Werner E. Kohler

Frequency content of randomly scattered signals

The statistical properties of acoustic signals reflected by a randomly layered medium are analyzed when a pulsed spherical wave issuing from a point source is incident upon it. The asymptotic analy...

Power statistics for wave propagation in one‐dimension and comparison with radiative transport theory. II

We consider a one‐dimensional medium with random index of refraction or a transmission line with random capacitance per unit length, allowing for impedance mismatch at the load and generator. We compute the expected value of the incident and reflected powers at any point between the generator and load in the limit of weak fluctuations and a long line. The results are compared with those of radiative transport theory and discrepancies show the limitations of that theory. Finally, we consider the spreading of pulses due to random fluctuations.

Asymptotic analysis of deterministic and stochastic equations with rapidly varying components

The asymptotic character of deterministic and stochastic equations whose solutions have a rapidly varying component is studied. Of particular interest is the class of problems for which the limiting behavior can be described in a contracted and simplified framework.

Reflection of waves generated by a point source over a randomly layered medium

We consider a randomly layered half space adjoined to a homogeneous half space at the plane interface z=0. An acoustic source in the homogeneous medium generates a time-limited pulse which is then multiply reflected and backscattered from the random medium. We compute here the time dependent statistics of the signals recorded at receivers located on the interface z=0

Power Reflection from a Lossy One-Dimensional Random Medium

We consider the problem of an electromagnetic plane wave normally incident upon a slab of material whose constitutive parameters are subjected to lossy random perturbations. A transmission line model is adopted, wherein the four distributed parameters are assumed to be strongly mixing random functions of distance along the line. We study the reflection of energy at the input in the diffusion limit, an asymptotic limit involving weak random perturbations and long transmission lines. In the presence of dissipation, the probability density function for the modulus of the reflection coefficient approaches a nontrivial limit as the line length approaches infinity. We compute the mean and fluctuations of the voltage and power reflection coefficients with respect to this limiting density as a function of the dissipation.

Reflection of pulsed electromagnetic waves from a randomly stratified half-space

We consider the reflection of temporally pulsed electromagnetic waves from a weakly dispersive and dissipative, randomly stratified half-space. The incident pulse width is chosen to be short relative to deterministic background variations in the constitutive parameters but broad relative to the rapid, fine-scale random fluctuations in these parameter values. We study both plane-wave excitation (normal and oblique incidences) and excitation by a point current source. We are interested in the coherent reflected field and the mutual coherence function of the reflected field at the surface. Explicit results are given in certain special cases. We also compare the theory with numerical simulations.

A transport‐theoretic analysis of pulse propagation through ocean sediments

The reflection of pulsed acoustic plane waves from ocean sedimentary layers is studied using a stochastic transport theory originally introduced by Barabanenkov et al. [Izv. Vyssh. Uchebn. Zaved., Radiofiz 15, 1852 (1972)]. The sediments are assumed to be a random medium in which the density and sound speed undergo small, highly laminated (pancakelike) fluctuations. Although the problem is formulated in a general context, the predictions of the theory are fully evaluated only in the special case of normal incidence and no refracting profile. However, even with these approximations, some reasonable qualitative agreement of theoretical predictions with measured data is achieved.

Reflection from a one-dimensional, totally refracting random multilayer

The interplay of random scattering and total internal reflection is studied in the context of an idealized one-dimensional problem arising in acoustic scattering from ocean sediments. Appropriate scalings are introduced; the asymptotic analysis of the Fokker–Planck equation for the reflection coefficient is performed and subsequently compared with the results of numerical simulations.

Random scattering in magnetotellurics

Typical well logs show substantial variations of formation electrical resistivity over small spatial scales, down to the resolution of the logging tool. Using a plane stratified earth model, we examine the effects of this fine‐scale microstructure on scattering of the naturally occurring electromagnetic (EM) waves used in magnetotellurics. We show how 1-D magnetotelluric (MT) data may be viewed as arising statistically from a smoothed effective medium version of the resistivity‐depth profile. The difference between the data produced by the true medium and the effective medium is attributable to random scattering noise. This noise is fundamental to magnetotellurics and other diffusive‐wave EM exploration methods since it arises from the very small spatial scales that are usually ignored. The noise has unique statistical properties, which we characterize. We show that if scattering is the dominant noise source, a thin layer of increased resistivity at depth can be reliably detected only if the noise statist...

A Markov process model of ocean sediments

Monochromatic plane‐wave illumination of a randomly stratified, laterally homogeneous sediment layer is considered. The deposition process creating the stochastic layering is assumed to be a continuous parameter, finite state Markov chain. A Riccati equation for the plane‐wave reflection coefficient is formulated and first‐order partial differential equations for relevant probability density functions are subsequently obtained. These equations are solved numerically for a two‐material turbidite model similar to the one considered by Gilbert [J. Acoust. Soc. Am. 68, 1454–1458 (1980)]. Statistical moments of the reflection coefficient are computed at 25 and 250 Hz as a function of overall sediment thickness. These equations are also used to derive the nonrandom or ‘‘smooth’’ geoacoustic model that is appropriate in the low‐frequency limit.