Random Medium Model Research Articles

Random medium model for producing optical coherence lattice.

Within the Markov approximation, we introduce a novel class of random media which can produce a scattered field with optical lattice patterns. It is shown that the array dimension, lobes intensity profile, and the periodicity of the optical lattice can be flexibly controlled by altering the correlation parameters of scattering potential of the random medium. In addition, a new method for designing random media is proposed. It is shown that the convolution of any two legitimate degrees of potential correlation can lead to a new degree of potential correlation corresponding to a new scattered intensity distribution. An example of a novel family of random media is cited to demonstrate the result.

Random medium model for cusping of plane waves.

We introduce a model for a three-dimensional (3D) Schell-type stationary medium whose degree of potential's correlation satisfies the Fractional Multi-Gaussian (FMG) function. Compared with the scattered profile produced by the Gaussian Schell-model (GSM) medium, the Fractional Multi-Gaussian Schell-model (FMGSM) medium gives rise to a sharp concave intensity apex in the scattered field. This implies that the FMGSM medium also accounts for a larger than Gaussian's power in the bucket (PIB) in the forward scattering direction, hence being a better candidate than the GSM medium for generating highly-focused (cusp-like) scattered profiles in the far zone. Compared to other mathematical models for the medium's correlation function which can produce similar cusped scattered profiles the FMG function offers unprecedented tractability being the weighted superposition of Gaussian functions. Our results provide useful applications to energy counter problems and particle manipulation by weakly scattered fields.

Anisotropic fractal model for the effective thermal conductivity of random metal fiber porous media with high porosity

An anisotropic fractal model for the effective thermal conductivity (ETC) of random metal fiber porous media saturated in a low thermal conductivity fluid is proposed in this paper. The representative elementary volume (REV) was introduced. The cross section and the tortuous fiber in REV were considered as a fractal surface and a fractal curve respectively. The thermal tortuosity, maximum and minimum ETCs of REV were derived based on the fractal description under the assumption of one-dimensional heat conduction along the tortuous fractal fiber. The space angles between the REV and the heat flow direction were determined based on the experimental data to deduce the anisotropic ETC model of random metal fiber porous media with high porosity. The model prediction results have good agreement with the experimental data in literature.

Plum pudding random medium model of biological tissue toward remote microscopy from spectroscopic light scattering.

Biological tissue has a complex structure and exhibits rich spectroscopic behavior. There has been no tissue model until now that has been able to account for the observed spectroscopy of tissue light scattering and its anisotropy. Here we present, for the first time, a plum pudding random medium (PPRM) model for biological tissue which succinctly describes tissue as a superposition of distinctive scattering structures (plum) embedded inside a fractal continuous medium of background refractive index fluctuation (pudding). PPRM faithfully reproduces the wavelength dependence of tissue light scattering and attributes the "anomalous" trend in the anisotropy to the plum and the powerlaw dependence of the reduced scattering coefficient to the fractal scattering pudding. Most importantly, PPRM opens up a novel venue of quantifying the tissue architecture and microscopic structures on average from macroscopic probing of the bulk with scattered light alone without tissue excision. We demonstrate this potential by visualizing the fine microscopic structural alterations in breast tissue (adipose, glandular, fibrocystic, fibroadenoma, and ductal carcinoma) deduced from noncontact spectroscopic measurement.

An analysis of intrinsic variations of low-frequency shear wave speed in a stochastic tissue model: the first application for staging liver fibrosis

Shear wave elastography is increasingly being used to non-invasively stage liver fibrosis by measuring shear wave speed (SWS). This study quantitatively investigates intrinsic variations among SWS measurements obtained from heterogeneous media such as fibrotic livers. More specifically, it aims to demonstrate that intrinsic variations in SWS measurements, in general, follow a non-Gaussian distribution and are related to the heterogeneous nature of the medium being measured. Using the principle of maximum entropy (ME), our primary objective is to derive a probability density function (PDF) of the SWS distribution in conjunction with a lossless stochastic tissue model. Our secondary objective is to evaluate the performance of the proposed PDF using Monte Carlo (MC)-simulated shear wave (SW) data against three other commonly used PDFs.Based on statistical evaluation criteria, initial results showed that the derived PDF fits better to MC-simulated SWS data than the other three PDFs. It was also found that SW fronts stabilized after a short (compared with the SW wavelength) travel distance in lossless media. Furthermore, in lossless media, the distance required to stabilize the SW propagation was not correlated to the SW wavelength at the low frequencies investigated (i.e. 50, 100 and 150 Hz). Examination of the MC simulation data suggests that elastic (shear) wave scattering became more pronounced when the volume fraction of hard inclusions increased from 10 to 30%.In conclusion, using the principle of ME, we theoretically demonstrated for the first time that SWS measurements in this model follow a non-Gaussian distribution. Preliminary data indicated that the proposed PDF can quantitatively represent intrinsic variations in SWS measurements simulated using a two-phase random medium model. The advantages of the proposed PDF are its physically meaningful parameters and solid theoretical basis.

