The conclusion that Pollack et al1 reach using borehole temperature logs, that the rate of temperature change in the current century is larger than that in any of the previous four, rests directly on the two following claims.
Figure 3 in their paper shows a mean global temperature increase over five centuries of 1.0 ± 0.1K. This borehole-derived result is consistent with, but twice as precise as, surface air temperature records spanning only 120 years. Despite this precision, a related study by Huang et al2, which considered only data from Eastern Canada and New England, did not resolve cooling trends clearly evident in borehole logs from the Greenland Ice Sheet3.
One may build a linear operator to calculate borehole temperatures from an integral equation for heat diffusion4, combined with a set of assumed basis functions5. If this operator, L, consists of century-long step pulses, a background temperature gradient, and a long term surface temperature as basis functions, then its inverse, L-1, provides least-squares estimates of average surface temperature in each century. When composed of triangular pulses instead of steps, L-1 produces estimates of temperature trend. Table 1, below, summarizes uncertainty resulting from measurement error of 0.01K per observation in a 400m deep borehole.
|Parameter||Std Error, triangle pulses||Std Error, step pulses|
|Long-term Surface Temp.||.001K||0.002K|
Even in a sample of 358 boreholes, the 18th Century error alone would make an uncertainty below ± 0.1K in average temperature rise difficult to attain. Why does resolution degrade so abruptly?
First, more ancient century-long pulses are not sufficiently independent of the long-term surface or background gradient temperatures, or of one another, to resolve precisely.
Second, Clow6 shows, for conditions appropriate to this analysis, that boreholes 450m deep provide minimal resolution, and optimal resolution requires boreholes deeper than 800m. Optimal resolution means no better than 50% of the age of the event. The typical borehole is too shallow and the basis pulses too brief to expect good resolution 500 years in the past.
The authors suggest reducing the number of parameters to untangle signals from noise. What matters, though, is the form of the parameters. Table 1 shows that a practical age limit for century-long pulses is perhaps three centuries. There is a trade-off between temperature and time resolution.
They also suggest smoothing the data or its parameterization. What this means in practice is discounting the small eigenvalues of L, or discounting the temperature observations associated with them7. Doing either will decrease parameter variance at the expense of introducing bias. Thus, there is also a trade-off between bias and variance in the inversion of L.
In light of these trades-off, an a priori assumption of zero temperature trend in each century does not seem neutral. If the data lack information for a Bayesian inversion to work with, then a priori assumptions become a posteriori results, and one ought to perform numerous such analyses using a range of initial conditions. Perhaps what happened in the 16th and 17th Centuries is still uncertain.