infrared species tomography of a transient flow field using kalman filtering

Infrared species tomography of a transientflow field using Kalman filtering

Kyle J. Daun,* Steven L. Waslander, and Brandon B. TullochDepartment of Mechanical and Mechatronics Engineering, University of Waterloo,

200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada

*Corresponding author: [email protected]

Received 14 September 2010; accepted 24 November 2010;posted 4 January 2011 (Doc. ID 134980); published 16 February 2011

In infrared species tomography, the unknown concentration distribution of a species is inferred from theattenuation of multiple collimated light beams shone through themeasurement field. The resulting set oflinear equations is rank-deficient, so prior assumptions about the smoothness and nonnegativity of thedistribution must be imposed to recover a solution. This paper describes how the Kalman filter can beused to incorporate additional information about the time evolution of the distribution into the recon-struction. Results show that, although performing a series of static reconstructions is more accurate atlow levels of measurement noise, the Kalman filter becomes advantageous when the measurements arecorrupted with high levels of noise. The Kalman filter also enables signal multiplexing, which can helpachieve the high sampling rates needed to resolve turbulent flow phenomena. © 2011 Optical Society ofAmericaOCIS codes: 100.3190, 110.6955, 120.1740, 300.1030, 300.6340, 300.6500.

1. Introduction

Improving the performance and reducing the ecologi-cal impact of combustion devices are predicated onachieving a highly controlled fuel/air mixture inthe combustion zone. This is especially important fordevices that operate close to the lean limit of stablecombustion. For example, ultralean low-NOx gas tur-bines are susceptible to localized, transient inhomo-geneities in mixture ratio induced by aeroacousticfeedback, which can cause unstable operation, exces-sive noise, reduced service life, and, in extreme cases,catastrophic failure of engine components.

Development of the next generation of combustiondevices will therefore rely on novel diagnostics tocharacterize the air/fuel distribution in the combus-tion zone; the temporal and spatial resolution neededto capture the turbulent flow phenomena of interestdisallows physical probing, so optical methods arepreferred. A powerful technique for measuring fuelconcentrations in experimental settings is planar

laser-induced fluorescence [1,2], or PLIF, but becauseit requires extensive optical access its usefulness foranalyzing practical combustion equipment is limited.

An alternative approach is infrared line-of-sightspectroscopy, which is based on the spectral absorp-tion of collimated light by hydrocarbon fuel mole-cules. The absorption of light along a single beamis governed by the Beer–Lambert law,

Iηðs; θÞ ¼ Iη0 exp�−

Z∞0

aη½rðs;θÞðuÞ�du�; ð1Þ

or

ln½Iη0=Iηðs; θÞ� ¼ bðs; θÞ ¼Z∞0

aη½rðs;θÞðuÞ�du; ð2Þ

where Iη0 and Iηðs; θÞ are the incident and trans-mitted intensities, s denotes the perpendiculardistance between a beam and the origin, θ is the an-gle formed between the beam and the y axis as shown

0003-6935/11/060891-10$15.00/0© 2011 Optical Society of America

20 February 2011 / Vol. 50, No. 6 / APPLIED OPTICS 891

in Fig. 1, rs;θðuÞ is a parameterized vector that pointsat a location on the beam, and bðs; θÞ is the negativelog of the beam’s spectral transmittance. The numberconcentration of the target molecules scales with thespectral absorption coefficient, aη, through its mole-cular absorption cross section [3].

Recovering the absorption concentration distribu-tion by deconvolving Eq. (2), a Fredholm integralequation of the first kind, is ill-posed due to the factthat an infinite set of candidate aη½rðuÞ� distributionsalong a given beam can substituted into Eq. (2) toobtain the same bðs; θÞ. To resolve this ambiguity,multiple beams over a variety of ðs; θÞ pairs mustbe shone though the tomography field. In the limit-ing theoretical case that bðs; θÞ is known as a contin-uous function over ½s; θ� domain, the unknown fieldvariable can be reconstructed unambiguously using aRadon transform [4]. If the sampling is sufficientlydense and uniformly spaced over ½s; θ�, as is the casein many medical tomography applications, the recon-struction is numerically stable (or mildly ill-posed)[5] and aηðrÞ can be reconstructed using filtered-backprojection (FBP) [6], based on the Fourier transformof the projected data. In steady-state laboratory-scale experiments, the dense and uniformly spacedattenuation data required for FBP can be obtainedby translating and rotating the optics and/or the to-mography field [7]. Unfortunately, this approach isnot suitable in an industrial setting where the num-ber and orientation of the beams are restricted by theoptical access afforded by the geometry. Moreover, ifthe goal is to investigate highly transient turbulentphenomena, the absorptance measurements must bemade simultaneously, which further limits the num-ber of projections available for reconstruction.

