Issue 
A&A
Volume 634, February 2020



Article Number  A104  
Number of page(s)  16  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201936966  
Published online  17 February 2020 
The effects of varying depth in cosmic shear surveys
^{1}
ArgelanderInstitut für Astronomie, Auf dem Hügel 71, 53121 Bonn, Germany
email: sven@astro.unibonn.de, peter@astro.unibonn.de
^{2}
The German Centre for Cosmological Lensing, Astronomisches Institut, RuhrUniversität Bochum, Universitätsstr. 150, 44801 Bochum, Germany
^{3}
Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK
^{4}
Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK
^{5}
Leiden Observatory, Leiden University, Niels Bohrweg 2, 2333 CA Leiden, The Netherlands
Received:
21
October
2019
Accepted:
2
December
2019
We present a semianalytic model for the shear twopoint correlation function of a cosmic shear survey with nonuniform depth. Groundbased surveys are subject to depth variations that primarily arise through varying atmospheric conditions. For a survey like the KiloDegree Survey (KiDS), we find that the measured depth variation increases the amplitude of the observed shear correlation function at the level of a few percent out to degreescales, relative to the assumed uniformdepth case. The impact on the inferred cosmological parameters is shown to be insignificant for a KiDSlike survey. For nextgeneration cosmic shear experiments, however, we conclude that variable depth should be accounted for.
Key words: gravitational lensing: weak / cosmology: miscellaneous
© ESO 2020
1. Introduction
The discovery of cosmic shear has provided us with a new and powerful cosmological tool to empirically test the standard model of cosmology and to determine its parameters. Contrary to the analysis of the cosmic microwave background (CMB, e.g., by Planck Collaboration VI 2018), cosmic shear is more sensitive to the properties of the lowredshift largescale structure and, thus, provides an excellent consistency check for the standard model. Current cosmic shear surveys are particularly sensitive to the parameter , where σ_{8} characterizes the normalization of the matter power spectrum and Ω_{m} is the matter density parameter. Constraints on S_{8} from the three current major cosmic shear results are all consistent with the CMB analysis by Planck Collaboration VI (2018). It is interesting to note, however, that they all favor values that are slightly lower than the Planck constraints of S_{8} = 0.830 ± 0.013. Hikage et al. (2019) report from an analysis of the Subaru Hyper SuprimeCam survey, Hildebrandt et al. (2020, hereafter H20) obtained from KiDS+VIKING data, and Troxel et al. (2018) constrain S_{8} = 0.782 ± 0.027 using the Dark Energy Survey (DES). Combined analyses of DES and KiDS data (Joudaki et al. 2019; Asgari et al. 2020) result in a ∼3σ tension with the CMB value for S_{8}. If this tension is not the manifestation of an unaccounted systematic effect, in either the cosmic shear surveys (Mandelbaum 2018) or the Planck mission (Addison et al. 2016), it certainly merits attention. It could be interpreted as a sign of new physics exemplified by massive neutrinos (Battye & Moss 2014), timevarying dark energy, or modified gravity (Planck Collaboration XIV 2016) and coupling within the dark sector (Kumar et al. 2019). It could also, however, prove to be a simple statistical coincidence.
For current cosmic shear surveys, the estimated systematic error is becoming comparable in magnitude to the statistical error, implying that for nextgeneration surveys, a significant reduction of systematic errors is necessary. With surveys like the Large Synoptic Survey Telescope (LSST, Ivezic et al. 2008) and Euclid (Laureijs et al. 2011) soon to begin, systematic effects in gravitational lensing have received a large amount of attention (see Mandelbaum 2018, and references therein).
In this paper, we focus on systematic effects induced by variation in survey depth that is so far unaccounted for in cosmic shear analyses (Vale et al. 2004). For a survey with a fixed exposure time, varying atmospheric conditions, dithering strategies and galactic extinction all contribute to an inhomogeneous limiting magnitude as a function of sky position. In order to assess the impact of variable depth for current and future surveys, we build an analytical model for the effect based on the survey specifications of the KiloDegree Survey (KiDS, Kuijken et al. 2015). To the first order, the depth variation in KiDS can be modeled by a piecewise constant depth function, which varies between each 1 deg^{2} square pointing. KiDS object detection is defined in the rband as these images were chosen to be significantly deeper in comparison to the other optical and nearinfra red filters. We, therefore, quantify survey depth with the limiting rband magnitude, as defined in de Jong et al. (2017). We defer the study of multiband variable depth and its impact on photometric redshift accuracy for a future work.
This work is complementary to the analysis of Guzik & Bernstein (2005), who investigate the effect of a general, positiondependent multiplicative shear bias on the shear power spectrum. In principle, the varying depth of the source galaxy sample that we study here could be recast as a varying effective shear bias. The inclusion of an inhomogeneous distribution of source galaxies has also been explored using mock catalogs of the Subaru Hyper SuprimeCam Survey (Shirasaki et al. 2019), with a focus on resulting estimation of the cosmic shear covariance matrix.
In Sect. 2, we will introduce two simple toy models to understand this effect and analyze the impact on the cosmic shear power spectrum. In Sect. 3, we will estimate the effect on the shear correlation functions ξ_{±} using a semianalytic model. We will present our results in Sect. 4. In Sect. 5, we will discuss our results and comment on the impact of our used simplifications. In the appendices, we present the full derivation of our model for finite field surveys. We assume the standard weak gravitational lensing formalism, a summary of which can be found in Bartelmann & Schneider (2001).
2. Simple, analytic toy models
For our first analysis, we assume that all the matter between the sources and observer is concentrated in a single lens plane of distance D_{d} from the observer. If we then distribute sources at varying distances D_{s}, two effects become apparent: firstly, the lensing efficiency D_{ds}/D_{s} varies, where D_{ds} is the distance between the lens plane and the respective source. Secondly, and more importantly, for a more distant source, more matter is concentrated between the source and the observer, leading to a stronger shear signal.
Assuming that the depth and, thus, the source redshift population, only varies between pointings of the camera, an observer will measure a shear signal that is modified by a steplike depthfunction, γ^{obs}(θ) = W(θ)γ(θ), where W is proportional to the mean of the lensing efficiency D_{ds}/D_{s} of one pointing and γ denotes the shear that this pointing would experience if it were of the average depth. We can parametrize W as W(θ) = 1 + w(θ). This implies that ⟨w(θ)⟩ = 0 holds, where ⟨ ⋅ ⟩ denotes the average over all pointings.
2.1. Modeling the power spectrum
In our first model, we describe the impact of varying depth on the power spectrum, following the simplifications described above. In accordance with the definition of the shear power spectrum
where denotes the Fourier transform of γ, we define the observed power spectrum via
We note that due to the depthfunction, both the assumptions of homogeneity and isotropy break down, which means that we can neither assume isotropy in the power spectrum, nor can we assume that vanishes for ℓ ≠ ℓ′. This estimator provides a natural extension to the definition of the regular power spectrum and, in the case of a homogeneous depth distribution, is reduced back to the original estimator. To model a constant depth on each individual pointing, α, we can choose random variables, w_{α}, that only need to satisfy ⟨w_{α}⟩ = 0. As we assume an infinite number of pointings, α can assume any twodimensional integer value ℤ^{2} and we can parametrize w(θ) as
with the boxfunction
where L is the sidelength of one pointing. Following the calculations in Appendix A.1, we derive
Here we have denoted as the dispersion of the depthfunction, since the statistical properties of this function do not depend on the pointing α. The Fourier transform of the box function, , is a 2dimensional sincfunction (see Appendix A.1). The observed power spectrum, P^{obs}, is thus composed of the original power spectrum P(ℓ) from Eq. (1), plus a convolution of the power spectrum with a sincfunction, scaling with the variance of the function w(θ).
2.2. Modeling the shear correlation functions
Measures that are more convenient for the inference of cosmological information from observational data are the shear correlation functions ξ_{±}, which are defined as
Here, γ_{t} and γ_{×}, denote the tangential and crosscomponent of the shear for a galaxy pair with respect to their relative orientation (see Schneider et al. 2002a). The shear correlation functions are the prime estimators to quantify a cosmicshear signal, since it is simple to include a weighting of the shear measurements into the correlation functions and, contrary to the power spectrum, one does not have to worry about the shape of the survey footprint or masked regions, or model the noise contribution. For this analysis, we follow the assumption that a deeper pointing shows a stronger shear signal γ^{obs}(θ) = W(θ)γ(θ) as described above. This assumption implies that a higher redshift just increases the amplitude of the shear signal, but as can clearly be seen by inspecting shear correlation functions of different redshift distributions, the change of the signal is extremely scaledependent and not just a multiplication with a constant factor. In other words, not just the average shear changes as a function of redshift, but also its entire twopoint statistics. However, this should serve as a reasonable first approximation for small variations in mean source redshift. Additionally, we assume that a greater depth does not only lead to a stronger average shear, but also to a higher galaxy number density, implying a correlation between those two quantities.
We denote by N^{i}(θ) the average weighted number of galaxies^{1} per pointing in redshift bin i and by W^{i}(θ) the weighting of average shear. The observed correlation functions now change from uniform depth, , via
where the average, ⟨ ⋅ ⟩, represents both an ensemble average as well as an average over the position θ′. Assuming that the depth of different pointings is uncorrelated, the only important property of a galaxy pair is whether or not they lie in the same pointing. We denote the probability that a random galaxy pair of separation θ lies in the same pointing by E(θ). This function is depicted in Fig. 1, and an analytic expression is derived in Appendix A.2.
Fig. 1.
Probability E(θ) that a random pair of galaxies of distance θ lie in the same 1 deg^{2} pointing. 

