# Climate Insensitivity

One of the biggest problems with this method is that it assumes that the climate system has only one “time scale,” and that time scale determines its long-term, equilibrium response to changes in climate forcing. But the global heat budget has many components, which respond faster or slower to heat input: the atmosphere, land, upper ocean, deep ocean, and cryosphere all act with their own time scales. The atmosphere responds quickly, the land not quite so fast, the deep ocean and cryosphere very slowly. In fact, it’s because it takes so long for heat to penetrate deep into the ocean that most climate scientists believe we have *not* yet experienced all the warming due from the greenhouse gases we’ve already emitted [Hansen et al. 2005].

Schwartz’s analysis depends on assuming that the global temperature time series has a single time scale, and modelling it as a linear trend plus an AR(1) process. There’s a straightforward way to test at least the *possibility* that it obeys the stated assumption. If the linearly detrended temperature data really do behave like an AR(1) process, then the autocorrelation at lag Δ*t* which we can call *r*(Δ*t*), will be related to the time constant *τ* by the simple formula

*r*(Δ*t*)= exp{-Δ*t*/τ}.

In that case,

*τ* = – Δ*t* / ln(*r*),

for any and all lags Δ*t*. This is the formula used to estimate the time constant *τ*.

And what, you wonder, *are* the estimated values of the time constant from the temperature time series? Using annual average temperature anomaly from NASA GISS (one of the data sets Schwartz uses), after detrending by removing a linear fit, Schwartz arrives at his Figure 5g:

Using the monthly rather than annual averages gives Schwartz’s Figure 7:

If the temperature follows the assumed model, then the estimated time constant should be the same for all lags, until the lag gets large enough that the probable error invalidates the result. But it’s clear from these figures that this is not the case. Rather, the estimated *τ* increases with increasing lag. Schwartz himself says:

As seen in Figure 5g, values of *τ* were found to increase with increasing lag time from about 2 years at lag time Δ*t* = 1 yr, reaching an asymptotic value of about 5 years by about lag time Δ*t*= 8 yr. As similar results were obtained with various subsets of the data (first and second halves of the time series; data for Northern and Southern Hemispheres, Figure 6) and for the de-seasonalized monthly data, Figure 7, this estimate of the time constant would appear to be robust.

If the time series of global temperature really did follow an AR(1) process, what would the graphs look like? We ran 5 simulations of an AR(1) process with a 5-year time scale, generating monthly data for 125 years, then estimated the time scale using Schwartz’s method. We also applied the method to GISTEMP monthly data (the results are slightly different from Schwartz’s because we used data through July 2007). Here’s how they compare:

This makes it abundantly clear that if temperature did follow the stated assumption, it would *not* give the results reported by Schwartz. The conclusion is inescapable, that global temperature cannot be adequately modeled as a linear trend plus AR(1) process.

You probably also noticed that for the simulated AR(1) process, the estimated time scale is consistently less than the true value (which for the simulations, is known to be exactly 5 years, or 60 months), and that the estimate *decreases* as lag increases. This is because the usual estimate of autocorrelation coefficients is a *biased* estimate. The word “bias” is used in its statistical sense, that the expected result of the calculation is not the true value. As the lag gets higher, the impact of the bias increases and the estimated time scale decreases. When the time series is long and the time scale is short, the bias is negligible, but when the time scale is any significant fraction of the length of the time series, the bias can be quite large. In fact, both simulations and theoretical calculations demonstrate that for 125 years of a genuine AR(1) process, if the time scale were 30 years (not an unrealistic value for global climate), we would expect the estimate from autocorrelation values to be less than half the true value.

Earlier in the paper, the AR(1) assumption is justified by regressing each year’s average temperature anomaly against the previous year’s and studying the residuals from that fit:

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