Discussion

 

Model-space Data-space

effective model m r ]$ effective data

0 ]$ d effective operator

I_m ]$ I_d ]$

$$\hat{r}^T \hat{r} \hat{m}^T \hat{m}$$ optimization problem formal estimate for m 3c

Model-space Data-space

effective model m r ]$ effective data

0 ]$ d effective operator

D ]$ I_d ]$

$$\hat{r}^T \hat{r}$$ optimization problem minimize
under the constraint

formal estimate for m ,
where C^-1 = D^T D ,
where C = P P^T.

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 W_m approximates , and W_d approximates .

Ryzhikov and Troyan (1991) present a curious interpretation of the operator L C L^T 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 = D^T 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 L^T 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 $\mbox{\unboldmath$\lambda$}$ 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 $\mbox{\unboldmath$\lambda$}$. Solving the optimization problem with different values of $\mbox{\unboldmath$\lambda$}$ 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 $\mbox{\unboldmath$\lambda$}$ if initial system (1) is essentially under-determined.