concentration of atmospheric CO2 increased by 4.7% between 1854 and 1918 ( 8 ), only one-quarter of the 1951–1980 rate. Third, chloroflourocarbons and similar ozone-depleting chemicals were not yet in use, and as there is considerable evidence ( 9 , 10 ) for stratospheric control of climate, it is desirable to use a reference period before changes in stratospheric chemistry by chloroflourocarbons began. Fourth, the major erratic changes in the timing of the seasons described in ref. 2 appear to have begun about 1920, so the early reference period is largely free of their effects. Opposed to these considerations are poorer spatial coverage of the temperature series and the presence of several major volcanic events (11) during the earlier period. There are no solar irradiance measurements without the confounding influence of the atmosphere in the early period and only a few near the end of the 1951–1980 period. Keeling’s CO2 measurements began in 1958, so the 1951–1980 period has better CO2 data than the earlier period. The average and median values of the 1854–1918 data are 171.9 mK and 180.0 mK (Northern Hemisphere) and 150.7 mK and 159.0 mK (Southern Hemisphere) below the 1951–1980 reference. Because the SD (s) of the raw Tg(t) series during the 65-year reference period is 57.7 mK, the 1990 temperature of 635 mK above the base temperature is, at a minimum, an 11 s increase.

The CO2 data are as listed in table A.6 of ref. 8 up to 1955, and the Mauna Loa averages through 1994 from the Oak Ridge National Laboratory data set ndp001r5 ( 12 ) since 1955. The early data has been interpolated from irregular and inhomogeneous observations and, statistically, is too smooth. I denote this series by C(t), and log2CO2(t) by CL(t). [Radiation theory predicts that temperature is a logarithmic function of atmospheric CO2 concentration ( 13 ). Use of the base 2 logarithm of the CO2 data gives coefficients that directly describe the effect of doubling CO2.]

I have also used Marland’s fossil-fuel production series ( 14 ), denoted F(t), as an adjunct to the CO2 measurements. This series starts in 1860, later than the temperature data, but the early data appear to be better than the corresponding CO2 data. The combination of these data with the CO2 data is described later.

For solar irradiance I used the Foucal–Lean ( 15 ) reconstruction. This series, L(t), is independent of the temperature data, matches direct solar irradiance measurements since they have been available, and is a reconstruction from other solar measurements before then. I emphasize, however, that this series is a proxy, not direct measurements, so that inferences drawn from it may not have the reliability of inferences from direct observations.

Changes in the period of the sunspot cycle were suggested ( 16 ) as a solar irradiance proxy. Although there is a high apparent correlation between this proxy and a heavily smoothed version of the Hansen temperature series ( 17 ), this correlation is not reliable. The jackknife variance (see below) of the low-frequency coherence between the temperature and the sunspot period is large, more than four times that expected under Gaussian theory. Because of this, lack of a physical basis for the proxy, and the failure of other statistical tests described in ref. 2, this proxy is not used here.

Least-Squares Fits

Both CO2 and changes in solar irradiance have been invoked to explain the observed increase in global temperature. Fig. 1 shows the filtered Jones–Wigley global temperature series, Tg(t) together with a least-squares fit to it using C(t) plus a constant, and a second least-squares fit to Tg(t) using L(t) plus a constant. Each of these fits explains more than 75% of the variance over the full 1854–1990 interval (the residual SDs are 74.2 mK and 83.8 mK, respectively). Including both C(t) and L(t) simultaneously as explanatory variables further reduces the residual SD to 62.5 mK. Nearly 87% of the variance is explained, and both partial F statistics ( 18 ) are highly significant, so neither variable can be dropped. Examining these residuals further, one finds that their autocorrelation at a 1-year lag is 0.914 so that the conditions required for the Gauss–Markov theorem (the basis of least-squares) to be valid ( 19 ) are not satisfied. This is not simply a technical mathematical quibble, but indicates serious fitting problems whose existence may be verified by repeating the fitting process over different time intervals and observing the change in the estimated coefficients. For example, a least-squares fit to Tg(t) with just the proxy solar irradiance L(t) and a constant the interval 1854–1918 gives a negative temperature response to increasing proxy solar irradiance. Consequently, one must use statistical time-series methods that are reliable when serially correlated residuals are present.

Calendar Date

FIG. 1. The Jones–Wigley global temperature series, low-pass filtered to a time resolution of 1 year, with independent least-squares fits to solar irradiance changes and to atmospheric CO2.

Mathematical Preliminaries

Many of the problems in the analysis of climate data require new methods for time-series analysis. Most of the commonly used time-series methods were derived under the assumption of stationarity, that is, their derivations assume that the statistics of the observed process are independent of the choice of time origin. The climate seems to be nonstationary; over a few years the annual cycle of the seasons makes cyclostationary (or periodically correlated) processes a better model than stationary processes, implying that Fourier transforms of frequencies offset by multiples of one cycle per year will be correlated. On longer time scales, evolution of the Earth’s orbit obviously results in vast shifts in the climate, and I recently have shown ( 2 ) that these changes in the orbit must be considered in the analysis of instrumental data series as well. Here these effects are accounted for in the way the data were filtered. Because both solar and CO2 effects alter the seasonal cycle as well as low frequencies, I have not removed common modulation terms at 0 and 1 cycle/year. Finally, anthropogenic changes are altering the composition of the atmosphere and the climate system. Thus, analysis methods that contain an implicit assumption of stationarity should be used with caution.

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