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

2.2. INVERSION OF LINEAR PROBLEMS 27
local methods will be identical to the generalised inverse solution. This solution nevertheless
requires the dual implementation of the SVD and Gauss-Newton in order to discriminate the
model null-space. An alternative is to damp the Hessian by adding a constant along its diagonal
such as
H d = H + κI (2.46)
where κ is a real, positive number. Because H d does not have eigenvectors with zero eigenvalues,
it is always invertible. Applying a damping term to the Hessian is equivalent to finding the
Gauss-Newton solution that minimises a new weighted misfit function defined as
E = 1 (
∆d t ∆d + κ∆m t ∆m ) (2.47)
2
which minimises both the data residuals and the model length update in proportion to κ.
Gauss-Newton solution and generalised inverse
It can easily be demonstrated that the matrix H −1
p Ft p is in fact the generalised inverse obtained
from the SVD since we have
H −1
p F t p = V p Λ −1
p
= G †
The Gauss-Newton solution is in fact equivalent to the solution given by the generalised inverse
U t p
when there is no a priori knowledge of the model (m o = 0).
2.2.2.4 Preconditioned gradient methods
We have seen that the component of the gradient along the space eigenvectors are the component
of the true model perturbation stretched by the eigenvalues of H. In order to mitigate this effect
without recourse of the inverse Hessian, it is possible to precondition the gradient direction by :
1. Adding a vector to the gradient
2. Weight the data residuals
3. Multiply the gradient by a matrix
The preconditioned gradient direction can always be associated with an equivalent misfit function
E, in which the preconditioned gradient is the steepest ascent. Therefore, an equivalence
exist between modifying the misfit function or the gradient direction.