An inversion technique is proposed to reconstruct an elevated point emission source of known height of atmospheric trace species from a finite number of concentration measurements in low-wind stable conditions. Observations from the SF6 diffusion experiment at Idaho National Engineering Laboratory in low-wind stable conditions are used for the point source reconstruction.
With growing attention to the contamination of the natural environment, the quantification of emission sources is necessary in regard to environmental monitoring and emergency response applications. Monitoring networks are deployed over the region of interest to detect and identify the emissions, and to this end it is required to establish a relationship between the source and the receptors. Often the concentrations are observed at the receptors and their origin of release is not known. The purpose is to identify unknown releases from the concentration measurements observed at the receptors. In low-wind stable conditions, the observed concentration distribution is generally multi-peaked and irregular (Sagendorf and Dickson, 1974). In these conditions, source reconstruction becomes complex because of relatively small and non-monotonic concentration gradients.
Source reconstruction in low-wind conditions (U <2 m s−1) is important but difficult. It is important because these relatively frequent conditions cause the pollutant to stagnate near the ground at relatively high and damaging concentrations. It is difficult because the stagnation also adversely affects the resolution of the source reconstruction. The diffusion of pollutant is irregular and indefinite in weak and variable wind. No single plume centreline is obvious and the observed concentration distribution is multi-peaked and non-Gaussian (Sagendorf and Dickson, 1974), especially in stable conditions. In these conditions, the state (turbulence and dispersion characteristics) of the lower atmosphere is not properly defined (Sharan et al., 2003; Kumar and Sharan, 2009). Thus it is necessary to explore the source reconstruction in low-wind stable conditions.
Yee (2008) used a Bayesian paradigm requiring a priori knowledge of sources and their expected distribution along with the form of background error covariance matrix which, in practice, is not known and is subject to assumptions. Recently, an inversion technique for the identification of a ground-level point source emission has been developed (Sharan et al., 2009) by modifying the methodology proposed by Issartel et al. (2007) for an areal source. The technique was evaluated with observations in low-wind convective conditions from a diffusion experiment conducted at the Indian Institute of Technology (IIT) Delhi. Concentration measurements are often sensitive to release height (Luhar, 2011) as well as height of the samplers above ground. Attempts have not yet been made for the reconstruction of an elevated point source in low-wind stable conditions.
The objective of the present study is to propose an inversion technique for the identification of an elevated point source in low-wind stable conditions.
2. Inversion technique
The methodology for source reconstruction is based on an inversion technique proposed by Issartel et al. (2007) for the retrieval of distributed areal emissions. This involves the computation of a distributed areal estimate that is linear with respect to the observations. Recently, Sharan et al. (2009) extended this methodology for the reconstruction of a ground-level point source using the fact that, in the case of observations generated from a point source, the maximum value of the estimate corresponds to its location. In the present study, this inversion technique is modified for the retrieval of elevated releases and a new formulation is proposed for the estimation of source intensity. In this section, we describe briefly the salient features of the inversion technique for the sake of completeness.
In this study, we are concerned with a continuous elevated point source generating a steady-state concentration field from which the n measurements µi (1, 2, …, n) are presumably taken at samplers having height zr above the ground. The point sources can be considered special cases among the vector space of functions and distributions s(x,y) representing a continuous flux of emissions in unit amount of tracer per unit area and time around the point of coordinates x and y on an area Σ which is the horizontal representation of the surface at an altitude z = h above ground, corresponding to the known height of the source. Here h is the height of the source above ground. The vector space is associated with a scalar product:
The scalar product (·,·)1 corresponding to the Euclidean geometry of area Σ is called the fundamental product of the inverse problem. It is chosen to reflect the priori assumption that all parts of Σ are equally likely to be the seat of emissions. Indeed, all portions of Σ in Eq. (1) are equally weighted without any special reference to a region as a likely location of a release.
