Seismic Ray Impedance Inversion

This thesis investigates a prestack seismic inversion scheme implemented in the ray parameter domain. Conventionally, most prestack seismic inversion methods are performed in the incidence angle domain. However, inversion using the concept of ray impedance, as it honours ray path variation following the elastic parameter variation according to Snell’s law, shows the capacity to discriminate different lithologies if compared to conventional elastic impedance inversion. The procedure starts with data transformation into the ray-parameter domain and then implements the ray impedance inversion along constant ray-parameter profiles. With different constant-ray-parameter profiles, mixed-phase wavelets are initially estimated based on the high-order statistics of the data and further refined after a proper well-to-seismic tie. With the estimated wavelets ready, a Cauchy inversion method is used to invert for seismic reflectivity sequences, aiming at recovering seismic reflectivity sequences for blocky impedance inversion. The impedance inversion from reflectivity sequences adopts a standard generalised linear inversion scheme, whose results are utilised to identify rock properties and facilitate quantitative interpretation. It has also been demonstrated that we can further invert elastic parameters from ray impedance values, without eliminating an extra density term or introducing a Gardner’s relation to absorb this term. Ray impedance inversion is extended to P-S converted waves by introducing the definition of converted-wave ray impedance. This quantity shows some advantages in connecting prestack converted wave data with well logs, if compared with the shearwave elastic impedance derived from the Aki and Richards approximation to the Zoeppritz equations. An analysis of P-P and P-S wave data under the framework of ray impedance is conducted through a real multicomponent dataset, which can reduce the uncertainty in lithology identification.Inversion is the key method in generating those examples throughout the entire thesis as we believe it can render robust solutions to geophysical problems. Apart from the reflectivity sequence, ray impedance and elastic parameter inversion mentioned above, inversion methods are also adopted in transforming the prestack data from the offset domain to the ray-parameter domain, mixed-phase wavelet estimation, as well as the registration of P-P and P-S waves for the joint analysis. The ray impedance inversion methods are successfully applied to different types of datasets. In each individual step to achieving the ray impedance inversion, advantages, disadvantages as well as limitations of the algorithms adopted are detailed. As a conclusion, the ray impedance related analyses demonstrated in this thesis are highly competent compared with the classical elastic impedance methods and the author would like to recommend it for a wider application

Constrained non-linear AVO inversion based on the adjoint-state optimization

Pre-stack AVO inversion of seismic data is a modeling tool for estimating subsurface elastic properties. Our focus is on the model-based inversion method where then unknown variables are estimated by minimizing the misfit to the observed data. Standard approaches for non-linear AVO inversion are based on the gradient descent optimization algorithms that require the calculation of the gradient equations of the objective function. To improve the accuracy and efficiency of these methods, we developed a technique that uses an implementation of the adjoint-state-based gradient computation. The inversion algorithm relies on three basic modeling components consisting of a convolution-based forward model using a linearized approximation of the Zoeppritz equation, the definition of the objective function, and the adjoint-computed gradient. To achieve an accurate solution, we choose a second-order optimization algorithm known as the Limited memory-BFGS (L-BFGS) that implicitly approximates the inverse Hessian matrix. This approach is more efficient than traditional optimization methods. The main novelty of the proposed approach is the derivation of the adjoint-state equations for the gradient of the objective function. The application of the proposed method is demonstrated using 1D and 2D synthetic datasets based on data from the Edvard Grieg oil field. The seismic data for these applications is generated by using both convolutional modeling and finite difference methods. The results of the proposed method are accurate and the computational approach is efficient. The results show that the algorithm reliably retrieves the elastic variables, P- and S-wave velocities and density for both convolutional and finite difference models.publishedVersio

