M. Asch, W. Kohler, G. Papanicolaou, M. Postel, and B. White, Frequency content of randomly scattered signals, SIAM Review 33, (1991) 519-626.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....changes the pulse beyond the geometrical e ects of the high frequency analysis in the smooth homogenized medium. In the 1960 s and early 1970 s mean pulse propagation over long distances was analyzed. More recently a mathematical theory has been developed that gives a more precise description [1, 8, 16] of pulse propagation. It deals with pulses in a particular realization of the random medium and explains why in many cases the evolution of the pulse shape is to leading order deterministic. We refer to this phenomenon as pulse stabilization. So far, two salient features of this pulse shaping ....

Asch M., W. Kohler, G. C. Papanicolaou, M. Postel and B. White, Frequency content of randomly scattered signals, SIAM Review, V 33, 519-625, (1991).

....changes the pulse beyond the geometrical effects of the high frequency analysis in the smooth homogenized medium. In the 1960 s and early 1970 s mean pulse propagation over long distances was analyzed. More recently a mathematical theory has been developed that gives a more precise description [1, 6, 11] of pulse propagation. It deals with pulses in a particular realization of the random medium and explains why in many cases the evolution of the pulse shape is to leading order deterministic. We refer to this phenomenon as pulse stabilization. So far, two salient feature of this pulse shaping ....

M. Ash, W. Kohler, G. C. Papanicolaou, M. Postel and B. White, Frequency content of randomly scattered signals, SIAM Review, V 33, 519-625, (1991).

....of its parameters. An important aspect of the estimation procedure is that it takes the tool into account. Realizing the importance of the approximation presented by O Doherty and Anstey [11] a number of authors have reexamined it and extended it to more general wave propagation scenarios [1, 3, 4, 6, 8, 9, 12, 13, 17]. Here we consider acoustic wave propagation in a layered medium with weak fluctuations. Based on the analysis in [8, 17] we obtain an approximation that reveals how the tool affects the propagating wave. Note that apparent attenuation becomes important only for relatively long propagation ....

....for the synthetic log, V k , dashed line. 4.3 Pulse shaping in the deconvolved medium Next, we illustrate how the smoothing of the medium by the tool can be compensated by deconvolution and also motivate our choice for the well log tool model. In [7] the tool model corresponding to w [1 1 1 1 1] is being used. The length of this filter is defined by the physical length of the tool. We first illustrate deconvolution with respect to these 6 values for the tool parameters. Based on the tool model (2.4) we design a least squares deconvolution filter as described in Appendix C. We ....

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M. Ash, W. Kohler, G. C. Papanicolaou, M. Postel, and B. White. Frequency content of randomly scattered signals. SIAM Review, 33:519--625, 1991.

....is, a discretely layered medium in which the travel or transit time of each layer is constant. Their motivation for studying pulse spreading was to explore whether scattering associated with fine scale layering in the earth could explain the observed damping of seismic waves. In a series of papers [1, 2, 3, 4, 5, 7, 12, 13, 20] the O Doherty Anstey approximation and its generalizations have been rigorously derived, under various conditions for the medium model. They all assume, however, a strictly layered medium. In [19, 21] and this paper we follow the line of research initiated by O Doherty and Anstey, but consider ....

....characterized asymptotically as a diffusion process through the application of a limit theorem for stochastic differential equations. This allows the characterization of the expected value of the transmission or reflection coefficient. Details of the stochastic theory can be found in Asch et al. [1] and references therein. In this paper we focus on the shallow water equations due to its analogy with the acoustic model, a shown below. The starting point are the Navier Stokes equations describing constant density, free surface flows of an incompressible fluid. The conservation of momentum ....

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M. Ash, W. Kohler, G. C. Papanicolaou, M. Postel and B. White, Frequency content of randomly scattered signals, SIAM Review, V 33, 519-625, (1991).

