A simple model for the concentration C of a gas emitted from a point source and carried in the wind is the Gaussian plume model, defined by 1C(x,y,z)=Qu12πσye-y22σy212πσze-(z-z0)22σz2, where (x,y,z) is a right-handed coordinate system with positive z normal to the Earth's surface and x increasing in the direction of the wind speed vector which was assumed constant in space and time in Eq. ()'s derivation . The plume origin is assumed to be at (x,y)=(0,0). Because we will simply integrate over z in the following section, we use an overly-simplistic vertical distribution that assumes that all the gas is above the surface, or z0≫σz, for convenience. The quantity Q is the gas emission rate, typically in units of kgh-1, and u is the wind speed.

The plume spreading parameters are denoted by σy and σz and are generally functions of the downwind position x. A variety of spreading models and parameterizations exist for Gaussian plume models . A particularly common model is to assume that 2σy(x)=axdb. Different authors find different values for parameters a, b, and d and often recommend that they change with wind speed or atmospheric state (like those of the Pasquill-Gifford atmospheric state models in, for example, ). In this work, we take d = 1000 m and allow the fitting procedure described below to find the values of a and b that best match the data. The choice of 1000 m follows but, based on the parameterization in Eq. (), this choice is arbitrary and only for convenience in comparing the retrieved parameters a and b to previous work . Importantly, the spreading model we propose does not include an explicit dependence on the wind speed. This means that the wind speed only appears in the ratio Q/u. As a result we cannot estimate Q separately from u, but only their ratio, which we call CQ below. For convenience, we use units of kgh-1 per ms-1 for CQ as Q is typically kgh-1 while u is ms-1. Again, we ignore σz as will become clear shortly.

Standard Gaussian plumes assume constant wind, but we can accommodate some wind direction variation by allowing the cross wind mean to vary as a function of the downwind position, as in C2D(x,y)→C2D(x,y-m(x)). Since many plumes appear to bend in the wind, this addition provides the model with the flexibility needed to obtain good fits, as will be seen below, and reduces the potential error associated with model-data mismatch. We leave the problem of relieving the inconsistency of allowing directional variation without a magnitude change to future work (but see Sect.  for a discussion regarding simultaneously estimating the wind magnitude). We parameterize this variation with a piecewise linear function defined by parameters mx spaced at 3 times the Ground Sampling Distance (GSD) in the along wind x-direction. Previous work with Gaussian plumes has allowed wind variation using Bezier curves , but we have found that the piecewise linear function works well for the relatively high-spatial-resolution data used below. We have also found that the emission estimate is not very sensitive to this choice. The number of mx parameters is the number that fit from the plume origin to the far end of the retrieval region.

We seek to estimate CQ by comparing the Gaussian plume model with the remote measurements. In the remainder of the paper, we assume that prior processing of the observations has estimated the concentration length l^i,j in ppm m, where i,j are spatial pixel indices, and that potential plumes have been detected and queued for emission rate estimation. Since this data is a 2D overhead map of the gas concentration, the comparison requires adapting the 3D model to the 2D remote measurement by integrating over the z direction, giving the 2D Gaussian model 3C2D(x,y)=Qu12πσye-y22σy2. In performing the integration, we assume that all the gas is above the surface (z0≫σz).

