Variance spectrum analysis

VOL.

3, NO. 3

WATER

RESOURCES

RESEARCH

THIRD

QUARTER

1967

VarianceSpectrum Analysis PAUL R. JULIAN

National Center •or AtmosphericResearch Boulder, Colorado Abstract. This paper is an annotated review of variance spectrum analysis.Faced with the impossibletask of reviewinghere the technique,providing illustrative examplesfrom the field of hydrology, and providing learned insight into the uses of these techniquesthat are likely to be productive, I make no apology for the material included and omitted. The researcherwith the appropriate problem and the ability to perceive that this or that particular tool is suited to the solution of that problem is the researcherwho advancesthe science.It is my hope that the discussion and examples contained in this review will stimulate workers to acquaint themselves with spectral techniques and thereby prepare themselves to choosethose problems in time (or space) seriesfor which the techniquesare suited. Spectrum and cross-spectrumanalysesare useful in analyzing the random or stochasticcomponents in a time (or space) series.For certain problems auto- or cross-correlationanalysis may be most attractive, and for others spectrum analysis might be more appropriate. (Key words: Hydrologic systems;statistics; time series)

comesavailable, there is immediate interest in the possibilitiesof application in all the sciences.Questionsof methodologyarisefrequently

as to the efficacyof spectrum analysis.Possibly the technique has on occasionbeen used when it was not appropriate. As with any mathematical tool, sufficientunderstandingof the technique on the part of the investigator is essential in order to select those problems for which it is appropriate. The geophysicisthas a compelling interest in time-series techniques, simply because in nature he cannot design experiments.He must employ statistical methods to investigate observations in time and space that are usually related in some unknown manner, rather than being independent of each other. As a result,

in such cases. Those inclined to be conservative

there

INTRODUCTION

Before undertaking a discussionof variance spectrum techniquesand some uses of these tools in the hydrologicalsciences,I would like to make somegeneral commentson the article and its place in thesesessions. The techniques to be discussedare, of course,statistical ones, and at the outset I will present my credentials and admit to being a meteorologist,not a statistician.

When a new mathematical research tool be-

are three

reasons for

the use of time-

series techniques: First, to allow the interdependenceof successiveobservationsto be taken into account in employing standard statistical tests that assumethe independenceof observahas been the casewith variance spectrum anal- tions; second,to account for the interdependence of the observationsin terms of the physysis. ics of the situation; third, to predict future The developmentof the theory of stochastic processes,the delineation of the central role values of the time series in those instances played in them by the spectrum,and the sug- where it is desired. Because I am a consumer, and my bias is gestionsfor ways in which the spectrum can be estimated have all occurred since about therefore dear, I am of the opinion that time1950. There is still disagreementamong both series analysis, in general, and spectrum analstatisticiansand the consumersof their prod- ysis, in particular, has been of great value in ucts (in the presentinstance,the geophysicists) geophysicalproblems.Certainly this has been

in their judgmentswant to know what the new techniquecan contributethat older techniques cannot,and the more progressivehurry to take advantage of the untried possibilities.Such

831

832

rAUL

•.

the case in oceanography and solid Earth physics. In meteorology and aeronomy the contribution of spectral techniqueshas, I would say, been of somewhatlessvalue, except in the field of atmospheric turbulence, where its use is essential.In hydrology any evaluation is at this time difficult to make and would be premature. The purpose of this review is to describe, in brief, the technique of variance spectrum analysisand to give some examplesof its application in the field of hydrology or in closely allied fields. SPECTRUM

ANALYSIS

The review of variance spectrum• analysis and cross-spectrum(covariance) analysis that follows will be brief. The preceding article has covered time-series analysis, particularly correlation analysis,and there is no need to repeat

