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E-raamat: Spectral Methods for the Estimation of the Effective Elastic Thickness of the Lithosphere

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Spectral Methods for the Estimation of the Effective Elastic Thickness of the LithosphereSuurem pilt
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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.
Part I Context
1 Isostasy, Flexure and Strength
3(34)
1.1 Isostasy
3(9)
1.1.1 Beginnings
3(4)
1.1.2 Pressure
7(2)
1.1.3 Airy-Heiskanen Isostasy
9(2)
1.1.4 Pratt-Hayford Isostasy
11(1)
1.2 Flexural Isostasy
12(8)
1.2.1 Regional Support
13(2)
1.2.2 Crust, Mantle, Lithosphere and Asthenosphere
15(5)
1.3 The Significance of Te
20(8)
1.3.1 Plate Strength
20(6)
1.3.2 Te---Causes and Effects
26(2)
1.4 What This Book Does Not Cover
28(2)
1.5 Conventions
30(1)
1.6 Summary
30(1)
1.7 Further Reading
31(6)
References
32(5)
Part II Spectra
2 The Fourier Transform
37(54)
2.1 Introduction
37(2)
2.1.1 Dimensionality
37(1)
2.1.2 Harmonics
38(1)
2.1.3 Continuous Signals Versus Discrete Sequences
38(1)
2.2 Fourier Theory
39(11)
2.2.1 Fourier Series
40(2)
2.2.2 The Continuous Fourier Transform
42(1)
2.2.3 Amplitude, Power and Phase Spectra
43(2)
2.2.4 Signal Translation
45(1)
2.2.5 Energy Conservation
46(1)
2.2.6 Resolution and the Uncertainty Relationship
46(1)
2.2.7 Differentiation
47(1)
2.2.8 Convolution
48(2)
2.3 Sampling a Continuous Function
50(11)
2.3.1 Delta and Comb Functions
50(2)
2.3.2 Sampling
52(2)
2.3.3 Aliasing
54(2)
2.3.4 The Nyquist Frequency
56(3)
2.3.5 Anti-Aliasing (Frequency) Filter
59(2)
2.4 Fourier Transforms of Discrete Data
61(8)
2.4.1 The Discrete-Time Fourier Transform
62(2)
2.4.2 The Discrete Fourier Transform
64(1)
2.4.3 The Fast Fourier Transform
65(4)
2.5 Artefacts and How to Avoid Them
69(12)
2.5.1 Signal Truncation
69(3)
2.5.2 Loss of Resolution
72(1)
2.5.3 Spectral Leakage
72(1)
2.5.4 The Gibbs Phenomenon
72(4)
2.5.5 Cyclic, Discrete Convolution
76(1)
2.5.6 Mitigation Methods
77(4)
2.6 The 2D Fourier Transform
81(7)
2.6.1 Spatial Frequency---Wavenumber
82(1)
2.6.2 Sampling Theory in the Space Domain
82(1)
2.6.3 The Non-Unitary ID Fourier Transform
83(1)
2.6.4 The 2D Continuous Fourier Transform
84(1)
2.6.5 The Hankel Transform
85(3)
2.6.6 The 2D Discrete Fourier Transform
88(1)
2.7 Summary
88(1)
2.8 Further Reading
89(2)
Appendix
89(1)
References
90(1)
3 Multitaper Spectral Estimation
91(36)
3.1 Introduction
91(1)
3.2 The Periodogram
92(4)
3.3 Slepian Tapers
96(11)
3.3.1 Time-Limited Concentration Problem
98(1)
3.3.2 Band-Limited Concentration Problem
99(3)
3.3.3 Discrete Prolate Spheroidal Sequences
102(2)
3.3.4 Resolution
104(2)
3.3.5 Eigenvalues
106(1)
3.4 Multitaper Spectral Estimation
107(4)
3.5 Moving Windows
111(2)
3.6 The 2D Multitaper Method
113(6)
3.6.1 2D Slepian Tapers
114(1)
3.6.2 2D Average Spectrum
115(2)
3.6.3 Radially Averaged Power Spectrum
117(2)
3.7 Summary
119(1)
3.8 Further Reading
120(7)
Appendix
120(4)
References
124(3)
4 The Continuous Wavelet Transform
127(44)
4.1 Introduction
127(1)
4.2 The ID Continuous Wavelet Transform
128(2)
4.3 Continuous Wavelets
130(6)
4.3.1 Properties of Continuous Wavelets
130(1)
4.3.2 The Derivative of Gaussian Wavelet
131(2)
