Amplitude Variation With Offset Inversion Research Articles

Nonlinear amplitude-variation-with-offset inversion for Lamé parameters using a direct inversion method

Prestack seismic inversion is widely used in fluid indication and reservoir prediction. Compared with linear inversion, nonlinear inversion is more precise and can be applied to high-contrast situations. The inversion results can be affected by the parameters’ sensitivity, so the parameterization of nonlinear equations is very significant. Considering the poor nonlinear amplitude-variation-with-offset (AVO) inversion results of impedance and velocity parameters, we adjust the parameters of the nonlinear equation, avoid the inaccuracy caused by parameters sensitivity and get the ideal nonlinear AVO inversion results of the Lamé parameters. The feasibility and stability of the nonlinear equation based on the Lamé parameters and method are verified by the model and the real data examples. The resolution and the lateral continuity of nonlinear inversion results are better compared with the linear inversion results.

Seismic inversion for geologic fractures and fractured media

Amplitude variation with offset (AVO) inversion attempts to use the available surface seismic data to estimate the density, P-wave velocity, and S-wave velocity of the earth model. Under linear slip interface theory, synthetic seismograms for models with fractures prove that fractures are also reflection generators. Consequently, observed reflections are not necessarily due to lithologic variations only, but they could be due in part to the effect of fractures. To obtain approximate equations for AVO inversion for fractured media, denoted by AVO with fracture (AVOF), we derived new equations for PP-wave reflection and transmission coefficients that are based on nonwelded contact boundary conditions. In particular, along with the fracture compliances, azimuth has also been taken into account in the equations because the fractures can have any orientation. The new approximate AVOF equations for a horizontally fractured medium with impedance contrast are developed by simplifying the equations for the new PP-wave reflection and transmission coefficients. In the new approximate AVOF equations, the reflection coefficients are divided into a welded contact part (a conventional impedance contrast part) and a nonwelded contact part (a fracture part). This makes the equations flexible enough to separately invert for the rock properties of the fracture and the background medium in the case of a fractured medium with impedance contrast. The new approximate AVOF equations state that fractures could cause the seismic reflectivity to be frequency dependent, and that the fractures not only influence the wave amplitude but also change the wave phase. The linear least-squares and nonlinear conjugate gradient inversion algorithms are applied to estimate the elastic reflectivity using the new approximate AVOF equations. The inverted results for seismic data for a horizontally fractured medium with impedance contrast are evaluated to find a more accurate delineation of the subsurface rock properties.

Broadband seismic amplitude variation with offset inversion

Seismic amplitude variation with offset (AVO) inversion is well-known as a popular and pragmatic tool used for the prediction of elastic parameters in the geosciences. Low frequencies missing from conventional seismic data are conventionally recovered from other geophysical information, such as well-log data, for estimating the absolute rock properties, which results in biased inversion results in cases of complex heterogeneous geologic targets or plays with sparse well-log data, such as marine or deep stratum. Broadband seismic data bring new opportunities to estimate the low-frequency components of the elastic parameters without well-log data. We have developed a novel AVO inversion approach with the Bayesian inference for broadband seismic data. The low-frequency components of the elastic parameters are initially estimated with the proposed broadband AVO inversion approach with the Bayesian inference in the complex frequency domain because seismic inversion in the complex frequency domain is helpful to recover the long-wavelength structures of the elastic models. Gaussian and Cauchy probability distribution density functions are used for the likelihood function and the prior information of model parameters, respectively. The maximum a posteriori probability solution is resolved to estimate the low-frequency components of the elastic parameters in the complex frequency domain. Furthermore, with those low-frequency components as initial models and constraints, the conventional AVO inversion method with the Bayesian inference in the time domain is further implemented to estimate the final absolute elastic parameters. Synthetic and field data examples demonstrate that the proposed AVO inversion in the complex frequency domain is able to predict the low-frequency components of elastic parameters well, and that those low-frequency components set a good foundation for the final estimation of the absolute elastic parameters.

