Identifiability of Geotechnical Site-Specific Trend Functions

This paper investigates the possibility of consistently identifying a site-specific geotechnical trend function $t(z)$ in the presence of spatial variability $ϵ(z)$, where $z$ denotes depth. The trend function $t(z)$ is represented as the linear combination of prescribed basis functions (BFs), whereas $ϵ(z)$ is modeled as a zero-mean stationary Gaussian stochastic process with unknown standard deviation and scale of fluctuation. It is found that $t(z)$ can be unidentifiable if the single exponential (SExp) autocorrelation model is adopted for $ϵ(z)$. Two mechanisms, overfit and poor fit, that cause unidentifiability are explored. The overfit happens when part of $ϵ(z)$ is falsely fitted by the BFs, whereas the poor fit happens when part of $t(z)$ is erroneously interpreted as spatial variability. Nonetheless, identifiability is also affected by the sounding depth (or data record length) and the number of (statistically independent) soundings. An important feature for the SExp autocorrelation model is that it produces rough realizations with local jitters. If an autocorrelation model that produces smooth realizations is adopted, the trend function can become identifiable. The reason the identifiability for $t(z)$ is related to the smoothness of the realizations is explored. Finally, it is found that the sparse Bayesian learning framework can effectively alleviate overfit but cannot alleviate poor fit.