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Calculating daily mean air temperatures by different methods:implications from a non-linear algorithm$

Albert Weiss*, Cynthia J. Hays

School of Natural Resources, University of Nebraska, Lincoln, NE 68583-0728, USA

Received 6 February 2004; accepted 30 August 2004

Abstract

Daily mean air temperature is used as an independent variable in algorithms describing many biological applications. These

algorithms are usually of a non-linear nature. The questions addressed in this study were: is there a difference in the daily mean

air temperature calculated by different methods and what is the impact of the various calculation methods on a non-linear

algorithm? The empirical coefficient in the non-linear algorithm used in this study was determined from a daily mean air

temperature based on the mean of 24 hourly mean temperature values. Daily mean air temperature was calculated by five

methods: mean hourly (Hourly); three equally spaced hourly mean observations weighted with the last observation (Weighted);

3 h mean temperatures (Mean 3 hour); the algorithm used in the CERES family of crop simulation models (CERES), and the

mean of the maximum and minimum daily temperatures (Max/Min). It was assumed that the Hourly method best represented the

daily mean air temperature and the other methods were compared to it. Two forms of air temperature were used in a non-linear

algorithm; a sequential approach where the algorithm was run as many times as the number of individual temperature values

used in each method, the results then averaged; and a single daily mean air temperature value. This non-linear algorithm was

evaluated over a wide range of locations, ranging in elevation from 2.4 to 1252 m and annual precipitation from 108 to 1820 mm.

There was little difference in daily mean air temperatures between the different methods. However, there were large differences

in responses from the non-linear algorithm when using any sequential approach when compared to the single daily mean

temperature values. The Mean 3 hour method worked well in all locations. The CERES method worked well except for two

locations characterized by high mean annual maximum and minimum temperatures. These results do not mean that the

sequential approaches are inappropriate, just that the temperature method used to determine empirical coefficients in the non-

linear algorithm must be consistently used in all applications. These results are a guide to different methods used to calculate

daily mean air temperature and the range of possible results when used in a non-linear algorithm. Although a specific example

was used in this study, the results are relevant to any non-linear algorithm containing empirically determined coefficients.

# 2004 Elsevier B.V. All rights reserved.

Keywords: Daily mean air temperature; Non-linear algorithm

www.elsevier.com/locate/agrformet

Agricultural and Forest Meteorology 128 (2005) 5765

$ A contribution of the University of Nebraska Agricultural Research Division, Lincoln, NE. Journal Series No. 14458. This research was

supported in part by funds provided through the Hatch Act.

* Corresponding author. Tel.: +1 402 472 6761; fax: +1 402 472 6614.

E-mail address: [email protected] (A. Weiss).

0168-1923/$ see front matter # 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.agrformet.2004.08.008

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1. Introduction

Daily mean air temperature is used as a driving

variable in many simulations of physical or biologicalprocesses and is calculated by various methods.

Hartzell (1919) used a planimeter to determine the

hourly mean temperature from thermograph charts,

summed the hourly means and divided by 24 to obtain

the daily mean air temperature for data collected at

Fredonia, NY for the year 1916. Regarding the best

method to calculate daily mean air temperature,

Hartzell (1919) concluded that if daily mean air

temperature was used for . . . temperature co-

efficients, for the study of thermal influence in botany

and zoology, thermograph averages alone should be

used . . .. He continued to state that daily mean air

temperature calculated as the mean of the maximum

and minimum temperatures was not suitable for these

purposes, because the daily mean air temperatures

calculated by the two methods did not always agree.

Even though measurement and temperature sensor

technologies have changed dramatically since this

study, the question of the best method to calculate

daily mean air temperature is still very relevant with

respect to the role of temperature as a driving variable

in the simulation of physical or biological processes.

At least two other methods have been used tocalculate the daily mean air temperature (aside from

the mean of the 24 hourly mean values and the mean

of the maximum and minimum temperatures). A

weighted mean of hourly mean temperature at three-

fixed observation times (e.g., 07:00, 14:00, 21:00 local

standard time) is determined by the sum of these tem-

perature values, where the last temperature observa-

tion is weighted twice, the sum then divided by four,

Landsberg (1958). The other method employs the

mean of equally spaced observations, e.g., the mean of

eight values, which represents the mean temperaturefor a 3-h period, can also be used to determine the

daily mean air temperature. A form of this method is

used in the CERES family of crop simulation models,

Jones and Kiniry (1986).

