Amplitude-variation-with-offset, elastic-impedence, and wave-equation synthetics — A modeling study

Geophysics ◽  
2007 ◽  
Vol 72 (1) ◽  
pp. C1-C7 ◽  
Author(s):  
Subhashis Mallick
Keyword(s):  
Wave Equation ◽  
Incidence Angle ◽  
Synthetic Data ◽  
P Wave ◽  
Equation Modeling ◽  
Parameter Estimates ◽  
Angle Of Incidence ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Wave Modes

Amplitude-variation-with-offset (AVO) and elastic-impedance (EI) analysis use an approximate plane P-wave reflection coefficient as a function of angle of incidence. AVO and EI both can be used in a three-term or a two-term formulation. This study uses synthetic data to demonstrate that the P-wave primary reflections at large offsets can be contaminated by reflections from other wave modes that can affect the quality of three-term AVO or EI results. The coupling of P-waves and S-waves in seismic-wave propagation through finely layered media generates the interfering wave modes. A methodology such as prestack-wave-equation modeling can properly account for these coupling effects. Both AVO and EI also assume a convolutional model whose accuracy decreases as incidence angles increase. On the other hand, wave-equation modeling is based on the rigorous solution to the wave equation and is valid for any incidence angle. Because wave interference is minimal at small angles, a two-term AVO/EI analysis that restricts input from small angles is likely to give more reliable parameter estimates than a three-term analysis. A three-term AVO/EI analysis should be used with caution and should be calibrated against well data and other data before being used for quantitative analysis.

Signal leakage in f-x deconvolution algorithms

Geophysics ◽  
2017 ◽  
Vol 82 (5) ◽  
pp. W31-W45 ◽  
Author(s):  
Necati Gülünay
Keyword(s):  
Field Data ◽  
New Technologies ◽  
Filter Design ◽  
Synthetic Data ◽  
Data Sets ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Backward Design ◽  
Use Of Data ◽  
Domain Prediction

The old technology [Formula: see text]-[Formula: see text] deconvolution stands for [Formula: see text]-[Formula: see text] domain prediction filtering. Early versions of it are known to create signal leakage during their application. There have been recent papers in geophysical publications comparing [Formula: see text]-[Formula: see text] deconvolution results with the new technologies being proposed. These comparisons will be most effective if the best existing [Formula: see text]-[Formula: see text] deconvolution algorithms are used. This paper describes common [Formula: see text]-[Formula: see text] deconvolution algorithms and studies signal leakage occurring during their application on simple models, which will hopefully provide a benchmark for the readers in choosing [Formula: see text]-[Formula: see text] algorithms for comparison. The [Formula: see text]-[Formula: see text] deconvolution algorithms can be classified by their use of data which lead to transient or transient-free matrices and hence windowed or nonwindowed autocorrelations, respectively. They can also be classified by the direction they are predicting: forward design and apply; forward design and apply followed by backward design and apply; forward design and apply followed by application of a conjugated forward filter in the backward direction; and simultaneously forward and backward design and apply, which is known as noncausal filter design. All of the algorithm types mentioned above are tested, and the results of their analysis are provided in this paper on noise free and noisy synthetic data sets: a single dipping event, a single dipping event with a simple amplitude variation with offset, and three dipping events. Finally, the results of applying the selected algorithms on field data are provided.

scholarly journals An Accurate Jacobian Matrix with Exact Zoeppritz for Elastic Moduli of Dry Rock

2019 ◽  
Vol 9 (24) ◽  
pp. 5485
Author(s):  
Xiaobo Liu ◽  
Jingyi Chen ◽  
Fuping Liu ◽  
Zhencong Zhao
Keyword(s):  
Jacobian Matrix ◽  
Incidence Angle ◽  
Inversion Method ◽  
Seismic Velocities ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Shear Moduli ◽  
Partial Derivatives ◽  
Solid Matrices ◽  
Derivatives Of

Seismic velocities are related to the solid matrices and the pore fluids. The bulk and shear moduli of dry rock are the primary parameters to characterize solid matrices. Amplitude variation with offset (AVO) or amplitude variation with incidence angle (AVA) is the most used inversion method to discriminate lithology in hydrocarbon reservoirs. The bulk and shear moduli of dry rock, however, cannot be inverted directly using seismic data and the conventional AVO/AVA inversions. The most important step to accurately invert these dry rock parameters is to derive the Jacobian matrix. The combination of exact Zoeppritz and Biot–Gassmann equations makes it possible to directly calculate the partial derivatives of seismic reflectivities (PP-and PS-waves) with respect to dry rock moduli. During this research, we successfully derive the accurate partial derivatives of the exact Zoeppritz equations with respect to bulk and shear moduli of dry rock. The characteristics of these partial derivatives are investigated in the numerical examples. Additionally, we compare the partial derivatives using this proposed algorithm with the classical Shuey and Aki–Richards approximations. The results show that this derived Jacobian matrix is more accurate and versatile. It can be used further in the conventional AVO/AVA inversions to invert bulk and shear moduli of dry rock directly.

