Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 94, NO. C8, PAGES 10,993-10,998, AUGUST 15, 1989 Statistics of Bicoherence and Biphase STEVE ELGAR AND GLORIA $EBERT Electrical and Computer Engineering Department, Washington State University, Pullman Statistics of estimates of bicoherence and biphase were obtained from numerical simulations of nonlinear random (harmonic) processeswith known true bicoherenceand biphase. Expressionsfor the bias, variance, and probability distributionsof estimatesof bicoherenceand biphaseas functions of the true bicoherenceand number of degreesof freedom (dof) used in the estimates are presented. The probability distributions are consistentwith theoretical distributions derived for the limit of infinite dof and are used to construct confidence limits on estimates of bicoherence and biphase. Maximum likelihood estimates of true values of bicoherence and biphase given observed values are also presented. 1. INTRODUCTION be used to statistically detect the presence or absence of Bispectral analysis has been used to study nonlinear interactions in a variety of ocean processes, including surface gravity waves in intermediate water depths [Hasselman et al., 1963], perturbations from the mean profiles of temperature, salinity and sound velocity [Roden and Bendiner, 1973], internal waves [Neshyba and Sobey, 1975; McComas and Briscoe, 1980], shoaling surface gravity waves [Elgar and Guza, 1985, 1986; Doering and Bowen, 1987], and temperature fluctuations [MMler, 1987]. A wide range of phenomenain other physical systemshas also been investigated with the bispectrum (see Nikias and Raghuveer [1987] for a recent review). In most of these studiesthe bispectrum was used to determine whether or not the process under investigationwas consistentwith linear dynamics. Specifically, nonlinear interactions are associated with nonzero values of the bicoherence, defined here as [Haubrich, 1965; Kim and Powers, 1979] nonlinear interactions, the statistics of estimates of bicoher- ence and biphase,/3(wj, w2), the phaseof B(w•, w2), for the case of nonzero true bicoherence have not previously been reported. The purpose of this study is to present such statistics.Brillinger [1965], Rosenblatt and Van Ness [1965], Brillinger and Rosenblatt [1967a, b], and others [see Nikias and Raghuveer, 1987] give some of the statistical properties of higher-orderspectra, including asymptotic distributions of the real and imaginary parts of the bispectrum. Haubrich [1965], Hinich and Clay [1968], Kim and Powers [1979], Hinich [1982], and Ashley et al. [1986] discussthe estimation of bicoherence. For the present study, harmonic random processes with true valuesof bicoherence (b2) between0.1 and 1.0 were numerically simulated (section 2), and the statistics of esti- matesof bicoherence (/•2)andbiphase (•) obtained fromthe simulated time series were calculated (section 3). The prob- abilitydistributions of/•2 and/3,including thebiasof/•2 and the variancesand confidence limits of /•2 and /3, were IB(wl, O92)12 asfunctionsof b2 anddof (sections3.1 and3.2). b2(wl,092)= (1) determined E[IA(rol)A(ro2)12]E[IA(rOl + ro2)l 2] These values are useful for experimental design. On the other hand, once an experiment has been conducted, maximum likelihood techniques may be used to estimate true where B is the bispectrum, B(wi, w2)= E[A(o•i)A(o•2)A*(wj + w2)] (2) A(w) is the complex Fourier coefficient of the time series at radian frequency w and E[ ] is the expected, or average, value. Alternately, the bispectrumcan be calculatedas the Fourier transform of the third-order correlation function of valuesof b2 and/3basedontheobserved values.Maximum likelihood estimates of bicoherence are presented in section 3.3. These results are then briefly applied to estimates of the bicoherence and biphase of narrow-band surface gravity waves observed in 9 m water depth (section 3.4). Conclusions follow the time series [Hasselman et al., 1963]. For a finite length time series, even a truly linear (e.g., Gaussian)processwill have nonzero bicoherence. Haubrich [1965] shows that for a Gaussianprocessthe bicoherence(whose true value equals in section 4. 