AERONET Inversions: Progress and Perspectives Oleg Dubovik (NASA / GSFC) Alexander Sinyuk (NASA /...
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Transcript of AERONET Inversions: Progress and Perspectives Oleg Dubovik (NASA / GSFC) Alexander Sinyuk (NASA /...
AERONET Inversions: Progress AERONET Inversions: Progress and Perspectivesand Perspectives
Oleg Dubovik (NASA / GSFC) Alexander Sinyuk (NASA / GSFC) Tatyana Lapyonok (NASA / GSFC) Brent Holben (NASA / GSFC) Tom Eck (NASA / GSFC)Alexander Smirnov (NASA / GSFC)Anne Vermeulen (NASA / GSFC) Teruyuki Nakajima (CCSR, Tokyo, Japan)Takashi Nakajima (CCSR, Tokyo, Japan)Charles Gatebe (NASA / GSFC)Michael King (NASA / GSFC)Francois-Marie Breon (CEA/DSM/LSCE, France) Michael Sorokin (NASA / GSFC)Ilya Slutsker (NASA / GSFC)AERONET/PHOTON…AERONET/PHOTON… (Word-Wide)
ObservationsObservations
Numerical inversion:Numerical inversion:-Accounting for noise-Solving Ill-posed problem- Setting a priori constraints
Forward model:Forward model:-Spectral and angular scattering by particles with different sizes, compositions and shapes- Accounting for multiple scattering in atmosphere
aerosol particle sizes,aerosol particle sizes, refractive index, refractive index,
single scattering albedosingle scattering albedo, etc.
(Dubovik and King, JGR, 2000(Dubovik and King, JGR, 2000)
Multiple ScatteringMultiple ScatteringMultiple scattering effects Multiple scattering effects are accounted by solving are accounted by solving scalarscalar radiative transfer equation with assuming radiative transfer equation with assuming Lambertian Lambertian ground reflectanceground reflectance (Nakajima – Tanaka code) (Nakajima – Tanaka code)
Aerosol scatteringAerosol scattering
Molecular scatteringMolecular scattering
Gaseous absorptionGaseous absorption
Surface reflectionSurface reflection
Single Scattering by Single ParticleSingle Scattering by Single Particle
Scattering and AbsorptionScattering and Absorption is modeled assuming aerosol is modeled assuming aerosol particle as particle as homogeneoushomogeneous sphere sphere with with spectrally dependentspectrally dependent complex complex refractive index refractive index ( m(( m()= n()= n() - i k() - i k()) - “Mie particles”)) - “Mie particles”
m()Radius
P(P()-)- Phase Function; -single scattering albedo() - extinction optical thickness; () absorptionoptical thickness
II00(())
IIscatscat(())
IItranstrans(())
Single ScatteringSingle Scattering
AERONETAERONET model of aerosol model of aerosolspherical:spherical:
Randomly orientedRandomly orientedspheroids :spheroids :
(Mishchenko et al., 1997)(Mishchenko et al., 1997)
Statistically optimized fittingStatistically optimized fitting:: (Dubovik and King, 2000)
Measurements:i=1 - optical thicknessi=2 - sky radiances-their covariances(should depend on and )-lognormal error distributions
a priori restrictions on norms of derivatives of:i=3 -size distr. variability;i=4 -n spectral variability; i=5 -k spectral variability;
Lagrange parameters
consistencyIndicator
weighting
ε0
2
εi2 fi
∗−fi x( )( )2
λ,θ( )i
∑ + ε0
2
εi2 fi
a −fi x( )( )2
i∑ → (Ntotal-Nx)ˆ ε 0
2
0.01
0.1
1
10
100
1000
0 20 40 60 80 100 120 140
Almucantar Fitting
Intensity
Scattering Angle (degree)
Int(0.44) * 1000
Int(0.67) * 100
Int(0.87) * 10
Int(1.02)
0
0.1
0.2
0.3
0.4
0.5
0.40 0.60 0.80 1.0
Fitting of optical thicknessin retrievals
MeasurementsFitting
Optical thickness
Wavelenths (micron)
0
0.05
0.1
0.15
0.2
0.25
0. 1 10
Retireved size distribution
Radius (microns)
dV/dlnR (
μm3/μm
2)
1.35
1.40
1.45
1.50
1.55
1.60
Wavelength (μ )m.44 .67 .87 1.2
Real Part
0.00
0.01
0.10
Wavelength (μ )m.44 .67 .87 1.2
Imarinary PartImaginary Part
Fitting as a retrieval strategy
The averaged optical properties of The averaged optical properties of various aerosol types various aerosol types (Dubovik et al., 2002, JAS)(Dubovik et al., 2002, JAS)
+
_
AERONET inversion developments
Forward model:- accounting for particle shape- using non-lambertian surface- modeling polarization
Retrieval flexibility:- additional spectral channels- different geometries
Inversion of combined data:- different geometries - combining with satellite- combining with aircraft
Output improvements:- detailed phase function- degree of polarization- flexible separation of modes- fluxes and forcing- details of fitting (biases and random)
Errors estimation:-for individual retrieval-for absorption optical thickness-for phase functions, etc.
