PhD Presentation 2

Slide1 : Heat Transfer and Thermal Power Laboratory Inverse Problems in Participating Media and Retrieval of Geophysical Parameters from Microwave Remote sensing data M. Deiveegan Research Scholar Heat Transfer and Thermal Power Laboratory, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, India Advisors Prof. S. P. Venkateshan Dr. C. Balaji


Slide2 : 2 Organization of Presentation Introduction – Inverse Problems in participating media Problem 1: Inverse Problem in Participating medium Direct Problem- Differential Discrete Ordinate Inverse Problem - Levenberg-Marquardt Method (LMM) Genetic Algorithm (GA) Artificial Neural Network (ANN) Bayesian Retrieval Algorithm Problem 2: Retrieval of Geophysical parameters from Artificial Data Direct Problem- Polarized Microwave model Inverse Problem– ANN, Bayesian Problem 3: Retrieval of Geophysical parameters from Actual Remote sensing data Direct Problem- Polarized Microwave model Retrieval of Geophysical Parameters from Remote sensing data – ANN, Bayesian Conclusions


Slide3 : 3 Introduction


Applications - Inverse Problems in Radiation : 4 Applications - Inverse Problems in Radiation To compute Radiative Properties & Temperature Fields inside Furnace For Optical computed Tomography - To obtain a mapping of the Absorption coefficient throughout the Tissue (Contain Information regarding both the Structure and Functional status of Tissue) Identifying the distribution of the Radiation Source - for applications including Thermal control in Space Technology, Combustion, High Temperature Forming and Coating Technology, High Temperature Engines To Retrieve the Constituents present in the Atmosphere from Satellite Data


Slide5 : 5 Y= F(x) Direct Problem Find the Effect Y of given a Cause x Inverse Problem Establish the Cause x that provides the Observed effect Y Non-invasive Techniques:– Allow Simultaneous Estimation of Properties Parameter estimation – Optimization Problem Objective:– To select Values of Unknown Parameters in such a way that an Objective Function is Minimized. Objective Functions:- Sum-of-squares functions which contain calculated and measured values. Inverse problems driven by Discrete Measurement Data - much more Practical than Direct problems Parameter Estimation Introduction Simple Example


Parametric Estimation : 6 Y - Observation vector Z - Modeled vector X - Unknown Parameter Ns- No of Channels No- No of observations Objective Function Parametric Estimation Accuracy of Parametric Retrieval depends on Accurate, Realistic Mathematical Representation of Physical System (Forward Model) Accuracy of Inverse Methodology Forward Model (Direct Problem) - Vital part of Parameter Retrieval Illustration:- Deiveegan et al., 2006 Introduction


Slide7 : 7 Solution to the problem must exist The solution must be unique The solution must be stable Difficulties associated with Inverse Problems in Radiative Transfer Inverse Problems tend to be ill posed Unique solution: Rare instance Very Sensitive to Disturbances in Parameter Field -Random error in Observations Observations do not provide Sufficient Information Well posed Unique solution Stable to perturbation in given data How accurately are the parameters Retrieved? Effect of Measurements error on Retrievals? Computational time required? Questions that need to be addressed: Well posed Problems Direct Problem Introduction


Slide8 : 8 Problem 1 Inverse Problem in Radiative Participating Media


Slide9 : 9 Parameters to be retrieved: x Absorption coefficient Scattering coefficient Top surface emissivity Bottom surface emissivity Measured Quantities : y [Artificial Data] Measured Temperature Profile Measured Heat Flux Minimize sum of square residuals between Simulated and Measurement vector Forward Model: Z(x) Simulated Temperature Profile Simulated Heat Flux Simultaneous Estimation of Gas Properties & Surface Emissivities for Plane Parallel Medium


Slide10 : 10 Plane Parallel Radiative Participating Media. Two parallel, Isothermal, Diffuse, Infinitely large plates. Objective Direct Problem To Find Temperature profile and Heat flux Plane parallel medium - Direct Problem Participating Medium


Slide11 : 11 Differential Discrete Ordinate Method Radiative Transfer Equation [RTE] Numerical Quadrature [Gauss Quadrature] Matrix Form Jacobian Divide into N Number of control volumes along z 2M Number of Ordinate Directions System of 2M first order ODE Plane parallel medium - Direct Problem Deiveegan et al., ASME-JHT, 2006


