Multi-Source Emission Determination Using an Inverse-Dispersion ...

Boundary-Layer Meteorol (2009) 132:11–30 DOI 10.1007/s10546-009-9387-1 ARTICLE

Multi-Source Emission Determination Using an Inverse-Dispersion Technique Thomas K. Flesch · Lowry A. Harper · Raymond L. Desjardins · Zhiling Gao · Brian P. Crenna

Received: 29 July 2008 / Accepted: 20 April 2009 / Published online: 8 May 2009 © Springer Science+Business Media B.V. 2009

Abstract Inverse-dispersion calculations can be used to infer atmospheric emission rates through a combination of downwind gas concentrations and dispersion model predictions. With multiple concentration sensors downwind of a compound source (whose component positions are known) it is possible to calculate the component emissions. With this in mind, a field experiment was conducted to examine the feasibility of such multi-source inferences, using four synthetic area sources and eight concentration sensors arranged in different configurations. Multi-source problems tend to be mathematically ill-conditioned, as expressed by the condition number κ. In our most successful configuration (average κ = 4.2) the total emissions from all sources were deduced to within 10% on average, while component emissions were deduced to within 50%. In our least successful configuration (average κ = 91) the total emissions were calculated to within only 50%, and component calculations were highly inaccurate. Our study indicates that the most accurate multi-source inferences will occur if each sensor is influenced by only a single source. A “progressive” layout is the next best: one sensor is positioned to “see” only one source, the next sensor is placed to see the first source and another, a third sensor is placed to see the previous two plus a third, and so on. When it is not possible to isolate any sources κ is large and the accuracy of a multi-source inference is doubtful. Keywords Condition number · Dispersion modelling · Emission rates · Inverse dispersion · Open path laser T. K. Flesch (B) · B. P. Crenna Department of Earth and Atmospheric Sciences, University of Alberta, Edmonton, Canada e-mail: [email protected] L. A. Harper Department of Poultry Science, University of Georgia, Athens, GA, USA R. L. Desjardins Agriculture and Agri-Food Canada, 960 Carling Ave., Ottawa, Canada Z. Gao College of Resources and Environmental Science, Agricultural University of Hebei, Baoding, China

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1 Introduction Inverse-dispersion calculations can be used to infer details of gas emissions to the atmosphere. A particularly successful application of this technique has been in the calculation of emissions from spatially well-defined sources, such as an industrial waste lagoon or a farm barn (Wilson et al. 1982; McInnes et al. 1985; Flesch et al. 2004). In these “single-source” problems the average gas concentration C (above background) is measured in the dispersing emission plume, and the emission rate (Q) is computed as: Q=

(C) , (C/Q)sim

(1)

where (C/Q)sim is a dispersion model prediction of the ratio of concentration at the sensor to the emission rate. For situations where C is measured close to the source (≤1 km) and the terrain is relatively uncomplicated, idealised dispersion calculations can accurately give (C/Q)sim ; Harper et al. (2009) summarizes several studies showing accuracies to within 15%. In these cases the inverse-dispersion technique for calculating emissions has substantial flexibility and economy compared to alternatives (e.g., flux gradient, eddy covariance). Inverse-dispersion calculations are more versatile than just a tool to calculate emissions from a single, spatially well-defined source. With a set of distributed concentration sensors it is possible to infer the location of an otherwise unknown source (e.g., Seibert and Frank 2004; Thomson et al. 2007) or the emission field of a spatially distributed source (e.g., Su et al. 2003). With enough concentration information one can infer more intricate source characteristics, e.g., the location history of moving sources and time dependent emission rates (e.g., Yee 2008). Here we investigate a problem that is a step-up in complexity from the single-source inference: inferring emissions from a compound source whose component locations are known. A real world example might be methane emissions from an industrial site having multiple emission points, or a livestock facility with ammonia emissions from several barns and waste lagoons. In these types of problems one can hope to concurrently calculate N component emission rates if there are at least N downwind sensors.1 This study follows the ideas of Crenna et al. (2008) who demonstrated (using synthetic data) that “multi-source” problems tend to be ill-conditioned, and the success of an emission inference depends on the geometry of sources and sensors, and on the atmospheric conditions. The focus of our investigation is a multi-source field experiment, where multiple concentration measurements are made near the sources. We will review the mathematical difficulties of this problem, examine how source–sensor geometry influences the success of our experiment, and speculate on the usefulness of the inverse-dispersion technique to real world multi-source problems. 2 Mathematical Challenge of Multi-Source Problems 2.1 Ill-Conditioned Systems Consider the hypothetical problem in Fig. 1, with two surface sources in short-grass terrain, and where we wish to concurrently calculate the two emission rates Q 1 and Q 2 . This requires 1 For N sensors and M sources one can construct a system of N equations to solve for the M emission rates. In

this study we look at the overdetermined case where N ≥ M. More complex problems are often underdetermined and ill-posed, requiring a more sophisticated treatment than discussed here (e.g., Pudykiewicz 1998; Yee 2008).