2. SIMULATION PROCEDURE Time series consisting of triads of sinusoids(with frequen- zero)isapproximately chi-square (X2)distributed in thelimit cies Wi, tOj,and Wi+j)were numericallysimulatedon the of large degrees of freedom (dof), and thus significance CRAY XMP at the San Diego Supercomputer Center. By adjusting the amplitudes of the component sinsuoids, some levels for zero bicoherence as a function of dof can be calculated. Elgar and Guza [1988] demonstrated that esti- of whose phases were chosen from a uniform random mates of bicoherence from a Gaussian process are also distribution (0 - 2rr), the true bicoherence value of the triad approximately X2 distributed forlowvaluesof dof(dof= 32) could be varied. For example, consider the case of a single and are not sensitive to smoothing procedures used to increase dof. Although the significancelevels for zero bicoherence can triad: x(t) = COS(wit + qbl)+ cos (w2t + 4>2)+ a cos (w3t + qb3) + COS(ro3t+ qbI + qb2) Copyright 1989by the American GeophysicalUnion. (3) Paper number 89JC01034. where w3 = wl + w2, &•, &2, and &3 are randomphases,and 0148-0227/89/89JC-01034505.00 a is an adjustable parameter. As a increases from 0 to o•, 10,993 10,994 ELGAR AND SEBERT: STATISTICSOF BICOHERENCEAND BIPHASE b2(wl,w2)decreases from1.0to 0.0.Fortheexample given 0.05 in (3), if a = 0, then/3(w•, w2) = 0. Time seriesconsistingof the following triads were simulated: a singletriad, (Wl, w2, w3); two triads with the same sum frequency, (w•, ws, w6) and (w3, w3, w6); and two triads with a commonfrequency other than the sum frequency, (w•, w3, w4) and (w3, ws, ws). True values of/3 = 0 and/3 = rr/4 were used. For each case 0.04 0.03 the true bicoherence of each triad could be adjusted. The statistics of estimates of bicoherence and biphase reported 0.02 herewerefoundto depend onlyontruevaluesof b2anddof; they did not depend on the number of triads, nor on how the triads were interrelated, nor on /3. Additional time series were simulated with Gaussian-distributed Fourier 0.01 coeffi- cients rather than uniformly distributed random phases [Elgar et al., 1985, and references therein]. The resulting 0.00 0.00 statistics of/•2and/3wereessentially identical tothoseofthe random phase simulations. Each simulation consisted of generating many realizations of 65,536-point time series, each of which was subdivided into short sectionsof 256 points. The short sectionswere not tapered. By ensemble averaging the bispectrum over all 256 of the short records, estimates of bicoherence and biphase with 512 dof were produced from each 65,536-point time series. Each 65,536-point record was also subdivided into 16 groups of 16 short records, each group producing estimates with 32 dof. Estimates with dof between 0.01 0.02 0.05 THEORETICAL 0.04 0.05 BIAS Fig. 1. Bias of bicoherence observed in simulated data versus the theoretical bias (equation (4)). Squares, dof = 32' triangles, dof = 64; diamonds, dof = 128' circles, dof = 256; and asterisks, dof = 512.The bicoherence, b2 = 0.1, 0.2, ..., 0.9 for eachdof. (!•)(b2)(v/2)-le-b2/ (6) 32 and 512 were where F( ) is the gamma function. The mean and variance randomvariableare av and 2a2v, point time series were generated, and thus 1000 (dof = 512) of an aX2 distributed to 16,000 (dof = 32) bicoherenceand biphaseestimateswere respectively.Thus the parametersa and v can be determined produced for each true value of bicoherence. Results from from the bicoherence as simulations using 100 realizations were indistinguishable E[b2] 2E2[b2] from those where averages of 1000 realizations were used. . = v= (7) obtained in a similar manner. One thousand of the 65,536- v 3. 