Methane (CH4) is an important transition fuel for decarbonisation of the global energy system . As countries increase the renewable energy mix into their existing electricity networks, CH4 can firm up network stability and supply . Utilisation of biogas or natural gas with carbon capture and storage offers a lower cost pathway to achieve deep decarbonisation targets . One of the disadvantages of CH4, however, is that its global warming potential is much greater than that of carbon dioxide (CO2), so that only a few percent of losses of CH4 into the atmosphere can negate any climate-change mitigation advantages from reducing conventional coal-fired power production . For this reason, it is critical that losses of CH4 along the supply chain are accurately accounted for to ensure public confidence in climate-change mitigation benefits of switching to natural gas. Unfortunately, while several types of instrumentation are available to aid the detection and estimation of fugitive emissions, harnessing acquired data for reliable emission detection and quantification remains a notoriously difficult problem.

Several controlled-release experiments of CH4 and CO2 have been conducted in order to improve techniques for estimating greenhouse-gas emissions . Building on this body of work, in 2015 a CH4 and CO2 controlled-release experiment was held at the Ginninderra Controlled Release Facility in Canberra, Australia . This large multidisciplinary, multi-institutional blind-release trial (i.e. the participants did not know the true release rate) simultaneously assessed eight different CH4 emission-rate estimation techniques, using data from both mobile and stationary instrumentation. These eight techniques included tracer ratio techniques, backwards Lagrangian stochastic modelling, forward Lagrangian stochastic modelling, Lagrangian stochastic footprint modelling, and atmospheric tomography techniques. A full description of the methods and results is given in .

Every group involved in the analysis presented in used a unique combination of instrumentation and estimation technique when carrying out the analysis, making it hard to establish the respective merits (or otherwise) of the employed techniques from the inversion results. Nonetheless, an interesting observation from the study is that none of the eight techniques deployed during the blind-release trial had a leakage uncertainty range (95 % interval) that included the true emission rate, while some estimates (including one obtained using atmospheric tomography) were factors of 2 or more off from the true value. Given that atmospheric methane concentration and meteorological instrument measurement uncertainty is generally low for each of the different approaches, it suggests that the techniques that were used did not adequately account for the variability of atmospheric measurements or the uncertainty introduced through parameterisation of atmospheric mixing conditions (e.g. Monin–Obukhov lengths and/or Pasquill stability classes; see Sect. ) and atmospheric dispersion/transport model uncertainty.

A number of studies have highlighted the importance of atmospheric-model error in estimating emission rates or fluxes e.g.. For example, showed that flux estimates are sensitive to the chosen spatio-temporal resolution of the fluxes and the chosen transport model. Uncertainty in the meteorological fields driving the transport model is also known to play a big role e.g.. While ensemble inversions are frequently used to highlight the sensitivity of the results to atmospheric models and meteorological fields, learning unknown parameters associated with transport concurrently with the emission rate is not often done. This is largely due to the computational implications of such an approach. Key here are the use of surrogate models (or emulators) to obtain simplified transport representations. For example, use decision/regression trees as a surrogate for FLEXPART-WRF (a sophisticated atmospheric transport model integrating weather research and forecasting into a Lagrangian particle dispersion model), which allows for quick simulation at various parameter settings that can in turn be used to make inference. For the Ginninderra data we employ the more traditional Gaussian plume model, which can be seen as a surrogate for a full-blown transport model. While known to work well in the small domain (an area of approximately 100m×100m) setting we consider e.g., importantly this plume model is quick to simulate from, giving us the opportunity to calibrate it while estimating the emission release rate e.g.. As we see in our sensitivity analysis of our results in Sect. , online plume-model calibration is crucial for obtaining accurate emission-rate estimates with our data.

