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2022 | Book

Isostasy, Flexure and Strength Read chapter Read first chapter

Spectral Methods for the Estimation of the Effective Elastic Thickness of the Lithosphere

Author: Jonathan Kirby

Publisher: Springer International Publishing

Book Series : Advances in Geophysical and Environmental Mechanics and Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik"

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About this book

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.

Table of Contents

Frontmatter

Context

Frontmatter
Chapter 1. Isostasy, Flexure and Strength
Abstract
Our current understanding of how loads are supported by the lithosphere is a combination of compensation by hydrostatic equilibrium, and mechanical support by bending stresses within a thin, elastic plate. The former mechanism is known as isostasy, the latter as flexure. Here, we chart the path taken from the beginnings of isostatic theory up to the current flexural isostatic models, though only in a qualitative, descriptive manner. We will be introduced to the effective elastic thickness of the lithosphere, \(T_e\), and learn how it relates to the strength and rigidity of tectonic plates via the yield strength envelope. We also touch on a few definitions of the lithosphere, and appreciate the importance of \(T_e\) to geodynamics and tectonic evolution.
Jonathan Kirby

Spectra

Frontmatter
Chapter 2. The Fourier Transform
Abstract
This chapter provides an introduction to the Fourier transform, focussing on its application to discretely sampled and truncated data. As such, it discusses the sampling of continuous signals and the associated problems of aliasing and introduces the discrete-time and discrete Fourier transforms in both one and two dimensions. However, these discrete versions do not provide a perfect spectral estimate of the data, so the chapter concludes with a discussion of the artefacts—spectral leakage, loss of resolution and ringing—and introduces techniques one can use to mitigate their influence.
Jonathan Kirby
Chapter 3. Multitaper Spectral Estimation
Abstract
With discretely sampled and truncated data, a power spectrum computed simply by squaring and summing the real and imaginary components of the Fourier transform provides a biased estimate of the true power spectrum. And while windowing with a single taper decreases spectral leakage, it does not reduce the variance of the spectrum. This chapter introduces a class of multiple tapers called discrete prolate spheroidal sequences, which are designed to reduce leakage by solving the so-called concentration problem. Furthermore, they form an orthogonal set meaning that their spectrum estimates are independent, allowing for averaging and an associated reduction of variance.
Jonathan Kirby
Chapter 4. The Continuous Wavelet Transform
Abstract
The wavelet transform was designed to estimate the power spectra of non-stationary signals, that is, those whose frequency content varies over time or space. This chapter introduces the continuous—as opposed to discrete—wavelet transform in one and two dimensions and explores some properties of wavelets, focussing on the 2D Morlet wavelet. This particular wavelet is used in the so-called fan wavelet method of elastic thickness estimation. The fan wavelet geometry enables the estimation of spectra that are both isotropic and complex-valued, which can then be used in the determination of the admittance and coherency.
Jonathan Kirby
Chapter 5. Admittance, Coherency and Coherence
Abstract
The estimation of effective elastic thickness from gravity and topography data using spectral methods requires the computation of the admittance and coherence between those data sets. The admittance is a frequency-domain transfer function—a filter—from topography to gravity. The coherence is derived from the coherency which provides the degree of correlation between gravity and topography as a function of the wavelength of features within the signals. Both are complex-valued quantities, and their imaginary parts provide useful information about certain types of noise. This chapter covers the estimation of admittance and coherency using both multitaper and wavelet methods, and discusses two methods to obtain their errors.
Jonathan Kirby
Chapter 6. Map Projections
Abstract
The spectral estimation methods discussed in this book are planar, in that the data under analysis are coordinated in the 2D Cartesian plane, in both space and wavenumber domains. However, the gravity and topography data from which effective elastic thickness is estimated are collected and coordinated on the surface of a planet—a curved surface which is best modelled as a sphere or ellipsoid. The procedure by which geodetically-coordinated data are represented on a plane map is called a map projection, but the formulae only permit one of three quantities to be faithfully preserved during the projection: angle, area or distance; the other two must be sacrificed. In this chapter we will learn about map projections and the concept of distortion, honing in upon the best projections for planar spectral analysis of data given on a curved surface.
Jonathan Kirby

