A late Quaternary climate reconstruction based on borehole heat flux
data, borehole temperature data, and the instrumental record
S. P. Huang,1 H. N. Pollack,1 and P.-Y. Shen2
Received 31 March 2008; revised 21 May 2008; accepted 27 May 2008; published 4 July 2008.

We present a suite of new 20,000 year reconstructions
that integrate three types of geothermal information: a
global database of terrestrial heat flux measurements,
another database of temperature versus depth
observations, and the 20th century instrumental record of
temperature, all referenced to the 1961–1990 mean of the
instrumental record. These reconstructions show the
warming from the last glacial maximum, the occurrence
of a mid-Holocene warm episode, a Medieval Warm Period
(MWP), a Little Ice Age (LIA), and the rapid warming of
the 20th century. The reconstructions show the temperatures
of the mid-Holocene warm episode some 1–2 K above the
reference level, the maximum of the MWP at or slightly
below the reference level, the minimum of the LIA about
1 K below the reference level, and end-of-20th century
temperatures about 0.5 K above the reference level.

T = Ln(t); where t is time before the moment of observation
X = Ln(x^2); where x is depth into sample (depth below ground).

affect the ill-conditioned nature of the matrix inversion? Is it still ill-conditioned after the transform?

Very interesting.

w.

T = Ln(t); where t is time before the moment of observation
X = Ln(x^2); where x is depth into sample (depth below ground).

Under this change:

Sinusoids in T map to sinusoids in X.

The amplitude and phase under the mapping depend only on the angular frequency w (in T).

The amplitude and phase are given by the Modulus and Argument of the
Complete Gamma Function for (1/2 + i*w) [normalised by division by Sqrt(PI)]

where w is the angular frequency in T and i is the Sqrt(-1) and * is
multiplication operator.

So the mapping function (from T to X) is complex multiplication by
G(w) = Gamma(1/2+i*w)/Sqrt(Pi).

To make this clear a temperature history of the form Sin(w*Ln(t)) with
constant amplitude will at the moment of observation produce a borehole
temperature record of the form Sin(k*Ln(x^2)) with an amplitude that is constant with depth.

This allows an observation, of temperature with depth, to be decomposed to its Fourier components each component divided by the mapping unction G(w) and recomposed to give the Temperature Record in T (and hence in t) directly.

In this way the inversion problem can be simplified. It is as easy to map from X to T as it is to map from T to X. No iterative relaxation is required to reclaim the temperature record. Fourier analysis and complex multiplication/division is all that is required.

Importantly one can construct an orthogonal basis in T that maps to an orthogonal basis in X and vice versa.

I did a lot of work on this a couple of years ago and could not raise any interest. If anyone would like more details please let me know.

Best Wishes

Alexander Harvey

You should write that up and submit it for publication in a suitable journal. That’s the only way it’s going get recognition. So the next question would be what are the suitable journals? Anyone?

So the Fourier components of the borehole record could be mapped back to the Fourier components of the surface temperature record. There is of course a multiplicative coeffiecient which is given by either the complete or incomplete Gamma Function (e.g. Gamma(1/2 + iw) as I recall).

Now not only does this make the inversion simpler to perform but it allows one to reason about such things as resolving power (which turns out to be pretty minimal one certainly cannot resolve the 1600s from the 1500s) and more generally the effects of noise in the data on the result in a quantifiable way.

Now I thought this might be of use so I wrote to anyone I thought might be interested. I started this in the first half of 2006 I got nowhere, so I thought I might take the opportunity to write this here to see if it can raise the interest that I think it deserves.

FWIW

I believe (and think I can show) that the methodology used in the reconstructions is badly flawed. Also that many of the borehole temperature records included in the reconstructions tell one more about their uneven geology than the surface temperature record and I believe one could show this from spectral analysis. I have also let people no this but sadly I rarely receive as much as a reply.

Best Wishes

Alexander Harvey

I didn’t read my equation correctly and left out distance. For a step change in temperature it’s deltaT(x,t) = deltaT0*erfc(x/sqrt(k*t)) where x is distance, t is time and k is the thermal diffusivity in m2/sec.

DeWitt, while the problems you refer to with the air-ground interface are real, I was just looking only at the ground part of the puzzle.

Thanks for the hint about the form of the profile, I’ll see what I can find.

w.

I suspect your method is more optimistic than reality. If you use the actual heat equation, which isn’t all that difficult to implement even in Excel, you have to worry about heat flux at the surface too. It is likely not infinite. That is, when you change the ambient temperature, the surface temperature won’t immediately become equal to it because heat transfer from the atmosphere to the surface is finite and proportional to the temperature difference. Then again, maybe the flux is so small because both the thermal conductivity and heat capacity or rock are low, and it doesn’t matter.

If your calculation is correct, the temperature profile with depth is described by the erfc(K*sqrt(t))) (error function complement function).