Distribution function of a spurious trend in finite-length dataset and detectability of the true trend

A linear trend estimated from a finite-length dataset with random internal variability has a spurious component which is the difference from the true trend caused by changes in external conditions or parameters. Some moments and distribution functions of the spurious trend depending on the length of data are derived theoretically under general statistical assumptions. When the internal variability has a normal distribution, the spurious trend also has a normal distribution. In general cases of non-normal distributions, we derived the distribution function of the spurious trend by the Edgeworth expansion.

In order to obtain the distribution function of a spurious trend, distribution function of the internal variability is necessary. We analyzed long time datasets generated by numerical experiments: a 15,200-year integration with a mechanistic circulation model under a purely periodic annual forcing and about 1,000-year integration with an atmosphere-ocean coupled general circulation model. With the obtained distribution function of the internal variability of the model atmosphere, detectability of the true trend was examined. In the stratosphere, the internal variability of zonal mean temperature has large seasonal variation due to occurrence of sudden warming events, and the annual mean of the data is not very appropriate to reduce the sample variance. We calculated necessary length of data for each month to detect the trend significantly. For 20-year data, we can detect the cooling trend of -0.5K/decade of zonal mean temperature in the polar stratosphere in summer with a statistical significance of 99%, while over 150 years are necessary in winter. In the troposphere, on the other hand, the distribution function of the internal variability is similar through a year, and the annual mean is appropriate. About 35 years are long enough to detect warming trend of 0.05K/decade in hemispheric and annual mean temperature.