Linear forward operator $({\bf L})$

Since in the conventional surface seismic experiment, geophones are located only at the surface, the data is the receiver wavefield at z=0. This can be represented in equation (9) as a truncation operation (transpose of zero padding ${\bf Z}$):  

$${\bf d}= {\bf Z}^t{\bf P}^-={\bf Z}^t{\bf B}^-{\bf P}^+\, \Sigma^t_{\omega s} \, {\bf r}.$$ (11)

Then the expression for linear shot-profile modeling one-way wave equation operator is  

$${\bf L}= {\bf Z}^t{\bf B}^-{\bf P}^+\, \Sigma^t_{\omega s},$$ (12)

a chain of linear operators.