ALBERT

Note: CONTENTS: PREFACE. - INTRODUCTION. - 1 DESCRIBING INVERSE PROBLEMS. - 1.1 Formulating Inverse Problems. - 1.2 The Linear Inverse Problem. - 1.3 Examples of Formulating Inverse Problems. - 1.4 Solutions to Inverse Problems. - 2 SOME COMMENTS ON PROBABILITY THEORY. - 2.1 Noise and Random Variables. - 2.2 Correlated Data. - 2.3 Functions of Random Variables. - 2.4 Gaussian Distributions. - 2.5 Testing the Assumption of Gaussian Statistics. - 2.6 Confidence Intervals. - 3 SOLUTION OF THE LINEAR, GAUSSIAN INVERSE PROBLEM, VIEWPOINT 1: THE LENGTH METHOD. - 3.1 The Lengths of Estimates. - 3.2 Measures of Length. - 3.3 Least Squares for a Straight Line. - 3.4 The Least Squares Solution of the Linear Inverse Problem. - 3.5 Some Examples. - 3.6 The Existence of the Least Squares Solution. - 3.7 The Purely Underdetermined Problem. - 3.8 Mixed-Determined Problems. - 3.9 Weighted Measures of Length as a Type of A Priori Information. - 3.10 Other Types of A Priori Information. - 3.11 The Variance of the Model Parameter Estimates. - 3.12 Variance and Prediction Error of the Least Squares Solution. - 4 SOLUTION OF THE LINEAR, GAUSSIAN INVERSE PROBLEM, VIEWPOINT 2: GENERALIZED INVERSES. - 4.1 Solutions versus Operators. - 4.2 The Data Resolution Matrix. - 4.3 The Model Resolution Matrix. - 4.4 The Unit Covariance Matrix. - 4.5 Resolution and Covariance of Some Generalized Inverses. - 4.6 Measures of Goodness of Resolution and Covariance. - 4.7 Generalized Inverses with Good Resolution and Covariance. - 4.8 Sidelobes and the Backus-Gilbert Spread Function. - 4.9 The Backus-Gilbert Generalized Inverse for the Underdetermined Problem. - 4.10 Including the Covariance Size. - 4.11 The Trade-off of Resolution and Variance. - 5 SOLUTION OF THE LINEAR, GAUSSIAN INVERSE PROBLEM, VIEWPOINT 3: MAXIMUM LIKELIHOOD METHODS. - 5.1 The Mean of a Group of Measurements. - 5.2 Maximum Likelihood Solution of the Linear Inverse Problem. - 5.3 A Priori Distributions. - 5.4 Maximum Likelihood for an Exact Theory. - 5.5 Inexact Theories. - 5.6 The Simple Gaussian Case with a Linear Theory. - 5.7 The General Linear, Gaussian Case. - 5.8 Equivalence of the Three Viewpoints. - 5.9 The F Test of Error Improvement Significance. - 5.10 Derivation of the Formulas of Section 5.7. - 6 NONUNIQUENESS AND LOCALIZED AVERAGES. - 6.1 Null Vectors and Nonuniqueness. - 6.2 Null Vectors of a Simple Inverse Problem. - 6.3 Localized Averages of Model Parameters. - 6.4 Relationship to the Resolution Matrix. - 6.5 Averages versus Estimates. - 6.6 Nonunique Averaging Vectors and A Priori Information. - 7 APPLICATIONS OF VECTOR SPACES. - 7.1 Model and Data Spaces. - 7.2 Householder Transformations. - 7.3 Designing Householder Transformations. - 7.4 Transformations That Do Not Preserve Length. - 7.5 The Solution of the Mixed-Determined Problem. - 7.6 Singular-Value Decomposition and the Natural Generalized Inverse. - 7.7 Derivation of the Singular-Value Decomposition. - 7.8 Simplifying Linear Equality and Inequality Constraints. - 7.9 Inequality Constraints. - 8 LINEAR INVERSE PROBLEMS AND NON-GAUSSIAN DISTRIBUTIONS. - 8.1 L1 Norms and Exponential Distributions. - 8.2 Maximum Likelihood Estimate of the Mean of an Exponential Distribution. - 8.3 The General Linear Problem. - 8.4 Solving L1 Norm Problems. - 8.5 The L [Infinity symbol] Norm. - 9 NONLINEAR INVERSE PROBLEMS. - 9.1 Parameterizations. - 