Selection biases in empirical p(z) methods for weak lensing

To measure the mass of foreground objects with weak gravitational lensing, one needs to estimate the redshift distribution of lensed background sources. This is commonly done in an empirical fashion, i.e. with a reference sample of galaxies of known spectroscopic redshift, matched to the source population. In this work, we develop a simple decision tree framework that, under the ideal conditions of a large, purely magnitude-limited reference sample, allows an unbiased recovery of the source redshift probability density function p(z), as a function of magnitude and color. We use this framework to quantify biases in empirically estimated p(z) caused by selection effects present in realistic reference and weak lensing source catalogs, namely (1) complex selection of reference objects by the targeting strategy and success rate of existing spectroscopic surveys and (2) selection of background sources by the success of object detection and shape measurement at low signal-to-noise. For intermediate-to-high redshift clusters, and for depths and filter combinations appropriate for ongoing lensing surveys, we find that (1) spectroscopic selection can cause biases above the 10 per cent level, which can be reduced to 5 per cent by optimal lensing weighting, while (2) selection effects in the shape catalog bias mass estimates at or below the 2 per cent level. This illustrates the importance of completeness of the reference catalogs for empirical redshift estimation.