We present a method to infer spatially and spatio-temporally correlated emissions of greenhouse gases from atmospheric measurements and a chemical transport model. The method allows fast computation of spatial emissions using a hierarchical Bayesian framework as an alternative to Markov chain Monte Carlo algorithms. The spatial emissions follow a Gaussian process with a Matérn correlation structure which can be represented by a Gaussian Markov random field through a stochastic partial differential equation approach. The inference is based on an integrated nested Laplacian approximation (INLA) for hierarchical models with Gaussian latent fields. Combining an autoregressive temporal correlation and the Matérn field provides a full spatio-temporal correlation structure. We first demonstrate the method on a synthetic data example and follow this using a well-studied test case of inferring UK methane emissions from tall tower measurements of atmospheric mole fraction. Results from these two test cases show that this method can accurately estimate regional greenhouse gas emissions, accounting for spatio-temporal uncertainties that have traditionally been neglected in atmospheric inverse modelling.

Emissions of greenhouse gases, ozone-depleting substances and air pollutants are increasingly inferred indirectly from atmospheric trace gas concentration observations and chemical transport models.
These “top-down” or “inverse” methods complement inventory- or process-model-based “bottom-up” techniques that are used, for example, in national reporting of greenhouse gas emissions to the United Nations Framework Convention on Climate Change

Top-down methods rely on some form of statistical inference, or inverse theory, to infer emissions at global

Recently, hierarchical Bayesian schemes have been developed to infer unknown uncertainties in the inversion framework

This work presents a computationally efficient hierarchical Bayesian framework for inferring spatio-temporally correlated trace gas emissions in a widely used regional atmospheric chemical transport modelling framework.
We use an integrated nested Laplacian approximation (INLA) for the Bayesian inference.
The spatial correlation structure with spatial Markov properties results from the Gaussian random field being a solution to a particular stochastic partial differential equation.
Kronecker product algebra allows efficient extension to spatio-temporal correlation.
Section

This section details an efficient approach to forming spatial and spatio-temporal correlation functions and outlines how this applies to the inference of regional trace gas emissions from measurements using fast inference for hierarchical models.
We limit the scope of this paper to the well-established problem of regional inference of long-lived trace gas emissions using a backward-running Lagrangian particle dispersion model

The aim is to infer some parameters of interest, here a spatial field of a priori emissions scaled by some factor,

The emissions scaling from its a priori value

A mesh constructed using constrained refined Delaunay triangulation, where the distribution of nodes is denser around the United Kingdom and Ireland.

A spatio-temporal extension to the forward model (Eq.

Estimating hourly emissions at each time

Inferring the emissions and the related uncertainties requires a hierarchical model to infer the quantities of interest from measurements while estimating some unknown parameters which are necessary for inference.
The main focus of this work is to estimate the posterior distribution of the emissions field

A Bayesian hierarchical model requires a method of inference to estimate the parameters of interest and any parameters that are not of direct interest but required, and uncertain, in order to infer the other parameters in the hierarchy.
Many methods of inference exist and have been applied to the problem of estimating emissions of trace gases (see references in Sect.

An integrated nested Laplacian approximation

While this method relies on calculating only the marginal posterior distributions of

This section presents two case studies to demonstrate how the method applies to inferring trace gas emissions.
The first uses simulated methane observations from four tall-tower measurement sites to infer simulated spatio-temporal emissions from the UK.
The second case study expands on the first case study by using real observations from the four tall towers to infer emissions of methane from the UK over four 3-month periods in 2014.
While the size of this problem is not particularly large, we demonstrate the method using UK methane emissions as a proof of concept as it is a well-studied test case

The case studies use the four UK DECC network measurement stations located on this map. RGL is Ridge Hill, TAC is Tacolneston, TTA is Angus and MHD is Mace Head station.

The case study observations are from four measurement sites: three in the UK and one in Ireland, which are part of the UK Deriving Emissions related to Climate Change (DECC) network

An atmospheric transport model calculates the sensitivity of hourly measurements to the emissions or boundary conditions, from which the matrices

Inventory data from the Emissions Database for Global Atmospheric Research (EDGAR) v4.3.2. provide the mean prior emissions

We test the method by performing an inversion using pseudo-data for four consecutive time periods of 1 month.
By creating a known-emissions field we are able to validate the method through comparing the inferred emissions to the known emissions, which is not possible in the real world.
We form a synthetic-emissions field by allowing the emissions to deviate from the prior mean emissions according to a Matérn field (see Sect.

