For an accurate assessment of the relative roles of natural variability and anthropogenic influence in the Earth's climate, reconstructions of past temperatures from the pre-industrial as well as the industrial period are essential. But instrumental records are typically available for no more than the past 150 years. Therefore reconstructions of pre-industrial climate rely principally on traditional climate proxy records, each with particular strengths and limitations in representing climatic variability. Subsurface temperatures comprise an independent archive of past surface temperature changes that is complementary to both the instrumental record and the climate proxies. Here we use present-day temperatures in 616 boreholes from all continents except Antarctica to reconstruct century-long trends in temperatures over the past 500 years at global, hemispheric and continental scales. The results confirm the unusual warming of the twentieth century revealed by the instrumental record, but suggest that the cumulative change over the past five centuries amounts to about 1 K, exceeding recent estimates from conventional climate proxies. The strength of temperature reconstructions from boreholes lies in the detection of long-term trends, complementary to conventional climate proxies, but to obtain a complete picture of past warming, the differences between the approaches need to be investigated in detail.

Analyses of underground temperature measurements from 358 boreholes in eastern North America, central Europe, southern Africa, and Australia indicate that, in the 20th century, the average surface temperature of Earth has increased by about 0.5 degreesC and that the 20th century has been the warmest of the past five centuries. The subsurface temperatures also indicate that Earth's mean surface temperature has increased by about 1.0 degreesC over the past five centuries. The geothermal data offer an independent confirmation of the unusual character of 20th-century climate that has emerged from recent multiproxy studies.

Under the assumption •that the Earth's thermM field is one-dimensionM and purely conductive, the temperature w is related to the Earth model m through a partiM differential equation (PDE), where m is the set of model parameters consisting of the ground surface temperature, the background heat flow density, the thermM conductivity, the specific heat capacity, and the rate of heat production; by setting the time origin sufficiently far in the past, the initiM temperature field may be taken as the steady state temperature field. Given data (do, Cd) on w and a priori information (m0, Gin) on m, where C d and Cm are covariance operators describing uncertainties in do and m0, respectively, the aim of the least squares inversion is to determine the most probable model that minimizes the 1 misfit function $ --1/2(C-(d-do), d-do)+l/2(C•n (m-too), m-too) We formulate this problem in the functmnM space as opposed to the conventtuna] d•screte formulatmn and solve it using iterative gradient methods. The formulation reduces the computation in each iteration to essentially two forward solutions of the PDE, the first for the primal problem: given m, solve for the actual field w, and the second for the dual problem: using the weighted data residuals as heat source, solve for the residual temperature field in the same medium, but with homogeneous boundary conditions and with time reversed. The correlation of the residual and the actual fields, then, gives the gradient and also the Hessian of $, the latter of which evaluated at the most probable model is the approximate a posterJori covariance operator. Because discretization is required only when solving the forward problems, we avoid the computing and storing of partial derivatives of d with respect to discretized m, which can be a prohibitive task when the number of data and the number of discretized ?• are large. 1991]. Traditional analysis of the T-z profiles, both forthe extraction of past climate and for the determination of background heat flow density (HFD), is based on forward calculations: a series of simple step or ramp models for the GST are tested, and the one that best fits the data under certain criterion is selected [Paper number 91JB01883. 0148-0227/91/91 JB-0188350 5.00 1984; Lindqvist, 1984; Beck et al., 1985; Lachenbruch and Marshall, 1986; Wang et al., 1986; Lachenbruch et al., 1988J.This method is satisfactory when we are interested only in obtaining rough estimates of the most recent GST trend; the transient perturbation due to this recent trend is so conspicuous that a reasonable estimate can be obtained with a simple analysis. A typical T-z profile, however, contains considerably more information about the GST than can be properly extracted this way. For example, data from a 600-m borehole contain information not only of the detailed structure of the recent warming but also of the general GST pattern for the past millenium. Considerable proxy evidence has indicated that the climate has varied significantly over this period. Following a generally warm peri...

All existing approaches to the analysis and interpretation of borehole temperatures in terms of a ground surface temperature history are based on the one-dimensional theory of heat conduction. Deviations from this idealization are manifest as noise in the interpretation. We present numerical experiments that explore the effects of three-dimensional subsurface heterogeneity on ground surface temperature histories inferred from borehole temperature profiles. Inversions of steady state temperature profiles containing such three-dimensional "noise" reveal that in an inversion formulation incorporating a priori information, spurious temperature histories can emerge when the a priori constraints on subsurface temperatures and thermophysical properties are too fight, i.e., when arbitrarily small variations in subsurface temperatures and thermophysical properties are interpreted to have significance for the derived climate history. Relaxation of the a priori constraints enables an effective muting of the spurious histories. Similar experiments conducted on synthetic transient signals confirm the steady state results but also reveal that relaxation of the a priori constraints will also lead to a loss of signal if the level of relaxation is excessive. Our experiments suggest that relaxation of constraints on thermal conductivity is more efficient than relaxation of borehole temperatures in suppressing artifacts arising from the threedimensional effects. We identify a range of constraints in which reasonable noise suppression and signal recovery are both achieved and then reprocess data from 22 Canadian boreholes for which surface temperature histories had earlier been derived by ourselves and others. Our reprocessed results reveal a remarkably simple surface temperature history common to much of eastern Canada. This history comprises a recent warming interval commencing in the nineteenth century, in which the surface temperature increased by some 1-4øC. Approximately 0.5-1.0øC of this increase was a recovery from a preceding cooler interval in which the temperature was below the long-term mean. The remainder of the increase represents warming that exceeds the long-term mean. IntroductionThe extraction of a geophysical signal in the presence of obscuring noise is a fundamental endeavor in virtually all analyses of geophysical observations. In this paper we address the special problem of inferring a time-varying surface boundary condition, i.e., the temperature history experienced by the surface of the solid Earth, from measurements of subsurface temperatures in boreholes and the thermophysical properties of the surrounding rock. Noise in the system is principally of two types: (1) the usual errors in the measurement of temperatures, depths, and thermophysical properties, and (2) representational errors, i.e., departures of