Instantaneous PAR estimated using long records of daily temperature and rainfall

Agricultural and Forest Meteorology 109 (2001) 47–59

Instantaneous PAR estimated using long records of daily temperature and rainfall Pekka Nöjd a,∗ , Pertti Hari b,1 a

b

The Finnish Forest Research Institute, Vantaa Research Center, Jokiniemenkuja 1, P.O. Box 18, FIN-01301 Vantaa, Finland Department of Forest Ecology, University of Helsinki, Latokartanonkaari 7, P.O. Box 27, FIN-00014 Helsingin yliopisto, Finland Received 17 November 2000; received in revised form 30 April 2001; accepted 11 May 2001

Abstract Photosynthetically active radiation (PAR) is a key variable needed for running process models that describe the functioning of agricultural crops or forest trees. Actual measured data that covers sufficiently long periods on PAR is usually not available for most research sites. Also, existing data usually consists of temporal averages, which — given the non-linear response of the photosynthetic rate to PAR — are not ideal for estimating photosynthetic production. The range of daily minimum and maximum temperatures is strongly correlated to the intensity of solar radiation. We present a method using a statistical relationship between these variables for generating a long time series of instantaneous PAR values for a given location. The method thus enables one to use photosynthetic production for analyzing growth on sites for which measured PAR is not available. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Photosynthesis; Photosynthetically active radiation; Process model; Temperature

1. Introduction The key process underlying plant growth is photosynthesis, which is strictly determined by the availability of photosynthetically active radiation (PAR). The rate of photosynthesis reacts to the intensity of radiation on a time-scale of seconds (Hari et al., 1988). Also, short-term variation of irradiance is often very strong because transmissivity of the atmosphere varies due to variation in cloud thickness. Because of this, data on short-term variation of PAR —

∗ Corresponding author. Tel.: +358-9-85-705-325; fax: +358-9-85-705-361. E-mail addresses: [email protected] (P. Nöjd), [email protected] (P. Hari). 1 Tel.: +358-9-19-158-135; fax: +358-9-19-158-100.

either measured or generated — is highly useful for approximating the rate of photosynthesis. Process models, which describe the functioning of crops on fields or forest trees, have been improving rapidly during the past few decades. As in any study where modeling is used, it is highly useful to test the performance of process models against various types of empirical data. Data on plant growth is obviously useful for the purpose, since growth is one of the main output variables of process models. A limited number of publications representing this approach are presently available (e.g. LeBlanc, 1993; Scuderi et al., 1993; Hari, 1999). Many existing growth data sets can be highly valuable for testing process models. Decades of agricultural research have produced long, valuable production statistics. In forestry, an obvious example of long time series on productivity are the so-called tree

0168-1923/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 1 9 2 3 ( 0 1 ) 0 0 2 5 8 - 1

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ring calendars, in which the growth rate of trees has been retrospectively measured by studying annual rings (e.g. Fritts, 1976; Nöjd et al., 1996). If such a long time series of growth data are available for testing, availability of input data may limit the testing of process-based growth models. Data on solar radiation is in a key role in modeling photosynthesis (De Wit et al., 1978), which in turn is the key process behind tree growth (e.g. Fritts, 1976; Nöjd and Hari, 2001). Systematic high-quality measurements on radiation are usually available since the late 1950s at best. The reason for this is that international standards for radiation measurements were established at that time. Prior to that, measurements are available from relatively few locations and the comparability of those measurements with subsequent ones may be dubious. Thus, the insufficiently long sets of radiation data frequently prevent one from testing process models against the existing long time series of plant production data. Another frequently occurring problem with radiation data is the fact that only temporal averages rather than primary measurements are available. For many purposes such averaged data is sufficient. The use of temporal averages for estimating photosynthesis is problematic, however (Hari et al., 1988; Palva et al., 1998). The instantaneous rate of photosynthesis is highly dependent on instantaneous irradiance, but the dependence follows a non-linear pattern. It is rather simple to show, that the potential bias caused by the use of mean PAR instead of instantaneous values in estimating photosynthetic production may be up to 30–40% over a prolonged period. Models describing photosynthesis such as the model of Farquhar et al. (1980) or the optimal stomatal control model (Hari et al., 1999) can be used for analyzing this type of bias. Today many weather stations record hourly averages for radiation, which considerably improves prospects for using the data for modeling photosynthesis. Berninger (1994) analyzed the use of hourly PAR for estimating photosynthesis; the resulting bias proved to be relatively small. However, hourly averages are not measured at every weather station even today. Also, if one aims to test models of photosynthetic production against historical plant production data, hourly radiation measurements are generally not available. And, as mentioned above, if one wishes to estimate photosynthetic production for the 1950s