These limitations make algebraic reconstructiontechniques better suited to infrared species tomogra-phy problems. In this approach, the tomography fieldis discretized into elements over each of which aη ismodeled as uniform. By doing this, the integral inEq. (2) becomes approximated by a “ray sum”:

bi ¼Z∞0

aη½rðs;θÞðuÞ�du ≈

Xnj¼1

Aijxj; ð3Þ

where xj is the concentration of the jth element andAij is the chord length of the ith beam subtended bythe jth pixel. Writing this equation for m beams re-sults in an ðm × nÞmatrix equation, Ax ¼ b. Providedenough beams are used and the grid is sufficientlycoarse, A is well-conditioned or only mildly ill-conditioned and x can be recovered with iterativeregularization [8,9]. Inmost practical scenarios, how-ever, restrictive optical access coupled with the needfor simultaneous measurements means that thenumber of elements far exceeds the number ofbeams. In this case, Ax ¼ b cannot be solved directlybecause A contains a nontrivial null space, meaningthat the information provided by the ray sums is in-sufficient to uniquely characterize x. It is thus neces-sary to specify extra information about the unknownconcentration distribution; for example, the distribu-tion must be nonnegative and most likely spatiallysmooth due to diffusion physics. Garcia-Stewartet al. [10] proposed a method for incorporating thisinformation into infrared species tomography basedon Landweber iteration [11], which is modified sothat smoothness and nonnegativity filters are passedover x between iterations. This approach has beenused successfully in subsequent limited-data infra-red species tomography experiments [12–14]. Daun[15] recently presented a more formalized methodfor incorporating presumed information to augmentthe beam attenuation data based on maximum a pos-teriori (MAP) inference [15], which finds the vector x�that maximizes the probability of the observed dataactually occurring, conditional on additional smooth-ness and nonnegativity priors. Subject to someconditions, it can be shown that this approach isequivalent to second-order Tikhonov regularization[11,16] with a nonnegativity constraint.

In the case of transient systems, it is possible toimpose additional priors about how the solution islikely to change in time; if the sampling rate is suffi-ciently high, the observed concentration field willoften evolve in a temporally smooth and somewhatpredictable way. The discrete time Kalman filter ap-pears well-suited for incorporating this informationinto the tomography problem. In this approach, theinverse problem is transformed into a state estima-tion problem, in which the state is predicted at eachtime step using a linearized model of the systemdynamics, and then corrected with an observationmodel. The relative influence of the evolution and ob-servation models on the predicted state are weightedby the state transition and measurement covariancematrices to deliver an optimal (minimal covariance)estimate. Because the measurements are still inade-quate by themselves to characterize the state, how-ever, the measurement model must be augmentedwith priors. This technique has been successfullyused to diagnose two-phase confined flows usingFig. 1. Geometry of a beam transecting the tomography field.

892 APPLIED OPTICS / Vol. 50, No. 6 / 20 February 2011

electrical impedance tomography (EIT) [17–19]. Fora survey of recursive estimation techniques and theirapplication to the EIT problem, see [20].

