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� Go back to the observations again and look at the deviation from the model.
Objective : we need to find a model that minimize δ t
Errors caused by the model and the source location are usually combined together, and the noise is
assumed to be white, with a Gaussian distribution.
If we know the 1D model, we can apply Snell’s law to estimate the geometry:
T
obs
dl
c x 3 D
Ray
Path
Note that
∇T x ( ) = k
c x( )
If c change, the ray path changes. We end up with a nonlinear problem. Thus, we should try to
linearize the inversion, using the Fermat’s Principle.
If we change a little bit the ray path around the optimum, we’ll end up with a small change in
travel time. We have two kinds of “deviations” from the reference ray:
- First contribution: effect of changes in velocity Δc.
- Second contribution: effect of change in the ray path. Fermat’s principle says that we can ignore
it.
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March 10/12, 2008 Adapted from notes 2/22, 3/28, 2005
Linearization of the travel time
Travel time residual :
Observation =δ t
3 D
= T
obs
− T
ref
dl −
dl 0
x true
c ( x)
reference
c 0
3 D 3 D
structure path
Fermat ' sPr inceple 1 1
dl 0
dl 0
c x c x reference
reference 0
3 D 3 D
path path
2
c
dl 0
( s^ ( ) −^ s
0
( x^ ))
0
x dl
c reference 0 reference
3 D 3 D
path path
= Δs x dl
( (^ ))
0
reference
3 D
path
In linearizing the problem, we get rid of the unknown ray. We can do our calculation in a reference
earth model.
� The travel time tomography is an iterative process:
� Create 1D model
� Ray tracing and get new rays in the model
� Update ray geometry
� Get the reference ray related to the 3D
t = T obs
(3D)
(The reference model does not have to be a 1D model.)
Linearization of the hypocenter mislocation
with t 0
the origin time, ( x y z, , ) the location of the earthquake.
Then we try to solve for Δs , δ t , δ x , δ y , δ z.
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March 10/12, 2008 Adapted from notes 2/22, 3/28, 2005
where
G
i
: Green functions , solution of a point source. We need to do a convolution with a point
perturbation in order to get the observations.
M : Number of model parameters.
N : Number of observations.
In general, M ≠ N. As a consequence, the matrix A is not square.
Multiplying the equation by A
T
, we can get the solution:
Back to our specific inverse problem:
One way is to take k
h as a series of cells/blocks, with a value for x inside the cell k and zero
otherwise. We have
i
t = Δsdl
1
M cell
k
k
s
=
ik
dl
where
i : event-station pair.
ik
dl : path length.
Rewrite the equation in the matrix form:
9
March 10/12, 2008 Adapted from notes 2/22, 3/28, 2005
⎡ Δl 11
… Δl 1 M
⎤ ⎡ Δs 1
⎤ ⎡^ δ^ t
1
A m = ik
⎢Δl … Δl ⎥ ⎢Δs ⎥ ⎢ δ t ⎥
⎣ N 1 NM ⎦ ⎣ M ⎦ ⎣ N ⎦
where each ray gives a row in the matrix.
We have an average wavespeed along the ray. In order to construct a model vector, we need to get
data from different rays crossing each other.
A is a sparse matrix. If we look at one ray:
T
A will have only ~100 elements non-zero. The good thing about sparse matrix is that A A is
approximately diagonal. The problem is that there are many singularities, which make the
inversion unstable (in that case, we need to add a damping factor or regularize the problem). One
possibility is to not use cells of the same size. Consequently, it reduces the number of cells; the
inverse matrix is less singular. Nevertheless, the computation time increases.
Another way is take h k
as spherical harmonics (in global seismology).
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