NRM – Module 8


Lecture notes



Temporal dynamics in weather and its influence on CO2 assimilation in crops















October 2004, V. Venus




Introduction. 3

Temperature. 3

Radiation. 4

CO2 assimilation in crops. 6


Figure 1. Observed and modeled average daily progression of air temperature. 4

Figure 2, Modeled daily progress of incident global radiation (daily total of 16 MJ m-2 d-1). 5

Figure 3. Generic AMAX-to-temperature response curves (Versteeg and van Keulen, 1986). 7



Daily totals or daily averages are convenient to work with and provide a first but rough impression of the weather conditions under which a given crop develops. For some plant processes this daily time interval not a bad match. For example, the development rate of a crop can be estimated quite well by making use of the average air temperature. After all, it is not possible to measure accurately the stages of development from day to day, which means that the effect of temperature must be integrated over at least a week. It is then usual to make use of the concept of 'temperature sum' (TSUM) or 'degree days'. This value is calculated as the integral of the value with which the (average) temperature exceeds a certain lower threshold value (e.g. 5 °C), below which no development occurs at all. This procedure implicitly assumes a linear response above the lower threshold.


However, for a number of reasons these summation methods may present a distorted image of the actual situation. Some of the most important reasons are:

  • The response is non-linear. One example is the photosynthesis-light response that becomes saturated at high light intensities, and another example is the temperature response of the development rate that may even deflect back at high temperatures.
  • Some factors work together to enhance each other's effects. An example is the rate of evaporation of open water, which increases with wind speed, but also with radiation, and with the dryness of the atmosphere. Because of this interaction, it is not permitted to simply enter the average values in the equation. Yet, in the calculation of the Penman evaporation average daily values are used (Penman, 1948). Its success can be attributed to the fact that diurnal courses are repetitive and that the values of empirical coefficients were found by calibration. The response has a time delay within the daily rhythm. An example is the plant's water content, which exhibits a minimum in the late afternoon. If the minimum should be very pronounced, the photosynthesis-light response will be higher in the morning than in the afternoon.


Because of these kinds of effects, it is often necessary to take account of the daily progress of the weather conditions when modeling crop growth.




At a standard height (1.5 m), the air temperature exhibits a characteristic daily cycle with a minimum at sunrise and a maximum at 13.30 h (solar time, Figure 3.3). In dry climates this maximum tends to occur a little later. Of course, deviations from this average pattern can occur as a result of, for example, the passage of a cold front. However, on average the daily cycle can be described reasonably well when the minimum and maximum temperatures are known. The accuracy with which this should be done depends on the purpose.


If the mutual position of the radiation and temperature wave is not important (little interaction between temperature and radiation), a simple linear progression between a maximum and a minimum, e.g. at 13.30 h in the afternoon and 6.00 h in the morning, would be sufficient. A further improvement is to round off the points by using a simple sine curve.


If we require a reasonable representation of both the phase and the form, we will have to use a sinusoidal progression during the daytime and a decreasing exponential curve at night (Parton and Logan, 1981; Geurts and Engelen, 1983). The transition from one curve to the next takes place at sunrise (minimum temperature) and sunset. The section in the period of light (daytime) follows practically one sine wave in time according to:


Ta = Tmin + (Tmax - Tmin) sin( π (th  - 12 + d/2) / (d + 2p) ) [°C] (3.9)



Ta   is the air temperature (12 - d(2 < th < 12 + d12) (°C),

Tmin   the daily minimum temperature (°C),

Tmax the daily maximum temperature (°C),

d    the daylength (Eqn 3.6) in hours,

th      the time in hours (solar time), and

p     time duration between solar noon and maximum temperature (h).


At sunrise (th  = 12 - d/2), the air temperature equals Tmin. The maximum temperature is reached when th  equals 12 +p h solar time. The default value for p is 1.5 h. The sinusoidal curve is followed until sunset. Then a transition takes place to an exponential decrease, proceeding to the minimum temperature of the next day. To plot this curve correctly, we first need the starting point, the temperature at sunset (Tsset). This can be found by substituting the correct time, 12 + d/2, in Eqn 3.9. The equation for the nocturnal temperature progression is now:


Ta = Tmin- Tssetexp(-n /TC) + (Tsset - Tmin) • exp(- (th- Tsset) /TC)        (3.10)

1 - exp(-n /TC)


In this equation, n stands for the duration of the night, i.e. 24 - d. The value of the nocturnal time coefficient (TC) is approximately 4 hours.



