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Models for estimation of hourly soil temperature at 5 cm depth and for degree-day accumulation from minimum and maximum soil temperature.

Introduction

The development of plants (He et al. 2010), insects (Dallwitz 1984), and soil microbes (Uchida et al. 2010) is temperature-dependent and is often calculated using degree-days or similar units (Vogt and Bedo 2001; Bryant et al. 2002). Since no development accrues below a threshold temperature, and development may reach a limit or fall to zero at high temperatures, it is necessary to calculate on an hourly basis and accumulate development over each 24-h period. The use of daily averages would result in errors due to periods below the threshold temperature or, in the case of non-linear development functions (Vogt and Bedo 2001), would incorrectly assess differences in rate between high and low temperatures.

In some models (triangular), a linear change in temperature between minimum and maximum is assumed, whereas other models use sine curves or other functions. A comparison of such models for regions of New South Wales (Watson and Beattie 1996) found that the most successful type of model varied between regions, although even the best model almost always overestimated the degree-day accumulation when compared with actual hourly measurements.

Some models allow for asymmetry in the temperature profile (Roltsch et al. 1999). For example, two sine functions can be used: an increasing function from minimum to maximum over 9h, and a decreasing function over 15 h. This corrects for asymmetry in the timing of maximum and minimum during the day, since the maximum is usually <12 h after the minimum. However, results in New South Wales (Watson and Beat-tie 1996) showed that these methods overestimate the average daily temperature and therefore overestimate the accumulation of degree-days.

A model has been described for the estimation of soil temperatures from air temperatures (Horton and Corkrey 2011). Although this model provides the average soil temperature in addition to the minimum and maximum, the average is not sufficient for accumulation of degree-days on days when the temperature falls below the threshold. Therefore, a method is needed for estimating hourly soil temperatures for calculation of development rates at 5 cm depth.

Seven different methods have been compared for air temperature (Roltsch et al. 1999), but there were limited differences and no advantages for more complex methods. However, all of these methods had symmetry in the period from minimum to maximum and also in the period from maximum to minimum (University of California 2003). Therefore, they all give a mean daily temperature equal to the average of the minimum and maximum, a situation that is usually not correct.

Alternative methods have separated the period from maximum to minimum into two phases--fast initial decrease followed by a slower decrease to the minimum. A square-root decay function has been used for air temperature (Cesaraccio et al. 2001), while an exponential function was applied to air temperature and soil temperature at 10 and 20 cm soil depth (Parton and Logan 1981). This exponential decay function matches the upper temperature correctly, at the transition point, but causes a discontinuity at the minimum temperature, although this might not be critical if only hourly temperatures are required. However, a more complex exponential decay function can be used (Goudriaan and van Laar 1994) that does not have a discontinuity at the transition point or at the minimum and this has been applied to air temperature.

The sine curve used for the period from the minimum up to the maximum is sometimes only the first quadrant of the sine curve (Parton and Logan 1981; Goudriaan and van Laar 1994), so the rate of increase in temperature in the model is most rapid immediately after the minimum, rather than the fourth quadrant of the sine curve for the minimum halfway to the maximum, then the first quadrant for the following period to the maximum. Inspection of actual data compared with the model suggests that the sine function for increasing temperature should cover two quadrants.

Several models (Parton and Logan 1981; Goudriaan and van Laar 1994; Roltsch et al. 1999; Cesaraccio et al. 2001) base the time of the minimum, maximum, or transition points on the time of sunrise, midday, or sunset. These models therefore require calculation of the time of sunrise and sunset for the latitude and time of year. This aspect was included in the models considered here.

For the period from the maximum to the transition point, the sine curve used is sometimes treated as a the second quadrant of the same sine curve used for the period up to the maximum (Parton and Logan 1981; Goudriaan and van Laar 1994), although others have preferred a separate sine curve with a different phase length for this section (Cesaraccio et al. 2001).

This study examined soil temperature available for each minute of the day over several years at three sites to examine the separate phases of the cycle and determine appropriate functions for each phase, for use where accurate estimates are required for temperatures throughout the day.