4.3.3 The 1D Morlet Wavelet
133(3)
4.4 Wavelet Scales
136(2)
4.5 Normalisation
138(2)
4.5.1 Time-Domain Normalisation
138(1)
4.5.2 Frequency-Domain Normalisation
139(1)
4.5.3 Practical Normalisation
140(1)
4.6 Equivalent Fourier Frequency
140(3)
4.7 Wavelet Resolution
143(2)
4.7.1 Time-Domain Resolution
143(1)
4.7.2 Frequency-Domain Resolution
144(1)
4.8 Wavelet Power Spectra
145(5)
4.8.1 Local Scalograms
145(2)
4.8.2 Heisenberg Boxes
147(3)
4.8.3 Global Scalograms
150(1)
4.9 Cone of Influence (Col)
150(2)
4.10 The 2D Continuous Wavelet Transform
152(1)
4.11 The 2D Morlet Wavelet
153(8)
4.11.1 Governing Equations
153(3)
4.11.2 2D Normalisation
156(3)
4.11.3 2D Resolution
159(2)
4.12 The Fan Wavelet Method
161(6)
4.12.1 The Fan Wavelet
161(3)
4.12.2 Fan Wavelet Transform
164(1)
4.12.3 Fan Wavelet Power Spectra
165(2)
4.13 Summary
167(1)
4.14 Further Reading
167(4)
Appendix
168(1)
References
168(3)
5 Admittance, Coherency and Coherence
171(42)
5.1 Introduction
171(1)
5.2 The Admittance
171(9)
5.2.1 The Earth's Response to Loading
171(3)
5.2.2 The Complex Admittance
174(4)
5.2.3 Admittance Phase
178(2)
5.3 The Coherency and Coherence
180(8)
5.3.1 The Coherency
180(2)
5.3.2 The Coherence
182(2)
5.3.3 The Complex Coherency
184(4)
5.4 Practical Estimation of the Admittance and Coherency
188(9)
5.4.1 Using Multitapers
188(3)
5.4.2 Using the Wavelet Transform
191(6)
5.5 Errors on the Admittance and Coherence
197(7)
5.5.1 Independent Estimates
197(2)
5.5.2 Errors from Analytic Formulae
199(1)
5.5.3 Jackknife Error Estimates
200(4)
5.6 Wavenumber/Wavelength Uncertainty
204(2)
5.6.1 Slepian Tapers
204(2)
5.6.2 Fan Wavelet
206(1)
5.7 Summary
206(1)
5.8 Further Reading
207(6)
Appendix
208(2)
References
210(3)
6 Map Projections
213(26)
6.1 Introduction
213(1)
6.2 Types of Map Projection
214(4)
6.2.1 Developable Surfaces
214(4)
6.2.2 Projection Classes
218(1)
6.3 Distortion
218(7)
6.3.1 Scale Factors
219(1)
6.3.2 Cylindrical Projections
219(3)
6.3.3 Tissot's Indicatrix
222(3)
6.4 Which Projection?
225(6)
6.5 Data Area Considerations
231(3)
6.5.1 Data Area Size
231(1)
6.5.2 Grid Spacing
232(2)
6.6 Summary
234(1)
6.7 Further Reading
235(4)
References
235(4)
Part III Flexure
7 Loading and Flexure of an Elastic Plate
239(40)
7.1 Introduction
239(1)
7.2 Thin, Elastic Plate Flexure
240(17)
7.2.1 Thin Plates
241(3)
7.2.2 Elasticity: Stress and Strain
244(3)
7.2.3 The Elastic Moduli
247(1)
7.2.4 Plane Stress
248(1)
7.2.5 Bending Moments
249(3)
7.2.6 Twisting Moments
252(3)
7.2.7 Flexural Equations
255(1)
7.2.8 Solving the Biharmonic Equation
256(1)
7.3 Buoyancy
257(2)
7.4 Surface Loading
259(5)
7.4.1 Two-Layer Crust Model
259(4)
7.4.2 Multiple-Layer Crust Model
263(1)
7.5 Internal Loading
264(6)
7.5.1 Loading at the Moho of a Two-Layer Crust
265(3)
7.5.2 Loading Within a Multiple-Layer Crust
268(2)
7.6 Combined Loading
270(2)
7.7 Flexural Wavelength
272(3)
7.8 Summary
275(1)
7.9 Further Reading
275(4)
Appendix
276(1)
References
276(3)
8 Gravity and Admittance of a Flexed Plate
279(36)
8.1 Introduction
279(1)
8.2 Gravity Anomalies
280(14)
8.2.1 Gravity and Gravitation
280(2)
8.2.2 Gravity Potential and the Geoid
282(2)
8.2.3 Normal Gravity
284(2)
8.2.4 Free-Air Anomalies
286(3)
8.2.5 Bouguer Anomalies
289(5)
8.3 Upward/Downward Continuation of Gravity
