Development of a Precise and Fast Multistream Scattering-Based DMRT model with Jacobian

Vancouver, CAJuly, 2011IGARSS 2011

Development of a Precise and Fast Multistream Scattering-Based DMRT

model with Jacobian

Miao TianA.J. Gasiewski

University of ColoradoDepartment of Electrical EngineeringCenter for Environmental Technology

Boulder, CO, USA


Motivation for a UMRTAttribute UMRT DOTLRT *

Fast, Stable Analytic Matrix Inversion Yes Yes

Fast Jacobian Yes Yes

Phase Matrix Reduced Mie or DMRT (4x4) Reduced HG (2x2)

Polarization Tri-polarization + 4th Stokes Single-polarization

Interface Refraction / Internal Reflection Yes No

Radiation Stream Interpolation Yes, cubic spline No

Thermal Emission Approximation. Linear dependence Constant

Level/Layer Centric Level Centric Layer Centric

DMRT Yes No

* Voronovich, A.G., A.J. Gasiewski, and B.L. Weber, "A Fast Multistream Scattering Based Jacobian for Microwave Radiance Assimilation,“ IEEE Trans. Geosci. Remote Sensing, vol. 42, no. 8, pp. 1749-1761, August 2004.


UMRT: Medium Model

(from Golden et al., 1998)

Upper-half, Sparse Medium

Lower-half, Dense Medium

• In UMRT, a planar-stratified model is used.• Categorized into

• Specular interface and Snell’s law are applied at each layer boundary.

1) Upper-half, Sparse medium:

• Not included yet: rough surface interfaces (ice ridges, rough soil, etc…)

a) General atmosphereb) Weakly homogeneous and slightly dissipativec) Independent scattering: scattering intensity is the sum of scattering intensities from each particle.d) Particle size distribution functione) Sparse medium radiative transfer

2) Lower-half, Dense medium: a) Ice, snow, soil and etc., b) Inhomogeneous and strongly dissipative c) Multiple volumetric scattering: scattering intensity is related to the presence of other particles.d) Pair distribution function (Percus-Yevick Approximation).

e) Dense medium radiative transfer (DMRT) by Tsang and Ishimaru


Parameters(m, x)

Scattering Angle (Θ)

Mie Coefficients

(an, bn)

Angular Functions

(πn, τn)

Modified Stokes matrix

L(Θ)Modified

Stokes Matrix L(θs,θi;Δϕ)

Rotation Matrix Lr(i1,2)

Interpretationof (i1,2)

2

0

d

0

)( dDDn

Isotropic Sphere

SphereReduced Mie Phase Matrix

P’(θs,θi)

Mie Phase Matrix

P(θs,θi; Δϕ)

Mie Phase Matrix: Flowchart

Under the assumptions: 1) Isotropy and 2) Sphericity

}Re{}Im{00}Im{}Re{00

000000

)(

0

2

2

hhvvhhvv

hhvvhhvv

hh

vv

hvvh

ffffffff

ff

L

ff


Mie Stokes MatrixFor Mie Scattering (van de Hulst, 1981; Bohren and Huffman, 1983),

1

1

)](cos)(cos[)1(

12)(

)](cos)(cos[)1(

12)(

nnnnnhh

nnnnnvv

bannn

kif

bannn

kif

where and are the Mie coefficients; and are the angle-dependent functions.nn ba nn

100002cos2sin2sin02sin5.0cossin02sin5.0sincos

22

22

rL

2) The Stokes rotation matrix isrL

1) Interpretation of i1 and i2

sincossincossincoscos

sincossincossincoscos

2

1

issi

siis

i

i

0)]acos[cos(2)]acos[cos(2

2,1

2,12,1 i

ii

2cos1sincossinsincoscoscos isis

where

and


The normal incident and scattered angles based Stoke matrix is

)()()();,( 12 iLLiLL rris

Stokes Matrix: Symmetry

);,();,( siis LL Is = ?

Symmetry of Stokes matrix is divided into two cases.

= );,();,( siis LL

3) Finally, the Rayleigh Stokes matrix is symmetric for all four Stokes parameters.

Thus, Stokes matrix is generally symmetric for the first two Stokes parameters.

2) For the Mie Stokes matrix, the above subtraction results has following formation:

1) The subtraction results of both cases have similar expressions for general Stokes matrices.

00

0000

434241

343231

2423

1413

SRSRSRSRSRSRSRSRSRSR

＝Subtraction Results (General)

000000

00000000

43

34SR

SR＝Subtraction Results (Mie)

Thus, the Mie Stokes matrix is symmetric for the first three Stokes parameters.