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Multi-Source Emission Determination

Emission Sources (zsrc = 0.01 m)

Q1

13

A

Concentration Sensors (zm = 1.5 m)

D E Wind

Q2

F B 50 m

G u* = 0.25 m s-1 L=∞m z0 = 0.01 m

Fig. 1 Hypothetical problem map. Two sources are emitting gas at rates Q 1 and Q 2 ; concentration is measured at locations A through G

at least two concentration sensors, and we chose sensor locations A and B. The concentration at these points can be decomposed as the sum of the contributions from each source, viz. CA,1 /Q 1 sim Q 1 + CA,2 /Q 2 sim Q 2 = CA , (2a) (2b) CB,1 /Q 1 sim Q 1 + CB,2 /Q 2 sim Q 2 = CB , (assuming no background gas concentration) or in matrix notation, Q1 CA (CA,1 /Q 1 )sim (CA,2 /Q 2 )sim = , (CB,1 /Q 1 )sim (CB,2 /Q 2 )sim Q2 CB

(3)

where the coefficients give the ratios of the concentration contributions from each source to each sensor, e.g., (CA,1 /Q 1 )sim gives the contribution at CA due to Q 1 . Using representative values for these coefficients2 (we ignore units for brevity): 0.000365 0 Q1 0.000365 = , (4a) 0 0.000365 Q2 0.000365 gives Q 1 = Q 2 = 1. Alternatively, one could use sensor locations D and E: 0.000900 0.000077 Q1 0.000977 = , 0.000840 0.000086 Q2 0.000926

(4b)

and the solution remains Q 1 = Q 2 = 1. Not all problem configurations allow solutions. The symmetry of sensors F and G, for example, results in linearly dependent equations and a singular coefficient matrix with no unique solution. In this study we focus on the mathematical solution to a system such as Eq. 3. We do not directly address details of the dispersion model, e.g., the choice of an appropriate model for calculating (C/Q)sim . However, an important caveat is the assumption of a linear relationship between concentration and emissions, i.e., the (C/Q) ratio for each source–sensor pair is constant for a particular period. This limits our calculation to linearly reactive gas species, or to cases where the source–sensor travel time is fast compared with the rate of reaction. The problem demonstrated by Eq. 3 can be generalised to more sources and sensors. With as many downwind sensors as sources we can create a set of linear equations and 2 Example (C/Q) sim coefficients calculated with the Lagrangian stochastic model are described in Flesch et al. (2004).

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directly solve for the source emission rates. But there is a concern with problems of this type. A linear system is said to be ill-conditioned if its solution is extremely sensitive to changes in the coefficients (or in the C measurements). Consider Eq. 4a: if the upper left coefficient (CA,1 /Q 1 )sim is increased by 10% we calculate Q 1 , Q 2 = 0.91, 1.0. This system is not ill-conditioned because the relative change in the solution is similar to the change in the coefficient. But consider Eq. 4b: if the lower left coefficient (CE,1 /Q 1 )sim is increased by 10% the solution is Q 1 , Q 2 = 2.03, −11.09. Here a 10% change in the dispersion coefficient results in a 1200% change in Q 2 . Sensor configuration D and E gives an ill-conditioned system. Ill-conditioning is important because in real problems we do not know either the model coefficients (C/Q)sim or the observed C without error. If we assume the dispersion model calculations for our short-range example have an uncertainty of 10%, then it is unlikely that source emissions can be accurately calculated using sensor configuration D and E (while this would be possible for configuration A and B). There is an interesting variant of the above problem, where one sensor is used to infer multiple emissions using concentration measurements made at different times. If one assumes emissions are time invariant, and if changing winds alter the source–sensor relationship (e.g., a wind direction change alters the source–sensor coefficients), then one can write a system of equations similar to Eq. 3. However, the time dependence of many real-world sources makes this a more limited approach. Over a time period where it may be realistic to assume constant emissions, there may not be sufficient wind variability to create a well-conditioned system of equations (in the extreme case where the wind is not changing, the result is a set of linearly dependent equations with no unique solution). Thus this approach is even more problematic than the multiple-sensor problem. 2.2 Condition Number The condition number provides a measure of ill-conditioning, and that of the (C/Q)sim matrix is calculated as (Gerald and Wheatley 1984): κ = (C/Q)sim (C/Q)−1 sim ,

(5)

or the product of the norm of the coefficient matrix and its inverse (double brackets are norms—we use spectral norms as given by the largest of the matrix row-sums). The larger is κ the greater the degree of ill-conditioning. For the two measurement scenarios above we calculate: κA&B = (C/Q)sim (C/Q)−1 sim = (0.00036)(2778) = 1, κD&E = (0.000977)(136792) = 134, which confirms that sensor configuration D and E gives an ill-conditioned system. The condition number is a useful metric because it has a physical interpretation. If the dispersion coefficients have model error εC/Q , and the concentration sensors have measurement error εC , then κ relates these errors to the error in the emission solutions εQ : εC/Q ε Q εC ≤κ . + (C/Q)sim Q C

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(6)


15

Thus the relative error in Q can be as large as the sum of the relative errors in (C/Q)sim and C multiplied by the condition number.3 The errors in Q from sensor configuration A and B should be of similar magnitude as the sum of the errors in (C/Q)sim and C. Locations D and E may give errors up to 130 times larger. The above examples are cases where there is an identical number of sensors and sources. However, these same principles extend to over-determined cases with more sensors than sources. This is the situation in our field experiment (calculating κ for this case is discussed in Sect. 4).

