scholarly journals Estimation and accounting for the modeling error in probabilistic linearized amplitude variation with offset inversion

Geophysics ◽  
2018 ◽  
Vol 83 (2) ◽  
pp. N15-N30 ◽  
Author(s):  
Rasmus Bødker Madsen ◽  
Thomas Mejer Hansen
Keyword(s):  
Probability Distribution ◽  
Signal To Noise Ratio ◽  
Noise Model ◽  
Modeling Error ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Zoeppritz Equations ◽  
Avo Inversion ◽  
Convolution Model ◽  
Modeling Errors

A linearized form of Zoeppritz equations combined with the convolution model is widely used in inversion of amplitude variation with offset (AVO) seismic data. This is shown to introduce a “modeling error,” compared with using the full Zoeppritz equations, whose magnitude depends on the degree of subsurface heterogeneity. Then, we evaluate a methodology for quantifying this modeling error through a probability distribution. First, a sample of the unknown probability density describing the modeling error is generated. Then, we determine how this sample can be described by a correlated Gaussian probability distribution. Finally, we develop how such modeling errors affect the linearized AVO inversion results. If not accounted for (which is most often the case), the modeling errors can introduce significant artifacts in the inversion results, if the signal-to-noise ratio is less than 2, as is the case for most AVO data obtained today. However, if accounted for, such artifacts can be avoided. The methodology can easily be adapted and applied to most linear AVO inversion methods, by allowing the use of the inferred modeling error as a correlated Gaussian noise model.

Linearized amplitude variation with offset (AVO) inversion with supercritical angles

Geophysics ◽  
2006 ◽  
Vol 71 (5) ◽  
pp. E49-E55 ◽  
Author(s):  
Jonathan E. Downton ◽  
Charles Ursenbach
Keyword(s):  
Reflection Coefficient ◽  
Critical Angle ◽  
Waveform Inversion ◽  
Phase Variation ◽  
Reflection Coefficients ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Zoeppritz Equations ◽  
Avo Inversion ◽  
Angle Data

Contrary to popular belief, a linearized approximation of the Zoeppritz equations may be used to estimate the reflection coefficient for angles of incidence up to and beyond the critical angle. These supercritical reflection coefficients are complex, implying a phase variation with offset in addition to amplitude variation with offset (AVO). This linearized approximation is then used as the basis for an AVO waveform inversion. By incorporating this new approximation, wider offset and angle data may be incorporated in the AVO inversion, helping to stabilize the problem and leading to more accurate estimates of reflectivity, including density reflectivity.

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.

Constrained nonlinear amplitude variation with offset inversion using Zoeppritz equations

Geophysics ◽  
2018 ◽  
Vol 83 (3) ◽  
pp. R245-R255 ◽  
Author(s):  
Ali Gholami ◽  
Hossein S. Aghamiry ◽  
Mostafa Abbasi
Keyword(s):  
Nonlinear Equations ◽  
Wave Velocity ◽  
Seismic Data ◽  
P Wave ◽  
Simultaneous Estimation ◽  
Model Parameters ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Zoeppritz Equations ◽  
Avo Inversion

The inversion of prestack seismic data using amplitude variation with offset (AVO) has received increased attention in the past few decades because of its key role in estimating reservoir properties. AVO is mainly governed by the Zoeppritz equations, but traditional inversion techniques are based on various linear or quasilinear approximations to these nonlinear equations. We have developed an efficient algorithm for nonlinear AVO inversion of precritical reflections using the exact Zoeppritz equations in multichannel and multi-interface form for simultaneous estimation of the P-wave velocity, S-wave velocity, and density. The total variation constraint is used to overcome the ill-posedness while solving the forward nonlinear model and to preserve the sharpness of the interfaces in the parameter space. The optimization is based on a combination of Levenberg’s algorithm and the split Bregman iterative scheme, in which we have to refine the data and model parameters at each iteration. We refine the data via the original nonlinear equations, but we use the traditional cost-effective linearized AVO inversion to construct the Jacobian matrix and update the model. Numerical experiments show that this new iterative procedure is convergent and converges to a solution of the nonlinear problem. We determine the performance and optimality of our nonlinear inversion algorithm with various simulated and field seismic data sets.