....shaped. For broad pulses di erent parts of the wave are 19 feeling di erent layers of the subsurface and therefore travelling at di erent speeds. This problem is linear but very dicult to analyse mathematically. Several research papers have been published in recent years, addressing this problem [1, 2, 10, 17, 29]. In some cases the theory involves the asymptotic analysis of stochastic di erential equations. This analysis also has been carried out for water waves [20, 22, 23, 24] Without getting into details, we will describe one interesting result. Consider that, if needed, a layer can be subdividied ....

.... p z = 0: Notice that for a homogeneous medium ( and K constants) we recover the second order wave equation as we know it. The surprising fact in the O Doherty Anstey problem is that the pulse di uses about its moving center due to the disordered multiple scattering of the wave energy [1, 2, 10, 17, 26, 28]. O Doherty and Anstey s motivation for studying pulse spreading was to explore whether the scattering associated with ne scale layering in the earth could explain the observed damping of 21 seismic (acoustic) waves used in the oil exploration industry. A similar phenomenon occurs for water ....

Asch M., Kohler W., Papanicolaou G., Postel M. and White B., (1991), Frequency content of randomly scattered signals, SIAM Review, Vol. 33, pp. 519{ 625.

....revised april 1997 Abstract This paper investigates the pressure field generated at the bottom of a high contrast randomly layered slab by an acoustical pulse emitted at the surface of the slab. This analysis takes place in the framework introduced by Asch, Kohler, Papanicolaou, Postel and White [1] where the incident pulse wave length is long compared to the correlation length of the random inhomogeneities, but short compared to the size of the slab. This problem has been studied in the one dimensional case simultaneously by Clouet and Fouque [4] and Lewicki, Burridge and Papanicolaou [6] ....

....results whereas the point source problem studied in this paper requires a non trivial combination of diffusion approximation results with stationary phase methods. The stationary phase method has been used by De Hoop, Chang and Burridge [5] for weakly fluctuating media and in [1] for the study of the reflected pressure. The main statement of this paper is that, in order to apply simultaneously diffusion approximation and stationary phase results, it is correct to apply them consecutively. We believe that this situation will be encountered again and again in this field and ....

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M. Asch, W. Kohler, G. Papanicolaou, M. Postel and B. White, Frequency content of randomly scattered signals, SIAM review, 33(4) (1991) pp. 519-625.

....0 (x) 1.2 The Reflected Signal The reflected signal B(0; t) obtained by a pulse scattered by a randomly layered medium in the regime of separation of scales described above has been studied by G. Papanicolaou and his coauthors. The one dimensional case is treated in a series of papers [4] 3] [2] and [14] the last paper being devoted to the one dimensional inverse problem. The first observation of this study is that, apparently, the information about the large scale variations of the medium is lost in the noise present in the reflected signal. This noise is due to the multiple scattering ....

M. Asch, G. Papanicolaou, M. Postel, P. Sheng and B. White, Frequency content of randomly scattered signals, Part I, Wave Motion, 12 (1990), pp. 429--450.

....the determination of the asymptotic probability distributions for the reflected and transmitted signals. Moreover it is possible to relate the local power spectral densities of the reflected signal to the large scale variations of the medium as shown by Asch, Kohler, Papanicolaou, Postel and White [1]; this is done through a system of hyperbolic transport equations and leads to a solution to the inverse problem which consists in recovering these large scale variations from the reflected signals. A time reversal method is applied to obtain consistent estimators for these local power spectral ....

....of stochastic processes or stochastic calculus with respect to Brownian motions are needed in this study. In the case of a uniform background (no large scale variations of the medium) this spectral density can be computed explicitely; this is very important for comparison to various simulations [1]. 1.2.1 The Inverse Problem. The inverse problem consists in the reconstruction of the large scale variations of the medium appearing as coefficients in the underlying transport equations. The practical solution to this problem requires good statistical estimators for the local covariances of the ....

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M.Asch, W. Kohler, G. Papanicolaou, M. Postel and B. White, Frequency content of randomly scattered signals, SIAM Review, 33 (1991), pp. 519--626.

