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

2.2. INVERSION OF LINEAR PROBLEMS 25
where Λ 2 is a (n m × n m ) diagonal matrix with the eigenvalues values of H on the diagonal
(which may contain some zeros if a model null space exists), and zero elsewhere. For a better
understanding of equation (2.42), it can be written as a sum of the model eigenvectors such as
∇ m E =
n m ∑
i=1
(
λ
2
i v i v t i ∆m true
)
. (2.43)
The product v i vi t is an operator in the model space of dimension (n m × n m ), that projects any
model vector on the eigenvector v i (Scales et al. (2001), section 4.9). As the eigenvectors form
an orthonormal basis, the sum of the projection of the vector ∆m true on each eigenvector v i is
simply the vector itself as we have (see equation (2.10))
∆m true =
n m ∑
i=1
(
vi v t i∆m true
)
(2.44)
where each of v i vi∆m t true is the orthogonal projection of the vector ∆m true along the direction
given by the eigenvector v i .
From equation (2.43), it is clear that the gradient vector is the sum of the projected component
of ∆m true along the eigenvectors v i , multiplied by the eigenvalue of v i . Therefore, the
components of the gradient vector on the eigenvector basis are the component of the true perturbation,
stretched by the eigenvalues values of H as shown Figure 2.5. The gradient direction
is the direction of the true model perturbation in a stretched model space.
Remarks on ∇ m E and the eigenvalues of H
• If ∆m true can be expressed as a unique component along one eigenvector (with a nonzero
eigenvalue), the gradient method will converge in one iteration.
• If the eigenvectors have all the same eigenvalues, the gradient method will converge in
one iteration.
• If ∆m true has components in the null space, the gradient will only contain information
about the component of ∆m true along eigenvectors with non-zero eigenvalues.
• The solution ∆m † is the projection of ∆m true on the model subspace V p .