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To compute the modified shear correlation functions, we parametrize the number densities N^{i}(θ) = ⟨N^{i}⟩[1 + n^{i}(θ)] and the weight W^{i}(θ) = 1 + w^{i}(θ) and, as in Eq. (4), interpret n^{i}(θ) as a function with average ⟨n^{i}⟩ = 0 that is constant on each pointing. We can see that ⟨n^{i}(θ′)n^{j}(θ′ + θ)⟩ = E(θ)⟨n^{i}(θ′)n^{j}(θ′)⟩ = E(θ)⟨n^{i}n^{j}⟩ holds and compute:
Ignoring correlations higher than second order in n^{i} and w^{i},^{2} and performing the same calculation for the denominator of Eq. (8) we find
A model correlation function for a cosmic shear survey is usually calculated by taking the average redshift distribution of a redshift bin, weighted by the number density. Ignoring that the depth is correlated on scales of one pointing (here at ) is equivalent to setting E(θ) ≡ 0. We note that there is still a correlation between N and W for the same galaxy. Performing the same calculations as above, this yields a relation between the correlation function of uniform depth, , and the one that is usually modeled, :
When an observer now calculates the model correlation functions without accounting for varying depth between pointings, the ratio between modeled and observed correlation functions becomes:
It is interesting to note that holds wherever E(θ) = 0, so we expect the observed and the modeled correlation functions to be equivalent on scales where the depth is uncorrelated. One thing left to determine is how to define the weightfunction W(θ). For this, we will refer the reader to the beginning of Sect. 4.
3. A semianalytic model
The previously derived analytic model describes, how varying depth between pointings modifies the correlation function due to the correlation between number density and the average redshift of source galaxies. While this model serves as an intuitive first approximation, it completely ignores any effects from the large scale structure (LSS) between the closest and the most distant galaxy. Therefore, we do not expect this model to yield accurate, quantitative results for cosmic shear surveys.
Below we derive a more sophisticated model that includes the effects of the LSS. While it is computationally more expensive, it improves the accuracy of the model for cosmic shear surveys, which are sensitive to the exact redshift distributions of sources as well as the underlying cosmology.
An inspection of KiDSdata showed that the redshift distribution of sources is highly correlated with the limiting magnitude in the rband. We thus chose to separate the survey into ten quantiles, sorted by rband depth, that is, if a pointing had a shallower depth than 90% of the other pointings, it would belong to the first quantile, and so on. For each quantile m and each tomographic redshift bin i we can extract a weighted number of galaxies and a source redshift distribution following the direct spectroscopic calibration method of H20. In Fig. 2, the average redshift and weighted number of galaxies are plotted for each quantile of each redshift bin, whereas a selection of source redshift distributions is depicted in Fig. 3. A table of the limiting magnitudes for each quantile can be found in Table C.1.
Fig. 2.
Weighted number of galaxies N and average redshift ⟨z⟩ in the KiDS+VIKING450 survey (KV450, Wright et al. 2019) in pointings of different depth for each of the five tomographic bins used in H20. Each color corresponds to one redshift bin of H20. A single point represents one quantile of the respective redshift bin, where the fainter points denote pointings of shallower depth. 

