500 -|—i——i—i—i—i—i—|- -|—i—i—i—i—i—i—i——---
Fig. 22. Steady-state temperature-depth anomaly (left) and geothermal gradient-depth anomaly (right) produced by the climatic temperature forcing corresponding to Central England (Chapter 1, Figure 8). Inset shows the enlarged segment of the gradient anomaly between 50 and 300m depth.
500 —1—'—'—'—i—'—■—1—'—i—'—'—1—i— 0 500 1000 1500
Fig. 23. The "thermal memory" of the Earth for the (climate) events on its surface: diffusion of the subsurface temperature perturbation caused by a sudden GST change of duration At (k = 10"6m2/s, AT = 1K).
Although high-frequency components of GST changes are suppressed by the heat diffusion, calculated temperature-depth profile contains a robust signal of more than three century long climatic history. Negative temperature anomalies and positive gradient in the depth range 50-300 m indicate generally cold conditions in the seventeenth to nineteenth centuries, while the noticeable curvature in the uppermost part of the calculated temperature-depth profile and negative gradients correspond to the rapid warming of the twentieth century.
The temperature disturbances propagate downward and slowly fade away. Figure 23 illustrates the downward propagation of the thermal front (1% of the surface temperature change) corresponding to the step-like surface temperature impulse of duration At. The AT = 1 K amplitude impulse with duration 10 years propagates to the maximum depth of 170 m and fades away in the underground after approximately 240 years since its cancellation on the surface; 50-year long change can penetrate to approximately 400 m depth and is preserved in the Earth's interior for approximately 1200-year long period. This example illustrates well the possible length of the surface climatic history that can be extracted from borehole temperature logs as well as the resolution capacity of the borehole climate reconstruction method.
Past variations of the GST propagate slowly downward and, although attenuated and smoothed, remain recorded in the subsurface as a perturbation to the steady-state temperature field. Because the transient climatic signals have significantly shorter living time than the geologic heat flow variations, these two signals operate in differing frequency domains and do not mix; thus, it is possible to use borehole temperature profiles for the extraction of the past GST variations. Present rock temperatures measured at depths of up to several hundred meters provide an archive of temperature changes that have occurred on the surface in the past. They can be recovered by an appropriate analysis of temperature-depth data and provide the possibility to reconstruct past GST history. The main advantages of this geothermal tool for climate reconstruction are
(1) The method is based on a direct temperature-temperature relation. The temperature-depth profiles reflect a direct relationship to the continuous GST forcing. The borehole climate reconstructions do not require any calibration against independent temperature data and contrary to most proxies such reconstruction is free of uncertainties of calibration. Inferred from geothermal data the GST histories can be used as themselves and present a powerful complementary source of information for the verification of proxy records and enable complex multiproxy reconstructions. Recently several groups of climatologists have combined the results of the GST reconstructions with high-resolution proxies such as tree-rings and/or ice core data and achieved both more accurate temperature estimates for the proxy methods as well as better resolution for the geothermal GST histories (Beltrami and Taylor, 1995; Harris and Chapman, 2001, 2005; Huang, 2004; for details see Section 3.3, Chapter 3).
(2) The process of heat conduction integrates GST changes continuously in the same manner and thus secures data homogeneity (unlike, e.g. the meteorological data that can suffer from inevitable modification of the recording equipment or station reorganization).
(3) Extensive time interval that could be recovered. Temperature-depth profiles contain a robust signal of the long-term surface temperature history. Resolution of the geothermal method covers one to two millennia, in some cases up to the last glacial (Bodri and Cermak, 1997b; Safanda and Rajver, 2001).
(4) Continuous rather than short-term sensitivity. The resolution of the geothermal method is essentially multi-decadal.
(5) Extensive geographical coverage. Once a borehole is available, temperature log can be obtained easily. The necessary equipments (borehole thermometer or data logger) are relatively inexpensive and common. At present thousands of the borehole temperature logs measured all over the world are available even when not all are useful and serious selection criteria must be applied. The ample worldwide catalog of temperature logs represents a valuable database for comprehensive paleo-climatic investigations. In many cases borehole temperature logs represent the only source of the climatic information for the region that otherwise appeared as a "white spot" on the paleoclimatic map.
(6) Minimal anthropogenic disturbances. Many existing meteorological stations were actually located in settlements which later grew to huge population centers; their records may suffer from the induced anthropogenic pollution when local climate became significantly warmer than its surroundings (urban heat island) and the observed data can be mistakenly used. On the contrary, boreholes are usually drilled far from the population centers in remote regions. Climate information stored here contains minimum of such anthropogenic influence.
(1) The decrease of the resolution into the past and progressive smoothing of the amplitude of the recovered temperatures when moving back to the past due to smearing of the climatic signal caused by thermal diffusion.
(2) Exact knowledge of the thermal properties of rocks at each borehole site is indispensable to determine uniquely the pattern of the past climatic changes, while this information is not always available in full. This disadvantage, however, is similar to the common insufficiency of the field information in many other geophysical branches.