3.1. SIMULATION RESULTS Var [b2] Combining (5) and (7), and using the biased value of bicoherence, Bicoherence (dof)b2 As shownbelow,/•2 is approximately chi-square distributed and thus will have a positive bias, similar to the ordinary coherence between two time series [Jenkins and Watts, 1968; Beningus, 1969]. Following Beningus [1969], the bias was empirically found to be approximately given by Bias[/•2]= (2/dof)(1- b2)2 •'=2(1- b2)3 As b2 --• 1 and/ordofincreases, theparameter •,increases, andfx•(b2/a) approaches aGaussian distribution. Probability (4) Equation (4) is similar to the correspondingexpressionfor coherence [Beningus, 1969] and is comparedto the observed bias in Figure 1. Guided by expressionsbased on somewhat heuristic theoretical arguments [Hinich and Clay, 1968; Kim 10-2 andPowers,1979],thevariance of/•2 wasempirically found Z to be approximately Var [/32]= (4b2/dof)(1- b2)3 (8) • 10 -3 • 10-4 (5) Equation (5) is similar to the analogous approximate theoretical equation for coherence [Jenkins and Watts, 1968], and the agreement between (5) and the simulated data is excellent, as shown in Figure 2. Haubrich[1965]suggests thatfor a process withb2 = 0.0, /32will asymptotically (largedof) approach a chi-square distributionwith parameter v = 2. Hinich [1982] showsthat bicoherenceis approximately asymptoticallydistributedas a noncentral chi-square random variable. The noncentral chi- 10-5 I IIIII I 10 -5 I I I IIIII I 10 -•' THEORETICAL Fig. 2. I I I I IIII I 10 -3 I I I IIII 10 -2 VARIANCE Variance of bicoherence observed in simulated data Each symbol represents different dof, as described in the caption to Figure 1. square distribution canbeapproximated by anaX2 distribu- versus the theoretical variance (equation (5)). tion [Abramowitz and Stegun, 1972], given by I ELGAR AND SEBERT: STATISTICS OF BICOHERENCEAND BIPHASE 3.5 0.030 10,995 - ,• p,i 0.025 : .,, , 2.5 o.oo i ,. t-• 0.015 ß• 0.010 1.5 o z 1.0 0.005 0.5 0.000 I 0.0 0.2 0.4 0.6 I I I I i I I I I 0.8 b2 Fig. 3. Probability distribution of/;2 observed in thesimulated I I I I I lO 2 DEGREES Fig. 4. OF FREEDOM The 90% confidence limits for bicoherence. The values the correspondingtrue value of shown have been normalized by datafordof= 32(dashed curve•) andtheoretical aX2 distribution (solid curve). From left to right, b • = 0.1,0.2, and 0.4. The bin width is A(b2) = 0.0067. bicoherence. Thus90%of thetime,/;2willbein therangegivenby theappropriate ordinate valuestimesb2. Solidcurvesaretheoret- ical values, and symbols are values observed in the simulated data. Squares, b2 = 0.1; triangles, b2 = 0.2; diamonds, b2 = 0.4; and distributions of the simulateddata are comparedto aX2 circles,b2 = 0.8. distributionsin Figure 3, with a and v obtained from (7) and (8),respectively. According to a X2 goodness of fit test,the hand,it is oftenthe casethat b2 mustbe estimatedfrom a theoretical and observed distributionswith 32 dof and b 2 --< limited set of data. In this section maximum likelihood 0.4 do not differ significantly at the 95% confidence level. estimates [Jenkins and Watts, 1968] of the true value of givenan observed value,/;2,withfinitedofare For 32 dof the probability distributionsof the simulateddata bicoherence are slightlyoffsettowardlowervaluesof b2 relativeto the presented. Freilich and Pawka [1987] present a similar aX2 distributions, withtheamount of offsetdecreasing with analysis for estimates of the cross spectrum, with results increasing b2 (Figure3). For highervaluesof doftheoffset analogousto those reported here. Maximum likelihood estiis virtually zero, and, consequently,no attempt was made to mation consistsof determiningparameters of the underlying account for it in the probability distributions with 32 dof probability distribution function (pdf) from samplesof the shown in Figure 3. The scatter in the probability distribu- distribution. The pdf is recast as a function of the parameters tions of the simulateddata shown in Figure 3 is due to finite to be determined, and those values of the parameters that record lengths and finite bin width. There is even better maximize this likelihood function are considered the maxiagreement between simulated dataandaX2 distributions as mum likely estimates (mle) of the true parameters of the b2 --) 1 and/or dof--) underlyingdistributiongiven the particular samplesat hand. thelikelihood function,L(b2),is Assuming/;2 is aX2 distributed witha andv givenby (7) Forthecaseof bicoherence (equation (6))andis shownas and(8), respectively, confidence limitsfor/•2 canbe con- givenby theaX2 distribution structed as functions of b 2 and dof. The 90% confidence limits are comparedto the correspondingvalues observedin the simulationsin Figure 4. 