Inverse Problem Of Wave Propagation Research Articles

Deep-learning-based surrogate model for forward and inverse problems of wave propagation in porous media saturated with two fluids

Deep-learning-based surrogate model for forward and inverse problems of wave propagation in porous media saturated with two fluids

Direct collocation method for identifying the initial conditions in the inverse wave problem using radial basis functions

ABSTRACTA direct collocation method associated with explicit time integration using radial basis functions is proposed for identifying the initial conditions in the inverse problem of wave propagation. Optimum weights for the boundary conditions and additional condition are derived based on Lagrange’s multiplier method to achieve the prime convergence. Tikhonov regularization is introduced to improve the stability for the ill-posed system resulting from the noise, and the L-curve criterion is employed to select the optimum regularization parameter. No iteration scheme is required during the direct collocation computation which promotes the accuracy and stability for the solutions, while Galerkin-based methods demand the iteration procedure to deal with the inverse problems. High accuracy and good stability of the solution at very high noise level make this method a superior scheme for solving inverse problems.

A meshfree method for inverse wave propagation using collocation and radial basis functions

A stable and highly accurate meshfree scheme based on strong form collocation associated with radial basis functions and explicit time integration is proposed to solve ill-posed inverse wave propagation problems. Tikhonov regularization technique and the L-curve criterion are employed to deal with the noise measurement data before solving the governing equation to get a stable solution. Appropriate weights are first derived for imposing boundary conditions and known conditions from the measurement data to yield the optimal convergence for inverse problems. Compared to the required iterative procedures in common methods, no iteration is needed in the proposed method which leads to the time-saving computation and avoids the instability resulting from the iterative methods. Dispersion analysis describes that this method has much less dispersion error than the classical finite element method (FEM) and popular weak form meshfree methods, and stability analysis is performed to predict the critical time step for the explicit time integration. Another outstanding advantage of this method is that it is effortless to extend the solution scheme of one-dimensional inverse problem to high-dimensional inverse problems. Furthermore, the influences of the high noise into measurement data are evaluated, which demonstrate that even handling with very high noise in the input data the proposed algorithm can achieve stable and high-accuracy solutions.

Ultrasound Shear Wave Viscoelastography: Model-Independent Quantification of the Complex Shear Modulus.

Ultrasound shear wave elastography methods are commonly used for estimation of mechanical properties of soft biological tissues in diagnostic medicine. A limitation of most currently used elastography methods is that they yield only the shear storage modulus ( G' ) but not the loss modulus ( G'' ). Therefore, no information on viscosity or loss tangent (tan δ) is provided. In this paper, an ultrasound shear wave viscoelastography method is developed for model-independent quantification of frequency-dependent viscoelastic complex shear modulus of macroscopically homogeneous tissues. Three in vitro tissue-mimicking phantoms and two ex vivo porcine liver samples were evaluated. Shear waves were remotely induced within the samples using several acoustic radiation force pushes to generate a semicylindrical wave field similar to those generated by most clinically used elastography systems. The complex shear modulus was estimated over a broad frequency range (up to 1000 Hz) through the analytical solution of the developed inverse wave propagation problem using the measured shear wave speed and amplitude decay versus propagation distance. The shear storage and loss moduli obtained for the in vitro phantoms were compared with those from a planar shear wave method and the average differences over the whole frequency range studied were smaller than 7% and 15%, respectively. The reliability of the proposed method highlights its potential for viscoelastic tissue characterization, which may improve noninvasive diagnosis.

Ultrasound viscoelasticity assessment using an adaptive torsional shear wave propagation method.

Different approaches have been used in dynamic elastography to assess mechanical properties of biological tissues. Most techniques are based on a simple inversion based on the measurement of the shear wave speed to assess elasticity, whereas some recent strategies use more elaborated analytical or finite element method (FEM) models. In this study, a new method is proposed for the quantification of both shear storage and loss moduli of confined lesions, in the context of breast imaging, using adaptive torsional shear waves (ATSWs) generated remotely with radiation pressure. A FEM model was developed to solve the inverse wave propagation problem and obtain viscoelastic properties of interrogated media. The inverse problem was formulated and solved in the frequency domain and its robustness to noise and geometric constraints was evaluated. The proposed model was validated in vitro with two independent rheology methods on several homogeneous and heterogeneous breast tissue-mimicking phantoms over a broad range of frequencies (up to 400 Hz). Viscoelastic properties matched benchmark rheology methods with discrepancies of 8%-38% for the shear modulus G' and 9%-67% for the loss modulus G″. The robustness study indicated good estimations of storage and loss moduli (maximum mean errors of 19% on G' and 32% on G″) for signal-to-noise ratios between 19.5 and 8.5 dB. Larger errors were noticed in the case of biases in lesion dimension and position. The ATSW method revealed that it is possible to estimate the viscoelasticity of biological tissues with torsional shear waves when small biases in lesion geometry exist.