2.1. Adjoint functions
The measurements are assumed to depend linearly on the emissions. Such dependence may be written as
in which ai(x,y) are the adjoint functions associated with each measurement for the scalar product (·,·)1 (Eq. (1)). The adjoint function ai is a key element (Marchuk, 1995) in Eq. (2), providing correspondence between a source s and the measurements µi. It is derived as ai(x,y) = ri(x,y,h), in which ri(x,y,h) is adjoint to µi among three-dimensional emission functions σ(x,y,z) with respect to the scalar product (Issartel etal., 2007). Air density ρ(x,y,z) does not really play a role in this study since it is practically constant in the thin air layer effectively involved in the local scale dispersion process. The function ri(x,y,h) is computed as a steady-state plume scattered back from the ith sampler located at reference height zr and thus it is termed an inverse plume. Since the sought source is located at z = h, the inverse plumes ri(x,y,h) are collected on the horizontal cross-sectional area passing through the plane z = h. In order that steady-state inverse plumes be well defined, an infinite domain is theoretically considered as for a finite domain; a constant source would not lead to steady state unless it is balanced by a sink or perturbations from the background are considered.
A symmetric relationship for forward transport (L(σ) = χ) and adjoint steady-state transport (L*(πi) = ri) can be described as (Marchuk, 1995)
in which χis tracer mixing ratio, L is a linear operator describing the forward steady-state transport, L* is its adjoint describing the inverse transport and πi(x,y,z) is the sampling distribution indicating the regions from which the sample for the ith measurement was taken (Issartel et al., 2007). As mentioned earlier, the inverse plume ri(x,y,h) at z = h corresponds to the adjoint function ai.
In view of Eq. (2), the measurement µi = (s,ai)1 is the scalar product of the sought source s with the adjoint function, which allows computation of its projection s| over the vector space generated by the ai. Indeed, the source can be decomposed as s = s| + s⟂ into components respectively parallel and perpendicular to ai. Since (i) s⟂ does not contribute to the measurements and (ii) s| has minimum norm (s||,s||)1/2 among the source functions compatible with the measurements, it seems natural to consider s| as a source estimate. Such estimation is, however, not acceptable. The adjoint functions ai associated with point detectors are singular at its corresponding locations, implying that s|, which is a linear combination of them, would have positive or negative infinite values at the location of the detectors. The problem with its mathematical description and physical consequences is described in Issartel et al. (2007).
To overcome this, a modified scalar product was introduced as follows:
where φ(x,y) are positive weights and aφ i are the adjoint functions associated with the weighted scalar product (·,·)ϕ.
Again, the source function can be decomposed as s = s||φ + s⟂φ with parts respectively parallel and orthogonal to all aφ i. Parallelism and orthogonality are considered according to the weighted scalar product (Eq. (4)). The part parallel s||φ is a linear combination of aφi:
in which the coefficients λi are derived by substituting Eq. (5) in the expression of the measurement; one obtains
where λ and are respectively the column vectors of the coefficients and measurements and Hφ is the Gram matrix of the weighted adjoint functions for the modified scalar product (Eq. (4)). The source estimate may also be written as
in which aφ(x,y) is the column vector of renormalized adjoint functions. The superscript T denotes the transposition.
In order to eliminate the singularities of the adjoint functions and minimize all inversion artefacts, the weight function φ(x,y) is chosen in such a way that (Issartel, 2005)
The weights φ subject to Eq. (8) are called renormalizing weights. The function φ(x,y)is also referred to as the visibility function as it is helpful in the physical interpretation of the extent of the regions seen by a monitoring network. The interpretation of φ as the visibility function associated with the monitoring network was explained by Sharan et al. (2009) and substantiated further by Issartel et al. (2012). In particular, φ(x,y)assumes large values close to and upwind of the samplers; it vanishes downwind of the monitoring network, corresponding to the impossibility of detecting a release there.