Harris and Pedersen (1995) compared daily mean

air temperature calculated by summing individual

20-min temperature observations and then dividing

by 72 to the mean of the maximum and minimum

temperatures at Calgary, Canada over a 24-month

period. They found the largest discrepancies (up to

7 8C) between these two methods during the winter

months.Cesaraccio et al. (2001)and Raworth (1994)

used different algorithms to determine thermal time

(growing degree days) on a daily basis from values ofmaximum and minimum temperatures. Raworth

(1994) found that observed hourly temperature data

provided a better estimate of thermal time than the

mathematical functionsfit to minimum and maximum

temperature data. The algorithm developed by

Cesaraccio et al. (2001) was found to be superior

over a wide range of California climates when

compared to other published thermal time algorithms.

Reicosky et al. (1989) evaluated five algorithms

used in simulation models that generate hourly air

temperature values from daily maximum and mini-

mum temperature observations. They noted that

although there were sometimes large differences

between observed and simulated hourly temperatures,

the ramifications of these differences in simulated

processes were not determined by Reicosky et al.

(1989)as this was not their main objective.Sadler and

Schroll (1997)had a similar objective toReicosky et

al. (1989), to evaluate the same algorithms used in this

earlier study plus a newly developed algorithm. This

newly developed algorithm was notrestricted by the

assumption that the shape of the daily pattern fits a

predefined curve, such as a sine wave. This algorithmperformed as well as or better than thefive algorithms

in about 50% of the cases but required the develop-

ment of a cumulative distribution function of normal-

ized temperature for a year at each location.

An important extension to these previous efforts is

to evaluate the use of observed air temperatures at

different time scales (e.g., maximum and minimum,

hourly mean, the mean of eight 3 h means, three

individual hourly mean observations, and a daily

value) in a non-linear process. Many simulated plant

related processes are of a non-linear nature, e.g.,photosynthesis and respiration. While different results

from using different methods to calculate daily mean

air temperature might be small on a daily basis, in a

non-linear process, these small differences can be

magnified if they are accumulated over any length

of time. An analogous situation occurred when

McMaster and Wilhelm (1997) used two accepted

protocols to calculate accumulated thermal time with

the same data set. Large monthly differences in accu-

mulated thermal time (in some cases exceeding 80%)

A. Weiss, C.J. Hays / Agricultural and Forest Meteorology 128 (2005) 576558

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were observed between the two methods. Differences

in accumulated thermal time over the growing season

varied starting in September with no difference, a 23%

difference in March, and with a 9% difference in Julyat physiological maturity.

In developing new applications, where daily mean

air temperature is a driving variable, it is important to

clearly define how this variable was calculated in order

to avoid any ambiguities in the calculation, which may

impact the results. When the application involvesa non-

linear algorithm, how is a single daily mean output

value determined with multiple input values of a para-

meter (e.g., maximum and minimum temperatures)?

Are multiple input values used in the algorithm and the

resulting outputs used to calculate a daily mean output

value or is a daily mean input value used to calculate a

daily mean output value? The objectives of this effort

were two-fold: (1) to evaluate different methods of

calculating daily mean air temperature from observed

values at different time scales and (2) to investigate the

responses of a non-linear algorithm using these differ-

ent methods of calculating daily mean air temperature

and using multiple input values. A consequence of the

second objective will provide an indication of the range

of possible results when using non-linear algorithms

over contrasting climates in the United States.

2. Material and methods

Hourly mean (local standard time) and daily

maximum and minimum air temperatures from seven

locations in the contiguous U.S. (Astoria, OR; Bishop,

CA; Brorson, MT; Caribou, ME; Del Rio, TX; Mead,

NE; and Tampa, FL) representing a wide range of

climate conditions (Fig. 1andTable 1), were used to

calculate daily mean air temperatures. The hourly mean

data obtained from the National Climatic Data Center

(NCDC) were calculated based on a hourly period that

ended during the last 10 min of the hour, this ending

time varied from station to station. For this study, the

time of observation was noted as the following hour

(e.g., the hourly mean temperature was calculated from

measurements takenfrom9:51 to 10:50 and this time of

observation was changed from 10:50 to 11:00). The

High Plains Regional Climate Center (HPRCC) hourly

mean data was calculated based on a hourly period thatbegan 1 min after the hour (e.g., the hourly mean

temperature was calculated from measurements taken

from 10:01 to 11:00 and the time of observation was

noted as 11:00). The time of observation of the data

from NCDC was changed in order to be consistent with

A. Weiss, C.J. Hays / Agricultural and Forest Meteorology 128 (2005) 5765 59

Fig. 1. Locations of the sites used in this study.