Methods

Minute-by-minute records

The Australian Bureau of Meteorology (BOM) provided data from Perth Airport, Western Australia (latitude -31.9275, longitude 115.9764), Geraldton Airport, Western Australia (latitude -28.7953, longitude 114.6975), and Longreach, Queensland (latitude -23.4373, longitude 144.2769), with records of soil temperature at 5 cm depth for every minute from October 2003, June 2005, or March 2007, respectively, to August 2010. All data were combined into a single file and the times of minimum and maximum temperatures were calculated and related to the times of sunrise and sunset at each location for each particular day, for analysis as a single dataset.

Hourly records

The University of Melbourne provided hourly soil temperatures for a site near Ballarat, Victoria (latitude -37.90, longitude 143.72), for the whole of 2005 and 2006, from a study of sheep blowfly juvenile development (De Cat et al. 2012). The probes for these measurements were at 5 cm depth under pasture. These records were used for validation of the degree-day aspects of the model that was developed using the BOM minute-by-minute data.

Three-hourly records

Records from BOM were available for 35 sites around Australia, with soil temperature recorded every 3 h. This dataset has been described previously (Horton and Corkrey 2011) and covers all states of Australia. Only days with a full set of eight records for temperature at 5 cm at that site were included in this analysis. These records were used for validation of the shape of the curve obtained by the model.

For each set of 3-hourly records from 00 : 00 to 21 : 00, the minimum and maximum temperatures were obtained. Since the model is derived from minute-by-minute data, the model was first used to estimate the expected minimum and maximum temperatures if minute-by-minute records had been available, compared with the minimum and maximum for the actual records taken at precisely 3-h intervals. This gave a small decrease to the estimated true minimum and an increase to the estimated true maximum compared with the 3-hourly minimum and maximum. The model was then used to estimate the temperature at each 3-h interval based on the revised minimum and maximum. For each site, the actual measured values at each 3-h time-point were compared with the estimates by comparing the correlation between measured and estimated values. In order to measure any bias, the mean of all estimates was compared with the mean of all actual temperatures for each individual site.

Minute-by-minute data analysis

The model used a sine function for the temperature rising from the minimum on the current day to the maximum for that day, a sine function for the temperature falling to a transition point, then a decay function from the transition point to the minimum of the following day. Several decay functions were compared, as described below.

The minute-by-minute BOM datasets for each site were processed to define points needed for fitted curves as follows.

Definition of values used

The [D.sub.min] and [D.sub.max] are the minimum and maximum temperatures for the day. The [H.sub.min] and [H.sub.max] are the times of day when the minimum and maximum temperatures occur. An additional subscript 0 is used for the previous day, 1 for the current day, and 2 for the following day. So [D.sub.max0] is the maximum on the previous day and [D.sub.min2] is the minimum on the following day. All times are expressed relative to the start of the current day, so [H.sub.min2] will be >1440 min.

Data analysis for each day

Although BOM records are commonly reported from 9 am to 9 am (09 : 00-09 : 00), the minimum daily soil temperature may occur at ~9 am, so it might not be clear whether the minimum reported for a given day had occurred before or after the maximum for that day. However, the minimum is rarely close to midnight, so all analysis was done using temperature records from 00:00 to 23:59. Time of day did not include daylight saving time, as all times reported for each temperature were in Eastern Standard Time (Longreach) or Western Standard Time (Perth and Geraldton).

The times of sunrise ([H.sub.rise]) and sunset ([H.sub.set]) for each day were calculated for each site using a Pascal version of a C program (Nautical Almanac Office 1990; Wouter 2011). The [H.sub.mid] was calculated as the time exactly halfway between sunrise and sunset, and [H.sub.night] was halfway between sunset and the following sunrise.

In the preliminary analysis, multiple linear regression was used to find the sun events most closely related to the timing of the temperature events. However, 5-10% of days may follow an atypical temperature pattern, giving a moderate number of points with high leverage, and these points distorted the multiple regression analysis. Therefore, for the final analysis, the median and the 33rd and 67th percentiles were used to define the normal range.

