A common requirement for spatial analysis is the modeling of the second-order structure. While the assumption of isotropy is often made for this structure, it is not always appropriate. A conventional practice to check for isotropy is to informally assess plots of direction-specific sample second-order properties, e.g., sample variogram or sample second-order intensity function. While a useful diagnostic, these graphical techniques are difficult to assess and open to interpretation. Formal alternatives to graphical diagnostics are valuable, but have been applied to a limited class of models. In this dissertation, we propose a formal approach testing for isotropy that is both objective and appropriate for a wide class of models. This approach, which is based on the asymptotic joint normality of the sample second-order properties, can be used to compare these properties in multiple directions. An L_2 consistent subsampling estimator for the asymptotic covariance matrix of the sample second-order properties is derived and used to construct the test statistic with a limiting $\\chi^2$ distribution under the null hypothesis. Our testing approach is purely nonparametric and can be applied to both quantitative spatial processes and spatial point processes. For quantitative processes, the results apply to both regularly spaced and irregularly spaced data when the point locations are generated by a homogeneous point process. In addition, the shape of the random field can be quite irregular. Examples and simulations demonstrate the efficacy of the approach.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1992. / Includes bibliographical references (p. [143]) and index.

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In most cases of seismic processing and interpretation, elastic isotropy is assumed. However, velocity anisotropy is found to exist in most subsurface media. Hence, there exists a fundamental inconsistency between theory on the one hand, and practice on the other. If not recognised, this can invalidate interpretation of seismic data. In this thesis, inversion methods for elastic parameters are developed to quantify the degree of velocity anisotropy of multi-layered transversely isotropic media. This primarily involves examining the velocity fields of layered media using anisotropic elastic wave propagation theory, and developing inversion programs to recover elastic parameters from those velocity fields. The resolved elastic parameter information is used in carrying out further studies on the effects of seismic anisotropy on normal moveout (NMO). Mathematical analyses, numerical simulations, and physical modelling experiments are used in this research for verification purposes before application to field survey data. Numerical studies show the transmission velocity field through layered media appears to be equivalent to that through a single-layered medium, within the practical offset limits in field surveys. The elastic parameters, which describe the property of such equivalent single-layered media, can be used as apparent elastic parameters to describe the collective mechanical property of the layered media. During this research, Snell's law was used in ray tracing to determine ray paths through the interface between any two component layers. By analyzing the signals recorded by any receiver in a walkaway VSP survey, the apparent transmission velocity field for the layered media above this receiver depth was inverted. / Software was developed to recover the apparent elastic parameters for the layered media above this receiver depth using the transmission velocity field as input. Based on a two-layered model, another method was developed to recover the interval elastic parameters for an individual layer of interest, using the signals recorded by receivers on the upper and lower surfaces of this layer. The recovered elastic parameters may be considerably different from the real values if a transversely isotropic medium with a tilted symmetry axis (TTI) is treated as a transversely isotropic medium with a vertical symmetry axis (VTI). A large angle of tilt of the symmetry axis significantly influences the recorded velocity field through the medium. An inversion program was written to recover the value of the tilt angle of a TTI medium, and the elastic parameters of the medium. Programs were also developed to combine information from P, SV, and SH-waves in an inversion procedure. This capability in inversion programs enables us to use the additional information provided by a multi-component VSP survey to obtain accurate estimates of the elastic parameters of geological formations. Software testing and development was carried out on numerically generated input data. Up to 10 milliseconds of random noise in travel time was added to the input to confirm the stability of the inversion software. Further testing was carried out on physical model data where the parameters of the model were known from direct measurements. Finally the inversion software was applied to actual field data and found to give plausible results. / In software testing in the physical modelling laboratory, other practical problems were encountered. System errors caused by the disproportionately large size of the transducers used affected the accuracy of the inversion results obtained. Transducer performance was studied, and it was found that reducing the size of transducers or making offset corrections would decrease the errors caused by the disproportionately large transducer dimensions. In using the elastic parameters recovered, it was found that the elastic parameter δ significantly influences the seismic records from a horizontal reflector. The normal moveout velocity was found to show variations from the zero-offset normal moveout velocity depending on the value and sign of elastic parameter δ. New approximate expressions for anisotropic normal moveout, phase and ray velocity functions at short offsets were developed. The value of anisotropic parameter δ was found to be the major factor controlling these relations. If the recovered parameter δ has a large negative value, analytical and numerical studies demonstrated that the new expression for moveout velocity developed herein should be used instead of Thomsen's normal moveout equation.

