Methods for Spatial Extrapolation of Methane Measurements in Constructing Regional Estimates from Sample Populations

Abstract

Methane emission estimates for oil and gas facilities are typically based on estimates at a subpopulation of facilities, and these emission estimates are then extrapolated to a larger region or basin. Basin-level emission estimates are then frequently compared with basin-level observations. Methane emissions from oil and gas systems are inherently variable and intermittent, which make it difficult to determine whether a sample population is sufficiently large to be representative of a larger region. This work develops a framework for extrapolation of emission estimates using the case study of an operator in the Green River Basin. This work also identifies a new metric, the capture ratio, which quantifies the extent to which sources are represented in the sample population, based on the skewness of emissions for each source. There is a strong correlation between the capture ratio and extrapolation error, which suggests that understanding source-level emissions distributions can mitigate error when sample populations are selected and extrapolating measurements. The framework and results from this work can inform the selection and extrapolation of site measurements when developing methane emission inventories and establishing uncertainty bounds to assess whether inventory estimates are consistent with independent large spatial-scale observations.

Short abstract

Uncertainties in extrapolating measurement-informed methane emission estimates from samplings of oil and gas sites to larger regions are estimated.

Introduction

Methane emissions from global petroleum and natural gas systems are estimated to be approximately 100 Tg/year,¹ equivalent over a 20-year period to the warming potential of 8 gigatons/year of carbon dioxide emissions. Methane emissions from US petroleum and natural gas facilities are currently reported in the Environmental Protection Agency’s (EPA) Greenhouse Gas Inventory (GHGI) and the Greenhouse Gas Reporting Program (GHGRP), in which reported emissions are generally derived from source-based emission estimates that rely on emission factors and equipment counts. Numerous studies find that measurement-based emission estimates are 1.5 to 4 times higher than GHGRP estimates,²⁻⁵ although the degree of under-reporting varies by region, and some studies have found that inventory estimation methods may be overestimating emissions in a limited number of cases.⁶

Recent actions from international, federal, and state regulatory agencies and voluntary initiatives are mandating a more extensive incorporation of emission measurements into emission inventories. At the international level, the United Nations’ Oil and Gas Methane Partnership 2.0 (OGMP 2.0) requires that members be actively working toward building an inventory that includes measurement and measurement reconciliation.⁷ Within the United States, the Environmental Protection Agency (EPA) proposed changes to current emissions reporting approaches through Subpart W and the GHGRP that incorporate a greater focus on emission measurements. The EPA has also proposed rules that would allow third-party agencies to conduct measurements of emissions from oil and gas sites.⁸ State-level agencies, such as Colorado’s Air Pollution Control Division, are proposing new rules that incorporate methane measurements into methane accounting and intensity calculations.⁹ The recent surge in voluntary and regulatory action represents a major and ongoing shift in the way methane emissions from oil and gas production are calculated, reported, and mitigated.

Methane measurements can occur across multiple spatial scales. Bottom-up inventories, such as a Level 4 inventory, as defined by OGMP 2.0 guidance, rely on source-level measurements. Bottom-up inventories are important because they identify the most material sources and can be used to guide mitigation strategies. Measurements can also occur at the site or multisite level, capturing a collection of sources within a single measurement. Certain frameworks, such as OGMP 2.0, require reconciliation between source-level measurements and independent site-level or multisite-level measurements, in order to validate or revise source-level inventories. Other frameworks may only require site-level or multisite-level measurements, but an understanding of emissions at the source level is necessary to optimize mitigation approaches.

Methane emissions are inherently variable across time and space and often follow highly skewed distributions.¹⁰⁻¹⁵ Independent methane measurements over wider areas such as a production basin also have a high degree of uncertainty due to differences in detection thresholds and limitations in spatial and temporal coverage. When comparing source-level emission inventories to independent observations, there is a need for uncertainty quantification in order to contextualize differences between inventories and measurements.¹³ The variability of emissions ensures that different measurement methods will never agree exactly, and uncertainty quantification can be used to design measurement campaigns to gather statistically significant data samples and guide measurement interpretation.