To facilitate the comparison of the model to the data, we seek a probabilistic treatment for the measurement following . In that work, it is shown that the concentration length in each pixel can be treated as a Gaussian random variable (not to be confused with the Gaussian plume model). It is also shown that the estimated concentration length is biased from the true concentration length by local albedo and atmospheric variation, an effect quantified by the “sensitivity”. Both the sensitivity and the pixel's concentration length uncertainty can be calculated from the observation. We adapt this model to create a maximum likelihood estimator for the parameters of the Gaussian plume model of Eq. (). To do so, we model the concentration length in each pixel as a Gaussian random variable and define its mean to be the Gaussian plume model in Eq. () scaled by the sensitivity. In this formulation, the concentration length measurement in each pixel is considered to be a realization of the corresponding random variable, with the measurement in highly uncertain pixels being more likely to deviate from the plume model than those in less uncertain pixels. We therefore obtain the probability distribution function in each pixel 4pC2Dl^i,j=12πSi,j2Ui,j2e-(l^i,j-2kSi,jC2D(xi,yj))22Si,j2Ui,j2, where Si,j and Ui,j are the sensitivity and uncertainty, at pixel (i,j). The factor k, which converts the model from area concentration [kgm-2] to concentration length [ppmm], is 106 (m3mol-1) (molkg-1). In the rest of this paper, we consider methane plumes near the surface at standard temperature and pressure (STP), yielding k≈ 1.4 × 10⁶ ppmm3kg-1. The factor of 2 in the exponent's numerator is to account for the light's path through the plume. Although this factor in principle depends on the look and solar angles along with the plume's cross-sectional shape and height above the ground, even including double plumes , we assume that the concentration length measurement in each pixel represents a double path through the plume: down from the sun then back up from the ground.

To estimate CQ, we form the likelihood of the plume model parameters θ∈Θ given the data, given by 5Λ(θ)=∏i,j∈PpC2Dl^i,j;θ, and then seek the values of those parameters that maximize the likelihood, as in 6θ^=arg maxθ∈ΘΛ(θ). The product in Eq. () is taken over all pixels inside the plume and surrounding area. In the results below, we define P to be all the pixels within the rectangular window that bounds the manually-delineated plume plus a 10 pixel buffer on all four sides. For the EMIT data used here, we rely on the plume boundaries that are part of the standard EMIT products . The AVIRIS data has no publicly available plume boundaries as of this writing, so we created them following the EMIT procedure. All the plume images below show the entire retrieval region P (except the right side of Fig. , which is zoomed for visual clarity). The advantage of including pixels beyond the manual delineation is twofold. First, we eliminate the subjectivity necessarily associated with drawing a boundary around the apparent plume by allowing the optimization to determine the plume shape without arbitrary restrictions. Second, we allow the method to take advantage of the region of gas enhancement beyond the apparent end of the plume. Although not visibly associated with the plume and therefore not included in the manual delineation, the model may predict gas enhancement here and its presence can be exploited by the likelihood formulation. We limit the buffer to 10 pixels both to limit computational time but also to reduce the likelihood that the plume model be degraded by inadvertently including scene artifacts that may be mistaken as true gas enhancements. The presence of the uncertainty in the denominator of the exponent in Eq. (), when combined over P in Eq. (), effectively downweights uncertain pixels. As the downweighting is relative to the neighboring pixels, this feature of SPIIRE is most helpful for plumes that span high and low-albedo regions (see Fig.  for example) rather than weak plumes that are only barely detectable. The parameter space Θ includes the wind-normalized emission rate CQ and the spreading parameters a and b, a rotation α about the plume origin to allow for arbitrary wind directions, and the mx plume bending parameters. Because the Gaussian plume model is derived for point sources, the concentration becomes infinite at the origin as the spreading model shrinks. In the implementation here, we work around this problem by forcing the origin to always lie on a pixel boundary that is excluded from P. However, a different spreading model σy or, ideally, a more physical plume model based on a finite-sized gas vent, could achieve a similar effect.

We use the SciPyBoundedMinimize optimizer provided in the jaxopt python library . The initial value for the wind direction is computed as the line from the manually-determined plume origin to the centroid of the plume boundary. In practice, we optimize the log likelihood to simplify the cost function and we estimate ln⁡CQ and ln⁡a to keep the numerical range of those parameters similar to that of the others.