3 Field Experiment 3.1 Emission Sources The experiment took place in a field of mowed grass at the Central Experimental Farm in Ottawa, Canada (Fig. 2). Four synthetic sources were used, each approximating a 3 × 3 m2 surface area source, with sources made of 19-mm diameter PVC pipe forming a manifold with nine outlets on a square grid of 1-m spacing. Methane (CH4 ) was our tracer gas, and gas cylinders were coupled to each source through a regulator and mass flow controller and CH4 released at rates between 7 and 70 l min−1 . The sources were arranged within a 30 × 30 m2 release area. The study took place during September 2006 and was organised into five gas release scenarios (see Figs. 3, 4, 5, 6, 7). These differed with regards to either the source or sensor locations, and during each scenario we released gas over many 15-min periods (referred to as repetitions). Releases occurred only when south-west winds were forecast (this was because of limited fetch conditions in other directions), and each release is designated by a scenario and repetition number (e.g. S3R2 refers to scenario 3, repetition 2). Gaps in the naming sequence are caused by unusable repetitions, e.g., poor wind direction. 3.2 Concentration Sensors Methane concentrations were measured with an 8-channel open-path laser spectrometer (Boreal Laser Inc, Edmonton, Alberta), where the collimated laser beams gave the lineaverage concentration (CL ) between the laser and a retro-reflector. The eight laser lines were placed in a horizontal plane over the release area at a height z laser = 0.75 or 1.5 m, and pathlengths were typically 100 m, although 25 and 40 m lengths were used. Concentrations were processed into 15-min averages. One laser path was positioned upwind of the emission sources and measured the background CH4 concentration (Cb = 1.5–1.85 ppmv during the experiment). Between gas release repetitions we calculated ratiometric calibration factors to force agreement in CL between all laser lines, assuming the true concentration was given by the “background” laser. These calibration factors were applied during the gas release periods.

3 Some problems suffer from “artificial” ill-conditioning, where a large κ results from a “badly scaled” matrix

(Stewart 1996). This does not necessarily indicate a sensitivity of the solution to perturbations.

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lasers

N 50 m

Grass

Sonic Mature Spruce Trees

Gas Release Area

in ds

low vegetation

W

Treed windbreak

SW

Source corner

Laser Lines

Sorghum Field

Harvested Crop

Fig. 2 Experiment site at the Central Experimental Farm, Ottawa, Canada. Four emission sources were placed within the gas release area. The position of the laser lines varied during the experiment

3.3 Wind Measurements When making dispersion calculations we represent the wind speed and turbulence using Monin–Obukhov similarity theory (MOST) relationships, defined by the friction velocity u ∗ , the Obukhov stability length L, and the surface roughness length z 0 (see Garratt 1992). A three-dimensional sonic anemometer (Gill Instruments Ltd., Lymington, UK) provided these properties. For each 15-min release period we calculated an average wind direction β, and then transformed the velocity and heat flux statistics into along/across wind coordinates using two coordinate rotations (yaw and pitch corrections, e.g., Kaimal and Finnigan 1994), with: 4 u ∗ = u w 2 + v w 2 , (7a) 3 u∗ T L=− , (7b) kv gw T z son , (7c) z0 = exp (U kv /u ∗ − ϕ)

where u w and u w are velocity covariances (see Garratt 1992), T is the average acoustic air temperature from the sonic (K), kv is von Karman’s constant (0.4), g is the accelera

tion due to gravity, w T is the vertical kinematic sensible heat-flux density, z son is the height of the sonic, U is the average horizontal wind velocity at z son , and ϕ is a stability correction (we used relations given by Paulson (1970), Dyer (1974)). The anemometer also provided velocity standard deviations (σu,v,w ) for each release period, which are inputs for our dispersion calculations.

