There is no means of explaining the nature of this circular argument without referring extensively to an earlier study (Study #1) which explains in detail how a particular inverse method behaves. Then in a later study, the method is applied in a way that is circular.
Study #1: The inverse method
The objective of this method is to use temperature measured in a borehole to figure what the ground surface temperature (GST) was like over the past 1000 years. To do this the authors make a model of temperatures in the subsurface, combined with a computer program that uses the observed temperatures to find the parameters of the model. The central idea is to begin with a initial set of parameters for the model, which I denote here symbolically as mo , and systematically adjust the parameters to minimize an objective function like the following...
S=misfit to observations + change from initial model
Or mathematically...
S=(d-do)tCd-1(d-do) +(m-mo)tCm-1(m-mo).
The reason for using this objective is as follows.
The penalty for misfit to data helps insure that whatever surface temperature history results from the analysis actually explains the observed borehole temperature. However, a common problem with obtaining the history of surface temperature from borehole temperatures is that heat conduction destroys information regarding long past temperature quite completely, and, therefore, many different different temperature histories explain the borehole data equally well. Quite a few of these histories oscillate in temperature wildly--far more, in fact, than the curve labelled "1" in Figure 1. By including a penalty for deviating from the initial model the objective function drives the final solution toward some unique result, and, if the initial model is smooth, the solution is also smooth.
Values in the matrices Cm and Cd provide an optimum balance between fitting the data (d) and adhering to an a priori model (mo). The matrix Cm deals with quantities in the a priori model; most specifically the initial estimate of ground surface temperature (GST) history while Cd deals with uncertainty in the temperature observations. Specifying small values for elements of these matrices implies that a person has great faith in the validity of the a priori model or the data. This is called a tight constraint. Specifying large values for these elements provides a loose constraint.
A presumption of the model is that there is a long-term steady state GST that may or may not equal the present day surface temperature, and a steady background thermal gradient which has to be removed before analysis. These are available from the data and the inverse method. The authors use loose constraints on these parameters, however; amounting to 100K and 500mW/m2, respectively. These are such loose constraints that they allow a background heat flow directed into the earth instead of out of it.
Constraining how far the final GST may stray from the initial temperature history also helps prevent an oscillating solution. The authors suggest a constraint that tightens more on older GST than on recent GST. Specifically, they allow recent GST history to stray about 4 times more from the initial model than they allow variation near 1000 years ago.
The tightening of constraint on GST implies that the authors have more faith in their initial ancient GST than they do in the recent GST. Certainly this is not consistent with the way borehole temperature behaves.
Three hypothetical GST histories. The dashed history shows no significant temperature variation until about 1600AD, exhibits a substantial cold period from then until 1800AD, vacillates over a few decades, and finally increases until the present time. By comparison, the dashed and dotted line is constant until approximately 1900AD and then begins the century-long temperature increase that we call "global-warming."
Three hypothetical GST histories. The dashed history shows no significant temperature variation until about 1600AD, exhibits a substantial cold period from then until 1800AD, vacillates over a few decades, and finally increases until the present time. By comparison, the dashed and dotted line is constant until approximately 1900AD and then begins the century-long temperature increase that we call "global-warming."
This analysis leads to unexpected results
Some of results of this first study are unexpected. For example, a figure in the original paper shows the analysis of actual borehole temperatures. The GST derived from these diverges from one borehole to another in the recent past, but converges toward a zero value near 1000 years ago. Certainly this is unexpected. The physics of the problem suggests that the results would diverge most near 1000 years ago.
Similar behavior occurs in the numerical simulations the authors used to illustrate how loose constraints on thermal conductivity of the soil at a borehole can suppress noise and oscillations. Random noise added to the simulations had its largest effect on GST in the time period 1600-1700 AD when constraints on conductivity were tight and 1800-1850 AD when the constraints were loose.
Summary of Study #1
The inverse method of this study suppresses large variations in GST for the most ancient time periods of an analysis in three ways.
There is no misfit penalty in the data because of thermal diffusion, there is only a penalty for deviating from the initial model.
Placing the largest penalty for deviation from initial model in the oldest time periods.
Absorbing very old climatic information into the temperature background.
As an example of the third point consider the figure below. The curve labelled "2" shows the difference between the dashed and dotted curves of GST in Figure 1 at depths between surface and 400m. The beginning of the little ice age is not well resolved in this T-Z curve. In fact, it is nearly a linear increase of temperature with depth. With a small amount of added measurement noise, the inverse method would remove this linear increase along with the background temperature field, leaving only the recent climb out of the little ice age for analysis. The so-called steady state GST would equal the temperature appropriate to the little ice age, and the only GST change remaining is an increase beginning about 1800AD.
The crux of the circular argument
No invalid argument is implied in Study #1. However, in a subsequent study the same authors use this inverse method to examine the following problem. Divide the past 500 years into century long segments. Analyze a set of borehole temperatures to determine what GST change occured in each century.
The approach to the problem involves the following three steps, which taken together form a circular argument.
First, the method suppresses ocillations in GST history for the three reasons cited above. The most extreme suppression will occur either as the effect of old climate is absorbed into the background temperature or as the penalty for deviating from the initial, smooth model takes effect.
Second, suppressing GST is exactly the same as suppressing past temperature change in each century long segment.
Third, the authors interpret the lack of temperature change for the oldest centuries in the analysis as having been derived, when it is possibly preordained by the method and its initial assumptions, almost independently of the observed data.
The authors interpret the zeroed GST as being characteristic of climate rather than being characteristic of assumptions and method. The conclusion may turn out to be perfectly correct for unrelated reasons, but the argument certainly appears circular and therefore invalid.
