This report deals with a notion of adjoint spectrum which is applicable to air-pollution monitoring by using pulsed lasers. Both Raman and absorption spectra of multiple gas complex are linear combinations of the spectra, each of which is specific to a gas species and its magnitude is proportional to the density. An extended formulation of the method of least squares is made in terms of a new notion, adjoint spectra, which visualizes the structure of the numerical filter. Applications of the derived numerical filters to the two methods are shown and features are described about the synthesized filters

Koga, Ryuji

Kosaka, Megumi

Sano, Hiroya

Tanada, Yoshihiro

Okayama University Scientific Achievement Repository

Memoirs of the School of Engineering,Okayama University VoI.13, February 1979A Notion of Adjoint Spectrum m the Applicationto Air-pollution MonitoringHiroya SANO*, Ryuji KOGA*, Yoshihiro TANADA*and Megumi KOSAKA*(Received February 7, 1979)SynopsisThis report deals with a notion of adjoint spectrumwhich is applicable to air-pollution monitoring by usingpulsed lasers. Both Raman and absorption spectra ofmultiple gas complex are linear combinations of thespectra, each of which is specific to a gas species andits magnitude is proportional to the density.An extended formulation of the method of leastsquares is made in terms of a new notion, adjoint spec-tra, which visualizes the structure of the numericalfilter. Applications of the derived numerical filtersto the two methods are shown and features are describedabout the synthesized filters.1. IntroductionAn optical method requires no chemical process to measure complexgas densities, and a quick response and remote measurement are possiblaIntroduction of a pulsed laser as the light source results in variousmerits on the atmospheric pollution detection. Raman scattering fromthe atmosphere exited by pulsed laser light is composed of disctetelines specific to the constituents of the atmosphere and the line in-tensities correspond to the gas densities 1 ). The absolute intensities* Department of electronics181182 Hiro~'a SAND. R~'uji KOGA, Yoshihiro TANADA and Megumi KOSAKAare, however, very weak even though a high powered pulsed laser is em-ployed, which implies overlaps of the line profile observed by a verysensitive spectrometric facility. For research use, a double-monochro-meter is employed to obtain necessary resolution. This is, however,impractical for the field application.Differencial absorption method utilizes the differences of theabsorption cross section at two wavelengths, one of which ~s "on" theresonance absorption line and, another, "off". Significance of thismethod lies in the point that the sensitivity is some orders largerthan the Raman scattering method 2 ) .In the ultraviolet region of 224-226 [nm] wavelength, NO, S02 andNH3 shares the absorption band and their spectra have clear distinc-tion 3,4). The overlap of the absorption lines also requires a schemeto separate each gas concentration. In the molecular correlation spec-troscopy, a couple of correlating masks which have multiple slits andlocate alternately on the slit position of a monochrometer 5 ) , or apolychrometers 6 ) are used. The attainable accuracy is not sufficientfor the pourpose of air-pollution monitoring because of the low signalto noise ratio in the photo-electric conversion process and the geo-metrical accuracy of the mask, though effort has been made to optimizethe mask design.Similar correlation procedure can be executed by a micro-computerinstalled in the measuring unit, which will produce more accurate re-sults than that by an analog method along with the employment of thepulsed dye laser as light sourses.This report deals with a mathematical filters which is generatedby a notion of adjoint spectra, though a part of this research has. . .. 