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17 Laser 6

Scenario 1

Laser 7 Laser 8

Laser 5 Laser 4

N

Laser 3

Gas Release: 7-8 Sept 2006 Laser 2 u* = 0.15 to 0.51 m s-1 L = +9 m (stable) to -46 m (unstable) z0 = 0.01 to 0.03 m zlaser = 1.5 m Q1 = 0.076 g s-1 Q2 = 0.152 g s-1 Q3 = 0.759 g s-1 Q4 = 0.434 g s-1 Range of Wind

Source 4 Source 1 Source 3 Source 2 30 m

Directions Laser 1 6

Q1 Q2 Q3 Q4

4

Ri

2 0 -2 -4

R3 S1

R4 S1

R5 S1

R6 S1

R7 S1

R8 S1

R9 S1

0 R1 S1

15-min Repetition Fig. 3 The layout of Scenario 1 (top) and the results of the emission inferences displayed in terms of the component recoveries R for each 15-min repetition (bottom). An R = 1 is a perfect inference (red dashed line)

4 Dispersion Model and the Singular Value Decomposition Emission Solution Our earlier discussion described using point concentration sensors to deduce emissions. In this study we instead measured line-average concentrations CL , and instead of calculating the relationship between Q and a point concentration, the dispersion model calculates the relationship to a line-average concentration (CL /Q)sim . This is a trivial change to the methodology. We used the software “WindTrax” (Thunder Beach Scientific, Nanaimo, Canada) to calculate the (CL /Q)sim coefficients. This software combines the backward Lagrangian stochastic (bLS) dispersion model described by Flesch et al. (2004) with an interface where sources and sensors are mapped. The (CL /Q)sim coefficient for each source–sensor pair is calculated by releasing model trajectories from each laser line. The important information for us is the trajectory intersections with ground (“touchdowns”), which are sorted according to the source area within which they occur, e.g., the (CL /Q)sim for laser line “A” due to source “1” is given by: 1 2 CL−A,1 /Q 1 sim = (8) w , N 0 Source-1

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Scenario 3

Laser 7 Laser 8

Laser 5 Laser 4

N

Laser 3

Gas Release: 8, 17 Sept 2006 Laser 2 u* = 0.27 to 0.42 m s-1 L = +49 m (stable) to -25 m (unstable) z0 = 0.01 to 0.04 m zlaser = 1.5 m Q1 = 0.076 g s-1 Q2 = 0.152 g s-1 Q3 = 0.759, 0.156 g s -1 Q4 = 0.434 g s-1 Range of Wind Directions

Source 3 Source 4 Source 1 Source 2

30 m

Laser 1 4

Q1 Q2 Q3 Q4

Ri

2

0

-2

S3

R1

S3

R2

R3 3R4 3R5 3R6 3R7 3R8 3R9 R10 S S S S S S S3 S3

15-min Repetition

Fig. 4 The layout of Scenario 3 (top) and the results of the emission inferences displayed in terms of the component recoveries R for each 15-min repetition (bottom). An R = 1 is a perfect inference (red dashed line)

where N is the number of trajectories released along laser-line A, w0 is the vertical velocity at touchdown, and the summation covers only touchdowns within the boundaries of source 1. In this experiment there are four sources and up to seven downwind lasers. The system of source–sensor equations is thus over-determined, with more equations than unknowns. A standard technique called singular value decomposition (SVD) (Press et al. 2007) is used to solve for the emissions; the SVD is an algorithm developed to minimise computational errors in matrix operations. The condition number κ remains an important characteristic in an SVD solution, although it is calculated somewhat differently than in our earlier examples. The SVD technique expresses the matrix of model-derived coefficients as a product of three matrices: (C L /Q)sim = UwV, in which U and V are rotations and w a stretching of the solution vector Q. The diagonal elements λi of w are called the singular values of (C L /Q)sim , and κ is defined as the ratio of the largest to smallest of these values i.e., κ = λmax /λmin . By explicitly calculating the singular values, SVD allows us to eliminate the smallest of them (whenever possible) and thereby decrease κ.

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19 Laser 8

Laser 6

Scenario 4 Laser 5 Laser 4

N

Laser 3

Gas Release: 18 Sept 2006 Laser 2 u* = 0.40 to 0.52 m s-1 L = +131 m (stable) to -113 m (unstable) z0 = 0.01 to 0.05 m zlaser = 0.75 m Q1 = 0.076 g s-1 Q2 = 0.152 g s-1 Q3 = 0.152 g s-1 Q4 = 0.434 g s-1 Range of Wind Directions

Source 3 Source 4

Source 1 Source 2

30 m

Laser 1 3

Q1 Q2 Q3 Q4

Ri

2

1

0

-1

S4

R1

R2 S4

R3 S4

S4

R4

S4

R5

R6 S4

S4

R8

S4

R9

0 R1 S4

15-min Repetition


5 Results 5.1 Accuracy of the Dispersion Model Here we briefly consider uncertainty in the (CL /Q)sim coefficients for our problem. Because we know the actual source emission rates we can compare the bLS predicted concentrations p CL to the actual observations CLo (assumed to be measured without error). Table 1 gives the p average bLS model errors for each scenario. Based on the fractional error |(CL −CLo )|/(CLo − 4 Cb ) of the five downwind lasers we conclude that the average uncertainty for our 15-min predictions of (CL /Q)sim is 10–20%. For a single-source problem this relative uncertainty translates to a similar level of uncertainty in the Q solution. In multi-source problems this (CL /Q)sim uncertainty will be amplified by the condition number. 4 For fractional errors we focused on the downwind lasers. The more upwind lasers often see only the edge

of the plume, and even a relatively small absolute error can correspond to a fractionally large error.

