Kjartansson's model for lateral variation in amplitude

The Grand Isle data is incomprehensible in terms of the model based on layered media theory. Kjartansson proposed an alternative model. Figure 2 illustrates a geometry in which rays travel in straight lines from any source to a flat horizontal reflector, and thence to the receivers.

 

kjidea
Figure 2 Kjartansson's model. The model on the top produces the disturbed data space sketched below it. Anomalous material in pods A, B, and C may be detected by its effect on reflections from a deeper layer.

The only complications are ``pods'' of some material that is presumed to disturb seismic rays in some anomalous way. Initially you may imagine that the pods absorb wave energy. (In the end it will be unclear whether the disturbance results from energy focusing or absorbing).

Pod A is near the surface. The seismic survey is affected by it twice--once when the pod is traversed by the shot and once when it is traversed by the geophone. Pod C is near the reflector and encompasses a small area of it. Pod C is seen at all offsets h but only at one midpoint, y₀. The raypath depicted on the top of Figure 2 is one that is affected by all pods. It is at midpoint y₀ and at the widest offset $h_{\rm max}$.Find the raypath on the lower diagram in Figure 2.

Pod B is part way between A and C. The slope of affected points in the (y,h)-plane is part way between the slope of A and the slope of C.

Figure 3 shows a common-offset section across the gas field. The offset shown is the fifth trace from the near offset, 1070 feet from the shot point. Don't be tricked into thinking the water was deep. The first break at about .33 seconds is wide-angle propagation.

 

kjcos
Figure 3 A constant-offset section across the Grand Isle gas field. The offset shown is the fifth from the near trace. (Kjartansson, Gulf)

 

kja
Figure 4 (a) amplitude (h,y), (b) timing (h,y) (c) amplitude (z,y), (d) timing (d,y)

The power in each seismogram was computed in the interval from 1.5 to 3 seconds. The logarithm of the power is plotted in Figure 4a as a function of midpoint and offset. Notice streaks of energy slicing across the (y,h)-plane at about a 45$^\circ$ angle. The strongest streak crosses at exactly 45$^\circ$ through the near offset at shot point 170. This is a missing shot, as is clearly visible in Figure 3. Next, think about the gas sand described as pod C in the model. Any gas-sand effect in the data should show up as a streak across all offsets at the midpoint of the gas sand--that is, horizontally across the page. I don't see such streaks in Figure 4a. Careful study of the figure shows that the rest of the many clearly visible streaks cut the plane at an angle noticeably less than $\pm$45$^\circ$.The explanation for the angle of the streaks in the figure is that they are like pod B. They are part way between the surface and the reflector. The angle determines the depth. Being closer to 45$^\circ$ than to 0$^\circ$, the pods are closer to the surface than to the reflector.

Figure 4b shows timing information in the same form that Figure 4a shows amplitude. A CDP stack was computed, and each field seismogram was compared to it. A residual time shift for each trace was determined and plotted in Figure 4b. The timing residuals on one of the common-midpoint gathers is shown in Figure 5.

 

kjmid Figure 5 Midpoint gather 220 (same as timing of (h,y) in Figure 4b) after moveout. Shown is a one-second window centered at 2.3 seconds, time shifted according to an NMO for an event at 2.3 seconds, using a velocity of 7000 feet/sec. (Kjartansson)

The results resemble the amplitudes, except that the results become noisy when the amplitude is low or where a ``leg jump'' has confounded the measurement. Figure 4b clearly shows that the disturbing influence on timing occurs at the same depth as that which disturbs amplitudes.

The process of inverse slant stack (not described in this book) enables one to determine the depth distribution of the pods. This distribution is displayed in figures 4c and 4d.