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Criticality control of random media such as fuel debris is one of the most important safety issues in postaccident management. A 1/fβ spectrum randomizing model is expected to simulate such random ...

In a homogeneous medium, the far-field generated by a localized source can be expanded in terms of multipoles; the coefficients are determined by the moments of the localized charge distribution. We show that this structure survives to some extent for a random medium in the sense of quantitative stochastic homogenization: In three space dimensions, the effective dipole and quadrupole - but not the octupole - can be inferred without knowing the realization of the random medium far away from the (overall neutral) source and the point of interest. Mathematically, this is achieved by using the two-scale expansion to higher order to construct isomorphisms between the hetero- and homogeneous versions of spaces of harmonic functions that grow at a certain rate, or decay at a certain rate away from the singularity (near the origin); these isomorphisms crucially respect the natural pairing between growing and decaying harmonic functions given by the second Green's formula. This not only yields effective multipoles (the quotient of the spaces of decaying functions) but also intrinsic moments (taken with respect to the elements of the spaces of growing functions). The construction of these rigid isomorphisms relies on a good (and dimension-dependent) control on the higher-order correctors and their flux potentials., Comment: 60 pages

In this paper, we study a macroscopic description on the ensemble of Kuramoto oscillators with finite inertia in a random media characterized by a white noise. In a mesoscopic regime, it is well kn...

The fluctuations of the number of scattering and multiplying particles in a random medium are investigated as functions of time. For this purpose, randomized Monte Carlo algorithms for estimating the probabilistic moments of the corresponding exponential asymptotic parameter are constructed.

We study a particular model of a random medium, called the orthant model, in general dimensions $d\ge 2$. Each site $x\in \Z^d$ independently has arrows pointing to its positive neighbours $x+e_i$, $i=1,\dots, d$ with probability $p$ and otherwise to its negative neighbours $x-e_i$, $i=1,\dots, d$ (with probability $1-p$). We prove a shape theorem for the set of sites reachable by following arrows, starting from the origin, when $p$ is large. The argument uses subadditivity, as would be expected from the shape theorems arising in the study of first passage percolation. The main difficulty to overcome is that the primary objects of study are not stationary, which is a key requirement of the subadditive ergodic theorem.

The Beer-Lambert-Bouguer Law of exponential attenuation is ubiquitous in the study of atmospheric radiative transfer. However, previous work has shown that adherence to the classical Beer-Lambert-Bouguer law requires the scatterers in the medium to be spatially uncorrelated. As particulates in the atmosphere are often statistically correlated/clustered, it is useful to identify the magnitude of the deviation from the classical expectation under different degrees of spatial clustering.Measuring this deviation is difficult in an experimental setting both because it is challenging to measure the spatial clustering and the deviations from the classical expectation are expected to be modest. Thus, we approach this question through a simplified “ballistic-photon” computational simulation.Here, we use a simplified numerical model to track the extinction of a collimated light source through correlated random media. The geometry is taken to mimic a sub-volume of the Michigan Technological University Pi Chamber, and the scatterers (cloud droplets) are explicitly resolved using a variety of increasingly realistic techniques for a frozen-field representation of the particle positions.We report on the anticipated deviations from the classical Beer-Lambert-Bouguer law through examination of the resulting intensity of the illumination leaving through different walls of the simulation domain.

We investigated the effect of β on effective multiplication factor(keff) in the 1/fβ spectrum random system. The random system was generated by the 1/fβ noise model. The model is a continuous space model based on the Randomized Weierstrass function and describes the component spatial distribution with a power spectrum of 1/fβ, where f and β are the frequency domain variable and the characteristic parameter related to randomness, respectively. In this work, the two-group Monte Carlo calculations were performed to obtain the keff for a simple cubic geometry that consisted of two materials (fuel burned to 12 GWd/t and concrete). A large number of replicas having different spatial distribution and characterized by the representative β values were generated using the model, and the distribution on keff was analyzed. We found the dependency on β of standard deviation, skewness, and kurtosis of keff distribution. This result is expected to help to predict the keff distribution due to the randomizing model.

