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1000 1200 1400 1600 1800 2000 TIME, years A.D.

Fig. 28. The effects of smoothing constraints in the case of a surrogate climate expressed by a sinusoidal model of the GST with period of 800 years in SVD (bottom), and FSI methods (top), respectively. The calculation was performed with a standard deviation a = 0.5 K. Estimated GST histories are marked by the applied values of cutoff and correlation time ic for SVD and FSI techniques, respectively. The inset shows corresponding noise-free temperature-depth profile.

itself in the reconstructed GST history. Estimations of this more complex GST history show that the T-z profile preserves significant information only about less remote course of the GST. As can be seen, the recent 400-500-year warming trend and the long-term mean temperature (zero-frequency component) can be reliably resolved by both techniques. Calculations show that the acceptable range for the smoothing constraint quantities in both methods depends on the actual GST variations. For the latter more complex GST history the instability of the solution occurs at higher cutoff values for SVD method and correspondingly for lower values of the standard deviation a for FSI technique. The interval of acceptable values for both parameters is smaller than in the previous case, e.g. optimal cutoff values lie in the range 10"2-10"4. The influence of the correlation time tc in FSI method is not so significant in this case. It may change within the same range as in the previous example without risk that the instability in the solution will arise.

The influence of the noise in the T-z profiles on the GST reconstructions is illustrated by the next set of reconstructed GST histories. Figure 29 shows GST reconstructions inferred from the temperature logs viewed through a noisy filter. Synthetic

Fig. 29. The effect of Gaussian noise on the inferred GST histories in SVD (bottom), and FSI (top) methods, respectively. The cutoff value for SVD is 10~4, and a = 0.5 K, ic = 100 years for FSI. The stationary Gaussian noise with standard deviations varying from 0.01 to 0.1 K and zero mean has been superimposed on the "gate" G1 model (Figure 22). For both methods, the results of inversion are practically independent on the noise level, up to standard deviation of 0.04-0.05 K.

Fig. 29. The effect of Gaussian noise on the inferred GST histories in SVD (bottom), and FSI (top) methods, respectively. The cutoff value for SVD is 10~4, and a = 0.5 K, ic = 100 years for FSI. The stationary Gaussian noise with standard deviations varying from 0.01 to 0.1 K and zero mean has been superimposed on the "gate" G1 model (Figure 22). For both methods, the results of inversion are practically independent on the noise level, up to standard deviation of 0.04-0.05 K.

temperature-depth profiles simulated for the "gate" model and included different levels of the Gaussian noise (Figure 25) were used as input data. As seen in Figure 29, both methods recovered the "true" gate model by the inversion reasonably well. The interval of the GST history most affected by the noise in the data appears to be the recent past, some 100-200 years. The noise-induced false temperature oscillations are clearly visible in this section of the GST histories reconstructed by both methods. This fact is an obvious consequence of the above-described physical nature of the heat conduction process, when GST variations are attenuated exponentially and smoothed with both depth and time. At the same time the best resolved recent parts of the reconstructed GST curves contain more noticeable fingerprints of noise.

The SVD method appears to be more stable to the noise in the temperature logs and is able to tolerate relatively strong noise contamination. As shown, the inversion results are practically independent of the noise level, up to standard deviations close to 0.1 K. The FSI technique appears to be more sensitive to the presence of noise. The amplitude of the GST change is overestimated by 0.5-0.8 K already for the noise s.d. of 0.05 K. The noise-induced instability of the solution occurs for the noise with s.d. values that are slightly above 0.05K. In principle, the FSI technique sensitivity to the strong noise contamination could be partly suppressed by an accounting of small value to a. However, in this case the amplitudes of the GST variations may be underestimated.

The role of the correlation time Tc (one of the constraints of FSI) increases in the case of the presence of noise in the data. Figure 30 shows a set of GST histories reconstructed for different values of a and Tc. The input temperature log was the "gate" model disturbed by the Gaussian noise of zero mean and 0.05 K standard deviation. Small value of a effectively suppresses the amplitude of detected GST changes. The longer correlation time Tc smoothes calculated climatic history and moves extremes of the GST variations to the past, while shorter correlation time enhances the effects of noise up to the instability of the solution and tends to move reconstructed extremes to the present. To reduce the effect of noise that is generally more prominent at short periods, large Tc value should be chosen. On the other hand, value of the correlation time cannot be too large, because in this case some important shorter period variations in the estimated GST history can vanish. In SVD, the data are assumed to have equal uncertainties. On the contrary, in FSI data are weighted in accordance with their uncertainties. This allows additional effective constraint on the noise-induced instability by employing greater uncertainties to the near-surface temperature data.

Summarizing above conclusions we can affirm that, notwithstanding that both methods vary significantly in their mathematical calculus, they gave generally similar results in a sense of the broad features of the reconstructed GST histories. The discrepancies in

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