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Abhiram Kukunuri M. Deiveegan Under the guidance of Prof. S. P. Venkateshan HEAT TRANSFER AND THERMAL POWER LABORATORY DEPARTMENT OF MECHANICAL ENGINEERING INDIAN INSTITUTE OF TECHNOLOGY MADRAS CHENNAI - 600036. May 2007 Contents: 2 Contents Introduction Objectives Forward Problem Formulation of DDOM Validation Inverse Problem Forward model without scattering Retrieval results Conclusions ReferencesIntroduction: 3 Introduction Radiation propagation through participating medium - Applications: Remote sensing of atmosphere – Microwave and Infrared Combustion systems. Infrared Remote sensing - Retrieval of Temperature and Humidity Profiles Forward Model and Inverse Problem [ involves repeated solution of forward problem] Slide4: 4 Forward model: Mathematical representation of physical model (or situation). Governing Equation - Radiative Transfer Equation (RTE) – Integro Differential Requirement: Very accurate, Fast Radiative Transfer Models Inverse problem: Retrieval of vertical atmospheric temperature profiles under clear sky conditions (scattering can be ignored). Accurate forward model is required. Objectives: 5 Objectives Forward problem Development of a multi layer differential discrete ordinates method to solve the spectral radiative transfer equation in inhomogeneous plane parallel media. Extensive validation of the model for homogeneous and inhomogeneous (multilayer) media with the exact methods and several benchmark cases available in the literature. Inverse problem To perform the retrievals of the vertical temperature profiles in the atmosphere using Bayesian retrieval algorithm from a data base of radiances and temperature profiles in the infrared spectral range under clear sky conditions. I. Forward Problem: 6 I. Forward Problem Formulation and validation of Differential Discrete Ordinates MethodFormulation – Governing equation: 7 Formulation – Governing equation The governing equation for radiation intensity field in an emitting, absorbing and scattering medium is given by, In terms of non dimensional co-ordinates, where Forward ProblemFormulation – Plane parallel medium (homogeneous): 8 where μ = cosθ Formulation – Plane parallel medium (homogeneous) Fig: schematic of a plane parallel medium The azimuthally symmetric (independent of φ ) form of above equation is given by, Forward ProblemFormulation – Differential Discrete Ordinates Method: 9 , i = -M,..., M, i0 Formulation – Differential Discrete Ordinates Method System of 2M coupled ordinary differential equations: In Matrix form, and Forward Problem The integral over μ’ is replaced by a suitable quadrature so that the equation reduces to a set of discrete ordinary differential equations.Formulation – Boundary conditions: 10 Formulation – Boundary conditions Diffusely emitting and reflecting surfaces: Top surface: Bottom surface: Forward ProblemFormulation – Scattering Phase function Φ: 11 Formulation – Scattering Phase function Φ Isotropic scattering: Equal amount of energy is scattered into all directions, Ф = 1 Anisotropic scattering: The phase function can be approximated as a truncated Legendre series as Considering first two terms of the Legendre series, → Linear anisotropic scattering g is the asymmetry factor in the range -1 ≤ g ≤ 1. The value of g indicates the strength of the scattering. Forward ProblemFormulation – Inhomogeneous medium: 12 Formulation – Inhomogeneous medium Absorption and scattering coefficients vary throughout the medium. Divide the medium into a number of homogeneous layers. Each layer interface is transparent. Intensities across the layer interfaces should be continuous. Hence, Fig: Schematic of a multilayer medium Forward ProblemValidation – isotropic scattering: 13 Validation – isotropic scattering Heat flux rate in a medium under radiative equilibrium and bounded by gray diffuse surfaces Non dimensional heat flux rate, Radiative equilibrium Forward Problem - homogeneous mediumValidation – Anisotropic scattering: 14 Validation – Anisotropic scattering Forward Problem - homogeneous medium Dimensionless radiative fluxes (ψ’) at the boundaries by different methods, for τL = 1.0, ω = 0.8 DG : Double Gauss, GL : Gauss Legendre †:% error in the results of DDOM for GL 24 with respect to FN method results Particle I: α = 1.0, m = 2.20+i1.12 Particle II: α = 4.0, m = 1.85+i0.22 Anisotropic scattering phase function, Medium with highly forward scattering particles [2, 3], Dimensionless radiative flux, I0 : incident radiationValidation – Two layer model: 15 Forward Problem - Inhomogeneous medium Validation – Two layer model No radiation sources in the two layers where * with 2M=8 Reflectivity and transmissivity of an isotropic scattering mediumValidation – Semi infinite atmosphere: 16 Validation – Semi infinite atmosphere Forward Problem - Inhomogeneous medium The exponentially varying single scattering albedo for an isotropic scattering semi infinite atmospheres can be expressed as, Albedo for semi infinite atmospheres for τ* = 30 Isotropic incidence at one boundary. Atmosphere is divided into a number of homogeneous layers. The expressions for scattering albedo for each layer ωr and layer optical thickness Δτr are given as, Validation – Atmosphere radiative transfer problem: 17 Validation – Atmosphere radiative transfer problem Fig: Schematic of the three layer atmosphere Brightness temperatures by various methods for different microwave frequencies Atmosphere is modeled as a three layer non gray medium. Each layer is characterized by different properties and anisotropic scattering. Brightness temperature (K) corresponding to the intensity leaving the top of the atmosphere at an angle of 500 is calculated. Forward Problem - AtmosphereII. Inverse Problem: 18 II. Inverse Problem Retrieval of vertical atmospheric temperature profiles under clear sky conditionsForward model – no scattering: 19 Forward model – no scattering Inverse Problem Fig: Schematic of an isothermal atmosphere model without scattering Clouds are opaque to the propagation of infrared radiation. Clouds present in the atmosphere are responsible for scattering. Under clear sky conditions, the scattering term in the radiative transfer equation can be ignored. The presence of scattering necessitates the use of numerical technique to solve RTE. DDOM can be effectively used in such a situation.Forward model - Solution: 20 Inverse Problem Forward model - Solution The governing equation of spectral radiative transfer for a plane parallel isothermal medium without scattering can be written as, where μ = cosθ The solution of above equation for any layer in the atmosphere is, The downwelling intensity at the top of the atmosphere the cosmic back ground radiation at 2.7 K. The bottom bounding surface is assumed to be a diffusely emitting and reflecting surface. Boundary conditions: where r=1, 2,…, L Forward model - Solution: 21 Forward model - Solution The expression for upwelling intensity at the bottom surface is given by, where T0 : bottom surface temperature Solve (II) for downwelling intensities at each layer by marching downwards from top layer Calculate upwelling intensity at bottom surface from (III) in the direction μsat The integral over I in eqn. (III) is replaced by summation using suitable quadrature. Inverse Problem Calculate upwelling intensity at each layer interface from (I) by marching upwards in the direction of μsat Obtain the Intensity at top of atmosphere in the direction μsat Calculation of absorption coefficients: 22 Calculation of absorption coefficients The radiative properties (absorption coefficients) of the medium are necessary to solve the forward model. The calculation of spectral absorption coefficients are done by several line by line models which use spectral line parameters database HITRAN (High Resolution Transmission Molecular Absorption Database). These calculations require more computation time. A database of Compressed look up tables is used to calculate the spectral absorption coefficients. Calculation of absorption coefficients from this database for an arbitrary temperature, pressure, and gas amount is orders of magnitude faster than using a line-by-line code. The database spans 605 cm-1 to 2805 cm-1 at a point spacing of 0.0025 cm-1 and this database is broken up into chunks that are 25 cm-1 wide. Spectral range selection: 23 Spectral range selection Fig. Upwelling radiance spectrum of CO2 for a 100 layer atmosphere in the vicinity of 690cm-1 wave number The source of emission must be an abundant gas of known and uniform distribution. The distribution of CO2 is fairly uniform throughout the atmosphere. (mixing ratio of about 370 ppm). CO2 has significant absorption in the vicinity of 2300cm-1 and 690 cm-1 wave number. Near the strong absorption band, the radiances reaching the top of atmosphere are only from upper layers. Decreasing or increasing the wave number beyond the 690 cm-1 will result in radiances that are completely different