Joint inversion of seismic PP- and PS-waves in the ray parameter domain

Seismic inversion is a quantitative analysis technique in reservoir geophysics to reveal subsurface physical properties from surface-recorded seismic data. But the most widely used inversion in oil and gas exploration for decades is PP-wave based. P-to-S converted wave, which has shown great success in the imaging of gas clouds, has a different response to rocks and pore-fluids from the PP-wave. A joint use of the PS-wave and PP-wave in the inversion can reduce the ill-posedness of the inverse problem and in particular enables simultaneous inversion for three independent elastic parameters. Conventionally, prestack seismic inversion is based on the incidence angle-dependent reflection coefficients. In my research, I define the seismic reflections and impedances along the ray paths of wave propagation, and these ray paths obey Snell’s law. I adopt the ray-impedance concept, which is a frequency-dependent parameter and is sensitive to fluid contents. Joined interpretation of PP- and PS-wave ray impedances can identify reservoirs, and also has potential in fluid discrimination. Joint inversion of PP- and PS-waves is performed on the constant ray parameter (CRP) profiles. For a constant ray parameter, a pair of PP- and PS-wave traces has exactly the same ray path between the source and the reflection point, which means the PP- and PS-wave reflection events represent exactly the same reflection point, in the horizontal direction. Therefore, PP and PS-wave calibration transforms PS-wave reflection events from PS-wave time to the corresponding PP-wave time, and reflections events in a pair of PP- and calibrated PS-wave traces with a constant ray parameter should correspond to each other, sample by sample, both horizontally and vertically. I also present a procedure which preserves the original wavelets in the transformed PS-wave trace. I use a bending ray-tracing method to construct the common image point (CIP) gathers in the ray-parameter domain. I estimate mixed-phase wavelets for each constant ray-parameter (CRP) profile through a frequency domain high-order statistical method, and then invert for the reflectivity series using weighted constraints. From the reflectivity sections, I estimate PP- and PS-wave ray impedances separately and also estimate three elastic parameters simultaneously in a joint inversion. I have applied the entire procedure to a couple of field data sets, to verify the robustness and effectiveness of the method, and to demonstrate the great potential of joint inversion in ray-parameter domain

Application of High Resolution Inversion of Ultrasonic Data to the Imaging of Multi-Layered Composite Structures

Ultrasonic imaging has evolved from its early application which utilized only amplitude C-scans to more complex techniques which make extensive use of digital signal processing. Techniques, such as one-and two-dimensional deconvolution processing and synthetic aperture focussing techniques (SAFT), are becoming more widely accepted for conventional applications. In general, each of these techniques aims to improve the interpretability of the ultrasonic image by increasing the resolution in one or more dimensions

Final report on estimation and statistical analysis of spatially distributed random processes

Includes bibliographical references.Final report;Supported by the NSF. ECS-8312921prepared by Alan S. Willsky, Bernard C. Levy, George C. Verghese

Modeling and inversion of seismic data using multiple scattering, renormalization and homotopy methods

