}<0.9$ sample ($\sigma_{\rm m}=0.061$), an extremely powerful addition to the dataset, particularly for studies of cosmic-shear where these data carry a very strong signal. We note that the value of $\eta_{3}$ increases slightly for the high-$z$ portion of the 9-band dataset, however this is primarily because the value of $\sigma_{\rm m}$ here is reduced by nearly a factor of two; the higher $\sigma_{\rm m}$ in the 4-band cases conceals the non-gaussianity of the distributions, artificially reducing the value of $\eta_{3}$ there.

We can further motivate the importance of having NIR data for computation of photo-$z$ by exploring how the statistics which describe the photo-$z$ vs spec-$z$ distribution vary under the addition of NIR data, as a function of spec-$z$. We note though, that these statistics as a function of spec-$z$ cannot be used for the quantification of photo-$z$ performance for sources selected in discrete bins of photometric redshift (such as tomographic cosmic shear bins). Rather, these can be used exclusively to demonstrate the influence the additional wavelength information has on the data as a function of true redshift.

Figure 8 shows the change in our 3 parameters of interest as a function of $z_{\rm spec}$, for changes in the prior (for the 4-band KiDS-450 data in grey) and under addition of NIR data (using only the R14 prior in colours). The 3 parameters in the figure are as follows: $\sigma_{\rm m}$, the median bias in $\Delta z/(1+z)$ ($\mu_{\Delta z}$), and $\zeta_{0.15}$. Each parameter is shown using a running median in $20$ equal-N bins of $z_{\rm spec}$. The equivalent figure with bins constructed as a function of $z_{\rm B}$ and $r$-magnitude are given in Figures 9 and 10.

The statistics as a function of $z_{\rm spec}$ demonstrate that it is the combination of all 9-bands which performs the best across both the full gambit of statistics and the redshift baseline. The addition of the $Z$-band causes a clear improvement, in all statistics, over the $4$-band case when we move beyond $z_{\rm spec}=0.9$. This is because at $z_{\rm spec}=0.9$ 4000Å-break flux enters the i-band, and in the $4$-band case is therefore poorly sampled and becomes sensitive to noise fluctuations. With the addition of the $Z$-band, however, sampling of this flux is more robust and the statistics unilaterally improve. There are further improvements with the addition of subsequent bands: the Y-band causes a large reduction in scatter at z¿1, because the same post-4000Å-break flux is now sampled by 2 or more bands (further decreasing the influence of noise). This benefit then saturates (subsequent bands do not improve the high $z_{\rm spec}$ scatter), however the story nonetheless continues. The $J$- and $H$-bands are primarily responsible for a reduction of outliers at $0.2<z_{\rm spec}<0.4$, where a model degeneracy (which is considerably worse after the inclusion of the $Z$-band) populates a cloud which can be seen in the photoz-specz distribution (Figure 7). Finally the $K_{\rm s}$-band helps bring the low-$z_{\rm spec}$ scatter down further, and also produces the lowest overall high-$z_{\rm spec}$ outlier rate. Indeed, the outlier rate at $z>0.9$ reduces continuously with the number of bands added. For these reasons, we conclude that the full complement of the 9-band data is what is required for the best performance, especially at $z_{\rm spec}>0.9$.

3.2.1 Binned by $z_{\rm B}$

Again, we note that these trends shown above and in Figure 8 are not directly transferable to a sample defined as a function of photo-$z$. We show the influence of the individual bands on photometrically defined samples in Figure 9.

Looking at the effect of the updated prior on the 4-band photo-$z$ statistics, we see that the new prior has the effect of greatly reducing scatter at low $z_{\rm B}$, while also reducing bias across essentially all $z_{\rm B}$. There is also a slight increase in the outlier rate with the new prior at intermediate and high $z_{\rm B}$, but this is minor compared to the significant decrease at $z_{\rm B}<0.4$.

