A Near-Field Gaussian Plume Inversion Flux Quantification Method, Applied to Unmanned Aerial Vehicle Sampling

Abstract

The accurate quantification of methane emissions from point sources is required to better quantify emissions for sector-specific reporting and inventory validation. An unmanned aerial vehicle (UAV) serves as a platform to sample plumes near to source. This paper describes a near-field Gaussian plume inversion (NGI) flux technique, adapted for downwind sampling of turbulent plumes, by fitting a plume model to measured flux density in three spatial dimensions. The method was refined and tested using sample data acquired from eight UAV flights, which measured a controlled release of methane gas. Sampling was conducted to a maximum height of 31 m (i.e. above the maximum height of the emission plumes). The method applies a flux inversion to plumes sampled near point sources. To test the method, a series of random walk sampling simulations were used to derive an NGI upper uncertainty bound by quantifying systematic flux bias due to a limited spatial sampling extent typical for short-duration small UAV flights (less than 30 min). The development of the NGI method enables its future use to quantify methane emissions for point sources, facilitating future assessments of emissions from specific source-types and source areas. This allows for atmospheric measurement-based fluxes to be derived using downwind UAV sampling for relatively rapid flux analysis, without the need for access to difficult-to-reach areas.

1. Introduction

Methane is the second most important greenhouse gas in terms of radiative forcing [1], with a direct radiative forcing of +0.61 W m⁻² [2]. Though a global mean methane concentration up to 800,000 years prior to 1850 was constrained between 300 ppb and 800 ppb [3], recent global mean measurements (2018) have exceeded 1850 ppb [4]. Following a hiatus at the turn of this millennium, a gradual annualised concentration increase has resumed since the year 2007 [5,6], at an annual growth rate of (5.7 ± 1.2) ppb yr⁻¹ up to the year 2014 [7]. Approximately 64% of global methane emissions are anthropogenic [8,9,10], with approximately one-fifth of all anthropogenic methane originating from landfills [11], representing the dominant source of methane waste emissions, produced primarily by the anaerobic microbial decomposition of organic municipal solid waste [6]. Though most methane emission source types are known, the quantification of source emission contributions to the global methane budget may be poor [12], especially on a facility-scale. This prioritises the retrieval of facility-scale methane emission fluxes using near-field (~100 m from the source) sampling to help constrain and improve emission inventories by top-down validation and ultimately their global extrapolation [6]. By carefully targeting potent anthropogenic methane emission sources, accurate facility scale flux quantification is crucial in efforts to mitigate climate change.

Existing facility-scale methane flux quantification methodologies require atmospheric methane concentration ([CH₄]) measurements, which can be obtained using either remote sensing or in-situ methods [13,14,15,16], although in-situ measurements are more commonly used [17]. Examples of previously used methane in-situ measurement-based techniques include eddy covariance [18], mass balance box models [17,19,20,21,22,23,24], and the tracer dispersion method (TDM), which compares the downwind ratio of methane concentration to an inert tracer [13,25,26,27,28,29,30,31]. Alternatively, inversion dispersion models may be used for flux quantification on a facility scale [17,21,32,33,34]. For example, Gaussian plume inversions have been applied using middle distance (~1 km from the source) [CH₄] sampling [33,35,36,37,38]. Foster-Wittig et al. [39] used near-field [CH₄] measurements from a stationary position, downwind of the source, to derive fluxes using a Gaussian plume inversion.

Far-field (defined here as > 1 km from source) in-situ sampling on a downwind two-dimensional vertical flux plane, roughly perpendicular to mean wind direction, has successfully been used to calculate fluxes using mass balance box modelling [40,41,42]. These previous far-field studies utilised geospatial kriging [43] to interpolate between sparse spatial sampling of a well-defined emission plume, characterised by dispersion [20]. Near-field sampling on a two-dimensional vertical flux plane cannot easily be treated in the same way, as turbulent fluctuations in wind direction through the flux plane are fast compared with the typical timescale of near-field sampling techniques, resulting in poorly defined plume morphology for short sampling times typical of single small unmanned aerial vehicle (UAV) flights. To address this challenge, we describe a near-field Gaussian plume inversion (NGI) flux quantification technique (Section 2) which is suitable for near-to-source sampling on a vertical flux plane.

The NGI method is ideally suited to derive fluxes from near field sampling, typical of small UAV platforms (<20 kg take-off mass). We define in-situ UAV sampling here to mean any sampling in which a UAV is used to obtain air samples for either live or subsequent analysis. Though aircraft-based [CH₄] measurements have been used to derive far-field methane fluxes in the past for strong emitters and area-sources such as urban environments [40,41], manned aircraft are unsuitable for near-field sampling due to their typically fast speed (>50 m s⁻¹) and flight restrictions on proximity and height [44]. They are significantly costlier to maintain and operate than small UAV platforms. Small UAVs have a recent and growing variety of scientific applications [44,45,46], especially in hazardous environments [47,48,49,50,51]. A key advantage of UAVs is their ability to fly in difficult-to-access areas quickly which, for example, stationary towers may struggle with, as they can be difficult to move on muddy terrain and can be logistically difficult to operate.

Here, we list some past examples of the use of UAVs in various applications, which is by no means an exhaustive list in this rapidly developing field. Vanegas et al. [52] used a suite of hyperspectral, multispectral and visible optical sensors on-board a 6 kg UAV platform to improve pest surveillance [52]. An algorithm was used to combine ground-based and airborne measurements to map the spread of a pest infestation in a vineyard [52]. Kim et al. [53] used an ultrasonic displacement sensor along with a camera module on-board a 1.8 kg UAV to detect cracks in concrete structures. They found that their algorithm was capable of measuring cracks thinner than 0.1 mm with an error of 7.3% [53]. Wang et al. [54] used a 1 kg light detection and ranging system mounted on-board a small UAV to measure biomass cover and canopy height over grassland. Arabi et al. [55] used a UAV to wirelessly recharge depleted batteries. The system was tested to recharge a network of depleted ground based data transmitters [55]. Schuyler et al. [56] used a 5.14 kg UAV, in the form of a balloon-launched glider, to derive measurements of temperature, pressure and humidity from up to 25 km in height, which were used to validate the results of a weather model. Nolan et al. [57] developed an Eulerian diagnostics wind velocity model suitable for single UAV flights, which they tested by simulating a UAV flying in circles. Elsewhere, Nolan et al. [58] measured wind speed, wind direction and temperature at 0.07 Hz using two UAVs combined with ground-based (tower) measurements to determine wind fields, which were then used to model the atmospheric transport of emissions.