5Ch•,•

Moon geometry. In the event that strictly periodic componentsor componentsin which the phase of the periodicity is important are of principal interest, variance spectrum analysis would not be the appropriate tool. A variance spectrum, in the case of a single series,or a crossspectrum,in the case of two series,partitions the variance or covarianceinto a number of intervals or bands of frequency. The quantity examined and plotted as the ordinate is nearly always the spectral (crossspectral)density,which is definedas the amount of variance (covariance) per interval of the quantity used as the abscissa.That is, if co

denotesangularfrequency,co= 2•-/p = 2•-[,

where p is the period in arbitrary time units, various choicesof partitioning the variance are, e.g., S(co),S(p), S(log co),S(log p). The abscissa scalesfor these exampleswould be linear frewhat is covered therein. There now exists a quency, linear period, logarithmic frequency, body of literature that explains with varying and logarithmic period, respectively. It is imdegreesof mathematical sophisticationthe con- portant to understandfor purposesof interprecept of a spectrum and the statistical estima- ting spectrathat there is a functionaldependence tion procedure, and that provides examples. of the spectral density upon the quantity emContributions on an introductory level are ployed as the abscissa.Thus, if it is desired to Gran9½r and Hatanaka [1964], the Appendix preserve in graphical form the proportionality in Munk and Macdonald [1960], Lumley and of area to variance, the followingtransformations Panofsky [1964], Lee [1960], and Bendat and of the estimated quantity•(co)are necessary' Piersol [ 1966]. •(co), (2•-/p•),•(co), co,•(co), (2•-/p)•(co).The The theory of variance spectrum analysis choice of log•(co)on a linearfrequency scale, postulates that the time series under examina- together with the commonlyused presentation tion is a sample from a population serieschar- of log •(co) on a logarithmic frequency scale, acterized by a continuousspectrum. That is, does not preserve the area-to-variancerelationthe observedtime seriesis a random sampleof ship. The choiceof presentationdependsupon a processin time (or space) that is made up the use to which the spectrumis put. However, of oscillationsof all possiblefrequencies.The care in interpretation is necessaryif the area-tobest explication of this postulate in mathe- varianceproportionalityis not satisfied. matical terms not requiring knowledgeof adOf fundamental importancein the interpretavanced mathematics (e.g. measure theory) I tion of variance spectra is the spectrum of a believe is in Lee [1960]. Opposed to this series of random uneorrelated numbers. Because postulate would be the assumption that the of the analogywith physicalopticsthis spectrum, time series is produced by a finite number of whichfunctionallyis S(•o) = constant,is termed sinusoldsand possesseswhat is known as a a 'white noise'spectrum.The randomnature of line spectrum. This view cannot be defended this seriesis expressed by the conditionthat no on physical grounds,except as it pertains to frequency interval containsany more variance a few individual harmonicsarising in geo- than any other interval, insteadof, as in correlaphysical time series because of Earth-Sun- tion analysis,by stating that the autocorrelation function is zero for all lags other than zero. x The use of variance spectrum is much to be Equating the interdependenceof elementsof a

preferred to the more commonly used power spectrum. To omit a qualifying adjective is con-

fusingto particle physicistsand spectroscopists as well as to ourselves.

series with

the

distribution

of variance

over

various intervals of frequencyof oscillationis a conceptualprocedurewhichdifferentindividuals

Symposiumon Analytical Methods

833

will effect with variousdegreesof understanding the spectrum to a discrete set of frequencies that this accounts to a is dictated solely by the discontinuoustime a continuousspecgreateror lesserdegreefor individualpreferences scale:the seriesstill possesses trum. for correlationor spectrumanalysis. Becausehigher frequenciesthan the Nyquist The mathematical developmentof the spectrum indicates the relation between the autofrequency nearly always occur naturally, the correlationfunctionand the spectrum.Thus, for analysis on the restricted set of frequencies 0 < to < to• -- •r will contain variance conthe variancespectrum tributed by thesehigher frequencies.This problem, which has been termed the aliasing probp(r) exp (-irw) dr lem, is treated in the literature. To concludethis section, I will briefly outline the extensionof the spectral equations to 71' include covariance,that is, the joint variance between two series.The appropriate transform = cos pairs for two seriesy (t) and x(t) are and success. I believe

1

p(r)cos cot dr

gives the appropriate Fourier integral transform pair. Here p(r) denotesthe autocorrelation function at lag interval ß units of time. Although for mathematical convenience the complex exponential is used with the necessity of introducingnegative frequencies,both p(,) and S (to) are real symmetric functions,and the convention of using only positive frequencies is nearly always followed in representing spectra.

