Geology 5670/6670 Inverse Theory 30 Mar 2015 © A.R. Lowry 2015 Last time: Bayesian Inversion Given an Uncertain Model Bayes’ theorem: suggests that a straightforward approach to accommodating an “inexact” relationship of model parameters to the data is to multiply the a priori probability density function p A (m,d) (i.e., expectation prior to modeling of the data) by the pdf of the model relationship p g (m,d) describing an inexact model g(m) = d. The probability density function dependent only on the model parameters (removing the data dependence) would be: Reading for Wed 1 Apr: posted on web

Double-difference location of earthquake hypocenters (Waldhauser & Ellsworth, 2000) Earthquake hypocenter location is a good example of an inversion problem with an “inexact” physical model, because triangulation for an earthquake source and origin time  from seismic arrival times T implies knowledge of the velocity structure of the Earth! Ray theory uses the path integral: for event i, station k, and slowness u = 1/V along the path s. Note that if velocity is non-uniform (guaranteed for any reasonable approximation of Earth structure, this problem is inherently nonlinear. The Taylor-series linearization of this is: where  m i is (  x i,  y i,  z i,  i ).

There are two significant sources of error associated with this approach: (1) The exact time of arrival of an earthquake phase is often challenging to pick, especially for small events and/or emergent phases. (2) Small errors in the model velocity structure can introduce large errors in the hypocentral estimate (especially in the estimate of depth!) One way to reduce uncertainty in the estimate of arrival time is to use cross-correlation of the arrivals. But if we do that, the observable is no longer arrival time T k i, but rather relative arrival time

In that case the first-order approximation to the inversion problem becomes where  m i is (  dx ij,  dy ij,  dz ij,  d  ij ). This is the double- difference problem, and not only does it have the advantage of smaller errors in the observable from cross-correlation picking, it also reduces the error from model velocity error to the extent events “see” the same structure.