Unsmooth Velocity

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.

Smooth Velocity

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.

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.

Tsunami Smooth Velocity - Article 3
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Smoothing Velocities for Depth Migration
Bill Kamps. Tsunami Development. August 2006