551.501.45 : 551.521 : 551.573 (University of California Agricultural Experiment Station, Davis, California}

Fourier Smoothing of Radiation Observations in Two Parts, Daytime and Nighttime By

F. A. Brooks With 2 Figures Summary. The problem in non-subjective smoothing of 4-minute readings concerns the elimination of the high harmonics usually needed to match the flat portion of net radiation curves at night. The higher harmonies would introduce spurious mathematical oscillations in the daytime part of the smoothed record. The conventional "reflected" pattern computer technique, which yields only a sine series, has been modified to show the times of m a x i m a by sine and cosine series in odd harmonies. Satisfactory smoothing is obtained at the 5th harmonic of 2 r: periods covering twice the time sunrise to sunset or sunset to sunrise. The data used are of interest in comparing net radiation for 24 hours with the resulting rate of evapotranspiration from a mowed turf of rye grass weighed in a ]ysimeter 6 m in diameter in a nearly uniform field 6 hectares in area. Evaporation continued all night because of the north wind.

Zusammenfassung. Die Problematik bei der objektiven Gl~ttung yon Vier-Minuten-Ablesungen aus Strahlungsbilanzaufzeichnungen durch Fourierreihen liegt in der Elimination dcr hohen Obersehwingungen, die gew6hnlieh gebraucht werden, mn den flachen Toil der Strahtungsbilanzkurven bei Naeht wiederzugeben; diese wiirden im Tagteil der gegl~tteten K u r v e nieht reelle, nur mathematiseh bedingte Sehwingungen einftihreu. Die gebr~uehliehe ,,Spiegelungsmethode", welche nur Sinusreihen ergibt, wurde abge~ndert, urn auch die Zeiten der Maxima yon Sinus- und Kosinusreihen in ungeraden Obersehwingungen erhalten zu kSnuen. Eine zufriedenstellende Gl~ttung wird erreicht mit der f~nften Harmonisehen VOlt Perioden, welche zweimal die Zeit Dedicated to Dr. W. M/31~It~OFEa on the occasion of his 70th b i r t h d a y . Partial report on research work in micrometeorology sponsored by The Signal Corps Electronic Proving Ground under Contract :~:DA-36-039-SC80334.

F. A. B~oo~rs: FOurier Smoothing of Radiation Observations

279

yon Sonnenaufgang bis -untergang oder von Sonnenuntergang bis -aufgang enthalten. Die verwendeten Beobachtungsdaten wurden fiir einen Vergleieh der Strahlungsbilanz in 24 Stunden mit der resultierenden Evapotranspiration yon einem gemiihten Rasen in einem Lysimeter mit 6 m Durehmesser in einem fast einheitlichen Feld yon 6 ha Grgge angewendet. Die Evapotranspiration hielt wegen des Nordwindes die ganze Naeht an. R6sum6. Dans l'6tablissement objectif de moyennes glissantes, au moyen de s6ries de Fourier, des eourbes d'enregistrement du bilan de rayonnement, courbes d@ouill6es de 4 en 4 minutes, le problgme principal eonsiste en l'61imination des harmoniques sup6rieures; eelles-ci sont g6n6ralement utilis6es pour earaet6riser les parties nocturnes plates des eourbes du dit bilan. Ceei introduit des oscillations dans la partie diurne des moyennes glissantes qui Re correspondent pas k la r6alit6, mais sont dues au proe6d6 math6matique utilis6. La ((m6thode de r6flexion~) utilis6e habituellement et qui ne donne que des s6ries sinusoidales a 6td modifi6e afin d'obtenir, pour les harmoniques sup6rieures impaires, les temps des maximums des s@ries sinusoidales et cosinuso~dales. On arrive ~ une approximation satisfaisante en utilisant la einqui6me harmonique des p6riodes eontenant deux fois le temps s6parant le lever du eoucher du soleil ou le temps du eoucher au lever du soleil. Ces observations furent utilis6es pour eomparer le bilan de rayonnement en 24 heures ~ l'6vapotranspiration d ' u n gazon fauch6 contenu dans u n lysimgtre de 6 m de diam~tre, lysim@tre plae6 dana u n champ presque homoggne de 6 ha. L'6vapotranspiration s'est maintenue durant route la nuit en raison du vent du nord.

