A New Time Domain Reflection Seismogram
Authors: Frasier Clint W. 
Published: October 2008

ABSTRACT
A 2-D point source seismogram is described which calculates marine profiles of reflections from a sequence of flat layers. The seismogram is expressed as a sum of plane wave reflection responses of the layers over a wide aperture of incident angles from the source. Each plane wave response is computed in the discrete time domain using a new recursion formula. The solutions include primaries, multiples, and all conversions between P and SV waves at the layer interfaces. Critical angle reflections and transmissions are included for propagating waves, but evanescent waves are not calculated. Each trace of the profile is obtained by delaying and summing the plane wave components over the spectrum of incident angles from the source. The time delay for each component equals the trace offset distance from the source divided by the horizontal phase velocity of the component. The summation produces a "wide band" stationary phase record in time, which shows the correct move out and amplitude variation of the reflections with offset.
This time domain method allows arbitrary offset distances and there is no wraparound in time and distance on the final profile, two common problems in frequency-wavenumber methods which use the 2-D FFT. The truncation of the incident angular spectrum to include only reflections causes some precursor energy as in the frequency-wavenumber solutions. This effect can be minimized by properly sampling and tapering the angular spectrum of plane wave components before summation.

INTRODUCTION
In this paper we describe a new method for generating synthetic marine seismograms by summing up the plane wave components directly in time over a spectrum of take off angles from the source.
The technique is a variation of the reflectivity method for computing the reflected P and SV waves in a multilayered medium. The purpose of the method is to calculate only the reflection response of the layers and omit the surface waves, which are often removed by signal processing methods in real data. For a 2-D point source the plane waves are summed over a range of real incident angles from -90 to +90 degrees. By neglecting the imaginary angles, which are used to synthesize the surface waves, the calculations are made simpler with no loss of information in the reflected phases, which are crucial in exploration. A recent paper which does this synthesis in the frequency domain is by Kennett (1979). In the 3-D problem the angular summation is also done over azimuthal angles. Fertig and Muller (1979) used this method to show that strong P to SV conversions occur for wide angle reflections from coal bed seams.
The plane wave components in this paper are calculated by a new layer recursion in discrete time which is adapted from a frequency domain recursion given by Kennett (1979). The recursion begins with the reflection response of the deepest interface of the model and progresses up through the layers to the free surface. In the recursion described below, 2 × 2 matrix filters in powers of (available in full paper) are used to propagate elastic waves through the layers. This discrete time approach to coupled P and SV waves was first given by Frasier (1970) in a reference we shall call Paper 1. Another plane wave synthesis of 2-D seismograms was presented by Aminzadeh and Mendel (1978). They used some results from Paper 1 to recast the wave propagation in terms of state-space models. In contrast to the recursive filter approach to wave propagation, the state-space method uses matrices which grow in dimension with the number of layers utilized, and requires a ray tracing scheme to keep track of the interface reflections and transmiss