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We introduce the inner theory or, more verbosely, isotropy Lawvere theory functor, which generalizes the isotropy group/monoid by assigning a Lawvere theory of coherently extendable arrows to each object of a category with finite powers. Then, we characterize the inner theory for categories of models of an algebraic (or, more generally, quasi-equational) theory, and note its relationship with a notion of definability for morphisms. Finally, we explore a variety of examples.

Collisionless turbulence is common in astrophysical magnetic fields and plasmas. The determination of the transport of high-energy charged particles both parallel and perpendicular to the average magnetic field in such a system is of considerable interest. It is recognized that the turbulent magnetic field has important effects on the transport of charged particles and that the properties of different turbulence models may significantly affect the resulting transport properties. A number of different magnetic turbulence models have been proposed in the last several decades. We present here the results of a study of charged particle transport in two new turbulent magnetic field models that have not been previously considered and include newly described characteristics. We investigate the effect of energetic charged particle motion in these new models. We use a method (D A Roberts, 2012) that optimizes phase angles of a set of circularly polarized, transverse modes with Kolmogorov power-law enveloped amplitudes to construct magnetic field fluctuations with nearly constant |B| but with large variances in the components Bx, By, Bz, as is observed in the Solar Wind. Charged particle scattering coefficients are determined through computer simulations. The results are compared with those from previous isotropic and composite turbulence models. We studied charged particle transport in the turbulent magnetic field with global anisotropy and compared with the isotropic turbulence. We found that the magnetic turbulence with global anisotropy and isotropy have similar effects on charged particle transport from 1𝑀𝑒𝑉 up to 10𝐺𝑒𝑉. We proposed a general framework for a multi-scale synthesis with the scale- dependent, localized anisotropic feature incorporated. We run test particle simula- tions in the field by the two-scale algorithm to calculate the transport coefficients for charged particles with different energies. We found that the local anisotropy has the significant difference from the previous model in the effect on charged particle transport. The parallel transport (𝑘∥) decreases while the perpendicular transport (𝑘⊥) increases compared with the isotropic and globally anisotropic field models, the difference is enhanced as the local anisotropy is enhanced, and there is an order of magnitude increase in the ratio of perpendicular to parallel transport coefficients. We proposed a simple power spectrum synthesis method based on the Fourier analysis to extract the large and small scale power spectrum from any single space- craft observation with a long enough period and a high sampling frequency. We applied the method to the solar wind measurement by the magnetometer onboard the ACE spacecraft and reconstructed the large scale isotropic 2D spectrum and the small scale anisotropic 2D spectrum.

1In this dissertation we work out in detail a new proposal to define rigorously a sector of loop quantum gravity at the diffeomorphism invariant level corresponding to homogeneous and isotropic cosmologies, and propose how to compare in detail the physics of this sector with that of loop quantum cosmology. The key technical steps we have completed are (a) to formulate conditions for homogeneity and isotropy in a diffeomorphism covariant way on the classical phase space of general relativity, and (b) to translate these conditions consistently using well-understood techniques to loop quantum gravity. To impose the symmetry at the quantum level, on both the connection and its conjugate momentum, the method used necessarily has similiarities to the Gupta-Bleuler method of quantizing the electromagnetic field. Lastly, a strategy for embedding states of loop quantum cosmology into this new homogeneous isotropic sector, and using this embedding to compare the physics, is presented. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2019. / FAU Electronic Theses and Dissertations Collection