Numerical Simulations of the Lunar Penetrating Radar and Investigations of the Geological Structures of the Lunar Regolith Layer at the Chang’E 3 Landing Site

In the process of lunar exploration, and specifically when studying lunar surface structure and thickness, the established lunar regolith model is usually a uniform and ideal structural model, which is not well-suited to describe the real structure of the lunar regolith layer. The present study aims to explain the geological structural information contained in the channel 2 LPR (lunar penetrating radar) data. In this paper, the random medium theory and Apollo drilling core data are used to construct a modeling method based on discrete heterogeneous random media, and the simulation data are processed and collected by the electromagnetic numerical method FDTD (finite-difference time domain). When comparing the LPR data with the simulated data, the heterogeneous random medium model is more consistent with the actual distribution of the media in the lunar regolith layer. It is indicated that the interior structure of the lunar regolith layer at the landing site is not a pure lunar regolith medium but rather a regolith-rock mixture, with rocks of different sizes and shapes. Finally, several reasons are given to explain the formation of the geological structures of the lunar regolith layer at the Chang’E 3 landing site, as well as the possible geological stratification structure.

Constraints on the heterogeneity spectrum of Earth's upper mantle

AbstractWe constrain the heterogeneity spectrum of Earth's upper mantle at scales from a few kilometers to tens of thousands of kilometers using observations from high‐frequency scattering, long‐period scattering, and tomography. Tomography and high‐frequency scattering constraints are drawn from previous studies, but constraints on mantle heterogeneity at intermediate scales (5–500 km) are lacking. To address this, we stack ∼15,000 long‐period P coda envelopes to characterize the globally averaged scattered wavefield at periods from 5 to 60 s and at ranges from 50 to 98°. To fit these observations, we consider models of random mantle heterogeneity and compute the corresponding global wavefield using both a ray theoretical “seismic particle” approach and full spectral element simulations. Von Kármán random media distributed throughout the uppermost 600 km of the mantle with a = 2000 km, ε = 10%, and κ = 0.05 provide a good fit to the time, range, and frequency dependence of the stacks, although there is a trade‐off between ε and the thickness of the assumed scattering layer. This random media model also fits previously published 1 Hz stacks of P coda and agrees with constraints on long‐wavelength structure from tomography. Finally, we explore geodynamically plausible scenarios that might be responsible for the RMS and falloff rate of the proposed spectrum, including a self‐similar mixture of basalt and harzburgite.

Solution of radiative transfer theory problems for ‘realistic’ models of random media using the Monte Carlo method

AbstractA methodology of efficient homogenization of a radiation model for a stochastic media is developed. In this connection, ‘realistic’ computational models of random fields are constructed, implementations of those fields are close to continuous ones, and the conditional one-dimensional distribution in the nonempty part of the medium is quite natural. It is shown that the corresponding homogenized probability of the passage of a quantum particle is actually determined by the correlation scale and the occupancy of the medium and also by the above conditional distribution.

Study on Multiphase Discrete Random Medium Model and its GPR Wave Field Characteristics

