Inverse Radiation Problem Research Articles

A one-dimensional inverse radiative transfer problem with time-varying boundary conditions

In this work are presented the formulation and solution of two inverse radiative transfer problems in one-dimensional homogeneous participating media with time-dependent boundary conditions. In the first one an energy balance for the incoming and exit radiation allows the direct determination of the absorption coefficient. In the second inverse problem we solve the simultaneous estimation of the total extinction and scattering coefficients using the source–detector method. We first consider a more general formulation of the direct radiative transfer problem, and then for the formulation of the inverse problems we use the particular cases of “cold medium” (no internal sources), transparent boundaries (no effects associated with differences on medium and environment refractive indices), and homogeneous media (constant radiative transfer properties). Test case results are presented.

Optimal regularisation for acoustic source reconstruction by inverse methods

An important inverse problem in the field of acoustics is that of reconstructing the strengths of a number of sources given a model of transmission paths from the sources to a number of sensors at which measurements are made. In dealing with this kind of the acoustical inverse problem, the strength of the discretized source distribution can be simply deduced from the measured pressure field and the inversion of corresponding matrix of frequency response functions. Hence, the accuracy of reconstruction of the source strength is crucially dependent on the conditioning of the matrix to be inverted. However, the problem of reconstructing acoustic source distributions from field measurement is very often ill-posed. In such cases, by using only the simple least-squares method, one cannot ensure a successful reconstruction of the acoustic source strength distribution. Therefore, Tikhonov regularisation is widely employed in order to produce reasonable solutions. However, determination of the amount of regularisation is not straightforward in practical applications without prior knowledge of either the strength of the acoustic sources or the contaminating measurement noise. Thus, two methods have been introduced, Generalised Cross Validation (GCV) and the L-curve method, which do not require prior information in order to determine the optimal regularisation parameter. In the present work, the abilities of the two methods are illustrated when these kinds of inverse sound radiation problems are dealt with using Tikhonov regularisation. Finally, through experimental demonstrations, some guidelines are proposed for determining the optimal degree of regularisation.

Sequential solution of a three-dimensional inverse radiation problem

A three-dimensional inverse radiation problem is solved sequentially using an extended Kalman filter, where the time-varying strength of a heat source is determined from temperature measurements in the participating media. By employing the Karhunen–Loève Galerkin procedure that reduces the set of governing equations composed of the radiative transfer equation and heat conduction equation to a small set of ordinary differential equations, the computational difficulties associated with the Kalman filtering for the integro-differential equation are overcome. The present technique allows a real-time implementation of the sequential identification of radiation field in participating media.

Spatial resolution limits for the reconstruction of acoustic source strength by inverse methods

One method for deducing the strength of an acoustic source distribution from measurement of the radiated field involves the inversion of the matrix of frequency response functions relating the field measurement points to the strengths of a number of point sources used to represent the source distribution. In practice, the frequency response function matrix to be inverted may very often be ill-conditioned. This ill-conditioning will also often result in an ill-posed problem and thus regularization algorithms are used to produce reasonable solutions. For this purpose, Tikhonov regularization has been applied, and generalized cross-validation (GCV) has been introduced as an effective method for determining the proper amount of regularization without prior knowledge of either the source distribution or the contaminating errors. In the present work, the emphasis is placed on the relationship between the spatial resolution of the reconstructed source distribution and the small singular values of the frequency response function matrix to be inverted. However, the use of Tikhonov regularization often suppresses the effect of small singular values and these are in turn often associated with high spatial frequencies of the source distribution. Thus, the process of regularization produces a useful estimate of the acoustic source strength distribution but with a limited spatial resolution. Furthermore, in the field of Fourier acoustics, the spatial resolution of the reconstructed source distribution is usually limited by the wavelength of the radiation. This paper expresses the relationship between estimation accuracy, spatial resolution, noise-level and source/sensor geometry, when a range of inverse sound radiation problems are regularised using Tikhonov regularization with GCV. The results presented form the basis of guidelines that enable the reconstruction of acoustic source strength with a resolution that is finer than the intrinsic half-wavelength limit.

