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31 August 2001

Random Numbers, Part III

An example by Kevin Kilty

Having a record of prior earth surface temperature might be veryimportant to the study of global warming. Unfortunately prior to 1600 orso, the only available temperatures are proxies which are not veryaccurate. There are some earth scientists who believe they can takecareful measurements of temperatures in boreholes and deduce from thisthe earth's temperature history. (If you wish to get a broad appraisalof this approach please refer to this paper. )

Researchers at the University of Michigan claim to have computed theaverage temperature over each of the previous 5 centuries in thismanner, accurate to 0.1K. A recent attempt of theirs is available as a short paper: Pollack and Shen, 1998, Science, 282, 279-280. Despite there being some circular reasoning involved in their conclusions, this provides an interesting use of random number generation and Monte Carlo technique. In fact, Dahl-Jensen, and others (1998, Science, 282, 268-271) put the method I describe below to use to infer past Earth temperature using a borehole in the Greenland ice sheet.

The effect on subsurface temperature from a unit change in surface temperature t years ago is...

Erfc( z/sqrt(4kt)), where; z=depth in meters,
 k=thermal =
diffusivity in m2/year,
 and Erfc is the complimentary =
error function.

If I assign 5 temperature changes, one for change in average temperature in each of the previous 5 centuries, then I can compute the expected temperature anomaly in a borehole by summing all contributions using the above formula. Some contributions, especially those in the 5th century ago, will be very small and difficult to resolve. At one point can I no longer resolve anything?

It is possible with care to measure a borehole temperature to a precision of 0.05K. If the temperature history I propose does not result in any borehole temperatures exceeding this value, then I cannot have much claim to having detected or resolved any variation in past temperature. I refer to this as a nil temperature history.

This provides me a way to calculate the resolving power of this method. I let each of 5 temperature values vary from -max to +max by small increments, and simply map out the domain over which I get nil temperature histories. If max=2K and the increment is 0.01K then the total number of calculations involved is about 10¹³. This is a lot of calculating; and there is no guarantee that using max=2K will cover the entire domain of nil temperature histories. Moreover, amount of calculating increases rapidly if I further subdivide the time scale.

An alternative is to generate a million or maybe a couple of million random temperature "vectors" (sets of 5 temperatures at a time), make a calculation with each one and see if the result is nil or not. The statistics of these million experiments will approximately map out the region of nil results in our 5-dimensional temperature space. This estimates the precision of past climate derived from borehole temperature, and also illustrates something about how average temperature in an earlier century can obscure high temperature in a later century and vice versa.