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Fig. 34. GST histories reconstructed for the temperature log measured in borehole Hearst (SVD method). The use of cutoff values of 10—3-10—4 resulted to GST histories showing the "Medieval Warm Period" centered near 1200 A.D. and the "Little Ice Age" at about 1650 A.D. Early reconstruction by Cermak (1971) based on the Monte Carlo solution is shown for comparison.

As mentioned above, diffusion process never retains sharp signals; thus, estimated GST histories are relatively smooth with increasing duration and decreasing amplitude of the climatic events into the past. Figure 34 shows GST histories inferred by SVD technique using different cutoff values. For comparison the earliest evaluation of the GST history by the Monte Carlo method (Cermak, 1971) is also shown. Except for the more pronounced appearance of the Medieval Warm Period in the Cermak's reconstruction, the coherence of results given by both methods is high. The use of cutoff values of 10—3-10—4 leads to similar GST histories with the "Little Climatic Optimum" centered near 1200 A.D. and the "Little Ice Age" near 1650 A.D. At smaller cutoff values (<10—6) the solution has unreliable amplitude and/or becomes unstable. The coincidence of the measured and a posteriori T-z profiles is quite high. The root mean square (rms) misfit equals to 0.01-0.015 K for different cutoff values. It is an essential feature of the GST reconstructions and reflects the underdetermined nature of the inverse problem. In other words, the measured and calculated (a posteriori) T-z profiles fall close to each other even for significantly differing GST histories.

Above GST reconstructions were performed without use of a priori additional information. The next calculation illustrates the advantages of including additional independent knowledge in the inversion procedure. As additional information we used the information on the decorrelation of the measured data and on the persistence of the climate changes (Section 2.3.4). As described in this section, the autocorrelation function for the most temperature logs approaches zero exponentially (short-range dependence): r(Az) = exp(—Az/D) where r(Az) is the autocorrelation function, Az the depth lag, and d the characteristic correlation distance (Figure 35). Parameter D corresponds to the depth lag at which the autocorrelation decreases to (1/e), i.e., it defines the distance at which the individual temperature values can be considered statistically independent. For the majority of the boreholes the D-value ranges from 100 to 300m (Bodri and Cermak, 1997a). Longest decorrelation distances are characteristic for boreholes drilled in ultrabasic rocks.

Fig. 35. Autocorrelation of measured temperatures at borehole Hearst in the depth interval 20-600m and its exponential fitting.

The effect of different factors on the correlation distance was studied in a number of synthetic examples in the work by Bodri and Cermak (1995). Decorrelation distance depends on the distribution of the thermophysical properties of the medium, and also on whether fluid circulation is present or not. Generally, an environment with fluid circulation has a smaller O-value compared with an environment with no circulation (Bodri and Cermak, 2005a). The presence of a strong climatic signal in measured underground temperatures can also significantly affect the O-value. The advantages of treating the additional information are illustrated in Figure 36. As seen, an augmentation with additional information does not significantly change inferred GST history especially in its more recent part; however, it helps to recover less smoothed GST history.

Under FSI runs, two sets of values for a priori constraints for a and tc were assigned (Figure 37). Two GST histories were calculated to reveal the uncertainty about the optimal values for the smoothing parameters. Applied values can be regarded as the upper and lower bounds for possible constraints. The GST reconstruction inferred by SVD technique with cutoff value of 10~4 is shown for comparison. Once parameters are appropriately chosen the two methods have provided very similar results. All reconstructions revealed cold conditions prior to 1800 A.D. There is also a slight decrease of temperature since about 1964 and 1976 for SVD and FSI reconstructions, respectively. The difference in the onset time of the recent cooling occurs because of the instability induced by noise.

