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GEOPHY SICS, VOL. 57, NO. 11 (NOVEM BER 1992); P. 152&1524, 2 FIGS.

Short NoteThe calculation of instantaneous frequency andinstantaneous bandwidth

Arthur E. Barnes*INTRODUCTION

A number of ways have been offered to calculate instan-taneous frequency, an important complex seismic traceattribute. The standard calculation follows directly from itsdefinition and requires two differentiations (Taner et al.,1979). By avoiding these differentiations, three formulas tha tapproximate instantaneous frequency are faster to compute.The first employs a two-point finite-impulse res ponse (FIR)differentiator in place of the derivative filter (Scheuer andOldenburg, 1988). The second s nearly the sam e as the first,excep t that it employs a three-point FIR differentiator(Boashashet al., 1991). The third takes a different approachand involves tw o approximations (Claerbout, 1976, p. 20;Yilmaz, 1987, p. 521). Ho w do these formulas compare, andwhich is best?

The results of the approximate formulas for instantaneousfrequency are comparable to that produced by the standardformula, and they are sufficient for most applications inreflection seismology. Of the three approximations, employ-ing the two-point differentiator is arguably the best. As anaverage of instantaneous frequency over a sample period, itis intuitively appealing and more closely related to thedefinition than C laerbouts mea sure. It p rodu ces a resultmore directly comparable to the standard than that of thethree-point filter. How ever, it causes a half-sample timeshift; the three-point filter d oes not cause a sh ift. Similarapproximations also apply to the calculation of instanta-neous bandwidth.

I review the definition o f instantaneous frequency and offour formulas for its computation and efficient approxima-tion. I apply these formulas to a seismic trace, com pare theresults, and draw conclusion s. I sho w that similar conclu-sions apply to the calculation of instantaneous bandwidth.

FORMULASInstantaneous frequency

Let x(t) be a seismic trace and let y(t) be its Hilberttransform, or the imaginary trace. An analytic trace z(t) isdefined as

z(t) = x(t) + iy(t) = R(t) exp [ie (t)], (1)whe re R(t) is the instantaneous amplitude (trace envelope)and O(t) is the instantaneous phase. Instantaneous fre-quency f(t) is defined as

f(t)=;;e(t), (2)(Taner et al., 1979; Cohen, 1989).An ambiguity inherent in the instantaneous pha se rendersequation (2) impractical for calculating instantaneous fre-quency: only the principal values of the phase are computed,which causes 27~phase discontinuities. Instantaneous fre-quency is instead calculated by another equation, directlyderived from equation (2):

1 x(t )y(t) - x(t)y (t)f(t) = 2, x(t)2 + y(t)2 (3)where the primes denote differentiation w ith respect to time(Taner et al., 1979). The instantaneous frequency of sam pleddata can be far in e xcess of Nyquist frequency.

INSTANTANEOUS FR EQUEN CY APPROXIMATIONSEquation (3) requires two differentiations to calculateinstantaneous frequency. By avoiding thes e differentiations,three formulas that approximate instantaneous frequencyare faster to compute.The first of the three approximations is developed asfollows. D efine an average instantaneous frequency f, (t) as

Manuscript eceivedby the Editor April 22, 1992; evisedmanuscripteceivedApril 22, 1992.*Genie Mineral, Ecole Polytechnique, .P. 6079,Succursale A, Montreal,quebec H3C 3A7, Canada.0 1992Society of Exploration Geophys icists.All rights reserved.1520

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Calculation of Instantaneous Frequencythe tempo ral averag e of instantaneous frequency in a tim einterval from t to t + T:

1521(12)

(e.g., Jones and Boasha sh, 1990). Substitution of the defini-tion given abov e for instantaneous frequency into equation(4) yields(5)

