Instantaneous spectral bandwidth and dominant frequency with applications to seismic reflection data

Transcription

1 GEOPHYSICS, VOL. 58, NO. 3 (MARCH 1993), P , 7 FIGS. Instantaneous spectral bandwidth and dominant frequency with applications to seismic reflection data Arthur E. Barnes* ABSTRACT 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. INTRODUCTION.. In reflection seismology, an entire frequency spectrum is sometimes usefully characterized by several average parameters, notably the center (or mean) frequency, the spectral bandwidth, and the dominant frequency (Anstey, 1977, p. 3-4; Widess, 1982; Kallweit and Wood, 1982; Berkhout, 1984, p. 28). Competing definitions for all three quantities exist; however, the center frequency is commonly defined as the average frequency of the power spectrum, and the spectral bandwidth as a standard deviation about the center frequency (Berkhout, 1984, p. 28). These definitions are intuitively reasonable and suggest corresponding instantaneous measures (Cohen and Lee, 1988, 1990; Cohen, 1989; Jones and Boashash, 1990). Dominant frequency lacks a comparable definition, but rather is defined in terms of the density of amplitude maxima (Kallweit and Wood, 1982; Widess, 1982; Yilmaz, 1987, p. 40; Sheriff, 1989, p. 212; Sheriff, 1991, p. 88). It is often referred to without being specified (e.g., Yilmaz, 1987, p. 160, 468; Sheriff and Geldart, 1989, p. 42; Lendzionowski et al., 1990; Tatham and McCormack, 1991, p. 114). The square root of the second moment of the power spectrum can serve as a practical measure of dominant frequency. It is an estimate of the zero-crossing frequency and is closely related to center frequency and spectral bandwidth. Equivalent instantaneous forms of these three measures are defined in terms of instantaneous power spectra; these are constrained such that the instantaneous quantities are computable with familiar complex seismic trace attributes. Instantaneous bandwidth and dominant fre- Manuscript received by the Editor March 23, 1992; revised manuscript received September 3, *Dépt de Génie Minéral, Ecole Polytechnique, CP 6079, Succursale A, Montreal, Québec, CANADA H3C 3A Society of Exploration Geophysicists. All rights reserved.

2 420 Barnes quency complement instantaneous frequency and find application in the analysis of the time-varying spectral properties of seismic reflection data. I review the concepts of the center frequency, spectral bandwidth, and dominant frequency of a Fourier power spectrum, and I present corresponding instantaneous measures expressed in terms of instantaneous power spectra. These are reduced to simple formulas in terms of complex trace attributes. Several examples illustrate applications of instantaneous spectral bandwidth and dominant frequency to reflection seismology. MATHEMATICAL DEVELOPMENT I review the definitions of center frequency, spectral bandwidth, and dominant frequency of a Fourier power spectrum to provide the basis for the introduction of corresponding instantaneous spectral measures. These quantities are then reduced to easily computable complex trace attributes. The objective in presenting these attributes in this manner is to emphasize their role as time-variant measures of average spectral properties. This lends them intuitively appealing meaning and suggests useful applications. The development considers one-sided power spectra only (e.g., Berkhout, 1984). This is justified because a seismic trace is a purely real function. Therefore, the negative frequency half of its spectrum is the mirror image of the positive frequency half and can be ignored, obviating the need to consider absolute values of frequency (e.g., McCarley, 1985). This is also justified if the seismic trace is made analytic, for then the negative half of its power spectrum is zero and can be discounted. Average spectral measures The mean or center frequency is defined as of a power spectrum Another average spectral measure is the second moment of the power spectrum, (Mandel, 1974; Papoulis, 1984, p. 110). The quantity can be called root-mean-square frequency. It is a kind of average measure of the zero-crossing frequency, and is introduced as a useful measure of the dominant spectral frequency (Papoulis, 1984, p. 348; see Appendix A). Inspection of equations (l), (2), and (3) leads to a result familiar from probability theory (Papoulis, 1984, p. 108; Cohen, 1989). Figure 1 illustrates these spectral measures with regard to the spectrum of a 30 Hz Ricker wavelet. Other definitions have been offered for center frequency, bandwidth, and dominant frequency. Anstey (1977, p. 3-4), Robertson and Nogami (1984), and Barnes (1991) take the average