What is an Inverse Problem?

Forward Problem

Given an initial mole fraction field \(\chi\) for some atmospheric trace gas, its surface flux field \(f\), and the wind field \(\vec{u}\), it is relatively straightforward to find the mixing ratio field at all future points in time if the trace gas is almost inert over the timescales of interest:

plus boundary conditions, at the surface relating \(f\) to the gradient of \(\chi\), and at the top of the domain of interest \(\frac{\partial\chi}{\partial z} = 0\) to ensure no flow through the top of the domain. If the domain of interest is not global, there will be additional boundary conditions at the lateral boundaries.

Since the PDE above is linear, one can prove the existence and uniqueness of the solutions.

Inverse problem

The problem this package tries to address is related: given the same equation for the evolution of \(\chi\) and a finite number of measurements at different points in time, find \(f\). Since this problem is nearly the reverse of the problem above, it is called an “Inverse Problem”.

Usually, for practical purposes, the flux field is projected onto a finite-dimensional subspace, allowing the flux to be described as a vector (\(\vec{f}\)). This vector is usually much larger than the vector of measurements, so that the inverse problem is underdetermined.

One way to deal with this is to introduce an estimate for the fluxes obtained before looking at the measurements, \(\vec{f}_b\), and minimize the difference between the measurements and the prediction using the flux estimate. If \(\vec{f}_b = \vec{0}\), this is equivalent to using the pseudo-inverse of the transport to find the fluxes.

This allows the problem to be solved, but the problem is often ill-posed and the solution unstable: small changes in the measurements can lead to large changes in the fluxes. Introducing uncertainties for the individual measurements and elements of the flux vector addresses some of this problem 1, but can still lead to discontinuities in the final flux field, which may not be realistic given the underlying physical processes. Introducing correlations between the fluxes can alliviate that problem. Correlations also make the problem larger and longer to solve.

Initial and Boundary Conditions

The forward problem requires specifying initial conditions, and if the domain is not global, lateral boundary conditions also need to be specified. These can also be included in the inverse problem as unknowns to be solved for, that is, as an additional part of the state vector. [Peylinetal2005] indicates that, for limited-area models, the influence of the initial conditions largely vanishes after a month or so, having been advected out of the domain, while the lateral boundary conditions become correspondingly more important.

As with the fluxes, the concentration field at the lateral boundaries of the spatial domain is usually discritized into a finite-dimensional vector, which is appended to the flux vector to form the full state vector.

The choice of basis functions is again arbitrary. One possibility for the lateral boundary conditions is to model the inflow as a constant times the concentration field of a global model. Another, which works better for small domains, is to model the inflow as a constant. A third possibility is to again use the concentration field of a global model but treat the edges of the domain (often north, south, east, and west) separatly. A fourth extends this idea, and splits the inflow farther. One might split the inflow into boundary-layer and free-troposphere values in any of these schemes, or choose a finer vertical resolution.

1

Incidenally, this moves the procedure from Ordinary Least Squares to Weighted Least Squares. The next step is Generalized Least Squares.

References

Peylinetal2005

Peylin, P.; P.J. Rayner; P. Bousquet; C. Carouge; F. Hourdin; P. Hinrich; P. Ciais; and AEROCARB contributors (2005). “Daily CO2 flux estimates over Europe from continuous atmospheric measurements: 1, inverse methodology” Atmospheric Chemistry and Physics vol. 5, no. 12 pp. 3173-3186 doi:10.5194/acp-5-3173-2005