Monte Carlo Simulations of Ultrasound Scattering and Absorption in Finite-Size Heterogeneous Materials

Determination of ultrasound scattering and intrinsic attenuations in heterogeneous media is of importance from material characterization to geophysical applications. Here, we present an efficient inverse method within a finite-size scattering medium, where boundary reflection plays a crucial role. To fit the energy profile of scattered coda waves, we solve the acoustic radiative-transfer equation by Monte Carlo simulations for cylinder and slab geometries, under the isotropic scattering approximation. We show that the fit with the simplistic radiative-transfer solution in an infinite medium may result in underestimated values of the scattering mean free path, $l_s$, and absorption, $Q_i^{-1}$, by up to 40%. Our main finding is anomalous transport behavior in thin slab samples, where the ballistic peak and the diffusionlike one are merged into one single peak. This anomalous behavior, related to a wave-focusing effect in the forward direction, can mislead the inverse process and lead to an overestimation of $l_s$ by more than 200%. We compare simulated energy profiles with ultrasound envelopes obtained in a polycrystal-like granite slab from the ballistic to the diffusive regime. The $l_s$ deduced from off-axis detections agrees with that estimated from the correlation length of the shear-wave velocity by structural imaging analysis.

Figure 1

Energy-density profiles in an infinite medium predicted by Eq. (3) at different source-receiver distances L (black curves) together with Monte Carlo simulations (red curves). t_b = L/V is the ballistic arrival time. Inset reveals the presence of two peaks separated by a minimum in the energy density (see text).

Figure 2

(a) Random walking of particles (or radiation pulses) in a random medium. (b) Reflections at a flat surface (top and bottom) and (c) at a curved surface (lateral wall).

Figure 3

MC simulations of radiation transport for a point source in a random medium with cylindrical geometry for three different scattering cylinders with (a) L/l_s = 1, (b) L/l_s = 3, and (c) L/l_s = 10. Snapshots are recorded at two instants, t₁ and t₂.

Figure 4

Comparison of temporal energy-density profiles obtained with simulations in finite-size media (red for cylinder and blue for slab) and with the RTE solution in infinite media (black) for a point source. Three panels correspond to different scattering samples with (a) L/l_s = 1, (b) L/l_s = 3, and (c) L/l_s = 10. Energy densities in cylindrical samples are about 1 order of magnitude higher than those in the slabs due to constrained energy spreading in the lateral direction. Inset (b), reverberations between ballistic and diffusion peaks.

Figure 5

Similar simulations to those in Fig. 3 but through a slab sample of width (diameter) w = 4L (L being thickness).

Figure 6

(a) Time profiles of energy density in optically thin (L/l_s = 3) and thick (L/l_s = 10) samples without absorption (Q_i = ∞). (b) Time profiles in the thin sample in the presence of absorption (Q_i = 500) for on-axis and off-axis detections.

Figure 7

(a) Finite-size source made of M = 20 concentric circles with a radius r = (D/2) = 12.5 mm. (b) Side view.

Figure 8

Simulations in infinite media for different source sizes at a fixed distance. Energy-density profiles for the point source (black) coinciding with those for a small source of D = L/4 (red) but are about 1 order of magnitude larger than those for a large source of D = 4L (blue). Three panels (a), (b) and (c) correspond to different scattering samples with $L/l_s=1$, 3 and 10, respectively (similar to Figs. 3 and 5).

Figure 9

Simulations of the energy-density profile in finite-size media for a plane-wave source. (a)–(c) Different scattering samples (similar to Figs. 3, 5, and 8) with $L/l_s=1$, 3 and 10, respectively, in which red curves refer to cylinders and black ones to slabs. Snapshots are recorded at instant t = 2t_b/3. Profiles are the same shape but of different magnitude due to radiation intensity.

Figure 10

(a) Schematic illustration of ultrasonic experiments in confined granular materials. (b) Ultrasonic signals transmitted through two different packing configurations. (c) Ensemble-averaged energy profiles measured in a large and a small cell (granular sample), with w = 60 (red curves) and 30 mm (black curves) and two different thicknesses L. (d) Comparison between experimental data obtained in the small cell (w = 30 mm) and the simulated energy-density profiles (red curves), as well as solutions of the diffusion solution with a plane-wave source (blue curves).

Figure 11

Ultrasonic experiment on a granite slab. (a) Photograph of the granite slab. (b) Mineral constituents of granite. (c1) Setup A, in which a detector is placed at the opposite end to the source axis. (c2) Setup B, in which a detector is placed on a concentric circle at the opposite end. (d1) Typical transmitted signal recorded with setup A. (d2) Typical transmitted signal recorded with setup B and r_off= 6 cm. (d3) Typical transmitted signal recorded with setup B and r_off= 15 cm.

Figure 12

(a) Ensemble-averaged energy profiles (black, blue, and red) obtained for geometries depicted in Fig. 11(d1)–(d3), respectively. (b) Normalized ensemble-averaged energy profiles (blue) truncated before t = L/V_s, fitted by the simulated profiles (red), and the RTE solution in an infinite medium (black).

Figure 13

Evaluation of correlation length from a photographic image of granite. (a) Grayscale image. (b) Segmented image with biotite (black), quartz (gray), and plagioclase (white). (c) Picture of the spatial distribution of shear-wave velocity fluctuations, ζ(r), around mean velocity V₀ ≈ 3800 m/s with standard deviation ε ≈ 0.1. (d) Estimated autocorrelation function (dots) from the map of shear-velocity fluctuations in (c) and best-fitted exponential-type ACF (solid line).