The objective of this paper is to evaluate the suit-ability of Kalman filtering for solving infraredspecies limited-data tomography problems. We firstreview the solution of the static infrared species to-mography problem through MAP inference and thenshow how Kalman filtering may be incorporated intothe analysis. This implementation uses a random-walk update model, while the observation model con-sists of the ray sums augmented by a second-ordersmoothness prior. The merits of this approach arethen evaluated by reconstructing the time-evolvingconcentration of a simulated buoyant methaneplume rising in a vertical duct. Comparison of thestatic reconstructions and Kalman filtering resultsshow that, when the signal-to-noise ratio is large,static reconstructions are more accurate than thosefound by Kalman filtering, due to the interactionbetween the second-order smoothness prior in theobservation model and the random-walk state tran-sition model. At smaller signal-to-noise ratios, on theother hand, the extra temporal regularization pro-vided by the Kalman filtering produces a more ac-curate solution than is obtainable through spatialregularization alone. Finally, we show that theKalman filter facilitates multiplexing of laser at-tenuation data in cases where the high sampling raterequired to resolve turbulent phenomena precludessimultaneous sampling of all channels.

2. Infrared Species Tomography through MAPInference

As noted above, the main difficulty in infrared spe-cies tomography is that the ray-summatrix equation,Ax ¼ b, does not provide enough information to un-iquely specify x; instead, there is an infinite set of so-lutions fx�g that minimizes ‖Ax − b‖. This is due tothe fact that, for a system ofm beams and n elementswith m < n, the maximum rank of A is m, so A has anontrivial null space containing an infinite set of so-lutions to the homogeneous matrix equation

NullðAÞ ¼ fxn ∈ Rn : Axn ¼ 0g: ð4Þ

Any solution that satisfies the consistent equationset is described by x� ¼ xLS þ xn, where xLS has thesmallest norm of fx�g. That such a situation existsis clear from Fig. 2(a), which shows that, in limited-data infrared species tomography problems, someelements are not transected by the beams and thushave no influence over Ax ¼ b. To demonstrate thispoint, consider the A matrix corresponding to thebeam and grid arrangement in Fig. 2(a), which has18 beams and 268 elements. The nullity of A canbe seen from its singular value decomposition (SVD),

A ¼ USVT ¼ U ·

266666666664

σ1σ2

. ..

σm0

. ..

0

377777777775· VT ;

ð5Þ

where S is a diagonal matrix containing RankðAÞ ¼m nontrivial singular values, σi, i ¼ 1; 2;…;m.Reconstructing x by truncated back-substitution,

xLS ¼Xmi¼1

uTi bσi

vi; ð6Þ

where ui and vi are column vectors ofU andV, resultsin a solution that satisfies Ax ¼ b but is also clearlynonphysical, as shown in Fig. 2(b).

The challenge in limited-data tomography, then, isto “span” the nullity of A with additional informationabout the assumed characteristics of x. A systematicway to do this is by MAP inference, which is based onBayes’ theorem,

pðxjbÞ ¼ pðbjxÞ · pprðxÞpðbÞ ; ð7Þ

where pðxjbÞ is the probability that a state specifiedby x is correct for a given observed data set b, pðbjxÞ isthe probability of a data set b occurring for a givenstate x, pðbÞ is the marginal probability of the data,and pprðxÞ is an additional probability of x being cor-rect based on prior knowledge and assumptionsabout x. MAP inference works by finding the valueof x that maximizes pðxjbÞ. If the laser absorptancedata is contaminated with normally distributednoise and Ax ¼ b is scaled so that the covarianceof b can be written as σm2I, then the probability ofb occurring for a specified state x is given by

Fig. 2. (Color online) Example tomography problem. (a) The Amatrix is rank-deficient since some elements are not transectedby the beams, so (b) solving Ax ¼ b directly through Eq. (6) resultsin xLS, a physically unrealistic solution. (c) Using MAP with Gibbssmoothness and nonnegativity priors generates a more accuratereconstruction.


pðbjxÞ ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffi2πσ2m

p Ymi¼1

exp�−ðbi − AixÞ2

2σ2m

�

∝ exp�−‖b − Ax‖2

2

2σ2m

�: ð8Þ

Maximizing Eq. (8) by itself is equivalent to minimiz-ing ‖b − Ax‖2

2, which is shown above to be insuffi-cient to determine the correct value of x. Instead,wemust add extra information to the solution by spe-cifying priors, which are multiplied onto Eq. (8)sequentially since they represent joint probabilities.By definition, the absorption coefficient/species con-centration must be strictly nonnegative, which isenforced by

pnon-negðxÞ ¼Ynj¼1

HðxjÞ; ð9Þ

where the Heaviside step function,HðxjÞ, equals zeroif xj < 0 and is otherwise unity. (In other words, theprobability of x being correct is zero if any element ofx is negative). Additionally, we expect the concentra-tion distribution to be spatially smooth due to turbu-lent diffusion, a condition promoted by a Gibbs prior:

pGibbsðxÞ ∝ exp½−βUðxÞ�: ð10Þ

In Eq. (10), β is an adjustable scalar and UðxÞ is thetotal energy function. We set UðxÞ equal to ‖Lx‖2