Seismic scattering theory plays an important role in seismic forward modeling and is the theoretical foundation for various seismic imaging methods. Full waveform inversion is a powerful technique for obtaining a high-resolution model of the subsurface. One objective of this thesis is to develop convergent scattering series solutions of the Lippmann-Schwinger equation in strongly scattering media using renormalization and homotopy methods. Other objectives of this thesis are to develop efficient full waveform inversion methods of time-lapse seismic data and, to investigate uncertainty quantification in full waveform inversion for anisotropic elastic media based on integral equation approaches and the iterated extended Kalman filter. The conventional Born scattering series is obtained by expanding the Lippmann-Schwinger equation in terms of an iterative solution based on perturbation theory. Such an expansion assumes weak scattering and may have the problems of convergence in strongly scattering media. This thesis presents two scattering series, referred to as convergent Born series (CBS) and homotopy analysis method (HAM) scattering series for frequency-domain seismic wave modeling. For the convergent Born series, a physical interpretation from the renormalization prospective is given. The homotopy scattering series is derived by using homotopy analysis method, which is based on a convergence control parameter $h$ and a convergence control operator $H$ that one can use to ensure convergence for strongly scattering media. The homotopy scattering scattering series solutions of the Lippmann-Schwinger equation, which is convergent in strongly scattering media. The homotopy scattering series is a kind of unified scattering series theory that includes the conventional and convergent Born series as special cases. The Fast Fourier Transform (FFT) is employed for efficient implementation of matrix-vector multiplication for the convergent Born series and the homotopy scattering series. This thesis presents homotopy methods for ray based seismic modeling in strongly anisotropic media. To overcome several limitations of small perturbations and weak anisotropy in obtaining the traveltime approximations in anisotropic media by expanding the anisotropic eikonal equation in terms of the anisotropic parameters and the elliptically anisotropic eikonal equation based on perturbation theory, this study applies the homotopy analysis method to the eikonal equation. Then this thesis presents a retrieved zero-order deformation equation that creates a map from the anisotropic eikonal equation to a linearized partial differential equation system. The new traveltime approximations are derived by using the linear and nonlinear operators in the retrieved zero-order deformation equation. Flexibility on variable anisotropy parameters is naturally incorporated into the linear differential equations, allowing a medium of arbitrarily anisotropy. This thesis investigates efficient target-oriented inversion strategies for improving full waveform inversion of time-lapse seismic data based on extending the distorted Born iterative T-matrix inverse scattering to a local inversion of a small region of interest (e. g. reservoir under production). The target-oriented approach is more efficient for inverting the monitor data. The target-oriented inversion strategy requires properly specifying the wavefield extrapolation operators in the integral equation formulation. By employing the T-matrix and the Gaussian beam based Green’s function, the wavefield extrapolation for the time-lapse inversion is performed in the baseline model from the survey surface to the target region. I demonstrate the method by presenting numerical examples illustrating the sequential and double difference strategies. To quantify the uncertainty and multiparameter trade-off in the full waveform inversion for anisotropic elastic media, this study applies the iterated extended Kalman filter to anisotropic elastic full waveform inversion based on the integral equation method. The sensitivity matrix is an explicit representation with Green’s functions based on the nonlinear inverse scattering theory. Taking the similarity of sequential strategy between the multi-scale frequency domain full waveform inversion and data assimilation with an iterated extended Kalman filter, this study applies the explicit representation of sensitivity matrix to the the framework of Bayesian inference and then estimate the uncertainties in the full waveform inversion. This thesis gives results of numerical tests with examples for anisotropic elastic media. They show that the proposed Bayesian inversion method can provide reasonable reconstruction results for the elastic coefficients of the stiffness tensor and the framework is suitable for accessing the uncertainties and analysis of parameter trade-offs

Signal processing techniques for the enhancement of marine seismic data

This thesis presents several signal processing techniques applied to the enhancement of marine seismic data. Marine seismic exploration provides an image of the Earth's subsurface from reflected seismic waves. Because the recorded signals are contaminated by various sources of noise, minimizing their effects with new attenuation techniques is necessary. A statistical analysis of background noise is conducted using Thomson’s multitaper spectral estimator and Parzen's amplitude density estimator. The results provide a statistical characterization of the noise which we use for the derivation of signal enhancement algorithms. Firstly, we focus on single-azimuth stacking methodologies and propose novel stacking schemes using either enhanced weighted sums or a Kalman filter. It is demonstrated that the enhanced methods yield superior results by their ability to exhibit cleaner and better defined reflected events as well as a larger number of reflections in deep waters. A comparison of the proposed stacking methods with existing ones is also discussed. We then address the problem of random noise attenuation and present an innovative application of sparse code shrinkage and independent component analysis. Sparse code shrinkage is a valuable method when a noise-free realization of the data is generated to provide data-driven shrinkages. Several models of distribution are investigated, but the normal inverse Gaussian density yields the best results. Other acceptable choices of density are discussed as well. Finally, we consider the attenuation of flow-generated nonstationary coherent noise and seismic interference noise. We suggest a multiple-input adaptive noise canceller that utilizes a normalized least mean squares alg orithm with a variable normalized step size derived as a function of instantaneous frequency. This filter attenuates the coherent noise successfully when used either by itself or in combination with a time-frequency median filter, depending on the noise spectrum and repartition along the data. Its application to seismic interference attenuation is also discussed

