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# Discussion

 Trivial regularization Non-trivial regularization Model-space Data-space effective model m r ]$effective data 0 ]$ d effective operator I_m ]$I_d ]$ optimization problem minimize , where minimize under the constraint formal estimate for m 3c Model-space Data-space effective model m r ]$effective data 0 ]$ d effective operator D ]$I_d ]$ optimization problem minimize , where minimize under the constraint formal estimate for m , where C-1 = DT D , where C = P PT.

I summarize the differences between model-space and data-space regularization in Table 1. Which of the two approaches is preferable in practical applications? In the case of trivial'' regularization (i.e., constraining the problem by the model power minimization), the answer to this question depends on the relative size of the model and data vectors: data-space regularization may be preferable when the data size is noticeably smaller than the model size. In the case of non-trivial regularization, the answer may additionally depend on the following questions:

• Which of the operators D or P (C-1 or C) is easier to construct and implement?
• Is an initial estimate for x available? In data-space regularization, it is difficult to start from a non-zero value of the model m.
• Is it possible to approximate or to compute analytically one of the inverted matrices in formula (27)?
The derivation of formula (27) suggests experimenting with the operator , where Wm approximates , and Wd approximates .

Ryzhikov and Troyan (1991) present a curious interpretation of the operator L C LT in ray tomography applications. In these applications, each data point corresponds to a ray, connecting the source and receiver pair. If the model-space operator C-1 = DT D has the meaning of a differential equation (Laplace's, Helmholtz's, etc.), then, according to Ryzhikov and Troyan, each value in the matrix L C LT has the physical meaning of a potential energy between two rays, considered as charged strings in a potential field. Such an interpretation leads to a fast direct computation of the matrix operator G.

The scaling parameter controls the relative amount of a priori information added to the problem. In a sense, it allows us to reduce the search for an adequate model null space to a one-dimensional problem of choosing the value of . Solving the optimization problem with different values of leads to the continuation approach, proposed by Bube and Langan (1994). As Nichols (1994) points out, preconditioning can reduce the sensitivity of the problem to the parameter if initial system (1) is essentially under-determined.

Next: Acknowledgments Up: Fomel: Regularization Previous: Stack equalization
Stanford Exploration Project
11/11/1997