Anisotropic Denoising in Functional Deconvolution Model with Dimension-free Convergence Rates

In the present paper we consider the problem of estimating a periodic $(r+1)$-dimensional function $f$ based on observations from its noisy convolution. We construct a wavelet estimator of $f$, derive minimax lower bounds for the $L^2$-risk when $f$ belongs to a Besov ball of mixed smoothness and demonstrate that the wavelet estimator is adaptive and asymptotically near-optimal within a logarithmic factor, in a wide range of Besov balls. We prove in particular that choosing this type of mixed smoothness leads to rates of convergence which are free of the "curse of dimensionality" and, hence, are higher than usual convergence rates when $r$ is large. The problem studied in the paper is motivated by seismic inversion which can be reduced to solution of noisy two-dimensional convolution equations that allow to draw inference on underground layer structures along the chosen profiles. The common practice in seismology is to recover layer structures separately for each profile and then to combine the derived estimates into a two-dimensional function. By studying the two-dimensional version of the model, we demonstrate that this strategy usually leads to estimators which are less accurate than the ones obtained as two-dimensional functional deconvolutions. Indeed, we show that unless the function $f$ is very smooth in the direction of the profiles, very spatially inhomogeneous along the other direction and the number of profiles is very limited, the functional deconvolution solution has a much better precision compared to a combination of $M$ solutions of separate convolution equations. A limited simulation study in the case of $r=1$ confirms theoretical claims of the paper.Comment: 29 pages, 1 figure, 1 tabl

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

A Primal-Dual Proximal Algorithm for Sparse Template-Based Adaptive Filtering: Application to Seismic Multiple Removal

Unveiling meaningful geophysical information from seismic data requires to deal with both random and structured "noises". As their amplitude may be greater than signals of interest (primaries), additional prior information is especially important in performing efficient signal separation. We address here the problem of multiple reflections, caused by wave-field bouncing between layers. Since only approximate models of these phenomena are available, we propose a flexible framework for time-varying adaptive filtering of seismic signals, using sparse representations, based on inaccurate templates. We recast the joint estimation of adaptive filters and primaries in a new convex variational formulation. This approach allows us to incorporate plausible knowledge about noise statistics, data sparsity and slow filter variation in parsimony-promoting wavelet frames. The designed primal-dual algorithm solves a constrained minimization problem that alleviates standard regularization issues in finding hyperparameters. The approach demonstrates significantly good performance in low signal-to-noise ratio conditions, both for simulated and real field seismic data

INVERSE ATTENUATION-FILTERING

When seismic waves propagate through the Earth, they are affected by numerous inelastic effects of the medium. These effects are usually characterized by the concept of the Q-factor and lead to variations of spectra of the signal and shapes of the waveforms, which further affect the results of reflection seismic imaging. Attenuation compensation, also often called the inverse Q filtering is a signal-processing procedure broadly used to compensate both of these effects of attenuation in reflection sections or volumes. The objective of this thesis is to present and investigate a new attenuation-compensation approach that is much more general than the conventional inverse Q filtering

Sources of uncertainties and artefacts in back-projection results

Back-projecting high-frequency (HF) waves is a common procedure for imaging rupture processes of large earthquakes (i.e. M_w > 7.0). However, obtained back-projection (BP) results could suffer from large uncertainties since high-frequency seismic waveforms are strongly affected by factors like source depth, focal mechanisms, and the Earth's 3-D velocity structures. So far, these uncertainties have not been thoroughly investigated. Here, we use synthetic tests to investigate the influencing factors for which scenarios with various source and/or velocity set-ups are designed, using either Tohoku-Oki (Japan), Kaikoura (New Zealand), Java/Wharton Basin (Indonesia) as test areas. For the scenarios, we generate either 1-D or 3-D teleseismic synthetic data, which are then back-projected using a representative BP method, MUltiple SIgnal Classification (MUSIC). We also analyse corresponding real cases to verify the synthetic test results. The Tohoku-Oki scenario shows that depth phases of a point source can be back-projected as artefacts at their bounce points on the earth's surface, with these artefacts located far away from the epicentre if earthquakes occur at large depths, which could significantly contaminate BP images of large intermediate-depth earthquakes. The Kaikoura scenario shows that for complicated earthquakes, composed of multiple subevents with varying focal mechanisms, BP tends to image subevents emanating large amplitude coherent waveforms, while missing subevents whose P nodal directions point to the arrays, leading to discrepancies either between BP images from different arrays, or between BP images and other source models. Using the Java event, we investigate the impact of 3-D source-side velocity structures. The 3-D bathymetry together with a water layer can generate strong and long-lasting coda waves, which are mirrored as artefacts far from the true source location. Finally, we use a Wharton Basin outer-rise event to show that the wavefields generated by 3-D near trench structures contain frequency-dependent coda waves, leading to frequency-dependent BP results. In summary, our analyses indicate that depth phases, focal mechanism variations and 3-D source-side structures can affect various aspects of BP results. Thus, we suggest that target-oriented synthetic tests, for example, synthetic tests for subduction earthquakes using more realistic 3-D source-side velocity structures, should be conducted to understand the uncertainties and artefacts before we interpret detailed BP images to infer earthquake rupture kinematics and dynamics

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