To create the synthetic-emissions field, the NAME sensitivities for each measurement at each grid cell, detailed in Sect.

Time-correlated deviation of emissions from the prior mean, simulated using a Matérn field for

The inference needs prior probabilities for the hyperparameters, which are known exactly here, but we set them to be deliberately incorrect, but feasible based on true prior knowledge, to check that the inversion method can still recover the correct emissions.
For the inversion we assign a prior probability for

The inferred mean deviation of emissions from the prior mean for

Figure

Hyperparameter estimation is less accurate than for the latent field.
The estimation of the noise

The inferred and true difference between UK emissions and a synthetic inventory value for each of the four time periods. The crosses show the inferred mean difference in total emissions with their associated 95 % uncertainty. The red line indicates one-to-one agreement.

This section presents methane emissions estimates for the UK in 2014.
The year is split into four time periods: January to March, April to June, July to September and October to December.
Based on the synthetic data set in Sect.

Inferred mean difference in methane emissions for the UK in 2014 compared to the EDGAR inventory for

Figure

Estimated total UK methane emissions for 2014. The prior mean comes from EDGAR v4.32 inventory data (red dashed).
For comparison, the figure also shows the posterior means for UK methane emissions from

The real benefit of the presented inversion method is speed, while still maintaining the idea that, in this application, uncertainties exist within the uncertainties that are inherent to hierarchical Bayesian inverse methods.
The computation of the marginal posterior distribution of the latent field is readily suited to run in parallel across multiple cores, making this approach scalable to problems with a larger parameter space.
This, however, requires that sufficient memory allocation is available.
The inference for the experiment in Sect.

The latent Gaussian field, crucial to this method, has the problem that it does not restrict emissions to strictly positive values, which is a physical requirement for many gas emissions.
In addition, the INLA method relies on the assumption of approximate multivariate normality of the posterior linear predictors.
MCMC algorithms do not suffer from this issue as any prior probability density function can be chosen, and no assumption is made about the posterior linear predictors.
This has to be traded off, however, with the speed and ease of implementation of the method and the scale of the problem that the user wishes to solve.
If strictly positive posterior emissions are an absolute requirement, another possible modification to the approach is to use a Taylor expansion around the nonlinear model

A potential extension to this work is global-scale modelling, for example a global study of methane emission from global satellite measurements.
Spatio-temporal estimation of global

This work presents a fast and efficient method using an integrated nested Laplacian approximation for hierarchical inference of trace gas emissions. This method is particularly well-suited to assimilating large data sets. We show that INLA with a stochastic partial differential equation approach for spatial correlation can reproduce synthetic emissions from pseudo-observations and benchmarked emissions using real data.

A real advantage over other hierarchical Bayesian inversion methods is the attractive convergence properties, which can be difficult to obtain using methods such as Markov chain Monte Carlo algorithms. As the method computes the marginal variance for each node, this allows for efficient parallel implementation and significant computational savings compared to other hierarchical methods. Computational speed will become increasingly important as more data from space-borne sensors become available, which will offer more measurements and increased spatial coverage.

Measurements of methane from the UK DECC network sites Tacolneston, Ridge Hill and Tall Tower Angus are available at

The supplement related to this article is available online at:

LW and ZS conceived and led the implementation of the study. MR, AG and JR supported and advised on the study. KS, SO and DY made the measurements from the UK DECC network. AM created the sensitivity footprints using the NAME model. LW led the writing of the manuscript, to which all authors have contributed and edited.

The authors declare that they have no conflict of interest.

We would like to thank Alfredo Farjat and an anonymous reviewer for their helpful reviews of the manuscript.

This work was funded by the Jean Golding Institute Seed Corn grant CHEM.HF8064. Luke Western was funded by grants NE/M014851/1 and NE/S016155/1 from the Natural Environment Research Council and a grant from the UK Department for Business, Energy and Industrial Strategy. Anita Ganesan was funded under a Natural Environment Research Council Independent Research Fellowship NE/L010992/1.

This paper was edited by Ignacio Pisso and reviewed by Alfredo Farjat and one anonymous referee.