Lattice basis of seismic wavefields

We choose to represent a seismic wavefield, W, by dissociating the underlying interpolated acquisition lattice (IAL), ${\cal L}$, from the discretely sampled values of the continuous wavefield, f^W. Lattice ${\cal L}$ is defined over experimental source and receiver coordinates, $\r$ and $s_{\xi}$, and time and depth coordinates, $\t$ and $z_{\xi}$. Importantly, the underlying grid function is only a construct of the experimental process and is associated, accordingly, with model space coordinates ${\cal L}(\r,s_{\xi},\t,z_{\xi})$.This stands in contrast to the continuous wavefield function defined by corresponding continuous variables, f^W(r,s,t,z). Written in this manner, it becomes natural to associate the act of observation with the mapping from physical (continuous) to experimental (discrete) variables,  

$$W(\r,s_{\xi},\t,z_{\xi})={\cal L}(\r,s_{\xi},\t,z_{\xi})\:f^W (r,s,t,z)\:\d(\r -r)\:\d(s_{\xi}-s)\:\d(z_{\xi}-z)\:\d(\t-t).$$ (6)

Note, that we define W as the entire experimental wavefield including traces from all source and receiver pairs. This volume is separable into many different subsets, but we will keep it in tact. One way to represent lattice ${\cal L}$ is with a 4D infinite sum over delta functions (i.e. a 4D Shah function),  

$${\cal L}(\r,s_{\xi},\t,z_{\xi})=\sum_{u_r,u_s,u_t,u_z=-\inft... ...(\t-a_t u_t \Delta t_{\xi}) \d(z_{\xi}-a_z u_z \Delta z_{\xi}).$$ (7)

In equation (7), variables (a_r,a_s,a_t,a_z) are subsampling factors over the fundamental discretization intervals $(\Delta r_{\xi},\Delta s_{\xi},\Delta t_{\xi},\Delta z_{\xi})$, and (u_r,u_s,u_t,u_z) are the associated summation indicies of the delta functions. It is assumed that for our ideal grid $\Delta r_{\xi}= \Delta s_{\xi}$ and any departures from this equality may be represented through the subsampling factors. Throughout the development, unless specified otherwise, the summation from $-\infty$ to $\infty$ in equation (7) is assumed. Due to the fact that no experiment is ever carried out with infinite extent, padding the wavefields to infinity with zero traces maintains the rigor of this evaluation.