Figure 1. Observed and modeled average daily progression of air temperature.

Legend: observed air temperature for the months December, March, June and September in De Bilt, The Netherlands is indicated by the solid lines. The broken line is the result of Eqns. 3.9 and 3.10 (Kropff and Laar, 1993).


Listing 3.3, Appendix I shows the procedure for reconstructing the diurnal course of ambient air temperature based on daily totals.



Light energy directly/indirectly influences many secondary plant processes. A list of these processes includes: plant growth, seedling regeneration, vertical structure and crown shape of forest stands, leaf morphology, and uptake and emission of trace gases that participate in biogeochemical cycling and atmospheric chemistry.


The amount of solar energy at the outer extremity of the atmosphere varies with the latitude of the site and the time of year. The daily progress of the incident global radiation has a sinusoidal shape which varies with the seasons (see Figure_4). The global radiation (S) from the sun (roughly between 350 and 2000 nm wavelength) incident on the earth's surface is the product of the incident radiation outside the atmosphere (So) and the atmospheric transmissivity (Ta) which is obviously very dependent upon the degree of cloudiness (see Figure_3).


Figure 2, Observed daily progress of incident diffuse and global radiation (daily total of 9.76 MJ m-2 d-1).


In application of radiation modeling to satellite data, incident values need to be interpreted to arrive at daily integrated values and visa versa in order to make rigid model input requirements commensurate with scant satellite observations. Listing 3.4, Appendix II shows the procedure for reconstructing incident global radiation based on a daily total. Based on this model the daily progress of incident global radiation has been reconstructed for Julian Day (JD) 110, 1988 as depicted in Figure 1.


Figure 3, Modeled daily progress of incident global radiation (daily total of 16 MJ m-2 d-1).



Table 1, Distribution of Solar energy by Waveband (Monteith, Unsworth et al.).


Waveband (Nm)

Energy (%)



300-400, ultra-violet


400-700, visible/PAR


700-1500, near infrared


1500 to infinity



Approximately half the total global radiation (Rg) is part of the photosynthetic photon flux density (PPFD) ranging from 0.38 to 0.71 µm, better known as photosynthetically active radiation (PAR). It provides the energy for photosynthesis. Light in this visible waveband contains energy across the primary colors of purple, blue, green, yellow, orange, red. In contrast to the indicative values in the above-listed table, the amount of solar radiation in this band is not constant and varies between 21 and 46 per cent.


The spectral quality of light incident to the ground is affected by selected absorption by water vapor gases and aerosols. Water vapor is a strong absorber of light with wavelengths of 100, 1400, 1600 and 1900 nm. Ozone absorbs ultraviolet light (λ < 300 nm), CO2 absorbs strongly in the 2750 and 4250 nm bands and oxygen absorbs 690 and 760 nm sunlight (Bonhomme, 1993). The spectral quality of sunlight is mainly determined by the direct/diffuse fraction of sunlight and the solar elevation angle. As a quick rule of thumb, the PAR:Rg, for total radiation, is rather conservative, being on the order of 0.46 to 0.50 (these computations assume that PAR conversion factor between quanta and energy is 4.6 umol quanta per J) In Figure 4, the zonal distribution of a five year mean  PAR values, as well as the corresponding conversion factors (CF), are illustrated.


Figure 4, Zonal distribution of five year averaged PAR (1983-1988), and corresponding conversion factors (Pinker and Laszlo, 1992).



Ross (1975) provides an algorithm for the conversion factor between PAR and Rg as a function of the direct and diffuse components (this algorithm is for daily integrated fluxes of solar energy):



On a clear day with 10% diffuse radiation, fpar:rg is 0.438. On a cloudy day with 90% diffuse radiation, fpar:rg is 0.582.