Uncertainty quantification has been conducted on emerging measurement technologies in the form of blind-release testing.¹⁶⁻²² These tests typically assess how well a measurement technology quantifies an instantaneous emission rate during a release test. These tests and uncertainty quantifications capture the error associated with the quantification of methane emissions, which can range from 18% to more than an order of magnitude.¹⁸

Uncertainty quantification in the context of emission inventories must go beyond instantaneous emission rate quantification uncertainty, as the application of these measurements to sites beyond where the measurements were made can also introduce significant sources of uncertainty. Measurements of emissions often have limited spatial and temporal coverage, which are then scaled to different spatial and temporal resolutions for comparison to other measurements. For example, emissions from millions of pneumatic controllers in oil and gas production facilities in the United States have been based on results from a few studies that collectively measured emissions from hundreds of these devices, with the measurement on each device lasting for only a few minutes. In the construction of a regional annual inventory, these limited measurements must be extrapolated to all devices at all sites for a year. When these extrapolated inventory estimates are compared to observations, operators must make assumptions about how spatially and temporally representative these measurements are. This extrapolation introduces significant sources of error that must be considered when comparing emission inventories to regional observations.

Temporal extrapolation, defined as the extrapolation of measurements with limited temporal coverage, can be a significant source of uncertainty, as reported by Schissel and Allen through a case study of the Barnett Shale in north central Texas.¹⁵ They found that average uncertainties due to the temporal extrapolation of short-duration measurements ranged from ∼10% to more than 50%, depending on the duration of emission events and the frequency of sampling, which is similar in magnitude to the quantification uncertainty in some measurement technologies.¹⁶⁻²² Extrapolation across space must also be considered, as the development of measurement-based inventories typically involves measuring a sample population of sites or sources and extrapolating the results from this sample population to a larger region and longer time scale. If only a subset of facilities is measured, it is important to understand how many and what types of sites are required to be in the sample population to effectively capture the complex and skewed behavior of methane emissions in a given region.

This work demonstrates an approach to quantify the uncertainty associated with spatial extrapolation from sample population measurements to the full population of sites, as a function of sample size and extrapolation method. A large set of natural gas production sites operated in the Green River Basin, WY, is used in the demonstration. While the results of the case study are specific to the Green River Basin facilities, the general approach has broad applications and can inform policy and industry decisions on measurement campaign design and calculation techniques when developing measurement-informed methane inventories for variable emission sources, such as from oil and gas production.

Methods

This work is divided into four phases of analysis (Figure 1). The first phase of analysis is emission simulations. In this step, stochastic simulations of emissions are conducted based on the input source-level data to generate representative emission time series with spatial information. Source-level data is classified in Table 2. Ten emissions simulations, each involving individual time series for each site in the region, are conducted to capture the variability in emission behavior within the full population. The second phase of analysis is the sampling simulation. For each emission simulation, 50 random samples are selected for each specified sample size. The sample sizes are 13 sites (∼1% of the entire population), 65 sites (∼5% of all sites), 230 sites (∼18% of all sites), and 650 sites (∼50% of all sites). For example, for a single emission time series, there would be 50 iterations of a 65-site sample. For each iteration, the 65 sites selected were varied randomly in order to capture the variability in site selection during random sampling. In the third phase of analysis, emissions from the sample population are extrapolated to the entire population. Source-based extrapolation and simple linear extrapolation are both assessed. Estimates of emissions for each sample population are compared with the true full population emissions, defined as the sum of emissions across all sites for each emission simulation. In the final stage of the analysis, a new variable, the capture ratio, was identified to define a relationship between expected uncertainty and sample population characteristics. More details on the final stage of analysis can be found in the Results and Discussion section.

Figure 1.

Schematic of analyses.

Table 2. Source Classifications and Data Source Descriptions for All Sources Considered in This Work.

emission source	source type	data source	data source classification	emission rate range per single source (kg/h)	duration data source	average event duration	persistence data source
engines/stationary combustion devices	continuous	stack testing	DM	2.1 × 10–4 to 6.7	N/A	N/A	N/A
compressors	continuous	GHGRP subpart W	GE	0.02	N/A	N/A	N/A
pneumatic pumps	continuous	GHGRP subpart W	GE	0.005a	N/A	N/A	N/A
dehydrators	continuous	engineering calculations	SE	0.01–0.37a	N/A	N/A	N/A
storage tanks	continuous	engineering calculations	SE	0.52–12.5a	N/A	N/A	N/A
unloadings	intermittent	engineering calculations	SE	8.8 × 10–4 kg/min, 82.5 kg/min	field data	8.6 min	field data
leaks/routine malfunction	intermittent	hi-flow measurement during LDAR survey	DM	1.2 × 10–3 kg/h, 115.2 kg/h	field data	26.7 days	field data
intermittent pneumatic controllers	intermittent	Allen et al.23	SE	7 × 10–3 kg/event	field data	1 min	field data

^aNote: These values correspond to 2% of the estimated emission rate because emissions from these sources are routed to an on-site flare, where a 98% combustion rate is assumed.

