Fractional Diffusion Modeling of Electromagnetic Induction in Fractured Rocks

The controlled-source electromagnetic (CSEM) technique is well-established for non-invasive geophysical survey. Due to the strong attenuation of earth materials to electromagnetic signals, the effective depth of most CSEM surveys is restricted to 1-2 km, a zone where pores and fractures over various length scales are highly complicated. Spatial confinement of fluid or electric charge transport by the fractal geometry gives rise to interesting dynamic processes within the pore space and fractures, such as anomalous diffusion. Conventionally, CSEM data are interpreted in terms of a 1-D, 2-D or 3-D piecewise constant geological structure with uniform conductivity and thickness of each cell. A very fine grid, and hence a lot of computation time, are needed to build and evaluate a model that can explain the Earths actual 3D CSEM response. Good accuracy may not be captured, using the conventional approach, in the presence of multi-scale hierarchical geoelectrical structure. Alternatively, the CSEM response of such structures are easily evaluated if the physics of anomalous diffusion of electromagnetic eddy currents is recognized and cast, for example, in terms of a continuous time random walk. Such a re-formulation leads to a generalization of Maxwell equations containing a fractional order time derivative. The fractional order of the derivative is equivalent to a roughening of the geological medium, introducing multi-scale variations of fractures and heterogeneities in a compact manner. This theory renders CSEM modeling and inversion much more efficient, as only a few model parameters are now required to be fit. However the EM fractional diffusion theory is far from perfect, e.g. the correlation between the roughness of a fracture model with its fracture properties. In this research, I use numerical modeling tool to answer this question and explore if classical piece-wise constant conductivity model can generate a fractional type response. In this thesis, I will review the fundamental theory of traditional CSEM survey technique and the continuous time random walk approach, and review the derivation of the generalized Maxwell equation. More importantly, I propose the finite difference method to discrete the generalized Maxwell equation in 2D and 3D. I explore a classical fractured model response created from the von Karman random media approach. I will show that the von Karman fractured model generates a classical type response which is inconsistent with the fractional diffusion response. It is difficult to generate a classical model numerically that is comparable with the rough natural model

A survey of uncertainty principles and some signal processing applications

The goal of this paper is to review the main trends in the domain of uncertainty principles and localization, emphasize their mutual connections and investigate practical consequences. The discussion is strongly oriented towards, and motivated by signal processing problems, from which significant advances have been made recently. Relations with sparse approximation and coding problems are emphasized

Idealized computational models for auditory receptive fields

This paper presents a theory by which idealized models of auditory receptive fields can be derived in a principled axiomatic manner, from a set of structural properties to enable invariance of receptive field responses under natural sound transformations and ensure internal consistency between spectro-temporal receptive fields at different temporal and spectral scales. For defining a time-frequency transformation of a purely temporal sound signal, it is shown that the framework allows for a new way of deriving the Gabor and Gammatone filters as well as a novel family of generalized Gammatone filters, with additional degrees of freedom to obtain different trade-offs between the spectral selectivity and the temporal delay of time-causal temporal window functions. When applied to the definition of a second-layer of receptive fields from a spectrogram, it is shown that the framework leads to two canonical families of spectro-temporal receptive fields, in terms of spectro-temporal derivatives of either spectro-temporal Gaussian kernels for non-causal time or the combination of a time-causal generalized Gammatone filter over the temporal domain and a Gaussian filter over the logspectral domain. For each filter family, the spectro-temporal receptive fields can be either separable over the time-frequency domain or be adapted to local glissando transformations that represent variations in logarithmic frequencies over time. Within each domain of either non-causal or time-causal time, these receptive field families are derived by uniqueness from the assumptions. It is demonstrated how the presented framework allows for computation of basic auditory features for audio processing and that it leads to predictions about auditory receptive fields with good qualitative similarity to biological receptive fields measured in the inferior colliculus (ICC) and primary auditory cortex (A1) of mammals.Comment: 55 pages, 22 figures, 3 table

Tomograms and other transforms. A unified view

A general framework is presented which unifies the treatment of wavelet-like, quasidistribution, and tomographic transforms. Explicit formulas relating the three types of transforms are obtained. The case of transforms associated to the symplectic and affine groups is treated in some detail. Special emphasis is given to the properties of the scale-time and scale-frequency tomograms. Tomograms are interpreted as a tool to sample the signal space by a family of curves or as the matrix element of a projector.Comment: 19 pages latex, submitted to J. Phys. A: Math and Ge