Determination of surface temperature history from borehole temperature gradients

The determination of surface temperature history from borehole temperature gradient profiles is commonly formulated as a linear least squares minimization problem. The surface temperature history is approximated by a series of intervals of constant temperature perturbation superimposed on a mean temperature and the best solution obtained by systematically adjusting the number, duration, and temperature of these intervals so as to minimize the least squares measure between the measured and calculated gradient profiles. However, this inversion problem is improperly posed. The gradient disturbances due to surface temperature perturbations in the series of time intervals are usually “almost linearly dependent.” Thus the accuracy of a numerical solution is dictated by the level of random noise in the data and rounding errors in the computation. Using simulated data, we have systematically examined the effects of different noise levels, sampling ranges (maximum usable borehole depth), and sampling intervals (depth increment) on the resolution and time span of the derived surface temperature history. The method of orthogonal polynomials in curve fitting is also used in an attempt to improve the time span and resolution. The results, however, indicate that under realistic conditions, only about four intervals of temperature perturbation can be used to approximate the history of surface temperature variations. The major implications are (1) large time span and high resolution are mutually exclusive; for example, with a resolution of 100 years, the maximum time span would be about 400 years, whereas if a resolution of 50 years is desired, the maximum time span would be only 200 years; (2) in order to have a reasonable resolution, the epochs at which surface temperature changes took place must be known from independent information because it is impossible to approximate adequately the surface temperature history with four arbitrarily chosen time intervals of constant temperature; and (3) any surface temperature changes which took place outside the time span of interest must be known and their effects removed from the gradient data; otherwise, interference from these temperature changes will lead to a distorted surface temperature history in the time span of interest.