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.

scholarly journals Data-driven wavefield focusing and imaging with multidimensional deconvolution: Numerical examples for reflection data with internal multiples

Geophysics ◽  
2014 ◽  
Vol 79 (3) ◽  
pp. WA107-WA115 ◽  
Author(s):  
Filippo Broggini ◽  
Roel Snieder ◽  
Kees Wapenaar
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Imaging Techniques ◽  
Velocity Model ◽  
Single Scattering ◽  
Data Driven ◽  
Multiple Reflections ◽  
Prestack Migration ◽  
Reflection Data ◽  
Internal Multiples

Standard imaging techniques rely on the single scattering assumption. This requires that the recorded data do not include internal multiples, i.e., waves that have bounced multiple times between reflectors before reaching the receivers at the acquisition surface. When multiple reflections are present in the data, standard imaging algorithms incorrectly image them as ghost reflectors. These artifacts can mislead interpreters in locating potential hydrocarbon reservoirs. Recently, we introduced a new approach for retrieving the Green’s function recorded at the acquisition surface due to a virtual source located at depth. We refer to this approach as data-driven wavefield focusing. Additionally, after applying source-receiver reciprocity, this approach allowed us to decompose the Green’s function at a virtual receiver at depth in its downgoing and upgoing components. These wavefields were then used to create a ghost-free image of the medium with either crosscorrelation or multidimensional deconvolution, presenting an advantage over standard prestack migration. We tested the robustness of our approach when an erroneous background velocity model is used to estimate the first-arriving waves, which are a required input for the data-driven wavefield focusing process. We tested the new method with a numerical example based on a modification of the Amoco model.

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.

On the fundamentals of the virtual source method

Geophysics ◽  
2006 ◽  
Vol 71 (3) ◽  
pp. A13-A17 ◽  
Author(s):  
Valeri Korneev ◽  
Andrey Bakulin
Keyword(s):  
Green’S Function ◽  
Direct Measurement ◽  
Green's Function ◽  
Velocity Model ◽  
Practical Approach ◽  
Virtual Source ◽  
Processing Stage ◽  
Source Method ◽  
Complex Part ◽  
Seismic Images

The virtual source method (VSM) has been proposed as a practical approach to reduce distortions of seismic images caused by shallow, heterogeneous overburden. VSM is demanding at the acquisition stage because it requires placing downhole geophones below the most complex part of the heterogeneous overburden. Where such acquisition is possible, however, it pays off later at the processing stage because it does not require knowledge of the velocity model above the downhole receivers. This paper demonstrates that VSM can be viewed as an application of the Kirchhoff-Helmholtz integral (KHI) with an experimentally measured Green’s function. Direct measurement of the Green’s function ensures the effectiveness of the method in highly heterogeneous subsurface conditions.

The isochron ray in seismic modeling and imaging

Geophysics ◽  
2004 ◽  
Vol 69 (4) ◽  
pp. 1053-1070 ◽  
Author(s):  
Einar Iversen
Keyword(s):  
Seismic Imaging ◽  
Model Updating ◽  
Velocity Model ◽  
Seismic Modeling ◽  
Prestack Depth Migration ◽  
Depth Migration ◽  
Geometric Spreading ◽  
Model Independent ◽  
Map Migration ◽  
Upper Level

The isochron, the name given to a surface of equal two‐way time, has a profound position in seismic imaging. In this paper, I introduce a framework for construction of isochrons for a given velocity model. The basic idea is to let trajectories called isochron rays be associated with iso chrons in an way analogous to the association of conventional rays with wavefronts. In the context of prestack depth migration, an isochron ray based on conventional ray theory represents a simultaneous downward continuation from both source and receiver. The isochron ray is a generalization of the normal ray for poststack map migration. I have organized the process with systems of ordinary differential equations appearing on two levels. The upper level is model‐independent, and the lower level consists of conventional one‐way ray tracing. An advantage of the new method is that interpolation in a ray domain using isochron rays is able to treat triplications (multiarrivals) accurately, as opposed to interpolation in the depth domain based on one‐way traveltime tables. Another nice property is that the Beylkin determinant, an important correction factor in amplitude‐preserving seismic imaging, is closely related to the geometric spreading of isochron rays. For these reasons, the isochron ray has the potential to become a core part of future implementations of prestack depth migration. In addition, isochron rays can be applied in many contexts of forward and inverse seismic modeling, e.g., generation of Fresnel volumes, map migration of prestack traveltime events, and generation of a depth‐domain–based cost function for velocity model updating.