2.3. Identification of a point release from measurements
A point emission source of intensity q, known height h located at position (x0,y0) is represented as an emission function qδ(x − x0)δ(y − y0) at z = h, where δ(·) is the Dirac delta function. The measurements corresponding to it are
where ai,φ and aφ i are defined at the point (x0,y0) lying on the area passing through height z = h. For these measurements, the estimate is derived from Eq. (7) as
The estimate s||φ is distributed on the whole domain passing through z = h. Using the Cauchy–Schwartz inequality, Sharan et al. (2009) have shown that the maximum of this source estimate (Eq. (11)) in the whole domain coincides with the location of the point source. Its location (x0,y0) is retrieved from a given set of measurements by maximizing the function s||φ(x,y) defined by Eq. (7). Once the location is obtained, the source intensity q is derived from Eq. (11) using the fact that s0(x0,y0) is unity in view of Eq. (7):
In Sharan et al. (2009), the intensity is estimated from the observed measurements as
which can be obtained from Eq. (7) using the relations (10) and (11).
Both these expressions (Eqs (12) and (13)) yield the same value for the intensity as long as the measurements are ideal and completely free from noise. However, in reality, the measurements are rarely free from random noise. Estimation of the location and intensity of a source is subject to noise due to (i) measurement errors associated with the sensors and (ii) representativity errors associated with the dispersion model. Owing to the noise, the location is estimated at the point (xe,ye), where the estimate becomes maximum. However, Eqs (12) and (13) are not equivalent at estimating the intensity of the release when the unknown true location (x0,y0) is replaced by (xe,ye). It is shown (Appendix A) that Eq. (13), used earlier, includes a positive bias and overestimates the intensity. The intensity of the source should be estimated from the real measurements as using Eq. (12) by evaluating both the numerator and denominator at the estimated location (xe,ye) in the area Σ passing through z = h.
3. Diffusion experiment
In the present study, we utilize the observations from a series of diffusion experiments conducted at Idaho Falls (Idaho, USA) with light wind stable conditions (Sagendorf and Dickson, 1974) over flat terrain for the retrieval of a point source. The tracer SF6 was released from a continuous point source at a height of 1.5 m above the ground and collected at a height of 0.76 m. The test criterion was a stable lapse rate with a wind speed of less than 2 m s−1. Sixty samplers were placed on each circular arc of radii 100, 200 and 400 m from the point of release, with an angular spacing of 6° between them (Figure 1).
Meteorological measurements were given at 2, 4, 8, 16, 32 and 61 m levels on a 61 m tower located on the 200 m arc. The range of hourly average wind speeds was 0.8–1.9 m s−1 at the 4 m level. Based on the average wind direction measured in runs 4, 5, 6, 7, 8, 9 and 10 every 2 min and 11, 13 and 14 every 1 min 20 s subinterval, the variability in the wind direction in each hour varies from about 40° in run 13 to 360° in run 8. These variabilities in the wind direction are consistent with the plume width in the measured concentrations in various runs (Sagendorf and Dickson, 1974). An hourly average plume, just like an instantaneous plume, is unlikely to be cone shaped due to lack of a well-defined plume central line. As a result, the observed concentration field normally has a reasonably large lateral spread and multiple peaks. Such an irregular behaviour of the plume is due to the usual lack of homogeneity and stationary in the dispersion process in low-wind conditions.
Of a total of 14 tests conducted, data for 11 (runs 4–14) were reported (Sagendorf and Dickson, 1974). A meandering phenomenon is observed in the concentrations measured in all the runs, but the case of run 12 corresponds to a large extent of concentration meandering such that observed concentrations are reported only on a 100 m arc in the third quadrant and not on 200 and 400 m arcs. Accordingly, this run is not considered in the present study and, out of 11 runs, only 10 are used in the analysis. In test 8, the concentrations were measured at all the samplers around 360° and so this can be considered as a case with a large variability in the wind direction.
Of the 10 tests, the release rate was g s−1 in runs 4, 5, 9, 10 and 14, whereas it was g s−1 in runs 7, 8, and 13. In the remaining runs, 6 and 11, the release rate was g s−1. In all the runs, the release point was located at the centre of the circular arcs.