Because of the heat diffusion and the uncertainties of measured temperatures, the time span for which the GST history can be reconstructed is limited. Its length is influenced by the depth and quality of measurements and also by the magnitude of the climatic signal that is to be reconstructed (Clow, 1992). The further we go back to the past, the less detail can be distinguished and a smoother course of the true temperature is obtained. This finding can be illustrated in terms of the resolving power of the SVD inversion

Ground Door Height
Fig. 36. Effect of the additional information used in the SVD method. An augmentation with additional information provides less smoothed GST history. (Demonstrated on the Hearst data.)
Setting Pinion Angle Diagrams
Fig. 37. The GST histories reconstructed for Hearst data by the FSI method using two sets of a priori constraints. The SVD reconstruction (cutoff = 10~4) is shown for comparison.

method (Section 2.3.4). As described in that the given section, the resolution matrix of the unknown parameters can be defined as R = VVT, where V is a matrix whose elements are the eigenvectors. The jth column of matrix R represents the least squares solution for maximizing the jth parameter. At proper choice of the discretization of time the resolution matrix exhibits delta-like behavior (compact resolution) when the column with the best resolving power is nearly always the column with the maximum diagonal element. Thus, the diagonal elements of the resolution matrix can be used as the measure of the resolving power. It can vary between 1 (perfect resolution) and 0 (no resolution). The resolution was shown to depend on the shape of the surface temperature history, and is also a complex function of many borehole specific parameters, such as accuracy and vertical spacing of the temperature measurements, distribution of thermal conductivity measurements, and the level of noise in the data (Clow, 1992; Bodri and Cermak, 1995); thus, it should be established for each borehole individually.

Fig. 38. Top: Resolving power of 100, 500, and 1000 year time intervals versus time (Hearst data). Bottom: Resolving power of 50-year intervals versus time without and with incorporating additional information on the interdependence of temperature measurements and on the climate changes (curves are labeled 1 and 2, respectively).

Fig. 38. Top: Resolving power of 100, 500, and 1000 year time intervals versus time (Hearst data). Bottom: Resolving power of 50-year intervals versus time without and with incorporating additional information on the interdependence of temperature measurements and on the climate changes (curves are labeled 1 and 2, respectively).

The diagrams of the variation of resolution back in time calculated for borehole Hearst (Figure 38 (top)) illustrate the most prominent property of resolving power of the geothermal method, namely the general fast decrease in resolution into the past. The variance of the jth parameter can be estimated by Eq. (21). A 100-year long event that occurred 300-500 years ago can be resolved with the relative variance of 10-15%. For as early as 2000-3000 years ago, it is only possible to resolve a 500-year interval with the same reliability, and the corresponding duration of event is 1000 years if it occurred 7000-9000 years ago. In other words, the further back we go into the past the less detail can be resolved and the smoother trend of the real temperature conditions on the Earth's surface can be obtained. However, as mentioned in Chapter 1, such diminishing of the resolution into the past represents a common property of the majority of proxy methods for the paleoclimatic reconstructions. Compared to the variety of proxy climatic reconstruction methods, the resolving power of the geothermal method is lower for the recent 50-100 years and is comparable with other paleoclimatic reconstructions when detecting more remote climatic events (Figure 7, Chapter 1).

Generally, the Hearst data permits to assess past climatic changes of one millennium or so. Figure 38 (bottom) illustrates the change of resolving power for 50-year time interval calculated for the Hearst hole, i.e. a rapid decrease in resolution with increasing time. The reliability to determine short GST change is about 10-15% for an event, which occurred 50 years ago; for an event that occurred 200 years ago, it is possible to resolve a 50-year interval with the same reliability, and a 200-year interval for an event that occurred 800 years ago. Incorporation of additional information can improve the resolving power of the SVD method. As demonstrated in Figure 38, the ability to resolve 50-year time intervals after incorporating the information about interdependence of temperature measurements and climate changes exhibits an improvement of almost 10% at a time of 0-50 year B.P., and of 23-35% at the interval of 100-150 year B.P.

We have compared two of the most powerful methods for the GST inversion using the subsurface temperature-depth profiles. Methods differ in both their parametrization and the technique of parameter estimation. The incorporation of a priori information to obtain a stable and unique solution is central for both techniques. In spite of the theoretical differences between both approaches, their application to synthetic and field examples gives generally similar results in the case of the appropriate choice of the stabilizing constraints. Summarizing conclusions are the next:

(1) Large part of discrepancies in the inverse results can be attributed to different constraints imposed on the GST to smooth and stabilize the inverse solution. In general similar results were obtained by two methods when equivalent assumptions were used.

(2) In principle, FSI technique allows incorporation of the thermophysical properties as the parameters to be estimated and weighting of the contribution of the data and a priori model and thus appears possessing potential to give better inversion results. However, exact knowledge of the weights/uncertainties of the data as well as a priori model is indispensable to realize this potential, while researchers generally have no sufficient a priori information in their disposal.

