## 1. Introduction

Detection of long-term trends in various types of atmospheric data has been studied extensively in the past 30 years (e.g., Hill et al. 1977; Reinsel et al. 1989; Tiao et al. 1990; Weatherhead et al. 1998; Von Storch and Zwiers 1999; Leroy et al. 2008a,b). There are many factors that affect the detection, including the magnitude of the trend to be detected, the temporal length of available observations, the magnitude of the variability and autocorrelation of the noise in the data, and uncertainty in the observations. During the past decade, two trend detection methods have been widely used: the method of Weatherhead et al. (1998, hereafter W98) and the method of Leroy et al. (2008b, hereafter L08). It was not clear if the uncertainty of climate trends and the time to detect them are sensitive to the method used and under what circumstances that one method is preferred over the other. That was the motivation for this study. The W98 method is detailed in section 1a. Section 1b details the L08 method. Comparison of both methods is given in section 1c.

### a. The W98 method

Let *Y*_{t} be the measurement of interest. The trend model assumes linearity, as expressed in *α* is a constant term, *S*_{t} is the seasonal component, *X*_{t} = *t*/12 is the linear trend function, *ω* is the magnitude of the trend per year, *N*_{t} is the unexplained noise of the measurement data, and *t* = 1, 2, 3, … , *T* months. The noise *N*_{t} can be obtained from the deseasonalized and detrended time series monthly data. A deseasonalized series is obtained by subtracting the monthly-mean value from each month. For example, the average of all January months is subtracted from each individual January month, with the same process used for all months in the year. This process removes the bulk of Earth’s very large and repeatable seasonal cycle from the climate record. Then the trend is removed from the deseasonalized series, generating the *N*_{t} deseasonalized and detrended time series.

*N*

_{t}follows a first-order autoregressive model (AR1). The uncertainty of trend estimate given by the W98 is

The estimate of the trend per year is *n* = *T*/12 is the number of years of time series data, *N*_{t} monthly time series, and

### b. The L08 method

*Y*

_{i}is the measurement,

*α*is a constant term,

*t*

_{i}is the linear trend variable,

*m*is the magnitude of the trend per month,

*N*

_{i}is the unexplained noise of the measurement data, and

*i*= 1, 2, 3, … ,

*T*months. The uncertainty of trend estimate given by the L08 is

*dt*is the interval between observations. In this study

*dt*= 1, and therefore

*T*.

The L08 takes theoretical autocorrelation at all lags (i.e., from −∞ to +∞) into consideration to obtain the uncertainty of the trend estimate [Eq. (2)]. However, we do not know the theoretical autocorrelation

### c. Method comparison

The obvious difference in the two uncertainties [Eqs. (2) and (3)] is the last term on the right-hand side of the equations, which indicates how the correlation structure is handled by the two methods. We define decorrelation time *τ* as the time needed for data to be uncorrelated (Von Storch and Zwiers 1999). The decorrelation time for W98 and L08 was set as

The W98 assumes the noise *N*_{t} follows the AR1 model, which requires only one estimated model parameter (i.e., the first lag sample autocorrelation coefficient *N*_{t} from the AR1 model will likely cause the W98 to perform poorly. The L08 does not assume any autoregressive model or any correlation structure for the *N*_{t} series. It takes into account theoretical autocorrelation coefficients at all lags in computing the uncertainty of trend estimate. It is therefore theoretically versatile. However, all theoretical coefficients are not known in practice, and rather are estimated by their corresponding sample autocorrelation coefficients. Mathematically it is not possible to include sample coefficients at all lags because

Method comparison summary.

The organization of the paper is as follows: Section 2 describes and compares the two methods using the Monte Carlo simulation with theoretically and randomly constructed time series. The findings from this section are applied to the reflected shortwave and emitted longwave irradiances at the top of the atmosphere (TOA), as estimated during March 2000 through June 2011 from observed Clouds and the Earth’s Radiant Energy System (CERES) instruments on the National Aeronautics and Space Administration (NASA) *Terra* satellite. This application is detailed in section 3. Section 4 concludes the study.

