One common way to overcome this problem is to use an asymptotic form of the solution. 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. These solutions are valid when wavelength of the velocity variations in the medium is much greater than the wavelength of the seismic data. Even when the scale of medium variation is large compared to the seismic wavelength, there are still problems associated with this approach. When there are multiple arrivals present in the data, calculating all the arrivals can become complicated. Many implementations calculate only the first arriving events. In particular, finite difference solutions to the eikonal equation can calculate the first arrivals efficiently, but they do not model the later arrivals. However, if the first arrival contains little energy, the solution obtained will be a poor approximation of the total Green's function.
In this paper I examine some methods for obtaining mono-frequency Green's functions. 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. However, this is likely to change rapidly. In addition, it may be informative to just compute a few Green's functions at selected frequencies and examine their dependence on frequency.
I use one-way extrapolators to downward or outward continue an initial condition until a solution is obtained in the whole space of interest. The one-way equations do not represent the full Green's function. However, they contain the information that is usually of greatest interest, the amplitude and phase of the primary arrivals. Arrivals due to multiple reflections may not be present, but these are often undesirable in geophysical applications. The multiply scattered events are very sensitive to the velocity model. If the model is not known accurately it can be beneficial to ignore multiples in the calculation of the Green's functions.
I show how to generate the Green's functions in a polar coordinate frame. Polar coordinates have been successfully used in finite difference travel time estimation van Trier and Symes (1991). They have the advantage of providing a coordinate frame that matches the constant velocity wavefronts. Even in variable velocity models, this results in wave propagation at low angles to the coordinate frame. Use of the polar coordinates could be regarded as a step towards using Gaussian beams Cervený et al. (1982). However, I feel that polar coordinates give many of the advantages of Gaussian beams without requiring a ray-tracing step to define the local coordinate frame.