An approximate empirical Bayesian method for large-scale linear-Gaussian inverse problems

We study Bayesian inference methods for solving linear inverse problems, focusing on hierarchical formulations where the prior or the likelihood function depend on unspecified hyperparameters. In practice, these hyperparameters are often determined via an empirical Bayesian method that maximizes the marginal likelihood function, i.e., the probability density of the data conditional on the hyperparameters. Evaluating the marginal likelihood, however, is computationally challenging for large-scale problems. In this work, we present a method to approximately evaluate marginal likelihood functions, based on a low-rank approximation of the update from the prior covariance to the posterior covariance. We show that this approximation is optimal in a minimax sense. Moreover, we provide an efficient algorithm to implement the proposed method, based on a combination of the randomized SVD and a spectral approximation method to compute square roots of the prior covariance matrix. Several numerical examples demonstrate good performance of the proposed method

Seismic Data Strong Noise Attenuation Based on Diffusion Model and Principal Component Analysis

Seismic data noise processing is an important part of seismic exploration data processing, and the effect of noise elimination is directly related to the follow-up processing of data. In response to this problem, many authors have proposed methods based on rank reduction, sparse transformation, domain transformation, and deep learning. However, such methods are often not ideal when faced with strong noise. Therefore, we propose to use diffusion model theory for noise removal. The Bayesian equation is used to reverse the noise addition process, and the noise reduction work is divided into multiple steps to effectively deal with high-noise situations. Furthermore, we propose to evaluate the noise level of blind Gaussian seismic data using principal component analysis to determine the number of steps for noise reduction processing of seismic data. We train the model on synthetic data and validate it on field data through transfer learning. Experiments show that our proposed method can identify most of the noise with less signal leakage. This has positive significance for high-precision seismic exploration and future seismic data signal processing research.Comment: 10 pages, 13 figures. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessibl

Expand Dimensional of Seismic Data and Random Noise Attenuation Using Low-Rank Estimation

Random noise attenuation in seismic data requires employing leading-edge methods to attain reliable denoised data. Efficient noise removal, effective signal preservation and recovery, reasonable processing time with a minimum signal distortion and seismic event deterioration are properties of a desired noise suppression algorithm. There are various noise attenuation methods available that more or less have these properties. We aim to obtain more effective denoised seismic data by assuming 3-D seismic data as a tensor in order three and increasing its dimension to 4-D seismic data by employing continuous wavelet transform (CWT). First, we map 3-D block seismic data to smaller blocks to estimate the low-rank component. The CWT of the tensor is calculated along the third dimension to extract the singular values and their related left/right vectors in the wavelet domain. Afterward, the effective low-rank component is extracted using optimized coefficients for each singular value. Thresholding is applied in the wavelet domain along the third dimension to calculate effective coefficients. Two synthetic and field data examples are considered for performance evaluation of the proposed method, and the results were compared with the competitive random noise suppression methods, such as the tensor optimum shrinkage singular value decomposition, the iterative block tensor singular value thresholding, and the block matching 4-D algorithms. Qualitative and quantitative comparison of the proposed method with other methods indicates that the proposed method efficiently eliminates random noise from seismic data

Modified Low Rank Approximation for Detection of Weak Target by Noise space Exploitation in Through Wall Imaging

Low dielectric materials referred as weak targets are very difficult to detect behind the wall in through wall imaging (TWI) due to strong reflections from wall. TWI Experimental data collected for low dielectric target behind the wall and transceiver on another side of the wall. Recently several researchers are using low-rank approximation (LRA) for reduction of random noise in the various data. Explore the possibilities of using LRA for TWI data for improving the detection of low dielectric material. A novel approach using modification of LRA with exploiting the noise subspace in singular value decomposition (SVD) to detect weak target behind the wall is introduced. LRA consider data has low rank in f-x domain for noisy data, local windows are implemented in LRA approach to satisfy the principle assumptions required by the LRA algorithm itself. Decomposed TWI data in the noise space of the SVD to detect the weak target adaptively. Results for modified LRA for detection of weak target behind the wall are very encouraging over LRA

Advancements in seismic tomography with application to tunnel detection and volcano imaging

