Detection and quantification of greenhouse-gas emissions is important for both compliance and environment conservation. However, despite several decades of active research, it remains predominantly an open problem, largely due to model errors and assumptions that appear at each stage of the inversion processing chain. In 2015, a controlled-release experiment headed by Geoscience Australia was carried out at the Ginninderra Controlled Release Facility, and a variety of instruments and methods were employed for quantifying the release rates of methane and carbon dioxide from a point source. This paper proposes a fully Bayesian approach to atmospheric tomography for inferring the methane emission rate of this point source using data collected during the experiment from both point- and path-sampling instruments. The Bayesian framework is designed to account for uncertainty in the parameterisations of measurements, the meteorological data, and the atmospheric model itself when performing inversion using Markov chain Monte Carlo (MCMC). We apply our framework to all instrument groups using measurements from two release-rate periods. We show that the inversion framework is robust to instrument type and meteorological conditions. From all the inversions we conducted across the different instrument groups and release-rate periods, our worst-case median emission rate estimate was within 36 % of the true emission rate. Further, in the worst case, the closest limit of the 95 % credible interval to the true emission rate was within 11 % of this true value.

Methane (

Several controlled-release experiments of

Every group involved in the analysis presented in

A number of studies have highlighted the importance of atmospheric-model error in estimating emission rates or fluxes

The transport model plays an important role in inverse modelling. Calibration of the transport model from observations can be done within the classic inverse theory framework of

Atmospheric tomography, a term inspired from medical imaging, combines data from a collection of measurement sites with Bayesian inversion to detect and quantify emissions. The primary contribution of this article is an extension of the atmospheric tomography technique described in Sect. 2.4.2 of

The remainder of the article is organised as follows. Section

A full description of the experimental setup, measurement techniques, and quantification methods used in the 2015 Ginninderra release experiment are given in

The gases were released at a height of 0.3 m, and the standard

The data set used to obtain the results presented in Sect.

Initial preprocessing was carried out to provide a complete data set without outliers. First, data containing missing values considered critical for emission-rate estimation (in particular, air temperature, air pressure, wind speed, and wind direction) were removed from the data set. Second, data points corresponding to upwind measurements that were more than three median absolute deviations away from the instrument's median upwind measured concentration were determined to be outliers and hence removed. A point measurement was classified as upwind if the angle subtended from the source by a line joining the instrument location to the plume centreline was more than

In this section we detail the plume model employed and how it is used to supply model-predicted concentrations for the path measurements.

As outlined in Sect.

Stability classes to which observations within the Ginninderra experiment are allocated, and the corresponding values of

The stability class to which an observation is allocated is classically based on (i) the Monin–Obukhov length (the theoretical height at which turbulence is produced by buoyancy and mechanical forces in equal amounts; see

The coefficients typically used for each stability class could be off by a factor of 2 or more

Predicted (blue) and observed (red) enhancements in parts per million (ppm) at EC.A between 21 May and 7 June 2015 when scaling

It is well known that the Gaussian plume model is less accurate for low wind speeds

Each wind speed

Therefore, conditional on all other terms in Eq. (

The plume model given by Eq. (

Let

We are ultimately interested in obtaining a range of plausible values for the emission rate,

Directed acyclic graph showing the conditional dependence relationships between the data (enhancements)

Let

Now, let

First, we capture instrument-specific measurement error characteristics and stability-condition-specific variation by introducing an auxiliary variable

Putting these two components together, we have that, conditional on the instrument type and stability class encoded in

The process of interest in this application is the emission rate,

While addressing nonnegativity, half-normal priors do not contain a point mass at zero and thus do not encode a prior belief that there is a possibility of having exactly a zero emission rate. As a consequence, a posterior estimate or even a credible interval that includes zero is not possible. A spike-and-slab distribution

Our parameter model is divided into two parts: one pertaining to the precision parameters

For conjugacy with the Gaussian likelihood, we model each

From separate studies into the reliability of the model values for

We use Gamma prior distributions for

Recall

Computation of the posterior distribution

In the case of

In this section we discuss results from applying our model to simulated data in an observing system simulation experiment (OSSE). To mimic the conditions in the real experiment, we simulated enhancements using the actual Boreal and EC instrument locations, meteorological observations from the Ginninderra data, and realistic variances for the random-error components. We considered the two release-rate periods separately, using a 6 g min

We made inference on

Posterior median emission rates in grams per minute (g min

In this section we discuss results from applying our model to enhancements from the compiled Ginninderra data. We considered several settings. In the first setting, we estimated the emission rate separately for each of the four instrument types and for each release-rate period (5.8 and 5.0 g min

As in the OSSE, we generated 60 000 MCMC samples, left out 20 000 of these as burn-in, and used a thinning factor of 10. In line with what we observed in the OSSE, our initial results showed that, more often than not,

The left panel of Fig.

