## Optimally damped rank-reduction method

A shortcoming of the traditional rank-eduction and damped rank-reduction methods is that the output perfomance is quite sensitive to the input parameter . The rank parameter varies greatly for different datasets, especially for field datasets. In order to alleviate the influence of the rank parameter, we take advantage of an adaptive singular-value weighting algorithm developed by Benaych-Georges and Nadakuditi (2012) and Nadakuditi (2013). An adaptive singular-value weighting matrix can be obtained by solving the following problem:

 (11)

where and denote th left and right singular vectors corresponding to the estimated signal component, denotes the th singular value in . is the adaptive weight vector, which can be used to construct the adaptive weighting matrix diag. The solution to the optimization problem can be obtained as:

 (12)

where represents a transform expressed as:

 Tr   Tr (13)

and denotes its derivative. The expression of can be expressed as:

 (14)

The symbol Tr denotes the trace of the input. The trace of a square matrix is defined to be the sum of elements on the main diagonal of .

 Tr (15)

where denotes the main diagonal elements of .

 (16)

To incorporate the adaptive weighting operator into the damped rank-reduction framework, we can introduce intermediate variables as

 (17) (18) (19)

then equation 16 turns into

 (20)

It can be derived that the damping formula (equation 8) also holds for equation 20 but with the damping threshold matrix expressed as

 (21)

where the subscript denotes a sufficiently large rank parameter, as required by the derivations detailed in Chen et al. (2019b). The resulted final form of the optimally damped rank-reduction method then can be expressed as

 (22)

The algorithm workflow for the optimally damped rank-reduction method (ODRR) is outlined in Algorithm 3. Due to the existence of a weighting matrix in the proposed method, it is more convenient to choose the input rank parameter empirically in practice. As mentioned previously, the rank parameter is in practice set to a relatively large value to forestall damaging the signal. But in the proposed algorithm, even if a large is used, the algorithm can adaptively shrink the singular-values. Thus, its performance is not sensitive to the input rank parameter, in contrast to other related approaches (Oropeza and Sacchi, 2011). Because of the insensitivity, one can use a sufficiently large in processing complicated datasets without leaving strong residual noise in the result. The convenience in tuning parameters makes the rank-reduction related methods more computationally feasible in large-scale data processing, e.g., the 5D reconstruction problem (Chen et al., 2019b), since one no longer needs to tune the parameters many times while one trial is already computationally demanding. A glossary describing the main mathematical notations are presented in Table 1. The three methods are all variations of how the singular-values are thresholded. Simply speaking, the DRR method improves the RR method by introducing a damping operation and the ODRR method improves DRR method by further introducing a weighting operation.

 Symbols Meanings 3D noisy seismic data or abbreviated notation of 2D frequency slice th Hankel matrix block Hankel matrix signal component estimated signal component left singular vector matrix singular-value matrix right singular vector matrix weighting matrix damping matrix damping threshold matrix temporary variable D-transform Hankelization operator thresholding operator averaging operator rank damping factor

 Tests Noisy (dB) RR (dB) DRR (dB) ODRR (dB) Linear synthetic (N=3) -8.39 6.57 10.29 11.27 Linear synthetic (N=6) -8.39 3.45 8.35 10.86 Hyperbolic synthetic (N=10) -2.17 8.27 9.58 9.65 Hyperbolic synthetic (N=20) -2.17 7.04 10.08 11.00

 Tests FK (s) RR (s) DRR (s) ODRR (s) Linear synthetic (N=3) 0.13 2.27 2.45 2.43 Linear synthetic (N=6) 0.17 3.02 3.04 3.17 Hyperbolic synthetic (N=10) 0.43 1.69 1.71 1.83 Hyperbolic synthetic (N=20) 0.49 2.75 2.69 2.83

2020-12-06