Appendix A: Review of $f-x$ deconvolution

$f-x$ deconvolution is one of the most widely used approaches for random noise attenuation. In this appendix, we briefly review the theory of $f-x$ deconvolution. Let $s_t(x)$ be the seismic signal located at trace $x$ and time $t$. If the slope of a linear event with constant amplitude in a seismic section is $\psi$, then:

$$s_t(x+1)=s_{t-x\psi\Delta h}(1),$$ (12)

where $\Delta h$ denotes the spatial interval. Equation 12 can be directly transformed into the frequency domain utilizing the time-shift property of the Fourier transform:

$$S_f(x+1)=S_f(1)e^{-i2\pi fx\psi\Delta h}.$$ (13)

For a specific frequency $f_0$, from equation 13 we can obtain a linear recursion, which is given by:

$$S_{f_0}(x+1)=a_{f_0}(1)S_{f_0}(x),$$ (14)

where $a_{f_0}(1)=e^{-i2\pi f_0\psi\Delta h}$. This recursion is also known as an auto-regressive (AR) model of order 1 (Canales, 1984). Similarly, superposition of $p$ linear events in the $t-x$ domain can be represented by an AR model of order $p$ (Harris and White, 1997; Tufts and Kumaresan, 1982) as the following equation:

$$S_{f_0}(x+1)=a_{f_0}(1)S_{f_0}(x)+a_{f_0}(2)S_{f_0}(x-1)+\cdots+a_{f_0}(p)S_{f_0}(x+1-p),$$ (15)

Equation 15 can be formulated as a convolutional form:

$$\mathbf{d}=\mathbf{f}*\mathbf{a},$$ (16)

where $\mathbf{d}$ denotes the vector composed of $S_{f_0}(x+1)(x=1,2,\cdots,X)$, $\mathbf{f}$ denotes the vector composed of $S_{f_0}(x)(x=1,2,\cdots,X)$, $\mathbf{a}$ denotes the vector composed of $a_{f_0}(x)(x=1,2,\cdots,X)$, and $X$ denotes the number of traces.

Equation 16 can be formulated as a matrix vector form:

$$\mathbf{d}=\mathbf{Fm},$$ (17)

where $\mathbf{F}$ is the covolution matrix composed by $\mathbf{f}$. Suppose $p=4$, the detailed form of equation 17 can be expressed as:

$$\left[ \begin{array}{c} f_2\\ f_3\\ \vdots\\ f_{X}\\ 0\\ 0\\... ...ight] \left[ \begin{array}{c} a_1\\ a_2\\ a_3\\ a_{4}\\ \end{array}\right].$$ (18)

However, equation 17 is based on clean signal model. In reality, the seismic data is composed of random noise. Thus, we have to solve $\mathbf{a}$ from the noise corrupted observation $\mathbf{d}$ based on some optimization schemes. Based on equation 15, we can formulate an optimization problem based on the minimum prediction error energy assumption. The predictive error filter can be solved by minimizing the following objective function:

$$J={\Arrowvert \mathbf{Fm}-\mathbf{d} \Arrowvert}_2^2,$$ (19)

where $\Arrowvert\cdot\Arrowvert_2^2$ denotes the squares of $L_2$ norm.

Taking derivatives of the cost function 19 with respect to $\mathbf{m}$, and setting the result to zero, we can obtain the following equation:

$$\mathbf{F}^T\mathbf{d}=\mathbf{F}^{T}\mathbf{Fm},$$ (20)

where $[\cdot]^T$ denotes transpose. Note that $\mathbf{F}^T\mathbf{F}$ is a Toeplitz form and thus can be efficiently solved using Levinson's recursion. In order to stabilize the recursion for solving $\hat{\mathbf{a}}$, we need to add a small perturbation to the diagonal of the Toeplitz matrix:

$$\hat{\mathbf{a}}= (\mathbf{F}^{T}\mathbf{F}+\mu\mathbf{I})^{-1}\mathbf{F}^T\mathbf{d}.$$ (21)

Finally, the estimated clean data (denoised data) can be expressed as:

$$\hat{\mathbf{d}}= \mathbf{F}\hat{\mathbf{a}}.$$ (22)

It is worth to be mentioned that, the $f-x$ deconvolution approach introduced here only applies a forward AR model to estimate the signal. A hybrid forward and backward version of AR model was proposed in Wang (1999).