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

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 analysis of time-lapse land seismic data in a gas field, Thrace Basin, Turkey

The Thrace Basin that is located in northwestern Turkey contains sandstone and carbonate reservoirs of Eocene and Oligocene age. Production and exploration activities are still underway. Mapping undrained sweet spots from seismic data is currently a challenge, so time lapse (4D) seismic is used to reduce the risk for new production and development drilling. We have evaluated the normalization and amplitude variation with offset (AVO) analysis of 3D-4D land seismic data in a gas producing field from which baseline and monitor surveys were acquired in 2002 and 2011, respectively. Through AVO analysis, intercept (A) and gradient (B) analysis was conducted, and fluid factor (FF) attribute maps were generated for the assessment of the remaining potential areas. Synthetic gathers were created for simulation of the AVO response, drained and undrained stages and compared with the corresponding 4D seismic data. The drainage of gas from the reservoir interval is evident from the difference maps between 2002 and 2011 seismic data. Both data sets were processed using an amplitude friendly processing sequence. This parallel processing was followed a mild data conditioning and crossequalization for reliable 4D interpretation. The 4D seismic data, especially land data, has low repeatability and requires conditioning to reduce the 4D noise. The 4D noise can be described as nonrepeatable noise, and any difference outside the reservoir zone is not related to production. A so-called crossequalization was applied to the base and the monitor data to bring out similarities so that they cancel out when differences of seismic data and its attributes indicated only the production results over the reservoir zones. As the available 4D data crossequalization software was implemented for stack data only, we created angle band stacks and crossequalized each angle band stack from the base and the monitor data cubes. Five angle band stacks from the base and the monitor prestack data cubes 0°–55° (0°–15°, 15°–25°, 25°–35°, 35°–45°, and 45°–55°) were crossequalized individually. The crossequalized angle band stacks were used in AVO analysis and AVO inversion to generate pore fill identifiers such as FF to map possible undrained zones after 10 years of production.

Application and performances of unconstrained optimization methods in seafloor AVO inversion

Seafloor elastic parameters are important for seafloor engineering and geophysical detection beneath the seafloor. We have proposed to use AVO (amplitude variation with offset) theory to estimate seafloor elastic parameters. However, the previous inversion methods are time-consuming. To improve the computing efficiency, we try to solve the inverse problem as an unconstrained optimization problem in this paper. Three kinds of classical unconstrained optimization methods are applied to seafloor AVO inversion, including the steepest descent method, the Newton’s method, and the conjugate gradient method. Then, we design different initial models to test the convergence behaviors of the three methods. Numerical tests show that the perturbation level of the initial models and the noise level of the observed data have a significant effect on the convergence performances of the three methods. Even for the same perturbation and noise levels, the convergence performances differ with different combinations of the perturbed initial elastic parameters. All three methods have higher computing efficiency than the previous methods. This research also offers a strategy to choose a proper optimization method for a specific case in real seafloor AVO inversion.

Amplitude variation with offset inversion using the reflectivity method

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.

AVO inversion based on inverse operator estimation in trust region

Amplitude variation with offset (AVO) inversion is widely utilized in exploration geophysics, especially for reservoir prediction and fluid identification. Inverse operator estimation in the trust region algorithm is applied for solving AVO inversion problems in which optimization and inversion directly are integrated. The L1 norm constraint is considered on the basis of reasonable initial model in order to improve effciency and stability during the AVO inversion process. In this study, high-order Zoeppritz approximation is utilized to establish the inversion objective function in which variation of with time is taken into consideration. A model test indicates that the algorithm has a relatively higher stability and accuracy than the damp least-squares algorithm. Seismic data inversion is feasible and inversion values of three parameters () maintain good consistency with logging curves.

Iterative reweighted least M-estimate AVO inversion

The l1 norm and its variants, such as the hybrid l1/l2 norm and the Huber norm, that are used to solve amplitude variation with offset (AVO) inversion optimisation problems, are mostly known to give a more robust solution than the classical least-squares (l2 norm) method by reducing the influence of outliers significantly, although never ignoring it. To deal with data having many outliers, biweight norm using iteratively reweighted least-squares (IRLS) as robust inversion method can improve robustness by ignoring outliers in computing the misfit measure. However, biweight norm uses a higher-order descending weighting on the measured data, which results in poor performance when dealing with the well measured data. Hampel’s three-part redescending M-estimate function as robust measure could be considered as a three-part combination of the l2 norm and l1 norm with excluding outliers, which could perform better. This paper describes an iterative reweighted least M-estimate (IRLM) algorithm as a robust AVO inversion. The synthetic and field seismic data tests show that the IRLM algorithm gives far more robust model estimates than the conventional Huber norm and biweight norm.