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Fig. 3.
Source redshift distributions for a selection of very shallow pointings (blue), average pointings (yellow) and very deep pointings (green). The percentage points in the legend denote to which quantile a pointing belongs, when all are ordered by their depth. 

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Given the two comoving distance probability distributions of sources, and , we can compute the shear correlation functions from the underlying matter power spectrum, P_{δ}(k, χ), via (Kaiser 1992)
Here, J_{n} denotes the nth order Bessel Functions, f_{K}(χ) is the comoving angular diameter distance and χ_{H} is the comoving distance to the horizon. The parameters H_{0} and c denote the Hubble constant and the speed of light.
Using Eq. (13), we can compute the model correlation functions, , for each pair of quantiles m, n and redshift bins i, j.^{3} When measuring the shear correlation functions of a survey, we take the weighted average of tangential and cross shears of all pairs of galaxies (see Hildebrandt et al. 2017). If, for a single pair of galaxies, one galaxy lies in the mth quantile of redshift bin i and the second one lies in the nth quantile of redshift bin j, then their contribution to the observed correlation functions is, on average, . This means that, if we know each of those single correlation functions, we can reconstruct the total correlation functions via a weighted average of the single functions. Formally, we define
where is a weighting of the correlation functions, which has to be proportional to the probability that a galaxy pair of separation θ comes from quantiles m and n. In this analysis, we will assume an uncorrelated distribution of depth and neglect boundary effects as well as the sample variance of the depthdistribution between pointings. We will later discuss the validity of these assumptions as well as possible mitigation strategies.
To calculate , we imagine two arbitrary (infinitesimally small) surface elements d^{2}θ_{1} and d^{2}θ_{2} of separation θ on the sky. For the case m ≠ n, we know that the two galaxies contributing to have to lie in different pointings, else they would automatically be in the same quantile. The probability that the surface elements are within different pointings is [1 − E(θ)]. Furthermore, the first element d^{2}θ_{1} has to lie in quantile m, the probability of which is 1/10. The pointing of the second element d^{2}θ_{2} has to be of quantile n; the probability of that is also equal to 1/10. The probability that a galaxy pair populates those surface elements is proportional to the weighted number of galaxies and . We get for n ≠ m:
For the calculation of , we have to account for a different possibility: In case that the galaxies lie in the same pointing, they automatically are in the same quantile. We therefore obtain
where δ_{mn} denotes the Kronecker delta. Inserting this into Eq. (16), we compute
with the normalization
A mathematically more rigorous derivation of this function can be found in Appendix A.3.
Computing this for all five redshift bins of the KV450survey, forces us to calculate and coadd 1275 correlation functions^{4}. Since the variation in depth is a relatively small effect, even tiny numerical errors can add up, skewing the calculations. Additionally, calculating 10^{3} correlation functions is computationally expensive. However, if we examine Eq. (15), we see that the comoving distance distribution of sources enters linearly. This, in turn, implies that in Eqs. (14) and (13), both source distance distributions enter linearly, meaning that, instead of adding correlation functions, we can add their respective redshift distributions and compute the correlation functions of that. In particular, we can define the combined number of galaxies N^{i} and average comoving distance probability distribution, ℒ^{i}(χ), of tomographic bin i as
Defining as the correlation functions between the average comoving distance distributions ℒ^{i}(χ) and ℒ^{j}(χ), we find:
Consequently, we can apply this to Eq. (19), yielding
For each pair of redshift bins we, thus, only have to compute eleven correlation functions, which reduces the number of functions to compute from 1275 to 165.
4. Results
We compare the analytic and semianalytic models for a variabledepth cosmic shear measurement in a KiDSlike survey. While the application of the semianalytic method is straightforward, for the analytic method we need to decide how to estimate the weight function W from the given redshift data. Following the separation of a survey into quantiles as in Sect. 3, we define W(θ) ≡ W_{n} whenever θ is in a pointing of quantile n. For the determination of W_{n} we test two approaches: As a first method, following Van Waerbeke et al. (2006), Bernardeau et al. (1997), we estimate
where ⟨z⟩_{n} is the average redshift of quantile n. As a second method, we define
where the denotes the model correlation function defined in Sect. 3, evaluated at a characteristic scale θ_{ref}, that needs to be chosen.
While the first method suffers from the fact that the powerlaw index only holds for sources of redshifts 1 ≲ z ≲ 2, the second method is sensitive to the angular range θ_{ref}, at which the shear correlation functions are evaluated, which is fairly arbitrary. For θ_{ref} ≈ 11′, which is roughly in the logarithmic middle between the range of the correlation functions, , the two calibration methods agree. The choice of other values for θ_{ref} leads to a different amplitude of the change , but does not affect its shape. Generally, a smaller θ_{ref} leads to a stronger effect, in particular, the highest amplitude of the change is at .
4.1. Effect on the shear correlation functions
In this section, we calculate both the analytic (Eq. (12)) and semianalytic (Eq. (23)) models for the shear correlation function from a KiDSlike variable depth survey. We adopt the tomographic bins defined in H20 and their resulting bestfit cosmological parameters to present. In Fig. 4, the ratio between our models for the observed correlation functions , and the standard theoretical prediction that assumes uniform depth are shown. We find that the level of variation in the depth of the KiDS survey increases the amplitude of the observed shear correlation function, on subpointing scales, by up to 5% relative to the uniformdepth case.
Fig. 4.
Ratio of correlation functions measured for a uniform depth survey, ξ_{±}, and a KiDSlike variable depth survey, , crosscorrelating five tomographic bins (as denoted in the upper left corner of each panel). The upper left triangle depicts the ratios of ξ_{+}, whereas the lower right triangle depicts the ratios of ξ_{−}. Results from mock KiDSlike data (red) can be compared to analytic models from Sect. 2 (average redshift weighting, green), and the semianalytic model from Sect. 3 (blue solid). Mock data is limited to the angular regime which is not significantly impacted by resolution effects. As the mocks only take galaxies with z < 2 into account, they slightly underestimate the effect. Applying the same redshiftcutoff to the semianalytical model (blue dashed) yields a nearperfect agreement on subpointing scales. Therefore, the seemingly better agreement between the mocks and the analytic method is purely coincidental. The models adopt the bestfit cosmology of H20. 

