A pragmatic Bayesian approach to wind field retrieval

The ERS-1 Satellite was launched in July 1991 by the European Space Agency into a polar orbit at about 800 km, carrying a C-band scatterometer. A scatterometer measures the amount of backscatter microwave radiation reflected by small ripples on the ocean surface induced by sea-surface winds, and so provides instantaneous snap-shots of wind flow over large areas of the ocean surface, known as wind fields. Inherent in the physics of the observation process is an ambiguity in wind direction; the scatterometer cannot distinguish if the wind is blowing toward or away from the sensor device. This ambiguity implies that there is a one-to-many mapping between scatterometer data and wind direction. Current operational methods for wind field retrieval are based on the retrieval of wind vectors from satellite scatterometer data, followed by a disambiguation and filtering process that is reliant on numerical weather prediction models. The wind vectors are retrieved by the local inversion of a forward model, mapping scatterometer observations to wind vectors, and minimising a cost function in scatterometer measurement space. This thesis applies a pragmatic Bayesian solution to the problem. The likelihood is a combination of conditional probability distributions for the local wind vectors given the scatterometer data. The prior distribution is a vector Gaussian process that provides the geophysical consistency for the wind field. The wind vectors are retrieved directly from the scatterometer data by using mixture density networks, a principled method to model multi-modal conditional probability density functions. The complexity of the mapping and the structure of the conditional probability density function are investigated. A hybrid mixture density network, that incorporates the knowledge that the conditional probability distribution of the observation process is predominantly bi-modal, is developed. The optimal model, which generalises across a swathe of scatterometer readings, is better on key performance measures than the current operational model. Wind field retrieval is approached from three perspectives. The first is a non-autonomous method that confirms the validity of the model by retrieving the correct wind field 99% of the time from a test set of 575 wind fields. The second technique takes the maximum a posteriori probability wind field retrieved from the posterior distribution as the prediction. For the third technique, Markov Chain Monte Carlo (MCMC) techniques were employed to estimate the mass associated with significant modes of the posterior distribution, and make predictions based on the mode with the greatest mass associated with it. General methods for sampling from multi-modal distributions were benchmarked against a specific MCMC transition kernel designed for this problem. It was shown that the general methods were unsuitable for this application due to computational expense. On a test set of 100 wind fields the MAP estimate correctly retrieved 72 wind fields, whilst the sampling method correctly retrieved 73 wind fields.

A critical assessment of the finite element method for calculating electric and magnetic fields

The present dissertation is concerned with the determination of the magnetic field distribution in ma[.rnetic electron lenses by means of the finite element method. In the differential form of this method a Poisson type equation is solved by numerical methods over a finite boundary. Previous methods of adapting this procedure to the requirements of digital computers have restricted its use to computers of extremely large core size. It is shown that by reformulating the boundary conditions, a considerable reduction in core store can be achieved for a given accuracy of field distribution. The magnetic field distribution of a lens may also be calculated by the integral form of the finite element rnethod. This eliminates boundary problems mentioned but introduces other difficulties. After a careful analysis of both methods it has proved possible to combine the advantages of both in a .new approach to the problem which may be called the 'differential-integral' finite element method. The application of this method to the determination of the magnetic field distribution of some new types of magnetic lenses is described. In the course of the work considerable re-programming of standard programs was necessary in order to reduce the core store requirements to a minimum.

Identification of visual field progression

The study developed statistical techniques to evaluate visual field progression for use with the Humphrey Field Analyzer (HFA). The long-term fluctuation (LF) was evaluated in stable glaucoma. The magnitude of both LF components showed little relationship with MD, CPSD and SF. An algorithm was proposed for determining the clinical necessity for a confirmatory follow-up examination. The between-examination variability was determined for the HFA Standard and FASTPAC algorithms in glaucoma. FASTPAC exhibited greater between-examination variability than the Standard algorithm across the range of sensitivities and with increasing eccentricity. The difference in variability between the algorithms had minimal clinical significance. The effect of repositioning the baseline in the Glaucoma Change Probability Analysis (GCPA) was evaluated. The global baseline of the GCPA limited the detection of progressive change at a single stimulus location. A new technique, pointwise univariate linear regressions (ULR), of absolute sensitivity and, of pattern deviation, against time to follow-up was developed. In each case, pointwise ULR was more sensitive to localised progressive changes in sensitivity than ULR of MD, alone. Small changes in sensitivity were more readily determined by the pointwise ULR than by the GCPA. A comparison between the outcome of pointwise ULR for all fields and for the last six fields manifested linear and curvilinear declines in the absolute sensitivity and the pattern deviation. A method for delineating progressive loss in glaucoma, based upon the error in the forecasted sensitivity of a multivariate model, was developed. Multivariate forecasting exhibited little agreement with GCPA in glaucoma but showed promise for monitoring visual field progression in OHT patients. The recovery of sensitivity in optic neuritis over time was modelled with a Cumulative Gaussian function. The rate and level of recovery was greater in the peripheral than the central field. Probability models to forecast the field of recovery were proposed.