A number of previous studies have measured boundary layer winds and turbulent parameters using UAV sampling [59,60,61]. Barbieri et al. [62] tested 23 different UAVs with a variety of atmospheric sensors designed to measure pressure, temperature, humidity and winds, by comparing UAV measurements to measurements made from a stationary tower on the ground. Lee et al. [63] calibrated a meteorological sensor (sampling at 1 Hz) on board two multi-rotor UAV platforms to characterise variability in moisture and temperature up to a height of 300 m. Alaoui-Sosse et al. [64] tested a 3.5 kg fixed-wing UAV (including payload) which made use of a fast inertial measurement unit to measure turbulence in the boundary layer. The platform was successfully tested by comparing UAV turbulence measurements to measurements from a stationary tower at a fixed height of 60 m at 10 Hz [64]. The UAV could also be used as a mobile data transmission base station [55]. Hemingway et al. [65] used a variogram analysis of temperature and humidity measurements, derived from a 1.2 kg UAV flying within the atmospheric boundary layer, to observe meteorological phenomena of a sub-kilometre spatial scale. Zhou et al. [66] made in-situ measurements of aerosol concentrations from on-board a small UAV in Nanjing, China during an air pollution episode. The position of the sensor was optimised to be on top of the UAV, using computational fluid dynamics to simulate downwash through the propellers [66].

Berman et al. [67] published the first high-precision (2 ppb at 1 Hz) UAV-derived in-situ [CH₄] measurements from the Arctic, albeit without flux calculation. Golston et al. [68] developed a lightweight (1.6 kg) open-path in-situ methane sensor with a relatively high precision (10 ppb at 1 Hz) which was mounted to a small UAV with a take-off mass of 4.9 kg. Brosy et al. [51] connected a small UAV to a high-precision in-situ sensor on the ground using 70 m of tubing to investigate methane altitude gradients. Andersen et al. [69] used their AirCore system to collect a time-continuous air sample on a small UAV platform, which they later analysed on the ground, resulting in a 0.5 Hz precision of 0.5 ppb, to investigate methane altitude profiles. Emran et al. [70] attached a cheap sensor to a small UAV platform to make remote sensing measurements between the ground and the UAV at a fixed height in order to map a landfill emission plume. Thus, UAVs can be used to measure methane in-situ by either collecting air samples for subsequent analysis, by using on-board instrumentation, or by connecting the UAV to a sensor on the ground. All three approaches have their advantages and drawbacks. Tethered sampling lacks flexibility, as the instrument must remain connected to the tether, potentially restricting spatial sampling extent. On-board sampling is ideal, but this requires sufficiently light-weight sensors with correspondingly high precision and resolution. At the time of writing, such instruments (less than 5 kg) were not commercially available, but rapid advances continue to be made in this regard [44].

UAVs have been used in previous near-field methane flux quantification work. Allen et al. [71] used UAV-derived carbon dioxide measurements to infer a landfill methane emission flux using characteristic landfill gas emission ratios: This yielded inaccurate results, as carbon dioxide was a poor proxy for landfill emissions due to extraneous sources of carbon dioxide [19,72], such as those expected in densely industrialised locations which were considered in this previous study. Nathan et al. [73] calculated a mass balance box model flux from a gas compressor station using sparse UAV-derived open path in-situ [CH₄] measurements of a relatively low precision (100 ppb at 10 Hz), where kriging was used. However, as near-field in-situ sampling was used in the Nathan et al. [73] study, the use of kriging as a form of interpolation could be subject to error due to effects of near-field turbulence. Yang et al. [74] used a downward facing remote sensing instrument attached to a small UAV platform to detect gas leaks, using a new interpretation of mass balance box modelling tailored for remote sensing sampling. Their UAV performed concentric circles around the presumed emission source at a fixed height [74]. Mass balance box modelling can be a highly suitable method for UAV sampling, as shown in a study by Yang et al. [74], whose method integrated the statistical average of remote sensing measurements from above an emission plume.

In this paper, an NGI mass balance method (see Section 2) is developed and tested using a preliminary sample dataset of geospatially mapped in-situ [CH₄] measurements obtained from eight UAV flights. The NGI method used here was specially adapted to allow fluxes to be quantified using near-field UAV sampling downwind of (assumed) point sources by tethering the UAV to a high-precision in-situ methane sensor on the ground. An advantage of UAV sampling is that it does not require access to difficult-to-reach areas (for example, access to industrial or restricted sites). In this study, controlled releases of methane were conducted by the UK National Physical Laboratory (NPL) for evaluation. The refinement of flux quantification techniques using known controlled emission fluxes, such as those provided by the NPL in this study, is crucial in order to legitimise the integrity of further work [21,25,34,35,74]. The NPL fluxes were also used to test simultaneously-derived TDM fluxes (described in detail in Fredenslund et al. [75]) by two TDM teams operating independently of the UAV team and each other. In Section 3, we then test the NGI method and account for inherent flux biases associated with the method, using simulated measurements acquired from a random walk simulation sampling a static plume. These simulations also serve as further guidance on improvements to the sampling approach in future use of the method.