?


θi

θs

dθi

dθs

Reduced Phase Matrix


The reduced Mie phase matrix can be numerically calculated.

4443

3433

2221

1211'

0000

0000

),();,(2

0

PPPP

PPPP

PP is

d

is

Mie

1) The 3rd and 4th Stokes parameters are independent of the first two Stokes parameters and can be calculated separately.

44

2221

1211'

00000000000

),(

P

PPPP

P is Rayleigh

Reduced Mie phase matrix: MP(rate)=10 mm/hr, <D>=2 mm, freq = 3 GHz (32x32 quadrature angles)

Reduced Rayleigh phase matrix (179x179 angles) Same <D> and frequency

oos

oi 01cos0,0 Each plot: Up-Left corner: , Forward Scattering

Up-Right corner: , Backward Scattering 1cos,0 so

i

Reduced Mie PM: Validation


Reduced Mie Phase Matrix

freq = 30 GHz

freq = 300 GHz freq = 1000 GHz

freq = 100 GHz


Reduced Henyey-Greenstein PM


DMRT-QCA Phase Matrix1) Incorporates latest version of DMRT-QCA by Tsang, et al., 2008. 2) A prominent advantage: simplification of the phase matrix calculation

Summary of the DMRT-QCA procedure:

q

ffffffff

ff

P

hhvvhhvv

hhvvhhvv

hh

vv


000000

)(2

2

max

max

1

)()()()(

1

)()()()(

)(cos)(cos)1(

1211

)(

)(cos)(cos)1(

1211

)(

N

nn

Mn

Mnn

Nn

Nn

rhh

N

nn

Mn

Mnn

Nn

Nn

rvv

XTXTnnn

kKRif

XTXTnnn

kKRif

a) Lorentz-Lorentz (L-L) law: effective propagation constant b) Ewald-Oseen theorem with L-L law: the average multiple amplitudes:c) The absorption coefficient is calculated as a function of d) Percus-Yevick approximation: structure factor e) Applying all above parameters, the DMRT-QCA phase matrix is calculated by

K)()( , M

nN

n XX)()( ,, M

nN

n XXK)(q

f) Applying same rotation and azimuthal integration procedure, the reduced DMRT-QCA phase matrix can be calculated.


PM Comparison: Mie vs. DMRT (1)

1) Non-sticky particle case mean diameter: 0.14cm fractional volume: 25% frequencies: 13.4GHz, 17.5GHZ, 37GHz 2) DMRT-QCA predicts more forward scattering than that of the Mie theory

Validation to “Modeling active microwave remote sensingof snow using DMRT theory with multiple scatteringeffects”, IEEE, TGARS, Vol.45, 2007, by L. Tsang et al.,


PM Comparison: Mie vs. DMRT (2)

1) Sticky particle case: mean diameter: 0.14cm fractional volume: 25% frequencies: 13.4GHz, 17.5GHZ, 37GHz 2) DMRT-QCA sticky case predicts much greater forward scattering than that of the Mie theory

1.0

Validation to “Modeling active microwave remote sensingof snow using DMRT theory with multiple scatteringeffects”, IEEE, TGARS, Vol.45, 2007, by L. Tsang et al.,


Reduced DMRT Phase Matrix

Sticky particle case: mean diameter: 0.14 cm fractional volume: 25%

1.0

DMRT-QCA phase matrix over 16 quadrature angles

10 GHz 30 GHz

100 GHz


UMRT: Discretizition

a) Separate the up- (+) and down- (-) welling components of radiation.b) Letc) Use the Gauss-Legendre quadrature with the Christoffel weights .

ii θcosi

jj and

1) Numerical DRTE for first two Stokes parameters

M

jBhjijhhj

M

jBhjijhhj

M

jBvjijhvj

M

jBvjijhvjaBhiehi

Bhii

M

jBhjijhhj

M

jBhjijhhj

M

jBvjijhvj

M

jBvjijhvjaBhiehi

Bhii

M

jBhjijvhj

M

jBhjijvhj

M

jBvjijvvj

M

jBvjijvvjaBvievi

Bvii

M

jBhjijvhj

M

jBhjijvhj

M

jBvjijvvj

M

jBvjijvvjaBvievi

Bvii

TPTPTPTPzTkTkdz

dT

TPTPTPTPzTkTkdz

dT

TPTPTPTPzTkTkdz

dT

TPTPTPTPzTkTkdz

dT

1_

1_

1_

1_

1_

1_

1_

1_

1_

1_

1_

1_

1_

1_

1_

1_

The boundary conditions are:

K73.2;