Ox• =_2 f•oo •x•(r) exp (--iwr) dr = < exp (-]-iwr) dw

(2)

where

•(r) = lim •T x(t)y(t q-r)dt where(•, •

are now complex.(The overbar

Sincep(0) = 1 and p(r) -- 0 for all• > 0, denotesa complexquantity.) This may be underp(o) -- fo© S(to) dto -- 1, so that the in- stood when it is noted that the cross-correlation tegral of the spectrumcontainsthe 'normalized function • is not necessarilysymmetric about variance' of the series. If the unnormalized r = 0. The real part of the crossspectrumis autocorrelation function (the autocovariance) denotedby C• and is called the cospectrum.It

•

p(•) is used instead, the integral of the

indicates

the

distribution

of covarianee

with

spectrum gives the total variance •2 of the frequency for oscillationsthat are in phase. series.The relationshipbetween an autocorre- The imaginary part of the crossspectrum is lation functionfor a nonrandomseries,p(•) denoted by Q• and is called the quadrature 0 for some • > 0, and S(to) =/= constant is spectrum,appropriate to the fact that it gives givenby the Fourier transformpair equation1. the distributionof covariancewith frequencyfor In nearly all meteorologicaland hydrological oscillations displacedby •/2 radiansin phase. time series the time scale is discrete rather Thus for each spectral band it is possibleto than continuous,owing either to instantaneous definea phaseangle measuringor to interval averagingof the natß (w) = tan-• Q(w)/C(w) (3) ural process. Spectral theory must then be modified

to allow

for

a discrete set of fre-

quenciesover which the variance is to be dis-

which expresses the averagelag betweenthe componentsof frequency in the two series. In addition, an important quantity to be defined is expressedby

tributed. For equally spacedobservationstaken at At time units apart, it is intuitively obvious that the oscillationwith the highestfrequency for which any information can be obtained is one with a frequencyf• ---- 1/(2 At), referred to in the literature as the Nyquist frequency. The right-hand quantity, variously defined in It is important to note that the restriction of the literature as the coherenceor the squared

Coh()=

& +&

(

834

rcur,

r•. J UL•AX

coherence, canbeshown toberestricted tothe

range0 to -51 andis analogous to a correlationcoefficient squared. Thisanalogy withthe

•(•)•' = E{[•(co) -- E(•(co))] •'}

(6)

= E{ [•(co)-- S(co) -- fi]•'}

correlationcoefficient apparentlyis the cause of the confusion in terminology. In the remainderof this paperthe quantityCoh• will be calledthe squaredcoherence. This statistic

Desirable,i.e., best,estimators thenare those that are unbiased and efficient,i.e., thosefor which/S = 0, and •(•)'- is a minimumfor a givensamplesizeand decreases as the sample

is quite important in spectral analysis,and an

size increases.

appreciationof the power of spectral techCurrent methodsof spectralestimationuse niquesdependsupon the appreciationof the a modificationof the historic periodogram conceptof a linear measureof association as [Schuster, 1898],whichis a plot of the squared a functionof frequency. amplitudesof a Fourier harmonicanalysisof Finally, I want to point out that all of the foregoingtheory applies to spaceas well as to time as the independentvariable. This should be clear enoughto the reader, and only one statement of caution needsto be added. When applied in the spatial domain,the regionunder consideration must not be closed,which would violatethe assumption of a continuous spectrum of spatial frequencies.Examples of the application of spectral techniquesin the spatial domainwill be referredto later.

the time series as a function of the Fourier frequencies. The modificationof the periodogram (e.g., that by Blackman and Tukey [1958]) employseither a 'faded' estimated autocorrelation functionor the transformof this, a smoothing of adjacent periodogramestimates,for a limited number of frequencies.Bartlett [1950] and othershave sinceshownthat the classical periodogramis not an efficientestimatorof the

continuous spectrum,althoughit is appropriate for a true line spectrum. Thus, the variability

of the•(co)'sderivedfromthe squared amplitudesof a fittedharmonic of frequency codoes

