## Zero offset seismic resolution theory for linear v(z)

### Gary F. Margrave

From the perspective of f-kx migration theory, the optimizing the ability of seismicdata to resolve earth features requires maximization of the spectral bandwidth after migration. The kx (horizontal wavenumber) bandwidth determines the lateral resolution and is directly proportional to the maximum frequency and the sine of the maximum scattering angle and inversely proportional to velocity. Constraints on the maximum scattering angle can be derived by examining the three effects of finite spatial aperture, finite recording time, and discrete spatial sampling. These effects are analyzed, assuming zero-offset recording, for the case of a linear (constant gradient) v(z) medium. Explicit analytic expressions are derived for the limits imposed on scatteringangle for each of the three effects. Plotting these scattering angle limits versus depth limits for assumed recording parameters is an effective way to appreciate their impact on recording. When considered in context with f-k migration theory, these scattering angle limits can be seen to limit spatial resolution and the possibility of recordingspecific reflector dips. Seismic surveys designed with the linear v(z) theory are often much less expensive than constant velocity theory designs.