University of ConnecticutOpenCommons@UConn

Special Reports Connecticut Institute of Water Resources

June 1967

Analysis of Quasi-Periodic Weather DataChelsey Johnston PoseyUniversity of Connecticut

Follow this and additional works at: https://opencommons.uconn.edu/ctiwr_specreports

Recommended CitationPosey, Chelsey Johnston, "Analysis of Quasi-Periodic Weather Data" (1967). Special Reports. 4.https://opencommons.uconn.edu/ctiwr_specreports/4

Report No. 3June, 1967

ANALYSIS OF QUASI-PERIODIC WEATHER DATA

Chesley J. Posey

INSTITUTE OF WATER RESOURCESof The University of Connecticut

ANALYSIS OF QUASI-PERIODIC WEATHER DATA

Chesley J. Posey

Professor of Civil Engineering

The University of Connecticut

June 1967

INSTITUTE OF WATER RESOURCESof The University of Connecticut

The work upon which this publication is based was supported in part by funds provided bythe United States Department of the Interior as authorized under the Water ResourcesResearch Act of 1964, Public Law 88-379.

ANALYSIS OF QUASI-PERIODIC WEATHER DATA

Daily temperatures follow a pattern of variationthat is familiar to us all. Although our very livesdepend upon its not varying too much, our plansare often thwarted by its unexpected turns. Theexperience of thousands of years, if not tens ofthousands, has shown that it is remarkably constantin some respects. We long to be able to predict itsvagaries, yet about all we can say with certaintyis that the temperature will be higher in summerthan in winter. Or, if you don't like the temperaturetoday, just wait, and it will change.

The purpose of the present study is to see whatcan be learned from long-time records taken atsingle stations, using a three-stage type of statisticalanalysis the second and third stages of which havenot previously been applied to this kind of data.The particular series chosen is that of the dailymaximum temperature, for which some of thelongest reliable records are available. When thestudy was first started, the necessary computationshad to be made by hand, but later high-speedcomputers became available, making possible thestudy of much longer records from data alreadyavailable on punched cards. Before describing themethod used, the limitations of two other methodswhich are often applied should be considered.

Fourier analysis is frequently used for recordswhich vary over a certain range, but which haveno tendency to continuously increase or decrease.There is plenty of geologic evidence of a long-timetrend in the mean temperature, but if we limit "long-time" to a period of 100 or even 1000 years thereis little reason to question the validity of Fourieranalysis on this basis. (A careful watch on thelong-time mean should be maintained, however,since it can be influenced by the addition of carbondioxide from fossil fuels into the atmosphere, aswell as whatever natural change may be in storefor us). If the pattern of variation from the meanis strictly periodic, Fourier analysis is ideal, but ifthe duration and magnitude of the swings awayfrom the mean differ from each other in a randommanner, interpretation of the results becomesdifficult. The coefficients of the corresponding series

terms determined for two such records of equallength will be different, even though the records arevery long. If in every record analyzed, for a givenstation, the coefficient of the term correspondingto a certain frequency was large and nearly constant,we would suspect that there was some physical reasonfor the frequency. We would surely find a significant3651/4 day period in maximum daily temperatures,for example, but we could not assume that thecoefficient for it would be the same in one recordperiod as in another because of the effect of the non-periodic variations on its evaluation.

In his study of weather of the Northeast, Bingham(1963) carries the Fourier analysis out to no morethan three terms, treating the remainder of thevariation as randomly distributed according tonormal error theory. Subject only to a presumablyvery slight inaccuracy due to the effect of therandom variations on the Fourier coefficientsdetermined, his report provides a valuable methodof predicting the possible range of the importanttemperature statistics for any week of the year atany location in the Northeast. Something is lackingif prediction into the near future is desired. If wewish to predict the possible range of next week'sweather, not that of a year from next week, we willsurely want to take into account what the weatherhas been for the past few weeks, and how rapidlyit may be expected to change. (It is assumed thatour source of information is restricted to the locationof our weather station).

