Figure 1

This figure shows realizations of the posterior distribution for the attenuation coefficient given in Eq. (24) for different values of \(\hat{\mu }\). Both the simulations and figure creation were done in Matlab 2019a²⁷, https://www.mathworks.com/. (a) This panel shows two unimodal reconstructed posterior distributions. With these distributions, the true parameter is much more likely than the DR estimate. This posterior was constructed with a layer mean of \(\langle \mu _{oct} \rangle = 0.4{\text { mm}}^{-1}\), \(\zeta = 6.87\times10^{-2}{\text { mm}}^{-1}\) and a DR estimates of \(\hat{\mu }= 0.08{\text { mm}}^{-1}\) and \(\hat{\mu }= 1.3 {\text { mm}}^{-1}\). (b) This panel shows a constructed posterior distribution which is Bi-modal and has two local maxima. For a given layer mean, the constructed distribution develops a second peak if the DR estimate used to construct the posterior is sufficiently small. This second peak can make the Maximum a Posteriori difficult due to non-convexity. In many cases, the maximum value of the Posterior distribution may sit very near the origin on this second peak. As demonstrated in this panel, often the total amount of probability mass under the addition peak is relatively small, meaning that while the initial peak is overwhelmingly the maximum likelihood. Thus, the Maximum of the posterior distribution is a poor representation for the distribution itself. In these cases an estimate for the mean is a better choice. This posterior was constructed with a layer mean of \(\langle \mu _{oct} \rangle = 0.4 {\text { mm}}^{-1}\), \(\zeta = 6.87\times10^{-2} {\text { mm}}^{-1}\), and a DR estimate of \(\hat{\mu }= 0.015 {\text { mm}}^{-1}\).