Band-limited lattices

In a seismic experiment, measurements are necessarily acquired at discrete sampling intervals. This fact requires placing a restriction on the frequency content representable on the 4-D lattice. One convenient manner to do this is to apply a 4-D Rect function in the frequency domain to cut off frequencies greater than predefined values [e.g. $(B_\r,B_s_{\xi},B_\t,B_z_{\xi})$]. To accomplish this, equation (7) is first Fourier transformed over all variables to yield,

$${\cal L}(k_{r_{\xi}},k_{s_{\xi}},\omega_{\xi},k_{z_{\xi}})=\... ...3\Delta t_{\xi}})\d(k_{z_{\xi}}-\frac{u_4}{a_4\Delta z_{\xi}}).$$ (8)

A 4D Rect function, $\Pi$, with arguments (in 1D)

$$\Pi(B_\r)=\left\{ \begin{array} {cc} 1 & \mbox{for} \;\; \v... ...ert \, \ge \frac{1}{2 B_\r} \end{array} \right. \nonumber \:,$$

is then applied to the infinite Fourier domain lattice ${\cal L}$ to yield a band-limited version ${\cal L}_{FL}$,

$${\cal L}_{FL}(k_{r_{\xi}},k_{s_{\xi}},\omega,k_{z_{\xi}})= {... ...r_{\xi}},B_s_{\xi}k_{s_{\xi}},B_\t\omega,B_z_{\xi}k_{z_{\xi}}).$$ (9)

Applying an inverse Fourier transform over all 4 dimensions to lattice ${\cal L}_{FL}$ yields,

$${\cal L}_{FL}(\r,s_{\xi},\t,z_{\xi})={\cal L}(\r,s_{\xi},\t,... ...ast_\r\ast_s_{\xi}\ast_\t\ast_z_{\xi}FL(\r,s_{\xi},\t,z_{\xi}),$$ (10)

where

$$FL(\r,s_{\xi},\t,z_{\xi})=\frac{{\rm sinc}(\frac{\r}{B_\r},\... ...\t}, \frac{z_{\xi}}{B_z_{\xi}})}{B_\r B_s_{\xi}B_\t B_z_{\xi}},$$ (11)

and the subscripts on the convolution symbol, $\ast$, delimit the coordinate over which the convolution is applied. Thus the seismic wavefield may be represented by,  

$$W(\r,s_{\xi},\t,z_{\xi})={\cal L}_{FL}(\r,s_{\xi},\t,z_{\xi})\: \d(z_{\xi}) f^W(\r,s_{\xi},\t,z_{\xi}).$$ (12)

It is important to emphasize that the lattice ${\cal L}_{FL}$ in equation (13) represents only the lattice structure on which the data wavefield is overlaid. For individual seismic experiments, the values at each location will vary, while the lattice structure remains invariant. Utilizing the crystal structure analogy again, any atom may inhabit a node in the lattice, but there must be one and only one atom present.