A. Derivation of Probability Density Function (PDF)

Consider an elastic (lossless) heterogeneous medium, which consists of multiple isotropic non-overlapping phases with elastic bulk moduli (K_j), shear moduli (G_j), mass density values (ρ_i) and the volume fractions of each phase (c_i; Σc_i = 1). The subscripts i denotes phase index, respectively. For simplicity, the multi-phase medium can be effectively treated as isotropic and macroscopically homogeneous. Therefore, this medium which can be characterized by effective bulk modulus (K), effective shear modulus (G) and effective mass density (ρ). Consequently, given a medium, K and G are two unknown random variables. Among most biological soft tissues, variations in mass density are small as compared to those in the shear modulus (Duck, 1990). We treat the effective mass density ρ as a constant to simplify the derivation.

Formally, we now assume that an arbitrary multi-phase medium can be specified by a set of physical properties J = (K_i, G_i, ρ_i, c_i). Given the multi-phase medium whose properties can be specified by the setJ, under the framework of effective medium theory, the macroscopic SWS of the medium (V_s) can be related to the effective shear modulus G by (Wells and Liang, 2011; Doherty et al., 2013),

where ρ is the effective mass density. Eqn. (1) is only valid with low-frequency shear wave propagation in elastic (lossless) media.

In information theory, Jaynes' information entropy (H) (Jaynes, 1957b) of the effective shear modulus G is given below,

where m(G) can be used to represent a prior information and p(G) is a (histogram) distribution of G. Based on the principle of ME, the Jaynes' entropy H reaches its maximum if p(G) reflects the data distribution in the best possible way, given the priori information m(G). Consequently, maximization of H leads to an optimized p(G) that best describes the given data G. In Eqn. (2), m(G) has been set to 1, meaning that no prior information regarding the distribution of G is available to us.

To ensure that the optimized p(G) is a good fit of the given data G, the maximization of H has to be constrained by the statistical moments of the data G as follows,

where G_E and G_D are mean and variance of the effective shear modulus G. It is important to note that G is a random variable and, now p(G) becomes the PDF of the given effective modulus data G. A Lagrange multiplier method (Bertsekas, 1995) was used to maximize H in conjunction with Eqn. (1). This process yields an analytical expression of PDF of the macroscopic SWS (V_s),

A complete derivation of Eqn. (6) is provided in Appendix A.