Jia Liu*, Tingjun Zhang
The problem of inferring Ground Surface Temperature History (GSTH) from borehole temperature-depth data, like virtually every other geophysical inverse problem, is characterized by instability due to presence of noise. Due to the different ways in which the problem may be parameterized and optimized the solution is method-dependent. In this work we attempt to analysis the results obtained by four methods, including currently widely used Functional Space Inversion (FSI) and Singular Value Decomposition (SVD), and also new developed Method of Fundamental Solutions (MFS), and Tikhonov method. All of four methods are based on the theory of 1-D heat conduction. To assess the effectiveness of various methods, synthetic ground temperature profile data with noise were prepared and used to compare different methods. We analyse five mathematical models describing the GSTH: (1) One-step signal, (2) Single-ramp signal, (3) Smooth single-ramp signal, (4) Sinusoidal signal, and (5) Mixed sinusoidal signal. We use the same forward solver and spatial and temporal discretization in the four methods in order to eliminate possible differences arising from these sources. The four inverse methods yield similar results of the variation trends of the GSTH that are concerned. However, the estimated GSTHs differ in details of the timing and the magnitude of changes. The effectiveness of four methods becomes signal dependent that sinusoidal signal can be inverted robust by MFS method, other types of signal are reconstructed exactly by Tikhonov method when adding small level of noise, and FSI is good at suppressing the noise.