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Laser 6

Scenario 5 Laser 5

N Laser 7 Gas Release: 25 Sept 2006 Source 4 Laser 2 u* = 0.21 to 0.29 m s-1 L = +143 m (stable) to -6 m (unstable) z0 = 0.01 to 0.05 m zlaser = 0.75 m Source 3 Q1 = 0.076 g s-1 Q2 = 0.152 g s-1 Q3 = 0.152 g s-1 Q4 = 0.434 g s-1 Range of Wind Directions

Source 1

Laser 3

Source 2

30 m

Laser 1 3 Q1 Q2 Q3 Q4

Ri

2

1

0

-1 S5

R1

S5

R2

S5

R3

S5

R6

S5

R9

S5

1 5 0 7 6 8 R1 5R1 5R1 5R1 5R1 5R1 S S S S S

15-min Repetition


5.2 Calculated Emission Rates We use a gas recovery ratio (R) as a measure of the accuracy of our emission calculations, and define

a recovery

ratio for each source Ri = Q bLS-i /Q i , and a total recovery ratio Rtotal = Q bLS-i / Q i (Q bLS-i is the calculated emission rate for source i, and Q i is the actual rate). A perfect calculation gives R = 1. 5.2.1 Scenario 1 The four sources were positioned in the corners of the release square (Fig. 3) and gas was released over two days during south-west winds. There was a wide range of meteorological conditions during the releases, from light to strong winds (u ∗ = 0.15–0.51 m s−1 ) and from strongly stable (L = 9 m) to moderately unstable stratification (L = −46 m). The accuracy of the inferred emissions varies greatly from one 15-min period to the next (Fig. 3), with recovery ratios Ri ranging from −3.5 to 5.5. A negative R implies that an emission sink that absorbs tracer within the “source” area best explains the observations (nothing precludes a negative Q in the solution). The potential for unrealistic negative emissions is a

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Scenario 6

Laser 7 Laser 8

Laser 5 Laser 4

N

Laser 3

Gas Release: 19 Sept 2006 Laser 2 u* = 0.20 to 0.34 m s-1 L = +29 m (stable) to -429 m (unstable) z0 = 0.01 to 0.05 m zlaser = 0.75 m Q1 = 0.076 g s-1 Q2 = 0.152 g s-1 Q3 = 0.152 g s-1 Q4 = 0.434 g s-1 Range of Wind Directions

Source 1

Source 2 Source 3 Source 4

30 m

Laser 1 Q1 Q2 Q3 Q4

30 20

Ri

10 0 -10 -20 -30 R S6

1

R S6

2

S6

R3

R S6

4

S6

R5

S6

R6

R S6

7

S6

R8

10 R9 S6 S6R

15-min Repetition

Fig. 7 The layout of Scenario 6 (top) and the results of the emission inferences displayed in terms of the component recoveries R for each 15-min repetition (bottom). An R = 1 is a perfect inference (red dashed line) Table 1 Average errors in bLS dispersion model predictions of laser concentrations Scenario

Average magnitude p error |CL – CLo | (ppm)

p

Average fractional error |CL – CLo |/(CLo − Cb ) All lasers

Five downwind lasers

1

0.30

0.30

0.15

3

0.18

0.62

0.17

4a

0.15

0.22

0.14

5a

0.43

0.26

0.12

6

0.48

2.92

0.19

a For scenario 4(5) we did not use laser 7(4) because of low light levels

problem (more on this later). The large period-to-period variability in calculated emissions is reflected in the standard deviations of Ri , with σR = 2.00, 2.24, 0.45, and 0.25 for the four sources (Table 2). This variability is much higher than found in similarly configured single-source experiments: e.g., Flesch et al. (2004) and McBain and Desjardins (2005) both found σR ≈ 0.2 for their single source studies.

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Table 2 Average gas recovery (R) for each scenario R1 (σR )

R2 (σR )

R3 (σR )

1

−0.69 (2.00)

2.08 (2.24)

0.91 (0.45)

1.51 (0.25)

1.13 (0.17)

3

1.06 (0.79)

0.76 (0.96)

0.74 (0.49)

1.17 (0.20)

1.04 (0.12)

Scenario

R4 (σR )

Rtotal (σR )

4

0.72 (0.28)

1.51 (0.20)

1.05 (0.16)

1.00 (0.07)

1.08 (0.05)

5

1.28 (0.61)

0.71 (0.30)

0.78 (0.29)

1.42 (0.37)

1.15 (0.18)

6

1.86 (15.6)

2.85 (3.92)

8.81 (10.4)

−1.58 (2.00)

1.51 (0.27)

The standard deviation of R for each scenario (σR ) is given in parenthesis; R = 1 is a perfect inference