Time of minimum temperature

The [H.sub.min] was determined by [H.sub.rise] and [H.sub.mid] using the following formula:

[H.sub.min] = [H.sub.rise] + [A.sub.min]([H.sub.mid] - [H.sub.rise]) + [K.sub.min] (1)

where [A.sub.min] is a factor adjusted to minimise the difference between the 33rd and 67th percentiles of the difference between the estimated time of the minimum compared with the actual time of the minimum temperature over all days included in the study. This is equivalent to minimising the standard deviation of the difference between the model and actual values. The term [K.sub.min] is an offset, adjusted after setting [A.sub.min], to ensure that the estimated time of the minimum temperature equals the median time of the minimum temperature.

Time of maximum temperature

The [H.sub.max] was determined from the times [H.sub.mid] and [H.sub.set]: [H.sub.max] = [H.sub.mid] + [A.sub.max] ([H.sub.set] - [H.sub.mid]) + [K.sub.max] (2)

where [A.sub.max] was adjusted to minimise the difference between the 33rd and 67th percentiles of the difference between the estimated time of the maximum and the actual time of the maximum temperature; and [K.sub.max] was adjusted to ensure that the estimated time of the maximum temperature equalled the median time of the maximum.

Increase from minimum to maximum

Two different methods have been used by others for the increase in temperature from minimum to maximum. A sine curve centred halfway between the minimum and maximum temperatures (Watson and Beattie 1996; Roltsch et al. 1999) provides a symmetrical change over the period, covering two quadrants of the sine curve. In contrast, a sine curve centred at the minimum (Parton and Logan 1981; Goudriaan and van Laar 1994; Cesaraccio et al. 2001) gives a rapid initial increase, with the rate of change slowing until the maximum is reached, covering only the first quadrant of the sine curve.

The shape of the increase in temperature between the minimum and maximum was initially examined by fitting two partial (single quadrant) sine curves to the temperatures over this period. If a two-quadrant sine curve is an appropriate model, then these sine curves would have similar length (in minutes). Whereas if the single-quadrant model was a better fit, then the first sine curve would have a much shorter period than the second. This method also examines whether there is any transition point during the rising temperature, similar to the transition point used for the decrease in temperature later in the day.

In Eqns 3-5, [H.sub.x] is a potential transition point that must be found by the model, and [D.sub.x] is the temperature at that transition point. The first sine curve was from [H.sub.min] to [H.sub.x] and the second from [H.sub.x] to [H.sub.max]; [D.sub.x] is the actual temperature at time [H.sub.x], while time H and temperature D define the points to be fitted.

When H [less than or equal to] [H.sub.x]:

D = [D.sub.x] + ([D.sub.x] - [D.sub.min]) sin ( ([3[pi]/2] + [[pi]/2] [(H - [H.sub.min])/([H.sub.x] - [H.sub.min])]) (3a)

and when H > [H.sub.x]:

D = [D.sub.x] + ([D.sub.max] - [D.sub.x]) sin (0 + [[pi]/2] [(H - [H.sub.x])/([H.sub.min] - [H.sub.x])]) (3b)

When H = [H.sub.x] both equations simplify to D = [D.sub.x], so this provides a continuous curve of temperature from the minimum to maximum. The estimation was tested by varying [H.sub.x] from 1 h after the time of the minimum to 1 h before the maximum.

For each value of [H.sub.x] within the permitted range, the actual temperature at each minute for the period from minimum to maximum was compared with the temperature estimated by the above functions. The sums of squares of deviations of actual from estimate were recorded and the value of [H.sub.x] was found for which the sums of squares were minimised. This procedure was repeated for each day for which temperatures were available, so each day in the records had a separate calculation for [H.sub.x], the corresponding [D.sub.x], and the minimised sums of squares. The median value of the optimised [H.sub.x], relative to the time of minimum and maximum temperatures, was determined to find the appropriate sine curves for the period from minimum to maximum.

Decrease from maximum to the minimum on the following day

The same procedure was used to examine the decrease in temperature after the maximum, by fitting two curves for this period. However, in this case the period from the transition point may not be a sine curve so exponential and square-root decay functions were considered.

