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.

Three‐dimensional primary zero‐offset reflections

Geophysics ◽  
1993 ◽  
Vol 58 (5) ◽  
pp. 692-702 ◽  
Author(s):  
Peter Hubral ◽  
Jorg Schleicher ◽  
Martin Tygel
Keyword(s):  
Ray Tracing ◽  
Large Scale ◽  
Initial Conditions ◽  
Fresnel Zone ◽  
Three Dimensional ◽  
Normal Incidence ◽  
Careful Analysis ◽  
Point Sources ◽  
Dynamic Ray Tracing ◽  
Zero Offset

Zero‐offset reflections resulting from point sources are often computed on a large scale in three‐dimensional (3-D) laterally inhomogeneous isotropic media with the help of ray theory. The geometrical‐spreading factor and the number of caustics that determine the shape of the reflected pulse are then generally obtained by integrating the so‐called dynamic ray‐tracing system down and up to the two‐way normal incidence ray. Assuming that this ray is already known, we show that one integration of the dynamic ray‐tracing system in a downward direction with only the initial condition of a point source at the earth’s surface is in fact sufficient to obtain both results. To establish the Fresnel zone of the zero‐offset reflection upon the reflector requires the same single downward integration. By performing a second downward integration (using the initial conditions of a plane wave at the earth’s surface) the complete Fresnel volume around the two‐way normal ray can be found. This should be known to ascertain the validity of the computed zero‐offset event. A careful analysis of the problem as performed here shows that round‐trip integrations of the dynamic ray‐tracing system following the actually propagating wavefront along the two‐way normal ray need never be considered. In fact some useful quantities related to the two‐way normal ray (e.g., the normal‐moveout velocity) require only one single integration in one specific direction only. Finally, a two‐point ray tracing for normal rays can be derived from one‐way dynamic ray tracing.

Zero-offset seismic amplitude decomposition and migration

Geophysics ◽  
2007 ◽  
Vol 72 (4) ◽  
pp. S187-S193
Author(s):  
Bjørn Ursin ◽  
Martin Tygel
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Normal Wave ◽  
Normal Incidence ◽  
Velocity Model ◽  
Transmission And Reflection Coefficients ◽  
Geometric Spreading ◽  
Map Migration ◽  
Zero Offset ◽  
And Migration

In an anisotropic medium, a normal-incidence wave is multiply transmitted and reflected down to a reflector where the phase-velocity vector is parallel to the interface normal. The ray code of the upgoing wave is equal to the ray code of the downgoing wave in reverse order. The geometric spreading, KMAH index, and transmission and reflection coefficients of the normal-incidence ray can be expressed in terms of products or sums of the corresponding quantities of the one-way normal and normal-incidence-point (NIP) waves. Here, we show that the amplitude of the ray-theoretic Green’s function for the reflected wave also follows a similar decomposition in terms of the amplitude of the Green’s function of the NIP wave and the normal wave. We use this property to propose three schemes for true-amplitude poststack depth migration in anisotropic media where the image represents an estimate of the zero-offset reflection coefficient. The first is a map migration procedure in which selected primary zero-offset reflections are converted into depth with attached true amplitudes. The second is a ray-based, Kirchhoff-type full migration. The third is a wave equation continuation algorithm to reverse-propagate the recorded wavefield in a half-velocity model with half the elastic constants and double the density. The image is formed by taking the reverse-propagated wavefield at time equal to zero followed by a geometric spreading correction.

Fresnel volume ray tracing

Geophysics ◽  
1992 ◽  
Vol 57 (7) ◽  
pp. 902-915 ◽  
Author(s):  
Vlastislav Červený ◽  
José Eduardo P. Soares
Keyword(s):  
Ray Tracing ◽  
High Frequency ◽  
Fresnel Zone ◽  
Body Wave ◽  
Low Frequency ◽  
Algebraic Manipulation ◽  
Ray Method ◽  
Propagator Matrix ◽  
Paraxial Ray ◽  
Fresnel Volumes

The concept of “Fresnel volume ray tracing” consists of standard ray tracing, supplemented by a computation of parameters defining the first Fresnel zones at each point of the ray. The Fresnel volume represents a 3-D spatial equivalent of the Fresnel zone that can also be called a physical ray. The shape of the Fresnel volume depends on the position of the source and the receiver, the structure between them, and the type of body wave under consideration. In addition, the shape also depends on frequency: it is narrow for a high frequency and thick for a low frequency. An efficient algorithm for Fresnel volume ray tracing, based on the paraxial ray method, is proposed. The evaluation of the parameters defining the first Fresnel zone merely consists of a simple algebraic manipulation of the elements of the ray propagator matrix. The proposed algorithm may be applied to any high‐frequency seismic body wave propagating in a laterally varying 2-D or 3-D layered structure (P, S, converted, multiply reflected, etc.). Numerical examples of Fresnel volume ray tracing in 2-D inhomogeneous layered structures are presented. Certain interesting properties of Fresnel volumes are discussed (e.g., the double caustic effect). Fresnel volume ray tracing offers numerous applications in seismology and seismic prospecting. Among others, it can be used to study the resolution of the seismic method and the validity conditions of the ray method.

