Methane is the second most important anthropogenic greenhouse gas in Earth's atmosphere, with additional indirect impacts as it affects both tropospheric ozone and stratospheric water vapor. Despite its significance, our understanding of global and regional CH4 budgets has remained inadequate due to the fact that the strength and distribution of CH4 emissions from various source types are not well-constrained (Houweling et al., 2017; Turner et al., 2017). Estimates of CH4 emissions from point sources (e.g., at facility scale) are particularly uncertain, since space-based observations lack sufficiently fine spatial resolutions while in situ measurements are too sparse and mostly representative of large-scale background concentrations. Improved estimates of the CH4 emissions at this point-source scale are critical in guiding emission mitigation efforts.

Recent developments in airborne imaging spectroscopy techniques to quantify CH4 plumes have opened the way for CH4 measurements at a sufficiently high spatial resolution needed to differentiate various local sources within regional scales (Frankenberg et al., 2016; Hulley et al., 2016; Thompson et al., 2015; Thorpe et al., 2016a, 2017; Tratt et al., 2014). A recent airborne campaign in the Four Corners region retrieved column methane enhancements at a resolution of 3 m (Frankenberg et al., 2016), enabling the observation of the plume shape in the direct vicinity of the point source. During the campaign, many plumes of various sizes ranging from a few tens of meters to hundreds of meters were detected across the region, with the majority of their source emission rates between 10 and 1000 kg (CH4) h-1 (Frankenberg et al., 2016). This allows for an effective way to remotely identify and locate CH4 emissions from point sources such as pipeline leaks or oil and gas facilities. The retrievals provide the quantification of a column enhancement (e.g., in molecule cm-2 above background), which can be integrated across the entire methane plume to derive the total amount of methane within the plume, denoted as integrated methane enhancement (IME, either in molecule or mass units, Frankenberg et al., 2016). In addition, the instrument observes the fine structure of the plume at an unprecedented spatial resolution. However, the flux inversion from the observed plumes to the actual emission rate at the source remains complicated due to the dependence on tropospheric boundary layer conditions such as wind speed and atmospheric stability during the overpass. To interpret the relationship between the observed plumes and flux rates, previous studies have relied on Gaussian plume inversion models (Krings et al., 2011, 2013; Rayner et al., 2014; Nassar et al., 2017; Schwandner et al., 2017) or an airborne in situ approach using a mass balance calculation based on the enhancement downwind of the source (Cambaliza et al., 2015; Conley et al., 2016; Gordon et al., 2015; Jacob et al., 2016; Lavoie et al., 2015). Frankenberg et al. (2016) used a simple linear scaling between the IME and flux rate, which allowed for a straightforward derivation of fluxes from the observed IME given an averaged wind speed across a large region for the campaign over several days. Varon et al. (2018) estimated the flux rate as the IME divided by the residence time of methane in the plume calculated based on the effective length of the plume from its area and the effective wind speed inferred from 10 m wind speed by in situ measurement or meteorological reanalysis data. All of these methods rely on knowledge of local wind speed, which is acquired through either in situ wind measurements or the estimation from meteorological forecast or reanalysis data. The former can be costly and time consuming without prior knowledge of source locations, while the latter can be inaccurate due to the rapid changes of a local plume over a much shorter temporal and spatial scale (minutes, hundreds of meters) than the typical atmospheric reanalysis products (a-few-hourly average, tens of kilometers).

In this work, we aim to improve our understanding of how the inferred emission rates change under different atmospheric conditions, e.g., the errors due to a lack of accurate wind measurements. To investigate this relationship and associated errors, we used large eddy simulations (LESs, Matheou and Bowman, 2016) to simulate the plume dynamics at high spatial resolution (5 m) with prescribed source rates under various background wind speeds and typical surface latent and sensible heat fluxes. Using 3-D LES model output for each snapshot, we simulated synthetic 2-D airborne measurements by applying the respective averaging kernels. Based on these synthetic measurements, we developed an algorithm to deduce the wind speed from the plume's spatial distribution and investigate the degree to which the flux rate can be inverted from only the remotely sensed CH4 retrievals. This allowed us to perform an end-to-end test of errors in inverted methane fluxes in both the absence and presence of ancillary information on the actual wind speed (Sect. 6.3).

This work was inspired by the use of IME to quantify methane single-point sources from field campaigns using airborne instruments. These plumes generally are of small-to-medium sizes (<2 km). The concept, nevertheless, can be applicable to larger sources as well as toward measurement of localized sources from space in the coming decade for satellite retrievals at a much finer spatial resolution (Thorpe et al., 2016b).