Sagendorf and Dickson (1974) reported the concentration measurements at the samplers operated in each run. In order to save costly analyses, the SF6 plume was made visible by simultaneously releasing oil fog. The samplers away from the plume were switched off (Sagendorf and Dickson, 1974). Accordingly, the number of concentration measurements in each run was different, depending on the spread of the plume. This spread is determined mainly by the wind speed and by the variability of its direction. Here, for computational purposes, the release is taken from an elevated source with an effective stack height h = 3 m, as reported in Sagendorf and Dickson (1974), and samplers are located at a height of zr = 0.76 m above the ground.
4. Numerical computations
The adjoint functions ai in the inversion technique are computed using a dispersion model developed by Sharan et al. (1996) for elevated releases in low-wind stable steady-state conditions. This requires the values of wind, dispersion parameters (σsx, σsy and σsz) and atmospheric stability. The atmospheric stability based on vertical temperature gradient is determined using the temperature measurements at the 8 and 32 m levels. The various schemes for the parametrization of dispersion parameters were evaluated (Sharan et al., 1995) for the treatment of low wind dispersion with the Idaho diffusion experiment. A split plume scheme is used in which horizontal stability is governed by horizontal wind direction fluctuations, whereas vertical stability is determined by the vertical temperature gradient. Thus, in the horizontal direction, the dispersion parameters σsx and σsy are parametrized in terms of σtheta (standard deviation of horizontal wind direction fluctuations) using the relations (Cirillo and Poli, 1992)
Notice that these are based on the formulations of σu and σv (standard deviation of the components u and v of horizontal wind) in terms of σtheta. Recently, Luhar (2011) pointed out the uncertainty involved in the streamwise turbulence component σu, and thus σsx = σsy is taken, in which σsy is calculated from Eq. (14). However, computations have also been carried out for source estimation with σsy≠σsx (Eq. 14) in each run and we obtained almost similar results. More precisely, the location is retrieved at the same grid point, whereas the variation in the intensity retrieved is within 5% in comparison to that with σsx = σsy in all the runs.
In the vertical direction, based on the stability of the vertical temperature gradient, σsz is estimated from the analytical expressions based on Pasquill–Gifford (P-G) curves (Green et al., 1980);
where b,a and d are constants depending on the atmospheric stability. Here, we have considered the values of wind speed and σtheta at the 4 m level.
The observed concentration field normally has a reasonably large lateral spread and multiple peaks. Hourly average concentration from a constant K or a variable K model, using hourly wind data, gives a single peak and a narrow lateral plume spread. Such a method fails to explain multiple peaks in the concentration field (Sharan et al., 1995). Segmented plume and short-term averaging approaches have the capability of overcoming the limitations and thus provide a significant improvement in reproducing the observed concentration field (Sharan et al., 1995, 1996). This is due to the fact that they account for the variability in the wind by using the solution or model for every subinterval of the test period for which observed mean wind data are available. In the observational dataset, the wind speeds, wind directions and wind direction fluctuations (σtheta) averaged over a period of every 2 min in runs 4, 5, 6, 7, 8, 9 and 10 and a period of every 1 min 20 s in runs 11, 13 and 14 are available. Accordingly, for our purpose, the hourly tests are divided into 30 subintervals of period 2 min or 45 subintervals of period 1 min 20s each, depending on the runs. The release is assumed continuous in each of the subintervals (Sharan et al., 1995). The dispersion model is used to compute the concentrations at the samplers in each of the subintervals and then the hourly averaged concentrations are obtained by taking an arithmetic averaging over the subintervals, implying that an equal weight is given to each subinterval plume.
For each run, two sets of computations are carried out by considering concentrations observed in the experiment and analogous synthetic measurements generated from the model (Sharan et al., 1996). The synthetic data minimize the errors associated with the dispersion model and measurements. This helps in evaluating the inversion process as well as in interpreting the results with real data. Source reconstruction mainly consists of the following steps: (i) generation of retro-plumes; (ii) computation of the renormalizing weight function; (iii) computation of the renormalized estimates; and (iv) estimation of the location and release rate of the source.
