Iterative Born inversions of the wavefield

The ideas of iterative Born inversion, introduced in Chapter 10 for an electromagnetic field, can be extended to wavefield inversion as well. Following the basic principles of this method, we first write the original integral equations for the acoustic or vector wavefields (14.20) and (14.81) as the domain equations for the wavefield inside the anomalous domain D  

We can now use similar equations to connect the scattered acoustic da = p = p — p ) or vector (dy — u = u — u ) wavefield observed on the surface S with the corresponding wavefield within the anomalous domain 

Note that each equation, (15.40), (15.41) and (15.42), (15.43), contains the product of unknown functions, As and p, or As and u. Therefore, these equations are bi-linear with respect to the corresponding unknowns. However, if we specify one of the unknowns, the equations become linear. For example, we can subsequently find As from equations (15.42), (15.43) for specified p or u, and then update p or u from equations (15.40) or (15.41) for predetermined As, etc. Within the framework of this method the Green s functions G and G and the incident wavefields p or u stay unchanged. 

As in the electromagnetic case, there is also another technique, the distorted-Born iterative method, which is based on updating the incident field and Green s functions after each iteration, according to the updated parameter As (Chew, 1990). 

Note in conclusion of this section that iterative Born inversion requires the application of the regularizing methods to make the solution stable. 

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