8.1 Theoretical framework
The Earthβs sub-surface thermal regime is governed by the temporal variations in the Ground Surface Temperature (GST) and the outflow of heat from the Earthβs interior. For a homogeneous subsurface with no GST variations and internal heat sources, the temperature in the subsurface increases linearly with depth. This temperature profile can be considered to be in a quasi-steady state relative to the timescale of recent climatic variations, since it depends only on heat flux from the Earthβs interior that varies over very long timescales.
Temporal variations in ground surface temperature propagate into the subsurface and are recorded as transient perturbations to this geothermal quasi-steady state. As a result of heat diffusion, the amplitude of the subsurface anomalies is proportional to the duration and magnitude of the GST perturbations. In addition, the amplitude of the subsurface anomalies decreases with time since their occurrence. To reconstruct the temporal variation in GST, the variation in the subsurface temperature as a function of depth is measured in boreholes.
8.1.1 Theory
For homogeneous, isotropic, source-free half space, the temperature perturbation due to temporal ground surface temperature variation, is solution of the diffusion equation in one dimension with initial and boundary conditions (Carslaw and Jaeger, 1959)
ππ2π
ππ§2 =ππ
ππ‘β¦ β¦ β¦ (1)
Where k is the thermal diffusivity of subsurface soil or rock, z is depth (positive towards down), and t is time. The one dimensional equation can be used if long term surface
41 temperature history. A perturbation T(z, t) induced by a periodic surface temperature change, T0(t)=cos(Οt) is expressed by (Mareschal and Beltrami, 1992)
π(π§, π‘) = cos (ππ‘ β π§βπ
2π ) ππ₯π (βπ§βπ
2π ) β¦ β¦ β¦ (3)
This perturbation transmission is similar to a wave and is propagates exponentially with depth. The wave is attenuated by a factor 1/e and the skin depth (Ξ΄) is dependent on wave frequency (Ο) and the rock thermal diffusivity (k) can be expressed as
πΏ = β2π
π β¦ β¦ β¦ (4)
As thermal diffusivity of rock is very low π β 10β6 m2 s-1 so the propagation of the thermal front into the rock is also very slow. For usual the Earth crust material, a thermal front can be propagated to a depth of a few meters in 1 year, 100 meters in 200 years, 500 meters in 1000 years and within 1000 to 2000 meters depth, the post glacial warming can be observed (Appleyard, 2005; Gosselin and Mareschal, 2003; Huang et al., 2000). For an instantaneous change of the surface temperature Ti at times ti before present, perturbation can be equilibrium heat flow is usually assessed in the deeper part of the temperature profile which is not affected by the recent surface temperature change. The equilibrium temperature is extrapolated upward and the perturbation is determined as the difference between the measured temperature and the upward continuation of the deep profile. The area between these curves represents the net heat absorbed by the ground and the shape of the perturbation
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can say about long time thermal history of the Earth surface. This history can be determined by assuming a model and iterating until a satisfactory fit is obtained or generally, it can be found directly by inversion (Beltrami et al., 1992; Beltrami and Bourlon, 2004; Beltrami and Mareschal, 1995; Gosselin and Mareschal, 2003; Mareschal and Beltrami, 1992)
8.1.2 Inversion
The inversion of borehole temperature profile is an operation that transforms a temperature versus depth profile at a given time (the time of measurement) into a temperature versus time profile at a given depth. Basically, inversion yields a ground surface temperature history. The link between depth and time is through the thermal diffusivity and other thermo-physical properties of the rock through which the climate signal is propagating.
The perturbation at depth z, T(z), due to temperature variation at the Earth surface, can be written considering the thermal conductivity variations, as the superimposition of the equilibrium temperature and the perturbation Tt(z) induced by temporal surface temperature condition (Beltrami et al., 1992)
π(π§) = π0+ π0π (π§) + ππ‘(π§) β¦ β¦ β¦ (6)
Where T0 is the equilibrium surface temperature, q0 is the surface heat flow density and R(z) is the thermal depth between the Earth surface and depth z. The effect of heat production is small and can be neglected. In general, short period variations are filtered out by the Earth.
The surface temperature can be estimated by the average surface temperature over k time intervals of equal duration Ξ, can be expressed as
π(π‘) = ππ(π β 1) β β€ π‘ β€ πβ β¦ β¦ β¦ (7) Equation (6) can then be modified as
ππ = π΄ππππβ¦ β¦ β¦ . (8)
Where ππ is the measured temperature at depth zj, Xi, is a vector encompassing the unknowns {T0, q0, T1,β¦β¦β¦β¦Tk}, and Ajk is a matrix, each row of which contains 1 in the first column,
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Mareschal and Beltrami, 1992; Menke, 1989). Individual inversions were performed using borehole temperature data. Greatest temperature change (GTC) in each borehole was calculated to analyse spatial variation of paleotemperature.
8.2 References:
Appleyard S. 2005. Late Holocene temperature record from southwestern Australia: evidence of global warming from deep boreholes. Aust. J. Earth Sci. 52:161β166.
Beltrami H, Bourlon E. 2004. Ground warming patterns in the Northern Hemisphere during the last five centuries. Earth Planet. Sci. Lett. 227:169β177.
Beltrami H, Jessop A, Mareschal JC. 1992. Ground temperature histories in eastern and central Canada from geothermal measurements: Evidence of climatic change.
Palaeogeogr. Palaeoclimatol. Palaeoecol. 98:167β184.
Beltrami H, Mareschal JC. 1995. Resolution of ground temperature histories inverted from borehole temperature data. Glob. Planet. Change 11:57β70.
Carslaw HS, Jaeger JC. 1959. Conduction of Heat in Solids (2nd edition). Oxford University Press, New York.
Gosselin C, Mareschal JC. 2003. Variations in ground surface temperature histories in the Thompson Belt, Manitoba, Canada: environment and climate changes. Glob. Planet.
Change 39:271β284.
Huang S, Pollack H, Shen P. 2000. Temperature trends over the past five centuries reconstructed from borehole temperatures. Nature 403:756β8.
Mareschal JC, Beltrami H. 1992. Evidence for recent warming from perturbed geothermal gradients: Examples from eastern Canada. Clim. Dyn.:135β143.
Menke W. 1989. Geophysical Data Analysis: Discrete Inverse Theory. International Geophysics Service.
Vasseur G, Bernard P. 1983. Holocene paleotemperatures deduced from geothermal measurements. Palaeogeogr. β¦ 43:237β259.
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