Equation (5) is applicable to sampled data with sampleperiod T. It is a type 4 linear-phase FIR filter (Parks andBurrus, 1987, p. 24), and it introduces a half-sample timeshift. Unlike the stan dard calculation of instantaneous fre-quency, this measu re is constrained to yield a m agnitude nogreater than N yquist frequency because the maximum dif-ference in the unwrapped phase hat can occur over a sampleperiod is 7~.Like equation (2), equation (5) is impractical for compu-tation, but it can be put into a practical form. Note fromequation (1) that

8 (t) = Zm [ln z(t)].Substituting this equality into equation (5) yields

(6)

fa(f) = & [Zm{ln z(t + T)} - Zm{ln z(t)>l, or (7)

f,(t)=&Zm{ln [F]}=&arg r%]. (8)

This is essentially the formula g iven by S cheu er and Olden-burg (1988). It c an be rewritten in a form m ore suitable forcomputer implementation:1f, (t) = - arctan x(t)y(t + T) - n(t + T)Y(l)2aT 1(t)x(t + T) + y(t)y(t + T) (9)

Boashashet al. (1991) offer a similar equation employing athree-point FIR differentiator. It is a type three linear-phaseFIR filter (Parks and Burrus, 1987, p. 24), and it does notintroduce a time shift. It is given byfb(t) = & e(t + T) - O(t - T)4nT (10)

This is an average of instantaneous frequency over twosamp le period s. It is also constrained to yield a ma gnitude nogreate r than Nyquist frequ ency. Following a similar devel-opm ent as given above results in

(11)and

1 rx(t - T)y(t + T) - x(t + T)y(t - T)jfb (t) = - arctan4,1rT x(t - T)x(t + T) + y(t - T)y(t + T) I

A third approxim ate formula for instantaneous frequencyis

fc(t) =$Z ;;;;;:;] (13)(Claerbou t, 19 76, p. 20; Yilmaz, 1987, p. 521); it involvestwo ap proxim ations in its derivation. Put into a practicalform, it is

fc(t) = L ]x(t)y(t + T) - x (t + T)y(t)

,ITT 1x(t) +x (t + T))2 + (y(t) +y(t + T))2 (14)This equation introduces a half-sample period time shift, andits results can be greater than Nyquist frequency.Comparison

To judge the three approximate formulas for instantaneousfrequency, equ ations (9), (12), and (14), I applied them to aseismic race and com pared the results with that obtained bythe standard ormula, equation (3); this is shown in Figure 1.All three approximate formulas produce results that arecomparable to that p roduced by the standard formula, andfor most applications any of the three would suffice. Sincethey are more efficient, they are p referable to the standardformula. With regard to frequency spikes, instantaneousfrequency computed with the two-point FIR differentiatormore closely resembles the standard instantaneous fre-quency than that computed by the three-point operator,whose spikes have a tendency to be negative. In mostseismic applications, this is unimportant since spikes aretypically ign ored. Instantaneous frequency pro duce d by thethree-point op erator is a little smo other becau se it involvestwice as much averaging. Both these measures are simplerand more intuitive to understand than Claerb outs approxi-mation, equation (14). Further, they produce results nogreater than Nyquist freque ncy, unlike bo th the standardformula and Claerbou ts approxim ation.

INSTANTANEOUS BANDWIDTHInstantaneous bandwidth, of(t), has been defined as

where R(t) is instantaneous amplitude and Rt) is its timederivative (Cohen and Lee, 1990). Given the instantaneousamplitude, instantaneous bandwidth can be calculated di-rectly from equation (15) with one differentiation. Thisdifferentiation can be avoided by employing efficient equa-tions similar to tho se developed for instantaneous frequencygiven above. Emp loying a two-point FIR differentiator in-stead of a sta ndard derivative filter yields

1 In R(t + T) - In R(t)of(t) = 2, Tx2(t + T) + y2(t + T)

x2(t) + y2(t) II 1

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1 ar

O_ ,,-.@&\/ r,J\q $pqp-50 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

z 100zh2 503kr& 00 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5time (s)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

R 1008GQ 50s

& 00 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5time (3)

time (s)FIG. 1. (a) Seismic trace, and instantaneous frequency calculated by (b) the stand ard ormula, (c) two-point filter app roximation(recom men ded), (d) three-point filter approximation , and (e) Claerbouts approxim ation.

