1 Introduction

In the last decade the simulation of synthetic Hertzsprung-Russell and colour-magnitude diagrams (hereafter respectively referred to as HRDs and CMDs) has advanced at a rapid pace. It has been applied successfully in various studies of young clusters in the Magellanic Clouds (Chiosi et al. 1989; Bertelli et al. 1992; Han et al. 1996; Mould et al. 1997; Vallenari et al. 1990, 1992, and 1994a,b), dwarf galaxies in the Local Group (Aparicioem&emGallart 1995, 1996; Aparicio et al. 1996, 1997a,b; Ferraro et al. 1989; Gallart et al. 1996a$\,$-$\,$c; Han et al. 1997; Tolstoy 1995, 1996; Tosi et al. 1991), open clusters in our Galaxy (Aparicio et al. 1990; Carraro et al. 1993, 1994; Gozzoli et al. 1996), and the structure of our Galaxy (Bertelli et al. 1994, 1995, 1996; Ng 1994, 1997a,b; Ngem&emBertelli 1996a,b; Ng et al. 1995, 1996a,b, 1997).

Generally all studies focus in the first place on matching the morphological structures at different regions in the CMDs (Gallart 1998). In the recent years a good similarity is obtained between the observed and the simulated CMDs. Unfortunately, the best fit is in some cases distinguished by eye. The morphological differences are large enough to do this and the eye is actually guided by a detailed knowledge of stellar evolutionary tracks. However, the stellar population technique has improved considerably and an objective evaluation tool is needed, to distinguish quantitatively one model from another.

Bertelli et al. (1992,em1995) and Gallart et al. (1996c) defined ratios to distinguish the contribution from different groups of stars. The ratios are defined so that they are sensitive to the age, the strength of the star formation burst and/or the slope of the initial mass function. Vallenari et al. (1996a,b) demonstrated that this is a good method to map the spatial progression of the star formation in the Large Magellanic Cloud. Robin et al. (1996), Han et al. (1997) and Mould et al. (1997) use a maximum likelihood method to find the best model parameter(s), while Chen (1996a,b) adopted a multivariate analysis technique. Different models can be quantitatively sampled through Bayesian inference (Tolstoy 1995; Tolstoyem&emSaha 1996) or a chi-squared test (Dolphin 1997).

In principle one aims with stellar population synthesis to generate a CMD which is identical to the observed one. The input parameters reveal the evolutionary status of the stellar aggregate under study. To obtain a good similarity between the observed and the simulated CMD one needs to implement in the model in the first place adequately extinction, photometric errors and crowding. It goes without saying that the synthetic population ought to be comparable with the age and metallicity (spread) of the stellar aggregate. Only with a proper choice of these parameters, one can start to study in more detail the stellar initial mass function and the star formation history for an aggregate.

This paper describes a quantitative evaluation method based on the combination of a chi-squared and Poisson merit function. It allows one to select the best model from a series of models. In the next section this method is explained and Monte-Carlo simulations are made, to display how the diagnostic diagrams of the residual points can be employed. It is demonstrated that the non-fitting, residual points provide a hint about the parameter that needs adjustment in order to improve the model. This paper ends with a discussion about the method and the diagnostic diagrams together with their limitations.

It is emphasized that this paper deals with a description of a quantitative evaluation method for CMDs. Aspects related with the implementation of an automated CMD fitting program and comparison with simulated or real data sets will not be considered, because they are not relevant for the general validity of the method described in the following section.