Description of the Vector Radiative Transfer Model MOMO

Andre Hollstein, Jonas von Bismarck, Jurgen Fischer, Rene Preusker

December 2, 2010

Abstract

In this document we describe the radiative transfer model MOMO which is able tocalculate the polarized light field in an atmosphere ocean system (AOS). We assumea horizontal homogeneous atmosphere consisting of layers with vertical uniform opticalproperties. The upward and downward directed light field is calculated at all inter layerboundaries and for all solar positions within the range of the user defined zenith Gaussianquadrature angles. The azimuthal dependence of the light field is internally expressed asFourier series and reconstructed at equidistant distributed azimuth angles. The modelis operated by several input files which govern the height profile of the atmosphere, thescatterer’s, the absorber and the atmosphere ocean interface and is therefore complex butvery flexible. The wind blown ocean surface is described by wave facets whose normalsare statistically distributed using the Cox and Munk model. Shadowing is included byusing an external function which has been derived for scattering by random media in theKirchoff approximation.

Contents

1 Introduction 21.1 Definition of Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Vector Radiative Transfer 42.1 Interaction Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Radiative Transfer Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Intermediate Recapitulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Atmosphere description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Wind ruffled ocean surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6 Ocean description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.7 Intermediate conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.8 Clear water Raman scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.9 Formalism for the inclusion of Raman scattering in the RTE . . . . . . . . . . 262.10 Optical input parameters for Raman scattering . . . . . . . . . . . . . . . . . 272.11 Raman source in matrix operator theory . . . . . . . . . . . . . . . . . . . . . 29

3 Implementation and Model Description 313.1 Numerical Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1.1 Sparse Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . 323.1.2 Phase matrix truncation . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1.3 Fourier Series Modification for the Conservation of Radiation . . . . . 33

1

3.1.4 Geometric Series of Reflection Matrices . . . . . . . . . . . . . . . . . 343.2 Computation Time Versus Accuracy . . . . . . . . . . . . . . . . . . . . . . . 35

1 Introduction

The problem that a vector radiative transfer system aims to solve is the forward problem ofcalculating the light field in an atmosphere ocean system for given atmosphere ocean system(AOS) properties. These include the vertical structure, the gaseous constituent parts, thescatterer such as aerosols and clouds, the atmosphere ocean interface, the relative refractiveindex of the ocean and atmosphere, scattering and absorption of the clear water body and theinherent optical properties of the hydrosols. The forward model may then be used to solvethe inverse problem of the characterization of the atmosphere from radiation measurementsfrom ground, aircraft or satellite[1].

The physical foundation of solving this problem is solving the Maxwell equations for agiven system. From these one can derive a Radiative transfer equation which is more suitablefor the treatment in a numeric system. Recently, an explicit derivation including the limitsof its applicability has been published by Mishchenko[2].

Figure 1: Plane parallel atmosphere ocean system (AOS) with plane parallel incidence ofsolar radiation.

To solve the radiative transfer equation problem numerically one need to make some as-sumptions about the system which affects the applicability of the results to the atmosphereof the earth. We assume that the system consists of plane parallel slabs with vertical homo-geneous optical properties. Since the Sun is much more far away from the earth than the twobodies are wide we can assume the source to be infinite far away hence the radiation incidenton the top of the earths atmosphere is plane parallel. The one remaining spacial dimensionof the system is height respective optical thickness (since it is a strict monotone function ofheight). At every interface of the slabs the light field is expressed as a four dimensional realvector depending on the cosine of the zenith angle µ and the azimuth angle φ. This sphericaldirectional space is divided into the upper and lower hemisphere and both cosines are takento be positive numbers.

2

1.1 Definition of Polarization

The Maxwell equations describe the electric and magnetic field of an given system. Sincethe earths atmosphere is homogeniuos and isotropic on the length scales of the consideredwavelengths it is sufficient to reduce the problem to the electric field ~E. Radiances may bedescribed by electric fields that doesn’t vary with respect to the direction of propagation ~ezbut may have a spacial dependency with respect to the two orthogonal directions ~ex and ~ey.These three unit vectors form a left handed Cartesian coordinate system.

Figure 2: Propagating electric field and it’s projection to the x-t and y-t plane. The correlationof these projections in time and or space define the polarization properties of the beam.

To perform a radiance measurement one needs the transportation of energy trough thefinite cross section of the detector which inherently needs the passing of some integrationtime. Every measurement of the radiance field implies averaging the field over time and,depending of the detector, space. This is expressed with the 〈.〉 symbols:

~Ez := 〈(eiϕxExeiϕyEy

)〉. (1)

Based on these definitions one can define the associated coherence matrix of the field Cij [3]which is by construction a hermitian 2 × 2 complex matrix. Transportation of radiation byJones matrices is analogous for field averages and coherence matrices:

Cij = EiE∗j (i, j = x, y). (2)

3

By defining a basis ~σ in the space of hermitian 2 × 2 complex matrices the coherencematrices may be expressed only by their coefficients in this basis representation. Here we willchoose the Pauli algebra ~σ which then leads to the expansion coefficients s0 to s3:

~σ :=1√2

((1 00 1

),

(0 11 0

),

(0 −ii 0

),

(1 00 −1

)). (3)

~S := (s0, s1, s2, s3)T ; C = ~S~σ. (4)

This expansion may be expressed as scalar product C =∑4

i=0 siσi. This shows that theinconsistent definition of polarization parameters in the literature is simply due to differentbasis definitions by the author. This formalism may be used to relate different definitions ofpolarization parameters to each other.

A commonly used definition is made in the form ~S = (I,Q, U, V )T since it is easily relatedto intensity measurements using ideal polarization filters and quarter wave plates. Somecommonly used definitions are given in reference[4].