## 2. Monte Carlo simulation study

In section 2a, the Monte Carlo simulation settings are described. Simulation results are given in section 2b.

### a. Simulation settings

We compare the W98 and L08 methods with the theoretically and randomly generated first- and second-order autoregressive (i.e., AR1 and AR2) time series. The AR1 series follows *t*, *t,* or *N*_{t} is a deseasonalized and detrended time series as described in section 1. Note that the white noise series can be thought of as AR1 with

The AR2 series follows *m*_{1} and *m*_{2} of the polynomial *m*_{1}| and |*m*_{2}| < 1, then the autocorrelation function (ACF) exhibits a mixture of two exponential decay patterns. In case 2, if both roots are complex conjugates in the form of *m*_{1} = *m*_{2}, then the ACF shows a single exponential decay pattern. We study all three cases of the AR2 series. We limit our simulation study to AR2 for simplicity because it already captures phenomena that are likely to complicate the W98.

Table 2 outlines the time series data that are used with the W98 and L08 methods in the first three columns. The sensitivity of both methods to record length (for 10, 20, …, 100 years) is included in this study. The exploration matrix is a 13 time series by two methods by 10 record lengths (13 × 2 × 10) matrix. The column “series number” will be used to facilitate all output figures. Figure 1 shows in detail how the computer simulation is done. The outer loop is repeated 13 times, one for each time series. The middle loop is repeated *M* times, one for each simulation. The determination of the number of simulations *M* will be given later in this section. For each simulation, the inner loop computes the *T* and *T*, the two methods could provide similar and very small uncertainties, but the similarity is primarily attributable to the effect of large *T*. Therefore, comparing the *T*.

Exploration matrix for the W98 and L08 methods. The record lengths for time series data are 10, 20, … , 100 yr.

Parameters *t* = 0, or *N*_{1} is obtained from *N*_{1} to *N*_{1200}) are obtained, representing a 100-yr monthly series. The ACFs of all chosen 100-yr monthly AR1 models are given in Fig. 2. Sample autocorrelation coefficients up to 10 log_{10}(*T*) lags, where *T* = 1200, are computed and shown in the ACFs (Venables and Ripley 2002). Any autocorrelations that are outside the dashed lines are considered statistically significant within a 95% confidence level. For the white noise series (Fig. 2a), only lag 0 is significant, which is the property of the serially uncorrelated series. Figures 2b to 2d give ACFs exhibiting exponentially decay patterns. As parameter

The ACF of all 100-yr monthly AR1 models with the series number from Table 2.

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

The ACF of all 100-yr monthly AR1 models with the series number from Table 2.

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

The ACF of all 100-yr monthly AR1 models with the series number from Table 2.

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

Parameters *N*_{−1} and *N*_{0} sampled from a Norm(0, 1) distribution (i.e., *N*_{1} is obtained from *N*_{1} to *N*_{1200}) are obtained, representing a 100-yr monthly series. The ACF of the first case has a mixture of two exponential decay patterns (Figs. 3a–c). The ACF of the second case exhibits a damped sinusoid pattern (Figs. 3d–f). The ACF of the third case has a single exponential decay pattern (Figs. 3g–i).

The ACF of all 100-yr monthly AR2 series with the series number from Table 2.

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

The ACF of all 100-yr monthly AR2 series with the series number from Table 2.

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

The ACF of all 100-yr monthly AR2 series with the series number from Table 2.

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

*M*in Fig. 1) needed for all chosen time series using the variance reduction technique (Ross 2006). The average

*M*increases. We evaluate the sensitivity of using different numbers of simulations to simulation error (the standard deviation of the average). We would expect simulation error to decrease as the number of simulations increases. Simulation error generally can be calculated from

*U*is the value of our simulated variable of interest, which is

*U*from

*M*simulations. Figure 4 shows the sensitivity of the number of simulations to simulation errors (as percent of the

Sensitivity of the number of simulations to the simulation errors (as percent of its simulated mean) for all 100-yr monthly series (i.e., AR1 and AR2 series) with the W98 method.