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 . This framework is in turn seated within a Bayesian paradigm, which underpins several of the inversion systems in place today (see, for example, ). Inference in such cases is often done using sampling techniques such as Markov chain Monte Carlo (MCMC) or importance sampling e.g.. Quick evaluation of the transport/dispersion model (or surrogate) is crucial when repeatedly evaluating it within an MCMC framework; the Gaussian plume model is hence a popular choice in these frameworks e.g.. MCMC is also our method of choice for Bayesian atmospheric tomography, because it allows relatively easy computation of posterior distributions of parameters that are deeply nested within a hierarchical model. It is also ideally suited for the case of point-source emissions, where the dimensionality of the latent space is low (unlike, for example, when performing regional emission quantification).

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 . In , atmospheric tomography was only used on one type of instrument and did not account for uncertainty in the transport model. The technique we propose accounts for uncertainty in our data, in our process models, and in our parameters; is applicable to both point- and path-sampling instruments; and takes into account instrument-specific bias. Inference is made on all unknown parameters using MCMC, and uncertainty in the transport-model parameters are propagated to our posterior inferences on the release rate. We demonstrate the efficacy and utility of the unifying Bayesian framework on data from point- and path-sampling instruments used in the Ginninderra experiment. A secondary contribution is the curated provision of a data set containing a large portion of the Ginninderra data at a 5 min resolution, which we hope will serve as a resource for other researchers to validate their own emission-rate estimation techniques on. The data and scripts required to reproduce the results in this article are available from https://github.com/Lcartwright94/BayesianAT (last access: 1 August 2019).

The remainder of the article is organised as follows. Section  gives an overview of the experimental setup and the data collected during the 2015 Ginninderra experiment. Section  describes the atmospheric transport model used, while Sect.  details the hierarchical model we employ and the Bayesian methodology we develop for emission-rate estimation. Section  gives the results from application of our Bayesian atmospheric tomography technique on the Ginninderra data. Section  examines how our results would change if certain components in our model (e.g. relating to the plume model) are (erroneously) assumed fixed and known. Section  concludes.

A full description of the experimental setup, measurement techniques, and quantification methods used in the 2015 Ginninderra release experiment are given in . Briefly, CH4 (together with CO2 and nitrous oxide) was released from a small chamber located in a fallow agricultural field from 23 April to 12 June 2015 and from 23 to 24 June 2015. A variety of CH4 sensors were placed around the release chamber. The measurement data considered in this study were obtained from two Picarro G2201-i analysers (positioned in the predominant upwind (NW) and downwind (SE) location of the release chamber, labelled Picarro.West and Picarro.East, respectively), four eddy covariance (EC) towers equipped with Li-COR 7700 open-path CH4 sensors (labelled EC.A, EC.C, EC.D, and EC.E, respectively), two scanning Fourier-transform infrared (FTIR) spectrometers with four retro-reflectors terminating six measurement paths (labelled P1 to P6, respectively), and a scanning GasFinder 2 Boreal laser with seven reflectors forming seven measurement paths (labelled R1 to R7, respectively); see the left panel of Fig. . Meteorological data were collected from EC.A equipped with a Vaisala HMP50 relative humidity and temperature sensor, a CSI EC150 CO2–H2O sensor, a Li-COR 7700 CH4 sensor, a Kipp and Zonen CNR4 radiometer, a CSI CSAT3 sonic anemometer, and a Gill WindSonic anemometer. Wind speed and wind direction were measured by the CSAT3 sonic anemometer and the Gill WindSonic anemometer. As part of data quality control, horizontal wind speed and wind direction data from the two instruments were compared, with no arising issues. Both sonic anemometers were using factory calibration. Wind directions were determined by manually aligning the sonic anemometers so that the reference direction was true north. Data from CSAT3 sonic anemometer were logged at 10 Hz and data from the Gill WindSonic anemometer at 1 Hz.

