Local seismic attributes

Definition of local frequency

The definition of the local frequency attribute starts by recognizing equation 5 as a regularized form of linear inversion. Changing regularization from simple identity to a more general regularization operator $\mathbf{R}$ provides the definition for local frequency as follows:

$$\mathbf{w}_{loc} = \left(\mathbf{D}+\epsilon\,\mathbf{R}\right)^{-1}\,\mathbf{n}\;,$$ (6)

The role of the regularization operator is ensuring continuity and smoothness of the local frequency measure. A different approach to regularization follows from the shaping method (Fomel, 2006). Shaping regularization operates with a smoothing (shaping) operator $\mathbf{S}$ by incorporating it into the inversion scheme as follows:

$$\mathbf{w}_{loc} = \left[\lambda^2\,\mathbf{I} + \mathb... ...^2\,\mathbf{I}\right)\right]^{-1}\, \mathbf{S}\,\mathbf{n}\;,$$ (7)

Scaling by $\lambda$ preserves physical dimensionality and enables fast convergence when inversion is implemented by an iterative method. A natural choice for $\lambda$ is the least-squares norm of $\mathbf{D}$.

Figure 2 shows the results of measuring local frequency in the test signals from Figure . I used the shaping regularization formulation 7 with the shaping operator $\mathbf{S}$ defined as a triangle smoother. The chirp signal frequency (Figure 2a) is correctly recovered. The dominant frequency of the synthetic signal (Figure 2b) is correctly estimated to be stationary at 40 Hz. The local frequency of the real trace (Figure 2c) appears to vary with time according to the general frequency attenuation trend.

This example highlights some advantages of the local attribute method in comparison with the sliding window approach:

• Only one parameter (the smoothing radius) needs to be specified as opposed to several (window size, overlap, and taper) in the windowing approach. The smoothing radius directly reflects the locality of the measurement.
• The local attribute approach continues the measurement smoothly through the regions of absent information such as the zero amplitude regions in the synthetic example, where the signal phase is undefined. This effect is impossible to achieve in the windowing approach unless the window size is always larger than the information gaps in the signal.

Figure  shows seismic images from compressional (PP) and shear (SS) reflections obtained by processing a land nine-component survey. Figure  shows local frequencies measured in PP and SS images after warping the SS image into PP time. The term ``image warping'' comes from medical imaging (Wolberg, 1990) and refers, in this case, to squeezing the SS image to PP reflection time to make the two images display in the same coordinate system. We can observe a general decay of frequency with time caused by seismic attenuation. After mapping (squeezing) to PP time, the SS image frequency appears higher in the shallow part of the image because of a relatively low S-wave velocity but lower in the deeper part of the image because of the apparently stronger attenuation of shear waves. A low-frequency anomaly in the PP image might be indicative of gas presence. Identifying and balancing non-stationary frequency variations of multicomponent images is an essential part of the multistep image registration technique (Fomel and Backus, 2003; Fomel et al., 2005).

Local seismic attributes