The dispersion model (Sharan et al., 1996) is used to compute the concentration at all the receptors having height z = zr, assuming the intensity of release as unity from z = h, and these computed concentrations are taken as measurements sampled at receptors and termed as synthetic measurements. This is also used to compute the concentrations in backward mode for defining the adjoint functions at height h by simply changing the wind direction by 180° considering the receptors at height zr. Note that adjoint functions for a given run are superposable on each other with reference to the sampler at the origin. Accordingly, in each given run all adjoint functions are deduced from a single computation (Sharan et al., 2009).
The renormalizing weight function φ is computed by using algorithm described by Issartel et al. (2007). A domain of size 1596 m ×1596 m is chosen and it is discretized into 799 × 799 grid points for the inversion process (Figure 1). Each mesh is a square of 2 m × 2 m. The centre of the circular arcs is at the grid point (400, 400). The adjoint function ai is singular at the position of the detectors at height zr. Since adjoint functions are collected at known source height z = h instead of sampler's height zr, singularity will not arise in the generation of adjoint function at source height h. Finally, convergence of the algorithm to compute φ with an error of 10−5 is attained within 20 iterations.
5. Results and discussion
The source estimation has been carried out for two types of data: (i) synthetic; and (ii) real. The monitoring network has 180 samplers (Figure 1) located on all three arcs. However, the samplers located away from the SF6 cloud visualized by means of oil fog were switched off (section 3). Samplers reporting concentrations normalized by a source strength larger than the threshold value of 10−7 s m−3 were considered. Thus the number of measurements taken in each run varies depending essentially on wind speed and wind direction variability. The corresponding visibility function φ is represented in Figure 2(a). From the computations and geometry, it can be shown that φ is peaked at each receptor at height zr and its geometry disperses widely as we move above the ground from height zr to h. Except in run 8, φ decays upwind of the monitoring network and becomes negligibly small after approximately 800 m. A source located downwind of the monitoring network cannot even be detected (Sharan et al., 2009) as the function φ vanishes immediately in that region. For run 8, Figure 2(a) shows that, in decaying away from the detectors, φ forms two lobes around the directions of approximately 225° and 350°, measured clockwise from the north. Thus the very irregular distribution of wind direction in that run cannot be described in terms of an average direction.
6. Synthetic data
In the case of synthetic data, intensity and location retrieved are similar to those prescribed in all runs. The global maximum of computed intensity in the whole domain is exactly at the same grid point (400, 400) prescribed as the original location of the release. For all runs, the estimated intensity is found to be almost exactly the same as the prescribed one: g s−1 in the case of runs 4, 5, 9, 10 and 14, g s−1 in case of runs 7, 8 and 13, and g s−1 in the case of runs 6 and 11. A very small deviation from the prescribed values occurs due to round-off errors (0.02%) involved in the computations. The exact retrieval of the prescribed values with synthetic data provides confidence in the inversion technique used for the source reconstruction in low-wind stable conditions.
6.1. Real data
With real data, source is reconstructed for all ten runs. In the case of all runs, the source estimate has its maximum close to the prescribed location (400, 400) within an average error of 20 m which is small compared to the 100 m radius of the innermost circle of samplers. In run 4, a maximum of the source estimate has been obtained at grid point (402, 396), 9 m from the original release; no secondary local maximum is observed. In runs 7, 9 and 11, reconstruction results a maximum of the source estimate around grid point (399, 404) in run 7 with an error of 8.2 m, at (400, 395) in run 9 with an error of 10 m and at grid point (408, 405) in run 11 with an error of 19 m in comparison to the original release location, respectively (Table I).
Table 1. Reconstruction results with the real data in all runs.
The experimental point releases (first and second rows) with renormalized estimates (third and fourth rows) are indicated in terms of location in grid units and intensity ( g s−1). In the fifth and sixth rows, the value of s||φ (g m−2 s−1) at real source location is compared to the maximum value corresponding to estimated source location. In the seventh and eighth rows, the value of the weight function φ (×10−6) is compared at real and estimated (Est.) source locations. The location error (m) and angular deviation (degrees) are shown in the ninth and tenth rows respectively.
