Abstract

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.

从材料表征到地球物理应用，确定异质介质中的超声散射和本征衰减都非常重要。

在这里，我们提出了一种在有限尺寸散射介质中的高效反演方法，其中边界反射起着至关重要的作用。

为了拟合散射尾波的能量曲线，我们在各向同性散射近似条件下，通过蒙特卡罗模拟解决了圆柱体和板块几何形状的声辐射传递方程。

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.

我们的研究表明，用无限大介质中的简化辐射传输解进行拟合，可能会导致散射平均自由程 $l_{s}$ 和吸收率 $Q_{i}^{-1}$ 的低估至少 $40$%。我们的主要发现是薄板样品中的异常传输行为，其中弹道峰和扩散峰合并为一个峰值。

Introduction

Experimental studies suggest that seismic (elastic) wave attenuation is much more sensitive to changes in rock properties than that of the wave velocity. Attenuation refers here to the intrinsic (i.e., anelastic) attenuation that converts seismic energy into heat, otherwise called absorption or dissipation.

Absorption is challenging to measure, since seismic amplitudes may be affected by many factors. This difficulty also exists in ultrasonic laboratory experiments on rock samples due to beam spreading, wave scattering, and boundary reflections.

实验研究表明，地震（弹性）波衰减对岩石性质变化的敏感程度远高于波速的。这里的衰减指的是将地震能量转化为热能的内在（即无弹性）衰减，也称为吸收或耗散。

由于地震振幅可能受到多种因素的影响，因此测量吸收率具有挑战性。由于波束扩散、波散射和边界反射等原因，在对岩石样本进行超声波实验室实验时也会遇到这种困难。

For quasi-homogeneous rocks, wave scattering may be ignored. Absorption can then be measured from the ballistic or reflected pulses, i.e., the coherent wave field. However, these methods may overestimate absorption in heterogeneous rocks, since ballistic wave attenuation also includes scattering by wavelength-scale heterogeneities. Moreover, in strong (multiple) scattering rocks, such as granite and gabbro, it is even challenging to identify correctly coherent pulses from scattered coda signals.

Similar situations are also encountered for nondestructive evaluation of heterogeneous materials in engineering applications in which diagnostic methods based on ultrasonic (or seismic) coda waves are developed, including diffusing-wave spectroscopy in the case of evolving media.

Here, we investigate the transition from the ballistic to the diffusive regime within acoustic (scalar) radiative-transfer (RT) theory, to disentangle scattering and intrinsic attenuations from ultrasound coda waves in finite-size heterogeneous media.

对于准均质岩石，可以忽略波的散射。这样就可以通过弹道或反射脉冲，即相干波场来测量吸收率。然而，这些方法可能会高估异质岩石的吸收率，因为弹道波衰减也包括波长尺度异质的散射。此外，在花岗岩和辉长岩等强（多重）散射岩石中，从散射尾波信号中正确识别相干脉冲甚至具有挑战性。

在工程应用中对异质材料进行非破坏性检测时也会遇到类似的情况，在这种情况下，开发了基于超声波（或地震波）尾波的诊断方法，包括在介质不断变化的情况下使用扩散波谱学。

在此，我们研究了声学（标量）辐射传递（RT）理论中从弹道机制到扩散机制的过渡，以区分有限尺寸异质介质中超声尾波的散射和本征衰减。

RT theory is a general scattering theory based on a particle description of wave propagation (see discussion in Sec. II A). It is usually expressed in terms of the specific intensity, defined as the radiant power per unit area; solid angle; and frequency. The specific intensity obeys a Boltzmann-type differential equation called the radiative-transfer equation (RTE), which can be solved through Monte Carlo (MC) simulations.

RT 理论是基于粒子阐释波传播的一般性的散射理论（见第 II 章 A 节的讨论）。它通常用比强度（定义为单位面积的辐射功率）、实体角和频率来表示。比强度服从波尔兹曼微分方程，即辐射传递方程（RTE），可通过蒙特卡罗（MC）模拟求解。

Interestingly, such simulations can handle complex boundary conditions, which would be difficult, if not impossible, to treat through analytical methods.

