Band-limiting root approximation

Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing

Series	Geophysical References Series
Title	Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
Author	Enders A. Robinson and Sven Treitel
Chapter	13
DOI	http://dx.doi.org/10.1190/1.9781560801610
ISBN	9781560801481
Store	SEG Online Store

Neidell (1991) set forth properties that apply to families of processed seismic wavelets, at least approximately. Most processed seismic data for which the wavelets have been treated or conditioned by wavelet processing satisfy these premises, at least to the necessary degree of approximation: (1) Seismic wavelet conditioning, such as spiking deconvolution, produces an effective band-limited wavelet with a 40- to 100-ms time duration and a smooth, unimodal amplitude spectrum characterized by a peak central frequency in the range of 15 to 30 Hz. (2) Effective digital sampling of such a processed wavelet, in view of the bandwidth, can be at 4-ms intervals with 10 to 25 samples. As a result, the Z-transform of the digitally sampled processed wavelet is a polynomial in Z of order 9 to 24, with real coefficients.

The factor 1 – Z (giving the root Z = 1) must be present in the Z-transform to have zero amplitude at zero frequency. This factor is the Z-transform of the two-term high-pass wavelet (1, –1). Figure 9a shows the amplitude spectrum of this wavelet.

Figure 9.  (a) Amplitude spectrum of a high-pass two-term wavelet. (b) Amplitude spectrum of a low-pass two-term wavelet.

Proper digital sampling requires effectively zero signal energy at the Nyquist frequency and beyond. The factor 1 + Z (giving the root Z = –1) must be present in the Z-transform if zero amplitude is required at the Nyquist frequency. This factor is the Z-transform of the two-term low-pass wavelet (1, 1). Figure 9b shows the amplitude spectrum of this wavelet.

Any processed seismic wavelet with a smooth and unimodal amplitude spectrum and short duration has its band-limited character described principally by multiple roots on the unit circle at Z = 1 and Z = –1. As a result, the Z-transform of the wavelet will contain the expression

${\begin{aligned}{\left(1-Z\right)}^{n}\ {\left({1}+Z\right)}^{m},\end{aligned}}$

(7)

where n (representing the number of high-pass factors) and m (representing the number of low-pass factors) are constant integers. The amplitude spectrum corresponding to expression 7 is smooth and unimodal, and it shifts its peak to higher frequencies as n, is increased and to lower frequencies as m is increased. For specific values of n and m, the resulting waveform is antisymmetric whenever n is odd. For even values of n, the waveform is symmetric. Neidell introduced expression 7 as the band-limiting root approximation (BLRA) to explain many properties of postprocessing seismic wavelets.

The requirements of restricted bandwidth, finite and short time duration, and appropriate digital sampling require that the geophysicist assign most of the wavelet roots to the band-limiting process. There are only a few uncommitted roots. As a result, methods that manipulate these few parameters often are remarkably successful in conditioning seismic waveforms to a desired behavior. Antisymmetric waveforms have odd values of n. Figure 10a shows such an antisymmetric wavelet, and Figure 10b shows its amplitude spectrum. Symmetric waveforms have even values of n. Figure 11a shows such a symmetric wavelet, and Figure 11b shows its amplitude spectrum. Waveforms that are largely symmetric or antisymmetric have most of their roots dedicated to band limiting.

Figure 10.  (a) A representative Neidell antisymmetric wavelet. (b) Its amplitude spectrum.

Figure 11.  (a) A representative Neidell symmetric wavelet. (b) Its amplitude spectrum.

Figure 12.  (a) A symmetric wavelet. (b) The symmetric wavelet with a $45^{\circ }$ Hilbert rotation. (c) The symmetric wavelet with a $45^{\circ }$ Neidell phase rotation.

Let us take a Neidell wavelet with n = 2, m = 23, with a peak frequency of 22.8 Hz. Because this wavelet actually has negative polarity or $180^{\circ }$ phase character, we reverse the sign to get a symmetric wavelet (Figure 12a). First we rotate the symmetric wavelet by $45^{\circ }$ with the complex-trace method (i.e., the Hilbert method) given earlier in the chapter (Figure 12b). Second, we rotate the symmetric wavelet by convolving it with the $45^{\circ }$ Neidell two-term wavelet (1, –0.7226), (Figure 12c). We can see that the approximation obtained by the Neidell rotation is close to that obtained by the more conventional complex-trace method.

Effective wavelet characterizations for seismic wavelet behavior are responsive to known physical properties and recurring physical measurements. Waveforms in processed data vary from causal to noncausal and can be described by attributes such as duration, characteristic frequency, bandwidth, and approximate quadrature or symmetric structure. The Neidell approximation explains the success of the constant-phase rotation operation and is valuable for an understanding of processed seismic waveforms. The success of this approximation shows that the processed seismic wavelet has fewer degrees of freedom than its duration alone implies.

Seismic reflection events are classified as being primary or multiple reflections. The standard view considers primary reflections to be the signal and multiple reflections to be a form of coherent noise. The basic model in seismic migration assumes that reflection data consist of primary reflections only. If multiple reflections are not removed, they can be misinterpreted as or they may interfere with primary reflections. Weglein (1999) gave an overview of methods for attenuating multiples and discussed two basic approaches to multiple attenuation: (1) methods that exploit a feature or property that differentiates a primary from a multiple and (2) methods that predict and then subtract multiples from seismic data. Weglein and Stolt (1999) presented migration-inversion (M-I) methods for processing primaries, as well as amplitude-variation-with-offset (AVO) methods that are useful in the case of curved and dipping reflectors in a multidimensional, heterogeneous, anisotropic earth.

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Also in this chapter

• Phase rotation
• Alternate approaches to phase rotation
• Appendix N: The energy-delay theorem

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