Slide12 : 12 Boundary Conditions Energy Equation Heat Flux Non Dimensional Heat Flux DOM continued… Plane parallel medium - Direct Problem


Slide13 : 13 Solution Procedure- DDOM Plane parallel medium - Direct Problem DOM continued… Programming : Compaq Visual Fortran Deiveegan et al., ASME-JHT, 2006 Variable Step size Finite Difference Method


Slide14 : 14 Parameters considered Isotropic Scattering Plane Parallel Medium with gray Boundaries Plane parallel medium - Direct Problem Validation continued… Validation – Direct Problem Grid sensitivity study


Slide15 : 15 Plane parallel medium - Inverse Problem Assumptions: The model is an accurate representation of the system The measurements will differ from simulated data by measurement noise The effect of measurement errors - small random perturbations Yi – Measurement Vector Zi – Simulated Data ξ – Normal Distributed Random number with zero mean unit standard deviation η – Measured error at 99% confidence Standard deviations of measured data 99% of a normally distributed population is contained within ±2.576 standard deviation of mean Generation of Measurement data


Slide16 : 16 Inverse Problem - Methods Used Levenberg-Marquardt Method (LMM) – Gradient Method Genetic Algorithm (GA) – Mechanics of natural selection Artificial Neural Network (ANN) – Data driven approach Bayesian Retrieval Algorithm – Data driven approach Plane parallel medium - Inverse Problem


Slide17 : 17 Levenberg-Marquardt Method Iterative method to Minimize - Sum of Squares of M functions in N variables. To Minimize - Differentiate with respect to X Vector F - expanded in a Taylor series - only the first order terms are retained To Improve Convergence - Damping parameter λ is added Elements of the Jacobian Matrix Iterative Procedure Evaluation or approximation of the Jacobian matrix in every iteration. The Gradients of M functions of N variables are approximated by finite differences. Yi - Measured data Zi - Simulated data Xi - Unknown Parameter ζ - Identity matrix F - (Yi - Zi) Plane parallel medium - Inverse Problem


Slide18 : 18 Convergence of Levenberg-Marquardt Algorithm Results continued… Plane parallel medium - Inverse Problem Mie scattering


Slide19 : 19 What are they? Random search of a defined N- dimensional solution space GA - Mimics processes in nature that led to evolution of higher organisms Natural selection (“survival of the fittest”) Reproduction - Crossover Mutation Do not require any gradient information How do they work? A population of genes is evaluated using a specified fitness measure Best members of population are selected for reproduction to form the next generation. Random mutations - To introduce new characteristics into new generation Genetic Algorithm Plane parallel medium - Inverse Problem


Slide20 : 20 Basic scheme Initialize Population Evaluate Fitness of each member Reproduce with fittest members Introduce Random Mutations in New Generation Continue 2-4 until pre specified number of generations are complete Rely heavily on random processes A random number generator will be called thousands of times Searches are inherently computationally intensive Usually will find the global max within the specified search domain GA works with only Maximization problems, Objective Function is modified Genetic Algorithm continued… Plane parallel medium - Inverse Problem


Slide21 : 21 Typical Genetic Algorithm Flowchart Genetic Algorithm continued… Simple Example Plane parallel medium - Inverse Problem


Slide22 : 22 Convergence history of Genetic Algorithm Results continued… Plane parallel medium - Inverse Problem Mie scattering Max generation = 400 Population size = 16 Uniform Crossover Micro GA is Implemented


Artificial Neural Network : 23 Artificial Neural Network Collection of Neurons Interconnected with Weights Act as Interpolative Functions for given Set of Data How do they work? Network Trained with Set of Data that cover solution space During Training weights in the network are adjusted until the correct answer is given for all data in the training set After Training, weights are fixed and network answers Testing data. These “Retrievals” are consistent with the training data Two Distinct phases - Training & Testing Plane parallel medium - Inverse Problem


Slide24 : 24 Plane parallel medium - Inverse Problem ANN continued… Training:- Feed-forward Multilayer Perceptron Networks Activation function:- Sigmoid Function Learning Algorithm:- Standard Back-Propagation updating


Bayesian Retrieval Algorithm : 25 Bayesian Retrieval Algorithm Database – Randomly chosen set of Parameters in possible range & corresponding pre-calculated vector of simulated values Integrates over the points in the database with Bayes theorem Bayes theorem Retrieval Parameter 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 Plane parallel medium - Inverse Problem