2. Method

The NGI method is a mass continuity model in principle, applied to near-field sampling, where in-situ [CH₄] measurements upwind and downwind of an emission source are differenced and combined with a wind vector, resulting in a measured flux density (q_me). This makes the NGI method suitable for near-field UAV sampling. Provided that upwind air is relatively homogenous in comparison to downwind enhancements and well mixed (regardless of whether it is enhanced in methane overall), the NGI method can be used to difference between upwind and downwind methane sampling. The NGI method requires near-to-source sampling (defined here to be ~100 m from the source) on a vertical two-dimensional plane roughly perpendicular to mean wind direction (θ), to define a net source flux into a volume of air advecting with the wind. The [CH₄] value upwind of the source (background) is assumed to be roughly constant compared with enhancements expected from the source of interest. This background variability can be measured and accounted for in error budgeting (see later). This assumption requires there to be no nearby interfering emission sources that are detectable from the point of sampling. Additionally, loss terms between the source and sampling plane are assumed to be negligible, as is the case for methane as its atmospheric lifetime (~9 years [9]) significantly exceeds the timescale of near-field advection downwind of a facility-scale source. For highly reactive trace gases, such considerations may become important if this method was adapted.

Under turbulent conditions, near-field sampling may not map out a characteristic conceptual time-invariant Gaussian plume for constant sources. Instead, a non-Gaussian distribution of q_me may be observed across the flux plane due to sampling different positions of an instantaneous plume, sampled at different times. However, provided that sampling on the vertical flux plane captures the centre of a time-averaged plume and extends sufficiently far outwards, a flux can be derived by assuming that the time-averaged plume also exhibits a Gaussian plume morphology. Sampling duration should also be sufficient to ensure unbiased measurement of the flux plane. This is the basis of the NGI approach, applicable to near-field UAV measurements of point source emissions. In the later discussion, we attempt to assess and account for the limitations of such sampling and examine sources of bias and error in the method using simulations.

2.1. Preparing the Sample Data

A DJI S900 hexacopter UAV (flying up to (24 ± 3) m above ground level) was used to obtain [CH₄] measurements using a Los Gatos Research, Inc. Ultra-portable Greenhouse Gas Analyzer (UGGA) on the ground, sampling air at a rate of 1 Hz, which was connected using a 150 m length of PFA tube to an inlet above the plane of the UAV rotors (see Appendix A for further experimental details including the sampling location, instrumental set-up, geolocation measurements and time corrections). These data served as a sample dataset of geospatially-mapped in-situ [CH₄] measurements, with corresponding measurements of altitude, latitude, and longitude, downwind of the NPL Controlled Release Facility (CRF). Eight flights were conducted for between 13 and 28 min each (see Table S2 for individual flight details). Each flight was conducted with the intention that it would be centred around the time-averaged plume as projected onto the flux plane and was adapted in-flight to capture the maximum extent of the plume. The UAV was directed towards the time-averaged plume mid-flight to maximize in-plume sampling based on mid-flight knowledge of its position estimated from smoke flares released near to the emission source. Such targeted sampling can result in a positive flux bias that (with hindsight) invalidates the assumptions of mapping a time-averaged plume.

Altitude, longitude, and latitude corresponding to each [CH₄] measurement were converted into height above ground level (z), the horizontal distance perpendicular to mean wind direction (y) and the distance from the source along the mean wind vector (x). To calculate z, the lowest in-flight altitude measurement was set to the height of the air inlet above the base of the UAV. Thus, the systematic accuracy of geospatial height measurements is not important, provided that they are of a sufficient precision (random uncertainty). To calculate x and y, longitude and latitude were converted into distances in metres (assuming a spherical Earth). These distances were then projected onto a sampling plane parallel to mean wind direction (θ) to calculate x and onto a plane perpendicular to θ to calculate y for the duration of sampling. The assumed y position of the emission source projected onto the sampling plane was set to 0 m as a reference, and the assumed x position of the source was set to 0 m. Figure 1a gives an example of a UAV flight track, with geospatially mapped measured [CH₄] as a function of x, y and z ([CH₄]_me) on the flux plane (see Figure A3 for all UAV flight tracks). The upwind background methane concentration ([CH₄]₀) and corresponding background uncertainty (σ_b) was assumed here to be constant for each flight. [CH₄]₀ was found by taking the peak of a log-normal distribution applied to a histogram of the lowest [CH₄] measurements (see supplement for full details). All sampling uncertainties in this paper are expressed as single standard deviations.

Anemometers positioned at various heights were used to infer continuous wind speed (U(z)) and wind direction profiles, along with corresponding wind speed variability (σ_U(z)) profiles (see Appendix A for further details of the anemometers used). A single mean value of θ was derived for each flight: This was calculated from the average of two-dimensional wind vectors at the height of each [CH₄]_me measurement. The atmospheric methane mass density (ρ), assuming all of air is methane (in units of kg m⁻³), was calculated using Equation (1), where M is the molar mass of methane (0.0164 kg mol⁻¹), R is the gas constant (8.314 m³ Pa mol⁻¹ K⁻¹), and T and P are the average temperature and pressure, respectively, for the duration of each UAV flight. Provided that changes in T and P are small, ρ can be assumed to remain constant for the duration of sampling.

$$\rho =\frac{PM}{TR}.$$

(1)

2.2. Flux Estimation Using the NGI Method

Here, we describe the procedure applied to each UAV flight from the sample data in order to estimate an initial flux using the NGI method. The q_me, as a function of y, z, and x, was calculated using Equation (2), assuming constant ρ. The q_me is the emission flux per unit area on the flux plane, derived from each individual [CH₄]_me measurement (see Figure 1b for flight 2, for example). As Equation (2) requires the concentration enhancement over the background, instrumental resolution (1 ppb for the sample data) can be a limiting factor in the magnitude of fluxes detectable: If fluxes are too low, concentration enhancements may be undetectable.