The first 10 rows of Table

The right panel in Fig.

The bottom 10 rows in Table

As detailed throughout Sect.

Grouping the precision parameters

On the other hand several components in our model appear to be crucial to obtaining reasonable emission-rate estimates. Using a single precision parameter to capture all observed variability due to measurement error and the stability-class categorisation clearly had a negative impact on our emission-rate estimates. Similarly, assuming the variability of the measurements is independent of wind speed when performing inversion resulted in 95 % posterior credible intervals on the emission rate that are considerably shifted in the negative direction. A similar observation was made by

The scaling factor

In this article we have proposed a fully Bayesian model for atmospheric tomography that takes into account uncertainty in the data measurement process, the physical processes, and parameters appearing in the transport model, when estimating the emission rate. We see that the model is robust to different instrument types and configurations, and it provides useful inferences on the emission rate and the plume dispersion model used. When applied to the Ginninderra data using a variety of instruments in different release-rate periods, we obtain 95 % posterior credible intervals on the emission rate that either encapsulate the true emission rate or have a limit which is no more than 11 % from the true value.

The methods developed in this study are ideal for quantifying local-scale leaks from industrial facilities or from the subsurface (e.g. well heads, buried pipelines, or gas leakage up geological fractures and faults) where a surface leak has been detected but needs to be quantified. It can be used where physical access to the source location is limited, e.g. gas bubbling from a creek or where measurement is hazardous. Depending on the circumstance, detection of leakage can take many different forms, from visible bubble detection, optical gas imaging, handheld sniffers, noise detection, helicopters equipped with lasers, drones equipped with gas sensors, to monitoring die-off in vegetation using remote sensing techniques. Surface leakage typically expresses as small, concentrated hotspots if sourced from the subsurface

In most applications neither the number of sources nor the source location is known. As such, the framework we construct should be seen as a foundational building block that needs to be extended appropriately for each specific application. For example, if the source location is not known, then source localisation can be incorporated into the Bayesian framework as discussed by

Our work is closely connected to other atmospheric tomography techniques but with some small, significant, differences.

Our results provide interesting insights into the design and monitoring of sensor networks for detecting and quantifying methane emissions. For example, our sensitivity analysis in Sect.

The fully Bayesian framework we adopt is adaptable to various scenarios. We envision, for example, that source localisation

Software code and data are available at

Posterior median emission rate in grams per minute (g min

Posterior 95 % credible intervals for the emission rates in grams per minute (g min

LC compiled the data with the help of all authors and ran all the analyses. LC and AZM conducted the research. AF conceptualised and supervised the study. LC, AZM, and AF wrote the manuscript. SB conducted an initial investigation using a simplified version of the proposed model. IS, FP, TC, KN, TN, MK, SZ, NW, and NMD acquired the field data for the study. All authors discussed the results and commented on the manuscript.

The authors declare that they have no conflict of interest.

This article is part of the special issue “The 10th International Carbon Dioxide Conference (ICDC10) and the 19th WMO/IAEA Meeting on Carbon Dioxide, other Greenhouse Gases and Related Measurement Techniques (GGMT-2017) (AMT/ACP/BG/CP/ESD inter-journal SI)”. It is a result of the 10th International Carbon Dioxide Conference, Interlaken, Switzerland, 21–25 August 2017.

Laura Cartwright acknowledges the support of the Australian Government Research Training Program Scholarship. Laura Cartwright, Andrew Zammit-Mangion, and Andrew Feit would like to acknowledge APR.Intern for facilitating the first 5 months of this modelling study. All authors thank Gareth Davies for reviewing an earlier version of the manuscript. The Ginninderra field site was supported by the Australian Government through the Carbon Capture and Storage – Implementation budget measure. The authors also acknowledge funding for the research provided by the Australian Government through the CRC programme and support from the CO2CRC. The National Geosequestration Laboratory is thanked for making the two Picarro instruments available for the study. We would like to thank Phil Dunbar and his staff (CSIRO Plant Industry) for maintaining the site and Dale Hughes (CSIRO) for his assistance with maintenance of the CSIRO EC tower. The authors also wish to acknowledge the assistance of Field Engineering Services at Geoscience Australia. Geoscience Australia and the Western Sydney University team would like to acknowledge Charles Jenkins (CSIRO) for early discussions about the atmospheric tomography line technique and the Australian Mathematical Sciences Institute. The University of Wollongong wishes to acknowledge Joel Wilson, Maximilien Desservettaz, and Ruhi Humphries for their assistance in the site operations. Andrew Feitz and Ivan Schroder publish with the permission of the CEO of Geoscience Australia.

This research has been supported by the Australian Research Council (grant nos. DE180100203 and FT180100327).

This paper was edited by Hubertus Fischer and reviewed by two anonymous referees.