Abstract This dissertation is a combination of four articles on the topic of scattering problems for a biharmonic operator. The operator of interest has two coefficients which may be complex-valued and singular. Each of the articles concerns a different aspect of the problem. Namely, the first article discusses the direct scattering problem in higher dimensions and culminates in a proof of Saito's formula, which yields a uniqueness result for the inverse scattering problem. The second paper is about a backscattering problem in two and three dimensions. We prove that the inverse Born approximation can be used to recover the singularities in the coefficients of the operator. The third article fills in an answer to the question about recovering the complex-valued coefficients in three dimensions that was left open in the second article. The final article studies the inverse scattering problem on the line for a quasi-linear operator. / Tiivistelmä Väitöskirjatyö koostuu neljästä artikkelista, jotka käsittelevät sirontaongelmia biharmoniselle operaattorille. Työn kohteena olevalla operaattorilla on kaksi kerrointa, jotka voivat olla kompleksiarvoisia ja singulaarisia. Kukin artikkeli käsittelee sirontaongelmaa eri näkökulmasta. Ensimmäinen artikkeli koostuu pääasiassa suorasta sirontateoriasta korkeammissa ulottuvuuksissa huipentuen lopulta Saiton kaavan todistukseen, jonka seurauksena saadaan yksikäsitteisyystulos käänteiselle sirontaongelmalle. Toisen artikkelin aiheena on takaisinsirontaongelma kahdessa ja kolmessa ulottuvuudessa. Todistamme, että käänteistä Bornin approksimaatiota voidaan käyttää paikantamaan kertoimien mahdolliset singulariteetit. Kolmas artikkeli vastaa toisessa artikkelissa avoimeksi jääneeseen kysymykseen kompleksiarvoisien kertoimien rekonstruoimisesta kolmessa ulottuvuudessa. Viimeisessä artikkelissa tutkitaan käänteistä sirontaongelmaa kvasilineaariselle operaattorille yhdessä ulottuvuudessa.

Since the introduction of the Born and Rytov approximations for use in random wave propagation some forty years ago, a controversy has boiled over the regions of validity and relative merits of the methods. Although the methods fail for strong fluctuations and distant path lengths, these two perturbation methods are the only approaches available for weak fluctuations in a random in homogeneous media. The approximations have also been applied to the inverse problem for optical and acoustical tomography. The intent of this thesis is to investigate the work of previous authors and attempt to clarify the distinctions of each method. The conclusion will be reached that neither approximation is necessarily better than the other in general for all applications. A careful consideration of the problem following the points given should point towards the use of one approximation over the other. / Master of Science

Thesis (PhD)--Stellenbosch University, 2012. / Please refer to full text for abstract.

This thesis presents a series of electron impact ionization measurements on the gas phase targets of argon and krypton. The (e,2e) coincidence technique has been employed to measure the triple differential cross section (TDCS) using a new coincidence spectrometer designed to operate in the low energy regime (2 to 5 times the ionization energy) and in the coplanar geometry. The spectrometer is a conventional device utilizing a non-energy selected electron gun and two 1800 hemispherical electron analysers fitted with channel electron multipliers for detection of the outgoing electrons. A series of TDCS measurements were performed on the 3s inner-valence and 3p valence orbitals of argon employing coplanar asymmetric kinematics. Measurements for both orbitals were performed at an incident energy of 113.5 eV, ejected energies of 10, 7.5, 5 and 2 eV and a scattering angle of -15°. The energy of the scattered electron in each case was chosen to satis~' energy conservation and is dependent on the ionization energies of the different orbitals. The experimental cross sections are compared to theoretical TDCS calculations using the distorted wave Born approximation (DWBA) and variations of the DWBA in an attempt to investigate the role that post collisional interaction (PCI), polarization and electron exchange play in describing the TDCS in the low energy regime. To further extend this analysis, a series of TDCS measurements were performed on the 3s and 4s. orbitals of argon and krypton, respectively, employing coplanar symmetric kinematics. Measurements were performed for the 3s orbital at outgoing energies of 50, 20, 10 and 4eV and for the 4s orbital at outgoing energies of 85, 50, 20 and 10 eV. The kinematics were chosen to coincide with several of the (e,2e) measurements made in the same geometry on the 3p orbital of argon by Rouvellou et al (1998). The experimental results were again compared to a DWBA calculation and similar variations to those employed for the asymmetric measurements.

