The zero-offset waveform

The first comparison involves the zero-offset response of the three methods. A window that includes the first two primary reflections (both PP) was selected from the zero-offset trace of the vertical components for the three simulated wavefields. Figure  shows the three waveforms. Except for small differences in the amplitude and the precursor lobe, the waveforms generated by the propagator-matrix and dual-operator methods are very similar. The dispersion is minimal and the kinematics are in accordance with theoretical expectations. The dispersive character observed in the wavefield of the traditional finite-difference scheme becomes more clear in the selected window. As a result of the dispersion, the traveltime difference between the two reflections (using the central picks as the measuring criterion) is slightly larger than the theoretical value.

 

zeroresp
Figure 5 Zero offset response generated by the three schemes. The three curves correspond to a window selected from the first traces of Figures -a, -a, and -a, which include the reflections from the first two interfaces of the model. The continuous line comes from the Haskell-Thomson scheme, the dotted-line comes from the dual-operator scheme, and the dashed-line comes from the traditional finite-difference scheme.

The theoretical ratio between the amplitudes of the two reflections is given by

$${\mbox{Amp}_2 \over \mbox{Amp}_1} = {R_2 \over R_1} \; (1-R_... ...[{t_1 v_1^2 \over (t_2 - t_1) v_2^2 + t_1 v_1^2} \right]^{1/2},$$

where the first term is the ratio between the normal incidence reflection coefficients, the second term is the two-way transmission coefficient, and the third term is the 2D-divergence relative correction. The value obtained from the propagator-matrix trace coincides with the predicted value, while the value from the dual-operator trace is 3.7 percent higher, and the value from the traditional finite-difference trace is 7.5 percent lower.