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

Sensitivity of the number of simulations to the simulation errors (as percent of its simulated mean) for all 100-yr monthly series (i.e., AR1 and AR2 series) with the W98 method.

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

Sensitivity of the number of simulations to the simulation errors (as percent of its simulated mean) for all 100-yr monthly series (i.e., AR1 and AR2 series) with the W98 method.

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

For the L08 method, the criterion for estimating the decorrelation time _{10}(*T*) lags, then only sum all coefficients of statistically significant lags to get the *M* (from Fig. 1) to 10 000 simulations for the exploration matrix in Table 2.

As in Fig. 4, but for the L08 method.

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

As in Fig. 4, but for the L08 method.

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

As in Fig. 4, but for the L08 method.

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

### b. Simulation results

*M*= 10 000 simulations for all 13 series. We draw our conclusion based on comparing each method’s

*T*− 1 degrees of freedom (Montgomery and Runger 1994). We can conclude that the distribution of simulated

*T*degrees of freedom. The tail regions of the distributions are possible but less probable.

The box plot of simulated

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

The box plot of simulated

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

The box plot of simulated

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

To focus on more critical information such as the quartile and median values, the vertical ranges of the figures are not extended out to the extreme values, which are usually on the tail regions of the distributions. These figures are intended to show the decreasing trend of ranges of

Table 3 summarizes the simulated performances obtained from the W98 and L08 methods for these five distinctive performances of the median *τ* estimate is dominated by the negative coefficients. The likelihood (as the percent of total 10 000 simulations) of the L08’s negative *τ* value is shown in Fig. 7. As the record length increases, the likelihood becomes smaller. For all series, except the series with damped sinusoid ACFs, the L08 has a likelihood of no more than 3% for the 10-yr record length to give negative *τ* values. For the series with damped sinusoid ACFs (Fig. 7c), the L08 has higher likelihood (as high as 28%) for the 10-yr record length.

The likelihood of negative *τ* values from the L08 for all simulated series, based on the percent total of the 10 000 computer simulations: (a) AR1 series; (b) AR2 series with ACF exhibiting a mixture of two exponential decay patterns (case 1, AR2_1); (c) AR2 series with ACF exhibiting damped sinusoid pattern (case 2, AR2_2); and (d) AR2 series with ACF exhibiting a single exponential decay pattern (case 3, AR2_3).

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

The likelihood of negative *τ* values from the L08 for all simulated series, based on the percent total of the 10 000 computer simulations: (a) AR1 series; (b) AR2 series with ACF exhibiting a mixture of two exponential decay patterns (case 1, AR2_1); (c) AR2 series with ACF exhibiting damped sinusoid pattern (case 2, AR2_2); and (d) AR2 series with ACF exhibiting a single exponential decay pattern (case 3, AR2_3).

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

The likelihood of negative *τ* values from the L08 for all simulated series, based on the percent total of the 10 000 computer simulations: (a) AR1 series; (b) AR2 series with ACF exhibiting a mixture of two exponential decay patterns (case 1, AR2_1); (c) AR2 series with ACF exhibiting damped sinusoid pattern (case 2, AR2_2); and (d) AR2 series with ACF exhibiting a single exponential decay pattern (case 3, AR2_3).

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

Thus far, we have used

Percent differences for uncertainty of trend estimate

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

Percent differences for uncertainty of trend estimate

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

Percent differences for uncertainty of trend estimate

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

## 3. Trend analysis for Clouds and the Earth’s Radiant Energy System

CERES instruments measure broadband radiances in three channels: the shortwave (wavelength from 0.2 to 5 *μ*m), the longwave (wavelength from 5 to 50 *μ*m), and the total (wavelength from 0.2 to 50 *μ*m). Observed radiances are converted to TOA irradiances using angular distribution models (ADMs) (Loeb et al. 2001, 2005). A detailed description of CERES and its instruments is provided in Wielicki et al. (1996).