A method that provides a way of taking thisup-to-the minute information into account is knownas the serial correlation method. The correlationcoefficient between records separated by a timeinterval is determined as a function of the lengthof that interval. Then if we know today's maximumtemperature, we can tell what the chances oftomorrow's maximum falling within various rangeswill be. We can't be as definite about day aftertomorrow's, and even less definite about the day afterthat. The method has nothing to say about thedistant future. The assumption is made that as soonas the correlation coefficient goes to zero the values

3

separated by that time interval are completely inde-pendent. Often, however, it is found that the cor-relation coefficient dips below zero. This evidence ofa cyclic tendency in the data, which means thatmeasurement of the decay of dependency shouldbe based upon the correlation ratio rather thanthe correlation coefficient, the use of which pre-supposes that the underlying relationship, uponwhich random fluctuations are imposed, is strictlylinear. Actual use of the correlation ratio is pre-vented by lack of knowledge of the form of theunderlying non-linear cyclic relationship.

The method applied in this study has little incommon with the above methods. Its immediategoal is the establishment of a means of identificationof quasi-periodic records. (Posey 1936, 1946, 1952)That is, we'd like to be able to say that one recordis or is not "equivalent" to another. Not equivalentin the sense of geometrical congruence, but in thatthey must have been taken at the same locationin the same manner, subject to the same kinds ofperiodic and non-periodic cyclic and non-cyclicvariations about a common mean value. Thus thestatistical measures to be determined might belikened to "fingerprints" of the time series, somethingfrom which it would surely be identified. Neitherof the previously described methods of analysis hasproved to be very useful for this purpose. Nowthere is no way to prove that the method of thisstudy can accomplish this goal, short of applicationto a very large number of records from differentsituations.

DEFINITION OF THE THREE STATISTICALMEASURES

The fluctuating quantity y is measured at regularconstant intervals of time. (If y varies continuouslywith time, the interval between measurements shouldbe short enough so that values linearly interpolatedbetween the measured values will agree closely withthe actual values.) If y is a discrete variable, theappropriate interval is of course that betweenadjacent values.

The first statistical measure of importance is thedistribution of the values of y. This may be obtainedeither in frequency form, or preferably (for conven-ience of computation) in cumulative distributionform, giving the percent of time that y exceeds eachvalue throughout its range. Let this distributionbe designated as Cd(y). If the values of y shouldprove to be normally distributed, the mean valueand standard deviation would suffice to characterize

it. Experience with natural phenomena, however,indicates that distribution according to Gaussianlaw cannot be depended upon. At present we shallhave to content ourselves with visual comparisonof graphical plots of the cumulative distributions.

The second important statistical measure is thefrequency distribution of values of the instantaneousrate of change of y. As a matter of practical com-putational procedure, it is approximated by thedistribution of the differences between the con-secutive values of y used in deriving the first measure.In cumulative distribution form it is designatedCd(Ay). If the time series had no long-time trendand the values of Ay proved to be normally dis-tributed, the standard deviation would suffice tocharacterize the distribution, since the mean wouldbe zero.

The third important statistic is the distributionof the rate of change of the rate of change of y,which may be approximated by the distributionof the successive differences of the successive dif-ferences of y. In cumulative distribution form itmay be designated C d(AAy).

The number of distributions could be carriedfurther, to consider higher derivatives of the basicvariable. The main reason for not doing so isthat by the time that the rate of change (slope)and rate of change of rate of change (curvature)have been considered, it seems likely that thefeatures of the variability of the series which mighthave important physical significance will have beeninventoried with sufficient completeness. Anotherreason is that because of the approximation neces-sarily involved in measurement and computation,the relative accuracy of the higher derivativesbecomes successively poorer.

A program which instructs an IBM' 7040 digitalcomputer to find the three distributions from dataon punched cards is given in the Appendix.

TESTS OF THE VALIDITY OF THE PROPOSEDMEASURES FOR DEFINITIVE

IDENTIFICATION

If the fluctuating measurements at a fixed locationin a basically unchanging physical situation are takenby an unvarying technique, if a sufficiently largenumber of cycles is included, and if there is nolong-time trend, it is our assumption that the dis-tributions of y, Ay, and AAy for different portionsof the record will be identical. This assumptioncannot, of course, be proven, and it may be that

4

exceptions will be found. It is obviously correctfor strictly periodic variations, one complete cyclebeing all that is necessary to characterize such aseries. This property can be used to test equipmentdesigned to evaluate the measures automatically.(Lonsdale, 1952). If a series is known to containan appreciable component of variation having afixed period the length of record chosen for analysisshould contain an integral number of said periods,or else be so long that the effect of the valuesthrough one fractional period is inconsequential.Thus in all of the records subsequently analyzed,the periods correspond to numbers of calendar years,close enough to integral multiples of the true lengthof the year. The effect of lunar cycles is ignoredas inconsequential, and that of sunspot cycles assmall and not strictly periodic.