~S =

ExE

∗x + EyE

∗y

EyE∗x + ExE

∗y

i(ExE

∗y − EyE∗x

)ExE

∗x − EyE∗y

:=

s0

s1

s2

s3

:=

IU−VQ

:=

sBW0

sBW2

−sBW3

sBW1

:=

sH0sH2−sH3−sH1

. (5)

For measurements of the polarization parameters of radiation one often assumes detectorsthat are only sensitive to the intensity (I or in general s0). At least as many independentmeasurements as to retrieved parameters have to be taken. In general the instrument isdescribed using Jones[5] or Mueller[6, 4] matrices and the polarization is retrieved by invert-ing a matrix describing the measurements and multiplying it with a vector containing themeasurements[7, 8].

2 Vector Radiative Transfer

In the following describe the used algorithms to calculate the polarized light field in anatmosphere ocean system. Any order of scattering is included using the matrix operatortechnique hence the model is also applicable to systems with high optical thickness. We willdescribe the three main parts of the model, the atmosphere, the atmosphere ocean interfaceand the ocean.

2.1 Interaction Principle

The main principle we are using to solve the problem of atmospheric radiation is the inter-action principle[9]. We state this as an almost obvious ad hoc principle and will show itsconnection to the radiative transfer equation. We will discuss its inherent limitations shortly.

We split the light field at any spatial level (height or optical thickness) into two fields, oneis the upward and the other one the downward directed light field (see fig (3)).

The interaction principle states, that the upward directed light field depends linear onthe transmitted light field from a layer at higher optical thickness and the downward directed

4

Figure 3: Interaction Principle.

intensity at the same level that is reflected by the atmosphere below. These dependencies arelinear and called reflection and transmission. The approach in this form is therefore limited tocases in which radiation only linearly interacts with the system. Analogous for the downwarddirected light field at the lower level:

L+ (τ2) = t21L+ (τ1) + r12L

− (τ2) + J+21, (6)

L− (τ1) = r21L+ (τ1) + t12L

− (τ2) + J−12. (7)

This principle may be conveniently stated in matrix form:(L+ (τ2)L− (τ1)

)= S12

(L+ (τ1)L− (τ2)

)+ J12, (8)

with the definition of a matrix S:

S12:=S (τ1, τ2) :=(t21 r12

r21 t12

). (9)

The statement of this principle is straight forward but doesn’t state how the reflection andtransmission operators can be derived or how they are related to the Maxwell or the radiativetransfer equation. We will come back to this later in this paper. Here the next logical step isto derive the algorithm of doubling and adding.

By stating the principle for two consecutive layers with three boundaries:

(L+ (τ3)L− (τ2)

)= S23

(L+ (τ2)L− (τ3)

)+ J23 (10)(

L+ (τ2)L− (τ1)

)= S12

(L+ (τ1)L− (τ2)

)+ J12 (11)

5

(L+ (τ3)L− (τ1)

)= S13

(L+ (τ3)L− (τ1)

)+ J13 (12)

we can eliminate the transmission and reflection operators of the intermediate layer. Bywriting the resulting equations in the same form as the original interaction equations thetransmission and reflection operators of the combined layers can be expressed as[10, 11, 12,13, 14]:

t31 = t32 (1− r12r32)−1t21, (13)

t13 = t12 (1− r32r12)−1t23, (14)

r31 = r21 + t12r32 (1− r12r32)−1t21, (15)

r13 = r23 + t32r12 (1− r32r12)−1t23. (16)

Applying the whole algorithm (Eq. (13) to Eq. (16)) for two layers with the same opticalproperties is known as doubling; applying it to layers with different optical properties asadding.

In Fig (4) we show how this concept can be applied to the atmosphere ocean system.We assume that the reflection and transmission operators for an elementary layer is known.We will show that the RTE can be used to calculate these operators for layers in which thesingle scattering assumption holds, hence optical very thin layers. When the reflection andtransmission operators of an elementary layer are known we use the doubling consecutively toconstruct the optical properties for the whole layer. This procedure is performed for all layers.Eventually these can be combined using the adding algorithm. The interaction principle fromequation (8) can be used to calculate the radiances at all inter layer boundaries.

Figure 4: Doubling and adding scheme.

In a series of papers by Grant and Hunt (1969) the mathematical foundation of thisprocedure is rigorously outlined. They showed that the adding procedure might be definedas a formal star product between two layers[9, 15]:

S13 = S12 ∗ S23. (17)

Applying this scheme one can show that the transition from a given adding:

t12 = t (τ1, τ2) , (18)

6

to an infinitesimal form where the difference in optical thickness vanishes:

∂τ

(L+ (τ1)L− (τ1)

)= S1

(L+ (τ1)L− (τ1)

)+ J1, (19)

S (τ1) = limτ2→τ1

S (τ1, τ2) = limτ2→τ1

(S (τ1, τ2)− S (τ1, τ1)

τ2 − τ1

). (20)

Hence the adding of an infinite thin layer converges to the unity operator:

S (τ1, τ1) =(1 .. 1

). (21)

Together with this procedure results in the differential form of the interaction principle inmatrix notation:

d

dτ

(L+

L−

)=

d

dτ

(t01 − 1 r10

r01 t10 − 1

)(L+

L−

)+

d

dτ

(J+

01

J−10

). (22)

2.2 Radiative Transfer Equation

The radiative transfer equation (23) states that the change in the light field depends onthe light field itself and the sources of radiation in the system. The direct sunlight can bedescribed using Lambert and Beers law (24):

L′ = −L+ J, (23)

S(τ) = e−τ/µs S0 (24)

Figure 5: Sources of radiation in an atmosphere ocean system.

Sources of radiation are the single scattered sun light and the scattering of diffuse lightwhich is shown in Fig. (5). These are given in the first and second term of the right handside of the RTE in Eq. (25).

(µd

dτ− 1)L(τ, µ, φ) =ω0e

−τ/µs P (µ, φ, µs, φs)S

+ ω0

∫dµ′dφ′P (µ, φ, µ′, φ′)L(τ, µ′, φ′)

(25)

7

This integro differential equation depends on the viewing geometry µ and φ and an inte-gration over the spacial dimension (µ, φ).