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We compare our models to mock KiDSlike data created using a modified version of the Fullsky Lognormal Astrofields Simulation Kit (FLASK, Xavier et al. 2016; Joachimi et al., in prep.; Lin et al., in prep.). Using FLASK lognormal fields, we generate galaxy mocks with coherent clustering and lensing signals. Adopting a linear relation between the limiting rband magnitude and the effective number density, fit to the KiDS data, we imprint the variable depth of the full KiDS1000 footprint (Kuijken et al. 2019) on a HEALPIX^{5} grid of Nside = 4096. Our resolution choice represents a compromise between minimising the mock computation time and maximising the accuracy of the recovered shear signal at small angular scales. With Nside = 4096, ξ_{+} is accurate to 7% above ∼1 arcmin, and ξ_{−} is accurate to 10% above ∼6 arcmin. For each HEALPIX pixel in the KiDS1000 data, the limiting rband magnitude defines the effective number density of sources, and the average source redshift distribution for each tomographic redshift bin (see Fig. 3). In the uniform depth case, redshifts and number densities are sampled from the average of these tomographic sets. The ratio of the lensing signals from the two mocks is computed, averaged over 2000 shapenoisefree realizations, and is shown in Fig. 4 (red).
The results can be seen in Fig. 4. We observe that for highredshift bins, all methods yield consistent results. For lowredshift bins, there are discrepancies between the different models. However, the averageredshift weighted analytic model, as shown here, is only valid for high redshifts, whereas the autocorrelationξ_{+}weighted model (not shown) is entirely dependent on the choice of θ_{ref}. For θ_{ref} = 11′, both analytic models agree very well for all redshift bins. For , the autocorrelation weighted method agrees with the semianalytic one pretty well for ξ_{+}. As ξ_{−} is affected much stronger by this effect^{6}, the analytic method is not able to trace this change for any choice of θ_{ref}. Furthermore we note, that the effect seems to be strongest in the first redshift bin, which is not surprising, as there the average redshift between pointings varies the most (compare Fig. 2).
The simulations and the models seem to be in relatively good agreement, but there are some differences. It is noticeable that in the simulations, the value consistently stays below unity at large scales, which can be attributed to the fact that the depth of different pointings is not completely uncorrelated, as was assumed in the models.
An additional difference between the models and simulations is, that in the models we neglect boundary effects and the samplevariance of the depthdistribution between pointings, whereas the simulations were performed with the KiDS1000 footprint. In Appendix B, we develop a model to extract the correction for a specific survey footprint. With this model, we can estimate the impact of a correlated distribution of depth, the sample variance of the depthdistribution and boundary effects. We find that for a square footprint of 450 deg^{2} or 1000 deg^{2} with an uncorrelated depthdistribution, finite field effects are negligible.
In general, the semianalytic model predicts a stronger effect than the mocks. This is due to the fact that the mocks are subject to a redshiftcutoff at z = 2, meaning that they do not take the highredshift tail of galaxies into account. This is of particular importance in the first redshift bin, where this feature is especially pronounced (compare Fig. 3). When the same cutoff is applied to the models, the agreement on subpointing scales is striking. In particular this means that the mocks slightly underestimate the effect of varying depth.
4.2. Impact on cosmological parameter constraints
As the next step, we assess how the observational depth variations propagate to cosmological parameters inferred from compared to . For this test, we choose a fiducial cosmology, Φ, and determine the relative change in Ω_{m} and σ_{8} compared to a reference setup with uniform depth. All other cosmological parameters are kept fixed. First, we compute the reference correlation functions, , for each pair of redshift bins i, j using NICAEA as described in Sect. 3. Then we derive the observed correlation functions, , from Eq. (23). Using the Markovchain Monte Carlo sampler EMCEE, we sample correlation functions for different cosmologies Φ′ and find the likelihood distribution, given the data vector and the covariancematrix computed in H20. This yields an estimate of the shift in Ω_{m} and σ_{8} introduced by varying depth.
As can be seen in Fig. 5, the impact of varying depth is insignificant compared to the uncertainties for a KiDS450like survey. To get a rough estimate for the impact on future surveys, we divide our covariancematrix by a factor of 30 to model a KiDS15 000like survey, which approximately accounts for the increased survey area of LSST and Euclid with respect to KiDS450. Here the impact on Ω_{m}, σ_{8} and S_{8} is significant at the level of approximately 1σ. As our modified covariancematrix does not account for the factor of ∼4 expected increase in galaxy number density for LSST and Euclid, we note that this is likely to be a lower estimate for the significance of the effect. Even though Euclid is a spacebased mission and, therefore, will not suffer from variable atmospheric effects, the key photometric redshift measurement uses data from several groundbased surveys, including KiDS. Placing a selection criteria on redshift estimation success, will therefore lead to depth variations in the source galaxy sample. While the data from LSST will be practically free of variations in depth after 10 years of observations, the first few years data will include significant depth variation. The impact may be even stronger than the KiDSlike analysis presented here as the multiband KiDS depth variation was minimised using seeingdependent data acquisition. This is in contrast to the seeingagnostic multiband cadence of LSST.
Fig. 5.
Recovered cosmological parameters for a variabledepth (Observed) and uniformdepth (Reference) KiDS450like (left) and KiDS15 000like survey (right). The KiDS450like figure was computed using the covariance matrix of H20. For the KiDS15 000like survey, we divided the covariance matrix of H20 by 30. This approximately accounts for the increased survey area of nextgeneration experiments, but does not factor in the increased number density and higher redshifts. Hence, this exercise provides a rough indication of the significance of varying depth effects in stage IV surveys. Both figures were computed using a fiducial cosmology of Ω_{m} = 0.25 and σ_{8} = 0.85. 