INVERSE DESIGN OF THREE-DIMENSIONAL ENCLOSURES WITH TRANSPARENT AND ABSORBING-EMITTING MEDIAL USING AN OPTIMIZATION TECHNIQUE

An inverse analysis is employed to determine the boundary conditions in 3-D enclosures that will provide both the specified radiative heat flux and temperature distribution on the design surface of the enclosure containing transparent or absorbing-emitting media. The inverse radiation problem is posed as a minimization problem of the least square criterion which is solved by the Levenberg-Marquart method. A boundary value problem approach is developed to solve the sensitivity problem. The discrete transfer method is employed to solve the radiative transfer equation. The effect of the spatial and the angular mesh refinements is studied through comparison with the exact solution of a 3-D benchmark problem. The performance of the present technique of inverse analysis is evaluated by numerical experiments in participating and transparent media

An inverse radiation problem of simultaneous estimation of heat transfer coefficient and absorption coefficient in participating media

AbstractAn inverse radiation problem is investigated where the spatially varying heat transfer coefficient h(z) and the absorption coefficient κ in the radiant cooler are estimated simultaneously from temperature measurements. The inverse radiation problem is solved through the minimization of a performance function, which is expressed by the sum of square residuals between calculated and observed temperature, using the conjugate gradient method. The gradient of the performance function is evaluated by means of the improved adjoint variable method that can take care of both the function estimation and the parameter estimation efficiently. The present method is found to estimate h(z) and κ with reasonable accuracy even with noisy temperature measurements. Copyright © 2002 John Wiley & Sons, Ltd.

Simultaneous estimation of extinction coefficient distribution, scattering albedo and phase function of a two-dimensional medium

In this work, we develop a scheme based on solving the general integro-differential equation of radiation transport to estimate simultaneously the distribution of the extinction coefficient, the scattering albedo and the phase function of a two-dimensional inhomogeneneous medium with less diffusive radiation. The forward problem for a cylindrical medium subjected to collimated incident radiation is solved by the discrete-ordinate method. The inverse radiation problem is formulated as a least square problem that minimizes the discrepancy between the measured and the calculated leaving radiative fluxes. The Levenberg–Marquardt algorithm is applied to the least square problems for a variety of cases. The results obtained show that this scheme can reconstruct accurate enough results for most of the cases considered. Comparisons of the results show that the accuracy of the estimated results decreases with the increase of the scattering albedo and we need more discrete ordinates to generate accurate enough estimated results for an optically thin case. The estimated results obtained from the measurement data with moderate errors are still acceptable.

Inverse radiation problem of axisymmetric turbulent sooting free flame

An iteration method is extended to reconstruct the time-averaged temperature distribution in turbulent axisymmetric sooting free flame by the multi-wavelength measured data of low time-resolution outgoing emission and transmission radiation intensities. A Gaussian probability density function is used to simulate the turbulent fluctuation of temperature. The reconstruction of time-averaged temperature profile consists of three steps. First, the time-averaged spectral absorption coefficient is retrieved from the time-averaged transmissivity data by iteration method. Then the time-averaged blackbody spectral radiation intensity is estimated from the low time-resolution outgoing spectral emission radiation intensities. Finally, the time-averaged temperature and its standard deviation are approximately reconstructed from the multi-wavelength time-averaged spectral emission radiation data by the least-square method. Both exact and noisy input data have been used to test the performance of the proposed inversion method. The results show that the time-averaged temperature profiles can be estimated with good accuracy by the presented inversion method, even with noisy input data, and the standard deviation of temperature is more sensitive to the measurement errors. In the case of large temperature fluctuation, the errors of estimation for time-averaged temperature profile are large if the turbulent fluctuation is not taken into account.

An inverse radiation problem of estimating heat-transfer coefficient in participating media

In the radiant cooler, where the hot gas from the pulverized coal gasifier or combustor is cooled to generate steam, the wall heat-transfer coefficient varies due to ash deposition. In the present work, we investigate an inverse radiation problem of estimating the heat-transfer coefficient from temperature measurement in the radiant cooler. The inverse radiation problem is solved through the minimization of a performance function, which is expressed by the sum of square residuals between calculated and observed temperature, utilizing the conjugate gradient method. The gradient of the performance function is evaluated by means of the improved adjoint variable method, which resolves the difficulty associated with the singularity of the adjoint equation through its inherent regularization property. The effects of the number of measurement points and measurement noise on the accuracy of estimation are also investigated.