Angular departure θe
In runs 5, 6, 10, 13 and 14, a maximum of the source estimates is elongated along the mean wind direction. In runs 10 and 13, reconstruction results in a flat maximum around grid point (384, 385) in run 10 and (407, 421) in run 13, shifted in the upwind direction of the monitoring network with an average distance of 45 m. The occurrence of a flat maximum making release identification relatively inaccurate is associated with weakly illuminated regions (Sharan et al., 2009). In run 8, the spread of concentration measurements is over 360° due to the variability in wind direction. In this run, the maximum of the source estimate coincides with the grid point (408, 393) with an error of 21 m from the original release.
In the reconstruction, the intensity of the source is retrieved within a factor of 2 in all the runs (Table I). Notice that the intensity q (Eq. (12)) depends on the source estimate s||φ and weight function φ at the estimated location of the source. The shifting of the estimated location from the prescribed one upwind/downwind of the monitoring network will change the value of weight function φ appearing in the denominator. This may result in over/under-prediction of intensity. In addition, the magnitude of the estimated source (s||φ) will also vary. However, the over/under-prediction of intensity will depend on the relative variation of s||φ and φ .
The physical reasons for deviations in the estimated intensity may be attributed to (i) large variability in the wind direction, (ii) meandering in the concentration measurements (Figure 3), (iii) the multi-peaked nature of the observed concentration distribution (Figure 3), (iv) uncertainties in meteorological and concentration measurements and (v) lack of representation of the observed concentration measurements by the atmospheric dispersion model used for generating the adjoint functions, which in turn influence s||φ and φ .
The computational time required for source estimation depends on the number of measurements and number of grid points in the domain in each run. In this study, the CPU time for source identification with 180 measurements in run 8 on an Apple (Intel® Core™ 2 Duo CPU E8135 at 2.66 GHz) 32-bit machine, was 23 min, and the corresponding time in run 6 with a minimum of 24 measurements was ∼ 2 min 20 s.
In the present study, the total number of grid points is taken as 799 in each direction with a resolution of 2 m. However, the computations have been carried out with finer resolutions of 1 m (1599 grid points) and 0.5 m (3199 grid points) and the computed results are found to be similar.
Henceforth, for the sake of brevity, results are presented for representative runs 4, 6 and 8 in Figure 2(a), (b) and (c). The source estimates are shown in (b) and (c) in the form of isopleths in the x–y plane, whereas panel (a) represents the function φ(x,y ).
6.2. Extension of the estimate upwind
The source estimates in the usual geometry (Figure 2(b) and (c) (runs 6 and 13)) are composed of mainly two parts: (i) the maximum of the estimate as a possible location of release spread in the central region of the domain and (ii) the extension of the estimate observed upwind of the monitoring network. To clarify this upwind extension, the features of the estimate are represented in a modified framework accounting for the modified geometry. After choosing a pole, transformation is made in polar rather than Cartesian coordinates (l,θ) → (l′,θ) such that dl′2 = φ(l,θ)dl2 so that the area is shrunk or extended in proportion to φ (Issartel, 2005). This transformation helps in (i) magnifying the central region and (ii) shrinking the information corresponding to the extended upwind estimate. In the transformed geometry, the pole is chosen at the centre of the domain so that the correspondence between the original and transformed geometries could be maintained along the radii. In the transformed geometry, the first part of the estimate at the centre is magnified, while the second part of the upwind extension shrinks. The samplers are distributed in the form of five to seven branches. From the transformed geometry, it becomes clear that the upwind extension of the estimate away from the central region is unimportant and does not contribute significantly to the measurements. Thus this information is insignificant, generated in the regions weakly illuminated due to the absence of receptors, and its influence becomes minimized in the transformed geometry. Notice that no such feature of extension of estimates is observed in run 8 as the release location is surrounded by all 179 operated receptors around the circular arcs. The similar features are observed with the analogous synthetic runs.
