SYNTHETIC DATA EXAMPLES

In this section the l₂ theory derived above is validated with two synthetic 2-D data sets. The first reflectivity model consists of five horizontal planar reflectors at depths of 1.0, 1.5, 2.0, 2.5, and 3.0 km respectively. The second reflectivity model consists of three planar reflectors which dip at 0, 15, and 30 degrees respectively. The depth of each reflector beneath the origin is 1.0, 2.0, and 3.0 km respectively. Each reflector is assigned the same angle-dependent reflectivity with the following parametric form:

 

$$\grave{P}\!\acute{P}(\Theta)= -0.26 + \sin\Theta \;,$$ (20)

which exhibits a strong polarity reversal at $\Theta = 15^{^{\circ}}$ reflection angle incidence. This model AVO behavior is a challenging synthetic test in that any error in the integral formulation will easily give a wrong result, since the strong negative and positive amplitudes along a given hyperbola will not constructively interfere correctly.

The background migration velocity model is a constant 2 km/s to give analytical accuracy to the synthetic seismogram amplitudes and various ray properties. However, the sources and receivers are given directivity and the geometrical spreading is based on a typical v(z) attenuation to make the synthetic amplitude more realistic.