In the following equations, D is the temperature in degrees at time H, and [D.sub.x] is the temperature at the transition point, [H.sub.x], where [H.sub.x] is determined by testing all time points from 1 h after the maximum to 1 h before the minimum. Equations 5a-d are alternative functions to be tested.

When H [less than or equal to] [H.sub.x]:

D = [D.sub.x] + ([D.sub.max] - [D.sub.x]) sin ([[pi]/2] + [[pi]/2] [(H - [H.sub.max])/([H.sub.x] - [H.sub.max])]) (4)

and when H > [H.sub.x]:

D = [D.sub.x] + ([D.sub.x] - [D.sub.min2]) sin ([pi] + [[pi]/2] [(H - [H.sub.max])/([H.sub.x] - [H.sub.max])]) (5a)

D = [D.sub.min2] + ([D.sub.x] - [D.sub.min2]) exp (-b [H - [H.sub.x])/[[H.sub.min2] - [H.sub.x]]]) (5b)

where b is a temperature coefficient of ~2.4 (Parton and Logan 1981). This function sets D = [D.sub.x] when H = [H.sub.x], but D > [D.sub.min2] when H = [H.sub.min2].

D = [[[D.sub.min2] - [D.sub.x] x exp(A) + ([D.sub.x] - [D.sub.min2]) exp(B)]/1 - exp(A)]

where A = [[[H.sub.min2] - [H.sub.x]]/TC] and B = [[H - [H.sub.x]] /TC] (5c)

where TC is a temperature coefficient of ~240 for air (Goudriaan and van Laar 1994). This function sets D = [D.sub.x] when H = [H.sub.x] and D = [D.sub.min2] when H = [H.sub.min2].

D = [D.sub.x] - ([D.sub.x] - [D.sub.min2]) [square root of ([[H - [H.sub.x]]/[[H.sub.min2] - [H.sub.x]]])] (5d)

In this function (Cesaraccio et al. 2001), D = [D.sub.x] when H = [H.sub.x] and D = [D.sub.min2] when H = [H.sub.min2].

For each value of [H.sub.x] within the permitted range, the actual temperature at each minute for the period from maximum to minimum was compared with the temperature estimated by the above functions. The sums of squares of deviations of actual from estimate were recorded and the value of [H.sub.x] was found for which the sums of squares were minimised. This determines the transition point for the decrease from maximum to minimum and the resulting time is [H.sub.tp] and the corresponding temperature is [D.sub.tp.]

Models tested

Single-sine model

The single-sine model (Roltsch et al. 1999) assumes that daily temperature follows a simple sine curve with minimum and maximum 12h apart (06:00 and 18:00, respectively). However, separate curves are used for the change from [D.sub.max0] to [D.sub.min1], then from [D.sub.min1] to [D.sub.max1], then from [D.sub.max1] to [D.sub.min2]. If the minimum and maximum are the same on consecutive days, then this forms a single sine curve for the 24-h period. However, the shape may be asymmetrical when the daily temperatures are increasing or decreasing.

Two-sine model

This is similar to the single-sine method, except that the time period of rising temperature is 9 h and the time period for falling temperature is 15 h, with the minimum at 06:00 and the maximum at 15:00 (Roltsch et al. 1999).

Triple-sine model

Equation 6 is used for the period from minimum to maximum:

D = [[D.sub.min1] + [D.sub.max1]/2] + [[D.sub.max1] - [D.sub.min1]/2]

x sin ([pi] [(H - [[H.sub.min1] + [H.sub.max1]/2])/[[H.sub.max1] - [H.sub.min1]]]) (6)

Equation 4 is then used to the transition point, and Eqn 5a from the transition point to the minimum.

Exponential1 model

The model is as for the triple-sine method but Eqn 5b is used instead of Eqn 5a.

Exponential3 model

The model is as for the triple-sine method but Eqn 5c is used instead of Eqn 5a.

Square root model

The model is as for the triple-sine method but Eqn 5d is used instead of Eqn 5a.

