The study area was the state of Nebraska, which is one of the main corn production states in the United States (USDA NASS, 2014; USDA NASS, 2015). Hourly averaged air temperature data were obtained from the High Plains Regional Climate Center’s Automated Weather Data Network (AWDN) through the online Climate Data Services (http://www.hprcc.unl.edu/services, accessed 10 December 2015). These data were quality controlled by the HPRCC staff with a spatial regression test; the advantages of this test were stated by Hubbard and You (2005), Hubbard et al. (2007), and You et al. (2008). A combination standard of data completeness and corn production representativeness was used to choose the study locations. From the beginning year of record to year 2015, the maximum acceptable amount of missing data for each station for this study was set at 5%. Missing data were replaced by reliable estimates, estimates based on weighted linear regression from surrounding stations, or unreliable estimates (HPRCC, 2015). Only two of the stations had no unreliable estimates of hourly temperature data, but that would be too few to represent the entire state’s climate. Therefore, stations with up to 0.03% unreliable estimates were included in this study; these unreliable estimates were manually checked to ensure that they are climatologically reasonable. By consulting with local agronomists, a total of 14 observing stations (40.08°–42.47°N and 96.48°–101.72°W, Fig. 1) that are located in active corn production areas were chosen for the analysis. Depending on the station, the beginning year of study spans from 1982 to 1991. The elevation of the stations ranges from 347 to 1029 m. In this study, active corn growing season was defined as from May 1 to September 30 based on the USDA reports (USDA NASS Agricultural Statistics Board, 1997; USDA NASS, 2010). The obtained hourly temperature data were used to compute daily temperatures, including maximum, minimum, and average temperatures. During a 24-hour period (i.e., from 0:00 to 23:59), the highest hourly temperature was considered as daily maximum temperature; the lowest hourly temperature was considered as daily minimum temperature; and the arithmetic mean of hourly temperature was considered as daily average temperature. This daily average temperature often differs from that derived from daily maximum and minimum temperatures alone.
In order to test the sensitivity of estimation methods to different temperature thresholds, three sets of lower and upper thresholds that are commonly used for corn were included in the analysis. They are: 8° and 29°C (Butler and Huybers, 2012), 10°C (predominantly used by seed companies) and 30°C (McMaster and Wilhelm, 1997), as well as 8° (used in crop models such as CERES-Maize and Hybrid-Maize) and 34°C (Kropff and van Laar, 1993). In addition to degree-days that are between lower and upper thresholds (i.e., DD8, 29, DD10, 30, DD8, 34), the performance of different estimation methods on degree-days that are above upper thresholds (i.e., DD29+, DD30+, DD34+) were also analyzed in this study. Accumulated above-upper-threshold temperatures have often been used to measure heat stress (Butler and Huybers, 2012; Lobell et al., 2011).
First, total growing season degree-days were calculated based on the observed hourly temperature data for each metric of thermal time at the study locations using Eqs. (2.1), (2.2), and (2.3), as described in Lobell et al. (2011):
Where N is the number of days (153) for crop development over the growing season spanning from May 1 to September 30, unitless; DDd is the daily degree-day, °Cžday; DDh is the hourly degree-day, °Cžday; Th is the hourly temperature, °C; Tlower is the lower threshold, °C; and Tupper is the upper threshold, °C.
Second, daily degree-days were estimated based on the calculated daily temperature data for each metric of thermal time at the study locations. A total of six estimation methods are evaluated in this study: Tavg-based rectangle method, adjusted Tmin and Tmax rectangle method, single-sine and double-sine methods with horizontal cut-off technique, and single triangulation and double triangulation methods with horizontal cut-off technique. For the two rectangle methods, Eqs. (2.4) and (2.5) were used to estimate daily degree-days, respectively. The detailed formulas to estimate daily degree-days for single-sine, double-sine, single-triangulation, and double-triangulation methods with horizontal cut-off technique are found at the UC IPM (2005). Eq. (2.1) was used to calculate total growing season degree-days for the six estimation methods.
Where Tavg_adj is the adjusted daily average temperature, °C; Tmax_adj is the adjusted daily maximum temperature, °C; and Tmin_adj is the adjusted daily minimum temperature, °C. They are adjusted to lower threshold if they are below the lower threshold, and to upper threshold if they are above the upper threshold.
For the six metrics of thermal time analyzed in this study, degree-days approximated with hourly temperature was taken as true. The differences between degree-days estimated with daily temperature and degree-days approximated with hourly temperature were considered as errors. According to the recommendations from Chai and Draxler (2014), a combination of statistical metrics of root mean square error (RMSE) and mean absolute error (MAE) was used to assess the performance of different estimation methods. At every study location, Eqs. (2.6) and (2.7) were used to calculate RMSE and MAE for each metric of thermal time during the study period, respectively.
In these equations, n is the number of total study years at the study location, unitless; ei is the error of total growing season degree-days, °Cžday.