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Calculating the correction for varying values of Ω_{m} and σ_{8}, reveals a nontrivial dependence on the cosmology, which can be seen in Fig. C.1. For various combinations of Ω_{m} and σ_{8} within the 95% confidence limit of KV450, we report a variation in of a few percentage points on small scales.
4.3. Variable depth contribution to Bmodes
To check for remaining systematics, a weak lensing signal can be divided into two components, the socalled E and Bmodes (Crittenden et al. 2002; Schneider et al. 2002b). To leading order, Bmodes cannot be created by astrophysical phenomena and are thus an excellent test for remaining systematics. Direct E and Bmode decomposition for cosmic shear surveys can be provided by Complete Orthogonal Sets of E and Bmode Integrals (COSEBIs, Schneider et al. 2010, hereafter S10), as they can easily be applied to real data. We note that the nonexistence of Bmodes does not necessarily imply that the sample is free of remaining systematics. To estimate the Bmodes created by this effect, we extract the COSEBIs from the correlation functions, , that have been modified under the semianalytic model, and from a reference set of correlation functions ξ_{±}. To be most sensitive to the effect of varying depth, we choose logarithmic COSEBIs with an angular range of to θ_{max} = 72′. As the Bmodes of the reference correlation functions are zero, they are a good test for numerical errors in the calculations. Motivated by the discussion in Asgari et al. (2017), we calculated the COSEBIs from correlation functions binned in 400 000 linear bins, which yielded neglible numerical errors. We report a consistent Bmode behaviour across all redshiftbins, which can be seen in Fig. 6. However, we compare the Bmodes to the ones measured by Asgari et al. (2020) in KV450, which were consistent with zero. Since the Bmodes created by varying depth are smaller than these by a factor of 50, we conclude that this effect cannot create measurable Bmodes in the KV450 survey. It should be noted that the difference in Emodes is as large as the Bmodes, which suggests that any significant change in the cosmological parameters due to varying depth will also yield a significant detection of Bmodes. Additionally, the created pattern is very characteristic, which makes it easy to recognize in a Bmode analysis of an actual survey (see Asgari et al. 2019).
Fig. 6.
Difference in the Emodes (top left) and Bmodes (bottom right) between the reference and the observed correlation functions. For comparison: Scaled total Emodes of the reference correlation function E_{n} and scaled Bmodes measured in the KV450 survey . All E and Bmodes were calculated using the logarithmic COSEBIs in S10 for an angular range of , θ_{max} = 72′. 

