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Examples

I first use a simulated synthetic example to demonstrate the performance of the proposed approach. I use the dithering approach following Chen et al. (2014a) to blend two sources in the common receiver domain. Fig 2a shows the unblended data. Fig 2b shows the blended data. I compare four approaches:

thresholding,

deconvolution, seislet thresholding, and the proposed approach. Both

thresholding and seislet thresholding assumes that the seismic data is composed of local plane-wave components and is sparse in the transform domains Yuan et al. (2015). Fig 3 shows the deblended data using the four methods. The proposed approach and the seislet thresholding obtain significantly better results. Fig 4 shows the deblending error sections (difference between the deblended data and unblended data) using four different approaches. It confirms the fact that the proposed approach obtains the least deblending error, which is followed by the seislet thresholding. Fig 5 shows the zoomed sections from each figures in Figs 2 and 3. The zoomed regions are highlighted by the frameboxes in Figs 2 and 3. One can see the highlighted difference from the zoomed sections. Fig 6a shows the convergence diagrams in terms of the signal-to-noise ratio (SNR). The definition of SNR follows Yang et al. (2015):

$\displaystyle SNR_n=10\log_{10}\frac{\Arrowvert \mathbf{m} \Arrowvert_2^2}{\Arrowvert \mathbf{m}-\mathbf{m}_n\Arrowvert_2^2},$

(11)

where $\mathbf{m}$ is the unblended data,

denotes the SNR after

th iteration. The convergence shows that while seislet thresholding can obtain very high SNR (around 24 dB), the proposed approach can obtain the highest SNR (above 30 dB). The proposed approach can also greatly accelerate the convergence. As shown in Fig 6a, it only takes around 15 iterations to achieve 25 dB, while the seislet thresholding takes more than 40 iterations to achieve a similar SNR.

Fig 6b shows the amplitude difference between 1.35s and 1.38s of the 25th trace in the simulated synthetic data example, as highlighted by the blue dash trace in Figs 2 and 3. The black solid line denotes the unblended trace (true trace). The blue double dot line corresponds to the proposed approach. The red dot dash line corresponds to seislet thresholding. The green dash line corresponds to thresholding. The yellow long dash line corresponds to deconvolution. The blue double dot line is the closest one to the black solid line. The red dot dash line is the second closest one to the black solid line. I conclude that, even though seislet thresholding can obtain a good deblending result, the proposed approach can further improve the performance. This superior performance is the same in all the profile. Here I only show a small portion of the trace in order to make the comparison clearer.


field,slet,ortho,fields,slet-e,ortho-e Figure 7. Simulated field data example. (a) Unblended data. (b) Deblended data using iterative seislet thresholding. (c) Deblended data using the proposed approach. (d) Blended data. (e) Estimation error using iterative seislet thresholding. (f) Estimation error using the proposed approach.

Fig 7 shows the deblending performance of the proposed approach on a simulated field data example. Figs 7a and 7d show the unblended and blended data, respectively. Figs 7b and 7c show the deblended data using seislet thresholding and the proposed approaches, respectively. Figs 7e and 7f show the estimation error sections using the two approaches. This example also shows that the proposed orthogonalization can improve the deblending performance of the traditional seislet thresholding.

I do not use local windows Yuan and Wang (2011) for all the aforementioned approaches and examples. Please note that when applied in local windows, all the approaches will work better. However, the seislet thresholding and the proposed approach does not need the local processing step and thus can be more convenient to implement. The limitation of the proposed approach is somewhat similar to the traditional iterative seislet thresholding approach, that is to say, the local slope need to be estimated correctly during the iterations and the data structure should not be too complicated. However, the orthogonalization can be combined with any existing iterative deblending approach and a combination of the orthogonalization strategy with other robust deblending approach can be a future investigation.

Next: Conclusion Up: Chen: Deblending with multiple Previous: Extra constraint: iterative orthogonalization

2015-09-15