We establish central limit theorems for the position and velocity of the charged particle in the mechanical particle model introduced by Fontes, Jordao Neves and Sidoravicius (2000).

This paper is concerned with the well-posedness and regularity of the distributional solutions for the stochastic acoustic and elastic scattering problems. We show that the regularity of the solutions depends on the regularity of both the random medium and the random source.

This paper deals with discrete-time semi-Markov random evolutions (DTSMRE) in reduced random media. The reduction can be done for ergodic and non ergodic media. Asymptotic approximations of random evolutions living in reducible random media (random environment) are obtained. Namely, averaging, diffusion approximation and normal deviation or diffusion approximation with equilibrium by martingale weak convergence method are obtained. Applications of the above results to the additive functionals and dynamical systems in discrete-time produce the above tree types of asymptotic results.

This work deals with the measurability of Fourier integral operators (FIOs) with random phase and amplitude functions. The key ingredient is the proof that FIOs depend continuously on their phase and amplitude functions, taken from suitable classes. The results will be applied to the solution FIO of the transport equation with spatially random transport speed as well as to FIOs describing waves in random media.

We prove well-posedness results and a priori bounds on the solution of the Helmholtz equation $\nabla\cdot(A\nabla u) + k^2 n u = -f$, posed either in $\mathbb{R}^d$ or in the exterior of a star-shaped Lipschitz obstacle, for a class of random $A$ and $n,$ random data $f$, and for all $k>0$. The particular class of $A$ and $n$ and the conditions on the obstacle ensure that the problem is nontrapping almost surely. These are the first well-posedness results and a priori bounds for the stochastic Helmholtz equation for arbitrarily large $k$ and for $A$ and $n$ varying independently of $k$. These results are obtained by combining recent bounds on the Helmholtz equation for deterministic $A$ and $n$ and general arguments (i.e. not specific to the Helmholtz equation) presented in this paper for proving a priori bounds and well-posedness of variational formulations of linear elliptic stochastic PDEs. We emphasise that these general results do not rely on either the Lax-Milgram theorem or Fredholm theory, since neither are applicable to the stochastic variational formulation of the Helmholtz equation., 36 pages, 2 figures

Models of population growth and extinction are an increasingly popular subject of study. However, consequences of stochasticity and noise in shaping distributions and outcomes are not sufficiently explored. Here we consider a distributed population with logistic growth at each location, subject to "seascape" noise, wherein the population's fitness randomly varies with {\it location and time}. Despite its simplicity, the model actually incorporates variants of directed percolation, and directed polymers in random media, within a mean-field perspective. Probability distributions of the population can be computed self-consistently; and the extinction transition is shown to exhibit novel critical behavior with exponents dependent on the ratio of the strengths of migration and noise amplitudes. The results are compared and contrasted with the more conventional choice of demographic noise due to stochastic changes in reproduction.

We develop the p-adic model of propagation of fluids (e.g., oil or water) in capillary networks in a porous random medium. The hierarchic structure of a system of capillaries is mathematically modeled by endowing trees of capillaries with the structure of an ultrametric space. Considerations are restricted to the case of idealized networks represented by homogeneous p-trees with p branches leaving each vertex, where p >1 is a prime number. Such trees are realized as the fields of p-adic numbers. We introduce and study an inhomogeneous Markov process describing the penetration of fluid into a porous random medium.