frequency weighted by the amplitude spectrum as a measure of center frequency, which has some usefulness in the analysis of constant-phase wavelets. Ha et al. (1991) present the definition of instantaneous frequency in an integral form to obtain the Fourier spectral centroid. Widess (1982) and de Voogd and den Rooijen (1983) introduce alternate definitions of spectral bandwidth, and Claerbout (1976, p. 68) discusses a quantity similar to spectral bandwidth. Dominant frequency also has several different definitions, as discussed in Appendix A. Nevertheless, the definitions given above are preferable because they are simple, consistent, usefully related, and independent of phase. Further, they lead directly to convenient instantaneous forms. (3) (1) (Berkhout, 1984, p. 29; Cohen, 1989; Cohen and Lee, 1990; Jones and Boashash, 1990). The variance of the frequency about this mean is given by (Berkhout, 1984, p. 28; Cohen, 1989; Cohen and Lee, 1990; Jones and Boashash, 1990). The quantity is the standard deviation about the center frequency and is taken as a measure of the bandwidth [or, as Gram-Hansen (1991) suggested, a measure of half bandwidth]. These two quantities find application in studies of seismic reflection resolution (Berkhout, 1984). FIG. 1. Power spectrum of a 30 Hz Ricker wavelet with a comparison of the center frequency, the bandwidth, and the root-mean-square frequency defined in the text. The freexpected value of the frequency of relative quency is the maxima, as discussed in Appendix A. The two dashed lines are equidistant from the center frequency, and the distance between them is twice the spectral bandwidth.

3 Instantaneous Bandwidth 421 Instantaneous spectral measures Instantaneous measures of center frequency, spectral bandwidth, and dominant frequency are defined by replacing the Fourier power spectrum,, in equations (l), (2), and (3) with the instantaneous power spectrum, where is instantaneous center frequency, is instantaneous spectral bandwidth, and I call instantaneous dominant frequency (Cohen and Lee, 1988, 1990; Cohen, 1989; Jones and Boashash, 1990). These measures are related by (8) which is the instantaneous equivalent to equation (4) above (Cohen, 1989). Reinforcing intuition, instantaneous center frequency, and dominant frequency are equal, in a suitably time-averaged way, to their Fourier spectral counterparts. The corresponding relationship for bandwidth is, unfortunately, more involved and less intuitive. The relationships are developed in Appendix B. These measures depend on the particular instantaneous power spectrum employed. However, the exact form of the spectrum need not be specified if two conditions are met, which I require to accord with intuition and for ease of computation. The first is that the instantaneous center frequency be instantaneous frequency itself (Tanner et al., 1979); ignoring several objections to this equivalence that are unimportant in the present context, this is true of many useful instantaneous power spectra (Cohen and Lee, 1988; Cohen, 1989). The second condition is that the instantaneous spectral bandwidth be always real and positive. This does not contradict the first and restricts to be of a class of instantaneous power spectra for which the instantaneous bandwidth can be expressed as where is the instantaneous amplitude (trace envelope) and is its derivative (Cohen and Lee, 1988, 1990; (5) (6) (7) Cohen, 1989). These two conditions are approximately satisfied by the spectogram, or running short time-window Fourier transform (Cohen and Lee, 1988, 1990), which at present is the most widely applied form of instantaneous power spectrum in reflection seismology (e.g., Martin and White, 1989). As defined in equation (9), instantaneous bandwidth is a measure of the rate of relative amplitude change (White, 1991), and consequently can be expressed meaningfully in units either of Hertz or of decibels per second, with the conversion being 1 Hz = 10 db/s = db/s. This is consistent with the concepts of rise-time and the uncertainty principle (e.g., Carlson, 1968, p. 103; Claerbout, 1976, p. 68; Poularikas and Seely, 1985, p. 246; Cohen, 1989). For example, large rates of change in relative amplitude, associated with narrow signals, result in large bandwidths, while small rates of change in relative amplitude, associated with broad signals, result in small bandwidths. Other formulas for instantaneous bandwidth exist, but they are generally more involved and less intuitive than that given as equation (9). For example, inserting the Wigner instantaneous power spectrum into equation (6) yields (10) (Claasen and Mecklenbraucker, 1980; Cohen, 1989; Jones and Boashash, 1990). This