Spectral Methods for the Estimation of the Effective Elastic Thickness of the Lithosphere (Advances in Geophysical and Environmental Mechanics and Mathematics)

Although several excellent works exist that describe the effective elastic thickness (Te) of the lithosphere―its theory, significance and relevance to Earth sciences in general―none cover the details of the methods for its estimation. This book brings together the disparate knowledge required to estimate Te in one handy volume: signal processing, harmonic analysis, civil engineering, and foundational mathematics and physics, in addition to the relevant geophysics and, to a lesser extent, geology. Its two principal focus areas are spectral estimation, covering various approaches to estimating the admittance and coherence between gravity and topography using Slepian multitapers and fan wavelets; and algebraic and finite difference solutions of the plate bending partial differential equation in a variety of geological settings. This book would be suitable for postgraduate students beginning their research, up to faculty professors interested in diversifying their skills. Preface Contents Abbreviations Symbols and Notation Part I Context 1 Isostasy, Flexure and Strength 1.1 Isostasy 1.1.1 Beginnings 1.1.2 Pressure 1.1.3 Airy-Heiskanen Isostasy 1.1.4 Pratt-Hayford Isostasy 1.2 Flexural Isostasy 1.2.1 Regional Support 1.2.2 Crust, Mantle, Lithosphere and Asthenosphere 1.3 The Significance of Te 1.3.1 Plate Strength 1.3.2 Te—Causes and Effects 1.4 What This Book Does Not Cover 1.5 Conventions 1.6 Summary 1.7 Further Reading References Part II Spectra 2 The Fourier Transform 2.1 Introduction 2.1.1 Dimensionality 2.1.2 Harmonics 2.1.3 Continuous Signals Versus Discrete Sequences 2.2 Fourier Theory 2.2.1 Fourier Series 2.2.2 The Continuous Fourier Transform 2.2.3 Amplitude, Power and Phase Spectra 2.2.4 Signal Translation 2.2.5 Energy Conservation 2.2.6 Resolution and the Uncertainty Relationship 2.2.7 Differentiation 2.2.8 Convolution 2.3 Sampling a Continuous Function 2.3.1 Delta and Comb Functions 2.3.2 Sampling 2.3.3 Aliasing 2.3.4 The Nyquist Frequency 2.3.5 Anti-Aliasing (Frequency) Filter 2.4 Fourier Transforms of Discrete Data 2.4.1 The Discrete-Time Fourier Transform 2.4.2 The Discrete Fourier Transform 2.4.3 The Fast Fourier Transform 2.5 Artefacts and How to Avoid Them 2.5.1 Signal Truncation 2.5.2 Loss of Resolution 2.5.3 Spectral Leakage 2.5.4 The Gibbs Phenomenon 2.5.5 Cyclic, Discrete Convolution 2.5.6 Mitigation Methods 2.6 The 2D Fourier Transform 2.6.1 Spatial Frequency—Wavenumber 2.6.2 Sampling Theory in the Space Domain 2.6.3 The Non-Unitary 1D Fourier Transform 2.6.4 The 2D Continuous Fourier Transform 2.6.5 The Hankel Transform 2.6.6 The 2D Discrete Fourier Transform 2.7 Summary 2.8 Further Reading References 3 Multitaper Spectral Estimation 3.1 Introduction 3.2 The Periodogram 3.3 Slepian Tapers 3.3.1 Time-Limited Concentration Problem 3.3.2 Band-Limited Concentration Problem 3.3.3 Discrete Prolate Spheroidal Sequences 3.3.4 Resolution 3.3.5 Eigenvalues 3.4 Multitaper Spectral Estimation 3.5 Moving Windows 3.6 The 2D Multitaper Method 3.6.1 2D Slepian Tapers 3.6.2 2D Average Spectrum 3.6.3 Radially Averaged Power Spectrum 3.7 Summary 3.8 Further Reading References 4 The Continuous Wavelet Transform 4.1 Introduction 4.2 The 1D Continuous Wavelet Transform 4.3 Continuous Wavelets 4.3.1 Properties of Continuous Wavelets 4.3.2 The Derivative of Gaussian Wavelet 4.3.3 The 1D Morlet Wavelet 4.4 Wavelet Scales 4.5 Normalisation 4.5.1 Time-Domain Normalisation 4.5.2 Frequency-Domain Normalisation 4.5.3 Practical Normalisation 4.6 Equivalent Fourier Frequency 4.7 Wavelet Resolution 4.7.1 Time-Domain Resolution 4.7.2 Frequency-Domain Resolution 4.8 Wavelet Power Spectra 4.8.1 Local Scalograms 4.8.2 Heisenberg Boxes 4.8.3 Global Scalograms 4.9 Cone of Influence (CoI) 4.10 The 2D Continuous Wavelet Transform 4.11 The 2D Morlet Wavelet 4.11.1 Governing Equations 4.11.2 2D Normalisation 4.11.3 2D Resolution 4.12 The Fan