AWhat about series lacking any definite periodicfrequencies? How "equivalent" can the distributionsfrom records of practical length be expected to be?How much, in comparison, will the distributionsfrom different situations differ? Some evidence offavorable answers to these questions comes fromapplications of the Cd(Ay) distribution to otherproblems. Ruhe (1950) found that it provided areliable means of quickly identifying different areasof glaciation in Iowa from highway profile datealone. A comparable statistic, the mean squaresuccessive difference, was described by Von Neumannet al (1941). Liederman and Shapiro (1962) foundthis measure useful in the quantification of time-ordered data in physiological and psychologicalresearch. Incidently, these investigations seem toindicate the effect of slow long-time trends of y onthe Cd(Ay) distribution is so small as to notinterfere much with its usefulness for identificationpurposes.

It seems likely that for every different applicationit will be necessary to find out by trial how longa record must be taken for each Cd distribution toconverge to a fixed curve characteristic of the series.At the present stage of investigation it seems sufficientto make a visual comparison in judging this. A moresophisticated criterion might be justified after moredata have been analyzed.

CONVERGENCE OF THE THREEDISTRIBUTIONS

Figure 1 shows curves representing the percentage

of days during which the indicated maximum dailytemperatures at Storrs, Connecticut, and Taipei,Taiwan, were not exceeded. Each curve representsthe distribution Cd(Tmax) for a single year. Thecorresponding ATmax and AATmax distributionsfor the same single years are shown in Figures 2and 3 respectively. (The curves for Taipei are notincluded on Figures 2 and 3 because they wouldoverlap those from Storrs too much.) Notice thatthe dispersion of the Cd(ATnmax) curves of Fig. 2is less than that of the C d(Tmax) and C d(AA Tmax)curves of Figures 1 and 3. One-year record periodsare the minimum practicable because of the annualperiodicity, but they are evidently far too short toyield equivalent or identical distributions.

Figures 4, 5, and 6 show Cd distributions witheach line representing the accumulated values fora five-year period. It is evident that lengtheningthe period has brought the lines closer together.The distributions of the different five-year periodsare not yet identical, but the fact that they aregrouped more closely shows that there is a definitetendency for convergence as the length of periodis increased. The lines representing the five-yearAy and ALy distributions for Taipei can be shownon the same chart as those for Storrs, without toomuch overlap. Figures 7, 8, and 9 show curves forthe distributions for longer periods at Storrs. Furtherconvergence is evident in each case.

Since it was thought that the tendency towardconvergence might have been due to the locationof both of these stations near the ocean, a similarstudy was made of maximum daily temperaturerecords from Burlington, Colorado, Elev. 4250,located on the plains some 150 miles east of the foot-hills of the Rocky Mountains, and Hayden, Colorado,Elev. 6300, in the Colorado mountains across theContinental Divide some 100 miles from the plains.The curves, covering periods of 4 and 14 years, areshown in Figs. 10-15. For both locations the dis-tributions of Tmax A Tmax, and A A Tmax con-verge as the length of period is increased. Toprovide most convincing evidence of the tendencytoward convergence, it would be necessary to compare36 single-year records, 36 five-year rcords, 36 ten-year records, etc. Data continuous over periodslong enough to carry this comparison very far maynot exist, since the temperature measuring stationsare affected by man's alterations of the environ-ment.

5

40

C30

0

6 FIGURE 2o1 20 30 40 so 60 70 80 90 /00

Percenf of doays

Percen of days FIGURE 3

/ 0 20 0 40 50 60 70 80 90 /00

Percent of days during which Maximum Temperature was nof exceeded FIGURE 4 7

"F30

20

10

x

11

qo

-20

-30

50 60Percent of days

qIq

FIGURE 6

8

0 /0 20 JO 40 50 60 70 80 90Percent of days during which Maximum Temperature was not exceeded FIGUR

FIGUR

C-

30

20

_ O

-(0

20

E 7'E 7

FIGURE 8 9

0oo

90

80

70t

L.