We continue the study of the homogenization of coercive non-convex Hamilton–Jacobi equations in random media identifying two general classes of Hamiltonians with very distinct behavior. For the first class there is no homogenization in a particular environment while for the second homogenization takes place in environments with finite range dependence. Motivated by the recent counter-example of Ziliotto, who constructed a coercive but non-convex Hamilton–Jacobi equation with stationary ergodic random potential field for which homogenization does not hold, we show that same happens for coercive Hamiltonians which have a strict saddle-point, a very local property. We also identify, based on the recent work of Armstrong and Cardaliaguet on the homogenization of positively homogeneous random Hamiltonians in environments with finite range dependence, a new general class Hamiltonians, namely equations with uniformly strictly star-shaped sub-level sets, which homogenize.

A novel time reversal approach for the point target imaging is proposed in this paper. This method requires only single measurement by the time reversal array, and it exploits the space-frequency decomposition plus the rotation invariance of the signal subspace for the target detecting. Numerical simulations are carried out to validate the imaging performance of the method for single target located at different positions in homogenous and continuous random medium respectively. The results indicate that the proposed method performs better than the conventional signal subspace based space-frequency algorithm both in imaging resolution and efficiency.

Fractional calculus is entering the field of nonlinear optics to describe unconventional regimes, as disorder biological media and soft-matter. Here we investigate spatiotemporal modulational instability (MI) in a fractional nonlinear Schrodinger equation. We derive the MI gain spectrum in terms of the Levy indexes and a varying number of spatial dimensions. We show theoretically and numerically that the Levy indexes affect fastest growth frequencies and MI bandwidth and gain. Our results unveil a very rich scenario that may occur in the propagation of ultrashort pulses in random media and metamaterials, and may sustain novel kinds of propagation invariant optical bullets.

Uniform regularity for second order elliptic equations in a highly heterogeneous random medium is concerned. The medium is separated by a random ensemble of simply closed interfaces into a connected sub-region with high conductivity and a disconnected subset with low conductivity. The elliptic equations, whose diffusion coefficients depend on the conductivity, have fast diffusion in the connected sub-region and slow diffusion in the disconnected subset. Without a stationary–ergodic assumption, a uniform Holder estimate in ω , ϵ , λ for the elliptic solutions is derived, where ω is a realization of the random ensemble, ϵ ∈ ( 0 , 1 ] is the length scale of the interfaces, and λ 2 ∈ ( 0 , 1 ] is the conductivity ratio of the disconnected subset to the connected sub-region. Results show that if external sources are small enough in the disconnected subset, the uniform Holder estimate in ω , ϵ , λ holds in the whole domain. If not, it holds only in the connected sub-region. Meanwhile, the elliptic solutions change rapidly in the disconnected subset.

The current paper is devoted to the investigation of wave propagation phenomenon in reaction-diffusion equations with ignition type nonlinearity in time heterogeneous and random media. It is proven that such equations in time heterogeneous media admit transition fronts or generalized traveling wave solutions with time dependent profiles and that such equations in time random media admit generalized traveling wave solutions with random profiles. Important properties of generalized traveling wave solutions, including the boundedness of propagation speeds and the uniform decaying estimates of the propagation fronts, are also obtained.

We consider a large number of randomly dispersed spherical, identical, perfectly conducting inclusions (of infinite conductivity) in a bounded domain. The host medium's conductivity is finite and can be inhomogeneous. In the dilute limit, with some boundedness assumption on a large number (proportional to the global volume fraction raised to the power of $-1/2$) of marginal probability densities, we prove convergence in $H^1$ norm of the expectation of the solution of the steady state heat equation, to the solution of an effective medium problem, where the conductivity is given by the Clausius--Mossotti formula. Error estimates are provided as well.

In this paper, we develop non-stationary martingale techniques for dependent data. We shall stress the non-stationary version of the projective Maxwell–Woodroofe condition, which will be essential for obtaining maximal inequalities and functional central limit theorem for the following examples: nonstationary $\rho$-mixing sequences, functions of linear processes with non-stationary innovations, locally stationary processes, quenched version of the functional central limit theorem for a stationary sequence, evolutions in random media such as a process sampled by a shifted Markov chain.