7)been In a preVlOUS paper wrltten In JapaneseThe notion "adjoint" already has been familiar to those who areengaged in solving neutron governing equation which determines powerdistribution in a nuclear power reactor 8). The mathematical foundationis on the least squares method.Two applications are made for the Raman and the resonance absorp-tion methods. In the Raman method, a synthesyzed numerical filterpickes up the densities of N2 and 02 in the atmosphere which is thecalibration signal to the pollutant densities from a photon-countingspectrum of the Raman scattering. Also from an absorption spectrumof a gas simplex of NO and S02, this filter can extract each densitieswithin enough accuracy. The numerical filter can also reduce an ef-fect of long term drift of instrument parameters.2. a numerical filter2.1 Adjoint spectraA Notion of Acljoint Spectrum 183Let an inner product <X(A) ,Y(A» be defined for two spectra X(A)and Y(A) on a closed set h of real numbers A, wavelength. In the fol-lowing, two set of integers i, k are denoted as I and K respectively.Also FI={s.(A) ,isI,Ash} and F*:{s1(A) ,isI,Ash} are two sets of~ ~spectra S.(A) and S1(A), isI. The spectra S1(A)sF* are called as the~ ~ ~ Iadjoint of s.sF if a bi-orthonormality~ I<S.(A) ,S~(A» = 6' j (Kronecker's)~ J ~ ( 1)holds.These spectra have several properties as follow.(a) The adjoint spectra siSF{ can be generated as linear combinationsof siSFI.(b) Given FI and FIr' Then the adjoint of F lur' gives a FI* of FI suchthatholds, that is, the adjoint F* of F is not unique.(c) A linear combinationof s. (A) gives~C.~<s1 (A), L(A».~(3 )(4 )This expression directly gives a numerical filter which gives a dens-ity c i from the obtained spectrum.2.2 A numerical filter for pulsed-laser system and the response fornoise.A general definition of the inner product is<X(A) ,Y(A» = LX(A)Y(A)dU(A) 'where the measure dU(A) should be substituted by(5)184 Hiroya SANO, Ryuji KOGA, Yoshihiro TANADA and Megumi KOSAKAdU(A) = 2 VkO(A-Ak)dAkEK(6)for a discretely tunable laser system, where 0 stands for Dirack'sdistribution.If a computer is installed in the instrumentation, the adjointspectra Si(Ak ), kEK can be retained in it, then a gas density of spe-cies i is calculated by Eq. (4) out of the spectrum f(A). The measuredspectrum f(A) inherently includes random noise n, then the result ofthe numerical filter becomesa. = 2 s1k f(A k )v k1- kEK 1-and the variance of c. is given by1-(7)(8)This is reduced from the assumption that the noise is independent ofthe wavelength. The notation sikis short for Si(Ak ).If the f value is averaged over mk laser shots retaining the wa-velength at Ak , the variance reduces toVar (a.) = Var (n) 2 (s 1k v k )2 .1: .1- kEK 1- mk (9)The deduction mentioned above is a version of the well-knownleast~squares method. But the notion of the adjoint clarifies thefunction of the method that prevents an error caused in the guess forone parameter propagates to others.2.3 Immunity of the adjoint spectra for interferencesMore than one adjoint spectrum exist for a Fl. Let the adjointhas been extracted f~om the Ft+J. The bi-orthonormality relation,Eq. (1), holds not only for a S.(A), iEI but also for a s .(A), jEJ1- Jwith thus synthesized adjoint spectra Si-(A) ,iEI. Immunity of F; froman interference is then extended from F I to FI + J • The numbers of theelements of FI+J, however, cannot exceed K that intrincically limitthe number of independent spectra. Further more the number I+J shouldbe enough smaller than K, because the round-off error also disturbesthe bi-orthonormality, Eg. (1).A Notion 0/ Adjoint Spectrum 1853. Optimization of the pulse-number allotment for discrete wavelengthGiven a set of adjoint spectra, the variance of the resultingdensities c. can be minimized by an proper allotment of the pulse shots1-to each wavelengths, Ak , under a restriction(10)where M is also a given number. Hereafter, the parameter vk is set tobe unity because it can be included in sik'3.1 The optimal solution for a single gas speciesThe optimal solution to minimize Var(c.) expressed by Eq.