Variability of the permetric response in normals and in glaucoma

This study investigated the variability of response associated with various perimetric techniques, with the aim of improving the clinical interpretation of automated static threshold perirnetry. Evaluation of a third generation of perimetric threshold algorithms (SITA) demonstrated a reduction in test duration by approximately 50% both in normal subjects and in glaucoma patients. SITA produced a slightly higher, but clinically insignificant, Mean Sensitivity than with the previous generations of algorithms. This was associated with a decreased between-subject variability in sensitivity and hence, lower confidence intervals for normality. In glaucoma, the SITA algorithms gave rise to more statistically significant visual field defects and a similar between-visit repeatability to the Full Threshold and FASTPAC algorithms. The higher estimated sensitivity observed with SITA compared to Full Threshold and FASTPAC were not attributed to a reduction in the fatigue effect. The investigation of a novel method of maintaining patient fixation, a roving fixation target which paused immediately prior lo the stimulus presentation, revealed a greater degree of fixational instability with the roving fixation target compared to the conventional static fixation target. Previous experience with traditional white-white perimetry did not eradicate the learning effect in short-wavelength automated perimetry (SWAP) in a group of ocular hypertensive patients. The learning effect was smaller in an experienced group of patients compared to a naive group of patients, but was still at a significant level to require that patients should undertake a series of at least three familiarisation tests with SWAP.

A method for the in-situ determination of the hydraulic conductivity of gravels as used in constructed wetlands for wastewater treatment

A new instrument and method are described that allow the hydraulic conductivities of highly permeable porous materials, such as gravels in constructed wetlands, to be determined in the field. The instrument consists of a Mariotte siphon and a submersible permeameter cell with manometer take-off tubes, to recreate in-situ the constant head permeameter test typically used with excavated samples. It allows permeability to be measured at different depths and positions over the wetland. Repeatability obtained at fixed positions was good (normalised standard deviation of 1–4%), and results obtained for highly homogenous silica sand compared well when the sand was retested in a lab permeameter (0.32 mm.s–1 and 0.31 mm.s–1 respectively). Practical results have a ±30% associated degree of uncertainty because of the mixed effect of natural variation in gravel core profiles, and interstitial clogging disruption during insertion of the tube into the gravel. This error is small, however, compared to the orders of magnitude spatial variations detected. The technique was used to survey the hydraulic conductivity profile of two constructed wetlands in the UK, aged 1 and 15 years respectively. Measured values were high (up to 900 mm.s –1) and varied by three orders of magnitude, reflecting the immaturity of the wetland. Detailed profiling of the younger system suggested the existence of preferential flow paths at a depth of 200 mm, corresponding to the transition between more coarse and less coarse gravel layers (6–12 mm and 3–6 mm respectively), and transverse drift towards the outlet.

The attenuation surface for contrast sensitivity has the form of a witch’s hat within the central visual field

Over the full visual field, contrast sensitivity is fairly well described by a linear decline in log sensitivity as a function of eccentricity (expressed in grating cycles). However, many psychophysical studies of spatial visual function concentrate on the central ±4.5 deg (or so) of the visual field. As the details of the variation in sensitivity have not been well documented in this region we did so for small patches of target contrast at several spatial frequencies (0.7–4 c/deg), meridians (horizontal, vertical, and oblique), orientations (horizontal, vertical, and oblique), and eccentricities (0–18 cycles). To reduce the potential effects of stimulus uncertainty, circular markers surrounded the targets. Our analysis shows that the decline in binocular log sensitivity within the central visual field is bilinear: The initial decline is steep, whereas the later decline is shallow and much closer to the classical results. The bilinear decline was approximately symmetrical in the horizontal meridian and declined most steeply in the superior visual field. Further analyses showed our results to be scale-invariant and that this property could not be predicted from cone densities. We used the results from the cardinal meridians to radially interpolate an attenuation surface with the shape of a witch's hat that provided good predictions for the results from the oblique meridians. The witch's hat provides a convenient starting point from which to build models of contrast sensitivity, including those designed to investigate signal summation and neuronal convergence of the image contrast signal. Finally, we provide Matlab code for constructing the witch's hat.