Application of the Non-Hermitian Singular Spectrum Analysis to the Exponential Retrieval Problem

Introduction. In practical signal processing and its many applications, researchers and engineers try to find a number of harmonics and their frequencies in a time signal contaminated by noise. In this manuscript we propose a new approach to this problem. Aim. The main goal of this work is to embed the original time series into a set of multi-dimensional information vectors and then use shift-invariance properties of the exponentials. The information vectors are cast into a new basis where the exponentials could be separated from each other. Materials and methods. We derive a stable technique based on the singular value decomposition (SVD) of lagcovariance and cross-covariance matrices consisting of covariance coefficients computed for index translated copies of an original time series. For these matrices a generalized eigenvalue problem is solved. Results. The original time series is mapped into the basis of the generalized eigenvectors and then separated into components. The phase portrait of each component is analyzed by a pattern recognition technique to distinguish between the phase portraits related to exponentials constituting the signal and the noise. A component related to the exponential has a regular structure, its phase portrait resembles a unitary circle/arc. Any commonly used method could be then used to evaluate the frequency associated with the exponential. Conclusion. Efficiency of the proposed and existing methods is compared on the set of examples, including the white Gaussian and auto-regressive model noise. One of the significant benefits of the proposed approach is a way to distinguish false and true frequency estimates by the pattern recognition. Some automatization of the pattern recognition is completed by discarding noise-related components, associated with the eigenvectors that have a modulus less than a certain threshold.Introduction. In practical signal processing and its many applications, researchers and engineers try to find a number of harmonics and their frequencies in a time signal contaminated by noise. In this manuscript we propose a new approach to this problem. Aim. The main goal of this work is to embed the original time series into a set of multi-dimensional information vectors and then use shift-invariance properties of the exponentials. The information vectors are cast into a new basis where the exponentials could be separated from each other. Materials and methods. We derive a stable technique based on the singular value decomposition (SVD) of lagcovariance and cross-covariance matrices consisting of covariance coefficients computed for index translated copies of an original time series. For these matrices a generalized eigenvalue problem is solved. Results. The original time series is mapped into the basis of the generalized eigenvectors and then separated into components. The phase portrait of each component is analyzed by a pattern recognition technique to distinguish between the phase portraits related to exponentials constituting the signal and the noise. A component related to the exponential has a regular structure, its phase portrait resembles a unitary circle/arc. Any commonly used method could be then used to evaluate the frequency associated with the exponential. Conclusion. Efficiency of the proposed and existing methods is compared on the set of examples, including the white Gaussian and auto-regressive model noise. One of the significant benefits of the proposed approach is a way to distinguish false and true frequency estimates by the pattern recognition. Some automatization of the pattern recognition is completed by discarding noise-related components, associated with the eigenvectors that have a modulus less than a certain threshold

Numerical Modeling Of Seasonally Freezing Ground And Permafrost

Thesis (Ph.D.) University of Alaska Fairbanks, 2007This thesis represents a collection of papers on numerical modeling of permafrost and seasonally freezing ground dynamics. An important problem in numerical modeling of temperature dynamics in permafrost and seasonally freezing ground is related to parametrization of already existing models. In this thesis, a variation data assimilation technique is presented to find soil properties by minimizing the discrepancy between in-situ measured temperatures and those computed by the models. The iterative minimization starts from an initial approximation of the soil properties that are found by solving a sequence of simple subproblems. In order to compute the discrepancy, the temperature dynamics is simulated by a new implementation of the finite element method applied to the heat equation with phase change. Despite simplifications in soil physics, the presented technique was successfully applied to recover soil properties, such as thermal conductivity, soil porosity, and the unfrozen water content, at several sites in Alaska. The recovered properties are used in discussion on soil freezing/thawing and permafrost dynamics in other parts of this thesis. Another part of this thesis concerns development of a numerical thermo-mechanical model of seasonal soil freezing on the lateral scale of several meters. The presented model explains observed differential frost heave occurring in non-sorted circle ecosystems north of the Brooks Range in the Alaskan tundra. The model takes into account conservation principles for energy, linear momentum and mass of three constituents: liquid water, ice and solid particles. The conservation principles are reduced to a computationally convenient system of coupled equations for temperature, liquid water pressure, porosity, and the velocity of soil particles in a three-dimensional domain with cylindrical symmetry. Despite a simplified rheology, the model simulates the ground surface motion, temperature, and water dynamics in soil and explains dependence of the frost heave on specific environmental properties of the ecosystem. In the final part, simulation of the soil temperature dynamics on the global scale is addressed. General Circulation Models are used to understand and predict future climate change, but most of them do not simulate permafrost dynamics and its potentially critical feedback on climate. In this part, a widely used climate model is evaluated and the simulated temperatures are compared against observations. Based on this comparison, several modifications to the Global Circulation Models are identified to improve the fidelity of permafrost and soil temperature simulations. These modifications include increasing the total soil depth by adding new layers, incorporating a surface organic layer, and modifying the numerical scheme to include unfrozen water dynamics

Pedestrian Travel-Time Maps for Homer, Alaska: An anisotropic model to support tsunami evacuation planning