An inversion method for seafloor elastic parameters

Amplitude-variation-with-offset (AVO) inversion is a common approach to estimate elastic parameters. In conventional AVO inversion, various approximations of the Zoeppritz equation have been used. However, the weak elastic contrast assumption of those approximations leads to big errors when dealing with the seafloor situation in which the contrast of the elastic parameters across the interface is very strong. To extend the application of AVO to seafloor elastic parameters estimation as a linear problem, we developed a high-contrast approximation of the P-wave reflection coefficient to the Zoeppritz equation via a series truncation procedure. Model tests determined that the approximation we derived was more applicable than conventional approximations when considering the seafloor case. Any approximation limits AVO inversion to a small incidence-angle range. A two-step inversion method using the joint application of approximation and exact equation was evaluated to overcome the angle limit. Then we determined the influence of P-wave reflection coefficient accuracy on the inverted elastic parameters, and we developed a new technique to invert for seafloor elastic parameters directly by amplitude-versus-angle data without estimating reflection coefficient first. Finally, we applied the technique to estimate the seafloor parameters from AVO data of the South China Sea.

Amplitude-variation-with-offset and prestack-waveform inversion: A direct comparison using a real data example from the Rock Springs Uplift, Wyoming, USA

Recent advances in seismic data acquisition and processing allow routine extraction of offset-/angle-dependent reflection amplitudes from prestack seismic data for quantifying subsurface lithologic and fluid properties. Amplitude-variation-with-offset (AVO) inversion is the most commonly used practice for such quantification. Although quite successful, AVO has a few shortcomings primarily due to the simplicity in its inherent assumptions, and for any quantitative estimation of reservoir properties, they are generally interpreted in combination with other information. In recent years, waveform-based inversions have gained popularity in reservoir characterization and depth imaging. Going beyond the simple assumptions of AVO and using wave equation solutions, these methods have been effective in accurately predicting the subsurface properties. Developments of these waveform inversions have so far been along two lines: (1) the methods that use a locally 1D model of the subsurface for each common midpoint and use an analytical solution to the wave equation for forward modeling and (2) the methods that do not make any 1D assumption but use an approximate numerical solution to the wave equation in 2D or 3D for forward modeling. Routine applications of these inversions are, however, still computationally demanding. We described a multilevel parallelization of elastic-waveform inversion methodology under a 1D assumption that allowed its application in a reasonable time frame. Applying AVO and waveform inversion on a single data set from the Rock Springs Uplift, Wyoming, USA, and comparing them with one another, we also determined that the waveform-based method was capable of obtaining a much superior description of subsurface properties compared with AVO. We concluded that the waveform inversions should be the method of choice for reservoir property estimation as high-performance computers become commonly available.

AVO inversion based on generalized extreme value distribution with adaptive parameter estimation

Amplitude Variation with Offset (AVO) inversion is very sensitive to noise and other uncertainties which are often far from Gaussian in the actual pre-stack seismic data, sometimes causing AVO inversion to give poor results. Since the generalized extreme value (GEV) distribution can efficiently fit any distribution with different parameters, to obtain steady and rational AVO inversion results, we use GEV distribution to model the distribution of the iterative residual error in the AVO inversion. The maximum likelihood method is used to re-evaluate the GEV distribution parameters of each iterative residual error to efficiently fit its non-Gaussian property and reduce the sensitivity of AVO inversion to non-Gaussian residual error. Quasi-Newton based Conjugate Gradient (QCG) AVO algorithm is derived from adaptive adjusted parameters GEV distribution to make sure the direction is always a descent direction for the objective function. Finally, a new choice of adaptive variable step length obtained by the Taylor expression of the residual vector can adaptively adjust the step length to accelerate the convergent speed. Both the synthetic and real seismic data tests illustrate the effectiveness of the proposed method.