294(4)
8.4 Gravity from Surface Loading
298(3)
8.4.1 Two-Layer Crust Model
298(2)
8.4.2 Multiple-Layer Crust Model
300(1)
8.5 Gravity from Internal Loading
301(4)
8.5.1 Loading at the Moho of a Two-Layer Crust
301(1)
8.5.2 Loading within a Multiple-Layer Crust
302(3)
8.6 The Admittance of Theoretical Models
305(7)
8.6.1 Surface Loading
306(3)
8.6.2 Internal Loading
309(3)
8.7 Combined Loading
312(1)
8.8 Summary
312(1)
8.9 Further Reading
313(2)
References
313(2)
9 The toad Deconvolution Method
315(52)
9.1 Introduction
315(2)
9.2 Combined Loading
317(1)
9.3 Combined-Loading Coherency, Coherence and Admittance
318(11)
9.3.1 Predicted Coherency, Coherence and Admittance
319(2)
9.3.2 The Load Ratio
321(1)
9.3.3 Theoretical Coherency, Coherence and Admittance
322(7)
9.4 Te Estimation with Load Deconvolution
329(16)
9.4.1 Overview of Load Deconvolution
329(8)
9.4.2 Load Deconvolution with Multitapers
337(4)
9.4.3 Load Deconvolution with Wavelets
341(4)
9.5 Load Versus Gravity Deconvolution
345(5)
9.5.1 Using Loads
345(2)
9.5.2 Using Gravity
347(3)
9.6 Model Noise
350(5)
9.6.1 Categorising Noise
350(1)
9.6.2 Unexpressed Loading
351(4)
9.7 Correlated Initial Loads
355(7)
9.7.1 Correlated-Load Theory
355(4)
9.7.2 Simulation with Synthetic Models
359(2)
9.7.3 Phase Relationships
361(1)
9.8 Some Theoretical Considerations
362(1)
9.9 Summary
363(1)
9.10 Further Reading
364(3)
References
364(3)
10 Synthetic Testing
367(32)
10.1 Introduction
367(1)
10.2 Fractal Surfaces
367(6)
10.3 The Initial Loads
373(2)
10.4 Uniform-Te Plates
375(4)
10.5 Variable-7; Plates
379(8)
10.6 Summary
387(1)
10.7 Further Reading
387(12)
Appendix
388(9)
References
397(2)
11 Practical Te Estimation
399(54)
11.1 Introduction
399(1)
11.2 Data
399(12)
11.2.1 Model Grid Spacing
400(1)
11.2.2 Topography Data
401(4)
11.2.3 Gravity Data
405(3)
11.2.4 Crustal Structure Data
408(2)
11.2.5 Sediment Data
410(1)
11.3 Equivalent Topography
411(7)
11.3.1 Calculation of the Equivalent Topography
412(2)
11.3.2 Effect on the Admittance and Coherency
414(3)
11.3.3 Equivalent Topography and the Bouguer Correction
417(1)
11.4 Depth and Density Tests
418(3)
11.4.1 Bouguer Reduction Density and Terrain Corrections
419(2)
11.4.2 Crustal Structure
421(1)
11.5 Estimation of the Load Ratio
421(3)
11.6 Deconvolution with the Admittance
424(1)
11.7 Uniform-ƒ Inversion
425(2)
11.8 Te Errors
427(2)
11.9 Noise Detection
429(3)
11.10 Accounting for Sediments
432(4)
11.11 Wavelet Versus Multitaper
436(8)
11.12 Other Considerations
444(3)
11.13 Summary
447(1)
11.14 Further Reading
448(5)
References
448(5)
Index 453
Jon Kirby is Associate Professor in the School of Earth and Planetary Sciences at Curtin University, located in Perth, Australia. Jon was born and educated in the UK, receiving a BSc in physics from Durham University, an MSc in exploration geophysics from the University of Leeds, and a PhD in geophysical geodesy from the University of Edinburgh. Although his post-doctoral career began with geoid determination, he soon gravitated towards isostasy and lithospheric flexure, and has been publishing on the topic since 2003, notably developing a wavelet-based method to estimate effective elastic thickness.
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