Time-lapse amplitude variation with offset inversion using Bayesian theory in two-phase media

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. N55-N79
Author(s):  
Longxiao Zhi ◽  
Hanming Gu
Keyword(s):  
Objective Function ◽  
Physical Model ◽  
Time Lapse ◽  
Bayesian Theory ◽  
Amplitude Variation ◽  
Two Phase ◽  
Amplitude Variation With Offset ◽  
Zoeppritz Equations ◽  
Avo Inversion ◽  
Rock Physical Model

In time-lapse seismic analysis, the Zoeppritz equations are usually used in the time-lapse amplitude variation with offset (AVO) inversion and then combined with a rock-physical model to estimate the reservoir-parameter changes. The real-life reservoir is a two-phase medium that consists of solid and fluid components. The Zoeppritz equations are a simplification, assuming a single-phase solid medium, in which the properties of this medium are estimated by effective parameters from the combined components. This means that the Zoeppritz equations cannot describe the characteristics of the seismic reflection amplitudes in the reservoir in an accurate way. Therefore, we develop a method for time-lapse AVO inversion in two-phase media using the Bayesian theory to estimate the reservoir parameters and their changes quantitatively. We use a reflection-coefficient equation in two-phase media, a rock-physical model, and the convolutional model to build a relationship between the seismic records and reservoir parameters, which include porosity, clay content, saturation, and pressure. Assuming that the seismic-data errors follow a zero-mean Gaussian distribution and that the reservoir parameters follow a four-variable Cauchy prior distribution, we use the Bayesian theory to construct the objective function for the AVO inversion, and we also add a model-constraint term to compensate the low-frequency information and improve the stability of the inversion. Using the objective function of the AVO inversion and the Gauss-Newton method, we derived the equation for time-lapse AVO inversion. This result can be used to estimate the reservoir parameters and their changes accurately and in a stable way. The test results from the feasibility study on synthetic and field data proved that the method is effective and reliable.

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.

Improving lateral and vertical resolution of seismic images by correcting for wavelet stretch in common-angle migration

Geophysics ◽  
2007 ◽  
Vol 72 (6) ◽  
pp. C95-C104 ◽  
Author(s):  
Gabriel Perez ◽  
Kurt J. Marfurt
Keyword(s):  
Signal To Noise Ratio ◽  
Incident Angle ◽  
Large Angle ◽  
Acoustic Medium ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Velocity Anisotropy ◽  
Time Migration ◽  
Reflection Angle ◽  
Seismic Reflections

Long-offset or high-incident-angle seismic reflections provide us with improved velocity resolution, better leverage against multiples, less contamination by ground roll, and information that is often critical when estimating lithology and fluid product. Unfortunately, high-incident-angle seismic reflections suffer not only from nonhyperbolic moveout but also from wavelet stretch during imaging, resulting in lower-resolution images that mix the response from adjacent lithologies. For an arbitrary acoustic medium, wavelet stretch from prestack migration depends only on the cosine of the reflection angle, such that the amount of wavelet stretch will be the same for all samples of a common-reflection-angle migrated trace. Thus, we are able to implement a wavelet stretch correction by applying a simple stationary spectral shaping operation to common-angle migrated traces. We obtain such traces directly by a prestack Kirchhoff migration algorithm. Correcting for stretch effectively increases the fold of imaged data, far beyond that achieved in conventional migration, resulting in improved signal-to-noise ratio of the final stacked section. Increasing the fidelity of large incident angles results in images with improved vertical and lateral resolution and with increased angular illumination, valuable for amplitude variation with angle (AVA) and amplitude variation with offset (AVO) analysis. Finally, such large-angle images are more sensitive to and therefore provide increased leverage over errors in velocity and velocity anisotropy. These ideas were applied to prestack time migration on seismic data from the Fort Worth basin, in Texas.

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.

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.

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