Degree-days and arbitrary development units (ADUs)

Degree-day models have a lower threshold below which no degree-days accumulate. For the calculations given here, 10[degrees]C was used, but tests with other thresholds gave similar comparisons between models. For each hour of the day, the number of degree-days (DD) corresponding to temperature D was: DD = (D - threshold)/24, if D>threshold, or zero if D [less than or equal to] threshold.

An upper threshold may also apply, with no additional gain above that limit. In the model tested here, if the temperature was >30[degrees] then D = 30. Other thresholds, including no threshold, gave similar comparisons between the models.

The ADUs were calculated using the method described for Lucilia cuprina development (Vogt and Bedo 2001) as follows:

ADU = exp(-6.18 + 0.03D - [0.0043D.sup.2]) (7)

If D < 0 or D > 37 then ADU = 0. Values are calculated for each hour and accumulated over 24h for daily ADUs. The arbitrary units have been adjusted to provide 100 units in total for juvenile L. cuprina development, and this provides a value approximately half of the degree-day method described here.

Results

Temperature profile

Figure 1 shows soil temperature at 5 cm for Perth, with mean values for November-January, February-April, May-July, and August-October. These periods were chosen to combine similar times of sunrise and sunset. A 48-h period is shown to clarify the daily temperature cycles.

The maximum occurs at approximately the same time throughout the year (14 : 30 in May-October and 14:57 in November-April). The minimum is earliest in November-January (06:06) when sunrise occurs early and latest in May-July (07:33) when the time of sunrise is the latest. February-April (07 : 00) and August-October (06 : 31) have intermediate times for the minimum, although the average temperatures are much higher in February April.

Figure 2 presents the actual data for the two most extreme periods (November-January and May-July), as well as sine curves adjusted to pass through the minimum and maximum for the relevant periods, since sine curves are normally used for the period from minimum to maximum. A sine curve with a minimum and maximum adjusted to match the temperature and time of day of the actual minimum and maximum was a good fit to the actual data for the period from minimum temperature to maximum temperature, although the actual temperatures led the sine curve slightly during this period. After the maximum was passed, the same sine curve initially followed the actual temperature, but fell too rapidly to be a satisfactory model. A sine curve appeared to be adequate for the first part of the period to the minimum and was used for further modelling.

A sine set to the maximum, but with a midpoint rather than minimum at the actual minimum, rose much more rapidly than the actual temperature and was clearly a poor fit for soil temperatures over this period, although it has been used for air temperature (Parton and Logan 1981; Goudriaan and van Laar 1994). The same curve initially descended too slowly after the maximum had been reached, but then dropped rapidly after the transition point had been passed. This curve was not used for further analysis.

Time of minimum

Using Eqn 1, the difference between the 33rd and 67th percentiles was minimised with a value of 0.19 for [A.sub.min] and an offset ([K.sub.min]) of-28.5 min. There were indications of a possible seasonal factor providing an additional effect in Perth, but not in the other locations further north. Without data from additional southern sites this was not included in the model.

The 33rd percentile was 6.5 min earlier than the median and the 67th percentile 7.6 min later, indicating that the minimum temperature was within 14 min of the estimate on two-thirds of all occasions.

Time of maximum

From Eqn 2, the optimum value was found with a value of 0.097 for [A.sub.max] and an offset ([K.sub.max]) of 118.5. The 33rd percentile was 13 min earlier and the 67th percentile 15 min later than the estimated time, giving a range for two-thirds of the days of 28 min.

Increase from minimum to maximum

The first step was to find the value of [H.sub.x] in Eqns 3a and b that minimised the sums of squares of the difference between the estimated and actual values. This would correspond to a transition point, if there is such a transition, in the period from minimum to maximum. This value for [H.sub.x] occurred at a time 0.5 4- 1.0 min (mean and standard error of all days in the dataset) before the time point exactly midway between [H.sub.min] and [H.sub.max]. Therefore, the rise from minimum to maximum is symmetrical and can be modelled by a single sine curve rather than requiring a pair of curves, and so Eqns 3a and b can be combined to the single expression in Eqn 6.