ESTIMATION OFSPECTRA

not decreaseas the length of the time series

Equations1 and 2 representthe population increases!The classof estimatorsthat either

spectrum andautocorrelation function asFourier multiplythe corre]ogram by a monotonically transform pairs.Whendealingwith a finite decreasing function (withincreasing r), which lengthof record, whichobviously mustalways becomes zeroat somer • T; or,equivalent]y, bethecase, a newsetofequations mustbeused smoothes the truncated periodogram estimates that relatesthe sample spectrum andsample by someweighted linearcombination, canbe autocorrelationfunction. A great deal of the

shownto be both unbiasedand efficient.

literature concerns itselfwith the estimation A greatdealof the literature, of which problem; thatis, givena sample timeseries,Jenkins[1961]and Blacfcman and Tukey whatisthebestalgorithm tousetoestimate the [1958]arerepresentative, concerns itselfwith population spectrum? The statisticians define thetypeof estimator that is mostdesirable. 'best' inquantitative terms, andit isinstructiveMy ownviewis that,although thisis anim-

in thepresent caseto examine thespectralportant andnecessary consideration, theconestimation problem, briefly, in these terms. If sumer-user in mostgeophysical applications

thecaret denotes anestimate, sothat•(co) is need notconcern himself with choosing among

theestimated spectral density, thenS(co) indi- thewealth of suggested estimation schemes. cares thepopulation spectrum. Twostatistical However, heshould haveanappreciation of

properties ofanaggregate ofsuch estimates are what isinvolved intheestimation process and relevant. Thebias/• ofthesampled distribution of theelementary properties of thevarious of•(co)isgiven bytheexpected value ofthe estimators.

difference oftheestimates andthepopulationIn thissurvey article there isnopoint. in

spectrum

/• = E'{•(co)S(co)}

going deeply intothesampling theory ofspec-

(5) Blackman traldensity estimates. Thetheory given by and Titkey [1958] prescribesthe

The variabilityof theseestimates is alsoimportantandis written

sampling limitsfor a randomsampleof spectral estimatesat a particularfrequencyby

Symposiumon Analytical Methods

stating that the estimatesare distributedabout the populationspectrumas X•' upon the num-

835

ysis for testing hypothesesin time seriesanalysis is that the theory is considerablysimpler berof degrees of freedom. The degree• of free- and more exact than is the theory of correlodom dependupon the total length of the record gram sampling. The exact distribution of a and upon the particular estimator of the serial correlationcoefficientobtainedby Anderspectral density. The most completediscussion son [1942], and extended by many people given in the literature on this samplirg theory since, uses the so-called circular definition of is in Jenkins [1961]. Cross-spectrumsampling the aut.oeorrelationfunction. This is apparently theory is discussed,e.g., by Jenkins [1963]. done for mathematical conveniencealone, and All of the spectralestimatorsrequire that the the exact physical interpretation of the circular autocorrelation function is obscure. experimenter select one or more constantsappearing in the expressionfor the estimated STATIONARITY spectral density. In the Tukey-elass of estimators, for example, the experimenter must A considerationfrequently relevant in geoselect the maximum auto- or cross-correlation physical time seriesis that of the stationarity eoefl%ientto retain, as well as the weighting of the series. Statisticianshave a precise defischeme for the Fourier-transformed eoeflqcients. nition of the condition of stationarity, but a These arbitrary parameters help to determine great deal of confusion is created when this the appropriate number of degreesof freedom definition is translated into other terms. A in the chi-square sampling theory. There is as seriesis said to be stationary when its expected a result a kind of indeterminaneyrelationship value (the mean) doesnot. vary with time and between the sampling variations or scatter of the autocorrelationfunction dependsonly upon the resulting spectral estimates and the num- time differencesand not upon the absolute ber of 'lags' (the number of auto- or cross- time scale.This doesnot imply, however,that correlations)usedor the resolutionin frequency all subportions of a sample record must look obtained. The finer the resolution on the 0 to the same or that, for example, averagesealcu•r frequency scale, the greater will be the ex- Iated from subsetsof the samplemust be equal. The condition of stationarity is frequently pected scatter of the estimated spectral points about the populationspectrum.Rules-of-thumb translated to imply the absenceof trend in the for selecting the maximum autocorrelation series.Two points shouldbe made concerning coefficient cut-off point to obtain reasonable this translation, since the presenceof trend is samplingvariability have been suggested(e.g., not a necessarycondition for nonstationarity, 10% of the total number of observationsor and, dependingupon the particular specificadata points). However, it shouldbe emphasized tion of trend, may not even be a sufficient that situations may occur in which sampling condition. The first point is that trend is selaccuracymay be sacrificed,and, with the inde- dom defined by the experiment. A decision as terminancy principle in mind, the experimenter to the presence or absence of trend must be need not always pay heed to these rules-of- based on either a sufficientknowledgeof the thumb. physical situation or on • statistical test that The major difficulty in the applicationof the permits testing against trend without specifysamplingtheory is that for mostkinds of hydro- ing the form of the trend. I believe it is correct logical time seriesonly one or at most a limited to say that no such test is known. The impornumber of samples are available. Since the tant point here is that movements in the population spectrum is not known, knowledge sample record with periods on the order of or of how random sample estimatesscatter about longer than the length of record are not necesthis unknown quantity, particularly with only sarily a manifestation of nonsta.tionarity. Ala singlesamplein hand, is not very helpful. Of though not all statisticians would agree on a course,testing against certain hypotheses,the procedurein the easewhere insuflqeientknowlnull hypothesis of a population white noise edge of the physical situation exists to assess spectrum, for example, can easily be accom- stationarity, it has been suggestedthat the experimenter assumestationarity with the proplished. The major advantageof usingspeetraIanal- vision that any interpretation of the resulting

