Time-frequency analysis of seismic data using local attributes |

Seismic instantaneous frequency is the derivative of the instantaneous
phase

Average frequency can be estimated from the time-frequency
map (Cohen, 1989; Steeghs and Drijkoningen, 2001; Sinha et al., 2009; Hlawatsch and Boudreaux-Bartels, 1992; Claasen and MecklenbrĂ¤uker, 1980).
Average frequency at a given time is

where is the time-frequency map. Average frequency measured by equation 13 is the first moment along the frequency axis of a time-frequency power spectrum. Saha (1987) and Brian et al. (1993) analyzed the relationship between instantaneous frequency and the time-frequency map in detail. Extraction of the attributes from the time-frequency map of the seismic trace leads to considerable improvement of the signal-to-noise ratio of the attributes (Steeghs and Drijkoningen, 2001). We therefore propose applying equation 13 to our time-frequency map to compute the time-varying average frequency.

ref-6,s-6
(a) Random reflectivity series. (b) Synthetic seismic trace.
Figure 3. |
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spec,st-6,proj-6
(a) Theoretical time-frequency map, which is scaled by
a maximum in the frequency axis. White and black lines indicate dominant
frequency and average frequency of Ricker wavelet, respectively. (b) Time-frequency map of the S-transform.
(c) Time-frequency map of the proposed method.
Figure 4. |
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lcf-6
Time-varying average-frequency estimation (blue solid
curve: estimated by our method; pink dashed curve: estimated
by S-transform; black dot-dashed curve: theoretical curve, which
is denoted by black line in Figure 9).
Figure 5. |
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We used a synthetic nonstationary seismic trace to illustrate our approach to estimating time-varying average frequency. Figure 3b shows a synthetic seismic trace generated by nonstationary convolution (Margrave, 1998) of a random reflectivity series (Figure 3a) using a Ricker wavelet, the dominant frequency of which is a function of time, . Figure 9 shows the scaled spectrum of the Ricker wavelet. Both the dominant frequency (white line in Figure 9 and the bandwidth increase with time. We computed the average frequency (black line in Figure 9 using equation 13 from the scaled spectrum of Ricker wavelets. We note that average frequency is larger than the dominant frequency at high frequencies for Ricker wavelets.

We generated the time-frequency map of the synthetic nonstationary seismic trace using the S-transform (Figure 4b) and the proposed method with a 10-point smoothing radius (Figure 4c). We observe that the time-frequency map by the proposed method has a bandwidth more similar to that of the time-frequency map of the Ricker wavelet (Figure 9), especially at the high frequencies. We estimated time-varying average frequency curves from the time-frequency map by the proposed method (blue solid curve in Figure 5) and S-transform (pink dashed curve in Figure 5), respectively. Compared with the theoretical curve (black dashed curve in Figure 5), which was computed using equation 13 on the scaled spectrum of Ricker wavelets (Figure 9), the time-varying average frequency estimated by the proposed method is closer to the theoretical one.

Time-frequency analysis of seismic data using local attributes |

2013-07-26