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Calculation of Instantaneous FrequencyThis introduces a half-sample time shift. For a three-pointFIR differentiator, the correspon ding equation is

1523particular m easure is equivalent to instantaneous frequencyaverag ed over a sa mple interval. This is equal to the differ-ence in instantaneous phas e over th at interval d ivided by thelength of the interval and is essentially a two-point FIRdifferentiator. It is a more intuitive mea sure than an approx-imation given by Claerb out (1976). It is arguably better thaninstantaneous frequency itself in tha t its values represe ntentire sample periods, whereas instantaneous frequencyrepresen ts only th e sam ple points. As a result, it has theappealing characteristic of constraining instantaneous fre-quency to be no greater than Nyquist frequency, unlike boththe standard formula and Claerbouts approxim ation. Ho w-ever, the two-point differentiator introduces a half-sampletime shift; this can be avoided by using a three-point FIRfilter, equivalent to an average over two sample intervals.The disadvan tage of using the three-point differentiator isthat its results look less like the standard because manyspikes become negative, though for most seismic applica-tions this is unimportant. Identical equations can be devel-oped for the efficient computation of instantaneous band-width. For most applications in reflection seismology, Irecommend the calculation of instantaneous frequency andbandwidth by the formulas employing a two- or three-pointdifferentiator.

ar(t) = In fW + T)2-Tln t - T)

=& InI [2(t + T) + yyt + ZJX2(t - T) + y2(t - T)II (17)This formula does not introduce a time shift.A comparison of instantaneous bandwidth calculated withequations (15), (16 ), and (17) is shown in Figure 2 . The threeresults are comparable. For most applications in reflectionseismology, instantaneous bandwidth can be satisfactorilyand efficiently computed by approximate formulas (16) or(17).

DISCUSSIONInstantaneous frequency is defined as the time derivativeof the instantaneous phase. The formula for its practicalcalculation, derived directly from this definition, requirestwo differentiations. Alternative formulas provide a closeapproximation to instantaneous frequency and are faster tocompute because they avoid these differentiations. One

tim e (s)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45time (s)FIG. 2. Instantaneous bandw idth of the seismic trace shown in Figu re 1, calculated by (a) the stan dard formula, (b) two-pointfilter approximation, and (c) three-point filter approximation.

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1524 BarnesACKNOWLEDGMENT

Partial support was provided by a Natural Sciences andEngineering Research Council of Canada (NSER C) strategicgrant awarded to A. Brown, M. Chouteau, C. Hub ert, J.Ludden, and M. Mareschal.REFERENCES

Boasha sh, B., OShea, P., an d Ristic, B., 199 1, Statisticalicompu-tation c omparison of some estimators for instantaneou s fre-quency: Proc. IEEE ICASSP-91, 3193-319 6.Claerbout, J. F., 1976, Fundamentalsof geophysicaldata process-

ing: With applications to p etroleum prospecting: McG raw-HillBook Co.Cohen, L., 198 9, Time-frequency distributions-A review: Proc.IEEE, 77, 941-981.Cohen, L., and Lee, C., 1990, nstantaneousbandw idth for sign alsand spectogram:Proc. IEEE ICASSP-90, 2451-2454.Jones, G., and Boashash,B., 199 0, nstantaneous requency, instan-taneous bandw idth, and the analysis of multicompo nent signals:Proc. IEEE ICASSP-90, 2467-2470.Parks, T. W ., and Burrus, C. S., 198 7, Digital filter design: JohnWiley & Sons, Inc.Scheuer, T. E., and Oldenburg, D. W., 1988, Local phase velocityfrom com plex seismic data: Geophysics, 53, 1503-15 1.Tane r, M. T., Koehler, F., and Sheriff, R. E., 197 9, Comp lexseismic. race analysis: Geophysics, 44, 1041-1063 .Yilmaz, O., 1987, Seismic data processing:Sot. Expl. Geophys.