Two-step joint PP- and PS-wave three-term amplitude-variation with offset inversion

2016 ◽  
Vol 4 (4) ◽  
pp. T613-T625 ◽  
Author(s):  
Qizhen Du ◽  
Bo Zhang ◽  
Xianjun Meng ◽  
Chengfeng Guo ◽  
Gang Chen ◽  
...  
Keyword(s):  
Reflection Coefficient ◽  
Wave Reflection ◽  
Coefficient Matrix ◽  
P Wave ◽  
Inversion Method ◽  
S Wave ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Avo Inversion ◽  
Pp Wave

Three-term amplitude-variation with offset (AVO) inversion generally suffers from instability when there is limited prior geologic or petrophysical constraints. Two-term AVO inversion shows higher instability compared with three-term AVO inversion. However, density, which is important in the fluid-type estimation, cannot be recovered from two-term AVO inversion. To reliably predict the P- and S-waves and density, we have developed a robust two-step joint PP- and PS-wave three-term AVO-inversion method. Our inversion workflow consists of two steps. The first step is to estimate the P- and S-wave reflectivities using Stewart’s joint two-term PP- and PS-AVO inversion. The second step is to treat the P-wave reflectivity obtained from the first step as the prior constraint to remove the P-wave velocity related-term from the three-term Aki-Richards PP-wave approximated reflection coefficient equation, and then the reduced PP-wave reflection coefficient equation is combined with the PS-wave reflection coefficient equation to estimate the S-wave and density reflectivities. We determined the effectiveness of our method by first applying it to synthetic models and then to field data. We also analyzed the condition number of the coefficient matrix to illustrate the stability of the proposed method. The estimated results using proposed method are superior to those obtained from three-term AVO inversion.

Amplitude variation with offset inversion using the reflectivity method

Geophysics ◽  
2016 ◽  
Vol 81 (4) ◽  
pp. R185-R195 ◽  
Author(s):  
Hongxing Liu ◽  
Jingye Li ◽  
Xiaohong Chen ◽  
Bo Hou ◽  
Li Chen
Keyword(s):  
Wave Velocity ◽  
P Wave ◽  
Transmission Losses ◽  
Amplitude Variation ◽  
Data Set ◽  
Amplitude Variation With Offset ◽  
Avo Inversion ◽  
Propagation Effects ◽  
Reflectivity Method ◽  
Internal Multiples

Most existing amplitude variation with offset (AVO) inversion methods are based on the Zoeppritz’s equation or its approximations. These methods assume that the amplitude of seismic data depends only on the reflection coefficients, which means that the wave-propagation effects, such as geometric spreading, attenuation, transmission loss, and multiples, have been fully corrected or attenuated before inversion. However, these requirements are very strict and can hardly be satisfied. Under a 1D assumption, reflectivity-method-based inversions are able to handle transmission losses and internal multiples. Applications of these inversions, however, are still time-consuming and complex in computation of differential seismograms. We have evaluated an inversion methodology based on the vectorized reflectivity method, in which the differential seismograms can be calculated from analytical expressions. It is computationally efficient. A modification is implemented to transform the inversion from the intercept time and ray-parameter domain to the angle-gather domain. AVO inversion is always an ill-posed problem. Following a Bayesian approach, the inversion is stabilized by including the correlation of the P-wave velocity, S-wave velocity, and density. Comparing reflectivity-method-based inversion with Zoeppritz-based inversion on a synthetic data and a real data set, we have concluded that reflectivity-method-based inversion is more accurate when the propagation effects of transmission losses and internal multiples are not corrected. Model testing has revealed that the method is robust at high noise levels.

Migration of P‐wave reflection data in transversely isotropic media

Geophysics ◽  
1994 ◽  
Vol 59 (4) ◽  
pp. 591-596 ◽  
Author(s):  
Suhas Phadke ◽  
S. Kapotas ◽  
N. Dai ◽  
Ernest R. Kanasewich
Keyword(s):  
Wave Equation ◽  
Dispersion Relation ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Limited Range ◽  
Horizontal Velocity ◽  
P Wave ◽  
Transversely Isotropic Media ◽  
Isotropic Media

Wave propagation in transversely isotropic media is governed by the horizontal and vertical wave velocities. The quasi‐P(qP) wavefront is not an ellipse; therefore, the propagation cannot be described by the wave equation appropriate for elliptically anisotropic media. However, for a limited range of angles from the vertical, the dispersion relation for qP‐waves can be approximated by an ellipse. The horizontal velocity necessary for this approximation is different from the true horizontal velocity and depends upon the physical properties of the media. In the method described here, seismic data is migrated using a 45-degree wave equation for elliptically anisotropic media with the horizontal velocity determined by comparing the 45-degree elliptical dispersion relation and the quasi‐P‐dispersion relation. The method is demonstrated for some synthetic data sets.