Table 1

Location, length of record, latitude, longitude, elevation, mean annual daily minimum and maximum temperatures for the length of record and

mean annual total precipitation

Location and length

of record

Latitude Longitude Elevation

(m)

Mean annual

daily minimumtemperature (8C)

Mean annual

daily maximumtemperature (8C)

Mean annual

total precipitation(mm)

Astoria, OR (19972002)a 468090N 1238530W 2.7 7.1 14.8 1820.4

Bishop, CA (19972002)a 378220N 1188220W 1252.4 3.0 23.6 108.3

Brorson, MT (19952002)b 478470N 1048150W 691.0 0.5 12.6 373.9Caribou, ME (19972002)a 468520N 688020W 191.1 0.6 9.9 868.0Del Rio, TX (19972002)a 298220N 1008550W 312.7 15.5 28.0 403.4

Mead, NE (19932002)b 418090N 968290W 366.0 3.8 16.6 665.5

Tampa, FL (19972002)a 278580N 828320W 2.4 18.5 27.8 1219.9

Superscripts denote source of data.a

National Climatic Data Center (NCDC) http://www.ncdc.noaa.gov/oa/ncdc.html.b

High Plains Regional Climate Center (HPRCC) http://www.hprcc.unl.edu/.
http://www.ncdc.noaa.gov/oa/ncdc.htmlhttp://www.hprcc.unl.edu/http://www.hprcc.unl.edu/http://www.hprcc.unl.edu/http://www.ncdc.noaa.gov/oa/ncdc.html

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the time of observation of the data from HPRCC. The

mean annual daily minimum temperatures ranged from

0.6 to 18.5 8C, while the mean annual daily maximum

temperatures ranged from 9.9 to 28.0 8C. The locationwith the largest difference between mean annual daily

maximum and minimum temperatures was Bishop,

CA (20.6 8C), the smallest Astoria, OR (7.7 8C). These

locations also had the precipitation extremes, 108.3

and 1820.4 mm, respectively.

The five methods used to calculate daily mean air

temperature T were:

Hourly:

T

P24i1Ti24 (1)

where i is the hour, Ti is the hourly mean air

temperature for hour i (8C).

Weighted:

TT07:00T14:002T21:00

4(2)

whereT07:00is the hourly mean air temperature for

07:00 local time (8C),T14:00is the hourly mean air

temperature for 14:00 local time (8C),T21:00 is the

hourly mean air temperature for 21:00 local time

(8C).

Mean 3 hour:

T

P8i1T3i

8(3)

wherei is every 3 h (03:00, 06:00, . . ., 21:00, 24:00

local time), T3 is the 3 h mean temperature for hour i

(8C).

CERES(Jones and Kiniry, 1986)

T P8i1Tci

8

T (4)

where i is 18, Tci= Tmin+ tmfaci(TmaxTmin) (8C),Tminis the daily minimum temperature (8C),Tmaxis

the daily maximum temperature (8C), tmfaci=

0.931 + 0.114i0.0703i2 + 0.0053i3; for i = 18. Max/Min:

TTmin Tmax

2(5)

whereTmin is the daily minimum temperature (8C)

and Tmax is the daily maximum temperature (8C).

The NCDC data were missing some air temperature

observations; the missing data could range from an

hour to several days. No attempt was made to estimate

these missing data and a daily mean air temperaturewas not calculated for any days that had missing data.

Mean bias errors (MBE) and root mean square

errors (RMSE) were calculated with respect to the

daily mean air temperature based on the Hourly

method using the following equation:

MBE

P Tm T

n(6)

RMSE

P Tm T

2

n

( )0:5(7)

where Tis the daily mean air temperature calculated

from the Hourly method (8C), Tmis the daily mean air

temperature calculated from the other methods (C8),

and n is the number of observations.

