1. Introduction

Tomographic techniques based on absorption spectroscopy represent one of the two major diagnostic tools (the other one being planar laser imaging) for combustion imaging [1,2]. Such tomographic techniques offer quite a few distinct advantages when compared to planar laser imaging techniques (e.g., planar laser induced fluorescence or Rayleigh scattering). The experimental implementation of such tomographic techniques is relatively straightforward (though maybe cumbersome) [2]. The absorption spectroscopy enjoys species-specific signals, usually with high signal strength [3]. Furthermore, these tomographic techniques can enable continuous and simultaneous imaging of multiple combustion parameters such as temperature and chemical species concentration [4,5].

Due to these unique features, over the past three decades or so, absorption-spectroscopy based tomography has been continuously researched, and demonstrated in a wide array of applications [1,2]. However, past efforts predominantly relied on the use of one of two wavelengths. As a result of such limited spectral information, a large number of projections (typically more than 100 [6,7]) is required for a satisfactory tomographic reconstruction. Such requirement severely limited the application of tomographic technique in practical reacting flows, and significantly restricts the imaging capability and applicable range of the tomographic technique.

The recent advances in optoelectronics enable a promising outlook to overcome these limitations. The rapid development in fiber optics and high-speed electronics greatly reduces the experimental difficulty of tomography techniques based on absorption spectroscopy, such that a relatively large number of projections can be implemented simultaneously and recorded rapidly. For example, by extensively using fiber optics and high-speed optoelectronics, Wright et al. have demonstrated a 32-projection tomographic sensor for applications in engine environments [8–10]. The sensor utilizes the absorption of hydrocarbon fuel in the near infrared, at one or two wavelengths (e.g., 1700 and 1651 nm), to image the concentration distribution of hydrocarbon fuel. Another example involves the use of water vapor absorption at 1396 nm to image the concentration of water vapor in a packed bed adsorber [11]. This example utilizes sheet-forming optics and photodiode arrays to facilitate the implementation of a large number of projections.

The development of hyperspectral laser sources (e.g., [12]) represents another direction of the advancements in optoelectronics. These laser sources enable one to acquire rich absorption spectra at rapid rates (the so-called hyperspectral absorption spectroscopy), and it is attractive to contemplate the use of such rich spectral information in tomographic techniques. Previous numerical studies [13–15] have suggested that rich spectra not only reduce the number of projections significantly (a five-fold reduction was demonstrated in [15]), but also offer several fundamental improvements over past schemes based on one or two wavelengths. For example, the rich spectra also enable 1) the simultaneous retrieval of temperature and concentration distributions (the examples cited in the previous paragraph only retrieved concentration distribution), and 2) superior stability of the tomographic reconstruction in the presence of measurement uncertainties [15]. Based upon these previous results, this paper reports the experimental demonstration of a tomographic sensor using hyperspectral absorption spectroscopy. This sensor utilized a hyperspectral laser source to scan the 1333-1377 nm wavelength range and measure many absorption features of water vapor (H₂O) in this spectral region. These measurements were then used as the inputs for a tomographic inversion algorithm to simultaneously reconstruct the distributions of temperature and mole fraction of H₂O. The sensor was demonstrated in a well-controlled near-adiabatic, atmospheric-pressure laboratory flame, and the reconstructed distributions are in good agreement with those obtained via independent methods such as coherent anti-Stoke Raman scattering (CARS) spectroscopy.

This experimental demonstration validates several of the key advantages enabled by the hyperspectral information content, including a significant reduction in the number of required projection measurements and an enhanced insensitivity to measurements/inversion uncertainties. These advantages are critical in the practical implementation and application of the tomography technique. The rest of this paper is organized as follows. Section 2 summarizes the physical and mathematical background of the hyperspectral tomography technique. Section 3 describes the experimental setup, followed by results and discussions in Section 4. Finally, Section 5 closes the paper with a summary.