Imaging of a fractal pattern with time and frequency domain topological derivative

While fractal boundaries and their ability to describe irregularity have been intensively studied, only a few studies are available on wave propagation in a medium where a fractal or quasi fractal pattern is embedded. Acoustical propagation in 1D or 2D domains can be modeled using Time Domain Finite Differences or in Frequency domain with Finite element methods. The fractal object is then considered as a subwavelength set of scatterers, and the problem becomes a multiple scattering one leading to acoustic localization. So, as some important part of the energy remains trapped inside the fractal pattern, imaging such a medium becomes difficult and imaging such complex media with classical tools as B-scan or comparable methods is not sufficient. The resolution of the inverse problem of wave propagation can then be achieved with the help of the more efficient imaging methods related with Time Reversal. Using the concept of topological derivative as defined in Time Domain Topological Energy and Fast Topological Imaging Method, respectively, in Time domain and Frequency domain is powerful in that case. Examples in 1D and 2D will be presented including the image obtained for a sliced sponge.

A Gauss–Newton full-waveform inversion for material profile reconstruction in viscoelastic semi-infinite solid media

An inversion framework employing a Gauss–Newton method is developed to reconstruct material profiles in heterogeneous, viscoelastic, semi-infinite domains. In particular, a full-waveform inversion approach is investigated to image the elastic and attenuating parameters of a layered media. To account for the viscoelasticity of the medium, a Generalized Maxwell Body with one spring and two Maxwell elements in parallel (GMB2) is adopted in the forward and inverse wave propagation problems. Perfectly-matched-layers were introduced as wave absorbing buffers to simulate the semi-infinite extent of the domain. Using transient wave equations endowed with the GMB2 constitutive relation and the PML, a partial-differential-equations-constrained optimization scheme was implemented that lead to classic KKT (Karush–Kuhn–Tucker) conditions including time-dependent state, adjoint, and time-invariant control problems. An optimal solution of the viscoelastic parameters was obtained using a reduced-space approach based on a line search algorithm where the search direction was computed by the Gauss–Newton method. Considerable improvements on the accuracy and convergence rate of solutions were made by the developed Gauss–Newton inversion procedure compared to previous research using the Fletcher–Reeves method.

Rayleigh wave propagation method for the characterization of a thin layer of biomaterials

An experimental method based on Rayleigh wave propagation was developed for quantifying the frequency-dependent viscoelastic properties of a small volume of expensive biomaterials over a broad frequency range. Synthetic silicone rubber and gelatin materials were fabricated and tested to evaluate the proposed method. Planar harmonic Rayleigh waves at different frequencies, from 80 to 4000 Hz, were launched on the surface of a sample composed of a substrate with known material properties coated with a thin layer of the soft material to be characterized. A transfer function method was used to obtain the complex Rayleigh wavenumber. An inverse wave propagation problem was solved and a complex nonlinear dispersion equation was obtained. The complex shear and elastic moduli of the sample materials were then calculated through the numerical solution of the obtained dispersion equation using the measured wavenumbers. The results were in good agreement with those of a previous independent study. The proposed method was found to be reliable and cost effective for the measurement of viscoelastic properties of a thin layer of expensive biomaterials, such as phonosurgical biomaterials, over a wide frequency range.

