Gaussian Random Media Research Articles

Numerical calculation of overlap distribution of (2 + 1)-dimensional directed polymer in random media

We investigate -dimensional discretized directed polymers in Gaussian random media. By numerically calculating the probability distribution function of overlap between two independent and identical systems on a common random potential, we show that there is no replica symmetry breaking. We also show that while the mean-squared displacement of one polymer end exhibits superdiffusive behavior, the mean-squared relative distance of two polymer ends exhibits subdiffusive behavior.

Constraints on coda wave interferometry estimates of source separation: the acoustic case

Synthetic experiments are used to test the applicability of coda wave interferometry (CWI) as a means for estimating distance between sources of nearby earthquakes. Acoustic waves for 45 sources are propagated through a Gaussian random medium. A pair-wise analysis of resulting waveforms illustrates the applicability of CWI as a tool for estimating source separation. Results suggest that, when the waveforms are filtered between 1 and 5 Hz, CWI provides accurate estimates of the separation for source-pairs separated by δ < 250 m. The technique provides a lower bound on the actual separation when δ > 250 m. The CWI breakdown distance of 250 m is likely to vary with frequency content in the waveforms.The interpretation of CWI source separation estimates is aided by the construction of a conditional probability density function (PDF) P(δt ∣ δCWI), which describes the probability of actual separation δt for given CWI estimates δCWI. The conditional PDF provides a constraint on event separation that is asymmetric. It can be used independently of, or combined with, standard travel-time techniques to improve earthquake location.

Asymptotic analysis of classical wave localization in multiple-scattering random media

In this work we consider the localization of classical waves propagating in random continuum. We apply the method of proper time for the transfer from the elliptic-type wave equation to the generalized parabolic one. Presenting the solution of the latter equation in the form of the Feynman path integral allows us to estimate the so-called wave correction terms. These corrections are related to coherent backscattering and recurrent multiple-scattering events, i.e., to phenomena that cannot be described within the framework of the conventional theories of radiative transfer or small-angle scattering. We evaluate the wave correction to the mean intensity of a point source located in a statistically homogeneous Gaussian random medium. Our results confirm that there is an essential difference between two- and three-dimensional systems. We consider both isotropic and anisotropic media and show in particular that in the latter case there is a critical value of the anisotropy parameter, below which the system behaves basically as a three-dimensional isotropic medium, i.e., the wave correction is positive for all observation angles. Above this critical value the properties of the medium are similar to those of a layered structure.

Long‐period ray parameters of the core diffraction Pdiff and mantle heterogeneity

In an attempt to map lateral velocity variations in the D” zone of the lowermost mantle of the Earth, long‐period Pdiff ray parameters are determined from digital data of the Global Seismic Network. Twenty‐six earthquakes between 1981 and 1986 provided 37 profiles in North America and Europe. Average profile length is 18°, and the number of stations per profile is 3 or 4. For accurate travel time determination of the Pdiff wave group peaks, seismograms are equalized to the same instrument characteristic. Ray parameter interpretation in terms of the velocity along the D′′ path of each profile requires correction of Pdiff travel times for mantle heterogeneity above D′′ on the upgoing Pdiff paths. The finite difference method of Podvin and Lecomte [1991] in two dimensions is used to calculate Pdiff travel time curves for the global tomography model of Inoue et al, [1990] and for related models, having lateral heterogeneity only in D′′ or only above D′′. From these calculations, Pdiff ray parameter corrections of up to 1.5% are found. Maps with uncorrected and corrected ray parameters are shown. Conflicting ray parameters on close D′′ profiles continue to exist after correction, which points to insufficient correction. Global tomography models underestimate upper mantle heterogeneity, which, according to regional tomographic studies, has scale lengths of a few hundred kilometers and velocity variations of up to ±3%. Such upper mantle structures are modeled by Gaussian random media with correlation distances of 250–1000 km and standard deviations of 0.5–1% of their slowness fluctuations. Pdiff ray parameter corrections for such structures can reach 3%. The results of this study show that regionalization of D′′ with Pdiff requires detailed knowledge of mantle heterogeneity, in particular of heterogeneity in the upper mantle. Currently, this information is only rarely available.

Path-integral analysis of scalar wave propagation in multiple-scattering random media.

