Smoothing velocities for depth migration

Depth migrations are often performed in areas where rapidly changing velocities occur. Users are encouraged to smooth the velocities before performing depth migrations, but they are also hesitant to do this because they fear the consequences. Specifically they are concerned about distorting the structure of the model, losing accuracy in the velocities, creating a poorer image, and causing a depth shift in the resulting image. These can all be valid concerns.

The velocity model needs to be smoothed because depth migration algorithms break down when velocities change too rapidly over short distances. If one is using ray tracing technology, the rays will scattered wildly when encountering a large velocity gradient. Even when using finite difference algorithms the model should be smoothed so that the algorithm remains stable.

Typical velocity modeling software gives the user the ability to smooth the model. The user specifies the length of the operator and the software smoothes the model. The user does not have any analytic means to determine how smooth the model is, or any way to accurately predict the degree of smoothness based on the length of the operator. They can often visually inspect the resulted smoothed model, but that is the limit of what can be done.

In this article we will demonstrate how using the Velocity Smoother in the Tsunami Imaging Suite can address the concerns about smoothing the model and provide much better feedback about the results of the smoothing operation. Further we demonstrate that smoothing the model will provide a better image, and greatly reduce the runtime of the imaging process.

Smoothing the model

The Tsunami Velocity Smoother was designed to allow users to specify the desired smoothness of the model, rather than the length of the operator. When smoothing a model the attribute of the model that needs changing is the rate of change of the velocities, or the velocity derivative. The velocity derivative is simply the change in velocity divided by the distance over which the change was measured. The smoothing utility allows the user to specify the maximum velocity derivative in three dimensions, x,y,z, and then constructs an operator that will achieve the desired smoothness.

The smaller the velocity derivatives are the smoother the model is. We suggest a maximum vertical velocity derivative of less than 50, and a horizontal one of less than 20. Since the distance part of the derivatives cancel, these values can be applied whether using models in feet/sec, or meters/sec so long as the distance is measure consistently. The examples in this article use the SEG Salt Model, and in this example the raw model has lateral velocity derivatives greater than 80, and a vertical velocity derivative greater than 600. These kinds of derivatives will cause problems in the imaging operation, and will cause the ray tracing run time to be excessive.

One of the user concerns is that smoothing the model will shift the depth of the resulting image. While this depth shift does take place, it is usually small. The larger issue is that in most velocity applications the user has no idea how much depth shift is likely to occur as a result of their smoothing the model. Therefore they have no idea whether the shift caused by smoothing is within acceptable limits. The Tsunami Velocity Smoother reports a table of cumulative shifts by depth in the output logfile of the utility.

Figure 1 is part of the logfile from the smoothing utility. The logfile displays the initial velocity derivatives (dx,dy,dz), the final derivatives after smoothing along with the length of the operator used for smoothing. The operator may be different in different directions to achieve the desired degree of smoothing. As we can see the derivatives have been reduced significantly.

Figure 2 shows the table of cumulative depth shifts at increasing depths of the model. For instance a shift of 8.01 meters at a depth of 3000 meters is shown. This is a shift of .26%. Quite small. If the shift is larger than the user would like to allow, the shift can be reduced by vertically resampling the model. The smoothing utility provides this vertical resampling capability. The second column of shifts shows the result of resampling the model from 10 meters to 5 meters. As is indicated by the chart this resampling reduces the cumulative depth shift. For both of these columns the resulting derivatives are the same. The user can also reduce the degree of vertical smoothing to reduce the cumulative depth shift.

Users normally datum the results of depth migrations using well control information. These final small adjustments, made after migration, make sure that the imaged horizons correspond correctly to the observed depths of the horizons in the field. Because the imaging velocities are somewhat different than velocities in the field no depth migration algorithm can locate an event more accurately than within a percent or so of the actual horizon depth.

Another concern of the user is that the smoothing process will distort the model. The output from the smoothing utility is a segy formatted file and a compressed block file for input into the Tsunami Ray Tracing software. The segy file can then be used to visually inspect the smoothed model, and it can also be used as input into a tomography program or other velocity analysis program. Figures 3 and 4 show the comparison of the original model with the smoothed model. It is apparent that while the velocity derivatives have been greatly reduced, the model shows little change, just a slight fuzziness at the salt boundary

Ray tracing results

The Tsunami Ray Tracer application is a paraxial ray tracing algorithm that uses wave front reconstruction methods to maintain the accuracy of the calculated travel time tables. Simple described the wavefront reconstruction algorithm creates fill-in rays, between existing rays, when the rays scatter. When the rays cross a large velocity gradient the rays scatter more rapidly and wildly. This causes more fill-in rays to be generated and increases the run-time of the application.

The Tsunami Ray Tracer gives the user full control of the shot increments, and travel time table increments. It also allows the user to adjust the rate at which the fill-in rays are propagated.

In the logfile of the ray tracer a report describes how many fill-in rays are generated for each shot, and what the relative accuracy of the shot is. By looking at Figure 5 we see a comparison of the two ray tracing jobs, one with no smoothing, and the other smoothed with the smoothing utility.

The job without smoothing created many more rays per shot. The “number of fill in rays” are the new rays created in the in the wave front reconstruction process. The job without smoothing created about five times as many new rays as the job with the smoothing. This is an indication of a large amount of ray scattering.

The “median ray difference” is a measure of the accuracy of the ray tracing results. When the travel times for the tables are calculated the time is calculated from multiple rays. This value, measured in seconds, is the median difference between the rays when calculating the travel times. Smaller differences are better. The log shows that the result with smoothing is far more accurate.

Runtimes are also much greater when no smoothing is applied. The ray tracing job without smoothing ran eight times longer than the job with the smoothing applied. This is because the job without smoothing created many more fill in rays. The application then created paths for each of these fill in rays through the model.

Imaging Results

In Figures 6 and 7 is a comparison of the imaging results. The results in Figure 7 have no smoothing applied and the left edge of the salt, and the base of the salt is not as well defined. In addition there is some noise below the salt that does not exist in the results using the smoothed model. In other examples steeply dipping sides of the salt often are not imaged well when the model is not smoothed.

Conclusions

Smoothing the velocity model before depth migration is required for achieving the best imaging results, and reducing the runtime of the ray tracing jobs. The Tsunami Velocity Smoother allows the user to target the important attribute of the model, the velocity derivatives, in order to achieve the desired degree of smoothing. The utility also provides feedback to the user regarding the cumulative depth shift that occurs as a result of the smoothing process. Our work suggests the user should reduce the vertical velocity derivative (dz) to less than fifty, and the horizontal derivatives (dx,dy) to less than twenty. As with all geophysical processes, the smoothing process is one of compromise so these numbers are not absolutes, but guidelines for the user to achieve the best results. The utility provides the necessary information to the user so that they can make the appropriate decisions on how best to smooth the model.