A sparse reconstruction method for the estimation of multi-resolution emission fields via atmospheric inversion

J. Ray, J. Lee, V. Yadav, S. Lefantzi, A.M. Michalak and B. van Bloemen Waanders


As the world explores avenues for climate change mitigation, it is becoming increasingly important to be able to evaluate the magnitude and spatial distribution of CO2 emissions from fossil fuel burning. Atmospheric observations of CO2 provide one key piece of the puzzle, as these observations reflect emissions at upwind locations. The strong spatial heterogeneity of fossil fuel emissions makes it challenging to disentangle signals coming from different geographic regions, however. This manuscript presents a sparse reconstruction methodology for estimating multi-resolution emission fields, and is related to the earlier Ray et al. (2014) manuscript. The approach leverages the information content of atmospheric observations, as well as the localized nature of fossil fuel emissions.

Figure: Plots of ffCO2 emissions during the 31st 8-day period. The units are micromoles of C m−2s−1. Emissions below 0.02 micromoles of C m−2s−1 are grayed out. Top left, we plot true emissions from the Vulcan inventory. Top right, the estimates from approach A. Bottom left and right figures contain the estimates obtained from approaches B and C, respectively. Each figure contains the measurement towers as white diamonds. Each figure is also divided into quadrants. We see that approach A, unconstrained by fpr provides low levels of (erroneous) emissions in large swathes of the western quadrants. Approach B reflects fpr very strongly. Approach C provides a balance between the influence of fpr and the information in yobs.


Atmospheric inversions are frequently used to estimate fluxes of atmospheric greenhouse gases (e.g., biospheric CO2 flux fields) at Earth's surface. These inversions typically assume that flux departures from a prior model are spatially smoothly varying, which are then modeled using a multi-variate Gaussian. When the field being estimated is spatially rough, multi-variate Gaussian models are difficult to construct and a wavelet-based field model may be more suitable. Unfortunately, such models are very high dimensional and are most conveniently used when the estimation method can simultaneously perform data-driven model simplification (removal of model parameters that cannot be reliably estimated) and fitting. Such sparse reconstruction methods are typically not used in atmospheric inversions. In this work, we devise a sparse reconstruction method, and illustrate it in an idealized atmospheric inversion problem for the estimation of fossil fuel CO2 (ffCO2) emissions in the lower 48 states of the USA.

Our new method is based on stagewise orthogonal matching pursuit (StOMP), a method used to reconstruct compressively sensed images. Our adaptations bestow three properties to the sparse reconstruction procedure which are useful in atmospheric inversions. We have modified StOMP to incorporate prior information on the emission field being estimated and to enforce non-negativity on the estimated field. Finally, though based on wavelets, our method allows for the estimation of fields in non-rectangular geometries, e.g., emission fields inside geographical and political boundaries.

Our idealized inversions use a recently developed multi-resolution (i.e., wavelet-based) random field model developed for ffCO2 emissions and synthetic observations of ffCO2 concentrations from a limited set of measurement sites. We find that our method for limiting the estimated field within an irregularly shaped region is about a factor of 10 faster than conventional approaches. It also reduces the overall computational cost by a factor of 2. Further, the sparse reconstruction scheme imposes non-negativity without introducing strong nonlinearities, such as those introduced by employing log-transformed fields, and thus reaps the benefits of simplicity and computational speed that are characteristic of linear inverse problems.

Ray, J., J. Lee, V. Yadav, S. Lefantzi, A.M. Michalak, B. van Bloemen Waanders, S. A. McKenna (2015) “A sparse reconstruction method for the estimation of multi-resolution emission fields via atmospheric inversion”, Geoscientific Model Development, 8, 1259-1273, doi:10.5194/gmd-8-1259-2015.