I'+- 6oIZ

~50

: 40

: 30

/O

0

STpORRio20-year period 3

I6-year period

I I I MAX

I I I I I I I

1 -f

- -

'd %

t

Percenf of do-w

Percent of days

1o

I

FIGURE 9

/00or

90

o8

70

460

4o

|8oI,

20

10

10

Percent of days

P So 4

APeren of doys FIGURE 12 11

t

I,

I

II

FIGURE 11

20

%

-3o

FIGURE 13

op.IF30 F

2.o0

oI,

-/20

-30

- /O

2o

0

12 FIGURE 14

Percent of days

Percent of days

Peret of days FIGURE 15

COMPARISON OF LONG-PERIOD RECORDS

Figs. 16, 17, and 18 permit comparison of the dis-tributions of Tmax , Tmax and A A Tmax forthree stations; Storrs, Connecticut, and Burlingtonand Hayden, Colorado. The curves for Storrsrepresent a total period of 36 years, while thosefor the two Colorado stations summarize the dailytemperature maximums for 28 years. The curvefor the 5-year period at Taipei has been includedon Fig. 10, but the corresponding curves for Taipeiare not shown on Figs. 17 and 18 since the 5-yearrecord is too short to warrant its being placed incomparison with the other curves when the differ-ences between curves are so small.

It seems evident from comparison of Figs. 16, 17,and 18 that the Tmax distribution alone is sufficientto distinguish between these different maximum dailytemperature records. This is to be expected becausethe periodic annual variation is strongly affectedby location, and has the greatest influence on theTmax: distribution. The differences between thedifferent locations on Fig. 16 are greater than the

differences on Figs. 17 and 18. For this particularquasi-periodic time series, then, the Cd(y) dis-tribution alone may be sufficient to establishidentification. The Tmax distribution is of in-cidental interest wth regard to whether one wouldenjoy living in the particular locality. Most of us,however, would also like to see the Tmin dis-tribution, and for a really complete investigation,the distribution of hourly temperatures.

Fig. 16 reflects the seasonal variation well, butsays little about how equable the day-by-day climateis. What are the chances that tomorrow's maximumwill differ from today's by 5, 10 or more degrees?Fig. 17 gives us this information. It shows that asfar as daily maximums go, Hayden has a moreequable climate than Storrs, which in turn is moreequable than Burlington. This is surprising, sinceit was originally thought that nearness to the oceanwould be the biggest factor. Apparently the highelevation of Hayden, the surrounding high moun-tains, and frequent clear skies produce moreuniformity than does proximity to the sea.

13

The A Tmax distributions shown in Fig. 17 donot differ nearly as much, in comparison, as theTmax distributions. This is probably due to thefact that they are mostly the product of the greatmacro-turbulence of the atmosphere's cyclonic storms,which may be remarkably regular in their patternof non-uniformity over much of the earth's surface.The comparison should be extended to includestations near the poles and nearer the equator.

The distributions of A ATmax, shown in Fig. 18,pertain to the changeability of the climate in adifferent way. How quickly do temperature trendsreverse themselves? The three stations shown rankin the same relative order as before. One cansympathize with the residents of Burlington, in themiddle of the bare plains, subject to easy attackby winds from every direction. Again, the dis-tributions are surprisingly similar for the differentlocations, probably for the same reason.

FUTURE APPLICATIONS

Now that the computations necessary for theanalysis of long time series by this method can bequickly made by the digital computer, the way isopen to investigate many types of phenomena, suchas the hourly temperatures previously mentioned,stream flows, etc. Perhaps the biggest difficultyis in obtaining reliable continuous records of suf-ficient length. The data must be on punched cardsor tape, ready for the machine, and must not haveany gaps.

GENERAL CONCLUSIONS

(1) Evidence is presented that the cumulativefrequency distributions of daily maximum temper-atures and of their first and second successivedifferences each tend to converge toward a fixedcurve as the length of period is increased.

