Optimal Modeling and Filtering of Stochastic Time Series for Geoscience Applications

Sequences of observations or measurements are often modeled as realizations of stochastic processes with some stationary properties in the first and second moments. However in practice, the noise biases and variances are likely to be different for different epochs in time or regions in space, and hence such stationarity assumptions are often questionable. In the case of strict stationarity with equally spaced data, the Wiener-Hopf equations can readily be solved with fast Fourier transforms (FFTs) with optimal computational efficiency. In more general contexts, covariance matrices can also be diagonalized using the Karhunen-Loève transforms (KLTs), or more generally using empirical orthogonal and biorthogonal expansions, which are unfortunately much more demanding in terms of computational efforts. In cases with increment stationarity, the mathematical modeling can be modified and generalized covariances can be used with some computational advantages. The general nonlinear solution methodology is also briefly overviewed with the practical limitations. These different formulations are discussed with special emphasis on the spectral properties of covariance matrices and illustrated with some numerical examples. General recommendations are included for practical geoscience applications

Least Squares for Practitioners

In experimental science and engineering, least squares are ubiquitous in analysis and digital data processing applications. Minimizing sums of squares of some quantities can be interpreted in very different ways and confusion can arise in practice, especially concerning the optimality and reliability of the results. Interpretations of least squares in terms of norms and likelihoods need to be considered to provide guidelines for general users. Assuming minimal prerequisites, the following expository discussion is intended to elaborate on some of the mathematical characteristics of the least-squares methodology and some closely related questions in the analysis of the results, model identification, and reliability for practical applications. Examples of simple applications are included to illustrate some of the advantages, disadvantages, and limitations of least squares in practice. Concluding remarks summarize the situation and provide some indications of practical areas of current research and development

Optimal Modeling and Filtering of Stochastic Time Series for Geoscience Applications

Sequences of observations or measurements are often modeled as realizations of stochastic processes with some stationary properties in the first and second moments. However in practice, the noise biases and variances are likely to be different for different epochs in time or regions in space, and hence such stationarity assumptions are often questionable. In the case of strict stationarity with equally spaced data, the Wiener-Hopf equations can readily be solved with fast Fourier transforms (FFTs) with optimal computational efficiency. In more general contexts, covariance matrices can also be diagonalized using the Karhunen-Loève transforms (KLTs), or more generally using empirical orthogonal and biorthogonal expansions, which are unfortunately much more demanding in terms of computational efforts. In cases with increment stationarity, the mathematical modeling can be modified and generalized covariances can be used with some computational advantages. The general nonlinear solution methodology is also briefly overviewed with the practical limitations. These different formulations are discussed with special emphasis on the spectral properties of covariance matrices and illustrated with some numerical examples. General recommendations are included for practical geoscience applications

Discrete Spherical Harmonic Transforms for Equiangular Grids of Spatial and Spectral Data

Article deposited according to Versita Open Access policy for Journal of Geodetic Science: http://www.degruyter.com/view/supplement/s20819943_Versita_Open_Access_License.pdf [July 24, 2013]Ye

Optimal Data Structures for Spherical Multiresolution Analysis and Synthesis

Ye

Some Reflections on Advanced Geocomputations and the Data Deluge

Ye

Randomness Characterization in Computing and Stochastic Simulations

Ye

Least Squares for Practitioners

In experimental science and engineering, least squares are ubiquitous in analysis and digital data processing applications. Minimizing sums of squares of some quantities can be interpreted in very different ways and confusion can arise in practice, especially concerning the optimality and reliability of the results. Interpretations of least squares in terms of norms and likelihoods need to be considered to provide guidelines for general users. Assuming minimal prerequisites, the following expository discussion is intended to elaborate on some of the mathematical characteristics of the least-squares methodology and some closely related questions in the analysis of the results, model identification, and reliability for practical applications. Examples of simple applications are included to illustrate some of the advantages, disadvantages, and limitations of least squares in practice. Concluding remarks summarize the situation and provide some indications of practical areas of current research and development.Peer Reviewe

Optimal Modeling and Filtering of Stochastic Time Series for Geoscience Applications

Sequences of observations or measurements are often modeled as realizations of stochastic processes with some stationary properties in the first and second moments. However in practice, the noise biases and variances are likely to be different for different epochs in time or regions in space, and hence such stationarity assumptions are often questionable. In the case of strict stationarity with equally spaced data, the Wiener-Hopf equations can readily be solved with fast Fourier transforms (FFTs) with optimal computational efficiency. In more general contexts, covariance matrices can also be diagonalized using the Karhunen-Loève transforms (KLTs), or more generally using empirical orthogonal and biorthogonal expansions, which are unfortunately much more demanding in terms of computational efforts. In cases with increment stationarity, the mathematical modeling can be modified and generalized covariances can be used with some computational advantages. The general nonlinear solution methodology is also briefly overviewed with the practical limitations. These different formulations are discussed with special emphasis on the spectral properties of covariance matrices and illustrated with some numerical examples. General recommendations are included for practical geoscience applications.Peer Reviewe