measure is not as simple as equation (9) and is difficult to interpret since it can be negative. Cohen (1989) gives a general formula for the instantaneous bandwidth for an important set of instantaneous power spectra, from which formula both equations (9) and (10) can be derived. Rihaczek (1968) proposed a definition for instantaneous bandwidth that is essentially the square root of the absolute value of the time derivative of the instantaneous frequency. This is not consistent with the definition given in equation (6) and it is more difficult to interpret. In all cases I prefer equation (9) because it is simpler and more in accord with intuition. The relationship between the instantaneous center frequency, spectral bandwidth, and dominant frequency can be visualized as a frequency triangle, shown in Figure 2. After a fashion, this triangle can also illustrate the relationship of these measures to the instantaneous quality factor, which is defined as (Barnes, 1990); a similar definition is given by Tonn (1989, 1991). The term is the instantaneous decay rate, defined as the derivative of the instantaneous amplitude divided by the instantaneous amplitude (Barnes, 1990, 1991). Except for a factor of and the fact that it can be negative, instantaneous decay rate is equal to instantaneous bandwidth given by equation (9). Hence, referring to Figure 2, instantaneous quality factor is the ratio of instantaneous frequency to 2 times instantaneous bandwidth, or to the tangent of the angle divided by 2. It can be thought of as the ratio of a frequency and an amplitude decay or as the ratio of a frequency and a bandwidth, which is consistent (11)

4 422 Barnes with standard definitions of the quality factor (e.g., Close, 1966, p. 296; Johnston and Toksöz, 1981, p. 2). APPLICATIONS Instantaneous bandwidth and dominant frequency can be used as complex seismic trace attributes complementing instantaneous frequency in the general analysis of seismic data (e.g., Taner et al., 1979). The comparison of these measures is illustrated by their application to a 30 Hz zero-phase Ricker wavelet (Figure 3). The peak of the instantaneous dominant frequency is much broader than that of the instantaneous FIG. 2. The relationship between instantaneous frequency. instantaneous bandwidth and instantaneous dominant frequency Instantaneous quality factor is related to the angle by frequency itself. The instantaneous bandwidth is zero at the time corresponding with the peak of the wavelet, where the rate of amplitude change is zero. At times beyond 20 ms from the wavelet peak, the Ricker wavelet resembles an overdamped oscillation, and the instantaneous bandwidth exceeds the instantaneous frequency. Here it may be more appropriate to consider the instantaneous bandwidth as a measure of relative amplitude decay rate. It reaches a maximum of 26 Hz, which is equivalent to about 1400 db/s. The two maxima roughly correspond to the effective beginning and ending of the wavelet. Figure 4 shows the application of these measures to a seismic trace. Instantaneous bandwidth and the instantaneous frequency are distinctly independent, though they tend to peak together. Instantaneous bandwidth is smaller than instantaneous frequency almost everywhere. Where it is larger, the instantaneous power spectrum is biased towards low frequencies, and the instantaneous frequency is usually small. Perhaps, as in the example of the Ricker wavelet given above, these places correspond to onsets of distinct wavelets. Instantaneous frequency has been employed to study wavelet interference patterns (Robertson and Nogami, 1984; Robertson and Fisher, 1988). Instantaneous bandwidth and instantaneous dominant frequency can be similarly employed, as shown in Figure 5. A synthetic seismic trace is constructed by convolving a zero-phase 30 Hz Ricker wavelet with three pairs of equal and opposite reflection coefficients with spacing of 40, 30, and 20 ms. Thus there are three pairs of progressively interfering wavelets, labeled a, b, and c; these are well resolvable, marginally resolvable, and poorly resolvable, respectively. The responses of the derived attributes differ markedly for the three wavelet pairs. In wavelet pair b, instantaneous bandwidth peaks while instantaneous frequency plunges sharply negative at the point where the second wavelet begins to dominate the first. This supports the idea that points where instantaneous bandwidth exceeds instantaneous frequency correspond to onsets of new arrivals. However, as the example FIG. 3. Zero-phase 30 Hz Ricker wavelet and envelope (dashed line) with derived (a) instantaneous frequency, (b) instantaneous bandwidth, and (c) instantaneous dominant frequency.