(9)1-under the restriction, Eq. (10), ism' k = M'l s1 k ll L Is1 k l.1- 1- k£K 1- (11)This solution is given through the Lagrange's method of indeterminatecoefficients regarding the integer mk as continuous variables, thoughit is an approximation.3.2 The optimal solution for multiple gas speciesFor the given data set f k , kEK, multiple densities c i ' i£I areobtained if necessary adjoint spectra sik' i£I, kEK are prepared, butcommon solution mk,kEK that minimize var(c i ) for every i€I at thesame time exists. The multi-objective programming technique9) whichrealizes a notion of the Pareto optimality can cope with this problem.A non-inferior or the Pareto optimal solution is such that6 {Var(c.)} > 0, iEI, (12)1- =holds for any mk , kEK that satisfies Eq.(lO).Another problem statement is made by the min-max problem withthe criterion186 Hiroya SANO, Ryuji KOGA, Yoshihiro TANADA and MeKumi KOSAKAmax{w . 'Var (a.) }iEI t, t,where w., iEI are weights. The solution of a min-max problem corres-t,ponds to a solution of the non-inferior solution, which is not unique.4. Application to an gas analyzer by Raman scatteringThis system gives molecular densities of pollutants from a Ramanscattering spectra of the polluted atmosphere excited by a pulsed Nd:YAG laser light. The scattered light is fed into a monochrometer at-tached with a photo-multiplier and a photon-counter.The Raman spectra is composed of discrete lines each of whichcorresponds to a specific gas species. The collected photon-countsspectrum is, however, broadened and the lines overlap each other,because of the inherently poor spectral resolution of sensitive ap-paratuses.The spectrum is expressed asn (A)s( 13)whereA :n :sK :N .:t,F. :t,the spectrometer wavelength setting,photon counts,a coefficient specific to the instrument,the molecular number density of the i-thgas species,the response of the instrument for asimplex of the i-th gas species,incident laser energy per pulse.Figure 1 shows a Raman spectrum of atmosphere excited by the Nd:YAG laser light. The Nd line is predominant and 02 and N2 lines areclearly separated. The N02 line is to locate at 572.6 nm where the02 trail overlaps. Thus distributed response must be reduced to theline intensities by the numerical filters.At first , the Nd, N2 and 02 spectra separated at 460 and 580 nmwere chosen. Other 23 base spectra were generated by Lagrange's.interpolation taking the center wavelengths with equal spacing as theA Notion of Adjoint Spectrum 187parameter. The result is shown in Fig.2 where 02 line is clearly re-constructed. The N2 line is, however, associated with spurious res-ponses around it. This is attributed to a grating anomaly of thespectrometer and the quantum noise of the photon counting method.NZNo"0Z300~20001000(J)~5 80860z 40g 20o5: 01dL-~~~:f\.f--~~-~~- 2 0 ~~-'-:~~--'----'-----"---'"-:-"~~530 550 600 620WAVE LENGTH [nm]Fig.l Raman scattering of atmosphereexited by the second harmonic ofNd:YAG laser light.-20 t-o Qf\ I \f \ I \I b ( \I \ I \I \ (\ /;>p ~ I \I \ ,'"P q A \P \ Y. \a Q ,</ ~\ O'~o 'T'/I I ~\o l:l v I II I---0--- CRUDE DATA(f)I-Z::> 60ouz 40~oIQ. 20Fig.2 A result obtained by the numeri-cal inverse filtering with the basespectra generated from three responsesshown in Fig.l using the Lagrange'sinterpolation.I FILTERED570 580 590 600 610 620WAVE LENGTH [nm]Figure 3 is the three adjoint spectra for Nd, 02 and N2 chosenout of the adjoint that were used to give the result of Fig.2. Eachof the adjoint spectra has violent profile swinging to both signs, anddouble word-length calculation was necessary to obtain these resultsthat retain the bi-orthonormality defined by Eq. (1) .188 Hiroya SANO, Ryuji KOGA, Yoshihiro TANADA and Megumi KOSAKAN202Ndw30.10:g0.08wO.06>~0.04<[iiJO.020:: 0 ~.pp.:J9b:P'lOl~~~~-0.02-0.04-0.06-0.08Fiq.3 Adjoint spectra for the excit-ation light of Nd:YAG laser and forRaman scattering from 02 and N2 mole-cules, which gives result of Fig.2.530 550 600 620WAVE LENGTH [nml5. Application to a gas analyzer by resonance absorptionThree gasses 502' NO and NH3 shares their absorption band on 217-227 nm. The small attenuation over a short path can be measured bya system of a pulsed dye laser and of a low noiRe electronic apparatusthat have been specifically developed by the authors.Let a light pulse with wavelength A and energy PI be attenuatedby the chromatic gasses of thickness L to PT' From the Lambert-Beer'slaw, the optical thickness, or the attenuation, ,(A) is expressed as(14)where c. and s. are the density and the absorption cross section of1- 1-the i-th gas species, respectively. Here the molecular number sensityis replaced by ac i .For small attenuation, Eq. (14) becomes, (A) "P -PI TPI( 15)The electronic circuit deals with two quantities PI-PT and PI insteadof PT and PI' which allows accurate signal processing without requir-ing intolerably large dynamic range.The absorption spectra of the three gases which had been givenby Thompson et al. 3l and Warneck and Marmo 4l are shown in Fig.4 alongA Notion of Adjoint Spectrum 189with the adjoint spectra for the wavelength region of 217-227 nm.Figure 5 is the solutions for optimal pulse number allotment problemfor three different criteria obtained by using the adjoint spectra ofFig.4.AffiQiguity of the resulting c. values due to a random noise that1.-is inherently included in the measured f is given byVar(c .L) = L {s. (Ak ) }2Var(n), iEI. (16)1.- kEK 1.-The non-inferior solution was discovered by a search method which con-sist of a random search and improving steps.I x IOZ8ADJOINT I,'ll ISPECTRA [m-I ]5xlo""224.5 217.0 224.5WAVE LENGTH [nm]o L..::._....bo,ae,.--.! '--"-__--'----J217.0os <1 I ABSORPTIONI SPECTRA[m"]-222xl03x ,crt·Fig. 4 Resonance absorption spectraof 502' NO and NH3 over 2l7~227.4[nmj4,sl, and the adjoint spectra.2000ILl~4000::>Q.Fig.5 Three optimal pulse numberallotment for the non-inferior,the crude min-max and theweighted min-max solutions.10000iffisooo \m \::E \~6000 \\\\\I\\I\\NON-INFERIOR .....--...CRUDE MIN-MAX 0-----0WEIGHTED MIN-MAX 6-----190 Hiroya SANO, Ryuji KOGA, Yoshihiro TANADA and Megumi KOSAKAThe crude min-max solution is for a criterionmax Var(e.L)i I '&which is not associated with weights. The method to obtain this isthe same as for the noninferior solution. In Fig. 5, the effort isexhausted to reduce the ambiguity of NH3 only, namely, the pulse num-bers are allotted to the left end, where the cross section of NH3 islarge.This solution is unreasonable and therefore weightsw.'&-1Var(e.L)] , isI'&(17)were introduced and the criterion was modified tomax[w .Var(e .L)].'& '&Each term of this criterion is normalized by the extreme value whichhad been the optimization only for i-th gas species. The solution forthis criterion shown in Fig.5 resembles to the non-inferior solutionvery well, which is compatible with a description in the multiobjec-tive optimization theory that a min-max solution with a set of wi cor-responds to an inferior solution. The weighted min-max solution ismore profitable to a practical use, because one can introduce his in-tuition.Table 1 Standard deviations of the results of thefiltering, a(e.L), for a noise of a(n)=l.'& [ppm·m/Nep]Pulse numberallotmentUniformOptimal fora specificgasNon-inferiorCrude min-maxHin-maxso11. 8 (1. 72)6.88 (1.0)8.20(1.19)11.4 (1.66)8.21 (1.19)NO9.77 (2.21)4.43 (1.0)9.69 (2.19)31.3 (7.07)9.65 (2.18)NH345.0,(1.71)26.3 (1.0)36.2 (1.38)31.3 (1.19)57.3 (2.18)A Notion of Adjoint Spectrum 191Standard deviations for different allotments are listed in Table1, where Var{n) is taken to be unity [Nep2]. For the pourpose of com-parison, a result for a uniform pulse number allotment and that forthe optimization for a single species are listed as well. The valuesin parenthesis are those normalized by w..1.