836

PAUL R. JULIAN

cipitation [Landsberg et al., 1959]; investigation of componentsof water pollution in tidal As a specificexample,considerthe measure- estuaries [Wastier, 1963]; the determination ment of temperature fluctuationson a meteor- of dominant meander wavelengths in stream ological tower.Samples of small-scale turbulent channels [Speight, 1965]; and the investiga-

spectrummay be incorrector subjectto revision.

fluctuationsextendingover a few hours will contain nonstationarycomponentsbecauseof the diurnal variation in temperature statistics. In this ease some attention must be paid to

the portionof the diurnalmovementcontained in the record. On the other hand, mean annual

temperaturescalculatedfrom the sameinstrument extendingover decadeswill undoubtedly contain movementswith time scaleslonger than decades.Here we have no physical knowledge of the form or cause of these oscillations (e.g.,

whether they are deterministicallyproduced or not) and, basedupon our current understandingof the atmosphere, we wouldbe justified in assumingthem to be oscillationsof a stationary process.

The secondpoint.,which has been presaged by the paragraphsabove,is that the stationarity not onlyappliesto the meanbut alsoto the higher momentsof the distribution.That the absenceof trend is not a necessary condition

for stationarityis easilyprovenby constructing a series with constant mean but in which the variance and therefore the autocorrelation function is a function of time.

Nonstationarity complicatesthe interpretation and samplingtheory of estimatedspectra, but in certain instances where the form of the

nonstationarityis known, allowance for the fact may be made [Granger and Hatanaka

tion

of

the

bed

forms

in

alluvial

channels

[Nordin and Algert, 1966]. A study principally of meteorologicalinterest which is relevant also in hydrology is that of Griffith et al. [1956] on the determination of the variance spectrum of atmospheric (surface) temperature. An example of spectral analysis in two-dimensional spatial domain has been provided by Leese and Epstein [1963]. Some'•examplesof spectra and crossspectr• of hydrological and meteorologicalseries are presentedhere, not only to illustrate their use in the detection of nonrandom components, but also to acqua.intthe reader with the properties and interpretation of spectra. Figures la and lb present,respectively,the estimatedcorrelogram and •he variancespectrum of monthly total precipitationfor a fairly long and homogeneous record(Pinal Ranch,Arizona). The most noticeablefeature of the correlogram is the tendency for •(r) to reach its highest valuesat lag valueswhich are multiplesof 12 months. Upon closerinspectiona six-monthly componen•is alsoplain in the correlogram.The spectrum of the monthly precipitation values reflects•hese,of course,by indicatingpeaksat periodsof 1 year and 6 months.This station was chosento illustrate an important feature of correlogramsand spectra.Anyone familiar with