)(;011

cbiB

s

M

jjiji

M

jjBijjiBi

TTHz

TsTsTz

where ),(),(),,(),( ____ jiijjiijjiijjiij PPPPPPPP

2) Symmetrizing variables BhiiihiBviiiviBhiiihiBviiivi TvTvTuTu ,,,


UMRT: SymmetrizationFor vertical polarization, let

,,,, _0_0_0_0 ijvh

ji

jiijijvh

ji

jiijijvv

ji

jiijijvv

ji

jiij

i

eviij PDPCPBPkA

,,,, ___0_ ijhh

ji

jioijijhh

ji

jiij

i

eoijijhv

ji

jiijijhv

ji

jioij PHPkGPFPE

Similarly, for horizontal polarization, let

In UMRT,

ff

ff

vvuu

UDDU

ff

ff

vvuu

GEHFCADBHFGEDBCA

vvuu

dzd

h

v

h

v

MM

h

v

h

v

MMM

h

v

h

v

DOTLRT

1414

44

0000

0000

0000

0000

14

Note: 1) The matrices and are symmetric.U D 2) Making use of Gershgorin’s circle theorem (see Voronovich et al., 2004), it was shown

that the matrices and are positive definite, since following condition always hold in RT:

se kk

DUDU

The argument of symmetric, positive definite (SPD) matrices and holds throughout the

entire UMRT algorithm.

DUDU


Discrete Ordinate-EigenanalysisStep 1. Make following linear transformation.

vuvvuu ''

,Step 2. The DRTE becomes

'

'

'

'

'

'

00

0)()(0

vu

BA

vu

DUDU

vu

dzd

, whereDUBDUA

Step 3. Decouple the two equations

''

2

'2''

''

2

'2''

,

,

vABdzudB

dzvduB

dzvd

uBAdzvdA

dzudvA

dzud

Step 4.1 There are two primary methods to solve equations of step 3. For example, Tsang

(L. Tsang, et al., 2000) uses: BA2,1Step 4.2 In UMRT, we use the solution given by A. Voronovich et al., 2004.

)0()0(

1111

1111

21

)()(

vu

csBAsc

zvzu

t

21

))(sinh()cosh(

BAzBAszBAc

by making use of the matrix identity for symmetric matrices: BABBAB )()( gg

where


UMRT: Solution for Single LayerApplying B.C, at z = h

ruinc

fDU ,,

uinc

tuinc

Δz

inc

inctt

tt

inc

uru

cAssBccAssBccAssBccAssBcut

21

0

)()(

)()(1

tt

tt

cAssBccAssBcr

cAssBcrcAssBc

1)(2 t

cAssBctSimilarly, t

MMA 111b) Positive definite: eigenvalues are non-negative, thus guarantees that exists. Details can be found in DOTLRT, Voronovich et al., 2004

21

1

))(()()( 12

1

1222

1

11 2

ttMMMMBA

gg

21

122

1

22

1

22

1

122

1

122

1

22

1

22

1

12

21

22

1

22

1

22

1

222

1

122

1

1

)5.0tanh(,)5.0tanh(

)sinh(),5.0coth(

tt

r

tt

t MzMbMzMb

zzMMa

To calculate the reflective and transmissive matrices,

a) Symmetry:

t

rt

t

t

MbbMtr

MabMPt

1

1

1

1

111

1

122

1) When x , tanh(x) and coth(x) are bounded to 1.2) When sinh(x) , sinh-1(x) is small (invertible).


UMRT: Solution for Single Layer (2)Inhomogeneous solution of the DRTE1) In DOTLRT, the up- and downwelling radiations are both assumed independent with height.

inhinhinh

inh vufA

fAvu

ff

vu

UDDU

vu

dzd

1

1

00

fDU ,,

uv

u

(b)

Under the assumption of mirror symmetry, the up- and downwelling self-radiation are also equal.

uinh

tuinh

fDU ,,

(c)

uinhuinh ruinh

fAtrIurutuu

urutuuinhinhinh

inhinhinh

1)(

emission

Layer Centric

Assume: 332211 ,, CztfCztvCztu

ff

vu

UDDU

tt

vu

dzd

2

1

2) In UMRT, the up- and downwelling radiations are both assumed to be linear with height.