When combining the NIR data (starting with the $Z$-band) with the 4-band photometry, we see an immediate improvement in the distribution scatter and outlier rate at high $z_{\rm B}>0.7$. In this range, when incorporating all NIR bands, we see decreases in scatter of between $30$ and $60$ per cent, over the 4-band R14-prior case. Of particular note is the effect of adding the NIR-bands to the outlier rate at $z_{\rm B}>0.7$. Here the added data reduce the observed outlier rate by a factor of ${\sim}2$. Overall, the distributions demonstrate that NIR data as a whole are extremely useful in constraining photo-$z$ for sources in the redshift range $0.7<z_{\rm B}<0.9$, and are invaluable for the estimation of photo-$z$ at $z_{\rm B}>0.9$.

3.2.2 Binned by $r$-magnitude

The introduction of the NIR data actually creates an increase in the observed scatter and outlier rate for sources at the brightest magnitudes. However beyond $r=22$, both the scatter and outlier rate reduce to levels superior to the $4$-band data. With the addition of subsequent NIR bands (i.e. ${Y\!J\!H\!K_{\rm s}}$), we see a essentially continual improvement in all statistics over the whole magnitude range. Otherwise, the distributions show the expected behaviour of photo-$z$ accuracy as a function of noise; the fainter (and so noisier) data exhibit higher scatter in their photo-$z$ and similarly higher outlier rates. We note, though, that this definition of outlier rate becomes somewhat nonsensical beyond $r\sim 25$, where the scatter of the distribution reaches $\sim 0.15$; i.e. the outlier criterion.

Of particular interest is the reduction in bias that is seen with the introduction of the $J$-band at $r\gtrsim 23$. The sources here show the largest bias in the $4$-band case, and this bias is only partially reduced in the $Z$ and $Y$ band cases. However the introduction of the $J$-band data causes the bias to reduce somewhat. The addition of the $H$- and $K_{\rm s}$-bands do not further reduce the scatter at the faint end, however do produce slightly lower biases at the brightest magnitudes. Again, we therefore conclude that the combined $9$-band dataset is therefore that which provides the best overall statistics.

3.2.3 Photo-$z$ distributions per field

Another quality check for the homogeneity of the data is to compare the distributions of photometric redshift in each of our 5 fields (shown in Figure 1). We compare the distribution of all photo-$z$ estimates for sources within our lensing sample (Figure 6) with $r\leq 23.5$. These two cuts allow us to compare the photo-$z$ distributions per field for samples of known non-stellar sources in a regime agnostic to the effects of variable depth from the comparison; a like-for-like comparison. These distributions per field are shown in Figure 11. We can see from the distribution that the fields are in very good agreement, with only GS appearing slightly deeper than the other 4 fields. As such, we conclude that the photo-$z$ among the different KV450 fields demonstrate satisfactory homogeneity.

4 Higher-order data products

We can subsequently utilise our photo-$z$ estimates to derive higher-order data-products. For this work, we choose to explore the rest-frame photometric properties of a selection of KV450 sources, as well as examine the fidelity of integrated properties, namely stellar masses. In order to explore these properties we perform template-fitting to the broad-band spectral energy distributions (SEDs) of each KV450 source, while maintaining a fixed redshift at the value of $z_{\rm B}$.

4.1 SED fitting

To estimate the rest-frame properties of our KV450 sources, we perform SED fitting with the Le Phare (Arnouts et al., 1999; Ilbert et al., 2006) template-fitting code, using a standard concordance cosmology of $\Omega_{\rm m}=0.3$, $\Omega_{\Lambda}=0.7$, H0 = $70$ km s*-1* Mpc*-1*, Chabrier (2003) IMF, Calzetti et al. (1994) dust-extinction law, Bruzual & Charlot (2003) stellar population synthesis (SPS) models, and exponentially declining star formation histories. Input photometry to Le Phare is as described in Sect. 3, including the per-band extinction corrections as used in BPZ. We fix the source redshift to be the value of $z_{\rm B}$ returned from BPZ. We opt to fit SEDs to all $>45$ million KV450 sources, regardless of masking, so that any/all subsequent subsamples of KV450 data may incorporate our stellar mass estimates. This requires that we also allow SEDs to be fit with QSO and stellar templates, for which we use the internal Le Phare defaults.

