I will use three different examples to show the performance of SGK in denoising multidimensional seismic data. Please note that when using equations 3 (or 8) and 4 for dictionary learning, the multidimensional seismic data is first mapped from the original form to a 2D matrix according to some patching criteria. Some details about the patching method can be found in Yu et al. (2015) or Chen et al. (2016a). After iteratively solving equations 3 (or 8) and 4 several times, the denoised data is expressed as

$\displaystyle \hat{\mathbf{D}} = \mathbf{F}^{Niter}\mathbf{M}^{Niter}.$ (16)

An inverse mapping is then applied to $\hat{\mathbf{D}}$ to output the finally denoised data.

For measuring the denoising performance of synthetic data examples, where one knows the clean data, I use the signal-to-noise ratio (SNR) (Huang et al., 2016a; Liu et al., 2009a) measurement and the formula is expressed as follows:

$\displaystyle SNR=10\log_{10}\frac{\Arrowvert \mathbf{D}_{true} \Arrowvert_F^2}{\Arrowvert \mathbf{D}_{true} -\hat{\mathbf{D}}\Arrowvert_F^2},$ (17)

where $\mathbf{D}_{true}$ denotes the clean data and $\hat{\mathbf{D}}$ denotes the denoised data.

In addition to the commonly used SNR measurement, one can also use local similarity (Fomel, 2007; Chen and Fomel, 2015b) as a convenient tool to evaluate denoising performance. The abnormal area in the local similarity map with high similarity indicates the area that contains significant signal leakage in the removed noise. A local similarity map with values that are close to zero, as well as the observed significant amount of removed noise, provides a valid support of a successful denoising performance. In addition, the local similarity measurement can be used in many cases since the input data of local similarity calculation are just the denoised data and removed noise. It provides an alternative in the case of field data processing, where the clean data is unknown and the SNR based evaluation becomes unavailable. Besides, local similarity is a local measurement of denoising performance, whereas the SNR is a global measurement, which cannot guide us to pick out the areas with poor denoising performances.