AVA analysis after velocity-independent DMO and imaging

Geophysics ◽  
1998 ◽  
Vol 63 (2) ◽  
pp. 686-691 ◽  
Author(s):  
Gerald H. F. Gardner ◽  
Anat Canning
Keyword(s):  
Reflection Coefficient ◽  
Peak Amplitude ◽  
Angle Of Incidence ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
The Past ◽  
Reflectivity Function ◽  
Velocity Of Propagation ◽  
Zero Offset ◽  
Amplitude Versus

A common midpoint (CMP) gather usually provides amplitude variation with offset (AVO) information by displaying the reflectivity as the peak amplitude of symmetrical deconvolved wavelets. This puts a reflection coefficient R at every offset h, giving a function R(h). But how do we link h with the angle of incidence, θ, to get the reflectivity function, R(θ)? This is necessary for amplitude versus angle-of-incidence (AVA) analysis. One purpose of this paper is to derive formulas for this linkage after velocity-independent dip-moveout (DMO), done by migrating radial sections, and prestack zero-offset migration. Related studies of amplitude-preserving DMO in the past have dealt with constant-offset DMO but have not given the connection between offset and angle of incidence after processing. The results in the present paper show that the same reflectivity function can be extracted from the imaged volume whether it is produced using radial-trace DMO plus zero-offset migration, constant-offset DMO plus zero-offset migration, or directly by prestack, common-offset migration. The data acquisition geometry for this study consists of parallel, regularly spaced, multifold lines, and the velocity of propagation is constant. Events in the data are caused by an arbitrarily oriented 3-D plane reflector with any reflectivity function. The DMO operation transforms each line of data (m, h, t), i.e., midpoint, half-offset, and time, into an (m1, k, t1) space by Stolt-migrating each radial-plane section of the data, 2h = Ut, with constant velocity U/2. Merging the (m1, k, t1) spaces for all the lines forms an (x, y, k, t1) space, where the first two coordinates are the midpoint location, the third is the new half-offset, and the fourth is the time. Normal moveout (NMO) plus 3-D zero-offset migration of the subspace (x, y, t1) for each k creates a true-amplitude imaged volume (X, Y, k, T). Each peak amplitude in the volume is a reflection coefficient linked to an angle of incidence.

Traveltime and relative geometric spreading approximation in elastic orthorhombic medium

Geophysics ◽  
2020 ◽  
Vol 85 (5) ◽  
pp. C153-C162 ◽  
Author(s):  
Shibo Xu ◽  
Alexey Stovas ◽  
Hitoshi Mikada
Keyword(s):  
Seismic Data ◽  
Real Data ◽  
P Wave ◽  
Form Expression ◽  
Amplitude Variation ◽  
Rational Form ◽  
Amplitude Variation With Offset ◽  
Practical Applications ◽  
Numerical Tests ◽  
Geometric Spreading

Wavefield properties such as traveltime and relative geometric spreading (traveltime derivatives) are highly essential in seismic data processing and can be used in stacking, time-domain migration, and amplitude variation with offset analysis. Due to the complexity of an elastic orthorhombic (ORT) medium, analysis of these properties becomes reasonably difficult, where accurate explicit-form approximations are highly recommended. We have defined the shifted hyperbola form, Taylor series (TS), and the rational form (RF) approximations for P-wave traveltime and relative geometric spreading in an elastic ORT model. Because the parametric form expression for the P-wave vertical slowness in the derivation is too complicated, TS (expansion in offset) is applied to facilitate the derivation of approximate coefficients. The same approximation forms computed in the acoustic ORT model also are derived for comparison. In the numerical tests, three ORT models with parameters obtained from real data are used to test the accuracy of each approximation. The numerical examples yield results in which, apart from the error along the y-axis in ORT model 2 for the relative geometric spreading, the RF approximations all are very accurate for all of the tested models in practical applications.

