## 1. IntroductionThis is the third and final paper in a series in which we have considered the reconstruction of the surface mass density of galaxy clusters from the (weak) image distortion it imposes onto faint background galaxies. Inspired by the pioneering work of Kaiser & Squires (1993, hereafter KS), who derived a parameter-free inversion equation for the surface mass density of the cluster in terms of the tidal deflection field [which has been discovered earlier by Tyson,Valdes & Wenk (1990); earlier work on cluster mass determination by weak lensing effects include Kochanek (1990) and Miralda-Escudé (1991)] we have started to generalize the KS inversion method to include also strong clusters, i.e., cluster which are - at least nearly - capable to produce giant arcs. In Paper I (Schneider & Seitz 1995) we have analyzed the basic observable from image distortions and pointed out a global invariance transformation of the surface mass density which leaves the observable distortion invariant. In Paper II (Seitz & Schneider 1995) we have then developed an iterative procedure to reconstruct the density field, which was then applied to synthetic data and shown to work well. In the present paper, we want to generalize the treatment of Paper II in two different ways. First, the inversion procedure constructed in Paper II is not unbiased and contains boundary artefacts in the same way as the original KS method, due to the fact that the observations are always limited to a finite data field. There are now several different inversion methods which are unbiased (Schneider 1995; Kaiser et al. 1995; Bartelmann 1995, Seitz & Schneider 1996) - all of them are based on a relation found by Kaiser (1995). As was demonstrated in Seitz & Schneider (1996), these finite-field methods work well and can be applied efficiently. Second, in the earlier papers cited above it was assumed that all sources have the same effective redshift; by that we mean that all sources have about the same ratio of the angular diameter distance as measured from the lens and from the observer. This assumption is fairly well justified if the cluster redshift is relatively small, , say. For weak cluster lenses, this assumption can be easily dropped since then the (linear) reconstruction proceeds equivalently to the case that all sources are at about the `mean redshift' of the population (this will be made more precise in Sect. 4.3 below). However, if the cluster is not assumed to be weak, the consideration of the source redshift distribution becomes more difficult. We shall discuss this problem in some detail below. The rest of the paper is organized as follows: in Sect. 2 we present basic equations and our notation. In Sect. 3 we derive the dependence of the local observables on the local values of the surface mass density and the shear. In Sect. 4, we describe the nonlinear reconstruction method; after briefly reviewing the unbiased finite-field inversion method developed by Seitz & Schneider (1996) for the case of a single source redshift, we generalize this technique to the case of a redshift distribution. As in the case of a single source redshift, we also have a global invariance transformation, which, however, cannot be written in closed form in general. However, restricting the consideration to moderately strong clusters, we construct the invariance transformation explicitly. As we show in Sect. 5 and in Appendix 3, the presence of a redshift distribution of the sources provides several methods to break the invariance transformation. However, from simulations we found that among the methods considered, only that is successful in practice which makes use of the magnification effect on the number density of galaxy images (see also Broadhurst, Taylor & Peacock 1995). In contrast to Broadhurst, Taylor & Peacock (1995) and Broadhurst (1995) we do not use the local magnification to derive the local mass density, but we use the magnification averaged over the field to constrain the mean mass density in the field (in order to combine information from the shear with local magnification information, a maximum likelihood approach seems to be the best strategy - see Bartelmann et al. 1996). Essential for the success of this method is that the source counts deviate sufficiently strongly from the behaviour, which seems to be the case for galaxies with red colour. Furthermore, the method depends strongly on the assumption that the faint background galaxies are distributed rather smoothly, i.e., that no strong correlations in their angular position is present, as seems to be justified by recent investigations (e.g., Infante & Pritchet 1995). We apply our methods to synthetic data in Sect. 6 to demonstrate its feasibility. The application to an HST exposure of the cluster Cl0939+4713 will be published elsewhere (Seitz et al. 1995c). We discuss our results in Sect. 7. © European Southern Observatory (ESO) 1997 Online publication: July 3, 1998 |