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5. Discussion
With our semianalytic model we describe the impact of varying depth in groundbased cosmic shear surveys. During our analysis we made several simplifications, which we discuss below.
In the most general terms, we analyze the effects of a positiondependent selection function on cosmic shear surveys. In our analysis, this selection function was governed by the KiDS rband depth of a pointing. This neglects a number of other effects: The depth in different bands and the seeing of a pointing will also modify the number densities and redshift distributions on the scale of a pointing (although those variations are also correlated with rband depth and thus at least partly accounted for), whereas dithering strategies as well as imperfections in the telescope and CCD cause modifications on subpointing scales. However, several tests showed that these effects are subdominant compared to the variations caused by the rband depth.
We assumed an uncorrelated distribution of the depthfunction and neglected boundary effects as well as the sample variance of the depthdistribution between pointings. While the boundary effects arising from a finite survey footprint have a small impact on the shape of the function E(θ)^{7}, the governing factor is the sample variance of the depthdistribution. We assumed that the probability for any pointing to be in quantile n is exactly the expectation value, namely 1/10. While this would be true for an infinitely large survey with an uncorrelated distribution of the depthfunction, it does not necessarily hold for a real survey. However, our analysis in Appendix B suggests that these effects are not significant for the KV450 survey. In the models, we also assumed an uncorrelated distribution of the depthfunction. As can be seen in Fig. 4, this approximation introduces a small error when compared to the simulations.
In our MCMC runs, we did not account for degeneracies with other cosmological parameters or observational effects. In particular, intrinsic alignments and baryon feedback also modify the correlation functions primarily on small scales, so they are likely to be degenerate with the effect of varying depth (Troxel & Ishak 2015). In an MCMC run that includes these nuisance parameters, we suspect that the parameters for intrinsic alignments and baryon feedback change to mitigate this effect, so that the impact on cosmological parameters will be smaller than in our results.
Despite these repercussions, we are confident to say that the effects of varying depth are not significant for the KV450 survey. In particular, this means that a varying depth cannot explain the tension between observations of the lowredshift Universe and results from analysis of the CMB.
For nextgeneration surveys like Euclid and LSST, we have demonstrated that if variable depth is unaccounted for in the analysis, the resulting bias is likely to be significant. A detailed LSST and Euclid study that uses a realistic variation of depth should therefore be conducted. If these studies reach a similar conclusion, variabledepth bias could be circumvented in likelihood analyses by including a cosmologydependent correction for this effect using the semianalytical model presented in this paper (see Fig. C.1).
Although the analytic model (Eq. (12)) does not fully describe the effect, it can be used as a valid approximation to estimate the importance of varying depth for an arbitrary survey: As can be seen in Fig. 2, in the KiDSsurvey the characteristic changes both in redshift and number density for the third redshift bin are about 0.1. Setting ⟨n^{i}n^{j}⟩=⟨w^{i}w^{j}⟩=⟨n^{i}w^{j}⟩ = 0.01 yields , which roughly agrees with the actual results (see Fig. 4). For a survey with only half the variation in depth, one gets . This method allows us to estimate a threshold for an acceptable variation of depth, given a required precision for the shear correlation functions.
Additionally, it is interesting to note that E(θ) is the azimuthal average of the function E(θ) derived in Appendix A.2, which is not isotropic. Therefore, it would be possible to observe a directiondependent correlation function in future surveys. An anisotropy in the observed correlation function could be a sign for the influence of varying depth.
The variations in depth will also affect the covariance of a survey, both because they modify the signal and because they introduce an additional term of sample variance in terms of the distribution of depth. This effect will be investigated in a forthcoming publication (Joachimi et al., in prep.; Lin et al., in prep.).
Instead of using the actual number of galaxies, we take the effective number density, as defined in Kuijken et al. (2015), scaled by the respective survey area. Due to this, we account for different weighting of galaxies in the shear correlation functions as well as in the average redshift distribution.
For the calculation of the shear correlation functions we use NICAEA (Kilbinger et al. 2009) For the power spectrum on nonlinear scales, we use the method of Takahashi et al. (2012).
The effect on ξ_{−} is much stronger due to the fact that in Eq. (13), ξ_{+} is computed by filtering the power spectrum with the 0th order Bessel function. This function peaks at ℓθ = 0, meaning that for all values of θ, the correlation function ξ_{+} is sensitive to small values of ℓ, corresponding to large separations θ. However, ξ_{−} is obtained by filtering with the 4th order Bessel function, which peaks at approximately ℓθ ≈ 5, so for different θ this function is sensitive to varying parts of the convergence power spectrum. A more detailed analysis of this can be found in the appendix of Köhlinger et al. (2017).
Acknowledgments
We thank the anonymous referee for very constructive and helpful comments. The results in this paper are based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme IDs 177.A3016, 177.A3017, 177.A3018 and 179.A2004, and on data products produced by the KiDS consortium. The KiDS production team acknowledges support from: Deutsche Forschungsgemeinschaft, ERC, NOVA and NWOM grants; Target; the University of Padova, and the University Federico II (Naples). We acknowledge support from the European Research Council under grant numbers 770935 (HH,JLvdB) and 647112 (CH,MA,CL). SH acknowledges support from the German Research Foundation (DFG SCHN 342/13). H. Hildebrandt is supported by a Heisenberg grant of the Deutsche Forschungsgemeinschaft (Hi 1495/51). CH acknowledges support from the Max Planck Society and the Alexander von Humboldt Foundation in the framework of the Max PlanckHumboldt Research Award endowed by the Federal Ministry of Education and Research. KK acknowledges support by the Alexander von Humboldt Foundation. TT acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie SklodowskaCurie grant agreement No 797794. Some of the results in this paper have been derived using the HEALPix (Górski et al. 2005) package. This research has made use of NASA’s Astrophysics Data System and adstex (https://github.com/yymao/adstex). Author contributions: all authors contributed to the development and writing of this paper. The authorship list is given in two groups: the lead authors (SH, PS, HHi) followed by key contributors to the scientific analysis in alphabetical order.
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Appendix A: Detailed calculations
A.1 Calculation of the power spectrum
Here we perform the calculation for the observed power spectrum P^{obs}(ℓ). For this, we assume an infinitely large field in order to perform our integration over ℝ^{2}. In reality, finite field effects would play a role. We begin with the calculation of the correlation for the Fourier transformed shear:
It is important to keep in mind, that the ensemble averages of the weight function are independent of the ensemble averages of the shear values, meaning ⟨W(θ)γ(θ)⟩ = ⟨W(θ)⟩⟨γ(θ)⟩. We can define W(θ) = 1 + w(θ) with ⟨w(θ)⟩ = 0, which leads to the expression
where in the final step we have used that the average vanishes. Up until now, we have not specified our weightfunction w. We parametrize it as
Here, the w_{α} are random variables, drawn from the random distribution describing the survey depths. For the Fourier transform we compute:
where
is a 2dimensional sinc function. Assuming an uncorrelated weightdistribution (⟨w_{α}w_{β}⟩ = 0 for α ≠ β) and setting for each α, we get
Using this result, we can obtain the observed power spectrum
by performing the ℓ′integration in (A.2):
which is a convolution of the power spectrum and the 2dimensional sinc function (A.5). We note that due to the statistical inhomogeneity of the field, many usually adapted conventions fail. In particular, ⟨γ(θ)γ^{*}(θ′)⟩ does not only depend on the separation vector θ′ − θ, but also on the position θ. For example, the Fourier transform of the observed power spectrum yields ⟨γ(0)γ^{*}(θ)⟩, but not the shear correlation function.
A.2 The function E(θ)
When computing the shear correlation between a pair of galaxies, it is of central importance whether or not those two galaxies lie in the same pointing. We want to model the probability that a pair of galaxies with separation θ lies in the same pointing by the function E(θ), which we will derive here: Given one square field of length L (in our case L = 60′) and a separation vector θ = (θ_{1}, θ_{2}), without loss of generality we can assume θ_{1}, θ_{2} ≥ 0. The dashed square in Fig. A.1 represents all possible positions that the first galaxy can take, such that the second galaxy is still within the same pointing. The volume of this square equals
Fig. A.1.
Graphic representation on how to obtain the function E(θ). For a separation vector θ, the dashed square represents the area of galaxies that have their partner in the same pointing. 