....are present (the scale is on the left) One can observe the small signal to noise ratio (of order 10 Gamma3 ) and guess that extracting information from the noisy reflected signal is not easy. The goal of this paper is to present recent complements to the theory elaborated in [4] 3] [2], 16] and presented in the review paper [1] 2.5. QUANTITIES OF INTEREST: TRANSMISSION AND REFLECTION 2.5.1. Transmission The transmitted signal is the right going wave at x = L, namely A(L; t) Defining the travel time in the deterministic macroscopic medium by (x) Z x 0 dy C 0 (y) ....

Asch, M., Papanicolaou, G., Postel, M., Sheng, P., and White, B. (1990) Frequency content of randomly scattered signals, Part I, Wave Motion 12, 429-450.

....pulses containing intermediate wavelengths can be used to probe the medium. This is done by an asymptotic analysis as these scales separate and by using diffusion approximation results. This approach has been initiated by G. Papanicolaou and his coauthors. We refer to the review paper [1] for a detailed presentation of the method. The goal of this contribution is to present in a condensed way the results obtained in these recent years using this technique. In Section 2 we present the simplest case of a one dimensional random medium. We recall the basic facts needed to describe ....

....in these recent years using this technique. In Section 2 we present the simplest case of a one dimensional random medium. We recall the basic facts needed to describe Markovian coefficients, we introduce the different scales and we write carefully the boundary conditions. Simulations borrowed from [1] show the transmitted pulse and the refected signal. Integral representations of these quantities of interest are given. Section 3 is devoted to the study of the transmitted front. The basic diffusion approximation result needed in this study is presented and applied to deduce the shape of the ....

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Asch, M., Kohler, W., Papanicolaou, G., Postel, M., and White, B. (1991) Frequency content of randomly scattered signals, SIAM Review 33, 519-626.

....Cedex France email : fouque paris.polytechnique.fr Received April 1996; revised November 1996 Abstract In the recent years a considerable amount of mathematical work have been devoted to the study of reflected signals obtained by the propagation of pulses in randomly layered media. We refer to [1] for an extensive survey and applications to inverse problems. The analysis is based on separation of scales between the correlation scale of the inhomogeneities present in the medium, the typical wavelengths of the pulse and the macroscopic variations of the medium. On the other hand, in the ....

.... been developed and their effects have been studied experimentally by Mathias Fink and his team at the Laboratoire Ondes et Acoustique (ESPCI Paris) We refer to [2] Our goal is to present a mathematical analysis of a time reversal method for analysing reflected signals in the model described in [1]. We restrict our analysis to the one dimensional case, the threedimensional layered case being the content of a forthcoming paper. It is noticeable that we do not introduce new mathematics in the problem but simply put together an already existing mathematical theory and a new device, the ....

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M. Asch, W. Kohler, G. Papanicolaou, M. Postel and B. White, "Frequency content of randomly scattered signals", SIAM review 33, No.4, 519-625 (1991).

....following question: how the shape of a pulse has been modified when it emerges from a randomly layered medium This analysis takes place in the general framework, based on separation of scales, introduced by G. Papanicolaou and his co authors (see for instance [5] for the one dimensional case or [1] for the three dimensional case) We consider here the problem of acoustic propagation when the incident pulse wavelength is long compared to the correlation length of the random inhomogeneities but short compared to the size of the slab. In this framework, it has already been proved in [1] see ....

....case or [1] for the three dimensional case) We consider here the problem of acoustic propagation when the incident pulse wavelength is long compared to the correlation length of the random inhomogeneities but short compared to the size of the slab. In this framework, it has already been proved in [1] (see also [7] for more details) that, when the random fluctuations are weak, the O Doherty Anstey theory is valid, i.e. the travelling pulse retains its shape up to a low spreading; futhermore, its shape is deterministic when observed from the point of view of an observer travelling at the same ....

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M. ASCH, W. KOHLER, G. PAPANICOLAOU, M. POSTEL and B. WHITE, Frequency content of randomly scattered signals, SIAM Review, Vol.33, No.4, 1991.

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M. Asch, W. Kohler, G. Papanicolaou, M. Postel and B. White, "Frequency content of randomly scattered signals", SIAM Review, vol. 33, No. 4, pp 519-625, December 1991.

