Making the most of imprecise measurements: Changing patterns of arsenic concentrations in shallow wells of Bangladesh from laboratory and field data

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.