Borehole temperature data are generally given at discrete zk points (k = 1,2,... M). Thus, the solution can be represented by a set of M linear equations in N unknowns Vi

where Tk is the temperature measured in borehole at depth zk and Aik a matrix formed by the values of series similar to Eq. (13) at depth zk for time interval (ti-1,ti). When parameters of initial (equilibrium) temperature field U0 and Qm are estimated simultaneously with the GST history V0(t), the vector V; will consist of (N + 2) unknowns, and the matrix A will contain 1 in its (N + 1) first column and the thermal resistance R(z) = j0(dz'lk(z')) to the depth zi in the (N + 2) column. At M > N this yields an underdetermined system of linear equations that can be solved for the unknown parameters Vt by the SVD (Jackson, 1972; Menke, 1989). This general least squares inversion method minimizes both the sum of the squares of deviations of the measured temperature profile from the theoretical model ere and the sum of the squares of estimated parameters V TV0 (s = AV0 — T).

Mathematically, the SVD procedure can be formulated as follows. Two sets of eigenvectors u and v can be found such that Avj = XjUj and ATui = Aivi, where Xt are the eigenvalues of the matrix A. One shall again assume that there are P non-zero eigenvalues common to the sets of eigenvectors u and v. The set of eigenvectors u represent complete massif of the orthonormal vectors in the data space, while the eigenvectors v are similar set in the "model" space. The P-value (P is less than or equal to the minimum of M and N) may be interpreted as the potential number of degrees of freedom in the data. The matrix A can be decomposed into the product A = UAVT, where U is the M X P orthonormal matrix whose columns are the eigenvectors ui: V an N X P matrix with the columns of the eigenvectors vi, and A the diagonal matrix of eigenvalues. The solution x can then be written as x = VA-1UTT, where x is the estimate of V0 and A-1 a diagonal matrix whose elements are X-According to our previous assumption all P eigenvalues are not zero; thus, the inverse matrix A-1 never ceases. However, definite problems may arise also if values of Xi are non-zero but very small.

The linear combination of model parameters that have weak impact on the data corresponds to the small singular values X. In the inversion, the data are divided by the smallest eigenvalues. Even for the noise- and error-free data numerical instabilities appear when the ratio of the largest to the smallest eigenvalue exceeds a critical limit (say, 10"6; see Section 2.4). The presence of noise in the data intensifies the problem. In this case the inverse matrix A"1 exists; however, the uncertainties in the estimated x-vector that for statistically independent data can be calculated as will be too large because of the reciprocal eigenvalue in the bracket. This variance can be interpreted as the amplification of the measurement errors in the solution. The order of magnitude of the variance is inversely proportional to the smallest eigenvalue. In practice, it is thus indispensable to eliminate the small eigenvalues from the solution. For the elimination of the parameters that have weak impact on the data the "sharp cutoff' approach is generally used (Wiggins, 1972). Under the sharp cutoff approach we select some dimensionless cutoff value and ignore all eigenvalues whose ratio to the largest eigenvalue is less than this limit. The cutoff value is thus the crucial parameter of the SVD method. It will be shown below that the large values of cutoff tend to smooth the reconstructed curve and move their extremes toward the present. At lower cutoff values, parameters that are weakly represented in solution are better resolved, but their errors will simultaneously grow. Too small cutoff values may lead to the unacceptable error level and to the instability of the solution. Some optimum value must be chosen. Wiggins (1972) suggested the procedure to establish the optimum cutoff. According to his approach one should set an upper limit on the standard deviation of the estimated parameters, and search for the largest number of eigenvalues associated with the solution for which each estimated variance is less than this limit. This then determines the number of degrees of freedom associated with the solution. In other words, the SVD naturally eliminates from the inverse all the oscillations of the surface temperature that the data cannot resolve and yields the smoothed course of the surface temperature.

The stabilizing and smoothing of the solution can be achieved also by adding a small constant to each singular value

This procedure does not affect the inversion of the larger singular values, but it stabilizes the inversion of smallest singular values by damping them to zero. The value of damping parameter s (and thus the GST resolution) is determined by the noise level in the data. This procedure generally gives smoother GST histories than the sharp cutoff technique. In principle, there are no exact arguments that could force the choice of either procedure.

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