Scattering of radiation is described by multiplying the radiation with the phase matrix ofsingle scattering for the given scatterer. To decouple the azimuth dependence of Eq. (25) intoa series of independent equations we expand the phase matrices into a Fourier series. Usinggeneral symmetry relationships[16] we expand the diagonal 2× 2 block sub matrices in a cosseries and the off diagonal block sub matrices in sin series:

P (µ, µ′, φ− φ′):=P (µ, φ, µ′, φ′)

:≈n∑i=0

Pm(µ, µ′) ∗(

Cos (m(φ− φ′)12×2) Sin (m(φ− φ′)12×2)Sin (m(φ− φ′)12×2) Cos (m(φ− φ′)12×2)

), (26)

Pm(µ, µ′):=1

π (1 + δ01)

∫dφP(µ, µ′, φ) ∗

(Cos(mφ)12×2 Sin(mφ)12×2

Sin(mφ)12×2 Cos(mφ)12×2

). (27)

Secondly by assuming that the light field is symmetric with respect to the principal plane(the plane that the nadir and solar direction span) we expand the light field in a sin and cosseries reflecting the internal symmetry of the system:

L(τ, µ, φ) ≈n∑

m=0

Lm(τ, µ) ∗(

Cos(mφ) 12

Sin(mφ) 12

). (28)

When inserting the two Fourier series from Eq. (26) and (28) into the RTE (25) theequation decouples into a series of equations in Fourier space that are now independent ofthe viewing azimuth angle:

(µd

dτ− 1)Lm(τ, µ) =ω0Pm (µ, µs) e−τ/µs

+ ω0π (1 + δ0m)∫dµ′Pm(µ, µ′)Lm(τ, µ′).

(29)

Equation (29) describes the light field in Fourier space. To derive the real field Eq. (28)must be used. Since in this numerical treatment we can only consider a limited number ofFourier terms this will limit the accuracy in the azimuth dependence of the RT calculations.With any number of Fourier terms the light field at any azimuth angle can be calculated butthe final accuracy of the result depends only on the number of terms used.

In the next steep we need do discretize Eq. (29) for suitable treatment on a computersystem. As in the interaction principle the light field is split into a upper and lower part:

L+m(τ, µ):=Lm(τ, µ > 0), (30)

L−m(τ, µ):=Lm(τ, µ < 0). (31)

Integrations are replaced by summing over the integrand at Gaussian quadrature pointsand multiplying with the Gauss Lobatto weights:∫

dµf(µ) ≈m∑i=1

f (µi) ci. (32)

8

By defining matrices that contain the Gaussian points and weights we can write Eq. (29)in form of a matrix equation:

c = diag (c1, . . . , cm) , (33)M = diag (µ1, . . . , µm) . (34)

By defining the matrices Γ++/−−/+−/−+m :

Γ++m = M−1

(1− ω0π (1 + δ0m)P++

l c), (35)

Γ+−m = M−1ω0π (1 + δ0m)P++

l c, (36)

Γ−+m = M−1ω0π (1 + δ0m)P++

l c, (37)

Γ−−m = M−1 (1− ω0π (1 + δ0m)P –l c) . (38)

The result can also be written form of a matrix equation:

d

dτ

(L+

L−

)=(−Γ++

m Γ+−m

−Γ−+m Γ–

m

)(L+m

L+m

)+(

Σ+m

−Σ−m

). (39)

By comparing Eq. (39) with the vector form of the interaction principle from Eq. (22)the transmission and reflection operators can be derived:

t10 = 1− Γ++m dτ , (40)

t01 = 1− Γ–mdτ , (41)

r01 = Γ+−m dτ , (42)

r10 = Γ−+m dτ . (43)

2.3 Intermediate Recapitulation

In the last two sections we demonstrated the two main arguments of the matrix operatormethod. First we stated the interaction principle that relates the upward and downwarddirected radiation fields at arbitrary spacial points in the AOS with at this point unknownreflection and transmission operators. We constructed the adding method that allows tocalculate the reflection and transmission properties of combined layers. Using the addingmethod with two identical layers is called the doubling method. We demonstrated thatany homogeneous layer can be constructed using the doubling iteratively starting from aneventually optical thin elementary layer. In the last step we reformulated and discretizethe radiative transfer equation and have shown a formal equality between the RTE andthe interaction principle for infinitesimal thin layers. Using this comparison we arrived atequations that relate the physical single scattering properties from the RTE to the beforeunknown reflection an transmission operators. Hence we can use all these building blocks tocalculate the radiance fields at layer boundaries.

9

2.4 Atmosphere description

The modelled features of the atmosphere are scattering and absorption of light by the definedconstituents. Scattering by molecules is described using the Rayleigh scattering functionincluding a depolarization factor δ (in this form from[17] but originally from[18] and[19]):

Pδ(θ) = ∆34

1 + cos(θ)2 − sin(θ)2 0 0− sin(θ)2 1 + cos(θ)2 0 0

0 0 2 cos(θ) 00 0 0 ∆′2 cos(θ)

− (1−∆)Diag(1, 0, 0, 0).

(44)The depolarization factor δ = 0.0279 which gives a King factor (6+3δ)/(6−7δ) = 1.0480 is

taken from the literature[20]. The Rayleigh optical thickness is calculated using the formulasgiven in reference[21].

Any other scatterer can be given in form of tabulated phase matrices which are thenexpanded in a Fourier series. These functions, including the given Rayleigh phase matrixfrom Eq. (44), are defined with respect to the plane of scattering which is shown in Fig. (6).

eÓx

eÓy

eÓz

Μ

Μr

Χr

Χi

Θ

Figure 6: Scattering angle θ and rotating angles χi and χr. The scattering plane is shown ingray.