4.2 Star-galaxy separation

One advantage of fitting all photometric sources in this way is that we are able to use the higher-order data products to assist with star-galaxy separation. In particular, by fitting all sources with templates for QSOs, stars, and galaxies, we are able to identify stellar contaminants that otherwise would make it into our overall sample. To do this, we identify all sources that are best fit by a stellar template in Le Phare and which have an angular extent that is point-like; specifically a flux-radius of $0.8$ arcseconds or smaller. Using this simple cut, we are able to produce an exceptionally clean galaxy-only sample (as shown in Fig. 6). We note, however, that this rejection has no effect on the lensing sample as the high-fidelity point-source rejection that is already performed during shape-fitting is very effective at removing stellar contaminants. Indeed, all sources that are identified as stars using our SED based selection are also flagged as stars during shape-fitting. Furthermore, for the sources with $g\leq 21$, we can cross-reference our stellar classification with that from the GAIA DR2 (Gaia Collaboration et al., 2018) point-source catalogue. Comparing to GAIA we find that $99.1\%$ of our sources classified as stars (and which are brighter than the $g\leq 21$ GAIA magnitude limit) are also classified as stars by GAIA222Should the GAIA catalogue have contamination by truly-extended galaxies, this would indicate a galaxy contamination within our star sample in the same proportion.. Again, this further increases our confidence in the stellar classification possible using our SED products.

4.3 Stellar mass estimates

For this work, we are primarily interested in assessing the fidelity of stellar masses that are estimated using the KV450 dataset. Our stellar masses, estimated by Le Phare, are calculated as the mass of stars required to produce the observed galaxy SED given the best-fit stellar population, assuming the combination of models given in Sect. 4.1. Therefore, in order to recover a fair estimate of the galaxy stellar mass, the observed SED must be representative of the total light emitted from the galaxy. Our aperture fluxes, however, have been intentionally optimised for high-fidelity colours, rather than for the recovery of total fluxes. This means that our mass estimates here will be systematically below what would be recovered with a total flux aperture, primarily as a function of source size. In order to remedy this systematic effect, we opt to use our quasi-total Source Extractor AUTO flux estimates (measured during our initial source extraction) to correct our masses. To do this, we implement a correction akin to the fluxscale correction discussed in Taylor et al. (2011); Wright et al. (2017), although the implementation there was designed to correct for systematic bias in Kron (1980) apertures for changing galaxy profile shapes.

Here our fluxscale factor, ${\mathcal{F}}$, is a multiplicative correction defined as the linear ratio of the quasi-total Source Extractor $r$-band AUTO flux to the non-total GAaP $r$-band flux: ${\mathcal{F}}=f_{\rm AUTO}/f_{\rm GAaP}$. This correction is applied post-facto to the Le Phare stellar mass estimates. The correction devised is such that our final SEDs will be fixed to the AUTO flux estimate, and our SEDs themselves will be reflective of the flux contained within the GAaP apertures. This can lead to systematic biases. For example, if there are significant colour gradients within the galaxies in our sample, such that the colours within and beyond our apertures differ considerably, then our SEDs will tend to be non-representative of the true integrated galaxy spectrum. Admittedly, however, this is only likely to be a significant effect for galaxies whose size is significantly larger than the PSF; i.e. low-redshift galaxies for which our analysis pipeline is already sub-optimal.

