-
8/10/2019 Inverse Problems in Short
1/12
Introduction to
Inverse ProblemsAlbert Tarantola
Institut de Physique du Globe de P
-
8/10/2019 Inverse Problems in Short
2/12
ProbabilityConsider a manifold M , with a notion of volume.
For
V (A) = A dv .A volumetric probability is a function f that to
any ciates its probability
P(A) = A dv f .
Example: If M
is a metric manifold endowed with so{ 1 n} h d d d 1 d n
-
8/10/2019 Inverse Problems in Short
3/12
A basic operation with volumetric probabilities is their
( f g)( P ) = 1
f (P ) g(P ) ,
where = M dv f (P ) g(P ) .
Example: Two planes make two estimations of the geocoordinates
of a shipwrecked man, represented by the tmetric probabilities f (
, ) and g( , ) . The volum bility that combines these two pieces of
information is
( f g)( , ) = f ( , ) g( , )
dS( ) f( ) g( )
-
8/10/2019 Inverse Problems in Short
4/12
This operation of product of volumetric probabilities ethe
following case:
there is a volumetric probability f (P ) denmanifold M ,
there is another volumetric probability (Q )second manifold N
,
there is an application P Q = Q(P ) from
Then, the basic operation is
g(P ) = 1 f(P ) ( Q (P ) )
-
8/10/2019 Inverse Problems in Short
5/12
Inverse Problems
In a typical inverse problem, there is:
a set of model parameters {m1, m2, . . . , mn
a set of observable parameters {o1, o2, . . .
a relation oi = oi(m1, m2, . . . , mn) predictcome of the
possible observations.
The model parameters are coordinates on the mo
manifold M while the observable parameters are
-
8/10/2019 Inverse Problems in Short
6/12
The three basic elements of a typical inverse problem
some a priori information on the model prepresented by a
volumetric probability pned over M ,
some experimental information obtained onable parameters,
represented by a volumetric p
ity obs (O ) dened over O , the forward modeling relation M
that we have just seen.
This leads to
-
8/10/2019 Inverse Problems in Short
7/12
Example I: Sampling Sample the a priori volumetric probability
to obtain (many) random models M 1 , M 2, .
For each model M i , solve the forward mproblem, O i = O i (M i
) .
Give to each model M i a probability of proportional to obs ( O
i (M i ) ) .
The surviving models M 1 M 2 are s
-
8/10/2019 Inverse Problems in Short
8/12
-
8/10/2019 Inverse Problems in Short
9/12
Then, the posterior volumetric probability for theparameters
is
post (m ) = k exp( S (m ) ) ,
where the mist function S (m ) is the sum of
2 S (m ) = ( m m prior ) t C m-1 (m m prio+ ( o (m ) o obs )
t C o-1 (o (m )
-
8/10/2019 Inverse Problems in Short
10/12
-
8/10/2019 Inverse Problems in Short
11/12
-
8/10/2019 Inverse Problems in Short
12/12