AbstractAsphalt concrete is a typical multiphase discrete random medium, which is composed of aggregate, asphalt mortar and air with different volume fraction. Aggregate, asphalt mortar and air usually have different sizes, shapes, dielectric properties, and distribute randomly in space. This paper is based on random medium theory, (1) we measured the dielectric constants of the asphalt concrete samples, and computed the statistical characteristics (such as mean values, standard deviations) of spatial random distribution of dielectric constants and the volume fraction of each component; (2) we calculated the autocorrelation function of asphalt concrete based on Wiener‐Khintchine theorem, and extracted its characteristic parameters (such as autocorrelation length, autocorrelation angle), and then classified the type of random media; (3) we developed the modeling algorithm of multiphase discrete random medium under quantization constrain, and constructed multiphase discrete random medium model based on intermixed elliptic autocorrelation function. Additionally, we studied the propagation characteristics of ground penetrating radar wave in multiphase discrete random medium model and compared the model with homogeneous medium model and continuous random medium model using numerical simulation. The calculated results show that the autocorrelation functions of a large number of asphalt concrete sections are approximate to ellipsoidal autocorrelation functions, which provide the foundation for using random medium theory to describe asphalt concrete. The multiphase discrete random medium model not only describes the statistical characteristics of spatial random distribution of asphalt concrete, but also describe the volume fractions of its compositions. For multiphase discrete random medium model and continuous random medium model, ground penetrating radar wave has strong scattering phenomenon. Random and disorderly scattering waves overlie and interfere with each other, which resulting random perturbations and noise in received waves. The reflected waves from anomalous body are with distortion and discontinuity and reduce the signal to noise ratio and resolution of ground penetrating radar data. When multiphase discrete random medium model and continuous random medium model have the same given model parameters, ground penetrating radar waves have stronger scattering in continuous random medium model. The study reveals that the multiphase discrete random medium model can describe asphalt concrete more comprehensive and precise than homogeneous medium model and continuous random medium model. The multiphase discrete random medium model also provides a new way for studying similar media or materials. The radar profile is more consistent with the field measured data, and more conducive to guide the interpretation of the ground penetrating radar profile data.

Fractional diffusion analysis of the electromagnetic field in fractured media — Part 2: 3D approach

We have proposed a 3D finite-difference (FD) approach to discretize the frequency-domain fractional-derivative Maxwell equation on a staggered grid. The Maxwell equation was reformulated to include a fractional-order time derivative that described multiscale electromagnetic (EM) induction in fractured formations exhibiting a fractal geometry. The roughness [Formula: see text] that appeared in the theory described the falloff of the power spectrum of the heterogeneity of a subsurface region in the wavenumber domain, disclosing the geologic model structure in an explicit way. The fractional-derivative Maxwell equation was transformed into the frequency domain and solved by the FD method. To further probe the controlled-source EM response of a power law length-scale distribution of natural fractures, a stochastic random medium model was generated using the von Kármán correlation function. The usual deterministic EM response to such a fractured block model was fitted by a zero-[Formula: see text] fractional EM response at multiple frequencies, indicating that the von-Kármán-type fractured model response is classical. This confirmed the expectation that a fractional diffusion EM response was not reproduced by piecewise constant models based on the classical Maxwell equation.

Correlated percolation models of structured habitat in ecology

Percolation offers acknowledged models of random media when the relevant medium characteristics can be described as a binary feature. However, when considering habitat modeling in ecology, a natural constraint comes from nearest-neighbor correlations between the suitable/unsuitable states of the spatial units forming the habitat. Such constraints are also relevant in the physics of aggregation where underlying processes may lead to a form of correlated percolation. However, in ecology, the processes leading to habitat correlations are in general not known or very complex. As proposed by Hiebeler (2000), these correlations can be captured in a lattice model by an observable aggregation parameter q, supplementing the density p of suitable sites. We investigate this model as an instance of correlated percolation. We analyze the phase diagram of the percolation transition and compute the cluster size distribution, the pair-connectedness function C(r) and the correlation function g(r). We find that while g(r) displays a power-law decrease associated with long-range correlations in a wide domain of parameter values, critical properties are compatible with the universality class of uncorrelated percolation. We contrast the correlation structures obtained respectively for the correlated percolation model and for the Ising model, and show that the diversity of habitat configurations generated by the Hiebeler model is richer than the archetypal Ising model. We also find that emergent structural properties are peculiar to the implemented algorithm, leading to questioning the notion of a well-defined model of aggregated habitat. We conclude that the choice of model and algorithm has strong consequences on what insights ecological studies can get using such models of species habitat.