2,where L is the Tikhonov smoothing matrix, whichhere is a discrete representation of the Laplacianoperator [15],

Lij ¼( 1−1=ni

0

if i ¼ jif jneighbors iotherwise

; ð11Þ

and ni is the number of elements neighboring the ithelement. The Gibbs prior reflects that the diffusionphysics (Fick’s law) drives the solution toward a uni-form concentration distribution.

Combining Eqs. (8)–(10) results in an objectivefunction equal to

pðxjbÞ ¼ pðbjxÞ · pGibbsðxÞ · pnon-negðxÞpðbÞ

∝ exp�−‖Ax − b‖2

2

2σ2m− β‖Lx‖2

2

�Yni¼1

HðxiÞ: ð12Þ

By defining λ2 ¼ 2βσm2 and taking the logarithm, onearrives at a log-likelihood objective function equal to

lnpðxjbÞ ∝ −ð‖Ax − b‖22 þ λ2‖Lx‖2

2Þ ð13Þ

for x ≥ 0 and negative infinity otherwise. It can beshown that the posterior probability pðxjbÞ is maxi-mized by solving the constrained weighted least-squares minimization problem

x� ¼ argmin��

A

L

�x −

�b

0

��T

×W��

A

L

�x −

�b

0

��; s:t: x ≥ 0; ð14Þ

with the weighting matrix

W ¼�I 00 λ2I

�; ð15Þ

which is the same as Tikhonov regularization with anonnegativity constraint.

A challenge of this approach is to select a suitablevalue of λ. As noted by Zdunek [21], traditional pa-rameter selection techniques like L-curve curvature[22], generalized cross-validation [23], and the dis-crepancy theorem [24] are not appropriate forlimited-data tomography since they all presume thatthe regularization methods are convergent to xLS andλ should be chosen to balance the perturbation andregularization error, whereas here the main role ofλ is to span the nullity of A. Daun [15] presented atechnique for choosing λ based on singular value de-composition, and found the recovered solution accu-racy to be robust over several orders of magnitude ofλ. Figure 2(c) shows the solution to the example to-mography problem obtained from Eq. (14) withλ ¼ 0:1.

3. Tomographic State Estimation throughKalman Filtering

The key to the limited-data tomography problem,then, is to augment the ray sums with additional in-formation about the assumed solution characteristicsso as to reduce the inherent ambiguity of the laser at-tenuation data associated with the null space of A. Inaddition to the assumption of a spatially smooth dis-tribution, as described above, further informationabout the concentration distribution should be avail-able from the natural, quasi-predictable evolution ofthe species concentration field.

The Kalman filter appears to be ideal for incorpor-ating this information into the reconstruction; in thisapproach, the tomography reconstruction problem isreformulated as a state estimation problem in whichthe state, xk (the vector of cell concentrations), is re-presented by a Gaussian distribution with mean μk

and covariance Σk. The filter is defined by assuminglinear models for both the evolution and the mea-surements of the state at each time step. The stateevolution is assumed to satisfy

xkþ1 ¼ Fkxk þ εk; ð16Þwith additive Gaussian disturbances εk ∼Nð0;QkÞ,and state transition matrix, Fk, defining the discretetime evolution of the state. It is sometimes possibleto specify a sophisticated time-evolution model whenthe flow field is well characterized [17] or in caseswhere diffusion dominates species transport. Since


this problem lacks detailed knowledge of the flowfield, however, a random-walk model is used in thiswork which assumes that state evolution is driven byrandom turbulent processes modeled as an additiveGaussian disturbance and results in an identitystate transition matrix, Fk ¼ F ¼ I, which is com-bined with a diagonal disturbance covarianceQk ¼ Q ¼ σt2I.

