Measurement of scattering parameters pdf

Jones, B. Curry, and K. Leong are with the Argonne improving these techniques. Quinn Brewster and M. Jones are with the Depart- scattering problem are classified as either analytical ment of Mechanical and Industrial Engineering, University of or empirical.

Because of the ill- 0. Therefore, the posed nature of inverse problems, most analytic available set of measurements consists of simulta- inversion techniques require the use of a priori neous measurements of the power scattered into the information regarding the distribution function or a solid angles subtended by detectors located at several careful optimization of the inputs.

Indeed, the pri- polar angles. In addition, most defined as an angular scattering cross section. Cj is related to the distribution of sizes and optical Empirical inversion techniques generally require properties by an inhomogeneous Fredholm equation that a parametric model of the light-scattering or of the first kind: extinction process be developed. The parameters are then adjusted within physically realistic bounds so that a least-squares fit of the measured data is obtained.

The optical strip- measurements and are assumed to be Gaussian dis- map technique has been successfully used to retrieve tributed. This dCj x, n, k h k. A description of the inversion technique ing volume. Fourth, the solid angles subtended by is given, with an emphasis on the mechanics of the the detectors are small, so the integral over fl can be inversion process.

The results from several inver- replaced by the product of the average of the differen- sions of simulated data sets are also presented. Two nephelometers were considered. However, the results of this study show that the real part of the refractive We can eliminate the unknown function h k from index is accurately obtained from the unconstrained Eq. Earlier investigations have shown that it is available measurement set.

We use information ob- beneficial to weight each measurement and scattering tained from the unconstrained solution in conjunc- kernel by the corresponding estimate of the experi- tion with a preliminary analysis of the data to choose mental error. The preliminary analy- angle subtended by each detector is the same. These sis of the data is a simplified version of the optical modifications simplify Eq. The value of the absorption index that results in a rms value of the residual errors that where is closest to the rms of the imprecision estimates is taken to be the retrieved absorption index.

Cavgi 3 made, it may be possible for one to retrieve the absorption index in a more direct manner. Shaw2 3 acj discussed an inversion procedure that uses a combina- tion of spectral extinction and angular scattering ' CaygACj measurements. We present an example inversion with the description of the mathematical formulation to illustrate the Inversion Process mechanics of the inversion process.

The simu- definition of a linear inverse problem with discrete lated data set used in the example inversion is data. The eigenfunction method is initially used for the retrieval of the unconstrained or general- ized solution. The eigenfunction method has been Preliminary Analysis of the Simulated Measurements applied to particle sizing by several authors and is One should calculate angular scattering cross sec- thoroughly discussed in the literature.

Simulated angular scattering cross sections. The ranges of sizes and optical properties considered in the example inversion are listed in Table 1. Angular scattering cross sections were in Table 2 indicates that the particle has a diameter of calculated for the channel nephelometer at 12 approximately pum a size parameter between 19 different sizes and three sets of optical properties.

Table 2 lists the average and angular scattering Input Selection cross sections. Table 2 can be considered to be a low-resolution optical strip map. The size and the optical properties of measurements. We use an algorithm similar distribution function were not known until after the to that used by Capps et al.

We added Gaussian distrib- set. We calculate the kernel covariance or Gram uted random noise to each angular scattering cross matrix corresponding to the complete set of measure- section to simulate experimental conditions. The ments, and the eigenvalues and eigenvectors of the angular scattering cross sections are plotted in Fig. The ele- and the imprecision estimates shown in the figure are ments of the kernel covariance matrix are defined by equal to the standard deviation of the random noise.

The average of these angular scattering cross sections is 0. The relative error Table2. Equation 13 is dominated 0.

Unconstrained PSDF. If the relative error given by Eq. A new kernel covariance matrix is then calculated, The value of the refractive index and the uncon- and the process is repeated until the relative error strained PSDF are calculated from the unconstrained calculated from Eq.