Flexure

Frontmatter
Chapter 7. Loading and Flexure of an Elastic Plate
Abstract
Effective elastic thickness (\(T_e\)) is estimated by inverting the admittance or coherency between observed gravity and topography data against predictions of flexural models. Hence, one needs to be able to determine a relationship between applied loads on the lithosphere and the amount of bending such loads generate. Studies have shown that, over very long timescales, the lithosphere flexes like a thin, elastic plate with thickness \(T_e\). Thus, this chapter introduces and develops elasticity theory when applied to thin plate flexure, deriving partial differential equations that model the flexure. It then provides solutions to those equations using scenarios where the loading occurs both at the surface of the lithosphere and within it.
Jonathan Kirby
Chapter 8. Gravity and Admittance of a Flexed Plate
Abstract
The estimation of effective elastic thickness (\(T_e\)) is undertaken by comparing two surfaces: the topographic surface and the relief on the crust-mantle boundary—the ‘Moho’—or some other intra-crustal interface. While topography is readily measured, the Moho is not, so its gravity signature is used as a proxy as gravity observations have near-global coverage. This chapter introduces and develops potential theory, the framework upon which estimates of the Moho relief may be made from surface gravity observations. It also shows how free-air and Bouguer gravity anomalies are constructed, and develops theoretical equations for the gravity anomalies and admittances of the loading scenarios introduced in Chap. 7.
Jonathan Kirby
Chapter 9. The Load Deconvolution Method
Abstract
In a seminal paper, Forsyth (1985) proposed that effective elastic thickness (\(T_e\)) estimates derived from the admittance under surface-only loading models would be underestimates of the true value. Instead, he proposed inversion of the coherence between Bouguer anomalies and the topography, against a predicted coherence constructed from a combined surface and internal loading model. This procedure—which involves recreating the two initial surface and internal loads from observed gravity and topography data and an assumed \(T_e\) value—is known as load deconvolution, or just ‘Forsyth’s method’. In this chapter we will derive the equations of the combined loading model, as well as those for the predicted coherence, and will learn, step-by-step, how best to implement load deconvolution using multitapers and wavelets. We will also investigate violations of the assumptions of the load deconvolution model—correlated initial loads—and how they might affect \(T_e\) estimates.
Jonathan Kirby
Chapter 10. Synthetic Testing
Abstract
Synthetic testing is a robust way to ascertain the accuracy of \(T_e\)-estimation methods. The equations of plate flexure enable the computation of post-flexure gravity anomalies and topography from known but synthetic initial loads and a known \(T_e\) distribution, which can be uniform or spatially-variable. Once obtained, the synthetic gravity and topography can be used to recover \(T_e\) using the chosen analysis method. The differences between known and recovered \(T_e\), perhaps averaged over many independent models, then provide a measure of the analysis method’s accuracy. Here we use random, fractal surfaces as the initial surface and internal loads. We investigate the case where \(T_e\) is constant—enabling use of the Fourier transform to generate synthetic gravity and topography—and where \(T_e\) is spatially variable, which requires a finite difference solution of the thin, elastic plate PDE.
Jonathan Kirby
Chapter 11. Practical Estimation
Abstract
This last chapter concerns the application to actual data of the theory discussed so far. Some different gravity and topography models are presented, accompanied by a discussion on how model resolution affects the minimum estimable \(T_e\). Auxiliary data sets such as sediment and crustal thickness and density are also addressed, as well as the conversion of topography to rock-equivalent topography. The effect of Bouguer reduction density and gravimetric terrain corrections upon \(T_e\) estimates is studied, and a brief analysis of the sensitivity of \(T_e\) to crust and sediment models is performed. The chapter then addresses the load ratio, load deconvolution using the admittance instead of coherence, \(T_e\) errors, noise detection techniques, and concludes with a comparison of multitaper and wavelet methods.
Jonathan Kirby
Backmatter
Metadata
Title
Spectral Methods for the Estimation of the Effective Elastic Thickness of the Lithosphere
Author
Jonathan Kirby
Copyright Year
2022
Electronic ISBN
978-3-031-10861-7
Print ISBN
978-3-031-10860-0
DOI
https://doi.org/10.1007/978-3-031-10861-7