Perspectives:- assuming bi-component aerosols- combining with polarimetric satellite observations- retrieval of shape distribution
Almucantar: ((), I(), I())= 0.38, 0.44, 0.5, 0.67, 0.87, 1.02, 1.64, μm
AERONET inversion scenarios
Principal Plane: ((), I(), I()) = 0.38, 0.44, 0.5, 0.67, 0.87, 1.02, 1.64, μm
Polarized Principal Plane: ((), I(), I(),P(),P()) = 0.87μm
spheres
spheroids
Inversion Products: dV/dln(rdV/dln(rii) )
n(n(k(k(
BRDFBRDFerrorserrors
satellite, aircraft, etc.
(P11(P12(
fine & coarsefine & coarsefluxes, …, …
Utilizing additional spectral channels
Potential enhancement of informationPotential enhancement of informationIncreased calibration effortsIncreased calibration efforts
0
0.05
0.1
0.15
0.2
0.1 1 10
4 channels5 channels (+ 1.637 μ )m
/ (dV dlnR
μm3 /μm
2 )
( )Particle Radius micron
21:2:24,5:21:34, Dhabi
1.35
1.4
1.45
1.5
1.55
1.6
0.4 0.6 0.8 1 1.2 1.4 1.6
4 channels5 channels (+ 1.637 μ )m
Real Part of Refractive Index
(Wavelength μ )m
21:2:24,5:21:34,Dhabi
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
0.4 0.6 0.8 1 1.2 1.4 1.6
4 channels5 channels (+ 1.637 μ )m
Single Scattering Albedo
(Wavelength μ )m
21:2:24,5:21:34,Dhabi
Desert Dust (Dhabi, UAI)
dV/dln(rdV/dln(rii)) n(n( (
Almucantar: ((), I(), I())= 0.44, 0.67,0.87, 1.02, 1.64, μm
Utilizing additional spectral channels
Potential enhancement of informationPotential enhancement of informationIncreased calibration effortsIncreased calibration efforts
GSFC aerosol
dV/dln(rdV/dln(rii)) n(n( (
Almucantar: ((), I(), I())= 0.38, 0.44, 0.67, 0.87, 1.02, μm
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.1 1 10
4 channels5 channels (+0.38 μ )m
/ (dV dlnR
μm3 /μm
2 )
( )Particle Radius micron
19:2:24,21:25:19, , ,33Almucantar GSFC
1.34
1.36
1.38
1.4
1.42
1.44
1.46
1.48
1.5
0.4 0.5 0.6 0.7 0.8 0.9 1
4 channels5 channels (+0.38 μ )m
Real Part of Refractive Index
(Wavelength μ )m
19:2:24,21:25:19, , ,33Almucantar GSFC
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
0.4 0.5 0.6 0.7 0.8 0.9 1
4 channels5 channels (+ 0.38 μ )m
Single Scattering Albedo
(Wavelength μ )m
19:2:24,21:25:19, , ,33Almucantar GSFC
0.04
0.06
0.08
0.1
0.3
0.5
0.4 0.5 0.6 0.7 0.8 0.9 1
measurementsfitted
Optical Thickness
Wavelength (μ )m
19:2:24,21:25:19, , ,33Almucantar GSFC
0.04
0.06
0.08
0.1
0.3
0.5
0.4 0.5 0.6 0.7 0.8 0.9 1 2
measurementsfitted
Optical Thickness
Wavelength (μ )m
2:3:24, 12:59:4, , ,131Almucantar GSFC
GSFC aerosolGSFC aerosol Dhabi dustDhabi dust
+ 0.38 μm+ 0.5, 1.64 μm+ 1.64 μm
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.4 0.5 0.6 0.7 0.8 0.9 1 2
measurementsfitted
Optical Thickness
Wavelength (μ )m
21:2:24, 5:21:34,Dhabi
Fitting additional spectral channels
+ 0.02 - 0.02
??? water vapor ? fine ?