Wavelet Method 4.12.1 The Fan Wavelet 4.12.2 Fan Wavelet Transform 4.12.3 Fan Wavelet Power Spectra 4.13 Summary 4.14 Further Reading References 5 Admittance, Coherency and Coherence 5.1 Introduction 5.2 The Admittance 5.2.1 The Earth's Response to Loading 5.2.2 The Complex Admittance 5.2.3 Admittance Phase 5.3 The Coherency and Coherence 5.3.1 The Coherency 5.3.2 The Coherence 5.3.3 The Complex Coherency 5.4 Practical Estimation of the Admittance and Coherency 5.4.1 Using Multitapers 5.4.2 Using the Wavelet Transform 5.5 Errors on the Admittance and Coherence 5.5.1 Independent Estimates 5.5.2 Errors from Analytic Formulae 5.5.3 Jackknife Error Estimates 5.6 Wavenumber/Wavelength Uncertainty 5.6.1 Slepian Tapers 5.6.2 Fan Wavelet 5.7 Summary 5.8 Further Reading References 6 Map Projections 6.1 Introduction 6.2 Types of Map Projection 6.2.1 Developable Surfaces 6.2.2 Projection Classes 6.3 Distortion 6.3.1 Scale Factors 6.3.2 Cylindrical Projections 6.3.3 Tissot's Indicatrix 6.4 Which Projection? 6.5 Data Area Considerations 6.5.1 Data Area Size 6.5.2 Grid Spacing 6.6 Summary 6.7 Further Reading References Part III Flexure 7 Loading and Flexure of an Elastic Plate 7.1 Introduction 7.2 Thin, Elastic Plate Flexure 7.2.1 Thin Plates 7.2.2 Elasticity: Stress and Strain 7.2.3 The Elastic Moduli 7.2.4 Plane Stress 7.2.5 Bending Moments 7.2.6 Twisting Moments 7.2.7 Flexural Equations 7.2.8 Solving the Biharmonic Equation 7.3 Buoyancy 7.4 Surface Loading 7.4.1 Two-Layer Crust Model 7.4.2 Multiple-Layer Crust Model 7.5 Internal Loading 7.5.1 Loading at the Moho of a Two-Layer Crust 7.5.2 Loading Within a Multiple-Layer Crust 7.6 Combined Loading 7.7 Flexural Wavelength 7.8 Summary 7.9 Further Reading References 8 Gravity and Admittance of a Flexed Plate 8.1 Introduction 8.2 Gravity Anomalies 8.2.1 Gravity and Gravitation 8.2.2 Gravity Potential and the Geoid 8.2.3 Normal Gravity 8.2.4 Free-Air Anomalies 8.2.5 Bouguer Anomalies 8.3 Upward/Downward Continuation of Gravity 8.4 Gravity from Surface Loading 8.4.1 Two-Layer Crust Model 8.4.2 Multiple-Layer Crust Model 8.5 Gravity from Internal Loading 8.5.1 Loading at the Moho of a Two-Layer Crust 8.5.2 Loading within a Multiple-Layer Crust 8.6 The Admittance of Theoretical Models 8.6.1 Surface Loading 8.6.2 Internal Loading 8.7 Combined Loading 8.8 Summary 8.9 Further Reading References 9 The Load Deconvolution Method 9.1 Introduction 9.2 Combined Loading 9.3 Combined-Loading Coherency, Coherence and Admittance 9.3.1 Predicted Coherency, Coherence and Admittance 9.3.2 The Load Ratio 9.3.3 Theoretical Coherency, Coherence and Admittance 9.4 Te Estimation with Load Deconvolution 9.4.1 Overview of Load Deconvolution 9.4.2 Load Deconvolution with Multitapers 9.4.3 Load Deconvolution with Wavelets 9.5 Load Versus Gravity Deconvolution 9.5.1 Using Loads 9.5.2 Using Gravity 9.6 Model Noise 9.6.1 Categorising Noise 9.6.2 Unexpressed Loading 9.7 Correlated Initial Loads 9.7.1 Correlated-Load Theory 9.7.2 Simulation with Synthetic Models 9.7.3 Phase Relationships 9.8 Some Theoretical Considerations 9.9 Summary 9.10 Further Reading References 10 Synthetic Testing 10.1 Introduction 10.2 Fractal Surfaces 10.3 The Initial Loads 10.4 Uniform-Te Plates 10.5 Variable-Te Plates 10.6 Summary 10.7 Further Reading References 11 Practical Te Estimation 11.1 Introduction 11.2 Data 11.2.1 Model Grid Spacing 11.2.2 Topography Data 11.2.3 Gravity Data 11.2.4 Crustal Structure Data 11.2.5 Sediment Data 11.3 Equivalent Topography 11.3.1 Calculation of the Equivalent Topography 11.3.2 Effect on the Admittance and Coherency 11.3.3 Equivalent Topography and the Bouguer Correction 11.4 Depth and Density Tests 11.4.1 Bouguer Reduction Density and Terrain Corrections 11.4.2 Crustal Structure 11.5 Estimation of the Load Ratio 11.6 Deconvolution with the Admittance 11.7 Uniform-f Inversion 11.8 Te Errors 11.9 Noise Detection 11.10 Accounting for Sediments 11.11 Wavelet Versus Multitaper 11.12 Other Considerations 11.13 Summary 11.14 Further Reading References Appendix Index Index