Mathematical Formalism for Atmospheric Flux Inversions¶
Atmospheric flux inversions attempt to find what fluxes \(\vec{x}_a\) are consistent with a set of atmospheric observations \(\vec{y}\) that are related by the observation operator \(H\) via \(\vec{y} \approx H \vec{x}_a\).
Since there tend to be fewer observations than there are fluxes of interest, we need a previous estimate of the fluxes, \(\vec{x}_b\), to regularize the problem so it is solvable.
Terminology¶
\(\vec{x}_a\) is also called the mean of the posterior distribution or the analysis.
The previous esitmate \(\vec{x}_b\) is also called the background, the mean of the prior distribution, or, in other fields, the forecast.
The rows of the observation operator \(H\) are sometimes called influence functions, since they indicate how much influence each flux has on a specific observation. \(H\) is also referred to as the transport matrix in some contexts, since it is based on how air is transported from the fluxes to where it forms the observations.
The estimate of the uncertainty in the previous estimate \(B\) is also called the background error covariance matrix or the covariance of the prior distribution.
The matrix \(K = B H^T (H B H^T + R)^{-1} = (B^{-1} + H^T R^{-1} H)^{-1} H^T R^{-1}\) is called the Kalman gain, and is called \(A_2\) in the derivation for the best linear unbiased estimator below
\(A\), the estimate of the uncertainty in \(\vec{x}_a\), is called the analysis error covariance matrix or the covariance of the posterior distribution.
Methodology¶
There are a few different ways to derive the equations used to optimize the fluxes.