Geology 5670/6670 Inverse Theory 18 Mar 2015 © A.R. Lowry 2015 Last time: Review of Inverse Assignment 1; Constrained optimization for nonlinear problems Quadratic & linear programming approaches are not applicable to iterative inversion of nonlinear problems Instead, may use Constrained Optimization approaches including:  Projection (“reflection”) at a bound (nonlinear programming)  Penalty functions added to the objective function:  Parameter transformations (Positivity constraints using log(m) or Bounds using the erf function)

Can we use stochastic inversion in the context of nonlinear inversion? Recall that linear stochastic inversion solves for the difference in model parameters from an a priori expected value,, given a model parameter covariance matrix and using the generalized inverse: One could do this also with e.g. a gradient search algorithm, using a starting model, but since both SI and the Taylor series nonlinear inversion calculate a model parameter perturbation  m, this would take the gradient search from an iterative algorithm to a one-step approach… Unless either the model expected values or covariance matrix was updated after each iteration!

Cooperative Inversion: (Also called “ Joint Inversion ”) Suppose we have two very different sets of data that can describe overlapping sets of model parameters. Geophysical example: A cylindrical body with known radius 10 m, from both gravity and magnetic data: Gravity: Magnetics: Then we create a new objective function: z x ,k,k

The trick lies in appropriately weighting the two pieces of the objective function. (The two contributions must be weighted somehow, because the relative “scaling” of the measurements inevitably is “arbitrary” unless the expected value and variance of each measurement is somehow independently known). Ideally the measurement variance is known, in which case we should use: If measurement errors are not known a priori, then have to define the weight-scheme empirically (i.e., trial & error)…

E.g. joint inversion of seismic P and S reflections: Synthetic seismograms from well logs using Zoeppritz’ equations Margrave et al., Leading Edge, 2001

E.g. joint inversion of seismic P and S reflections: Inverted P-wave impedance at top of a producing channel: