March 1979 / Vol. 4, No. 3 / OPTICS LETTERS 75

Two-dimensional remote air-pollution monitoring viatomography

Robert L. Byer

Applied Physics Department, Stanford University, Stanford, California 94305

Lawrence A. Shepp

Bell Laboratories, Murray Hill, New Jersey 07974

Received October 30, 1978

We propose to apply computerized tomography to measure a two-dimensional pollutant-concentration map overan area that may contain several potential sources of pollution. A tunable-laser source at the center of the areagenerates secondary or virtual light sources around the perimeter of the area that play the role of x rays in conven-tional computerized tomography of the human body.

The use of tunable-laser sources for remote pollutionmeasurements is now well known.1"2 Depth-resolvedabsorption measurements along a line can be obtainedby using atmospheric backscattering and time resolvingthe return signal. Because of the weak backscatteredreturn signal, high transmitted laser energies and largereceiver diameters are required in addition to high-speed detection and signal digitization to obtaindepth-resolved measurements over a few kilometers'range to a resolution of 10 m.

We propose to use computerized tomography 3' 4 togenerate a two-dimensional map of pollutant concen-tration. Air-pollution measurements by computerizedtomography provide several advantages over the dif-ferential-absorption method. These include a signifi-cantly lower required transmitted laser energy, in-creased range, and the ability to monitor an area con-taining several pollutant sources so numerous as tomake it impractical to monitor each of them separately.The measurement is implemented by using a single lasersource located at the center of the area to generate anumber of secondary light sources around the perimeterthat are analogous to x-ray sources in conventional to-mography of the human body.

Specifically, we suppose that a tunable-laser sourceL is mounted in the center of a circle of radius R, asshown in Fig. 1, such that its beam can easily be rotatedand directed toward the circumference of the circle. Ateach of n points equally spaced on the circumference,we mount a cylindrical mirror Mi (i = 1 ... n) so that thecollimated incident laser beam is reflected in a fan beamover an angle y across the circle. The beams from Mitraverse the circular region A and strike a set of m fixeddetectors Dj(j = 1 ... m). The mirrors and detectorslie in a common plane P, which may be parallel to theground at a sufficient height h so that the beams are notobscured by hills, chimneys, or buildings.

In traversing the path from Mi to Dj, the laser beamsare attenuated exponentially so that

P=- PQ- exp[ -L aA(xyX)dsJ,where Pr, and PQj are the received and transmittedpowers over the path Lij and aA(x,y,X) = aR(x,y) +aM(X,y) + aabs(XYX) is the atmosphere extinctioncoefficient composed of terms due to Rayleigh and Miescattering and to molecular absorption. For moleculardensity N with absorption cross section a-, aabs(XyX)= N(x,y) - (X).

We define a projection number Pij to be used as inputto the tomographic reconstruction algorithm as the ratioln(PQ-/Pr;), where P7, is the normalized intensity atmirror Mi. Thus

Pij = X aA(X,y,X)ds,Lij (1)

where the integral is over the path from Mi to Dj.To make a measurement, the tunable-laser beam

rotates and strikes mirrors Mi, which illuminate de-tectors Dj. The projection numbers Pij are measured,stored, and used to form the input to the mathematicalalgorithm,5 which allows the approximate reconstruc-tion of aA (X,y, X) for all (x,y) in the monitored area A.To recover the pollutant density, two successive mea-surements are made with the laser tuned on and off thepollutant absorption, and the difference in extinctioncoefficients is taken to yield N(x,y)cr(X). The recon-structed pollution field is then gray coded and displayedas a picture.

If aA (x,y, X) is a continuous function f, the Radontheorem guarantees that an exact reconstruction ofaA (x,y, X) can be found, given all projection numbers.In practice, only a finite number of measurements canbe made, and an approximation of -aA (X,y, X) toXA (x,y, X) is formed. The closeness of the approxima-

tion of -aA (X,y, X) to aA (x,y, X) depends on the smooth-ness of f and on the sampling numbers n and m.

Simulations3' 4' 6 made for the problem of designingan x-ray tomographic device for human-body sections

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76 OPTICS LETTERS / Vol. 4, No. 3 / March 1979

these assumptions, the fraction of power collected bythe receiver is v1T/irLij.

For a dark-current-limited detector likely to be usedin the infrared, the minimum detectable power at asignal-to-noise level S/N is

Pmin = NEP(S/N)v2f, (3)

where Af is the detector or electronics bandwidth andNEP = 4/a-/D* is the detector noise equivalent power,which is related to the detectivity, D*, by the squareroot of the detector area a.

By setting Pi = PMin, we determine that the requiredtransmitted power for the measurement at a given(S/N) level is

preq = NEP(SIN)x\17&if(4)

K (V exp|- aL A (xyX)dsI

Fig. 1. Schematic of a computerized air-pollution tomog-raphy measurement using fan-beam geometry. The lasersource at L rotates and illuminates mirrors at Mi, which actas virtual sources and in turn illuminate detectors Dj.Monitored pollutant sources Q are located within an areadetermined by the fan-spread angle y.

involved the use of superpositions of several ellipses tosimulate body parts. These simulations should alsoserve well for the air-pollution case because, if Q is asource of pollutant, say a chimney or leak, then by thetime the pollutant has attained a height h in the mea-surement plane, the pollutant source cloud has assumeda circular shape because of diffusion; if there is a windfield, the cloud has assumed an elliptical shape.

Based on the simulations3' 4' 6 and on Nyquist's theo-rem, it is a useful rule of thumb that, to detect clearlya circular pollutant cloud, which may be only a few percent more absorbing than the surrounding air, it isnecessary that at least two or three line integrals fromeach fan of measurement cross the cloud. Thus, as-suming that the two rays adjacent to the central ray arejust tangent to a circular pollutant cloud of radius r =10 m, where the circle A has radius R = 1000 m, we findthat m - 27r/2 sin-' (10/1000) = 314 detectors areneeded. Since for this application the mirrors Mi aremounted on poles or buildings and it is natural to mountthe detectors adjacent to Mi, we take n = m.

We can estimate the required transmitted laser powerneeded to carry out the tomographic measurement bysetting the power received at the detector equal to theminimum detectable power at a given signal-to-noiseratio. Conservation of energy gives us the followingexpression for the power received at detector Dj (Ref.2):

Pr =KPQ VX exp - aA(x>yX)dsI (2)

where K is the system's optical efficiency, P? and Pj arethe transmitted and received powers, A is the receiveraperture area, and Lij is the path length. Here we haveassumed that mirror Mi fans the incident laser beamuniformly over a 1800 field in the plane and that thecircular receiver aperture is large enough to accept thediffraction-limited laser beam out of the plane. Under

For a pulsed-laser source, the required transmittedenergy is found by multiplying Eq. (4) by r, the systemresponse time, and assuming that Afr = 2, to give

hpreq - 2 NEP(S/N) - IJK V-) exp - fL aA(Xy, X)ds]

Note that the required power and energy scale linearlywith Lij rather than with L?., as in the differential-absorption or topographical target-measurement case.Also note that the detector bandwidth can be reducedto lower the cost of the detector and digitizing elec-tronics without increasing the 'required laser powersignificantly. Here we have also made the simplifyingassumption that the transmitted power is not signifi-cantly depleted when tuned onto the absorption reso-nance. An extension to the case of significant depletionis straightforward and has been treated previously byByer and Garbuny.7

As a first example, we assume that the measurementis made over a 2-km-diameter area with a 10-m depthresolution at S/N = 1000, K = 0.1, and exp -fLi-aA(x,y,X)ds = e- 1 . If a time t is taken to rotate thelaser beam and make the measurement, then a mirrorMi is illuminated for a time ri = Wmt/(2irR), where Wmis the mirror width. The detector must operate at abandwidth Af = 2/Tr to resolve the signal. For t = 10sec, R = 1 km, and Wm = 10 cm, we find that Af = 1.2X 104 Hz. To be conservative we assume inexpensiveroom-temperature detectors with NEP = 10-11 WHz-1/2 operating at 104-Hz bandwidth. The calculatedrequired power and energy from Eqs. (4) and (5) is

preq = 3.0 W (cw source),

EJeq = 0.50 mJ (pulsed source)

for a 10-cm receiver diameter at a 2-km range. Thesepower and energy levels are well within the range ofavailable laser sources and are well below the mega-watt-peak-power and 0.1-J energy levels required fordepth-resolved measurements by the differential ab-sorption method.7 With appropriate hardware of thetype now used in advanced medical computerized to-mography scanners, the function -aA(x,y,X) can be re-constructed in approximately 10 sec, thus permitting

