3-D crosswell transmissions: Paraxial ray solutions and reciprocity paradox

Geophysics ◽  
1995 ◽  
Vol 60 (3) ◽  
pp. 810-820 ◽  
Author(s):  
Jianguo Sun
Keyword(s):  
Seismic Waves ◽  
Fresnel Zone ◽  
Matrix Decomposition ◽  
Decomposition Theorem ◽  
Wave Theory ◽  
Transformation Matrix ◽  
Reciprocity Relation ◽  
Earth Model ◽  
Vector Calculus ◽  
Paraxial Ray

Transmission of seismic waves through a 3-D earth model is of fundamental importance in seismology. If the model consists of many layers separated by curved interfaces, the only feasible solution to the transmitted waves is the one given by the geometrical optics approximation. Transmitted rays, transmitted wavefield, and the first Fresnel zone associated with a transmission point can be expressed by four 2 × 2 constant matrices constituting the 4 × 4 linearized ray transformation matrix. Generally, the ray transformation matrix can be constructed by dynamic ray tracing. However, if the layers are homogeneous, it can be formulated in closed form by using elementary vector calculus and coordinate transformations. Using the symplecticity of the ray transformation matrix, transmissions in opposite directions can be formulated by the ray transformation matrix for only one direction, and a reciprocity relation can be established. After the decomposition theorem for the ray transformation matrix, the transmitted wavefield in a model with many curved interfaces can be computed in a cascaded way. Using the [Formula: see text]‐matrix decomposition theorem, the normalized geometrical spreading factor can be expressed by means of the area of the first Fresnel zone of a transmission point. If seismic waves propagate through a locally spherical interface, the reciprocity relation may not hold. Using ray theory, this fact is shown by formulating the transmitted wavefield with the two principal radii of curvature of the transmitted wavefront at the transmission point under consideration. Using wave theory, this fact is shown by analyzing the Debye integral with the method of stationary phase.

DIVERGENCE EFFECTS IN A LAYERED EARTH

Geophysics ◽  
1973 ◽  
Vol 38 (3) ◽  
pp. 481-488 ◽  
Author(s):  
P. Newman
Keyword(s):  
Normal Incidence ◽  
Wave Theory ◽  
Diagnostic Value ◽  
Earth Model ◽  
Correction Factors ◽  
Amplitude Correction ◽  
Multiple Attenuation ◽  
Seismic Recording ◽  
Zero Offset ◽  
Special Case

Of the various factors which influence reflection amplitudes in a seismic recording, divergence effects are possibly of least direct interest to the interpreter. Nevertheless, proper compensation for these effects is mandatory if reflection amplitudes are to be of diagnostic value. For an earth model consisting of horizontal, isotropic layers, and assuming a point source, we apply ray theory to determine an expression for amplitude correction factors in terms of initial incidence, source‐receiver offset, and reflector depth. The special case of zero offset yields an expression in terms of two‐way traveltime, velocity in the initial layer, and the time‐weighted rms velocity which characterizes reflections. For this model it follows that information which is needed for divergence compensation in the region of normal incidence is available from the customary analysis of normal moveout (NMO). It is hardly surprising that NMO and divergence effects are intimately related when one considers the expanding wavefront situation which is responsible for both phenomena. However, it is evident that an amplitude correction which is appropriate for the primary reflection sequence cannot in general be appropriate for the multiples. At short offset distances the disparity in displayed amplitude varies as the square of the ratio of primary to multiple rms velocities, and favors the multiples. These observations are relevant to a number of concepts which are founded upon plane‐wave theory, notably multiple attenuation processes and record synthesis inclusive of multiples.

Amplitude, Fresnel zone, and NMO velocity for PP and SS normal-incidence reflections

Geophysics ◽  
2006 ◽  
Vol 71 (2) ◽  
pp. W1-W14 ◽  
Author(s):  
Einar Iversen
Keyword(s):  
Phase Shift ◽  
Ray Tracing ◽  
Explicit Knowledge ◽  
Fresnel Zone ◽  
Normal Wave ◽  
Normal Incidence ◽  
Geometric Spreading ◽  
Dynamic Ray Tracing ◽  
The One ◽  
Paraxial Ray