$$q_{me}=\left({\left[{\mathrm{CH}}_4\right]}_{me}{-[\mathrm{CH}}_4{]}_0\right)U\left(z\right)\rho .$$

(2)

Equation (3) was used to calculate the uncertainty in q_me (Δq_me), using the instrumental Allan deviation precision (σ_AV, specifically 0.00127 ppm for the sample data) and an instrumental uncertainty factor (σ_i([CH₄])) associated with each [CH₄]_me measurement (see Appendix A for details on σ_i([CH₄])).

$${\mathsf{\Delta }q}_{me}{=q}_{me}\sqrt{\frac{{{\sigma }_{AV}}^2+{\left({\left[{\mathrm{CH}}_4\right]}_{me}{{\sigma }_i\left(\right[\mathrm{CH}}_4\left]\right)\right)}^2{+{\sigma }_b}^2}{{\left({\left[{\mathrm{CH}}_4\right]}_{me}{-[\mathrm{CH}}_4{]}_0\right)}^2}+{\left(\frac{{\sigma }_U\left(z\right)}{U\left(z\right)}\right)}^2+{\left(\frac{{\sigma }_{\rho }}{\rho }\right)}^2}.$$

(3)

The NGI method assumes that spial variability in the time-averaged plume is Gaussian. As a consequence, the NGI method also assumes that the plume is not capped in the z direction by an atmospheric temperature inversion. Conceptually, over an infinite period of randomised sampling, and assuming that the prevailing meteorology does not vary, a Gaussian morphology of q_me would be observed as an average of instantaneous sampling. This essentially smooths out the manifestation of turbulence in the time-averaged plume morphology. However, in practice, limited sampling time constraints (particularly for UAV sampling) may potentially lead to the observation of non-Gaussian q_me across the sampling plane due to turbulence (see Figure A3 for the example of the sample data). The NGI method is therefore a trade-off between the need to sample for a long enough time to characterise the time-averaged plume as much as possible and the need to minimise the systematic effect of any potential dynamic changes in atmospheric stability, source strength and prevailing meteorology over the course of measurement (parameters which may be derived from complementary specialised UAV sampling [57]). Such competing needs would need to be considered on a case-by-case basis by considering weather forecasts, the local topography and the nature of the expected sources being studied.

An initial flux estimate (F_e) can be derived by fitting q_me values to corresponding modelled flux density (q_mo) values using an adaptation of the two dimensional Gaussian plume model [76] given by Equation (4). Equation (4) does not include wind speed, as this has been incorporated into q_me in Equation (2). An advantage of incorporating wind speed into q_me in this way is that variations in wind speed with altitude are incorporated into the measured plume morphology in flux space. An advantage of this method is that it does not require us to assume any stability class, as plume spatial variance is implicitly measured.

$$q_{mo}=\frac{F_e}{{2\pi \sigma }_y{\left(x\right)\sigma }_z\left(x\right)}\exp \left(\frac{{-(y-y}_c{)}^2}{{2\sigma }_y{\left(x\right)}^2}\right)\left(\exp \left(\frac{{-(z-h)}^2}{{2\sigma }_z{\left(x\right)}^2}\right)+\exp \left(\frac{{-(z+h)}^2}{{2\sigma }_z{\left(x\right)}^2}\right)\right).$$

(4)

σ_y(x) is the crosswind mixing factor, σ_z(x) is the vertical mixing factor, y_c is centre of the plume in the y direction (based on the actual sampled q_me measurements) and h is the emission height.

On an idealised downwind vertical flux plane, perpendicular to θ, σ_y(x) and σ_z(x) would be constant for all q_me values. However, as σ_y(x) and σ_z(x) are both functions of x, they may vary depending on the orientation of the sampling plane, if not perfectly perpendicular to θ. In reality, it is difficult to sample on a perfect perpendicular plane, especially with UAV sampling (see Figure S2 for the example of the sample data), as θ may vary throughout the sampling period. To account for this variability in sampling distance from the source, we assume σ_y(x) and σ_z(x) to increase linearly with x (i.e., q_me decreases with increasing distance from the source). Though σ_y(x) and σ_z(x) may not increase linearly with x over large distances, they can be approximated to be linear functions of x under near-field conditions and where the variation in x remains small (less than 40 m). σ_y(x) and σ_z(x) for each stability class were plotted up to 500 m, using the equations found in Turner [76], to test the validity of this assumption (see Figures S4 and S5). This assumption of linearity allowed us to define a crosswind mixing factor evaluated at a distance of 1 m (τ_y) using Equation (5) and a vertical mixing factor evaluated at a distance of 1 m (τ_z) using Equation (6). τ_y and τ_z are both then scalable factors that can be used to derive σ_y and σ_z as a function of x and are therefore applicable to the variable sampling typical of UAV measurement.

$${\tau }_y=\frac{{\sigma }_y\left(x\right)}{x}.$$

(5)

$${\tau }_z=\frac{{\sigma }_z\left(x\right)}{x}.$$

(6)

As τ_y and τ_z are both independent of x, Equation (4) can now be expressed as a three dimensional Gaussian plume, using Equation (7).

$$q_{mo}=\frac{F_e}{{2\pi \tau }_y{\tau }_z{x}^2}\exp \left(\frac{{-(y-y}_c{)}^2}{2{\left({\tau }_{y}x\right)}^2}\right)\left(\exp \left(\frac{{-(z-h)}^2}{2{\left({\tau }_{z}x\right)}^2}\right)+\exp \left(\frac{{-(z+h)}^2}{2{\left({\tau }_{z}x\right)}^2}\right)\right).$$

(7)