or prior to that, most likely no comparable radiation measurements are available. Therefore, testing possibilities for models of photosynthetic production would be considerably improved, if one were able to generate reasonably accurate instantaneous PAR values. To achieve that, one would obviously need to utilize other types of meteorological records that cover longer periods. The longest existing meteorological records generally include daily minimum and maximum temperatures as well as precipitation. Daily temperature variation is due to the fact, that incoming solar radiation warms the atmosphere during the day, while during the night outgoing infrared radiation outweighs it, thus cooling the atmosphere. Both these fluxes are smaller during cloudy weather than under clear sky conditions. Therefore, the difference between daily maximum and minimum temperatures is likely to be smaller under cloudy weather than during clear episodes. The difference is exemplified by measurements of net radiation during a clear day (Fig. 1a) and a cloudy one (Fig. 1b) at a forest site in southern Finland. Empirical research has shown that the daily range of minimum and maximum temperatures is quite strongly correlated with daily solar global radiation, which reaches the earth surface (Bristow and Campbell, 1984; Meza and Varas, 2000). The aim of our article is to explore possibilities of predicting instantaneous PAR on the basis of daily minimum and maximum temperatures. The statistical relationship between the two is analyzed using data from southern Finland. We propose a method for using the difference of daily maximum and minimum temperatures for generating instantaneous PAR values. 2. Materials and methods 2.1. Climatic data Test data for studying the relationship between daily minimum and maximum temperatures and PAR obviously has to include measurements on all those quantities from the same site. We used meteorological data from two considerably differing locations, the Värriö research station in northern Finland (SMEAR I) and the Hyytiälä forestry field station (SMEAR II) in southern Finland (Table 1). SMEAR I is located on

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Fig. 1. Two examples of daily net radiation (the difference of incoming and outgoing global radiation; 0.3–40 ␮m) during a clear (a) 28 August 1997 and a cloudy (b) 25 August 1997 day. Measurements from the SMEAR II station (see chapter on climatic data).

the top of a steep hill at 400 m a.s.l. in a fairly infertile Scots pine (Pinus silvestris L.) forest. The density of the forest stand is low. The mean height of trees is 8 m. The measurement system for SMEAR I is described in more detail by Hari et al. (1994). SMEAR II is located on top of a smaller hill (181 m a.s.l.) in a Scots pine stand with a tree density of 2500 trees per hectare

and a mean height of 13 m. Haataja and Vesala (1997) describe the measurements system for SMEAR II. For the purposes of this study air temperature, measured at the height of 2.2 m above ground level, had been measured at SMEAR I, whereas air temperature at 4.2 m above ground level was available from SMEAR II. Platinum resistance thermometers

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Table 1 The locations of measurement stations, measured variables, measurement intervals and periods as well as data sets sizes for the climatic data used Location

Measurements

Measurement interval

Measurement periods

Data set size (total number of observations)

SMEAR I

67◦ 46 N 29◦ 35 E 400 m a.s.l.

Air temperature (2.2 m) PAR

Every 5 min

1 June–11 August 1996

19584

SMEAR II

61◦ 51 N 24◦ 17 E 181 m a.s.l.

Air temperature (4.2 m) PAR Net radiation

Every 3 min

18 June–31 August 1997 1 June–29 August 1998 30 June–31 August 1999

108480

(PT-100, T. Pohja, Juupajoki, Finland) were used at both stations. The sensors are protected against solar radiation and ventilated by fans. At both locations, PAR was measured using a quantum sensor (LI-190 SB, LI-COR Inc., Lincoln, NE, USA), that was placed above tree canopy layer. The test data covers the summer months, which are the vital period for plant production in Finland where the growing season is very short. Even though the day length changes considerably during that period, the daily temperature range is rather similar on clear days (Heino, 1973). The net radiation presented in Fig. 1 is the difference between incoming and outgoing radiation (wavelengths 0.3–40 ␮m, measured in units of W m−2 ). Net radiometer (Reemann MB 1, Tartu observatoorium, Toravere, Estonia) was used for the purpose.

