# Climate Insensitivity

To estimate those parameters, Schwartz uses observed climate data. He assumes that the time series of global temperature can effectively be modeled as a linear trend, plus a one-dimensional, first-order “autoregressive” or “Markov” or simply “AR(1)” process [an AR(1) process is a random process with some ‘memory’ of its previous value; subsequent values y_t are statistically dependent on the immediately preceding value y_(t-1) through an equation of the form y_t = ρ y_(t-1) + ε, where ρ is typically required to be between 0 and 1, and ε is a series of random values conforming to a normal distribution. The AR(1) model is a special case of a more general class of linear time series models known as “Autoregressive moving average” models].

In such as case, the *autocorrelation* of the global temperature time series (its correlation with a time-delayed copy of itself) can be analyzed to determine the time constant *τ*. He further assumes that ocean heat content represents the bulk of the heat absorbed by the planet due to climate forces, and that its changes are roughly proportional to the observed surface temperature change; the constant of proportionality gives the heat capacity. The conclusion is that the time constant of the planet is 5±1 years and its heat capacity is 16.7±7 W • yr / (dec C • m^2), so climate sensitivity is 5/16.7 = 0.3 deg C/(W/m^2).

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:

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