2. Theoretical background

The physical and mathematical background of the hyperspectral tomography technique has been detailed in [15]. A brief summary is provided here for convenience. Figure 1 schematically illustrates the mathematical formulation of the problem. A hyperspectral laser beam is directed along the line of sight, denoted by 1, to probe the domain of interest as shown in the left panel. Absorption by the target species will attenuate the probe laser beam, and the absorbance at a certain wavelength (e.g., λ_i) generally contains contributions from multiple transitions centered at various wavelengths (including that centered at λ_i itself), as schematically shown in the right panel. Here, we use p(L_j, λ_i), termed a projection, to denote the absorbance at a projection location L_j and a wavelength λ_i. The projection, p(L_j, λ_i), is expressed by the following integral:

where a and b the integration limits determined by the line of sight and the geometry of the domain of interest, S(λ_k, T(1)) is the line strength of the contributing transition centered at a wavelength λ_k and depends nonlinearly on temperature (T) [15]; T(1) and X(1) the temperature and mole fraction profile of the absorbing species along the line of sight, respectively; Φ the Voigt lineshape function [16]; and P the pressure, assumed to be uniform. The summation runs over all the transitions with non-negligible contributions. In this work, the domain of interest is discretized by superimposing a square mesh in the Cartesian coordinate, as shown in the left panel of Fig. 1; and the integration in Eq. (1) is also discretized accordingly.

 

Fig. 1. The mathematical formulation of the hyperspectral tomography problem.

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With the above understanding, the hyperspectral tomography problem seeks to determine the distributions of T and X over the discretized domain with a finite set of projections as described in Eq. (1). Hence, mathematically, the hyperspectral tomography problem is an inverse problem, which has been studied extensively. However, due to the inclusion of multiple wavelengths and the nonlinear dependence of the line strength on temperature, the hyperspectral tomography problem poses distinct challenges and algorithms designed in the past cannot be readily applied. A new inversion algorithm was therefore developed to address the special challenges of the hyperspectral tomography problem [13,15]. The algorithm casts the inversion problem into a nonlinear optimization problem, where the T and X distributions are retrieved by minimizing the following function:

where p_m(L_j, λ_i) denotes the measured projection at a location L_j and a wavelength λ_i; p_c(L_j, λ_i) the computed projection based on a reconstructed T and X profile (denoted by T^rec and X^rec, respectively); and J and I the total number of wavelengths and projection locations used in the tomography scheme, respectively. This function, D, provides a quantitative measure of the closeness between the reconstructed and the actual temperature and concentration profiles. The contribution from each wavelength to D is normalized by the projection at this wavelength itself, such that projections measured at all wavelengths are weighted equally in the inversion. When T^rec and X^rec match the actual profiles, D reaches its global minimum (zero). Note that the formulation in Eq. (2) is designed specifically for the hyperspectral tomography problem. For a general tomography problem, this formulation may encounter singularity issues (i.e., D(T^rec, X^rec) → ∞ as p_m → 0). In the hyperspectral tomography problem (and other sensing techniques based on absorption spectroscopy), the technique is designed such that the minimal absorbance (i.e., the projections, p_ms) is above a certain level, and consequently, the singularity issue will not occur.

However, the problem is a nonlinear optimization problem due to the nonlinear temperature-dependence of the line strength. The nonlinearity of the problem resulted in two difficulties: 1) the problem is ill-posed, i.e., the existence, uniqueness, and stability of the solution are not simultaneously ensured [17]; and 2) typical minimization methods based on the derivatives (or gradients) of the objective function are generally not able to converge to the global minimum [13,15,18]. These issues can be addressed, respectively, by 1) applying a regularization technique, and 2) using a stochastic minimization algorithm, the simulated annealing algorithm. More specifically, the following new target function (F) is minimized instead of D:

where R_T and R_X are the regularization factors for temperature and concentration, respectively; γ_T and γ_X are positive constants (regularization parameters) to scale the magnitude of R_T and R_X properly such that they do not dominate the D(T^rec, X^rec) term. In Eq. (3), the regularization factors represent the a priori information (e.g., smoothness of the T and X distributions); and the magnitudes of λ_T and λ_X reflect the relative weights of the a priori information and the a posteriori knowledge (i.e., measurements). The master function, F, is then minimized using a stochastic algorithm, the simulated annealing algorithm [19]. More details of the use of regularization factors, the determination of the optimal regularization parameters, and the simulated annealing algorithm can be found in [13–15]. Finally, the solution of the minimization problem described in Eq. (3) provides the tomographic reconstruction of the T and X distributions.

