Sirgue, Laurent, 2003. Inversion de la forme d'onde dans le ...

2.2. INVERSION OF LINEAR PROBLEMS 15
The resolution of the waveform inversion problem, implemented in the frequency domain will
be introduced in the final section.
2.2 Inversion of linear problems
For linear system, the forward problem is a linear combination of model parameters and can be
expressed in discretised form as
Gm = d (2.4)
where m is the model vector of dimension (n m × 1), and d is the data vector of dimension
(n d × 1). G is the (n d × n m ) forward operator matrix of coefficients independent of m, that
projects a model vector into the data space. Finding the solution to equation (2.4) consists in
finding the vector m † that explains the observed data d obs .
The direct method consists in finding the inverse matrix G −1 of the matrix G such that the
inverse solution is
m † = G −1 d. (2.5)
The resolution of the inverse problem using direct methods thus implies that the matrix inverse
can be found or at least estimated. The inverse matrix G −1 is defined as
and/or
G −1 G = I nm for n m ≤ n d
GG −1 = I nd for n m ≥ n d
(2.6)
where I n is the identity matrix of dimension (n × n).
The inverse matrix G −1 will not exist if the matrix G is not invertible in which case a
solution of the type given by equation (2.5) cannot be obtained. Furthermore, the existence of
the inverse matrix does not assure the solution to be unique as there may be none, one or an
infinite number of solutions as shown Table 2.1 (p. 48 of Scales et al. (2001)). If G is invertible
and of dimension (n m × n d ), direct solver such as LU decomposition may be used (Press et al.,
1992) in the case of a well posed inverse problem.
In the next section, I will introduce the technique of Singular Value Decomposition (SVD)
which allows one to obtain an estimate of the inverse matrix called the generalized inverse
G † . The SVD is a powerful method since it also diagnoses the problem by providing useful
information on how well the problem is posed. It also yields an estimate of the solution m † ,
even in the case where G is not invertible.