Slide26 : 26 Calculate the RMS error for each database profile Calculate the weight (wi) for each profile based on Normal distribution of database values over measurement Retrieved Parameter For channel, Calculate the difference between simulated and measured quantity wj - Weights for each measurement Nchannel - Number of Measurements xi – Database profile Ndata – Number of Database data σ – Gaussian of standard deviation Bayesian – Continued… Plane parallel medium - Inverse Problem Deiveegan et al., ASME-JHT, 2006


Slide27 : 27 Bayesian – Continued… χ2 - measure of Difference between Measurement and Database vector To speed up Algorithm - Bayes integration is to stop χ2 summation and not calculate posterior PDF when, χ2 > 30. Bayesian algorithm interpolates points in database that have a reasonable match with the observations. Plane parallel medium - Inverse Problem


Slide28 : 28 Parameter Estimation- Single layer Plane parallel medium- Mie scattering Plane parallel medium - Inverse Problem Results continued… Deiveegan et al., ASME-JHT , 2006 Measurement Error in % Accuracy Speed Results - Mie scattering


Slide29 : 29 Levenberg-Marquardt Method (LMM) Genetic Algorithm (GA) Artificial Neural Network (ANN) Bayesian Retrieval Algorithm Simultaneous Estimation of Parameters Accurately Accurate Estimation with Noise in the Measurement Real Time Estimation (Faster Convergence) Retrieval Algorithm – Requirements Deiveegan et al., ASME-JHT, 2006 Results continued… Plane parallel medium - Inverse Problem


Slide30 : 30 Plane parallel medium Conclusions – Problem 1 Forward Problem DOM - To Analyze Radiative Transfer in 1-D – Participating Medium- Planar Slab - Radiative Equilibrium Validation - Three Test cases - Agree well with benchmark cases DOM - Efficient & Accurate Inverse Problem Parameters can be Estimated with Good Accuracy using Measured Temperatures and Heat Flux Levenberg-Marquardt – Presences of Noise reduces the Accuracy of Estimation GA - Not Very Accurate - But Quite Robust Bayesian and ANN - Robust and yield Accurate Estimation - Even for Noisy Input Time taken for Parameter Retrieval Levenberg-Marquardt, Genetic Algorithm – Slow Bayesian, ANN – Real Time Retrieval [Very Fast]


Slide31 : 31 Forward Modeling Polarized Microwave Model


Slide32 : 32 Illustration: Swaminathan et al.


Slide33 : 33 Atmospheric Radiation Atmosphere Modeled as.. Multilayer, Plane Parallel Participating Medium with Absorption, Emission & Scattering. Ocean Surface: Emission, Reflection Gases: (Water vapor, CO2) Emission, Absorption Cloud Liquid Water: Emission, Absorption Cloud Ice Water: Emission, Absorption, Scattering Liquid Hydrometeor: (Rain) Emission, Absorption, Scattering Ice Hydrometeor: Emission, Absorption, Scattering Source of Polarized Signal: Ocean Surface (Reflection) Hydrometers (Scattering) Radiation Interaction with Atmosphere Illustration: Deiveegan et al., 2006


Slide34 : 34 Polarization -Direct Problem Atmospheric Parameters Precipitation water (Rain) & Precipitation Ice Cloud Liquid water & Cloud Ice water Atmospheric water vapor & Oxygen Humidity Profile Temperature & Pressure Profile Ocean Surface Parameters Ocean Salinity Sea surface temperature Wind speed in mm/hr in g/m3 in %RH in K & bar in ppt in K in m/s Forward Modeling


Slide35 : 35 Generation of the Atmospheric Environment Generation of the Open Ocean Surface Environment Generation of the Atmosphere Interaction Parameters Generation of the surface Interaction Parameters Generation of Physical Environment Solution of Vector Radiative Transfer Equation Doubling & Adding Method Generation of Interaction Parameters for Surface & Atmosphere Polarized Microwave Model I II III Polarization -Direct Problem


Vector Radiative Transfer Equation : 36 Monochromatic Plane parallel Polarized RTE for Randomly Oriented Particles Source of Diffuse Radiation- Thermal Emission + Solar Radiation Diffuse Radiance M- Scattering Matrix μ – Cosine of Zenith angle Φ– Azimuth Angle τ – Optical Depth – Single-scatter Albedo F0–Unpolarized Solar Flux Vector Radiative Transfer Equation The Plane Parallel Solution is Sufficient to Cover most Applications for Radiation Scattering in Planetary Atmospheres Polarization -Direct Problem Why Plane Parallel medium Approximation ? Difference between Polarized and Unpolarized Model