The uncertainty in the emission rate Q depends on many factors, including instrument noise and wind speed. These two can be combined using standard error propagation methods based on the expression Q=uCQ, yielding 7σQ2=σCQ∂Q∂CQ2+σu∂Q∂u2. We calculate σCQ using the Cramer-Rao bound and σu as the standard deviation of the surrounding wind samples following . Another source of error is due to the mismatch between the true plume and the Gaussian model, but this is difficult to quantify. The use of the sensitivity and the mx help to reduce this error, but the uncertainty persists, meaning that the above expression for uncertainty likely underestimates the true uncertainty. For further discussion, see Sect.  for empirical evidence regarding the uncertainty.

The principal limitations of SPIIRE are the challenge of estimating wind speed and the difficulty in fully quantifying the detrimental effects of applying SPIIRE to non-Gaussian plumes. The former is discussed below in Sect. . Regarding the latter, we suggest a path forward in Sect.  in which we generalize the model to adapt to apparently non-Gaussian plumes. Other limitations include the requirement for a parameterized downwind spreading model σy, the unknown 3D shape of the plume and the light's path through it (the 2 in Eq. ) , and numerical difficulties associated with the optimization.

A main advantage of SPIIRE is that the optimization region P is large and extends beyond the visible plume enhancement, allowing the model to smoothly respond to local variation and reducing the estimator's susceptibility to measurement noise. To demonstrate SPIIRE's resilience to measurement noise, we simulated 600 plumes using the 2D Gaussian plume model with random CQ and spreading parameters but with no bends (the simulated mx are all 0). We choose to simulate Gaussian plumes because they satisfy the assumptions of both IME methods and SPIIRE and therefore provide a test of the enhanced noise performance. The simulated CQ are uniformly distributed between 200 and 3000 kgh-1 per ms-1 to span a typical range of emission rates and wind speeds observed with EMIT. The plumes are injected (added) into random locations in 263 randomly chosen EMIT methane enhancement scenes that have previously been determined to have no clouds or true emissions (see Figure 8 of for injection details). We include the local sensitivity when injecting the simulated plume into the noisy data. Figure  shows the simulation results by comparing the IME method on the left with SPIIRE on the right. As expected, SPIIRE has superior noise performance (reduced uncertainty) as demonstrated by its narrower vertical spread and is unbiased over the full range. We note that it may be difficult in practice to realize the full noise improvement as data-model mismatch will inevitably occur due to unmodeled effects like changing winds, local topography variation, intermittent sources, and so on. However, these effects often plague IME-based methods too, making SPIIRE's noise resilience an important contribution.

We demonstrate the accuracy of SPIIRE using controlled-release experimental data . Following the approach in , methane is released and accurately measured from a point 7.3 m above ground at two different sites: Casa Grande, AZ at (32.822° N, 111.785° W) and Evanston, WY at (41.275° N, 110.930° W). AVIRIS-3 performed 110 passes at Evanston on 11–14 September 2024 and 34 passes at Casa Grande on 4–5 November 2024. EMIT observed the site during 7 overpasses. Wind speeds and directions are measured at 2 and 10 m above ground using a sonic anemometer.

An example plume is shown in Fig.  using AVIRIS-3 data from 4 November 2024 at Casa Grande, Arizona (AV320241104t183250). As mentioned previously, the retrieval region P includes all pixels in the rectangular region within 10 pixels of the widest part of the plume in all directions (all pixels shown in Fig. a). The adapted Gaussian plume model is shown in Fig. b with the sensitivity correction applied. The bending of the plume model to accommodate apparent wind direction changes in the plume is achieved through the mx. A key feature of the method is the modification of the model by the sensitivity to account for surface- and atmospheric-related biases, although this effect is modest at the mostly-uniform experimental site.

