Sparse inversion

For noise attenuation and data interpolation with radon transforms, numerous authors have shown that the inclusion of sparseness constraints during transformation into the radon domain improves the final result Herrmann et al. (2000); Sacchi and Ulrych (1995); Trad et al. (2003). The goal is to obtain a solution with minimum entropy Burg (1975). This property is important because radon transforms suffer from decreases in resolution due to the limited aperture of the data, creating transformation artifacts known as butterfly patterns Kabir and Marfurt (1999). The sparse inversion attenuates these effects by minimizing energy that does not focus well in the radon domain.

In this paper, we use the method of Sacchi and Ulrych (1995) to estimate a sparse model m. This technique imposes a Cauchy form probability-density function to the model parameters. This long-tailed probability-density function isolates the most energetic components of the radon domain and ignores the smallest, thus giving a minimum entropy solution. Note that other techniques such as stochastic inversion Thorson and Claerbout (1985) are also available.

To obtain a sparse radon domain m, a regularization term, i.e., the Cauchy function, is introduced in equation (7) as follows:  

$$f({\bf m})=\Vert{\bf r_d}\Vert^2 + \epsilon^2 \sum_{i=1}^{N} \mbox{ln} \left ( 1+\frac{m_i^2}{\gamma^2} \right ),$$ (8)

where N is the number of parameters to be estimated, $\epsilon$ the Lagrange multiplier and $\gamma$ a parameter controlling the amount of sparseness in the model. Both $\epsilon$ and $\gamma$ are estimated by trial and error. Whereas solving for m in equation (7) is a linear problem, solving for m equation (8) is not. Therefore, we use a quasi-Newton method to minimize iteratively the objective function in equation (8) Guitton and Symes (2003).

To demonstrate the effectiveness of the sparse radon transform for signal noise separation, we show the representation of the data panel from Figure in the f-k domain (a), the radon domain (b), and the sparse radon domain (c). Much of the energy in the synthetic data panel follows linear moveout. Therefore, the dominant factor in real data for signal/noise separation becomes the moveout of the scattered arrival compared to the direct arrival. f-k filtering is effective at isolating energy with differing dips with proper spatial sampling but is unable to isolate the energy in time regardless of sampling geometry. The side reflection and the direct arrival are colocated in f-k space and can not be separated. In the radon domain, the side reflection and the direct arrivals map to distinctly different dips for different $\tau$'s. But, without sparseness constraints many spurious artifacts are introduced from unwanted arrivals. The shaded region near zero slope represents an example radon mute that would remove the side reflection without reducing energy following the moveout of the direct arrival.

 

complarse-fp-tp
Figure 3 Figure showing representation of the synthetic teleseismic wavefield shown in in the f-k domain (a), the radon domain (b), and the sparse linear radon domain (c). The letters denote arrival location identified in Figure . The f-k domain allows the isolation of distinct dips but does not offer a way to isolate different dips in time (e.g. time variable dip filter). The standard linear radon domain offers the possibility of separating dips as a function of time but introduces artifacts related to the transform. The sparse linear radon transform (c) allows us to isolate energy in time as in (b) while minimizing artifacts introduced from the transform. The shaded region indicates the radon mute used for the synthetic example shown in Figure .