By applying conventional block (2x2) matrix inversion,

21

31

113

111

2

1

))((

))(( tttDUDUDUI

tDUDUDUItt


UMRT: Solution for Single Layer (3)and

Validation of above solutions can be done by reducing the case to DOTLRT: if t3 = 0, then t1 = t2 = 0.From where, we find

11

11

31

111

111

31

11

13

13

1

2

1

))(())((

))(())((

tDUDUDUICDUDUDUI

tDUDUDUICDUDUDUItCtC

UDDU

CC

111

111

11

111

)())(())((

]))()[((

ADUDUDUDUIIDUDUDUDU

DUDUDUIDU

111

111

11

111

)())(())((

]))()[((

BDUDUDUDUIIDUDUDUDU

DUDUDUIDU

Similarly,

Finally, the inhomogeneous solution in UMRT is

3

13

1

2

1

tAtA

tt

ff

vu

UDDU

tt

vu

dzd

2

1

inhinhinh

inh vutABfA

tABfAvu

3

1113

111


UMRT: Recursive Solution for MultilayerIn UMRT, the up- and downwelling self-radiations of a single layer are not the same, thus:

DU ,v

u

(a)

-vinh -rvinh

-uinh

-tuinh

DU ,

(b)

uinh

0z

hz tr,

-vinh

-uinh -ruinh

DU ,

(c)

vinh -tvinh

layer theof at toplayer theof bottomat layer theof bottomat

layer theof at toplayer theof bottomat layer theof at top

inhinhinh

inhinhinh

vturvv

vrutuu

)()0()0()()0()(

hzutzurzvvhzvrzuthzuu

inhinhinh

inhinhinh Level Centric

Example Type Thickness Bot. Temp.

Top Temp.

RainRate

Particle dia. Freq.

Case I Rain 1 km 300 K 273 K 10 mm/hr 1.4 mm 13.4 GHz

Case II Rain 1 km 273 K 300 K 10 mm/hr 1.4 mm 13.4 GHz


UMRT: Recursive Solution for Multilayer (2)


UMRT: Recursive Solution for Multilayer (3)

tRrRItrRnnn )(

1)()1(

)(

)()( *

)()(

*1

)(

*

)1(

* vRUrRItuUnnnn

Note:

1) Matrices and for all individual layers should be first obtained.2) The initial conditions are

Finally, we have following recursive solutions in UMRT:

vutr ,,,

ijji

ji sSRfU

)0()0()0(

* and

Critical angle and Interpolation:1) Only the incident streams that are inside the critical angle will pass through the interface.

2) Such refractive streams are bent from the quadrature angles. UMRT employs the cubic spline interpolation to compensate them back to the quadrature angles

3) The incident streams whose angle are greater than critical angle will be remove for upwelling radiation streams and added back to the corresponding downwelling radiation streams.Finally, the UMRT solutions are modified as

)()()()(1

)()1(cai

nnn

cain

vRUrRItuU

tRrRItrRnnn )(

1)()1(

)(


UMRT: Jacobian ProcedureUMRT Jacobian Procedure

)()(,

nntr

)(nu

)(nv

''v

''u

'v

'u

)1(nR

)1( n

V

)1(nR

)1( n

U

Key:

, , ,

, w.r.t: , , ,

, ,

, , ,

, ,

, , ,


Summary UMRT is developed based on the DOTLRT concept, however, ithas following key improvements:1) The symmetry property of the polarized reduced Mie phase matrix

is exploited so that the applicability of the fast and stable matrix operation (based on symmetry and positive definiteness) is applicable to both sparse and dense media.

2) Mie phase matrix is applied so that radiation coupling is included in a fully polarimetric solution.

3) DMRT-QCA phase matrix is included for dense medium layers describing (e.g.) vegetation, soil, ice, seawater, etc…

4) The physical temperature of a layer is linear in height, allowing the precise solutions for piecewise linear temperature profiles, thus extending the applicability of DOTLRT to a level-centric grid.

5) The refractivity profile is accounted for by including the critical angle effect and applying cubic spline interpolation to a refractive transition matrix (not discussed).


QUESTIONS?


Replica of Fig. 3 in “Modeling active microwave remote sensing of snow using DMRT theory with multiple scattering effects”,IEEE, TGARS, Vol.45, 2007, by L. Tsang et al.,

Normalized Mie Stokes matrix elements for single particle: particle diameter =1.4 mm, material

permittivity (ice) = 3.15-j0.001: a) freq = 13.4 GHz; b) freq = 37 GHz.

Mie Stokes Matrix: Validation

(a)

(b)


P33 is asymmetric to 90o and P33(at 0o) is greater than P33(180o). P34 is symmetric to 90o.

Qualitative validation by the description in Thermal Microwave Radiation Applications for RemoteSensing, 2006, ch.3, A. Battaglia et al., edited by C. Matzler.