3. Experimental setup

The schematic of the experiments and the laser used is shown in Fig. 2. In these experiments, a prototype hyperspectral sensor was use to image the distribution of temperature and H₂O concentration over a Hencken flame (top panel of Fig. 2). To facilitate the discussion, we name the laser used here “fiber Fabry-Perot tunable filter laser” (FFP-TFL) (bottom panel of Fig. 2). The fiber-coupled source output was split into 8 similar fiber outputs using a fiber tree coupler. Two of these outputs were used for monitoring: one fed to a Mach-Zehnder interferometer to track the FFP-TFL wavelength and the other fed to a photoreceiver to monitor the FFP-TFL intensity (the so-called reference intensity, I_O). The six remaining fibers delivered free-space beams (labeled as Beam 1 through 6) to perform projection measurements across the flame region of interest. The flame is produced by a square H₂/air Hencken burner (2.54 cm × 2.54 cm) and is surrounded by a N₂ co-flow. Surrounding the co-flow was a dry N₂ purge flow used to eliminate the interference of H₂O in room air. The thickness of the co-flow was 1.27 cm, and that of the purge flow ~2 mm. The transmitted laser beams were registered by six photoreceivers (labeled as Detector 1 through 6), and the signals collected converted into absorption spectra for use in the tomographic reconstruction. Two sets of spectra measured at the 6th beam location are shown in Fig. 3, one measured at an equivalence ratio of 1.0 (Φ = 1.0) and the other atΦ = 0.5, respectively. Note the variations in the relative strengths of the absorption peaks between the spectra at these equivalence ratios. Such variations form the basis of temperature sensing and the tomographic reconstruction.

 

Fig. 2. Schematic of the experimental setup and the hyperspectral laser source. Top panel: experimental arrangement and a seven -zone tomography scheme used to perform the tomographic reconstruction. The dimensions of the zones are not drawn to scale. Bottom panel: schematic of the FFP-TFL used to measure the flame spectra. The design is among the most basic possible for a swept-wavelength laser based on a fiber Fabry-Perot tunable filter.

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The laser cavity is composed of a semiconductor-based booster optical amplifier (SOA, Covega, BOA1036), two fiber isolators (ISO, Opto-link OLISO-I-D131-300-90), a polarization controller (PC), a fiber Fabry-Perot tunable filter (FFP-TF, Micron Optics, TF06EN) and an output coupler. By controlling the FFP-TF with a function generator (FG, Stanford Research Systems, DS360), the laser was swept over the 1333 – 1377 nm range at up to ~200 Hz rates. This hyperspectral probe beam continuously scans the 1333-1377 nm spectral region, sampling many H₂O absorption transitions (most belong to the R branch of the ν ₁ × ν ₃ band, and some belong to the 2ν ₁ and 2ν ₃ bands). The FFP-TFL is similar to a standard external-cavity tunable diode laser (ECDL), with three key differences. First, it is composed entirely of fiber-optic components. Second, the FFP-TF is used to set the instantaneous wavelength rather than a free-space grating. Finally, because standard fiber pigtails were present on all laser components, the laser cavity length is significantly longer than a typical external-cavity laser, ultimately preventing single-mode operation. The cavity length of the laser was ~17 m, which set the cavity mode spacing to ~12 MHz. We operated the FFP-TFL in a multiple-spectral-mode fashion for all the measurements presented here, with the FFP-TF enforcing an instantaneous linewidth of ~5GHz. The multimode operation is not especially desirable, because competition and beating among the multiple (up to ~500 at any instant) modes results in an intensity noise that is higher than in a traditional ECDL;

 

Fig. 3. Example spectra measured by the laser at beam location 6.

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however, unlike many ECDLs, the FFP-TFL exhibits no obvious discontinuities in its wavelength sweeps, it is simple to build, and it is rugged. Furthermore, it is adequate for many spectral measurements such as the ones reported in this paper.