One could attempt to solve Equation (7) iteratively for the four unknown constant parameters (F, y_c, τ_y and τ_z). However, such a solution is not always well constrained by the sampled data. Instead, by utilising the separability of Equation (7), we can place constraints on y_c and τ_y by equating the mean and standard deviation (in the y dimension) of $q_{mo}$ to the equivalent values for $q_{me}$. This requires us to remove the vertical (z) component of $q_{me}$ using Equation (8) to define q_me with the z Gaussian component removed (q_me,y). As Equation (8) depends on τ_z, good spatial sampling in the z direction is required in order to derive τ_z.

$$q_{me,y}{=q}_{me}\frac{x\sqrt{2\pi }}{\left(\exp \left(\frac{{-(z-h)}^2}{2{\left({\tau }_{z}x\right)}^2}\right)+\exp \left(\frac{{-(z+h)}^2}{2{\left({\tau }_{z}x\right)}^2}\right)\right)}.$$

(8)

y_c and τ_y were then derived analytically from measured data using Equations (9) and (10), respectively, which require each “j” value of z, x and q_me,y. Foster-Wittig et al. [39] used an equation similar to Equation (10) to derive σ_y(x), rather than τ_y, using downwind concentration measurements acquired at a single position on the ground. As τ_z in Equation (8) was unknown, Equation (9) and (10) were solved simultaneously along with Equations (7) and (8) as part of the model. A key advantage of calculating y_c in this way is that, provided geospatial measurements of a sufficient precision are recorded, the overall systematic accuracy of geospatial sampling is not crucial, as the centre of the plume is derived from spatial measurements relative to each other.

$$y_c=\frac{{{\displaystyle \sum }}_j\left(q_{me,y}{}_j{y}_j\right)}{{{\displaystyle \sum }}_j\left(q_{me,y}{}_j\right)}.$$

(9)

$${\tau }_y=\sqrt{\frac{{{\displaystyle \sum }}_j\left(q_{me,y}{}_j{\left(\frac{y_j{-y}_c}{x_j}\right)}^2\right)}{{{\displaystyle \sum }}_j\left(q_{me,y}{}_j\right)}}.$$

(10)

Equations (9) and (10) require good spatial sampling in the y direction, as y_c weights the flux contributions to the position of the centre of the measured plume (as sampled), independent of z. This means that if q_me at any point in the y direction was equal to 0, it would have no influence in changing the centre of the calculated plume morphology in the y direction. By calculating y_c and τ_y in this way we constrain the model and optimise the flux calculation procedure, such that the only remaining unknown variables in Equation (7) are τ_z and F_e.

The initialised values of τ_z and F_e were then evaluated computationally to minimise the sum of squared residuals between all q_me and q_mo values by solving Equations (7), (9), and (10) simultaneously (as y_c and τ_y are both derived from values of q_me,y, which ultimately depend on τ_z). This was achieved here using the lsqcurvefit function of Matrix Laboratory (MATLAB) 2016a, which applies a least-squares solution to the input data. During the optimisation procedure τ_z and F_e were both constrained to ensure good parameterisation, as described below, without which the model may produce unrealistic flux results. It was clear that the maximum constraining value of τ_z (τ_z,max) had an impact on F_e: If τ_z,max was too small, τ_z may have been equal (or almost equal) to τ_z,max (as illustrated in Figure A4 for example). Furthermore, F_e sometimes failed to stabilise as τ_z,max increased, regardless of how high τ_z,max was set to (see Appendix C for further discussion). Therefore, before a final constant F_e result could be produced, the flux evaluation procedure had to be repeated at various values of τ_z,max. A constant value of F_e was assumed to be produced once incremental changes in τ_z,max no longer resulted in changes in F_e. Figure 1c gives an example of final constant modelled q_mo values for flight 2 from the sample data. Final τ_z and F_e values were then substituted into Equations (9) and (10) to establish definite values of y_c and τ_y. All resulting y_c, τ_y and τ_z values for the sample data are given in Table S4.

Once constant values of τ_z and F_e were acquired for each flight, a measurement uncertainty in F_e (ΔF_m) could be calculated through Equation (11), by using each “j” value of q_me and Δq_me from Equation (3), assuming each Δq_me term is uncorrelated.

$${\mathsf{\Delta }F}_m{=F}_e\sqrt{\frac{{{\displaystyle \sum }}_j\left({\mathsf{\Delta }q}_{me}{{}_j}^2\right)}{{{\displaystyle \sum }}_j\left(q_{me}{{}_j}^2\right)}}.$$

(11)

A total minimum uncertainty (ΔF₋) was then calculated using Equation (12) from residuals between each “j” calculation of q_me and q_mo (see Figure 1d for flight 2, for example). Equation (12) finds the average q_me residual, weighted by q_me.

$${\mathsf{\Delta }F}_{-}{=-F}_e\left|\sqrt{\frac{{{\displaystyle \sum }}_j\left({\left(q_{me}{}_j{-q}_{mo}{}_j\right)}^2\right)}{{{\displaystyle \sum }}_j\left(q_{me}{{}_j}^2\right)}}\right|.$$

(12)

The residuals between q_me and q_mo in Equation (12) incorporate uncertainty in wind direction. Under stable wind conditions, at a constant wind direction, and with infinite spatial sampling, the residual ($q_{me}{}_j{-q}_{mo}{}_j$) term in Equation (12) would converge to zero. However, under turbulent (or variable) wind conditions, regardless of sampling spatial extent and sampling duration on the flux plane, the residual ($q_{me}{}_j{-q}_{mo}{}_j$) term may be non-zero due to the potential for overlap of various q_me measurements at the same point in space at different times. Furthermore, the limited extent of spatial sampling on the flux plane can lead to a flux under-estimate, which must be incorporated into the total maximum uncertainty (ΔF₊). This is described in Section 3.

