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

2.2. INVERSION OF LINEAR PROBLEMS 21
The minimisation of the function E (m) is achieved by finding the model parameter m † such
that E ( m †) is minimum. The minimisation is local: the quantities involved in the optimisation
of the misfit function depend on the starting model m o . All local methods require the calculation
of the local gradient of the misfit function ∇ m E, indicating the directions of increase of the
misfit function : the steepest ascent. The gradient vector contains the first derivatives of the
misfit function E (m) with respect to the model parameters m and is defined as
∇ m E = ∂E
∂m . (2.25)
By deriving the misfit function expressed in equation (2.24), the gradient is given by
∇ m E (m) = F t ∆d (2.26)
where F is the Fréchet derivative matrix: the derivative of the data with respect to the model
F = ∂d
∂m . (2.27)
For linear problem, the Fréchet derivative matrix is constant with respect to m and expressed as
F = ∂d
∂m = ∂ (Gm) = G, (2.28)
∂m
is simply the forward operator G and is therefore independent of m.
2.2.2.1 The Gauss-Newton method
Since the forward problem is linear, the misfit function is exactly quadratic and can be expressed
as a Taylor series of the second order
E ( m †) = E (m o + ∆m) = E (m o ) + ∇ m E t ∆m + 1 2 ∆mt H∆m (2.29)
where H is the Hessian, the second derivative matrix defined as
H = ∂ (∇E)
∂m
= ∂Ft (∆d · · · ∆d)
∂m } {{ }
+F t F. (2.30)
n m
For linear problem, as the Fréchet derivative matrix is independent of m, the Hessian is given
by
H = F t F. (2.31)