On an hourly basis, the PAR:Rg ratio, for direct radiation, ranges from 0.2 to 0.43 as the solar elevation (Y) goes from 0 to 40 degrees. It then remains relatively constant with higher elevation angles. For diffuse radiation, this ratio ranges from about 0.6 to 0.75. Analytically, these functions take the following 3rd order polynomial form:


ƒ PARdirect:Rdirect      = 0.1714 Y3 - 0.5917 Y2 + 0.6535 Y + 0.2002

ƒ PARdiffuse:Rdiffuse:   = 0.1118 Y3 - 0.3859 Y2 + 0.4262 Y + 0.6001


where Y is the solar elevation angle in radians.

CO2 assimilation in crops


The photosynthetic activity of a leaf is a function of the light intensity and the maximum rate of assimilation (AMAX).


At canopy level, the daily gross CO2 assimilation (DTGA, kg CO2 ha-1 d-1) can be approximated by integrating the above-mentioned instantaneous leaf processes at different heights in the canopy at different times of the day. DTGA is


Light intensity


Light use efficiency at low light intensity (EFF) decreases by only 1% for every degree of temperature increase in C3 plants, and even less in C4 plants. For practical purposes EFF is assumed constant with a value of some 0.5 kg ha-1 h-1/J m-2 s-1 (de Wit, Goudriaan et al., 1987).


Maximum rate of assimilation


The maximum rate of assimilation, AMAX (in kg ha-1 h-1), is co-determined by leaf temperature and the CO2 assimilation capacity at light saturation (AMX, kg CO2 ha-1 h-1). AMX varies with crop species (and even cultivars) and is much greater for C4 crops (AMX = 88) than for C3 crops (AMX = 27 - 50) because of enzymatic differences. Whether stomata are active or inactive (in assimilating CO2) depends also on the nitrogen content of a leaf, which is indirectly related to the age of a leaf.


Influence of temperature on the CO2 assimilation capacity at light saturation


The CO2 assimilation capacity at light saturation AMAX (in kg ha-1 h-1) is illustrated by the figure below and strongly temperature dependent:































Figure 5. Generic AMAX-to-temperature response curves (Versteeg and van Keulen, 1986).

Legend: I = C3 crops in cool and temperate climates; II = C3 crops in warm climates; III = C4 crops in warm climates; IV = C4 crops in cool climates.


These highly simplified AMAX-to-temperature (AMAXTEMP) response curves can be analytically solved and for instantaneous processes the following (polynomial) approximations hold:


ƒ AMAXTEMPI  : = -0.0046 Tc2 + 0.1646 Tc  - 0.4506

ƒ AMAXTEMPII : = -0.0024 Tc2 + 0.1364 Tc  - 0.9376

ƒ AMAXTEMPIII: =   6E-05 Tc3 - 0.008  Tc2 + 0.3229 Tc  - 2.9443

ƒ AMAXTEMPIV : =  -2E-06 Tc4 + 0.0003 Tc3 - 0.0152 Tc2 + 0.3525 Tc - 1.996


Where Tc is the canopy temperature in Celsius.


Influence of plant and leaf age on the maximum rate of assimilation


The photosynthetic capacity of individual leafs and plants are also affected by their age. For maize (Zea Mays L.), Dwyer and Stewart (1986) showed that as leaf age increased past full leaf expansion, photosynthesis decreased and the curves flattened to approach an asymptote at lower irradiance levels. Linear regressions of photosynthesis at irradiance (I) = 2000 [mu] E m-2 s-1 against plant and leaf age had high correlations (r = 0.91 and 0.92, respectively). Nitrogen (N2) also plays an important role in enzymatic processes by which plants are able to extract CO2 from the atmospheric boundary layer. This ageing process can be explained by the efficiency by which nitrogen is redistributed; over the life duration of a leaf this process involves costs that are higher than the benefits in increased photosynthesis (Field, 1983). The effect of ageing of the canopy is introduced by a multiplication factor (AMAXRDS) defined as a function of the development stage. After emergence this factor increases from 0 to 1 as leafs are fully expanded. Hereafter, AMAXRDS decreases gradually until a minimum of 0,5 is attained. In other words, the moment when AMAX attains a less then optimal value is determined by the rate at which a given crop develops as determined by the weather conditions until the moment of simulation. This can take the following form:





Exercise: sensitivity analysis


In the up-scaling of crop growth simulation using RS we would like to prioritize our activities by identifying the most important driving and/or state variable within a model of crop growth. Your task is to investigate the sensitivity of two versions of the same CO2 assimilation model, both using a 3-point Gaussian integration of the canopy but one for instantaneous and the other for describing daily integrated CO2 assimilation processes. This sensitivity analysis will reveal which of the driving variables (temperature, PAR, etc.) or state variables (LAI) are most ‘important’ and at what time-step remote sensing should be able to provide estimate of this driving variable.


