Stochastic Models In Geosystems

This IMA Volume in Mathematics and its Applications STOCHASTIC MODELS IN GEOSYSTEMS is based on the proceedings of a workshop with the same title and was an integral part of the 1993-94 IMA program on "Emerging Applications of Probability." We would like to thank Stanislav A. Molchanov and Wojbor A. Woyczynski for their hard work in organizing this meeting and in edit ing the proceedings. We also take this opportunity to thank the National Science Foundation, the Office of N aval Research, the Army Research Of fice, and the National Security Agency, whose financial support made this workshop possible. A vner Friedman Willard Miller, Jr. v PREFACE A workshop on Stochastic Models in Geosystems was held during the week of May 16, 1994 at the Institute for Mathematics and Its Applica tions at the University of Minnesota. It was part of the Special Year on Emerging Applications of Prob ability program put together by an organiz ing committee chaired by J. Michael Steele. The invited speakers represented a broad interdisciplinary spectrum including mathematics, statistics, physics, geophysics, astrophysics, atmo spheric physics, fluid mechanics, seismology, and oceanography. The com mon underlying theme was stochastic modeling of geophysical phenomena and papers appearing in this volume reflect a number of research directions that are currently pursued in these areas.

This volume contains edited proceedings of a workshop on Stochastic Models in Geosystems held at the Institute for Mathematics and Its Applications at the University of Minnesota. The authors represent a broad interdisciplinary spectrum including mathematics, statistics, physics, geophysics, astrophysics, atmospheric physics, fluid mechanics, seismology, and oceanography.

The study of the ocean is almost as old as the history of mankind itself. When the first seafarers set out in their primitive ships they had to understand, as best they could, tides and currents, eddies and vortices, for lack of understanding often led to loss of live. These primitive oceanographers were, of course, primarily statisticians. They collected what empirical data they could, and passed it down, ini tially by word of mouth, to their descendants. Data collection continued throughout the millenia, and although data bases became larger, more re liable, and better codified, it was not really until surprisingly recently that mankind began to try to understand the physics behind these data, and, shortly afterwards, to attempt to model it. The basic modelling tool of physical oceanography is, today, the partial differential equation. Somehow, we all 'know" that if only we could find the right set of equations, with the right initial and boundary conditions, then we could solve the mysteries of ocean dynamics once and for all.

A collection of articles written by mathematicians and physicists, designed to describe the state of the art in climate models with stochastic input. Mathematicians will benefit from a survey of simple models, while physicists will encounter mathematically relevant techniques at work.

The recent emergence and widespread use of remote sensing and geographic information systems (GIS) has prompted new interest in scale as a key component of these and other geographic information technologies. With a balanced mixture of concepts, practical examples, techniques, and theory, Scale in Remote Sensing and GIS is a guide for students and users of remote sensing and GIS who must deal with the issues raised by multiple temporal and spatial scales. Sixteen pages of full-color photographs help demonstrate key points made in the text.

Fluctuating parameters appear in a variety of physical systems and phenomena. They typically come either as random forces/sources, or advecting velocities, or media (material) parameters, like refraction index, conductivity, diffusivity, etc. The well known example of Brownian particle suspended in fluid and subjected to random molecular bombardment laid the foundation for modern stochastic calculus and statistical physics. Other important examples include turbulent transport and diffusion of particle-tracers (pollutants), or continuous densities (''oil slicks''), wave propagation and scattering in randomly inhomogeneous media, for instance light or sound propagating in the turbulent atmosphere. Such models naturally render to statistical description, where the input parameters and solutions are expressed by random processes and fields. The fundamental problem of stochastic dynamics is to identify the essential characteristics of system (its state and evolution), and relate those to the input parameters of the system and initial data. This raises a host of challenging mathematical issues. One could rarely solve such systems exactly (or approximately) in a closed analytic form, and their solutions depend in a complicated implicit manner on the initial-boundary data, forcing and system's (media) parameters . In mathematical terms such solution becomes a complicated "nonlinear functional" of random fields and processes. Part I gives mathematical formulation for the basic physical models of transport, diffusion, propagation and develops some analytic tools. Part II and III sets up and applies the techniques of variational calculus and stochastic analysis, like Fokker-Plank equation to those models, to produce exact or approximate solutions, or in worst case numeric procedures. The exposition is motivated and demonstrated with numerous examples. Part IV takes up issues for the coherent phenomena in stochastic dynamical systems, described by ordinary and partial differential equations, like wave propagation in randomly layered media (localization), turbulent advection of passive tracers (clustering), wave propagation in disordered 2D and 3D media. For the sake of reader I provide several appendixes (Part V) that give many technical mathematical details needed in the book. For scientists dealing with stochastic dynamic systems in different areas, such as hydrodynamics, acoustics, radio wave physics, theoretical and mathematical physics, and applied mathematics The theory of stochastic in terms of the functional analysis Referencing those papers, which are used or discussed in this book and also recent review papers with extensive bibliography on the subject

Fluctuating parameters appear in a variety of physical systems and phenomena. They typically come either as random forces/sources, or advecting velocities, or media (material) parameters, like refraction index, conductivity, diffusivity, etc. The well known example of Brownian particle suspended in fluid and subjected to random molecular bombardment laid the foundation for modern stochastic calculus and statistical physics. Other important examples include turbulent transport and diffusion of particle-tracers (pollutants), or continuous densities (''oil slicks''), wave propagation and scattering in randomly inhomogeneous media, for instance light or sound propagating in the turbulent atmosphere. Such models naturally render to statistical description, where the input parameters and solutions are expressed by random processes and fields. The fundamental problem of stochastic dynamics is to identify the essential characteristics of system (its state and evolution), and relate those to the input parameters of the system and initial data. This raises a host of challenging mathematical issues. One could rarely solve such systems exactly (or approximately) in a closed analytic form, and their solutions depend in a complicated implicit manner on the initial-boundary data, forcing and system's (media) parameters . In mathematical terms such solution becomes a complicated "nonlinear functional" of random fields and processes. Part I gives mathematical formulation for the basic physical models of transport, diffusion, propagation and develops some analytic tools. Part II sets up and applies the techniques of variational calculus and stochastic analysis, like Fokker-Plank equation to those models, to produce exact or approximate solutions, or in worst case numeric procedures. The exposition is motivated and demonstrated with numerous examples. Part III takes up issues for the coherent phenomena in stochastic dynamical systems, described by ordinary and partial differential equations, like wave propagation in randomly layered media (localization), turbulent advection of passive tracers (clustering). Each chapter is appended with problems the reader to solve by himself (herself), which will be a good training for independent investigations. · This book is translation from Russian and is completed with new principal results of recent research. · The book develops mathematical tools of stochastic analysis, and applies them to a wide range of physical models of particles, fluids, and waves. · Accessible to a broad audience with general background in mathematical physics, but no special expertise in stochastic analysis, wave propagation or turbulence

This volume contains a selection of papers on recent developments in fields such as stochastic processes, multivariate data analysis and stochastic models in operations research, earth and life sciences and information theory, from an applicative perspective. Some of them have been extracted from lectures given at the Department of Statistics and Operations Research at the University of Granada for the past two years (Kai Lai Chung and Marcel F Neuts, among others). All the papers have been carefully selected and revised.