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To be estimated

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"Relevant only for deep boreholes.

answer the question, what is the effective number of degrees of freedom in the data and what parameters can be independently estimated with an acceptable variance.

Every method takes the steady-state GST, the equilibrium surface temperature U0, and the basal heat flow Qm as unknown parameters. Together with the thermal conductivity of the medium these quantities control the steady-state temperature profile, and such formulation permits to estimate the parameters characterizing the steady state even for relatively shallow boreholes where estimation of these quantities from the lowermost undisturbed parts of the measured temperature-depth profiles is problematic.

From the point of view of the accuracy of the GST reconstruction the SVD and FSI are more effective since they use more complex temporal discretization and incorporate additional information in the analysis, and thus allow the reconstruction of far more detailed GST histories than the ramp/step method. The parametrization scheme applied in the ramp/step method and in SVD uses analytical expressions for the temperature field T(z,t), while in the FSI it can be expressed only numerically. It should be mentioned, however, that the computational efficiency of the methods does not strongly depend on whether the analytical or numerical approximations are used and represent the complex output of the whole parametrization scheme.

The ramp/step method and SVD are limited to the problems when the thermal properties of the medium are known. At a first glance it seems to be serious restriction. General insufficient thermal conductivity data in boreholes is a well-known fact. However, in the most field examples errors in K(z) and pc(z) are not systematic. In the SVD approach, their effect can be extenuated by imposing appropriate smoothing constraints on the GST history. Thus, notwithstanding the known thermal properties assumption, SVD technique is applicable to a large number of practical cases. The principal distinction from the ramp method and SVD and/or merit of FSI is that the thermal properties of the medium are all formulated as unknown parameters and estimated simultaneously with the quantities describing the initial thermal state of the medium and the past climate history. Thus, this method is preferable when the errors in K(z) and pc(z) are significant and expected to be systematic. When the thermophysical properties are included in the model as parameters, the problem becomes nonlinear. However, as shown by Shen and Beck (1991) and Wang (1992), for the cases when these quantities are reasonably well known with small uncertainties, the problem is only mildly nonlinear and permits the application of iterative gradient methods to solve the optimization problem described above. It should be also mentioned that for the field examples with exactly known thermal properties the theory becomes strictly linear.

Methods of the inversion of the borehole temperature data are all based on the least squares inverse approach. The problem is ill posed; thus, in the real cases the misfit function itself is insufficient to determine the unique and stable solution. The SVD and FSI put definite constraints on the model and employ the concept of incorporating a priori (additional) information to avoid this problem. As will be shown in the next section, a large part of discrepancies in the inverse results obtained by both methods can be attributed to the kinds of constraints and/or the nature of additional information used in the analyses. Unfortunately, the optimal values for the applied constraints depend on the real GST history. Thus, at least some knowledge of the amplitude and timing of the climatic variations that should be reconstructed is indispensable to obtain consistent inversion results.

All described methods are based on the least squares technique. Cooper and Jones (1998) have performed a comparison between the effectiveness of the least squares approach and other popular techniques, namely the minimization of the absolute difference between measured data and estimated parameters for inversion of borehole temperature logs that is performed for the purposes of the GST reconstruction. The authors have found that the latter technique requires approximately half the number of iterations to reach the possible minimum error compared to the least squares procedure. According to the above-cited work, the inversion of borehole temperature data in some cases can be significantly improved by the use of techniques other than the standard least squares approach. According to their calculations, exact choice of the inversion technique depends on the statistics of the data. Anyhow, it was used by Cooper and Jones (1998) during all trial runs the best results were obtained by the latter approach combined with some additions that accelerate the procedure in the damped intervals where the model improves only slowly by subsequent iterations.

2.4 Comparison of Ground Surface Temperature (GST) Reconstruction Methods

As many geophysical problems, the GST reconstruction involves the estimation of a number of unknown parameters that bear definite relationship to experimental data. These data are generally contaminated by various kinds of random and/or systematic noise as well as may be inconsistent and insufficient for estimation of the unknown parameters. Thus, generally we have strongly underdetermined system; in other words, we would like to draw out the infinity of the details about unknown function from very limited amounts of data. One of the usual ways to overcome this problem is to calculate a family of the trial inversions, compare the interdependence between resolution and variance for each case of inversion, and select the run that appears to be most appropriate for the interpretation of the solution. All three above-described methods of the GST reconstruction have treated this problem in their own manner.

Existing methods for the GST reconstruction from subsurface temperature-depth profiles are based on the same theory of 1-D heat conduction in a layered medium; thus, we can expect similarity of obtained results in terms of their general features. On the other hand, the GST reconstruction is an ill-posed inverse problem. Its finer output may be method dependent. The inconsistency between GST histories reconstructed by different inverse methods may arise both from the differences in the mathematical/physical approaches used and from the manner in which different kinds of noise (uncertainties) are treated. For comparison and/or joint use of the GST histories inferred by different methods at first the methods themselves should be compared and evaluated. Such comparison including both synthetic and field results has been widely performed at an early stage of the development of "geothermal" method (Beck et al., 1992; Shen et al., 1992). Below we present some examples to illustrate conclusions obtained in above and similar works. We applied two most powerful methods of the GST reconstruction, namely SVD and FSI, to some standard data sets and field data, to illustrate the effects of various constraints on the inferred GST histories.