....phenomenon and the way the modifications relate to the characterization of the inhomogeneities. We model the fine scale medium as random. The key question that we consider is how these random heterogeneities affect the propagating pulse. Our analysis is partly based on the framework set forth in [2, 19]. The modification of the acoustic pulse is well known in the one dimensional purely layered case. We consider a generalization to more realistic three dimensional wave propagation problems, to media that are locally layered. Such media have general, three dimensional, smooth, background ....

....source over a layered medium in [9] by decomposing the source in terms of plane waves and using a stationary phase argument. Berlyand et al. 5] derive a correction estimate for O Doherty Anstey approximation based on an equal travel time or Goupillaud representation of the medium. Asch et al. [2] presents the first rigorous derivation of the formula in a continuous framework using invariant imbedding and by applying a limit theorem for stochastic ordinary differential equations. This analysis was generalized to reflections, rather than only the directly transmitted pulse, in [25] by ....

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M. Ash, W. Kohler, G. C. Papanicolaou, M. Postel, and B. White. Frequency content of randomly scattered signals. SIAM Review, 33:519--625, 1991.

....avoid the global search we must have a rough estimate of the size of the intervals of stationarity, which for the aerothermal data set we get from a variogram analysis of the wavelet coecients [20] Another case where the intervals of stationarity are known approximately is analyzed in Asch el al. [2] using the local Fourier transform. Finally we show some simulations from the estimated model that assess the overall relevance of our analysis. The main point of our analysis is that we are able to identify the local variations of the power law parameters, the Kolmogorov turbulence law, which ....

Asch, M., Kohler, W., Papanicolaou, G., Postel, M., and White, B., (1991), \Frequency Content of Randomly Scattered Signals," in Siam Review, 33, pp. 519-626.

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M. Asch, W. Kohler, G. Papanicolaou, M. Postel, and B. White. Frequency Content of Randomly Scattered Signals. SIAM Review, 33:519-626, 1991.

....propagates through an inhomogeneous medium its shape and travel time are modi ed by ne scale heterogeneities. We will analyze in detail this phenomenon and the way the modi cations depend on the inhomogeneities, which we model as random. Our analysis is partly based on the framework set forth in [1, 25]. The modi cation of the acoustic pulse is well known in the one dimensional purely layered case. We consider a generalization to more realistic three dimensional wave propagation problems, to media that are locally layered. Such media have general, three dimensional, smooth, background variations ....

....source over a layered medium in [9] by decomposing the source in terms of plane waves and using a stationary phase argument. Berlyand et al. 5] derive a correction estimate for O Doherty Anstey approximation based on an equal travel time or Goupillaud representation of the medium. Asch et al. [1] presents the rst rigorous derivation of the formula in a continuous framework using invariant imbedding and by applying a limit theorem for stochastic ordinary di erential equations. This analysis was generalized to re ections, rather than only the directly transmitted pulse, in [33] by Lewicki ....

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M. Ash, W. Kohler, G. C. Papanicolaou, M. Postel, and B. White. Frequency content of randomly scattered signals. SIAM Review, 33:519-625, 1991.

....the underlying random layering The difficulty inherent in providing an answer to this question lies in the fact that such geometric perturbations, no matter how modest, destroys transverse homogeneity and forces one to deal with stochastic partial, rather than ordinary, differential equations. In [15] we studied extensively the problem of acoustic wave propagation in a rapidly varying plane layered slab. A small parameter was used to delineate three relevant length scales. The largest scale or macroscale (e.g. the slab thickness L) was assumed to be O(1) The acoustic wavelength defined an ....

....reasonably be expected to play an important role. On the other hand, the acoustic wavelength was assumed to span many correlation lengths and one might further expect this scaling to produce a meaningful probabilistic limit asymptotically as 0. These expectations were indeed realized in [15]. The multiple scattering induced by the random layering created localization phenomena which profoundly affected the reflection and transmission properties of the slab. The O Doherty Anstey phenomenon was also shown to follow from a weak fluctuation variant of this model. In [17] we studied the ....