Case Study Region

This work focuses on an oil and natural gas production asset in the Green River Basin in Wyoming. This asset consists of 1254 simple well pad sites (containing multiple well pads) and 128 central delivery points (CDPs). CDPs are more complex sites where production fluids are routed for processing. Well pad sites are limited to more simple equipment configurations and wellhead equipment, with a small number (<10) of well pads containing more complex equipment. The configuration of CDPs ranges in complexity, but larger equipment such as tanks, dehydrators, engines, and compressors are found on these types of sites. The equipment considered in this work are well heads, pneumatic pumps, pneumatic controllers, engines and stationary combustion devices, compressors, dehydrators, and tanks. Liquid unloading emissions are also considered at gas well pads, and emissions due to leaks are considered across all equipment types. The operator provided equipment counts and detailed site configurations for all sites in the basin. Average equipment types disaggregated by site type are shown in Table 1.

Table 1. Average Count of Equipment Types for Well Pad Sites and Central Delivery Point Sites.

	well pad	central delivery point
pneumatic controllers	0.1	10.5
pneumatic pumps	0.01	1.1
wellhead	1.7	1.8
engine	0.01	0.7
compressors	0	0.4
dehydrators	0	1
tanks	0	5.5

This region was selected because of extensive source-level measurement campaigns and the availability of data. Reconciliation between bottom-up and top-down estimates is difficult because bottom-up inventories typically fail to capture the skewed distribution of emissions that arise due to unknown leaks or equipment malfunctions.¹¹⁻¹⁵ The operator in this case study revisits every site at least once per month, meaning that all leak and long-duration malfunction emissions at all sites are captured in the emission inventory. This allows the results of the analysis to be more meaningful, as they are representative of regions with skewed emissions distributions, which are commonly found in oil and gas operations, yet coverage is for all sites. The distribution of emission rates in the source-level input data varies over more than 3 orders of magnitude, as described in the Supporting Information.

Stochastic Emissions Simulation

An emission model was developed to simulate a site-by-site time series of methane emissions based on the best available data for each source (Table 2). The time step and duration for all simulations in this work were set to 1 h and 10 days, respectively. The operator provided detailed data on all sites in the basin, consisting equipment counts and, when available, source-level emission measurements for all 1254 simple well pad sites and 128 CDPs. As is typical in the industry, the type of source-level emissions data that is available varies among sources. Three distinct forms of emission data were identified in this case study: (1) emission estimates based on activity counts and generic emission factors, (2) emission estimates derived from equipment-specific emission factors, and (3) emission estimates based on direct measurement.

• Generic emission factor (GE): Emission rates are calculated by multiplying equipment or source count by a nationally averaged or general emission factor, such as provided by the GHGRP or other large-scale studies.

• Equipment-specific emission factor (SE): Emission rates are calculated by multiplying each equipment or source count by a specific emission factor, that incorporates some form of measurement that is specific to that equipment type, such as using engineering software to predict emissions based on equipment dimensions and mass-balance approaches.

• Direct methane measurement (DM): Emission rates are measured directly at the source.

These three tiers of data are parallel to the levels of reporting outlined in the United Nations’ Oil and Gas Methane Partnership (OGMP) 2.0 protocol, where Level 3 estimates are derived from average emission factors by source, and Level 4 estimates are derived from source-specific quantification techniques, based on individual facility design data, through either computational efforts or direct measurement. The tier of data per source used in this work is based on the best available information for this case study. More than 90% of emissions are simulated using Level 4 data, which is considered the best available form of data for source-level reporting through OGMP 2.0.⁷

Beyond the data tier classification, emissions are also characterized as intermittent or continuous. Intermittent emissions are characterized as emissions with distinct start and stop times. Continuous emissions are defined as emissions that persist for the entire simulation duration. Whether an emission is modeled as continuous or intermittent may be limited by the estimation approach. For example, pneumatic controller emissions may be known to be intermittent by the operator. However, if the best estimation approach for pneumatic controller emissions is a national average emission factor, which assumes constant emissions, then pneumatic controller emissions would be classified as a “continuous emission event”, consistent with the assumptions used in inventory reporting.

