GEO7600 Inverse Theory 09 Sep 2008 Inverse Theory: Goals are to (1) Solve for parameters from observational data; (2) Know something about the range of models that can fit the data within uncertainties Recall from last time that model parameter uncertainty can be estimated from the model covariance matrix C m : ***** Parameter variance for each model parameter is then: And:

What should we expect E min = e T e to be? Can substitute ; after lots of algebra and using the identity: We get: And: If we assume uncorrelated errors: Then:

Useful take-home points:  One can always fit the data exactly if N = M.  If your measurement errors are unknown and N–M is “large”, The latter is useful because often-times when  2 is estimated independently, we find that ! This generally indicates either (1) unanticipated noise in the measurements, (2) correlated errors or (3) (& very likely) the model is under-parameterized. Hence we define a chi-squared parameter

The  2 is commonly used to evaluate data fit and 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 : Assume: zero-mean, jointly normal, uncorrelated errors Estimate: Confidence intervals expressed as %: 100(1–  )% 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

Suppose we want the 95% confidence interval: Typically use the F distribution = 1–  /2 95% conf  1–  = –  /2 = Looking up on a standard table, find F(z) = 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).

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