Comparison of weights in prestack amplitude‐preserving Kirchhoff depth migration

Geophysics ◽  
1997 ◽  
Vol 62 (6) ◽  
pp. 1812-1816 ◽  
Author(s):  
Christian Hanitzsch
Keyword(s):  
Point Source ◽  
Green’S Function ◽  
Ray Tracing ◽  
Green's Function ◽  
Weight Functions ◽  
Depth Migration ◽  
Theoretical Approaches ◽  
Geometric Spreading ◽  
Dynamic Ray Tracing ◽  
Kirchhoff Depth Migration

Three different theoretical approaches to amplitude‐preserving Kirchhoff depth migration are compared. Each of them suggests applying weights in the diffraction stack migration to correct for amplitude loss resulting from geometric spreading. The weight functions are given in different notations, but as is shown, all of these expressions are similar. A notation that is well suited for implementation is suggested: entirely in terms of Green's function quantities (amplitudes or point‐source propagators). For the most common prestack configurations (common‐shot and common‐offset) and 3-D, 2.5-D, and 2-D migrations, expressions of the weights are given in this notation. The quantities needed for calculation of the weights can be computed easily, e.g., by dynamic ray tracing.

Inversion of normal incidence seismograms

Geophysics ◽  
1982 ◽  
Vol 47 (5) ◽  
pp. 757-770 ◽  
Author(s):  
A. Bamberger ◽  
G. Chavent ◽  
Ch. Hemon ◽  
P. Lailly
Keyword(s):  
Simulated Data ◽  
Normal Incidence ◽  
State Equation ◽  
Inversion Algorithm ◽  
Multiple Reflections ◽  
Common Depth Point ◽  
Adjoint State ◽  
Nmo Correction ◽  
The One ◽  
Practical Feasibility

The well‐known instability of Kunetz’s (1963) inversion algorithm can be explained by the progressive manner in which the calculations are done (descending from the surface) and by the fact that completely different impedances can yield indistinguishable synthetic seismograms. Those difficulties can be overcome by using an iterative algorithm for the inversion of the one‐dimensional (1-D) wave equation, together with a stabilizing constraint on the sums of the jumps of the desired impedance. For computational efficiency, the synthetic seismogram is computed by the method of characteristics, and the gradient of the error criterion is computed by optimal control techniques (adjoint state equation). The numerical results on simulated data confirm the expected stability of the algorithm in the presence of measurement noise (tests include noise levels of 50 percent). The inversion of two field sections demonstrates the practical feasibility of the method and the importance of taking into account all internal as well as external multiple reflections. Reflection coefficients obtained by this method show an excellent agreement with well‐log data in a case where standard estimation techniques [deconvolution of common‐depth‐point (CDP) stacked and normal‐moveout (NMO) correction section] failed.

scholarly journals Study on the Velocity Analysis Method of Low SNR Seismic Data

2021 ◽  
Vol 11 (1) ◽  
pp. 78
Author(s):  
Jianbo He ◽  
Zhenyu Wang ◽  
Mingdong Zhang
Keyword(s):  
Seismic Data ◽  
Fresnel Zone ◽  
Signal To Noise Ratio ◽  
Normal Wave ◽  
Curvature Radius ◽  
Velocity Spectrum ◽  
Velocity Analysis ◽  
Signal To Noise ◽  
Noise Ratio ◽  
Zero Offset

When the signal to noise ratio of seismic data is very low, velocity spectrum focusing will be poor., the velocity model obtained by conventional velocity analysis methods is not accurate enough, which results in inaccurate migration. For the low signal noise ratio (SNR) data, this paper proposes to use partial Common Reflection Surface (CRS) stack to build CRS gathers, making full use of all of the reflection information of the first Fresnel zone, and improves the signal to noise ratio of pre-stack gathers by increasing the number of folds. In consideration of the CRS parameters of the zero-offset rays emitted angle and normal wave front curvature radius are searched on zero offset profile, we use ellipse evolving stacking to improve the zero offset section quality, in order to improve the reliability of CRS parameters. After CRS gathers are obtained, we use principal component analysis (PCA) approach to do velocity analysis, which improves the noise immunity of velocity analysis. Models and actual data results demonstrate the effectiveness of this method.

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