2-D AMO operator

When the input-offset vector ${\bf h}_{1}$ is parallel to the output-offset vector ${\bf h}_{2}$, the triangle ${\bf m}_1$-${\bf m}_0$-${\bf m}_2$, formed by the midpoints of the input trace, zero-offset trace, and output trace, degenerates to a line. The location of the zero-offset midpoint ${\bf m}_0$ is not constrained by the input and output midpoints and can take different values on the line. The cascade of DMO and inverse DMO becomes a convolution on that line. To find the summation path of 2-D AMO (offset continuation), one needs to consider the envelope of the family of traveltime curves (where m₀ is the parameter of a curve in the family):  

$$t_1 = t_2\,\sigma_{12}\left(m_1,m_2,h_{1},h_{2}\right) = t_2... ...t(m_2-m_0\right)^2} \over {h_{1}^2-\left(m_1-m_0\right)^2}}\;.$$ (49)

Solving the envelope condition ${\partial \sigma_{12} \over \partial m_0}=0$for the zero-offset midpoint m₀ produces  

$$m_0={{\left(\Delta m\right)^2+h_{2}^2-h_{1}^2+ \mbox{\rm sig... ...)^2- 4\,h_{1}^2\,h_{2}^2}} \over {2\,\left(\Delta m\right)}}\;,$$ (50)

where $\Delta m=m_1-m_2$. Substituting (51) into (50), we obtain the explicit expression (3) of the offset continuation summation path. C