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Optical coatings formed from amorphous oxide thin films have many applications in precision measurements. The Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO) and Advanced Virgo use coatings of$\mathrm{S}\mathrm{i}{\mathrm{O}}_2$(silica) and$\mathrm{T}\mathrm{i}{\mathrm{O}}_2:\mathrm{T}{\mathrm{a}}_2{\mathrm{O}}_5$(titania-doped tantala) and post-deposition annealing to 500°C to achieve low thermal noise and low optical absorption. Optical scattering by these coatings is a key limit to the sensitivity of the detectors. This paper describes optical scattering measurements for single-layer, ion-beam-sputtered thin films on fused silica substrates: two samples of$\mathrm{T}{\mathrm{a}}_2{\mathrm{O}}_5$and two of$\mathrm{T}\mathrm{i}{\mathrm{O}}_2:\mathrm{T}{\mathrm{a}}_2{\mathrm{O}}_5$. Using an imaging scatterometer at a fixed scattering angle of 12.8°, in-situ changes in the optical scatter of each sample were assessed during post-deposition annealing to 500°C in vacuum. The scatter of three of the four coated optics was observed to decrease during the annealing process, by 25–30% for tantala and up to 74% for titania-doped tantala, while the scatter from the fourth sample held constant. Angle-resolved scatter measurements performed before and after vacuum annealing suggest some improvement in three of the four samples. These results demonstrate that post-deposition, high-temperature annealing of single-layer tantala and titania-doped tantala thin films in vacuum does not lead to an increase in scatter, and may actually improve their scatter.

This corrigendum improves upon the size-dependent representation of graupel and hail terminal velocities, kinetic energies, and mass fluxes that were reported in the Heymsfield et al. (2018) study. In particular, representation of these dependencies on diameter over the full range of particle sizes is improved upon by correcting minor errors and by developing representations that cover different size ranges.

There is substantial interest in assessing how exposure to environmental mixtures, such as chemical mixtures, affects child health. Researchers are also interested in identifying critical time windows of susceptibility to these complex mixtures. A recently developed method, called lagged kernel machine regression (LKMR), simultaneously accounts for these research questions by estimating the effects of time‐varying mixture exposures and by identifying their critical exposure windows. However, LKMR inference using Markov chain Monte Carlo (MCMC) methods (MCMC‐LKMR) is computationally burdensome and time intensive for large data sets, limiting its applicability. Therefore, we develop a mean field variational approximation method for Bayesian inference (MFVB) procedure for LKMR (MFVB‐LKMR). The procedure achieves computational efficiency and reasonable accuracy as compared with the corresponding MCMC estimation method. Updating parameters using MFVB may only take minutes, whereas the equivalent MCMC method may take many hours or several days. We apply MFVB‐LKMR to Programming Research in Obesity, Growth, Environment and Social Stressors (PROGRESS), a prospective cohort study in Mexico City. Results from a subset of PROGRESS using MFVB‐LKMR provide evidence of significant and positive association between second trimester cobalt levels andz‐scored birth weight. This positive association is heightened by cesium exposure. MFVB‐LKMR is a promising approach for computationally efficient analysis of environmental health data sets, to identify critical windows of exposure to complex mixtures.

Exposure to environmental mixtures can exert wide‐ranging effects on child neurodevelopment. However, there is a lack of statistical methods that can accommodate the complex exposure‐response relationship between mixtures and neurodevelopment while simultaneously estimating neurodevelopmental trajectories. We introduce Bayesian varying coefficient kernel machine regression (BVCKMR), a hierarchical model that estimates how mixture exposures at a given time point are associated with health outcome trajectories. The BVCKMR flexibly captures the exposure‐response relationship, incorporates prior knowledge, and accounts for potentially nonlinear and nonadditive effects of individual exposures. This model assesses the directionality and relative importance of a mixture component on health outcome trajectories and predicts health effects for unobserved exposure profiles. Using contour plots and cross‐sectional plots, BVCKMR also provides information about interactions between complex mixture components. The BVCKMR is applied to a subset of data from PROGRESS, a prospective birth cohort study in Mexico city on exposure to metal mixtures and temporal changes in neurodevelopment. The mixture include metals such as manganese, arsenic, cobalt, chromium, cesium, copper, lead, cadmium, and antimony. Results from a subset of Programming Research in Obesity, Growth, Environment and Social Stressors data provide evidence of significant positive associations between second trimester exposure to copper and Bayley Scales of Infant and Toddler Development cognition score at 24 months, and cognitive trajectories across 6‐24 months. We also detect an interaction effect between second trimester copper and lead exposures for cognition at 24 months. In summary, BVCKMR provides a framework for estimating neurodevelopmental trajectories associated with exposure to complex mixtures.