Experimental methods for the characterization of the frequency-dependent viscoelastic properties of soft materials

A characterization method based on Rayleigh wave propagation was developed for the quantification of the frequency-dependent viscoelastic properties of soft materials at high frequencies; i.e., up to 4 kHz. Planar harmonic surface waves were produced on the surface of silicone rubber samples. The phase and amplitude of the propagating waves were measured at different locations along the propagation direction, which allowed the calculation of the complex Rayleigh wavenumbers at each excitation frequency using a transfer function method. An inverse wave propagation problem was then solved to obtain the complex shear/elastic moduli from the measured wavenumbers. In a separate, related investigation, dynamic indentation tests using atomic force microscopy (AFM) were performed at frequencies up to 300 Hz. No systematic verification study is available for the AFM-based method, which can be used when the dimensions of the test samples are too small for other existing testing methods. The results obtained from the Rayleigh wave propagation and AFM-based indentation methods were compared with those from a well-established method, which involves the generation of standing longitudinal compression waves in rod-shaped test specimens. The results were cross validated and qualitatively confirmed theoretical expectations presented in the literature for the frequency-dependence of polymers.

Dynamic inverse problem in a weakly laterally inhomogeneous medium: theory and numerical experiment

An inverse problem of wave propagation into a weakly laterally inhomogeneous medium occupying a half-space is considered in the acoustic approximation. The half-space consists of an upper layer and a semi-infinite bottom separated with an interface. An assumption of a weak lateral inhomogeneity means that the velocity of wave propagation and the shape of the interface depend weakly on the horizontal coordinates, x = (x1, x2), in comparison with the strong dependence on the vertical coordinate, z, giving rise to a small parameter ε ≪ 1. Expanding the velocity in power series with respect to ε, we obtain a recurrent system of 1D inverse problems. We provide algorithms to solve these problems for the zero and first-order approximations. In the zero-order approximation, the corresponding 1D inverse problem is reduced to a system of non-linear Volterra-type integral equations. In the first-order approximation, the corresponding 1D inverse problem is reduced to a system of coupled linear Volterra integral equations. These equations are used for the numerical reconstruction of the velocity in both layers and the interface up to O(ε2).

Damage Modelling and Simulation of Composite Materials using Ultrasonic Measurements

The monitoring of the elastic properties of Al2O3/Al2O3 composites during the exposure at high temperature environment that simulates the working conditions of a gas turbine has been performed non-destructively using ultrasonics. The applied methodology is based on velocity measurements of the elastic waves that propagate in an orthotropic medium. These were estimated experimentally using a custom pulser-receiver setup which allows control of the angle of the incident pulse on the sample, while the latter is immersed in a water bath. The derivation of the elastic constants in order to reproduce the stiffness matrix of the composite is an inverse wave propagation problem described by the Christoffel equation. The damage initiation and propagation as depicted by the deterioration of the moduli of the material was described using deterministic and stochastic approaches. Finally, the damage accumulation process was simulated as a Markov process.

Inversion of Reservoir Parameters Based on the BISQ Model in Partially Saturated Porous Medium

AbstractThe Biot‐flow mechanism and the squirt‐flow mechanism are the two most important mechanisms of pore fluids in a fluid‐saturated porous medium. Based on the BISQ model including the two mechanisms simultaneously, we perform the inversion of reservoir parameters (e.g., porosity, permeability, and saturation) by using Niche Genetic Algorithms (NGA). Numerical experiments indicate that during the process of reservoir parametric inversion the convergence of objective function is fast, and our present algorithms have the property of strongly protecting the noises of observation. The results of reservoir parametric inversion are very precise while the errors of observation are less than 5%. Finally, applying to the practical inversion of experimental data proves that our present algorithms are effective in inverse problems of wave propagation in partially saturated reservoir media affected simultaneously by the Biot and squirt‐flow mechanisms.

Fermat's principle for non–dispersive waves in non–stationary media

Fermat9s principle of stationary travel time serves as a powerful tool for solving direct and inverse problems of wave propagation in media with time–independent parameters. In this paper, Fermat9s principle is extended to non–stationary media that support waves with frequency–independent velocity. The approach used to prove the stationarity of travel times with respect to deformation of the actual ray trajectory is based on a comparison of the rays that follow from the variational principle with those from the eikonal equation. Inhomogeneous, moving and anisotropic media are considered. The identities that relate phase and group velocities and their derivatives in general anisotropic, inhomogeneous, non–stationary media are established. It is shown that not only the travel time but also the eikonal is stationary on the actual ray in media with time–dependent parameters if all trial rays are required to arrive at the receiver simultaneously. Some corollaries and applications of Fermat9s principle in non–stationary media are discussed briefly.