Slide37 : 37 Polarization- Direct Problem Polarized Microwave Model – Over view


Slide38 : 38 Comparison with Airborne measurements Measurements by Airborne radiometers 155 m above Bearing sea Single layer 155 m- Absorption by gases considered, No precipitations Surface- Specular sea surface model Sea Salinity = 33 ppt (Assumed) , Sea surface Temperature = 288.0 K (Assumed) Frequency = 37.0 GHz Viewing angle = 380 Webster et al. Polarization -Direct Problem - Validation


Slide39 : 39 Validation with Tropical Cyclone (FANOOS) Data Atmosphere – Polarization-case study JAXA/EORC Tropical Cyclone Database TRMM – Tropical Rain rate Measuring Mission TRMM


Slide40 : 40 Forward Model – Inputs Number of Layers: 14 Frequencies: 10.7, 19.4, 21.3, 37.0, 85.5 GHz, Polarization: Vertical, Horizontal (2 Stokes Parameters, only Thermal Source) Number of Quadrature Angles: 12 Profiles of Liquid Cloud, Ice Cloud & Hydrometeors (Liquid & Ice phase) - Taken from TRMM Data Temperature, Pressure and Humidity Profiles – from GCE Profiles Ocean Surface conditions Unknown, Surface Parameters are Assumed SST=300 K, Salinity = 35 ppt, Wind Speed = 18 m/s Total No. of Data Profiles over ocean surface: 28912 No. of profiles with rain rate >0.015 mm/hr : 6815 Output: Vertical, Horizontal Polarized Brightness Temperatures Steps involved Profile Generation Generation of Interaction Parameters Solution of V-RTE


Slide41 : 41 Simulated Brightness Temperature Measured Brightness Temperature (TRMM) Forward Model Simulations Vertical Polarized, 85.5 GHz


Slide42 : 42 Horizontal Polarized, 85.5 GHz Simulated Brightness Temperature Measured Brightness Temperature (TRMM) Forward Model Simulations


Slide43 : 43 Problem 2 Inverse Problem Retrieval of Geophysical parameters from Artificial Data


Problem 2 : Parameter Retrieval in Two layer Model : 44 Inverse Problem: Forward Problem: To Find: Brightness Temperatures at given frequencies - 12 Known : Atmospheric Constituents [rain rate, ice content, surface albedo] - 3 To Find: Atmospheric Constituents [rain rate, ice content, surface albedo] - 3 Known: Brightness Temperatures at given frequencies [Artificial Data] - 12 Problem 2 : Parameter Retrieval in Two layer Model Forward Problem – Polarized Microwave Model using Adding & Doubling Method Inverse Problem – Bayesian Retrieval Algorithm, Artificial Neural Network Parameter retrieval for Two Layer Atmosphere Model Data base: Number of Data for Training = 30000 Number of Data for Testing = 1000


Slide45 : 45 Two layer precipitating Atmosphere – Absorbs, Emits & Scatters radiation Unpolarized, Diffuse, Cosmic black body radiation at 2.7 K Surface- Lambertian, Albedo: 0.01-0.8, Temperature=300 K Rain Layer: 0.02- 49 mm/hr, Ice Layer: 0.01- 24.5 mm/hr Frequencies = 6.6, 10.7, 18.0, 37.0, 85.6, 183.0 GHz Atmosphere –Direct Problem Direct Problem – Two layer Model


Slide46 : 46 Results- Parameter Retrievals Sea Surface Albedo (Reflectivity) Case 1 : With out Measurement Error ANN Bayesian Atmosphere -Inverse Problem


Slide47 : 47 Results – Continued… First Layer Rain Rate Case 1 : With out Measurement Error ANN Bayesian Atmosphere -Inverse Problem


Slide48 : 48 Results – Continued… Second Layer Ice Content Case 1 : With out Measurement Error ANN Bayesian Atmosphere -Inverse Problem


Slide49 : 49 Results – Continued… ANN Bayesian Case 2 : Measurement Error within ±2% Sea Surface Albedo (Reflectivity) Atmosphere -Inverse Problem