I. Introduction H e a t balance at a grass-covered soil surface involves so m a n y small oscillations a n d irregular time lags t h a t the i n s t a n t a n e o u s heat a n d m o i s t u r e balances are difficult to generalize over short-time periods. The five comp o n e n t s of energy have, however, very strong half-day periodicity due to solar energy coming as a n e a r l y perfect half sine wave on cloudless days. I t is most useful, therefore, to smooth out small irregularities b y a n automatic, n o n - s u b j e c t i v e method. True F o u r i e r h a r m o n i c analysis is the most effective m e t h o d now t h a t electronic computers can include the influence of each of h u n d r e d s of observations in the d i u r n a l cycles. There is need, however, to keep w i t h i n a few harmonics i n order to avoid waviness i n the " s m o o t h " curve. This can be done b y t r e a t i n g the 24-cycles in two parts: daytime and nighttime. F o u r types of d a t a acquisition systems are i n use for recording micrometeorological factors in a basic research project in energy a n d m o i s t u r e fluxes near the ground. The 50-ton weight of a lysimeter 6 meters i n diameter a n d 1 meter deep is p r i n t e d a u t o m a t i c a l l y every 4 m i n u t e s . The electrical responses of radiometers a n d heat-flux meters are recorded on electronic p o t e n t i o m e t e r strip charts either once or three times every four minutes. T e m p e r a t u r e s are p u n c h e d on I B M cards once or more times every two m i n u t e s , a n d cup a n e m o m e t e r s have been c o n t i n u o u s l y recorded on strip charts. The c o m m o n time i n t e r v a l of all these systems a t present is 4 m i n u t e s , n a m e l y 360 intervals i n 24 hours. This is a c o n v e n i e n t n u m b e r

280

F.A. B~ooKs:

for Fourier harmonic smoothing by the available IBM ~ 1620 electronic eompul~er. In fact the computer can handle up to 900 intervals in a time series so it is possible to use a two-minute round trip of the scanning switch. Because with atmospheric turbulence instantaneous observations are inherently irregular the repetitive observations must be treat;ed statistically to obtain meaningful results. This poses two questions: 1) Is it of most interest to interpret the scatter of instantaneously correct readings at short time intervals, which might lead to interpretation of eddy transfers in a system of air parcels, or 2) Is it only necessary to determine true mean magnitudes over a considerable time interval, a half-hour being typical, as is needed for aerodynamic interpretation by gradients ? Maybe both methods should be pursued. However, for the second procedure truer mean magnitudes would be determined if the response time constant of each transducer was made to suit the round-trip time of the scanning switch. This would provide practically a continuous record for each scan point but would transfer some of the statistical interpretation of instantaneous observations to the thermal lagging of transducers. Secondly, because the research concerns outdoor energy and mass balances, all the parameters vary with time mainly in response to a nearly half sine wave of solar energy in the daytime. The timing of the various fluxes of heat and moisture are not coincident, but all the factors are continuous functions and vary with distance from the surface of the ground. Thus the ultimate smoothing to obtain mean profiles is at least two-dimensional in distance and time. I n studies of advection horizontal gradients are also determined. These call for threedimensional smoothing for each factor preferably carried out before trying 4- and 5-part energy balances. For short test periods of a few hours, it may be sufficient to smooth the irregular succession of observations by finding a parabolic running mean. This procedure, however, does not systematically determine the time lags between various factors.

