Page 197 - Physical Principles of Sedimentary Basin Analysis

6.19 Variable surface temperature 179

a data set when just one parameter is considered. An improved match requires a more
complex surface temperature history with more parameters, and a non-trivial optimization
method.
Measurements of the borehole temperatures are an important source of information
about the climate in the past few centuries (Vasseur et al., 1983, Beltrami and Mareschal,
1992, 1995, Huang et al., 2000). A large number of boreholes have been drilled and mea-
sured (Mareschal, 2005) all over the planet. Numerical techniques have been developed and
applied in the reconstruction of the past surface temperature (Vasseur et al., 1983, Wang,
1992, Beltrami and Mareschal, 1995, Chouinard and Mareschal, 2007). An extensive dis-
cussion of methods and results of borehole climatology is given by Gonzalez-Rouco et al.
(2008).

Note 6.11 Duhamel’s theorem. The transient solution (6.282) is derived from Duhamel’s
theorem. It gives the transient solution of the temperature equation

2
∂T ∂ T
− κ = 0 (6.284)
∂t ∂z 2
for the infinite half-space (z ≥ 0), when the boundary condition at z = 0 is the surface
temperature T surf (t) as a function of time. It assumes that the initial condition is T = 0 and
that T surf (t) = 0for t < 0. The second boundary condition is T = 0 at infinite depth. The
solution T is written in terms of the simpler solution U of the same problem, where U is
the temperature in the case of instantaneous heating of the surface with a unit step-function,
T surf = 1for t ≥ 0. Duhamel’s theorem then says that the temperature solution T is
t ∂U
T (z, t) = T surf (λ) (z, t − λ) dλ. (6.285)
0 ∂t
A direct way to verify the theorem is to insert the solution (6.285) into temperature equa-
tion (6.284). (See Exercise 6.32.) The remaining task is to verify that the solution (6.285)
fulfills the boundary condition at the surface, where we have

t ∂U
T (z=0, t) = T surf (λ) (z=0, t − λ) dλ
0 ∂t
t
= T surf (λ) δ(t − λ) dλ = T surf (t). (6.286)
0
The function U is the unit step function of time at z = 0, and we used that ∂U/∂t is equal
to Dirac’s delta function δ(t − λ) for z = 0. The derivation of the step function and Dirac’s
delta function are covered in detail in text books on mathematical methods in physics, see
for instance Riley et al. (1998). From Section 6.14 we have that function U is

∞
z 2 −μ 2
U(z, t) = erfc √ = √ √ e dμ (6.287)
2 κt π z/2 κt