the optimal profile that explains the data.

However, since the inverse problem is ill-posed, a unique solution does not exist. We generate a probability density plot for the transmission coefficients using Gibbs sampling to explore adjacent regions of the optimal transmission coefficient profile. We first randomly generate around 1000 profiles by perturbing the optimal profile with a smooth function defined using the following formula $$\delta = A[r_1\text{cos}(2\pi\omega/\omega_{max}r_2+2\pi r_3)+r_4\text{sin}(2\pi\omega/\omega_{max}r_5+2\pi r_6)], \label{eq:perturbation}$$ where the amplitude of the perturbation $A$ is 0.1, and $r_1$, $r_2$, $r_3$, $r_4$, $r_5$ and $r_6$ are random numbers between 0 to 1. We evaluate the objective function at all the perturbed profiles and recorded the values. Then, we start the Gibbs sampling process. At each iteration, we randomly draw a profile, $a$, from the stored population and compare the value of its corresponding objective function, $f_{n}$ to the one from the previous step, $f_{n-1}$ evaluated at profile $b$. If $f_{n}$ is less than $f_{n-1}$, we accept $a$ and kept $f_{n}$. If not, a random number $r$ is drawn and compared to $u = p/(1+p)$, where $$p = \text{exp}\left(\frac{f_{n}-f_{n-1}}{T_0}\right). \label{eq:GibbsSampling}$$ If $r$ is smaller than $u$, then we accept $a$ and kept $f_{n}$. If not, we reject $a$ and 