2.2.2. Global Space-Varying Deconvolution

Equation (8) shows that the point spread function is exactly the image of the scattered wave radiated from the scattering point, which has a limited distribution in the space. Thus, the image-domain LSM can be based on the framework of the region division strategy, where the global point spread function can be divided into *n* blocks and the real impact of

each point spread function is constrained within local neighborhoods *sn* to the scattering point *xi*, *i* = 1, . . . ,*sn*,

$$K(\mathbf{x}\_{\text{tr}}, \mathbf{x}\_{i}, \boldsymbol{\omega}) = L(\mathbf{x}\_{\text{tr}}, \boldsymbol{\omega}; \mathbf{x}\_{\text{s}}, \mathbf{x}\_{\tau})^{T} [L(\mathbf{x}\_{i}, \boldsymbol{\omega}; \mathbf{x}\_{\text{s}}, \mathbf{x}\_{\tau}) \delta(\mathbf{x} - \mathbf{x}\_{i})].\tag{14}$$

Furthermore, within the neighborhood, the media velocity and illumination conditions are slowly varied, thus the PSFs for each position in *sn* can be assumed to be the same, and the reflectivity model for the neighborhood can be expressed as

$$m\_n = \sum\_{i \in s\_n} I\_i \otimes \mathbf{K}\_i = I\_n \otimes \mathbf{K}\_i. \tag{15}$$

Equation (15) means the regional reflectivity model for the neighborhood *sn* can be obtained by performing local deconvolution of the regional RTM image and the point spread function. The entire model can be obtained by stacking all the regional parts, and the global space-varying PSF can be computed using only one-time demigration and migration.