Input data: primary and derived parameters


The Excel worksheet “input data.xls” lists both the instantaneous (green columns), as well as the derived daily total (red colums) radiation measurements. The column “Qs_mast” contains the Global short wave radiation, which is the incident short wave radiation (radiation between 0,2 and 3 µm) and comprises the direct and diffuse components. It is measured using a Pyranometer.  Qdiff_mast” contains the diffuse short wave radiation, which is only the diffuse component of the incident short wave radiation. It also is measured using Pyranometer, but now combined with a shadow ring. The Direct short wave radiation can be inferred by subtracting the diffuse component from the Global short wave radiation.


Model: instantaneous and daily integrated CO2 assimilation


For daily integrated CO2 assimilation use the “DAYASS.exe”.

For instantaneous CO2 assimilation use the “INSTASS.exe”.




Bonhomme, R. (1993). "The solar radiation: characterization and distribution in the canopy." Crop Structure and Light Microclimate. INRA.: 17-28.


de Wit, C. T., J. Goudriaan, H. H. van Laar, F. W. T. Penning de Vries, R. Rabbinge, H. van Keulen, W. Louwerse, L. Sibma and C. de Jonge (1987). Simulation of assimilation, respiration and transpiration of

crops. Wageningen, Pudoc.


Dwyer, L. M. and D. W. Stewart (1986). "Effect of leaf age and position on net photosynthetic rates in maize (Zea Mays L.)." Agricultural and Forest Meteorology 37(1): 29-46.


Field, C. (1983). "Allocating leaf nitrogen for the maximization of carbon gain: leaf age as a control on the allocation program." Oecologia 56(2-3): 341-347.


Geurts, H. A. M. and A. F. V. v. Engelen (1983). Een rekenmodel dat het verloop van de temperatuur over een etmaal berekent uit drie termijnmetingen van de temperatuur. De Bilt, Knmi.


Kropff, M. and H. H. V. Laar (1993). Crop-Weed Interactions. [S.l.], C. A. B. Internat.


Monteith, J. L., M. H. Unsworth and A. Webb "Principles of environmental physics." Quarterly journal of the Royal Meteorological Society, Vol. 120, No. 520 (1994), p. 1699.


Parton, W. J. and J. A. Logan (1981). "A model for diurnal variation in soil and air temperature." Agricultural Meteorology 23: 205-216.


Pinker, R. T. and I. Laszlo,1992b. Global distribution of photosynthetically active radiation as observed from satellites. J.Climate, 5, 56-65.


Ross, J. (1975). "Radiative transfer in plant communities." Monteith J.L., ed. Vegetation and the atmosphere. London: Academic Press.: 13-55.


Versteeg, M. N. and H. van Keulen (1986). "Potential crop production prediction by some simple calculation methods, as compared with computer simulations." Agricultural Systems 19(4): 249-272.

APPENDIX I Listing 3.3, Diurnal course of temperature



TITLE Exercise 3.3, Diurnal course of temperature (L33.FST)



CONSTANT PI = 3.1415926 PARAM LATT = 52.

FUNCTION TMAXTB = 150.,20., 151.,22., 153., 21., 154., 21.7,

155.,25., 156.,25.

FUNCTION TMINTB = 150.,12., 151.,13., 153., 13., 154., 12.,

155.,13., 156.,13.


*       RAD converts from degrees into

         RAD     = PI/180.

*       Sine and cosine of latitude:

         SINLAT  = SIN(RAD*LATT)

         COSLAT  = COS(RAD*LATT)

*       Maximal sine of declination:

         SINDCM  = SIN(RAD*23.45)


TIMER STTIME = 151., FINTIM = 155., PRDEL = 0.04, DELT = 0.04




• Sine and cosine of declination (Eqns 3.4, 3.5): SINDEC = -SINDCM*COS(2.*PI*(TIME+10.)/365.) COSDEC = SQRT(l.-SINDEC*SINDEC)

The terms A and B according to Eqn 3.3: A = SINLAT*SINDEC


Daylength according to Eqn 3.6:

DAYL  = 12.*(1.+(2./PI)*ASIN(A/B) )

HOUR  = MOD(TIME,1.)*24. - hour/24.

