{"title":"Elastic Solutions for Strike-Slip Faulting","authors":"D. Sandwell","doi":"10.1017/9781009024822.011","DOIUrl":"https://doi.org/10.1017/9781009024822.011","url":null,"abstract":"Today's lecture will be the mathematical development of the deformation and strain pattern due to strike-slip deformation on a partially locked fault. The notes come from Chapter 8 of Turcotte & Schubert but I'll focus on section 8-6 through 8-9. While I'll follow the overall theme of Chapter 8, I'll deviate in two respects. First I'll use a coordinate system with the z-axis pointed upward to be consistent with my previous notes on gravity, magnetics, and heat flow. Second I'll develop the solution using fourier transforms to be consistent with my previous notes.","PeriodicalId":120442,"journal":{"name":"Advanced Geodynamics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130256334","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Gravity Field of the Earth, Part 1","authors":"D. Sandwell","doi":"10.1017/9781009024822.013","DOIUrl":"https://doi.org/10.1017/9781009024822.013","url":null,"abstract":"This chapter covers physical geodesy-the shape of the Earth and its gravity field. This is just electrostatic theory applied to the Earth. Unlike electrostatics, geodesy is a nightmare of unusual equations, unusual notation, and confusing conventions. There is no clear and concise book on the topic although Chapter 5 of Turcotte and Schubert is OK. The things that make physical geodesy messy include: • earth rotation • latitude is measured from the equator instead of the pole; • latitude is not the angle from the equator but is referred to the ellipsoid; • elevation is measured from a theoretical surface called the geoid; • spherical harmonics are defined differently from standard usage; • anomalies are defined with respect to an ellipsoid having parameters that are constantly being updated; • there are many types of anomalies related to various derivatives of the potential; and • mks units are not commonly used in the literature. In the next couple of lectures, I'll try to present this material with as much simplification as possible. Part of the reason for the mess is that prior to the launch of artificial satellites, measurements of elevation and gravitational acceleration were all done on the surface of the Earth (land or sea). Since the shape of the Earth is linked to variations in gravitational potential, measurements of acceleration were linked to position measurements both physically and in the mathematics. Satellite measurements are made in space well above the complications of the surface of the earth, so most of these problems disappear. Here are the two most important issues related to old-style geodesy. Elevation Prior to satellites and the global positioning system (GPS), elevation was measured with respect to sea level-orthometric height. Indeed, elevation is still defined in this way however, most measurements are made with GPS. The pre-satellite approach to measuring elevation is called leveling.","PeriodicalId":120442,"journal":{"name":"Advanced Geodynamics","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127371081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Gravity/Topography Transfer Function and Isostatic Geoid Anomalies","authors":"D. Sandwell","doi":"10.1017/9781009024822.018","DOIUrl":"https://doi.org/10.1017/9781009024822.018","url":null,"abstract":"This lecture combines thin-elastic plate flexure theory with the solution to Poisson's equation to develop a linear relationship between gravity and topography. This relationship can be used in a variety of ways. (1) If both the topography and gravity are measured over an area that is several times greater then the flexural wavelength, then the gravity/topography relationship (in the wavenumber domain) can be used to estimate the elastic thickness of the lithosphere and/or the crustal thickness. There are many good references on this topic including Dorman and Lewis [1972], McKenzie and Bowin, [1976]; Banks et al., [1977]; Watts, [1978]; McNutt, [1979]. (2) At wavelengths greater than the flexural wavelength where features are isostaticallycompensated, the geoid/topography ratio can be used to estimate the depth of compensation of crustal plateaus and the depth of compensation of hot-spot swells [Haxby and Turcotte, 1978]. (3) If the gravity field is known over a large area but there is rather sparse ship-track coverage, the topography/gravity transfer function can be used to interpolate the seafloor depth among the sparse ship soundings [Smith and Sandwell, 1994].","PeriodicalId":120442,"journal":{"name":"Advanced Geodynamics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130831614","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Flexure of the Lithosphere","authors":"","doi":"10.1017/9781009024822.009","DOIUrl":"https://doi.org/10.1017/9781009024822.009","url":null,"abstract":"These lecture notes are basically a supplement to Turcotte and Schubert, Chapter 3. The results of the first derivation are the same as equation 3-130 in T&S but rather than guessing the general solution, the solution is developed using fourier transforms. The approach is similar to the solutions of the marine magnetic anomaly problem, the lithospheric heat conduction problem, the strike-slip fault flexure problem and the flat-earth gravity problem. In all these cases, we use the Cauchy integral theorem to perform the inverse fourier transform. Later we'll combine this flexure solution with the gravity solution to develop the gravity-to-topography transfer function. Moreover, one can take this approach further to develop a Green's function relating temperature, heat flow, topography and gravity to a point heat source (e.g., Sandwell, Thermal Isostasy: response of a Moving Lithosphere to a Distributed Heat Source, In addition to the constant flexural rigidity solution found in the literature, we develop an iterative solution to flexure with spatially-variable rigidity. Before going over these notes, please re-read section 3-9 in Turcotte and Schubert on the development of moment versus curvature for a thin elastic plate. q(x) x w(x) F","PeriodicalId":120442,"journal":{"name":"Advanced Geodynamics","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124621233","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Fourier Transform Methods in Geophysics","authors":"D. Sandwell, January","doi":"10.1017/9781009024822.003","DOIUrl":"https://doi.org/10.1017/9781009024822.003","url":null,"abstract":"1. Fourier Transforms Fourier transform are use in many areas of geophysics such as image processing, time series analysis, and antenna design. Here we focus on the use of fourier transforms for solving linear partial differential equations (PDE). Some examples include: Poisson's equation for problems in gravity and magnetics; the biharmonic equation for problems in linear visco-elasticity; and the diffusion equation for problems in heat conduction. We do not treat the wave equation in this book because there are already many excellent books on seismology. For each of these problems we search for the Green's function that represents the response of the model to a point load. There are two approaches to solving this class of problem. In some cases one can derive a fully analytic solution, or Green's function, to the point-load problem. Then a more general model can be constructed by convolving the actual distribution of sources with the Green's function. A familiar example is the case of constructing a gravity anomaly model given a 3-D density anomaly structure. The second semi-analytic approach can be used to solve more complicated problems where the development of a fully analytic Green's function is impossible. This involves using the derivative property of the fourier transform to reduce the PDE and boundary conditions to algebraic equations that can be solved exactly in the transform domain. A more general model can be constructed by taking the fourier transform of the source, multiplying by the transform domain solution, and performing the inverse transform numerically. Indeed the only difference between the two methods is that in the first case the final model is generated by direct convolution while in the second case the convolution theorem is used for model generation. When dealing with spatially complex models the second approach can be sometimes orders of magnitude more computationally efficient because of the efficiency of the fast fourier transform algorithm. In this chapter we introduce the minimum amount of fourier analysis needed to understand the solutions to the PDE's provided in the following chapters. If the reader is not familiar with fourier transforms and analysis, they should first study any of the excellent books on the topic. We recommend the first 6 chapters of the book by Bracewell [1978] for a more complete discussion of the material presented in Chapter 1.","PeriodicalId":120442,"journal":{"name":"Advanced Geodynamics","volume":"85 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115303117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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