Slide50 : 50 Results – Continued… ANN Bayesian Case 2 : Measurement Error within ±2% First Layer Rain Rate Atmosphere -Inverse Problem


Slide51 : 51 Results– Continued… ANN Bayesian Case 2 : Measurement Error within ±2% Second Layer Ice Content Atmosphere -Inverse Problem


Slide52 : 52 Results– Continued… Atmosphere -Inverse Problem Comparison between ANN & Bayesian


Conclusions – Problem 2 : 53 Conclusions – Problem 2 Inverse Problem presented- Bayesian Retrieval Algorithm, ANN. Simultaneous Retrieval of Surface Albedo, Rain Rate, Ice content. Input to algorithms:- Large pre calculated database - Randomly chosen properties, corresponding simulated Brightness temperatures. Bayesian and ANN are Robust & yield Accurate Estimation of parameters. Bayesian & ANN –Fast Retrieval Atmosphere -Inverse Problem


Slide54 : 54 Problem 3 Inverse Problem Retrieval of Geophysical parameters from Remote sensing data


Problem 3 : Geophysical Parameter Estimation : 55 Inverse Problem: Forward Problem: To Find: Brightness Temperatures at given frequencies - 9 Known : Atmospheric Constituents [rain rate – 14 , ice content – 14, cloud Liquid content –14, Cloud Ice–14, Ground Rain rate – 1, Convective rain rate – 1] To Find: Atmospheric Constituents [rain rate – 14 , ice content – 14, cloud Liquid content –14, Cloud Ice–14, Ground Rain rate – 1, Convective rain rate – 1] Known: Brightness Temperatures at given frequencies - 9 [Satellite measured Data] Problem 3 : Geophysical Parameter Estimation Forward Problem – Polarized Microwave Model using Adding & Doubling Method Inverse Problem – Bayesian Retrieval Algorithm, Artificial Neural Network Parameter retrieval - Raining Atmosphere – 14 Layers Data base: Number of Data for Training = 43000 Number of Data for Testing = 7000


Slide56 : Prior Knowledge: TRMM-TMI Profiles


Slide57 : Prior Knowledge: TRMM-TMI Profiles


Slide58 : 58 Precipitation Retrieval Algorithm For rain pixel over ocean


Slide59 : 59 Screening for Rain / No rain Retrieval algorithm valid only for Rain pixel over Ocean Surface Classification depends - Measured brightness Temperatures (TRMM) Test 1: Screening for No-Rain condition Test 2: Screening for No-Rain condition Test 3: Screening for No-Rain condition Test 4: Screening for Rain condition and Ref: Hargens, 1994 LWP - Integrated cloud water and or and or


Slide60 : 60 Emission and Scattering Index Emission and Scattering Indexes For both TRMM measured and Forward Models Simulations


Artificial Neural Network : 61 Training:- Feed-forward Multilayer Perceptron Networks Activation function:- Sigmoid Function Learning Algorithm:- Standard Back-Propagation updating Artificial Neural Network


Slide62 : 62 ANN Training – Correlation coefficient Ground rain rate - 0.7921 ; Convective rain rate – 0.7149


Artificial Neural Network - Results : 63 Testing Training Artificial Neural Network - Results


Bayesian Retrieval Algorithm : 64 Bayesian Retrieval Algorithm Database :– TRMM Profiles of Rain, CLW, CIW, Ice, Ground Rain Rate & corresponding Simulated Brightness Temperatures at Various Frequencies Integrates over the points in the database with Bayes theorem Bayes theorem Retrieval Parameter 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


Slide65 : 65 Calculate the RMS error for each database profile Calculate the weight (wi) for each profile based on Normal distribution of database values over measurement Retrieved Parameter For each frequency, Calculate the difference between simulated and TRMM measured brightness temperatures wj - Weights for each channel from TRMM Ndata - Number of TRMM Channels xi – Database profile Ndata – Number of Database Profiles σ – Gaussian of standard deviation [8K]


Slide66 : 66 Weights for each channel


Slide67 : 67 Retrieval of Geophysical Parameters Application to Tropical Cyclone (FANOOS, December 2005)


Slide68 : 68 Tropical Cyclone FANOOS - Data Atmosphere – Polarization-case study JAXA/EORC Tropical Cyclone Database FANOOS - continued.. Rainfall Rate & Cloud Image


