## 1. Introduction

[2] Earth's thermosphere has been predicted [*Roble and Dickinson*, 1989; *Akmaev et al.*, 2006] to cool and contract in response to increasing concentrations of atmospheric CO_{2}, which is the dominant ultimate cooling agent of the thermosphere (downward molecular conduction connects the upper thermosphere with the lower thermosphere; CO2 is the primary loss mechanism below about 140 km altitude) [*Roble*, 1995; *Mlynczak et al.*, 2010]. Consequently, thermospheric density at fixed geometric altitudes should be gradually decreasing, and there is evidence, derived from the orbits of objects in low-Earth orbit, that this is occurring at a rate of 2 to 5% per decade (dependent on the phase of the solar cycle) near 400 km altitude [*Keating et al.*, 2000; *Emmert et al.*, 2004, 2008; *Marcos et al.*, 2005]. Long-term changes in O_{3}, H_{2}O, and CH_{4} also affect thermospheric densities [*Akmaev et al.*, 2006; *Roble and Dickinson*, 1989], and it appears that the entire upper atmosphere (including the mesosphere, thermosphere, and ionosphere) is responding to this radiative forcing [*Laštovička et al.*, 2008].

[3] Various methods have been used to estimate long-term upper atmospheric trends and their statistical uncertainties. The objective of most studies is to detect and quantify long-term changes that are not driven by solar forcing, which is the dominant source of thermospheric variations. We here define “trends” as long-term changes that may or may not be monotonic, and use the term “linear trends” to refer to the linear secular component of such changes. A single linear trend is usually assumed, but some studies [e.g., *Merzlyakov et al.*, 2009; *Liu et al.*, 2010] also consider piecewise linear trends. The trend analysis is usually performed on monthly [e.g., *Bremer*, 1992] or yearly [e.g., *Laštovička*, 2001] average anomalies, but daily or higher-resolution data are sometimes used [e.g., *She et al.*, 2009], particularly when the trend term is computed simultaneously with other climatological parameters such as seasonal terms. The selection of temporal resolution typically has little effect on the value of the trend estimate, but if not done properly it can greatly affect the uncertainty estimate, as we demonstrate in the following sections.

[4] Most studies have applied standard ordinary least squares (OLS) techniques to estimate the trend and its uncertainty, but these standard estimates assume that each data point is an independent measurement. If there is autocorrelation in the residual time series, then there are fewer degrees of freedom, and the estimated statistical uncertainty of the trend value will be biased low. *Tiao et al.* [1990] and *Weatherhead et al.* [1998] discuss the analytical effect of a first-order autoregressive (AR(1)) process on trend uncertainty, and some studies [e.g., *Marcos et al.*, 2005; *Liu et al.*, 2010] have applied Monte Carlo techniques to simulate (usually low-order) autocorrelation effects. However, the residual autocorrelation of upper atmospheric data can be quite complex [e.g., *Emmert and Picone*, 2010], and low-order AR models may not adequately account for the correlated data. We also note that resampling techniques such as bootstrapping and jackknifing [*Efron*, 1982] assume independence of the data points, and therefore do not account for autocorrelation.

[5] In this paper, we build on the AR(1) approaches of *Tiao et al.* [1990] and *Weatherhead et al.* [1998] by generalizing the trend uncertainty estimation to account for AR processes of arbitrary order. We employ direct parametric statistical inference techniques, which supersede Monte Carlo simulation. We apply the method to global average thermospheric density data, including daily, monthly, and yearly averages, to obtain realistic and internally consistent estimates of thermospheric density trends and their statistical uncertainty. We also examine the distribution of the residuals around the AR models and test whether they are consistent with the assumption of normally and identically distributed random errors, which facilitates construction of confidence intervals.