Many hydrocarbon reservoirs have sufficient porosity but low permeability (for example, tight gas sands and coal beds). However, such reservoirs are often naturally fractured. The fracture patterns in these reservoirs can control flow and transport properties, and therefore, play an important role in drilling production wells. On the scale of seismic wavelengths, closely spaced parallel fractures behave like an anisotropic media, which precludes the response of individual fractures in the seismic data. There are a number of fracture parameters which are needed to fully characterize a fractured reservoir. However, seismic data may reveal only certain fracture parameters and those are fracture orientation, crack density and fracture infill. Most of the widely used fracture characterization methods such as Swave splitting analysis or amplitude vs. offset and azimuth (AVOA) analysis fail to render desired results in laterally varying media. I have conducted a systematic study of the response of fractured reservoirs with laterally varying elastic and fracture properties, and I have developed a scheme to invert for the fracture parameters. I have implemented a 3D finite-difference method to generate multicomponent synthetic seismic data in general anisotropic media. I applied the finite-difference algorithm in both Standard and Rotated Staggered grids. Standard Staggered grid is used for media having symmetry up to orthorhombic (isotropic, transversely isotropic, and orthorhombic), whereas Rotated Staggered grid is implemented for monoclinic and triclinic media. I have also developed an efficient and accurate ray-bending algorithm to compute seismic traveltimes in 3D anisotropic media. AVOA analysis is equivalent to the first-order Born approximation. However, AVOA analysis can be applied only in a laterally uniform medium, whereas the Born-approximation does not pose any restriction on the subsurface structure. I have developed an inversion scheme based on a ray-Born approximation to invert for the fracture parameters. Best results are achieved when both vertical and horizontal components of the seismic data are inverted simultaneously. I have also developed an efficient positivity constraint which forbids the inverted fracture parameters to be negative in value. I have implemented the inversion scheme in the frequency domain and I show, using various numerical examples, that all frequency samples up to the Nyquist are not required to achieve desired inversion results.

The problem of subsurface imaging with Ground Penetrating Radar (GPR) is a challenging one. Due to the low-pass nature of soil sensors must utilise wave-lengths that are of the same order of magnitude as the object being imaged. This makes imaging difficult as straight ray approximations commonly used in higher frequency applications cannot be used. The problem becomes even more challenging when the target is shallowly buried as in this case the ground surface reflection and the near-field parameters of the radar need to be considered. This thesis has investigated the problem of imaging shallowly buried targets with GPR. Two distinct problems exist in this field radar design and the design of inverse scattering techniques. This thesis focuses on the design of inverse scattering techniques capable of taking the electric field measurements from the receiver and providing accurate images of the scatterer in real time. The thesis commences with a brief introduction to GPR theory. It then provides an extensive review of linear inverse scattering techniques applied to raw GPR data. As a result of this review the thesis draws the conclusion that, due to its strong foundations in Maxwell's equations, diffraction tomography is the most appropriate approach for imaging shallowly buried targets with GPR. A three-dimensional diffraction tomographic technique is then developed. This algorithm forms the primary contribution of the thesis. The novel diffraction tomography technique improves on its predecessors by catering for shallowly buried targets, significant antenna heights and evanescent waves. This is also the first diffraction tomography technique to be derived for a range of antenna structures. The advantages of the novel technique are demonstrated first mathematically then on synthetic and finally practical data. The algorithm is shown to be of high practical value by producing accurate images of buried targets in real time.

No description available.

The purpose of this work was to examine the effect of the use the assumption κω2K/ΕCM «1 in calculating K-shell ionization cross sections in the plane wave Born approximation (PWBA) where κω2K is the observed binding energy of the K-shell and ECM is the energy of the incident particle in the center of mass system. Avoiding this assumption produces a threshold for ionization at Ecm = κω2K. Calculations employing the assumption, which leads to the use of approximate limits of integration, do not go to zero for even the .Lowest values of the incident energy.

Diplomová práca sa zaoberá rozptylom elektromagnetického žiarenia na biologickej bunke použitím metódy konečných diferencií v časovej oblasti (FDTD), Bornovej aproximácie a Rytovovej aproximácie. Metóda FDTD dáva presné výsledky v širokej škále problémov. Je spravené porovnanie Bornovej aproximácie a Rytovovej aproximácie prostredníctvom FDTD metódy. Ďaľšia časť práce zahrnuje krátky popis koherenciou riadeného holografického mikroskopu CCHM. Záverečná časť sa venuje zobrazeniu rozptýleného poľa získaného z jednotlivých simulácií pomocou simulácie objektového ramena mikroskopu CCHM.

Under current investigation is the re-implementation of the Distorted-Wave Impulse Approximation (DWIA), originally formulated in FORTRAN by N.S. Chant and P.G. Roos, with the intention of developing it in a portable Python environment. This will be complimented by developing a GEANT4 detector simulation application. These two techniques will be used to model the (p,2p) proton knock-out reaction 40Ca(p; 2p)39K (2.52 MeV)1 2 + first excited state, at intermediate incident energies of 150 MeV. This study is a test-bed that lays the foundation and platform from which one may develop an interactive workbench and toolkit in GEANT4 which: (i.) accurately models an accelerator-detector experimental set-up, such as those found at iThemba Labs, and (ii.) incorporates the DWIA formalism as a built-in physics process within the framework of GEANT4. Furthermore the Python modules developed for the specific proton knock-out reaction studied here, can be generalized for an arbitrary set of nuclear scattering reactions and packaged as a suite of scientific Python codes. / Theoretical and Computational Nuclear Physics / M. Sc. (Theoretical and Computational Nuclear Physics)