Mie Stokes Matrix: Validation (2)

(a)

(b)


UMRT: Recursive Solution for Multilayer

nU n

R

nU n

R

r

1nR

tr,

vinc

tvinc"

v"

u

tRrRItrR

utvrvRvturv

vRu nnn

incinc

ninc

n

)(1

)()1(

")1(

""

")("

)(

If there exists an external downwellingradiation vinc, it will result in the presenceof two additional up- and downwelling

field

and , which must satisfy:

""vu

v*=u*

u*

tr,

)()()()( ')(

*'

'*

)(')('')(

**)1(

uUrvvuRvvRuuUtuU n

nnnn

and

In DOTLRT, the self-radiation of each layer is same in the up- and downwelling directions, ( )

thus:

vu

where and are the multiple reflections between the extra top layer and the stack.''

vu

)()( *

)()(*

1)(

*)1(

* uRUrRItuUnnnn

1n

U

'v

'u


UMRT: Critical Angle and InterpolationCritical angle: the angle of incidence above which total internal reflection occurs. It is given by

1

2arcsinnn

C

Atmosphere(natm)

Ice(nice)

Snow(nsnow) …

… …

… ……

1) In both UMRT and DOTLRT, once the number M is chosen, the quadrature angles are then fixed in each layer (for self-radiation).

Superposition to the self-radiation of this

layer

…

snow

airsaC n

narcsin_

ice

snowisC n

narcsin_

…

2) Only the refractive streams whose incident streams are inside the critical angle will successfully pass through the interface.

3) Such refractive streams are bended and generally away from the quadrature angles. Therefore need to be correctly compensated back to the fixed quadrature angles.

4) The incident streams whose angle are greater than critical angle will be remove for upwelling radiation streams and added back to the corresponding downwelling radiation streams.


b) Natural raindrop distribution: Marshall and Palmer (MP) SDF c) Ice-sphere distribution: Sekhon and Srivastava SDFd) The absorption coefficient is sea kkk

3) Polydispersed spherical particles

a) With size distribution function (SDF): QDP eDNDn 0

0

2

0

2 )(),(4

,)(),(4

dDDnDmxkdDDnDmxk ssee

Mie Scattering and Extinction

Details can be found in M. Jansen, Ch.3 by A. Gasiewski, 1991.

1) Spherical Mie scattering

2) Monodispersed spherical particle

)]'()[()]'()[()]'()[()]'()[(

)]'()[()]'()[()]'()[()]'()[(

2

2

xxhmxjmmxmxjmxhxxjmxjmmxmxjxjb

mxmxjxhxxhmxjmxmxjxjxxjmxja

nnnn

nnnnn

nnnn

nnnnn

where and are spherical Bessel and Hankel functions of the 1st kind.)()( nn hj is the size parameter and ax 2 m

}Re{)12(2}Re{)12(2 22

12

12 nn

nsnn

ne ban

xx,mη,ban

xx,mη


Validation of Mie Absorption and Scattering

Replica of Fig.3.6-7 in Chapter 3 (right side), originally produced by A. Gasiewski, Atmospheric Remote Sensing byMicrowave Radiometry, edited by M. Janssen, 1993.


Mie Stokes Matrix

Figure from Scattering of Electromagnetic Waves, vol. I, p.7 by L. Tsang, et al., 2000

Under the assumptions: 1) Isotropy and 2) Sphericity


000000

)(

0

2

2

hhvvhhvv

hhvvhhvv

hh

vv

hvvh

ffffffff

ff

L

ff

For Mie Scattering (van de Hulst, 1981; Bohren and Huffman, 1983),

1

1

)](cos)(cos[)1(

12)(

)](cos)(cos[)1(

12)(

nnnnnhh

nnnnnvv

bannn

kif

bannn

kif

where and are the Mie coefficients; and are the angle-dependent functions asnn ba

cos,1,0

)1(,11

12

00

121

nnnnnn nnn

nnn

nn


Reduced Mie PM: 10 GHz1) Below ~60 GHz: Forward scattering ~ Backward scattering2) Above ~60 GHz: Forward scattering > Backward scattering

3) P12 is 90o rotation to P21.

4) P34 = -P43

5) P44 is greater than P33


incut

incu incur

Δz fDU ,, Δz

)( zzu

)0( zv

)0( zu)( zzv

fDU ,,

)( zzu )0( zut ihn

fDU ,,

)( zzvr inh

fDU ,,

)( zzu )( zzv inh

)0( zu inh

tr,

)( zzu ihn

fDU ,,

Development of a Precise and Fast Multistream Scattering-Based DMRT model with Jacobian

Documents

Transcript of Development of a Precise and Fast Multistream Scattering-Based DMRT model with Jacobian