Finite-dimensional filters with nonlinear drift. V: solution to Kolmogorov equation arising from linear filtering with non-Gaussian initial condition

Despite its usefulness, the Kalman-Bucy filter is not perfect. One of its weaknesses is that it needs a Gaussian assumption on the initial data. Recently Yau and Yau introduced a new direct method to solve the estimation problem for linear filtering with non-Gaussian initial data. They factored the problem into two parts: (1) the on-line solution of a finite system of ordinary differential equations (ODEs), and (2) the off-line calculation of the Kolmogorov equation. Here we derive an explicit closed-form solution of the Kolmogorov equation. We also give some properties and conduct a numerical study of the solution.     

Finite-dimensional filters with nonlinear drift. III: Duncan-Mortensen-Zakai equation with arbitrary initial condition for the linear filtering system and the Benes filtering system

We consider the Duncan-Mortensen-Zakai (DMZ) equation for the Kalman-Bucy filtering system and Benes filtering system. We show that this equation can be solved explicitly with an arbitrary initial condition by solving a system of ordinary differential equations and a Kolmogorov-type equation, Let n be the dimension of state space. We show that we need only n sufficient statistics in order to solve the DMZ equation.     

Source processes in the high-latitude ionosphere

Space Science Reviews -     

Theories and Observations of Ion Energization and Outflow in the High Latitude Magnetosphere

A review is given of several mechanisms causing outflow at high latitudes of ionospheric ions to the terrestrial magnetosphere. The upward ion motion along the geomaagnetic field can be divided into several categories, including polar wind, bulk ion outflow in the auroral region, upwelling ions and ion conics and beams. More than one ion energization mechanism can be operating within each category, and a combination of categories is important for the total ion outflow.     

Finite-dimensional filters with nonlinear drift. XV. New directmethod for construction of universal finite-dimensional filter

We consider the explicit solution of Duncan-Mortensen-Zakai (DMZ) equation for the finite-dimensional filtering system. We show that under certain conditions, the nonlinear filtering system can be solved explicitly with an arbitrary initial condition by solving a system of ordinary differential equations and a Kolmogorov-type equation. Let n be the dimension of state space. We show that we need only n sufficient statistics in order to solve the DMZ equation     

Nonlinear filtering and time varying Schrodinger equation

Based on our previous work we have successfully reduced the nonlinear filtering problem for Yau filtering system to the time-varying Schrodinger equation. In order to solve the nonlinear filtering problem, one needs to solve the time-varying Schrodinger equation with an arbitrary initial condition. We then solve the time-varying Schrodinger equation by constructing the fundamental solution explicitly via a system of nonlinear ODES in case the potential is quadratic in state variables. This system of nonlinear ODES is solved explicitly by the power series method.     

Sources of Ion Outflow in the High Latitude Ionosphere

Ion composition observations from polar-orbiting satellites in the past three decades have revealed and confirmed the occurrence of a variety of ion outflow processes in the high-latitude ionosphere. These processes constitute a dominant source of ionospheric plasma to the Earth's magnetosphere. We review the current state of our observational knowledge on their occurrence, energy, composition, variability, interrelationships, and quantitative contributions to the overall mass input to the magnetosphere. In addition, we identify the prevalent sources and the gaps of our current understanding of these sources.     

The Freja F3C Cold Plasma Analyzer

The F3C Cold Plasma Analyzer (CPA) instrument on theFreja spacecraft is designed to measure the energy per unit charge (E/Q) of ions oe electrons in the range 0<E/Q<200 V and complements the observations made by the F3H Hot Plasma Experiment. The CPA sensor, which is deployed on a boom, is an electrostatic analyzer which produces angle/energy images of particles incident on the sensor in a plane perpendicular to the boom axis. Charged particles incident normal to the CPA sensor housing axis of symmetry, which coincides with the boom axis, pass through collimators and enter a semi-spherical electrostatic analyzer which disperses particles in energy and azimuthal angle of arrival onto an imaging MCP detector thus producing images of the particle distributions in a plane perpendicular to the boom axis. Measurements are transmitted either as discrete 16×16 (angle/energy) images or as parameters related to the incident particle distribution function. Pixels in the discrete images are separated approximately equally in azimuthal angle while the 16 energy bins are separated approximately geometrically in energy. The ratio of the maximum to minimum energy imaged is programmable up to a maximum of more than a factor of ten, and the energy range itself is also under the control of the processor and can be varied by more than an order of magnitude. The density dynamic range of the sensor is increased by the introduction of an electrostatic gating system between the entrance aperture and the analyzer which can be used to duty-cycle low-energy electrons into the sensor thus keeping the count rate within appropriate levels. To reduce the effects of spacecraft induced perturbations on the lower-energy particle distributions, the sensor portion of the instrument is deployed on a 2 m long boom, perpendicular to the spacecraft spin axis. Spacecraft rotation is used to recover complete (4) angle/energy distributions every half spin period. In addition, the sensor skin may be biased with respect to the spacecraft ground to offset effects due to spacecraft charging. Current to the skin is monitored, making the exterior of the sensor equivalent to a large cylindrical Langmuir probe. Two separate processing paths for signals from the MCP anode may be chosen; slow and rast. The slow pulse processing path provides discrete angle/energy images at a nominal rate of 10 images per second and a peak burst mode rate of 100 images per second. The fast analog or current mode path provides crude parameterized estimates of densities, temperatures and drift velocities at nominal rates of up to 1000 parameters per second with a burst rate near 6000 parameters per second. Observations of cold ions and electrons in an unperturbed ionospheric plasma are presented which demonstrate the functionality of the instrument. Suprathermal ion observations in a transverse ion energization or acceleration region are also shown which demonstrate many of the small-scale features of these events.The Canadian Government's right to retain a non-exclusive, royalty free licence in and to any copyright is acknowledge.     

Ptolemy – an Instrument to Measure Stable Isotopic Ratios of Key Volatiles on a Cometary Nucleus

A fundamental goal of cometary studies is to determine the exact relationship between these bodies and the Solar System – the question(s) can be summarised as follows: did comets originate during the same events that spawned the Sun and planets, are they more primitive bodies that record a pre-solar history, or are they interstellar materials collected in relatively more recent times? Now, whatever the origin of comets, it is entirely possible that they could, in part, contain interstellar or pre-solar components – indeed, it seems rather likely in light of the fact that primitive meteorites contain such entities. These particular components are likely to be refractory (dust, macromolecular organic complexes, etc.). Of more relevance to the issues above are the volatile constituents, which make up the bulk of a comet's mass. Since these materials, by their very nature, volatilise during perihelion passage of a comet they can, in some instances, be detected and measured spectroscopically. Perhaps the most useful species for isotopic investigations are C₂, HCN and CN. Unfortunately, spectroscopic measurements can only currently be made with accuracies of ±10 to ±20%. As such it is very often not practical to conclude anything further than the fact that isotopic measurements are compatible with ‘`solar’' values, which tends to imply an origin from the margins of the solar accretion disk. But there is another problem with the spectroscopic measurements – since these are made on gaseous species in the coma (and relatively minor species at that) it is impossible to be certain that these represent the true nuclear values. In other words, if the processes of sublimation, active jetting, and photochemistry in the coma impart isotopic fractionation, the spectroscopic measurements could give a false impression of the true isotope ratios. What is required is an experiment capable of measuring isotopic ratios at the very surface of a comet. Herein we describe the Ptolemy instrument, which is included on the Philae lander as part of the Rosetta mission to 67P/Churyumov-Gerasimenko. The major objective of Ptolemy is a detailed appraisal of the nature and isotopic compositions of all materials present at the surface of a comet.