Smooth inversion for ground surface temperature histories: estimating the optimum regularization parameter by generalized cross-validation

The inversion of recent borehole temperatures has proved to be a successful tool to determine ancient ground surface temperature histories. To take into account heterogeneity of thermal properties and their non-linear dependence on temperature itself, a versatile 1-D inversion technique based on a finite-difference approach has been developed. Regularization of the generally ill-posed problem is obtained by an appropriate version of Tikhonov regularization of variable order. In this approach, a regularization parameter has to be determined, representing a trade-off between data fit and model smoothness. We propose to select this parameter by generalized cross-validation. The resulting technique is employed in case studies from the Kola ultradeep drilling site, and another borehole from northeastern Poland. Comparing the results from both sites corroborates the hypothesis that subglacial ground surface temperatures as met in Kola often are much higher than the ones in areas exposed to atmospheric conditions (Poland).