Oscillatory integral operators have been of interest to both mathematicians and physicists ever since the emergence of the work Theorie Analytique de la Chaleur of Joseph Fourier in 1822, in which his chief concern was to give a mathematical account of the diffusion of heat. For example, oscillatory integrals naturally arise when one studies the behaviour at infinity of the Fourier transform of a Borel measure that is supported on a certain hypersurface. One reduces the study of such a problem to that of having to obtain estimates on oscillatory integrals. However, sub-level set operators have only come to the fore at the end of the 20th Century, where it has been discovered that the decay rates of the oscillatory integral I(lambda) above may be obtainable once the measure of the associated sub-level sets are known. This discovery has been fully developed in a paper of A. Carbery, M. Christ and J.Wright. A principal goal of this thesis is to explore certain uniformity issues arising in the study of sub-level set estimates.

To interpret the data from spaceborn lidar measurements, one must have a basic understanding of the backscattering of oriented ice particles. The conventional raytracing method is not applicable to the scattering of light by oriented particles. In this study, the dipole approximation (DDA) method is employed to the scattering of light on oriented hexagonal ice columns and plates with various tilting angles. It is found that the oriented hexagonal ice particles tend to have strong backscattering intensity with low depolarization ratios which are strongly dependant on the tilting angle of oriented particles. The present results show that the e ffect of particle orientation plays an important role in determining the optical properties of ice clouds consisting of horizontally oriented ice crystals.

We obtain an upper bound for decay rate of p-adic oscillatory integrals of with analytic phase function and constructible amplitude map. / Thesis / Doctor of Philosophy (PhD)

Remote sensing is the collection of information about an object from a distance without physically being in contact with it. The type of remote sensing of interest here is in the form of digital images of water bodies acquired by satellite. The advantage over traditional sampling techniques is that data can be gathered quickly over large ranges, and be available for immediate analysis. Remote sensing is a powerful technique for the monitoring of water bodies. To interpret the remotely sensed data, however, knowledge of the optical properties of the water constituents is needed. One of the most important of these is the volume scattering function, which describes the angular distribution of light scattered by a sample. This thesis presents the first measurements of volume scattering functions for Australian waters. Measurements were made on around 40 different samples taken from several locations in the Gippsland lakes and the Great Barrier Reef. The measurements were made by modifying an existing static light scattering spectrometer in order to accurately measure the volume scattering functions. The development of the apparatus, its calibration and automation, and the application of a complex series of post-acquisition data corrections, are all discussed. In order to extrapolate the data over the full angular range, the data was analysed using theoretical curves calculated for multi-modal size distributions using Mie light scattering theory applied to each data set. From the Mie fits the scattering and backscattering coefficients were calculated. These were compared with scattering coefficients measured using in situ sensors ac-9 and Hydroscat-6, and with values from the literature. The effect of chlorophyll a concentrations on the scattering coefficients was examined, and a brief investigation of the polarisation properties of the samples was also undertaken. Finally the angular effects on the relationship between the backscattering coefficient and the volume scattering function were investigated. This is important as in situ backscattering sensors often assume that measuring at a single fixed-angle is a good approximation for calculating the backscattering coefficient. This assumption is tested, and the optimal measurement angle determined.

This dissertation focuses on the global investigation of optically thin cirrus cloud optical thickness (tau) and microphysical properties, such as, effective particle size (D_(eff)) and ice crystal habits (shapes), based on the global satellite observations and fast radiative transfer models (RTMs). In the first part, we develop two computationally efficient RTMs simulating satellite observations under cloudy-sky conditions in the visible/shortwave infrared (VIS/SWIR) and thermal inferred (IR) spectral regions, respectively. To mitigate the computational burden associated with absorption, thermal emission and multiple scattering, we generate pre-computed lookup tables (LUTs) using two rigorous models, i.e., the line-by-line radiative transfer model (LBLRTM) and the discrete ordinates radiative transfer model (DISORT). The second part introduces two methods (i.e., VIS/SWIR- and IR-based methods) to retrieve tau and D_(eff) from satellite observations in corresponding spectral regions of the two RTMs. We discuss the advantages and weakness of the two methods by estimating the impacts from different error sources on the retrievals through sensitivity studies. Finally, we develop a new method to infer the scattering phase functions of optically thin cirrus clouds in a water vapor absorption channel (1.38-µm). We estimate the ice crystal habits and surface structures by comparing the inferred scattering phase functions and numerically simulated phase functions calculated using idealized habits.