The gases were released at a height of 0.3 m, and the standard CH4 release rate was 5.8 g min-1, limited mostly to daylight hours. On brief occasions, the CH4 release rate was varied between 2.9 and 20 g min-1 to enable testing of mobile CH4 sensor platforms. Towards the end of the experiment (8–12 June), the CH4 release rate was decreased from 5.8 to 5.0 g min-1, and the setup for the Boreal laser measurements was modified with the number of retro-reflectors and paths reduced to six (labelled R8 to R13, respectively; see the right panel of Fig. ). The location of all other CH4 sensors did not change over the duration of the experiment. The Picarro analysers were not deployed until 21 May, and the CH4 release rate on 23 and 24 June was constantly varied. Hence, in this article we only consider data between 21 May and 12 June, excluding 26 and 27 May where the release rate was also constantly varied.

The data set used to obtain the results presented in Sect.  was compiled by pooling together the separate meteorological and concentration data sets used in the Ginninderra experiment. A common resolution of 5 min was chosen; that is, all measurements of concentration and meteorological variables were averaged over a regular set of 5 min intervals. Measured CH4 concentrations were then matched with corresponding meteorological measurements by time and placed into long-table format, with each row corresponding to a unique data point. For path measurements, two extra columns were used to denote the end-point coordinates of the paths.

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 45∘. A path measurement was classified as upwind if the angles subtended at every point along the path were more than 45∘.

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

We are ultimately interested in obtaining a range of plausible values for the emission rate, Q, a posteriori, (i.e. after we have observed some data). In this section we present a hierarchical statistical model that relates Q to the observed concentrations via the Gaussian plume model. Although Q itself is univariate, the model contains several other unknown parameters that capture our uncertainty about the physical and the measurement processes; inferences on these parameters and Q are made simultaneously. For ease of exposition we adopt the terminology of to describe the model, which we also summarise graphically in Fig. . The top layer in the hierarchy is the data model (the model for the observations, Y, Sect. ), the middle layer is the process model (the model for Q, Sect. ), and the bottom layer is the parameter model (the unknown parameters not of direct interest, τ, ωy,ωz, Sect. ). In Sect.  we outline the MCMC strategy we use to make inference with the model.

As detailed throughout Sect. , the Bayesian model we employ contains many parameters that are updated using MCMC. A natural question to ask is whether all these parameters do need to be updated and what the effects on the emission rate inferences are when instead some of these are assumed fixed and known. Specifically, we are interested in seeing what happens when (i) considering only one single precision parameter τ for all of the data regardless of stability class and/or instrument group, (ii) considering one τmi per instrument group only, (iii) not accounting for plume-model variability in low wind speeds (i.e. setting U^=1), (iv) not updating ωy when using point measurements, (v) not updating ωz, and (vi) not updating both ωy and ωz when using point measurements. The 95 % credible intervals for Q in grams per minute for all these settings and for each of the 10 groupings considered in Sect.  are given in Table .

Grouping the precision parameters {τmi} by instrument only (instead of by instrument and stability class) had a slightly negative impact on the emission-rate estimates obtained during the second release-rate period but less so during the first release-rate period. Assuming (and fixing) ωz=1 for both the point and path measurements also did not have a serious impact on the emission-rate estimates. Note that this does not mean that these components are not relevant in the general model – for example, from our estimates of ωz in Table  we see ωz=1 would be a plausible choice for this experiment if one opted to fix ωz (while ωy=1 would not be).

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 p. 207 when analysing data from the Boreal lasers. There, observations with wind speeds below 1.5 m s-1 were removed to mitigate this effect.

The scaling factor ωy is clearly also crucial for obtaining emission-rate estimates of practical significance for point measurements, with the ensuing emission-rate estimates often being off by nearly a factor of 2 when ωy=1 is assumed. As expected, the width of the credible intervals on the emission rate decreased substantially when ωy=ωz=1 was assumed, indicating that ωy and ωz play a big role in quantifying uncertainty on the emission rate. Therefore, as noted in other studies discussed in Sect. , incorporating uncertainty in the transport model by treating parameters within the model itself as uncertain (note that this is different from adding another component of variability in the data model, as is often done) is likely to have a positive impact on emission-rate estimates and uncertainty quantification.