-The crude min-max solution and the weighted min-max solution givesalmost equivalent results for NO and NH3. This is because the spectrumof S02 overlaps with both spectra of NH 3 and NO, whereas, spectra ofNO and NH3 are independent and scramble for the pulse numbers.5.2 NO and 502 complex measurementDespite the spectrum in Fig.4, the NO spectrum of y{O,O) band near260 nm is composed of discrete lines each of which stands for the ro--1tational sublevel though they are broadened to 1.2 cm FWJM. As forthe spectrum of S02 in this band, no discrete lines are resolved evenif a laser diode spectrometer is employedll ). The lines are dense andcollapsed due to the pressure broadening. These contrasting spectralprofiles suggests that a rather narrow width of wavelength band isenough to separate the S02 and the NO densities by using the numericalfilter stated above.Figure 6 shows absorption spectra of NO and S02 in the designatedband. Each gases are contained in a cell of 3 cm long. The gases arefor span calibration use and are diluted in N2 gas. Also the complexspectra which was obtained by arranging these two cells in tandem con-figuration, is shown in the figure. Employment of the numerical filt-er which is composed of the NO and 802 spectra as the base gives thedensities within an accuracy of 0.58% for NO and 3.2% for S02.Fiq.6 Absorption spectra of NO and802 under 1 atm. The cell lengthis 3 cm. The (a)+(b) spectrum isthat of the complex of 802 and NO.I(0)502 3680ppmo~~~----c::;;:;-~------;~~----=:,=-~--=,-.".J225.76 .78 .80 .82 .84WAVELENGTH [nm]192 Hiroya SANO, Ryuji KOGA, Yoshihiro TANADA and Megumi KOSAKAThree different adjoint spectra for NO with different base spec-tra are shown in Fig.7. In that wavelength region, the 502 spectrafor NO and S02 as the base is smaller than that associated with twoechelons, which represents the trade-off between the specificity andthe sensitivity or the signal to noise ratio.Fig.7 Example of the adjointspectra. The 502 spectrumcan be replace with the2-echelon.6. Conc1uding remarksXI03r---.,----r--r--r--r--r--.---,>t ~ 3 Base spectrol-- NO,S02,2echelons~ -.-.- NO,S02~ g. ---- NO,2 echelonsr£l ZQ .--fIN I 1\S g 'f ~p:: • \E-< =~ §; 0 k-4--+--+A-+--+-~¥v-"-+-"""',>.,-<-I--+--;>',+-~--<\P-< ~UlE-<ZHg - 2 ,:-:~~,.---,-=--="==-,,""-,J-=-,--=,=----,""""'-",,=---'~ 225.84.85 .86 .87 .88 .89 .90225.91WAVELENGTH [nm]The notion of the adjoint spectra and the features are stated.Successful applications were made to the Raman method and to theresonance absorption method both for atmospheric air pollution. Thedensities of constituent gases are extracted from their diffused andoverlapping spectra.This function is realized when a microcomputer is installed inan instrument, which will be a common situation for a sophisticatedmeasuring apparatus in near future.A Notion of Adjoint SpectrumReferencesl)H. Inaba: Laser Monitoring of the At~osphere(E. o. Hinkley, Ed.),(Springer-Verlag, Berlin 1976), p.153.2)E. o. Hinkley, R. T. Ku, and R. L. Kelly:ibid., p.237.1933)B. A. Thompson, P. Harteck, and R. R. Reeves, Jr.: J. Geophs, Res.,68 (1963) 6431.4)P. Warneck, F. F. Marmo, and J. o. Sullivan: J. Chern. Phys., 40(1970),1597.5)0. T. Williams and B. L. Kolitz: Appl. Opt., 7(1968) 607.6)K. Ito et al.: Jap. J. App1. Phys., 14(1975), Supp1. 14-1, 131.7)R. Koga, Y. Tanada, and H. Sano: J. Spectroscop. Soc. Jap. 27(1978)297.8)W. M. Stacey, Jr.: Reactor Tech., 15(1972) 210.9)J. L. Cohen, O. H. Marks: Water Resourses Res., 11(1975) 208.10)0. T. Williams and R. N. Hager, Jr.: Appl. Opt., 9(1970) 1597.11)G. A. Antcliffe and J. S. Wrobel: Appl. Opt., 11(1971) 1543.

A Notion of Adjoint Spectrum in the Application to Air-pollution Monitoring

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