An Artificial Immune Algorithm and its Application in AVO Inversion

An improved artificial immune algorithm is proposed for geophysical P-wave amplitude variation with offset (AVO) inversion. In this paper, the algorithm is described and implemented. The orthogonal crossover is used to generate the initial population and the elitist-crossover is adopted to add the good patterns of the population. The hybrid mutation method is presented to increase the ability of local and global optimization. The improved immune algorithm is then applied to earth interface models of Mexican gulf for AVO inversion. The experimental results show that the improved algorithm is of high precision than the traditional immune algorithm.

AVO inversion and poroelasticity with P- and S-wave moduli

The fluid term in the Biot-Gassmann equation plays an important role in reservoir fluid discrimination. The density term imbedded in the fluid term, however, is difficult to estimate because it is less sensitive to seismic amplitude variations. We combined poroelasticity theory, amplitude variation with offset (AVO) inversion, and identification of P- and S-wave moduli to present a stable and physically meaningful method to estimate the fluid term, with no need for density information from prestack seismic data. We used poroelasticity theory to express the fluid term as a function of P- and S-wave moduli. The use of P- and S-wave moduli made the derivation physically meaningful and natural. Then we derived an AVO approximation in terms of these moduli, which can then be directly inverted from seismic data. Furthermore, this practical and robust AVO-inversion technique was developed in a Bayesian framework. The objective was to obtain the maximum a posteriori solution for the P-wave modulus, S-wave modulus, and density. Gaussian and Cauchy distributions were used for the likelihood and a priori probability distributions, respectively. The introduction of a low-frequency constraint and statistical probability information to the objective function rendered the inversion more stable and less sensitive to the initial model. Tests on synthetic data showed that all the parameters can be estimated well when no noise is present and the estimated P- and S-wave moduli were still reasonable with moderate noise and rather smooth initial model parameters. A test on a real data set showed that the estimated fluid term was in good agreement with the results of drilling.

The effect of interface curvature on AVO inversion of near-critical and postcritical PP-reflections

Widely exploited in the industry, amplitude-variation-with-offset (AVO) inversion techniques are based on weak-contrast approximations of the plane-wave reflection coefficients. These approximations are valid for plane waves reflected at almost flat interfaces with weak contrasts in seismic parameters and for reflection angles below the critical angle. Regardless of the underlying assumptions, linearized coefficients provide a simple and physically adequate tool to accurately invert AVO data for seismic parameters at precritical angles. However, the accuracy of linearized coefficients drastically decreases with increasing incidence angle. Limitations occur around and beyond the critical ray, where the effect of wavefront curvature becomes prominent and thus can no more be neglected. The effective reflection coefficients generalize the plane-wave reflection coefficients for waves generated by point sources and reflected at curved interfaces. They account for the wavefront curvature and are adequate at any incidence angle. Our previous studies have shown that including the reflections around and beyond the critical angle in the AVO inversion significantly improves the accuracy of estimated parameters. However, the interface curvature also must have its contribution to the long-offset AVO inversion. We find that the interface curvature affects the energy propagation along the ray tube and the energy diffusion across the ray tube. The energy propagation along the tube is characterized by the geometrical spreading, which is strongly affected by interface curvature. The transverse diffusion is captured by the effective reflection coefficients which are less influenced by interface curvature. The long-offset AVO inversion is thus sensitive to interface curvature through a combination of several wave propagation factors.

Long-offset AVO inversion of PP reflections from plane interfaces using effective reflection coefficients

