Modeling and migration can be expressed in terms of frequency-dependent Green's functions of the wave equation. However, practical implementations do not use the Green's functions explicitly. The reason for this is the prohibitive cost of calculating and storing all the required functions.
One common way to overcome this problem is to use an asymptotic form of the Green's functions. These are usually of the form where can be found by solving the eikonal equation and A can be found by solving the transport equation. When these approximate functions are substituted into the migration equations they reduce the calculation to a line or surface integral form. This is computationally very efficient but there may be problems when the asymptotic form is not a good approximation to the Green's function.
In a previous paper Nichols (1993) I discussed calculating mono-frequency Green's functions using one way wave equations in polar coordinates. The complete Green's function is the superposition of all the mono-frequency Green's functions. Using currently available computers, it may be too expensive to routinely calculate these functions for all frequencies. In this paper I examine the possibility of calculating the Green's functions at selected frequencies and estimating the other frequencies.
Once the Green's functions at all frequencies have been estimated they can be used for pre-stack and post-stack modeling and migration. If the medium has velocities that are independent of the horizontal coordinate, only one set of Green's functions needs to be calculated. Because the Green's functions are calculated in polar coordinates they are not limited to waves traveling at less that 90 degrees to the vertical. They can be used to image reflections due to overturned waves. I use a single set of Green's functions for modeling and migration in a V(z) medium with overturned waves.
Since the Green's functions are calculated using the wave equation they contain all the correct amplitude and phase information for the waves, this is difficult to do with Green's functions calculated from the eikonal equation, since the eikonal equation does not carry any phase information.
The true amplitude nature of the wave equation Green's functions reveals a problem with using the adjoint of the modeling operator as an imaging operator. In the noise free case, an inverse operator would be an ideal imaging operator that would invert both the amplitude and phase information. The adjoint operator correctly inverts the phase but is squares the amplitude. An alternative operator is one that removes the phase and attempts to approximately recover the original amplitudes. This process is sometimes referred to as migration/inversion Lumley and Beydoun (1991).