(3) Climate change is related to the mean air temperature, but the inversion of the borehole data provides the GST, which is not identical to the air temperature. The soil-air temperature coupling is complex and dependent on the type of surface topography, albedo, type of bedrock, micro-vegetation cover (and its temporal changes), snow cover, precipitation, changes in water table, etc. The time variations in all of these factors and their reaction to climate and terrain changes make the interpretation of borehole data more complicated. In general, ground temperature is always higher than the air temperature; however, both follow in principle the same trend and the basic features of the climate reconstruction can be well substituted by the GST reconstruction.
2.3 Geothermal Method of Climate Reconstruction: Principles, Resolution, Limitations (Forward and Inverse Techniques, Sources of Perturbation)
The effect of the past climate changes, namely the last Ice Age, on the temperature gradient was actually recognized by the early heat flow workers (Lane, 1923), and a general mathematical formulation was thoroughly discussed by Carslaw and Jaeger (1962). Only much later it was realized that the problem could be reversed, i.e. from the detailed measured T-z data to infer the past climate. The first attempt dates back to Beck and Judge (1969) who speculated on recent surface temperature variations using data from a borehole drilled on the campus of the University of Western Ontario. Cermak (1971) used T-z profiles from three holes in the Kapuskasing area and reconstructed GSTs for the past millennium in northeastern Ontario, Canada, when the Monte Carlo type statistical method was used to alleviate the calculation instability problems. However, it was not before the late 1980s when Lachenbruch and Marshall (1986) presented clear geothermal evidence from a number of boreholes in Alaska for a recent global warming. This work can be considered as the beginning of the worldwide attention paid to the importance of the temperature logs for the paleoclimate studies.
Even when the very first attempt to infer past climate changes from measured temperature-depth profile in inverse problem dates back to Hotchkiss and Ingersoll (1934), the application of the modern geophysical inverse theory to the reconstruction of the GST
changes from the measured temperature-depth profiles has started with the work by Vasseur et al. (1983), using Backus-Gilbert formalism (Backus and Gilbert, 1967). Since then numerous inversion methods have been developed. The first compilation of the research results inferring wide assessment and comparative study of various methods for the past temperature history reconstruction from underground temperatures has appeared in 1992 (Lewis, 1992). Generally, three basic groups of the inversion methods won widespread popularity: (1) ramp and step models, sometimes referred as the one or last event analysis, (2) inversion techniques employing singular value decomposition (SVD) algorithm, and (3) functional space inversions (FSI). All three algorithms are based on the theory of 1-D heat conduction.
The problem of heat conduction is formulated for a source-free composite medium and is defined over the time interval [t0 = 0, tn] and depth interval [z0 = 0, zm].
We assume that within the medium heat is transferred exclusively by conduction; thus, basic heat equation takes the form of Eq. (4). The surface temperature and temperature at the depth zm (great enough not to be affected by the surface conditions) are, respectively
Within the solid Earth the temperature field is governed by the heat flow from the depth and by the distribution of the thermophysical parameters. All responses to the changing surface conditions are superimposed over the steady-state (initial) internal temperature field U(z) as transient thermal perturbation V(z,t). Thus, the solution of Eq. (4) can be represented as a superposition of two functions (Carslaw and Jaeger, 1962)
The choice of the value for t0 depends on available T-z data. Anyhow, it should be moved sufficiently back into the past such that the thermal regime prior to t0 could be regarded as the steady state. Further, we assume that at the depth zm the climatic perturbation vanishes, i.e., it is not measurable. In this case, the bottom boundary condition will be Tm = U(zm). Strictly speaking, this assumption fulfills exactly only at zm ^ oo, but this will not make much difference if the depth zm is large. The equilibrium (initial) temperature U(z) is taken as steady-state temperature for boundary conditions U(z = 0) = T0(t = 0) = U0, KdUldz = Qm (z = zm), where Qm represents the undisturbed steady-state heat flow at depth zm. In the case of the stratified medium with constant thermophysical properties in each layer U(z) takes the form of Eq. (5). Function V(z,t) represents the transient temperature field due to changing surface conditions; in other words, it is extracted climate signal. The surface temperature perturbation is propagated downward with amplitude attenuation and time delay that increases with depth. The heat equation for V(z,t) is identical to Eq. (4) for the modified surface boundary condition V0(z = z0,t) = T0(t)-U0, and for zero bottom and initial temperatures.
The parameterization of the problem (model) is defined by the set of five parameters/functions: K(z), pc(z), U0, Qm, and V0(t). Assumption of the source-free medium is sufficient in the most practical cases. If necessary, the rate of the heat production per unit volume can be added to the steady-state equation for the initial temperature U(z).
After the choice of the model one can solve the equations for quantities U(z) and V(z,t). It is so-called forward problem. Its solution requires discretization procedure that transforms primary partial differential equations into a set of algebraic equations relating the discretized model m to the vector of the GST values G
Depending on the discretization, the forward problem can be solved as accurately as desired. Solution takes analytical form in the case of the layered medium with constant known thermophysical parameters of each layer (Bodri and Cermak, 1995). In this case, K(z) and pc(z) can be excluded from m, and Eq. (9) can be written as G = Dm, where matrix D is generally referred to as the data kernel. The Laplace transformation can be used to integrate Eq. (4). Let T and Q*;-1 be the transforms of the temperature and heat flow at the depth z = Zj_ 1, corresponding to the base of the jth layer, and Tj, Q* be the analogous values at the depth z = Zj. In the case of perfect thermal contact between the layers we have where
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