Reservoir-oriented wave-equation-based seismic amplitude variation with offset inversion

2017 ◽  
Vol 5 (3) ◽  
pp. SL43-SL56 ◽  
Author(s):  
Dries Gisolf ◽  
Peter R. Haffinger ◽  
Panos Doulgeris
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Time Lapse ◽  
Background Model ◽  
Nonlinear Inversion ◽  
Elastic Wave Equation ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Avo Inversion ◽  
Object Domain

Wave-equation-based amplitude-variation-with-offset (AVO) inversion solves the full elastic wave equation, for the properties as well as the total wavefield in the object domain, from a set of observations. The relationship between the data and the property set to invert for is essentially nonlinear. This makes wave-equation-based inversion a nonlinear process. One way of visualizing this nonlinearity is by noting that all internal multiple scattering and mode conversions, as well as traveltime differences between the real medium and the background medium, are accounted for by the wave equation. We have developed an iterative solution to this nonlinear inversion problem that seems less likely to be trapped in local minima. The surface recorded data are preconditioned to be more representative for the target interval, by redatuming, or migration. The starting model for the inversion is a very smooth (0–4 Hz) background model constructed from well data. Depending on the data quality, the nonlinear inversion may even update the background model, leading to a broadband solution. Because we are dealing with the elastic wave equation and not a linearized data model in terms of primary reflections, the inversion solves directly for the parameters defining the wave equation: the compressibility (1/bulk modulus) and the shear compliance (1/shear modulus). These parameters are much more directly representative for hydrocarbon saturation, porosity, and lithology, than derived properties such as acoustic and shear impedance that logically follow from the linearized reflectivity model. Because of the strongly nonlinear character of time-lapse effects, wave-equation based AVO inversion is particularly suitable for time-lapse inversion. Our method is presented and illustrated with some synthetic data and three real data case studies.

Amplitude variation with offset inversion using acoustic-elastic local solver

Geophysics ◽  
2020 ◽  
Vol 85 (3) ◽  
pp. R251-R262 ◽  
Author(s):  
Ligia Elena Jaimes-Osorio ◽  
Alison Malcolm ◽  
Ali Gholami
Keyword(s):  
Elastic Material ◽  
Point Sources ◽  
P Wave ◽  
Reflection Coefficients ◽  
Local Domain ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Zoeppritz Equations ◽  
Avo Inversion ◽  
Full Domain

Conventional amplitude variation with offset (AVO) inversion analysis uses the Zoeppritz equations, which are based on a plane-wave approximation. However, because real seismic data are created by point sources, wave reflections are better modeled by spherical waves than by plane waves. Indeed, spherical reflection coefficients deviate from planar reflection coefficients near the critical and postcritical angles, which implies that the Zoeppritz equations are not applicable for angles close to critical reflection in AVO analysis. Elastic finite-difference simulations provide a solution to the limitations of the Zoeppritz approximation because they can handle near- and postcritical reflections. We have used a coupled acoustic-elastic local solver that approximates the wavefield with high accuracy within a locally perturbed elastic subdomain of the acoustic full domain. Using this acoustic-elastic local solver, the local wavefield generation and inversion are much faster than performing a full-domain elastic inversion. We use this technique to model wavefields and to demonstrate that the amplitude from within the local domain can be used as a constraint in the inversion to recover elastic material properties. Then, we focus on understanding how much the amplitude and phase contribute to the reconstruction accuracy of the elastic material parameters ([Formula: see text], [Formula: see text], and [Formula: see text]). Our results suggest that the combination of amplitude and phase in the inversion helps with the convergence. Finally, we analyze elastic parameter trade-offs in AVO inversion, from which we find that to recover accurate P-wave velocities we should invert for [Formula: see text] and [Formula: see text] simultaneously with fixed density.

Efficient acoustic scalar wave equation modeling in VTI media

Geophysics ◽  
2020 ◽  
pp. 1-93
Author(s):  
Yury Nikonenko ◽  
Marwan Charara
Keyword(s):  
Wave Equation ◽  
Differential Operator ◽  
Acoustic Wave ◽  
Stress Component ◽  
Finite Difference Methods ◽  
P Wave ◽  
Computational Time ◽  
Equation Modeling ◽  
Pseudo Differential Operator ◽  
Spatial Operator

We present a new approach for acoustic wave modeling in transversely isotropic media with a vertical axis of symmetry. This approach is based on using a pure acoustic wave equation derived from the basic physical laws – Hooke’s law and the equation of motion. We show that the conventional equation noted as pure quasi-P wave equation computes only one stress component. In our approach, there is no need to approximate the pseudo-differential operator for decomposition purposes. We make a discrete inverse Fourier transform of the desired frequency response contained in the pseudo-differential operator to build the corresponding spatial operator. We then cut off the operator with a window to reduce edge effects. As a result, the obtained spatial operator is applied locally to the wavefield through a simple convolution. Consequently, we derive an explicit numerical scheme for a pure quasi-P wave mode. The most important advantage of our method lies in its locality, which means that our spatial operator can be applied in any selected region separately. Our approach can be combined with classical fast finite-difference methods when media are isotropic or elliptically anisotropic, therefore avoiding spurious fields and reducing the total computational time and memory. The accuracy, stability, and the absence of the residual S-waves of our approach were demonstrated with several numerical examples.

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