In this solution. For the angular scattering cross sections shown in Fig. The variation in the number of inputs was Inf fxf m due to variations in the range of real refractive indices, as we discuss in the next section. The unconstrained solution We use the weighting function W n to increase the is obtained by the expansion of the distribution sensitivity of the unconstrained solution to changes function as a linear combination of the Schmidt- in the real refractive index.

In order to be effective, Hilbert eigenfunctions. A weighting function that proved to be useful in this study is n - 1 2. This function has the same depen- The unconstrained expansion coefficients are calcu- dence on the refractive index as the phase shift lated from squared, which is an approximation to the extinction efficiencyfor large size parameters. When inverting the example data Fj set we first considered the entire range of refractive indices 1.

The range of refractive indices was kernel covariance matrix and Xjis the eigenvalue that then narrowed to 1. The following index was 1. I 0 Fig. Constrained PSDF. Retrieval of the Particle Size Distribution Function known that the measurements are of light scattered The unconstrained solution satisfies Eq. Based on this set of inputs, and therefore it is a mathematically fact, the form of the trial function is chosen to be correct solution. These unre- The preliminary analysis of the measurements alistic characteristics are due to the ill-posed nature indicated that xt should be in the range We of the problem and must be eliminated through the obtained the initial value of xt used in Eq.

In general, the nature of the trial func- Therefore, xt was chosen to be The present study will demonstrate of orthonormal basis functions. However, only the that if the data consist of measurements of the light portion of the trial function that lies in the space scattered by a single particle or by an ensemble of spanned by the scattering kernels can be represented nearly identical particles, the unconstrained solution by the use of the Schmidt-Hilbert eigenfunctions.

In our example inversion, it is way to form a more nearly complete set in the solution space. We obtain the additional basis CalculatedAngularScatteringCross Sections functions by orthogonalizing a set of supplemental SimulatedAngular ScatteringCross Sections orthonormal functions with respect to the Schmidt- Hilbert eigenfunctions. The supplemental orthonor- mal functions used in the example inversion were 4. TypicalResultsfromthe Inversionof a Channel o.

Comparison of the simulated measurements and the Optical properties 1. An approximation function constraint. A performance function is de- for n - n8 was obtained in the same manner. JzLQv JiJi aicIi x, n dQ xs n, Therefore, the number of supplemental basis func- 2 [ nfxfm tions p will be less than the number of 4,j x,n. The x dxdn-c J 2Y aic'i x, n number of supplemental basis functions used in the example inversion was Comparison RefractiveIndices 0.

Particle size distribution functions. The relatively large discrepancy the inversion process. Shaw2 3 demonstrated that between the calculated and measured 20 angular rms deviation between the retrieved and actual distri- scattering cross sections indicated that the size param- butions has a minimum with respect to the Lagrange eter selected for the trial function was too large. The largest peak at a size and the partial derivative of the RRV with respect to y parameter less than A new is given in Eq.

These re- constrained expansion coefficients can be calculated sults are representative of a number of inversions from Eq. We then obtain the PSDF by integrat- performed with simulated data sets. Table 4 shows typical results from these inversions. To test the inversion process further, we were provided with six sets of simulated Retrieval of the Absorption Index light-scattering measurements in a blind test.

An initial guess of the absorption index is light scattered by narrow distributions of nonabsorb- made and the scattering pattern is calculated through ing spheres, but no other information was given.

Calculation of the angular scatter- The results of the inversions are shown in Table 5. In the data in the blind test is plotted in Fig. Although example inversion, it is assumed that the scattering the PSDF for case 4 has the same geometric standard pattern is due to a single particle, so N, is known.

These results show that the tech- the value of the particle number concentration are nique is successful when the distributions are narrow varied until the calculated and measured scattering cases 1, 2, and 6 but has difficulty when the distribu- patterns match.