For validation purposes, we compare our fluxscale-corrected stellar mass estimates to those also estimated by GAMA (Wright et al., 2017) and G10-COSMOS (Andrews et al., 2017; Driver et al., 2018) in Fig. 12. Both of these studies utilise spectroscopic redshifts, and implement the same cosmology, SPS models, dust-law, and IMF as used in this work when estimating stellar masses. They also use total matched aperture fluxes. These similarities allow direct comparison of our mass estimates, despite the use of different algorithms and wavelength bandpasses for the mass estimation. We perform this comparison both for the KV450 masses described above and for masses estimated in the same way but utilising only 4-band photometric information (i.e. the KiDS-450 equivalent masses). The GAMA dataset here is sky-matched to our KiDS-450/KV450 dataset within a $1$ arcsec radius, for GAMA galaxies with redshift $z\geq 0.004$, GAMA redshift quality flag nQ$>2$, and for KiDS-450/KV450 sources with $z_{\rm B}<0.7$ (so as to avoid spurious matches to the much deeper KiDS-450/KV450 catalogues). The G10-COSMOS sample is subset such that it contains only sources with spectroscopic redshifts (i.e. those with G10-COSMOS flag $z_{\rm use}\leq 3$) and is also sky-matched to KiDS-450/KV450 with a $1$ arcsecond radius. Note that there is no requirement for consistency between matched sources photo-$z$ and spec-$z$ values. As such, the scatter here is a reflection of the scatter in the mass estimates due to, jointly, systematics in our photometric data and photo-$z$ estimation.

We see that the KiDS-450 masses show significant scatter in the comparison distributions (Fig.12, left panels), particularly for the COSMOS dataset which extends to significantly higher redshift than the GAMA sample ($\sigma=0.464$). Conversely, we see very good agreement with the same sample when using masses derived with KV450; $\sigma=0.202$. We note that the scatter in the mass comparison with the GAMA sample increases slightly when moving from KiDS-450 to KV450. This increase in scatter between masses estimated in KV450 and by GAMA is slightly larger than the typical scatter induced by slightly different mass estimation methods (${\sim}0.2$ dex; see Wright et al., 2017, for a detailed discussion of such comparisons and systematic effects), and is induced by the updated photo-$z$ prior implemented here (Sect. 3.2). This is not surprising, given that this prior is optimised for analysis of the full KiDS sample, which is COSMOS-like. The variation between KV450 and GAMA is highly correlated with systematic differences between the GAMA spec-z and KV450 photo-$z$, which shows roughly a factor of two stronger bias than we see in the main survey spectroscopic calibration sample (i.e. Fig. 7), again due largely to our updated prior. Importantly, we see no such systematic variations in our comparisons with G10-COSMOS (in mass or photo-$z$) for KV450. This is in stark contrast to the significant bias and scatter that is evident in the KiDS-450 to G10-COSMOS comparisons. In particular, we note that the bias in the G10-COSMOS comparison decreases by nearly an order of magnitude when moving from KiDS-450 ($\mu_{\Delta}=-0.213$) to KV450 ($\mu_{\Delta}=0.041$). Furthermore, we note that the scatter in the comparison between KV450 and G10-COSMOS is reduced to $\sigma_{\Delta}=0.208$; consistent with the $0.2$ dex typical uncertainty induced by different mass estimation methods agnostic of variations in input photometry and redshifts. As such, we conclude that, for our KV450 sample, the 9-band stellar mass estimates are equivalent in quality to those that can be estimated using significantly more accurate spectroscopic redshift surveys.

5 Stellar Mass Function

Given the accuracy of our observed stellar mass estimates when compared to the G10-COSMOS survey, we are prompted to explore whether we can reproduce complex redshift-dependent mass functions using these estimates. Such mass functions typically require spectroscopic redshift estimates and/or high-accuracy photo-$z$ estimates derived from $20+$ broad and narrow photometric bands (see, e.g., Andrews et al., 2017; Davidzon et al., 2017; Wright et al., 2018). However, given the apparent fidelity of our mass and photo-$z$ estimates, we wish to explore whether we can derive sensible mass-evolution distributions from our relatively low-resolution photo-$z$ estimates alone.

Fluxscale-corrected stellar masses from Le Phare are shown in Fig. 13 for all galaxies in the KV450 footprint, as a function of $z_{\rm B}$. The distribution shows an underdensity of high-mass sources at low-redshift, and also a considerable amount of structure as sources approach the detection limit. This structure is a form of redshift focussing, and is caused by sources systematically dropping below the detection limit in particular bands as a function of galaxy SED shape. Otherwise, the distribution is well bounded and fairly uniform, showing little evidence of photo-$z$ dependent biases.

