Two-dimensional earth

In the general case, modeling multiples becomes more expensive. Equation (12) is not valid anymore (except for smoothly varying media), and the convolution becomes nonstationary (shot gathers are different from one location to another). Hence, the wavefield is not only a function of offset, h, but also depends on another spatial coordinate such as shot location s. Under this parameterization, equation (10) can be written as  

$$u_1(h,s) = \int u_0(h-h',s+h') \; u_0(h',s) \; dh'.$$ (16)

Now, following Dragoset and Jericevic (1998), we introduce some amplitude corrections in the previous equation:

$$\begin{eqnarray} u_0(h-h',s+h') & = & F_{t\rightarrow \omega}[\sqrt{t}u_0(h-h',s... ...{\omega}{4\pi}} F_{t\rightarrow \omega}[\sqrt{t}u_{0g}(h',s,t)].\end{eqnarray}$$ (17)