3. Flux Sensitivity Random Walk Simulations

The capability of the NGI method to derive a prescribed modelled point source target flux (F_t) was tested here by simulating randomised near-field sampling of a time-averaged plume. As well as evaluating ΔF₊, these simulations can be used for guidance to better plan future sampling, to yield fluxes within some threshold of acceptable uncertainty in operational settings. The simulations were designed to replicate real sampling in the y, z, and x directions (see Figure S6 for an example). Each simulation was based on parameters retrieved when calculating F_e from the NGI model (y_c, h, τ_y and τ_z), the orientation of the q_me sampling plane in the x direction and the spatial distribution of q_me sampling. Each simulation was constrained by the maximum bounds of q_me sampling.

Each simulation was initialised from a random y and z position on the sampling plane (point 1 in Figure 2, where Figure 2 is a schematic diagram of a random walk). The distance of each step in the random walk was constant (on a plane perpendicular to x), with a length (L) equal to the average distance between successive real measurement points. In order to replicate changes in sampling direction on the plane perpendicular to x, an exponential probability distribution was fitted to the absolute change in angle between each successive measurement step (see insert in Figure 2). This probability distribution was used to model the change in angle (in magnitude but not in sign) between each successive step in the random walk (δθ), from an initial random direction (θ_R). At the sampling plane boundary, the random walk direction was deflected by an angle of either 135° or 225°, provided that the random walk remained within the spatial confines of sampling. The orientation of the simulated sampling plane in the x and y direction, relative to mean wind direction, was chosen to match the orientation of the sampling plane from real sampling (see supplement for details). Simulated q_me values were then generated for each step in the random walk using Equation (7), where F_e was replaced by F_t and q_mo was replaced by q_me.

For each test described in the following sections, multiple independent random walks (and simulated q_me values) were generated from scratch, for either a single fixed simulated sampling time (t) or for various values of t. For each random walk in each test, t was adjusted by altering the total number of steps in the random walk. Each individual random walk sampled the flux plane following a different route, making each random walk unique. A virtual flux estimate (F_e,v) was calculated from each random walk (which is a function of t), following the NGI method. As modelled fluxes can fail to stabilise when τ_z,max is equal to model output value of τ_z, five conditions were put in place to ensure that the model had stabilised. These conditions are described in detail in Appendix B.

3.1. Upper Flux Uncertainty Bounds

The difference between F_t and simulated flux (F_e,v) was assessed at a fixed simulated sampling time corresponding to the duration of actual sampling for each flight in the sample data. Fluxes for this analysis were derived by continuously repeating a random walk simulation to accumulate a large number (180 for the sample data) of virtual fluxes (see green lines in Figure 3a for example). The average value of F_e,v ($\overline{F_{e,v}}$, solid green line in Figure 3a) was negatively biased from F_t due to three independent effects.

• The simulated sampling extent was restricted as a consequence of the managed sampling strategy, resulting in an inadequate characterisation of the entire emission plume.
• Sampling was physically restricted in the z direction due to the non-zero height of the air inlet (as was the case for our UAV platform), resulting in an under-sampled area very close to the ground.
• The random walk was time-limited, resulting in an incomplete exploration of the available flux plane and hence, sampling gaps, leading to a residual negative flux bias.

These effects collectively result in a lack of spatial sampling across the flux plane and thus insufficient sampling with which to calculate τ_y and τ_z, which are derived here from measurements of q_me across the sampling plane. In theory, these effects can be alleviated with infinite spatial sampling on an infinitely sized sampling plane, but this cannot realistically be achieved from a UAV (or any) sampling platform. These three effects were incorporated into ΔF₊ by adding the deviation of F_e,v from F_t to ΔF₋, using Equation (13).

$${\mathsf{\Delta }F}_{+}=+\left(\left|{\mathsf{\Delta }F}_{-}\right|+\left(\frac{F_e}{F_t}\left(F_t-\overline{F_{e,v}}\right)\right)\right).$$

(13)

Thus, Figure 3a shows that as a consequence of the three effects outlined above, the NGI method can result in a large variation in flux estimation with the potential for a large negative flux bias.

3.2. Uncertainty Sampling Thresholds

By gradually increasing t in each simulation, the convergence of F_e,v towards F_t was assessed (blue crosses in Figure 3b, for example). This makes it possible to analyse the limiting uncertainty that exists in F_e,v as a function of t, due to the random error associated with the NGI method itself rather than due to the quality of sampling. This test is only valid for relatively non-variable atmospheric conditions, as it assumes the time-averaged plume to be static. For each flight from the sample data, independent random walks of up to four hours were generated. A simulated scale factor as a function of t was applied to these fluxes to correct for the inherent negative bias that occurs as a function of t (see previous section). This scale factor was derived by repeatedly performing random walks at a range of fixed sampling times to quantify the average negative flux bias as a function of t (see supplement for details). Without applying this scale factor, F_e,v gradually increased towards F_t, as a function of t (see Figure S7 for example). Following the application of this scale factor, the absolute difference between bias-corrected F_e,v and bias-corrected $\overline{F_{e,v}}$ was used to find the flux anomaly (A) as a function of t (red diamonds in Figure 3b, for example). The decrease of A with increasing t represents a reduction in the fundamental methodological uncertainty due to a greater understanding of the time-averaged plume morphology. The flux anomalies in Figure 3b show that with a sufficient sampling duration, the fundamental methodological uncertainty can gradually be reduced until it converges towards zero.

The flux anomaly was assumed to decrease exponentially with t, from which a fit could be produced using Equation (14) to define an anomaly coefficient, α, and an anomaly decay constant, β (see Table S5 for values for the sample data). This fit was weighted to take into account fewer simulations at longer t, such that the fit was evenly applied as a function of t.

$${A=F}_t\alpha \exp \left(\frac{-t}{\beta }\right).$$

(14)

By setting t in Equation (14) to the duration of each UAV flight in the sample data and by setting F_t to F_e, the limiting uncertainty (A_f) associated with each flight could be derived. A_f represents the inherent limiting uncertainty, determined by the sampling strategy (assuming non-variable atmospheric conditions), which decreases as a function of t. Alternatively, Equation (14) could be rearranged to make t the subject. Then, by setting A to a specified target fraction of F_t, the minimum time required to surpass a target limiting uncertainty threshold could then be derived. This may be useful to decide future UAV sampling durations within a desired minimum confidence level. For example, setting A to 1% of F_t yielded the 1% fundamental uncertainty sampling threshold (t_1%). The use of limiting confidence timescales is only valid if non-variable atmospheric conditions are assumed. Therefore, the threshold time must be comparably smaller than the timescale of prevailing meteorological change.