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M. Asch, W. Kohler, G. Papanicolaou, M. Postel, and B. White, Frequency Content of Randomly Scattered Signals, SIAM Review 33, 519-625 (1991).

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M. Asch, W. Kohler, G. Papanicolaou, M. Postel and B. White, "Frequency content of randomly scattered signals", SIAM Review, vol. 33, No. 4, pp 519-625, December 1991.

....by effective medium theory, 1.45 as opposed to 1.41. A similar phenomenon was observed for weak variations by O Doherty and Anstey [18] and has been extensively studied since [2,4,5,6,9,16,17] On the other hand the coda of the wave has been studied thoroughly even when the variations are finite [1,7,8,15,19] and that work already contained some numerical examples showing the pulse like head of the wave. In our analysis we regard our rapidly varying medium as a single (typical) realization of a stochastic process, whose correlation length is on the same scale as the medium variations. We then find an ....

....probability measure. The analysis is based on invariant imbedding equations for the time harmonic reflection and transmission coefficients. The recognition of the importance of the invariant imbedding technique to the analysis of multiple scattering came with the series of papers mentioned above [1,7,8,15,19], devoted to the behavior of the coda which follows the main pulse. Here, a new insight into the versatility of the invariant imbedding technique enabled us to find a moving reference frame relative to which the pulse stabilizes to a deterministic waveform. This co moving frame moves at the ....

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M.Ash, W.Kohler, G.Papanicolaou, M.Postel, B.White, "Frequency content of randomly scattered signals", SIAM Review 33, 519-625, (1991).

....signal R f (t) in addition to (2.9) 2.13) It is that as ffl tends to zero R f (t) becomes approximately a Gaussian process. We have not been able to prove this but we have some good heuristic indications that it is true [9] and extensive numerical simulations that corroborate it very well [11] [1] From the Gaussian property of R f (t) we conclude that 1 j f( j 2 Z e i t R f t ffl t 2 R f t Gamma ffl t 2 d t = t; 2.20) is approximately, when ffl is small, an exponential random variable with mean (t; given by (2.9) 2.13) when c(z) is ....

M. Asch G. Papanicolaou M. Postel P. Sheng and B. White. Frequency content of randomly scattered signals. Part I. Wave Motion, 12:429--450, 1990.

....AND B. WHITE 1. Introduction. Waves reflected by randomly layered media are very noisy because of the intense multiple scattering they undergo before returning to the surface. As a result, it is difficult to extract useful information about the medium from observations of reflected waves. In [1] and the references cited there we showed that if the large scale variations of medium properties, such as density and sound speed, can be distinguished well from their small scale fluctuations due to the layering, then it is possible to use reflection data to solve some statistical inverse ....

....the two scales acting as a small parameter in an asymptotic analysis. Within the framework of this three scale theory (where the scale of macroscopic variations is much larger than the pulse width which in turn is much larger than the scale of the random layering) we formulated and solved in [1] and [2] statistical inverse problems for plane wave pulses. That is, from the reflected acoustic pressure or velocity measured at the surface of the randomly layered medium we recovered the large scale variations in the medium properties. In the plane wave case it is necessary to have available ....

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M. Asch W. Kohler G. Papanicolaou M. Postel and B. White. Frequency content of randomly scattered signals. SIAM Review, 33:pp 526--629., 1991.

....case (x3.7) it is not possible to solve them explicitly, so a numerical solution is necessary. This is particularly important in implementing the inversion algorithms of x6. Although we could use the probabilistic representation of x3.5 to construct Monte Carlo solutions of the W equations [11], we have found that finite difference methods are better in calculating the amplitude of the solutions near the singular front, which is what we need. This has to be done after an appropriate regularization of the equations using a progressing wave representation. The main objective after that is ....

....2. Formulation of the problem. 2.1. The acoustic equations. We consider acoustic wave propagation in three space dimensions using the linearized equations of momentum and mass conservation for the velocity u(t; x) and pressure p(t; x) ae(z) u t oeu rp = F(t; x) 2.1a) 2 See Fig. E. 5 and [11]. FREQUENCY CONTENT OF RANDOMLY SCATTERED SIGNALS 10 1 K(z) p t r Delta u = 0; 2.1b) where ae = ae(z) and K = K(z) are the material properties (density and bulk modulus, respectively) of the medium layered in the z direction, x = x; y; z) is the position vector, and oe = oe(z) is the ....

