Inverse Problem for Paleo-Temperature Reconstruction Based on the Tree-Ring Width and Glacier-Borehole Data

摘要

There is studied the inverse problem to determine solution {T(z, t), μ(t)} of the equation ρ(z)C(z)T_t = (k(z)T_z)_z-ρ(z)C(z)w(z)T_z, (t, z) ∈ Q ≡ [0, t_f] × [0,H], with initial condition T(z, 0) = U(z), z ∈ [0, H], and boundary conditions T(0, t) = U_s + μ(t), -k(H)T_z(H, t) = q, t ∈ [0, t_f], and redetermination condition T(z, t_f) = χ(z), z ∈ [0, H], where w(z) is the advection rate of glacier layers, ρ(z), C(z), and k(z) are the density, specific heat, and thermal conductivity of ice, respectively, q is the geothermal heat flux, U_s is the steady-state surface temperature in the past, t_f is the present time, H is the borehole depth, x(z) is the measured temperature-depth profile. The solution of the inverse problem μ(t) is looking for in the finite Fourier series form where periods correspond to the climatic signals retrieved by the annual tree-ring width index. It is derived that the solution is unique and stable and can be found out by the Tikhonov's regularization method. This method is applied for the Kamchatka region.