Measurements are assumed to be a linear combi-nation of the state variables corrupted by additiveGaussian measurement noise of known covariance,δk ∼Nð0;RkÞ,

yk ¼ Gkxk þ δk; ð17Þ

where Gk is the measurement matrix. For the tomo-graphy problem, the measurement model is definedas above, combining Eq. (3) and the Tikhonovsmoothing matrix of Eq. (11):

Gk ¼ G ¼�AL

�; ð18Þ

where we consider the smoothness prior a “pseudo-measurement.” The covariance matrix, Rk, is definedconsistently with the weighting matrix defined inEq. (15):

Rk ¼ σ2m�I 00 1=λ2I

�: ð19Þ

As in the static case, λ determines the influence of thesmoothness prior pseudomeasurement relative tothe laser attenuation measurements. A small λmeans that the prior has a large assumed covarianceand is accordingly de-emphasized relative to the ac-tual beam measurements, while a large λ increasesthe influence of smoothing relative to the ray sums.The measurement noise parameter, σm, representsthe standard deviation of the actual laser attenua-tion measurements and premultiplies the covariancematrix to allow for the relative weighting of measure-ment and prior distributions.

This pair of linear models can be used to define astate estimator that solves the MAP inference pro-blem at each time step, known as the Kalman filter.At time step kþ 1, the mean of the state probabilitydistribution is predicted using the linearized modelof the system dynamics and the state distributionfrom time step k,

�μkþ1 ¼ Fkμk; ð20Þ

where the overbar denotes the predicted state prob-ability distribution. The covariance of the predictedstate distribution is found by propagating the priordistribution through the linear transformation withthe additive Gaussian disturbance covariance,

�Σkþ1 ¼ FkΣkðFkÞT þQk: ð21Þ

Once new measurements have been taken, the statedistribution is corrected through a linear measure-ment update. The state distribution is determinedby the difference between the actual observed dataset and the one expected from the predicted state

μkþ1 ¼ �μkþ1 þKkþ1ðykþ1−G �μkþ1Þ: ð22Þ

Here, the Kalman gain, Kkþ1, is selected so that theupdate results in the MAP state distribution (themean-squared error of the state estimate is mini-mized as before) and is defined by

Kkþ1 ¼ �Σkþ1ðGkþ1ÞT ½Gkþ1 �Σkþ1ðGkþ1ÞT þ Rkþ1�−1:ð23Þ

The covariance of the state distribution is updated by

Σkþ1 ¼ ðI −Kkþ1Gkþ1Þ �Σkþ1; ð24Þ

which follows from the derivation of the MAP updatefor the unconstrained state estimation problem. It isalso possible to include nonnegativity constraints inthe Kalman filter by either projecting the uncon-strained solution (i.e., passing a nonnegativity filterover the solution at each time step [10]) or by directnumerical solution of a bound-constrained quadraticprogramming problem [25]; while the latter methodis a more rigorous treatment of the bound constraint,it is also computationally intractable for the problemof interest. Results demonstrating the projectionmethod are presented in Section 5.

The Kalman filter can also be interpreted in aBayesian context in which the objective is to maxi-mize the conditional posterior probability:

pðxkjykÞ ¼ pðykjxkÞ · pðxkjxk−1ÞpðykÞ

∝ exp�−12ðxk − �xkÞTð �ΣkÞ−1ðxk − �xkÞ

−12ðyk −GkxkÞTðRkÞ−1ðyk −GkxkÞ

�

¼ exp�−12ðxk − Fxk−1ÞTðF �ΣkFT þQkÞ−1

× ðxk − Fxk−1Þ

−12ðyk −GkxkÞTðRkÞ−1ðyk −GkxkÞ

�: ð25Þ

This view of the Kalman filter highlights its struc-ture as the combination of a temporal prior withthe latest measurement information, which incorpo-rates the spatial prior. In the tomography problem,the block diagonal nature of the measurement covar-iance, Rk, results in


