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The conclusion that Pollack et al^{1} 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.

- Designating a fixed averaging period fixes the temperature resolution.
- Century-long trends of temperature are well resolved over the previous five centuries.

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
al^{2}, which considered only data from Eastern Canada
and New England, did not resolve cooling trends clearly evident
in borehole logs from the Greenland Ice Sheet^{3}.

One may build a linear operator to calculate borehole
temperatures from an integral equation for heat
diffusion^{4}, combined with a set of assumed basis
functions^{5}. 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 |

Background Gradient | .005K/km | 0.009K |

1900-2000 Rate/Average | .06K/100year | 0.027K |

1800-1900 Rate/Average | .325K/100yr | 0.13K |

1700-1800 Rate/Average | 2.94K/100yr | 0.99K |

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, Clow^{6} 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
them^{7}. 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.

- H. N. Pollack et al. Science 282, 279-281, (1998).
- S. Huang et al. Geophysical Research Letters 23, No. 3, 257-260, (1996).
- D. Dahl-Jensen et al. Science 282, 268-271, (1998).
- E. Butkov, Mathematical Physics,(Addison-Wesley Publishing Co., 1968).
- R.F. Harrington, Field computation by moment methods,(IEEE Press. 1992 Ed.)
- G. D. Clow, Global and Planetary Change 98, 81-86, (1992).
- C. Lanczos, Linear differential operators,(Van Nostrand and Company, 1961).