The generality of RT theory is critical for the attenuation inverse problem, since ultrasound scattering in rocks may lead to an intermediate regime between two limiting cases: the weak-scattering regime, which is well described by a single-scattering analytical model, and the strong-scattering regime, where the RTE is asymptotically equivalent to the diffusion equation. Nevertheless, applying MC simulations to describe wave propagation in finite-size rocks requires the boundary condition and the geometry of the source to be treated carefully.

有趣的是，这种模拟(MC)可以处理复杂的边界条件，如果换作是分析方法来处理这些条件, 即使并非不可能也困难重重。

RT 理论的泛用性对于衰减逆问题至关重要，因为超声波在岩石中的散射可能会导致两种极限情况之间的中间机制：弱散射机制和强散射机制，前者可以用单一散射分析模型很好地描述，后者的 RTE 与扩散方程近似等效。然而，应用 MC 模拟来描述波在有限尺寸岩石中的传播需要仔细处理边界条件和波源的几何形状。

衰减逆问题(Attenuation Inverse Problem)

衰减逆问题的一般目标是根据已知的传播数据，例如波测量或散射数据，推断介质中的吸收或衰减分布。这通常是一种反问题，因为它要求从传播数据反推介质的性质，而不是直接测量这些性质。

Unlike the layered half-space model used in seismology, rock samples used in laboratory experiments are often shaped in cuboid, slab, or cylinder geometries to facilitate the experimental control of pressure, temperature, saturation, etc.. As shown in optics, and later in seismology, boundary reflections from such a finitesize sample have to be considered properly to determine the transport parameters.

For slablike polycrystals, this can be done based on the existing steady-state RTE solution, but the time-dependent RTE needs to be solved numerically. For ultrasonic measurements in granite and gabbro, only the reflection of coherent elastic waves from the boundary was previously taken into account. Here, we investigate the effect of boundary reflections in the multiple-scattering regime to infer adequately intrinsic and scattering attenuations (or elastic mean free path).

与地震学中使用的层状半空间模型不同，实验室实验中使用的岩石样品通常为立方体、板状或圆柱体形，以便于对压力、温度、饱和度等进行实验控制。正如光学以及后来的地震学所显示的那样，要确定传输参数，必须正确考虑来自这种有限尺寸样品的边界反射。

层状半空间模型(Layered Half-Space Model)

这个模型假设地下介质是由一系列分层组成的，每一层具有不同的物理性质，例如密度、波速、波阻抗等。这些分层通常是水平分布的，并且可以延伸到无限深度，因此被称为"半空间"。

层状半空间模型的使用是为了简化地震波的传播分析。地震波通常会在地下的不同层之间传播，并在每个界面上发生反射和折射，这些界面可以是地壳中的岩石界面或不同地质层之间的分界面。通过使用层状半空间模型，地震学家可以更容易地模拟和分析地震波的传播路径和性质。

对于板状多晶体，可以根据现有的稳态 RTE 解法来实现，但对于含时 RTE 就需要进行数值求解。对于花岗岩和辉长岩中的超声波测量，以前只考虑了相干弹性波从边界的反射。在此，我们研究了多重散射机制中边界反射的影响，以充分推断本征衰减和散射衰减（或弹性平均自由程）。

Furthermore, the ultrasonic source transducer is often a finite-size source (instead of a point source). When the source is large relative to the propagation distance, planewave source geometry can be assumed. In this case, the energy profile can noticeably be broadened, and the peak value time is delayed. When the source is smaller than the travel distance, it may instead be treated as a point source. Here, we consider the source-size effect precisely in our MC simulations and study its influence on inversed attenuations.

此外，超声波源传感器通常是有限尺寸源（而不是点源）。当声源相对于传播距离较大时，可以假定声源的几何形状为平面波。在这种情况下，能量曲线会明显变宽，峰值时间也会延迟。当声源小于传播距离时，可将其视为点声源。在这里，我们在 MC 模拟中精确考虑了源尺寸效应，并研究了它对反向衰减的影响。

In Sec. II, we review the scalar RTE and its timedependent analytical solution and then describe the numerical resolutions of the RTE using MC simulations in the case of infinite media and finite-size media with boundary reflection. In Sec. III, we study the effect of a finitesize source. Particular attention is paid to the impact of boundary reflections and source sizes on the intrinsic attenuation, $Q_{i}^{-1}$, and the scattering mean free path, $l_{s}$, inferred from MC simulations.