Source type classifications and data tier classifications used in this work are summarized in Table 2. A range of emission rates is provided for sources with variable emission rates. Equipment-specific emission ranges (engines, dehydrators, storage tanks, and unloadings) are representative of differences in equipment operation and emissions from site to site. Measured emission source ranges (unloadings and leaks) are derived from the ranges in observations. The duration is the length of time that a single event emits (e.g., the duration of a single unloading event). The persistence is the number of times or the length of time that an emission is active over the course of a year (e.g., number of unloadings in a year).

A complete inventory of all sources at each site was developed to simulate emissions for the entire population of sites. Time series of emissions at individual sites were developed at a 1 h resolution. Continuous emission sources began at the first time step of the simulation and persisted until the last. Emission rates for continuous emission sources were either randomly selected from the database of population-wide emission rates or were selected based on equipment ID if the emission data were specific to certain pieces of equipment. Intermittent emission events were simulated based on the following approaches.

Unloadings

Emissions from liquid unloadings occur at large rates over short times, on the order of minutes, For each site in the simulation, data from all unloadings that occurred over the course of a year at that site were used to estimate the probability of an unloading event or multiple unloading events occurring over the course of the simulation. The probability of an unloading occurring during a specific day, P_{unloading_day} was calculated for site i using eq 1.

1

For automated unloadings, the start times of unloading events are distributed throughout a day. The probability (p) of an unloading occurring on a given day and starting at minute m is therefore given by eq 2.

2

A binomial distribution was then used to calculate the number of unloadings that would occur during the simulation duration. The binomial distribution is constructed using the np.random.binomial package in python.²⁴ The output of this tool is a list representing a distribution of possible outcomes from a set number of Bernoulli trials, n, each with k number of successes, a success rate of p, and a failure rate of (1 – p). The probability mass function for the binomial is as follows:

3

The probability of a “success”, or an unloading starting during a given minute for a specific site, is given in eq 3. An unloading can begin anytime during the simulation or, if t = the duration of the unloading, up until t – 1 min before the simulation period. Therefore, the number of chances in the binomial distribution is given as n = (simulation duration (min) + t – 1). From this binomial output, a random number is selected from the distribution, representing the number of unloadings that will occur during the simulation at that specific site. For each unloading selected to occur, a start time ranging from (t – 1, simulation duration) is randomly selected, and the emission rate and duration are selected based on the unloadings database for that site. The unloading persists for the appropriate number of time steps or until the simulation is over if the (start time + duration) > (length of simulation). For example, if an unloading lasts 10 min, it would appear for 10 consecutive time steps at a minute duration and only one time step during an hourly or day resolution. The emission rate for the unloading is also scaled by temporal resolution according to the following formula:

4

This process was repeated for each site with unloading in the simulation. For the simulations in this work, the time step was defined as 1 h and the simulation duration was set to 10 days.

Intermittent Pneumatic Controllers

All pneumatic controllers are assumed to have the same emission rate when operating (Table 3) and the total time in operation per device is assumed to be 7200 min (1.4% of the time) based on quantitative field observations provided by the operator. The duration of these events was assumed as 1 min based on qualitative field observation. The emission rate while emitting was derived from observed emissions from properly functioning pneumatic controllers in the Rocky Mountain region.¹⁷ This emission rate was calculated by estimating the average emission per actuation for controllers with observed actuations (∼0.3 scf/actuation). Assuming an emission event lasts 1 min, and only one actuation occurs per hour, this yields an emission rate of 0.007 kg/h for the simulation hour in which the actuation occurs. If more than one actuation occurs per hour, then this emission rate is scaled linearly to reflect the increase in emissions.

Table 3. Average Absolute Percent Error as a Function of the Sample Size and Extrapolation Error^a,b.

sample size	absolute average error in linear extrapolation (bias)	absolute average error in source-based extrapolation (bias)
13 sites	90% (−28%)	78% (−76%)
65 sites	62% (−2%)	50% (−3%)
230 sites	29% (−9%)	21% (0%)
650 sites	13% (−4%)	8% (3%)

^aBias is shown in parentheses. Results below are for emission simulation A; the other nine simulation results are available in the Supporting Information.

