1. Geology 5670/6670 Inverse Theory 26 Jan 2015 © A.R. Lowry 2015 Read for Wed 28 Jan: Menke Ch 4 (69-88) Last time: Ordinary Least Squares (   Statistics) The expected value of is: Based on this, if (N – M) is “large” we can estimate unknown data variance as: If data variance is known a priori, we can calculate the chi-squared parameter of fit as: (so called because a sum of squared r.v.’s follows a   distribution). This can be used to evaluate fit and adjust parameterization…

2. The   2 parameter is commonly used to evaluate data fit & optimize the choice of number of parameters : 1) If, can safely add more model parameters 2) If, too many parameters (model is fitting noise). Solution appraisal : What is the range of solutions? Assume: zero-mean, Gaussian, uncorrelated errors Estimate: Confidence intervals expressed as %: 100(1–  )% (i.e.,  is probability the true value falls in the conf interval). Case 1: Data error variance is known ( =   2 ) -z+z 1-  /2 Desired confidence interval is ±z of the normal ( z ) distribution function Can get this from standard statistical tables or codes

3. Suppose we want the 95% confidence interval: Typically we use the F -distribution for F = 1 –  /2 95% conf  1 –  = 0.95 1 –  /2 = 0.975 Looking up on a standard table, find F(z) = 0.975 when z = 1.96 (i.e. not quite 2  ). Case 2: Use estimated error variance from (Look up the t -distribution as you would z -distribution in math probability tables, or use corresponding routines in Matlab or other stat codes).

4. For a multi-parameter linear model, in reality we have confidence regions : hyperellipsoids in a multidimensional model parameter space m1m1 m2m2 E min E.g. Mars…

5. Another example: Gravity modeling from the Lowry & Pérez-Gussinyé approach to joint inversion of crustal thickness & V P /V S. Here, confidence intervals were estimated by varying H & K at one seismic site (a nonlinear problem, so not perfectly elliptical).

6. Note that some nonlinear problems can have rather pathological misfit error functions… Especially if sampling is sub-optimal.

7. Confidence on Linear sol’ns… To estimate confidence regions from contours of (G  m) T G  m : Example 1 : Given known   2, the confidence region is defined by where is the inverse F -distribution with M,  DOF Example: to get 95% confidence for M = 10,