If it is not possible to bring the tions are broad cases These results also show measured and calculated scattering patterns into the need to obtain reliable a priori information agreement by adjusting the value of the absorption regarding the PSDF in order for one to invert light- index, the inversion process should be repeated with a scattering measurements successfully.

In this study, different trial function. It should be noted that the it was assumed that the height of each distribution calculated angular scattering cross sections were was greater than its width, and the trial functions required to match the measurements that were not used to constrain the solution were selected accord- used as inputs in the inversion for the PSDF and the ingly.

In cases the assumption of a narrow real part of the refractive index as well as the PSDF was not valid, and the retrieved PSDF did not measurements that were used. If more reliable informa- In the example inversion, the absorption index that tion regarding the PSDF was available, the inversion gave the rms residual error closest to the rms of the process presented in this paper would probably give estimated experimental errors, Cj, was The more accurate results in all cases.

This conclusion is in agreement with Cavg, Normalized and imprecision Koo,2 6 who recommended the complementary use of weighted average angular scat- laser light scattering and mechanical collection tech- tering cross section cm-2 niques after he reviewed particle sizing techniques C, Angular scattering cross sec- used in the analysis of the metallic oxide smoke tions cm2 produced by the combustion of solid rocket propel- c, Normalized and imprecision lants.

Similar conclusions were reached by Bot- weighted angular scattering tiger2 7 after he compared five different inversion cross sections cm-2 techniques. Finally, it is interesting to note that f x, n, k , Distribution of sizes and optical even when the retrieved distributions differed from properties the actual distributions, the refractive index was f x , Particle size distribution func- retrieved accurately. We use the orthogonal- tions cm 2 ity properties of the basis functions to find the expansion coefficients that minimize the residual Differential scattering cross sec- d x, n, k , tions that have been averaged errors subject to a trial function constraint.

The technique is shown to be capable of retrieving the size over the solid angle subtended and optical properties from simulated measurements by the detectors cm2 dCavg of the light scattered by a weakly absorbing sphere.

Attempts to M, Kernel covariance matrix retrieve the PSDF were less successful when the n, Real part of the refractive index distributions were not narrow, but the refractive N,, Particle number density cm-3 index was accurately retrieved in all cases. Because p, Number of supplemental basis of the ill-posed nature of the inverse light-scattering problem, accurate a priori information regarding the functions PSDF must be available for the inversion technique RRV, Residual relative variance developed in this study to be applied successfully to u, Eigenvectors of the kernel co- broad size distributions.

If a priori information variance matrix cannot be obtained from an analysis of the particular V, Scattering volume cm3 environment in which the light-scattering measure- x, Size parameter ments are made, the complementary use of collection Greek techniques is recommended. Further research is needed to investigate the possibility that the trial 8C, Experimental errors cm2 function could be selected with minimal reliance on a 8c, Normalized and imprecision priori information.

One possibility is the use of an weighted experimental errors interative procedure that begins with a nonprejudi- cm-2 cial trial function. Further modifications of the 8 x , Dirac delta function inversion technique will focus in this area.

Q, Solid angle sr Curry, "Constrained eigenfunction method for the Subscripts inversion of remote sensing data: application to particle size determination from light scattering measurements," Appl. King, D. Byrne, B. Herman, and J. Reagan, i, Initial or index "Aerosol size distributions obtained by inversion of spectral j, Index optical depth measurements," J. Bertero and E. Pike, "Particle size distributions from Fraunhofer diffraction I. However, only the that if the data consist of measurements of the light portion of the trial function that lies in the space scattered by a single particle or by an ensemble of spanned by the scattering kernels can be represented nearly identical particles, the unconstrained solution by the use of the Schmidt-Hilbert eigenfunctions.

In our example inversion, it is way to form a more nearly complete set in the solution space. We obtain the additional basis CalculatedAngularScatteringCross Sections functions by orthogonalizing a set of supplemental SimulatedAngular ScatteringCross Sections orthonormal functions with respect to the Schmidt- Hilbert eigenfunctions.

The supplemental orthonor- mal functions used in the example inversion were 4. TypicalResultsfromthe Inversionof a Channel o. Comparison of the simulated measurements and the Optical properties 1. An approximation function constraint. A performance function is de- for n - n8 was obtained in the same manner.

JzLQv JiJi aicIi x, n dQ xs n, Therefore, the number of supplemental basis func- 2 [ nfxfm tions p will be less than the number of 4,j x,n. The x dxdn-c J 2Y aic'i x, n number of supplemental basis functions used in the example inversion was Comparison RefractiveIndices 0. Particle size distribution functions. The relatively large discrepancy the inversion process. Shaw2 3 demonstrated that between the calculated and measured 20 angular rms deviation between the retrieved and actual distri- scattering cross sections indicated that the size param- butions has a minimum with respect to the Lagrange eter selected for the trial function was too large.

The largest peak at a size and the partial derivative of the RRV with respect to y parameter less than A new is given in Eq. These re- constrained expansion coefficients can be calculated sults are representative of a number of inversions from Eq. We then obtain the PSDF by integrat- performed with simulated data sets. Table 4 shows typical results from these inversions. To test the inversion process further, we were provided with six sets of simulated Retrieval of the Absorption Index light-scattering measurements in a blind test.

An initial guess of the absorption index is light scattered by narrow distributions of nonabsorb- made and the scattering pattern is calculated through ing spheres, but no other information was given. Calculation of the angular scatter- The results of the inversions are shown in Table 5.

In the data in the blind test is plotted in Fig. Although example inversion, it is assumed that the scattering the PSDF for case 4 has the same geometric standard pattern is due to a single particle, so N, is known. These results show that the tech- the value of the particle number concentration are nique is successful when the distributions are narrow varied until the calculated and measured scattering cases 1, 2, and 6 but has difficulty when the distribu- patterns match.

If it is not possible to bring the tions are broad cases These results also show measured and calculated scattering patterns into the need to obtain reliable a priori information agreement by adjusting the value of the absorption regarding the PSDF in order for one to invert light- index, the inversion process should be repeated with a scattering measurements successfully.

In this study, different trial function. It should be noted that the it was assumed that the height of each distribution calculated angular scattering cross sections were was greater than its width, and the trial functions required to match the measurements that were not used to constrain the solution were selected accord- used as inputs in the inversion for the PSDF and the ingly.

In cases the assumption of a narrow real part of the refractive index as well as the PSDF was not valid, and the retrieved PSDF did not measurements that were used. If more reliable informa- In the example inversion, the absorption index that tion regarding the PSDF was available, the inversion gave the rms residual error closest to the rms of the process presented in this paper would probably give estimated experimental errors, Cj, was The more accurate results in all cases.

This conclusion is in agreement with Cavg, Normalized and imprecision Koo,2 6 who recommended the complementary use of weighted average angular scat- laser light scattering and mechanical collection tech- tering cross section cm-2 niques after he reviewed particle sizing techniques C, Angular scattering cross sec- used in the analysis of the metallic oxide smoke tions cm2 produced by the combustion of solid rocket propel- c, Normalized and imprecision lants.

Similar conclusions were reached by Bot- weighted angular scattering tiger2 7 after he compared five different inversion cross sections cm-2 techniques. Finally, it is interesting to note that f x, n, k , Distribution of sizes and optical even when the retrieved distributions differed from properties the actual distributions, the refractive index was f x , Particle size distribution func- retrieved accurately. We use the orthogonal- tions cm 2 ity properties of the basis functions to find the expansion coefficients that minimize the residual Differential scattering cross sec- d x, n, k , tions that have been averaged errors subject to a trial function constraint.

The technique is shown to be capable of retrieving the size over the solid angle subtended and optical properties from simulated measurements by the detectors cm2 dCavg of the light scattered by a weakly absorbing sphere.

