Optimizing Techniques and Cramer-Rao Bound for Passive Source Location Estimation

This work is motivated by the problem of locating potential unstable areas in underground potash mines with better accuracy more consistently while introducing minimum extra computational load. It is important for both efficient mine design and safe mining activities, since these unstable areas may experience local, low-intensity earthquakes in the vicinity of an underground mine. The object of this thesis is to present localization algorithms that can deliver the most consistent and accurate estimation results for the application of interest. As the first step towards the goal, three most representative source localization algorithms given in the literature are studied and compared. A one-step energy based grid search (EGS) algorithm is selected to address the needs of the application of interest. The next step is the development of closed-form Cram´er-Rao bound (CRB) expressions. The mathematical derivation presented in this work deals with continuous signals using the Karhunen-Lo`eve (K-L) expansion, which makes the derivation applicable to non-stationary Gaussian noise problems. Explicit closed-form CRB expressions are presented only for stationary Gaussian noise cases using the spectrum representation of the signal and noise though. Using the CRB comparisons, two approaches are proposed to further improve the EGS algorithm. The first approach utilizes the corresponding analytic expression of the error estimation variance (EEV) given in [1] to derive an amplitude weight expression, optimal in terms of minimizing this EEV, for the case of additive Gaussian noise with a common spectrum interpretation across all the sensors. An alternate noniterative amplitude weighting scheme is proposed based on the optimal amplitude weight expression. It achieves the same performance with less calculation compared with the traditional iterative approach. The second approach tries to optimize the EGS algorithm in the frequency domain. An analytic frequency weighted EEV expression is derived using spectrum representation and the stochastic process theory. Based on this EEV expression, an integral equation is established and solved using the calculus of variations technique. The solution corresponds to a filter transfer function that is optimal in the sense that it minimizes this analytic frequency domain EEV. When various parts of the frequency domain EEV expression are ignored during the minimization procedure using Cauchy-Schwarz inequality, several different filter transfer functions result. All of them turn out to be well known classical filters that have been developed in the literature and used to deal with source localization problems. This demonstrates that in terms of minimizing the analytic EEV, they are all suboptimal, not optimal. Monte Carlo simulation is performed and shows that both amplitude and frequency weighting bring obvious improvement over the unweighted EGS estimator

Identification of audio evoked response potentials in ambulatory EEG data

Electroencephalography (EEG) is commonly used for observing brain function over a period of time. It employs a set of invasive electrodes on the scalp to measure the electrical activity of the brain. EEG is mainly used by researchers and clinicians to study the brain’s responses to a specific stimulus - the event-related potentials (ERPs). Different types of undesirable signals, which are known as artefacts, contaminate the EEG signal. EEG and ERP signals are very small (in the order of microvolts); they are often obscured by artefacts with much larger amplitudes in the order of millivolts. This greatly increases the difficulty of interpreting EEG and ERP signals.Typically, ERPs are observed by averaging EEG measurements made with many repetitions of the stimulus. The average may require many tens of repetitions before the ERP signal can be observed with any confidence. This greatly limits the study and useof ERPs. This project explores more sophisticated methods of ERP estimation from measured EEGs. An Optimal Weighted Mean (OWM) method is developed that forms a weighted average to maximise the signal to noise ratio in the mean. This is developedfurther into a Bayesian Optimal Combining (BOC) method where the information in repetitions of ERP measures is combined to provide a sequence of ERP estimations with monotonically decreasing uncertainty. A Principal Component Analysis (PCA) isperformed to identify the basis of signals that explains the greatest amount of ERP variation. Projecting measured EEG signals onto this basis greatly reduces the noise in measured ERPs. The PCA filtering can be followed by OWM or BOC. Finally, crosschannel information can be used. The ERP signal is measured on many electrodes simultaneously and an improved estimate can be formed by combining electrode measurements. A MAP estimate, phrased in terms of Kalman Filtering, is developed using all electrode measurements.The methods developed in this project have been evaluated using both synthetic and measured EEG data. A synthetic, multi-channel ERP simulator has been developed specifically for this project.Numerical experiments on synthetic ERP data showed that Bayesian Optimal Combining of trial data filtered using a combination of PCA projection and Kalman Filtering, yielded the best estimates of the underlying ERP signal. This method has been applied to subsets of real Ambulatory Electroencephalography (AEEG) data, recorded while participants performed a range of activities in different environments. From this analysis, the number of trials that need to be collected to observe the P300 amplitude and delay has been calculated for a range of scenarios

Boosting the sensitivity of continuous gravitational waves all-sky searches using advanced filtering techniques

