INFORMATION RETRIEVAL IN ELECTROMAGNETIC SCATTERING AND REMOTE SENSING

By MARK THOMAS

CONTENTS CHAPTER VECTOR RADIATIVE TRANSFER (VRT) THEORY OF IN HOMOGENEOUS RANDOM MEDIA 1-1 ITERATIVE SOLUTION OF MULTIPLE SCATTERING AND EMISSION FROM INHOMOGENEOUS DENSE SCATTER MEDIA 1-2 ITERATIVE SOLUTION OF HIGH-ORDER SCATTERING SOLUTION FOR INHOMOGENEOUS NON-SPHERICAL SCATTER MEDIA 1-3 AN APPROACH OF THE THREE-DIMENSIONAL VECTOR RADIATIVE TRANSFER EQUATION (3D-VRT) 1-4 REFERENCES CHAPTER FULLY POLARIMETRIC SCATTERING OF NON-SPHERICAL SCATTER MEDIA 2-1 POLARIMETRIC SCATTERING INDEXES AND INFORMATION ENTROPY OF THE SAR IMAGERY FOR SURFACE MONITORING 2-2 TEMPORAL MUELLER MATRIX SOLUTION FOR POLARIMETRIC SCATTERING FROM INHOMOGENEOUS RANDOM MEDIA 2-3 PULSE ECHOES FROM A INHOMOGENEOUS LAYERED CANOPY AND RANDOM TARGETS BENEATH THE CANOPY 2-4 RETRIEVALS OF UNDERLYING SURFACE ROUGHNESS AND MOISTURE FROM 2-5 STATISTICS OF FOUR STOKES PARAMETERS IN MULTI-LOOK POLARIMETRIC SAR IMAGERY 2-6 IMAGE SIMULATION OF HIGH-RESOLUTION RADAR FOR FULLY POLARIMETRIC SCATTERING FROM HETEROGENEOUS CANOPY SURFACE 2-7 REFERENCES POLARIMETRIC PULSE ECHOES SAR IMAGERY AND APPLICATIONS 3-1 TERRAIN TOPOGRAPHIC INVERSION FROM SINGLE-PASS POLARIMETRIC SAR IMAGEDATA 3-2 CHANGE DETECTION OF MULTI-TEMPORAL ERS-2 SAR IMAGES BY USING TWO-THRESHOLDS EM AND MRF ALGORITHMS 3-3 DEORIENTATION THEORY OF POLARIMETRIC SCATTERING TARGETS AND APPLICATION TO TERRAIN SURFACE CLASSIFICATION 3-5 AN IMPROVED METHOD OF THE MINIMUM ENTROPY FOR REFOCUSING MOVING TARGET IMAGE 3-6 REFERENCES REMOTE SENSING DATA VALIDATION AND APPLICATIONS 4-1 TERRAIN SURFACE MOISTURE MAPPING BASED ON MULTI-YEAR PASSIVE MICROWAVE REMOTE SENSING 4-2 MONITORING THE SANDSTORM USING THE SSM/I DATA AND GETIS STATISTICS 4-3 A GENETIC ALGORITHM TO SIMULTANEOUSLY RETRIEVE THE LAND SURFACE ROUGHNESS AND SOIL WETNESS AND NUMERICAL SIMULATION 4-5 CORRELATED OBSERVATIONS BY THE ERS-1 AND SSM/I OVER OCEAN AND NUMERICAL SIMULATION 4-6 DATA FUSION OF RADARSAT SAR AND DMSP SSM/I FOR MONITORING SEA ICE OF CHINA’S BOHAI SEA 4-8 A HYBRID BP-ANN/GA ALGORITHM FOR DATA FUSION OF LANDSAT ETM+ AND ERS-2 SAR IN CLASSIFICATION OF URBAN TERRAIN SURFACES 4-9 REFERENCES 3-4 AN ALGORITHM FOR SHIP WAKE DETECTION FROM THE SAR IMAGES 4-4 CORRELATION OF THE ERS AND SSM/I OBSERVATIONS OVER SNOWPACK STATISTIC 4-7 SPATIAL AUTO-CORRELATION USING OPTIMAL MULTI-LAYERING MULTI-SCALE GETIS CHAPTER NUMERICAL FORWARD-BACKWARD METHOD (FBM) FOR SCATTERING FROM RANDOMLY ROUGH SURFACE AND TARGET 5-1 BISTATIC SCATTERING FROM A COMPOSITE MODEL OF ROUGH SURFACE AND A TARGET AT LOW GRAZING ANGLE INCIDENCE USING THE GFBM/SAA METHOD 5-2 NUMERICAL SIMULATION OF RADAR SURVEILLANCE FOR THE SHIP TARGET AND 5-3 PARAMETERIZATION OF THE TAPERED INCIDENT WAVE FOR NUMERICAL SIMULATION OF EM SCATTERING FROM ROUGH SURFACE 5-4 BISTATIC SCATTERING FROM A FRACTAL ROUGH DIELECTRIC SURFACE USING THE FBM/SAA 5-5 BISTATIC SCATTERING