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

Geophys. J. Int. (2001) 147, 29–40 The use of broad-band prior covariance for inverse palaeoclimate estimation Delia Zemira Şerban1,2 and Bo Holm Jacobsen1 1 2 Department of Earth Sciences, University of Aarhus, Finlandsgade 8, DK-8200 Aarhus N, Denmark. Email: . The Institute of Geodynamics, 19–21, Jean-Louis Calderon Str., R-70201, Bucharest-37, Romania. Accepted 2001 May 5. Received 2001 May 7; in original form 2000 December 28 Key words: ground surface temperature history, inversion of temperature logs, palaeoclimate, power-law fractals INTRODUCTION Climate varies at all scales of time (Beck 1992), and these changes imply ground surface temperature variations. Fig. 1 illustrates the types of change that can occur at different periods of interest. Fig. 1(a) shows daily temperature averages at a meteorological station in Bucharest, Romania, for the year 1997 (Constanta Boroneant, personal communication 2000), Fig. 1(b) shows annual averages from Lake Duparquet in Quebec, based on tree ring measurements (Beltrami et al. 1995), and Fig. 1(c) shows 100 year averages at the timescale of glaciations, based on deuterium measurements from ice cores (Jouzel et al. 1996). All timescales show variation of the order of degree K. Due to the relatively low thermal diffusivity of rocks, surface temperature variations are recorded in the subsurface as delayed and filtered transient perturbations to the steady-state temper# 2001 RAS ature field (Čermàk 1971; Vasseur et al. 1983). The perturbation propagates downwards, the depth of penetration depending on the frequency of the temperature signal (Clauser & Mareschal 1995). Short wavelength (high frequency) temperature variations reach only shallow depths, whereas longer wavelength (low frequency) perturbations, although attenuated and smoothed, are expressed deeper in the subsurface. Thus, the Middle Ages climatic optimum, the Little Ice Age and the recent warming trend can be studied in relatively short boreholes (e.g. Nielsen & Beck 1989; Shen & Beck 1992; Beltrami et al. 1992; Clauser & Mareschal 1995), whereas the warming at the end of the last glaciation is revealed only by longer boreholes (e.g. Şerban et al. 2001; Dahl-Jensen et al. 1998; Rajver et al. 1998). Inverse methods are generally used to extract the GSTH from borehole temperature measurements. The first inverse method (Vasseur et al. 1983) used the formulation by Backus & 29 SUMMARY 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. 30 Delia Zemira Şerban and Bo Holm Jacobsen (a) Temperature(˚C) 30 20 10 0 -10 0 40 80 120 160 200 Day 240 280 320 360 400 1240 1280 1320 1360 1400 Year 1440 1480 1520 1560 1600 360 320 280 240 200 160 Time bp (kYears) 120 80 40 0 (b) 1 0 -1 -2 1200 Temperature(˚C) 2 (c) 0 -2 -4 -6 -8 400 Figure 1. Examples of temperature variations at different timescales. (a) Daily averages at Baneasa meteorological station, Bucharest, Romania for the year 1997 (The data bank of the Romanian Institute of Meteorology and Hydrology). (b) Annual averages from Lake Duparquet in Quebec, Canada, based on tree ring measurements (Beltrami et al. 1995); (c) Hundred year averages at the timescale of glaciations, based on deuterium measurements on ice cores (Jouzel et al. 1996). Gilbert (1967). More recently, the great majority of researchers use inversion methods based on the least-squares inverse theory of Tarantola & Valette (1982a,b) (e.g. Nielsen & Beck 1989; Shen & Beck 1991, 1992; Wang 1992; Lewis 1992; Huang et al. 1996). Estimation of GSTH from borehole temperatures defines an ill-posed problem. These works quantify both the measurement noise and the variability of the GSTH in terms of covariances which then, in principle, define a unique and optimal regularization of the ill-posed problem. Still, the comparative studies of Beck et al. (1992) and Shen et al. (1992) showed that various model parametrizations and various a priori constraints applied produced different GSTH estimates for the same set of synthetic temperature data. It is not satisfactory when results depend on technique and/or researcher. The need to standardise the procedure for the reconstruction of GSTH was noted by Huang et al. (1996) and Shen et al. (1996). Such a standardisation should allow the comparison of GSTH estimates derived by different research groups from temperature measurements at different well depths, different history intervals and different model parametrizations. Many of the inconsistencies between previous works originate in the interplay between different discretizations and different choices of typical correlation length in the single-scale prior covariances applied to the GSTH. We shall demonstrate that the extension of the prior to a multiple-scale or broad-banded covariance model lifts these inconsistencies and leads to more stable and robust results. A PRIORI COVARIANCE FUNCTIONS. SINGLE-SCALE OR MULTIPLE-SCALE STOCHASTIC MODELS The a priori information which represents our knowledge on GSTH before in (...truncated)