In this paper we describe STECMAP (STEllar Content via Maximum A Posteriori), a flexible, non-parametric inversion method for the interpretation of the integrated light spectra of galaxies, based on synthetic spectra of single stellar populations (SSPs). We focus on the recovery of a galaxy's star formation history and stellar age-metallicity relation. We use the high-resolution SSPs produced by PÉGASE-HR to quantify the informational content of the wavelength range λλ= 4000-6800. Regularization of the inversion is achieved by requiring that the solutions are relatively smooth functions of age. The smoothness parameter is set automatically via generalized cross validation.

A detailed investigation of the properties of the corresponding simplified linear problem is performed using singular value decomposition. It turns out to be a powerful tool for explaining and predicting the behaviour of the inversion, and may help designing SSP models in the future. We provide means of quantifying the fundamental limitations of the problem considering the intrinsic properties of the SSPs in the spectral range of interest, as well as the noise in these models and in the data. We demonstrate that the information relative to the stellar content is relatively evenly distributed within the optical spectrum. We show that one should not attempt to recover more than about eight characteristic episodes in the star formation history from the wavelength domain we consider. STECMAP preserves optimal (in the cross validation sense) freedom in the characterization of these episodes for each spectrum.

We performed a systematic simulation campaign and found that, when the time elapsed between two bursts of star formation is larger than 0.8 dex, the properties of each episode can be constrained with a precision of 0.02 dex in age and 0.04 dex in metallicity from high-quality data [R= 10000, signal-to-noise ratio (SNR) = 100 per pixel], not taking model errors into account. We also found that the spectral resolution has little effect on population separation provided low- and high-resolution experiments are performed with the same SNR per Å. However, higher spectral resolution does improve the accuracy of metallicity and age estimates in double-burst separation experiments. When the fluxes of the data are properly calibrated, extinction can be estimated; otherwise the continuum can be discarded or used to estimate flux correction factors.

The described methods and error estimates will be useful in the design and in the analysis of extragalactic spectroscopic surveys.