We explore quantitatively the low but sufficient sensitivity of oscillation modes to probe both the core composition and the details of the chemical stratification of pulsating white dwarfs. Until recently, applications of asteroseismic methods to pulsating white dwarfs have been far and few, and have generally suffered from an insufficient exploration of parameter space. To remedy this situation, we apply to white dwarfs the same double-optimization technique that has been used quite successfully in the context of pulsating hot B subdwarfs. Based on the frequency spectrum of the pulsating white dwarf R548, we are able to unravel in a robust way the unique onion-like stratification and the chemical composition of the star. Independent confirmations from both spectroscopic analyses and detailed evolutionary calculations including diffusion provide crucial consistency checks and add to the credibility of the inferred seismic model. More importantly, these results boost our confidence in the reliability of the forward method for sounding white dwarf internal structure with asteroseismology.