maximum potential PAR for a given moment and location in clear sky conditions. The ratio of observed PAR and a theoretical PAR under maximum transmittance can be used for describing transmissivity. In our paper both observed and the theoretical maximum PAR are integrated over a period of 1 day. Following the approach of Gates (1980), the daily transmittance j coefficient (Tt ) for PAR is calculated as the ratio of observed and theoretical maximum PAR. The ratio is related to the difference of daily minimum and maximum temperatures (T) using ordinary least squares regression. The solar elevation angle (β) can be used for calculating a fairly accurate estimate of PAR under clear sky conditions. It can be calculated for a given moment and location using the following formula (Gates, 1980)

2.2. PAR under optimum transmittance conditions

sin(β) = cos(l) cos(h) cos(D) + sin(l) sin(D)

In order to study the relationship between daily minimum and maximum temperatures and instantaneous PAR in a meaningful way, one needs to be able to compare observed PAR with the potential maximum PAR for that particular instant. The latter is equivalent to PAR, that would reach the earth surface at that instant under optimum transmittance (clear sky conditions). That quantity is of course highly dependent not only on the time of day, but also on the time of year. This is especially so at the high northern latitudes, where day length varies greatly. PAR at a given moment is dependent mainly on two factors: position of the sun and transmissivity of the atmosphere. The position of the sun follows a highly regular pattern, that can be used for estimating

where β is the solar elevation angle (rad), l the latitude (rad), h the true solar time (given in angular distance from the meridian of the observer) and D the declination angle (rad), determined as

(1)





 23.5 2π(j − 172) cos ; 360 365 j = j th day of the year D = 2π

The equation for the declination angle approximates tables of solar declination by List (1963). We tested the relationship between the solar elevation angle and PAR at the locations of SMEAR I and SMEAR II. At these high northern latitudes, the following heuristic regression equation gives numerical values that are

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rather close to measured PAR on cloudless summer days, regardless of the time of day  1.9β(t) − 0.1, if β > 0.0526 j (2) Qp,p (t) = 0, if β < 0.0526 j

where Qp,p (t) is the potentially available PAR for time t during day j (mmol m−2 s−1 ), β(t) the solar elevation angle for time t (rad). An example of how the theoretical maximum potential PAR matches observed PAR during a period of 5 days is given in Fig. 2. A fairly sunny period was chosen for the purpose. The first day, for which net global radiation was shown in Fig. 1b, was cloudy. Also the morning of 26 August 1997 was obviously cloudy, and there were also some clouds around noon the next day. Apart from these periods, the observed values are rather close to the predicted ones. About 1.30 PM every day the observed values are invariably well below the predicted ones. This is simply because the PAR sensor is partly shadowed by the measurement mast at that time. Notice also that the observed values sometimes exceed the theoretical values, which describe the PAR in clear sky conditions (e.g. afternoon 30 August 1997). This can be explained by the fact that even though there are no clouds directly between the sun and the sensor, some clouds may exist. In such condi-

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tions the radiation reaching the sensor consists partly of radiation scattered by the clouds, and the measured intensity may exceed the PAR of clear sky conditions. Naturally results for such a short period do not prove, that the model performs equally well throughout the year. However, it is quite difficult to produce statistical results on overall performance of the model, as totally cloudless days occur quite rarely, and their occurrence is somewhat difficult to define objectively. On the basis of our test data from SMEAR I and SMEAR II, the method presented above appears to be reasonably accurate for predicting intensity of direct solar PAR during totally cloudless periods. 2.3. The daily transmission coefficient Since the PAR under clear sky conditions can be estimated for a given instant, it can be integrated over a day  tj +1 j j Pp = Qp,p (t)dt (3) tj

j

where Pp is the daily totals of potentially available j PAR for day j (mmol m−2 ), Qp,p (t) the potential instantaneous PAR for time t during day j (mmol m−2 s−1 ) and tj the beginning instant of day j.

Fig. 2. Potential PAR under clear sky conditions (solid line) and observed (dotted line) PAR for the period 25–30 August 1997. The measured values are from the meteorological station at Hyytiälä (SMEAR II). The potential PAR values were calculated using formula (2). Time is given as true solar time.