OF A FRACTAL ROUGH SURFACE WITH HIGH PERMITTIVITY USING THE PBTG-FBM/SAA 5-6 REFERENCES NUMERICAL FINITE ELEMENT METHOD (FEM) FOR SCATTERING FROM RANDOMLY ROUGH SURFACE AND TARGET IN A FEM APPROACH 6-2 NUMERICAL SIMULATION OF BISTATIC SCATTERING FROM A TARGET ABOVE ROUGH SEA SURFACE UNDER AN EM WAVE INCIDENCE AT LOW GRAZING ANGLE BY USING THE FEM 6-3 BISTATIC SCATTERING FROM THE COMPREHENSIVE MODEL OF A SHIP ON AND A OCEANIC SURFACE BY USING THE FEM-DDM METHOD 6-5 THE DIFFERENCE FIELD OF SCATTERING FROM THE TARGET ABOVE A ROUGH SURFACE 6-6 REFERENCES OCEANIC CLUTTERS ROUGH SURFACE 6-1 ANGULAR CORRELATION FUNCTION OF SCATTERING FROM AN OBJECT OVER A RANDOMLY TARGET ABOVE LARGE-SCALE ROUGH SEA SURFACE USING THE FEM-DDM 6-4 NUMERICAL SIMULATION OF THE DOPPLER SPECTRUM OF A FLYING TARGET ABOVE DYNAMIC CHAPTER INVERSE SCATTERING AND PARAMETER RETRIEVALS MEASUREMENT AT GRAZING INCIDENCE 7-2 ITERATIVE INVERSION OF THE SCALAR RADIATIVE TRANSFER EQUATION FOR A LAYER OF RANDOM SPHEROIDS 7-3 ITERATIVE INVERSION OF THE VECTOR RADIATIVE TRANSFER EQUATION FOR A LAYER OF RANDOM SPHEROIDS MULTI-ORDER MUELLER MATRIX SOLUTIONS 7-5 REFERENCES ELECTROMAGNETICS OF COMPLEX PARTICULATE MEDIA 8-1 POLARIMETRIC SCATTERING AND TRANSMITTING OF THE STOKES VECTOR FROM A LAYER OF CHIRAL SMALL SPHEROIDS 8-2 POLARIMETRIC SCATTERING FROM A LAYER OF SPATIALLY ORIENTED METAMATERIAL SMALL SPHEROIDS 8-3 TIME-FREQUENCY ANALYSIS OF EM PULSE RESPONSE FROM A SPHERICAL TARGET RANDOM SPHERICAL PARTICLES 8-5 REFERENCES 7-1 RECONSTRUCTION OF ROUGHNESS PROFILE OF FRACTAL SURFACE FROM SCATTERING 7-4 ITERATIVE INVERSION OF CANOPY PARAMETER AND SURFACE MOISTURE BY USING 8-4 MONTE CARLO SIMULATION FOR CORRELATED SCATTERING OF VERY DENSELY CHAPTER VECTOR RADIATIVE TRANSFER (VRT) THEORY OF INHOMOGENEOUS RANDOM MEDIA 1-1 ITERATIVE SOLUTION OF MULTIPLE SCATTERING AND EMISSION FROM INHOMOGENEOUS DENSE SCATTER MEDIA THE VECTOR RADIATIVE TRANSFER (VRT) EQUATION IS AN INTEGRAL-DEFERENTIAL EQUATION TO DESCRIBE MULTIPLE SCATTERING, ABSORPTION AND TRANSMISSION OF FOUR STOKES PARAMETERS IN RANDOM SCATTER MEDIA FROM THE INTEGRAL FORMAL SOLUTION OF VRT EQUATION, THE LOWER ORDER SOLUTIONS, SUCH AS THE FIRST-ORDER SCATTERING FOR A LAYER MEDIUM OR THE SECOND ORDER SCATTERING FOR A HALF SPACE, CAN BE OBTAINED [JIN 1994, TSANG ET AL 1985] THE LOWER ORDER SOLUTIONS ARE USUALLY GOOD AT LOW FREQUENCY WHEN HIGH-ORDER SCATTERING IS NEGLIGIBLE IT IS NOT FEASIBLE TO CONTINUE ITERATION FOR OBTAINING A HIGH ORDER SCATTERING SOLUTION BECAUSE TOO MANY FOLDS OF INTEGRATION WOULD BE INVOLVED FOR EXAMPLE, IN THE SPACE-BORNE MICROWAVE REMOTE SENSING THE DEFENSE METEOROLOGICAL SATELLITE PROGRAM (DMSP) SPECIAL SENSOR MICROWAVE/IMAGER (SSM/I) EMPLOYED SEVEN CHANNELS OF 19, 37 AND 85GHZ WITH DUAL POLARIZATION AND 22 GHZ WITH ONLY VERTICAL POLARIZATION MULTIPLE SCATTERING FROM THE TERRAIN SURFACES SUCH AS SNOWPACK CANNOT BE NEGLECTED AT THESE CHANNELS THE