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Next: Interpolating with nonstationary filters Up: Curry and Shan: Near-offset Previous: Introduction

Generating Pseudo-primaries

Multiple reflections are typically viewed as undesired noise to be removed from reflection seismic data. One way to do this is first to predict the multiples and then subtract them from the data. Free-surface multiples can be predicted by autoconvolving data so that the convolution of primaries with themselves creates events at arrival times that are the same as those of multiple reflections with a single bounce point at the free surface. This approach creates free-surface multiple reflections with correct kinematics without the need for any additional subsurface information except the recorded data (Verschuur et al., 1992; Reiter et al., 1991; Riley and Claerbout, 1976).

Autocorrelation can be used to extract synthetic active source data from incoming waves that reflect at the free surface and return within the recording array (Schuster, 2001; Claerbout, 1968; Cole, 1995; , ,). This has traditionally been thought of in a passive context, where random noise is assumed to be arriving from all locations. The reflecting waves from active-source experiments can be treated in the same manner, where the primary reflections correlate with the free-surface multiples (Shan, 2003; Berkhout and Verschuur, 2003; Reiter et al., 1991; Berkhout and Verschuur, 1994). Autocorrelating data, $ d$, (implemented as multiplication of complex-conjugates in the $ f$ domain) for two receiver points, $ r_1$ and $ r_2$, both for a single shot, $ s$, gives the pseudo-primaries, $ p$,

$\displaystyle p(s,r_1,r_2,f)= d(s, r_1,f)\bar{d}(s,r_2,f).$ (1)

One of the receiver coordinates, $ r_1$, becomes the virtual source location, while the other receiver coordinate, $ r_2$, remains the receiver location, or vice-versa, and the $ \bar{d}$ denotes the complex-conjugate of $ d$. This is similar to 2D surface-related multiple prediction, where instead of cross-correlation the data are convolved with itself, so that the convolution of primaries with primaries produces surface-related multiples (Verschuur et al., 1992).

sigsbee1shot
sigsbee1shot
Figure 2.
Crosscorrelation of a single shot. (a): Original fully-sampled split-spread shot; (b): The same shot recreated from crosscorrelating the traces in (a) using equation 1. (c): correlations are made for $ 496$ shots and are then summed using equation 2. The quality of the pseudo-primaries is poor for a single shot, but improves after the summation of many shots. All data are scaled by $ t$. $ [{\bf CR}]$
[pdf] [png]

Figure 2a shows an example of a fully-sampled split-spread shot from the Sigsbee2B data set, while Figure 2b shows a slice of the pseudo-primary output of autocorrelating the zero-offset trace with all of the other traces within that single shot. Clearly, the original shot and the pseudo-primaries generated from the autocorrelation are different. The water-bottom reflection at zero offset is present as are the first group of diffractors, but they are not present at the more distant offsets. The many other correlations between events produce undesired noise in the output. Primaries can correlate with other primaries, multiples with multiples, and noise with other noise or signal. Much of these undesired correlations, however, vary as a function of source position, $ s$, so when the correlations for the same receiver pair $ r_1$ and $ r_2$ are performed for many different source locations, $ s$, and are then summed, the unwanted events destructively interfere, while the correct pseudo-primaries constructively interfere.

$\displaystyle p(r_1,r_2,f)=\sum_{s} d(s,r_1,f)\bar{d}(s,r_2,f).$ (2)

As shown in Figure 2c, where $ 496$ sources were used, this summing over multiple source positions thus can greatly improve the pseudo-primary signal. The range of usable offsets is greatly improved, as is the signal-to-noise ratio. The pseudo-primaries now look more similar in character to the original data.

Pseudo-primaries generated from recorded multiples are interesting in part because they have different illumination than do the recorded primaries. Figure 3a illustrates a desired near-offset primary ray-path that is not recorded because of the gap between the source and the nearest receiver. In Figure 3b, the source is positioned such that the raypath first travels from the source to the water-bottom and back up to the water surface within the recording array. This ray then reflects back into the subsurface and eventually returns into the recording array at another receiver. This four-segment raypath is a free-surface multiple, which can be reconsidered as two distinct events: the first, a recorded primary event from the source $ s$ to the first receiver $ r_1$ and the second, another primary from the virtual source $ r_1$ to another receiver $ r_2$. Crosscorrelating the multiple recorded at $ r_2$ with the primary recorded at $ r_1$ produces a pseudo-primary trace with a virtual source at $ r_1$ and a receiver at $ r_2$. Comparing Figures 3a and 3b, we see that we can transform multiples that we record into pseudo-primaries with virtual source locations where we did not originally record data. This is most useful at unrecorded near offsets.

nearoffsetcartoon
nearoffsetcartoon
Figure 3.
Raypaths of a primary and multiple reflection. (a): an unrecorded primary reflection that hits the surface at near offset. (b): a recorded multiple that first reflects at the surface within the recording array, then returns within the recording array. $ [{\bf NR}]$
[pdf] [png]

The generated pseudo-primaries differ from the recorded data in several ways. First, the crosscorrelation of the data squares the amplitude spectrum of the pseudo-primaries relative to the original data, meaning that the wavelet of the pseudo-primaries will become zero-phase, different than that of the input data. Second, the amplitudes of the pseudo-primaries will differ from the amplitudes of the actual primaries, as the pseudo-primaries are a correlation and summation of different events. This amplitude difference will be both on a global scale difference between the pseudo-primaries and the primaries, as well as different relative amplitudes within each data set. Third, spurious events from other correlations may still exist in the data as the number of sources is limited.

Because of these differences between the pseudo-primaries and primaries, a direct substitution of the created near-offset pseudo-primaries is not adequate. The pseudo-primaries, however, can be used as training data for a nonstationary prediction-error filter, as the PEF is insensitive to the phase of the training data. This PEF is then used to interpolate the missing near offsets, so that the negative aspects of the pseudoprimaries, such as the incorrect wavelet and extra noise, do not affect the PEF, while the positive aspects of the pseudo-primaries, contained in the autocorrelation of these data, are used.


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Next: Interpolating with nonstationary filters Up: Curry and Shan: Near-offset Previous: Introduction

2009-04-13