(3) The computational advantages when incorporating additional information are obvious. Including additional information can improve the resolution and significantly enlarge the extent of the climatic history that can be recovered by the inversion. The more complete is a priori knowledge about past climatic changes from independent complementary sources, the more reliable GST histories can be inferred from borehole data.

2.4.4 Recent testing of borehole inversion methods in simulated climates

The reconstruction of the past temperature variations on the global/hemispheric scales is performed by three principal approaches: proxy methods, inversion of borehole temperature logs, and modeling. Different proxy techniques are the oldest and traditional, while the "borehole" method and simulations of the past climate with the state-of-art GCM represent recent developments. The first attempts to decipher certain information on the GST changes from underground temperatures dates back to the early 1970s, and the corresponding inversion methods become generally known in the mid-1980s. The first compilation of the studies inferring past climatic variations from underground temperatures edited by T. Lewis has appeared in 1992 (Lewis, 1992). Together with other topics it gathered numerical comparisons/testing of different GST inversion techniques. The goal of these investigations was to prove the ability of the "borehole" method for reliable reconstruction of the past climate change (e.g. Beck et al., 1992; Shen et al., 1992). The attempts to validate and to refine "borehole" method and/or answer numerous questions arising during further development of the techniques and drawing more and more field data in the processing continued permanently for the recent two-three decades. Simultaneously numerous attempts were undertaken to bring together/compare/combine results of different approaches and to integrate them into the complex multi-dimensional paleoclimatic network.

Probably the most recent testing of the possibility of the GST reconstruction from borehole temperature logs was performed in the work by Gonzalez-Rouco et al. (2006; see also the references therein). This attempt has been inspired by the recent comparative studies of various global/hemispherical paleoclimatic reconstructions and somewhat different magnitudes of the past temperature changes (especially in the earlier parts of the records from the sixteenth to eighteenth centuries) of the averaged GST histories in comparison with climatic trends defined from proxy records (Briffa and Osborn, 2002). The global and/or hemispheric scale temperature histories for the several past centuries based on borehole measurements suggest colder past conditions than the reconstructions based on the multiproxy data. For example, tentative hemispheric GST history by Huang et al. (2000) (Figure 94, Chapter 3) revealed a much colder Little Ice Age of approximately -0.8 to —1.0K in comparison with -0.2K given by the Mann et al.'s (1998, 1999) multiproxy compilation (for details see Section 3.3, Chapter 3; Mann et al., 2000; www.ngdc.noaa.gov/paleo/ei/ei_cover.html). Results by Briffa et al. (2001; see also Figure 11, Chapter 1) are somewhat closer to the Huang et al.'s (2000) conclusions and give temperatures of the Little Ice Age by 0.3-0.6K lower than the present, while Crowley and Lowery (2000) proposed, rather, a warming of ~0.2K

from 1000 to 1400 A.D., cold conditions of--0.3K up to 1900, and rapid warming of

0.4-0.8 K in the twentieth century. Two reconstructions for Europe using independent proxies by Luterbacher et al. (2004) and Guiot et al. (2005) have detected almost similar twentieth century warming of 0.25 and 0.27 K, respectively. Mann et al. (2003) have tried to re-assess the coupling of the borehole and traditional proxy data and have re-calibrated the GST history using the twentieth century SAT data. This procedure somewhat increased possible warming to 0.2-0.4 K. Another distinction of the "borehole" and proxy reconstructions is that the amount of warming obtained by Huang et al. (2000) is more regularly distributed over the past five centuries, while in other works the twentieth century warming appears as a continuation of the trend that started only in the nineteenth century. Similar inconsistency was found also among different proxy series. Figure 39 shows comparison of three multiproxy SAT anomaly series for the Northern Hemisphere. Pattern by Esper et al. (2002) represents tree-ring temperature reconstruction, while the compilation by Mann et al. (1998) is based on multiproxy data (tree-rings, ice cores, corals, historical documents, and instrumental data). Reconstruction by Huang (2004) merges multiproxy and borehole sources (for details see Section 3.3, Chapter 3). It was proposed that the inconsistency in the pre-instrumental period between "geothermal" and

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