NUMERICAL EXPERIMENTS

Using traveltime relations, I generated synthetic constant offset sections that contain events reflected from plane reflectors. The velocity is 2000 m/s. I chose five offsets ranging from 0 to 2000 m. The dips of the reflectors vary from 0 to 80 degrees in increments of 20 degrees. I assume that these sections can be processed with spherical divergence and attenuation corrections so that the peak amplitudes of the wavelets are proportional to the reflection coefficients at the depth points from which these wavelets were reflected. It is important to preserve this proportionality after NMO and DMO corrections. In my examples, I further assume that the reflection coefficients are constant along the reflectors and are independent of the angles of rays. (Although this assumption is unrealistic, it simplifies the explanation of the results.) In accordance with these assumptions, the ideal outputs of the DMO operator should have a constant peak value for all the wavelets.

Figure displays constant offset sections with a fixed offset of 2000 meters. The top panel shows the result after the NMO corrections: the peak amplitudes of the wavelets should be constant. This is because NMO is a time-stretching operation that changes the shapes of wavelets but not their peak values. The middle panel shows the output of Hale's DMO operator: the amplitudes of the wavelets decrease as the dips of the reflector increase. Because the sections are small windows near the centers of large sections, the possibility of boundary effects can be ruled out. Therefore, we can conclude that this operator attenuates events with large dips. The bottom panel shows the result after applying uniform-amplitude DMO operator: the peak amplitudes of the wavelets are constant.

 

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Figure 2 Common offset sections. Top: after NMO correction; middle: after the conventional DMO correction; bottom: after the uniform-amplitude DMO correction. The offset is 2000 meters, and the dips of the reflectors are 0, 20, 40, 60, 80 degrees.

A better way to visualize the contrast between the outputs of the two DMO operators is to plot the peak amplitude of the wavelets as a function of offset and dip, as shown in Figures a and b. Evidently, uniform-amplitude DMO works as suggested. The amplitude curves are almost constant for all dips and offsets. With Hale's DMO, however, the amplitude curves drop down to about 50% when the dip is 80 degrees or the offset is 2000 meters.

 

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Figure 3 The peak amplitude of the wavelets as the function of (a) the dip of a reflector with an offset of 2000 meters, (b) the offset of the experiment with a dip angle of 60 degrees. The samples are picked from the wavelets at the centers of the sections. The circles indicate samples associated with uniform-amplitude DMO; the squares indicate samples associated with conventional DMO. The continuous curves are calculated with the analytical formulae of Black and Schleichor (1990).