The FFP-TFL can also be compared to the Fourier-domain mode-locked (FDML) laser, such as one used previously for engine gas thermometry [20]. The FFP-TFL contains no delay spool and therefore has a cavity length approximately 2 orders shorter than the FDML. The FFP-TFL is not intended to be driven at the correspondingly high (~MHz range) mode-locking frequency, and the benefits of narrow spectral linewidth and low intensity noise possible with the FDML [21] are thereby sacrificed. Also, it is not possible to run this laser near typical FDML repetition rates (~30 kHz) because every wavelength in the sweep must be built-up each time from spontaneous emission. However, because the sweep repetition rate is not resonant with the cavity length, the FFP-TFL can be easily run at any low repetition rate (below ~200 Hz in this case) without any physical adjustments to the cavity. This scan rate is sufficient for the present experiments because the flame investigated is steady. For applications which require higher temporal resolution, other types of laser sources featuring higher repetition rates can be used. For instance, the time-division-multiplexed (TDM) laser described in [12] can cycle through a many wavelengths in 1333-1377 nm spectral range every 15 μs, therefore enabling one to monitor engineering variables at rates greater than 50 kHz.

To perform the tomography reconstruction, the measurement region was discretized into seven zones, as shown in the top panel of Fig. 2, with a uniform temperature and H₂O mole fraction assumed in each zone. The flame was divided into five zones: a 0.85 cm × 0.85 cm square zone at the center, and four symmetric zones around the edges. The N₂ co-flow and purge flow were considered as the sixth and seventh zones, respectively. Note that even though the 7-zone scheme shown in Fig. 2 is symmetric, our tomographic technique does not invoke the assumption of symmetry (analogously, setting up a tomographic scheme by imposing a 4-by-4 square mesh does not invoke the symmetry assumption). The assumption of symmetry is only invoked when the tomographic inversion algorithm assumes/forces the values of temperature/concentration of certain zones to be equal; and our technique made no such assumption.

4. Results and discussions

4.1 Tomographic imaging of temperature and chemical species

The signal obtained at each beam was processed to yield absorption spectra using the procedure detailed in [22]. Only selected portions of the absorption spectra were used in the analysis performed here for two reasons. The first reason involves the consideration of measurement uncertainties. The relative measurement uncertainty is minimal when the absorbance is strong; and vice versa. For this reason, the use of the portions of the spectra with relatively strong absorbance is preferred. The second reason involves the consideration of the computational cost. This work used a spectroscopic database named BT2 [23], which contains more than 25,000 transitions in the spectral range scanned by our sensor. For each transition in the database, there are about 2,000 transitions with non-negligible contributions (see Eq. (1)). Therefore, calculating the spectra once at one beam location with all these transitions considered requires ~50 million evaluations of the complicated Voigt lineshape function; and minimizing F to obtain T and X requires calculating the spectra at all six beam locations for a large number of times. Such a computational cost is beyond our existing capability. With both reasons considered, here the strongest 100 absorption peaks in the measured spectra were selected and used in the analysis. This selection reduced the computational cost by a factor of ~3000, and the peaks selected can still provide a reasonable overall representation of the spectra. Though this current method is simple and practical, research is underway to refine the selection of transitions for use in the hyperspectral tomography technique. Besides the measurement uncertainty and computational cost, the selection method should also consider the response of the line strength of the selected transitions with respect to temperature (which is mainly controlled by a parameter named the lower state energy). Intuitively, transitions with line strength varying more sensitively to temperature change should provide more accurate temperature measurements.

With these transitions selected, a line-of-sight-averaged analysis was first performed using the spectra measured at each beam location, with results shown in Fig. 4. Two sets of data are show here, one set obtained at an equivalence ratio of 1.0 (Φ = 1.0) and the other at Φ = 0.5. As can be seen, the line-of-sight-averaged analysis yields a different temperature and concentration when the measured spectra at different beam locations are used. Such variation is partly due to measurement uncertainties, but primarily due to the nonuniformity over the measurement region (including the purge flow and co-flow zones).

 

Fig. 4. T and X obtained from line-of-sight analysis using the measured spectra at each beam location. The adiabatic flame temperature and concentration obtained from equilibrium calculations are: T = 2379 K and X = 0.347 at Φ = 1.0; T = 1647 K and X = 0.190 at Φ = 0.5. The flame temperatures measured by CARS are: T = 2400 K at Φ = 1.0; T = 1625 K at Φ = 0.5.

