# Methane Emission Estimates by the Global High-Resolution Inverse Model Using National Inventories
## 2 Materials and Methods
### Inverse Modeling System--NTFVAR
#### 2.1.2 The Inverse Modeling Scheme  
The same inversion scheme with high resolution (0.1\({}^{\circ}\times 0.1^{\circ}\)) fluxes and variational optimization was applied to estimate CO\({}_{2}\) fluxes by [PERSON] et al. [34], who also demonstrated its capability to capture the observed atmospheric CO\({}_{2}\) time series. In this study, the high-resolution variant of the transport model and its adjoint described by [PERSON] et al. [30] were combined with the optimization scheme proposed by [PERSON] et al. [46] and [PERSON] et al. [47]. Following the approach by [PERSON] et al. [46], flux corrections were estimated independently for two categories of emissions (anthropogenic and natural). Variational optimization was applied to obtain flux corrections as two sets of scaling factors to vary prior uncertainty fields on a monthly basis at a 0.1\({}^{\circ}\times 0.1^{\circ}\) resolution separately for anthropogenic and natural wetland emissions with bi-weekly time steps. Anthropogenic emission corrections were proportional to the monthly climatology emissions provided by EDGAR, and the wetland emission corrections were proportional to the monthly climatology wetland emissions provided by the Vegetation Integrative Simulator for Trace Gases (VISIT) model (details of prior fluxes given in Section 2.2), with both given as prior uncertainty fields. The grid scale flux uncertainty was defined as 30% of the EDGAR climatology for the anthropogenic flux category and 50% of the VISIT climatological emissions for the wetland emissions category. Other flux categories, such as biomass burning, geological, remites, and soil sink, were not corrected by inversion. The spatial correlation of 500 km and temporal correlation of two weeks were used to provide smoothness constraints on the resulting fields of the scaling factors. The inverse modeling problem was formulated (e.g., [PERSON] [48], [PERSON] et al. [46]) and solved to find the optimal value of \(x\)--vectors of corrections to prior fluxes at the minimum of the cost function \(\ J(x)\):  
\[J(x)=\frac{1}{2}(H\cdot x-r)^{T}\cdot R^{-1}\cdot(H\cdot x-r)+\frac{1}{2}x^{T} \cdot B^{-1}\cdot x \tag{1}\]  
where \(\ r\) is the residual (difference between the observed concentration and the forward simulation made with prior fluxes without correction), \(R\) is the covariance matrix of observations, and \(B\) is the covariance matrix of fluxes. In our design of matrix \(B\), we follow [PERSON] et al. [46] in representing matrix \(B\) as a multiple of the non-dimensional covariance matrix \(C\) and the diagonal flux uncertainty \(D\) as \(B=D^{T}\cdot C\cdot D\). Matrix \(C\) is commonly implemented as the band matrix with non-diagonal elements declining as \(\sim\exp\bigl{(}-x^{2}/l^{2}\bigr{)}\) with the distance \(x\) between the grid cells. To achieve computational efficiency for the flux covariance operator, the implicit diffusion technique was used to implement a spatial covariance matrix, as in [PERSON] and [PERSON]'s study [48]. To construct symmetric matrix \(C\), we first implemented the square root of \(C\) by implicit diffusion and constructed the adjoint of it by applying an adjoint code compiler, Tapenade, to the Fortran code of modules approximating the square root of \(C\). The temporal covariance matrixes were of low dimension and were designed as in [PERSON] et al., 2008 by deriving the square root of the Gaussian-shaped time covariance matrix with direct Singular Value Decomposition (SVD) [49]. The optimal solution was calculated as the minimum of the cost function \(J\). This was done iteratively with the efficient Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, as implemented by [PERSON] and [PERSON] [50]. We limited the number of iterations to 40, as cost function reduction slows down by the time that this number is reached.