the meteorologyof Arizonawouldexpectto see a semiannualpeak in precipitationbecauseof the presenceof winter and summer maxima. APPLICATIONS OF SPECTRU1V[ ANALYSIS Indeed,perusalof the raw recordof precipitation IN I-IYDROLOGY clearly shows these maxima. It is possible, The applicationsthat have been made of however,to obtainpeaksat harmonicfrequencies spectraltechniquesin hydrologywill be divided of the annual frequencybecauseof •he distorted into two groups and discussedseparately, al- wave shapeof the fu.ndamental. The peak at 4 though someoverlap will occur.A third group months obviously is causedby a distortion of of applications,to my knowledgewithout exam- the 12 month or combined 12- and 6-month of a periodic(seasonal) ples in hydrology,will be discussedat the end pattern.In theseinstances of this section. componentwithin the data, care must be taken The first group of examplescomprisesprob- in the interpretation of the estimatedspectra: lems involving the detection o)e nonrandom it should be recalled that the estimators of componentsin the time series. To mention a continuousspectraare not designedfor strictly few that have been studied: the properties of periodicor nonlinearcomponents. If attention is focused on the continuum precipitation and streamflow [Julian, 1961]; the investigation of quasiperiodicitiesin pre- spectrum a slight inflation in the lowest fre[ 1964], Chaps.8 and 9].

Symposiumon Analytical Methods

837

1.0

5

p(k) 4

o

o

o

o

o •

o

o

øo o

9

o

o o

o

g

o

oo

o

o

o

o o

•

o

o•

o

o

o

o

o

o

•

o

o

o

-I

o

o

o

o

øo0

o

øo

o

o

o

o

o

o

o

o

o

o

o

o

-2

-5

0

4

8

12 16 20 24 28 32 36 40 44 48 5;) 56 60 64

Lag k (months) Fig. la.

Estimated correlogram of monthly total precipitation, Pinal Ranch, Arizona, 1896-1960.

5.5 $.0

^ 2.5 S(f) 2.0

x x

1.5 1.0

x

--

xx

X

0.5

•

x

X

x

X

XXX

XXXXX

XX x Xx xXxx Xx

0

.0833

cO

12

.1667 6

X

Xx xx xXX XxxxxxX x xXx

XxXXX

0.250 4

x

X

0.333

X

0.4167

x1(x

X 3(

0.5 f (month -•) 2 P (months)

Fig. lb. Estimated variance spectrumof monthly total precipitation, Pinal Ranch, Arizona, 1896-1960.The abseissais presentedon a linear frequencyscale,with reciprocalperiod •qale (in units of months) also shown.

838

PA•L

R. JULIAN

x =Colville, Wash., 1900-1963, monthly temperature anomalies o=Tatoosh Is., Wash.,1884-1960, monthlyprecipitation anomalies 1.0

.9 xxx

x

,,.6

xx x

S

xx

x

x x

x

x

.5

x xx xx

x

x

x

x

x

x

xX Xxx

x x

x

x

xx x

xx

x

.4

.3 o

.2

o o o oo o

o

o0

øo

o

o o0 o o0 00 00 00 0 • O0 00000000 ofO00000 000 0000000 000O0 0

I

I I

4 4 2

I I I

I

I

I I

I I

I

I

I

8 12 16 20 24 28 I yeors 8

6

I

I

I

I I

I

I

I

I

32 36 40 44 48

4

log

2 months

Fig. 2. Estimated variance spectra of average monthly temperature, 1900-1963, for Colville, Washington, and monthly total precipitation, 1884-1960,for Tatoosh Island, Washington. The data have been standardized by removing from each value the sample mean and dividing by the sample standard deviation for that calendar month.