Decrease from maximum to the minimum on the following day

For the period from maximum to minimum, a transition point was expected (from Fig. 2) and the requirement was to find which of several possible models was most appropriate for the period after the transition. One variable to be determined was the time of the transition point ([H.sub.x]), but in the case of the exponential decay curves (Eqns 5b and c) it was also necessary to find the optimum value of the temperature coefficient corresponding to those curves.

Using Eqns 4 and 5a-d, the optimum value of [H.sub.x] was found separately for each version of the model, together with temperature coefficients where required. The optimum values are shown in Table 1 with coefficients required to calculate the time of [H.sub.tp] and the corresponding value for [D.sub.tp]. All models gave a mean time for the transition point within an hour after sunset, although this varied throughout the year, since each model has different weightings for sunset and midnight. This time is about 3 h before the time point halfway between the maximum and the following minimum. This confirms that the decrease in temperature is strongly asymmetrical, with a rapid initial decrease followed by a much slower decline to the minimum.

The preceding calculations were based on separate analyses optimised for each day. However, if the model is to be used, then it is necessary to use an average model for all days. Table 1 shows the median sums of squares for each of the models covering the period from the maximum to minimum.

The exponential3 model had the smallest deviation from the actual values, although the triple-sine and exponential1 models were only slightly inferior, whereas the square-root model had a high deviation.

Summary of the recommended method

The preceding results provide the data needed to use the method as follows.

The time of the minimum temperature at 5 cm depth in soil occurred at:

[H.sub.min] = [H.sub.rise] + 0.19 ([H.sub.mid] - [H.sub.rise]) + 28.5 (8)

The maximum temperature occurred at:

[H.sub.max] - [H.sub.mid] + 0.097 ([H.sub.set] - [H.sub.mid]) + 118.5 (9)

Using the Exponential3 model, Eqn 5c, the transition point of the decrease in temperature occurred at:

[H.sub.tp] = [H.sub.set] + 0.26 ([H.sub.night] - [H.sub.set]) - 41 (10)

[D.sub.tp] = [D.sub.min] + 0.55 ([D.sub.max1] - [D.sub.min2]) (11)

These values can be calculated for any given location and date, and then entered into Eqns 4, 5c, and 6 to estimate the soil temperature at any required time of day.

Temperature during the period from the minimum to the maximum can be calculated using Eqn 6. Temperature from the maximum to [H.sub.tp] is calculated using Eqn 4. Temperature from [H.sub.tp] to [H.sub.min] is calculated using Eqn 5c.

Figure 3 shows the actual temperatures compared with the exponential3 model for the two most extreme periods from Fig. 1, November January and May-July.

Validation of the shape of the temperature curve

When the models were tested on 3-hourly data for 35 Australian sites, the simple single-sine and two-sine models differed substantially from the actual temperatures, whereas the models allowing a temperature transition between maximum and minimum were closer in absolute values and gave better correlations.

Table 2 shows that the median temperature was overestimated (by 0.69-0.76[degrees]C) by models that assume that the mean temperature over 24 h will be the same as the average of the maximum and minimum. The single-sine and two-sine models have symmetry from minimum to maximum and from maximum to minimum, and the difference between actual and estimated values was greatest during the period from maximum to minimum. In contrast, the triple-sine method provided an average temperature over the 24-h period that was within 0.22[degrees]C of the actual average of the eight 3-hourly values and had a correlation of 0.993. The exponential3 model was also satisfactory, with an average deviation of 0.31[degrees]C and a correlation of 0.992. The other functions deviated further from the correct values, particularly the exponential l model, which was no better than the two-sine model.

Validation of the degree-day model

The method was validated using hourly soil temperature data collected during 2005 and 2006 at an experimental site near Ballarat. For each hour of the day, the degree-day value for that hour was calculated for each model based on the actual minimum and maximum temperatures for each day. The development rate models used were a linear model (with minimum threshold 10[degrees]C and maximum threshold 30[degrees]C) and the Vogt Bedo model (Vogt and Bedo 2001).