Any days with missing air temperatures were

omitted from the MBE and RMSE calculation. The

RMSE was calculated monthly and for the length of

record.

While many non-linear algorithms of plant

processes could be used, the non-linear algorithm

(Streck et al., 2003) describing winter wheat devel-

opment rate (Rdev, d1) was selected. This algorithm

was selected because it has both positive and negative

slopes providing a very demanding test for the

objectives of this research effort, whereas other

non-linear algorithms may have either have positive

or negative slopes. It is obvious that winter wheat is

not grown at all the locations in this study. However,

the objective of this study was to evaluate a tem-

perature driven non-linear algorithm, not the feasi-

bility of growing winter wheat in different geographic

locations. The use of the same algorithm allows for the

evaluation of the different non-linear responses fromthe different methods of calculating daily mean air

temperature. The following relationships describe the

non-linear algorithm for winter wheat phenological

development:

fT 2T Tn

aToptTna T Tn

2a

ToptTn2a

;

if Tn TTx

(8)

fT 0; if T< Tn or T> Tx (9)


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a ln 2

ln TxTn=ToptTn (10)

Rdev RmaxfT (11)

where T is the temperature (8C), Tn is the minimum

cardinal temperature (8 8C for July and August and

0 8C for all other months),Txis the maximum cardinal

temperature (35 8C for July and August and 30 8C for

all other months), Topt is the optimum cardinal tem-

perature (24 8C for July and August and 19 8C for all

other months),Rmaxis the maximum development rate

(0.0381 d1 for July and August and 0.0241 d1 for all

other months).

Rmax was empirically determined based on the

comparison offield observations of plant development

relatively near Mead, NE for a wheat cultivar and a

daily mean air temperature calculated from 24 hourly

mean air temperatures (Streck et al., 2003). The value

of Rmax will change if determined by the different

temperature methods or if a different cultivar was

selected. The change in Rmax associated with the

different single value temperature methods would

probably be small. However, the values of Rmaxassociated with the different sequential methods will

probably be quite different then those of the single

value temperature methods. For comparison purposes

in this study, the original value found in Streck et al.(2003)was used. The development rate was calculated

for each day and accumulated for three different

periods representing early season plant development,

mid-season plant development and fall sowing

(starting on April 1, July 1 and September 1,

respectively). The development stage equals the

accumulated development rate and has a value of 0

at emergence, 1 at anthesis, and 2 at physiological

maturity.

The first approach to the calculation of the daily

development rate (Eqs.(8)(11)) used the single valueof the daily mean temperature determined from Eqs.

(1)(5). The second approach took into account the

non-linear nature of the daily development rate by

using the individual temperatures from Eqs.(1)(5)in

a sequential process in Eqs. (8)(11) (which will be

referred to as sequential) and then averaged the

results of Eq.(11). For example, using the sequential

Hourly method, Eqs.(8)(11)were ran 24 times and

the daily development rate was the average of Eq. (11).

When the air temperature data were missing, the

development rate was estimated as the mean of the

development rate 5 days prior and 5 days after the

missing data. The reason for using 5 days in

comparison to1 days were to smooth out any abrupttemperature changes that might have occurred. The

development stage based on the Hourly method was

compared to the development stages calculated using

the other approaches. The day of year when the

development stage based on the Hourly method

equalled 0.75 (DOY24) was compared to the day of

year when the other approaches reached 0.75 (DOY).

The development stage of 0.75 was selected instead of

1.0, since for some locations a development stage of

1.0 was not achieved. This comparison was made by

subtracting DOY24 from DOY (i.e., a positive value

means that the time for the development stage to reach

0.75 was longer than the time using the Hourly

method). The mean of the differences for each period

(early or mid-season plant development and fall

sowing) were calculated.

3. Results

In general, there was very good agreement (in terms

of annual RMSE values) between the Hourly method

and the other methods for the calculation of dailymean air temperature for the seven locations (Table 2).