Quantifying the effect of capillarity on attenuation and dispersion in patchy-saturated rocks

Waves in patchy-saturated rocks are attenuated through the mechanism of wave-induced pressure diffusion. Previous studies reveal that attenuation and phase-velocity dispersion depend on the fluid patch size and distribution. These patch characteristics in turn can be influenced by capillary forces. The effect of capillarity on wave attenuation in patchy-saturated rocks is not fully understood. We studied the combined effects of wave-induced pressure diffusion and capillarity on acoustic signatures. To do so we made use of the concept of patch membrane stiffness as a macroscopic expression of capillarity. We incorporated the membrane stiffness into the continuous random media model of patchy saturation. The membrane stiffness is associated with a pressure discontinuity at patch interfaces. This pressure discontinuity impedes wave-induced pressure diffusion and, therefore, reduces wave attenuation. Conversely, the phase velocity increases due to additional capillarity reinforcement. We applied this capillarity-extended random media model to interpret velocity-saturation relations (VSR) and attenuation-saturation relations (ASR) retrieved from an ultrasonically monitored core flooding experiment. Because the fluid distribution is approximately known from accompanying computerized tomographic images, all but one required model input parameters can be inferred. The elusive input parameter is a shape factor quantifying the geometric irregularity of the pore channels. We found, however, that the experimental data can be consistently modeled only if the capillarity effect is accounted for. The results suggested that wave-induced fluid-pressure diffusion at mesoscopic patches in conjunction with capillary action can have important implications for interpreting ultrasonic VSR and ASR in patchy-saturated rocks.

Estimation of 2D Stationary Random Medium Parameters from Post‐Stack Seismic Data

AbstractThe key to apply the random medium theory to seismic exploration is the estimation of random medium parameters (RMP) from seismic data. In this paper, we propose an approach to estimate parameters of 2D stationary random medium from post‐stack seismic data (PSD). Firstly, the relationship between PSD and the impedance model of random medium is established, and the principle and method of parameters estimation of autocorrelation function (ACF) are introduced. Then, by using power spectrum method, the algorithms and procedures for estimating RMP are discussed with an example; afterward, the feasibility and correctness of this approach is tested using several 2D theoretical models; finally, this approach is applied to the field seismic data, and the results indicate that the estimated parameters can provide an important reference for further dividing delta sedimentary facies, which reflects the estimation method of RMP is very promising. Compared with peer‐research work, the estimation method in this paper is a real 2D algorithm, and in particular it can estimate the autocorrelation angle, and this approach based on power spectrum method is obviously efficient and easily understandable. However, we could not ignore the relatively large errors in our approach, and it needs to be further improved and amended in the later researches.

Constructing Von Karman Random Media Model with Power Spectrum Method

In this paper, Von Karman random media is constructed with power spectrum method. The random media with Von Karman autocorrelation cannot be constructed in time domain but in frequency domain. So power spectrum method is utilized to solve the problem. The spectrum of random function is generated in frequency domain with random field and auto-correlation function. With inverse Fourier transformation, the Von Karman random media model is constructed. With different correlation lengths, six group of Von Karman random media are constructed.

3D elastic full‐waveform inversion of small‐scale heterogeneities in transmission geometry

ABSTRACTThree‐dimensional elastic full‐waveform inversion aims to reconstruct elastic material properties of 3D structures in the subsurface with high resolution. Here we present an implementation of 3D elastic full‐waveform inversion based on the adjoint‐state method. The code is optimized regarding runtime and storage costs by using a time‐frequency approach. The gradient is computed from monochromatic frequency‐domain particle‐velocity wavefields calculated with a time‐domain velocity‐stress finite‐difference scheme. The 3D full‐waveform inversion was applied to data of a complex random medium model, which resembles a realistic crystalline rock environment. We show synthetic inversion results of P‐wave and S‐wave velocities for two transmission geometries: (1) a 3D acquisition geometry with planes of sources and receivers and (2) a 2D geometry with two lines of sources and receivers, resembling a realistic two‐borehole geometry. The 3D inversion of data acquired with 3D source‐receiver geometry is capable to reconstruct differently sized 3D structures of shear and compressional velocities with resolution of about a wavelength. The 3D random medium data recorded with 2D acquisition geometry were inverted using 3D inversion and 2D full‐waveform inversion for comparison. The 2D inversion suffers from strong artefacts that are caused by 3D scattering. The multiparameter 3D inversion, by contrast, is capable to invert the 3D scattered waves and to reconstruct 3D structures up to about 1–2 wavelengths adjacent to the plane between sources and receivers. The resolution is lower compared to the 3D acquisition geometry result. Still, a 3D inversion of cross‐hole data can be beneficial compared to a 2D inversion in the presence of complex 3D small‐scale heterogeneities, as it is capable to resolve 3D structures next to the source‐receiver plane.