Utilizing principal plane
Enhanced range of scattering anglesEnhanced range of scattering anglesSensitivity to vertical structure of aerosolSensitivity to vertical structure of aerosolChallenging cloud screeningChallenging cloud screening
Desert Dust (Dhabi, UAI)
dV/dln(rdV/dln(rii)) n(n( (
Principal Plane: ((), I(), I()) = 0.44, 0.67, 0.87, 1.02, 1.64, μm
0
0.05
0.1
0.15
0.2
0.25
0.1 1 10
almucantarprinciple plane (4 channels)principle plane (5 channels: + 1.64 μ )m
/ (dV dlnR
μm3 /μm
2 )
( )Particle Radius micron
21:2:24,5:47:17, _ , ,132Principal Plane Dhabi
1.3
1.35
1.4
1.45
1.5
1.55
1.6
1.65
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
AlmucantarPrinciple Plane (4 channels)Principle Plane (5 channels)
Real Part of Reffactive Index
Wavelengths (μ )m
21:2:24,5:47:17, _ , ,132Principal Plane Dhabi
0.6
0.7
0.8
0.9
1
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
AlmucantarPrinciple Plane (4 channels)Principle Plane (5 channels)
Single Scattering Albedo
Wavelengths (μ )m
21:2:24,5:47:17, _ , ,132Principal Plane Dhabi
Utilizing principal plane
Enhanced range of scattering anglesEnhanced range of scattering anglesSensitivity to vertical structure of aerosolSensitivity to vertical structure of aerosolChallenging cloud screeningChallenging cloud screening
dV/dln(rdV/dln(rii)) n(n( ((
Principal Plane: ((), I(), I()) = 0.44, 0.5, 0.67, 0.87,1.02, 1.64, μm
GSFC aerosol
0
0.02
0.04
0.06
0.08
0.1
0.1 1 10
almucantar (4 channels)principle plane (4 channels)principle plane (6 channels: + 0.5, +1.64 μ )m
/ (dV dlnR
μm3 /μm
2 )
( )Particle Radius micron
2:3:24,12:59:4, , ,131Almucantar GSFC
1.3
1.35
1.4
1.45
1.5
1.55
1.6
1.65
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
AlmucantarPrinciple Plane (4 channels)Principle Plane (6 channels)
Real Part of Reffactive Index
Wavelengths (μ )m
21:2:24,5:47:17, _ , ,132Principal Plane Dhabi
0.6
0.7
0.8
0.9
1
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
AlmucantarPrinciple Plane (4 channels)Principle Plane (6 channels)
Single Scattering Albedo
Wavelengths (μ )m
2:3:24,12:59:4, , ,131Almucantar GSFC
((), I(), I(),P(),P()) Numerical inversion:Numerical inversion:-Accounting for uncertainty (F(F1111; -F; -F1212/F/F11 11 !!!)!!!) - Setting a priori constraints
aerosol particle sizes,aerosol particle sizes, refractive index, refractive index,
single scattering albedosingle scattering albedo
AERONET Polarized InversionAERONET Polarized Inversion
€
P11
,P12
,P22
,P33 ,,P34 ,,P44 , ≈ K ip λ;θ ;n;k( )
i;p( )
∑ ω p V ri( )
Forward Model:Forward Model:
Single Scat:Single Scat:
Multiple Scat:Multiple Scat: DEUZE JL, HERMAN M, SANTER R, JQSRT, 1989
Successive Orders of Scattering CodeSuccessive Orders of Scattering Code
Utilizing polarizationEnhanced range of scattering anglesEnhanced range of scattering anglesSensitivity to vertical structure of aerosolSensitivity to vertical structure of aerosolChallenging cloud screeningChallenging cloud screeningCalibration verificationCalibration verification
dV/dln(rdV/dln(rii)) n(n( ((
Principal Plane: ((), I(), I()) = 0.44, 0.5, 0.67, 0.87,1.02, 1.64, μmPolarization : ((), I(), I(),P(),P()) = 0.87μm Cape Verde aerosol
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.1 1 10
almucantarprinciple plane 0.87 μ ( )m with polarization
(4 + )principle plane channels polarization
/ (dV dlnR
μm3 /μm
2 )