5 Instantaneous Bandwidth 423 of wavelet pair a shows, instantaneous bandwidth does not generally exceed instantaneous frequency when the second wavelet begins to dominate the first. Instantaneous bandwidth and instantaneous dominant frequency complement instantaneous frequency in the analysis of low-frequency shadows (e.g., Taner et al., 1979). Consider a gas sand with a quality factor of 25 buried in a homogeneous earth with a quality factor of 250. The gas sand has a thickness corresponding to a reflection record time of 25 ms, and it is at a depth corresponding to a time of 1.5 s. The expected drop in instantaneous frequency from reflections beneath this gas sand is the low-frequency shadow, and is matched by corresponding drops in instantaneous bandwidth and instantaneous dominant frequency (Figure 6; see Appendix C for details of the calculation). Using both instantaneous frequency and instantaneous bandwidth to recognize a shadow lends more confidence to an interpretation. Instantaneous frequency is an excellent tool for the analysis of the effects of normal moveout stretch on seismic data (Vermeer, 1990, p. 91; Barnes, 1992). Such analysis is aided with instantaneous bandwidth and instantaneous dominant frequency, as illustrated in Figure 7. Similar study of other data-processing algorithms can lead to a better understanding of their effects on the data. DISCUSSION Fourier power spectra can be usefully characterized by average measures. Of particular importance in reflection seismology are center frequency, spectral bandwidth, and FIG. 4. Seismic trace (a) with derived instantaneous frequency (b), instantaneous bandwidth (c), and instantaneous dominant frequency (d).

6 424 Barnes 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. Dominant frequency serves as an estimate of the zero-crossing frequency. Numerous competing definitions for these quantities can be found, but these definitions are preferred because they are simple, internally consistent, independent of phase considerations, and have insightful physical analogs. Further, they suggest convenient and closely related instantaneous counterparts defined with instantaneous power spectra in place of Fourier power spectra. These instantaneous measures represent averages in frequency that are instantaneous in time. Their exact form depends on the particular instantaneous power spectrum; intuitively reasonable requirements yield specific forms that can be computed with familiar complex seismic trace attributes. Instantaneous center frequency is simply instantaneous frequency. Instantaneous bandwidth is the absolute value of the rate of change of the instantaneous amplitude divided by the instantaneous amplitude. Instantaneous dominant frequency equals the square root of the sum of the squares of the instantaneous frequency and instantaneous bandwidth, which is always positive. Instantaneous bandwidth and instantaneous frequency are completely independent of one another, while instantaneous dominant frequency is a function of both and can substitute for either. Together these attributes find application in the FIG. 5. A synthetic seismic trace (a) with derived instantaneous frequency (b), instantaneous bandwidth (c), and instantaneous dominant frequency (d). The seismic trace is constructed by convolving a zero-phase 30 Hz Ricker wavelet with three pairs of equal and opposite reflection coefficients labeled a, b, and c, with spacing 40, 30, and 20 ms, respectively. The dashed line is instantaneous amplitude.

7 Instantaneous Bandwidth 425 detailed analysis of seismic reflection data. Instantaneous bandwidth tends to be smaller than instantaneous frequency, and the places where it is larger may indicate arrivals of new wavelets. These instantaneous spectral measures are perhaps better suited for tracking the time-varying spectral content of the data. For example, both instantaneous frequency and instantaneous bandwidth should record a low frequency shadow beneath a highly attenuating zone, such as a gas sand, and so the use of both attributes together can lend more confidence to an interpretation. Similarly, both can be used in the analysis of time-varying frequency change due to effects of data processing, such as that caused by normal moveout stretch. Instantaneous bandwidth and instantaneous dominant frequency complement instantaneous frequency and should find routine application in the analysis of seismic reflection data. ACKNOWLEDGMENTS Erick Adam assisted with computer applications. Partial support was provided by a Natural Sciences and Engineering FIG. 6. Theoretical decay curves for instantaneous frequency and instantaneous spectral bandwidth for seismic energy that has propagated through a highly attenuating zone. The top of the zone