{ "metadata": { "name": "", "signature": "sha256:8f2150f8c116d400ac97daf6d79ffccac1c0924a1b19f62f8934661208f3ef15" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Module 2: Inversion\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In the previous module we started with a continuous distribution of a physical property and discretized it into many cells, then we performed a forward simulation that created data from known model parameters. Inversion, of course, is exactly the opposite process. Imagine each model parameter that we had represents a layer in a 1D layered earth. At the surface of the earth we measure the data, and when we invert we do so for the model parameters. Our goal is to take the observed data and recover models that emulate the real Earth as closely as possible. \n", "\n", "You may have noticed that the act of discretizing our problem created more cells than data values. In our latter example we produced 20 data points from 1000 model parameters, which is only a few data points and many model parameters. While this was not much of a problem in the forward simulation, when we want to do the inverse process, that is, obtain the model parameters from the data, it is clear that we have many more unknowns than knowns. In short, we have an underdetermined problem, and therefore infinite possible solutions. In mathematical terms, geophysical surveys represent what are called \"ill-posed\" problems. \n", "\n", "An \"ill-posed\" problem is any problem that does not satisfy the requirements for the definition of \"well-posed\" problem. A *well-posed* problem is a problem in mathematics that must satisfy all three of the following criteria:\n", "\n", "