The uncertainties present in various dielectric materials may have an effect on the properties of the electromagnetic components. Deterministic schemes are unable to capture the statistics of the examined quantities and as a result, reliable outcomes cannot be extracted. In this paper, we present a novel two-dimensional (2D) finite-difference time domain (FDTD) method, that manages to calculate the statistics of the involved fields in a single simulation. The proposed algorithm is a variation of the Stochastic FDTD method and is suitable for assessing random media uncertainties, as well as to estimate the variation in the width and height of a random rectangular dielectric. Numerical results indicate the validity of our scheme compared with the Monte Carlo approach.

Nous etudions le comportement de la constante ergodique associee a une equation de Hamilton–Jacobi avec un hamiltonien convexe dans un environnement periodique, lorsque le hamiltonien est perturbe soit par une perturbation periodique avec une periode tendant vers l'infini, soit par une perturbation de type Bernoulli avec un petit parametre. Nous donnons un developpement de Taylor de la constante ergodique qui depend de la dimension. Lorsque d = 1 , le premier terme est non trivial, tandis que si d ≥ 2 , il est toujours nul. Bien que des questions similaires aient ete etudiees dans le cadre de l'homogeneisation d'equation elliptiques lineaires, notre resultat est le premier de cette nature dans un cas non-lineaire. Nos arguments, qui reposent sur des techniques de solutions de viscosite et de theorie KAM faible, soulevent de nombreuses questions nouvelles et stimulantes.

This project was supported by Sapienza Visiting scholarship scheme. A.D.F. and A.F. thank EPSRC (EP/M508214/1). C.C. acknowledges H2020 QuantERA Quomplex (grant number 731743). Random media with tailored optical properties are attracting burgeoning interest for applications in imaging, biophysics, energy, nanomedicine, spectroscopy, cryptography and telecommunications. A key paradigm for devices based on this class of materials is the transmission matrix, the tensoriallink between the input and the output signals, that describes in full their optical behavior. The transmission matrix has specific statistical properties, as the existence of lossless channels, that can be used to transmit information, and are determined by the disorder distribution. In nonlinear materials, these channels may be modulated and the transmission matrix tuned accordingly. Here wereport the direct measurement of the nonlinear transmission matrix of complex materials, exploiting the strong optothermal nonlinearity of scattering Silica Aerogel (SA). We show that the dephasing effects due to nonlinearity are both controllable and reversible, opening the road to applicationsbased on the nonlinear response of random media. Postprint Postprint

This work is devoted to the stability/resolution analysis of several imaging functionals in complex environments. We consider both linear functionals in the wavefield as well as quadratic functionals based on wavefield correlations. Using simplified measurement settings and reduced functionals that retain the main features of functionals used in practice, we obtain optimal asymptotic estimates of the signal-to-noise ratios depending on the main physical parameters of the problem. We consider random media with possibly long-range dependence and with a correlation length that is less than or equal to the central wavelength of the source we aim to reconstruct. This corresponds to the wave propagation regimes of radiative transfer or homogenization.

This paper deals with a reconstruction of random media via multi-objective optimization. Two statistical descriptors, namely a two-point probability function and a two-point lineal path function, are repetitively evaluated for the original medium and the reconstructed image to appreciate the improvement in the optimization progress. Because of doubts of the weights setting in the weighted-sum method, purely multi-objective optimization routine Non-dominated Sorting Genetic Algorithm~II is utilized. Three operators are compared for creating new offspring populations that satisfy a prescribed volume fraction constraint. The main contribution is in the testing of the proposed methodology on several benchmark images.