Forward and inverse modeling in porous media

Techniques for solving forward and inverse wave propagation problems involving poroelastic layers will be discussed. Parabolic equation techniques are efficient for solving problems in laterally varying media. A parabolic equation for poroelastic media has been developed and applied to problems in ocean acoustics [J. Acoust. Soc. Am. 98, 1645–1656 (1995)]. This approach has been generalized to the anisotropic case. The smallness of shear speeds in many ocean sediments has motivated the study of poroacoustic media [J. Acoust. Soc. Am. 104, 783–790 (1998)], which is a limiting case of Biot theory in which the rigidity vanishes. Although parabolic equation techniques have not been fully generalized from acoustics to poroelasticity, studying the intermediate case of poroacoustics has helped to bridge the gap. The parabolic equation techniques have been used as tools for solving inverse problems. This approach is presently being applied to field data. One of the issues that arises in solving the inverse problem is the mapping between the coefficients of the wave equation and the wave speeds. The inverse of this mapping can be used to define problems in terms of natural parameters rather than moduli. [Work supported by ONR.]

Direct and inverse problems of wave propagation through a one-dimensional inhomogeneous medium

This paper deals with a wave process induced by an acoustic impulse in a vertically inhomogeneous half-space. The solvability of the corresponding direct initial boundary value problem is proved. We also consider a class of one-dimensional inverse problems including, in particular, the classical inverse problem of the scattering theory, the transmission and reflection problems of seismology, etc. It is shown that all these inverse problems are equivalent in a certain sense, and therefore can be solved by identical methods. Theoretical results are illustrated by numerical examples.

A method for approximately solving inverse problems of wave propagation in a dissipative layered medium

A method for reducing one-dimensional inverse problems of wave propagation theory to Cauchy problems for nonlinear systems of ordinary differential equations is proposed. This approach is applicable to a special class of inverse data, which is dense in the whole set of data equipped with any natural topology. Bibliography: 2 titles.

Transform operators in a two-dimensional inverse problem for a finite domain

An approach to the solution of inverse problems of wave propagation at a fixed frequency based on using transform operators is suggested. A system of integral equations of the Gelfand-Levitan type is obtained. The uniqueness theorem for the refraction index in the analytic case is proved. Bibliography: 6 titles.

Coherent scattering indicatrix for arbitrary medium inhomogeneities the general mathematical solution and feasibility of a fast computer code

A general solution of Equation (1) in a three-dimensional inhomogeneous medium is found. Use is made of the available ray-optical approximate solution of (1) and the Riemann geometry of wave rays. This allows an exact solution in series form, where the first term coincides with the approximate solution. The second term of the exact solution contains the major part of the scattered wave and its computation gives the right value of the coherent scattering. The second term and its transformation leading to easier computation are analysed. The main result of this work is that the mathematical approach used is convenient for a description of the coherent scattering in inhomogeneous media when solving direct and inverse problems of wave propagation. The description is of a very general character and contains Bragg scattering as a particular case. The application of this approach to the development of the computer codes can provide a powerful tool for studying inhomogeneous media. Developing these codes would be time consuming but provide a fast solution of many difficult problems.

Determination of elastic constants of thin films from phase velocity dispersion of different surface acoustic wave modes

We have carried out a systematic investigation of the achievable accuracy in the determination of the independent elastic constants c11 and c12 and the related constant c44 of thin isotropic films from the phase velocity dispersion of surface acoustic waves (SAWs). As a model system SiO2 and Au films deposited on Y-cut LiNbO3 were considered. The phase velocity dispersion was calculated for real elastic constants in dependence on the film thickness. These input data were used to determine the least-squares fits of the input dispersion and the phase velocity obtained by modifying c11 and c12. The minimum of this error field describes the solution of the inverse wave propagation problem. Error fields of the least-squares fits were calculated with ±5% variation for c11 and ±30% for c12. Different functional behavior and various magnitudes of the dispersion were compared. When simulating a small measuring uncertainty for the phase velocity the solution of the inverse problem becomes unstable which results in an insufficient accuracy of the interesting elastic constants. By superposition of two SAW modes the inaccuracy was significantly reduced. For the model system each independent SAW mode offers the ability to determine one elastic constant or one relation between different elastic parameters.

Inverse problem of wave propagation over a membrane in acoustic media

Inverse problem of wave propagation over a membrane in acoustic media