  We develop a method for mapping the anharmonic lattice potential using the
time-dependent electric field of the transmitted pulse through thin sample
supported by a substrate of non-negligible thickness. Assuming linear
propagation in the substrate we fully take into account internal reflection in
it while the sample can show arbitrary nonlinear response. We examine the
effect of frequency averaging appropriate for broad-band pulse and compare the
results taking into account the full frequency dependence. We illustrate the
procedure applying it to a model based on recently observed ferroelectric
soft-mode nonlinearity in SrTiO$_3$.
