Laser-absorption tomography beam arrangement optimization using resolution matrices Matthew G. Twynstra* and Kyle J. Daun Department of Mechanical and Mechatronics Engineering, University of Waterloo, 200 University Avenue West Waterloo, Ontario N2L 3G1, Canada *Corresponding author:
[email protected] Received 6 June 2012; accepted 27 August 2012; posted 11 September 2012 (Doc. ID 170042); published 9 October 2012
Laser-absorption tomography experiments infer the concentration distribution of a gas species from the attenuation of lasers transecting the flow field. Although reconstruction accuracy strongly depends on the layout of optical components, to date experimentalists have had no way to predict the performance of a given beam arrangement. This paper shows how the mathematical properties of the coefficient matrix are related to the information content of the attenuation data, which, in turn, forms a basis for a beamarrangement design algorithm that minimizes the reliance on additional assumed information about the concentration distribution. When applied to a simulated laser-absorption tomography experiment, optimized beam arrangements are shown to produce more accurate reconstructions compared to other beam arrangements presented in the literature. © 2012 Optical Society of America OCIS codes: 110.6955, 220.4830, 280.2490, 120.1740, 300.6360, 110.6960.
1. Introduction
In many applications involving gas-phase chemistry, it is essential to obtain a controlled (often homogeneous) species concentration distribution within the flow field. For example, the performance of nextgeneration internal combustion engines depends strongly on obtaining an engineered fuel–air dispersion within the combustion chamber to minimize emissions and maximize fuel efficiency [1]. Likewise, highly controlled gas concentrations are essential to avoid the problem of “ammonia slip” from selective catalytic and noncatalytic reduction systems in power plants [2]. Applications like these create a pressing need for a diagnostic that can provide real-time concentration distributions of gas species in the flow field in order to pinpoint potential problems and improve the performance of these devices. Measuring concentration through physical probing has a number of drawbacks: physical probes have limited temporal and spatial resolution, especially 1559-128X/12/297059-10$15.00/0 © 2012 Optical Society of America
for turbulent flows; the sampling probe often perturbs the measurement; and in many applications it is difficult to physically access the flow field. One technology that overcomes the drawbacks of physical measurements is planar laser-induced fluorescence (PLIF). PLIF relies on the fluorescent properties of materials to provide real-time flow time visualization and species distribution analysis of the flow field. In this technique, the target species is excited by a laser sheet shone through the flow field. A detector camera then records the fluorescence intensity from a second optical plane, which in turn is proportional to the concentration of the target species [3]. Though PLIF has been successfully implemented in industry for process monitoring and diagnostics, the method has several limitations. First, the flow field must contain a chemical species with resonant wavelengths accessible to the laser; otherwise a fluorescent tracer must be added to dope the target species. This, in turn, can lead to measurement error as complete mixing between the target species and tracer may not always occur. Second, obtaining a second optical access plane is difficult or infeasible for many problem geometries. 10 October 2012 / Vol. 51, No. 29 / APPLIED OPTICS
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An emerging alternative to PLIF is laser absorption tomography, which reconstructs the flow field distribution using laser line-of-sight attenuation (LOSA) measurements. In this method, the flow field is discretized into an n pixel grid, and the target species concentration in each pixel is modeled as uniform. The flow field is then transected with m coplanar lasers having wavelengths corresponding to an absorption line of the target species. The concentration distribution x is related to the LOSA measurements in b by an m × n matrix equation, Ax b. Because the number of pixels far exceeds the number of beams, however, A is rank-deficient, and consequently the laser attenuation data b must be supplemented with additional assumed information about the concentration distribution attributes, such as smoothness and nonnegativity [4]. Experimental [5–7] and numerical [4,8,9] laserabsorption tomography experiments have shown that a good beam arrangement is essential for obtaining accurate reconstruction results. Moreover, the extreme cost and complexity of the laser and detection optics demands that these components be used to their maximum potential. Nevertheless, to date only Terzija et al. [6] have proposed a heuristic strategy for optimizing the beam arrangement. This paper presents a more rigorously defined algorithm for optimizing the beam arrangement using the properties of the resolution matrix, R, which is often used to assess how regularization limits reconstruction accuracy in discrete geophysical tomography applications [10]. We assess the merit of the proposed fitness function by investigating how closely it correlates with reconstruction error of simulated limited-data absorption tomography experiments under both noise-free and noisy data conditions. The objective function is then minimized for a given number of sensors and constraints using a genetic algorithm, and the optimized beam array is shown to outperform other arrangements reported in the literature in terms of reconstruction accuracy. To quantify the effects of beam misalignment