( )Particle Radius micron
28:9:23,18:7:54, _ , _ ,47Principal Plane Capo Verde
1.3
1.35
1.4
1.45
1.5
1.55
1.6
1.65
0.4 0.5 0.6 0.7 0.8 0.9 1
almucantarprinciple plane 0.87 μ ( )m with polarization
(4 + )principle plane channels polarization
Real Part of Reffactive Index
(Wavelengths μ )m
28:9:23,18:7:54, _ , _ ,47Principal Plane Capo Verde
0.6
0.7
0.8
0.9
1
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
almucantarprincipal plane (radiances)principal plane (polarization, 0.87 μ )m
(4 + )principal plane channels polar
Single Scattering Albedo
(Wavelengths μ )m
28:9:23,18:7:54, _ , _ ,47Principal Plane Capo Verde
Fitting polarizationEnhanced range of scattering anglesEnhanced range of scattering anglesSensitivity to vertical structure of aerosolSensitivity to vertical structure of aerosolChallenging cloud screeningChallenging cloud screeningCalibration verificationCalibration verification
RadianceRadiance Linear PolarizartionLinear Polarizartion
Principal Plane: ((), I(), I()) =0.87μmPolarization : ((), I(), I(),P(),P()) = 0.87μm
Cape Verde aerosol
0.01
0.1
1
10
0 30 60 90 120 150
Measurements
Fitting
Radiance in Principle Plane
Scattering Anlge (degrees)
0
0.1
0.2
0.3
0.4
0.5
0.6
50 75 100 125 150
Measurements
Fitting
Linear Polarization (-F
12/F
11)
Scattering Anlge (degrees)
Fine and Coarse modes separations
RadianceRadiance
Beijing aerosol
0.1
1
10
100
0 45 90 135 180
totalfine modecoarse mode
Phase Function
Particle Radius (micron)
02:05:2003,09:27:51,PolarPP,Beijing,14
0
0.3
0.6
0.9
0 45 90 135 180
totalfine modecoarse mode
Linear Polarization (-F
12/F
11)
Particle Radius (micron)
02:05:2003,09:27:51,PolarPP,Beijing,14
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.1 1 10
dV/dlnR (
μm3 /μm
2 )
( )Particle Radius micron
Coarse
Fine
Coarse
2:5:23,9:27:51, , ,14PolarPP Beijing
Beijing Aerosol
Flexible separation betweenfine and coarse modes
(curently: ~0.6 μm)
0.45μm
Retrieval using combinations of up-looking Ground-based and down-looking satellite
observations
Retrieved:
Aerosol Properties:- size distribution- real ref. ind. - imag. ref. ind(AERONET sky channels)
Surface Parameters:-BRDF (MISR channels)-Albedo (MODIS IR channels)
AERONET / POLDER-2 retrievalPOLDER-2 fit
Size distribution
BRDF
Biomass burningMongu, Zambia, June, 2003
AERONET Ground-based Sun-sky radiometer:AERONET Ground-based Sun-sky radiometer:
± 0.02 at± 0.02 at 6 channels6 channels: 0.34, 0.38, 0.44, 0.67, 0.87, 1.02, : 0.34, 0.38, 0.44, 0.67, 0.87, 1.02, 1.651.65 μμmm
± 0.05% at± 0.05% at4 channels4 channels: : 0.380.38, 0.44, 0.67, 0.87, 1.02, , 0.44, 0.67, 0.87, 1.02, 1.651.65 μμmm3° ≤ scattering angles ≤ ~70°3° ≤ scattering angles ≤ ~70°- PP± 0.02% at ± 0.02% at 0.870.87
MISR MISR Reflectance atReflectance at4 channels4 channels: 0.45, 0.55, 0.67, 0.87 : 0.45, 0.55, 0.67, 0.87 μμmm9 viewing angles9 viewing angles: ±70.5: ±70.5oo, ± 60, ± 60oo,, ± 45.6± 45.6oo, ± 26.1, ± 26.1oo, 0, 0oo
MODISMODISReflectance atReflectance at7 channels7 channels: 0.47, 0.55, 0.66, 0.87,1.2, 1.6, 2.1: 0.47, 0.55, 0.66, 0.87,1.2, 1.6, 2.1
AERONET/ MISR/ MODIS retrieval
Retrieval using combinations of up- and down-looking observations
Retrieved:
Aerosol above plane:- size distribution- real ref. ind.- imag. ref. ind
Aerosol below plane:- size distribution- real ref. ind.- imag. ref. ind
Surface Parameters:- BRDF, albedo, etc.
