Reciprocity Rules!

Summary:
In one section of a three-part article about subsalt illumination published in The Leading Edge (Muerdter, Kelly and Ratcliff, July 2001), we reported a lack of reciprocity in the raytracing results of a simple salt wedge model. We suggested that reciprocity did not hold in this case and backed it up with amplitude calculations. After receiving feedback from various colleagues, we discovered that the raytracing program did not calculate the amplitudes accurately and our calculations did not consider a spreading factor.  Therefore, reciprocity is not violated and the raytracing software program has been fixed. Further investigation showed that the software flaw had a very small impact on the illumination calculations and did not change any of the conclusions in the article, except the bold reciprocity statements.

Details:
We wish to thank the authors of the letters to the editors in the September and November issues of The Leading Edge for constructively relating the theory and references of reciprocity and providing insights into our calculations. The letters were in response to one section of our three-part paper titled "Understanding Subsalt Illumination" published in the June, July and August issues

Our "Illumination" paper was a collection of observations about subsalt imaging discovered by raytracing both simple and complex subsalt models. The purpose was to catalogue some of the imaging problems and to make the exploration community more aware of illumination problems that can still exist in 3-D pre-stack seismic data. Examples in the paper showed that large subsalt zones are not well illuminated, while other zones appear anomalously bright, compared with what would be expected from rock property considerations. This observation makes the common use of amplitudes extracted from 3-D seismic, usually a reliable indicator of anomalous and possibly hydrocarbon related rock properties, very risky unless accompanied by a careful study of illumination and amplitude distortion using some type of modeling tool.

The section of the paper that precipitated the letters to the editor concerned the lack of amplitude symmetry observed when switching source and receiver in some subsalt models and our assigning a lack of reciprocity as the probable reason. We constructed a simple salt wedge model that exhibited the lack of amplitude symmetry effect (refer to Muerdter et al., July 2001, The Leading Edge, Figures 5 and 6). The Sierra raytracing program showed this effect, as did our independent calculations as described in the paper. After receiving feedback about reciprocity, we designed an operation code named 'Codswallop' in honor of the "geophysical" term introduced by Francis Muir in his letter to the editor. The operation had four parts: 1) to test the same simple model with other commercial raytracing codes, 2) to recalculate the amplitudes including the effects of spreading, 3) to determine if the Sierra raytracing codes could be modified to produce amplitudes that followed reciprocity, and 4) to understand the effect of this problem on previously run subsalt models. Here are the results of operation 'Codswallop' that confirm reciprocity does indeed hold true.

1. Raytracing Results

The original model and raytracing results using QUIKSHOT+ version 1998.2 are shown below as Figures 1 and 2. Note that the left to right shooting orientation is called down-dip, while the right to left is termed up-dip.


Figure 1:


Figure 2:

The same 2-D salt wedge model was created and raytraced in GX Technology's GXII Software. The ray travel times for reciprocal raypaths are equal indicating that the raypaths are the same in both directions. The amplitude results for the same three CRPs in Figure 1A also show equivalent amplitudes for reciprocal raypaths. GXT's raytracing produces results consistent with reciprocity.

Norsar's Norsar2D software was also tested with the same salt wedge model. The model is shown in Figure 2 and the equal amplitudes for reciprocal rays shown in Figure 1B. Norsar raytracing also is consistent with reciprocity. Further tests with Norsar software showed on other models reported in the 3-part article, showed no difference in fold results and only very minor amplitude differences. The amplitude differences did not change any of the conclusions in the article except for the bold reciprocity statements.


Figure 3: A) Simple salt wedge model built in Norsar2D software. Raypath with 5400 m offset and reflection point in 6000m CRP bin is shown as black line. The blue lines are the search rays that are shot at many angles and then examined to find the raypaths that surround a receiver. Note how many of the search rays do not penetrate the salt because they strike the salt wedge at greater than critical angle.

Figure 4: Raytrace modeling results from the three programs tested. All produce results consistent with reciprocity for the three CDPs tested.
A) GX2 B) Norsar2D C) QUIKSHOT v.2003.

2. Recalculation of Amplitudes for reciprocal rays including spreading effects
Given a series of N uniform, isotropic, planer layers, the spreading correction (SC) to the amplitude for a ray from the surface, reflected at the Nth layer, and returning to the surface becomes:

where a and b are the incident and refracted angles and the subscripts d and u are the downward and upward portions of the raypath. It should be noted that for a series of parallel, horizontal layers ( a d ) i = ( b u ) i , and in this case SC = 1.

Rock properties used in the model and transmission coefficients at the model's interfaces for reciprocal rays with an offset of 5400m and a reflection point in the 6000m bin on the subsalt reflector are shown in Tables 2 and 3 in Muerdter et al., TLE, July 2001. Using the equation for spreading correction given above for these 5600m offset reciprocal rays, a spreading correction of .837 is calculated for the up-dip path relative to the down-dip path. The transmission and spreading effects when combined have values of 0.519  (calculation: 0.519*1.0) for the down-dip path while for the up-dip path also produces 0.519 (calculation: 0.620*0.837).  So, even though the transmission amplitudes are different for the two directions along the same path, the spreading effects exactly compensate to make the observed amplitudes from the two directions the same. It is this spreading correction that had not been considered by our initial calculations and by the raytrace modeling package, and hence resulted in the apparent amplitude differences.

3. Investigation of QUIKSHOT+ raytracing codes

The error was identified deep in the code and corrected. The current release of QUIKSHOT+ codes contains a fix for the amplitude reciprocity defect.

Testing repaired code and comparing output

A trial of the corrected code on the simple salt wedge model (Figure 1C) shows the results are reciprocal and match the results of Norsar and GXT.

Some of the subsalt models were rerun on the corrected QUIKSHOT+ code. The results of two of the test are shown in Figure 5 through 8 There are just subtle changes in the results indicating that the flaw caused only small variations in the results and the error would not change any of the conclusions made from the illumination studies.

Figure 5: Above) Original results (QUIKSHOT version 1998) of raytrace illumination study of the subsalt area in the area of the Tanzanite field in Gulf of Mexico.
Above right)Results of same model reshot with the corrected QUIKSHOT program (version 2003).
Right)The percent differences ( [v2003 - v1998]/v2003 ) indicate that the largest differences were less than 20% and most of those differences were in  areas of extremely low illumination. Therefore the error in amplitude made relatively small differences in results and do not change the overall illumination conclusions.

Conclusions:

  1. Reciprocity is a well-established theory, one that modeling software should be tested against.

  2. The raytracing codes GX2 by GX Technology and Norsar2D by Norsar are consistent with reciprocity.

  3. The QUIKSHOT raytracing code has been modified and does produce reciprocity accurate results.

  4. Very minor differences were found between results of subsalt raytracing using the original QUIKSHOT code (v1998) and the corrected version (v2003). Therefore the conclusions drawn in the article are still valid with the exception of the bold reciprocity statements.

We wish to thank Todd Jones of GX Technology who ran the GXII software and Bill Pramik of PGS Research who produced the results using Norsar2D software. Gregg Hofland and Andreas Rueger of Landmark Graphics discovered the problem and corrected the QUIKSHOT+ software.

 David Muerdter, November 2001

Copyright 2002 by Emerald Geoscience Research Corporation