Given the enormous galaxy data bases of modern sky surveys, parametrizing galaxy morphologies is a very challenging task due to the huge number and variety of objects. We assess the different problems faced by existing parametrization schemes (CAS, Gini, M20, Sérsic profile, shapelets) in an attempt to understand why parametrization is so difficult and in order to suggest improvements for future parametrization schemes.
We demonstrate that morphological observables (e.g. steepness of the radial light profile, ellipticity, asymmetry) are intertwined and cannot be measured independently of each other. We present strong arguments in favour of model-based parametrization schemes, namely reliability assessment, disentanglement of morphological observables and point spread function modelling. Furthermore, we demonstrate that estimates of the concentration and Sérsic index obtained from the Zurich Structure & Morphology catalogue are in excellent agreement with theoretical predictions. We also demonstrate that the incautious use of the concentration index for classification purposes can cause a severe loss of the discriminative information contained in a given data sample. Moreover, we show that, for poorly resolved galaxies, concentration index and M20 suffer from strong discontinuities, i.e. similar morphologies are not necessarily mapped to neighbouring points in the parameter space. This limits the reliability of these parameters for classification purposes. Two-dimensional Sérsic profiles accounting for centroid and ellipticity are identified as the currently most reliable parametrization scheme in the regime of intermediate signal-to-noise ratios and resolutions, where asymmetries and substructures do not play an important role. We argue that basis functions provide good parametrization schemes in the regimes of high signal-to-noise ratios and resolutions. Concerning Sérsic profiles, we show that scale radii cannot be compared directly for profiles of different Sérsic indices. Furthermore, we show that parameter spaces are typically highly non-linear. This implies that significant caution is required when distance-based classification methods are used.