This work deals with the inverse heat diffusion problem applied to the characterization of graded materials. It capitalizes on recent advances on analytical modeling of transfer phenomena in heterogeneous materials exhibiting continuous one-dimensional variations of their properties. A method was described to build infinite sequences of analytically solvable profiles of the thermal effusivity together with the associated temperature (exact) solutions. The expressions of both are in closed-form and involve only elementary functions of a specific scale, namely the square root of the diffusion time down to the running depth. Some of these analytically solvable profiles exhibit simultaneously high versatility and parsimony which make them candidates for use as solvable splines. Splines of this type can be fused one to the other to build a composite profile aimed to fit a real effusivity profile of any shape, with the distinct advantage over classical splines (i.e. polynomial splines) that they are compatible with the heat equation in graded materials. This means that to each of these elementary profile can be associated a quadrupole that provides an exact transcription of the diffusion process therein. This simplifies considerably the temperature calculation in graded materials. A description is given of the potential of these analytical tools for the task of evaluating the depth profile of effusivity in a graded material from the measured temperature on the free surface when submitted to a time-dependent heat flux. This inversion task is encountered in different contexts. The first one is photothermal non-destructive testing of industrial materials. Inversion results are presented based on theoretical temperature data corrupted by artificial noise. This allows evaluating the robustness of the inverse method. Another example is soil subsurface characterization. In this case, the Martian surface temperature as measured by the infrared radiometer onboard the InSight lander during one Martian day is used to evaluate the effusivity profile (alternatively called thermal inertia) from the uppermost layer of regolith down to the bulk. The expectation on depth-resolved uncertainty regarding effusivity is presented.