CAR - Cloud Absorption Radiometer
8 spectral channels: 0.34, 0.38, 0.47, 0.68, 0.87, 1.03, 1.19, 1.27 μm
Measures radiation transmitted* and reflected:
0° ≤ Obs. Zenith ≤ 180°0° ≤ Obs. Azimuth ≤ 360°
* Stray light problems for
scattering angles ≤ 10°
Flown by CV-580 aircraftat ~ 700 m above ground
Univ. of Washington CV-580
Optical thickness on September 6, 2000
AERONETAERONETdaily variationsdaily variations
0
0.5
1
1.5
2
2.5
5:00 7:00 9:00 11:00 13:00 15:00
Mongu/Zambia, September 6, 2000
1.02 μm.87μm.67μm.44μm.38μm.34μm
Optical Thickness
Time of DayCAR
AERONETalmucantar
AATS-14 AATS-14 versus AERONETAERONET
0
0.5
1
1.5
2
0.2 0.4 0.6 0.8 1 1.2
AERONET (ground-based)
AATS-14 (airborne)
"AERONET - AATS-14"
Optical Thickness
Wavelengths (μ )m
α=1.82
α=1.74α=2.12
Aerosol retrieved from combinedCAR - AERONET - AATS-14 obs.
0.4
0.5
0.6
0.7
0.8
0.9
0.4 0.6 0.8 1.0 1.2
Single Scattering Albedo
Below PlaneAbove PlaneAERONET
Single Scattering Albedo
Wavelength (μ )m
1.3
1.35
1.4
1.45
1.5
1.55
1.6
1.65
0.4 0.6 0.8 1 1.2
Real Part of Ref. Index
BelowAboveAERONET
Real Part of the Ref. Index
Wavelengths (μ )m
0
0.05
0.1
0.15
0.2
0.1 1.0 10
Volume Size Distribution
Below PlaneAbove PlaneBelow + AboveAERONET
dV/dlnR (
μm3 /μm
2 )
(Radius μ )m
Comparison of model retrieved BRDFwith corrected direct BRDF
Gatebe et al. 2003
0
0.1
0.2
0.3
0.4
0.5
-50 0 50
Vegetation model BRDF (lines)
6S corrected BRDF (points)
1.22 μm.87μm.67μm.47μm.34μm
( )Angle degree
Reflectance
BRDF constrains model:- positive and smooth;- PP symmetrical
Comparison of retrieved surface reflectance with other observations
Mongu, September 6, 2000
LambertianLambertian approximation
Mongu, Zambia
0
0.1
0.2
0.3
0.4
0.5
0.6
0.2 0.4 0.6 0.8 1 1.2 1.4
Lambertian Surface Albedo
AERONET/ POLDER (June, 2003)
CAR (Gatebe et al., 2000)
MODIS (Sept. 2 , Sept. 10)
Albedo
Wavelengths (μ )m
0
0.1
0.2
0.3
0.4
0.5
0.6
0.2 0.4 0.6 0.8 1 1.2 1.4
Lambertian Surface Albedo
CAR + AERONET inversion
CAR (Gatebe et al., 2000)
MODIS (Sept. 2 , Sept. 10)
Albedo
Wavelengths (μ )m
New Inversion Options:- Almucatars (any numbers of spectral channels)Almucatars (any numbers of spectral channels)- Principal planes (any numbers of spectral channels)Principal planes (any numbers of spectral channels)- Polarized Principle Planes Polarized Principle Planes (0.87 mm)(0.87 mm)
- Combined Principle Planes: Combined Principle Planes: Polarized (0.87 mm) + Regular (0.44 - 1.02mm)Polarized (0.87 mm) + Regular (0.44 - 1.02mm)
- Other Combined Cimel Data:Other Combined Cimel Data:-Almucantar + Principle Plane (?)Almucantar + Principle Plane (?)- Several Almucantars + Principle Planes (??)Several Almucantars + Principle Planes (??)