Thesis (Ph.D.) University of Alaska Fairbanks, 1998Practical geotomography is an inverse problem with no unique solution. A priori information must be imposed for a stable solution to exist. Commonly used types of a priori information smooth and attenuate anomalies, resulting in 'blurred' tomographic images. Small or discrete anomalies, such as tunnels, magma conduits, or buried channels are extremely difficult imaging objectives. Composite distribution inversion (CDI) is introduced as a theory seeking physically simple, rather than distributionally simple, solutions of non-unique problems. Parameters are assumed to be members of a composite population, including both well-known and anomalous components. Discrete and large amplitude anomalies are allowed, while a well-conditioned inverse is maintained. Tunnel detection is demonstrated using CDI tomography and data collected near the northern border of South Korea. Accurate source and receiver location information is necessary. Borehole deviation corrections are estimated by minimizing the difference between empirical distributions of apparent parameter values as a function of location correction. Improved images result. Traveltime computation and raytracing are the most computationally intensive components of seismic tomography when imaging structurally complex media. Efficient, accurate, and robust raytracing is possible by first recovering approximate raypaths from traveltime fields, and then refining the raypaths to a desired accuracy level. Dynamically binned queuing is introduced. The approach optimizes graph-theoretic traveltime computation costs. Pseudo-bending is modified to efficiently refine raypaths in general media. Hypocentral location density functions and relative phase arrival population analysis are used to investigate the Spring, 1996, earthquake swarm at Akutan Volcano, Alaska. The main swarm is postulated to have been associated with a 0.2 km\sp3 intrusion at a depth of less than four kilometers. Decay sequence seismicity is postulated to be a passive response to the stress transient caused by the intrusion. Tomograms are computed for Mt. Spurr, Augustine, and Redoubt Volcanoes, Alaska. Relatively large amplitude, shallow anomalies explain most of the traveltime residual. No large amplitude anomalies are found at depth, and no magma storage areas are imaged. A large amplitude low-velocity anomaly is coincident with a previously proposed geothermal region on the southeast flank of Mt. Spurr. Mt. St. Augustine is found to have a high velocity core

Seismic reverse-time migration in viscoelastic media

Seismic images are key to exploration seismology. They help identify structures in the subsurface and locate potential reservoirs. However, seismic images suffer from the problem of low resolution caused by the viscoelasticity of the medium. The viscoelasticity of the media is caused by the combination of fractured solid rock and fluids, such as water, oil and gas. This viscoelasticity of the medium causes attenuation of seismic waves, which includes energy absorption and velocity dispersion. These two attenuation effects significantly change the seismic data, and thus the seismic imaging. The aim of this thesis is to deepen the understanding of seismic wave propagation in attenuating media and to further investigate the method for high-resolution seismic imaging. My work, presented in this dissertation, comprises the following three parts. First, the determination of the viscoelastic parameters in the generalised viscoelastic wave equation. The viscoelasticity of subsurface media is succinctly represented in the generalised wave equation by a fractional temporal derivative. This generalised viscoelastic wave equation is characterised by the viscoelastic parameter and the viscoelastic velocity, but these parameters are not well formulated and therefore unfavourable for seismic implementation. The causality and stability of the generalised wave equation are proved by deriving the rate-of-relaxation function. On this basis, the viscoelastic parameter is formulated based on the constant Q model, and the viscoelastic velocity is formulated in terms of the reference velocity and the viscoelastic parameter. These two formulations adequately represent the viscoelastic effect in seismic wave propagation. Second, the development of a fractional spatial derivatives wave equation with a spatial filter. This development aims to effectively and efficiently solve the generalised viscoelastic wave equation with fractional temporal derivative, which is numerically challenging. I have transferred the fractional temporal derivative into fractional spatial derivatives, which can be solved using the pseudo-spectral implementation. However, this method is inaccurate in heterogeneous media. I introduced a spatial filter to correct the simulation error caused by the averaging in this implementation. The numerical test shows that the proposed spatial filter can significantly improve the accuracy of the seismic simulation and maintain high efficiency. Moreover, the proposed wave equation with fractional spatial derivatives is applied to compensate for the attenuation effects in reverse-time migration. This allows the dispersion correction and energy compensation to be performed simultaneously, which improves the resolution of the migration results. Finally, the development of reverse-time migration using biaxial wavefield decomposition to reduce migration artefacts and further improve the resolution of seismic images. In reverse-time migration, the cross-correlation of unphysical waves leads to large artefacts. By decomposing the wavefield both horizontally and vertically, and selecting only the causal waves for cross-correlation, the artefacts are greatly reduced, and the delicate structures can be identified. This decomposition method is also suitable for reverse-time migration with attenuation compensation. The migration results show that the resolution of the final seismic image is significantly improved, compared to conventional reverse-time migration.Open Acces

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