Before expanding them in a Fourier series the scattering matrix must be rotated accordingto the angles χi and χr (see Fig. (6)). The scattering matrix that relates µ and µr is thengiven by[13]:

10

S(µ, µr, φ) = R(π − χr)S(α)R(χi), (45)

where L is a rotation matrix:

R(χ) =

1 0 0 00 cos(2χ) sin(2χ) 00 − sin(2χ) cos(2χ) 00 0 0 1

. (46)

Deriving the angles χi and χr involves some vector algebra but should be redone sincethe actual signs depend on the definitions of the actually used unit vectors. We define unitvectors like:

~e(µ, φ) = (√

1− µ2 cos(φ),√

1− µ2 sin(φ), µ)T , (47)

where µ is positive. To derive χi one must calculate the two unit vectors ~eiz,~eir that lie inthe ~µ− ~ez and ~µ− ~µr plane and are orthogonal to µ itself. Since ~eiz is located in the ~µ− ~ezplane it can be constructed as a linear combination of ~ez and ~ei with two free constants αiz1and αiz2 , hence ~eiz = αiz1 ~ei + αiz2 ~ez. With the constraints ~eiz~µi = 0 and |~eiz| = 1 we derive(and analog for eir):

~eiz =(−µr cos(φr),−µr sin(φr),

√1− µr2

)T, (48)

~eir =

(α√

1− µi2 cos(φi) +√

1− µr2 cos(φr)√1− α2

,α√

1− µi2 sin(φi) +√

1− µr2 sin(φr)√1− α2

,µr − αµi√

1− α2

)T,

(49)

α = µiµr −√

1− µi2√

1− µr2 cos(φi − φr). (50)

Taking the the scalar product the cosine of the angle χi becomes:

cos(χi) =αµi − µr√

1− α2√

1− µi2. (51)

These angles and definitions also apply for the scattering and reflection matrices we willderive for the atmosphere ocean interface in section 2.5.

2.5 Wind ruffled ocean surface

Main results of this section can be found in papers from Nakajima and Tanaka[22],Kattawarand Adams[13] and Sancer[23]. The influence of the water surface is governed by the reflectionand transmission due to an abrupt change in the refractive index of the media, the distributionof surface facet normals due to the surface roughening caused by the blowing wind over thesurface and the shadowing of the surfaces facts caused by each other.

We model the ocean surface by assuming statistically distributed normals of wave facetsthat reflect and transmit light according to the Fresnel matrices and are refracted accordingto Snell’s law. One should keep in mind that this is a rather abstract representation of thesurface. In the model world the surface is not some smooth timely varying interface likein the upper part of Fig. (7). Instead it consists of a point (since the model is vertically

11

Figure 7: Abstract wind ruffled surface in a horizontal homogeneous world.

homogeneous) at which all possible wave facets exist at the same time and transmit andreflect incident light.

Diffraction can be described by Snell’s law in angle form:

sin θisin θt

=ntni

:= n, (52)

and in terms of cosines:

µ′ =

√1 +

µ2 − 1n2

. (53)

The reflection function of the surface can then be expressed as the product of the shad-owing function G, the probability of existence for a given wave facet, the Jacobean and therotated Fresnel matrix[22]:

R(µ, µ′, φ− φ′) =1µµn

G(νµ, νµ′)p(µrn)r(cosωr, n). (54)

Where µn is the normal of a wave facet that reflects light from ~µi to ~µr. For every pair of~µi and ~µr there exists only one solution for µn:

µrn =µ+ µ′

2 cosωr, (55)

cosωrn :=

√1 + α

2, (56)

α := µµ′ +√

1− µ2√

1− µ′2 cos(φ− φ′). (57)

We derive the wave facet normal direction for incident and reflected radiation. It must bethe normalized median line of the parallelogram spanned by the directions ~µ and ~µ′ (see Fig.8):

~n =~µ+ ~µ′

|~µ+ ~µ′|. (58)

12

Μn

Μ'

Μ

Ωnr

eÓx

eÓy

eÓz

Μ

Μr

n

Μt

Ωnr

Ω

Ωnr

Ωnt

Figure 8: Angle definition of reflection and transmission.

The occurrence of ocean wave facet normals can be described using a Gaussian probabilitydensity function whichs σ depends on the wind speed. These dependency’s have been derivedby analyzing the sun glitter in aerial photographs (or satellite images by independent authors)with the use of known wind speed and direction[24, 25]:

p(µn) =1

πσ2µ3n

e− 1−µ2

nσ2µ2

n . (59)

The width σ depends on the wind speed in 10m height:

σ2 = 0.003D0 + 0.00512 ∗ w. (60)

The νµ and νµ′ are normalized wave slopes and are used for the correction due to multiscattering and shadowing[23, 22]:

13

G(νµ, νµ′) = (1 + F (νµ) + F (νµ′)), (61)

νµ =µ

σ√

1− µ2, (62)

F (ν) =12

(e−ν

2

√πν− 2√

π

∫ ∞ν

dt e−t2

). (63)

The real 4× 4 matrix r(µ, n) is the Fresnel reflection matrix (e.g. see[13]):

r(µ;µ2 > 1− n2, n) =12

r2

1 + r22 r2

1 − r22 0 0

r21 − r2

2 r21 + r2

2 0 00 0 2r1r2 00 0 0 2r1r2

. (64)

With:

r1 =n2µ−

√n2 + µ2 − 1

n2µ+√n2 + µ2 − 1

, (65)

r2 =µ−

√n2 + µ2 − 1

µ+√n2 + µ2 − 1

. (66)

When n2 + µ2 − 1 < 0 the value becomes unphysical and one has to use:

r(µ;µ2 < 1− n2, n) =

1 0 0 00 1 0 00 0 Re(γ) −Im(γ)0 0 Im(γ) Re(γ)

. (67)

The transmission is given as:

T (µ, µ′, φ− φ′) =n cos2 ωtt

µµn(cosωt − n cosωtt)2G(νµ, νµ′)p(µtn)t(cosωtt)

H((n− 1)(mµ− µ′))H(α− 1/n).(68)

With:

µtn =µ− nµ′√

n2 − 2nα+ 1, (69)

cosωt :=1− nα√

n2 − 2nα+ 1, (70)

cosωtt :=|n− α|√

n2 − 2nα+ 1. (71)

The angles may be calculated using the vector form of Snell’s law:

14

nµ~µ× ~n = nµ′~n× ~µ′, (72)

and the ray vectors in spherical coordinate form: ~µ =(√

1− µ2 cosφ,√

1− µ2 sinφ, µ)

while solving the n3 to calculate to cosine of the wave facet normal. When applying thenormalizing condition

√n2 = 1 all the normal vector components can calculated:

~n =

1−nα−µ2+nµµt√1+n2−2nα

√1−µ2

n√−1+α2+µ2−2αµµt+µt2

(1+n2−2nα)(−1+µ2)µ−nµt√

1+n2−2nα

(73)

The angle between the incident ray and the wave facet normal cosωt can than be expressedas cosωt = ~µ~n (since they are normal vectors). The angle between the wave facet normaland the transmitted and refracted ray can be calculated using Snell’s law from Eq. (52):

cosωtt =√

1 + cos(ωt)2−1n2 .

With t as the Fresnel transmission matrix:

t(µ, n) =12

t21 + t22 t21 − t22 0 0t21 − t22 t21 + t22 0 0

0 0 2t1t2 00 0 0 2t1t2

. (74)

With the definitions:

t1 =2nµ

n2µ+√n2 + µ2 − 1

, (75)

t2 =2µ

µ+√n2 + µ2 − 1

. (76)

(77)

For the reflection and transmission coefficient the following normalization constraintsholds:

12

(r21 + r2

2) +12

(t21 + t22)

√n2 + µ2 − 1

µ= 1, (78)

where the correction term for the transmission describes the altered solid angle of thetransmitted radiation.

The interface reflects and transmits light and also change its spacial distribution butmust conserve the intensity of the light field. A numeric model may suffer from energydeficiency due to resolution and numerical effects and also approximation and simplificationsmade by the model. Here the multi scattering and shadowing is handled by the externalshadowing function[23] and its derivation is based on ray tracing like assumptions (Kirchhoff’sapproximation) but not a solution to the Maxwell equations. We define a deficiency ε:

ε(µ) = 1− 1µ

∫ 1

0dµ′∫ 2π

0dφ(R(µ, µ′, φ) + T (µ, µ′, φ)

). (79)

15

The deficiency term ε(µ) for all incidence directions can display such errors. A distinctionbetween numeric effects and modeling effects can be found using a sensitivity analysis withrespect to the numeric resolution.

In Fig. (9) we show the hemispheric reflection and transmission for wind speeds of 5,10and 15m/s for relative refractive indexes 1.33 and 1.33−1. The values have been calculatedwith our independent implementation in Mathematica. The refractive index greater oneshows the values for light incident from the atmosphere and the one smaller than one forincident light from the ocean. The window of total internal reflection (Snells cone) can beclearly seen in part (b) of Fig. (9). The deficiency in this implementation is generally largerfor larger wind speeds. This may be due to the stronger multi scattering and shadowingeffects for the larger deviations from the flat surface which are only approximately describedby the shadowing function G (see Eq. (61))

In Fig. (10) we show dependency of the hemispheric reflectivity with refractive index(incident from atmosphere) and angle of incidence for the two wind speeds 5 and 15m/s. TheAngle of incidence is measured in cosines so larger angles are expressed by smaller numbers.For both wind speed cases it can be seen that the hemispheric reflectivity increases withconstant angle of incidence but increasing with relative refractive index. This effect increaseswith increasing wind speed.

The total hemispheric reflectivity Rn,v = 1/µ′∫dµµR(µ, µ′, n, v) can be defined by a angle

weighted integral over the hemispheric reflectivity and only depends on the refractive indexand wind speed. In Fig. (11) we show these values for a wind speed of 5m/s. At constantwind speed the total hemispheric reflectance is increasing with increasing relative refractiveindex.

16

0 20 40 60 80

0.02

0.05

0.10

0.20

0.50

1.00

angle of incidence in °

v=5v=10

v=15

v=5v=10

v=15

v=5

v=10

v=15

reflection

transmission

Ε=1-r-t

n=1.34

(a) Radiation incident from atmosphere.

0 20 40 60 80

0.02

0.05

0.10

0.20

0.50

1.00

angle of incidence in °

v=5v=10

v=15

v=5

v=10v=15

v=5

v=10

v=15

reflection

transmission

Ε=1-r-t

n=1.34-1

(b) Radiation incident from ocean.

Figure 9: Hemispheric reflectivity and transmissivity for wind speeds 5,10,15 ms for a relative

refractive index of 1.33 (from atmosphere)and 1.33−1 (from ocean). The deficiency ε is showngray dashed lines for all three wind speeds.

17

(a) V=5m/s (b) V=15m/s

Figure 10: Dependency of the hemispheric reflectance with relative refractive index and angleof incidence for the two wind speed 5 and 15ms .

18

(a) V=5m/s

Figure 11: Total hemispheric reflectivity with dependence on wind sped and relative refractiveindex.

19

2.6 Ocean description

Scattering of radiation in the ocean can be described using a Rayleigh like phase matrix(see Eq. (44)) with a depolarization factor of δ = 0.039. Since we model the ocean as anhomogeneous water bulk the scattering doesn’t involve the water molecules (as the gaseousmolecules in the atmosphere) but rather fluctuations in the density and concentration of thewater body and salt ions due to a nonzero temperature and hence Brownian motion in theocean bulk[26].

