Mathematical Problems in Engineering

Volume 2015, Article ID 162530, 11 pages

http://dx.doi.org/10.1155/2015/162530

## Logical Inference for Model-Based Reconstruction of Ultrasonic Nonlinearity

Nondestructive Evaluation Laboratory, Department of Structural Mechanics, University of Granada, 18071 Granada, Spain

Received 9 April 2015; Revised 25 June 2015; Accepted 7 July 2015

Academic Editor: Giacomo Innocenti

Copyright © 2015 Carlos Rus and Guillermo Rus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Quantifying the constitutive nonlinearity parameter in fluids is of key interest for understanding ultrasonic propagation and its wide implications in medical and industrial applications. However, current methods for ultrasonically measuring it show large limitations in that the signal is only valid at a reduced and unjustified spatial range away from the transducer. This is not consistent with the fact that should be constant everywhere in the fluid and independently of the ultrasonic experimental setup. To overcome this, the nonlinear wave propagation equations are rigorously derived and the ensuing differential equation is numerically solved. As a second contribution, the experimental and model information sources are treated under the information theory context to probabilistically reconstruct , providing not only its value, but also the degree of confidence on it given both sources of data. This method is satisfactorily validated testing the repeatability of in water varying distances, energies, frequencies, and transducer setups, yielding values compatible with = 3.5.

#### 1. Introduction

The acoustic nonlinearity observed as appearance of harmonics in ultrasound propagation is a consequence of the deviation from perfect linear elasticity of the compressional mechanical constitutive law.

The continuum mechanics foundation of constitutive nonlinearity was put forth by Landau and Lifshitz [1] and detailed for acoustics and in particular for ultrasonic harmonic generation by Hamilton and Blackstock [2] and others [3–6]. The second harmonic generation has indeed been observed and predicted to depend linearly on distance and quadratically on the amplitude of the fundamental, according to a proportionality constant that has been called .

Experimental quantifications of this parameter based on ultrasonic harmonic generation have been reported in the literature [7, 8], yielding values between and but with large degrees of variability and uncertainty. These methods for ultrasonically quantifying the constitutive nonlinearity parameter show large limitations in that the signal is only valid at a reduced spatial range away from the transducer, whose limits are furthermore not justified. For instance, those authors estimate from a set of measurements between 60 and 150 mm from the transducer, neglecting the full set of measurements from 0 to 300 mm without justification. In addition, although usually not explicitly specified in the literature, the measured values are highly sensitive to transducer-hydrophone alignment and focusing conditions. This is not consistent with the fact that should be constant everywhere in the fluid and independently of the geometry of the transducer that generates the ultrasonic field. On the other hand, few theoretical proposals of the value of have been derived, such as that from the atomistic anharmonicity [9], yielding a theoretical estimate of .

To shed light on the degree of knowledge on the value of , an information theory based probabilistic inverse problem [10–12] is formulated and used to reconstruct not only its value but also its range of reliability. This reconstruction problem was historically first solved in a deterministic way, providing unique answer to the unknown parameters [13–15]. However, if the degree of certainty and reliability of the parameters are relevant, a probabilistic approach is required. This was introduced using the framework of Bayesian statistics by Jaynes [16] based on Cox’s postulates [17] and still being developed [11, 18–24]. An alternative theoretical framework was posed by Tarantola [10] based on the idea of conjunction of states of information (theoretical, experimental, and prior information, generally on model parameters). The axioms of probability theory apply to different situations: the Bayesian one is the traditional statistical analysis of random phenomena, whereas the information states one is the description of (more or less) subjective states of information on a system.

In this paper, an information theory reconstruction framework is built on new metrics of information density that drops Cox’s normalization in favor of strong simplifications. This metric is used with the concept of combining information density functions from two independent sources: (i) experimental measurements of harmonic acoustic pressures and (ii) the differential equation that governs the harmonic generation along the propagation, over the same data (observations and model parameters) under the idea of finding which ones are all true at the same time.

In the next section, the experimental method for obtaining several sets of independent measurements is described, along with the derivation of the mathematical model of the harmonic generation and the formulation of the information theory based probabilistic inverse problem that combine both. The results compare the reconstruction with the current method as compared to the basic method used in the literature, showing how it overcomes the limitations and thus validating it.

#### 2. Methods

##### 2.1. Experimental Measurements

The experimental setup consists in a transducer that emits an ultrasonic wave of excitation frequency and a hydrophone that records the propagated wave in the center of the beam at a distance , as shown in Figure 1. Degassed water at stabilized temperature is used in an immersion tank with digital controlled motion. A monochromatic sine burst signal of >40 cycles is generated with an Agilent 33250 generator and amplified with an Amplifier Research 150A100B (150 watts, 10 kHz–100 MHz) amplifier at 46 dB (200) gain programmed to generate an amplitude of 1–128 V, so that the peak acoustic pressure ranges from 1 kPa to 100 kPa, depending on the transducer. A range of transducers, focussed and nonfocussed, with various central frequencies between 1 MHz and 10 MHz was tested to discard any dependency on the hardware and transducer design.