Section 2 illustrates the plume observations and the instrument specifications. Section 3 will give a brief overview of Gaussian plume modeling. The setup of the LES and application of instrument operators to simulate airborne measurements are described in Sects. 4 and 5 respectively. Section 6 shows simulated plumes under different atmospheric scenarios and the relationship between observed IME and actual emission rates. The error analysis of flux inversion based on the IME method is also provided. The final section provides a discussion and conclusion.

Figure 1 shows examples of observed methane plumes using the next-generation Airborne Visible/Infrared Imaging Spectrometer (AVIRIS-NG) and the Hyperspectral Thermal Emission Spectrometer (HyTES) during the Four Corners flight campaign (Frankenberg et al., 2016). The iterative maximum a posteriori differential optical absorption spectroscopy (IMAP-DOAS) method (Thompson et al., 2015) and clutter matched filter (CMF) were used to retrieve the scenes from AVIRIS-NG and HyTES respectively. In this case, the aircraft repeatedly flew over a coal mine venting shaft, with approximately 10 min revisit time. Evidently, the plume is changing in time and exhibits fine-scaled features due to atmospheric turbulence. Quantifying the source rate from detected plumes using atmospheric simulations to understand their behavior and variations in space and time is the main subject of this work. In order to compare our simulations with actual observations, we need to take the measurement characteristics of the remote sensing instrument into account. This relates to both measurement precision, which determines detection thresholds which mark and define the detected plume, as well as vertical sensitivity, which affects what parts of the plume structure can actually be observed. Depending on the techniques being used, both can vary widely.

The left column in Fig. 1 shows scenes that are retrieved from the AVIRIS-NG instrument, which measures reflected solar radiation between 0.35 and 2.5 µm at 5 nm resolution and sampling (Hamlin et al., 2011; Thompson et al., 2015). To first order, it has a uniform vertical sensitivity (averaging kernel) of 1 at each height (see Fig. 2). Another instrument that was used in the Four Corners campaign is HyTES, which enables the detection of CH4 plumes due to its absorptions in the thermal infrared around 7.65 µm (Hulley et al., 2016). Its varying sensitivity in the vertical can be calculated as the derivative of the retrieved total column amount with respect to the change in a particular layer. These vertical sensitivities are formally called column averaging kernels. They inform us on how well methane deviations from the prior at each height can be measured, which determines whether they will be visible in retrieved column enhancements. Mathematically, we can express this relationship as 1Ei,j=∑k(ΔxΔyΔh)⋅Ci,j,k⋅CAK(k), where Ei,j is the observed total column enhancement (mass or molecules) at the horizontal grid cell i,j. Δx,Δy, and Δh are grid sizes in i^,j^, and k^ respectively; C is the concentration (mass or molecules per volume); and CAK(k) denotes the column averaging kernel evaluated at level k. Technically, the CAK can also be a function of location i,j, but for the purpose of producing synthetic measurements from our simulations in this work, we apply the CAK only as a function of height.

Figure 2 illustrates the difference between the column averaging kernels that we use to model AVIRIS-NG and HyTES synthetic measurements. The distinct column averaging kernels of both instruments hold significant importance, each with its advantages and disadvantages. The column averaging kernel of AVIRIS-NG is approximately uniform across all vertical levels, which implies that the retrieved column enhancement accurately reflects the actual column enhancement. On the other hand, the sensitivity of HyTES is almost zero near the surface but increases with height, becoming even larger than 1 at a certain height. This means that the instrument is almost blind to methane near the ground but amplified the actual methane amount at certain heights in the column. This distinction is evident in Fig. 1 where the observed methane plume remains more consistent from AVIRIS-NG scenes, whereas more variations appear in the HyTES scenes potentially due to changes in plume vertical structures. It should also be noted that the HyTES averaging kernel strongly depends on the temperature profile as well as the surface temperature, which can vary within and between scenes. In contrast, averaging kernels using shortwave reflected light are less variable.