3.2. Biphase 10-1 - 10-:• _ Similar to the phase of the cross spectrum [Jenkins and Watts, 1968], the biphase is approximately Gaussiandistributed, unbiased, and has variance (in units of radians) The theoretical variance and confidence limits (calculated from a Gaussiandistribution with variance given by (9)) for 10-3 - /3 are compared to simulated valuesin Figures5 and6, respectively. i i i i i i iii i i i I i i iii 10 -3 THEORETICAL 3.3. Maximum Likelihood i i i 10 -2 I i i iii I i i i 10 -I VARIANCE Estimation Fig. 5. Variance of biphase observed in simulated data versus The results of the previous sectionsdescribethe statistics the theoretical variance (equation (9)). The units are radians of bicoherence giventhe true value,b2, thusenablingthe designof experiments to measurebicoherence. On the other squared. Each symbol representsdifferent dof, as described in the caption to Figure 1. 10,996 ELGAR AND SEBERT: STATISTICS OF BICOHERENCE AND BIPHASE 6O TABLE 5O -r 20 •2 dof 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.3 0.3 0.4 32 64 128 256 32 64 128 32 64 32 1. Parameters for the Likelihood b2mle 0.129 0.116 0.108 0.104 0.217 0.209 0.204 0.309 0.304 0.403 2 bmele 0.146 0.125 0.113 0.107 0.223 0.212 0.206 0.310 0.305 0.404 Functions s.d. (7.5:1) 0.047 0.041 0.032 0.024 0.054 0.042 0.031 0.047 0.037 0.045 (0.018, 0.237) (0.026, 0.201) (0.039, 0.172) (0.054, 0.151) (0.078, 0.314) (0.109, 0.284) (0.135, 0.261) (0.191, 0.387) (0.224, 0.364) (0.323, 0.462) lO Theparameter/; 2istheunbiased observed bicoherence, dofisthe degrees of freedomassociated withthe measurement, b2mle is the 0 -I I I I I I I I I lo :• DEGREES OF FREEDOM Fig. 6. The 90% confidencelimits for biphase.Solid curves are theoreticalvalues, and symbolsare values observedin the simulated maximum of thelikelihood function(L), bmele 2 is the mean of L, s.d. is the standarddeviationof L, and (7.5:1)indicatesvaluesof b2 correspondingto likelihood values down by a factor of 7.5 from the maximumof L (if L is Gaussian,these values correspondto 95% confidence limits). data. Each symbolrepresentsdifferentbe as describedin the caption to Figure 4. that an observed value comes from the center of the distri- a functionof dofand/;2in Figure7. For largevaluesof/;2 butionwith high •, (i.e., high b2) than from the uppertail of and/ordof,L(b2) is symmetrical aboutits maximum. Howa distributionwith lower •,. Table 1 and Figure 8 both show ever,for small/;2and/ordof,L(b2) haslongtailsextendingthat exceptfor small/;2 and/ordof, bmele 2 does not differ towardhigh valuesof /;2 (Figure7). In order to better substantially from/;2. account for the asymmetrical shape of the likelihood funcL(b2)canalsobeusedto calculate confidence intervalsfor tion, Jenkinsand Watts [ 1968]suggestusingthe meanof the the estimated true values of bicoherence. The likelihood likelihoodfunction(mele)as an estimateof the true paramfunctionspresentedhere (Figure7) are approximatelyGauseter. Both mle and mele estimates of bicoherence are listed sian,andthusthe meleandthe varianceof L(b2) determine in Table 1. As L(b2) becomes moresymmetrical, 2 --> the entire likelihoodfunction [seeJenkinsand Watts, 1968]. bmele b2mle __>/;2 (Table1 andFigure8), although thelikelihoodThe standarddeviationsof L(b2) and the b2 valueswhose estimates of b2 arealwaysgreaterthanthecorresponding likelihoodsare down by a factor of 7.5 from the mle (95% measured values (Figure 8). This can be understoodby considering theX2 pdfsshownin Figure9. It is morelikely confidence limitsonb2if L(b2)isGaussian) arepresented in Table 1. From the table it can be seen that the standard deviation of L(b2) is nearlyindependent of/;2 anddepends primarily on dof. Since the distributionof biphaseis approximatelyGaus- 0.20 sian,•mele•-' •mle• •' 0.15 O.lO 1.4 0.05 0.00 .