Flux representation for inversions

The relationship between the state vector and the flux in each transport model grid cell can vary greatly between inversions.

Influence functions \(H\) for direct and scaling factor inversions are most easily found by some variety of adjoint transport model. Those for larger basis functions can often be found more readily by repeated forward integrations of the transport model, as in TRANSCOM.

Direct

Each entry in the state vector \(\vec{x} = \vec{f}\) is the flux in a different grid cell. The influence function \(H = H_{flux}\) determines the change in each measurement due to a one-unit change in flux for each grid cell.

Also called “additive”.

Scaling Factor

Each entry in the state factor is a multiplier for the fluxes in each grid cell. One obtaines this by making the prior state vector a vector of ones (i.e. \(\vec{x} = \vec{1}\)) and moving the prior fluxes to the influence functions (\(H = H_{flux} diag(\vec{f}_{prior})\)). The fluxes can be found as \(\vec{f} = \vec{x} \otimes \vec{f}_{prior}\)

Larger Basis Functions

The basis functions are often chosen not to nonoverlap in flux space, with spatial boundaries corresponding to TRANSCOM regions or ecoregions. Another method for choosing basis functions is to find Empirical Orthogonal Functions for the fluxes.

A matrix \(F\) is chosen to transform the state vector into the fluxes: \(\vec{f} = F \vec{x}\). The usual choice is to make \(\vec{x}\) much smaller than \(\vec{f}\), and to make \(F^T F = I\). The influence functions can then be found as \(H = H_{flux} F\).