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Accurate specification of the error statistics required for data assimilation remains an ongoing challenge, partly because their estimation is an underdetermined problem that requires statistical assumptions. Even with the common assumption that background and observation errors are uncorrelated, the problem remains underdetermined. One natural question that could arise is as follows: can the increasing amount of overlapping observations or other datasets help to reduce the total number of statistical assumptions, or do they introduce more statistical unknowns? In order to answer this question, this paper provides a conceptual view on the statistical error estimation problem for multiple collocated datasets, including a generalized mathematical formulation, an illustrative demonstration with synthetic data, and guidelines for setting up and solving the problem. It is demonstrated that the required number of statistical assumptions increases linearly with the number of datasets. However, the number of error statistics that can be estimated increases quadratically, allowing for an estimation of an increasing number of error cross-statistics between datasets for more than three datasets. The presented generalized estimation of full error covariance and cross-covariance matrices between datasets does not necessarily accumulate the uncertainties of assumptions among error estimations of multiple datasets.

Increasing evidence suggests that residential exposures to natural environments, such as green spaces, are associated with many health benefits. Only a single study has examined the potential link between living near water and mortality. We sought to examine whether residential proximity to large, natural water features (e.g., lakes, rivers, coasts, \"blue space\") was associated with cause-specific mortality. Our study is based on a population-based cohort of nonimmigrant adults living in the 30 largest Canadian cities [i.e., the 2001 Canadian Census Health and Environment Cohort) (CanCHEC)]. Subjects were drawn from the mandatory 2001 Statistics Canada long-form census, who were linked to the Canadian mortality database and to annual income-tax filings, through 2011. We estimated associations between living within of blue space and deaths from several common causes of death. We adjusted models for many personal and contextual covariates, as well as for exposures to residential greenness and ambient air pollution. Our cohort included approximately 1.3 million subjects at baseline, 106,180 of whom died from nonaccidental causes during follow-up. We found significant, reduced risks of mortality in the range of 12-17% associated with living within of water in comparison with living farther away, among all causes of death examined, except with external/accidental causes. Protective effects were found to be higher among women and all older adults than among other subjects, and protective effects were found to be highest against deaths from stroke and respiratory-related causes. Our findings suggest that living near blue spaces in urban areas has important benefits to health, but further work is needed to better understand the drivers of this association. https://doi.org/10.1289/EHP3397.

We present a parametric Kalman filter data assimilation system using GOSAT methane observations within the hemispheric CMAQ model. The assimilation system produces forecasts and analyses of concentrations and explicitly computes its evolving error variance while remaining computationally competitive with other data assimilation schemes such as 4-dimensional variational (4D-Var) and ensemble Kalman filter (EnKF). The error variance in this system is advected using the native advection scheme of the CMAQ model and updated at each analysis while the error correlations are kept fixed. We discuss extensions to the CMAQ model to include methane transport and emissions (both anthropogenic and natural) and perform a bias correction for the GOSAT observations. The results using synthetic observations show that the analysis error and analysis increments follow the advective flow while conserving the information content (i.e., total variance). We also demonstrate that the vertical error correlation contributes to the inference of variables down to the surface. In a companion paper, we use this assimilation system to obtain optimal assimilation of GOSAT observations.

We applied the parametric variance Kalman filter (PvKF) data assimilation designed in Part I of this two-part paper to GOSAT methane observations with the hemispheric version of CMAQ to obtain the methane field (i.e., optimized analysis) with its error variance. Although the Kalman filter computes error covariances, the optimality depends on how these covariances reflect the true error statistics. To achieve more accurate representation, we optimize the global variance parameters, including correlation length scales and observation errors, based on a cross-validation cost function. The model and the initial error are then estimated according to the normalized variance matching diagnostic, also to maintain a stable analysis error variance over time. The assimilation results in April 2010 are validated against independent surface and aircraft observations. The statistics of the comparison of the model and analysis show a meaningful improvement against all four types of available observations. Having the advantage of continuous assimilation, we showed that the analysis also aims at pursuing the temporal variation of independent measurements, as opposed to the model. Finally, the performance of the PvKF assimilation in capturing the spatial structure of bias and uncertainty reduction across the Northern Hemisphere is examined, indicating the capability of analysis in addressing those biases originated, whether from inaccurate emissions or modelling error.

We examine how passive and active observations are useful to evaluate an air quality analysis. By leaving out observations from the analysis, we form passive observations, and the observations used in the analysis are called active observations. We evaluated the surface air quality analysis of O3 and PM2.5 against passive and active observations using standard model verification metrics such as bias, fractional bias, fraction of correct within a factor of 2, correlation and variance. The results show that verification of analyses against active observations always give an overestimation of the correlation and an underestimation of the variance. Evaluation against passive or any independent observations display a minimum of variance and maximum of correlation as we vary the observation weight, thus providing a mean to obtain the optimal observation weight. For the time and dates considered, the correlation between (independent) observations and the model is 0.55 for O3 and 0.3 for PM2.5 and for the analysis, with optimal observation weight, increases to 0.74 for O3 and 0.54 for PM2.5. We show that bias can be a misleading measure of evaluation and recommend the use of a fractional bias such as the modified normalized mean bias (MNMB). An evaluation of the model bias and variance as a function of model values also show a clear linear dependence with the model values for both O3 and PM2.5.