pðxkjykÞ∝ exp�−12ðxk−Fxk−1ÞTðFΣk−1FTþσ2t IÞ−1

× ðxk−Fxk−1Þ−12ðbk−AxkÞTðσ2mIÞ−1ðbk−AxkÞ

−12ðLxkÞT

�σ2mλ2 I

�−1ðLxkÞ

�: ð26Þ

It is clear that the Kalman filter is now combiningthree sets of information during each time step: atemporal prior; a spatial prior; and the newly ac-quired measurement data. The three parameters,σt, λ, and σm, determine the relative influence eachof the three distributions has on the final solution.The measurement variance, σm2, can be determinedbased on the true system characteristics, while λ andσt must be chosen heuristically to determine the em-phasis of the smoothness and random-walk updatepriors in the reconstruction.

4. Problem Description

To validate the Kalman filter approach, syntheticdata sets are generated from a large eddy simulation(LES) [26,27] of a mixed buoyant turbulent methaneplume, which are then contaminated with measure-ment noise of various degrees. Using this data, weevaluate and compare the quality of reconstructionsobtained using the Kalman filter formulation withthose found from independent static MAP inference.

The computational domain consists of a 1:5m ×1:5m× 6m enclosure, shown in Fig. 3, filled withair at atmospheric pressure and 298K. A square0:2m× 0:2m duct is centered at the bottom, whileopen boundary conditions are applied to the topand sides. Methane is injected upward through theduct at 2m=s, corresponding to a jet Reynolds num-ber of 3000 based on the hydraulic diameter of theduct. The resulting plume has a densimetric Froudenumber of 2.2, indicating that the buoyant and mo-mentum forces within the plume are comparable inmagnitude.

The simulation is carried out with a Smagorinskyconstant of 0.1, which has been shown to be suitablefor thermally driven plumes [28–30], and a turbulentSchmidt number of 0.5 is used to model the diffusionof methane into the surrounding air. The governingmass, momentum, and species transport equationsare discretized on a 128 × 128 × 512 mesh, using sec-ond-order finite differences, and then integrated intime using a second-order adaptive Runge–Kuttascheme [27].

Line-of-sight attenuation data is collected at the3mheight cross section using a 32 beam array shownin Fig. 4, inspired by a configuration proposed inTerzija et al. [14]. The tomography domain is a 1m ×1m square centered in the cross section, discretizedinto 400 square elements. Beam attenuation data issampled at the rate of 100Hz, or at 0:01 s intervals,for 20 s. Since the width of most laser beams used ininfrared species tomography is small relative to boththe tomography grid spacing and the major turbu-

lent flow artifacts, the beams are modeled as in-finitesimal. The methane absorption coefficient isrelated to the simulated molar concentration by

aηðrÞ ¼ σηNðrÞ ¼ σηχCH4ðrÞP

R°T; ð27Þ

where N is the molecular number density, ση is themolecular absorption cross section, P and T are thepressure and temperature of the gas mixture, respec-tively, and R° is the universal gas constant. Since thereconstructions in this study are done using a simu-lated χCH4ðrÞ profile instead of experimental data,

Fig. 3. (Color online) LES simulation of a mixed buoyantmethane plume.

Fig. 4. (Color online) Beam configuration in the tomographyplane, and methane molar concentration at 13 s. The tomographydomain is defined by the 1m× 1m white box.


the tomography procedure is independent of ση,although in actual tomography experiments thisparameter can be found either by shining a laserthrough a “test cell” of known concentration [3] orfrom line-by-line spectral absorption data [31]. Theintegral in Eq. (2) is evaluated using a high-orderGaussian quadrature, while a continuous methanemole fraction distribution χCH4ðrÞ is approximatedover the tomography domain through bilinear inter-polation between the LES grid points. Figure 5shows the uncontaminated time-resolved attenua-tion measurements for the set of beams highlightedin Fig. 4.

Each data point is then contaminated with un-biased Gaussian “white” noise,

~biðkÞ ¼ biðkÞ þ δbk; ð28Þwhere δbk is sampled from an unbiased normaldistribution having a standard deviation of σm. In ex-periments, δbk is due to photonic “shot” noise, con-taminant droplets, and particles in the gas streamand to electronic noise in the data collection system.