•ce 1.2 0.0 0.1 0.2 0.3 0.4 b2 1.1 Fig. 7. LikelihoodfunctionsL(b2) for variousmeasured bicoherenceanddof values.Solidcurves,unbiased/;2= 0.1; longest dashes, /;2 = 0.2; mediumlengthdashes, /;2 = 0.3; andshortest dashes,/;2 = 0.4.Foreach/;2theoutermost curvehas32dof,and the dof increaseby factorsof 2 for eachsucceeding innercurve.•The 1.0 I O. 10 I 0.20 I I 0.30 I I 0.40 I I 0.50 numberof suchcurvesdecreases forincreasing/; 2. Unbiased values are obtainedby subtractingthe bias(equation(4)) from the observed value, where the observed value of bicoherence is used on the Fig. 8. Ratioof bmele 2 to/;2 versus/; 2. Eachcurveisfordifferent right-handside of (4). Replacing/;2 with /;2 melein (4) results in dof, increasingby factorsof 2 from dof - 32 (top curve) to dof - 256 negligible differences in the bias. (bottom curve). ELGARAND SEBERT:STATISTICSOF BICOHERENCEAND BIPHASE 10,997 simulations(Figure 4). The statisticsof biphaseare similar to those of the phase of the cross spectrum (equation (9) and Figure 5), and confidencelimits for biphase estimatesbased 0.18 0.16 onb2anddofalsoagreewithcorresponding valuesobserved 0.14 in the simulations (Figure 6). Maximum likelihood estimates of the true value of bicoherence given an observed value •0.12 0.10 wereobtained fromtheaX2 distribution (Table1 andFigure 7). The ratio of maximum likelihood estimates of bicoher- 0.08 ence to observedvaluesapproaches1 as /•2 and/ordof increases, withlessthan10%differences if/•2 > 0.25(fordof -> 32)and/ordof > 150(for/•2 _>0.1) (Figure8). 0.06 0.04 0.02 Acknowledgments. This research was supportedby the Coastal SciencesBranch of the Officeof Naval Researchand by the Physical Oceanography Program of the National Science Foundation. The computations were performed at the San Diego Supercomputer Center (funded by NSF). R. T. Guza made helpful comments. 0.00 0.0 1.0 2.0 :.3.0 4.0 X2/v Fig. 9. Chi-squareprobability distributionfunctions.From outermostto innermostcurves, v = 4, 12, 20, and 28, respectively.The abscissa has been normalized by v so that the mean of each distributionis 1, and the scaleof the ordinateis arbitrary. 3.4. Application to Shallow Water Surface Gravity Waves REFERENCES Abramowitz, M., and I. Stegun, Handbook of Mathematical Functions, p. 1046, Dover, New York, 1972. Ashley, R. A., D. M. Patterson, and M. J. Hinich, A diagnostictest for nonlinear serial dependence in time series fitting errors, J. Time Set. Anal., 7, 165-178, 1986. Beningus, V. A., Estimation of the coherence spectrum and its confidence interval using the fast Fourier transform, IEEE Trans. Audio E!ectroacoust., 17, 145-150, 1969. Using bispectral analysis, Elgar and Guza [1985, p. 432] concludedthat narrow-bandswell (peak period about 16 s) Brillinger, D. R., An introductionto polyspectra,Ann. Math. Stat., • '•' 30, 10..J 1--10'• / •'t, 1965 . observedin 9 m water depth was "... qualitatively consistent with Stokes-like nonlinearities." Their conclusion was basedon observedvalues of biphasefor triads consistingof thepowerspectral primarypeakfrequency (fp) andits first few harmonics.Based on the results presentedabove, bispectral estimatesfrom these data can now be interpreted quantitatively. For dof - 256 the observed values of bicoherence and biphase for the two lowest-order triads were Q2(,fp, fp)= 0.161, •OCp, fp)= --7øand/•2OCp, f2p)= 0.067, 13(fp, f2p) = -35ø. (Bicoherences at othertriadswere not significantlygreater than zero.) Accounting for the bias, equation (4),b2meleOCp ,fp2) 160andb2meleOCp, f2p)= 0.070. =O. For dof = 256 and bmeleas given above, 90% confidence limits for biphase can be constructed. For the self-self interaction (fp, fp, f2p)the90%limitsare _+13.5ø.Thusthe Brillinger, D. R., and M. Rosenblatt, Asymptotic theory of estimates of k-th order spectra, in Advanced Seminar on Spectral Analysis of Time Series, edited by B. Harris, pp. 153-188, John Wiley, New York, 1967a. Brillinger, D. R., and