Benjamin S. White

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...

Field tests of electroseismic hydrocarbon detection

Geophysicists, looking for new exploration tools, have studied the coupling between seismic and electromagnetic waves in the near-surface since the 1930s. Our research explores the possibility that electromagnetic-to-seismic (ES) conversion is useful at greater depths. Field tests of ES conversion over gas sands and carbonate oil reservoirs succeeded in delineating known hydrocarbon accumulations from depths up to 1500 m . This is the first observation of electromagnetic-to-seismic coupling from surface electrodes and geophones. Electrodes at the earth’s surface generate electric fields at the target and digital accelerometers detect the returning seismic wave. Conversion at depth is confirmed with hydrophones placed in wells. The gas sands yielded a linear ES response, as expected for electrokinetic energy conversion, and in qualitative agreement with numerical simulations. The carbonate oil reservoirs generate nonlinear conversions; a qualitatively new observation and a new probe of rock properties. The...

Probing a random medium with a pulse

This paper studies the reflection of pulses from a randomly layered half space. It characterizes the statistical properties of the reflected signals at the surface in a suitable asymptotic limit in...

Statistics for pulse reflection from a randomly layered medium

We consider reflection of a pulse incident on a layered halfspace whose density and bulk modulus vary randomly. We show that when the pulse width is long compared to the average time it takes to travel over one correlation length, the reflected signal is approximately a Gaussian random process. The parameters of this process change slowly on a scale long compared to the pulse width. We give a full characterization of the power spectrum of the Gaussian process in terms of a universal function that does not depend on the medium.

Localization and backscattering spectrum of seismic waves in stratified lithology

Using four sonic well logs from diverse geological environments, we analyze the statistics of lithological layers relevant to seismic wave propagation. The autocorrelation functions are found to be well approximated by exponentials with correlation lengths generally in the 1.5 to 3 m range. We use localization theory to calculate the apparent attenuation caused by random scattering as a function of frequency. This attenuation has a peak at 50 to 150 Hz with a corresponding localization length of 1.6 to 4.8 km (1 to 3 mi) and an apparent Q of 120 to 450. At seismic frequencies (20 Hz) the attenuation is smaller, with localization lengths in the 16 to 32 km (10 to 20 mi) range and an apparent Q of 300 to 700. These values of apparent Q are consistent with observations of previous authors who used well‐log calculations. Using finite differences we compute synthetic traces for a 20-Hz pulse with multiple backscattering from the lithology given by each of the well logs. The power spectral density of the trace ...

Gaussian Wave Packets in Inhomogeneous Media with Curved Interfaces

Time-dependent particle-like pulses are considered as asymptotic solutions of the classical wave equation. The wave packets are localized in space with gaussian envelopes. The pulse centres propagate along the rays of the wave equation, and the envelope parameters satisfy evolution equations very similar to the ray equations for time-harmonic disturbances. However, the present theory contains an extra degree of freedom not found in the time-harmonic theory. Explicit results are presented for media with constant velocity gradients, and interesting new phenomena are identified. For example, a pulse that is initially long in the direction of propagation and comparatively narrow in the orthogonal direction, maintains its initial spatial orientation even as the propagation direction rotates. The reflection and transmission of a pulse incident upon an interface are also discussed. The various theoretical results are illustrated by numerical simulations. This method of solution could be very useful for fast forward modelling in large-scale structures. It is formulated explicitly in the time domain and does not suffer from unphysical singularities at caustics.

Electroseismic Prospecting in Layered Media

Electroseismic (ES) prospecting is an experimental method that seeks to use the conversion of electromagnetic (EM) waves to seismic waves in the earth to explore for oil and gas. The wave conversion occurs through the phenomenon of electrokinetics, for which a complete set of partial differential equations was derived by S. Pride. In this paper, we show how Pride’s equations in plane layered media can be written in a convenient mathematical form suggested by B. Ursin, who used this form to give a unified treatment of EM waves, acoustic waves, and the waves of isotropic elasticity in plane layered media. We use Ursin’s formalism, which we develop and simplify for the case of a stack of homogeneous layers, to derive explicit formulas that can be made the basis of an efficient computer code. Numerical results are presented for spatially extended electrode sources that have been used in field tests of ES prospecting. More generally, the methods developed are applicable to any system that can be put into Ursin...

Localization and mode conversion for elastic waves in randomly layered media I

Abstract This paper is Part I of a two-part work in which we derive localization theory for elastic waves in plane-stratified media, a multimode problem complicated by the interconversion of shear and compressional waves, both in propagation and in backscatter. We consider the low frequency limit, i.e., when the randomness constitutes a microstructure. In this part, we set up the general suite of problems and derive the probability density and moments for the fraction of reflected energy which remains in the same mode (shear or compressional) as the incident field. Our main mathematical tool is a limit theorem for stochastic differential equations with a small parameter. In Part II we will use the limit theorem of Part I and the Oseledec Theorem, which establishes the existence of the localization length and other structural information, to compute: the localization length and another deterministic length, called the equilibration length, which gives the scale for equilibration of shear and compressional energy in propagation; and the probability density of the ratio of shear to compressional energy in transmission through a large slab. This last quantity is shown to be asymptotically independent of the incident field. We also extend the results to the small fluctuation, rather than the low frequency case.

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

Asymptotic Theory of Electroseismic Prospecting

In a porous medium such as the earths subsurface, electromagnetic (EM) waves and mechanical waves are coupled through the phenomenon of electrokinetics, for which a complete set of partial differential equations was derived by S. Pride. In this paper, we derive from Prides equations an asymptotic theory that enables forward modeling of the seismic response to an EM source in fully three-dimensional geometries on a scale that is relevant to exploration. For simplicity, we consider piecewise homogeneous media separated by interfaces which are curved surfaces in three dimensions. The following physical picture emerges: An EM source excites an EM wave which propagates into the earth, stirring up local mechanical movement. At an interface, EM energy is converted to seismic waves, which may be described by ray theory. Instantly, on the seismic time scale, every interface becomes a wavefront for both compressional and shear waves; that is, seismic P- and S-waves explode from both sides of each interface, at ev...