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March 1979 / Vol. 4, No. 3 / OPTICS LETTERS 77

a 10-m spatially resolved measurement over an areabounded by a 2-km-diameter circle every 10 sec with thepulsed-laser source operating at 30 pulses/sec.

As a second example, consider a 10-km-radius circlewith a spatial resolution of 100 m and a measurementtime of 10 sec. The illumination time for each detectoris ri = 1.6 X 10-5 sec, so that Af = 1.2 X 105 Hz. For aliquid-N2-cooled detector with NEP = 10-12 W Hz1/2and aA = 0.1 km-1 , a value typical in the infrared, wefind that

preq = 25 W (cw source),

EJ = 0.4 mJ (pulse source)

for a receiver diameter of 10 cm at a 20-km range. Inthis case, the improved detector sensitivity offset theorder-of-magnitude increase in range. The requiredcw source power can be reduced by increasing the re-ceiver aperture or the measurement time.

In the first example, the transmitted laser energy isalmost two orders of magnitude below that required fora depth-resolved differential absorption measurementat 1-km range. 7 In the second case, a depth-resolvedabsorption measurement in the infrared would requireover six orders of magnitude more transmitted energyand thus be practically infeasible. These examples il-lustrate the advantage of the VA/Lij scaling depen-dence for this measurement method.

The examples clearly show that the required trans-mitted laser power and energy for computerized air-pollution tomography are well within the present tun-able-laser state of the art and also well within acceptedeye-safety levels. In a number of monitoring situations,the disadvantage of erecting 300 mirror-detector loca-tions around a perimeter may be more than offset by thecapability of computerized air-pollution tomographyto provide a spatially resolved two-dimensional map ofnumerous potential pollutant sources.

In conclusion, we have shown that two-dimensionalair-pollution monitoring over a wide area is possiblewith a cw or low-energy pulsed tunable-laser source andmultiple-mirror virtual sources using conventionalcomputerized tomography geometry and inversionmethods.

We would like to acknowledge helpful discussionswith P. Switzer, director of the SIMS Seminar atStanford University. This work was partially sup-ported by the National Science Foundation.

References

1. R. L. Byer and E. R. Murray, "Remote Monitoring Tech-niques," in Handbook of Air Pollution Analysis, R. Perryand R. Young, eds. (Chapman and Hall, London, 1977), pp.406-447.

2. R. L. Byer, "Review: remote air pollution measurement,"Opt. Quantum Electron. 7, 147-177 (1975).

3. L. A. Shepp and B. F. Logan, "The Fourier reconstructionof a head section," IEEE Trans. Nucl. Sci. NS-21, 21-43(1974).

4. L. A. Shepp and J. B. Kruskal, "Computerized tomography:the new medical x-ray technology," Am. Math. Monthly25, 420-437 (1978).

5. A. V. Lakshminarayanan, Reconstruction from DivergentRay Data, SUNY Tech. Rep. 92 (Computer Science De-partment, Buffalo, N.Y., 1975).

6. L. A. Shepp and J. A. Stein, "Simulated artifacts in com-puterized tomography," in Reconstructive Tomographyin Diagnostic Radiology and Nuclear Medicine, M. Ter-Pogossian et al., eds. (University Park Press, Baltimore,Md., 1977), pp. 33-48.

7. R. L. Byer and M. Garbuny, "Pollutant detection by ab-sorption using Mie scattering and topographic targets asretroreflectors," Appl. Opt. 12, 1496-1505 (1973).