The simplest way to simulate plumes is Gaussian plume modeling, which assumes a steady and uniform wind U along the x axis and orthogonal spreading of the plume in crosswind (y axis) and vertical (z axis) directions. The spreading of the plume depends on the dispersion functions σy(x) and σz(x). The dispersion functions depend on the atmospheric stability (Pasquill, 1961). For instance, convective conditions favor vertical dispersion, whereas in a stable atmosphere the plume primarily disperses in the horizontal directions (Briggs, 1973; Matheou and Bowman, 2016; Sutton, 1931). The three-dimensional Gaussian plume equation is given by (Matheou and Bowman, 2016) 2C(x,y,z)=12πσy(x)σz(x)⋅QU⋅exp⁡-y22σy2(x)∑m=0∞exp⁡-z-2mzi22σz2(x)+exp⁡-z+2mzi22σz2(x), where C(x,y,z) is the (equilibrium) concentration at each point in the three-dimensional space within the atmospheric boundary layer with inversion height zi. The model assumes a reflective boundary condition where the parameter m multiplied by zi indicates the height at which the reflection occurs and the summation over this parameter m represents the equivalent concentration within 0 to zi. Q is the source flux rate at the origin. The variances σy(x) and σz(x) are given by empirical relations based on atmospheric stability following the Pasquill classification (Matheou and Bowman, 2016; Pasquill, 1961).

By integrating Eq. (2) in the z direction, the methane column enhancement can be modeled in analytical form as 3C¯(x,y)=12πσy(x)⋅QU⋅exp⁡-y22σy2(x). Based on this model, we can vary the source rate, wind speed, and stability category to simulate the 2-D integrated concentration field. We then apply a device detection threshold to illustrate how the synthetic Gaussian plume column enhancement may change under distinct atmospheric conditions. Examples of the simulated Gaussian plumes with a flux rate of 300 kg h-1 are shown in Fig. 3. The left column of Fig. 3 shows the Gaussian plumes under different wind speeds for a fixed stability category, while the right column demonstrates those under a fixed wind speed at 4 m s-1 but different stability regimes.

The wind speed U influences the column enhancement, which, based on Eq. (1), is proportional to the ratio Q/U. Thus, the Gaussian plume model suggests a strong dependence of the IME on wind speed, which in turn does not explicitly affect the shape of the plume. One way of quantifying a plume shape is using an aspect ratio in the x–y plane. In the Gaussian plume model, the aspect ratio of the plume only changes when the stability switches from one category to another. Thus, the wind speed is only implicitly linked to the shape of the plumes by affecting the stability categories and changing the crosswind variances (as can be seen in Eq. 3).

The stability categories in this model, nonetheless, are based on empirical formulae. In reality, the wind speed can influence the shape and distribution of the plumes more directly through advection of the tracer along the flow. The actual plume observations from the Four Corners campaign (Fig. 1) demonstrate that the plumes are of turbulent nature – at times being discontinuous – and cannot be modeled as Gaussian when only one plume snapshot in time is recorded. Therefore, we utilize an LES model, which yields a realistic realization of the turbulent flow and the methane plume, to quantify the effect of wind speed on the plume structure.

Realistic modeling of CH4 plumes is a prerequisite for this study. We use LES to model the time-resolved three-dimensional CH4 distribution in the boundary layer under different atmospheric conditions at resolutions currently available from aircraft measurements (1–5 m). The LES model setup for the simulation of plumes emanating from point sources is as described in Matheou and Bowman (2016). Further details of the model formulation, including the turbulence parameterization, are in Matheou and Chung (2014). A methane surface point source with a specific emission rate in a cloud-free convective atmospheric boundary layer is simulated. The buoyancy of methane is currently being ignored – a good approximation for the present methane concentrations away from the source.

The atmospheric boundary layer is initialized with a mixed layer inversion free troposphere with an initial inversion height zi=800 m. The initial potential temperature and specific humidity in the mixed layer are θ=298 K and qt=6.6 g kg-1. The lapse rate is Δθ/Δz=0.12 Km-1. The flow in the boundary layer is driven by a constant geostrophic wind in the x direction, ug. Different values of the geostrophic wind from 1 to 10 m s-1 are used. The surface sensible and latent heat fluxes are 400 and 40 W m-2. These values are based on typical field campaign data. Additional simulations with other sensible and latent heat fluxes are also performed later in Sect. 6.4. Surface momentum fluxes are estimated using Monin–Obukhov similarity theory (MOST).

The model domain is 10.24×2.56×1.5 km3 in the x, y, and z direction, and the grid resolution is uniform and isotropic Δx=Δy=Δh=5 m. The model computational time step is 1 s. Following 1 h of model spin-up, where fully developed three-dimensional turbulence is established in the boundary layer, the three-dimensional concentration at each location at 1 min intervals (snapshots are written out at every minute) is used to construct the synthetic observations. Furthermore, the 10 and 2 m wind speeds are extracted from the model output to compare with the large-scale geostrophic wind value in each run.

