Integral approximations of the acoustic wavefield

We consider again the basic integral equation of the acoustic forward modeling problem (14.19), written for the scattered wavefield 

The conventional Bom approximation p rj,Lj) for the scattered field can be obtained from (14.34), if inside D the scattered field is negligibly small in comparison with the incident field  

One can see that the Born approximation reduces the forward modeling solution to simple quadrature calculation. However, this approximation holds only for a weaJi scatterer (when As (r) is relatively small). We can improve the accuracy of integral approximation significantly if, following the ideas described in Chapter 9 for an electromagnetic field, we introduce a quasi-linear approximation. 

Using expression (14.9), we can represent the total field in the form 

Substituting expression (14.36) for the total wavefield in the integral formula (14.34), we obtain 

---