The solution of inverse radiation problems using an efficient computational technique

In our previous work (Park, Kim, JQSRT 58 (1) (1997) 115), an efficient computational technique for solving the radiative transfer equations for participating media has been devised, which is as accurate as the S 4 method but consumes much less computer time. In the present investigation, we employ this technique to solve an inverse radiation problem of determining the time-varying strength of a heat source, which mimics flames in a furnace, from temperature measurements in three-dimensional participating media where radiation and conduction occurs simultaneously. The present technique is found to identify the strength of the heat source efficiently without a priori information about the unknown function to be estimated.

An inverse radiative transfer problem of simultaneously estimating profiles of temperature and radiative parameters from boundary intensity and temperature measurements

A parallel-plane space filled with absorbing, emitting, isotropically scattering, gray medium is studied in this paper. The boundary intensity and boundary temperature profiles are calculated for the inverse analysis. For the simultaneous estimation of temperature, absorption and scattering coefficient profiles in the medium, the sum of residuals of boundary intensity and temperature after being weighted by a balance factor is minimized through using a Newton-type iteration algorithm and the least-squares method. To avoid over-updating for the parameters, the relative updating magnitude during the iteration process is constrained not to be >0.5. It is shown that the boundary intensity measurement alone is not enough to estimate simultaneously the temperature (source) and the radiative properties (both absorption and scattering coefficients) when the measurement data contain sensitive random errors. The boundary temperature measurement can serve as a necessary supplementation to the boundary intensity to make this kind of inverse radiative transfer problem resolvable. It was shown that a compensation relationship between absorption and scattering coefficients makes it difficult to fix them accurately. Parabolic profiles for the three parameters are used to validate the estimation method. When the optical thickness approaches 4.0, the results for the radiative properties are not acceptable, although the result for temperature profile is reasonable. This means the method needs further improvements.

Simultaneous estimation of the temperature and heat rate distributions within the combustion region by a new inverse radiation analysis

A new inverse radiation analysis is presented for estimating the heat rate and temperature distributions in the combustion region from the information of the temperature and heat flux profiles of wall elements in the system. The Monte Carlo method is employed to solve the radiative heat transfer equation. The inverse radiation problem is posed as a minimization problem of the least squares criterion, which is solved by the conjugate gradient method. The performance of the present technique of inverse analysis is evaluated and the effects of the errors of the absorption coefficient, emissivity and convective heat transfer coefficient on the inverse analysis are investigated. The results show that the present technique is robust and yields accurate estimation even with noisy measurement.

Solution of the inverse radiative load problem in a two-dimensional system

An inverse radiation analysis is presented for estimating the temperature and the heat load distributions of the heating surface from the temperature and the heat flux measurements of the heated object. The Monte Carlo method is employed to solve the direct radiation problem. The inverse radiation problem is solved using the conjugate gradient and singular value decomposition methods. The measured data are simulated by adding random errors to the exact solution of the direct problem. The effects of the measurement errors on the accuracy of the inverse analysis are investigated. The study shows that the heat load distribution of the heating surface can be estimated accurately for the exact and noisy data. And the conjugate gradient method is better than the singular value decomposition method since the former can obtain more accurate results if the measurement errors are the same.

Comparison of three regularized solution techniques in a three-dimensional inverse radiation problem

Inverse methods provide a good alternative to traditional trial-and-error methods for design of thermal systems. The inverse boundary condition estimation problem in radiating enclosures involves the solution of an ill-posed system that requires regularization to obtain a reasonable physical solution. This study compares three regularized solution techniques that can be used in the inverse boundary condition estimation problems in a three-dimensional radiating enclosure. The regularized solution techniques covered in this study are the conjugate gradient method, bi-conjugate gradient method and truncated singular value decomposition.

A multidimensional inverse radiation problem of estimating the strength of a heat source in participating media

We consider an inverse radiation problem of determining the time-varying strength of a heat source, which mimics flames in a furnace, from temperature measurements in three-dimensional participating media where radiation and conduction occur simultaneously. The inverse radiation problem is posed as a minimization problem of the least-squares criterion, which is solved by a conjugate gradient method employing the adjoint equation to determine the descent direction. The discrete ordinate S 4 method (M.F. Modest, Radiative Heat Transfer, McGraw-Hill, New York, 1993) is employed to solve the radiative transfer equation and its adjoint equation accurately. The performance of the present technique of inverse analysis is evaluated by several numerical experiments, and it is found to solve the inverse radiation problem accurately without a priori information about the unknown function to be estimated.

