Complex seismic trace analysis treats a seismic trace as the product of two independent and separable functions: instantaneous amplitude and cosine of the instantaneous phase. At any given time, instantaneous amplitude is the maximum value the seismic trace can attain under a constant phase rotation, and instantaneous phase is the phase angle required to rotate the trace to the maximum. Defining these two attributes in this way permits complex seismic trace analysis to be founded without reference to the complex trace. All other complex seismic trace attributes derive from amplitude and phase through differentiation, averaging, combination, or transformation. The chief derived attributes are frequency, relative amplitude change, wavelength, dip, and azimuth. At any point on a seismic trace, the instantaneous attributes describe a sinusoid that locally matches the trace. Most instantaneous attributes are improved through filtering or weighted averaging. The phase wavenumber vector provides a convenient basis for quantifying 3D seismic properties such as reflection parallelism.
Fourier power spectra are often usefully characterized by average measures. In reflection seismology, the important average measures are center frequency, spectral bandwidth, and dominant frequency. These quantities have definitions familiar from probability theory: center frequency is the spectral mean, spectral bandwidth is the standard deviation about that mean, and dominant frequency is the square root of the second moment, which serves as an estimate of the zero-crossing frequency. These measures suggest counterparts defined with instantaneous power spectra in place of Fourier power spectra, so that they are instantaneous in time though they represent averages in frequency. Intuitively reasonable requirements yield specific forms for these instantaneous quantities that can be computed with familiar complex seismic trace attributes. Instantaneous center frequency is just instantaneous frequency. Instantaneous bandwidth is the absolute value of the derivative of the instantaneous amplitude divided by the instantaneous amplitude. Instantaneous dominant frequency is the square root of the sum of the squares of the instantaneous frequency and instantaneous bandwidth. Instantaneous bandwidth and dominant frequency find employment as additional complex seismic trace attributes in the detailed study of seismic data. Instantaneous bandwidth is observed to be nearly always less than instantaneous frequency; the points where it is larger may mark the onset of distinct wavelets. These attributes, together with instantaneous frequency, are perhaps of greater use in revealing the time-varying spectral properties of seismic data. They can help in the search for low frequency shadows or in the analysis of frequency change due to effects of data processing. Instantaneous bandwidth and dominant frequency complement instantaneous frequency and should find wide application in the analysis of seismic reflection data. In reflection seismology, an entire frequency spectrum is
A number of ways have been offered to calculate instantaneous frequency, an important complex seismic trace attribute. The standard calculation follows directly from its definition and requires two differentiations (Taner et al., 1979). By avoiding these differentiations, three formulas that approximate instantaneous frequency are faster to compute. The first employs a two‐point finite‐impulse response (FIR) differentiator in place of the derivative filter (Scheuer and Oldenburg, 1988). The second is nearly the same as the first, except that it employs a three‐point FIR differentiator (Boashash et al., 1991). The third takes a different approach and involves two approximations (Claerbout, 1976, p. 20; Yilmaz, 1987, p. 521). How do these formulas compare, and which is best?
Many seismic attributes are redundant or useless and confuse seismic interpretation more than they help. They are easily recognized given a few guidelines and tools. Discarding them leaves an attribute list that is both more manageable and more honest.
The Grenville Front is a major tectonic boundary exposed on the Canadian Shield. The front is defined as the northwestern limit of Grenvillian deformation, on the basis of geochronological and metamorphic data. This boundary is also evident in some geophysical data sets. The Lithoprobe Abitibi–Grenville transect crosses the Grenville Front near Lac Témiscamingue in western Quebec. A new 114 km long deep seismic reflection line shows a crustal structure quite different from that seen on previous surveys across the Grenville Front. The Archean foreland (Pontiac Subprovince) has a pattern of reflectivity similar to that seen in most of the Superior Province. This pattern continues for some 30 km south of the surface exposure of the Grenville Front. There is no evidence for a band of dipping reflectors truncating the horizontal Pontiac reflectors; in fact, the leading edge of the Grenville Province is difficult to identify on the seismic section. The Moho is well defined and reveals that the crust thins under the Grenville Front. The magnetotelluric survey shows that the upper crust is resistive across the entire profile, but the resistivity is higher within a Grenvillian allochthonous terrane at the southern end of the profile. The mid-crustal low-resistivity layer and the upper mantle electrical anisotropy are also continuous across the Grenville Front. The Grenville Front is highly variable in its character along the Grenville Orogen, and this character may be strongly controlled by the nature of the foreland to the northwest.
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