9.2 Linearizing Parameterizations. - 9.3 The Nonlinear Inverse Problem with Gaussian Data. - 9.4 Special Cases. - 9.5 Convergence and Nonuniqueness of Nonlinear L2 Problems. - 9.6 Non-Gaussian Distributions. - 9.7 Maximum Entropy Methods. - 10 FACTOR ANALYSIS. - 10.1 The Factor Analysis Problem. - 10.2 Normalization and Physicality Constraints. - 10.3 Q-Mode and R-Mode Factor Analysis. - 10.4 Empirical Orthogonal Function Analysis. - 11 CONTINUOUS INVERSE THEORY AND TOMOGRAPHY. - 11.1 The Backus-Gilbert Inverse Problem. - 11.2 Resolution and Variance Trade-off. - 11.3 Approximating Continuous Inverse Problems as Discrete Problems. - 11.4 Tomography and Continuous Inverse Theory. - 11.5 Tomography and the Radon Transform. - 11.6 The Fourier Slice Theorem. - 11.7 Backprojection. - 12 SAMPLE INVERSE PROBLEMS. - 12.1 An Image Enhancement Problem. - 12.2 Digital Filter Design. - 12.3 Adjustment of Crossover Errors. - 12.4 An Acoustic Tomography Problem. - 12.5 Temperature Distribution in an Igneous Intrusion. - 12.6 L1, L2, and L [infinity symbol] Fitting of a Straight Line. - 12.7 Finding the Mean of a Set of Unit Vectors. - 12.8 Gaussian Curve Fitting. - 12.9 Earthquake Location. - 12.10 Vibrational Problems. - 13 NUMERICAL ALGORITHMS. - 13.1 Solving Even-Determined Problems. - 13.2 Inverting a Square Matrix. - 13.3 Solving Underdetermined and Overdetermined Problems. - 13.4 L2 Problems with Inequality Constraints. - 13.5 Finding the Eigenvalues and Eigenvectors of a Real Symmetric Matrix. - 13.6 The Singular-Value Decomposition of a Matrix. - 13.7 The Simplex Method and the Linear Programming Problem. - 14 APPLICATIONS OF INVERSE THEORY TO GEOPHYSICS. - 14.1 Earthquake Location and the Determination of the Velocity Structure of the Earth from Travel Time Data. - 14.2 Velocity Structure from Free Oscillations and Seismic Surface Waves. - 14.3 Seismic Attenuation. - 14.4 Signal Correlation. - 14.5 Tectonic Plate Motions. - 14.6 Gravity and Geomagnetism. - 14.7 Electromagnetic Induction and the Magnetotelluric Method. - 14.8 Ocean Circulation. - APPENDIX A: Implementing Constraints with Lagrange Multipliers. - APPENDIX B: L2 Inverse Theory with Complex Quantities. - REFERENCES. - INDEX

Note: Geophysics of the Caledonian-Appalachian Orogen --- R. T. Haworth, R. Hipkin, R. D. Jacobi, M. Kane, J. P. Lefort, M. D. Max, H. G. Miller, and F. Wolff: Geophysical framework and the Appalachian-Caledonide connection / Geological Society, London, Special Publications, 38:3-20, doi:10.1144/GSL.SP.1988.038.01.01 --- F. A. Cook, D. H. Matthews, and A. W. B. Jacob: Crustal and upper mantle structure of the Appalachian-Caledonide orogen from seismic results / Geological Society, London, Special Publications, 38:21-33, doi:10.1144/GSL.SP.1988.038.01.02 --- J. C. Briden, D. V. Kent, P. L. Lapointe, J. L. Roy, R. A. Livermore, A. G. Smith, M. K. Seguin, R. Van der Voo, and D. R. Watts: Palaeomagnetic constraints on the evolution of the Caledonian-Appalachian orogen / Geological Society, London, Special Publications, 38:35-48, doi:10.1144/GSL.SP.1988.038.01.03 --- J. P. Lefort, M. D. Max, and J. Roussel: Geophysical evidence for the location of the NW boundary of Gondwanaland and its relationship with two older satellite sutures / Geological Society, London, Special Publications, 38:49-60, doi:10.1144/GSL.SP.1988.038.01.04 --- Pre-Arenig Activity in the