In this paper, we present algorithms for reconstructing an unknown compact scatterer embedded in a random noisy background medium, given measurements of the scattered field and information about the background medium and the sound profile. We present six different methods for the solution of this inverse problem using different amounts of scattered data and prior information about the random background medium and the scatterer. The different inversion algorithms are defined by a combination of stochastic programming methods and Bayesian formulation. Our basic results show that if we have data for just one instance of the random background medium the best strategy is to invert for both random medium and unknown scatterer with appropriate regularization. However, if we have data for multiple instances of the medium it may be worth solving a coupled set of multiple inverse problems. We present several numerical results for inverting for various scatterer geometries under different inversion scenarios. The main take-away of our study is that one should invert for both unknown scatterer and random medium, with appropriate, prior-information based regularization. Furthermore, if data from multiple realizations of the background medium is available, then it may be beneficial to combine results from multiple inversions., 40 pages

The main approximations for calculation of effective parameters of inhomogeneous media are Bruggeman effective medium approximation and Maxwell Garnett model, which are currently widely used in the description of composites containing various nanostructured objects. In this paper we consider a simple approach to these approximations, allowing to embrace the most general case of statistically anisotropic media with particles of arbitrary internal structure, and also to clarify their physical meaning. The type of result essentially depends on the choice of "effective cells" used in estimating the field and induction values averaged over the volume.

The question addressed by this paper is tackled through a continuum micromechanics model of a 2D random checkerboard, in which one phase is linear elastic and another linear viscoelastic of integer-order. The spatial homogeneity and ergodicity of the material statistics justify homogenization in the vein of the Hill–Mandel condition for viscoelastic media. Thus, uniform kinematic- or traction-controlled boundary conditions, applied to sufficiently large domains, provide macroscopic (RVE level) responses. With computational mechanics, this strategy is applied over the entire range of the relative content of both phases. Setting the volume fraction of either the elastic phase or the viscoelastic phase at the critical value (≃0.59) results in fractal patterns of site-percolation. Extensive simulations of boundary value problems show that, for a viscoelastic composite having such a fractal structure, the integer (not fractional) calculus model is adequate. In other words, the spatial randomness of the composite material—even in the fractal regime—is not necessarily the cause of the fractional order viscoelasticity.

We study the homogenization of a stationary random maximal monotone operator on a probability space equipped with an ergodic dynamical system. The proof relies on Fitzpatrick's variational formulation of monotone relations, on Visintin's scale integration/disintegration theory and on Tartar-Murat's compensated compactness. We provide applications to systems of PDEs with random coefficients arising in electromagnetism and in nonlinear elasticity., Added references

This paper collects the four contributions which were presented during the session devoted to Stochastic Geometry at the journées MAS 2014. It is focused in particular on several questions related to the transmission of information in a general sense in different random media. The underlying models include the Boolean model, simplicial complexes or geometric random graphs induced by a point process.

A single server queue with unreliable server is considered. The server breakdown probability depends on the state of a random medium described by a Markov chain. An ergodic theorem is proved for this system.

This paper presents a multiple length-scale asymptotic analysis for transport problems in 1-D diffusive random media. This analysis shows that the Levermore-Pomraning (LP) equations can be adjusted in order to achieve the correct asymptotic behavior. This adjustment appears in the form of a rescaling of the Markov transition functions by a factor $\eta$, which can be chosen in a simple way. Numerical results are given that (i) validate the theoretical predictions; and (ii) show that the adjusted LP equations greatly outperform the standard LP model for this class of transport problems., Comment: V.2.: We have added around 7 pages of new text; three new sets of numerical results; 6 new figures; and two new references [13, 17]. Current version has 18 pages, 10 figures. The changes correspond to an expansion of Sections 3 and 4. Section 3 now has more detailed and complete analysis; Section 4 has new numerical results

Divergence-form operators with stationary random coefficients homogenize over large scales. We investigate the effect of certain perturbations of the medium on the homogenized coefficients. The perturbations that we consider are rare at the local level, but when occurring, have an effect of the same order of magnitude as the initial medium itself. The main result of the paper is a first-order expansion of the homogenized coefficients, as a function of the perturbation parameter., 32 pages. V3: revised introduction, some corrections and clarifications

