Velocity analysis

Inversion schemes for various Radon transforms Artman and Guitton (2005); Artman and Sacchi (2003); Guitton (2004), have been proposed to provide sparse model spaces for data analysis and noise removal. Figure  shows the result of 20 iterations of hyperbolic Radon transform performed on gather 08 from the Yilmaz data collection. Coordinates selected by LA are overlain. I purposefully parameterized the model space to include slowness values twice those necessary. The picks remain stable with or without including this aphysical part of the model domain. With a threshold of 1% of the maximum value in the model space, the modified algorithm considered only 500 out of 43,000 coordinates for potential selection. The algorithm started with 200 evenly distributed coordinate pairs and returned with 14 optimally representative coordinates.

 

shot Figure 7 Hyperbolic Radon transform model domain of shot 08 from the Yilmaz data collection. 20 iterations of least-squares inversion were performed. 14 coordinates selected by LA are overlain. Without inclusion of the inappropriate high slowness values on the right side of the plot, the remaining picks remain stable.

I introduce Lloyd's algorithm as a tool to optimally represent the statistics of the model space from incomplete inversions. The modified algorithm selects coordinates with high amplitude surrounded by substantial energy. Thus isolated, powerful outliers are neglected. By optimally parsing potentially large multi-dimensional model spaces, the algorithm can cut short costly inversion iterations and focus an interpreter's attention to important locations within potentially large model domains. The algorithm returns stable solutions even in the presence of substantial noise.

The algorithm is very simple, easy to modify, has few parameters, and very fast. Using the algorithm depends on parameters being correlable. For multidimensional cases, uncorrelated parameters can simply be concatenated to an existing axis. Thus hypercubes of correlable and uncorrelable parameters can be evaluated simultaneously.

The next step in evaluating the effectiveness of using the algorithm to select optimal parameters would be to migrate data with a velocity model derived from RMS velocities selected by LA. This could potentially dovetail with the velocity uncertainty analysis presented in ().

Planewave decomposition of passive data to characterize non-obvious sources does not work. Until traces have been correlated, analysis transforms will simply redistribute the random character of the raw data. Unfortunately, correlating the wavefield destroys all the unique character of individual sources including timing, waveform, location and much of the spectral content information.

Thanks to Bob Clapp for the introduction of LA to the group and good discussions with Jeff Shragge on potential uses.