Inspired by recent ray-theoretical developments, the theory of normal-incidence rays is generalized to accommodate P- and S-waves in layered isotropic and anisotropic media. The calculation of the three main factors contributing to the two-way amplitude — i.e., geometric spreading, phase shift from caustics, and accumulated reflection/transmission coefficients — is formulated as a recursive process in the upward direction of the normal-incidence rays. This step-by-step approach makes it possible to implement zero-offset amplitude modeling as an efficient one-way wavefront construction process. For the purpose of upward dynamic ray tracing, the one-way eigensolution matrix is introduced, having as minors the paraxial ray-tracing matrices for the wavefronts of two hypothetical waves, referred to by Hubral as the normal-incidence point (NIP) wave and the normal wave. Dynamic ray tracing expressed in terms of the one-way eigensolution matrix has two advantages: The formulas for geometric spreading, phase shift from caustics, and Fresnel zone matrix become particularly simple, and the amplitude and Fresnel zone matrix can be calculated without explicit knowledge of the interface curvatures at the point of normal-incidence reflection.

Interpretative lessons from three‐dimensional modeling

Geophysics ◽  
1982 ◽  
Vol 47 (5) ◽  
pp. 784-808 ◽  
Author(s):  
Fred J. Hilterman
Keyword(s):  
Fresnel Zone ◽  
Specular Reflection ◽  
Three Dimensional ◽  
Wave Theory ◽  
Three Dimensional Modeling ◽  
Seismic Resolution ◽  
Maximum Resolution ◽  
Dimensional Modeling ◽  
Complex Models ◽  
Concave Boundary

Three‐dimensional (3-D) seismic modeling has been accomplished by describing geologic surfaces with triangular plates and then computing the seismic response by Kirchhoff wave theory. The resulting time sections illustrate many interesting 3-D phenomena which are useful in interpreting geologic structures. Three‐dimensional resolution studies relate the concept of Fresnel zone reflection to seismic resolution. If high resolution is desired both horizontally and vertically, then not only is a dense field survey required, but also a detailed amplitude study. The dense seismic coverage is required to map the focal line of concave boundary edges, which are difficult to delineate with conventional seismic data. Additional studies on complex models, such as grabens and 3-D permeability traps, associate interpretational pitfalls to a wandering specular reflection path, that is, “side‐swipe.” In each geologic model, maximum resolution is obtained on a principal plane line (dip line). If a seismic dip line is not available, the necessity of doing 3-D migration is emphasized, even if it is a migration of the time map.

High‐frequency reflections in granite? Delineation of the weathering front in granodiorite at Piñon Flat, California

Geophysics ◽  
1999 ◽  
Vol 64 (6) ◽  
pp. 1828-1835 ◽  
Author(s):  
Stanley J. Radzevicius ◽  
Gary L. Pavlis
Keyword(s):  
High Frequency ◽  
Seismic Waves ◽  
Lower Layer ◽  
Specular Reflection ◽  
Earth Model ◽  
Remarkable Feature ◽  
Reflection Data ◽  
Weathered Layer ◽  
High Frequency Seismic Waves ◽  
Seismic Lines

We analyze data from two orthogonal seismic lines 336 m in length collected at Piñon Flat, California, over weathered granodiorite bedrock. Each line was made up of 10 reversed segments 84 m in length. We analyzed the first arrivals from these data and found dramatic variations in velocity along the profiles. An upper layer (approximately 2-m thick) known from trenching to be composed of soil and sandy grus had measured velocities ranging from 400 to 700 m/s. Velocities inferred from refraction analysis of first arrivals of the reversed lines revealed a heterogeneous lower layer below the soil with measured velocities of 1600–2700 m/s by a depth of 15 m. We interpret these data to be measuring velocities of a deeply weathered unit characterized by granodiorite corestones embedded in a matrix of saprolite. The most remarkable feature of these data emerged from attempting to process the same data as reflection data. Simple bandpass filtering in the 250–400 Hz band revealed a bright, impulsive arrival with three characteristic properties: (1) irregular velocity moveout that is inconsistent with that expected from a layered earth model, (2) the arrival is at a nearly constant time‐depth on all data, and (3) the arrival tends to be followed by a ringing coda whose frequency varies from trace to trace. This arrival ties exactly with a velocity discontinuity measured in a borehole located on one of the profiles that we interpret as the base of the weathered layer. We suggest this arrival is a specular reflection from a weathering front that occurs along horizontal sheeting joints at a fixed depth below the surface. This surface acts as an effective mirror for high‐frequency seismic waves which are then channeled upward through an intact, high-Q path of unaltered blocks of granodiorite to define the observed signals at the surface.