The emission rates from SPIIRE are plotted against the metered emission rate for the two AVIRIS-3 controlled release experiments in Fig. . We have removed several overpasses where the plumes are too weak to be credibly detected. The linear fit is shown in dashed gray with the fit statistics inset. The wind speeds used are from the 2 m sonic anemometer. Both the wind speeds and the metered emission rates are the mean over the preceding 30 s relative to the overpass time. We chose 30 s because it represents a reasonable propagation time for the range of plumes included in the experiment while providing sufficient smoothing. For both experimental locations, the estimated emission rates are accurate in the aggregate with a reasonable spread about the true emission rate. The IME method, shown in Fig. , yields similar but slightly worse performance, driven in part by outliers whose estimates are sensitive to the particular choices of implementation and parameter. We note that uncertainty in the wind speed and decisions about how it is processed affect the comparisons too.

The estimated emission rates for the EMIT controlled release experiment overpasses are shown in Fig.  for SPIIRE and Fig.  for IME. EMIT did not detect the first two despite relatively large metered emission rates, likely due to high winds during those collections (5.1 and 5.2 ms-1, respectively). Because the releases only started a few minutes before the overpass, they are short compared to typical plumes, all between 8 to 16 EMIT pixels (< 1 km). This means that the assumptions guiding the decision to use a buffer region surrounding the visible plume do not hold here and we assume instead that the end of the apparent plume actually represents the end of the true gas enhancement. Accordingly, we modified the cost function for the five examples shown in Fig.  to include only the pixels within a circular region surrounding the emission source and whose radius matches the last visible portion of the plume. This choice reduces the number of pixels used in the retrieval and likely degrades the performance relative to that observed in the AVIRIS-3 data in Fig. . Nevertheless, the estimator's performance is good, with four of the five points within their uncertainty of the metered emission rate.

Section  noted that the uncertainty calculation in Eq. () and used in Figs.  and  neglects the uncertainty due to model-data mismatch because it is difficult to quantify. However, the experimental results provide a means to empirically assess the degree to which Eq. () underestimates the true uncertainty, at least for the point-source plumes provided by the experiment. We find that the emission rates fall within their uncertainty range of the true value in 50 %, 62 %, and 80 % of the overpasses at the AVIRIS Arizona, AVIRIS Wyoming, and EMIT Arizona sites, respectively. Since Eq. () represents a 1σ uncertainty, we would expect approximately 68 % of the overpasses to be within their uncertainty of the true emission rate if the uncertainty were not biased. The results indicate that neglecting model-data mismatch in the uncertainty calculation does not significantly bias the resulting uncertainty estimate.

We note that the IME method used to produce the estimates quoted throughout this paper requires a different set of user-defined parameters for the AVIRIS-3 results versus the EMIT results. This is to address the differences in spatial resolution and noise performance between the two instruments. It is possible to tune these parameters for AVIRIS-3 using the controlled release data, but the limited availability of such data for EMIT means that it is difficult to project the IME performance quoted above to EMIT observations. In contrast, the SPIIRE results throughout this work use the same set of parameters for both instruments, meaning that the validation shown here using AVIRIS-3 and EMIT controlled release data is more likely to apply to the full EMIT database, especially for point-source plumes like those in the experiment.

A major benefit of SPIIRE is the improved understanding of visually-complex plumes. Figure  and the third example in Fig.  exhibit EMIT plumes whose initial appearance suggests an intermittent source or turbulent winds. We find instead that these plumes can be well modeled with a steady-state Gaussian plume that includes albedo effects through the sensitivity, suggesting that in some cases intermittency or turbulence need not be considered to understand complicated plume shapes. The cross-wind transects on the right side of the Fig.  demonstrate the quality of the plume modeling in this case. Although no ground truth emission rate is available for this plume, SPIIRE estimated Q at 3.4 times that of the IME method. This plume provides an example of the two methods diverging despite the plume appearing to be due to a point emission source. In this case the model-data agreement is good, suggesting SPIIRE's estimate is likely a more accurate representation of the true emission rate.