Attempts to M, Kernel covariance matrix retrieve the PSDF were less successful when the n, Real part of the refractive index distributions were not narrow, but the refractive N,, Particle number density cm-3 index was accurately retrieved in all cases. Because p, Number of supplemental basis of the ill-posed nature of the inverse light-scattering problem, accurate a priori information regarding the functions PSDF must be available for the inversion technique RRV, Residual relative variance developed in this study to be applied successfully to u, Eigenvectors of the kernel co- broad size distributions.

If a priori information variance matrix cannot be obtained from an analysis of the particular V, Scattering volume cm3 environment in which the light-scattering measure- x, Size parameter ments are made, the complementary use of collection Greek techniques is recommended.

Further research is needed to investigate the possibility that the trial 8C, Experimental errors cm2 function could be selected with minimal reliance on a 8c, Normalized and imprecision priori information. One possibility is the use of an weighted experimental errors interative procedure that begins with a nonprejudi- cm-2 cial trial function. Further modifications of the 8 x , Dirac delta function inversion technique will focus in this area.

Q, Solid angle sr Curry, "Constrained eigenfunction method for the Subscripts inversion of remote sensing data: application to particle size determination from light scattering measurements," Appl. King, D. Byrne, B. Herman, and J. Reagan, i, Initial or index "Aerosol size distributions obtained by inversion of spectral j, Index optical depth measurements," J. Bertero and E. Pike, "Particle size distributions from Fraunhofer diffraction I.

An analytic eigenfunction ap- This research was supported by the U. Army proach," Opt. Acta 30, Chemical Research, Development, and Engineering Viera and M.

Box, "Information content analysis of Center. Jones is grateful for the continued sup- aerosol remote-sensing experiments using an analytic eigen- port of the Division of Educational Programs at function theory: anomalous diffractionapproximation," Appl.

Argonne National Laboratory. Box, K. Sealey, and M. Box, "Inversion of Mie References extinction measurements using analytic eigenfunction theory," 1. Twomey, Introduction to the Mathematics of Inversion in J. Phillips, "A technique for the numerical solution of dam, Bertero, C.

De Mol, and E. Pike, "Linear inverse Mach. I: General formulation and Mathews and R. Walker, Mathematical Methods of singular system analysis," Inv.

Physics, 2nd ed. Benjamin, New York, , Chap. Pike, "Linear inverse Bertero, P. Boccacci,and E. Pike, "Particle-size distribu- problems with discrete data. II: Stability and regularisa- tions from Fraunhofer diffraction: the singular value spec- tion," Inv.

Bohren and E. Hirleman, eds. Del Mol, and E. Pike, "Particle-size distribu- particle sizing, Appl. Twomey, "Information content in remote sensing," Appl. Arridge, P. Delpy, and M. Cope, "Particle 6. Capps, R. Henning, and G. Hess, "Analytic inversion of remote-sensing data," Appl.

Ben-David, B. Reagan, "Inverse Shaw, "Inversion of optical scattering and spectral problem and the pseudoempirical orthogonal function method extinction measurements to recover aerosol size spectra," of solution. Quist and P. Wyatt, "Empirical solution to the Turchin, V. Koslov, and M. Malkevich, "The use of inverse-scattering problem by the optical strip-map techique," mathematical-statistical methods in the solution of incorrectly J.

A 2, Bottiger, "Sizing spheres with the submicron particle Bottiger, U. Obscuration and Aerosol Research, D. Clark, J. Rhodes, personal communication, Claunch, eds. Koo, "A review of particle sizing methods in rocket Operations, Edgewood, Md. Jones, K. Brewster, and B. Bottiger, "Intercomparison of some inversion methods on Bohren and D.

Deepak, H. Fleming, and M. Cha- Deepak, Hampton, Va. Related Papers. By Eugene Yee. By Alain Royer.