Automatic rank selection

In order to obtain a satisfactory result, an appropriate $L$, referred to the rank, should be cautiously selected. A big value of $L$ will lead to preserving all components, while a small value of the rank will cause damage on the preserved reflections by the rank-reduction filter. In the first case, negligible diffraction energy are separated. In the second case, only the most coherent wave components, e.g., horizontal waves, are preserved (Chen et al., 2017,2019). Thus, there will be a strong mixture between the separated reflections and diffractions. Therefore, the determination of rank parameter is difficult and traditional methods that exploit distinct event slowness (Vicente and Mauricio, 2011) cannot perform well in complicated situations.

For the rank-reduction filtering, an alternative method is to adaptively and automatically select the rank parameter based on the features and complexity of the data. In the singular spectrum, the singular values of signals and noise are discrepant. The cut-off rank in singular value spectrum can indicate the separation of signal and noise energy. On this basis, we use an adaptive strategy to optimize the rank (Chen et al., 2017,2019). At first, we define a singular value ratio (SVR):

$$m_i=\dfrac{\sigma_i}{\sigma_{i+1}},i=1,2,\cdots,N-1,$$ (8)

where $\{m_i\}$ denotes the SVR sequence, and $N$ is the length of the singular spectrum. Then, the rank $L$ can be calculated by maximizing the SVR sequence, i.e.,

$$\hat{L}= \max\limits_i\quad m_i.$$ (9)

A detailed analysis of the LRRA method on the diffraction separation problem is presented in the next section. In addition, we will aslo comprehensively analyze the reliability of the adaptive rank selection strategy in accurately separating diffractions from reflection waves in the next section.