Bayesian inference for the ultrasonic characterization of rigid porous materials using reflected waves by the first interface

Abstract

The purpose of this paper is to present a method for the ultrasonic characterization of air-saturated porous media, by solving the inverse problem using only the reflected waves from the first interface to infer the porosity, the tortuosity, and the viscous and thermal characteristic lengths. The solution of the inverse problem relies on the use of different reflected pressure signals obtained under multiple obliquely incident waves, in the time domain. In this paper, the authors propose to solve the inverse problem numerically with a first level Bayesian inference method, summarizing the authors' knowledge on the inferred parameters in the form of posterior probability densities, exploring these densities using a Markov-Chain Monte-Carlo approach. Despite their low sensitivity to the reflection coefficient, it is still possible to extract the knowledge of the viscous and thermal characteristic lengths, allowing the simultaneous determination of all the physical parameters involved in the expression of the reflection operator. To further constrain the problem and guide the inference, the knowledge of a particular incident angle is used at one's advantage in order to more precisely define the thermal length, by effectively yielding a statistical relationship between tortuosity and characteristic length ratio.

Inverse identification of a higher order viscous parameter of rigid porous media in the high frequency domain

“…In practice, it has been observed that foams with higher tortuosities might have lower characteristic viscous lengths, thus denoting a negative correlation between those two parameters, as opposed to the positive one observed during the inference. The same finding was reported in [28], where a correlation between φ and α ∞ was evidenced when the inference was performed with reflected signals.…”

“…We preferred to give what we thought was a reasonable value of Σ . Furthermore, we know [21,38,22,25,23] that the thermal characteristic length Λ is usually a factor 2 − 3 greater than Λ, and that its identification is problematic due to a low sensitivity [39,28]. As a result, Λ is set to 3Λ in the following, as was done in Refs.…”

“…This showed the possibility of taking into account the uncertainty of the measurement during the identification procedure. A similar Bayesian approach was then followed [28], where ultrasonic reflected signals from the first interface were used to retrieve the posterior probability densities of the porosity, tortuosity and the viscous and thermal characteristic lengths, thus extending the work presented in Ref. [9].…”

Multiple Waves Propagate in Random Particulate Materials

“…Still, the relative error of Im (R M −R O ) ≈ 10%. The imaginary part of the reflection coefficient, and where it changes sign, can be key for characterising random microstructure [48]. The real part of the reflection coefficients is not shown, as the relative errors for the real part are even smaller.…”

How Do Acoustic Materials Work?

“…The relationship between the acoustic impedances does not only have to do with the modification of the amplitude of the incident wave: it is also related to the fraction of the incident energy that will be reflected or transmitted. Most of the acoustic properties of materials can be studied by the application of new laboratory and numerical methods as ultrasonic characterization [4], inverse characterization with basis on the acoustic impedance measurement [5], or ensemble averaged scattering [6].…”