14

2 PsoPercen+ of days

I

0FIGURE 18

15

(2) The three cumulative frequency distributionsobtained from daily maximum temperatures con-verged rapidly enough that curves from the differentlocations studied could be definitely distinguished,even though the length of period was as short asfive years.

(3) The distributions of the first and secondsuccessive differences provide insight into the patternof variability of the daily maximum temperature.

(4) The fact that the distributions of the firstand second successive differences of daily maximumtemperatures for the widely separated locationsstudied show more uniformity than the temperaturedistributions themselves suggests the possibility ofa world-wide similarity in the atmospheric macro-turbulence that causes the quasi-random quasi-periodic fluctuations. This should be tested byincluding a comparison of records from oceanicequatorial and near-polar stations.

ACKNOWLEDGEMENTS

Initially, this study was supported by the Depart-ment of Civil Engineering of the University ofConnecticut, with computer time provided by theUniversity's Computer Center. The Water ResourcesCommission of the Republic of China assisted byanalyzing the records from Taipei. A brief reportwas presented at the spring 1964 meeting of theAmerican Geophysical Union. One of the anony-mous reviewers of that report suggested that muchmore data were necessary to support the conclusionsreached, and raised questions which showed the needof a more thorough explanation. Study was recom-menced in 1965 as a project of the ConnecticutInstitute of Water Resources. The writer is indebtedto many individuals, particularly to WV. C. Kennard,J. J. Brumbach, Byron Janes, Joseph Breen, BijoyBhatacharya, W. B. Moeller, and Gerald Gromko,for assistance in various stages of the work.

BIBLIOGRAPHY

Bingham, Christopher, Probabilities of Weekly Aver-ages of the Daily Temperature Maximum, Minimum,and Range, Bulletin 659, Connecticut AgriculturalExperiment Station, New Haven, September, 1963.

Leiderman, P. Herbert, and David Shapiro, Ap-plication of a Time Series Statistic to Physiologyand Psychology, Science, 138, 141, 1962.

Lonsdale, Edward N., Development of a StatisticalAnalyzer for Random Waveforms, Dissertation,

University of Iowa, Department of ElectricalEngineering, 1952.

Posey, Chesley J., Discussion of Modern Conceptionsof the Mechanics of Fluid Turbulence, ProceedingsAmerican Society of Civil Engineers, 62, 626, 1936.

Posey, C. J., Measurement of Surface Roughness,Mechanical Engineering, 305, 1946.

Posey, C. J., Fluctuation Analysis of TurbulenceMeasurements, Proceedings of the second MidwesternConference on Fluid Mechanics, 49, The Ohio StateUniversity, 1952.

Ruhe, Robert V., Graphic Analysis of Drift Topo-graphies, American Journal of Science, 248, 435,443, 1950.

Ruhe, Robert V., Reclassification and Correlationof the Glacial Drifts of Northwestern Iowa andAdjacent Areas, Dissertation, University of Iowa,Department of Geology, 1950.

von Neumann, J., R. H. Kent, H. R. Bellinson,and B. I. Hart, The Mean Square Successive Dif-ference, Ann. Math. Stat. 12, 153 (1941).

APPENDIX

Computer Program

by

Bijoy K. Bhattacharya

The program described here is for the purposeof obtaining cumulative frequency distributions ofa large number of successive values of a quasi-periodicvariable, and their first and second differences. Asgiven, it is for the use in the IBM 7040 (FORTRANIV). It could be modified for IBM 1620 (FORTRANII). The latter, however, will take much morecomputing time.

In the present example the successive values aredaily maximum temperatures, covering a period of14 calendar years.

Let

N = Daily maximum temperature

L = Total number of days

N(I) = Maximum temperature of the "I"th day(I varies from 1 to L according to thesequence of the dates.)

The first difference of the daily maximum temper-ature is

16

NDEL (I) = N(I+1) - N(I).

In the above equation I varies from 1 to (L-l).In the program (L-l) has been replaced by anothervariable "LP".

The second difference of the daily maximumtemperature is

NNDEL (I) = N(I+2) - 2N (I+1) + N(I).In this equation I varies from 1 to (LP-1).