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The daily transmission coefficient is calculated by j comparing actual measured PAR to Pp . In order to do this, the measured PAR also needs to be integrated over the day j Pa

 =

tj +1 tj

j

Qp,a (t)dt

(4)

j

where Pa is the daily totals of actual, measured PAR j for day j (mmol m−2 ), Qp,a (t) the actual measured instantaneous PAR for time t during day j (mmol m−2 s−1 ). Following the approach of Gates (1980), we j calculate the daily transmission coefficient Tt , which is the ratio of the daily measured PAR and the calculated potential daily maximum PAR. It is used for describing the proportion of daily PAR that has not been intercepted by clouds j

Tt =

j

Pa

j

Pp

j

(5)

where Tt is the daily mean transmission coefficient for day j.

2.4. The statistical relationship between temperature, rainfall and daily mean transmissivity In order to test the possibilities of predicting the daily mean transmissivity by temperature variation, j the daily transmission coefficient (Tt ) from Eq. (5) was related to the difference of daily maximum and minimum temperatures (T), measured at the same site. Ordinary least squares regression was used. Bristow and Campbell (1984) suggest modifications to this basic procedure. Whenever daily maximum temperature is measured, also daily rainfall is generally available. As suggested by Bristow and Campbell (1984), we modified the daily temperature range T by subtracting 25% of it for rainy days. The procedure accounts for the reduced radiation loads under rainy conditions. For SMEAR II the use of this modification produced a stronger statistical relationship between the meteorological variables and the daily transmission j coefficient (Tt ) than the use of unmodified daily temperature range as regressor. For SMEAR I the difference between the two methods was small. For SMEAR I (northern Finland, 67◦ 46 ) the modified difference of daily maximum and minimum

Fig. 3. Daily transmission coefficient for PAR vs. the difference between daily maximum and minimum temperatures (T) using data from SMEAR I in northern Finland (latitude 67◦ 46 N). The linear regression line is shown with the coefficient of determination, r2 and number of days, N.

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Fig. 4. Daily transmission coefficient for PAR vs. the difference between daily maximum and minimum temperatures (T) using data from SMEAR II in southern Finland (latitude 61◦ 51 N). The linear regression line is shown with the coefficient of determination, r2 and number of days, N.

temperatures explains 63% of the variation of the daily transmission coefficient (Fig. 3). The respective result using the unmodified T was very similar (64%). The dependence appears to be linear; the following linear model below is quite adequate for describing it (cf. Bristow and Campbell, 1984). j

Tt = β0 + β1 T

(6)

j

where Tt is the daily transmission coefficient (ratio of measured and theoretical maximum PAR) for day j, β0 = 0.140, β1 = 0.071◦ C−1 , T the difference of daily maximum and minimum temperature (25% reduction for rainy days), ◦ C. For our data from southern Finland (SMEAR II, latitude 61◦ 51 ), the modified T produced a stronger statistical relationship with the daily transmission coefficient (r 2 = 0.67) than the unmodified one (r 2 = 0.61). The dependence is described by the following regression equation j Tt

= β0 + β1 T j

(7)

where Tt is the observed daily transmission coefficient (for PAR) for day j, β0 = 0.171, β1 = 0.058◦ C−1 .

The scatter plot describing the data and the regression model (Fig. 4) looks rather similar as that for SMEAR I. Note, however, that the variation of daily temperature range is considerably larger. At SMEAR II T varied from 2 to almost 17◦ C, while at SMEAR I only one observation exceeded 12◦ C. The residuals of the two regressions did not show any clear pattern. A slightly different method for determining the daily temperature range, used by Bristow and Campbell (1984), was also tested. Using it, the daily temperature range for day j was defined as the difference of maximum temperature for day j and the mean of minimum temperature for days j and j + 1. Using that method as a predicting variable resulted in slightly j lower r2 -values in predicting Tt . 2.5. The method for generating instantaneous PAR Since a clear statistical relationship exists between the difference of daily maximum and minimum temperatures and transmissivity, it becomes reasonable to use this knowledge for simulating PAR. Several reasonable approaches are available. One needs to calculate the theoretical maximum PAR of direct