The large variability in calculated emissions encourages averaging over all the 15-min periods. However, even after averaging there are large errors (Table 2), with a negative Q 1 and Q 2 having twice the actual rate. Although Q 3 is calculated to within 10%, Q 4 is overestimated by 50%. Perhaps the most interesting result is that, while Ri for the source components is highly variable and inaccurate, the total gas recovery Rtotal is only overestimated by 13%. This divergence between component and total accuracy is a reoccurring trend in our study. 5.2.2 Scenario 3 The sources were spaced along a diagonal of the release area (Fig. 4), and gas releases took place over two days during moderate south-west winds. Atmospheric stratification ranged from moderately stable to moderately unstable. The emission inferences from Scenario 3 are more accurate than in Scenario 1, with the large period-to-period variability σR reduced, and only the second repetition period (S3R2) shows the high level of Ri error observed in the earlier scenario. The average accuracy is also improved, with individual emission rates calculated to within 30% (Table 2). Total emissions are calculated to within a very accurate 4%. 5.2.3 Scenario 4 The layout of Scenario 4 is only marginally different from Scenario 3 (Fig. 5). Only the laser path heights were reduced from z laser = 1.5 to 0.75 m, and gas releases took place during moderate south-west winds and near-neutral stratification. The light levels of Laser 7 (diagnostic indicating laser alignment) fell below the manufacturers’ recommendation, and this laser was not used. This is the most successful scenario for inference accuracy. The period-to-period variability in Ri is significantly reduced: for Q 1 , Q 3 , and Q 4 the σR is 1/3 that of the next best scenario (Table 2). We no longer calculate any negative Q nor have any emission errors of more than 100%, and the only disappointment is that Q 2 is consistently calculated as having 50% more emissions than actual. The average calculation of total emissions is a good Rtotal = 1.08, and the standard deviation σR is a very small 0.05 (Table 2). 5.2.4 Scenario 5 In Scenario 5 we returned the four sources to the corners of the release area (Fig. 6) just as in Scenario 1, but with laser paths lowered to z m = 0.75 m and the position of laser paths 3 and 7 shortened. Gas releases took place on 25 September under light west to south-west winds,

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and atmospheric stability ranged from near-neutral to strongly unstable. The light levels of Laser 4 (indicates laser alignment) were inadequate and so this laser was not used. Compared with the consistent (and accurate) inferences in Scenario 4, there is a return to the more variable results of earlier scenarios. We again see errors of more than 100% for individual repetitions, and the largest inaccuracy is the average 42% overprediction of Q 4 (Table 2). Compared with the previous scenarios there is slightly less accuracy in the calculations of total emissions (average Rtotal = 1.15). 5.2.5 Scenario 6 The sources were placed near the centre of the release area (Fig. 7), and gas releases took place on 19 September during light to moderate south-west winds. Atmospheric stability ranged from near-neutral to moderately stable. Scenario 6 is our least successful configuration; for any one repetition the calculated emissions are in error by a factor of five or more, with many instances of emission sinks (Fig. 7). Not only is there tremendous period-to-period variability in the calculations, but the average calculations are in error. The average emission rate for Source 3, for example, is almost nine times the actual emissions. Total emissions are more accurate, although we still calculate 50% more emissions than actually occur (Table 2). 6 Discussion There are large differences in the accuracy of our emission inferences for the different scenarios. For example, the calculations for Scenario 4 are very accurate while those for Scenario 6 are very inaccurate. Even within a given scenario there is great variability from one 15-min repetition to the next. What explains these differences? We consider how meteorological conditions and the condition number relate to the results. 6.1 Role of Meteorology in Inference The foundation of the bLS dispersion model is Monin–Obukhov similarity theory (MOST), where statistical properties of the wind are described by u ∗ , L , z 0 , and β. But MOST relationships are less accurate in certain conditions, and in these cases we should expect less accurate dispersion calculations: • Low winds (low u ∗ ): Flesch et al. (2004) suggested u ∗ = 0.15 m s−1 as a threshold below which bLS calculations are inaccurate, and McBain and Desjardins (2005) suggested u ∗ = 0.19 m s−1 . • Extreme stability (small |L|): a traditional view is that MOST is inaccurate when |z/L| > 1. The limiting height z will be a maximum altitude that a significant number of source trajectories reach before intersecting the sensors. For our leeward most laser (fetch ≈ 100 m) it is possible that many trajectories reach z = 10 m, which implies inaccuracy when |L| < 10 m. • Disturbed winds: terrain obstacles can “disturb” the airflow and invalidate MOST wind relationships in their lee. At our site there were windbreaks to the south and north-west as well as some subtle inhomogeneities (a field of sorghum south-west of our site). We were surprised to see little evidence of a relationship between gas recovery and meteorological conditions. In fact, our least accurate results occur in conditions that should be conducive to accurate dispersion model predictions: moderate winds and near-neutral stratification. We believe two factors explain this: first, because we avoided releasing gas in the most