The Fresnel volume and transmitted waves

Geophysics ◽  
2004 ◽  
Vol 69 (3) ◽  
pp. 653-663 ◽  
Author(s):  
Jesper Spetzler ◽  
Roel Snieder
Keyword(s):  
Wave Equation ◽  
Fresnel Zone ◽  
Wave Theory ◽  
Three Dimensions ◽  
Ray Theory ◽  
Finite Frequency ◽  
Frequency Wave ◽  
Reflected Waves ◽  
Band Limited ◽  
Propagation Of Waves

In seismic imaging experiments, it is common to use a geometric ray theory that is an asymptotic solution of the wave equation in the high‐frequency limit. Consequently, it is assumed that waves propagate along infinitely narrow lines through space, called rays, that join the source and receiver. In reality, recorded waves have a finite‐frequency content. The band limitation of waves implies that the propagation of waves is extended to a finite volume of space around the geometrical ray path. This volume is called the Fresnel volume. In this tutorial, we introduce the physics of the Fresnel volume and we present a solution of the wave equation that accounts for the band limitation of waves. The finite‐frequency wave theory specifies sensitivity kernels that linearly relate the traveltime and amplitude of band‐limited transmitted and reflected waves to slowness variations in the earth. The Fresnel zone and the finite‐frequency sensitivity kernels are closely connected through the concept of constructive interference of waves. The finite‐frequency wave theory leads to the counterintuitive result that a pointlike velocity perturbation placed on the geometric ray in three dimensions does not cause a perturbation of the phase of the wavefield. Also, it turns out that Fermat's theorem in the context of geometric ray theory is a special case of the finite‐frequency wave theory in the limit of infinite frequency. Last, we address the misconception that the width of the Fresnel volume limits the resolution in imaging experiments.

THE METHOD OF GENERALIZED REFLECTION AND TRANSMISSION COEFFICIENTS

Geophysics ◽  
1960 ◽  
Vol 25 (3) ◽  
pp. 625-641 ◽  
Author(s):  
T. W. Spencer
Keyword(s):  
Integral Representation ◽  
Wave Theory ◽  
Earth Model ◽  
Reflection And Transmission ◽  
Localized Source ◽  
Transmission Coefficients ◽  
The Laplace Transform ◽  
Normal Mode Theory ◽  
Ray Path ◽  
Irrotational Wave

The objective of this work is to provide a method for predicting the surface response of a stratified half space to the radiation from a localized source when neither the assumptions of the plane wave theory nor the assumptions of the normal mode theory are valid. The earth model consists of a finite number of perfectly elastic, homogeneous, isotropic layers separated by interfaces which are plane and parallel to one another. The method leads to an infinite series for the Laplace transform of the response function (displacement, velocity, stress, etc.) in a multi‐interface system. Each term in the series describes all the energy which traverses a particular generalized ray path between the source and the receiver. The specification of the mode of propagation across each stratum (either as an irrotational wave or as an equivoluminal wave) and of the sequence in which the strata are traversed serve to define a generalized ray path. A prescription is given for constructing the integral representation for the disturbance which has traversed such a path directly from the integral representation for the source radiation. The method therefore obviates the necessity for solving a tedious boundary value problem. The time function associated with each term can be obtained by using Cagniard’s method.

Seismoelectric numerical modeling on a grid

Geophysics ◽  
2006 ◽  
Vol 71 (6) ◽  
pp. N57-N65 ◽  
Author(s):  
Seth S. Haines ◽  
Steven R. Pride
Keyword(s):  
Electric Fields ◽  
Electric Potential ◽  
Potential Distribution ◽  
Seismic Waves ◽  
Fresnel Zone ◽  
Near Field ◽  
Fluid Pressure ◽  
Heterogeneous Porous Media ◽  
Time Step ◽  
Electric Potential Distribution

