Sigma-point particle filter for parameter estimation in a multiplicative noise environment

A pre-requisite for the "optimal estimate'' by the ensemble-based Kalman filter (EnKF) is the Gaussian assumption for background and observation errors, which is often violated when the errors are multiplicative, even for a linear system. This study first explores the challenge of the multiplicative noise to the current EnKF schemes. Then, a Sigma Point Kalman Filter based Particle Filter (SPPF) is presented as an alternative to solve the issues associated with multiplicative noise. The classic Lorenz '63 model and a higher dimensional Lorenz '96 model are used as test beds for the data assimilation experiments. Performance of the SPPF algorithm is compared against a standard EnKF as well as an advanced square-root Sigma-Point Kalman Filters (SPKF). The results show that the SPPF outperforms the EnKF and the square-root SPKF in the presence of multiplicative noise. The super ensemble structure of the SPPF makes it computationally attractive compared to the standard Particle Filter (PF)

Konzept der Glückseligkeit in Summa Theologiae in Bezug auf zeitgenössische psychologische Studien

Thomas Aquinas (1224-1274)1 is one of the greatest thinkers in Western philosophy and theology. He is even called the angelic Doctor by the Roman Catholic Church. One of his marvelous works is Summa Theologiae which is the primary source of my research to understand the concept of happiness in Summa Theologiae with reference to contemporary psychological studies. Aquinas was a Dominican monk, philosopher, theologian, saint and contemplator, and continues to be important and significant in particular for students of philosophy and theology. Pope Paul VI says that `in order that the students may illumine the mysteries of salvation as completely as possible, they should learn to penetrate them more deeply with the help of speculation, under the guidance of Aquinas, and to perceive their interconnections´. It is also concern of the church regarding studies in schools and universities that by their very constitution individual subjects be pursued according to their own principles, method, and liberty of scientific inquiry, in a way that an ever deeper understanding in these fields may be obtained and that, as questions that are new and current are raised and investigations carefully made according to the example of the doctors of the Church and especially of Aquinas, so that there may be a deeper realization of the harmony of faith and reason.3 Aquinas constructs a vast system of integrating Greek philosophy with the Christian faith in his masterpiece Summa Theologiae. The encounter with the philosophy of Aristotle opened up a new perspective for Aquinas´s synthesis and distinction between philosophy and theology. This was different from the way of the Fathers of the Church because they were confronted by different philosophies of a platonic type in order to get a complete vision of world and of human life including religion; they mainly used Platonism in the light of faith to respond to question of human being whereas Aquinas convincingly explained Aristotelian works. In one of the general audiences pope Benedict XVI explained the relevance of Aquinas who explored the relation of reason and faith in the philosophy of Aristotle and explained it convincingly. A "philosophy" existed that was complete and convincing in itself, a rationality that preceded the faith, followed by "theology", a form of thinking with the faith and in the faith. The pressing question was this: are the world of rationality, philosophy conceived of without Christ, and the world of faith compatible? Or are they mutually exclusive? Elements that affirmed the incompatibility of these two worlds were not lacking, but St Thomas was firmly convinced of their compatibility indeed that philosophy worked out without the knowledge of Christ was awaiting, as it were, the light of Jesus to be complete. This was the great "surprise" of St Thomas that determined the path he took as a thinker. Showing this independence of philosophy and theology and, at the same time, their reciprocal relationality was the historic mission of the great teacher. And thus it can be understood that in the 19th century, when the incompatibility of modern reason and faith was strongly declared, Pope Leo XIII pointed to St Thomas as a guide in the dialogue between them. In his theological work, St Thomas supposes and concretizes this relationality. Faith consolidates, integrates and illumines the heritage of truth that human reason acquires. The trust with which St Thomas endows these two instruments of knowledge faith and reason may be traced back to the conviction that both stem from the one source of all truth, the divine Logos, which is active in both contexts, that of creation and that of redemption. According to Aquinas sacred doctrine is a science. However he says that there are two kinds of sciences. One proceeds from a principle known by the natural light of intelligence, such as arithmetic and geometry and the like. Second kind proceeds from principles known by the light of a higher science. Thus the science of perspective proceeds from principles established by geometry, and music from principles established by arithmetic. Sacred doctrine is a science because it proceeds from principles established by the light of a higher science, the science of God and the blessed. According to Pope Benedict XVI Aquinas presents to us a broad and confident concept of human reason: broad because it is not limited to the spaces of the empirical-scientific reason but open to the whole human being, to the fundamental and inalienable questions of human life; and confident because human reason, especially if it accepts the inspirations of Christian faith, is a promoter of a civilization that recognizes the dignity of the person, the intangibility of rights and the cogency of duties. The most perfect in all nature to be found is a subsistent individual of a rational nature´. Aquinas wrote commentaries on the works of Aristotle to make sense of Aristotle´s philosophy, and not to set out a philosophy of his own. The appreciation of his outstanding value as a philosopher depends on seeing his ostensibly theological works as also fundamentally philosophical and an extrapolation of Aristotle´s view in the light of catholic theology, and from his own contemplation of truth. Under the guidance of his enlightening thoughts, and following his spirit I just try to recreate Aquinas in our times. My major concern is the theory of happiness, and I have restricted myself to explore and understand Aquinas´s view based on Summa Theologiae and psychological studies. Before he begins treatise on human being, he discusses the existence of God, substance of Angels and the work of creation; these three treatises consist of 74 questions, and lay a strong foundation for treatise of human being

