This paper addresses the problem of characterizing the statistical uncertainties associated with the estimation of a depth-dependent trend function using limited site-specific geotechnical data. Specifically, the statistical uncertainties associated with the following elements of the problem are considered: (1) the functional form of the trend function, (2) the parameters of the trend function (e.g., intercept and gradient), and (3) the random field parameters, namely standard deviation (a) and scale of fluctuation (8). The problem is resolved with a two-step Bayesian framework. In Step 1, a set of suitable basis functions that parameterize the trend function is selected using the sparse Bayesian learning. In Step 2, an advanced Markov chain Monte Carlo method is adopted for the Bayesian analysis. The two-step approach is shown to be consistent in the well-defined sense that the resulting 95% Bayesian confidence interval (or region) contains the actual trend (or actual a & 8) with a chance that is close to 0.95.
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