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where ϕ represents the polar angle of the vector θ. The function E(θ) then simply equals V(θ,ϕ)/L^{2}. To exclude negative volumes (which could occur when θ > 1 holds), we need to add the Heaviside step function ℋ:
As E(θ) is not isotropic, in order to obtain the function E(θ) = E(θ), we need to calculate the azimuthal average of Eq. (A.10) over all angles ϕ. While the case θ_{1}, θ_{2} ≥ 0 certainly does not hold for all angles ϕ, we can omit the other cases by making use of the symmetry of the problem.
A.3 Calculation of the shear correlation functions
Given a set of galaxies, we calculate the shear correlation function via
Here, w represents the lensing weight of the galaxy, whereas ϵ is its (complex) ellipticity and θ its position on the sky. We have defined the function Δ as
where we assume dθ ≪ θ. We define 𝒩 as the number of pointings in the survey and as the set of galaxies in pointing k and tomographic redshift bin i. The numerator in Eq. (A.12) then transforms to:
When we denote the probability that pointing k is of quantile m by and assume that the product always equals its expectation value, we can set the numerator as
The term (A.15.a) denotes all galaxies that lie within distance interval [θ, θ + dθ] of galaxy a, and are in the same pointing as galaxy a. This term is equal to the (weighted) number density of galaxies in the pointing multiplied by 2πθ dθ E(θ).
The term (A15.b) denotes all galaxies within distance interval [θ, θ + dθ] of galaxy a, that are not in the same pointing as galaxy a. This is equal to the number density of galaxies in the respective pointings multiplied by 2πθ dθ [1 − E(θ)].
If we assume that said number density in a pointing is equal to the number density in the quantile it belongs to, , and set , the numerator becomes
The term denotes the number of galaxies in pointing k, which we set as the number density of galaxies in the respective quantile multiplied with the area A of the pointing. Applying this and setting , the numerator reads
The same line of argumentation can be applied to the denominator, which then reads:
Taking the ratio of the two quantities, and setting , we see that Eqs. (A.12) and (19) are the same.
Appendix B: Finite field effects
In this appendix, we outline how to calculate the correction of the correlation functions for a finite survey with a potentially correlated distribution of depth between pointings. Essentially, this boils down to the calculation of from Eq. (16). We calculate this weighting by the geometrical probability that a pair of galaxies of separation θ is of quantiles m and n, 𝒫(m, nθ), weighted by the respective number of galaxies in the quantiles :
At first, we define functions E_{ab}(θ) as the probabilty that a galaxy pair of separation θ is in pointings of distance (a, b). This situation is depicted in Fig. B.1. Due to symmetry, for the azimuthal average of the functions, E_{ab}(θ) = E_{−ab}(θ) = E_{ba}(θ) holds for all combinations of a and b. We note that E_{00}(θ) = E(θ) and ∑_{a, b}E_{ab}(θ) ≡ 1.
Fig. B.1.
Graphic representation of the definitions of E_{ab}(θ). When the first galaxy is in the bottom left pointing, the probability to find the second galaxy in a pointing of distance (a, b) is E_{ab}(θ). 

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Let 𝒫^{*}(m, na, b) denote the probability that two pointings of distance (a, b) are of quantile m and n (which is directly calculable from a given survey footprint). Then the following equation holds:
Note that the expectation value of 𝒫^{*}(m, na, b) for uncorrelated distributions is
where δ_{mn} denotes the Kronecker delta. Keeping in mind that
we can use the expectation value (B.3) to calculate (B.2) as a consistency check. In that case, we receive the same value for the coefficients in (B.1) as we have in Eq. (18) in Sect. 3 for the case of an infinite footprint and uncorrelated distribution of depth.
The E_{ab} can all be calculated analytically, similar to our method in Appendix A.2. We again assume a selection of square fields with side length L, and later set L = 60′ to adapt to the KV450 survey. As an example, for E_{01} we have several possible situations, depicted in Fig. B.2. Setting E_{ab}(θ) = V(θ, ϕ)/L^{2}, we define
Fig. B.2.
Representation of how to calculate E_{01}(θ) for different values of θ. For θ sin (ϕ) < L, as depicted in the left part, the volume of the dashed rectangle is V(θ, ϕ) = θ sin (ϕ)[L − θ cos (ϕ)]. For θ sin (ϕ) > L, as depicted in the right part, the volume of the dashed rectangle is V(θ, ϕ) = [2L − θ sin (ϕ)] [L − θ cos(ϕ)]. 

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With some geometric considerations, we compute:
Naturally, to calculate those functions for all possible combinations would be rather tedious, however they are simple to determine numerically (compare Fig. B.3). A plot of these functions can be found in Fig. B.4.
Fig. B.3.
Visualization of the numerical computation for E_{01}(θ). For a circle of radius θ, the length of the red arc divided by 2π represents the fraction of galaxies within the respective pointing. This value needs to be integrated for all possible centers of the circle in the pointing. That procedure is straightforward to expand for other E_{ab}(θ). 

