  Observations of groundwater pollutants, such as arsenic or Perfluorooctane
sulfonate (PFOS), are riddled with left censoring. These measurements have
impact on the health and lifestyle of the populace. Left censoring of these
spatially correlated observations are usually addressed by applying Gaussian
processes (GPs), which have theoretical advantages. However, this comes with a
challenging computational complexity of $\mathcal{O}(n^3)$, which is
impractical for large datasets. Additionally, a sizable proportion of the data
being left-censored creates further bottlenecks, since the likelihood
computation now involves an intractable high-dimensional integral of the
multivariate Gaussian density. In this article, we tackle these two problems
simultaneously by approximating the GP with a Gaussian Markov random field
(GMRF) approach that exploits an explicit link between a GP with Mat\'ern
correlation function and a GMRF using stochastic partial differential equations
(SPDEs). We introduce a GMRF-based measurement error into the model, which
alleviates the likelihood computation for the censored data, drastically
improving the speed of the model while maintaining admirable accuracy. Our
approach demonstrates robustness and substantial computational scalability,
compared to state-of-the-art methods for censored spatial responses across
various simulation settings. Finally, the fit of this fully Bayesian model to
the concentration of PFOS in groundwater available at 24,959 sites across
California, where 46.62\% responses are censored, produces prediction surface
and uncertainty quantification in real time, thereby substantiating the
applicability and scalability of the proposed method. Code for implementation
is made available via GitHub.