On the normal incidence of linear waves over a plane incline partially covered by a rigid lid

2009 ◽  
Vol 623 ◽  
pp. 209-240 ◽  
Author(s):  
ULF T. EHRENMARK
Keyword(s):  
Green’S Function ◽  
Standing Wave ◽  
Green's Function ◽  
Radiation Condition ◽  
Normal Incidence ◽  
Wave Pattern ◽  
Linear Waves ◽  
Finite Difference Discretization ◽  
Arbitrary Location ◽  
Uniformly Bounded

The effect is examined on infinitesimal standing waves over a plane beach when restricted by the arbitrary placing of a finite rigid (or permeable) lid of length ℓ on the undisturbed surface. A uniformly bounded solution for the potential function is obtained by a Green's function method. The Green's function is derived and manipulated, for subsequent computational expedience, from a previously known solution for the problem of an oscillating line source placed at an arbitrary location in the sector. Applications are made to both the case of plate anchored at the origin and the case of plate anchored some distance at sea (drifted plate problem). In both cases water column potentials and equipotentials are constructed from the numerical solution of a Fredholm equation of the second kind by finite difference discretization. Solutions are further extended to include the logarithmically singular standing wave, combination with which allows the construction of progressing waves. Computation of initially incoming progressing wave envelopes demonstrates the emergence of a partially standing wave pattern shoreward of the plate. There is no difficulty, in principle, to extend the theory to any number of plates, and this is verified by computation for the case of two plates. A new shoreline radiation condition is constructed to allow formulation, in the usual way, of the reflection/transmission problem for the plate, and results are in good qualitative agreement with a similar model on a horizontal plane bed. It is argued that the Green's function constructed here could be used in a number of diverse problems, of this linear nature, where all, or part, of the submerged boundary is that of a plane incline.

Common‐reflection‐surface stack: Image and attributes

Geophysics ◽  
2001 ◽  
Vol 66 (1) ◽  
pp. 97-109 ◽  
Author(s):  
Rainer Jäger ◽  
Jürgen Mann ◽  
German Höcht ◽  
Peter Hubral
Keyword(s):  
Optimal Parameter ◽  
Normal Wave ◽  
Normal Incidence ◽  
Prestack Migration ◽  
Computational Costs ◽  
Reflection Data ◽  
Common Reflection Surface ◽  
The Common ◽  
Zero Offset ◽  
Independent Parameters

The common‐reflection‐surface stack provides a zero‐offset simulation from seismic multicoverage reflection data. Whereas conventional reflection imaging methods (e.g. the NMO/dip moveout/stack or prestack migration) require a sufficiently accurate macrovelocity model to yield appropriate results, the common‐reflection‐surface (CRS) stack does not depend on a macrovelocity model. We apply the CRS stack to a 2-D synthetic seismic multicoverage dataset. We show that it not only provides a high‐quality simulated zero‐offset section but also three important kinematic wavefield attribute sections, which can be used to derive the 2-D macrovelocity model. We compare the multicoverage‐data‐derived attributes with the model‐derived attributes computed by forward modeling. We thus confirm the validity of the theory and of the data‐derived attributes. For 2-D acquisition, the CRS stack leads to a stacking surface depending on three search parameters. The optimum stacking surface needs to be determined for each point of the simulated zero‐offset section. For a given primary reflection, these are the emergence angle α of the zero‐offset ray, as well as two radii of wavefront curvatures [Formula: see text] and [Formula: see text]. They all are associated with two hypothetical waves: the so‐called normal wave and the normal‐incidence‐point wave. We also address the problem of determining an optimal parameter triplet (α, [Formula: see text], [Formula: see text]) in order to construct the sample value (i.e., the CRS stack value) for each point in the desired simulated zero‐offset section. This optimal triplet is expected to determine for each point the best stacking surface that can be fitted to the multicoverage primary reflection events. To make the CRS stack attractive in terms of computational costs, a suitable strategy is described to determine the optimal parameter triplets for all points of the simulated zero‐offset section. For the implementation of the CRS stack, we make use of the hyperbolic second‐order Taylor expansion of the stacking surface. This representation is not only suitable to handle irregular multicoverage acquisition geometries but also enables us to introduce simple and efficient search strategies for the parameter triple. In specific subsets of the multicoverage data (e.g., in the common‐midpoint gathers or the zero‐offset section), the chosen representation only depends on one or two independent parameters, respectively.

