Continental-scale Contributions To The Global CFC-11 Emission ...

2.1 Overview

To infer regional CFC-11 emissions from observed atmospheric mole fractions, we used a Bayesian inverse modeling framework following the method described in previous studies (Hu et al., 2015, 2016, 2017). In brief, the inverse modeling method assumes a linear relationship between measured atmospheric mole fraction enhancements and emissions upwind of the measurement locations. The linear operator, termed footprint, is the sensitivity of atmospheric mole fraction enhancements to upwind emissions, and it was computed for each sample using the Hybrid Single Particle Lagrangian Integrated Trajectory (HYSPLIT) model described in Stein et al. (2015). Bayesian inverse models (Rodgers, 2000) require initial assumptions about the magnitudes and distributions of emissions or prior emissions. By assuming that errors between the “true” and prior emissions and errors between atmospheric mole fraction observations and simulated mole fractions (using the computed footprints) follow Gaussian distributions, we construct a cost function (L) (Eq. 1) based on Bayes' theorem:

(1) L = 1 2 z - H s T R - 1 z - H s + 1 2 s - s p T Q - 1 s - s p ,

where, z represents the observed atmospheric enhancement relative to the upwind background atmosphere (Sect. 2.2.3), and sp and s represent the prior and posterior CFC-11 emissions. H represents the Jacobian matrix or the first-order partial derivatives of z to s. R and Q stand for the model–data mismatch covariance and prior flux error covariance. The values given to R and Q determine the relative weight between the prior emission assumptions and atmospheric observations in the final solution. Here, we used the maximum likelihood estimation method (Hu et al., 2015; Michalak et al., 2005) and atmospheric observations to directly solve for site-dependent model–data mismatch errors and prior flux errors. For the aircraft campaigns (HIPPO and ATom), we derive separate model–data mismatch errors, one for each campaign.