Garden with Insight v1.0 Help: Weather - Air temperature and solar radiation

The model developed by Richardson (1981) was selected for use in EPIC because it simulates temperature and radiation, which are mutually correlated with rainfall. The residuals of daily maximum and minimum air temperature and solar radiation are generated from a multivariate normal distribution.

The multivariate generation model used implies that the residuals of maximum temperature, minimum temperature, and solar radiation are normally distributed and that the serial correlation of each variable may be described by a first-order linear autogregressive model. Details of the multivariate generation model were described by Richardson (1981). The dependence structure of daily maximum temperature, minimum temperature, and solar radiation were described by Richardson (1982).

The temperature model requires monthly means of maximum and minimum temperatures and their standard deviations as inputs. If the standard deviations are not available, the long-term observed extreme monthly minima and maxima may be substituted. The model estimates standard deviation as 0.25 of the difference between the extreme and the mean for each month. For example: [Equation 100] whre SDTMX is the standard deviation of the daily maximum temperature, TE is the extreme daily maximum temperature, and Tbar is the average daily maximum temperature for month k.

Equation 100:

SDTMX = 0.25 * (TE - Tbar)
Code:
equation for max temp is the same. equation for min temp is
SDTMX = 0.25 * (Tbar - TE)
Variables:
SDTMX = StdDevMaxTempFromMeanAndExtremeForMonth_degC or ...Min...
TE = extremeDailyMaxTempForMonth_degC or ...Min...
Tbar = dailyMeanMaxTempForMonth_degC or ...Min...

The solar radiation model uses the extreme approach extensively. Thus, only the monthly means of daily solar radiation are required as inputs. The equation for estimating standard deviation is [Equation 101] whre SDRA is the standard deviation of daily solar radiation i MJ/m2, RAMX is the maximum daily solar radiation at midmonth, and RA is the mean daily solar radiation for month k.

Equation 101:

SDRA = 0.25 * (RAMX - RA)
Code:
same
Variables:
SDRA = StdDevRadiationFromMeanAndExtremeForDay_MJPm2
RAMX = maxPossibleRadiation_MJPm2
RA = meanRadiationForDay_MJPm2

Maximum temperature and solar radiation tend to be lower on rainy days. Thus, it is necessary to adjust the mean maximum temperature and solar radiation downward for simulating rainy day conditions (note not min temp). For T(mx) this is accomplished by assuming thta wet day values are less than dry day values by some fraction of T(mx) - T(mn): [Equation 102] where TW is the daily mean maximum temperature for wet days in degrees C in month k, TD is the daily mean maximum temperature for dry days, omega(T) is a scaling factor ranging from 0.0 to 1.0, T(mx) is the daily mean maximum temperature, and T(mn) is the daily mean minimum temperature. Choosing omega(T) = 1.0 provides highest deviations on wet days and omega(T) = 0.0 ignores the wet day effect. Observed data indicate that omega(T) usually lies between 0.5 and 1.0.

Equation 102

TW(mx) = TD(mx) - omega(T) * (T(mx) - T(mn))
Code:
same (omega(T) = 0.5)
Variables:
TW(mx) = DailyMeanMaxTempWetDaysForMonth_degC
TD(mx) = dailyMeanMaxTempDryDaysForMonth_degC
omega(T) = tempResponseToWetDays_frn = 0.5
T(mx) = dailyMeanMaxTempForMonth_degC
T(mx) = dailyMeanMinTempForMonth_degC

Since equation 102 gives lower mean maximum temperature values for wet days, a companion equation is necessary to slightly increase mean maximum temperature for dry days. The development is taken directly from the continuity equation

T(mx) * ND = TW(mx) * NWD + TD(mx) * NDD (Equation 103)

where ND is the number of days in a month, NWD is the number of wet days, and NDD is the number of dry days. The desired equation is obtained by substituing equation 102 into equation 103 and solving for TD [Equation 104]. Use of the continuity equation guarantees that the long-term simulated value for mean maximum temperature agrees with the input value of T(mx).

Equation 104

TD(mx) = T(mx) + (NWD / ND) * omega(T) * (T(mx) - T(mn))
Code:
same
probWetDayForMonth_frn = NWD / ND
omega(T) = 0.5
Variables:
TD(mx) = DailyMeanMaxTempDryDaysForMonth_degC
NWD = numWetDaysForMonth
ND = numDaysInMonth
omega(T) = tempResponseToWetDays_frn = 0.5
T(mx) = dailyMeanMaxTempForMonth_degC
T(mx) = dailyMeanMinTempForMonth_degC

The method of adjusting solar radiation for wet and dry days is similar to that of adjusting maximum temperature. The radiation on wet days is a fraction of the dry day radiation [Equation 105] where RAW is the daily mean radiation on wet days in MJ/m2, omega(R) is a scaling factor ranging from 0.0 to 1.0, and RAD is the daily mean solar radiation on dry days. An omega(R) value of 0.5 gives satisfactory results for many locations.

Equation 105

RAW = omega(R) * RAD
Code:
same (omega(R) = 0.5)
Variables:
RAW = DailyMeanRadiationWetDaysForMonth_MJPm2
omega(R) = radiationResponseToWetDays_frn = 0.5
RAD = dailyMeanRadiationDryDaysForMonth_MJPm2

The dry day equation is developed by replacing temperature with radiation in equation 103 and substituting equation 105 for RAW. Then, [Equation 106] where RA is the daily mean solar radiation for month k in MJ/m2.

Equation 106

RAD = (RA * ND) / (omega(R) * NWD + NDD)
Code:
NDD = ND - NWD so
RAD = (RA * ND) / (omega(R) * NWD + (ND - NWD))
divide all terms by ND so
RAD = RA / (omega(R) * (NWD/ND) + ND/ND - NWD/ND)
probWetDayForMonth_frn = NWD / ND so
RAD = RA / (omega(R) * probWetDayForMonth_frn + 1 - probWetDayForMonth_frn)
omega(R) = 0.5 so
RAD = RA / (0.5 * probWetDayForMonth_frn + 1 - probWetDayForMonth_frn)
RAD = RA / (1.0 + (0.5 * probWetDayForMonth_frn - probWetDayForMonth_frn))
RAD = RA / (1.0 - 0.5 * probWetDayForMonth_frn)
Variables:
RAD = DailyMeanRadiationDryDaysForMonth_MJPm2
RA = dailyMeanRadiationForMonth_MJPm2
ND = numDaysInMonth
omega(R) = radiationResponseToWetDays_frn = 0.5
NWD = numWetDaysForMonth
NDD = numDryDaysInMonth (= ND - NWD)