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solar radiation for each instant and use the estimated j transmissivity (Tt ) for approximating how large proportion of the theoretical maximum PAR has been cut off by clouds. The simplest way would be to multiply the theoretical maximum PAR for each instant with j the daily transmission coefficient (Tt ) from Eq. (5). This would hardly be satisfactory, since the estimates would not include a realistic amount of variation. If such estimates were used for running process models, the result would be an overestimation of photosynthetic production (cf. Berninger, 1994). A better approach would be to introduce the typically strong variation of instantaneous PAR into  j   Qp,p (t) Qr (t), j Qp,g (t) = Qrp,p (t) p,a   j j Qp,p (t)Tt , the simulation. This can be done in a number of ways. Berninger (1994) presents a method, which generates stochastic values for instantaneous irradiance. When the method was used for estimating photosynthetic production, the estimates were rather accurate. The method, however, assumes that the daily total radiation that reaches earth’s surface is known. If one only has historical weather data, and needs to generate radiation values on the basis of it, this criteria is not fulfilled. We propose a simple approach for generating instantaneous PAR values. It can be applied even in situations where the total daily radiation has not been measured. Assuming that one has some empirical data on instantaneous PAR as well as longer time series on daily minimum and maximum temperatures, that very data can be used for simulating instantaneous PAR. The procedure would go as follows: 1. Determine the statistical dependence of the daily j transmission coefficient (Tt ) on the range of daily temperature range (T) using the data set that includes measurements on both temperature and PAR. j 2. Use the statistical relationship for estimating Tt for each day j for which PAR will be generated. 3. For each day j, select a reference day r, for which measured data on PAR exists. The reference day j with a daily transmission coefficient (Tt ) as close

as possible to that one estimated for day j should be selected. 4. The instantaneous PAR values of the reference day r are used for generating instantaneous PAR values for day j. However, they need to be adjusted: each instantaneous PAR value for day r is multiplied with the ratio of instantaneous clear sky PAR values for day j and the reference day r. One cannot use the measured PAR values of the reference day as such, because day length will be different, as well as the solar angle at any given time (unless exactly the same calendar day happens to be chosen). A simple way to adjust them is as follows: if Qrp,p (t) > 0.2 mmol m−2 s−1

(8)

if Qrp,p (t) < 0.2 mmol m−2 s−1 j

where Qp,g (t) is the generated PAR for day j, time t; j Qp,p (t) the instantaneous PAR in clear sky conditions for day j, time t; Qrp,p the instantaneous PAR in clear sky conditions for the reference day r, time t; Qrp,a (t) the actual measured instantaneous PAR for the reference day r, time t. For those periods shortly after sunrise and before sunset, when the solar elevation angle is small, a rather unsophisticated method is suggested. The procedure accounts for problems caused by differences in day length between the day for which PAR is to be generated (j) and the reference day (r). Using such average figure is hardly a major handicap, since the intensity of PAR will in any case be very low during those periods, and, consequently, also the rate of photosynthesis will be low. If one is generating PAR for a long period, one should create a additional procedure that ensures that the time difference between day j and the reference day r is not overly long, i.e. they do not represent a totally different time of year. This is especially true for regions like Scandinavia, where day length varies greatly. Using this approach, it is clear that the generated instantaneous radiation values may strongly deviate from actual ones. If they are used for calculating instantaneous photosynthetic rate, the estimate will often be severely biased. Thus, the method is applicable for

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Fig. 5. Measured (solid line) and simulated (dotted line) PAR radiation for 18 July 1998 (time is given as true solar time). The theoretical PAR under clear sky conditions, calculated using Eq. (6), is presented by a dashed line.

estimating photosynthetic production over sufficiently long periods. However, when process models are tested against historical plant production statistics, the estimated photosynthetic rate is definitely integrated over long periods. In such cases, it is more important that the estimated variation of PAR follows a pattern that is likely to occur on that particular site, both in terms of expected value and the variation around it. Our method fulfills these requirements, if the generation is based on data from the same region. 2.6. The performance of the model Using data from SMEAR II, we simulated instantaneous PAR values for the 3 days: 18 July 1998 (sunny), 24 July 1998 (half-cloudy) and 20 July 1998 (cloudy). We picked those 3 days of the summer of 1999, which j happened to have Tt values closest to them. They were (respectively): 26 August 1999, 23 July 1999 and 12 August 1999. Instantaneous PAR values for those days were calculated using Eq. (8). The following graphs describe the results that were achieved. The generated PAR values for the fairly sunny day 18 July 1998 — based on measured values of the reference day 26 August 1999 — happen to follow

