  We study the elastic scattering time $\tau_\mathrm{s}$ of ultracold atoms
propagating in optical disordered potentials in the strong scattering regime,
going beyond the recent work of J. Richard \emph{et al.} \textit{Phys. Rev.
Lett.} \textbf{122} 100403 (2019). There, we identified the crossover between
the weak and the strong scattering regimes by comparing direct measurements and
numerical simulations to the first order Born approximation. Here we focus
specifically on the strong scattering regime, where the first order Born
approximation is not valid anymore and the scattering time is strongly
influenced by the nature of the disorder. To interpret our observations, we
connect the scattering time $\tau_\mathrm{s}$ to the profiles of the spectral
functions that we estimate using higher order Born perturbation theory or
self-consistent Born approximation. The comparison reveals that self-consistent
methods are well suited to describe $\tau_\mathrm{s}$ for Gaussian-distributed
disorder, but fails for laser speckle disorder. For the latter, we show that
the peculiar profiles of the spectral functions, as measured independently in
V. Volchkov \emph{et al.} \textit{Phys. Rev. Lett.} \textbf{120}, 060404
(2018), must be taken into account. Altogether our study characterizes the
validity range of usual theoretical methods to predict the elastic scattering
time of matter waves, which is essential for future close comparison between
theory and experiments, for instance regarding the ongoing studies on Anderson
localization.