3.3. Testing the NGI Method

Finally, virtual flux simulations were used to test the NGI method in principle. First, the flux plane was extended to mitigate biases due to the limited size of the sampling plane. The maximum sampling extent in the z direction was doubled, and the left and right x bounds were also doubled (resulting in an increase in the size of the original sampling plane by a factor of approximately 6). Furthermore, the flux plane was extended down to ground level. The total simulated sampling time (t) was then set to an arbitrary (approximately 72 h) sampling time across the extended flux plane to mitigate the effects of limited spatial sampling coverage, as well as to minimise the limiting uncertainty discussed in the previous section. The simulation was then repeated 60 times for each flux. Average F_e,v results deviated from F_t by (−0.7 ± 0.5)%, (see Table S6 for results using the sample data). This test validates the utility of the NGI method by confirming that it accurately yields F_t, in principle.

4. Results and Future Guidance

The sample data used to develop the NGI flux quantification method described here was acquired using the controlled release of natural gas (supervised by the NPL) at two flux rates ((1.5 ± 0.1) g s⁻¹ and (3.0 ± 0.2) g s⁻¹). The NGI fluxes for each of eight UAV flights are displayed in Figure 4, along with flux uncertainty bounds, expressed as a percentage of the NPL controlled release facility (CRF) reported emission flux (see Table S1 for individual flux values for each flight). All values of F_e, ΔF₊, and ΔF₋ are given in Table S7. As this was a preliminary study, regardless of uncertainty, we note that most NGI fluxes calculated for the sample data exceeded CRF fluxes due to the biased nature of our sampling. Though potential sources of negative bias due to sampling limitations are incorporated into the NGI flux uncertainty range, the effects of positive bias are clearly dominant here. One such source of positive bias is the targeted sampling strategy described in Section 2.1, resulting in plume chasing. The NGI method inherently assumes that the measurements constitute an unbiased sample of the time-averaged Gaussian plume. However, during the acquisition of the sample data, the UAV was directed towards the perceived position of the instantaneous plume to maximise in-plume sampling. Future studies adopting this technique should avoid chasing the plume in this way, as it results in the over-representation of in-plume measurements, which biases NGI fluxes. Another possible source of bias may be wind measurements; if the winds at the location of the anemometers were systematically different from those at the UAV sampling locations, this would have a direct impact on the calculated fluxes.

The flux biases in this work can be compared to a previous study by Brantley et al. [35], which quantified the controlled release of methane using Gaussian plume modelling of stationary measurements; their method both overestimated and underestimated fluxes by between −87% and +184%, respectively. The wide variation in fluxes acquired by Brantley et al. [35] highlights issues similar to those experienced in other studies using near-field concentration measurements in turbulent (non-ideal) conditions with time-limited sampling, i.e. that sampling a non-disperse emission plume can result in large flux uncertainty.

Though seven out of eight F_e values were above the CRF flux due to the biased sampling, all reported CRF fluxes fell within the full UAV flux uncertainty range, which spanned between (−76 ± 5)% and (+132 ± 15)% of F_e. Flight 6 had a particularly high NGI flux uncertainty range due to a high quantity of sampling at the edge of the plume (see Figure A3), rather than the centre. Failure to adequately sample the centre of the plume can result in a much larger false plume being extrapolated, using the NGI method, in certain cases. It is clear from Figure 4 that the ΔF_m uncertainty range was dwarfed by the extended uncertainty range, which was higher in all cases due to the dominating effect of the uncertainty in wind direction. To assess the various sources of UAV sampling uncertainty in ΔF_m, a sensitivity test was performed (see Table S8). The principal source of uncertainty in ΔF_m was from uncertainty in wind speed (see Table S8). An improved understanding of the temporal variability of the two-dimensional wind field on the flux plane, either by on-board UAV wind measurement or by other means, would serve to reduce this dominant source of uncertainty. However, the combined effects of instrumental uncertainty and uncertainty in the background were small compared with effects associated with natural variability in the wind field, which suggests that instruments such as the UGGA are suitable for measurement of fluxes similar to those in our study at similar downwind distances.

We compared the UAV flux uncertainties from the sample data to other methods. Seven of the eight TDM flux estimates which were acquired simultaneously during this study (see Table S1) agreed with the CRF, within uncertainty, with an average bias of (−2 ± 10)%. The average uncertainty in individual TDM fluxes (through comprehensive uncertainty budgeting) was ±21% [75]. In a different study using mass balance box modelling, Krautwurst et al. [42] derived the emission flux from a single landfill site based on both remote sensing column-concentration aircraft measurements and in-situ aircraft concentration measurements, with average flux uncertainties of ±34.5% and ±19.7%, respectively. Elsewhere, the uncertainty in fluxes obtained by Nathan et al. [73], using mass balance box modelling of polar coordinates and kriging to quantify the emission flux from a natural gas compressor station, was ±55%, although we dispute the use of kriging as an appropriate tool in that case. Yacovitch et al. [37] used a Gaussian plume inversion of ground based measurements to quantify emission fluxes from oil and gas infrastructure facilities, which used σ_y(x) and σ_z(x) values based on stability classes (rather than derived from q_me), with asymmetric 95% confidence uncertainties of between −33.4% and +334%. Therefore, we conclude that the NGI method using UAV sampling can be a competitive tool for rapid flux assessments of emitter facilities, assuming sampling from point source emitters. While the uncertainties we derived are comparatively large, the NGI method is able to derive rapid flux estimates from time-limited UAV sampling.