is at 1.5 s and the zone has a thickness of 25 ms. At zero time, the seismic energy has a boxcar spectrum from 10 to 90 Hz. FIG. 7. The effect of normal moveout correction stretch on the average spectral properties of data that initially had a constant instantaneous center frequency of 50.0 Hz and a constant instantaneous bandwidth of 23.1 Hz. The offset is 1 km and the correction velocity is a constant 2.5 km/s. Research Council of Canada (NSERC) strategic grant awarded to A. Brown, M. Chouteau, C. Hubert, J. Ludden, and M. Mareschal. Comments made in review by L. Cohen and A. J. Berni were very helpful; to the extent that I was able to respond to their suggestions, the paper was improved. REFERENCES Anstey, N. A., 1977, Seismic Interpretation: The physical aspects. Internat. Human Res. Develop. Corp. Barnes, A. E., 1990, Analysis of temporal variations in average frequency and amplitude of COCORP deep seismic reflection data: 60th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, , Instantaneous frequency and amplitude at the envelope peak of a constant-phase wavelet: Geophysics, 56, , Another look at NMO stretch: Geophysics, 57, Berkhout, A. J., 1984, Seismic resolution: Resolving power of acoustical echo techniques: Geophysical Press. Carlson, A. B., 1968, Communication systems: An introduction to signal and noise in electrical communication: McGraw-Hill Book co. Claasen, T. A. C. M., and Mecklenbrtäucker, W. F. G., 1980, The Wigner distribution- Atool for time-frequency signal analysis: Part I: Continuous-time signals: Philips J. Res., 35, Claerbout, J. F., 1976, Fundamentals of geophysical data processing: With applications to petroleum prospecting: McGraw-Hill Book Co. Close, C. M., 1966, The analysis of linear circuits: Harcourt, Brace & World, Inc. Cohen, L., 1989, Time-frequency distributions-a review: Proc. IEEE, 77, Cohen, L., and Lee, C., 1988, Instantaneous frequency, its standard deviation and multicomponent signals: Proc. SPIE, 975, , Instantaneous bandwidth for signals and spectogram: Proc. IEEE ICASSP-90, de Voogd, N., and den Rooijen, H., 1983, Thin-layer response and spectral bandwidth, Geophysics, 48, Gram-Hansen, K., 1991, A bandwidth concept for CPB timefrequency analysis: Proc. IEEE ICASSP-91, Gupta, M. S., 1975, Definition of instantaneous frequency and frequency measurability: Am. J. Phys., 43, Ha, S. T. T., Sheriff, R. E., and Gardner, G. H. F., 1991, Instantaneous frequency, spectral centroid, and even wavelets: Geophys. Res. Lett., 18, Johnston, D. H., and Toksöz, M. N., 1981, Definitions and terminology, in Toksöz, M. N., and Johnston, D. H., Eds., Seismic wave attenuation: Soc. Expl. Geophys., Geophysics Reprint Series no. 2, l-5. Jones, G., and Boashash, B., 1990, Instantaneous frequency, instantaneous bandwidth, and the analysis of multicomponent signals: Proc. IEEE ICASSP-90, Kallweit, R. S., and Wood, L. C., 1982, The limits of resolution of zero-phase wavelets: Geophysics, 47, Kjartansson, E., 1979, Constant Q-wave propagation and attenuation: J. Geophys. Res., 84, Kreyszig, E., 1972, Advanced engineering mathematics: John Wiley & Sons, Inc. Lendzionowski, V., Walden, A. T., and White, R. E., 1990, Seismic character mapping over reservoir intervals: Geophys. Prosp., 38, 95 l-969. Mandel, L., 1974, Interpretation of instantaneous frequencies: Am. J. of Physics, 42, Martin, J. E., and White, R. E., 1989, Two methods for continuous monitoring of harmonic distortion in Vibroseis signals: Geophys. Prosp., 37, McCarley, L. A., 1985, An autoregressive filter model for constant Q attenuation: Geophysics, 50, Papoulis, A., 1984, Probability, random variables, and stochastic processes: McGraw-Hill Book Co. Poularikas, A. D., and Seely, S., 1985, Signals and systems: PWS Publishers. Rihaczek, A. W., 1968, Signal energy distribution in time and frequency, IEEE Trans. Information Theory, IT-14, Robertson, J. D., and Nogami, H. H., 1984, Complex seismic trace analysis of thin beds: Geophysics, 49, Robertson, J. D., and Fisher, D. A., 1988, Complex seismic trace attributes: The Leading Edge, 7, no. 6, Saha, J. G., 1987, Relationship between Fourier and instantaneous