^bNote: Absolute average percent error was calculated by taking the average across all sampling simulations of the absolute value of E_MC,S (eq 7). The bias was calculated by taking the average across all Monte Carlo simulations of E_MC,S. A negative bias indicates underestimation, and a positive bias indicates overestimation.

The number of emitting time periods per pneumatic controller per year is given by the operator as 7200 min. 7200 min were randomly selected from the possible 525,600 min in a year. If the emitting minutes intersected with the simulation time frame, then the emissions from the pneumatic device are included in the simulation.

Leaks

For this case study, leak information is gathered through Leak Detection and Repair (LDAR) surveys, where operators visit all sites to survey for leaks and measure leaks if found. Operators revisit sites monthly on average. Information on the leak source is also collected by operators, and leaks are disaggregated into 11 types of leaks based on the equipment on which they occur: combustors, compressors, dehydrators, flares, heaters, pneumatics, separators, tanks, unit components, and well heads. For each selected facility, the number of each type of leak over the course of a year is calculated using eq 5.

5

where X is the selected facility and Y is the equipment of interest, such as a compressor. The number of annual leaks per year per equipment are randomly assigned start times from [−(t – 1), 365 days], where t represents the average leak duration for leak type Y. If the start time falls into the range [−(t – 1), simulation duration], then the leak is included in the simulation. For leaks that are included in the simulation, an emission rate and duration are randomly selected from the distribution of emission rates for that particular type of leak. The emission rate and duration are appropriately scaled to the time step of interest, using the same methodology as eq 5. This process is iterated over all possible leak types and all selected facilities.

Emissions Simulations

Each emission simulation includes a random selection of emission rates, locations, duration, and start times from the input data for each source. Different emission simulations will therefore have different underlying behaviors, even at the aggregate (full population) level. To account for this variability in the emissions themselves, 10 stochastic simulations for all sites at a 1 h resolution for 10 days were simulated.

Sampling Simulations

Each sampling strategy (ie. 65 sites, 230 sites, etc.) involves random selection of sample population sites of a prescribed size. There is inherent variability in sample population selection as the sample size may include different types of sites in different quantities. To account for the variability that arises during sample population selection, 50 samples per emission simulation per sample size were selected and analyzed.

Extrapolation Analysis

Construction of a basin-wide inventory from measurements across a sample population requires spatial extrapolation. In this work, spatial extrapolation is defined as taking emission information for a randomly selected subset of sites and scaling this work to apply to all 1254 simple well pad sites and 128 CDPs. Two types of spatial extrapolation are explored: simple linear extrapolation and source-based extrapolation. The extrapolated emissions estimates for the entire population are compared against summation of all emissions in the full population simulation, termed the true total value.

Simple Linear Extrapolation

Simple linear extrapolation is the most straightforward extrapolation technique. The only a priori knowledge required is the total number of sites in the basin and the number of sites in the sample population, with no classification between sites. Simple well pad sites and CDPs are not segregated.

6

where TE_Total,S is the extrapolated emissions for the whole population based on population size S, SE_S is the summed emissions for the sample population of size S, N_T is the total number of sites in the region, and N_s is the total number of sites in the sample population. The extrapolation error for emission simulation and sample selection is calculated using eq 7, where the true emissions are the summed total emissions for the emission simulation I.

7

Source-Based Extrapolation

Source-based extrapolation is a more data-intensive technique of extrapolation. This method requires an understanding of the source count within the sample population and full population.

8

where y denotes each source category, SE_y,S is the summed emissions in the sample population from source type y, N_y,S denotes the count of source type y in the sample population of size S, and N_y,T denotes the total count of source type y in the total population. Percent error is computed using eq 7.

The analyses presented in this work address the issue of how to determine a sufficiently large sample size such that extrapolation across measurements from the subset of sites is representative of a larger basin. Two types of variability are accounted for in these analyses: emissions variability and sampling variability. The emissions variability, quantified through the emissions simulations, was found to be minimal. To focus this discussion of the results on sampling variability, a representative emission simulation (emission simulation A), out of 10 simulated (emission simulations A–J) was used. Results from all of the emissions simulations can be found in the Supporting Information.

