Inference of ground surface temperature history from borehole temperature data: a comparison of two inverse methods

The problem of inferring ground surface temperature history (GSTH) from borehole temperature-depth data, like virtually every other geophysical inverse problem, is characterized by nonuniqueness because of an inadequate amount of data, and instability due to presence of noise. As such, the inverse solution for a given data set depends on model representation (parameterization), optimization method, and the subjective view of the researcher. In this work we attempt to identify the cause of differences that arise from the application of two inverse methods that are currently widely used in inferring a GSTH: a generalized non-linear Bayesian formulation based on the method of least squares called Functional Space Inversion (FSI), and a linear formulation solved by the method of singular value decomposition (SVD). We use the same forward solver and spatial and temporal discretizations in the two methods in order to eliminate possible differences arising from these sources. Experiments with synthetic noise-free data show that when noise is not a factor both methods are essentially equal in their ability to identify and represent both the pre-existing reference state and the transient excursion therefrom. However, in the presence of noise, differences arise between the different model representations, particularly when the borehole site has experienced a net warming or cooling in the recent millennium. For such a site, the two methods can give significantly different estimates for the prior baseline or reference surface temperature, and for the subsequent changes in surface temperature. SVD is less effective at recovering the steady-state reference temperature when the mean of the transient deviates significantly from the prior steady-state temperature. In FSI, the use of an a priori null hypothesis for the deviation from the steady-state, along with the imposition of smoothing constraints, yields a robust estimate of the prior steady-state, but a conservative estimate of the transient.