Fig. 30. The effect of bounding constraints governed by a priori standard deviation a on the inferred GST history (FSI method). Used T-z data are synthetically generated; "gate" model contains Gaussian noise with zero mean and standard deviation of 0.05 K. Small values of a tend to over-smooth GST history, while large values could drive to instability of the solution. The individual curves are marked by the value of the correlation time rc.

the inverse results obtained by both methods are not essential and a large part of them can be attributed to the kinds of constraints and/or the nature of additional information used in the analyses. When the optimal values for constraints are applied, the incoherence between solutions obtained by different methods is minimal. On the other hand, unproved choice of parametrization schemes and a priori constraints can significantly affect both the timing of estimated climatic changes and their amplitude. Unfortunately, the optimal values for essential constraints depend on the real GST history. A priori null GST hypothesis (no a priori knowledge of the GST history to be estimated) was used for the above GST reconstructions. Since the estimated GST models were relatively simple, obtained results reproduced the true climate change sufficiently well. For the real field examples at least some knowledge of the amplitude and timing of the climatic variations that should be reconstructed is indispensable to obtain consistent inversion results. Thus, to achieve optimal results information from other available climatic reconstructions should be incorporated into inversion procedure.

More cunning testing of the effectiveness of any technique for the GST reconstruction should include the simulation of the subsurface thermal regimes driven by the state-of-art General Circulation Models (GCM) of surface temperature (Section 2.4.4) and/or perturbed by influences additional to the GST changes and the use of simulated T-z profiles to retrieve the real surface forcing. The sensitivity of the inversion techniques to various non-climatic uncertainties is illustrated in the next section.

2.4.2 Effect of systematic errors in thermal conductivity

There may be different kinds of the systematic noise in borehole temperature logs; however, one of them is almost common in the geothermal data, namely more or less poor knowledge of the thermal conductivity. Above synthetic examples dealt with the homogeneous strata. As shown in Section 2.2 under pure conductive heat transfer conditions the geothermal gradient is inversely proportional to thermal conductivity; thus, its variations with depth produce corresponding variations from the otherwise linear T- z profile (Eq. (4)) that can be misinterpreted in terms of a transient surface temperature. In principle, the GST reconstruction methodology can be readily extended to include thermal conductivity variations. However, most of the temperature logs are accompanied by a few measurements of thermal conductivity and/or the conductivity measurements can be available for some sections of the borehole. Thus, the geophysicists are compelled to treat the subsurface as a homogeneous medium and/or assume its significantly simplified model. For example, erroneous values can be accepted when extrapolating these data on "empty" intervals.

The real Earth's subsurface generally contain significant conductivity variations that can be caused by mineralogy changes and stratigraphic heterogeneity at different crustal levels. Compaction of sedimentary rocks with depth leads to an increase in thermal conductivity through reduction of porosity. Similar changes can be caused by the changes of the fluid saturation of subsurface rocks with depth (low conductivity in unsaturated rock near the surface and higher conductivity in the water-saturated rock below the water table), etc. Poor knowledge of the thermal conductivity of the medium manifests itself as a noise in the interpretation biasing the reconstructed GST history. As shown in the previous section random noise does not affect seriously the results of inversion. However, what is about systematic noise? It is clear that the perturbation caused by this kind of disturbance can be removed from the solution only if its exact pattern is known. Of course, in the real field situations such procedure is unreal.

Synthetic example below illustrates the influence of the systematic noise on the results of the GST reconstruction. For the simulation below we have assumed three-layer thermal conductivity K(z) = {2.5 W/mK in the depth interval of 0-100m; 3.0W/mK between 100 and 200m; 2.5W/mK between 200-500m depth}. As previously, the noise free T-z profile was calculated for the "gate" GST change by 1-D forward heat transfer modeling with 5 m intervals to a depth 500 m. Other parameters remained the same as in the previous examples. The Gaussian noise of zero mean and s.d. of 0.03K typical for the field temperature measurements was superimposed on the above model. When formulating inverse problem we have taken the medium as homogeneous with constant thermal conductivity of 2.5 W/mK. Figure 31 shows the results obtained from this mistaken assumption. As seen, the systematic bias in temperature conductivity pattern has only a weak impact on the GST histories reconstructed by SVD method. The noise free and contaminated with Gaussian noise T-z profiles gave quite coherent inversion results. Influence of the systematic errors is somewhat stronger in the case of FSI. The GST histories

Fig. 31. The effect of a systematic error in thermal conductivity on the inferred GST history. The true model is 2.5 W/mK except between 100 and 200 m where it is 3 W/mK. A uniform value of 2.5 W/mK was assumed in the inversion: SVD method (bottom) and FSI method (top). Both noise-free T-z profile and a profile contaminated with Gaussian noise with zero mean and s.d. = 0.03 K were used for the inversion. The GST history for homogeneous medium (grey line) reconstructed by FSI method is included for comparison (top).