Our finite-difference algorithm provides a new method for simulating how seismic waves in arbitrarily heterogeneous porous media generate electric fields through an electrokinetic mechanism called seismoelectric coupling. As the first step in our simulations, we calculate relative pore-fluid/grain-matrix displacement by using existing poroelastic theory. We then calculate the electric current resulting from the grain/fluid displacement by using seismoelectric coupling theory. This electrofiltration current acts as a source term in Poisson’s equation, which then allows us to calculate the electric potential distribution. We can safely neglect induction effects in our simulations because the model area is within the electrostatic near field for the depth of investigation (tens to hundreds of meters) and the frequency ranges ([Formula: see text] to [Formula: see text]) of interest for shallow seismoelectric surveys.We can independently calculate the electric-potential distribution for each time step in the poroelastic simulation without loss of accuracy because electro-osmotic feedback (fluid flow that is perturbed by generated electric fields) is at least [Formula: see text] times smaller than flow that is driven by fluid-pressure gradients and matrix acceleration, and is therefore negligible. Our simulations demonstrate that, distinct from seismic reflections, the seismoelectric interface response from a thin layer (at least as thin as one-twentieth of the seismic wavelength) is considerably stronger than the response from a single interface. We find that the interface response amplitude decreases as the lateral extent of a layer decreases below the width of the first Fresnel zone. We conclude, on the basis of our modeling results and of field results published elsewhere, that downhole and/or crosswell survey geometries and time-lapse applications are particularly well suited to the seismoelectric method.

Fresnel zone plate solved by the boundary-diffraction-wave theory

1970 ◽  
Vol 48 (15) ◽  
pp. 1799-1805 ◽  
Author(s):  
J. W. Y. Lit ◽  
R. Tremblay
Keyword(s):  
Finite Number ◽  
Wave Amplitude ◽  
Fresnel Zone ◽  
Wave Theory ◽  
Zone Plate ◽  
Fresnel Zone Plate ◽  
Physical Picture ◽  
Diffraction Wave

The Fresnel zone plate is studied by the boundary-diffraction-wave theory (BDWT). Expressions are given for the fields in the transverse planes and along the axis; in particular, the wave amplitudes at the foci are considered in some detail. The BDWT provides a clear physical picture of the problem. The wave amplitude at any point is given by the sum of a finite number of rays determined by the number of zones in the plate. The foci are the points where the singly diffracted rays are in phase.

Coseismic electric and magnetic signals observed during 2017 Jiuzhaigou Mw 6.5 earthquake and explained by electrokinetics and magnetometer rotation

2020 ◽  
Vol 223 (2) ◽  
pp. 1130-1143
Author(s):  
Yongxin Gao ◽  
Guoze Zhao ◽  
Jiajun Chong ◽  
Simon L Klemperer ◽  
Bing Han ◽  
...  
Keyword(s):  
Magnetic Fields ◽  
Electric Fields ◽  
Seismic Waves ◽  
Western China ◽  
Magnetic Data ◽  
Earth Model ◽  
Induction Effect ◽  
Jiuzhaigou Earthquake ◽  
Electrokinetic Effect ◽  
Motional Induction

SUMMARY Very clear coseismic electric and magnetic signals accompanying seismic waves were observed during the 2017 Mw 6.5 Jiuzhaigou earthquake, which took place in western China. In order to understand the generation mechanism of these observed signals, we simulate electric and magnetic responses to this specific earthquake based on three mechanisms, namely, the electrokinetic effect, the motional induction effect and the rotation effect of the coil-type magnetometer. We conduct the simulations using a point source model and a realistic layered earth model and compare to the observed data in the frequency band 0.05–0.3 Hz. Our results show that the electrokinetic effect can explain the observed electric fields in both waveform and amplitude, but it cannot explain the magnetic signals accompanying the Rayleigh wave. The motional induction effect cannot explain either the coseismic electric or magnetic data because it predicts much weaker coseismic electric and magnetic fields than the observed data. The magnetic fields resulting from the rotation of the magnetometer agree with the observed data in the waveforms though their amplitudes are two to four times smaller than the observed data. Our simulations suggest that the electrokinetic effect is responsible for the generation of coseismic electric fields and that rotation of the coil magnetometer is likely the main cause of coseismic magnetic fields. The results improve our interpretation of the coseismic electromagnetic (EM) phenomenon and are useful for understanding other kinds of earthquake-associated EM phenomena.

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