A divergence method approach utilizing gaussian processes for carbon dioxide emission estimation

With ongoing climate change and rising global temperatures, monitoring and quantifying anthropogenic greenhouse gas (GHG) emissions has become increasingly critical. One of the recent activities responding to the needs of accessing the effectiveness of strategies for Carbon Dioxide (CO₂) emission reduction is the upcoming Copernicus CO₂ Monitoring mission (CO2M), scheduled to launch in 2027. 

To support the CO2M, the data-driven emission quantification (ddeq) - Python library was developed as a shared library of various lightweight approaches focusing on quantifying CO₂ and NO_x emissions from synthetic CO2M data. One of these lightweight approaches is the divergence method, which was originally used for estimating NO_x emissions from TROPOMI NO₂ retrievals and later applied also for CO₂. The divergence method is based on the continuity equation in steady state and requires computing the flux using differentiation. However, unlike the other methods in the ddeq, the divergence method requires temporal averaging to mitigate the noise gained from the numerical differentiation over a noisy data. Unfortunately this makes cross-validation between the divergence and the other methods in the ddeq a challenge.

In the divergence method, it suffices to compute the quantity called advection defined as a dot product between the wind vector and the spatial gradient of the total vertical column density (TVCD).

Traditionally, the gradient is computed using some numerical differentiation scheme, such as a finite difference, but they often unable to produce reasonable estimates for the derivatives in a noisy environment. To solve this issue, we utilized a Gaussian process (GP) to estimate flux of the TVCD. Due to the properties of the GP, the partial derivatives can be computed analytically based on the optimized GP and the chosen kernel function.

Given a noisy measurement z of our signal f at locations s^*, we can model the signal f as a zero-mean GP with a covariance kernel K. We require that the kernel function defining the positive definite kernel matrix is (at least) twice differentiable and that the noise is i.i.d. Gaussian with mean and variance . Mathematically, this can be expressed as

Our objective is to study a linear transformation of the signal, where is some linear operator. Due to properties of Gaussian processes, is also a Gaussian process. As a consequence, the mean and the covariance matrix of the transformed signal conditionally to the observed data can be computed analytically.

Assuming that the observations of the TVCD near a source are prominent enough, we are able to optimize hyper parameters of a GP. This GP can then used to estimate the advection which should be elevated at the immediate proximity of the source. As per Gauss' divergence theorem, the emission rate of a source can be computed by integrating the advection field over the vicinity of the point source.