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Geology 5670/6670 Inverse Theory 13 Feb 2015 Last time: Nonlinear Inversion • Given a nonlinear problem, i.e. of the form F(m) = d, we can solve by one of several approaches: (1) Apply a linearizing transformation. Note this same transformation will also be applied to errors! (2) Grid search (or Monte Carlo; simulated annealing). Brute force; this can be computationally expensive for M large. (3) A gradient search method. Iteratively solve m k1 G k d k Fi m k in which Gk consists of sensitivity k Gij coefficients: m j Read for Tue 17 Feb: Menke Ch 9 (163-188) © A.R. Lowry 2015 The gradient search approach to solution of a nonlinear inverse problem, F(m) = d, can be summarized by the following algorithm: 0. Choose a starting model, m0 1. Calculate model misfit dk = F(mk) – d m k1 G k d k 2. Evaluate Fi m k k where Gk is the kth sensitivity matrix Gij 3. Update the model mk+1 = mk + mk+1; 4. Set k = k + 1 and iterate m j This approach uses the first-order term of the Taylor series to approximately linearize the problem… Note that most of the same tools we used for the linear problem (e.g., estimates of parameter error from the parameter covariance matrix; model resolution and covariance matrices; parameter etc.) still apply to the iterative solution for a nonlinear model… The main difference being that we apply these metrics to the final (iterated) model estimate using the sensitivity matrix G of the best-fitting model. Complications of the general nonlinear problem include multiple minima (resulting in starting-model dependence of the solution), and nonconvergence: Example: Solving for fault slip from displacement time series at just one GPS site: Could see evidence for transient fault slip; wanted to know where and how it moved as a function of time… So modeled as a slip patch with length L, total slip U, moving along-strike with velocity V, centered at a distance y from the GPS instrument… Using grid search. It’s not clear that a slip pulse initial model with an eastward propagation and/or seaward centroid would ever converge to the global minimum. These kinds of problems are common (particularly when measurement sampling is less-than-ideal as in this case). Here the global minimum is found because a fine-mesh grid-search was used, but an iterative (gradient) method would have encountered problems for most starting models. The algorithm we derived last time: Choose a starting model m0 d k d F m k Gijk G Fi m k m j T k T m k F m k T 1 T m k G k G k G k d k is often referred to as the Gauss-Newton algorithm. m k 1 m k m k Note this approach may get divergent answers (i.e., the method will move the model in directions that produce larger model misfits) if the sensitivity matrix is unstable (i.e., the smallest eigenvalues are extremely small). One way to “stabilize” the solution is to control (reduce) the parameter step so that the iterative method does not “overstep” into another minimum domain… E.g., a truncated Gauss step: T 1 T m k 1 m k k G k G k G k d k in which k < 0.5 is a scalar step parameter. Problem: We would like to travel in the direction of “true steepest descent”. However that may be very different from the direction of a derivative at a point! Recall the Generalized Inverse: Can instead use 1 T m k1 m k V p p U p d k where p is the number of “non-zero” eigenvalues. This is more robust but also more time-consuming… Of course can also stabilize using something similar to the “other approach” used previously to stabilize linear inverse problems, damping: We call this Levenberg Damping: T 1 T m k 1 m k G k G k k I G k d k Or Levenberg-Marquardt damping: 1 T T T Gk Gk kdiagGk Gk Gk d k mk 1 mk Here we choose k heuristically based on trial & error (i.e., pick a that maximally decreases the residual norm). Note large results in a smaller step size (because the determinant of the matrix being inverted will be larger); small is more similar to Gauss-Newton…