The 136 months of reflected shortwave and emitted longwave irradiances at TOA estimated from observed CERES broadband radiances from March 2000 through June 2011 [Energy Balanced and Filled (EBAF) edition 2.6r; Loeb et al. 2009] are used in this study. The reflected shortwave and emitted longwave irradiances monthly series are globally averaged and deseasonalized. No statistically significant trend was found in either deseasonalized series. This is an indication that the current record length may be insufficient to detect any irradiance change. Figure 9 shows time series plots for deseasonalized and detrended global average reflected shortwave and longwave data. The standard deviations ^{−2} for shortwave and longwave, respectively. These values are consistent with those reported in Kato (2009).

Time series plots of monthly deseasonalized and detrended global averages for (a) reflected shortwave irradiances and (b) emitted longwave irradiances.

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

Time series plots of monthly deseasonalized and detrended global averages for (a) reflected shortwave irradiances and (b) emitted longwave irradiances.

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

Time series plots of monthly deseasonalized and detrended global averages for (a) reflected shortwave irradiances and (b) emitted longwave irradiances.

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

Figure 10 gives autocorrelation functions (Figs. 10a,b) for both monthly series. Figures 10c and 10d provide autocorrelation functions for both monthly series if they were to behave as the perfect AR1 series, using the estimated lag-1 sample autocorrelation coefficient from the shortwave (Fig. 10c) and longwave (Fig. 10d). The ACFs of both series (Figs. 10a,b) do not exhibit purely damped sinusoid.

Autocorrelation functions of deseasonalized and detrended monthly series for (a) reflected shortwave irradiances and (b) emitted longwave irradiances. Autocorrelation functions if the (c) reflected shortwave and (d) emitted longwave irradiances were a perfect AR1 time series using lag-1 sample autocorrelation coefficients estimated from the corresponding series.

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

Autocorrelation functions of deseasonalized and detrended monthly series for (a) reflected shortwave irradiances and (b) emitted longwave irradiances. Autocorrelation functions if the (c) reflected shortwave and (d) emitted longwave irradiances were a perfect AR1 time series using lag-1 sample autocorrelation coefficients estimated from the corresponding series.

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

Autocorrelation functions of deseasonalized and detrended monthly series for (a) reflected shortwave irradiances and (b) emitted longwave irradiances. Autocorrelation functions if the (c) reflected shortwave and (d) emitted longwave irradiances were a perfect AR1 time series using lag-1 sample autocorrelation coefficients estimated from the corresponding series.

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

Using Eqs. (3) and (2), the uncertainties of trend estimate for the shortwave series are 1.27 and 1.48 mW m^{−2} for the W98 and L08, respectively. For the longwave series, the uncertainties of trend estimate for the longwave are 1.40 and 2.31 mW m^{−2} for the W98 and L08, respectively. Uncertainty of trend estimate for the longwave series is higher than that for the shortwave series. By examining Figs. 10a and 10b, the longwave irradiance series has more significant lags (at a 95% confidence level) than the shortwave series. Consequently, the longwave series takes a longer time to have monthly data uncorrelated (i.e., larger decorrelation time), and, in turn, its estimated uncertainty is higher than that of the shortwave series.

The shortwave ACF looks similar to the perfect AR1 (Figs. 10a,c). From section 2, the simulations of the AR1 series for 10-yr record length (Fig. 8a) have shown that the W98 method produces better uncertainty (in terms of smaller percent differences from the theoretical value) than that from the L08 method. Therefore, we recommend using the W98 method for the shortwave series. The longwave ACF does not exhibit a purely exponential decay nor damped sinusoid pattern, but rather a mixture of both patterns, making the exponential decay slower than a purely exponential decay pattern. This behavior can be observed by comparing Figs. 10b and 10d. The W98, which is based on exponential decay of the ACF pattern similar to Fig. 10d, underestimates the uncertainty of trend estimate for the longwave series with its ACF as in Fig. 10b, while the L08 provides a more reasonable value. Therefore, we recommend using the L08 method for the longwave series.

*n*needed to detect a trend (per year) of magnitude

^{1}as estimated by the W98:

Number of years required to detect various

Citation: Journal of Climate 27, 9; 10.1175/JCLI-D-13-00400.1

Number of years required to detect various

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Number of years required to detect various

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As an example, based on our earlier recommendation of using the W98 to estimate the uncertainty for the reflected shortwave irradiances, if the trend per year is about 5.06 mW m^{−2} [i.e., ^{−2} and the ^{−2} [i.e., ^{−2} and the ratio of 0.01], it will take about 84.4 yr to detect it using the L08 method.