INPUT DATA FOR THE PROGRAM

First the machine will read the values of L, M1and M2 where, L = total number of data (or days).(M1 and M2 will be discussed later.)

Then the machine will read the data N as N(1),N(2), N(3) ... N(L), according tothe "FORMAT" specified. In this program theFORMAT used for the daily maximum temperatureis (2013).

A schematic viewshown in Figure 19.

of the deck of input cards is

particular value of the first and second differenceoccurs in the set of "NDEL" and "NNDEL", re-spectively.

3. Here the machine will compute the percentageof occurrence of:

(a) each and every value in the set of "N"

(b) each and every value of first difference inthe set of "NDEL" already computed

(c) each and every value of second difference inthe set of "NNDEL" already computed.

VALUES OF "Ml" AND "M2"

The values of M1 and M2 will depend upon thetype of problem. The values of M1 and M2 indicatethe range of numbers in which all the values of N,NDEL and NNDEL will lie.

For example if all the numbers lie between (-80)and (150) then

M1 = 150 + 81 = 231

M2 = 81

From the following two statements

DO 60 K=l, M1

J (K) = K -M2

With the above values of Ml and M2 we find that:

J(K) = 1-81 = -80

J(K) = 231-81 = 150

FIGURE 19 SCHEMATIC VIEW OF THE DECKOF INPUT CARDS

EXPLANATION OF THE STEPS OFCOMPUTATIONS

1. The machine will first compute the first difference(A) as "NDEL" and the second difference (AA) as"NNDEL".

2. The machine will count the number of times eachparticular value occurs in the given set of dataand it will also count the number of times each

when K=l

when K=231

That is the machine will compute the percentageof occurrence of N, NDEL and NNDEL in whichall the values of N, NDEL and NNDEL lie between-80 and 150.

Hence, the values of M and M2 are to beconsidered according to the type of problem.

The daily maximum temperatures of the followingtwo stations will serve as an example:

1. Burlington - Daily maximum temperatures from1932 to 1959

2. Hayden - Daily maximum temperatures from1934 to 1961

In the case of Burlington the minimum temper-ature throughout the period as mentioned was-8°F and maximum temperature was 106°F. Thevalues of M1 and M2 were taken as 251 and 81,respectively, to have a range of -80° to 170°F.The increased range is necessary to avoid the dangerof getting a total of less than 100%. It has to go

17

far beyond the Tmax range to take care of the

A Tmax range.

In the case of Hayden the minimum temperaturethroughout the period was -3°F and the maximumwas 100°F. The values of M1 and M2 were takenas 231 and 81, respectively, to have a range of-80 to 150.

So far, it has been found that the followingrelation avoids the danger of getting a total of lessthan 100% in any of the three distributions:

M1-M2 = maximum value of the set of data

Note that the "DIMENSION" statement in theprogram depends upon the type of the problem.

The dimensions of all subscripted variables areset according to the values of L, MI and M2. Thereare four subscripted variables and the dimensionsof them will be as follows:

N (value of L)

j (value of M1)

NDEL (value of L)

NNDEL (value of L)

For example, if L = 1500 and Ml = 201 then thedimension statement will be DIMENSION N(1500),

J(201), NDEL(1500), NNDEL(1500).

Hence, we have the following input data cardsafter the "END" card of the main program.

1st DATA CARD - containing the values of L, M1and M2 according to theFORMAT (316).

After the 1st card the rest of the cards will containthe data (e.g. Maximum temperatures, etc.) accordingto the FORMAT (2013) (or as required).

Note that this program is applicable only fordata which are all integral values.

The main program with a sample output for theyears 1948 - 1961 at Hayden is given in the followingpages.