5. Results

The first task in implementing the Kalman filter is tochoose a value of λ, which we do using the SVD of theaugmentedmeasurementmatrix [15] generated withdifferent values of λ as shown in Fig. 6. The λ ¼ 0 casehas the 32 nontrivial singular values ofA; as noted inSection 2, Eq. (6) produces xLS, the smallest two-normminimizer of ‖Ax − b‖2

2, but this solution is ob-viously nonphysical since the information providedby the ray sums is inadequate to uniquely specifythe unknown null space component, xn�, of the truesolution, xexact ¼ xLS þ xn�. Increasing λ to moderatevalues results in a well-conditioned but overdeter-mined system of equations; no x exists that solvesboth Ax ¼ b and Lx ¼ 0 since the Gibbs prior is max-imized only by a uniform or bilinear distribution.Nevertheless, at moderate values of λ, the informa-tion provided by the Gibbs prior should promote a so-lution closer to the true solution, xexact, than xLS.Making λ too large, however, results in singularvalues that overwhelm the original singular values

of A, suggesting that the information provided bythe ray sums has become obscured by the Gibbs prior.Based on the singular values in Fig. 6, then, wewould expect a good choice for λ to be around 0.1.

We confirm this choice by evaluating the recon-struction accuracy at different values of λ for the con-centration distribution at t ¼ 13 s. Reconstructionaccuracy is defined as

εðxλÞ ¼ ‖xλ − xexact‖=‖xexact‖ave; ð29Þwhere xλ is the solution found from Eq. (14), xexactcontains the exact cell-centered values from theLES simulation, and ‖xexact‖ave is the average ofthe exact solution norms between 2:5 s, which is ap-proximately when the rising methane plume reachesthe tomography plane, and the end of the measure-ment period at 20 s. The time-averaged exactsolution norm is used to avoid overemphasizing re-construction errors corresponding to the beginningof the process, when xexact is small. Figure 7 showsthe reconstruction error for static MAP with σm ¼10−2 for different values of λ; the inset reconstruc-tions correspond to the case where λ ¼ 10−5 (underregularized), λ ¼ 10 (over regularized), and λ ¼ 0:1(which is close to the optimal value of regularization).(The true distribution is shown in Fig. 4.) This resultvalidates our strategy of choosing λ based on the SVDof the augmented coefficient matrix.

We now turn to a full implementation of theKalman filter. With λ fixed 0.1 for both the staticMAP and Kalman filter reconstructions, σt becomesa tuning parameter that adjusts the influence therandom-walk prior has over the ray-sum data andspatial prior in the Kalman filter. Figure 8 showsa series of representative time slices of the true solu-tion, the static MAP solution obtained using datacontaminated with noise having σm ¼ 10−2, andKalman filter solutions found using σt ¼ 10 andσt ¼ 10−3. Three consecutive time slices are includedto show that the random-walk temporal prior, whichhypothesizes that the solution changes very littlebetween consecutive time steps, is reasonable.

Fig. 5. (Color online) Sample beam attenuation data, correspond-ing to the highlighted beams in Fig. 4.

Fig. 6. Singular values from the augmented matrix equation, forvarious values of λ.


Qualitatively, both the static MAP and Kalman filteralgorithms capture the main flow artifacts; we canalso observe that the Kalman filter reconstructionobtained using σt ¼ 10 closely resembles the staticMAP result, which we would expect since the largestate transition variance means that the Kalman fil-ter is ignoring the random-walk model. When moreemphasis is placed on the temporal prior by loweringσt to 10−3, Fig. 8 shows that the Kalman filter recon-struction appears oversmoothed.

A quantitative assessment of performance is pro-vided in Fig. 9, which shows the average errors of sta-tic MAP and projected Kalman filter reconstructionsfor different values of σm, ranging from 10−4 (lownoise) to 10−1 (high noise), and σt ranging from 10−4

to 102. Representative ray-sum data for σm ¼ 10−2

and 2 × 10−2 are presented in Fig. 10 to give a betterunderstanding of the measurement noise levels.Perhaps surprisingly, the Kalman filter outperformsthe static MAP only at high levels of measurementnoise, with σm > 10−2; at lower noise levels, incor-porating the temporal prior actually degrades the re-construction accuracy.

