Wave equation migration methods are quickly becoming the standard for high-end imaging. These methods are more effective in handling complex wave behavior than traditional Kirchhoff methods Workshop (2001). The downside of these methods, specifically ones based upon the one-way wave equation, are that they are generally more expensive than Kirchoff methods and are only accurate for v(z) media. To overcome the latter weakness extensions such as phase screen Le Rousseau and de Hoop (1998), split-step Stoffa et al. (1990), and Fourier Finite Difference (FFD) Ristow and Ruhl (1994) have been introduced. These methods generally rely on downward continuing the wave-field with multiple reference velocities and then applying an approximate correction to the wavefield for the true velocity. The larger the deviation of the reference from the true velocity the less accurately the wavefield will be modeled. The trade-off is that the cost of migration increases almost proportionally with the number of reference velocities. Many different methods are currently employed to select the reference velocities, each with varying levels of success, dependent on the input velocity.
Electrical engineers and image processors face similar problems. In speech compression it is important to accurately describe a signal in as few bytes as possible. In image processing it is often important to reduce the number of colors in image with as little loss in image quality as possible. These problems have led to the field known as quantization. One family of method often employed is based on Lloyd's method Lloyd (1982), an iterative technique that allows for variable rate quantization.
In this paper I propose a new method for selecting reference velocities based on a generalized Lloyd's method. I show that the method is effective in choosing appropriate reference velocities for a large variety of models. On two synthetics I demonstrate that it produces a higher quality image with fewer reference velocities.