2.4.1 Effects of smoothing constraints in different methods and the noise in the data: Synthetic examples

The most effective way to compare different inversion techniques is to simulate a series of perturbed subsurface thermal regimes using synthetic GST histories, i.e., the true result is known, and to apply the available techniques on those data. Such attempts have been undertaken in numerous works for the simple case of 1-D forward pure conductive model driven by different variants of GST forcing. An inversion approach has been applied to infer GST histories from the simulated profiles and to compare them with the true surface forcing used. The results of such trial numerical experiments have been widely discussed (Beck et al., 1992; Shen and Beck, 1992; Shen et al., 1992). Below we present some illustrations of this approach.

The basic synthetic T_ z data, with which we would like to assess the most effective methods of the GST reconstruction, SVD and FSI, are calculated with the following parameters: K(z) = 2.5W/(mK), pc(z) = 2.5MJ/(m3K), Qm = 0, U0(z) = 4°C. Constant and known thermophysical parameters are chosen to illustrate the effect of the smoothing constraints and not to digress on other influences. The "gate" model of the GST temperature is used to demonstrate the effect of the smoothing constraints in the case of a sudden temperature change. This model has a shape V0(t) = {4°C at t > 1600 A.D.; 3°C at 1600 < t >1900 A.D., 4°C at t > 1900 A.D.} and roughly corresponds to the Little Ice Age conditions, followed by subsequent warming. Temperature logs were calculated at 5 m interval to a depth of 500 m. The first generated data set G1 is completely noise free, while other profiles were randomly perturbed by a noise with Gaussian distribution with zero mean and standard deviations of 0.01, 0.03, 0.05, and 0.1K, respectively. Typical measurement error is 0.03 K. The standard deviations were assumed to be independent of the depth. Simulated in this manner T-z profiles are shown in Figure 25.

Fig. 25. Temperature-depth profile for the "gate" model free of noise (G1) and two synthetic profiles perturbed with Gaussian noise of zero mean and different s.d. of ±0.3 K and/or ±1.0K, respectively added to G1. Typical measurement error is usually about 0.03 K.

The GST histories were approximated by a series of individual intervals of constant temperature, when the mean value of temperature in each time interval is an unknown parameter. It is these temperature values that represent the direct result of the inversion procedure. When demonstrating the results graphically, these values were ascribed to the midpoints of the corresponding time intervals and were found as approximated by cubic spline technique (Bodri and Cermak, 1997a).

Figure 26 shows the results of GST reconstruction by SVD and FSI for the noise-free G1 data using different smoothing constraints. A priori null GST hypothesis was assumed in the GST reconstructions by FSI technique (no a priori knowledge of the GST history to be estimated). As seen, the true GST history is reasonably well recovered by both inversion techniques. Clearly, the solution depends critically on the cutoff value for SVD and on the values of a and tc for FSI. The large values of cutoff tend to smooth the reconstructed curve and move its extremes slightly toward the present (Figure 26a). Too small values may lead to instability of the solution (Figure 26b) with frequent small false extremes. It should be mentioned, however, that these false oscillations do not fog significantly general GST pattern; the long scale course of the GST history is preserved even in the reconstructions calculated with the smallest possible cutoff value. Obviously, some optimum cutoff value

Fig. 26. The effect of smoothing constraints on the GST histories inferred from G1 temperature-depth profile: (a), (b) SVD, and (c) FSI, respectively. The real GST history is the "gate" model (dashed line). Smoothing constraints are imposed in the form of the cutoff value in SVD and the correlation time rc in FSI (a = 0.5K). Too small cutoff value may cause the instability of the solution (Panel b).

Fig. 26. The effect of smoothing constraints on the GST histories inferred from G1 temperature-depth profile: (a), (b) SVD, and (c) FSI, respectively. The real GST history is the "gate" model (dashed line). Smoothing constraints are imposed in the form of the cutoff value in SVD and the correlation time rc in FSI (a = 0.5K). Too small cutoff value may cause the instability of the solution (Panel b).

must be chosen. Wiggins (1972) suggested powerful procedure for its establishment. According to a suggestion of this author that is based on the results of numerous experimental runs, one should set upper limit on the standard deviation of the estimated parameters and search for the largest number of the eigenvalues associated with the solution for which each estimated variance (Eq. (21)) is less than this limit. This then determines the number of degrees of freedom associated with the solution.

In FSI method, the constraint imposed on the GST history depends on a priori standard deviation a and correlation time Tc (Eq. (27)). The former is responsible for the amplitude of the detected GST changes. and the latter controls the smoothness of the quantity [V0(t) —V(t)] (Eq. (27)). Both constraints operate together. Too large values assigned to a may turn the autocovariance function into inoperative regime. As will be

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