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obvious problem periods—low u ∗ and |L|, and wind directions other than south-westerly— we may simply have missed the obvious sensitivity. Secondly, as we will show, problem geometry appears to be a dominant factor in determining the accuracy of our inferences. Either the geometry effect is so large as to render secondary meteorological effects unnoticeable, or, as indicated in Crenna et al. (2008), these two effects maybe intertwined so that the meteorological effect is not evident in our simple analysis. 6.2 Role of Problem Geometry 6.2.1 Condition Number Each 15-min gas release period has a unique κ determined by the source–laser geometry and wind conditions; in this experiment κ ranges from 3.2 to 275. Figure 8 illustrates how total gas recovery (Rtotal ) and the recovery for Source 4 (R4 ) vary with κ across all scenarios. Two relationships are suggested: greater R variability as κ increases, and increasing bias in R as κ increases. These are especially visible if we remove four outliers in Scenario 5 (all occurring during the late afternoon transition from unstable to stable stratification, and typically associated with MOST inaccuracy). These trends are consistent with κ being an error amplification factor. 6.2.2 Detailed Look at Scenario 4 The accurate results of Scenario 4 make it a good choice for a detailed examination of the relationship between inference accuracy and κ. This is done by systematically removing lasers from the calculations (we will refer to the different lasers as L1, L2, etc.). We begin by using all six downwind lasers (L1 provides Cb , and L7 was not working), then ignore L2, then L3, then L4, and then ignore both L3 and L4, and finally L3 and L5. Each combination results in different κ values. The results of this exercise are shown in Fig. 9, where we see similarity with the earlier results: at higher κ we find extreme uncertainty and bias in the inferred emissions. It is interesting that instead of a continuous trend of increasing inaccuracy with increasing κ, there appears to be a threshold κ that divides accurate and inaccurate inferences. It is not clear why this would be so, but total emissions are calculated accurately if κ < 20 (i.e., Rtotal is calculated to within 25% in every case). Once κ > 50 the errors become large. The pattern for R3 and R4 shows a similar threshold behaviour (Fig. 9): for κ < 10 the component emissions are deduced to within 40% (and usually within 20%), but when κ > 10 the calculations become inaccurate. It appears the threshold for κ, above which the component inferences become inaccurate (κlimit ≈ 10), is lower than the threshold for inferring total emissions (κlimit ≈ 50). This is not surprising since total emissions are more tightly constrained by the aggregate of the concentration observations than are the components, i.e., component errors tend to cancel when they are summed to give the total emissions. 6.2.3 Geometry and Condition Number—Scenario 4 Condition number is clearly related to inference accuracy. Qualitatively, what explains a high κ and how can it be avoided in real world problems? For example, during Scenario 4 why is κ sensitive to removing L2 but not L4?

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When all lasers are included in the solution for Scenario 4 there is a progressive hierarchy of single-source calculations (Fig. 5). For south-west winds L2 “sees” only emissions from Source 1, and Q 1 can be determined with the accuracy of a single-source problem. With Q 1 accurately known, the contribution of Source 1 to L3 can be accurately calculated (and “removed”) so that Source 2 and L3 effectively form another single-source calculation. There is a similar progression up to Q 4 . This “progressive” layout results in a low κ and good accuracy. If L4 is removed from the calculations the situation does not appreciably change (κ remains low): it is still a progressive problem as L5 is positioned to take the place of L4 in isolating Source 3. But if L2 is removed the nature of the problem changes. Laser 3 “sees” the blended plume from Source 1 and 2, and Source 1 cannot be isolated (so κ increases); removing L2 and L3 exasperates the problem. Source isolation is the key to a small κ. 6.2.4 Hypothetical Case Consider the source layout in Scenario 1 (sources in the corners of release area). If charged with finding the four unknown emission rates, how would one best arrange four lasers to

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Scenario 4 All Lasers Remove L4 Remove L2 Remove L3 Remove L2 & L3 Remove L3 & L4 Remove L3 & L5

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accomplish this task? Minimizing κ is critical for success. Figure 10 shows four hypothetical layouts. The best design is Fig. 10a with a very low κ = 1.2. This configuration nearly creates a set of independent single-source problems. With the exception of the north-easterly laser, each laser “sees” the plume from only one source. This is the ideal for determining component emissions. In the real world this layout has disadvantages, and the short laser pathlengths constrain inferences to a narrow range of wind directions. And it might not be possible to place lasers/reflectors inside an industrial site due to management activity. Figure 10b shows an alternative layout where laser paths fully span the source boundaries (a possible advantage if there is management activity at the site). In this layout κ = 6.6, and the qualitative explanation for the κ increase compared to the prior layout is that we no longer have a set of independent single-source calculations. Instead we have the “progressive” layout