a rather similar pattern as the actual measured values: almost uninterrupted radiation in the morning, followed by somewhat cloudy conditions in the afternoon (Fig. 5). The cloudy episode could equally well have taken place in the morning of the reference day (unless sunny mornings and cloudy afternoons happen to be typical for the region). For the cloudy day (20 July 1998) both the measured and simulated values — based on actual data from 12 July 1999 — are naturally well below the theoretical PAR under clear sky conditions (Fig. 6). The variation of actual PAR appears to be larger than the variation of the generated PAR. The difference evidently reflects differences in the type of cloud cover during these 2 days. During a day of intermittent cloudiness (such as in Fig. 7), the intensity of radiation may change very rapidly. This means that the generated instantaneous irradiance may greatly differ from the true one. This is exemplified by the rather large differences of generated and measured PAR in the afternoon and evening (after 16.00 h). However, the daily mean PAR and the magnitude of variation are rather similar. On the basis of these results, the model appears to have the potential for generating instantaneous PAR values. There are, however, limitations that need to

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Fig. 6. Measured (solid line) and generated (dotted line) PAR radiation for 20 July 1998 (time given as true solar time). The theoretical PAR under clear sky conditions, calculated using Eq. (6), is presented by a dashed line.

Fig. 7. Measured (solid line) and generated (dotted line) PAR radiation for 24 July 1998 (time given as true solar time). The theoretical PAR under clear sky conditions, calculated using Eq. (6), is presented by a dashed line.

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be considered when applying the outlined method and calibrating it to different locations and sites. It is known that the dependence between radiation and daily temperature range (T) is specific to a region or even site-specific (Meza and Varas, 2000). Our two test sites produced clearly different results. At SMEAR I the variation of daily temperature range (T) was rather small: almost invariably less than 12◦ C. This is partly due to local topography: the measurement station is on top of a steep hill (400 m a.s.l.) which is surrounded by valleys at 200–300 m a.s.l. The measurement station SMEAR II is also on top of a small hill, but the difference with surrounding areas is very small compared to that at SMEAR I. As a result, T varies more: between 2 and 17◦ C. It is, therefore, understandable that the slope parameter of the regression line was smaller for SMEAR II than for SMEAR I. Another reason for the small differences of daily minimum and maximum temperature at SMEAR I is the fact, that summer nights are very short in the north. In fact, the sun never sets for a couple of weeks around midsummer — the station is 200 km north of the Polar Circle. It is also of interest that in our data the relationship j between T and Tt for PAR appears to be linear. In the data of Bristow and Campbell (1984) the statistical relationship between daily temperature range j and daily transmission coefficient (Tt ) of extraterrestrial insolation followed a clearly non-linear pattern. They used a non-linear modified Weibull equation for describing the dependence between the two. The data presented by Bristow and Campbell (1984) covers a wider daily temperature range than the data measured at the two stations in Finland. For summer months in Finland T values between roughly 2–17◦ C were measured, while the data of Bristow and Campbell (1984) includes values up to 22◦ C. In their data j the dependence between T and Tt also appears to be linear for the smaller values of T. In regions where the daily temperature range can be large, it is likely that the dependence between T and the daily transj mission coefficient for PAR (Tt ) follows a non-linear pattern — such as the one observed by Bristow and Campbell (1984). In such cases the simple regression model used by us obviously has to be substituted by a non-linear one. In conditions of Finland, the models using daily temperature range as predictor variable predicted 63

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and 67% of the variation of daily PAR. This is considerably less than the high proportion of explained variance (70–90%), achieved in predicting daily measured irradiance by Bristow and Campbell (1984). Even though the statistical relationship between daily temperature range and transmissivity is specific to a region, it is relatively easy to determine this type of statistical relationship for any given site, if even a limited number of observations on both radiation and temperature are available. Even if one does not have such data (daily maximum and minimum temperatures and PAR, measured at short intervals for some months), those measurements can be produced with simple equipment. One only needs a thermometer and a PAR sensor, both connected to a data logger, which registers the measurements at short intervals. The statistical link between daily temperature range and PAR can be used in a number of ways for simulating PAR. Several mathematical and statistical techniques will probably produce a reasonable result. It is not easy to judge, which of such techniques would be the best choice, and the answer might even be quite different for climatically differing regions. The variation pattern of PAR is largely determined by the type of cloud cover that typically exists in the area. Those can be rather different in different climatic zones. The method we proposed for the simulation is simple, yet the simulation results are likely to follow a pattern that is typical to the region from where the data originates. On a bright day the actual irradiance values will follow a pattern determined mainly by the position of the sun and instantaneous deviations from that pattern will be very small. On such days the simulated PAR should not differ greatly from the actual one. On cloudy days the variation of PAR can follow much more varying patterns. This also means that the simulated instantaneous PAR values can deviate greatly from the true ones. For modeling daily photosynthesis it is not essential that the simulated PAR values are close to the actual PAR ones for each instant. It is more important, that the pattern of variation of the simulated PAR is realistic. It is also worth noting, that near sunrise and sunset our method will produce estimates that are slightly biased. Specifically, during those moments the variation of generated PAR is unnaturally small. Also, the method based on calculating the solar angle will result in no irradiance when the solar angle is below

