Filter estimation

The PEFs are time-space domain non-stationary filters to cope with the variability of seismic data with time, offset and shot position. The basic equations for non-stationary PEFs estimation are Guitton (2003b):

$$\begin{array} {rcccl} \bf{0} &\approx& {\bf r_y} &=& \bf{YK... ...bf{y}, \\ \bf{0} &\approx& {\bf r_a} &=& \bf{Ra}, \end{array}$$ (8)

where ${\bf Y}$ is a non-stationary combination matrix Margrave (1998), ${\bf K}$ is a masking operator, ${\bf a}$ a vector of the unknown PEFs coefficients, ${\bf y}$ the data vector from which we want to estimate the PEFs and ${\bf R}$ a regularization operator.

Often with seismic data, the amplitude varies across offset, midpoint and time. These amplitude variations can be troublesome with least-squares inversion because they tend to bias the final result Claerbout (1992). Therefore, it is important to make sure that these amplitude variations do not affect our processing. One solution is to apply a weight to the data like Automatic Gain Control (AGC) or a geometrical spreading correction. However, a better way is to incorporate this weight inside the inversion by scaling the residual Guitton (2003a). Introducing a weighting function ${\bf W}$ in the PEFs estimation leads to:

$$\begin{array} {rcccl} \bf{0} &\approx& {\bf r_y} &=& \bf{W(YKa+y)}, \\ \bf{0} &\approx& {\bf r_a} &=& \bf{Ra}. \end{array}$$ (9)

As shown by Guitton (2003a), this weighting improves the signal/noise separation results and can incorporate a mute zone where no PEFs are to be estimated. This weight can be different for the noise and the signal PEFs. Estimating ${\bf a}$ in a least-squares sense gives:

$$f({\bf a})=\Vert{\bf r_y}\Vert^2+\epsilon^2\Vert{\bf r_a}\Vert^2,$$ (10)

which leads to the least-squares estimate of ${\bf a}$ 

$$\hat{\bf{a}}=-({\bf K'Y'W}^2{\bf YK}+\epsilon^2{\bf R'R})^{-1}{\bf K'Y'W}^2{\bf y}.$$ (11)

Because many filter coefficients are estimated, ${\bf a}$ is estimated iteratively with a conjugate-gradient method.

Now, prior to the signal estimation in equation (5), ${\bf S}$ and ${\bf N}$ need to be computed from a signal and noise model, respectively. The multiple model is often (but not necessarily) derived by auto-convolving the recorded data Verschhur et al. (1992), thus obtaining a prestack model of the multiples later used to estimate a bank of non-stationary PEFs ${\bf N}$.

At this stage, a key assumption is that the relative amplitude of all order of multiples is preserved. In theory, an accurate surface-related multiple model can be derived if (1) the source wavelet is known, (2) the surface coverage is large enough, and (3) all the terms of the Taylor series that model different orders of multiples are incorporated Verschhur et al. (1992). In practice, however, a single convolution is performed (first term of the Taylor series) which gives a multiple model with erroneous relative amplitude for high-order multiples Guitton et al. (2001); Wang and Levin (1994). In addition, the surface coverage might not be sufficient. This can generate wrong amplitudes for short offset traces and complex structures. Because PEFs estimate patterns, wrong relative amplitude can affect the noise estimation. However, as we shall see later, 3D filters seem to better cope with noise modeling inadequacies.

The signal PEFs are more difficult to estimate since the signal is usually unknown. However, Spitz (1999) shows that for uncorrelated signal and noise, the data PEFs ${\bf D}$ can be approximated with  

$$\begin{array} {rcl} \bold D &=& \bold S \bold N. \end{array}$$ (12)

As demonstrated by Claerbout and Fomel (2000), equation (12) is a good approximation for the data PEFs because PEFs are important where they are small. Both ${\bf N}$ and ${\bf D}$ can be estimated from the model of the multiples and the data (primaries plus multiples), respectively. Estimation of the signal PEFs involves a deconvolution in equation (12) that can be unstable with non-stationary filters. To avoid the deconvolution step, the noise PEFs are convolved with the data:

$${\bf u = Nd}.$$ (13)

Then the PEFs ${\bf U}$ are estimated for ${\bf u}$ such that

$${\bf Uu = UNd \approx 0}.$$ (14)

From Spitz's approximation in equation (12), the following relationships hold:

$${\bf UN = D = SN},$$ (15)

and ${\bf U = S}$. Therefore, by convolving the data with the noise PEFs, signal PEFs consistent with the Spitz approximation can be computed. Equation (12) insures that the PEFs ${\bf S}$ and ${\bf N}$ will not span similar components of the data space.

Thanks to the Helix Claerbout (1998); Mersereau and Dudgeon (1974), the PEFs can have any dimension. In this paper, I use 2D and 3D filters and demonstrate that 3D filters lead to the best noise attenuation result. When 2D filters are used, the multiple attenuation is performed on one shot gather at a time. When 3D filters are used, the multiple attenuation is performed on one macro-gather at a time. A macro-gather is a cube made of fifty consecutive shots with all the offsets and time samples. When the multiple attenuation is done, the macro-gathers are reassembled to form the final result. There is an overlap of five shots between successive macro-gathers. In the next section, I show a prestack multiple attenuation example with the synthetic Sigsbee2B dataset.

 

stratigraphy
Figure 1 Stratigraphic interval velocity model of the Sigsbee2B dataset.

 

datasignal
Figure 2 Two constant offset sections (h=1125 ft) of the Sigsbee2B dataset with (a) and without (b) free surface condition. The multiples are very strong below 5 s.