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.
Author(s): Jonathan Kirby
Series: Advances in Geophysical and Environmental Mechanics and Mathematics
Publisher: Springer
Year: 2022
Language: English
Pages: 471
City: Cham
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