6.3. Errors in retrieval
Ideally, in view of Eq. (7), the non-noisy measurement vector and the adjoint vector a(x0,y0) at the release location are parallel. However, in reality the noisy measurement vector may not be exactly parallel to the adjoint vector a(xe,ye) at the estimated release location. The angular deviation (θe) between these two can be computed and provides an indication of the intensity of the noise. A brief description for the estimation of angular departure between measurement and adjoint vectors is given in Appendix B. This departure is naturally formulated in the form of angle θe such that . For non-noisy data, the accuracy of the source identification in respect of the intensity can be justified by the minimum value of angular deviation. The angle θe becomes close to zero when the measurement vector lies along or exactly coincides with the adjoint vector. This is observed and verified with the synthetic data, where the renormalized estimates of intensities are computed almost exactly to the corresponding prescribed parameters. Similarly, error estimates are also computed for real data in all runs (Table I). With real data, it is observed that the adjoint vector does not coincide with the measurement vector exactly and departs by an angular distance of 42° on average in all the runs.
6.4. Sensitivity analysis
A sensitivity analysis is carried out for analysing the influence of background concentration. Here the word ‘background’ refers to the prevailing concentration of the tracer in ambient air before the actual release. This is also used in the context of the Bayesian theory in which the background emissions designate the first guess of source emissions prior to the observations and to inversion. In the present text, the word is used for denoting the ambient background concentration of a tracer in the atmosphere.
The monitoring network in the Idaho diffusion experiment includes, in principle, a total of 180 samplers, arranged on arcs of 100 m, 200 m and 400 m with release at the centre. As mentioned in section 3, the SF6 plume was made visible by simultaneously releasing oil fog. As a result, the detectors away from the plume were switched off. The number of samplers operating or reporting the concentrations normalized by a source strength larger than 10−7 s m−3 varies from run to run depending on the variability in the wind direction. However, in the dataset, at all other samplers, a threshold value of normalized concentration is recorded as 10−7 s m−3. In the computations in section 5.2, while considering the monitoring network, the samplers with normalized concentrations larger than the threshold value are taken into account. Now we examine the impact of the presence of the remaining samplers in the monitoring network with a background concentration equal to threshold value on the source reconstruction.
For this purpose, the computation has been carried out with all 180 samplers by considering the measured normalized concentrations at the remaining samplers as 10−7 s m−3. The inversion is performed for all ten runs. The extension of the upwind estimates disappears (Figure 4(i), (ii), panel (b) and (c)) in the presence of the samplers with background concentrations in the monitoring network and the source estimate becomes more focused around the estimated release point. From the computations (Table II), it is observed that there is a slight variation in the estimated release location in runs 4, 7, 10, 11and 13. The intensity is still predicted within a factor of 2 in all the runs; however, they are becoming relatively closer to those originally prescribed in most of the runs except runs 4, 7 and 11. The average angular departure is estimated as 42.9°, compared to 42.0° obtained in the absence of samplers with background concentration.
Table 2. Sensitivity results with respect to the background concentrations in all runs.
In our earlier study (Sharan et al., 2009), we have highlighted the issues and limitations related to the inversion technique. Here we would like to point out additional issues related to data and dispersion model.
The observed concentration distribution in the Idaho diffusion experiment is multi-peaked and exhibits a meandering phenomenon in all the runs dependent on the extent of variability in the wind direction. Occurrence of large wind variability and meandering causes an irregular diffusion of the pollutant. In spite of this, we have used this dataset in the present study for source reconstruction as the diffusion experiments in low-wind stable conditions are very limited. However, this approach for source reconstruction needs to be further evaluated with the availability of diffusion data in low-wind stable conditions.
