Regularization of Instantaneous Frequency Attribute Computations
Abstract:We compare two different methods of computation of a temporally local frequency:
1) A stabilized instantaneous frequency using the theory of the analytic signal.
2) A temporally variant centroid (or dominant) frequency estimated from a time-frequency decomposition.
The first method derives from Taner et al (1979) as modified by Fomel (2007) and utilizes the derivative of the instantaneous phase of the analytic signal. The second method computes the power centroid (Cohen, 1995) of the time-frequency spectrum, obtained using either the Gabor or Stockwell Transform. Common to both methods is the necessity of division by a diagonal matrix, which requires appropriate regularization.
We modify Fomel’s (2007) method by explicitly penalizing the roughness of the estimate. Following Farquharson and Oldenburg (2004), we employ both the L curve and GCV methods to obtain the smoothest model that fits the data in the L2 norm.
Using synthetic data, quarry blast, earthquakes and the DPRK tests, our results suggest that the optimal method depends on the data. One of the main applications for this work is the discrimination between blast events and earthquakes
Fomel, Sergey. " Local seismic attributes." , Geophysics, 72.3 (2007): A29-A33.
Cohen, Leon. " Time frequency analysis theory and applications." USA: Prentice Hall, (1995).
Farquharson, Colin G., and Douglas W. Oldenburg. "A comparison of automatic techniques for estimating the regularization parameter in non-linear inverse problems." Geophysical Journal International 156.3 (2004): 411-425.
Taner, M. Turhan, Fulton Koehler, and R. E. Sheriff. " Complex seismic trace analysis." Geophysics, 44.6 (1979): 1041-1063.