The work presented in this PhD thesis has been done in the context of gravitational-waves searches. Since the first detection on the 14th September 2015 by the LIGO-Virgo collaboration, a growing number of gravitational-wave events has been detected, all emitted by the coalescence of binary systems involving black holes and/or neutron stars. My work is focused on the search for continuous gravitational waves, which still miss the first detection. These signals are expected to be emitted, for instance, by spinning neutron stars with an asymmetric shape with respect to the rotation axis, and are at least five orders of magnitude weaker than the typical amplitude of detected binary coalescences. In this PhD thesis I report on the work done in four different projects, with the common purpose of increasing the sensitivity of continuous-wave searches, involving both data analysis and instrumental aspects. The first project is a contribution to the commissioning of the Virgo interferometer in view of the next observing run, O4, which will start in May 2023. My contribution has been mainly devoted to the noise hunting activity, focused on the identification and mitigation of instrumental-noise sources that can degrade the sensitivity of continuous-wave searches. The other three projects are related to data analysis. I have focused, in particular, on all-sky searches for sources without electromagnetic counterpart and long-lasting signals from rapidly evolving newly-born neutron stars. I have studied in great detail the robustness of an all-sky data analysis method in the case of overlapping signals. This is relevant for some exotic classes of continuous wave sources and, more generally, in view of third generation detectors, like Einstein Telescope. I have developed a two-dimensional filter, called triangular filter, to be applied to the search for long-lasting gravitational waves from unstable neutron stars, showing that thanks to this method an increase of the search sensitivity of about $20\%$ is achievable. Finally, I describe the first steps of a wide work to develop a new procedure for all-sky continuous-wave searches, exploiting a statistics based on the sidereal modulation, that affects astrophysical signals, due to the Earth rotation

Changement de masse des glaciers à l’échelle mondiale par analyse spatiotemporelle de modèles numériques de terrain

Les glaciers de la planète rétrécissent rapidement, et produisent des impacts qui s'étendent de la hausse du niveau de la mer et la modification des risques cryosphériques jusqu'au changement de disponibilité en eau douce. Malgré des avancées significatives durant l'ère satellitaire, l'observation des changements de masse des glaciers est encore entravée par une couverture partielle des estimations de télédétection, et par une faible contrainte sur les erreurs des évaluations associées. Dans cette thèse, nous présentons une estimation mondiale et résolue des changements de masse des glaciers basée sur l'analyse spatio-temporelle de modèles numériques de terrain. Nous développons d'abord des méthodes de statistiques spatio-temporelles pour évaluer l'exactitude et la précision des modèles numériques de terrain, et pour estimer des séries temporelles de l'altitude de surface des glaciers. En particulier, nous introduisons un cadre spatial non stationnaire pour estimer et propager des corrélations spatiales multi-échelles dans les incertitudes d'estimations géospatiales. Nous générons ensuite des modèles numériques de terrain massivement à partir de deux décennies d'archives d'images optiques stéréo couvrant les glaciers du monde entier. À partir de ceux-ci, nous estimons des séries temporelles d'altitude de surface pour tous les glaciers de la Terre à une résolution de 100,m sur la période 2000--2019. En intégrant ces séries temporelles en changements de volume et de masse, nous révélons une accélération significative de la perte de masse des glaciers à l'échelle mondiale, ainsi que des réponses régionalement distinctes qui reflètent des changements décennaux de conditions climatiques. En utilisant une grande quantité de données indépendantes et de haute précision, nous démontrons la validité de notre analyse pour produire des incertitudes robustes et cohérentes à différentes échelles de la structure spatio-temporelle de nos estimations. Nous espérons que nos méthodes favorisent des analyses spatio-temporelles robustes, en particulier pour identifier les sources de biais et d'incertitudes dans les études géospatiales. En outre, nous nous attendons à ce que nos estimations permettent de mieux comprendre les facteurs qui régissent le changement des glaciers et d'étendre nos capacités de prévision de ces changements à toutes échelles. Ces prédictions sont nécessaires à la conception de politiques adaptatives sur l'atténuation des impacts de la cryosphère dans le contexte du changement climatique.The world's glaciers are shrinking rapidly, with impacts ranging from global sea-level rise and changes in freshwater availability to the alteration of cryospheric hazards. Despite significant advances during the satellite era, the monitoring of the mass changes of glaciers is still hampered by a fragmented coverage of remote sensing estimations and a poor constraint of the errors in related assessments. In this thesis, we present a globally complete and resolved estimate of glacier mass changes by spatiotemporal analysis of digital elevation models. We first develop methods based on spatiotemporal statistics to assess the accuracy and precision of digital elevation models, and to estimate time series of glacier surface elevation. In particular, we introduce a non-stationary spatial framework to estimate and propagate multi-scale spatial correlations in uncertainties of geospatial estimates. We then massively generate digital elevation models from two decades of stereo optical archives covering glaciers worldwide. From those, we estimate time series of surface elevation for all of Earth's glaciers at a resolution of 100,m during 2000--2019. Integrating these time series into volume and mass changes, we identify a significant acceleration of global glacier mass loss, as well as regionally-contrasted responses that mirror decadal changes in climatic conditions. Using a large amount of independent, high-precision data, we demonstrate the validity of our analysis to yield robust and consistent uncertainties at different scales of the spatiotemporal structure of our estimates. We expect our methods to foster robust spatiotemporal analyses, in particular to identify sources of biases and uncertainties in geospatial assessments. Furthermore, we anticipate our estimates to advance the understanding of the drivers that govern glacier change, and to extend our capabilities of predicting these changes at all scales. Such predictions are critically needed to design adaptive policies on the mitigation of cryospheric impacts in the context of climate change