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M. ASCH, G. PAPANICOLAOU, M. POSTEL, P. SHENG, and B. WHITE, Frequency content of randomly scattered signals. Part I, Wave Motion, 12 (1990), pp. 429-450.

....local spectral characteristics can be obtained efficiently using an adaptive windowed Fourier transform. We illustrate with some examples from seismology the use of our methods and the accompanying software that we have developed. 1. Introduction. In many applications, in seismology in particular [1], it is necessary to estimate spectral densities of signals that are not stationary but do not differ much from stationarity most of the time. The spectral analysis of non stationary signals [4] is too general and inefficient to be useful in practice so it is important that the idea of approximate ....

....variable window Fourier (cosine) transform [2] provides an approximate spectral decomposition. We introduce an optimally adaptive algorithm for estimating the spectral properties of locally stationary signals based on the windowed Fourier transform and apply it to a class of simulated seismograms [1]. The results are quite good, primarily because the window sizes in the transform are adapted optimally to the signal properties. The implementation of the spectral estimation is done with a collection of software tools that we developed and tested extensively and are, moreover, relatively easy to ....

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M. Asch, W. Kohler, G. Papanicolaou, M. Postel, and B. White. Frequency content of randomly scattered signals. SIAM Review, 33:519--625, 1991.

....between the waves and the inhomogeneities, which is the most interesting and difficult case to analyze. In addition to these three conditions, the inhomogeneities must not be too anisotropic because in layered random media wave localization occurs even with weak fluctuations, instead of transport [14]. When the fluctuations are strong, wave localization can occur even when the inhomogeneities are isotropic [15] 16] We shall also analyze the diffusive behavior of solutions of (1.1) which emerges at times and distances that are long compared to a typical transport mean free time 1= Sigma and ....

M.Asch, W.Kohler, G.Papanicolaou, M.Postel and P.Sheng, Frequency content of randomly scattered signals, SIAM Review33, 1991, 519-625.

....in plane parallel structures. I mean random layering. If radiative transport were valid in this case, then the differential scattering cross section would be singular, concentrated in only two (in the simplest case) directions, up and down or forwards and backwards propagation. In the long paper [48] we deal in detail with the point source case, that is, the propagation of an acoustic pulse generated by a point source over a layered random medium. Here I will describe only the reflection of acoustic plane wave pulses. 4.1 Pulse reflection from randomly layered media The acoustic pressure ....

....the reflection of acoustic plane wave pulses. 4.1 Pulse reflection from randomly layered media The acoustic pressure p(t; z) and velocity u(t; z) satisfy the continuity and momentum equations 1 K p t u z = 0 aeu t p z = 0 (19) Here ae is the material density and K the bulk modulus. As in [48] we assume for simplicity that the density has no random variation ae(z) ae ae 0 ; z 0; ae 1 ; z 0 (20) Documenta Mathematica Delta Extra Volume ICM 1998 Delta 1 1000 Mathematical Problems in Geophysical Wave Propagation 15 with ae 0 and ae 1 constants. For the bulk modulus we ....

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M. Asch, W. Kohler, G. Papanicolaou, M. Postel and B. White, Frequency Content of Randomly Scattered Signals, SIAM Review, 33, pp. 526-629, 1991.

....problem. Second, if we try to use (8) as an estimate for j f( j 2 ( t) it will depend sensitively on how we truncate the infinite Fourier integral. With enough experimenting with the data we can make a suitable version of (8) which provides useful estimates for ( t) This was done in [5] as well as in [6] and [7] with data generated by direct numerical simulation of the scattering problem. The question is then: what is a good way to estimate ( t) from R ffl f (t) when we only know the parameter ffl By good we mean primarily robust, that is, by an algorithm that does not need ....