Fig. 31. The effect of a systematic error in thermal conductivity on the inferred GST history. The true model is 2.5 W/mK except between 100 and 200 m where it is 3 W/mK. A uniform value of 2.5 W/mK was assumed in the inversion: SVD method (bottom) and FSI method (top). Both noise-free T-z profile and a profile contaminated with Gaussian noise with zero mean and s.d. = 0.03 K were used for the inversion. The GST history for homogeneous medium (grey line) reconstructed by FSI method is included for comparison (top).

calculated for mistaken one-layer thermal conductivity and using real three-layer model are comparable only for the noise-free data. To avoid instability during FSI of the T~z profile contaminated with Gaussian noise, the imposing of stronger bounding and smoothing constraints on the GST history (a = 0.1 K, Tc = 500 years) was indispensable. Resulting GST curve appears to be over-smoothed.

On the other hand, FSI method provides the possibility to treat systematic errors in thermal conductivity that is unattainable for SVD technique. Namely, it can formulate thermal conductivity as a parameter with uncertainty that should be estimated together with the GST history. On the contrary, the ramp/step method and SVD are limited to the problems when the thermal properties of the medium are assumed to be known. Synthetic example below illustrates the potential advantage of formulating the thermal properties of the medium as the model parameters to be estimated. The noise-free T-z profile was calculated for the above three-layer model ("real conductivity" in Figure 32). When reconstructing GST history by FSI method (Figure 31, top), a priori constant conductivity of 2.5 W/mK was assigned and the thermal conductivity distribution was formulated as the parameter to be estimated simultaneously with the GST history. Figure 32 shows the results of inversion obtained for the thermal conductivity profile. As seen, estimated course of the thermal conductivity approaches the real variations of this quantity well.

Summarizing above results illustrating the use of SVD and FSI approaches, we should mention that as it was expected, both methods gave generally similar GST histories. Applied in the above models systematic uncertainty in the thermal conductivity values was sufficiently small and its effect was to a degree absorbed by the influence of the T-z noise. Including K(z) as a parameter to be estimated can somewhat improve the GST history. The advantages of this approach may appear more obviously, when the systematic noise is stronger.

All existing techniques of the GST reconstruction from subsurface T-z profiles are based on the 1-D theory of conductive heat transfer. At given depth the medium is assumed to be horizontally homogeneous, and the variations of the thermophysical parameters are supposed to be only vertical. Deviations from this assumption manifest themselves as noise in interpretation. Lewis and Wang (1992) described effects of spatial and temporal variations of the terrain (upper boundary conditions) and have concluded that potentially such kind of noise can give erroneous GST estimations. According to Lewis and Wang (1992), it is these effects that may be responsible for observed deviations in the GST histories inferred from temperature logs measured in closely spaced boreholes and occurrence of so-called "spaghetti diagrams" - tangling of the superposed GST curves that reflects high regional variability of the results (Figure 33, top; see also Section 3.1.1, Chapter 3).

Shen et al. (1995) investigated an influence of possible spatial heterogeneity of the thermal properties of the medium (3-D thermal conductivity structure). The authors performed a set of numerical experiments with synthetic T-z profiles. The 3-D subsurface model was obtained on the cubic 10 X 10m grid by perturbing initially homogeneous subsurface by Gaussian noise with zero mean and s.d.= 0.25W/mK. Synthetic T-z profiles were calculated for above subsurface structure using 3-D heat conduction equation. The first set of profiles ("without signal") was constructed for zero surface temperature boundary conditions to reveal the properties of noise misinterpreted as signal. These profiles have helped to recognize how strong is the influence of the subsurface heterogeneity on the GST history as well as to reveal most effective constraints (standard deviation of the measured temperatures, uncertainties in a priori values of thermal conductivity, etc.) that can suppress this noise. Numerical trial runs have shown that possibility to obtain "spaghetti diagrams" increases when a priori constraints are too severe; thus, small variations in measured temperatures and thermophysical properties are taken as significant for the reconstructed GST history. The authors have shown that extending constraints on thermal conductivity is a more effective way to suppress the influence of noise arising from the 3-D effects rather than change of constraints on the borehole temperatures. They also determined a range of constraints that appear the best for effective noise suppression.

Annual temperatures reconstructed for North America from subsurface data were used as the surface boundary conditions for second set of T-z profiles ("with signal"). This experiment was essential, because too wide a priori constraints may lead to a loss of signal and smoothing of inverted GST history. Numerous inversions of profiles "with signal" under a wide range of values assigned to a priori conditions supported conclusions based on the calculations using "without noise" data and have identified final range of constraints for the reasonable suppression of noise and effective signal recovery. The inversions performed using an appropriate range of constraints significantly attenuated the tangling of the GST histories, although small variations still remain.

