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35 calculation procedure. However, this thermal gradient bound- from the daily maximum temperature to the daily minimum ary condition is not accurate if extrapolated to too great a in about 15 hours. A more accurate air temperature prediction depth. method would incorporate this non-sinusoidal pattern. In order to obtain a more representative pattern of daily air temperatures, data over an entire year were obtained from the Numerical Solution of the Model Automatic Weather Station (AWS) in the LTPP database and This model was solved numerically using a finite difference analyzed using a seasonal trend decomposition time series approximation method, together with required input data analysis. The trend trace is a moving average of the measured (i.e., hourly solar radiation, air temperature, wind speed, and data, which represents the daily average temperature through- model parameter values). In the numerical solution, the out the year. The "seasonal" trace is obtained by subtracting pavement thickness was divided into cells, which are thinner the trend line from the measured data and finding a local poly- near the surface and thicker at deeper levels. Each cell is given nomial which best fits the result. This trace represents the reg- a temperature (equal to air temperature) at the start of the ular pattern of daily air temperature, which is used instead calculation as an initial condition. The model then calculates of a sinusoidal function. a new temperature for each cell (several times for each simu- With a known daily pattern of air temperature, the hourly lated hour) at each time step. air temperatures can be reconstructed from daily maximum and minimum measured data. First, the daily average air tem- perature data are taken from the trend trace. Then, the trend Obtaining Hourly Climatic Input Data and the seasonal traces obtained from the time series analysis For any pavement site, model calculation requires accurate are added together. Finally, the result is linearly transformed site-specific hourly climatic data and model parameters, to fit the measured data, day by day. The non-sinusoidal pre- including hourly solar radiation, hourly air temperature, and dicted daily temperature patterns at six different sites are daily average wind speed data in an hourly format. shown in Figure 32. Hourly solar radiation can be collected from the National More details on this temperature model are presented in Solar Radiation Database (NSRDB). Hourly solar radiation Appendix B, including comparisons of the predicted pave- data are modeled using State University of New York at Albany ment temperatures at different depths with the measured (SUNY) or Meteorological-Statistical (METSTAT) models temperatures. Also presented are maps of the geographic dis- based on satellite images, covering nearly all parts of the coun- tributions of the temperature variables of albedo, emissivity, try from 1990 to 2005. and absorptivity, all of which were shown by a sensitivity Daily average wind speed can be collected directly from the analysis to be important input variables. Virtual Weather Station program in the LTPP. Additionally, daily wind speed can be obtained directly from the National Stiffness, Tensile Strength, Climatic Data Center (NCDC) or the meteorological network Compliance, and Fracture in each state. Hourly wind speed is preferred, but such data Properties of Mixtures are difficult to obtain and more vulnerable to environmental conditions, adding difficulty in the interpolation. However, The properties of a HMA mixture in an overlay must be the model is not overly sensitive to the wind speed such that estimated both accurately and with computational efficiency daily values are adequate. to achieve an overlay design resistant to reflection cracking. Hourly air temperature data are not commonly available, The stiffness and compliance of the mixture must be calcu- yet reasonable estimates of these hourly temperatures are lated at widely different temperatures and loading rates (ther- needed for accurate temperature calculations. In order to mal and traffic). The tensile strength must also be calculated estimate hourly wind speed data, a method was developed to over the same wide ranges of temperature and loading rates. estimate hourly air temperatures from daily maximum and The fracture properties (i.e., Paris and Erdogan's Law coeffi- minimum air temperatures. Recorded daily maximum and cients) must be calculated. These coefficients are also sensi- minimum air temperatures can be obtained easily from the tive to temperature and loading rates. For these reasons, ANN Virtual Weather Stations in the LTPP database or NCDC. algorithms which reproduce Witczak's 1999 (2) and 2006 (3) A conventional method to impute hourly air temperatures Complex Modulus models were developed to form the basis fits a sinusoidal function to daily maximum and minimum air for calculating the overlay stiffness under traffic loads and temperatures (e.g., 32, 33, 34). However, the daily profile of computing the viscoelastic thermal stress for thermal reflec- air temperature is not exactly sinusoidal. Typically, air tem- tion cracking. The method used to construct ANN algorithms perature rises from the daily minimum temperature to the is described in the literature (22). The accuracy with which daily maximum temperature in about 9 hours, and decreases these algorithms reproduce the Witczak Complex Modulus