In a seismic profile, the amplitude of time
and space
can be expressed as:
![$\displaystyle A(t,x) = \sum_{j=0}^{M-1} C_j(t) P_j(x),$](img12.png) |
(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:
![$\displaystyle P_j(x)P_i(x)=\delta_{ij},$](img17.png) |
(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
![$\displaystyle \mathbf{A} = \mathbf{C}\mathbf{P},$](img21.png) |
(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
![$\displaystyle \mathbf{C}=\mathbf{A}\mathbf{P}^H(\mathbf{P}\mathbf{P}^H)^{-1},$](img26.png) |
(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:
![$\displaystyle \mathcal{M}_{\tau}(C_j(t)) = \left\{ \begin{array}{ll}
C_j(t) & \text{for}\quad j \le \tau \\
0 & \text{for}\quad j > \tau
\end{array}\right.,$](img31.png) |
(5) |
where
denotes the mask operator,
denotes the orthogonal polynomial coefficients at time
and order
.
The coefficients after applying the mask operator 5 become
![$\displaystyle \hat{\mathbf{C}}=\mathcal{M}_{\tau}(\mathbf{C}).$](img34.png) |
(6) |
The useful signals can be reconstructed by
![$\displaystyle \hat{\mathbf{A}} = \hat{\mathbf{C}}\mathbf{P},$](img35.png) |
(7) |
where
denotes the denoised data.
2020-03-27