Making Models from Data: A (very) Basic Overview of Parameter Estimation and Inverse Theory
A (one might say the) fundamental problem in science is finding a mathematical representation, hopefully with predictive or insightful benefits, that describes a physical phenomenon of interest. In seismology, one such classic problem is determining the seismic velocity structure of the Earth from travel time or other measurements made on seismograms (seismic tomography). The document below provides lecture notes and an associated exercise to be completed in Matlab.
Resource 4 Related Resources Animations Fact-Sheets Webinars Making Models from Data: A (very) Basic Overview of Parameter Estimation and Inverse Theory 1h 15min Advanced Resource Files Lecture Notes & Exercise Download All 1MB A (one might say the) fundamental problem in science is finding a mathematical representation or model (m), hopefully with predictive or insight benefits, that describes observable data (d). If we knew the velocity structure of the Earth exactly (m), we could use basic physics principles (e.g. Snell's Law) to trace the path a seismic wave would take from a source (for instance an earthquake or explosion) to a seismometer. The calculation of the travel time (d) from the source to the seismometer is called a "Forward Problem," and can be calculated by integrating the inverse velocity (or "slowness") along the ray-path. However, we are not able to physically sample much of the Earth's interior to get seismic velocity estimates (the deepest hole ever drilled is only ~0.2% of the Earth's radius!). One of the classic problems in seismology, called "seismic tomography," is to take travel times from seismic sources to seismometers (d) and "invert" for the velocity structure of Earth (m). Seismic tomography is of fundamental importance to Earth Science as understanding the velocity structure of Earth gives us valuable information about the composition, temperature, and phase of material within the planet. A exercise introduces inverse theory by having students solve for the velocity structure of a 4 blocks in a 2x2 grid. Author: Rick Aster, Professor of Geophysics, Colorado State University (Rick.Aster "at" colostate.edu) Objectives: Students will be able to: Gain an understanding of what an inverse problem is and the fundamental steps needed to solve it Describe a simple tomography problem as a system of equations Represent a system of equations with matrices/vectors Further develop MATLAB/coding skills by setting up and solving a simple tomography example Related Animations Seismic Tomography (CT scan as analogy) Seismic tomography is an imaging technique that uses seismic waves generated by earthquakes and explosions to create computer-generated, three-dimensional images of Earth's interior. CAT scans are often used as an analogy. Here we simplify things and make an Earth of uniform density with a slow zone that we image as a magma chamber. Animation Novice Related Fact-Sheets Seismic Tomography Introductory description of how seismic tomography is used to determine Earth structure. Fact-Sheet Intermediate Related Webinars New Constraints on the Nature of the Gutenberg Discontinuity From Anisotropic Seismic Tomography This webinar, presented by Caroline Beghein of the University of California-Los Angeles, explores the origin of the lithosphere-asthenosphere boundary (LAB) and the Gutenberg (G) seismic discontinuity. Webinar Expert Seismic Tomography and Interferometry: from Shallow to Deep Seismic waves that propagate through the earth's interior are important tools to study earth structure. Large scale seismic arrays promote the development of new data analysis methods, like surface-wave tomography and seismic interferometry. This webinar, presented by Fan-Chi Lin of Caltech, demonstrates how different types of seismic maps are determined based on array analysis and how different seismic phases can be extracted from seismic noise to reveal detailed earth structure. Webinar Expert