Inverse radiation problem of source term in three-dimensional complicated geometric semitransparent media

A method is presented to identify the three-dimensional source term distribution in complicated geometric systems of known radiative properties from the knowledge of the radiative intensities exiting the boundaries. The inverse radiation problem is formulated as an optimization problem, and solved by the conjugate gradient method that minimizes the errors between the exit radiative intensities calculated and the experimental measurements. The analysis consists of the direct problem, the gradient equation, and the sensitivity problem. In this approach, the discrete ordinates method is employed to solve the direct and the sensitivity problems in general body-fitted coordinates. The effects of the measurement errors on the accuracy of the inverse analysis are investigated. The study shows that the three-dimensional source term distribution in complicated geometric systems can be estimated accurately, even with noisy data.

Absorption coefficient estimation in heterogeneous media using a domain partition consistent with divergent beams*

An inverse radiative transfer problem is solved for the estimation of the absorption coefficient in a purely absorbing two-dimensional heterogeneous medium. A natural base built with a domain partition that takes into account the possibility of having divergent radiation beams originated at external sources is used in combination with a family of action by line reconstruction algorithms, built within the framework of Lebesgue measure, with Bregman distances based on a q-discrepancy functional. The domain partition and the assembly of the corresponding system of linear algebraic equations, whose unknowns are the absorption coefficients for each element of the partition, are described in detail. Results are presented for test cases using synthetic experimental data generated with a Monte Carlo simple integration technique.

Inverse radiation problem for reconstruction of temperature profile in axisymmetric free flames

An inverse analysis is presented for the reconstruction of temperature profile in axisymmetric free flames from the knowledge of the outgoing emission and transmission radiation intensities. The cross section of flame is divided into a series of concentric rings, and the temperature and absorption coefficient profiles are expressed as a set of discrete values, respectively. The reconstruction of temperature profile consists of two steps. First, the absorption coefficient profile is retrieved from the transmission radiation data, and then based on the previous retrieval of absorption coefficient profile, the temperature profile is reconstructed from the emission radiation data. The effects of measurement errors on the accuracy of the estimations are examined. The results show that the temperature profiles can be estimated accurately, even with noisy data.

Inverse radiation problem in three-dimensional complicated geometric systems with opaque boundaries

An inverse radiation analysis is presented to identify the three-dimensional source term distribution in complicated geometric systems of known radiative properties from the knowledge of the exit radiative intensities. The inverse radiation problem is formulated as an optimization problem, and solved by the conjugate gradient method that minimizes the errors between the calculated exit radiative intensities and the experimental data. The measured data are simulated by adding random errors to the exact solution of the direct problem. The analysis consists of the direct problem, the gradient equation, and the sensitivity problem. In this approach, the discrete ordinates method is employed to solve the direct and the sensitivity problems in general body-fitted coordinates. The effects of the measurement errors, single scattering albedo, and scattering asymmetry parameter on the accuracy of the inverse analysis are investigated. The study shows that the three-dimensional source term distribution in complicated geometric systems with opaque and diffuse-gray boundaries can be estimated accurately for the exact and noisy data.

Solution of inverse radiation problems using the Karhunen–Loève Galerkin procedure

The low-dimensional model obtained by the Karhunen–Loève Galerkin procedure is employed to solve the inverse radiation problem of estimating radiative parameters from the temperature measurement in three-dimensional participating media, where radiation and conduction occur simultaneously. The inverse radiation problem is solved through the minimization of a performance function, which is expressed by the sum of square residuals between calculated and observed temperature, utilizing the conjugate gradient method. The low-dimensional model based on the Karhunen–Loève Galerkin procedure can represent a given system with a minimum degree of freedom, and consequently the number of equations to be solved in the estimation of parameters is minimized when the present technique is used. It is shown that the present technique of solving inverse radiation problems employing the Karhunen–Loève Galerkin procedure yields results as accurate as those using the S 4 method at a drastically reduced computational cost.