DISCRETE ORDINATE AND EIGEN-ANALYSIS METHOD HAS BEEN STUDIED TO TAKE INTO ACCOUNT MULTIPLE SCATTERING AND APPLY IT TO REMOTE SENSING OF ATMOSPHERIC PRECIPITATION, SNOWPACK, ETC [JIN 1994] A SNOWPACK WAS MODELED AS A LAYER OF RANDOM AND DENSE SPHERICAL PARTICLES, AND THE VRT FOR A LAYER OF UNIFORMLY DENSE SPHERICAL PARTICLES HAS BEEN NUMERICALLY STUDIED BY THE DISCRETE ORDINATE METHOD [JIN 1994, TSANG ET AL 1985] HOWEVER, DUE TO SURFACE MELTING AND REFROZEN CRUST PRESENCE, THE SNOWPACK UNDERGOES STRATIFYING AND MIGHT HAVE INHOMEGENEOUS PROFILES OF THE ICE GRAIN SIZE, FRACTIONAL VOLUME AND PHYSICAL TEMPERATURE ETC IT BECOMES NECESSARY TO STUDY MULTIPLE SCATTERING AND EMISSION FROM STRATIFIED SNOWPACK OF DENSE PARTICLES HOWEVER, THE DISCRETE ORDINATE AND EIGEN-ANALYSIS METHOD CANNOT BE SIMPLY APPLIED TO MULTI-LAYERS MODEL, BECAUSE NUMERICALLY SOLVING A SET OF THE VRT EQUATIONS IS DIFFICULT BY STRATIFYING THE INHOMOGENEOUS MEDIA INTO MULTIPLE THIN SLABS AND EMPLOYING THE FIRST ORDER MUELLER MATRIX OF EACH THIN SLAB, AN ITERATIVE METHOD IS DEVELOPED TO DERIVE HIGH ORDERS SCATTERING SOLUTIONS OF THE WHOLE SCATTER MEDIA HIGH-ORDER SCATTERING AND EMISSION FROM INHOMOGENEOUS STRATIFYING MEDIA OF DENSE SPHERICAL PARTICLES ARE NUMERICALLY OBTAINED [JIN AND LIANG 2003] THE BRIGHTNESS TEMPERATURES AT LOW FREQUENCY, SUCH AS 5 3 GHZ, WITHOUT HIGH-ORDER SCATTERING AND AT SSM/I CHANNELS WITH HIGH ORDER SCATTERING ARE DISCUSSED THIS APPROACH IS ALSO COMPARED WITH THE CONVENTIONAL DISCRETE ORDINATE METHOD FOR A UNIFORM LAYER MODEL NUMERICAL SIMULATION FOR AN INHOMOGENEOUS SNOWPACK IS ALSO COMPARED WITH THE MEASUREMENTS OF MICROWAVE REMOTE SENSING 1-1 1 ITERATIVE SOLUTION OF VRT EQUATION FOR DENSE SCATTERERS MEDIA THE EFFECTIVE PROPAGATION CONSTANT K IS DEFINED TO TAKE INTO ACCOUNT THE SCATTERING COHERENCE OF RANDOM DENSE PARTICLES [JIN 1994, TSANG ET AL 1985] IT HAS BEEN DERIVED AS ] ) 1( ) 1( F F B KA I FB K K (1-1 1) WHERE ) K F K K K K B S S (1-1 2) THE RADIUS OF SPHERICAL PARTICLES IS A , THE FRACTIONAL VOLUME IS F , AND 0K , S K ARE THE WAVE NUMBERS OF THE FREE SPACE AND PARTICLES, RESPECTIVELY EQ (1-1 2) YIELDS THE EFFECTIVE PERMITTIVITY OF THE MEDIUM AS THEORY AND APPROACH OF INFORMATION RETRIEVALS 2 K K F H (1-1 3) THE VRT EQUATION OF THE STRATIFIED MEDIA OF RANDOM SPHERICAL PARTICLES IS WRITTEN AS [JIN 1994] ) ( ) , ( ' SIN ' ) ,' ( ) ,' , ( ) , ( ) , ( ) , ( COS Z T C Z D Z I Z P Z I Z Z I DZ D A E ˜ ˜ ˜ ³ T N T T T T T T T N T T S (1-1 4) WHERE A E N N , ARE THE EXTINCTION AND ABSORPTION MATRICES, RESPECTIVELY P IS THE PHASE MATRIX FOR MULTIPLE SCATTERING, AND ) ( 0 Z T IS THE PHYSICAL TEMPERATURE PROFILE OF THE SCATTER MEDIA THE BOUNDARY CONDITIONS AT Z AND D Z ARE WRITTEN AS ) ˜ Z I R Z I F T T T S (1-1 5A) U FU FU T C T D Z I R D Z I ) ( ) , ( ) ( ) , ( ˜ ˜ T T S T T (1-1 5B) WHERE ) ( 0 T F R , ) (T FU R ARE THE REFLECTIVITY MATRICES AT THE TOP AND BOTTOM (UNDERLYING, U) BOUNDARIES, RESPECTIVELY ) (T FU T IS THE TRANSMITTIVITY MATRIX OF THE UNDERLYING SURFACE AT Z=-D, AND ) ( 0 Z T AND U T0 ARE THE PHYSICAL TEMPERATURES OF THE SCATTER MEDIA AND UNDERLYING SURFACE, RESPECTIVELY IN THE MULTI-LAYERED SNOWPACK, IT IS ASSUMED THAT ALL ICE PARTICLES WITHIN EACH LAYER (SEPARATED BY THE SOLID LINES IN FIG 1-1 1) ARE IDENTICAL AND THAT FRACTIONAL VOLUME IS CONSTANT THE REFLECTION FROM AND TRANSMISSION THROUGH THOSE INTERFACES BETWEEN THE LAYERS ARE CALCULATED BY THE EFFECTIVE PERMITTIVITIES OF THE LAYERS FIG 1-1 1 GEOMETRY OF INHOMOGENEOUS SNOWPACK HIGH-ORDER SCATTERING, COMPARED WITH LOW-ORDER SCATTERING, IS RELATED TO THE ALBEDO AND OPTICAL DEPTH OF THE LAYER IN OTHER WORDS, THE SCATTERING CAN BE APPROXIMATED BY LOW ORDER IF THE OPTICAL DEPTH AND ALBEDO OF THE LAYER ARE SMALL ENOUGH THUS, IF EACH LAYER OF THE WHOLE SCATTER MEDIA IS STRATIFIED INTO MANY THIN SLABS WHOSE OPTICAL DEPTHS ARE SMALL (SEE THE SEPARATION BY THE DOTTED LINES IN FIG 1-1 1), SCATTERING IN EACH ISOLATED SLAB CAN BE WELL APPROXIMATED BY ITS LOW-ORDER SOLUTION TAKING L SLABS FROM ˜˜˜ A FROM THE TOP TO BOTTOM, THE ZERO-TH ORDER EMISSION AND LOW ORDER MUELLER MATRIX CAN BE EMPLOYED FOR EACH ISOLATED SLAB [JIN AND LIANG 2003] H ) (Z F H Z D Z ) (T B T C D ' D Z D Z U H ' ' D EA N W IN EACH THIN SLAB?THE EMISSION OF THE SLAB IS APPROXIMATED AS D D A A A A A A |)], SEC | EXP( 1[ ) ( ) ( ( D ' | T T N N N T T D T C T C I E E A N S (1-1 6) WHERE THE SUBSCRIPT S DENOTES THE EMISSION COMING FROM THE ISOLATED SLAB SELF ALSO, THE FIRST ORDER MUELLER MATRIX OF THIS SLAB IS WRITTEN AS ° ° ° ° ° ¯ ° ° ° ° ° ® ­ ' ' ' ' ' ' | D D D D A D D D D A A D D D D A A D D D D A A A A A )] SEC (SEC EXP[ ) SEC EXP( ) SEC EXP( ) SEC (SEC SEC ) , ( ) , ( ) , ( )1( I I E I E I E I E I E I I E I E I I I D D D D D D P M M T T T T N T T T N T N T T T N T N T T T T N T T N T T S T T T T T (1-1 7) IF THE OPTICAL DEPTH IS LARGE, YIELDING TOO MANY THIN SLABS ACCORDING TO OF (1-1 6,7), THE THICKNESS OF EACH SLAB D ' CAN BE CHOSEN SUCH THAT ' ' D EA N W , AND THE ZERO-TH ORDER EMISSION IS NOW APPROXIMATED AS ) ( ) ( ) ( )1( ) ( T T T A A A T C T C I N S | (1-1 8) WHERE ' ' SIN )] ' , ( ) ( )[ ' , ( )] ' SEC EXP( )' , ( ) ( )[ ' , ( [ SEC ) )1( T T T T T T S T T N T T T T T T N T S D D D P D D D P T C T C E A ' ³ A A A A A D A (1-1 9A) ' ' SIN )]} ' SEC EXP( )' , ( ) ( )[ ' , ( )] ' , ( ) ( )[ ' , ( { SEC ) )1( T T T N T T S T T S T T T S T T T T N T S D D D D P D D P T C T C E A ' ! ³ A A A A A D A (1-1 9B) HERE )' SEC (SEC )] ' SEC (SEC EXP[ T T N T T ' N T T A A E E D D (1-1 10A) )' SEC (SEC )] ' SEC (SEC EXP[ T T N T T ' N T T A A E E D D (1-1 10B) T N T ' N T SEC ) SEC EXP( A A E E D D (1-1 10C) CORRESPONDINGLY, THE MUELLER MATRIX OF EACH SLAB IS FIRST APPROXIMATED BY ) , ( ) , ( ) , ( ) ( I I I M M M T T T T T T A A A | (1-1 11) WHERE °° ° ° ¯ °° ° ° ® ­ ˜ ˜ ˜ ˜ ³ ³ D D A A A A D D A A A A D A ' ' SIN )] ,' , ( ) ,' ( )' , ( ) ,' , ( ) ,' ( )' , ( [ ' ' SIN )] ,' , ( ) ,' ( )' , ( ) ,' , ( ) ,' ( )' , ( [ | COS | ( T T T T T T S T S T T T T T T S T S T S T S T T T T T T T T S T T T T T T T S T S T S T T T T S S D A P P A P P D A P P A P P M I I I I I I I I I (1-1 12A) °° ° ° ¯ °° ° ° ® ­ ˜ ˜ ˜ ˜ ! ³ ³ D D A A A A D D A A A A D A ( T T T T S T T S T S T T T T S T T S T S T S T S T T T T T S T T T S T T T T S T T T S T S T S T T T T S S D A P P A P P D A P P A P P M I I I I I I I I I (1-1 12B) WHERE ) SEC ' (SEC )]} SEC ' (SEC EXP[ )' , ( ) , ( {' SEC ) ,' , ( I E I E I I D D D A T T N T T N T T T T T T T T ' A A (1-1 13A) ) SEC ' (SEC )] , ( )' , ( [' SEC ) ,' , ( I E I I D D A T T N T T T T T T T T A (1-1 13B) ) SEC ' (SEC )] ' , ( ) , ( )[ SEC EXP( ' SEC ) ,' , ( I E I I E I D D D A T T N T T T T T N T T T T ' A A (1-1 13C) ) SEC ' (SEC )] ' SEC EXP( )' , ( ) SEC EXP( ) , ( [' SEC ) ,' , ( I E E I E I I D D D D A T T N T N T T T N T T T T T T ' ' A A A (1-1 13D) THE EMISSION AND MUELLER MATRIX IN EQS (1-1 6,8) AND EQ (1-1 11) ARE, RESPECTIVELY, EMPLOYED FOR CALCULATION OF SCATTERING INTENSITY IN EACH ISOLATED SLAB TO OBTAIN SCATTERING AND EMISSION FROM WHOLE MEDIA, IT SHOULD TAKE INTO ACCOUNT ALL INTERACTIONS OF ALL SLABS FOR WHICH SCATTERING AND EMISSION HAVE BEEN KNOWN THE TRANSMISSION AND REFLECTION ARE TAKEN INTO ACCOUNT ONLY THROUGH THE BOUNDARIES OF THE SUBLAYERS (E G IN FIG 1-1 1, THE SOLID LINES AT D Z AND D Z )?AND T A A T AND T A A R AT THE INTERFACES OF THOSE THIN SLABS INDICATED BY THE DOTTED LINES THE DOWNWARD EMISSION IN THE A -SLAB FROM THE ( A )-SLAB IS WRITTEN AS ) SEC EXP( )] ( ) ( ) ( ) ( [ ) ( ) ( ) ( , ) ( ) ( T N T S T T T T S T S D I T I R I I E N N N S N ' ˜ ˜ A A A A A A A A A (1-1 14A) THE UPWARD EMISSION IN THE A -SLAB FROM THE ( , ) ( ) ( T N T T T S T T T D I T I R I I E N N N S N ' ˜ ˜ A A A A A A A A A (1-1 14B) WHERE ) ( D D  T ?THE SUPERSCRIPT N DENOTES THE ITERATION NUMBER? ) ( ) ( T N S IA IS FIRST CALCULATED BY EQ (1-1 6) OR EQ (1-1 8) ( N ) AND THEN BY EQ (1-1 15) ( ! N ) AS FOLLOWS THE EMISSION FROM Z AND D Z , ) ( ) ( T S N IA AND ) ( ) ( 1 T N L I A ?ARE DEFINED BY EQS (1-1 5A,B) AS N I R N I N F N T T T S A A (1-1 15A) °¯ °® ­ ! ˜ ˜ ˜ N I R N T C T I R I N L FU U FU N L FU N L T S T T T S T T A A A (1-1 15B) FROM EQS (1-1 13,14), THE N-TH ORDER EMISSIONS, UPWARDS AND DOWNWARDS, TO TAKE INTO ACCOUNT THE INTERACTIONS OF THE SLABS, ) ( ) ( T N IA AND ) ( ) ( T S N IA , CAN BE OBTAINED FURTHER, THE (N+1)-TH ORDER EMISSION OF THE A -SLAB-SELF CAN BE CALCULATED BY ITS MUELLER MATRIX AS ' ' SIN )' ( )' ( )' , ( ' ' SIN )' ( )' ( )' , ( ) ( ( ³ ³ ˜ ˜ ˜ ˜ S S S T T T T T T T T T T T T T D I T M D I T M I N N N S A A A A A A A A A (1-1 16) THUS, WE OBTAIN A HIGH-ORDER ITERATIVE SOLUTION ITERATION PROCESS IS SUMMARIZED AS FOLLOWS (1) STRATIFYING THE MEDIA INTO MULTI-SLABS, CALCULATE ) ( ) ( T N S IA , IS CALCULATED BY ) ( ) ( T N IA (4) REPEATING THE STEPS (2) AND (3) TO A FINAL CONVERGENCE, CALCULATION IS FINISHED THE HIGH-ORDER SCATTERED EMISSION IS OBTAINED TAKING ) ( 1 T AI AS THE SUM OF ALL ORDERS ) ( ) ( 1 T N I A , THE BRIGHTNESS TEMPERATURE OBSERVED AT THE REGION 0 IS OBTAINED AS ) ( ) ( T T T ˜ AI T C T F B (1-1 17) IT SHOULD MENTION THAT IN EACH ITERATION, ONLY ONE REFLECTION BETWEEN THE INTERFACES OF THE STRATIFIED LAYERS IS TAKEN INTO ACCOUNT MULTI-REFLECTIONS BETWEEN THOSE LAYERS INTERFACES ARE CALCULATED DURING MULTI-ORDER ITERATIONS THUS, ) ( ) ( 1 T N I A IS DIFFERENT FROM THE N-TH ORDER INTEGRAL FORMAL SOLUTION EMISSION OF EACH SLAB-SELF TAKES ACCOUNT OF THE ZEROTH ORDER SOLUTION (AS 'W ) OR THE ZEROTH AND FIRST-ORDER SOLUTIONS (AS 'W ) CORRESPONDINGLY, THE FIRST-ORDER OR THE FIRST- AND SECOND-ORDER MUELLER MATRICES ARE EMPLOYED TO TAKE ACCOUNT OF INTERACTIONS OF THE SLABS TOTAL EMISSION FROM THE SCATTER MEDIA, ) ( ) ( 1 T N I A , INCLUDES THE EMISSIONS OF ALL SLABS AND N TIMES INTERACTIONS AMONG THEM 1-1 2 NUMERICAL RESULTS A UNIFORM SNOWPACK CONSIDER AN UNIFORM SNOWPACK OF THE DEPTH CM D THE ICE PARTICLES ARE IDENTICAL SPHERES WITH RADIUS CM A F , AND DIELECTRIC CONSTANT ( H H I S THE UNDERLYING MEDIUM HAS ( H H I U THE PHYSICAL TEMPERATURES OF BOTH THE SNOWPACK AND UNDERLYING SURFACE ARE SIMPLY ASSUMED AS K BASED ON EQ (1-1 2), THE EFFECTIVE PROPAGATION CONSTANT OF THE SNOWPACK IS CALCULATED AS 1( H H I F TAKING CM D ' OF EACH SLAB IN OUR LAYERING METHOD AND UNIFORMLY DISCRETIZING THE SCATTERING ANGLE T BY EACH D 6 EQUALLY, EQS (1-1 6,7) AND THE MUELLER MATRIX ARE CALCULATED ITERATION IS CARRIED OUT TO THE FOURTH ORDER FIG 1-1 2 PRESENTS THE BRIGHTNESS TEMPERATURE OF GHZ AND COMPARISON WITH THE RESULT OF THE DISCRETE ORDINATE AND EIGEN-ANALYSIS METHOD [JIN 1994] (GAUSSIAN QUADRATURE NUMBER IS N ) THE RESULTS OF TWO METHODS ARE ALMOST EXACTLY IDENTICAL TB ( K ) T ( O ) TBV TBH THIS METHOD TBV TBH EIGEN METHOD FIG 1-1 2 BRIGHTNESS TEMPERATURE OF AN UNIFORM SNOWPACK AT GHZ 1 0 1 2 ' T V ITERATIVE ORDER N F RE Q U E N C Y(G H Z ) N 'T V ( K ) FIG 1-1 3 ITERATION NUMBER N AND COMPARISON WITH THE DISCRETE ORDINATE EIGEN-ANALYSIS FIG 1-1 3 SHOWS THE ITERATION NUMBER N FOR DIFFERENT FREQUENCY, AND COMPARISON WITH THE DISCRETE ORDINATE AND EIGEN-ANALYSIS EIGEN ITER V T T T ' THE OBSERVATION ANGLE IS D T , CM D , CM A , AND OTHER PARAMETERS ARE THE SAME AS FIG 1-1 2 IT CAN BE SEEN THAT AS THE FREQUENCY INCREASES, HIGH-ORDER ITERATION IS NEEDED DUE TO LARGER ALBEDO ANDSTRONGER SCATTERING FOR EXAMPLE,THE ALBEDO AT GHZ Z , WHILE IT BECOMES Z AT GHZ IT IS OBVIOUS THE ITERATION ORDERS AT SUCH HIGH FREQUENCY INCREASES IF ONE CAN SEE 90 | B INHOMOGENEOUS SNOWPACK CALCULATE THE BRIGHTNESS TEMPERATURE OF INHOMOGENEOUS SNOWPACK OF M D THE ICE PARTICLE RADIUS IS CM A ?AND THE PARTICLE DENSITY IS ASSUMED TO HAVE THE PROFILE AS (IN FIG 1-1 4) ) EXP( 1[ (  R IS A RANDOM NUMBER UNIFORMLY (0 1) THE PROFILE OF FRACTIONAL VOLUME IS