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 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

Total Time: 1h 15min
Level: Advanced

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