8 426 Barnes Toksöz, M. N., and Johnston, D. H., 1981, Seismic wave attenua- tion: Soc. Expl. Geophys. Tonn, R., 1989, Comparison of seven methods for the computation of Q: Phys. Earth Plan. Int., 55, , The determination of the seismic quality factor Q from VSP data: A comparison of different computational methods: Geophys. Prosp., 39, l-27. Vermeer, G., 1990, Seismic wavefield sampling: Soc. Expl. Geophys. White, R. E., 1991, Properties of instantaneous attributes: The Leading Edge, 10, no. 7, Widess, M. B., 1982, Quantifying resolving power of seismic systems: Geophysics, 47, 1160-l 173. Yilmaz, O., 1987, Seismic data processing: Soc. Expl. Geophys. frequency: 57th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, Sheriff, R. E., and Geldart, L. P., 1989, Exploration seismology, volume 1: History, theory, and data acquisition: Cambridge Univ. Press. Sheriff, R. E., 1989, Geophysical methods: Prentice-Hall, Inc. 1991, Encyclopedic dictionary of exploration geophysics, third edition: Soc. Expl. Geophys. Spencer, T. W., Sonnad, J. R., and Butler, T. M., 1982, Seismic Q-Stratigraphy or dissipation: Geophysics, 47, Taner, M. T., Koehler, F., and Sheriff, R. E., 1979, Complex seismic trace analysis: Geophysics, 44, 104 l Tatham, R. H., and McCormack, M. D., 1991, Multicomponent seismology in petroleum exploration: Soc. Expl. Geophys. APPENDIX A ZERO-CROSSING FREQUENCY AND DOMINANT FREQUENCY Zero-crossing frequency is the number of zero crossings that occur on a seismic trace divided by twice the length of the trace (Gupta, 1975); it is a function of the amplitude spectrum and the phase spectrum of the data. If the data were a random normal process with zero mean and with power spectrum then its expected zero-crossing frequency is the root-mean-square frequency of equation (3), which is independent of the phase (Papoulis, 1984, p. 348). Stated differently, for a trace length At there will be on the average points whre equals zero, with the factor 2 accounting for the two zero-crossings per period. Root-mean-square frequency can thus be interpreted as an expected zero-crossing frequency; this lends it an intuitive and appealing meaning. It is also simple, independent of phase, and closely related to the center frequency and spectral bandwidth. This close relationship facilitates computation and affords insightful comparisons with analogous equations in physics. For example, in dynamics the center of mass corresponds to center frequency, the radius of gyration with respect to the center of mass corresponds to the bandwidth, and the radius of gyration with respect to the origin corresponds to the rootmean-square frequency (Papoulis, 1984, p. 110). A more relevant though imperfect example is in damped harmonic oscillation, where the frequency of oscillation is similar to the center frequency, the natural (undamped) frequency is similar to the root-mean-square frequency, and the decay coefficient is similar to the spectral bandwidth (Close, 1966, p. 201; Kreyszig, 1972, p. 65). These reasons justify root-mean-square frequency as a measure of dominant frequency. Dominant frequency, however, is usually estimated by counting the number of relative maxima within some interval (Kallweit and Wood, 1982; Widess, 1982; Yilmaz, 1987, p. 40; Sheriff, 1989, p. 212; Sheriff, 1991, p. 88); these estimates depend on the phase of the data and tend to be higher than the zero-crossing frequency. The formula for the expected value of the frequency of relative maxima for the spectrum of a random normal process with zero mean and power spectrum is 2 (A-1) (Papoulis, 1984, p. 350); this is to be compared with equation (3). For the spectrum of a 30 Hz Ricker wavelet, the center frequency is 31.9 Hz, the spectral bandwidth is 10.2 Hz, the root-mean-square frequency is 33.5 Hz, and the average frequency of relative maxima is 39.7 Hz (Figure 1). The dominant frequency as defined by Kallweit and Wood (1982) is 39.0 Hz, which is nearly the same as the average frequency of relative maxima. The measures of dominant frequency cited above do not lend themselves readily to instantaneous equivalents. While the measure defined by equation (A-l) does suggest an instantaneous equivalent, it is more complex than rootmean-square frequency defined by equation (3), and it lacks the convenient relationship with the center frequency and the spectral bandwidth defined by equations (1) and (2). I conclude that root-mean-square frequency is a more useful measure of dominant frequency. APPENDIX B RELATIONSHIPS BETWEEN THE AVERAGE FOURIER SPECTRAL MEASURES AND THEIR INSTANTANEOUS COUNTERPARTS Useful relationships exist between the average Fourier spectral properties introduced in the text and weighted time-average values of their instantaneous counterparts. Those for center frequency and dominant frequency are particularly simple and intuitively appealing; that for bandwidth is unfortunately more involved and less intuitive. These are developed below. Let be an instantaneous power spectrum of a seismic trace with the following properties (Cohen, 1989): (B-1)