OBTAINED FROM ) ( ) ( Z Z F U AND THE PHYSICAL TEMPERATURE PROFILE (IN K) IS ALSO ASSUMED AS ) EXP( Z K Z Z T (1-1 20) IT IS REASONABLE TO HAVE HIGHER TEMPERATURE AT THE TOP LAYERS DUE TO THE SUNSHINE 100 120 140 160 180 U (G/CM 3) Z (CM) FIG 1-1 4 DENSITY PROFILE OF AN INHOMOGENEOUS SNOWPACK THE INHOMEGENEOUS SNOWPACK OF M IS SEEN AS THE STRATIFIED LAYERS WITH EACH LAYER CM EVERY LAYER HAS DIFFERENT DENSITY (AS SHOWN IN FIG 1-1 4), PHYSICAL TEMPERATURE, AND THE MUELLER MATRIX EACH LAYER IS NOW DIVIDED INTO THIN MULTI-SLABS OF THICKNESS CM D ' FOR ITERATION CALCULATION DUE TO RANDOM NOISE R IN EQ (1-1 18), CALCULATIONS OF FIVE REALIZATIONS ARE AVERAGED TO OBTAIN FINAL RESULT FIG 1-1 5(A) SHOWS THE BRIGHTNESS TEMPERATURE AT 5 3GHZ DISCRETE POINTS ARE THE MEASUREMENTS OF TSANG ET AL (2001) THE TEMPERATURE OF UNIFORM SNOWPACK IS OBTAINED FROM EQ (1-1 19) AND THE DENSITY IS IS LOW, MULTIPLE SCATTERING IS ACTUALLY NEGLIGIBLE TO TAKE INTO ACCOUNT OF MULTIPLE SCATTERING, FIG 1-1 5(B) PRESENTS THE BRIGHTNESS TEMPERATURE AT GHZ AND GHZ UNDER THE SAME SITUATION IT BECOMES SIGNIFICANT TO TAKE INTO ACCOUNT MULTIPLE SCATTERING AT THESE HIGH FREQUENCIES PARTICLE SCATTERING CAN DARKEN OR SHADOW THE EMISSION FROM UNDERLYING MEDIA AS THE FREQUENCY BECOMES HIGHER OR THE LAYER DEPTH BECOMES THICKER, THIS DARKENING BECOMES STRONGER AND YIELDS A LOWER BRIGHTNESS TEMPERATURE THUS, IT USUALLY YIELDS B B B T T T ! ! HOWEVER, IT MIGHT BECOME ANOMALOUS FOR A INHOMOGENEOUS SNOWPACK [ANDREW AND MARK 2000] FOR EXAMPLE, FIG 1-1 6 GIVES VERTICALLY POLARIZED BRIGHTNESS TEMPERATURE OF THE SUMMER 2001 AT THE LOCATION ( E S D D ) OBSERVED BY THE SSM/I AS F15 AT 19-?37- AND GHZ CHANNELS IT CAN BE SEEN THAT V B V B T T DURING THE DAYS 210 300, AND ANOMALOUS V B V B T T 85 ! DURING THE DAYS 300 365 THIS IS MAINLY DUE TO INHOMOGENEOUS STRUCTURES OF SNOWPACK, SUCH AS ICE CRUSTS OR COARSER GRAINS FORMED DURING MELTING, REFROZEN STRATIGRAPHY THE RADIOMETERS AT DIFFERENT FREQUENCIES SEEM TO BE SENSITIVE TO THE SNOWPACK WITH DIFFERENT EFFECTIVE DEPTHS, THAT IS, THICK DEPTH FOR THE LOW-FREQUENCY SENSOR AND THIN DEPTH FOR HIGH- FREQUENCY ONE HOMOGENEOUS INHOMOGENEOUS TBV TBH DATA TB5 3 ( K ) (A) T ( O ) FIG 1-1 5 (A) BRIGHTNESS TEMPERATURE OF HOMOGENEOUS AND INHOMOGENEOUS SNOWPACKS TB ( K ) (B) T ( O ) 19GHZ SINGLE 19GHZ MULTIPLE 37GHZ SINGLE 37GHZ MULTIPLE FIG 1-1 5(B) BRIGHTNESS TEMPERATURE OF SINGLE AND MULTIPLE SCATTERING FIG 1-1 7 SHOWS THE BRIGHTNESS TEMPERATURE OF 1-, 3- AND 5-LAYERS SNOWPACK THE 1-LAYER MODEL TAKES CM A F ? CM D THE ICE CRUST IN THE 3- AND 5-LAYERS MODELS HAS A THICKNESS CM AND CM A THE ~ ~ THEORY AND APPROACH OF INFORMATION RETRIEVALS BOTTOM LAYER IS COARSER SNOW WITH THE THICKNESS CM F , AND ICE RADIUS CM A THE MIDDLE LAYER OF THE 5-LAYER MODEL HAS THICKNESS CM ? CM A THE UNDERLYING BOTTOM MEDIUM IS PURE ICE THE SNOWPACK HAS TEMPERATURE K THE OBSERVATION ANGLE IS D T NUMERICAL RESULTS OF