For data over the whole period of 2005 and 2006, the single-sine and two-sine models overestimated development rates by 6-7%, whereas all the methods that allowed a transition point in the period from maximum to minimum estimated development rate within 1.6% of calculations based on actual hourly temperatures (Table 3). The exponential3 model had the lowest percentage error for both degree-days (0.66%) and ADUs (0.15%).

Discussion

Soil temperature at 5 cm depth increases rapidly during the day until a few hours after midday, then initially there is a rapid decrease, followed by a slower rate of decrease until the minimum is reached and the cycle is repeated. As a result, the temperature is at below-average values for a longer period than it is at above-average levels. This affects a range of air temperature models (Watson 1980), almost all of which overestimated the accumulated degree-days for New South Wales, unless local correction factors were applied. In California, a comparison of several different degree-day models for air temperature all gave overestimates of 3-12% (Roltsch et al. 1999).

Models using simple functions for estimation of hourly temperatures will overestimate development rates if the method assumes that the average daily temperature (mean of all times throughout the day and night) is the same as the average of the maximum and minimum temperature. Other methods have been tested here using a transition point to allow the temperature to be below the average of maximum and minimum for a longer period than it is above this average. These methods were developed from data at three different locations, but have been validated using 3-hourly data from many sites around Australia and for hourly data used to test degree-day models.

The exponential3 model gave satisfactory results on all sets of data and the best fit on the minute-by-minute data and hourly degree-day validation datasets. The triple-sine model performed moderately well on the minute-by-minute data and the degree-days, and was the best fit for the 3-hourly data. The exponential 1 model was a relatively poor fit for the 3-hour test data, and the square-root model had the greatest error of these four models for the hourly data.

Although the exponential3 model has been reported to be suitable for air temperature (Goudriaan and van Laar 1994) and the exponential1 model has been used for air temperature and soil temperature at 10cm (Parton and Logan 1981), no other models were compared in those reports. The exponential3 model is generally the best for soil temperature at 5 cm, although the triple-sine method could be satisfactory and is a simpler model that does not require a temperature coefficient. Unlike air temperature, soil temperature is subject to a damping effect from deeper soil levels, so although an exponential decay curve may be suitable for air temperature and shallow soil depths, it is possible that sine functions may be more appropriate at deeper levels. None of the functions tested was a perfect fit to the data, so more complex functions may be appropriate. Closer examination of soil temperature at deeper levels may determine whether a general model can be found that is suitable for all soil depths.

At deeper soil levels, the temperature at any given time of day is more dependent on the average soil temperature and whether this is increasing or decreasing over periods of days or weeks (Horton and Corkrey 2011), whereas at 5cm, the primary factor is the air temperature for the current day. Therefore, more complex models may be necessary to calculate degree-days at deeper soil levels by including the temperature on preceding days in the model.

In all the models which use a transition point for maximum to minimum (Parton and Logan 1981; Goudriaan and van Laar 1994; Roltsch et al. 1999; Cesaraccio et al. 2001), the time of the minimum, maximum, and transition point are related to the time of sunrise, midday, and sunset. This requires calculation of these times based on the latitude, longitude, and time of year. The results confirm that this is necessary, since the minimum and transition point in particular will vary depending on the time of year. The factor 0.19 for Amin indicates that [H.sub.mid] contributed only 19% and [H.sub.rise] contributed 81% to the timing of the minimum, so the time of the minimum is highly variable depending on the season of the year and the time of sunrise. In contrast, the factor 0.097 for Areax indicates that [H.mid] contributes 90.3% and the time of sunset only 9.7% to the timing of the maximum temperature, so the timing of the maximum is relatively constant throughout the year because there is only limited variation of [H.sub.mid]. The factor 0.31 used to obtain [H.sub.tp] indicates that the time of sunset contributes 69% of the timing for the transition point, so this varies substantially with season.

http://dx.doi.org/ 10.1071/SR12165

Received 29 February 2012, accepted 8 August 2012, published online 19 September 2012

Acknowledgments

I thank the Bureau of Meteorology for supplying the minute-by-minute data and the 3-hourly soil temperature data for Australian sites, and Sandra de Cat, John Larsen, and Norman Anderson for the hourly soil temperature data from Victoria.