Caledonian-Appalachian Orogen --- Derek Powell, T. B. Andersen, A. A. Drake, Jr, Leo Hall, and J. D. Keppie: The age and distribution of basement rocks in the Caledonide orogen of the N Atlantic / Geological Society, London, Special Publications, 38:63-74, doi:10.1144/GSL.SP.1988.038.01.05 --- Frederic L. Schwab, Johan P. Nystuen, and Linda Gunderson: Pre-Arenig evolution of the Appalachian-Caledonide orogen: sedimentation and stratigraphy / Geological Society, London, Special Publications, 38:75-91, doi:10.1144/GSL.SP.1988.038.01.06 --- S. Conway Morris and A. W. A. Rushton: Precambrian to Tremadoc biotas in the Caledonides / Geological Society, London, Special Publications, 38:93-109, doi:10.1144/GSL.SP.1988.038.01.07 --- Nicholas Rast, B. A. Sturt, and A. L. Harris: Early deformation in the Caledonian-Appalachian orogen / Geological Society, London, Special Publications, 38:111-122, doi:10.1144/GSL.SP.1988.038.01.08 --- Ben Harte: Lower Palaeozoic metamorphism in the Moine-Dalradian belt of the British Isles / Geological Society, London, Special Publications, 38:123-134, doi:10.1144/GSL.SP.1988.038.01.09 --- Inge Bryhni: Early Palaeozoic metamorphism in the Scandinavian Caledonides / Geological Society, London, Special Publications, 38:135-140, doi:10.1144/GSL.SP.1988.038.01.10 --- Jo Laird: Pre-Arenig metamorphism in the Appalachians / Geological Society, London, Special Publications, 38:141-147, doi:10.1144/GSL.SP.1988.038.01.11 --- D. W. Rankin, Harald Furnes, A. C. Bishop, B. Cabanis, D. J. Milton, S. J. O’Brien, and R. S. Thorpe: Plutonism and volcanism related to the pre-Arenig evolution of the Caledonide-Appalachian orogen / Geological Society, London, Special Publications, 38:149-183, doi:10.1144/GSL.SP.1988.038.01.12 --- James W. Skehan: Evolution of the Iapetus Ocean and its borders in pre-Arenig times: a synthesis / Geological Society, London, Special Publications, 38:185-229, doi:10.1144/GSL.SP.1988.038.01.13 --- Arenig-Wenlock Activity in the Caledonian-Appalachian Orogen --- R. A. Fortey and L. R. M. Cocks: Arenig to Llandovery faunal distributions in the Caledonides / Geological Society, London, Special Publications, 38:231-246, doi:10.1144/GSL.SP.1988.038.01.14 --- David L. Bruton and David A. T. Harper: Arenig-Llandovery stratigraphy and faunas across the Scandinavian Caledonides / Geological Society, London, Special Publications, 38:247-268, doi:10.1144/GSL.SP.1988.038.01.15 --- Robert B. Neuman: Palaeontological evidence bearing on the Arenig-Caradoc development of the Iapetus Ocean basin / Geological Society, London, Special Publications, 38:269-274, doi:10.1144/GSL.SP.1988.038.01.16 --- C. J. Stillman: Ordovician to Silurian volcanism in the Appalachian—Caledonian orogen / Geological Society, London, Special Publications, 38:275-290, doi:10.1144/GSL.SP.1988.038.01.17 --- Leo M. Hall and David Roberts: Timing of Ordovician deformation in the Caledonian-Appalachian orogen / Geological Society, London, Special Publications, 38:291-309, doi:10.1144/GSL.SP.1988.038.01.18 --- Jo Laird: Arenig to Wenlock age metamorphism in the Appalachians / Geological Society, London, Special Publications, 38:311-345, doi:10.1144/GSL.SP.1988.038.01.19 --- G. J. H. Oliver: Arenig to Wenlock regional metamorphism in the Paratectonic Caledonides of the British Isles: a review / Geological Society, London, Special Publications, 