881 research outputs found
Three-dimensional stochastic modeling of radiation belts in adiabatic invariant coordinates
A 3-D model for solving the radiation belt diffusion equation in adiabatic invariant coordinates has been developed and tested. The model, named REM (for Radbelt Electron Model), obtains a probabilistic solution by solving a set of Itô stochastic differential equations that are mathematically equivalent to the diffusion equation. This method is capable of solving diffusion equations with a full 3-D diffusion tensor, including the radial-local cross diffusion components. The correct form of the boundary condition at equatorial pitch-angle α0 = 90° is also derived. The model is applied to a simulation of the October 2002 storm event. At α0 near 90°, our results are quantitatively consistent with GPS observations of phase-space density (PSD) increases, suggesting dominance of radial diffusion; at smaller α0, the observed PSD increases are overestimated by the model, possibly due to the α0-independent radial diffusion coefficients, or to insufficientelectron loss in the model, or both. Statistical analysis of the stochastic processes provides further insights into the diffusion processes, showing distinctive electron source distributions with and without local acceleration
Estimation of the parameters of a stochastic logistic growth model
We consider a stochastic logistic growth model involving both birth and death
rates in the drift and diffusion coefficients for which extinction eventually
occurs almost surely. The associated complete Fokker-Planck equation describing
the law of the process is established and studied. We then use its solution to
build a likelihood function for the unknown model parameters, when discretely
sampled data is available. The existing estimation methods need adaptation in
order to deal with the extinction problem. We propose such adaptations, based
on the particular form of the Fokker-Planck equation, and we evaluate their
performances with numerical simulations. In the same time, we explore the
identifiability of the parameters which is a crucial problem for the
corresponding deterministic (noise free) model
Learning the temporal evolution of multivariate densities via normalizing flows
In this work, we propose a method to learn multivariate probability
distributions using sample path data from stochastic differential equations.
Specifically, we consider temporally evolving probability distributions (e.g.,
those produced by integrating local or nonlocal Fokker-Planck equations). We
analyze this evolution through machine learning assisted construction of a
time-dependent mapping that takes a reference distribution (say, a Gaussian) to
each and every instance of our evolving distribution. If the reference
distribution is the initial condition of a Fokker-Planck equation, what we
learn is the time-T map of the corresponding solution. Specifically, the
learned map is a multivariate normalizing flow that deforms the support of the
reference density to the support of each and every density snapshot in time. We
demonstrate that this approach can approximate probability density function
evolutions in time from observed sampled data for systems driven by both
Brownian and L\'evy noise. We present examples with two- and three-dimensional,
uni- and multimodal distributions to validate the method
Interplanetary particle transport simulation for warning system for aviation exposure to solar energetic particles
Solar energetic particles (SEPs) are one of the extreme space weather
phenomena. A huge SEP event increases the radiation dose received by aircrews,
who should be warned of such events as early as possible. We developed a
warning system for aviation exposure to SEPs. This article describes one
component of the system, which calculates the temporal evolution of the SEP
intensity and the spectrum immediately outside the terrestrial magnetosphere.
To achieve this, we performed numerical simulations of SEP transport in
interplanetary space, in which interplanetary SEP transport is described by the
focused transport equation. We developed a new simulation code to solve the
equation using a set of stochastic differential equations. In the code, the
focused transport equation is expressed in a magnetic field line coordinate
system, which is a non-orthogonal curvilinear coordinate system. An inverse
Gaussian distribution is employed as the injection profile of SEPs at an inner
boundary located near the Sun. We applied the simulation to observed SEP events
as a validation test. The results show that our simulation can closely
reproduce observational data for the temporal evolution of particle intensity.
By employing the code, we developed the WArning System for AVIation Exposure to
Solar energetic particles (WASAVIES).Comment: 23 pages, 11 figures, accepted for publication in Earth, Planets and
Spac
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