Probability Distributions of Monthly Degree Day Variables at U.S. Stations. Part I: Estimating the Mean Value and Variance from Temperature Data

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  • 1 Climate Analysis Center, National Meteorological Center, NWS/NOAA, Washington, DC 20235
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Abstract

This paper examines the question of how strongly the skewness of the daily temperature variable (t) affects estimating the mean value and variance of the corresponding degree day variable (q) at U.S. stations where the q(t) relationship is nonlinear. Mean and variance values for monthly q were estimated from t statistics for monthly periods by use of two t models, one using skewness data and one not, and the results were compared with observed data. When q(t) is nonlinear, i.e., for months when the average daily temperature (μ) differs by fewer than 10°C (18°F) from the degree day base temperature (b), the accuracy of estimation was improved from about 5–10% to 1–3% when a one-parameter gamma function model instead of a Gaussian model was used. The gamma function model provided estimates that fall within the sampling error of the verification data for both daily and monthly average q parameters. The results suggest that when |b−μ|<10°C, q parameters can be estimated accurately only by use of models that take t skewness into account. Data are also presented suggesting that the variation of (a) skewness and (b) number of unique temperature “events” in a month between neighboring stations and from month to month at the same station is gradual. This opens the possibility of accurate estimation of daily and monthly average q parameters at intermediate locations and periods for which temperature data do not exist.

Abstract

This paper examines the question of how strongly the skewness of the daily temperature variable (t) affects estimating the mean value and variance of the corresponding degree day variable (q) at U.S. stations where the q(t) relationship is nonlinear. Mean and variance values for monthly q were estimated from t statistics for monthly periods by use of two t models, one using skewness data and one not, and the results were compared with observed data. When q(t) is nonlinear, i.e., for months when the average daily temperature (μ) differs by fewer than 10°C (18°F) from the degree day base temperature (b), the accuracy of estimation was improved from about 5–10% to 1–3% when a one-parameter gamma function model instead of a Gaussian model was used. The gamma function model provided estimates that fall within the sampling error of the verification data for both daily and monthly average q parameters. The results suggest that when |b−μ|<10°C, q parameters can be estimated accurately only by use of models that take t skewness into account. Data are also presented suggesting that the variation of (a) skewness and (b) number of unique temperature “events” in a month between neighboring stations and from month to month at the same station is gradual. This opens the possibility of accurate estimation of daily and monthly average q parameters at intermediate locations and periods for which temperature data do not exist.

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