In Sec. IV, we compare the simulated energy profiles to experimental data obtained both in a confined dry granular medium and on granite (i.e., a polycrystal rock) for the purpose of the inverse problem.

在第 II 节中，我们回顾了标量 RTE 及其含时解析解，然后介绍了在无限介质和具有边界反射的有限尺寸介质情况下使用 MC 仿真对 RTE 的数值解析。在第 III 节中，我们研究了有限尺度源的影响。我们特别关注边界反射和源尺寸对 MC 模拟推断出的本征衰减 $Q_{i}^{-1}$ 和散射平均自由路径 $l_{s}$ 的影响。

在第四节中，我们将模拟的能量曲线与在封闭的干燥粒状介质和花岗岩（即多晶体岩石）上获得的实验数据进行比较，以解决逆向问题。

MC SIMULATIONS OF RADIATION TRANSPORT FOR A POINT SOURCE

Radiative-transfer theory

Wave scattering in random heterogeneous media is common in natural materials and has been extensively modeled in many areas of physics, such as solid-state physics, optics and acoustics, as well as seismology. The basic idea is to model wave propagation in an ensemble (instead of specific) of random heterogeneous media.

RT theory can statistically model multiply scattered waves. This theory is based upon the assumption that randomly scattered waves have uncorrelated random phases. The superposition of these scattered waves may be incoherent, leading to a description of wave propagation, not in terms of field quantities, such as stress or displacement, but in terms of average intensity. As such, it, of course, cannot provide a full description of wave propagation, but it can accurately describe the ensemble-averaged energy densities.

随机异质介质中的波散射在天然材料中很常见，物理学的许多领域，如固体物理、光学和声学以及地震学，都对其进行了广泛建模。其基本思想是模拟波在随机异质介质系综（而非特定）中的传播。

RT 理论可以对多重散射波进行统计建模。该理论基于随机散射波具有不相关随机相位的假设。这些散射波的叠加可能是不相干的，从而导致对波传播的描述不是用应力或位移等场量，而是用平均强度。因此，它当然无法全面描述波的传播，但可以准确描述系综平均能量密度。

The RTE can be derived in either of two ways. The simple phenomenological method relies upon energy-conservation considerations in a representative volume, consisting of discrete scatterers.

In this approach, the wave equation is used only to determine wave velocities and properties for single-scattering events that constitute the multiple-scattering process. One may also derive the RTE directly from the wave equation based on the Bourret approximation to the Bethe-Salpeter equation for the second moment of the wave field.

Therefore, unlike the Dyson equation for describing the mean field of scattered waves, the RTE only accounts for the intensity evolution of scattered waves and ignores their phase information, which matters in wave-interference phenomena.

RTE 可以通过两种方法之一得出。简单的现象学方法依赖于由离散散射体组成的代表性体积中的能量守恒考虑。

在这种方法中，波方程仅用于确定构成多重散射过程的单次散射事件的波速和特性。也可以根据波场第二矩的 Bethe-Salpeter 方程的 Bourret 近似，直接从波方程推导出 RTE。

因此，与描述散射波平均场的 Dyson 方程不同，RTE 只考虑了散射波的强度演变，而忽略了其相位信息，而相位信息在波干涉现象中非常重要。

The scalar RTE governs the dependence of position $\mathbf{r}$ and time $t$ on the average intensity, $I(\mathbf{r},t;\hat{\mathbf{s}})$, radiated in the direction given by the unit vector $\hat{\mathbf{s}}$(the so-called average specific intensity), which can be written as follows for isotropic scattering and a pulsed point source:

$$ \frac{1}{V}\frac{\partial }{\partial t}I(\mathbf{r},t;\hat{\mathbf{s}}) + \hat{\mathbf{s}}\cdot\nabla I(\mathbf{r},t;\hat{\mathbf{s}})\\ =-\left(\frac{1}{l_{s}} + \frac{1}{l_{a}}\right)\nabla I(\mathbf{r},t;\hat{\mathbf{s}}) + \frac{1}{l_{s}}\int\frac{\mathrm{d}\hat{\mathbf{s}}’}{\Omega_{d}}I(\mathbf{r},t;\hat{\mathbf{s}}’) + \frac{1}{V}\delta(\mathbf{r})\delta(t) $$