Some temperature and salinity measurement at 5m below the sea surface from aroundthe globe have been compiled in Fig. 12. Global salinity values seem almost constant around35PSU apart from the most northern measurements east of Iceland. The water temperatureis varying from almost 0C (and below) to 30C.

-60 -40 -20 20 40 60 80lon

5

10

15

20

25

T in °C

-60 -40 -20lat

5

10

15

20

25

T in °C

-60 -40 -20 20 40 60 80lon

25

30

35

S in PSU

-60 -40 -20lat

25

30

35

S in PSU

Atlantic OceanNorwegian SeaPacific Ocean

(a) Temperature and salinity measurements on POLARSTERN cruises in 2010[27, 28, 29].Measurements have been taken in 5m water depth.

(b) Cruise track inthe Atlantic Ocean

(c) Cruise track in the Pacific Ocean (d) Cruise track in theNorwegian Sea

Figure 12: Temperature and salinity measurements from 2010 in 5m water depth from theGerman research ship POLARSTERN. Map images taken from maps.google.com.

In the actual implementation we also keep the legacy form for the bulk scattering coeffi-

20

cient since it may be used for comparison studies. In his paper[30] Morel describes a modelwith exponential spectral dependency with two constants that have been derived from mea-surements in open oceans. The dependency of scattering with temperature and salinity is notincluded in this model:

βmoreli = zi+1,i ∗ 0.00288 ∗(

λ

500nm

)−4.32

; zi+1,i := zi+1 − zi. (80)

The absorption coefficients of pure sea water is taken from the WATERRADIANCEATBD[31] with data from[32, 33, 34, 35, 36, 37, 38]. The dependency with respect to wave-length, salinity and temperature is included in this model:

aw(T, S, λ) := aw(T0, S0, λ) + (T − T0)ΨT (λ) + (S − S0)ΨS(λ). (81)

The model consists of measurements taken at constant temperature and salinity and a listfor the spectral varying temperature and salinity coefficients. This model therefore assumesthese effects to be linear (in fact this is a multidimensional Taylor expansion up to the firstorder).

The refractive index of air relative to sea water[31] can be calculated using the followingmodel (with values of the ni from the ATBD):

nairsw (T, S, λ ∈ [300nm, 800nm]) =

n0 +(n1 + n2T + n3T

2)S + n4T

2 +n5 + n6S + n7T

λ+n8

λ2+n9

λ3.

(82)

The refractive index of pure air (relative to vacuum)[31] can be calculated using: (withvalues of the ki from the ATBD):

nvacair (λ) = 1 +108k1

k0 − λ−2+

108k3

k2 − λ−2. (83)

The relative refractive of air and sea water above 800nm is modeled by:

nairsw (T, S, λ ∈ [800nm, 4000nm]) :=nvacsw (λ)nvacair (λ)

+[nsw(T, S, 800nm)− nvacsw (800nm)

nvacair (800nm)

]. (84)

Where nvacsw (λ) is the refractive index measured at T = 27C and S = 0PSU from theWATERRADIANCE ATBD.

The Rayleigh phase matrix with polarization (in this form from[17] but originally from[18]and[19]) is used to describe the Einstein Smoluchowski scattering:

Pδ(θ) = ∆34

1 + cos(θ)2 − sin(θ)2 0 0− sin(θ)2 1 + cos(θ)2 0 0

0 0 2 cos(θ) 00 0 0 ∆′2 cos(θ)

− (1−∆)Diag(1, 0, 0, 0)

(85)With ∆ = 1−δ

1+δ/2 and ∆′ = 1−2δ1−δ/2 and δ as the depolarization factor which is varying with

wavelength and atmospheric compositions see[21], for air we use 0.035.

21

Relative differences of the volume scattering coefficient with dependency on temperatureand salinity to the model proposed by Morel (in 1977) are shown in Fig. (13). The modelwith salinity and temperature dependence is shown as a set of curves by varying the twoparameters. At approximately 600nm the models agree with each other. When going to tothe UV the differences rise up to 10%.

Figure 13: Volume scattering coefficient with log scale according to the Morel model (in black)and varying with temperature and salinity (in gray). The dashed line represents (right scale)relative differences between the model for a temperature of 20 and salinity of 20PSU.

The volume scattering model is build from the two independent pairs of scattering; onedue to density and one due to temperature fluctuations and have been discussed in detail byZhang[26]:

βsw(λ, T, S) = βdf (λ, T, S) + βcf (λ, T, S). (86)

The part due to density fluctuations is given by:

βdf (λ, T, S) =π2

2λ4(ρ∂ρn2)2

TkTkβT (T, S)f(δ). (87)

With the following definitions:

(ρ∂ρn2)T = (n2 − 1)

(1 +

23

(n2 + 2)(n2 − 1

3n

)2), (88)

n(λ, T, S) = d0 + (d1 + d2T + d3T2)S + d4T

2 +d5 + d6S + d7T

λ+d8

λ+d9

λ, (89)

βT (T, S) = (B0 +B1T +B2T2 +B3T

3 +B4T4)

+ (C0 + C1T + C2T2 + C3T

3 + C4T4)S + (C4 + C5T + C6T

2)S1.5,(90)

f(δ) =6 + 6δ6− 7δ

. (91)

22

The part due to concentration fluctuations of salt ions is given by:

βcf (λ, T, S) =π2M0S

2λNAρ(S, T )(∂Sn2)2(−∂S ln a0)−1f(δ). (92)

With the following definitions:

ρ(S, T ) = (b0 + b1T + b3T2 + b4T

4 + b5T5) + (c0 + c1T + c2T

2 + c3T3 + c4T

4)S

+ (c5 + c6T + c7T2)S1.5 + c8S

2,(93)

∂S ln a0 = (a0 + a1T + a2T2 + a3

3) + 1.5(a4 + a5T + a6T2 + a7T

3)√S

+ 2(a8 + a9T + a210)S.