With the output from the LES simulations, we can create synthetic measurement of a plume instance that would enable simulation of observations from any instrument. The procedure is that we apply vertical integration as described by Eq. (1) to the 3-D concentration at a given time step, using the column averaging kernel of the instrument of interest. We apply the column averaging kernel of AVIRIS-NG as well as that of HyTES to produce synthetic measurements for these instruments. The detection thresholds of the AVIRIS-NG and HyTES instruments can potentially be dependent on the surface properties such as surface reflectance and surface temperature respectively. However, given the typical scale of the plumes of our interest, we assume an average uniform detection threshold across the scene. Here, we use a constant threshold of 500 ppm m-1 (or about 1.341018 molecules cm-2), which is a common value for AVIRIS-NG. As for HyTES, we used the same threshold to exemplify the differences due to averaging kernels only, as opposed to thresholds. This allows us to understand to what extent each instrument can detect CH4 plumes under various wind speeds.

The output from the LES run provides a more realistic simulation, compared to the Gaussian model, of the plume dynamics as shown in Fig. 4 for AVIRIS-NG synthetic measurements. The left column of Fig. 4 shows single snapshots of the plume, while the right column shows the time-averaged plume snapshots over 60 time steps, spanning a duration of 60 sequential minutes in total, under distinct background wind speeds but with a constant flux rate. Based on this simulation, we see that the plume varies rapidly in shape and orientation from snapshot to snapshot due to turbulence. The temporal averages in the right column also still exhibit some structure as we only averaged 60 individual snapshots. Overall, the simulated plumes from the LES closely resemble actual plumes from remotely sensed observation as shown in Fig. 1. The instantaneous plumes exhibit non-Gaussian behavior; sometimes the plume can even be discontinuous as eddies can rupture the plume structure. However, we found that the total enhancement across the scene (the IME) remains rather constant over time for a given wind speed and flux rate, making it a reliable variable for performing the flux inversion of the source. In addition, we also found that the plumes have distinct features in both magnitude and spatial characteristics for different wind speeds, which are evident in the plume snapshots as well as their ensemble means shown in Fig. 4.

Figure 5 illustrates the differences between the synthetic measurements for AVIRIS-NG and HyTES over the same plume for three different wind speed conditions. Because the column averaging kernel of the HyTES is close to zero near the ground, the synthetic measurements for HyTES miss parts of the plume near the surface and detect only the parts of the plume that have risen high enough. This is consistent with the averaging kernels shown in Fig. 2. This is especially apparent for the case of high wind speed where the majority of the CH4 is advected horizontally, resulting in a plume remaining near the ground. The result in Fig. 5 is in accord with the comparison between the observed AVIRIS-NG and HyTES scenes in Fig. 1 during the first overpass. This potentially indicates that the plume at this time remains mostly near the ground, which may not always happen in the same way for the coal mine venting shaft, which is emitting above the ground surface. The insensitivity of HyTES near the ground makes it complicated to locate the source accurately, and there are additional uncertainties in the methane retrievals associated with averaging kernels that vary with environmental conditions (Kuai et al., 2016). The advantage of the HyTES instrument, on the other hand, is the fact that in principle it can operate at night when there is no sunlight, which is a prerequisite for the AVIRIS-NG instrument. For AVIRIS-NG, the total column CH4 enhancement in each pixel is also better constrained given the averaging kernel is approximately one throughout the column. For these reasons, we proceed to focus only on AVIRIS-NG results in the current study, while we will study the information content of joint measurements in the future.

Multiple LES runs from a combination of typical point-source flux rates and wind speeds enable us to quantify the relationship between the actual source rate and the resulting IME for a given wind speed. This gives us the first step to invert the flux rate. Furthermore, we show how different wind speeds affect this relationship for the flux inversion. The output from the LES gives us not only the IME but also the spatial distribution of the plume snapshots that correspond to a given pair of flux rate and wind speed. We analyze how the morphology of the plumes is linked with the underlying background wind speeds. This helps us understand how we can use the remotely sensed airborne imagery of the plume to predict the wind, and thus ultimately the flux rate, together with its associated errors.

In our analysis, we primarily refer to the wind speed in each scene from our model runs by using the geostrophic wind speed, as opposed to the instantaneous wind at 2 m (U2) or 10 m (U10) above ground which is usually used in literature. For reference, the average U10 across the horizontal domain in our run ranges approximately from 0.4 to 0.7 of the background geostrophic wind speed in the run. The main reason is that our output snapshots from each LES run is written out every minute; thus we only have the information of the U10 and the plume structure at every minute, which can change rapidly in direction and magnitude. However, the overall structure of the plume at any given instance could be influenced by the average wind cumulatively from the past minute. The constraint on the output that we have makes it ambiguous to choose what values of near-surface winds should be applied when making the prediction of the flux rate from the spatial structure of a plume snapshot. We thus resort to using a background wind speed, which, in turn, is one of the key governing drivers for U10 itself. While using the large-scale background wind speed might not be as accurate as the ideal case of having continuous U10 output, it provides a robust correlation with the overall pattern of the plume (see Sect. 6.2). In other words, in the following, we are using the shape of the plume to predict the value of background geostrophic wind speed that underlies the wind that has driven CH4 from the point source into the detected plume over that geographical location, and we use that background wind speed to quantify the source rate.