7.2. Representativity error
Primarily, representativity error refers to the lack of representation of the observed concentration distribution by the chosen atmospheric dispersion model. In the present study, the representativity of the model is certainly limited by large variability in the wind direction and meandering, resulting in the multi-peaked and non-Gaussian nature of observed concentrations. Representativity errors are expected in view of the uncertainties in the meteorological measurements and of the formulation of dispersion parameters (Luhar, 2011). However, we are able to retrieve the source within a reasonable error. Thus alternative approaches need to be evolved for minimizing the model representativity errors to improve the retrieval of the source.
The present study is concerned with the reconstruction of an elevated single point emission source from the concentration measurements in low-wind stable conditions corresponding to a continuous release. An earlier proposed inversion technique for a ground-level point source (Sharan et al., 2009) is modified for the source reconstruction of an elevated point source of known height along with improved formulation for the estimation of source intensity.
The source reconstruction is evaluated with real data taken from the Idaho diffusion experiment and analogous synthetic data generated from the atmospheric dispersion model. With the synthetic data, source location and release rate are exactly retrieved in all the runs. However, with the real data, the position of the source is retrieved within an average error of 20 m and intensity is retrieved within a factor of 2 in all the runs. The incurred errors in the retrieval are justified by estimating the angular departure between the vectors of measurement and adjoint function.
The inclusion of the samplers with the background concentration having a threshold value 10−7 s m−3 is shown to minimize the artificial information distributed in the domain and makes the source reconstruction more focused. This study raises the issue that the error in the source reconstruction is not only subject to noise in the measurements but also to the model representativity and wind variability.
Estimation of Intensity
The location x0 = (x0,y0) and intensity q of a continuous point source can be estimated using Eqs (10) and (11) from ideal measurements free from noise. If s||φ is the source estimate computed (Eq. (7)) from such an ideal measurement vector = q0φ(x0)aφ(x0) (Eq. (10)), it can be easily shown from Eqs (7, 10, 11, 12 and 13) that the following four quantities are equal:
In this, all these identities will lead to the same value of intensity.
However, the measurement vector r really observed is noisy and can be expressed as
where Δ is the noise vector decomposed in to an instrumental error associated with the sensors and representativity errors associated with the model. Owing to the noise, r may not be ideally associated with a point source, either at (x0,y0) or at any other location. It is, however, possible to determine a location xe = (xe,ye) and an intensity qe such that the ideal measurement vector
corresponding to xe and qe is the closest one to r.
can be rewritten using Eqs (A1) and (A3) as
in which s||φr is the estimate computed from real measurements. Notice that the least value of the function f(α,β) = α2 − 2αβ for a fixed β will occur at α = β and its least value is f(β,β) = −β2. Using this fact in Eq. (A5) by taking α = qeφ(xe) and β = s||φr(xe), the expression on the right-hand side of Eq. (A5) vanishes. This provides the conclusion that becomes minimal when (i) the estimate s||φr(xe) is maximal and (ii) s||φr(xe) = qeφ(xe ).
It is interesting to note from Eqs (A2) and (A3) that r = e + ε, and from geometrical representation of the vectors one gets
This inequality can also be obtained by the property of norm in (A5). This implies that the intensity from the real measurements using Eq. (13) is overestimated.
Thus intensity corresponding to the real measurements should be computed from
Estimation of Errors
The location of the source is estimated from the actually observed noisy measurements r at a location xe = (xe,ye) such that is a maximum. This is equivalent to maximizing . Since vectors and aφ(xe) both have norm 1, the estimated location xe of the release is that such that the vector aφ(xe) minimizes the angular distance from the observations r.
Notice that θe provides an indication of the intensity of the noise contained in the data for the following reasons. First, if there was no noise in the data, i.e. r = = q0φ(x0)aφ(x0), we would obtain xe = x0 and thus θe = 0. Second, it is smaller than the angle between the observed and ideal measurement vectors from the location, i.e. and, third, θe can be computed from the observations whereas, in principle, θr remains unreachable. In controlled experiments where the real location of the source is in fact well known, it is possible to determine both angles θe and θr.
The authors wish to thank the reviewers for their valuable comments/suggestions.