M. Asch, G. Papanicolaou, M. Postel, P. Sheng and B. White. Frequency content of randomly scattered signals. Part I. Wave Motion 12, 429-450, (1990).

....not stationary. This work has been motivated by signals obtained in the context of wave propagating in random media. A one dimensional model of random media is given by a layered medium whose properties are random variables in each layer independent from one layer to another. It has been shown ,[1], that if a pulse of typical wavelengths ffl is sent in such a medium with small layers of order ffl 2 , the reflected signal is a non stationary stochastic process which converges in distribution, as ffl goes to 0, to a locally stationary gaussian process. This convergence will be the key ....

....section we show how this wavelet method enables us to get a fairly good estimation of the small parameter ffl when this one is not known a priori. Spectral Analysis of Randomly Scattered Signals 1 Introduction 2 1 Introduction We will summarize briefly the mathematical results detailed in [1]. Their scope is a class of reflected signals propagated in a medium satisfying two main hypotheses: ffl The medium is layered. ffl The incident pulse is broad compared to the typical width of the layers but narrow compared to the large scale variations in the properties of the scattering half ....

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M. Asch, W. Kohler, G. Papanicolaou, M. Postel and B. White. Frequency content of randomly scattered signals. SIAM Review 33, 526-629, (1991).

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M.Asch, W.Kohler, G.Papanicolaou, M.Postel and P.Sheng, Frequency content of randomly scattered signals, SIAM Review 33 no. 4 (1991), 519-625.

....media, waves AMS Classification 60H15, 35R60, 70B35 1 Introduction Wave propagation in random media leads to many interesting and difficult problems in stochastic processes and differential equations. Several such problems arose in the study of pulse reflection from randomly layered media [4,2,15,10]. In this paper we give a more detailed mathematical analysis of the basic limit theorem used in [4] in the framework of reflected signal functionals introduced there. A comprehensive review of our work is given in [1] In section 2 we formulate the acoustic pulse reflection problem for normally ....

Asch M., Papanicolaou G., Postel M., Sheng P. and White B., Frequency content of randomly scattered signals. Part I, Wave Motion vol. 12, 1990, pp. 429-450.

....arose in the study of pulse reflection from randomly layered media [4,2,15,10] In this paper we give a more detailed mathematical analysis of the basic limit theorem used in [4] in the framework of reflected signal functionals introduced there. A comprehensive review of our work is given in [1]. In section 2 we formulate the acoustic pulse reflection problem for normally incident plane waves on a randomly layered half space and show how the study of the quantities of physical interest leads to the asymptotic analysis of a class of distribution valued stochastic process, the reflection ....

Asch M., Kohler W., Papanicolaou G., Postel M. and White B., Frequency Content of Randomly Scattered Signals, SIAM Review, vol. 33, 1991, 519-625.

....is not closed, as it involves the interactions between the reflected pulse and the backscattering of the transmitted one. Consequently, the averaging principle used in the papers cited above does not seem to apply. We use here the approach based on invariant imbedding that was used successfully in [1]. One expects that the interactions between the reflected pulse and the backscattering of the transmitted one do not change the evolution of the reflected wavefront. Consequently, this evolution is like the one for the transmitted pulse in the undisturbed medium that originates at the reflector ....

....f( j 2 W 11 (0; t; d = 1 2 i 1 Z j f( j 2 W 1 (0; t; d (50) where W N is the solution of the equation (40) We need a closed formula for the solution of equation (40) if we want to apply (50) successfully in our asymptotic analysis. However, as it was already noted in [1], the W equation (40) allows for the probabilistic representation of its solution. This is because the operator in the right hand side of W equation (40) DeltaW ) N = 2a RN 2 h W N Gamma1 Gamma 2W N W N 1 i (51) is a infinitesimal generator of a Markov chain N(oe) with the ....

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M.Ash, W.Kohler, G.Papanicolaou, M.Postel, B.White, "Frequency content of randomly scattered signals", SIAM Review 33, 519-625, (1991).