Agreement between the methods was within 1 8C for

all locations except Bishop, CA. The MBEs varied

from 0.52 8C for the Max/Min method for Bishop,CA to +0.52 8C using the CERES method at Tampa,

FL. The location with the best agreement between the

methods was Astoria, OR. As would be anticipated,

there was no difference between the Hourly and Mean

3 hour methods, since the 3 h mean value is the mean

of three of the mean hourly values. The other three


Table 2

MBE/RMSE values of daily mean temperature calculated using the

Max/Min, Weighted, and CERES methods

Location Max/Min (8C) Weighted (8C) CERES (8C)

Astoria, OR 0.06/0.63 0.04/0.57 0.13/0.64Bishop, CA 0.52/1.31 0.28/1.10 0.35/1.22Brorson, MT 0.13/0.94 0.12/0.94 0.26/0.97

Caribou, ME 0.19/0.88 0.14/0.88 0.08/0.85Del Rio, TX 0.26/0.76 0.25/0.74 0.38/0.80

Mead, NE 0.01/0.92 0.03/0.92 0.14/0.92Tampa, FL 0.43/0.72 0.09/0.54 0.52/0.78

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methods provided very similar results; no method was

clearly superior to another at all locations.

Only the monthly temperature RMSE values for

Astoria, OR and Bishop, CA will be discussed in detailsince they represent the best and worst agreement

between the different methods. Astoria, OR usually

has a small daily variation in temperature, while

Bishop, CA has a relatively large daily variation in

temperature. The other locations have daily tempera-

ture variations that are between these two locations.

For Astoria, OR, monthly RMSE values for the Max/

Min and CERES methods were similar while the

Weighted method showed a seasonal trend with the

lowest RMSE values in JuneJuly (Table 3a). In

general, for the Max/Min and Weighted method for the

fall and winter months the MBE was negative (ranging

from 0.21 to +0.02 8C) while for the spring andsummer months the MBE was positive (ranging from

0.02 to 0.23 8C). Using the CERES method all MBE

values were positive with the largest values in June

through September. For the majority of months in

Bishop, CA, the Max/Min method had the largest

RMSE and absolute MBE values and the Weighted

method the lowest, Table 3b. In contrast to the trendwith the Weighted method at Astoria, OR, all methods

at Bishop, CA had the highest RMSE and absolute

MBE values during the period MaySeptember.

There were few differences between calculating

daily mean air temperature by the different methods.

However, there were large differences in the number

of simulated days when the development stages

reached 0.75 for the three different periods beginning

on April 1, July 1, and September 1; (Table 4ac). As

would be expected from the previous analysis (Table

2), in general Bishop, CA and Astoria, OR represent

the extremes in differences of simulated days of

development stage for these three periods. Since

Astoria, OR; had the lowest RMSE and absolute MBE

values (Table 2) all methods worked equally well for

the first two periods (Table 4a and b). For the last

period, there was a difference between the single value

and sequential methods (Table 4c). Mean differences

for Bishop, CA ranged from 0.7 to 1.8 days for thesingle value methods and from 4.8 to 39.8 days for the

sequential methods during the April 1 early season

plant development (Table 4a). During the July 1 mid-

season plant development, the mean difference in daysranged from 0.5 to 1.2 days for the single valuemethods and from 5.8 to never achieving 0.75 using

the sequential methods (Table 4b). Mean differences

ranged from 0.5 to 1.8 days using the single valuemethods and 11.5 to 52.5 days using the sequential

value methods for the September 1 sowing (Table 4c).

The sequential Max/Min method had the poorest

agreement among the other methods at all locations,

since Eq.(11)was sometimes evaluated as 0. The best

temperature method was the Mean 3 hour method with

mean differences ranged from0.8 to 0.0 days for thethree different periods (Table 4ac). The Weighted andCERES methods usually performed satisfactorily,

with the magnitude of the differences depending on

the period and location.

4. Discussion

Reicosky et al. (1989) investigated five different

subroutines to simulate hourly mean air temperatures:


Table 3

Monthly MBE/RMSE values of daily mean temperature calculated

using the Max/Min, Weighted, and CERES methods

Month Max/Min (8C) Weighted (8C) CERES (8C)

(a) Astoria, ORJanuary 0.04/0.63 0.02/0.67 0.02/0.63February 0.03/0.60 0.12/0.66 0.04/0.61March 0.02/0.54 0.11/0.62 0.06/0.54April 0.05/0.63 0.02/0.49 0.13/0.64

