2- and 3-D earth

In the general case, modeling multiples becomes more expensive. Equation () 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. In 3-D, the integral in equation spans the entire acquisition plane van Dedem (2002), which makes the prediction very expensive. Introducing the nonstationary convolution, equation () can be written as  

$$u_1(h,s) = \int u_0(h-h',s+h') \; u_0(h',s) \; dh'.$$ (149)

Now, following Dragoset and Jericevic (1998) for 2-D prediction, we introduce some amplitude corrections in the previous equation:

$$\begin{eqnarray} u_0(h-h',s+h') & = & F_{t\rightarrow \omega}[\sqrt{t}u_0(h-h',s... ...{\omega}{4\pi}} F_{t\rightarrow \omega}[\sqrt{t}u_{0g}(h',s,t)].\end{eqnarray}$$ (150)

Replacing u₀ by the data with primaries and multiples, equation () with the amplitude correction is used throughout this thesis to model surface-related multiples in 2-D.