IME-based methods are currently being used operationally and their emissions estimates are publicly available, including for example , subject to quality control thresholds . SPIIRE could be readily incorporated into such workflows to augment the estimates and to provide the additional benefits of plume interpretation using the adapted 2D Gaussian model. As described in the following subsections, SPIIRE may also help to relieve the quality control thresholds by providing a new method to quantify otherwise challenging plumes in difficult environments. Three more SPIIRE examples from the EMIT database are shown in Fig. .

IME-based approaches to plume emission rate estimation do not perform well in plume complexes having multiple apparent sources whose plumes merge downwind. In such cases it is impossible to know how to allocate CH4 enhancement pixels to any particular source. However, it is straightforward to adapt SPIIRE to plume complexes by simultaneously inverting multiple plumes, with the model to be optimized simply the sum of each plume. Figure  shows an example EMIT scene having multiple apparent sources. Originally labeled CH4_PlumeComplex_4191, it is clear from a close examination of the data image at left along with high resolution base-layer imagery from Google Maps that there are in fact several point sources. The wind direction in this case appears to be coincidentally aligned so that the plumes overlap. We identify five independent point sources and simultaneously invert five corresponding Gaussian plumes. It is not necessary to assign the CH4 enhancement in a particular pixel to any particular plume as the simultaneous inversion automatically adapts each plume to the data to provide a combined CH4 model over the entire set of pixels in the figure.

Distributed, or area, sources are also challenging for IME-based quantification methods as they often do not appear to satisfy the assumptions needed to calculate the emission rate based on their IME. In particular, these sources often appear as amorphous enhancement regions spanning many pixels but without any apparent point source or long downwind tail. The plume in Fig. , showing a landfill in northwest China, is a good example. Using the Gaussian plume, we can model distributed sources as a collection of point sources spread over a predefined area and then allow the optimizer to individually determine each plume's CQ. We assume that the wind speed is the same over the entire region, so we force each point plume to share the same spreading parameters. In the example shown in Fig. , we selected a polygon that covers the landfill and surrounding area using high-resolution Google Maps imagery. This region is approximately 800 m wide. The point sources within the polygon, one per pixel, are shown in the green inset on the right side of the figure with color indicating relative intensity. Specifically, white represents no emission and dark blue relatively high emission. The resulting optimized model is shown in the middle panel of Fig. . Intriguingly, the optimizer found that the data is best represented by a relatively few sources concentrated near the bottom left and middle of the enhancement region. The lack of downwind tail, a common feature of distributed sources, is also modeled well using the collection of point sources. The IME method yields an estimated emission rate of approximately 6200 kgh-1 with a wind speed of 3.3 ms-1 while the distributed point source model yields approximately 11 000 kgh-1. Although it is possible that this method is susceptible to artifacts not compensated for with the sensitivity, we hope that future work can expand on this idea and demonstrate good performance over a large class of amorphous plumes commonly observed in plume databases.

A major challenge with emission rate quantification is in the estimation of the wind speed. Historically the wind speed has been derived from weather reanalysis data, like HRRR and ERA5 , but recent work using multiple observations separated by a few seconds provides an alternative based on the observation itself . A seemingly straightforward improvement to the method described above would be to simply simultaneously estimate the wind speed along with the other parameters, but, as mentioned in Sect. 2, the functional form of the standard plume model in Eq. () is degenerate in Q and u. attempt this, which may partially explain the poor performance that they obtained in their Gaussian plume inversions. Alternative models with more explicit dependence on wind speed exist, as in Eq. (4) of , but these either have other degeneracies or are apparently nonphysical. We have not been able to generate meaningful results with them. We have also considered establishing a correlation between the retrieved spreading parameters a and b with the various studies that predict them using atmospheric stability classes (Pasquill-Gifford, for example, as in ). Unfortunately, the retrieved values for these parameters often exceed their published ranges making the correlation difficult to establish. We therefore leave simultaneous wind speed estimation for future work, especially the development of a physical plume model with explicit wind speed dependence that cannot be compensated by a nuisance parameter.