DATE  = MOD(TIME, 365.)







$     TEMP)


PARAMETER (PI=3.14159, TC=4., P=1.5)


C     Errors and warnings





SUNRIS = 12.-0.5*DAYL SUNSET = 12.+0.5*DAYL



C     Period A: Hour between midnight and sunrise





$     (1.-EXP(-NIGHTL/TC))


C     Period B: Hour between sunrise and normal time of TMAX



C     Period C: Hour between normal time of TMAX and sunset



C     Period D: Hour between sunset and midnight





$     (1.-EXP(-NIGHTL/TC))



APPENDIX II Listing 3.4, Diurnal course of radiation


TITLE Exercise 3.8, Diurnal course of radiation (L34.FST)



CONSTANT PI     =  3.1415926

PARAM    LATT   = 52.

*        Daily global radiation in MJ/m2:

PARAM    RDSUM   = 9.75E6


*       RAD converts from degrees into

         RAD     = PI/180.

*       Sine and cosine of latitude:

         SINLAT  = SIN(RAD*LATT)

         COSLAT  = COS(RAD*LATT)

*       Maximal sine of declination:

         SINDCM  = SIN(RAD*23.45)


TIMER STTIME = 229.; FINTIM = 230.; PRDEL = 0.01; DELT = 1.





*        Sine and cosine of declination (Eqns 3.4, 3.5):

         SINDEC  = -SINDCM*COS(2.*PI*(TIME+10.)/365.)



*        The terms A and B according to Eqn 3.3:

         A      = SINLAT*SINDEC

         B      = COSLAT*COSDEC


*        Daylength according to Eqn 3.6:

         DAYL   = 12.*(1.+(2./PI)*ASIN(A/B) )

         HOUR   = AMOD(TIME,1.)*24.


*        Sine of solar height:

         SINB    = A*B*COS(2.*PI*(HOUR-12.)/24.)


*        Integral of sine of solar height (Eqn 3.7):

         SININT  = A*DAYL + (24.*B/PI)*COS((PI/2.)*((DAYL/12.)-1.))


*        Solar constant fluctuations (W/m2):

         SC      = 1367.*(1.+0.033*COS(2.*PI*(TIME-10.)/365.))


*        Daily average atmospheric transmissivity:

         ATMTR   = RDSUM/(SININT*3600.*SC)


*        Incident global radiation in W/m2:

          S      = MAX(0.,ATMTR * SC * SINB)*10.







This is termed "photosynthetic photon flux density" (PPFD) or "photosynthetically active radiation" (PAR) and is given in units of µmoles photons/m2 of surface area/s.  You may also see reference to µEinsteins/m2/s, which is equivalent because an Einstein is a mole of photons.

Measurements of the temperature relations of plants require a measure of the energy content of the incident radiation, rather than the PPFD.  The pyranometer is filtered to measure the total energy content (Watts/m2) of incident shortwave radiation (350 - 2000 nm).  The relationship between the energy content and photon content of radiation depends upon the spectral distribution of the light source in question.  Thus, appropriate conversions must take in account what light source is being measured (see table below).  The same goes for conversions of lux (commonly seen in older literature) to PPFD.


Approximate conversion factors for various light sources for photosynthetically active (visible) radiation (400-700 nm).


Light source

Daylight                        Metal                Sodium             Mercury                        White            Incand.

Halide                 (HP)                                        fluor.

To convert                     Multiply by

W m-2 (PAR) to               4.6                  4.6                      5.0                     4.7                 4.6                  5.0

µmol m-2 s-1 (PAR)


klux to                            18                   14                       14                      14                  12                    20

µmol m-2 s-1 (PAR)


klux to W m-2 (PAR)        4.0                  3.1                      2.8                     3.0                 2.7                  4.0


data courtesy of Li-Cor Co., Lincoln, Nebraska