Slide69 : 69 Inverse Problem– Inputs Number of Layers: 14 Frequencies: 10.7, 19.4, 21.3, 37.0, 85.5 GHz, TRMM-TMI Measured Vertical, Horizontal Polarized Brightness Temperatures Total No. of Database Profiles over ocean surface: 43000 Output: Profiles of Liquid Cloud, Ice Cloud & Hydrometeors (Liquid & Ice phase) Ground Rain rate


Ground Rain rate – Comparison with TRMM-TMI : 70 TRMM -TMI Present - ANN Present - Bayesian Ground Rain rate – Comparison with TRMM-TMI


Ground Rain rate – Comparison with TRMM-TMI : 71 TMI Vs Present (ANN) Ground Rain rate – Comparison with TRMM-TMI TMI Vs Present (Bayesian)


Ground Rain rate - Comparison with Precipitation Radar (PR) : 72 Ground Rain rate - Comparison with Precipitation Radar (PR) TRMM - PR TRMM - TMI Present - ANN Present - Bayesian


Ground Rain rate - Comparison with Participation Radar : 73 Ground Rain rate - Comparison with Participation Radar PR Vs Present (ANN) PR Vs TMI PR Vs Present (Bayesian)


Slide74 : 74 Rain Ice Cloud Ice Cloud Liquid Retrieval of Vertical Profiles using Bayesian


Retrieval of Vertical Profiles using ANN : 75 Retrieval of Vertical Profiles using ANN Rain Ice Cloud Ice Cloud Liquid


Slide76 : 76 Retrieval - Ground Rain Rate Ground Rain rate Source of Error: Effect of Melting Layer Beam filling Problem Cloud-model microphysical assumptions


Slide77 : 77 Effect of Measurement error on Retrieval To study the effect of measurement error on retrieval Random noise between -1 to +1 K added with TMI measured brightness temperatures at all frequencies and polarization Effect of measurement error on retrieval – Significant at Lower rain rates Perturbed TB at all frequencies


Slide78 : 78 Effect of Measurement error on Retrieval Effect of measurement error - significant at higher frequency channels


Conclusions – Problem 3 : 79 Conclusions – Problem 3 Simultaneous Estimation of Ground Rain Rate with Vertical Profiles of Rain Rate, Ice content, Cloud Liquid water and Cloud Ice water from Microwave Remote sensing Data. Development of Retrieval Algorithm - Bayesian, ANN. Prior Knowledge:- Large pre calculated database TMI retrieved profiles of 6 cyclones Corresponding Brightness temperatures using Polarized Microwave Model. Demonstrated – Retrieval of Geophysical Parameters – using Bayesian & ANN Test case: Retrieval of Precipitation related parameters for Tropical Cyclone Atmosphere -Inverse Problem


Slide80 : 80 Thank you Illustration: NASA webpage


Slide81 : 81 Backup slides


Simple Example: Inverse Problem in Radiation : 82 Inverse Problem: Forward Problem: To Find: Temperature History Known : Ambient Temperature (Te), Mass (m), Specific Heat (Cp), Emissivity (ε) To Find: Emissivity (ε) Known: Ambient Temp (Te), Mass (m), Specific Heat (Cp), Temperature History Simple Example: Inverse Problem in Radiation Vacuum Chamber Isothermal Plate Cooling Experiment: Main Presentation


Tropical Rainfall Measuring Mission (TRMM) : 83 Tropical Rainfall Measuring Mission (TRMM) Joint mission between NASA and Japan Aerospace Exploration Agency (JAXA). TRMM : 1997 - Present To measure the Rainfall TRMM Instruments TRMM Microwave Imager (TMI) Precipitation Radar (PR) Cloud and Earth Radiant Energy Sensor (CERES) Visible and Infrared Scanner (VIRS) Lightning Imaging Sensor (LIS) TRMM Continued… Illustration: NASA webpage TMI frequencies: 10.7, 19.4, 21.3, 37, 85.5 GHz Main Presentation


Megha Tropiques (MT) : 84 Megha Tropiques (MT) Joint mission between ISRO and CNES-France To measure the Rainfall TRMM Instruments MADRAS - Microwave Analysis and Detection of Rain and Atmospheric Structures SAPHIR - Sounding Instrument to measure Water Vapor ScaRaB - Scanning Radiative Budget Instrument Illustration: CNES -France MADRAS