In this study, we showed that Gaussian plume modeling cannot be used for a meaningful comparison with observed methane plumes from a point source. Thus, large eddy simulations (LESs) were used to generate realistic synthetic measurements of methane plumes under different background wind speeds and source flux rates. This allowed a comparison of the performances of two considered instruments, one measuring in the shortwave infrared (AVIRIS-NG) and the other in the thermal infrared (HyTES), resulting in widely different vertical sensitivities towards methane enhancements. The AVIRIS-NG was found to provide an unambiguous identification and quantification of the methane source as it is sensitive to methane throughout the air column. While the HyTES instrument has the potential for nighttime observations, variations in the integrated methane enhancements depended highly on vertical plume structure, rendering the interpretation more challenging. While we attempt to make use of the vertical information in the future, we focus this study on results from the AVIRIS-NG synthetic plume measurements. Using the IME method and a large ensemble, we derived the relationship between the detected IME of a plume and its source flux rate. This relationship is found to be nonlinear because of the device detection threshold, which causes a variable fraction of the true IME to fall below the detection limit. In addition, the inversion of IME to an accurate flux rate depends strongly on the wind speeds during the measurements. This finding is expected and confirms the significance of wind speeds on the methane point-source flux estimations from remote sensing data. To study whether we can gain additional information from the plume shape itself, we performed an analysis on a large ensemble of plume snapshots from wide-ranging source flux rates and wind speeds. We found that the angular width of the plume negatively correlates with the wind speed, allowing us to constrain the effective wind speed from the shape itself. The angular width is defined based on the plume angular distribution around its main axis and is found to be effectively independent of the source rates.

Using the relationship between the IME and the flux rates for different wind speeds together with the connection between plume shape and the wind speed, we can disentangle the source flux rate based on an observed snapshot of the plume which provides both the IME and the spatial distribution. Our error analysis of this method applied on randomly generated snapshots of various flux rates in the range of 10–1000 kg h-1 showed an error of around 30 % on average for an individual point-source estimate. Given that point sources are highly uncertain and also fluctuate in time, this single measurement error appears acceptable. More important than single measurement precision is accuracy for larger ensemble averages, which informs regional emission estimates. Thus, we also performed an error analysis for aggregated flux estimates from 30 plumes. We used bootstrap sampling and found the aggregation error estimate to be in the range of less than 10 %. This provides a significant improvement from other preexisting approaches that rely on wind data, for which reliable meteorological reanalysis data might not be available at high spatial resolution everywhere.

Furthermore, our method is validated by the application of this method on an actual scene from a controlled-release experiment from a natural gas pipeline in 2017, which demonstrated an error of 32 % from the controlled flux rate of 89 kg h-1, a notable accuracy given the simplicity of our algorithm that does not require wind speed data. This provides added value in quantifying methane-point-source emissions especially in locations where atmospheric reanalysis products and surface meteorological observations are not available.

It should be noted that altering the device detection threshold level in our synthetic modeling to higher values does impact the robustness of the correlation between the plume width and the wind speed. In this study, we set the threshold to 500 ppm m-1 to match the capabilities of the current instrumentations. Future instruments with improved gas sensitivity (Thorpe et al., 2016b) will likely improve our ability to estimate emission rates. Repeat overflights that result in multiple snapshots of the same source can also further reduce uncertainties from transient variations of the plume due to turbulence. Another aspect is that our current LES does not yet model direct emission that could be released at height above the ground. Incorporating this feature into our future analysis may provide even more realistic methane plume simulations. Despite these limitations, this current study is a first step proving the potential of the method.

In this study, we have demonstrated the ability to estimate flux rates of methane point sources based solely on the remotely sensed column methane enhancement without the need for ground measurements or weather reanalysis data. This method could be applied to recent large-scale flight campaigns to improve previous emission rate estimates. This also has immediate implications for future AVIRIS-NG flight campaigns, in particular over parts of the world lacking available wind data. The methodology described in this study could also be applied to anticipated satellites that will provide methane measurements at finer spatial resolutions than currently available. A path towards an improved understanding of the regional methane budget as well as insights into methane source distributions by categories is made possible.