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

Linear statistics and pushed Coulomb gas at the edge of beta random matrices: four paths to large deviations

The Airy$_\beta$ point process, $a_i \equiv N^{2/3} (\lambda_i-2)$, describes the eigenvalues $\lambda_i$ at the edge of the Gaussian $\beta$ ensembles of random matrices for large matrix size $N \to \infty$. We study the probability distribution function (PDF) of linear statistics ${\sf L}= \sum_i t \varphi(t^{-2/3} a_i)$ for large parameter $t$. We show the large deviation forms $\mathbb{E}_{{\rm Airy},\beta}[\exp(-{\sf L})] \sim \exp(- t^2 \Sigma[\varphi])$ and $P({\sf L}) \sim \exp(- t^2 G(L/t^2))$ for the cumulant generating function and the PDF. We obtain the exact rate function $\Sigma[\varphi]$ using four apparently different methods (i) the electrostatics of a Coulomb gas (ii) a random Schr\"odinger problem, i.e. the stochastic Airy operator (iii) a cumulant expansion (iv) a non-local non-linear differential Painlev\'e type equation. Each method was independently introduced to obtain the lower tail of the KPZ equation. Here we show their equivalence in a more general framework. Our results are obtained for a class of functions $\varphi$, the monotonous soft walls, containing the monomials $\varphi(x)=(u+x)_+^\gamma$ and the exponential $\varphi(x)=e^{u+x}$ and equivalently describe the response of a Coulomb gas pushed at its edge. The small $u$ behavior of the excess energy $\Sigma[\varphi]$ exhibits a change at $\gamma=3/2$ between a non-perturbative hard wall like regime for $\gamma<3/2$ (third order free-to-pushed transition) and a perturbative deformation of the edge for $\gamma>3/2$ (higher order transition). Applications are given, among them: (i) truncated linear statistics such as $\sum_{i=1}^{N_1} a_i$, leading to a formula for the PDF of the ground state energy of $N_1 \gg 1$ noninteracting fermions in a linear plus random potential (ii) $(\beta-2)/r^2$ interacting spinless fermions in a trap at the edge of a Fermi gas (iii) traces of large powers of random matrices.Comment: Main text : 8 pages. Supp mat : 49 page

Automatic Modulation Classification Using Cyclic Features via Compressed Sensing

Cognitive Radios (CRs) are designed to operate with minimal interference to the Primary User (PU), the incumbent to a radio spectrum band. To ensure that the interference generated does not exceed a specific level, an estimate of the Signal to Interference plus Noise Ratio (SINR) for the PU’s channel is required. This can be accomplished through determining the modulation scheme in use, as it is directly correlated with the SINR. To this end, an Automatic Modulation Classification (AMC) scheme is developed via cyclic feature detection that is successful even with signal bandwidths that exceed the sampling rate of the CR. In order to accomplish this, Compressed Sensing (CS) is applied, allowing for reconstruction, even with very few samples. The use of CS in spectrum sensing and interpretation is becoming necessary for a growing number of scenarios where the radio spectrum band of interest cannot be fully measured, such as low cost sensor networks, or high bandwidth radio localization services. In order to be able to classify a wide range of modulation types, cumulants were chosen as the feature to use. They are robust to noise and provide adequate discrimination between different types of modulation, even those that are fairly similar, such as 16-QAM and 64-QAM. By fusing cumulants and CS, a novel method of classification was developed which inherited the noise resilience of cumulants, and the low sample requirements of CS. Comparisons are drawn between the proposed method and existing ones, both in terms of accuracy and resource usages. The proposed method is shown to perform similarly when many samples are gathered, and shows improvement over existing methods at lower sample counts. It also uses less resources, and is able to produce an estimate faster than the current systems

Cumulant based identification approaches for nonminimum phase FIR systems

Cataloged from PDF version of article.In this paper, recursive and least squares methods for identification of nonminimum phase linear time-invariant (NMP-LTI) FIR systems are developed. The methods utilize the second- and third-order cumulants of the output of the FIR system whose input is an independent, identically distributed (i.i.d.) non-Gaussian process. Since knowledge of the system order is of utmost importance to many system identification algorithms, new procedures for determining the order of an FIR system using only the output cumulants are also presented. To illustrate the effectiveness of our methods, various simulation examples are presented

Higher-order statistics for DSGE models

Closed-form expressions for unconditional moments, cumulants and polyspectra of order higher than two are derived for non-Gaussian or nonlinear (pruned) solutions to DSGE models. Apart from the existence of moments and white noise property no distributional assumptions are needed. The accuracy and utility of the formulas for computing skewness and kurtosis are demonstrated by three prominent models: Smets and Wouters (AER, 586-606, 97, 2007) (first-order approximation), An and Schorfheide (Econom. Rev., 113-172, 26, 2007) (second-order approximation) and the neoclassical growth model (third-order approximation). Both the Gaussian as well as Student's t-distribution are considered as the underlying stochastic processes. Lastly, the efficiency gain of including higher-order statistics is demonstrated by the estimation of a RBC model within a Generalized Method of Moments framework