Orthogonal polynomial transform

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

$$A(t,x) = \sum_{j=0}^{M-1} C_j(t) P_j(x),$$ (1)

where $\{P_j(x),j=0,1,2,\cdots,M-1 \}$ is a set of orthogonal polynomials and $M$ is the number of basis functions and $\{C_j(t),j=0,1,2,\cdots,M-1 \}$ is a set of coefficients. The $P_j(x)$ is a unit basis function that satisfies the condition:

$$P_j(x)P_i(x)=\delta_{ij},$$ (2)

where $\delta_{ij}$ denotes the Kronecker delta. The spectrum defined by $C_j(t)$ denotes the energy distribution of the $t-x$ 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

$$\mathbf{A} = \mathbf{C}\mathbf{P},$$ (3)

where $\mathbf{A}$ is constructed from $A(t,x)$, $\mathbf{C}$ is constructed from $C_j(t)$, $\mathbf{P}$ is constructed from $P_j(x)$. $\mathbf{A}$ is known and $\mathbf{P}$ can be constructed using the way introduced in Appendix A. The unknown is $\mathbf{C}$. $\mathbf{C}$ can be obtained by inverting the equation 3

$$\mathbf{C}=\mathbf{A}\mathbf{P}^H(\mathbf{P}\mathbf{P}^H)^{-1},$$ (4)

where $[\cdot]^H$ denotes matrix tranpose. In this paper, we choose $M=20$, which indicates that we select 20 orthogonal polynomial basis function to represent the seismic data. Hence, inverting equation $\mathbf{P}\mathbf{P}^H$ is simply inverting a $20\times 20$ 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:

$$\mathcal{M}_{\tau}(C_j(t)) = \left\{ \begin{array}{ll} C_j(t) & \text{for}\quad j \le \tau \\ 0 & \text{for}\quad j > \tau \end{array}\right.,$$ (5)

where $\mathcal{M}$ denotes the mask operator, $C_j(t)$ denotes the orthogonal polynomial coefficients at time $t$ and order $j$.

The coefficients after applying the mask operator 5 become

$$\hat{\mathbf{C}}=\mathcal{M}_{\tau}(\mathbf{C}).$$ (6)

The useful signals can be reconstructed by

$$\hat{\mathbf{A}} = \hat{\mathbf{C}}\mathbf{P},$$ (7)

where $\hat{\mathbf{A}}$ denotes the denoised data.