from those obtained in the vicinity of 690 cm-1 wave number. Inverse ProblemSynthetic data base creation: 24 Synthetic data base creation Fig. Typical temperature profile over the tropical region Database contains several temperature profiles and corresponding radiances at the top of the atmosphere at various wavelengths, for each profile. The prior information to generate random temperature profile database is obtained from various temperatures profiles retrieved from Atmospheric Infrared Sounder (AIRS) instrument observed radiances. A random set of 20,000 temperature profiles are generated. Gas amounts are assumed to be same for all profiles. Surface temperature is randomly varied between 300 K and 305 K. These random profiles are used to calculate the radiative properties (absorption coefficients). Inverse ProblemBayesian algorithm: 25 Bayesian algorithm Integrates over the points in database using Bayes theorem Inverse Problem ppost(x|y) -Posterior Probability Density Function pf(y|x) -Conditional Probability Density Function ppr(x) -Prior Probability Density Function x - State Vector y - Vector of Observations Zj(x) – Vector of Simulated values Bayes theorem Retrieval ParameterSlide26: RMS error for each database profile, Calculate the weight (wi) for each profile based on Normal distribution of database values over measurement Retrieved Parameter where wj - Weights for each channel = 1 (for all channels) Ndata - Number of Channels (wave numbers) xi – Database profile Ndata – Number of Database Profiles σ – standard deviation Inverse ProblemRetrieval Results: 27 Retrieval Results Inverse Problem Retrievals for two different databases: Atmosphere is divided into 100 layers. Hence, a total of 101 temperatures are to be retrieved. The wavelength range used for calculating radiances is between 650 cm-1 and 720 cm-1. A set of 300 equally spaced channels (wave numbers) in the range 650 cm-1-720 cm-1 with an interval of 0.25cm-1 is chosen for radiances calculation. Retrievals are performed using Bayesian algorithm.Retrieval Results: 28 Retrieval Results Inverse Problem Retrievals for different random errors in measurement The retrievals are better when the instrument random error (noise) is less. Here number of radiances measured is more than the number of parameters to be retrieved In case of low spectral resolution sensors, the number of channel measurements is less that of the parameters retrieved, thus making the retrieval problem ill posed.Conclusions: 29 Conclusions Forward Problem A Multi layer differential discrete ordinates method has been proposed and several validation cases are reported. This method is very accurate for any range of optical depth and for any degree of scattering anisotropy. No complex mathematics and avoids tedious programming effort. Once formulated, it can easily incorporate any variations in boundary conditions. In three layer atmosphere model, calculation of brightness temperature using the Rayleigh-Jeans approximation leads to an error of about 5K at higher frequencies (at 183 GHz). Slide30: Inverse Problem Retrieval of few temperature profiles is performed using Bayesian algorithm from a synthetic database of random generated temperature profiles and radiances. Retrieval results are more accurate when the database has more number of radiances and temperature profiles. The instrument noise should be less to obtain a better retrieval. Research output: 31 Research output International Conference Abhiram, K. T. V. S., Deiveegan, M., Balaji, C. and Venkateshan, S. P., A Multilayer Differential Discrete Ordinate Radiative Transfer Model for Atmospheric Applications, 5th International Conference on Computational Heat and Mass Transfer, Canmore, Alberta, Canada, June 18-22, 2007 (accepted).References: 32 References Coelho, P. J., Fundamentals of a new method for the solution of the radiative transfer equation, International Journal of Thermal Sciences, 44, 2005, pp 809-821. Mengüc, M. P., and Viskanta, R., Comparison of radiative transfer approximations for highly forward scattering planar medium, Journal of. Quantitative Spectroscopy and Radiative Transfer, 29, 1983, pp 381-394. Kumar, S., Majumdar, A., Tien, C. L., The differential discrete ordinate method for solutions of the equation of radiative transfer, Journal of Heat Transfer, 112, 1990, pp 424-429. Ozisik, M. N., Shouman, S. M., Source function expansion method for radiative transfer in a two layer slab, Journal of. Quantitative Spectroscopy and Radiative