## Orthogonal polynomial transform

In a seismic profile, the amplitude of time and space can be expressed as:

 (1)

where is a set of orthogonal polynomials and is the number of basis functions and is a set of coefficients. The is a unit basis function that satisfies the condition:

 (2)

where denotes the Kronecker delta. The spectrum defined by denotes the energy distribution of the domain data in the orthogonal polynomials transform domain. Besides, the low-order coefficients represent the effective energy and the high-order coefficients represent the random noise energy. We provide a detailed introduction about how we construct the orthogonal polynomial basis function in Appendix A.

In a matrix-multiplication form, equation 1 can be expressed as the following equation

 (3)

where is constructed from , is constructed from , is constructed from . is known and can be constructed using the way introduced in Appendix A. The unknown is . can be obtained by inverting the equation 3

 (4)

where denotes matrix tranpose. In this paper, we choose , which indicates that we select 20 orthogonal polynomial basis function to represent the seismic data. Hence, inverting equation is simply inverting a matrix and is computationally efficient.

In the OPT method, we need to define the order of coefficients we want to preserve, the process of which corresponds to applying a mask operator to the orthogonal polynomial coefficients. Mask operator can be chosen to preserve low-order coefficients and reject high-order coefficients. It takes the following form:

 (5)

where denotes the mask operator, denotes the orthogonal polynomial coefficients at time and order .

The coefficients after applying the mask operator 5 become

 (6)

The useful signals can be reconstructed by

 (7)

where denotes the denoised data.

2020-03-27