ALBERT

Note: CONTENTS Abstract Preface Introduction Scope of work Ice discharge from Lake Huron into St. Clair River Water velocity profiles at Port Huron Ice conditions Physical model Basis for selection Description Instrumentation Model ice control structure Open water calibration Open water tests Experimental procedures and techniques Ice cover calibration Ice control structure orientation Analysis of ice discharge due to ship transits Natural ice Synthetic ice Forces on the ice control structure Static measurements Dynamic force measurements Potential additional shear stresses Anticipated ice conditions with ICS Conclusions Literature cited Appendix A. Application of model results Appendix B. Suggested additional studies Appendix C. Derivation of ice discharge

Note: CONTENTS Abstract Preface List of symbols Introduction Theoretical models Case 1 Case 2 Case 3 Experimental program Experimental facility Scaling factors Experimental results Analysis of wind data for lower Lake Huron Summary and conclusions Release of ice through the opening of an ICS Ice forces on the ice control structure Ice forces on ice control structure from a large unconsolidated ice cover Literature cited Appendix A. Equation for the stress resultants and velocities of the ice cover Appendix B. Monthly summary of wind data at Port Huron

Note: CONTENTS Abstract Preface Introduction Data collection and processing Analysis General Variations in ridging Ridge height distributions Occurrence of high ridges The tail of the distribution Extreme values Applications to offshore design Conclusions Literature cited Appendix A. Tabulated ice ridge data

Note: Contents Abstract Preface Summary Introduction Previous work Results Models for predicting thickness from ice freeboard Comparison between measured and predicted thicknesses Spectral behavior of measured and predicted profiles Comparisons of ice thickness using airborne laser profilometry Conclusions Literature cited Appendix A: Misgivings on isostatic imbalance as a mechanism for sea ice cracking

Note: CONTENTS Abstract Preface Introduction and statement of problem Review of relevant analyses and tests Determination of the largest ice force on an isolated structure Preliminary remarks The buckling analysis of a floating wedge Proposed method to determine Literature cited