Transfer, 24, 1980, pp 441-449. Garcia, R. D. M., Siewert, C. E., Radiative transfer in inhomogeneous atmospheres –numerical results, Journal of. Quantitative Spectroscopy and Radiative Transfer, 25, 1981, pp 277-283.References: 33 Kummerow, C., On the accuracy of the Eddington approximation for the radiative transfer in the microwave frequencies, Journal of Geophysical Research, 98, 1993, pp 2757-2765. Deiveegan, M., Balaji, C., and Venkateshan, S. P., Comparison of discrete ordinate and finite volume methods in the solution of Radiative Transfer Equation, 18th National and 7th ISHME – ASME Heat and Mass Transfer Conference, IIT Guwahati, India, 2006. Deiveegan, M., Balaji, C., and Venkateshan, S. P., A Polarized microwave radiative transfer model for passive remote sensing, atmospheric research [under review]. Liou, K.N., An Introduction to Atmospheric Radiation. 2nd edition, Academic Press, 2002. Web page: http://disc.gsfc.nasa.gov/data/datapool/AIRS/02_L2_Products/ ReferencesSlide34: 34Literature review: Forward problem: 35 Literature review: Forward problemLiterature review: Inverse problem: 36 Literature review: Inverse problem Propagation of radiation in participating medium: 37 Propagation of radiation in participating medium Blackbody intensity, (Planck’s function) Rayleigh-Jeans approximation,Gauss Quadrature: 38 Gauss Quadrature The integral over a function is replaced by summation as, μj and wj are determined by considering the function f(μ) to be a polynomial of degree N having (N+1) coefficients Gauss scheme : 2M weights. Degree of polynomial, N = 2M-1. Lobatto scheme : Boundary points μ = -1 and μ = 1 are fixed. Degree of polynomial, N = 2M-3. Gauss Quadrature: 39 Gauss Quadrature Fiveland scheme : Weights are held fixed to 2/M. Degree of polynomial, N= M-1. Double Gauss scheme: Gaussian formula is applied separately to the half-ranges : -1<μ<0 and 0<μ<1. Quadrature points are clustered both toward │μ│ = 1 and μ = 0. The intensity varies rapidly around μ = 0 and hence the clustering toward μ =0 will give accurate results. Medium energy and temperature conditions: 40 Medium energy and temperature conditions Cold medium: emission from medium can be neglected Isothermal medium : constant temperature of the medium and hence the black body intensity (IB) is known. Coupled with other modes of heat transfer Radiative equilibrium :Validation - Temperature distribution in isotropic scattering medium: 41 Validation - Temperature distribution in isotropic scattering medium Non-dimensional emissive power or temperature, Medium bounded by black surfaces Medium bounded by gray diffuse surfaces Forward Problem - homogeneous mediumValidation – isotropic scattering: 42 Validation – isotropic scattering Effect of albedo on a cold medium for external isotropic incidence where Forward Problem - homogeneous medium Non radiative equilibriumValidation – isotropic scattering: 43 Forward Problem - homogeneous medium Validation – isotropic scattering Non radiative equilibrium Heat flux and incident radiation flux at both boundaries (non emitting and non reflecting) Optical thickness of the medium, τL = 10.0 Validation – Linear anisotropic scattering: 44 Validation – Linear anisotropic scattering Forward Problem - homogeneous medium Radiative equilibrium Non dimensional heat flux rate, Heat flux rate for different asymmetry factors Heat flux rate for various emissivities of top surface, ε0 = 0.8 and gω = +0.7 g: asymmetry factor ω : scattering albedo MC : Monte Carlo method Validation – Linear anisotropic scattering: 45 Validation – Linear anisotropic scattering Forward Problem - homogeneous medium Diffuse Hemispherical reflectance of a scattering plane parallel slab with transparent boundariesValidation – Six layer model: 46 Validation – Six layer model Forward Problem - Inhomogeneous medium Optical depth and scattering albedo for six layers All layers follow same scattering law No emission from top boundary Cold medium Incident radiation on the bottom boundary, Assumptions: Albedo for anisotropic scattering medium Phase function coefficients Phase function coefficients : 47 Phase function coefficients Particle I: α = 1.0, m = 2.20+i1.12 Particle II: α = 4.0, m = 1.85+i0.22 Six layer modelInteraction parameters for 3 layer atmosphere: 48 Interaction parameters for 3 layer atmosphere ReturnSlide49: Schematic of the three layer atmosphere You do not have the permission to view this presentation. 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