Adjoint-state method for seismic AVO inversion and time-lapse monitoring

This dissertation presents seismic amplitude versus offset (AVO) inversion methods to estimate water saturation and effective pressure quantitatively in elastic and viscoelastic media. Quantitative knowledge of the saturation and pore pressure properties from pre- or post-production seismic measurements for reservoir static or dynamic modeling has been an area of interest for the geophysical community for decades. However, the focus on the existing inversion methodologies and explicit expressions to estimate saturation-pressure variables or changes in these properties due to production or fluid injection has been based on elastic AVO models. These conventional methods do not consider the seismic wave attenuation effects on the reflection amplitudes and therefore can result in biased prediction. Numerous theoretical rock physics models and laboratory experiments have demonstrated the sensitivity of various petrophysical and seismic properties of partially fluid-filled porous media to seismic attenuation. This makes seismic wave attenuation a valuable time-lapse attribute to reliably measure the saturation (Sw) and effective pressure (Pe) properties. Therefore, in this work, I have developed two AVO inversion processes i.e., the conventional AVO inversion method for elastic media and the frequency-dependent amplitude versus offset (FAVO) inversion technique for the viscoelastic media. This dissertation first presents the inversion strategies to invert the pre-stack seismic data for the seismic velocities and density by using the conventional AVO equation and for the seismic velocities, density, and Q-factors by using the frequency-dependent AVO method. These inversion methods are then extended to estimate the dynamic reservoir changes e.g., saturation and pressure variables, and can be applied to predict the saturation and pressure variables at any stage e.g., before and during production, or fluid injection, or to estimate the changes in saturation (ΔSw) and pressure (ΔPe). The first part of the dissertation describes the theory and formulation of the elastic AVO inversion method while in the second half, I have described the viscoelastic inversion workflow. FAVO technique accounts for the dependence of reflection amplitudes on incident angles as well as seismic frequencies and P and S waves attenuation in addition to seismic velocities and density. The fluid saturation and pressure in the elastic and inelastic mediums are linked to the reflection amplitude through seismic velocities, density, and quality factors (Q). The inversion process is based on the gradient-descent method in which the least-square differentiable data misfit equation is minimized by using a non-linear limitedmemory BFGS method. The gradients of the misfit function with respect to unknown model variables are derived by using the adjoint-state method and the multivariable chain rule of derivative. The adjoint-state method provides an efficient and accurate way to calculate the misfit gradients. Numerous rock physics models e.g., the Gassmann substitution equation with uniform and patchy fluid distribution patterns, modified MacBeth’s relations of dry rock moduli with effective pressure, and constant Q models for the P and S wave attenuation are applied to relate the saturation and effective pressure variables with elastic and an-elastic properties and then forward reflectivity operator. These inversion methods have been defined as constrained problems wherein the constraints are applied e.g., bound constraints, constraints in the Lagrangian solution, and Tikhonov regularization. These inversion methods are quite general and can be extended for other rock physics models through parameterizations. The applications of the elastic AVO and the FAVO methods are tested on various 1D synthetic datasets simulated under different oil production (4D) scenarios. The inversion methods are further applied to a 2D realistic reservoir model extracted from the 3D Smeaheia Field, a potential storage site for the CO2 injection. The inversion schemes successfully estimate not only the static saturation and effective pressure variables or changes in these properties due to oil production or CO2 injection but also provide a very good prediction of seismic velocities, density, and seismic attenuation (quantified as the inverse quality factor). The partially CO2-saturated reservoir exhibits higher P wave attenuation, therefore, the addition of time-lapse P wave attenuation due to viscous friction between CO2-water patches helps to reduce the errors in the inverted CO2/water saturation variables as compared to the elastic 4D AVO inversion. This research work has a wide range of applications from the oil industry to carbon capture and storage (CCS) monitoring tools aiming to provide control and safety during the injection. The uncertainty in the inversion results is quantified as a function of the variability of the prior models obtained by using Monte Carlo simulation