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Fig. 10 Layouts of a hypothetical multi-source problem (with the associated condition number κ). The filled squares represent four sources, and the dashed lines are four laser paths

described earlier, where laser placement enables a progression of single-source calculations. The κ = 6.6 is still reasonably low, and based on our earlier results we would expect to accurately infer the component emissions. The layout in Fig. 10c is a poorer choice. The increase to κ = 26 is explained by the fact that we no longer have a progressive layout, and while L1 “sees” only Source 1, L2 now “sees” the blended plume from Sources 1, 2, and 3. There is no longer an isolation of Sources 2 and 3. From our earlier results, we might expect that with κ = 26 there should be a reasonable estimate of total emissions, but probably not the component emissions. Figure 10d shows a layout with an important practical advantage. The laser paths are located away from the emission site so as to avoid interference with potential management activities. However, with κ = 3, 090 we do not expect to accurately infer either the total or the component emissions. The high κ occurs because no sources can be isolated, and most lasers “see” a blending of three emission plumes. The above examples again illustrate the importance of source isolation, and suggest there is an advantage in using point concentration sensors for multi-source problems. Point sensors make it easier to isolate a source–sensor pair and create a layout having a low κ (as with the short pathlength example in Fig. 10a). However, this is balanced by the disadvantage of a more limited range of wind directions that place point sensors in a useful position. It is interesting that while long line-averaging sensors provide important advantages when looking at single-source problems—an expanded range of useful wind directions and a reduced sensitivity to dispersion modelling errors (Flesch and Wilson 2005)—this becomes a disadvantage in multi-source problems.

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6.3 Alternative Solution to Eliminate “Sinks” Nothing requires our mathematical solutions to give positive emission rates: emission “sinks” are valid solutions. But most real world problems focus on emission sources. Can we avoid the unrealistic outcome of emission sinks? An iterative procedure is an alternative to a direct (or best-fit) calculation. One can iteratively search for emission solutions within bounds decided by the user, e.g., no negative emissions. In this case one defines a metric like the sum of the square errors, and iteratively finds an “in-bounds” solution that maximizes or minimizes that quantity. We apply this technique to re-calculate our scenario solutions without allowing negative emissions. We minimize the sum of square errors, using a technique called “annealing” (Press et al. 2007) to rapidly reach an iterative solution. Figure 11 compares the scenario gas recoveries calculated earlier with those from the constrained iterative method. The iterative solutions are better in several respects. By prohibiting negative emissions there is an increase in the accuracy of the component results and a reduction in the period-to-period variability in Ri . While for most scenarios there is

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Scenario Fig. 11 Comparison of two alternative calculation methods. The top graph gives the direct solution for gas recovery R (for the four components and total emissions) averaged by scenario (error bars are the periodto-period standard deviation σR ). The bottom graph gives the results for an iterative solution that prevents negative emissions. R = 1 is a perfect inference

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little difference in the total gas recovery Rtotal between the alternative solutions, there is improvement in our worst case Scenario 6 using the iterative approach: the average Rtotal is reduced from 1.51 to 1.34. For most situations the iterative approach should be superior as it allows extra constraints to the solution, better reflecting the physical reality of the problem. However, from an application standpoint it requires additional processing and complexity. 7 Summary and Conclusions A field study with four synthetic emission sources and eight concentration sensors was performed to examine the accuracy of multi-source emissions inferences. The accuracy depended strongly on the condition number κ, and the most successful layout (Scenario 4) was characterised by an average κ of 4.2. Here the total emissions were deduced to within 10% on average, although there was less accuracy in the component inferences. Our least successful layout (Scenario 6) had an average κ of 91, and in this case the total emissions were calculated to within only 50% and the component inferences were highly inaccurate. We also found: • The uncertainty in our dispersion model predictions of (C/Q)sim for any one 15-min period was 10–20%. This level of uncertainty is an important factor when judging the utility of a multi-source inference (because this error is amplified by κ). In real-world situations, where the terrain may not be ideal, the level of model uncertainty may be higher. • The quality of our inferences did not depend on meteorological conditions. We believe the lack of a meteorological effect was due to our avoidance of troublesome conditions (e.g., low winds). • Calculations of total emissions were inaccurate (errors > 50%) when κ exceeded approximately 50. Accurate inferences of component emissions exhibited a lower κ threshold of 10–20. In real-world problems, where non-ideal terrain increases dispersion model errors, the expectation would be for lower threshold values of κ. • Better results were found using an iterative solution technique that prohibited negative emissions; the improvement was most significant for our least successful Scenario having a large κ. We conclude that for problems with a smaller κ, the simpler direct calculation is appropriate. An iterative approach is recommended for problems with a large κ. The most successful problem layouts (minimum κ) occur when sources can be isolated (i.e., each sensor “sees” only one source). In many cases it is not possible to isolate sources, perhaps because of limitations in sensor locations or the need to accommodate a range of wind directions. The next best choice is a “progressive” layout, where one begins by isolating one source with a sensor, then adds the next sensor to see the first source plus another, then adds a third sensor to see the previous two plus a third, and so on. This layout (Scenarios 3, 4, and 5) gave reasonably good results. The accuracy of our inferences declined rapidly for layouts where it was not possible to isolate any source. Fortunately, it is possible to simulate problems in advance in order to determine the general magnitude of κ one will experience. If κ is low enough (our results suggest