Spiritual advancements in the educational perspective of pastoral community development with reference to Vincent de Paul and Thomas Aquinas

Spiritual pedagogy implies commitment to create spiritual interests and habits of spiritual exercises and thus enlarge the catholic culture of living the faith in Jesus Christ. Participating actively in the pedagogical process of spiritual activity will develop a practical understanding of theological precepts and, as a result, both pastoral leaders and the faithful will profit from the act of preaching, teaching, and on-going formation. There exists in the human person a capacity of restructuring the method of their thinking and developing a suitable attitude towards spiritual commitments. This spiritual commitment of people who are involved in the field of pastoral care in the name of Jesus Christ, creates an atmosphere in and around them for the process of learning and transformation. Christology remains at the center of catholic spiritual pedagogy and, I would even say that the climax of Christian spiritual pedagogy is Christology

Nonlinear measurement function in the ensemble Kalman filter

The optimal Kalman gain was analyzed in a rigorous statistical framework. Emphasis was placed on a comprehensive understanding and interpretation of the current algorithm, especially when the measurement function is nonlinear. It is argued that when the measurement function is nonlinear, the current ensemble Kalman Filter algorithm seems to contain implicit assumptions: the forecast of the measurement function is unbiased or the nonlinear measurement function is linearized. While the forecast of the model state is assumed to be unbiased, the two assumptions are actually equivalent. On the above basis, we present two modified Kalman gain algorithms. Compared to the current Kalman gain algorithm, the modified ones remove the above assumptions, thereby leading to smaller estimated errors. This outcome was confirmed experimentally, in which we used the simple Lorenz 3-component model as the test-bed. It was found that in such a simple nonlinear dynamical system, the modified Kalman gain can perform better than the current one. However, the application of the modified schemes to realistic models involving nonlinear measurement functions needs to be further investigated

Sigma-Point Particle Filter for Parameter Estimation in a Multiplicative Noise Environment

A pre-requisite for the “optimal estimate” by the ensemble-based Kalman filter (EnKF) is the Gaussian assumption for background and observation errors, which is often violated when the errors are multiplicative, even for a linear system. This study first explores the challenge of the multiplicative noise to the current EnKF schemes. Then, a Sigma Point Kalman Filter based Particle Filter (SPPF) is presented as an alternative to solve the issues associated with multiplicative noise. The classic Lorenz '63 model and a higher dimensional Lorenz '96 model are used as test beds for the data assimilation experiments. Performance of the SPPF algorithm is compared against a standard EnKF as well as an advanced square-root Sigma-Point Kalman Filters (SPKF). The results show that the SPPF outperforms the EnKF and the square-root SPKF in the presence of multiplicative noise. The super ensemble structure of the SPPF makes it computationally attractive compared to the standard Particle Filter (PF)

Satellite-Observed Soil Moisture as an Indicator of Wildfire Risk

Wildfires are a concerning issue in Canada due to their immediate impact on people’s lives, local economy, climate, and environment. Studies have shown that the number of wildfires and affected areas in Canada has increased during recent decades and is a result of a warming and drying climate. Therefore, identifying potential wildfire risk areas is increasingly an important aspect of wildfire management. The purpose of this study is to investigate if remotely sensed soil moisture products from the Soil Moisture and Ocean Salinity (SMOS) satellite can be used to identify potential wildfire risk areas for better wildfire management. We used the National Fire Database (NFDB) fire points and polygons to group the wildfires according to ecozone classifications, as well as to analyze the SMOS soil moisture data over the wildfire areas, between 2010–2017, across fourteen ecozones in Canada. Timeseries of 3-day, 5-day, and 7-day soil moisture anomalies prior to the onset of each wildfire occurrence were examined over the ecozones individually. Overall, the results suggest, despite the coarse-resolution, SMOS soil moisture products are potentially useful in identifying soil moisture anomalies where wildfire hot-spots may occur

Comparing the Assimilation of SMOS Brightness Temperatures and Soil Moisture Products on Hydrological Simulation in the Canadian Land Surface Scheme

Soil moisture is a key variable used to describe water and energy exchanges at the land surface/atmosphere interface. Therefore, there is widespread interest in the use of soil moisture retrievals from passive microwave satellites. In the assimilation of satellite soil moisture data into land surface models, two approaches are commonly used. In the first approach brightness temperature (TB) data are assimilated, while in the second approach retrieved soil moisture (SM) data from the satellite are assimilated. However, there is not a significant body of literature comparing the differences between these two approaches, and it is not known whether there is any advantage in using a particular approach over the other. In this study, TB and SM L2 retrieval products from the Soil Moisture and Ocean Salinity (SMOS) satellite are assimilated into the Canadian Land Surface Scheme (CLASS), for improved soil moisture estimation over an agricultural region in Saskatchewan. CLASS is the land surface component of the Canadian Earth System Model (CESM), and the Canadian Seasonal and Interannual Prediction System (CanSIPS). Our results indicated that assimilating the SMOS products improved the soil moisture simulation skill of the CLASS. Near surface soil moisture assimilation also resulted in improved forecasts of root zone soil moisture (RZSM) values. Although both techniques resulted in improved forecasts of RZSM, assimilation of TB resulted in the superior estimates