## 4. Conclusions

In this paper, two trend analysis methods, the methods of W98 and L08, are compared. The two methods are based on linear regression models with covariance structures of the time series data properly accounted for under different underlying assumptions. The W98 assumes that the time series data follow the first-order autoregressive model (AR1), where its autocorrelation function (ACF) decays exponentially with lags. In practice, only the lag-1 sample autocorrelation coefficient is used to estimate the uncertainty of the trend estimate making it simple to use. However, with any departure from the exponential decay pattern, the W98 is expected to perform poorly.

The L08 method, on the other hand, does not assume any autoregressive model or any correlation structure. It uses theoretical autocorrelation coefficients at all lags (i.e., *μ* autocorrelation coefficient) in the analysis making it ideally versatile. In practice, the theoretical coefficients *T* months. Taking sample coefficients at all available lags in estimating L08’s uncertainty of the trend will provide zero uncertainty; specifically, _{10}(*T*) lags in the summation.

Because the

For the AR2 with damped sinusoid ACFs, the L08 gives reasonable medians, while the W98 gives significantly higher medians than the theoretical values. The ranges of *τ* value, which, in turn, makes a negative

From the Monte Carlo simulation study, the white noise monthly series has a maximum percent difference from the theoretical (truth) of 1% and 2% from the W98 and L08, respectively. The AR1 series has a maximum difference of 26% using the W98 and 38% using the L08. The AR2 series with a mixture of two exponential decay ACF patterns has a maximum difference of 17% using the W98 and of 26% using the L08. The AR2 series with single exponential decay ACF pattern has a maximum difference of 22% using the W98 and of 20% using the L08. The AR2 series with the damped sinusoid ACF pattern has a maximum of 73% using the W98 and 30% using the L08.

Both methods are then applied to the time series of deseasonalized anomalies derived from monthly and globally averaged reflected shortwave and emitted longwave irradiances at the top of the atmosphere (TOA), as estimated from CERES broadband radiances measurements during March 2000 through June 2011. Examination of the autocorrelation structure shows that the reflected shortwave series has an exponential decay pattern, while the longwave series has a mixture of exponential decay and damped sinusoid patterns. Therefore, we recommend using the W98 with the reflected shortwave series and the L08 with the longwave series for trend analysis. The uncertainty of trend estimate for the shortwave series is approximately 1.27 mW m^{−2}. Its month-to-month standard deviation is 0.51 W m^{−2}, and if the trend per year is 100 times smaller than this standard deviation (i.e., trend per year of approximately 0.0051 W m^{−2}), it will take about 52.3 yr to detect it. For the longwave series, the uncertainty of the trend estimate is about 2.31 mW m^{−2}. Its month-to-month standard deviation is 0.45 W m^{−2}, and if the trend per year is 100 times smaller than this standard deviation (i.e., trend per year of approximately 0.0045 W m^{−2}), it will take about 84.4 yr to detect it. Note that while the results in this paper are applied to radiation data as an example, they apply in general to all climate change time series.

One potential for future study is to explore other more accurate methods for estimating

We have shown that estimates of climate trend uncertainties can be sensitive to the methods used. The tests performed in this paper can be applied to any climate time series to determine if the results are sensitive to the method used, either the W98 or L08. If the results are sensitive, then the choice of the method should be based on the physical nature shown in the autocorrelation structure of the climate time series.

## Acknowledgments

We thank Dr. Stephen Leroy for useful discussions and all reviewers’ comments for valuable insights. We also thank Ms. Amber Richards and Dr. Joe A. Walker for proof reading the manuscript. This work is supported by the NASA CLARREO project.

## APPENDIX

### Derivation for the AR2 Theoretical Value Given in Section 2a

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^{1}

The testing power is the power to detect correctly 90% of the time a trend that is at least twice as large as its uncertainty (i.e.,