The results from the sampling variability analysis are presented in four sections: (1) understanding the impacts of extrapolation technique and sample size on estimate error and bias, (2) defining a metric that can be used to inform sample population selection, (3) defining a function to relate extrapolation uncertainty and capture ratio, and (4) broad considerations for sample population selection and extrapolation for future applications.

Impacts of Extrapolation Technique and Sample Size

This work explores two variables and their relationship to the overall error in the extrapolated emissions estimate: the extrapolation technique and sample size. Spatial extrapolation error arises when emissions from the sample population are extrapolated to the entire region. The average spatial extrapolation error across 50 sampling simulations is given by the difference between the extrapolation of the subset of sites to the full population over the 10-day period and the emissions from all sites for the 10-day period. The simulations were the same across all extrapolation techniques, meaning that the only difference between the 65-site simple linear estimate and the 65-site source-based estimate was the extrapolation method, as all underlying emissions simulations remained the same. Based on average error, source-based spatial extrapolation estimates were more accurate than simple linear estimates in every case (Table 3). Results from all emission simulations are listed in the Supporting Information.

There is significant bias and average error associated with smaller sampling sizes. Bias and average error can be mitigated marginally in smaller sample sizes by using source-based extrapolation, but error and bias are still significant. Smaller sample sizes have large negative biases because it is more probable that the sample population will miss sources in the fat tail of the skewed distribution, which are less common but contribute significantly to overall emissions. It is important to note that the smaller bias in the linear extrapolation of the 13-site population is due to scenarios where the extrapolated emissions were overestimated, not because the linear extrapolation method has lower uncertainty (Figure 2 and Table S4.1). Larger sample sizes (sample populations ≥230 sites) had bias within ±10%, regardless of the extrapolation method. These results suggest that small sample populations introduce significant negative biases. This bias can be mitigated by using source-based extrapolation or increasing the sample size.

Figure 2.

Box and whisker plot for error in extrapolated emissions across sampling simulations for emission simulation A as a function of sample size by using (A) simple linear extrapolation and (B) source-based extrapolation. Outliers have been excluded from the figure.

There is significant variability in error across sampling simulations for a given sample population size. Results from all 50 sample simulations for a representative emission simulation are shown in Figure 2. Results from all emission simulations are available in the Supporting Information.

The results of sampling simulations (Figure 2) show large ranges of error within the sample sizes. The range of observed error declines as population size increases, with narrow error intervals for the 650-site sample population in both extrapolation scenarios. However, other relatively large sample population sizes, such as the 65-site and 230-site sizes, still demonstrated significant variability in expected error conditions. For this representative scenario, the 65-site sample population absolute extrapolation error ranges from 192 to 6% for source-based extrapolation, and from 215 to 1% for simple linear extrapolation. In the 230-site scenario, source-based absolute extrapolation errors ranged from 68 to 2%, and simple linear absolute extrapolation errors ranged from 76 to 0.5%. Measurement of 50% of sites, as represented by the 650 sample populations, may not be feasible for most operators due to logistical constraints. If an operator seeks to measure ∼5% (65 sites) or 18% of sites (230 sites), the range of expected error is too large to make reasonable recommendations about the uncertainty associated with a specific random selection of sites. In other words, there is limited application in reporting that spatial extrapolation error from a 230-site sample may vary between 0.5 and 293%. The next stage of analysis seeks to leverage existing data from the source-based inventory to define a new metric that can better inform sample population selection and can explain why certain estimated emissions from a sample population are significantly more accurate than others.

Defining a Metric to Determine Sample Size

The results from this analysis demonstrate that while sample size is important in minimizing the error associated with spatial extrapolation to a larger population, there are other metrics that can be used to estimate how well a sample population can be extrapolated to a larger region. Since emissions occur at the source level, it is important to consider how well a sample population represents emissions at the source level. In other words, a larger sample size is not necessarily better if it still fails to capture the population-wide behavior of emissions. In contrast, a smaller sample size may be sufficient in estimating emissions across a larger region if the sources within that population are representative of the overall behavior. Certain sources will be more difficult to accurately capture than others, and this difficulty is largely dependent on the variability in the range of emission rates for each source.