We wish to use this distribution of stellar masses to estimate a series of volume-complete galaxy stellar mass functions (GSMFs) for the KV450 dataset. To do this, we first define the mass limit of the dataset as a function of photo-$z$. We take the same method of estimating the mass limits as described in Wright et al. (2017), using the turn-over points in both number counts and photo-$z$ to estimate the mass-completeness limit. Briefly, the mass limits as a function of photo-$z$ are constructed assuming that any observed down-turn in number-density is due exclusively to incompleteness; i.e. that the mass function, over the redshifts and masses probed here, has no true down-turn. Using this assumption we estimate the completeness limit as a function of photo-$z$ as being the point at which either comoving number density and/or stellar mass number density starts to fall. The procedure is shown graphically in Figure C1 of Wright et al. (2017). The calculation of the completeness limit is done in a series of overlapping bins of photo-$z$ and stellar mass, and the resulting limit estimates are fit with a fifth-order polynomial. This derived mass limit is shown in Fig. 13 as a dashed red line. The mass limit can be seen to effectively select against sources in the redshift-focused low-SNR portions of the distribution, and suggests that the mass estimates of KV450 can be considered to be volume complete down to $M_{\star}\geq 10^{10}M_{\odot}$ for sources with $z_{\rm B}\leq 1$.

Using these mass limits, we define a series of volume-complete bins in stellar mass and redshift, and calculate the resulting mass functions in these bins. These are shown in Fig. 14. For our binning, we choose to use the same tomographic redshift limits as are implemented in our cosmological analysis (Hildebrandt et al., 2018), out to $z_{\rm B}=1.2$. The mass functions are calculated using a simple volume calculated using the survey area and the redshift limits annotated in each bin, and we show the mass functions derived with and without the implementation of the fluxscale correction, for reference. For comparison, we also show the model evolutionary mass functions presented in Wright et al. (2018), derived using a compilation of consistently analysed GAMA, G10-COSMOS, and 3D-HST data over the redshift range $0.1\leq z\leq 5$. For demonstration, the Wright et al. (2018) model is shown both as the model expectation at the mean redshift of the bin (grey line), and as the range of model values (grey shading) that would be expected when allowing for: photo-$z$ bias $|\Delta z_{\rm B}|\leq 0.2$, additional systematic bias in our stellar mass estimation ($|\Delta M_{\star,\rm{sys}}|\leq 0.2$ dex), and Eddington bias ($|\Delta M_{\star,\rm{edd}}|=0.2$ dex).

The first photo-$z$ bin shows a mass function that has a clear deficit in number density for the highest mass sources. This deficit, we argue, is again caused by our pipelines optimisation for small-angular scale sources: the largest sources on sky will also be the most massive at low redshift, and our analysis methods are biased against accurate extraction of these sources. In the subsequent bins, however, the mass functions from our sample are in good agreement with the evolutionary model of Wright et al. (2018). This is particularly noteworthy, given the coarseness of our photo-$z$ estimation and that no correction for the redshift distribution bias (such as is done in cosmic shear analyses; see Hildebrandt et al., 2017) has been attempted. The mass functions, however, clearly suffer from considerable Eddington bias in their masses (i.e. our mass functions are biased toward higher masses).

6 Summary

In this work we present a new photometric dataset for astrophysics and cosmology, KiDS+VIKING-450. The dataset builds on the optical dataset of KiDS-450 with the inclusion of 5-band near-IR data from the VIKING survey, reduced and analysed in a way entirely consistent with the optical dataset.

We discuss the reduction of the VIKING dataset, and the derivation of relevant data products such as photometry. We demonstrate that the products derived are robust, consistent with, and superior to previous photometric estimates of sources from overlapping surveys such as 2MASS.