May 0.09/0.56 0.06/0.51 0.16/0.57

June 0.13/0.51 0.05/0.43 0.21/0.52

July 0.23/0.62 0.02/0.41 0.30/0.64

August 0.20/0.61 0.02/0.43 0.29/0.64September 0.24/0.82 0.18/0.60 0.34/0.86October 0.04/0.69 0.21/0.65 0.05/0.70November 0.04/0.72 0.07/0.67 0.03/0.73December 0.06/0.61 0.02/0.57 0.00/0.61

(b) Bishop, CA

January 0.42/1.07 0.28/1.02 0.60/1.16February 0.02/0.84 0.20/0.87 0.18/0.87March 0.57/1.06 0.06/0.92 0.37/0.96April 0.86/1.23 0.58/1.11 0.69/1.07May 1.24/1.59 0.78/1.22 1.02/1.41June 1.35/1.72 0.91/1.25 1.13/1.53July 1.54/1.88 1.06/1.28 1.13/1.68August 1.36/1.61 0.88/1.21 1.12/1.40September 0.89/1.21 0.55/1.06 0.67/1.05October 0.12/0.87 0.19/1.00 0.10/0.86November 0.17/0.97 0.41/1.15 0.37/1.03December 0.62/1.09 0.36/0.94 0.80/1.22

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(1) WAVE in ROOTSIMU V4.0 (Hoogenboom andHuck, 1986), (2) WCALC from the crop simulation

model SoyGRO V5.3 (Wilkerson et al., 1983), (3)

WEATHER in GLYCIM (Acock et al., 1983), (4)

TEMP (Parton and Logan, 1981), and (5) SAW-

TOOTH (Sanders, 1975).Reicosky et al. (1989)found

that WCALC provided the best results. The hourly

mean temperatures estimated by WCALC were

obtained by dividing a day into three segments;

midnight to sunrise +2 h, sunset to midnight, and the

daylight hours. WCALC also requires maximum and

minimum temperature for the day before, theminimum temperature for the following day, the

times of sunrise and sunset of the day before, and the

time of sunrise of the following day.

As part of this current study, the daily mean air

temperature from WCALC was compared to the

Hourly and CERES methods for calculating daily

mean air temperature at Mead, NE. This analysis was

preformed to document why the CERES method was

used in this study instead of WCALC. The Mead, NE

location was selected because it had a long period of


Table 4

Mean differences in days between the sequential temperature methods, the single daily mean temperature methods and the Hourly method in the

non-linear algorithm (Eqs. (8)(11))

Method Astoria, OR

(days)

Bishop, CA

(days)

Brorson, MT

(days)

Caribou, ME

(days)

Del Rio, TX

(days)

Mead, NE

(days)

Tampa, FL

(days)

(a)a

Sequential Hourly 1.0 9.3 2.6 1.5 4.0 3.7 3.8

Sequential Max/Min 2.2 39.8 7.1 4.8 12.5 10.9 14.5

Sequential Weighted 0.5 4.8 1.0 0.2 10.7 3.0 3.2Sequential Mean 3 hour 0.8 9.0 2.2 1.7 3.7 3.7 3.7

Sequential CERES 0.7 12.5 2.2 2.0 8.0 4.7 7.7

Max/Min 0.3 1.8 0.4 0.7 4.2 0.3 1.5Weighted 0.2 0.7 0.4 1.2 8.2 0.1 0.3Mean 3 hour 0.0 0.0 0.0 0.0 0.0 0.0 0.0

CERES 0.5 1.2 0.6 0.3 5.0 0.1 1.7(b)

b


Sequential Max/Min 1.2 NV0 12.4 6.0 4.8 7.8 5.0Sequential Weighted 0.7 5.8 2.1 1.3 5.7 2.0 1.5

Sequential Mean 3 hour 0.7 11.0 3.6 2.0 1.7 2.3 1.5


Max/Min 1.0 0.5 0.0 0.2 1.0 0.2 0.8Weighted 0.0 1.2 0.0 0.2 2.0 0.1 0.2Mean 3 hour 0.0 0.0 0.0 0.0 0.0 0.0 0.0

CERES 1.2 0.5 0.0 0.3 2.0 0.1 0.8(c)

c


Sequential Max/Min 5.5 52.2 29.97

8.45

7.0 20.1 10.8

Sequential Weighted 2.5 11.5 11.8 7.3 3.2 6.4 1.3

Sequential Mean 3 hour 1.7 15.5 12.1 9.2 2.0 6.0 3.3


Max/Min 0.0 0.7 1.1 3.8 2.0 0.2 5.0Weighted 0.7 1.8 0.8 0.2 0.8 0.5 0.7Mean 3 hour 0.0 0.0 0.0 0.0 0.0 0.0 0.8CERES 0.5 0.5 2.1 0.7 2.0 0.1 5.5a

