The forward model

After preprocessing for multiple reflections, the monochromatic 2-D forward model for a seismic shot record can be written as follows Berkhout (1985):  

$${\bf g}_j(z_0) = F(z_0,z_0) {\bf s}_j(z_0)$$ (1)

with  

$$F(z_0,z_0) = \sum_{n=1}^{N} W(z_0,z_n) R(z_n) W(z_n,z_0)$$ (2)

In equation () source vector ${\bf s}_j(z_0)$and measurement vector ${\bf g}_j(z_0)$refer to one seismic experiment at the surface z=z₀ with source location at x=x_j. In equation (), propagation operators W(z_n,z₀) and W(z₀,z_n) quantify the full propagation effects (down and up, respectively) between depth levels z₀ and z_n, and reflection matrix R(z_n) defines the elastic angle-dependent reflection properties caused by inhomogeneities at depth level z_n. All matrices and vectors refer to one temporal Fourier component. Equations  () and  () are valid for single-component as well as for multi-component measurement of 2-D as well as 3-D data.