next up previous [pdf]

Next: - domain regularized nonstationary Up: Liu et al.: Noise Previous: Introduction

Review of $f$-$x$ domain stationary autoregression

We first consider a seismic section $S(t,x)$ that consists of a single linear event with the slope $p$ and constant amplitude. The frequency domain representation of $S(t,x)$ is given by

\begin{displaymath}
S(f,x)=A(f){{e}^{j2\pi fxp}},
\end{displaymath} (1)

where $A(f)$is the wavelet spectrum, $f$is the temporal frequency, and $x$is the spatial variable. We assume $x=n\Delta x$, where $n=1,2,...,N$, $N$is the number of traces in the whole section. The relationship between the n-th trace and (n-1)-th trace can be easily shown as
\begin{displaymath}
{{S}_{n}}(f)={{a}_{1}}(f){{S}_{n-1}}(f),
\end{displaymath} (2)

where ${{a}_{1}}=\exp (j2\pi fp\Delta x)$. This recursion is a first-order differential equation also known as an AR model of order 1 and represents a single complex-valued harmonic (Bekara and van der Baan, 2009). If there are $M$ linear events in x-t domain, we can have a difference equation of order $M$ (Sacchi and Kuehl, 2001)
\begin{displaymath}
{{S}_{n}}(f)=\sum\limits_{i=1}^{M}{{{a}_{i}}(f){{S}_{n-i}}(f)}.
\end{displaymath} (3)

The recursive filter $\left\{ {{a}_{i}}(f) \right\}$can be found for predicting a noise-free superposition of complex harmonics. Considering seismic data with additive random noise and non-causal prediction with order $2M$which includes both forward and backward prediction equations (Spitz, 1991; Naghizadeh and Sacchi, 2009), we can obtain
\begin{displaymath}
{{\varepsilon }_{n}}(f)={{S}_{n}}(f)-\sum\limits_{i=1}^{M}{...
...n-i}}}(f)-\sum\limits_{i=-1}^{-M}{{{a}_{i}}{{S}_{n-i}}(f)},
\end{displaymath} (4)

where ${{\varepsilon }_{n}}(f)$is a complex noise sequence. Canales (1984) argues a causal estimate of signal $\sum\limits_{i=1}^{M}{{{a}_{i}}(f){{S}_{n-i}}(f)}$ is the predictable part of data obtained by an AR model. This operation is usually called $f$-$x$ deconvolution (Gulunay, 1986). Noise-free events that are linear in the $t$-$x$ domain manifest as a superposition of harmonics in the $f$-$x$ domain and these harmonics can be perfectly predicted using AR filter. If seismic events are not linear, or the amplitudes of wavelet are varying from trace to trace, they no longer follow Canales’s assumptions (Canales, 1984). One needs to perform $f$-$x$ deconvolution over a short sliding window in time and space. This leaves the choice of window parameters (window size and length of overlapping between adjacent windows). Bekara and van der Baan (2009) discuss some limitations of conventional $f$-$x$ deconvolution in detail.


next up previous [pdf]

Next: - domain regularized nonstationary Up: Liu et al.: Noise Previous: Introduction

2013-11-13