The central role of resolution in inversion has been emphasized by Backus and Gilbert (1968), who base their general inversion methods on techniques designed to optimize the resolution of the resulting geophysical model obtained from processed data. Indeed, any inverse method based on incomplete or imperfect data should be expected to produce a filtered or blurred image of the object or region of interest. Apparent resolution is therefore an important figure of merit for imaging and inversion schemes, since the user of such schemes will eventually want to know what is the smallest object that can be distinguished.
Tomographic reconstruction schemes have well-defined resolution properties (Jackson, 1972). Yet, these properties are seldom computed because of the common misconception (Nolet, 1985) that resolution matrices can only be computed using singular value decomposition, which is generally the most computationally intensive of all matrix inversion methods. Other matrix inversion methods such as conjugate gradients, conjugate directions, Lanczos, and LSQR (van der Sluis and van der Vorst, 1987) are more commonly used in tomography codes, because of their smaller storage and computing requirements. It would therefore clearly improve the state of our knowledge about tomographic resolution in practice if methods for computing resolution matrices were available for all (or at least for the most common) iterative matrix inversion schemes.
I show in this paper how the resolution can be computed for two matrix inversion schemes (Lanczos and LSQR). The techniques developed here can be generalized to other iterative inversion schemes as well.