quencies is noticeable. This decrease in the spectral densities from low to high frequency is not statistically significantin this case.Nevertheless,this property is characteristicof many meteorological spectra (as further examples will demonstrate) and is produced by varying degrees of persistence in the data. This can be understood qualitatively by noting that the cosinetransform of a smoothly decreasingautocorrelation function will also decreasesmoothly from low to high frequency. In Figure i the autocorrelationfunction of daily precipitation valuesis apparently still differentfrom zero for lags of, say, 4 or 5 days. Arbitrary, nonoverlapping 30-day intervals will therefore have an automatic correlationwhich, although small, is neverthelessmeaningful. Figure 2 gives the variance spectrum of

monthly standardizedprecipitation and tem-

perature records from two stations in the Pacific Northwest. The standardization procedure used removes the annual

variation

and

any of its harmonics from the series. The method used here might not be the most desirable, but it has the virtue of being simple: it consists of removing the sample mean and dividing by the sample standard deviation for each month. The temperature spectrum• shows a decreasein spectral density with increasing frequency, indicating a persistent structure in monthly temperature anomalies. (The term 'anomaly' is used here to denote a deviation

from an average.) The precipitationspectrum, however,exceptat the very lowestfrequencies, is a white noise spectrum characteristic of a z I am indebted to Dr. Gunnar Roden, Department of Oceanography, Univer•ty of Washington, for this spectrum.

839

Symposiumon Analytical Methods random uncorrelated

series. This latter

increase

causedperhapsby gagingerror or consumptive

of spectral density at the very lowest frequencies is either indicative of climatic changes (or relatively long period changes) or of a nonstationary componentin the Tatoosh precipitation record. The persistence of monthly temperature anomaliesand lack of persistence of precipitation anomalies seem to be characteristicof a great many suchtime series. Figures 3, 4, and 5 give the variance spectra, cross spectra, and coherenceof the monthly

use.

We may neglect (4) on the groundsthat it cannot quantitatively account for the observed persistence.Figure 2 suggeststhat (1) is not important either, but that (2) might well be, becausemonthly precipitation amounts apparently exhibit little persistence,whereasmonthly temperaturestend to be persistent.It is intuitively obvious,and can also be demonstrated mathematically, that basin storagewill produce standardized anomalies of the streamflow of the a persistence in streamflow records. We are John Day and Umpqua rivers in Oregon.Both thus left with two factors, of unknown relative variance spectra indicate that a significantper- importance, to account for the linkage in sistenceis present in the monthly streamflow. monthly streamflowanomalies. A potentially productive use of variance This linkage is not causedby the water-year variation in runoff, because this component spectrum analysismight be an analysisof lake has been eliminatedby the standardizationpro- level and groundwater fluctuationsin an effort cedure. The hydrological explanation of this to study natural regulatory processes. persistencemust be that (1) monthly precipiFigure 4 presents the cross spectra of the standardized anomalies in the two basins. The tation anomalies are persistent in time; (2) evapotranspirativelossesare persistentin time; cospectrum C(/) indicates that most of the (3) storage capacity in the basins is large covariance is concentrated in the lowest frequenciesand is positive at all frequencies.The enough to produce monthly carry-over; (4) nonstationary movementsexist in the records, quadrature spectrumQ(/) is also positive (ex-

x: John Day River at McDonald Ferry,

3.5•x

1906-1960 1906-1960

^

monthly anomalies

o: Umpqua River near Elkton,

3.0-

Oregon

Oregon

monthly

anomalies

2.5-

s(f) 2.0 ø

xx

øoø ox

1.0-

oø

oX o xo

XxO 0o

oX

0.5-

oo

oo

X•xx Ooø•" XxxXx

I

0

0

.0833 12

.1667 6

o,

o

o

•i•x øøøøx•i•o,•oo oøooøOooo I

0.250

xxxxx'lx•'i•xXX 0.333

4

0.4167

xxih•ø"m• 0.5 f (month 'm) 2 P (months)

Fig. 3. F•stimated variance spectra of monthly total streamflow, John Day Ri•er, at McDonald Ferry, Oregon, 1906-1960; and Umpqua River near FJkton, Oregon, 1906-1960. The data have been standardizedby removingfrom each value the samplemean and divid-

ing by the sample standard deviation for that calendar month.

