The use of broad-band prior covariance for inverse palaeoclimate estimation

The determination of ground surface temperature history (GSTH) from present borehole temperature defines an ill-posed inverse problem for which the required regularization must reflect the stochastic properties of both measurement noise and ground surface temperature. The timescales of interest in the study of climate changes range from a few decades to hundreds of thousands of years. This makes climate a very broad-banded process. This paper presents a multiple-scale stochastic prior model for GSTH, which overcomes several problems in previous commonly applied single-scale modes of regularization. The von Karmann power-law stochastic processes come out as special cases. Should other information warrant uneven prior variation bounds on different frequency intervals, the multiple-scale formulation can readily incorporate this. The practical computation of the prior covariances between past temperature node values is achieved through an elementary function space projection formulation. This projection approach is generally applicable for arbitrary base functions underlying the discretization. The superior robustness of this multiple-scale prior model is demonstrated by comparison to previously used single-scale models for the classic test case by Beck ( 1977) where the multiple-scale prior model allows Simultaneous estimation of temperature history from decade scale to glaciation scale. Moreover, the function space projection method ensures that GSTH estimates are insensitive to discretization, provided that discretization is finer than the inherent resolution limit. A test on two very different experimental data sets confirms the merits of the proposed stochastic model. For maximum consistency between GSTH estimates across timescales, borehole depths scales and groups of investigators, we propose that a covariance function with a uniform variance of (5 K)(2) per frequency decade be used as a standard prior for inverse ground surface temperature history problems.