38:347-363, doi:10.1144/GSL.SP.1988.038.01.20 --- R. D. Dallmeyer: Polyphase tectonothermal evolution of the Scandinavian Caledonides / Geological Society, London, Special Publications, 38:365-379, doi:10.1144/GSL.SP.1988.038.01.21 --- David R. Wones and A. K. Sinha: A brief review of early Ordovician to Devonian plutonism in the N American Caledonides / Geological Society, London, Special Publications, 38:381-388, doi:10.1144/GSL.SP.1988.038.01.22 --- W. E. Stephens: Granitoid plutonism in the Caledonian orogen of Europe / Geological Society, London, Special Publications, 38:389-403, doi:10.1144/GSL.SP.1988.038.01.23 --- W. S. McKerrow: The development of the Iapetus Ocean from the Arenig to the Wenlock / Geological Society, London, Special Publications, 38:405-412, doi:10.1144/GSL.SP.1988.038.01.24 --- Wenlock to Mid-Devonian Activity in the Caledonian-Appalachian Orogen --- M. F. Thirlwall: Wenlock to mid-Devonian volcanism of the Caledonian-Appalachian orogen / Geological Society, London, Special Publications, 38:415-428, doi:10.1144/GSL.SP.1988.038.01.25 --- David Roberts: Timing of Silurian to middle Devonian deformation in the Caledonides of Scandinavia, Svalbard and E Greenland / Geological Society, London, Special Publications, 38:429-435, doi:10.1144/GSL.SP.1988.038.01.26 --- W.S. McKerrow: Wenlock to Givetian deformation in the British Isles and the Canadian Appalachians / Geological Society, London, Special Publications, 38:437-448, doi:10.1144/GSL.SP.1988.038.01.27 --- P. H. Osberg: Silurian to Lower Carboniferous tectonism in the Appalachians of the USA / Geological Society, London, Special Publications, 38:449-452, doi:10.1144/GSL.SP.1988.038.01.28 --- Peter Robinson, R. J. Tracy, D. S. Santallier, P.-G. Andreasson, and J. I. Gil-Ibarguchi: Scandian-Acadian-Caledonian sensu strictu metamorphism in the age range 430–360 Ma / Geological Society, London, Special Publications, 38:453-467, doi:10.1144/GSL.SP.1988.038.01.29 --- D. V. Kent and J. D. Keppie: Silurian-Permian palaeocontinental reconstructions and circum-Atlantic tectonics / Geological Society, London, Special Publications, 38:469-480, doi:10.1144/GSL.SP.1988.038.01.30 --- N. J. Soper: Timing and geometry of collision, terrane accretion and sinistral strike-slip events in the British Caledonides / Geological Society, London, Special Publications, 38:481-492, doi:10.1144/GSL.SP.1988.038.01.31 --- J. Chaloupský: Caledonian folding in the Bohemian Massif / Geological Society, London, Special Publications, 38:493-498, doi:10.1144/GSL.SP.1988.038.01.32 --- Robert D. Hatcher, Jr: The third synthesis: Wenlock to mid-Devonian (end of Acadian orogeny) / Geological Society, London, Special Publications, 38:499-504, doi:10.1144/GSL.SP.1988.038.01.33 --- Mid-Devonian-Permian Activity in the Caledonian-Appalachian Orogen --- Robert D. Hatcher, Jr: Basement-cover relationships in the Appalachian-Caledonian-Variscan orogen: mid-Devonian (end of Acadian orogeny) to end of Permian / Geological Society, London, Special Publications, 38:507-514, doi:10.1144/GSL.SP.1988.038.01.34 --- William A. Thomas and Paul E. Schenk: Late Palaeozoic sedimentation along the Appalachian orogen / Geological Society, London, Special Publications, 38:515-530, doi:10.1144/GSL.SP.1988.038.01.35 --- K. C. Allen and D. L. Dineley: Mid-Devonian to mid-Permian floral and faunal regions and provinces / Geological Societ