We consider a discretized version of the quenched Edwards-Wilkinson model for the propagation of a driven interface through a random field of obstacles. Our model consists of a system of ordinary differential equations on a $d$-dimensional lattice coupled by the discrete Laplacian. At each lattice point, the system is subject to a constant driving force and a random obstacle force impeding free propagation. The obstacle force depends on the current state of the solution and thus renders the problem non-linear. For independent and identically distributed obstacle strengths with exponential moment we prove ballistic propagation (i.e., propagation with a positive velocity) of the interface if the driving force is large enough. For a specific case of dependent obstacles, we show that no stationary solution exists, but still the propagation of the front is not ballistic., 13 pages, 1 figure

Deterministic schemes are usually unable to predict random media uncertainties and as a result produce unreliable outcomes in this kind of problems. In this work, we present a novel finite difference time domain (FDTD) technique that manages to calculate the statistics of the involved field quantities, considering that the variability of random media is known. The proposed method is applied in polar grids and can efficiently deal with such cases in a single realization. Comparisons with the Monte Carlo method show that reliable results may be obtained, thus making our proposed technique much more faster.

SpringerReference; International audience; The statistical inverse problem for the experimental identification of a non-Gauss-ian matrix-valued random field that is the model parameter of a boundary value problem, using some partial and limited experimental data related to a model observation, is a very difficult and challenging problem. A complete advanced methodology and the associated tools are presented for solving such a problem in the following framework: the random field that must be identified is a non-Gaussian matrix-valued random field and is not simply a real-valued random field; this non-Gaussian random field is in high stochastic dimension and is identified in a general class of random fields; some fundamental algebraic properties of this non-Gaussian random field must be satisfied such as symmetry, positiveness, invertibility in mean square, boundedness, symmetry class, spatial-correlation lengths, etc; the available experimental data sets correspond only to partial and limited data for a model observation of the boundary value problem. The developments presented are mainly related to the elasticity framework, but the methodology is general and can be used in many areas of computational sciences and engineering. The developments are organized as follows. The first part is devoted to the definition of the statistical inverse problem that has to be solved in high stochastic dimension. The second one deals with the construction of two possible parameterized representations for a non-Gaussian positive-definite matrix-valued random field that models the model parameter of a boundary value problem. The third part is devoted to the description of all the steps of the methodology allowing the statistical inverse problem to be solved in high stochastic dimension. These steps are based on the identification of a prior stochastic model of the Non-Gaussian random field by using the maximum likelihood method and then, on the identification of a posterior stochastic model of the Non-Gaussian random field by using the Bayes method.The fourth part presents the construction of an algebraic prior stochastic model of the model parameter of the boundary value problem, for a non-Gaussian matrix-valued random field. The generator of realizations for such an algebraic prior stochastic model for a non-Gaussian matrix-valued random field is presented.

International audience; Eigenvalue problems for neutron transport in random geometries are key for many applications, ranging from reactor design to criticality safety. In this work we examine the behaviour of the reactivity and of the kinetics parameters (the effective delayed neutron fraction and the effective neutron generationtime) for three-dimensional UOX and MOX assembly configurations where aportion of the fuel pins has been randomly fragmented by using various mixingstatistics. For this purpose, we have selected stochastic tessellations of the Poisson, Voronoi and Box type, which provide convenient models for the randompartitioning of space, and we have generated an ensemble of assembly realizations; for each geometry realization, criticality calculations have been performedby using the Monte Carlo code TRIPOLI-4 developed at CEA. We have then examined the evolution of the ensemble-averaged observables of interest as afunction of the average chord length of the random geometries, which is roughlyproportional to the correlation length of the fuel fragmentation. The methodology proposed in this work is fairly general and could be applied, e.g., to theassessment of re-criticality probability following severe accidents.