18

DIMENSION N(5500),J(250),NDEL(5500),NNDEL(5500)READ (5,7) LMlM2

7 FORMAT (316)READ (5,2) (N(I),I=l,L)

2 FORMAT (2013)C CALCULATION OF FIRST AND SECOND DIFFERENCES

LP=L-1DO 8 I=1,LPNDEL (I)=N(I+1)-N(I)IF (I-LP) 9,8,8

9 NNDEL(I)= N(I+2)-2*N(I+1)+N(I)8 CONTINUE

C CALCULATIONS OF NUMBER OF OCCURRENCESKOUNT=OKNT=OKT=OWRITE (6,100)

100 FORMAT (lX,51HTEMPERATURE PERCENTAGE PWRITE (6,101)

101 FORMAT (4X,51HT(MAX) OF DAYS DURING OFWRITE (6,102)

102 FORMAT (12X,49HWHICH T(MAX) DELTA T(MAWRITE (6,103)

103 FORMAT (12X,40HWAS NOT XEDED NOT XEDINGWRITE (6,104.)

104 FORMAT (29X,25HGIVEN VALUE GIVEN VALUE)DO 60 K=1M1J(K)=K-M2DO 55 I=1,LIF (J(K)-N(I)) 55,54,55

54 KOUNT=KOUNT+155 CONTINUE

DO 5 I=1,LPIF (J(K)-NDEL(I)) 5,6,5

6 KT=KT+15 CONTINUE

LPP=LP-1DO 12 I=1,LPPIF (J(K)-NNDEL(I)) 12,13,12

13 KNT=KNT+112 CONTINUE

C CALCULATION OF PERCENTAGE OF OCCURRENCESA=KOUNTB=KTC=KNTD=LE=LPF=LPPX=(A/D)*100.Y=(B/E)*100*Z=(C/F)*100.WRITE (6,120) J(K) ,XYZ

120 FORMAT (1X,I OXF10 ,IO.55X,F10 XF1.55XF105)60 CONTINUE

STOPEND

ERCENTAGE PERCENTAGE)

DAYS WITH OF DAYS WITH)

X) DELTA DELTA T(MAX))

NOT XCDING)

FIGURE 20 MAIN PROGRAM

19

TEMPENATURE PERCEi: L- c6T(MAX) OF DAYS .3URING

WHICH T(MAX)WAS NOT XEDED

C .CCCO0 .OcCCc0.CCCCOO.CCCCC0.CCCCO

O.cCCCOo .CCCCOo.CCCOC.OCCCCC . c COO.occooo.occcO0.COQO

O.OCCO0C.OCCCOO. CCCO0.CCCCOC.CCCCOC.OC00OCO CCC0. cCCO

C. CC CO. CCCc0.01955C.019550.019550.058660.058660.09777C.136880.15643

5.1C3645.807596.804857.938999.07313

10.26594

99.94 13499.9413499.9804499.9804499.98044

1CO.OCCCOlCO.OCCCO1CO.OCCCO

PERCENTAGE PERCENTAGEOF DAYS WITH OF DAYS WITHDELTA T(MAX)

NOT XEDINGGIVEN VALUE

O.OCCOO

.0OCCco

o.00cccoO.OCCCO, .OCCCO

0.0ccco

. OcCOCO.OCOCOO . 0 C C 0C. OCCCO

0.0195CO6

0.0CC00

.OCCCO

O.OCCO0

0.01956

0.019560.01956

0.039120.039120.03912

21.3182125.9338930.8038336.8863743.6143251.3397259. 1238066.282C373.0295378.17328

99.94133S 9 . 9 6 8 38399.96088S99.98044

99.98044

1CO.00. oCLCO.OCOOO

1Co.CCCOO100 .oCOOo1CO.000001CO. 000C0O100. 00000

DELTA DELTA T(MAX)NOT XEDINGGIVEN VALUE

O.OCCCCO.CCCCCO.CCOCCO.CCOCoO.OCCCC

O.CCCCCC .CCoCOC.019560.019560.019560.019560.019560.01956

0.01956C.019560.C19560.01956C.019560.019560.039120.03912

29.5579C34.1353739.0649444.5422549.687C154.8317759.9374C64.7887368.9554072.84820

98.9632299.1197299.3153499.4522799.5109599.66745

100.CCCCC1CO.COOO100.CCCCO10C.OCCCO1CO.CCCCO1CO.OCCOO1CO.OCOCO1CO.OCOO0

Fig. 21 Print-out. Record from Hayden, Colorado(1948-1961, condensed to single page by omitting 166 lines)

20

-8C-79-78-77-76

-63-62-61-eC-59

-58-57-56

-43-42-41-4C-39-38-37-36

-5-4-3-2-101234

25262728

30

9596979899

1G01C21C2