Note: Contents: List of Figures. - List of Tables. - 1. Introduction. - a. An Overview of Principal Component Analysis (PCA). - b. Outline of the Book. - c. A Brief History of PCA. - d. Acknowledgments. - 2. Algebraic Foundations of PCA. - a. Introductory Example: Bivariate Data Sets. - Monterey, California air temperatures. - Centering and rotating the data set. - Variances in the rotated frame. - Principal angles. - Principal variances. - Principal covariance. - Principal directions. - Principal components; principal directions as basis vectors. - Matrix representation. - The PCA property. - Invariance of the total variance under rotation. - Principal variances for standardized data sets. - PCA and estimates of the statistical parameters of normal populations. - PCA and the construction of Monte Carlo experiments. - Eigenvalues and eigenvectors of the covariance and scatter matrices. - b. Principal Component Analysis: Real-valued Scalar Fields. - t-centering the data set. - The scatter probe and the scatter matrix. - The eigenstructures of PCA. - The basic data set representations; analysis and synthesis formulas. - The PCA property. - Second-order properties of PCA; the total scatter . - The singular value decomposition (SVD) of a data set. - Second-order properties of PCA; correlations. - PCA characterized by the PCA property. - The asymptotic PCA property and dynamical systems. - PCA of spatial composites of data sets. - PCA of temporal composites of data sets. - c. Principal Component Analysis: Complex-valued Scalar Fields, and Beyond. - PCA of complex-valued data sets (C-PCA). - Complex algebra conventions. - The scatter probe and scatter matrix for C-PCA. - Derivation of the eigenstructures of C-PCA. - The fundamental formulas of C-PCA. - Generalization of PCA to quaternion-valued data sets (Q-PCA). - Matrix representations of complex and quaternion numbers. - PCA of matrix-valued data sets (M-PCA). - Reduction of M-PCA to C-PCA form. - d. Bibliographic Notes and Miscellaneous Topics. - Alternate interpretation of the scatter probe. - Numerical calculations of eigenstructures of a scatter matrix. - Some elementary properties of eigenstructures of a scatter matrix. - Sample space vs. state space: choosing the dual computation. - PCA for continuous domains. - PCA for continuous domains: the viewpoint of empirical orthogonal functions. - The sixteen possible domain pairs for PCA: abstract PCA. - 3. Dynamical Origins of PCA. - a. One-dimensional Hannonic Motion. - A spring-linked-mass model; general form. - A spring-linked-mass model; special form. - A numerical example of the asymptotic PCA property. - Further investigations of the asymptotic PCA property and of EOF's. - b. Two-dimensional Wave Motion. - Solution of a two-dimensional damped-wave model. - Demonstration of the asymptotic PCA property (forcing and friction absent). - Demonstration of the asymptotic PCA property (forcing and friction present). - Physical basis for eigenframe rotations. - c. Dynamical Origins of Linear Regression (LR). - From continuous to discrete solutions to the regression model. - The linear regression procedure. - Comparison of LRA and PCA. - d. Random Processes and Karhunen-Loeve Analysis. - Origins of random processes in linear settings. - Karhunen-Loeve representation of random data sets and comparison with PCA. - e. Stationary Processes and PCA. - Derivation of the PCA representation of a one-dimensional stationary process via a simple wave model. - Connections between PCA and stationary processes: the case of one dimension. - Connections between PGA and stationary processes: extension to two dimensions. - f. Bibliographic Notes. - 4. Extensions of PCA to Multivariate Fields. - a. Categories of Data and Modes of Analysis. - Examples. - Generalized notation: the concepts of "individual" and "variable" in PCA. - b. Local PCA of a General Vector Field. - The PCA formalism. - Squared correlations. - Variational origin of the scatter matrix. - Examples. - c. Global PCA of a General Vector Field: Time-Modulation Form. - The PGA formalism. - Squared correlations. - Degeneracy of global PGA to local PGA. - Variational origin of the scatter matrix. - d. Global PCA of a General Vector Field: Space-Modulation Form. - The PCA formalism. - Squared correlations. - Variational origin of the scatter matrix. - e. PCA of Spectral Components of a General Vector Field. - Fourier analysis of the vector field components. - The scatter matrix in the spectral setting. - Example of spectral PCA of a windfield. - f. Bibliographic Notes and Miscellaneous Topics. - The eight modes of analysis and Cattell's classifications. - Time-modulation PGA as a special case of matrix-valued PGA. - Applications to the PGA of wind fields. - Distinction between time-modulation PGA and complex PGA. - Applications to the PGA of storm tracks. - 5. Selection Rules for PCA. - a. Random Reference Data Sets. - b. Dynamical Origins of the Dominant-Variance Selection Rules. - A dynamical model. - Rationale for selection rules. - c. Rule A4. - Statistical basis and discussion. - Choice of λ0. - d. Rule N . - Statistical basis and discussion. - Adjustments for correlated data: effective sample size. - Asymptotic eigenvalues for large data sets. - e. Rule M. - f. Comments on Dominant-Variance Rules . - g. Dynamical Origins of the Time-History Selection Rules. - h. Rule KS2. - The white spectrum and the cumulative periodogram. - Statement of Rule KS2. - i. Rules AMPλ. - Fisher's test. - Siegel's test. - Statement of Rules AMPλ. - j. Rule Q. - k. Selection Rules for Vector-Valued Fields. - Local PCA rules. - Global PCA (time-modulated) rules. - Global PCA (space-modulated) rules. - I. A Space-map Selection Rule. - Canonic direction angles. - Differential relations between unit vectors and canonic direction angles. - An r-tile metric for comparing canonic direction angles. - Statistical aspects: critical values for class errors. - Statement of the selection rule. - m. Bibliographic Notes and Miscellaneous Topics. - Puzzles and problems underlying Rule N; the logarithmic eigenvalue curve. - Numerical intractability of the classical formulas for the eigenvalues of a random matrix. - Monte Carlo approaches to the eigenvalue distribution problem. - Comparison of Monte Carlo methods and asymptotic formulas for eigenvalue distributions. - The problem of closely spaced eigenvalues; tests for equal eigenvalues. - The generalized basis for dominant variance selection rules. - Parallel work in atomic physics. - 6. Factor Analysis (FA) and PCA. - a. Comparison of PCA, LRA, and FA. - Similarities between PCA, LRA, and FA. - Dissimilarities between PCA, LRA, and FA. - The usual algebraic form of FA; its PC and LR interpretations. - b. The Central Problems of FA. - The matrix formulation of FA. - The detailed sub-problems of FA. - c. Bibliographic Notes. - The selection rule problem in FA. - The parameter estimation problem in FA. - 7. Diagnostic Procedures via PCA and FA. - a. Dual Interpretations of a Data Set: State Space and Sample Space. - b. Interpreting E-frames in PCA State Space. - Example: graphical display of eigenvectors. - Rationales for interpreting eigenmaps and time series. - PCA as a means, rather than an end. - c. Informative and Uninformative E-frames in PCA State Space. - d. Rotating E-frames in PCA State Space (varimax). - A two-dimensional example of the varimax procedure. - The general varimax procedure. - The loss of the PCA property for rotated E-frames. - e. Projections onto E-frames in PCA State Space (procrustes). - Derivation of the procrustes technique. - Some observations on the generality of the procrustes technique. - f. Interpreting A-frames in PCA Sample Space. - g. Rotating A-frames in PCA Sample Space (varimax). - h. Projections onto A-frames in PCA Sample Space (procrustes). - i. Detecting Clusters of Points in PCA State or Sample Spaces. - Minimal spanning trees. - Defining cluster pairs, and te

Note: CONTENTS Abstract Preface Summary Introduction Description of the model Ice thickness equations Heat budget and oceanic boundary layer Analytic examination of the ridge redistribution process Theoretical framework Some specific redistributors Comparison to ridge morphological data Ice strength for different redistributors Numerical simulation results Basin-wide ice thickness and velocity characteristics Ice edge evolution and sensitivity Ice thickness characteristics off the Canadian Archipelago Comparison of observed and simulated ice drift Mass balance characteristics Concluding remarks Literature cited Appendix A: Mechanical redistributor Appendix B: Heat budget code Appendix C: Thickness finite difference code

Note: CONTENTS Abstract Preface Introduction Bering Strait Field reconnaissance Estimation of ice forces on Fairway Rock 1. Creep deformation 2. Crushing failure 3. Flexural failure 4. Forces required to form floating or grounded pressure ridges along therock or to pile ice on the beaches 5. Buckling failure Driving forces Angle of internal friction of sea ice Summary Literature cited Appendix A: April 1982 field observations at Fairway Rock