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The tomographic inversion method described in Section 2 was then applied to reconstruct the T and X distributions over the measurement region, using measured spectra at all six beam locations as inputs. Two sets of results are shown in Fig. 5, again at Φ = 1.0 and Φ = 0.5, respectively. In this figure, T_i and X_i represent the temperature and H₂O concentration in the ith zone, respectively. The results confirm that a certain degree of nonuniformity exists in the test region, and also the insulating effects of the purge flow and co-flow. The results shown are in reasonable agreement with the adiabatic flame calculations and past measurements using CARS [24]. The Hencken burner used in this experiment has been well characterized in the past [24] to produce adiabatic flame conditions. The discrepancy between the tomography results and the adiabatic flame calculations/CARS measurements are largely due to the mismatch between the BT2 database and the measured spectra. Research efforts are ongoing to measure the absorption spectra in well-controlled environments to resolve this mismatch.

 

Fig. 6. Comparison of the fitting residuals between line-of-sight-averaged and tomography analysis.

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Fig. 5. Tomographic reconstructions of T and X over the seven zones. Left panel: Φ=1.0. Left panel: Φ=0.5. Treatment of the Lorentzian width was the same as that in Fig. 4. The adiabatic flame temperature and concentration obtained from equilibrium calculations are: T=2379 K and X=0.347 at Φ=1.0; T=1647 K and X=0.190 at Φ=0.5. The flame temperatures measured by CARS are: T=2400 K at Φ=1.0; T=1625 K at Φ=0.5. A constant Lorentzian width (ν_L) was used in evaluating the Voigt lineshape: ν_L =0.13 cm^-1 for Φ=1.0 and ν_L =0.14 cm^-1 for Φ=0.5. The dimensions of the zones are not drawn to scale.

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Another verification of the nonuniformity (and the validity of the tomographic analysis) is that the above tomographic analysis also reduces the fitting residual between the simulated and measured spectra, as shown in Fig. 6. Here, the fitting residual was calculated by 1) subtracting the measured spectra from the spectra calculated using the T and X obtained from tomographic/line-of-sight-averaged analysis to obtain the difference, and 2) integrating the square of the difference obtained in step 1 over the spectral range scanned. To make the residuals comparable, about the same number of fitting parameters were used in the tomographic and the line-of-sight-averaged analysis. More specifically, the line-of-sight-averaged analysis assumed different Lorentz widths for different parts of the spectra, and a total of 11 Lorentz widths were used in the fitting. Therefore, a total of 13 fitting parameters (11 Lorentz widths, 1 temperature, and 1 concentration) were used. In the tomographic analysis, the Lorentz width was taken to be constant in the entire spectra range as shown in the caption of Fig. 5, and a total of 14 fitting parameters were used (7 temperatures and 7 concentrations). Ideally, the Lorentz width should not be a fitting variable; and it should be determined independently either by controlled measurements or computation. The fitting residual therefore serves as a quantitative indicator of the closeness between the fitted and measured spectra. A smaller fitting residual indicates that the analysis reproduces spectra that better match the measured spectra. As shown in Fig. 6, at Φ = 1.0, the tomographic analysis results in smaller fitting residual at four beam locations (1, 4, 5, and 6) and slightly larger fitting residual at two beam locations (2 and 3); at Φ = 0.5, the tomographic analysis results in smaller fitting residual at all six beam locations. At Φ = 1.0, though the tomographic analysis results in slightly larger fitting residual at beam locations 2 and 3, the summation of the fitting residuals at all six beams locations is reduced, indicating a better overall reproduction of the spectra.

Before leaving this section, note that for such a seven-zone tomographic problem, traditional tomographic techniques based on limited wavelengths (e.g., two wavelengths) generally require more than seven projection measurements. In contrast, with the hyperspectral information, six projection measurements are more than sufficient to retrieve the distributions of temperature and concentration of water vapor. When only four projection measurements (e.g., the two diagonal and horizontal ones) were used, the T and X distributions reconstructed are in good agreement (within ~7%) with those reconstructed using all six projection measurements. The reduction in the number of projections becomes more dramatic when the scale of the problem increases [15].