One possible metric for characterizing the representativeness of a subpopulation is a capture ratio, which quantifies the ratio of each source that is captured in the sample population, weighted by the skewness of the emissions per source (Table 4). The capture ratio increases as a larger fraction of sources are included in the sample population and sources with highly skewed emission distributions are weighted more heavily in the metric. Skewness is a metric that quantifies the lack of symmetry in a data set (eq 9). Statisticians have studied this field of research for over a century, leading to a variety of statistics that can be used to detect skewness. This work defines skewness using the adjusted Fisher–Pearson standardized moment coefficient (G₁), which is used across many major software packages.^18,25

9

where N_y,T is the number of source type y emission measurements across the total population, x_i is the ith value in the emission contribution measurement data set for source type y, x̅ is the average value across all x_i values, and s is the standard deviation across all x_i values. Emission contribution is defined as the total emissions per emission event, or the emission rate multiplied by the duration.

10

where G_1,y is the Fisher–Pearson standardized moment coefficient (a measure of skewness) for source type y, N_y,S is the number of source type y present in the sample population, and N_y,T is the number of source type y in the total population. The capture ratio was calculated for each sampling simulation and plotted against extrapolation error for representative emission simulation A (Figure 3). Results from each emission simulation are available in the Supporting Information.

Table 4. Skewness Metric for Each Source Type.

	unloading	dehydrator	tank	engine	leak	sources with static emission factors (compressors, controllers, pumps)
skewness	2.7	0.8	2.01	1.1	30.8	0

Figure 3.

Correlation between absolute error in emission estimates and the capture ratio using simple linear extrapolation (A) and source-based extrapolation (B).

The capture ratio has a strong negative correlation with absolute error, with a Spearman correlation coefficient of −0.69 and −0.77 for the simple linear and source-based extrapolation methods, respectively, which means the relationship between the capture ratio and absolute error is well defined by a monotonic function. This relationship is true across all emission simulation scenarios (see the Supporting Information).

Capture ratios are defined based on all emissions but could also be calculated separately for intermittent and continuous emissions (Figure 4). The trend of increasing the capture ratio to decrease the extrapolation error is present in both types of emissions, though this trend is much stronger in the case of continuous emissions. One possible explanation for this behavior is that intermittent emission sources have more uncertainty in the temporal dimension, such as estimating their duration and persistence. This information cannot necessarily be gathered through increasing site-level sampling alone, and so there is more residual error despite large values of the capture ratio. Details on this analysis are available in the Supporting Information.

Figure 4.

Correlation between absolute error in emissions estimates using source-based extrapolation (blue) and simple linear extrapolation (green) methods for only continuous sources (A) and intermittent sources (B).

Defining a Function to Relate Extrapolation Uncertainty and Capture Ratio

The capture ratio is a useful metric in defining a correlation between the extrapolation error and characteristics of the sample population. The results from this work can be used to calculate the average extrapolation uncertainty as a function of the capture ratio. To define this function, the capture ratio data points were binned based on data density; each bin was defined such that there were approximately 20 data points per bin. The average extrapolation error per bin was calculated and plotted against the average capture ratio per bin. The results for each emission simulation (simulations A–J) are also plotted below (Figure 5). The average simple linear extrapolation error closely follows a power-law function (R² = 0.85) and the average source-based extrapolation error closely follows a logarithmic function (R² = 0.98). The data used in this analysis are available in the Supporting Information.

Figure 5.

Extrapolation error as a function of the capture ratio across different emissions simulations for simple linear extrapolation (A) and source-based extrapolation (B). Simulation A, the representative simulation selected for analysis in the earlier discussion of results, is highlighted.

Figures 3 and 5 illustrate two main ideas of this manuscript. The first is that, in smaller sample populations, error can be mitigated by incorporating source count information through source-based extrapolation. In scenarios with a capture ratio <0.1, source-based extrapolation error does not exceed 120%, whereas simple linear extrapolation error (Figure 3) can be as large as over 350%. The second main idea is that extrapolation error can be rapidly reduced through informed selection of a relatively small sample population, as there is a steep decline in error as the capture ratio increases up until about 0.2, followed by a plateau in error reduction. In the case of source-based extrapolation, average error for capture ratio <0.2 is 61%. This average error decreases to 13% for capture ratios >0.2. In simple linear extrapolation, average error for capture ratios <0.2 is 71%, which decreases to 19% for capture ratios <0.2.