The ultimate application of this work is constraining the optical properties of cirrus particles, which are poorly understood, by providing an automatic method, using all-sky cameras and an infrared radiometer, to identify the occurrence of the 22° halo formed by cirrus. This is done by interpreting all sky images in terms of a scattering phase function (SPF), from which the halo ratio (HR) is calculated, and by implementing a cirrus detection algorithm to associate HR measures to ice cloud occurrences. Cirrus reflectivity at solar wavelengths is inversely related to the HR which, being an indirect measure of the regularity of the shape of the ice crystals forming the cloud, relates in turn inversely to the asymmetry parameter g. Therefore, the method proposed here to derive statistics of HRs is expected to reduce the uncertainty over the optical and microphysical properties of cirrus. The light intensity measured by the all sky camera is transformed into a scattering phase function, from which the halo formation is identified. This is done by developing image transformations and corrections needed to interpret all sky images quantitatively in terms of scattering phase function, specifically by transforming the original image from the zenith-centred to the light-source-centred system of coordinates and correcting for the air mass and for vignetting. The SPF is then determined by averaging the image brightness over the azimuth angle and the HR by calculating the ratio of brightness at two scattering angles in the vicinity of the 22° halo peak. The instrument transformation and corrections are performed using a series of Matlab scripts. Given that the HR is an ice cloud characteristic and since the method needs additional temperature information if the halo observation is to be associated with cirrus, a cirrus detection algorithm is necessary to screen out non-ice clouds before deriving reliable HR statistics. Cloud detection is determined by quantifying the temporal fluctuations of sky radiance, expressed as brightness temperature (BT), through De-trended Fluctuation Analysis and setting a clear sky fluctuation threshold. Cloud phase discrimination instead is achieved through first constructing an analytic radiative transfer model to obtain an estimate for average molecular absorption cross-section of water vapour within the spectral window of the radiometer. This is done to model the down-welling clear sky radiance, which is in turn used to correct cirrus emissivity and ultimately determine a dynamic BT threshold for the transition from ice to liquid-containing clouds. In addition to the molecular cross section the screen level air temperature and integrated water vapour are used as input parameters to the model. The utilisation of the all sky camera for such quantitative measurement was the particularly novel aspect of this work; this has not been done previously to the best of my knowledge. The cirrus detection method proposed is also innovative in that with respect to previous works it does not rely on the use of additional techniques such as LIDAR or microwave radiometry for discriminating cloud phase. Furthermore, the cirrus threshold proposed is not fixed but accounts for the attenuating properties of the atmosphere below the cloud. Once the cirrus detection algorithm is validated and cirrus occurrences determinable, the HR could be extended to estimating the asymmetry parameter and crystal roughness. These are retrievable, for instance, from in-situ observations of single ice crystal 2D scattering patterns from cloud probes of the SID (Small Ice Detector) type. This would be significant for the constraining of the optical and microphysical properties of cirrus.

Nonstationary signals are common in many environments such as radar, sonar, bioengineering and power systems. The nonstationary nature of the signals found in these environments means that classicalspectralanalysis techniques are notappropriate for estimating the parameters of these signals. Therefore it is important to develop techniques that can accommodate nonstationary signals. This thesis seeks to achieve this by firstly, modelling each component of the signal as having a polynomial phase and by secondly, developing techniques for estimating the parameters of these components. Several approaches can be used for estimating the parameters of polynomial phase signals, eachwithvarying degrees ofsuccess.Criteria to consider in potential estimation algorithms are (i) the signal-to-noise (SNR) ratio threshold of the algorithm, (ii) the amount of computation required for running the algorithm, and (iii) the closeness of the resulting estimates' mean-square errors to the minimum theoretical bound. These criteria will be used to compare the new techniques developed in this thesis with existing techniques. The literature on polynomial phase signal estimation highlights the recurring trade-off between the accuracy of the estimates and the amount of computation required. For example, the Maximum Likelihood (ML) method provides near-optimal estimates above threshold, but also incurs a heavy computational cost for higher order phase signals. On the other hand, multi-linear techniques such as the high-order ambiguity function (HAF) method require little computation, but have a significantly higher SNR threshold than the ML method. Of the existing techniques, the cubic phase (CP) function method is a promising technique because it provides an attractive SNR threshold and computational complexity trade-off. For this reason, the analysis techniques developed in this thesis will be derived from the CP function. A limitation of the CP function is its inability to accurately process phase orders greater than three. Therefore, the first novel contribution to this thesis develops a broadened class of discrete-time higher order phase (HP)functions to address this limitation.This broadened class is achieved by providing a multi-linear extension of the CP function. Monte Carlo simulations are performed to demonstrate the statistical advantage of the HP functions compared to the HAFs. A first order statistical analysis of the HP functions is presented. This analysis verifies the simulation results. The next novel contribution is a technique called the lower SNR cubic phase function (LCPF)method. It is an extension of the CP function, with the extension enabling performance at lower signal-to-noise ratios (SNRs). The improvement of the SNR threshold's performance is achieved by coherently integrating the CP function over a compact interval in the two-dimensional CP function space. The computation of the new algorithm is quite moderate, especially when compared to the ML method. Above threshold, the LCPF method's parameter estimates are asymptotically efficient. Monte Carlo simulation results are presented and a threshold analysis of the algorithm closely predicts the thresholds observed in these results. The next original contribution to this research involves extending the LCPF method so that it is able to process multicomponent cubic phase signals and higher order phase signals. The LCPF method is extended to higher orders by applying a windowing technique as opposed to adjusting the order of the kernel as implemented in the HP function method. To demonstrate the extension of the LCPF method for processing higher order phase signals and multicomponent cubic phase signals, some Monte Carlo simulations are presented. Finally, these estimation techniques are applied to real-worldscenarios in the fields of Power Systems Analysis, Neuroethology and Speech Analysis.

