Seismic dip estimation based on the two-dimensional Hilbert transform and its application in random noise attenuation |

From equations 4 and A-5, we obtain the difference as . For

and , the Taylor series of at center c is expressed

where , . Consequently, the signum function sgn is expressed

We substitute sin for , based on sgn =sgn(sin ) for , truncate the series at the first M terms, and obtain the sinusoidal power series of the signum function as

The series in A-9 converges for ; that is, has to be larger than 1/2. On the other hand, the expansion center in the -domain is associated to the frequency center in the -domain via the relation . Therefore, must be less than or equal to 1. Accordingly, is constrained by and the corresponding is within the range . Clearly, the ideal frequency response is well approximated within the middle frequency band. Multiplying A-9 by and substituting for sin , the transfer function for the zero phase FIR of the Hilbert transform is expressed as

To obtain the causal transfer function, is multiplied by and the resultant transfer function of the FIR Hilbert transform of the (2 +2)th-order is

For =0, the transfer functions of equations A-4 and A-11 are approximated as

We compare equations A-12 and A-13, and we conclude that these two transfer functions in middle frequency band of the frequency domain differ by the constant coefficient .

Seismic dip estimation based on the two-dimensional Hilbert transform and its application in random noise attenuation |

2015-05-07