M. Rosenblatt, Computation and interpretation of k-th order spectra, in Advanced Seminar on Spectral Analysis of Time Series, edited by B. Harris, pp. 189-232, John Wiley, New York, 1967b. Doering, J. C., and A. J. Bowen, Skewness in the nearshore zone: A comparisonof estimatesfrom Marsh-McBirney current meters and collocated pressure sensors, J. Geophys. Res., 92, 13,17313,183, 1987. Elgar, S., and R. T. Guza, Observations of bispectra of shoaling surface gravity waves, J. Fluid Mech., 161,425-448, 1985. Elgar, S., and R. T. Guza, Nonlinear model predictionsof bispectra of shoaling surface gravity waves, J. Fluid Mech., 167, 1-18, 1986. include the observed value (-35ø). Consequently,with a 10% possibility of being incorrect, it is concluded that the Elgar, S., and R. T. Guza, Statistics of bicoherence, IEEE Acoust. Speech Signal Process., 36, 1667-1668, 1988. Elgar, S., R. T. Guza, and R. J. Seymour, Wave group statistics from numerical simulationsof a random sea, Appl. Ocean. Res., 7, 93-96, 1985. Freilich, M. H., and R. T. Guza, Nonlinear effects on shoaling surface gravity waves, Philos. Trans. R. Soc. London, Ser. A, 311, 1-41, 1984. biphase of this triad is not consistent with Stokes-like Freilich,M. H., andS.S. Pawka,Statistics of Sxyestimates, J. nonlinearities Phys. Oceanogr., 17, 1786-1797, 1987. Hasselman, K., W. Munk, and G. MacDonald, Bispectra of ocean waves, in Time Series Analysis, edited by M. Rosenblatt, pp. 125-139, John Wiley, New York, 1963. Haubrich, R. A., Earth noises, 5 to 500 millicycles per second, 1, J. Geophys. Res., 70, 1415-1427, 1965. Hinich, M. J., Testing for Gaussianity and linearity of a stationary time series, J. Time Ser. Anal., 3, 66-74, 1982. Hinich, M. J., and C. S. Clay, The application of the discrete Fourier transform in the estimation of power spectra, coherence, and bispectra of geophysical data, Rev. Geophys., 6, 347-363, observed value (-7 ø) is within the 90% limits of the Stokes biphase,/3 = 0ø, and the data are consistentwith Stokes-like nonlinearities. On the other hand, 90% confidence limits for thebiphase of thetriad(fp,f2p,f3p)are +21.6øanddo not and that shallow water resonance effects [Freilich and Guza, 1984]are important. 4. CONCLUSIONS Numerical simulationsof random processeswith nonzero bicoherence were used to investigate the statisticsof estimates of bicoherence and biphase. Bicoherence is biased (Figure1) andis approximately aX2 distributed (Figure3), with parameter v a function of the true bicoherence and the 1968. degreesof freedom associatedwith the estimate (equation Jenkins, G. M., and D. G. Watts, Spectral Analysis and Its (8)). Confidence limits for estimates of bicoherence can be constructed fromtheaX2 distribution (i.e.,fromb2 anddof) and agree with the correspondingvalues observed in the Applications, Holden-Day, San Francisco, Calif., 1968. Kim, Y. C., and E. J. Powers, Digital bispectral analysis and its application to nonlinear wave interactions, IEEE Trans. Plasma Sci., 1, 120-131, 1979. 10,998 ELGAR AND SEBERT: STATISTICSOF BICOHERENCEAND BIPHASE McComas, C. H., and M. G. Briscoe, Bispectra of internal waves, J. Fluid Mech., 97, 205-213, 1980. Mfiller, D., Bispectra of sea-surface temperature anomalies, J. Phys. Oceanogr., 17, 26-36, 1987. Neshyba, S., and E. J. C. Sobey, Vertical crosscoherenceand cross bispectrabetween internal waves measuredin a multiple layered ocean, J. Geophys. Res., 80, 1152-1162, 1975. Nikias, C. L., and M. R. Raghuveer, Bispectrum estimation: A digital signal processing framework, Proc. 1EEE, 75, 867-891, with depth off northeasternJapan, J. Phys. Oceanogr., 3, 308317, 1973. Rosenblatt, M., and J. W. Van Ness, Estimation of the bispectrum, Ann. Math. Stat., 36, 1120-1136, 1965. S. Elgar and G. Sebert, Electrical and Computer Engineering Department, Washington State University, Pullman, WA 991642752. 1987. Roden, G. I., and D. J. Bendiner, Bispectra and cross-bispectraof temperature, salinity, sound velocity, and density fluctuations (Received December 20, 1988; accepted March 27, 1989.)