4.2 Insensitivity of the tomography algorithm to initial guesses

The increased spectral information content also enhances the insensitivity of the technique to measurement and inversion uncertainty. The enhanced insensitivity to the uncertainty in the projection measurements has been studied and documented in [15], where the hyperspectral method was compared to a two-wavelength method. Here, we report the study of the insensitivity to an inversion uncertainty, namely the initial guesses used in the inversion algorithm.

In the minimizing of the master function F in Eq. (3) using the simulated annealing technique, an initial guess of the T and X distributions must first be made to start the algorithm. Therefore, a practical concern is whether the tomographic reconstruction is sensitive to the initial guesses. This section uses both numerical simulation and the experimental results obtained above to address this concern.

 

Fig. 7. The T and X phantoms used to investigate the sensitivity of the tomographic algorithm to initial guesses.

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In the numerical simulation, T and X phantoms were generated over a 10-by-10 square grid, with a set of samples shown in Fig. 7. These distribution phantoms were created by superimposing two Gaussian peaks on a paraboloid to simulate a representative multi-modal and asymmetric temperature distribution of H₂O that could be encountered in practical combustion devices. Other distribution phantoms have been tested and the results obtained are similar to those obtained with the phantoms shown here. A hypothetical tomography sensor with 20 beams is applied to probe the phantom T and X distributions. Each beam contains 10 wavelengths, probing 10 different transitions of H₂O. More details of the setup of the simulation can be found in [15]. A set of simulated projections was then generated, and the algorithm described in Section 2 applied to inverse the projections to retrieve the phantom T and X distributions with different initial guesses. The evolutions of the value of F at three different initial guesses (a Gaussian, a uniform, and a parabolic distribution) were recorded and plotted in Fig. 8. As can be seen from Fig. 7, the values of F were considerably different at the beginning of the minimization (at the first iteration), because of the different initial guesses used. As the tomographic algorithm proceeded, the values of F converged; and at the end of the algorithm, the values of F became within 2% across all three initial guesses.

 

Fig. 8. The evolution of F_min during the minimization with different initial guesses.

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Two of the T distributions, corresponding to that obtained with a uniform initial guess and that with a parabolic initial guess, respectively, are shown in Fig. 9, together with their differences. Figure 9 indicates that the retrieved T distributions agree well with each other, and the maximum difference between these two distributions is within ± 60 K. The same observations were made with the X distributions. Also note that according to Fig. 8, the uniform and parabolic initial guesses resulted in the largest F discrepancy, and consequently the largest discrepancy in T and X distributions. Therefore, the level of difference shown in Fig. 9 represents the worst scenario for the cases simulated here.

 

Fig. 9. The T distributions reconstructed with different initial guesses.

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The insensitivity of the algorithm was also confirmed by processing the experimental results with different initial guesses. Again, several distinctively different initial guesses were used and the retrieved T and X distributions compared. Due to the relative simplicity of the problem compared with the 10-by-10 problem simulated above, the retrieved T and X distributions are essentially the same across all the different initial guesses. For example, when a uniform, a Gaussian, and a random initial guess were used to process the experimental data at Φ = 1.0 and Φ = 0.5, the retrieved T and X distributions agree with those shown in Fig. 5 to within 0.1%.

5. Summary

A new method has been developed to exploit the hyperspectral information provided by new laser sources for tomographic imaging. Such hyperspectral information enables several key advantages for practical implementation and application. These advantages include a significant reduction in the number of required projections and enhanced stability of the reconstruction. These advantages, initially shown via numerical studies, are corroborated by the experimental results obtained with a prototype sensor. With the recent advancement in hyperspectral laser sources, the prototype sensor can be readily modified to provide tomographic images with rapid temporal response (on the order tens of kHz), and the distributions of temperature and key chemical species can be monitored simultaneously and continuously. For instance, another sensor is being assembled to provide tomographic measurements of the temperature and concentration of H2O over a 15 × 15 discretization at 30 kHz. Such sensing capability is expected to contribute critically to the resolution of combustion instability, a long-standing issue in the design and operation of practical combustors.