....As we noted earlier, locally stationary processes arise when the mechanism that generates them changes slowly. Stochastic differential equations with slowly varying coefficients will often generate processes that are locally stationary. Many examples of geophysical interest are considered in [2, 13]. The processes depend on a parameter ffl which is the ratio of a typical fast scale to a typical slow one. Locally stationary time series that depend on a parameter in a similar way are also considered by Dahlhaus [6] We will explain briefly some of the ideas in [2] with a simple example. ....

....interest are considered in [2, 13] The processes depend on a parameter ffl which is the ratio of a typical fast scale to a typical slow one. Locally stationary time series that depend on a parameter in a similar way are also considered by Dahlhaus [6] We will explain briefly some of the ideas in [2] with a simple example. Spectral estimation for processes that vary on two widely separated time scales can take advantage of this with the use of asymptotics, as discussed in appendix E of [2] To generate simple examples of locally stationary processes with separation of scales, we start with a ....

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M. Asch, W. Kohler, G. Papanicolaou, M. Postel and B. White, "Frequency content of randomly scattered signals", SIAM Review, vol. 33, No. 4, pp 519-625, December 1991.

.... z ffl 2 ) Gamma ( z ffl 2 ) 61) The reflected signal R ffl f (t) is given by R ffl f (t) A(t; 0 ) 1 2 Z 1 Gamma1 e Gammai t=ffl f( A ffl (t; 0 )d : 62) The asymtotic local power spectrum 0 (t; as ffl Gamma 0 for constant 0 (z) and ae 0 (z) is given by [10] 0 (t; ff 2 (1 ff 2 t) 2 ; 63) where ff 0 is a parameter depends on c 0 and the statistics of the noise (z) and j(z) Fig. 13 shows one realization of the density ae(z) and the sound speed choosen in this experiment. The absorption coefficients oe u and oe p are choosen to be ....

M. Asch, W. Kohler, G. Papanicolaou, M. Postel and B. White, "Frequency content of randomly scattered signals", SIAM Review, vol. 33, No. 4, pp 519-625, December 1991.

....than typical propagation distances while the wavelength of the incident waves is intermediate between these two length scales. The random fluctuations in the elastic parameters are not, however, assumed to be small. We studied in detail acoustic wave reflection and transmission in this regime in [1] and pointed out its relevance to exploration geophysics, discussed extensively in [2] For waves in random media this asymptotic regime is interesting because localization phenomena [3] are fully developed. In its simplest form localization means that time harmonic plane waves are exponentially ....

.... for low frequencies or weak fluctuations, as well as some other cases, is known [4,5] Multiple scattering attenuation is seen clearly in numerical experiments [6] and its separation from intrinsic attenuation is an important problem for which a lot of progress has been made in the acoustic case ([1] and the references therein) This paper is the first one, as far as we know, where localization phenomena for elastic waves in randomly layered media are analyzed in detail, in particular, modal wave energy conversion and Lyapounov exponents. Earlier attempts [7,8] dealt with a more general, ....

[Article contains additional citation context not shown here]

M. Asch, W. Kohler, G. Papanicolaou, M. Postel and B. White, "Frequency content of randomly scattered signals", SIAM Review 33, 519--625 (1991).

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M. Asch, W. Kohler, G. Papanicolaou, M. Postel, and B. White, Frequency content of randomly scattered signals, SIAM Review 33, (1991) 519-626.

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M. Asch, W. Kohler, G. Papanicolaou, M. Postel and B. White, Frequency content of randomly scattered signals, SIAM Rev., vol 33, pp 519-625, 1991.

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M. Asch, W. Kohler, G. Papanicolaou, M. Postel, and B. tity characterizing the phenomenon. The Lyapunov expo- White, Frequency content of randomly scattered signals, nent calculation gives us a robust numerical procedure for SIAM Rev. 33, 519 (1991). computing the frequency-dependent details of localization

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Asch M., Kohler W., Papanicolaou G., Postel M., and White B., 1991, Frequency content of randomly scattered signals, SIAM Review, Vol. 33, pp. 519--625

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Asch M., Kohler W., Papanicolaou G., Postel M., and White B., 1991, Frequency content of randomly scattered signals, SIAM Review, Vol. 33, pp. 519-- 625

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