Estimating the sensitivity of the climate to CO2 is of paramount importance in predicting the potential effects of human activity on global temperature. Estimates vary widely and are bounded by large uncertainties. Estimates derived from climate models are of dubious value given their generally poor performance when compared to empirical observation. We undertake here an estimate using only empirical data.

One of the first difficulties encountered in estimation is the large but unknown variation in the empirical temperature data due to natural causes. Slope estimates are influenced greatly by low-frequency quasi-periodic signals due to external forcing and climate system dynamics. Presented here is a method for removing these cyclical components.

The dataset used in this analysis is the Hadcrut4v2 data with corrections (Cowtan and Way). The raw monthly data is shown below.


The first thing we need to do is remove the seasonal periodicity by calculating a yearly mean for each of the 163 years spanned by the data.


Next we remove the obvious outlier near 1878 by interpolation.


We can began to see a hint of cyclicality by filtering the data slightly using a Gaussian (r=5) filter to remove some of the noise. The unfiltered data though is used for the analysis that follows.

hadcrutYearFiltIn order to isolate the cyclical components it is first necessary to remove the gross trend. We judge which order of polynomial fit to use by examining the auto-correlation for various orders of fit.

acfplotsAs the order of the polynomial increases, the width of the main lobe narrows (implying a whiter residual) and the cyclical component increases, until at order 5 at which point the polynomial begins to follow the periodicity we are trying to isolate reducing its contribution to the residual. We select order 4 as the best choice for detrending.

The plot  below shows (left to right) the ACF with order 4 detrending , the power spectral density (Fourier transform of the ACF), the order=4 residual  and the histogram of the residual.

Note residGrid2Note the strong spectral peak at f = 0.01583 (T ~ 63 years)  and its 3rd harmonic (T  ~ 21.5 years). These are the cyclical components we seek to remove from the residual.

Using a non-linear fitting algorithm, we fit the residual to a sum of sinusoids model which finds the best fit to be:

-0.0405767 sin(0.167108 -0.29583 n)-0.0829206 sin(1.34258 -0.0986101 n)


Removing the cyclical component from the time series yields the trend plot used in part 2  corrected