1. Geology 5670/6670 Inverse Theory 16 Mar 2015 © A.R. Lowry 2015 Last time: Review of Inverse Assignment 1 Expect Assignment 2 on Wed or Fri of this week!

2. Assignment 1: Key to the assignment Q1 : OLS inversion for mantle heat flow Q m & radiogenic length l rad : 1.i. Use ordinary least squares to find the model parameters for Q s = Q m + A 0 l rad : Let from column 1, ; ; and If the units are correctly scaled, this gives m 1 = Q m = 43.701 mW/m 2 and m 2 = l rad = 19.457 km. 1.ii. Assume variance from given values and estimate formal parameter uncertainties: Note there are three issues in what I gave you! (1) Should use mean variance, not mean of  ! (2) from column 4, not column 3! (3) And these errors are expressed in mW m -2 so must also be converted to MKS units. But, using that and, uncertainty in Q m is mW/m 2 and in l rad is km.

3. 1.iii. Assume measurement error is unknown and estimate from the misfit: We calculate a residual and use   statistics to estimate the data variance via: This yields   = 21.1 mW m -2 (as compared to 37.1 mW m -2 if we use the mean of variance from the inverse of values in column 4 of the file). Using that estimate of data variance, the model uncertainties are mW m -2 and km. If we use the a priori estimate of data variance, the   parameter of misfit is. This probably reflects a “correlation” that has been introduced by interpolating heat flow to a large number of locations from a small number of measurements (and that is not reflected in our representation of error).

4. 1.iv. We’ve already looked at the square-root of the model parameter variances; in addition, the model covariance matrix exhibits a negative covariance. It is difficult to interpret what that means (other than a negative cross-correlation) without normalizing. The correlation matrix is: That is, the parameter estimates are highly cross-correlated (and so, very suspect!) From these results, I would say that (1) the parameter variances are surprisingly small (primarily because of the very large N ) but (2) that doesn’t help us much, because the parameters are so strongly cross-correlated that the parameter variance is only a conditional uncertainty (i.e.  of one parameter is conditioned on the premise that the other parameter estimate is correct!)

5. Q2 : WLS inversion for mantle heat flow Q m & radiogenic length l rad : 2.i. Use weighted least squares to find Q m & l rad : Now the pseudoinverse is given by where the measurement covariance matrix C   has 1/  i 2 (column 4 of the data file) along its main diagonal. (This can be done in Matlab/Octave with smaller memory & computation using C_eps=sparse(diag(dat(:,4)))*1e6; where dat is the name of the variable the file was loaded in). This gives m 1 = Q m = 45.476 mW/m 2 and m 2 = l rad = 19.049 km (compared with 43.7 mW/m 2 and 19.5 km for OLS!) 2.ii. Using,  uncertainty in Q m is ±0.311 mW/m 2 and in l rad is ±0.263 km. The   parameter of misfit is now 0.778, much closer to 1 we got using an averaged estimate of variance, and showing that using an accurate representation of data uncertainty is important to getting trustworthy results.

6. Although the changes resulting from using WLS instead of OLS were small (~1.8 mW m -2 and 0.4 km for Q m and l rad respectively), they were large relative to the formal uncertainties in each estimate. The improvement of the  2 parameter (along with basic theory) suggests the WLS is a better estimate, but the problematic high parameter cross- correlations are about the same in both cases (–0.9327 for OLS; –0.9325 for WLS). Q3 : The scatter is very large on the plot and the lines don’t pass through the densest part of the cloud, suggesting model inaccuracies and outlier effects… But there does appear to be a significant relationship!

7. Q4 : Grid-search analysis of parameter error (using WLS): The figures at left color-contour the WRMS misfit as a function of model parameters near the best-fit case. The grey contours show  and  confidence intervals for (top) the likelihood ratio method: and (bottom) the model length: for  & . The contours are identical (as expected). Also shown are the  bounds from C m as white bars (top); interestingly the l rad bounds stop at the contour but the Q m bound is larger…

8. … Probably reflecting the coordinate transformation from the error ellipse axes to the parameter coordinate axes. Regardless these closely match the expected elliptical form. These plots were made by first plotting E WRMS using surf, setting “hold on”, plotting EWRMS or the model length using contour within the same axes, printing to an epsc file format and editing in Illustrator (to remove the opaque background of the contour plot). The matlab/ octave shellscript is provided on the course website for reference.

9. Constrained Optimization: Suppose we have inequality constraints on a nonlinear problem (!). Quadratic programming applied to a Taylor-series approximation (iterative) approach can be both computationally expensive and subject to breakdown! Instead, may adopt a nonlinear programming approach: (1) Projection Methods : mimi uiui mjmj If model update would cross constraint, “reflect” back… When minimum is outside permissible region, converges to the minimum on the boundary.

10. (2) Penalty Functions (“ Barrier Functions ”) Modify the objective function such that where e.g. mimi mjmj uiui Minimization becomes: Suppose k = 1 : where Then

11. Can also use parameter transformations : (1) Positivity constraints  m i ≥ 0 Transform as, e.g., Then minimize the misfit error norm E with respect to: and transform the m ’ vector back to get m.

12. (2) Bounds Transform as where: P(m ’ i ) has the property: Again, we minimize the misfit error E with respect to and transform back to m after we’ve found m ’.