Transcript Slide 1

Geology 5670/6670
Inverse Theory
30 Mar 2015
Last time: Bayesian Inversion Given an
Uncertain Model
Pd | m Pm
• Bayes’ theorem: Pm | d 

Pd
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 pA(m,d)
(i.e., expectation prior to modeling of the data) by the
pdf of the model relationship pg(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:
     
pm 
pA m,d pg m,d dN d 
Reading for Wed 1 Apr: posted on web
  
p m,d dN d
© A.R. Lowry 2015
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:
Ti   
k
i

k
i
uds
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:
i
i
tki
m  tobs  tmod k where mi is (xi, yi, zi, i).
m
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 Tki, but rather relative
obs
ij
i
j
t

t

t
arrival time k  k k 
In that case the first-order approximation to the inversion
problem becomes
ij
ij
tkij
m  tobs  tmod k where mi is (dxij, dyij, dzij, dij).
m
This is the doubledifference 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.