  Millions of people in Bangladesh drink well water contaminated with arsenic.
Despite the severity of this heath crisis, little is known about the extent to
which groundwater arsenic concentrations change over time: Are concentrations
generally rising, or is arsenic being flushed out of aquifers? Are spatially
patterns of high and low concentrations across wells homogenizing over time, or
are these spatial gradients becoming more pronounced? To address these
questions, we analyze a large set of arsenic concentrations that were sampled
within a 25 km$^2$ area of Bangladesh over time. We compare two blanket survey
collected in 2000/2001 and 2012/2013 from the same villages but relying on a
largely different set of wells. The early set consists of 4574 accurate
laboratory measurements, but the later set poses a challenge for analysis
because it is composed of 8229 less accurate categorical measurements conducted
in the field with a kit. We construct a Bayesian model that jointly calibrates
the measurement errors, applies spatial smoothing, and describes the
spatiotemporal dynamic with a diffusion-like process model. Our statistical
analysis reveals that arsenic concentrations change over time and that their
mean dropped from 110 to 96 $\mu$g/L over 12 years, although one quarter of
individual wells are inferred to see an increase. The largest decreases
occurred at the wells with locally high concentrations where the estimated
Laplacian indicated that the arsenic surface was strongly concave. However,
well with initially low concentrations were unlikely to be contaminated by
nearby high concentration wells over a decade. We validate the model using a
posterior predictive check on an external subset of laboratory measurements
from the same 271 wells in the same study area available for 2000, 2014, and
2015.