9 Instantaneous Bandwidth 427 and (B-2) (B-3) where is the magnitude of the trace and is its power spectrum. If the trace is analytic, corresponds to instantaneous amplitude. Equation (B-3) is Parseval s theorem. The relationship between instantaneous center frequency,, averaged over time, and the average Fourier spectral frequency follows readily from their definitions given in the text and the above properties. From equation (5) and equation (B-l) is derived Hence (B-4) (B-5) Performing the outside integration first and dividing each side of equation (B-5) by one side of equation (B-3), and recalling equation (1) gives (B-6) Thus, the weighted average over time of the instantaneous center frequency is equal to the average Fourier spectral frequency (Cohen, 1989). When is equated with instantaneous frequency and with trace envelope, this is equal to a result obtained many times before (e.g., Mandel, 1974; Saha, 1987). Denoting the weighted average over time by brackets and recalling equation (1) permits equation (B-6) to be expressed in a succinct and useful form: (B-7) In a like manner, the relationship between the instantaneous dominant frequency and root-mean-square frequency is found to be I dt 0 0 I (B-8) (B-9) Interestingly, and perhaps unfortunately for intuition, the corresponding relationship between instantaneous bandwidth and Fourier spectral bandwidth is more involved. It is readily derived by inserting equation (B-9) and equation (8) into equation (4) to get = + (B-10) + (B-l 1) (Cohen and Lee, 1988; Cohen, 1989). Thus, the weighted average value over time of the instantaneous bandwidth squared is not equal to the Fourier spectral bandwidth squared, and in fact is always less. This is true whether the instantaneous bandwidth is defined by equation (9) in the text, or by equation (10), or by some other equation consistent with the given definitions and the properties of With as instantaneous frequency and defined by equation (9), equation (B-11) is identical to a result derived by Mandel(1974) and Berkhout (1984, p. 29), among others. APPENDIX C A SIMPLE MODEL OF INSTANTANEOUS FREQUENCY AND BANDWIDTH CHANGE DUE TO Q LOSSES A simple model is developed to compute the changes in the instantaneous center frequency and spectral bandwidth of seismic energy propagating through an attenuating earth. I assume that in some average way the resultant equations also apply to reflected energy. Such an assumption is implicit in the use of instantaneous frequency to find low-frequency shadows beneath bright spots (e.g., Taner et al., 1979). The limitations of the assumption are beyond this discussion, but they do not detract from the general argument that spectral changes due to attenuation losses can be observed with instantaneous bandwidth as well as with instantaneous frequency. The quality factor Q is commonly defined by (C-1) where is the power spectrum at time t of the propagating seismic energy (Johnston and Toksöz, 1981, p. 2; McCarley, 1985). This can be expressed as (C-2) with defined as the power spectrum of the wavelet at time 0 (McCarley, 1985). Let be a factor representing the frequency-independent energy losses suffered by the wavelet as a result of wavefront spreading (e.g., Toksoz and Johnston, 1981, p. 9). Then the spectral power at time t, accounting for Q losses and wavefront spreading, is (Toksozand Johnston, 1981, p. 9). By defining a quantity as

10 428 Barnes (C-3) (C-6) (e.g., Kjartansson, 1979; Toksöz and Johnston, 1981, p. 9; Spencer et al., 1982), where Q(t) is the quality factor as a function of propagation time, this can be written as At t = 0 this equation is undefined, but application of L Hôpital s rule gives The corresponding equations for instantaneous dominant frequency are (C-7) (C-4) This is an expression for the power spectrum of seismic energy propagating through the earth as a function of time, in other words, an instantaneous power spectrum. Given this expression, the initial power spectrum, and the quality factor as a function of traveltime, the instantaneous center frequency and bandwidth of the propagating energy can be calculated from the definitions given in the text. The resulting quantities prove to be independent of b(t). Let the initial power spectrum E a (0, f) be an ideal band-pass spectrum from f e to f h defined by (C-5) The instantaneous power spectrum is now defined; performing the integrations required for instantaneous center frequency yields and 1 (C-8) The instantaneous bandwidth can be found from equations (C-6) and (C-7) and equation (8) in the text. Equations (C-6) and (C-7) govern the change in instantaneous center frequency and instantaneous dominant frequency for seismic energy propagating through the earth as a function of total traveltime t. If, after accounting for the doubling of traveltimes, and if in an average sense the power spectrum of the reflected energy equals that of the transmitted energy, then these equations can also be employed to estimate the corresponding instantaneous frequency and instantaneous bandwidth of the reflected energy.