Tsunami-induced pedestrian evacuation for the community of Homer is evaluated using an anisotropic modeling approach developed by the U.S. Geological Survey. The method is based on path-distance algorithms and accounts for variations in land cover and directionality in the slope of terrain. We model evacuation of pedestrians to the tsunami hazard zone boundary and to predetermined assembly areas. The pedestrian travel-time maps are computed for two cases: for travel across all variable terrain or by roads only. Results presented here are intended to provide guidance to local emergency management agencies in tsunami inundation assessment, evacuation planning, and public education to mitigate future tsunami hazards. This report was funded by the National Tsunami Hazard Mitigation Program grant to the Alaska Division of Homeland Security and Emergency Management and University of Alaska Fairbanks from the Department of Commerce/National Oceanic at Atmospheric Administration (NOAA). This does not constitute an endorsement by Alaska Earthquake Center (AEC) or NOAA

Pedestrian Travel-Time Maps for King Cove, Alaska: An anisotropic model to support tsunami evacuation planning

Tsunami-induced pedestrian evacuation for the community of King Cove is evaluated using an anisotropic modeling approach developed by the U.S. Geological Survey. The applied method is based on path-distance algorithms and accounts for variations in land cover and directionality in the slope of terrain. We model evacuation of pedestrians to the tsunami hazard zone boundary and to predetermined assembly areas. The pedestrian travel-time maps are computed for two cases: for travel across all viable terrain or by roads only. Results presented here are intended to provide guidance to local emergency management agencies for tsunami inundation assessment, evacuation planning, and public education to mitigate future tsunami hazards. This report was funded by the National Tsunami Hazard Mitigation Program grant to the Alaska Division of Homeland Security and Emergency Management and University of Alaska Fairbanks from the Department of Commerce/National Oceanic at Atmospheric Administration (NOAA). This does not constitute an endorsement by Alaska Earthquake Center (AEC) or NOAA

Application of the Non-Hermitian Singular Spectrum Analysis to the exponential retrieval problem

We present a new approach to solve the exponential retrieval problem. We derive a stable technique, based on the singular value decomposition (SVD) of lag-covariance and crosscovariance matrices consisting of covariance coefficients computed for index translated copies of an initial time series. For these matrices a generalized eigenvalue problem is solved. The initial signal is mapped into the basis of the generalized eigenvectors and phase portraits are consequently analyzed. Pattern recognition techniques could be applied to distinguish phase portraits related to the exponentials and noise. Each frequency is evaluated by unwrapping phases of the corresponding portrait, detecting potential wrapping events and estimation of the phase slope. Efficiency of the proposed and existing methods is compared on the set of examples, including the white Gaussian and auto-regressive model noise

Pedestrian Travel-Time Maps for Unalaska/Dutch Harbor, Alaska: An anisotropic model to support tsunami evacuation planning

Tsunami-induced pedestrian evacuation for the community of Unalaska/Dutch Harbor is evaluated using an anisotropic modeling approach developed by the U.S. Geological Survey. The method is based on pathdistance algorithms and accounts for variations in land cover and directionality in the slope of terrain. We model evacuation of pedestrians to the tsunami hazard zone boundary and to predetermined assembly areas. Pedestrian travel-time maps are computed for two cases: for travel across all viable terrain or by roads only. Results presented here are intended to provide guidance to local emergency management agencies for tsunami inundation assessment, evacuation planning, and public education to mitigate future tsunami hazards. This report was funded by the National Tsunami Hazard Mitigation Program grant to the Alaska Division of Homeland Security and Emergency Management and University of Alaska Fairbanks from the Department of Commerce/National Oceanic at Atmospheric Administration (NOAA). This does not constitute an endorsement by Alaska Earthquake Center (AEC) or NOAA

Using in-situ temperature measurements to estimate saturated soil thermal properties by solving a sequence of optimization problems

International audienceWe describe an approach to find an initial approximation to the thermal properties of soil horizons. This technique approximates thermal conductivity, porosity, unfrozen water content curves in horizons where no direct temperature measurements are available. To determine physical properties of ground material, optimization-based inverse techniques are employed to fit the simulated temperatures to the measured ones. Two major ingredients of these techniques are an algorithm to compute the soil temperature dynamics and a procedure to find an initial approximation to the ground properties. In this article we show how to determine the initial approximation to the physical properties and present a new finite element discretization of the heat equation with phase change to calculate the temperature dynamics in soil. We successfully apply the proposed algorithm to recover the soil properties for the Happy Valley site in Alaska using one-year temperature dynamics. The determined initial approximation is utilized to simulate the temperature dynamics over several consecutive years; the difference between simulated and measured temperatures lies within uncertainties of measurements

PEDESTRIAN TRAVEL-TIME MAPS FOR WHITTIER, ALASKA: An anisotropic model to support tsunami evacuation planning

Tsunami-induced pedestrian evacuation for the community of Whittier is evaluated using an anisotropic modeling approach developed by the U.S. Geological Survey. The method is based on path-distance algorithms and accounts for variations in land cover and directionality in the slope of terrain. We model evacuation of pedestrians to exit points from the tsunami hazard zone boundary. The pedestrian travel is restricted to the roads only. Results presented here are intended to provide guidance to local emergency management agencies for tsunami inundation assessment, evacuation planning, and public education to mitigate future tsunami hazards