Comparison of results from Figures 2 to 3 and 5 suggests that sample size alone is not a sufficient metric on which to estimate extrapolation error. Extrapolation error is also dependent on the underlying complexity of the emission sources within the region. Sufficiently large sample sizes are important in improving the precision of an extrapolated estimate (Figure 2), but this alone is not enough to guarantee accuracy (Table 3). A deeper understanding of source-level variability and skewness can lead to the calculation of a parameter such as the capture ratio, which is a better proxy for estimating error than sample size and can inform more granular extrapolation. Sample populations that are selected to maximize the capture ratio could lead to significant reductions in the extrapolation error.

Future Applications

This work has broad applications in policy decisions and industry efforts surrounding measurement-informed methane emission reporting. This work identifies and quantifies the error of spatial extrapolation of emissions measurements to construct regional emission inventories, and proposes metrics and methods that can be used to estimate and minimize this error. Results from this case study suggest that extrapolation of measurements across space can be significant, and should be considered. Extrapolation methods can also have significant impacts on uncertainty, as source-based extrapolation performs better than simple linear extrapolation, especially in scenarios with limited coverage (capture ratio <0.3), though this discrepancy diminishes as the capture ratio increases.

While the results from this research are specific to the case study region, the methodology outlined in this work can in concept be applied to any region or operator. The data required for this type of analysis is an understanding of source-level behavior and source-level measurements to characterize the distribution of its emissions. This understanding of the underlying source-level behavior can be used to compute a metric such as the capture ratio, which can explain the variability of error across sample population sizes. To effectively extrapolate measurements from a sample population, source-level emissions within the sample and total populations should be accounted for. In many regions, source-level variability is too significant to assume emissions within a small sample population are representative of the entire region, unless selections of sample populations are informed by overall knowledge of the emissions distribution. Future work should focus on validating the relationship between the capture ratio and the spatial extrapolation error in other regions.

Measurement campaigns can be optimized if they are informed by emission characteristics at the source level. This work examines emission extrapolation from random selections of sites, finding that an accurate representation of emission characteristics is crucial for representative site-level measurements. Future work will examine if a stratified selection of sites, based on groupings of sites determined by equipment configurations, can be used to minimize both extrapolation uncertainty and the number of sites in the sample population. This introduces a new type of extrapolation, a site-level extrapolation that is based on measurements of sites within subgroupings.

This work also has implications for reconciling bottom-up and top-down measurement-based inventories. For example, a major part of the United Nations’ OGMP 2.0 initiative is reconciliation between Level 4 inventories, constructed using source-level measurements or source-level estimates, and Level 5 measurements, which occur at regional scales, such as basin-level aircraft of satellite measurements.²⁶⁻²⁸ The inherent variability of methane emissions ensures that these inventory estimates will never be an exact match; however, the framework presented in this work can be used to predict thresholds of uncertainty that can contextualize measurements. As emission reporting efforts continue to shift toward a measurement-informed basis, it will become increasingly important to understand the extrapolation uncertainty associated with the construction of emission inventories. Quantification of this uncertainty can inform the sample population selection, measurement campaign size, and measurement reconciliation. This work highlights the importance of extrapolation uncertainty in the context of measurement-informed inventories and its dependence on source-level behavior. The methodology used in this work has direct applications in advancing current regulatory and industry efforts to improve methane emission reporting.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.est.3c08185.

• Detailed information on emissions data and full simulation results (PDF)

The authors declare the following competing financial interest(s): One of the authors (D.T.A.) has served as chair and is currently a member of the Environmental Protection Agency, Science Advisory Board; in this role, he is a Special Governmental Employee. D.T.A. has current research support from the National Science Foundation, the Department of Energy, the Texas Commission on Environmental Quality, the GTI Energy, the Collaboratory to Advance Methane Science, the ExxonMobil Upstream Research Company, Pioneer Natural Resources, Cheniere, EQT, Williams and the Environmental Defense Fund. He has also worked on methane emission measurement projects that have been supported by multiple natural gas producers and the Environmental Defense Fund. D.T.A. has done work as a consultant for multiple companies, including Cheniere and SLR International. C.S. has worked as an intern at the Gas Technology Institute and Scientific Aviation. C.S. has also been an employee of the United Nations Environment Program, working on the Oil and Gas Methane Partnership 2.0. One of the authors (H.D.) works for an oil and natural gas production company.