where $V$ is the acoustic wave velocity, $l_{s}$ is the scattering mean free path, $l_{a}$ is the absorption length, and $\Omega_{d}$ is the surface area of the unit sphere. Under the isotropic scattering assumption, the average energy density, $E(\mathbf{r},t) = I(\mathbf{r},t)/V$, can be derived from the specific intensity, $I(\mathbf{r},t;\hat{\mathbf{s}})$, integrated over all directions $\hat{\mathbf{s}}$.

标量 RTE 控制着位置 $\mathbf{r}$ 和时间 $t$ 对平均强度 $I(\mathbf{r},t;\hat{\mathbf{s}})$（即所谓的平均比强度）的依赖关系，在各向同性散射和脉冲点源的情况下，可以写成下面的形式:

$$ \frac{1}{V}\frac{\partial }{\partial t}I(\mathbf{r},t;\hat{\mathbf{s}}) + \hat{\mathbf{s}}\cdot\nabla I(\mathbf{r},t;\hat{\mathbf{s}})\\ =-\left(\frac{1}{l_{s}} + \frac{1}{l_{a}}\right)\nabla I(\mathbf{r},t;\hat{\mathbf{s}}) + \frac{1}{l_{s}}\int\frac{\mathrm{d}\hat{\mathbf{s}}’}{\Omega_{d}}I(\mathbf{r},t;\hat{\mathbf{s}}’) + \frac{1}{V}\delta(\mathbf{r})\delta(t) $$

其中，$V$ 是声波速度，$l_{s}$ 是散射平均自由程，$l_{a}$ 是吸收长度，$\Omega_{d}$ 是单位球体的表面积。在各向同性散射假设下，平均能量密度 $E(\mathbf{r},t) = I(\mathbf{r},t)/V$，可以从比强度 $I(\mathbf{r},t;\hat{mathbf{s}})$ 得出，并在所有方向 $\hat{\mathbf{s}}$ 上进行积分。

To obtain $I(\mathbf{r},t)$ or $E(\mathbf{r},t)$ from integrodifferential Eq. (1), it is useful to consider the contribution from different scattering events($n = 0,1,2,\dots$), as for the Boltzmann equation in the kinetic theory of gases:

$$ E(\mathbf{r},t) = E_{0}(\mathbf{r},t) + \sum_{n}E_{n}(\mathbf{r},t)(n\geq 1), $$

in which the first term, $E_{0}(\mathbf{r},t) = W_{0}G_{0}(\mathbf{r},t)$, indicates the incident ballistic wave energy at receiver point $\mathbf{r}$, and the second term is the sum of scattered wave energies from all possible scatterer points, $\mathbf{r}’$, to receiver $\mathbf{r}$. Here, $W_{0}$ is the source energy and

$$ G_{0}(\mathbf{r},t) = \frac{e^{-\frac{Vt}{l_{s}}}}{4\pi Vr^{2}}\delta(t-\frac{r}{V}) $$

is the pulsed Green function; $n = 1$ corresponds to the firstorder or single scattering and $n> 1$ to multiple scattering. In three-dimensional infinite media, an approximate solution is deduced analytically for the energy density:

$$ E(\mathbf{r},t) = \frac{W_{0}e^{-Vt\left[\frac{1}{l_{s}} + \frac{1}{l_{a}}\right]}}{V4\pi r^{2}}\delta\left(t-\frac{r}{V_{0}}\right) + \frac{\left(1-\frac{r^{2}}{V^{2}t^{2}}\right)^{\frac{1}{8}}}{\left(\frac{4}{3}\pi l_{s}Vt\right)^{\frac{3}{2}}}\times G\left\{\frac{Vt}{l_{s}}\left[1-\frac{r^{2}}{V^{2}t^{2}}\right]^{\frac{3}{4}}\right\}H\left(t-\frac{r}{V}\right), $$

where $G(x) \approx e^{x}(1 + \frac{2.026}{x})^{\frac{1}{2}}$ and $H$ is the Heaviside function.