Radar wave scattering loss in a densely packed discrete random medium: Numerical modeling of a box-of-boulders experiment in the Mie regime

Abstract In rough geologic media such as alluvial gravels, glacial tills, talus or colluvium, the grain sizes may span the range of GPR in situ wavelengths. Here we experimentally and numerically modeled the scattering loss from both rough-surface and subsurface dielectric scatterers. The combination of the selected radar frequency and the dimension of the scatterers placed the scattering within the Mie regime. We compared the GPR signal amplitude and waveform reflected from the metal sheet on the bottom of a large box filled with boulders with the numerically computed response from a discrete random medium (DRM) model. The DRM consists of a collection of densely packed ellipsoids. The size and orientation of the ellipsoids are randomized; the size has a Gaussian distribution similar to the physical experiment. The dielectric permittivity of the ellipsoids is constant and their electric conductivity is negligible. The starting in situ dominant pulse wavelength at 900 MHz was about 17 cm, as was about the average rock dimension. Experimentally, the 900-MHz radar pulse underwent most dispersion within the first in situ wavelength of depth, and then, at 500–700 MHz dominant frequency, the pulses underwent a near inverse range dependency loss rate, as if the media were a pure dielectric. The numerical model agrees well with the experimental data. Both experimental and numerical results support a significant scattering loss in Mie regime. Besides the scattering attenuation loss, velocity dispersion has also been observed from both observation and simulation. However, the scattering attenuation and dispersion cannot be fit by the Kramers–Kronig relation that is commonly found in intrinsic attenuation and worth further theoretical investigations.

Study on the Scattering Effect of Micro-Scale Inhomogeneity to the Elastic Wave

Some research on the wave propagation in random medium with Von Karman correlation has been developed in this paper. It focuses on the seismic record of circular disturbance in random medium with Von Karman autocorrelation function. Six different kinds of random medium become the background of the dielectric object. The study of the impact to the responds of the dielectric objects can be measured by applying the FDTD to random background medium model. The numerical results show that the random media make the most obvious effect when the scale of imhomogeneity is close to the wave length.

Simulation and analysis of GPR signal based on stochastic media model with an ellipsoidal autocorrelation function

Target detection using ground penetrating radar (GPR) is based on the contrast between the electrical parameters of the target and the background medium, such as dielectric permittivity, conductivity and permeability. The application mainly concentrates on the detection of the medium interface and the target shape. In any theoretical study, a simulation model is built with a homogeneous medium . However, real detection encounters heterogeneous media which might produce scattering and diffraction at electrical interfaces and distort the radar pulse shape and affect the detection resolution. In this paper, we build multi-scale random media model with an ellipsoidal autocorrelation function and use FDTD method to simulate the GPR signal response. We then estimate and analyze the arrival time, layer thickness, permittivity and the physics relation in different scale random models according to the S transform method and the transmission wave method. The results demonstrate that we can use GPR to obtain geophysical information of multi-scale heterogeneous media, and provide a foundation for real media detection and complex media inversion. • Simulation and analysis of GPR signal based on stochastic media model. • Build the multi-scale stochastic media model with an ellipsoidal autocorrelation function. • Estimate the random media parameter with transmission wave method.

A random medium model for simulation of concrete failure

A random medium model is developed to describe damage and failure of concrete. In the first place, to simulate the evolving cracks in a mesoscale, the concrete is randomly discretized as irregular finite elements. Moreover, the cohesive elements are inserted into the adjacency of finite elements as the possible cracking paths. The spatial variation of the material properties is considered using a 2-D random field, and the stochastic harmonic function method is adopted to simulate the sample of the fracture energy random field in the analysis. Then, the simulations of concrete specimens are given to describe the different failure modes of concrete under tension. Finally, based on the simulating results, the probability density distributions of the stress-strain curves are solved by the probability density evolution methods. Thus, the accuracy and efficiency of the proposed model are verified in both the sample level and collection level.

Comparison of Detection Probability for Conventional and Time-Reversal (TR) Radar Systems

We compare the detection probabilities of the time-reversal(TR) detection system and the conventional radar system. The target is assumed to be hidden inside a random medium such as a forest. We propose a TR detection system based on the SAR(Synthetic Aperture Radar) algorithm. Unlike the conventional SAR images, the proposed TR-SAR system has an interesting property. Specifically, the target-related signal components due to the time-reversal refocusing characteristics, as well as some of clutter-related signal components are concentrated at the time-reversal reference point. The remaining clutter-related signal components are scattered around that reference point. In this paper, we model the random media as a collection of point scatterers to avoid unnecessary complexities. We calculate the detection probability of the TR radar system based on the proposed simple random media model.