1. Geology 5670/6670 Inverse Theory 21 Jan 2015 © A.R. Lowry 2015 Read for Fri 23 Jan: Menke Ch 3 (39-68) Last time: Ordinary Least Squares Inversion Ordinary Least Squares (OLS) solves for parameters m that minimize the L 2 -norm of misfit residuals Solution for an overdetermined ( N > M ) linear problem is given by: The matrix is the pseudoinverse of G The misfit residual provides some intuitive insight into measurement error… *****

2. What does this tell us about uncertainty? First, if #obs N is “large enough” relative to M, misfit tells us something about errors in measurements! We use that to estimate parameter uncertainties. Statistical Properties: First denote pseudoinverse: And recall. Thus, Inverse Theory : Goals include (1) Solve for parameters from observational data ; (2) Determine the range of models that fit the data within uncertainties

3. Quick review of Gaussian distributions & statistics : Central Limit Theorem : The sum of N independent, random variables approaches a Gaussian distribution for N sufficiently large. Univariate case: Where we define: Mean :, the expected value of x Variance : is the Probability Density Function

4. For the multivariate case, the probability density function is Mean : Data Covariance Matrix : (Note outer product yields a matrix!) Independent random variables: Uncorrelated random variables: (Strong condition)

5. Based on these definitions, (1) If we assume errors are zero-mean, (i.e., if measurements are unbiased, the model parameters are unbiased) (2) The model covariance matrix *****

6. For the ordinary least squares case, (Note because it is symmetric, and for symmetric A If measurement errors are uncorrelated, constant variance: and *****

7. So we can estimate a parameter variance for each model parameter: And we write