An investigation into the light scattering properties of Saharan dust grains is presented. An electrodynamic trap has been used to levitate single dust particles. By adjusting the trap parameters, partial randomisation of the particle orientation has been introduced. While levitated, the particles were illuminated by a laser, and a rotating half-wave retarder enabled selection of vertically or horizontally polarized incident light. A laser diffractometer and linear photodiode array have been used to measure intensity at scattering angles between 0.5° and 177°. Combining these measurements with Fraunhofer diffraction as calculated for a range of appropriately-sized apertures allows the calculation of the phase function and degree of linear polarization. The phase functions and degree of linear polarisation for four case study particles are presented - the phase functions are found to be featureless across most of the scattering region, with none of the halo features or rainbow peaks associated with regularly shaped particles such as hexagonal columns or spheres. Particle models comprised of large numbers of facets have been constructed to resemble the levitated particles. Utilizing Gaussian random sphere methods, increasing levels of roughness have been added to the surfaces of these models. A Geometric Optics model and a related model, Ray Tracing with Diffraction on Facets, have been modified to calculate scattering on these particle reconstructions. Scattering calculations were performed on each of these reconstructions using a range of refractive indices and two rotation regimes – one where the orientations of the reconstructed particle were limited to match those observed when the particle was levitated, and one where the orientation was not limited. Qualitative comparisons are performed on the phase functions and degree of linear polarization, where it is observed that the addition of roughness to the modelled spheroids causes the computed phase functions to increasingly resemble those from the levitated particles. Limiting the orientation of the particles does not affect the scattering noticeably. The addition of a very small absorption coefficient does not change the comparisons considerably. As the absorption coefficient is increased, however, the quality of the comparisons decreases rapidly in all cases but one. The phase functions are quantitatively compared using RMS errors, and further comparison is performed using the asymmetry parameter.

An investigation was carried out into light scattering by Gaussian rough ice crystals. Gaussian rough crystal geometries were generated using roughness parameters derived from mineral dust grains, which have been reported to be suitable proxies for rough ice crystals. Light scattering data for these geometries was computed using the discrete dipole approximation (DDA) method. Phase functions, 2D scattering patterns, degree of linear polarisation patterns and asymmetry parameters were computed for smooth, moderately rough and highly rough crystals with a variety of orientations and size parameters. A sodium fluorosilicate ice analogue crystal with three partially roughened prism facets was created using focused ion beam (FIB) milling and 2D scattering patterns were collected from it using the small ice detector (SID) 3 cloud probe. It was found that roughness reduces features in the phase function compared to scattering by smooth hexagonal prisms, particularly when the roughness features were horizontally much larger than the wavelength. However, the most effective roughness model also takes account of horizontal features whose size is closer to that of the wavelength. Horizontal features smaller than the wavelength have very little effect.

<p>To estimate the propagation of light in tissue-like optical phantoms (fat emulsions), this thesis utilized the diffuse reflectance spectroscopy in combination with Monte Carlo simulations. A method for determining the two-parametric Gegenbauer-kernal phase function was utilized in order to accurately describe the diffuse reflectance from poly-dispersive scattering optical phantoms with small source-detector separations. The method includes the spectral collimated transmission, spatially resolved diffuse reflectance spectra (SRDR) and the inverse technique of matching spectra from Monte Carlo simulations to those measured. An absolute calibration method using polystyrene micro-spheres was utilized to estimate the relation between simulated and measured SRDR intensities. The phase function parameters were comparable with previous studies and were able to model measured spectra with good accuracy. Significant differences between the phase functions for homogenized milk and the nutritive fat emulsions were found.</p><p> </p>