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or equal to zero. Obviously there will be some reflected light shortly before sunrise and after sunset. It would naturally be possible to create simulation methods that would describe this phenomenon more accurately than the one we proposed. We consider it as an unnecessary complication for the purpose of modeling daily photosynthesis, since photosynthetic production during these periods is negligible. The method used for calculating potential PAR under clear sky conditions is based on well-known properties about the irradiance of direct solar radiation. It appears to be reasonably accurate. There are some sources of inaccuracy, however. For example, the procedure does not take into account the fact, that the distance between the sun and the earth varies annually within ±1.7% (Ross, 1981). However, variation in the irradiance of direct solar radiation, caused by this factor, is fairly small. For the sake of simplicity it was not taken into account.

3. Conclusions Process-based models describing photosynthetic production use data on instantaneous PAR as a vital input variable. It would be most useful to test these models against plant production records, which cover long periods. However, long records of measured instantaneous PAR are rarely available. It is, therefore, common practice to use generated data on PAR for running process models and testing their performance against data on plant growth. As daily maximum and minimum temperatures are reasonably strongly correlated with transmissivity, it is evident that they can be useful for generating instantaneous PAR values. Also, early meteorological measurements usually include very few variables, but daily minimum and maximum temperatures are generally among them. In the article we presented a simple method, which uses commonly available meteorological variables (daily minimum and maximum temperatures, rainfall) for generating instantaneous PAR. In addition, one has to have measurements of instantaneous PAR from a comparable site covering a period of some months at minimum. The latter requirement is not a severe handicap, because such measurements are relatively easy to produce at a reasonable cost today.

The method includes two phases: daily temperature range and rainfall are used for producing an estimate of the daily mean transmissivity. Being able to estimate daily mean transmissivity reasonably well enables one to predict also the expected value of daily PAR reasonably well. In the second phase, measured variation of instantaneous PAR within a day is used to convert the daily value of PAR into instantaneous values. Of course, these instantaneous values will not be the true values, but they will exhibit a realistic pattern of variation within the day. Test results for the performance of the model were presented. These suggest that the method is able to produce estimates of PAR with a realistic pattern of variation. Valuable plant production statistics covering extensive periods of time are available for both agricultural and forest research. Process models are an increasingly popular approach for modeling plant production. Therefore, there is an obvious need for long time series’ of PAR. If the presented method is properly calibrated for local conditions, it appears to have potential for improving possibilities for testing process models against long period data on plant productivity in many parts of the world.

Acknowledgements The comments and suggestions from Academecian Juhan Ross are gratefully acknowledged. We also thank Dr. J.B. Stewart, the editor of Agricultural and Forest Meteorology, for numerous helpful comments. Our thanks are also due to the other, anonymous reviewer. The research has been supported by the European Union (LTEEF II: ENV4–CT97–0577). References Berninger, F., 1994. Simulated irradiance and temperature estimates as a possible source of bias in the simulation of photosynthesis. Agric. Forest Meteorol. 71, 19–32. Bristow, K., Campbell, G., 1984. On the relationship between incoming solar radiation and daily maximum and minimum temperature. Agric. Forest Meteorol. 31, 159–166. De Wit, C.T., Goudriaan, J., van Laar, H.H., 1978. The simulation of photosynthetic systems. In: Prediction and Measurement of Photosynthetic Productivity. Proceedings of the IBP/PP Technical Meeting, Trebon, 14–21 September 1969, Pudoc, Wageningen, pp. 47–70.

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