INHOMOGENEOUS SNOWPACK SHOW THAT ANOMALOUS BRIGHTNESS TEMPERATURE B B B T T T FIG 1-1 6 VERTICALLY POLARIZED BRIGHTNESS TEMPERATURE OBSERVED BY THE SSM/I 5-LAYERS 3-LAYERS 1-LAYER TBV( K ) T( O ) FIG 1-1 7 BRIGHTNESS TEMPERATURE OF INHOMOGENEOUS SNOWPACK VS FREQUENCY 1-2 ITERATIVE SOLUTION OF HIGH-ORDER SCATTERING SOLUTION FOR INHOMOGENEOUS NON-SPHERICAL SCATTER MEDIA THE VRT TAKES INTO ACCOUNT MULTIPLE SCATTERING, ABSORPTION AND TRANSMITTING OF FOUR STOKES PARAMETERS IN RANDOM MEDIA ITERATIVE APPROACH OF THE VRT CAN DEMONSTRATE SCATTERING MECHANISM IN DIFFERENT ORDERS THE FIRST-ORDER SOLUTION OF THE MUELLER MATRIX FOR A SINGLE LAYER OF RANDOM MEDIUM, AND SECOND SOLUTION OF THE MUELLER MATRIX FOR A HALF-SPACE HAVE BEEN DERIVED [JIN 1994, TSANG ET AL 1985] HOWEVER, IT SEEMS VERY DIFFICULT TO CONTINUE ITERATIONS TO FIND HIGHER ORDER SCATTERING SOLUTION BECAUSE MULTI-FOLD INTEGRATIONS OVER THE SCATTERING ANGLES HAVE TO BE INVOLVED IN NUMERICAL APPROACH, THE DISCRETE ORDINATE AND EIGEN-ANALYSIS METHOD [JIN 1994, TSANG ET AL 1985, ISHIMARU ET AL 1982], INVARIANT IMBEDDING METHOD [TSANG ET AL 1985], THE MONTE CARLO METHOD ETC [JIN 1994, TSANG ET AL 1990] HAVE BEEN STUDIED THESE NUMERICAL APPROACHES USUALLY NEED MORE COMPUTATIONS MOREOVER, THE SINGULARITY APPEARED WHEN THE INCIDENT ANGLE HAPPENS BE EQUAL TO THE DISCRETE GAUSSIAN ANGLE [SIEWERT 2000, BARICHELLO ET AL 2000] IN ACTIVE VRT PROBLEM HAS NOT BEEN WELL DISCUSSED THE INVARIANT IMBEDDING METHOD WAS ONLY STUDIED FOR PASSIVE VRT PROBLEM [TSANG ET AL 1985] AS DISCUSSED IN THE LAST SESSION 1-1, BY STRATIFYING A RANDOM SCATTER MEDIA INTO MULTIPLE THIN-LAYERS IN THE Z DIRECTION, THE FIRST-ORDER SCATTERING SOLUTION OF EACH THIN LAYER IS EMPLOYED TO DERIVE HIGH-ORDER SCATTERING SOLUTION OF WHOLE RANDOM MEDIA AN ITERATIVE APPROACH TO SOLVE HIGH-ORDER SCATTERING SOLUTION FROM THE MUELLER MATRICES IN DISCRETE ORDINATES IS DEVELOPED NUMERICAL RESULTS DEMONSTRATE THAT THIS NEWLY ITERATIVE APPROACH AS FEASIBLE, EFFECTIVE AND ESPECIALLY APPLICABLE TO HIGH-ORDER SOLUTION OF VRT FOR BOTH BISTATIC SCATTERING AND THERMAL EMISSION OF INHOMOGENEOUS NON-SPHERICAL SCATTER MEDIA 1-2 1 STRATIFYING THE RANDOM MEDIA AND FIRST-ORDER ITERATIVE SOLUTION AS AN ELECTROMAGNETIC WAVE, ) ( ) COS (COS M G T T G I , IS INCIDENT UPON THE SCATTER MEDIA, THE VRT EQUATION FOR A LAYER OF INHOMOGENEOUS SCATTERING MEDIA IS WRITTEN AS ³ ³ ˜ ˜ S S M T M T M T M T T M T M T N M T T ) ,' ,' ( ) ,' ,' ; , ( ' ' SIN ' ) , , ( ) , , ( ) , , ( COS Z I Z P D D Z I Z Z I DZ D E (1-2 1) THE BOUNDARY CONDITIONS ARE WRITTEN AS ¯ ® ­ ˜ ) , , ( ) ( ) , , ( ) ( ) COS (COS ) D Z I R D Z I I Z I M T S T M T M G T T G M T S (1-2 2) WHERE 0I IS INCIDENT STOKES VECTOR, ( T S , M ) ARE THE INCIDENT ANGLES, ) ( 12 T R IS THE REFLECTIVITY MATRIX OF THE UNDERLYING SURFACE, P IS THE PHASE MATRIX, E N IS THE EXTINCTION MATRIX AND IS WRITTEN AS " " " " ¼ º " " " " " ¬ ª ! ! ! ! ! ! ! ! ! ! ! ! ! !