In this work we consider the general problem of scalar wave propagation in a continuously inhomogeneous random medium, applying the approach originally proposed by Fock for the integration of quantum-mechanical equations. The principal idea of the method is based on the introduction of an additional pseudotime variable and the transfer to a higher-dimensional space, in which the propagation process is described by the generalized parabolic equation similar to the nonstationary Schr\"odinger equation in quantum mechanics. We present its solution in a form of Feynman path integral, the asymptotic evaluation of which in the far field allows us to estimate the so-called wave correction terms. These corrections are related to coherent backscattering and repeated multiple-scattering events on the same inhomogeneities, i.e., to the phenomena that are not described in the framework of the conventional theories of radiative transfer or small-angle scattering. As an example of the approach we consider the first statistical moments of the field for a point source located in a statistically homogeneous Gaussian random medium. The correction term obtained for the mean field coincides exactly with the classical result of the Bourret approximation for the Dyson equation, but with a much weaker restriction on the value of wave number k, which allows us to analyze the correction as a function of k. The main feature of the result obtained for the second moment of the field is that the normalized mean intensity is not equal to unity. We relate such behavior to the localization phenomenon. The dependence of the correction term does not differ significantly from that obtained in works concerning the localization of classical waves in discrete random media. \textcopyright{} 1996 The American Physical Society.

A comparison between phase screen, finite difference, and eigenfunction expansion calculations for scalar waves in inhomogeneous media

Abstract Phase screen, fourth-order finite difference (FD), and eigenfunction expansion calculations of scalar wave propagation in two-dimensional (2D) inhomogeneous media are compared to assess the accuracy of the phase screen method. The phase screen method is a forward propagation (one-way wave) algorithm. The finite difference and eigenfunction expansion calculations, which are solutions of full wave equation, are chosen as references in this study. Comparison of synthetic seismograms by phase screen and finite difference methods is made for four kinds of models: (1) multi-uniform-cylinder model, (2) Gaussian random media, (3) exponential random media, and (4) flicker-noise random media. Results show good agreement for weak random media (velocity perturbations ≦10%). For discrete heterogeneities, such as the multi-uniform-cylinder model, the results agree well for up to 50% deviation in velocities. The computer CPU time of the phase screen program for a problem of grid size 1024 by 512 is 367 sec in a SUN SPARC station II, about 57 times faster than the FD program we used. For large 3D problems the time saved is expected to be much greater. For a single cylinder scatterer with and without absorption, we compare synthetic seismograms by the phase screen method and by the eigenfunction expansion method (exact solution). The agreement between the two methods demonstrates that the phase screen method can also give good results for inhomogeneous absorbing media.

Focusing and twinkling: critical exponents from catastrophes in non-Gaussian random short waves

The critical exponent of the intensity moment is computed from the Thorn-Arnol'd classification of caustics as catastrophes. The caustics are studied on the torus whose coordinates are 16 N random phases theta 1... theta N for members of the ensemble describing the phase screen or inhomogeneous medium responsible for the disorder of the wave. The results indicate that when N to infinity (Gaussian random medium) the exponents depend only on whether the waves propagate in two or three space dimensions.

Theory of waves in a homogeneous random medium

A novel theory is developed to cope with the difficulty of the multiple-scattering problem in a random medium (RM). The theory is given for a one-dimensional homogeneous RM which is represented by a strictly stationary random process. Some possible forms of the stochastic solution are determined by a group-theoretic consideration based on the shift-invariance property of the homogeneous RM. The form of the solution has some analogy with Floquet's solution for a periodic medium. It is shown that there are two kinds of solutions in the one-dimensional RM: a travelling-wave mode and a cutoff mode. The former exists only when the power spectrum of the medium becomes zero at nearly double the wave number. Otherwise the wave is in the cutoff mode which is almost a standing wave whose envelope increases or decreases exponentially with distance. For a Gaussian RM with small fluctuations, an approximate stochastic solution given in the possible form is obtained in terms of multiple Wiener integrals with respect to the Brownian-motion process. The average and variance are calculated for the phase and amplitude of the wave in terms of the power spectrum of the RM. The law of large numbers is shown to hold concerning the fluctuations of the phase and amplitude. The average value of the wave and the transmission coefficient of a medium with finite thickness are also studied using the stochastic solution.