Regularization of the AVO inverse problem by means of a ...

CHAPTER 3. BAYESIAN INVERSION APPROACH AND ALGORITHMS 36
solution is given by
m = (L T C −1
d L + µQ(m))−1L T C −1
d
d. (3.19)
This problem can be solved by setting a certain number of iterations to update the weighting
term from previous iteration and treating the problem as a weighted least square at each
iteration.
Table 3.3: IRLS Algorithm
1: Input data, d.
2: Construct the operator, L.
3: Select µh.
4: Initialize the solution, m.
5: Set the maximum iteration to up-date Q ( ∼ = 3 to 5 iterations).
6: Calculate Q(mk ),
7: use the result in step 6 to find,
m k+1 = (L T L + µQ(m k )) −1 L T C −1
d d,
8: repeat step 6 and 7 until the maximum iteration is reached.
In the above three possible least square problems, all involves solving the inverse of a matrix.
If the size of the matrices is small to be manipulated easily, one can set up the problem in
matrix form and use any matrix inverse solver such as Gauss-Seidel or Gaussian-Elimination
or matrix form of the Conjugate Gradient algorithm shown in Table 3.4 (Shewchuk, 1994). If
the size of the problem is large, solving the problem by setting the operators in matrix form
needs a lot of computer memory. This kind of problems can be solved using iterative inverse
solvers such as Conjugate Gradient algorithms which uses forward and adjoint operators in
function form to replace the matrix operators.
3.5 Target oriented AVO inversion
In most geophysical problems, the targets could be identified from the acquired data. If this
is the case, target oriented inversion is advantageous to reduce computational time of the
inversion process and to clearly understand the result at region of the target. If inversion
is important through out the available data set, window by window inversion can be done.
This is same as breaking up a large problem in to pieces of small problems which can be
easily manipulated in matrix form. One advantage of using window by window inversion
is to avoid problems resulting from treatment of very large and small model parameters in