A migration approach based on a local application of the Born approximation
within each extrapolation interval contains a singularity that can make di
rect application unstable. Previous authors have suggested adding an imagin
ary part to the vertical wavenumber to eliminate the singularity. However,
their method requires that the reference slowness must be the maximum slown
ess of a given layer; consequently, the slowness perturbations are larger t
han those when the average slowness is selected as a reference slowness. Th
erefore, its applicability is limited. We develop an extended local Born Fo
urier migration method that circumvents the singularity problem of the loca
l Born solution and makes it possible to choose the average slowness as a r
eference slowness. It is computationally efficient because of the use of a
fast Fourier transform algorithm. It can handle wider angles (or steeper in
terfaces) and scattering effects of heterogeneities more accurately than th
e split-step Fourier (SSF) method, which accounts for only the phase change
as a result of the slowness perturbations but not amplitude change. To han
dle large lateral slowness variations, we introduce different reference slo
wnesses in different regions of a medium to ensure the condition of small p
erturbation. The migration result obtained using the extended local Born Fo
urier method with multiple reference slownesses demonstrates that the metho
d can produce high-quality images of complex structures with large lateral
slowness variations.