in the experimental setup, a study was conducted in which the positions of the beams were altered within expected ranges associated with optics installation. It was shown that the system was robust to misalignment effects. Finally, a plot of reconstruction accuracy obtained from optimized beam arrangements versus the number of beams in the array demonstrates, for the first time, the trade-off between reconstruction accuracy and experimental cost and complexity. This result provides a framework for establishing the number of beams needed to achieve a given level of reconstruction accuracy and thus reduce associated experimental costs. While this research is motivated by near-IR laser absorption tomography, the algorithm presented in this paper could easily be extended to other tomographic experiments, such as x-ray and gammaray tomography, with similarly limited sensor geometries. 7060
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2. Laser Absorption Tomography
The attenuation of the ith laser beam transecting a participating medium is governed by the Beer– Lambert law, ln
I η0;i bΦi I η;i
Z
∞ 0
κ η rΦi udu;
(1)
where I η0;i and I η;i are the incident and exiting intensities respectively, η is the wavenumber, rΦi u is the position vector, and u is a parametric distance along the beam. This attenuation model neglects the effect of scattering because the molecular scattering cross section is much smaller than the absorption cross section at mid-IR wavelengths [11]. The position of each beam is specified by Φi si ; θi T, where si is the perpendicular distance from the laser beam to the origin and θi is the angle formed between the beam and the y axis, as shown in Fig. 1. The absorption coefficient, κη , scales with the mole fraction of the absorbing species, χ, through κη σ η N σ η
χrP ; Ro T
(2)
where σ η is the molecular absorption cross section, N is molecular number density, P and T are the pressure and temperature of the gas mixture (which are assumed to be uniform), and Ro is the universal gas constant. The molecular absorption cross section can be found either from tabulated data [12] or by performing absorption experiments on a test cell containing a known concentration [13]. In absorption spectroscopy experiments, selection of the monochromatic wavelength is paramount to successful species detection. Typically a wavelength is chosen that corresponds to one of the resonant absorption overtones of the target species and has no absorption signature in the other species present. These techniques have also been extended to dual-wavelength absorption spectroscopy [1,5,6], which can account for species scattering effects, and hyperspectral absorption [14] spectroscopy techniques, which can allow
i
y x
I
i
u du
si u=0 I
0,i
Fig. 1. Geometry of the ith beam transecting the tomography field.
both concentration and temperature distribution reconstructions. Equation (1) is a Fredholm integral equation of the first kind. Deconvolving this equation for κ η rs;q u is mathematically ill-posed because an infinite set of candidate distributions could be substituted into the integral to produce the observed attenuation data [15]. Simultaneously solving these equations for multiple beams reduces, but does not eliminate, this ambiguity. By discretizing the flow field into a set of n pixels, each of which is assumed to contain a uniform concentration of the target species, Eq. (1) can be approximated as a “ray-sum,” bi
n X
Aij xj ;
(3)
j1
where Aij is the chord length of the ith beam subtended by the jth pixel and xj is the mole fractionnormalized absorption coefficient for the jth pixel. Writing Eq. (3) for all m beams results in an m × n matrix equation, Ax b, where A ∈ ℜm×n . In principle, given enough projections, RankA n and the concentration profile, x, could be uniquely solved via least squares minimization of ‖b − Ax‖. Unfortunately, due to the cost of detection equipment and spatial constraints imposed by most practical geometries, the number of projections is typically far less than the required number of pixels to achieve a reasonable level of spatial resolution (m < n). Thus, laser absorption tomography is usually a limiteddata tomography problem and A has a nontrivial null space. Therefore, the true concentration profile is a sum of two components, x xLS xn , where xLS arg min‖b − Ax‖ belongs to the range of A, while xn is a nontrivial vector belonging to the null space of the problem, fxn ∈ NA: xn jAxn 0g [15]. To span this null space, it is necessary to supplement the sensor data with additional assumed information, such as smoothness and nonnegativity. In a Bayesian context, these assumed attributes are called “priors” and correspond to the prior probabilities used to condition an indefinite likelihood function in Bayes’ equation. McCann’s group at Manchester University [1,5–7,16,17] use a modified Landweber iterative regularization scheme to carry out the reconstruction. Landweber regularization is designed to exploit the semiconvergence property of full-rank but illconditioned linear problems [15]. In this application, the nontrivial null space is “filled” by passing a median filter [1,5–7,16] or wavelet filter [17] through the solution between Landweber passes, followed by a nonnegativity filter. The solution stops once a prescribed minimum level of change between iterations is satisfied. Salem et al. [18] used the iterative algebraic reconstruction technique with discrepancy between theoretical data (from the candidate reconstruction) and measurement data serving as stopping criteria.
Daun [4] used Tikhonov regularization to span the null space of A; in this technique, an additional n set of equations, λLx 0, is introduced into the analysis, where L is a discrete approximation of the ∇ operator, 8
