I would just like to say that there is a very good reason why the last 500 year borehole reconstructions should not be added to longer reconstrctions and it is in the design of the borehole reconstruction and nothing to do with its other well known problems.

In order to make the reconstruction a design decision had to be made concerning the period prior to the 16th century and that decision was that nothing happened.

The reconstruction could be called what does the data tell us if the temperature prior to 1500 was the same as the year 1500.

If you re-worked the reconstruction with say a MWP in the prior period you would get a different reconstruction.

To me at least this precludes tacking it on to a longer spaghetti graph.

If the period prior to 1500 was marked by a continuous straight line that would be more acceptable but it would still be rather questionable.

Anyway I have said what I need to. To me it seems a ridiculous thing to do but I as it does not seem to bother anyone else I will try to move on.

Alex

I think that you’re talking about the same thing that I had in mind when mentioning the Chlandni functions which are cos-driven as well. But I’ll have to think about this.

The main problem is that two boreholes only a few miles apart can show very divergent temperature “histories”. How can that be? Here’s an example from Utah:

These two boreholes are only 5 miles apart … hardly reassuring regarding the inherent accuracy of the borehole thermometer.

The two main problems are inhomogeneity and water. The inhomogeneous nature of the subsurface means that the heat conduction is far from uniform. Water also greatly complicates the calculation problems.

Drilling in the permafrost eliminates the water problem … at least in the upper layers. But even in the report you cite, there are a number of boreholes which show warming, while a smaller number show cooling … he handwaves away the discrepancy, but I’d like to understand it.

w.

I will give it a go at the end, but to recap, I would say that I would be surprised if the PCs that Steve produced are not related to functions of the form t^(a+iw) as his PCs are the eigenvectors of the equivalent covariance matrix wheras t^(a+iw) are similarly eigenfunctions of the equivalent continous linear operator.

In the continous case you take the partial derivative of erfc(x/Sqrt(4dt)) with respect to t (call this B) and create the following operator which is the continous equivalent of the covariance matrix:

(A) = (Integral(B*Integral(B*.)dt)dx) where . is a place holder for the function to be operated on, then

(A)t^(a+iw) = (1/sin((a+iw)*pi))t^(a+iw) that is

functions of the form t^(a+iw) are eigenfunctions of the operator (A) with eigenvalues given by 1/(sin((a+iw)*pi)).

Functions of the form t^(a+ib) can be rewriten as t^a*e^(iw*ln(t)) the real part of which which generalises to t^a*(cos(w*ln(t)+a) where a is a phase angle. That is they are periodic in ln(t).

Now the measurement positions down the hole are limited to the range (17.32,389.17) so one might expect the only the eigenfunctions corresponding to harmonics on the dimensionless “aperture” ln(389.17/17.32) to be candidate PCs. Much like harmonics on a string. The fundamanetal (first harmonic) has no crossings, the 2nd harmonic has two crossings etc.

That is if you regard the borehole as a string then the PCs are equivalent to its harmonics but the middle of the string would be the geometric mean of the end points not their arithmetic mean.

Now as always, nothing is a simple as that, not only is the borehole limited but Steve’s choice of time-slots are also bounded this gives rise to another potentially conflicting “aperture”, also the use of a small number of discrete datapoints and time-slots will have an effect.

Alexander Harvey

http://vzj.scijournals.org/cgi/content/full/6/3/591

Combined Effects of Urbanization and Global Warming on Subsurface Temperature in Four Asian Cities. Makoto Taniguchia, Takeshi Uemurab and Karen Jago-ona

Therefore, no thermal free convection is expected (Taniguchi et al., 1999). The logged boreholes were drilled and cased mostly before the 1980s. Thus, the water temperatures in boreholes represent the temperature of groundwater surrounding the boreholes.

(My bold)

There is huge variability among individual temperature profiles in each city.

and on UHI,

Nevertheless, the averaged temperature–depth profiles in all cities show strong evidence of surface warming. The differences between (old) surface temperature extrapolated from geothermal gradients and (new) surface temperature extrapolated from the average of observed subsurface temperature are estimated to be 2.8°C in Tokyo (Fig. 2), 2.2°C in Osaka (Fig. 3), 2.5°C in Seoul (Fig. 4), and 1.8°C in Bangkok

I would not be at all surprised if you found that the PCs corresponded to functions of the form COS(ln(t)w+a) where “w” acts like a angular frequency (albeit dimensionless) and “a” is a phase angle. If so you may find that w increases in steps derived from a dimensionless “aperture” given by 2*ln(389.17/17.32) =~ 6.22.

Indicating steps of 2*pi/6.22 =~ 1, giving 0, 1, 2 crossings etc.

Simply throwing the graph into ln(t) should show if such sinusoids are present.

More generally functions of the form t^z (z a complex constant) tend to dominate in such reconstructions irrespective of method.

It may be of interest that for functions of this form t^z map to x^(2*z) * Gamma(1/2 – z)/Sqrt(pi()) the sharp decline of Gamma(1/2 – z) with increasing w (imaginary part of z) give the magnification commonly seen in the higher order components of such reconstructions.

*****

If no one has mentioned it you seem to used flux in the graphic rather than temperature.

*****

The flux parameter you are looking for is indicated in the borehole data. From the temperature gradient (/1000 to get metres) * the conductivity we get ~0.021 w/m^2.

Alexander Harvey

Please note my caveat in #22 that conditions can vary widely.

It would be possible to select sites that are far less troublesome than others. Maybe some would be so good as to allow inversion maths to be tested on them. That is partly the reson for for doing this work on ice, the other part being time calibration from ice bands.

To answer one question, depending on location, groundwaters can be influenced like tides. They are probably everywhere influenced by seasonal evaporation/precipitation patterns.