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 , for which the quantification approach outlined in this article is ideally suited. Equipment placement can be optimised around the leakage site (i.e. prevailing upwind/downwind) for optimal quantification.

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 . If there are multiple possible sites, and these locations are not known, then the framework needs to be further extended to incorporate multiple Gaussian plume models (one for each site), and joint localisation–inversion will be required. While these extensions are straightforward both mathematically and computationally, in practice they are unlikely to be effective for detection of leakage over large spatial scales. Gas fields or geological storage sites can cover areas of tens to hundreds of square kilometres. Unless there is a high density of sensors ≈100 m scale,, the sensitivity of detection will be poor . It is however relatively straightforward to effectively extend the methodology to when the emission is from an area rather than a point source.

Our work is closely connected to other atmospheric tomography techniques but with some small, significant, differences. used a backward Lagrangian particle model to simulate the trajectories of methane and carbon dioxide backwards in time to localise the source and estimate the emission rates. Their approach yielded good quality estimates for the methane emission rates but highly uncertain estimates for the carbon dioxide emission rates and source location parameters. Twenty-three runs of the Lagrangian model required approximately 1 h of computing time, and therefore their framework becomes problematic with thousands of observations as we have in our study. More pertinently, online calibration of the atmospheric transport model would be virtually impossible without the construction and use of a surrogate model or emulator e.g.. In the study of , carbon dioxide and nitrous oxide emission rates and source locations were estimated relatively well. We do not consider the localisation problem but otherwise extend their method to handle various instrument types and a number of extra levels of uncertainty. The case in our sensitivity analysis in which we fix ωy=ωz=1 yields a model that is structurally very similar to that of ; we see from our results that having this hard constraint is not a tenable assumption in practice. Our work also has close connections with that of where the Pasquill stability class for an observation is chosen from a subset of appropriate stability classes, based on the best fit of model-predicted values to observed values. While this may help fit the Gaussian plume dispersion model to the data, it does not take into account the uncertainty arising from stability-class choice. Further, if all plume model standard deviations are off by a factor of 2 or more, there is a distinct possibility that no stability class yields a good fit. Online calibration of these standard deviations is needed to account for lack-of-fit arising from the inherently simple Gaussian plume model.

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.  showed that estimates using the two Picarro analysers were particularly sensitive to assumptions made on the model plume parameters. Moreover, when uncertainty on these parameters was considered, the release-rate estimates from these instruments tended to be uncertain. This is despite the Picarro analysers being among the more accurate and expensive instruments used in the study. Uncertainty in our experiment is, as is often the case, dominated by that in the transport model. Hence, the number of instruments used, the proximity of the instruments to the source, and their configuration around the source appear to be more important design criteria than instrument accuracy when the inferential target is emission-rate quantification of a point source. In particular, having more (less expensive) instruments set up to cover many more possible wind directions is better than having only one or two more expensive instruments with which to monitor emissions. If one is limited to using a small number of instruments, then those giving path measurements are preferable to those giving point measurements, as the former will be able to capture a larger range of wind directions. Our results also provide insight on the transport model used. For example, close inspection of our posterior inferences for τ indicated that, across all instrument groups and for both release-rate periods, the model–data mismatch was much lower for the more neutral stability classes C and D than for the more stable/unstable classes A and F.

The fully Bayesian framework we adopt is adaptable to various scenarios. We envision, for example, that source localisation e.g. could be done in tandem with plume-model calibration within an inversion framework, provided several instruments in suitable configurations (as in the Ginninderra experiment) are available. Future work will also investigate how uncertainty in other meteorological variables such as wind-direction, as well as the stability-class categorisation adopted (possibly via z0), could be incorporated within the model.