II. Fourier Harmonic Smoothina" The 24-hour diurnal curves of most micrometeorologieal factors can be well represented in 3 or 4 harmonics in cloudless weather and this provides the most powerful smoothing technique for all conditions because for each harmonic the method of determining amplitude and epoch angle provides a least-squares best fit using all the observations in the time series. The epoch angles are also useful in that by using the same procedure on all observed factors the major time lags between each are revealed closely, e.g. the timing of heat-flux in the soil relative to net radiation. The Fourier fitting could, of course, be carried out by machine to 30 or more harmonics, but in coming closer to observed magnitudes the oscillations in first derivative increase. Also ~he higher harmonics needed for fitting the daytime curve better may introduce spurious oscillations in the part of the 24-hour curve which is essentiMly smooth as between midnight and an hour before sunrise.

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Short-cut harmonic analyses such as using a reflected pattern to eliminate cosine terms may provide a nice fit by a single sine series, but it sacrifices physical reality by obliterating the epoch angles of the true Fourier harmonic analyses. The point of reflection is the zero point of the sine series and different solutions are obtained for different choice of

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reflection points if these points do not coincide with the true Fourier harmonic analysis of the degree chosen. To apply a nearly true Fourier smoothing without going to the high harmonics which would be needed to fit a 24-hour cycle, it is convenient to divide the day into two parts: daytime, sunrise to sunset, covering the predominant half wave of solar energy; and the nearly constant nighttime, sunset to sunrise9 Fig. 1 shows acceptable smoothing of very scattered observations of daytime net radiation using only 3 harmonies. The imaginary second half of the 2 ~ period is constructed by a translation and inversion of the daytime observations. The only forced distortion in the Arch.

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smoothed fit is that the slopes at sunset and sunrise have to be equ&l and opposite. These oeeur at times of least importance. When the slopes of the curve are of the same sign at sunrise and sunset &s for lysimeter weight Fig. 2, the ordinary technique is used of normalizing and putting the whole Fourier 2 ~ period between sunrise and sunset. Again there is a small error in forcing the slopes at sunrise and sunset to be identical, but the fit in the main part of the curve is very good. I n this ease it is essential to use only & few harmonies because with higher hat95180 [

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Fig. 2. Four-minute observations of lysimeter weight with 3-harmonic Fourier smoothing a~ld the resulting 1st derivative in units of energy rate compatible with other heat fluxes shown in Fig. 1 monies the successive derivative curves become very rough. A smooth first derivative is needed from the lysimeter weight/time curve in order to specify the rate of energy use in latent heat of vaporization. This curve in Fig. 2 is unusual in showing evaporation loss all night long caused by the continual north wind. The heat flow rate in the soil at, 1 cm depth averaged for 3 locations is remarkably smooth as recorded and needs no Fom'ier smoothing. This is plotted in Fig. 1 and is unusual in showing no heat penetration in midday because of the strong, cold wind. From the above three energy rate curves it is possible to calculate from the energy balance equation the convective heat flux. This is also plotted in Fig. 1, the two figures together thus giving all the usual four components of heat balance. The fifth component, metabolic heat, has been neglected in this presentation because it is small and has not been

Fourier Smoothing of !~adiation Observations

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determined yet. In generM it is to be noted that in spite of the irregularity in soIar radiation due to scattered clouds in the morning very positive relationships between the four modes of heat transfer have been established by using parttime Fourier harmonic smoothing. III. Extension of Half-cycle Fourier Curves Away from Surface The above application of Fourier analysis in half periods produces some discontinuities when, for example, the temperature waves are followed down into the soil. The main effect is separation in time because the temperature waves in the shorter daytime period penetrate more rapidly than the longer nocturnal period temperature waves. The gap is easily closed, however, with tangential interpolation. With this connection Mr. A. ]~OICAYE?r has found that two-part curve fits the exact 24-hour analysis almost perfectly. Acknowledgements This report has been possible only because of the cooperation of many staff members principally Dr. D. A. PoP~ and I~. B ~ o ~ of the Computer Center, Messrs. W. O. PRUIT~ and F. LAu~E~o~ of the Department of Irrigation Science and T.V. CaAWFO~D, W.B. GODDA~D, and H.B. SC~ULTz of the Department of Agricultural Engineering.

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