We previously introduced the parametric variance Kalman filter (PvKF) assimilation as a cost-efficient system to estimate the dynamics of methane analysis concentrations. As an extension of our development, this study demonstrates the linking of PvKF to a 4D-Var inversion aiming to improve on methane emissions estimation in comparison with the traditional 4D-Var. Using the proposed assimilation–inversion framework, we revisit fundamental assumptions of the perfect and already optimal model state that is typically made in the 4D-Var inversion algorithm. In addition, the new system objectively accounts for error correlations and the evolution of analysis error variances, which are non-trivial or computationally prohibitive to maintain otherwise. We perform observing system simulation experiments (OSSEs) aiming to isolate and explore various effects of the assimilation analysis on the source inversion. The effect of the initial field of analysis, forecast of analysis error covariance, and model error is examined through modified 4D-Var cost functions, while different types of perturbations of the prior emissions are considered. Our results show that using PvKF optimal analysis instead of the model forecast to initialize the inversion improves posterior emissions estimate (~35% reduction in the normalized mean bias, NMB) across the domain. The propagation of analysis error variance using the PvKF formulation also tends to retain the effect of background correlation structures within the observation space and, thus, results in a more reliable estimate of the posterior emissions in most cases (~50% reduction in the normalized mean error, NME). Our sectoral analysis of four main emission categories indicates how the additional information of assimilation analysis enhances the constraints of each emissions sector. Lastly, we found that adding the PvKF optimal analysis field to the cost function benefits the 4D-Var inversion by reducing its computational time (~65%), while including only the error covariance in the cost function has a negligible impact on the inversion time (10–20% reduction).

This contribution addresses the characterization of the model-error covariance matrix from the new theoretical perspective provided by the parametric Kalman filter method which approximates the covariance dynamics from the parametric evolution of a covariance model. The classical approach to obtain the modified equation of a dynamics is revisited to formulate a parametric modelling of the model-error covariance matrix which applies when the numerical model is dissipative compared with the true dynamics. As an illustration, the particular case of the advection equation is considered as a simple test bed. After the theoretical derivation of the predictability-error covariance matrices of both the nature and the numerical model, a numerical simulation is proposed which illustrates the properties of the resulting model-error covariance matrix.

BACKGROUND:Diabetes is infrequently coded as the primary cause of death but may contribute to cardiovascular disease (CVD) mortality in response to fine particulate matter (PM2.5) exposure. We analyzed all contributing causes of death to examine susceptibility of diabetics to CVD mortality from long-term exposure. METHODS:We linked a subset of the 2001 Canadian Census Health and Environment Cohort (CanCHEC) with 10 years of follow-up to all causes of death listed on death certificates. We used survival models to examine the association between CVD deaths (n=123,500) and exposure to PM2.5 among deaths that co-occurred with diabetes (n=20,600) on the death certificate. More detailed information on behavioral covariates and diabetes status at baseline available in the Canadian Community Health Survey (CCHS) - mortality cohort (n=12,400 CVD deaths, with 2,800 diabetes deaths) complemented the CanCHEC analysis. RESULTS:Among CanCHEC subjects, co-mention of diabetes on the death certificate increased the magnitude of association between CVD mortality and PM2.5 (HR=1.51 [1.39-1.65] per 10 μg/m) – versus all CVD deaths (HR=1.25 [1.21-1.29]) or CVD deaths without diabetes (HR=1.20 [1.16-1.25]). Among CCHS subjects, diabetics who used insulin or medication (included as proxies for severity) had higher HR estimates for CVD deaths from PM2.5 (HR=1.51 [1.08-2.12]) relative to the CVD death estimate for all respondents (HR=1.31 [1.16-1.47]). CONCLUSIONS:Mention of diabetes on the death certificate resulted in higher magnitude associations between PM2.5 and CVD mortality, specifically amongst those who manage their diabetes with insulin or medication. Analyses restricted to the primary cause of death likely underestimate the role of diabetes in air pollution-related mortality.

A computational simplification of the Kalman filter (KF) is introduced - the parametric Kalman filter (PKF). The full covariance matrix dynamics of the KF, which describes the evolution along the analysis and forecast cycle, is replaced by the dynamics of the error variance and the diffusion tensor, which is related to the correlation length-scales. The PKF developed here has been applied to the simplified framework of advection-diffusion of a passive tracer, for its use in chemical transport model assimilation. The PKF is easy to compute and computationally cost-effective than an ensemble Kalman filter (EnKF) in this context. The validation of the method is presented for a simplified 1-D advection-diffusion dynamics.

We present a general theory of estimation of analysis error covariances based on cross-validation as well as a geometric interpretation of the method. In particular, we use the variance of passive observation-minus-analysis residuals and show that the true analysis error variance can be estimated, without relying on the optimality assumption. This approach is used to obtain near optimal analyses that are then used to evaluate the air quality analysis error using several different methods at active and passive observation sites. We compare the estimates according to the method of Hollingsworth-Lönnberg, Desroziers et al., a new diagnostic we developed, and the perceived analysis error computed from the analysis scheme, to conclude that, as long as the analysis is near optimal, all estimates agree within a certain error margin.