Fig. 33. Transient GST histories of 22 boreholes from central and eastern Canada calculated by the FSI algorithm. Top: The relatively narrow constraints on a priori information resulted in the "spaghetti diagram". Bottom: Effective attenuation of tangling was achieved by the use of optimal constraints that took into account the possible spatial conductivity heterogeneities. (Redrawn from Shen et al. (1995).)

Fig. 33. Transient GST histories of 22 boreholes from central and eastern Canada calculated by the FSI algorithm. Top: The relatively narrow constraints on a priori information resulted in the "spaghetti diagram". Bottom: Effective attenuation of tangling was achieved by the use of optimal constraints that took into account the possible spatial conductivity heterogeneities. (Redrawn from Shen et al. (1995).)

The authors applied this information for the re-processing of the temperature logs of 22 boreholes from central and eastern Canada. As seen in Figure 33 (top), merged together earlier GST reconstructions by different authors exhibit real chaos and are not easy for comparison and determination of averaged climate history in the area. Shen et al. (1995) suggested that at least a part of the disorder observable in Figure 33 (top) is attributable to the insufficient suppression of the representational noise. Their re-processing with detected optimal constraints provided more consistent results. The higher coherency obtained for closely located boreholes as well as for combined GST histories of all holes (Figure 33, bottom) revealed considerably simpler picture than the previous pattern. Average GST history for central and eastern Canada consists of some 1-4K warming that began in nineteenth century. Part of this warming may be interpreted as the recovery from the earlier colder period.

2.4.3 Using additional information: Field example

The testing of the inversion procedures using real temperature logs is somewhat difficult because of absence of exact knowledge of the GST history in the borehole site against which inferred results could be calibrated. If inversion methods gave different results, it is quite difficult to define whether a technique has provided better estimation of the GST history. Thus, further field examples will be used not for the testing of the methods, but only for the illustration of the effectiveness of incorporation of different kinds of additional a priori information to stabilize and uniquely determine the solution.

Both SVD and FSI methods differ in their theoretical approach, parametrization, and the way of parameter estimation. Common characteristic of both algorithms is that they incorporate a priori (additional) information to achieve optimal results. For a final test of the inverse methods we use a real temperature log measured in borehole Hearst (eastern Canada). This borehole was chosen not only because of the high-quality temperature logs and heat conductivity data available, but also because of its "historical" value. In the 1970s this borehole enabled one of the first practical attempts to assess the past climate history from subsurface temperature data (Cermak, 1971). The GST reconstructions were repeated further in numerous works (Nielsen and Beck, 1989; Shen and Beck, 1992; Cermak et al., 2003).

Three holes including Hearst site were drilled in northeastern Ontario in 1968 as a part of the heat flow project of the Dominion Observatory. The sites were carefully selected in a flat terrain and in geologically uniform strata. The 600 m deep borehole Hearst (49.69°N, 83.54°W) is located in a slightly elevated, bushed terrain at the boundary of large forested and cleared fields. A small nearby lake and swampy area affect the temperatures insignificantly. The site is apparently free of the groundwater disturbances. The first incremental log was measured in 1969 (Cermak and Jessop, 1971). Further virtually continuous loggings were performed in 1985 and in 2000 (Nielsen and Beck, 1989). Temperature measurements are highly precise with the absolute accuracy of less than 20mK for the incremental logs and as small as 10mK for the continuous logs. Figure 17 (Chapter 1) shows results of these measurements (Cermak et al., 2003). As seen, all temperature logs are quite similar with the clear positive "U-shape" curvature in their uppermost parts. Figure 17 and all similar diagrams below present the temperature log not only on the measured, but also on a reduced scale obtained by subtracting from the measured temperatures a temperature value = gradient X depth (see also Eq. (5), Section 2.2). This representation enhances the nonlinearities. The shape of the reduced temperature-depth profiles is more complex than that occurring in the case of the single warming event (Figure 13). The waves of the opposite sign in the reduced temperature profiles hint the presence of the recent warming that may be amplified by the environmental effect of the forest clearing that occurred approximately 100 years ago (Wang et al., 1992), subsequent cooling, warming, and cooling again. The 192 measurements for thermal conductivity, rather regularly distributed over the length of the hole, are available. The conductivity is almost constant at 3.23 ± 0.09W/mK. The specific heat capacity is also relatively constant with an estimated mean value of about 2.5MJ/(m3K). The mean rate of the heat production is 0.86 ^W/m3 (Jessop and Lewis, 1978). For the purpose of the present analysis heat production of this rate has negligible effect on the inversion results.

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