The simulations started on April 1 and ended when the development stage reached 0.75.b

The simulations started on July 1 and ended when the development stage reached 0.75 or August 31. In this latter case, the superscript

represents the number of years that the development stage reached 0.75 out of the total years of record. NV denotes no value, in this case there

were no years that the development stage reached 0.75.c

The simulations started on September 1 and ended when the development stage reached 0.75 or December 31. In this latter case, the

superscript represents the number of years that the development stage reached 0.75 out of the total years of record.

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record (10 years) and a data set with no missing values

as compared to the other locations. In this comparison,

it was found that the CERES method was superior

to WCALC (RMSE values of 0.92 and 1.17 8C,respectively; MBE values of 0.14 and 0.66 8C for theperiod of record) and did not require data from

previous and future days.

This study demonstrated that daily mean air

temperature could be relatively accurately calculated

by Eqs. (2)(5) over a wide range of climates and

usually incorporated into a non-linear algorithm.

However, when sequential methods were used in the

non-linear algorithm there were large differences

when compared to the single value methods. These

differences were due to the empirically defined

coefficient (Rmax), which was developed based on

the Hourly method. This result indicates the impor-

tance of the consistent use of the same temperature

method as associated with the non-linear algorithm.

The consistent use of the same non-linear algorithm,

representing phenological development in winter

wheat, across different climates was necessary to

ensure that the different responses were due solely to

the different temperature methods employed in the

non-linear model even though winter wheat is not

grown in all locations.

In general, the best predictions were made forAstoria, OR and the worst for Bishop, CA, regardless

of the temperature method employed. No doubt these

results are due to the uniformity, or lack of uniformity,

of the temperature regimes at these locations. Astoria,

OR had the smallest difference between mean annual

daily maximum and minimum temperatures, while

Bishop, CA had the greatest difference.

Brorson, MT; Caribou, ME; and Mead, NE have

similar RMSE values (Table 2) for the different

methods of calculating daily mean air temperatures as

compared to Del Rio, TX and Tampa, FL. These fivelocations can also be characterized by the mean annual

daily minimum and maximum temperatures, the

former locations having lower values than the latter

locations.

The Mean 3 hour method was the best across all

locations in the non-linear algorithm. Given that these

eight values are a composite of the 24 hourly values,

this result is not surprising. In general, the sequential

methods did not agree as well as the single value

methods in this non-linear algorithm, especially at

locations in which there was a large range in daily

temperatures (i.e., Bishop, CA). The Weighted and

CERES methods compared equally well over all the

locations. The sequential Max/Min method performedpoorly. An advantage of the CERES method was that it

only required daily maximum and minimum tem-

peratures compared to the Mean 3 hour and Weighted

methods, which requires hourly mean temperature

observations. A disadvantage of the CERES method is

that it does not work as well for locations with high

mean annual daily maximum and minimum air

temperatures, such as Del Rio, TX and Tampa, FL.

This limitation may be overcome by modifying the

tmfac algorithm (Eq.(4)) to handle such situations.

5. Conclusion

In this study, the single value temperature methods

generally performed well in the non-linear algorithm.

The reason for this result is that the empirical

coefficient used in the non-linear algorithm was

developed using a single value temperature. These

results do not mean that the sequential methods are

inappropriate. They just mean that the temperature

method used to determine the empirical coefficients

must be consistently used in all applications and thesemethods should be thoroughly described in any

publication. One could argue that the sequential

methods make more sense in a non-linear algorithm,

but the application of any of these sequential methods

must be balanced by the necessary accuracy of the

results and the availability of the input data. These

results are a guide to different methods used to

calculate daily mean air temperature and the range of

possible results when used in a non-linear algorithm.

Although a specific example was used in this study, the

results apply to any non-linear algorithm containingempirically determined coefficients.

Acknowledgements

Drs. G.S. McMaster and W.W. Wilhelm provided

valuable comments on an earlier version of this

manuscript. The responses to the questions raised by

the two anonymous reviewers further helped to clarify

the contents of this manuscript.


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