John Day-Umpqua 1906-1960

3.0

Streamflow

monthly anomalies

o = Cospectrum,

2.5

C(f)

x = Quadrature Spectrum, Q(f) 2.0 oo )

o --

1.5

0

X

o

1.0

X

ø0

o o

X XOo X XXo o o oO

0.5

X

0 7.

X 0

xx

000XX

XxxxxO

-0.5 0

0

xoooo

x•

x

00

Oo

ooo o

"Xxxxxx•t•øvx• •o•i"-o9ø• •9•e'e• •-oøO•e• o•øg•-

I

I

I

I

I

.0833

.1667

0.250

0.333

0.4167

I

12

6

0.5 f (month 'l)

4

2

P (months}

Fig. 4. Estimated cross spectrum for the d•t• in Figure3. The cospectrum •(f) •nd the quadrature spectrum•(f) •re plottedtogether.

1.0 .g

.8

XxX

x x

xX

xX

x

.7

x

x

x

x x

xx

x

Coh

xx

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x xx

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xX X

.6

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x

x xx

x

x

x x x

x

x x

xX

x x x

x

x

x

x

x x

00

I .0833

I .1667

I 0.250

OD

12

6

4

I 0.333

I 0.4167

I f (month 0.5 'l) 2

P (months)

Fig. 5. The coherenceCoh, as defined by equation 4 in the text for the monthly streamflow of the John Day and the Umpqua, 1906-1960.

Symposium on A•alytical Methods

cept at a period of one year) everywherebut somewhatsmaller in magnitude than the cospectrum.The phase angle •I,(.o)), indicating the relative timing of oscillationsin the basins, is thereforealwaysin the first quadrantwhich, with the conventionused in the spectral estimation algorithm, indicatesthat features occur on the Umpqua recordbeforethe John Day at (nearly) all frequencies.The interpretationof this result may be open to question,but the Umpquaat the gagingpoint is the smallerbasin, and perhapsthis fact is hydrologicallyrelevant. The coherence,Figure 5, indicatesa high degreeof associationbetweenthe basins,with slightly higher valuesat the lower frequencies. Although the va.riationin the coherencestatistic is not as pronouncedhere as I would have liked in an illustrative case, a general comment is

841

eration method must be used.) This type of information, of course,is not containedin the statisticaldistributionor time-structureparameters. The contributionof spectralanalysisto the data generation problem is direct and rather fundamental. The goal is a large amount of data that possesses the same statistical properties as the samplerecord.By the application of spectral analysisto the historical data the processof selectingan appropriate model to duplicate the time-dependentstructure is simplified. In random generation schemesperformed on digital computers,formulas of the type known as moving average and autoregressive schemes are convenient. These formulasmay be written

appropriate. Theobserved decrease inthecowith theinterpretation that secular changes in

xt= l(•t et--l''') xt- •(xt_• xt-2"')

herenee with increasing frequency is consistent

(6)

the runoffpatternare moreconsistent over in whichtheet arerandom uncorrelated numwider horizontal distancesthan are runoff bers generatedfrom the appropriatedistribu-

amounts produced by atmospheric circulationtion,andthefunctions involved arerestricted changes for shorterintervals, i.e., stormsor to linearcombinations. A particular modelto stormgroupsof limitedhorizontal extent. represent the time-dependent structure of the Anexample of a problem in whicha partition streamflow values maybeselected by comparing of variance,covariance, and correlation into the estimated streamflow spectrum with the frequency intervalswouldhavevaluewould spectrumof the xt's calculated analytically be that of extending or interpolating stream- fromthe formof the equation used.For exflowrecords. Someof theaspects of thisprob- ample,a simpleautoregressive schemeof lem have been treated by Fiering [1964]. The secondgroup o)e applications o)e spectral analysis pertains to the digital simulation o)e streamflow records. The technique of synthetic hydrologic data generation for use in

order n

(7) has a spectrum

economic basindevelopment models hasbeen

S•

described inThomas and Fiering [1962] andSx(co)=[l_alexp(_ico)_a2exp(_ briefly in Chow [1964]. The basic idea is to derive from a limited

(8)

historical sample the appropriate parameterswhichcanbe simplified to givea polynomial describing the statistical (probability) distri- in cos