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Fig. B.4.
The functions E_{ab}(θ) for the first few possible combinations. 

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We sample several realizations of a random depthdistribution for a 100 deg^{2}field, a 450 deg^{2}field and a 1000 deg^{2}field. For each realization we extract the Function 𝒫^{*}(m, na, b) and, using Eq. (B.2), calculate the ratio . Afterwards, we compute the variance of these ratios. As can be seen from Fig. B.5, the effect is quite significant for a 100 deg^{2}field, but almost negligible for a 1000 deg^{2}field. This leads to the assumption that both for the KV450 survey as well as for all nextgeneration cosmic shear surveys, finite field effects do not need to be accounted for. However, if the distribution of depth is correlated in the surveys, that might have a noticeable impact on the results.
Fig. B.5.
2σcontours of the corrections for the correlation functions for a 100 deg^{2} field (blue), a 450 deg^{2} field (red) and a 1000 deg^{2} field (green). As can be seen, the variance of the variation is small for a 450 deg^{2} field and barely noticeable for a 1000 deg^{2} field. 

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Appendix C: Additional figure and table
Limiting magnitudes for the ten quantiles.
Fig. C.1.
Correction to the correlation functions in varying cosmologies. Depicted here are three flat sample cosmologies, where values within the 98% CL of the KV450 survey were sampled. 

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All Tables
All Figures
Fig. 1.
Probability E(θ) that a random pair of galaxies of distance θ lie in the same 1 deg^{2} pointing. 

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In the text 
Fig. 2.
Weighted number of galaxies N and average redshift ⟨z⟩ in the KiDS+VIKING450 survey (KV450, Wright et al. 2019) in pointings of different depth for each of the five tomographic bins used in H20. Each color corresponds to one redshift bin of H20. A single point represents one quantile of the respective redshift bin, where the fainter points denote pointings of shallower depth. 

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In the text 
Fig. 3.
Source redshift distributions for a selection of very shallow pointings (blue), average pointings (yellow) and very deep pointings (green). The percentage points in the legend denote to which quantile a pointing belongs, when all are ordered by their depth. 

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In the text 
Fig. 4.
Ratio of correlation functions measured for a uniform depth survey, ξ_{±}, and a KiDSlike variable depth survey, , crosscorrelating five tomographic bins (as denoted in the upper left corner of each panel). The upper left triangle depicts the ratios of ξ_{+}, whereas the lower right triangle depicts the ratios of ξ_{−}. Results from mock KiDSlike data (red) can be compared to analytic models from Sect. 2 (average redshift weighting, green), and the semianalytic model from Sect. 3 (blue solid). Mock data is limited to the angular regime which is not significantly impacted by resolution effects. As the mocks only take galaxies with z < 2 into account, they slightly underestimate the effect. Applying the same redshiftcutoff to the semianalytical model (blue dashed) yields a nearperfect agreement on subpointing scales. Therefore, the seemingly better agreement between the mocks and the analytic method is purely coincidental. The models adopt the bestfit cosmology of H20. 

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In the text 
Fig. 5.
Recovered cosmological parameters for a variabledepth (Observed) and uniformdepth (Reference) KiDS450like (left) and KiDS15 000like survey (right). The KiDS450like figure was computed using the covariance matrix of H20. For the KiDS15 000like survey, we divided the covariance matrix of H20 by 30. This approximately accounts for the increased survey area of nextgeneration experiments, but does not factor in the increased number density and higher redshifts. Hence, this exercise provides a rough indication of the significance of varying depth effects in stage IV surveys. Both figures were computed using a fiducial cosmology of Ω_{m} = 0.25 and σ_{8} = 0.85. 

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In the text 
Fig. 6.
Difference in the Emodes (top left) and Bmodes (bottom right) between the reference and the observed correlation functions. For comparison: Scaled total Emodes of the reference correlation function E_{n} and scaled Bmodes measured in the KV450 survey . All E and Bmodes were calculated using the logarithmic COSEBIs in S10 for an angular range of , θ_{max} = 72′. 

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In the text 
Fig. A.1.
Graphic representation on how to obtain the function E(θ). For a separation vector θ, the dashed square represents the area of galaxies that have their partner in the same pointing. 

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In the text 
Fig. B.1.
Graphic representation of the definitions of E_{ab}(θ). When the first galaxy is in the bottom left pointing, the probability to find the second galaxy in a pointing of distance (a, b) is E_{ab}(θ). 

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In the text 
Fig. B.2.
Representation of how to calculate E_{01}(θ) for different values of θ. For θ sin (ϕ) < L, as depicted in the left part, the volume of the dashed rectangle is V(θ, ϕ) = θ sin (ϕ)[L − θ cos (ϕ)]. For θ sin (ϕ) > L, as depicted in the right part, the volume of the dashed rectangle is V(θ, ϕ) = [2L − θ sin (ϕ)] [L − θ cos(ϕ)]. 

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In the text 
Fig. B.3.
Visualization of the numerical computation for E_{01}(θ). For a circle of radius θ, the length of the red arc divided by 2π represents the fraction of galaxies within the respective pointing. This value needs to be integrated for all possible centers of the circle in the pointing. That procedure is straightforward to expand for other E_{ab}(θ). 

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In the text 
Fig. B.4.
The functions E_{ab}(θ) for the first few possible combinations. 

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In the text 
Fig. B.5.
2σcontours of the corrections for the correlation functions for a 100 deg^{2} field (blue), a 450 deg^{2} field (red) and a 1000 deg^{2} field (green). As can be seen, the variance of the variation is small for a 450 deg^{2} field and barely noticeable for a 1000 deg^{2} field. 

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In the text 
Fig. C.1.
Correction to the correlation functions in varying cosmologies. Depicted here are three flat sample cosmologies, where values within the 98% CL of the KV450 survey were sampled. 

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In the text 
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