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

2.2. INVERSION OF LINEAR PROBLEMS 19
Equation (2.18) shows that the SVD allows the reduction of the forward operator G to its
efficient state of information. The forward problem may use the portion of the model space
constrained to V p to model the data contained in U p only. The other portion of the data U o
cannot be modeled.
Solution estimate of the inverse problem
We have seen that the SVD allows the diagnosis of the forward problem by identifying the
portion of the data and model spaces U p , V p . The SVD technique may also be used to estimate
a solution of the inverse problem. By reformulating the state of information with the perspective
of inversion, the sub-space
• V p is constrained by U p only
• V 0 is undetermined
• U 0 is useless
Since V p can only be determined using the information contained in U p only, we define the
generalised inverse G † as
G † = V p Λ −1
p Ut p . (2.19)
The generalised inverse G † is an estimate of the inverse matrix G −1 (which may not exist, see
table 2.1). The estimate of the inverse problem solution m † is then defined as
m † = G † d. (2.20)
In order to evaluate how well the model is resolved, equation (2.4) may be included into equation
(2.20) yielding
m † = G † Gm
= R m m (2.21)
where R m = G † G is the model resolution matrix. We can also measure how well the estimated
solution explained the data by defining d † , the data predicted by the generalised inverse solution
d † = Gm †
= GG † d
= R d d (2.22)