(94)

The factor (∂Sn2) may be derived from equation (89). Equation (89) is valid from 300nmto 800nm and its temperature and salinity dependency at 800nm is used to model the de-pendency at all wavelengths. The derivatives may be derived from this equation alone.

The bulk scattering coefficient for the radiative transfer model is then given by:

βsw(λ, S, T, θ) = βsw(λ, S, T )Pδ(θ). (95)

Since the bulk scattering coefficient is build from two independent parts we show therelative behavior according to them in Fig. (14).

400 600 800 1000 1200 1400

8.5

9.0

9.5

10.0

10.5

11.0

T=1°C

T=6°C

T=11°C

T=16°C

T=21°C

T=26°CT=31°C

Λ in nm

Βdf

HΛ,T

,S=

10PS

UL

Βcf

HΛ,T

,S=

10PS

UL

relative bulk scattering coefficients

(a) Constant salinity at 10 PSU.

400 600 800 1000 1200 1400

4

6

8

10

12

S=10PSU

S=15PSU

S=20PSU

S=25PSU

S=30PSUS=35PSU

Λ in nm

Βdf

HΛ,T

=15

°C,S

LΒ

cfHΛ

,T=

15°C

,SL

relative bulk scattering coefficients at constant

(b) Constant temperature at 15C.

Figure 14: Relative scattering coefficient of sea water with variations from the two independentparameters salinity and temperature.

The variation of the relative refractive index with respect to wavelength, salinity andtemperature is shown in Fig. (15). On the temperature and salinity scales relevant to theearth oceans the changes with salinity seem more pronounced then those with temperature.In contrast the salinity is probably regional more stable than temperature.

The spectral variation of the relative refractive index and pure water absorption are shownin Fig. (16) for a salinity of 35PSU and temperature of 15.

To put these variations into the radiative transfer perspective we show in Fig. (17) therelative variations in hemispheric reflection and transmission at 600nm for a change in salin-ity from 5PSU to 35PSU at a constant temperature of 15C. The relative change in thetransmission is slightly rising when going to the horizon while the relative change in thehemispheric reflectance is largest for nadir direction. Deviation are as big as 2.5%.

23

400 600 800 1000 1200 1400

1.325

1.330

1.335

1.340

1.345

1.350

1.355

S=5PSUS=15PSUS=25PSUS=35PSU

Λ in nm

nHΛ

,T,S

L

refractive index

T=10°C

T=20°C

T=15°C, varying SS=20 PSU, varying T

Figure 15: Relative refractive index with respect to wavelength, salinity and temperature.Variations due to salinity changes at constant temperature are shown in black, those withtemperature at constant salinity are shown in gray dashed lines.

Figure 16: Relative refractive index and absorption coefficient at S = 35PSU and T = 15C.

24

20 40 60 800.97

0.98

0.99

1.00

1.01

Θ

r 1r

2an

dt 1

t2

relative changes in reflection and transmission

Figure 17: Relative change of hemispheric reflection and transmission using the model dataat 600nm, salinity of 5PSU and 35PSU and a temperature of 15C.

25

2.7 Intermediate conclusion

To compute the light field any level of the treated atmosphere one has to compute the reflectionand transmission matrices for every layer which represents the main computational burdenof the method. Once these are computed it is very fast, almost negligible when comparedto the doubling and adding, to compute the light field at any internal layer for all azimuthdirections. These matrices can be easily stored and reused when only a smaller part in theAOS changes within some model runs, which can significantly improve the computationalburden of a study. For example when only the light field at the lower boundary changes (eg.for the fluorescence of plants) the properties of the atmosphere have to be computed onlyonce.

The second main advantage is that the method is very well suited for optical thick media- only the far field scattering assumption must hold since the RTE wouldn’t be applicablein this case. The optical thickness reached with linear doubling steps grows exponentially sothat the computational burden is very well under control.

2.8 Clear water Raman scattering

Raman scattering of the solar light-field due to energy absorption by vibrational modes ofwater molecules may contribute significantly to the signals observed by ocean remote sensingsatellites. The inelastic fraction of the water-leaving radiance for clear water can reach valuesof several percent [39], depending on the observed wavelength and the incident angle. Further-more, inelastic scattering due to chlorophyll and yellow substance fluorescence adds to thisfraction. For these reasons the inclusion of inelastic scattering sources into radiative-transfermodels used in ocean remote sensing applications can be important.

The following chapters describe the inclusion of a module for clear water Raman scatteringinto the scalar version of MOMO. The calculation of the light field including Raman scatteringat the model layer boundaries requires prior model runs with a high spectral resolution todefine the depth dependent excitation radiances to calculate the corresponding Raman sourceoperators. A final model run for every observation wavelength then yields radiances at themodel boundary including the first order Raman scattering contribution. Higher orders ofRaman scattering can be incorporated by iteratively repeating the last step with a highspectral resolution with the resulting radiances of the last step as input. Due to computingtime issues and the only marginal contribution of higher orders (see [40]), this study is limitedto the first order contribution of Raman scattering.

2.9 Formalism for the inclusion of Raman scattering in the RTE

The general formalism for the incorporation of Raman scattering into a radiative-transfermodel will be introduced in this Section.