Slide85 : 85 Database for Precipitation Retrieval Number of profiles. surface type; 0=ocean, 1=land, 2=coast layer heights [km] For each profile: Freezing level[m] Surface Rain [mm/hr] Convective Rain[mm/hr] Rain cover Convective cover Surface wind speed[m/s] For each Layer: Layer 1 CLW, RAIN, CIW, ICE ... Layer 14 CLW, RAIN, CIW, ICE Tb at each frequency (10H,10V,19H,19V,21V,37H,37V,85H,85V) Tb at each frequency for clear-air background Input: Output:


Slide86 : 86 Extinction = absorption + scattering Single scattering albedo = scattering / (absorption + scattering) Radiative Transfer Equation [RTE] Phase Functions Isotropic scattering: Anisotropic scattering: Radiatively Participating Medium Solution Methods Eddington Method Discrete Ordinate Methods – Fiveland et al. Finite Volume Method – Karthikeyan et al.(2003), Swaminathan et al.(2004) Differential Discrete Ordinate Method – Kumar et al., Deiveegan et al.(2006)


Error due to plane parallel approximation has two very different natures depending on pixel size : 87 Why Plane Parallel medium Approximation ? (H / R) << 1 ; H  Scale Height (~10 km), R Distance from centre of Planet (6380×10+10 km) Concentration of Scatterers and Absorbers does not vary in Horizontal Boundary Conditions do not depend on the Horizontal Polarization- Direct Problem Error due to plane parallel approximation has two very different natures depending on pixel size Megha Tropiques & TRMM Sensor Resolutions Main Presentation


Slide88 : 88 Comparison – Polarized, unpolarized models Main Presentation


Slide89 : 89 Genetic Algorithm Simple Example


Slide90 : 90 Why GA ?


Slide91 : 91 Combinatorial Multimodal Unimodal When robustness is desired, nature does it better


Slide92 : 92 Differences Between GAs and Traditional Optimization Methods GAs work with the coding of the parameter set, not the parameters themselves GA’s search for a population of points, not a single point GA’s use the objective function information and not the derivative or other auxiliary knowledge GA’s use probabilistic transition rules, not deterministic rules


Slide93 : 93 Example: Spherical Reactor Problem Case B: Two Variable Problem Case A: Single Variable Problem Optimum Values :


Slide94 : 94 We are looking at Minimizing Q GA is only for Unconstrained Maximization Problem So Y = 800 – Q (say) Now we want to Max (Y) 800 – a number that ensures Y is positive Now let us say that D < 6.0 m


Slide95 : 95 A population of 2n to 4n trial design vectors (rather than just one) is used initially in a problem with n design parameters Each design vector (solution) is represented by a string of binary variables, corresponding to the chromosomes in genetics Case A : Single Variable Problem


Slide96 : 96 The string length is usually determined according to the desired solution accuracy The numerical value of the objective function corresponds to the concept of fitness in genetics. Therefore GAs are naturally suitable for solving maximization problems


Slide97 : 97 Reproduction - Good strings (“fittest”) in a population are selected and assigned a large number of copies to form a mating pool


Slide98 : 98 Crossover – New strings are created by exchanging information among strings of the mating pool Mutation – Changes 1 to 0 and vice versa to create a point in the neighborhood


Slide99 : 99 Sum 1695.9 Maximum 689.6 Average 423.9 Minimum 130.3 Sum 2253.0 Maximum 696.0 Average 563.3 Minimum 364.9


Slide100 : 100 D = 0.1388 m Q = 104.34 W Converged Values:


Slide101 : 101 Case B: Two Variable Problem D = 0.1388 m θ = 144.0922 K Q = 104.34 W Converged Values:


Slide102 : 102 Results - Optimum Values


Slide103 : 103 Works best in problems where the function is very complex GA implementation is still an art Involves lot of tweaking Easy to get started and get an approach that is working Often requires lot of effort to make it work well Features of GA Main Presentation


Slide104 : 104 Validation – Direct Problem Parameters considered Isotropic Scattering Plane Parallel Medium with Black Boundaries Plane parallel medium - Direct Problem


Slide105 : 105 Precipitation Characteristics Water in Clouds Emits Radiation, producing Cold areas that can be seen against a Radiatively Cold Background Ice in Clouds Scatters Radiation downward, producing Cold areas that can be seen against a Radiatively Warm Background Introduction Illustration: Deiveegan et al.