Though we report large NGI flux uncertainties for the sample data, alternative techniques such as mass balance box modelling cannot be used reliably to quantify fluxes using off-site downwind UAV sampling. To explain these conclusions, we have assessed the suitability of two mass balance box modelling approaches here. Mass balancing using geospatial kriging cannot be used, as kriging requires the interpolation of time-invariant downwind sampling. In this work, the q_me plume sampled by a UAV near-to-source was not time invariant due to turbulent fluctuations in the position of the plume on the sampling plane. Though kriging may be used in principle in stable (non-turbulent) wind conditions, stable winds are unlikely when sampling with a UAV near-to-source. Alternatively, mass balancing may be applied by averaging q_me into equally sized grid squares on a plane perpendicular to mean wind direction. However, this requires the entire extent of the time-averaged plume to be captured by the UAV, down to ground level. It also requires dense spatial sampling without any large sampling voids. While the method of grid square averaging may again be possible in principle, in order to produce sufficiently dense spatial sampling, a longer UAV sampling duration would be required compared to the sampling presented here. Furthermore, sampling down to ground level can be problematic, as it relies on the position of the air inlet on a UAV being close to the base of the UAV, and effects or prop-wash may be expected. Thus, we suggest that neither mass balance box modelling approach can be used for the type of near-field UAV sampling presented here—rather, the NGI method is a suitable flux quantification technique for near-field UAV sampling.

Future UAV flight sampling should be planned to minimise flux uncertainties, including potential sources of bias. The random walk simulations were used to generate values of A_f for each flight from the sample data, which are given in

Table 1. A_f as a fraction of F_e and t_0.01 for each unmanned aerial vehicle (UAV) flight in the sample data.

UAV Flight	Af/Fe	t0.01 (Hours)
1	(57 ± 3)%	14.0 ± 1.8
2	(22 ± 2)%	5.1 ± 0.5
3	(41 ± 3)%	5.8 ± 0.5
4	(26 ± 2)%	5.0 ± 0.5
5	(71 ± 4)%	6.8 ± 0.5
6	(123 ± 41)%	7.3 ± 2.8
7	(68 ± 4)%	12.7 ± 1.2
8	(53 ± 4)%	8.2 ± 0.8

. They show that regardless of measurement uncertainty, wind variations and the spatial extent of sampling, an average (58 ± 30)% limiting threshold uncertainty remains due to limited spatial coverage on the sampling plane. The random walk simulation could also be applied to obtain t_0.01 required to surpass a 1% uncertainty threshold, assuming good randomised sampling of a static plume. It is clear that the theoretical t_0.01 values in Table 1 (of at least 5 h) are unrealistic for UAV sampling, which cannot practically be achieved by a single light UAV platform over the timescale of a single flight (typically < 30 min using current battery technology constraints). However, using multiple UAV flights or tethered power (for uninterrupted sampling) may increase sampling time. The t_0.01 values are only valid when sampling in stable atmospheric conditions with a relatively non-variable wind direction. In reality, over the t_0.01 times given in Table 1, non-variable atmospheric conditions are unlikely, leading to the potential for large NGI uncertainties from UAV sampling. Therefore, if additional UAV sampling is used, a compromise sampling time must be reached over which atmospheric conditions change very little while the sampling period is maximised to minimise uncertainty.

5. Conclusions

A near-field Gaussian plume inversion was used to quantify methane emission fluxes from eight UAV flights downwind of a controlled emission source. The new NGI method takes into account the imperfect orientation of a two-dimensional vertical sampling plane perpendicular to mean wind direction. The number of unknown parameters in the inversion was reduced by deriving constraining parameters in the y direction from flux density measurements. Uncertainties were derived from measurement uncertainty as well as the difference between measured and modelled flux density results. The upper flux uncertainty bound was further enhanced by taking into account the lack of spatial sampling extent, using a random walk simulation of sampling. The random walk sensitivity tests also highlight areas of future improvements to the sampling strategy, which will be a focus of future experimental work. They also highlight the importance of evenly-distributed sampling of an adequate duration and the need for a sufficiently large sampling plane.

We suggest that the NGI method is a highly suitable technique that can be used to accurately quantify emission fluxes from static point sources of both methane and other trace gas sources, with measurement potential by UAVs. This allows for the full potential of near-field in-situ spatial sampling that UAVs provide to be realised. Though we acknowledge the magnitude of uncertainty may be high over a single UAV flight, the NGI method allows for rapid flux estimation with a defined maximal uncertainty. Mass balance box modelling may not be suitable for UAV in-situ sampling as it requires either the sampling of a time invariant plume or dense spatial sampling of a time-averaged plume. Though the NGI method yielded highly successful results when using a random walk simulation, the preliminary fluxes from the sample data resulted in high uncertainties. It is suggested that future studies that utilise the NGI method should adopt a sampling strategy that addresses the potential sources of bias highlighted here by ensuring that plume-chasing is avoided and by ensuring that sampling is sufficient to capture the entire extent of the time-averaged plume.

With the possibility of in-situ UAV wind measurements in future, there is further scope to improve flux accuracy. Future progress may be made by mounting an in-situ methane sensor directly onto a UAV, to eliminate issues associated with the use of a tether. Such improvements and further near-field sampling studies using the NGI method can play an important role in efforts to improve emission inventories. Additionally, UAVs may be used to add a vertical sampling dimension to the TDM technique using in-situ tracer gas detection. To conclude, a Gaussian plume inversion flux quantification method, using in-situ [CH₄] measurements of a sub-ppm precision, has been characterised here and future improvements are viable. This permits future studies of unknown well-defined sources (such as landfill sites, oil and gas infrastructure facilities, or herds of cattle) to be conducted using the method, depending on the precision of instrumentation.