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.

Efficient loop antenna modeling for zero-offset, off-ground electromagnetic induction in multilayered media

Geophysics ◽  
2010 ◽  
Vol 75 (4) ◽  
pp. WA125-WA134 ◽  
Author(s):  
Davood Moghadas ◽  
Frédéric André ◽  
Harry Vereecken ◽  
Sébastien Lambot
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Electromagnetic Induction ◽  
Dynamic Range ◽  
Transfer Functions ◽  
Signal To Noise Ratio ◽  
Loop Antenna ◽  
Multilayered Media ◽  
Fast Evaluation ◽  
Zero Offset

Retrieval of the subsurface electrical properties from electromagnetic induction (EMI) data using inverse modeling relies in particular on the accuracy of the considered EMI model. We have developed a new EMI approach whereby a zero-offset, off-ground loop antenna is efficiently modeled using frequency-dependent, complex linear transfer functions and the air subsurface is described by a Green’s function for wave propagation in 3D multilayered media. To ensure proper calibration of the system, vector network analyzer (VNA) technology is used as the transmitter and receiver. An optimal integration path is proposed for fast evaluation of the spatial Green’s function from its spectral counterpart. We validated the antenna model in laboratory conditions with measurements performed with a loop antenna in free space and at different heights above a perfect electrical conductor. Provided that the loop antenna is high enough above the reflector (off-ground condition), the measured and modeled Green’s functions agreed remarkably well. In addition, inversion of the EMI data resulted in accurate estimates of the antenna heights. Yet, as expected, signal-to-noise-ratio issues occurred for the higher antenna heights and frequencies away from the loop resonant frequency. The method appears to be promising for accurate and robust soil characterization, but needs high VNA dynamic range and antenna gain.

Fast Inversion of the Earthquake Rupture Processes with Complicated Velocity Structure: An Application to the Earthquake of 2017 Mw 6.5 Jiuzhaigou, China

2021 ◽  
pp. 2150030
Author(s):  
Jiemin Wang ◽  
Haitao Yin ◽  
Zhijun Feng ◽  
Pifeng Ma ◽  
Liang Wang
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Green's Functions ◽  
Velocity Structure ◽  
Green’S Functions ◽  
Velocity Model ◽  
Rupture Process ◽  
Earthquake Rupture ◽  
Earthquake Rupture Process ◽  
Complex Structural

Due to the limitation of seismic station coverage or the network transport interrupted when the earthquake occurred, an accurate seismic shakemap may not be released to the public quickly. When the near-source observed waveforms for the intensity prediction technology used are incomplete, we synthesize the seismic waveform into observation waveforms. An accurate seismic rupture process is necessary to synthesize virtual station observations. So, we should release the rupture process as soon as possible after a large earthquake. Most large earthquakes occur at the junction of two or three tectonic terranes. With violent tectonic movements, fault basins and uplift zones are distributed on the edge of the plateau. With complex structural conditions, the 1D layered half-space velocity structure model could not meet the requirement of earthquake rupture process inversion. It takes much time to calculate 3D Green’s function with a 3D velocity model for the complete waveform inversion of the earthquake rupture process. To rapidly invert the rupture process as accurately as possible, according to the geological conditions of the station, we calculated several Green’s function libraries in advance. We extracted Green’s functions from these libraries for each site based on the sites’ coordinates once an earthquake occurs. The time we spend in extracting Green’s functions from several Green libraries equals that we spend in extracting Green’s functions from one single library. The applicability of this method was tested in the 2017 Jiuzhaigou M6.5 earthquake with complex structural conditions in the mountain uplift zone. With our model, the time we spent in calculating the rupture process was almost the same as that we spent with the 1D velocity structure model, which was far less than that we could have spent in calculating 3D Green’s function. The degree of fitting between the synthetic data and the observation data of our model was much higher than the fitting of the 1D velocity model, which means that the earthquake rupture process we determined was more reliable.

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