A conventional amplitude variation with offset (AVO) inversion is based on geometrical seismics which exploit plane-wave reflection coefficients to describe the reflection phenomenon. Widely exploited linearizations of plane-wave coefficients are mostly valid at pre-critical offsets for media with almost flat and weak-contrast interfaces. Existing linearizations do not account for the seismic frequency range by ignoring the frequency content of the wavelet, which is a strong assumption. Plane-wave reflection coefficients do not fully describe the reflection of seismic waves at near-critical and post-critical offsets, because reflected seismic waves are typically generated by point sources. We propose an improved approach to AVO inversion, which is based on effective reflection coefficients (ERCs). ERCs generalize plane-wave coefficients for seismic waves generated by point sources and therefore more accurately describe near-critical and post-critical reflections where head waves are generated. Moreover, they are frequency-dependent and incorporate the local curvatures of the wavefront and the reflecting interface. In our study, we neglect the effect of interface curvature and demonstrate the advantages of our approach on synthetic data for a simple model with a plane interface separating two isotropic half-spaces. A comparison of the inversion results obtained with our approach and the results from an AVO inversion method based on the exact plane-wave reflection coefficient suggests that our method is superior, in particular for long-offset ranges which extend to and beyond the critical angle. We thus propose that long offsets can be successfully exploited in an AVO inversion under the correct assumption about the reflection coefficient. Such long-offset AVO inversion shows the potential of outperforming a conventional moderate-offset AVO inversion in the accuracy of estimated model parameters.

Reflection coefficients in attenuative anisotropic media

Such reservoir rocks as tar sands are characterized by significant attenuation and, in some cases, attenuation anisotropy. Most existing attenuation studies are focused on plane-wave attenuation coefficients, which determine the amplitude decay along the raypath of seismic waves. Here we study the influence of attenuation on PP- and PS-wave reflection coefficients for anisotropic media with the main emphasis on transversely isotropic models with a vertical symmetry axis (VTI). Concise analytic solutions obtained by linearizing the exact plane-wave reflection coefficients are verified by numerical modeling. To make a substantial contribution to reflection coefficients, attenuation must be strong, with the quality factor [Formula: see text] not exceeding 10. For such highly attenuative media, it is also necessary to take attenuation anisotropy into account if the magnitude of the Thomsen-styleattenuation-anisotropy parameters is relatively large. In general, the linearized reflection coefficients in attenuative media include velocity-anisotropy parameters but have almost “isotropic” dependence on attenuation. Our formalism also helps evaluate the influence of the inhomogeneity angle (the angle between the real and imaginary parts of the slowness vector) on the reflection coefficients. A nonzero inhomogeneity angle of the incident wave introduces additional terms into the PP- and PS-wave reflection coefficients, which makes conventional amplitude-variation-with-offset (AVO) analysis inadequate for strongly attenuative media. For instance, an incident P-wave with a nonzero inhomogeneity angle generates a mode-converted PS-wave at normal incidence, even if both half-spaces have a horizontal symmetry plane. The developed linearized solutions can be used in AVO inversion for highly attenuative (e.g., gas-sand and heavy-oil) reservoirs.

Two-term AVO inversion: Equivalences and new methods

Most amplitude-variation-with-offset (AVO) studies use two-parameter inversion methods that are approximations of a more general three-parameter method based on the Aki-Richards approximation. Two-parameter methods are popular because the three-parameter inversion is often plagued by numerical instability. Reducing the dimensionality of the parameter space stabilizes the inversion. A variety of constraints can accomplish this, and these lead to the multiplicity of current two-parameter methods. It would be useful to understand relationships between various two-parameter methods. To this end, we derive formal expressions for inversion errors of each method. Using these expressions, conversion formulas are obtained that allow the flexibility to convert results of any two-parameter method to those of any other two-parameter method. The only requirement for the equivalence of methods is that the maximum angle of incidence be at least a few degrees less than the critical angle. In addition, error expressions result in a new formulation for a two-parameter AVO tool that combines strengths of two commonly used methods. The expressions also suggest a simple way to incorporate information from well-log calibration into legacy AVO inversions. These results should be helpful in resource exploration.

Lithofacies detection through simultaneous inversion and principal component attributes

Seismic attributes derived from amplitude variation with offset (AVO) analysis and inversion of prestack seismic data allow discrimination of hydrocarbon-bearing lithofacies. Commonly derived AVO inversion attributes are acoustic impedance (Zp), elastic impedance (EI), shear impedance (Zs) and the ratio of compressional-wave and shear-wave velocity (Vp/Vs).

Density extraction from P-wave AVO inversion: Tuscaloosa Trend example

Density extraction from amplitude-variation-with-offset (AVO) inversion is thought to be unstable and difficult. Recent research has, however, shown that it is possible to reliably extract density from P-wave reflection data if the interface has significant contrast.