On a generalization of Filon's method and the computation of the oscillatory integrals of seismology

Summary. We review Filon's method (FM) for the quadrature of oscillatory integrals and then introduce a generalization of Filon's method (the GFM) which enables us to treat a large class of oscillatory integrals to which FM cannot be directly applied. One member of this class is the integral ( p ) exp [ sg ( p )] dp which occurs in the spectral WKBJ and Cagniard-de Hoop methods of seismogram synthesis. Another large class of integrals can be treated directly with FM but is better treated with the GFM since, for a given error tolerance, the GFM is simpler and faster. This class consists of integrals of the form ( p ) J ( s, p ) dp in which J ( s, p ) is a special function with an asymptotic expansion valid for large s. Such integrals occur in the reflectivity method. In general, every non-Filon formula for the quadrature of integrals from either class has an associated GFM formula (called the GFM analogue) which reduces to the original formula as s approaches zero but is more efficient than the original formula wher, s is large. We show how the GFM can be applied to the computation of synthetic seismograms in the reflectivity method and the spectral WKBJ method.
Although reflectivity integrals can, in theory, be computed with FM the GFM is easier to code and more economical. For reflectivity computations where: (a) the source and receiver are many wavelengths apart, or (b) the depth to the reflectivity zone is much greater than its thickness, the GFM approach is much more efficient than any non-Filon quadrature technique. Some test calculations are presented for wavefields containing only body waves and for wavefields containing both body waves and locked modes.
In the spectral WKBJ method the GFM permits the use of a much greater step size in the quadrature than would otherwise be possible. Each quadrature step contains a stationary point so no advantages accrue from deforming the contour of integration over the saddle points of the integrand.