- AERONET + satellite data (MODIS, MISR, POLDER …)AERONET + satellite data (MODIS, MISR, POLDER …)- AERONET + aircraft (CAR) + …satelliteAERONET + aircraft (CAR) + …satellite
- Spherical & Nonspherical modelSpherical & Nonspherical model ( (for all retrievalsfor all retrievals))
Perspectives:Perspectives:- assuming bi-component aerosolsassuming bi-component aerosols- combining with polarimetric satellite observationscombining with polarimetric satellite observations- retrieval of shape distribution- retrieval of shape distribution
Sensitivity to instrumental offsetsSensitivity to instrumental offsets
Offsets were considered in:
- optical thickness: - optical thickness:
- sky-channel calibration:- sky-channel calibration:
- azimuth angle pointing: - azimuth angle pointing:
- assumed ground reflectance:- assumed ground reflectance:
Aerosol models considered (bi - modal log-normal):(bi - modal log-normal):
- Water-soluble aerosol for 0.05 ≤ - Water-soluble aerosol for 0.05 ≤ (440) ≤ 1;(440) ≤ 1;
- Desert dust for 0.5 ≤ - Desert dust for 0.5 ≤ (440) ≤ 1;(440) ≤ 1;
- Biomass burning for 0.5 ≤ - Biomass burning for 0.5 ≤ (440) ≤ 1;(440) ≤ 1;
Δτ λ( )=±0.01; ±0.02;
ΔI λ;Θ( ) I λ;Θ( ) 100% = ±5%;Δφ=0.5o; 1o;
ΔA λ( )/ A λ( ) 100%= ±30%; ±50%;
Results summary:
--(440) ≤ 0.2(440) ≤ 0.2 - - ddV/V/ddlnr (lnr (++), ), nn(() () (--), ), kk(() () (--), ), (() () (--))
--(440) > 0.2(440) > 0.2 - - ddV/V/ddlnr (lnr (++), ), nn(() () (++),), k k(() () (++), ), (() () (++))
- - Angular pointing accuracy is critical for Angular pointing accuracy is critical for ddV/V/ddlnr of dustlnr of dust
((++) ) CAN BECAN BE retrievedretrieved ( (--) ) CAN NOT BECAN NOT BE retrieved retrieved
bias influence at 0
bias:
€
˜ ω 0 =τscat
τext+Δτ
0.7
0.75
0.8
0.85
0.9
0.95
0 0.2 0.4 0.6 0.8 1
0(real)
ω0(real)-0.03
τ+0.01
τ+0.02
τ+0.03
ω0
(retrieved)
τ
Sky Radiance bias:
€
˜ ω 0 =τscat 1-Δ( )
τext
=ω0 1-Δ( )
Random ERRORS in AERONET Random ERRORS in AERONET retrievalsretrievals
ASSUMPTIONS:- measurements have Normal Noise: - optical thickness: - optical thickness: 0.01 0.01
- sky-radiances: - sky-radiances: 5% 5%
0.00
20.00
40.00
60.00
80.00
0.1 1 10
(.44)=.5(.44)=.2(.44)=.5(.44)=1.
(%)Errors
(Radius μ )m
CONCLUSIONS:- the retrievals stable- important tendencies outlined
Rigorous ERRORS estimates:Rigorous ERRORS estimates: General caseGeneral case: : large number of unknownslarge number of unknowns
andand redundant measurements redundant measurements
U - matrix of partial derivatives in the vicinity of solution €
ˆ x i( )2
≈ Δˆ x irandom
( )2
+ Δˆ x ibias
( )2
€
CΔˆ x random = UTC-1U( )
-1
€
ˆ x bias = UTC-1U( )−1
UTC-1ΔIbias
ˆ x
Above is valid:- in linear approximation - for Normal Noise
- no a priori constraints
ERRORS estimates withERRORS estimates with a priori constraints a priori constraints
€
CΔˆ x random = UTC-1U + U a
T Ca−1Ua( )
-1
€
ˆ x bias = UTC-1U + U aT Ca
−1Ua( )−1
UTC-1ΔIbias + U aT Ca
−1ΔIabias
( )
ISSUES:- in linear approximation - for Normal Noise
- strongly dependent on a priori constraints- very challenging in most interesting cases
ERROR ERROR Factors:Factors:
Important Factors:- Aerosol Loading - Scattering Angle Range
- Number of Angles (homogeneity)- Aerosol Type
etc.
Examples of error estimates
0
0.05
0.1
0.15
0.1 1 10
dV/dlnR (
μm3 /μm
2 )
(Radius μ )m
,GSFC (.44)=.7
1.25
1.3
1.35
1.4
1.45
0.4 0.6 0.8 1
GSFC, (.44)=.7
Real Part of Refractive Index
(Wavelength μ )m
0.6
0.7
0.8
0.9
1
0.4 0.6 0.8 1
GSFC, (.44)=.7
Single Scattering Albedo
(Wavelength μ )m
0.8
0.9
1
0.4 0.6 0.8 1
GSFC, (.44)=.7
Single Scattering Albedo
(Wavelength μ )mhigh
loading
low loading
flaming combustionRio Branco, Brazil
smoldering combustion Quebec fires, July 2002
ABSORPTION of SMOKE