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Brian Horton

Tasmanian Institute of Agriculture, University of Tasmania, PO Box 46, Kings Meadows, Tas. 7249, Australia.

Email: brian.horton@utas.edu.au

Table 1. Optimised coefficients for the four models compared for
the period from the transition point (Htp) to the minimum
temperature and median sums of squares of the difference between
the model and actual for each day of the period studied

The time of the transition point, [H.sub.tp] = [H.sub.set] +
TimeFactor x ([H.sub.night--[H.sub.set])--TimeOffset, where
[H.sub.set] is the time of sunset and [H.sub.night] is the time
of midnight (halfway between sunset and the next sunrise). The
temperature at the transition point, [D.sub.tp] = [D.sub.min2] +
TemperatureFactor x ([D.sub.max 1 -[D.sub.min2]). where Dmax, is
the maximum temperature and [D.sub.min2] is the minimum
temperature the following morning

Model             Time     Time     Mean minutes    Temperature
                 Factor   Offset    after sunset    Coefficient
                                   for [H.sub.tp]

Triple-sine       0.3       54           45             --
Exponential1      0.58     192           40            2.56
Exponentia13      0.26      41           43             422
Square-root       0.50     157           52             --

Model            Temperature   Median sums
                   Factor      of squares


Triple-sine         0.54          0.682
Exponential1        0.62          0.662
Exponentia13        0.55          0.612
Square-root         0.60          2.563

Table 2. Median difference between estimated and actual 3-hourly
temperatures over 24 h at 5 cm under soil for 35 Australian
locations and  correlation between estimate and actual

Values are medians of results from 35 locations
(33rd percentile to 67th  percentile)

Model               Estimate--actual            Correlation

Single-sine      0.76 (0.89-0.71)           0.959 (0.925-0.963)
Two-sine         0.69 (0.81-0.65)           0.986 (0.976-0.989)
Triple-sine     -0.219 (-0.190 to -0.296)   0.993 (0.987-0.994)
Exponential1    -0.722 (-0.648 to -0.853)   0.987 (0.983-0.988)
Exponential3    -0.312 (-0.303 to -0.388)   0.992 (0.987-0.993)
Square root     -0.403 (-0.365 to -0.488)   0.989 (0.984-0.991)

Table 3. Mean ([+ or -] standard deviation) degree-days or arbitrary
development units (ADUs) during 2005 and 2006, based on actual
hourly temperatures, or temperatures estimated by different models
and the percentage error of each model

%Error=(estimate--actual)/actual; expressed as a percentage.
All results for %error are significantly different from zero

Model                Degree-days               %Error

Actual            5.36 [+ or -] 4.61
  temperature
Single-sine       5.72 [+ or -] 4.83     6.86 [+ or -] 9.71
Two-sine          5.69 [+ or -] 4.82     5.90 [+ or -] 8.50
Triple-sine       5.31 [+ or -] 4.68    -0.72 [+ or -] 4.51
Exponential1      5.31 [+ or -] 4.66    -0.90 [+ or -] 4.63
Exponential3      5.32 [+ or -] 4.70    -0.66 [+ or -] 5.71
Square root       5.27 [+ or -] 4.64    -1.61 [+ or -] 4.59

Model                    ADUs                  %Error

Actual           2.163 [+ or -] 1.614
  temperature
Single-sine      2.316 [+ or -] 1.744   7.05 [+ or -] 10.95
Two-sine         2.303 [+ or -] 1.737    6.04 [+ or -] 9.58
Triple-sine      2.156 [+ or -] 1.649   -0.32 [+ or -] 4.75
Exponential1     2.150 [+ or -] 1.638   -0.63 [+ or -] 4.85
Exponential3     2.160 [+ or -] 1.657   -0.15 [+ or -] 5.26
Square root      2.136 [+ or -] 1.626   -1.27 [+ or -] 4.74
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Publication:Soil Research
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Geographic Code:8AUST
Date:Sep 1, 2012
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