Using our photometry, we derive new 9-band photometric redshifts for the full KV450 sample, and compare these new photo-$z$ to those presented previously in Hildebrandt et al. (2017). We find that the new photo-$z$ exhibit a reduced scatter in $\Delta z/(1+z)$ (especially at high photo-$z$; down by $\sim 40\%$ compared to the ${ugri}$-only case), a lower overall bias (down $50\%$), and allow us to dramatically improve our ability to accurately estimate photo-$z$ beyond $z_{\rm B}=0.9$, with the outlier rate reducing by over $40\%$. The improvement is sufficiently dramatic as to motivate the inclusion of a higher-redshift bin in KiDS cosmic shear studies using this dataset (Hildebrandt et al., 2018), and to motivate us to explore whether our photo-$z$ alone are able to be used to constrain galaxy evolution parameters of interest (such as the stellar mass function) out to high redshift.

Using the SED fitting code Le Phare, we estimate stellar masses for all sources in the KV450 footprint. We compare these mass estimates to previous samples from GAMA (Wright et al., 2017) and G10-COSMOS (Andrews et al., 2017), finding good agreement between the datasets. Our comparison to G10-COSMOS (a sample that matches the overall KV450 dataset well) demonstrates negligible bias in our mass estimates ($\mu_{\Delta}=0.041$) and a scatter that is equivalent to that seen inherent to stellar mass estimates agnostic of changes to photometry and redshift ($\sigma_{\Delta}=0.202$; Wright et al., 2017; Taylor et al., 2011). Furthermore, we demonstrate that the SED fits allow us to perform a high-fidelity star-galaxy separation, and thereby clean the full sample of contaminating sources.

Using our mass estimates, we calculate the mass-completeness limit of the dataset, deriving an empirical mass limit that suggests the sample is volume complete above $M_{\star}\geq 10^{10}M_{\odot}$ at $z_{\rm B}\leq 1$. We bin the data into eight volume complete samples spanning $0.1\leq z_{\rm B}\leq 2$ and plot the resulting galaxy stellar mass functions for these bins. Comparing these bins to the evolutionary model of the GSMF from Wright et al. (2018), we find agreement in the range of $0.3\leq z_{\rm B}\leq 2$. The lowest photo-$z$ bin shows considerable incompleteness at high-masses, which we attribute to our extraction pipeline being optimised for small-angular-size sources. In the regime where our pipeline is optimised, we demonstrate that we are able to reproduce the results of previous studies which utilised spectroscopic redshifts and/or significantly more photometric data than we use here. Future KiDS+VIKING releases, containing three times the on-sky area utilised here, will further push the boundaries of studies that are possible with photometric-only data.

Acknowledgements.

This work is been supported by European Research Council Consolidator Grants: 770935 (AHW,HH) and 647112 (CH). HH is also supported by Emmy Noether (Hi 1495/2-1) and Heisenberg grants (Hi 1495/5-1) of the Deutsche Forschungsgemeinschaft. KK acknowledges support by the Alexander von Humboldt Foundation. CH acknowledges support from the Max Planck Society and the Alexander von Humboldt Foundation in the framework of the Max Planck-Humboldt Research Award endowed by the Federal Ministry of Education and Research. AC acknowledges support from NASA grant 15-WFIRST15-0008. JTAdJ is supported by the Netherlands Organisation for Scientific Research (NWO) through grant 621.016.402. ACE acknowledges support from STFC grant ST/P00541/1. This work is supported by the Deutsche Forschungsgemeinschaft in the framework of the TR33 ‘The Dark Universe’. Based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme IDs 179.A-2004, 177.A-3016, 177.A-3017, 177.A-3018, 298.A-5015. This work was performed in part at Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611.

*Author Contributions: All authors contributed to the development and writing of this paper. The authorship list is given in three groups: the lead authors (AHW, HH, KK), followed by two alphabetical groups. The first alphabetical group includes those who are key contributors to both the scientific analysis and the data products. The second group covers those who have either made a significant contribution to the data products or to the scientific analysis. *