Some insight into this behavior can be gained fromthe singular value decomposition of the ray-sum ma-trix augmented with the Gibbs spatial prior. If thelaser attenuation data in b is contaminated withnoise, substituting b ¼ bexact þ δb into Eq. (6),

x ¼ xexact þ δx ¼Xnj¼1

uTbexact

σivi þ

Xnj¼1

uTδbσi

v; ð30Þ

Fig. 7. (Color online) Error of static reconstructions at 13 s. Set-ting λ too small fails to fill the null space of A, while setting λ toolarge oversmooths the solution. The optimal value is aroundλ ¼ 0:1, consistent with the singular values shown in Fig. 6.

Fig. 8. (Color online) Comparison of the static MAP and Kalmanfilter reconstructions at various process times, for σm ¼ 10−3.

Fig. 9. (Color online) Average errors produced for the static MAPand Kalman filter solutions for different measurement noise andstate variable covariance.

Fig. 10. (Color online) Sample beam attenuation measurementsfor low- and high-noise scenarios.


shows that the error amplification characteristic ofdiscrete ill-posed problems is caused when δb is am-plified by small singular values. Inspection of Fig. 6reveals, however, that the singular values of the aug-mented ray-sum matrix are sufficiently large forλ ¼ 0:1, so no additional temporal regularization isrequired when ‖δb‖ ≪ ‖b‖; in fact, the additionalfiltering introduced by the random-walk model over-smooths the solution as can be seen in the rightmostcolumn of Fig. 8. Only when δb is large does the sec-ond term in Eq. (30) overwhelm the solution, inwhich case the additional temporal regularizationin the Kalman filter provides more accurate solu-tions compared to those found using the spatial prioralone. Close inspection of Fig. 9 shows that, even inthis scenario, overemphasizing the random-walkmodel by choosing too large a value for σt will de-grade the reconstruction accuracy.

Kalman filtering also presents an important ad-vantage in cases where simultaneous measurementof the laser attenuation data is difficult, especially atthe high sampling rates required to resolve turbulentphenomenon. In this scenario, the Kalman filtercan sample a subset of the laser attenuation dataat each time step and then combine them throughthe update. In the beamarrangement shown in Fig. 4,for example, the beams can be sampled in subsets ofeight at 25Hz out of phase instead of full sets of 32beams at 100Hz. A likely multiplexing patternwould be to simultaneously sample the beams point-ing in one direction, which would be especially con-venient if a single linear photodiode array were usedin place of eight individual detectors. The error re-sults for this scenario are presented in Fig. 11, forboth static calculations and the Kalman filter. Whilethe Kalman filter shows almost no degradation inperformance between using all 32 measurements si-

multaneously and using only eight from each timestep, the static solution deteriorates appreciably.In this instance, the static MAP would never providea reasonable reconstruction since at any instant theattenuation of a set of parallel beams provides am-biguous data. It is interesting to note that, for thefirst three time steps, the Kalman filter performanceis also seriously degraded, but once sufficient mea-surements are made, the Kalman filter solutionsnaps to a reasonable solution, due to the correct pro-pagation of covariance information.

6. Conclusions

Reconstructing the species concentration distribu-tion from laser attenuation measurements involvessolving a rank-deficient linear inverse problem,which can only be done by incorporating presumedinformation about the nature of the concentrationdistribution, called priors, into the reconstructionthrough MAP estimation. While static reconstruc-tions can be carried out using smoothness and non-negativity priors, in the case of a dynamicallyevolving concentration field, the Kalman filter allowsspecification of additional temporal priors based onhow the flow field changes between measurementtimes. This technique was demonstrated by recon-structing the concentration cross section of a turbu-lent buoyant methane plume. At low levels ofmeasurement noise, the static MAP estimation pro-vides more accurate reconstructions because the ad-ditional transient prior of the Kalman filter (in thiscase a random-walk model) overregularizes the solu-tion. At higher levels of measurement noise, however,the Kalman filter provides superior reconstructions.In contrast to static MAP reconstructions, theKalman filter also facilitates multiplexing, whichmay enable the high sampling rates needed toresolve turbulent phenomena.

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infrared species tomography of a transient flow field using kalman filtering

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