Streaming orthogonal prediction filter in - domain for random noise attenuation |

where is the identity matrix and is the given data sample at the point . and are the scale parameters controlling the filter variability on the two axes.

Consider a 2D noncausal prediction filter with 20 prediction coefficients :

where `` '' denotes zero, and the data vector and the filter vector in equation 1 are shown as follows:

where represents the translation of in both time and space directions with time shift and space shift . and are the temporal and spatial lengths of the prediction filter, e.g., =2 and =2 in equation 2.

The least-squares solution of equation 1 is

where

Fomel and Claerbout (2016) propose the Sherman-Morrison formula to directly transform the inverse matrix in equation 4 without iterations. The Sherman-Morrison formula is an analytic method for solving the inverse of a special matrix (Hager, 1989). If both matrices and are invertible, then is invertible and

In the special case where matrix is a column vector and matrix is a row vector , equation 6 can be rewritten as

The direct derivation of the Sherman-Morrison formula (equation 7) is described in Appendix A.

Applying the Sherman-Morrison formula to equation 4, the - streaming PF coefficients and prediction error can be calculated as

and

For seismic random noise attenuation, we assume the residual of prediction filtering is the random noise at the point . For calculating 2D - streaming PFs, we need to store one previous time-neighboring PF, , and one previous space-neighboring PF, , both and will be used when the stream arrives at its adjacency.

One can compare a streaming PF with a stationary PF. The autoregression equation for a traditional PF takes the following form:

The least-squares solution of equation 10 at each point is

The matrix in equation 11 is similar to that in equation 4. The comparison of equation 4 and equation 11 indicates that the results of the streaming PFs become gradually more accurate as the scale parameter decreases. However, according to equation 9, a small may cause the residual to also be small, which means that there is too much noise in the signal section. To solve this problem, we use a two-step strategy. First, we choose a relatively large to get a large residual . The first step amounts to an ``over-filtering'', which generates an approximately ``clean'' signal. Next, the signal leakage in the noise section can be extracted by applying signal-and-noise orthogonalization.

We derive the definition of the streaming orthogonalization weight (SOW) from the global orthogonalization weight (GOW) (Chen and Fomel, 2015). Assume that the leaking signal has a linear correlation with the estimated signal section in the first step,

where is the GOW. The ideal signal and noise can then be estimated by

Under the assumption that the noise is orthogonal to the signal ,

Substituting equation 13 and 14 into equation 15, one can get the GOW as

To get the orthogonalization weight for each data value, one possible definition of the SOW is:

where is the SOW for the data point . and represent the signal and noise values at the point , respectively. In the SOPF, and correspond to the predictable part and the prediction residual produced in the first step. Direct point-by-point division of the values may produce unstable values. One common method to solve this problem is the iterative least-squares method (Chen and Fomel, 2015). Here, we propose a streaming method to calculate the SOW.

Suppose that the SOW gets updated with each new data point . The new SOW, , should stay close to the previous time-neighboring SOW and the previous space-neighboring SOW . Equation 17 can be rewritten as

where and are the scale parameters controlling the variability on the two axes.

The least-squares solution of equation 18 is

where

Applying equation 13, one can get the denoised data value after the SOPF

where and are the predictable signals and the prediction residuals calculated in the first step.

We implement the two-step strategy within the streaming method and obtain the denoised data value as each new noisy data value arrives. Table 1 compares the computational cost between - deconvolution, - regularized nonstationary autoregression (RNA) (Liu et al., 2012), and the proposed method. In general, the cost of SOPF is minimal.

Method | Filter storage | Cost |

- deconvolution | ||

- RNA | ||

- SOPF |

Streaming orthogonal prediction filter in - domain for random noise attenuation |

2019-05-06