Статья посвящена двухточечным моментам решений, возникающих в простых лагранжевых моделях для уравнения индукции в случае конечного корреляционного времени случайной среды. Рассматривается вопрос о связи коммутационных свойств соответствующих алгебраических операторов с минимальным объемом выборки независимых случайных реализаций, который необходим в численном эксперименте для моделирования двуточечного коррелятора решения. Показано, что, как и для одноточечных моментов, численное исследование двуточечного коррелятора в случае коммутирующих операторов (случайные числа) требует существенно меньших объемов выборки, чем в случае, когда они не коммутируют (случайные матрицы)., This paper is devoted to the two-point moments of the solutions arising in simple Lagrange models for the induction equations in the case of finite correlation time of a random medium. We consider the question on the connection between the commutative properties of the corresponding algebraic operators and the minimal sample size of independent random realizations necessary in numerical experiments for modeling the two-point correlator of the solution. It is shown that, as for the one-point moments, the numerical study of the two-point correlator in the case of commutating operators (random numbers) requires a much smaller sample size than in the case when they do not commute (random matrices)., ВЫЧИСЛИТЕЛЬНЫЕ МЕТОДЫ И ПРОГРАММИРОВАНИЕ: НОВЫЕ ВЫЧИСЛИТЕЛЬНЫЕ ТЕХНОЛОГИИ, Выпуск 3 2017

Well-established methods for the solution of stochastic partial differential equations (SPDEs) typically struggle in problems with high-dimensional inputs/outputs. Such difficulties are only amplified in large-scale applications where even a few tens of full-order model runs are impracticable. While dimensionality reduction can alleviate some of these issues, it is not known which and how many features of the (high-dimensional) input are actually predictive of the (high-dimensional) output. In this paper, we advocate a Bayesian formulation that is capable of performing simultaneous dimension and model-order reduction. It consists of a component that encodes the high-dimensional input into a low-dimensional set of feature functions by employing sparsity-enforcing priors and a decoding component that makes use of the solution of a coarse-grained model in order to reconstruct that of the full-order model. Both components are represented with latent variables in a probabilistic graphical model and are simultaneously trained using Stochastic Variational Inference methods. The model is capable of quantifying the predictive uncertainty due to the information loss that unavoidably takes place in any model-order/dimension reduction as well as the uncertainty arising from finite-sized training datasets. We demonstrate its capabilities in the context of random media where fine-scale fluctuations can give rise to random inputs with tens of thousands of variables. With a few tens of full-order model simulations, the proposed model is capable of identifying salient physical features and produce sharp predictions under different boundary conditions of the full output which itself consists of thousands of components., Comment: 31 pages, 12 figures

We consider scattering of a pulse propagating through a three-dimensional random media and study the shape of the pulse in the parabolic approximation. We show that, similarly to the one-dimensional O’Doherty-Anstey theory, the pulse undergoes a deterministic broadening. Its amplitude decays only algebraically and not exponentially in time, due to the signal low/midrange frequency component. We also argue that the parabolic approximation captures the front evolution (but not the signal away from the front) correctly even in the fully threedimensional situation.

The 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.

In this article we address the problem of the depinning transition for driven interfaces in random media. We introduce a fractional Kardar–Parisi–Zhang equation with quenched noise, in which the normal diffusion term is replaced by a fractional Laplacian accounting for long-range interactions through quenched disorder. The critical values of the external driving force and nonlinear term coefficient evidently depend on the system size at the depinning transition. For a fixed value of the external driving force, the fractional order much determines the value of the nonlinear term coefficient that leads to a depinned interface. Near the depinning threshold, the critical exponent obtained numerically is nonuniversal, and weakly depends on the fractional order.

We consider a medium composed of randomly dispersed spherical, identical inclusions in a bounded domain, with conductivity different than that of the host medium. We study the limit where the number inclusions tends to infinity but their volume fraction remains fixed. For small volume fractions, we prove convergence, in the $W^{1,p}$ norm ($1