All optical processes affecting the redistribution of radiation by the water body are math-ematically described by the radiative-transfer Equation(23). In addition to the terms for thespatial redistribution of light by absorption and elastic scattering, a source term JR for theangular and spectral redistribution of daylight by Raman scattering needs to be introduced:

JR(µ, ϕ, λ) =1

4π

∫Λ

∫4πβR(µ′, µ, φ′, φ, λ′, λ)L′(µ′, φ′, λ′)dΩ′dλ′. (96)

26

Here, the apostrophe denotes variables of the exciting light field (So, for instance, λ′ is awavelength of the incoming radiation that has not yet been inelastically scattered, whereasλ denotes an observed wavelength of the resulting light field after the wavelength shift dueto inelastic scattering.). The Raman volume scattering function βR can be written as theproduct of the Raman phase function PR describing the angular redistribution and the Ramanscattering coefficient βRsca, which describes the re-emission of radiation after the wavelength-shift:

βR(µ′, µ, φ′, φ, λ′, λ) = βRsca(λ′, λ)PR(µ′, µ, φ′, φ). (97)

The water Raman phase function PR for scattering by an angle θ can be expressed in theform [41]:

PR(µ′, µ, φ′, φ) =3

16π1 + 3ρ1 + 2ρ

(1 +

1− ρ1 + 3ρ

cos2 θ

), (98)

where ρ is the depolarization factor for water Raman scattering (see Section 2.10 forvalues). The scattering angle θ can be calculated from the zenith and azimuth angles withthe following relation:

cos θ = µµ′ +√

1− µ2√

1− µ′2 cos (φ− φ′). (99)

The spectraly shifted re-emission of light may be described as a product of the water Ra-man absorption coefficient and a spectral redistribution function for water Raman scattering:

βRsca(λ′, λ) = βRabs(λ

′)fR(λ′, λ). (100)

The wavelength dependency of Raman absorption in water follows a power law:

βRabs(λ′) = aR0 (λ′0)

(λ′0λ′

)n. (101)

The values used for the parameter aR0 are discussed in Section 2.10.The spectral redistribution function fR can be expressed as the superposition of four

Gaussian curves for the different vibrational modes [42]:

fR(λ′, λ) =107

λ′21

(2π)1/2∑4

i=1 αiσi

4∑i=1

exp[

(107(λ′−1 − λ−1)− δνi)2

2σ2i

]. (102)

In this form of fR the input for the distribution parameters have to be in cm−1 whereasthe output is in nm−1. A Table containing the distribution parameters αi, δνi and σi can befound in Table 1. Figure 18 shows the redistribution function for a fixed excitation wavelengthof λ′ = 500nm (a) and the contributing excitation wavelengths λ′ for different observationwavelengths λ (b).

2.10 Optical input parameters for Raman scattering

This Section enlists the sources of the optical parameters and their values needed to performthe formalism summarized in Section 2.9.

27

0 50 100 150θ in °

0.06

0.07

0.08

0.09

0.10

0.11

PR

(a) Raman phase function.

400 600 800 1000λ’ in nm

0.1

1.0

10.0

β absR (λ

’) in

m−

1

(b) Raman absorption coefficient.

For the calculation of the depth dependent excitation energy within the water body, thewater model introduced in Section 2.6, depending on salinity and temperature, has beenintroduced to the scalar version of MOMO. The wavelength dependence of the refractiveindex, however, can not be taken into account for the calculation of the zenith angle changedue to refraction and the atmosphere ocean interface, since constant angles throughout thespectrum are necessary to perform the spectral integration in Equation 96. Therefore, aconstant value of 1.34 is assumed for the determination of the zenith angles in water. Inthe scalar version of MOMO, transmission through, and reflection by the wind ruffled oceansurface, is calculated by the method described in [14].

The spectral dependence of the Raman depolarization factor ρ in the Raman phase func-tion Equation (98) is neglected, since a constant average value of 0.17 is an acceptable ap-proximation for the contributing spectral range [43]. This leads to a phase function quitesimilar to the one describing elastic Rayleigh scattering in pure sea water.

580 590 600 610 620λ in nm

0.00

0.02

0.04

0.06

0.08

0.10

fR(λ

’=50

0nm

) in

nm

−1

(c)

300 400 500 600 700λ’ in nm

0.00

0.05

0.10

0.15

0.20

fR in

nm

−1

λ=400nm

λ=500nm

λ=600nm

λ=700nmλ=800nm

(d)

Figure 18: (a): fR(λ) (black line) and it’s components (dotted lines) for an excitationalwavelength of 500nm. (b): fR(λ′) for different output wavelengths.

Values for aR0 in the Equation 101 for the Raman absorption coefficient in literatureshow big variations in older publications. Measurements in the past two decades, however,

28

i αi δνi(cm−1) σi(cm−1)1 0.41 3250 89.1792 0.39 3425 74.3173 0.1 3530 59.4534 0.1 3625 59.453

Table 1: Parameters of the spectral redistribution function.

show a much better accordance[39, 43]. For this study the values of Bartlett et al. [44]are used (aR0 (λ′0 = 488nm) = (2.7 ± 0.2)10−4m−1 and n = 5.5). The resulting Ramanscattering coefficient is roughly one tenth of the scattering coefficient for pure water, thereforeapproximately every tenth photon scattered by water molecules sees a wavelength shift.

For the parameters in the spectral redistribution function fR (Equation 102), publishedvalues in Walrafen [45, 42] and Haltrin et al. [46] where used. They can be found in Table 1.

400 600 800 1000λ’ in nm

0.5

1.0

1.5

2.0

L’s

in W

m−

2 nm

−1 s

r−1

Figure 19: Extraterrestrial solar radiance spectrum.

Due to the spectral integration over the incoming radiances L′(λ′) in Equation 96, theincoming solar irradiance at the top of the atmosphere can not be set to 1. A spacingof 1nm was chosen for the input radiances computed in the first MOMO run to providean accurate sampling of the spectral redistribution function. For the exoatmospheric solarradiance spectrum L′s(λ

′), data from Neckel and Labs [47] are used (Figure 19).

2.11 Raman source in matrix operator theory

To incorporate the Raman source term JR into the RTE form of a matrix operator model,an accurate and efficient discretization of its components has to be performed. Approachesbased on azimuthally averaged radiances, for instance as described in [48, 39] based on anexpansion in Legendre polynomials, are only valid for the computation of fluxes and nadiror zenith radiances when treating non isotropic Raman scattering. For the implementationof Raman scattering in this study, a separation of azimuth and zenith dependence by anexpansion of the RTE and its components in a Fourier series (see Equations 27 and 26) waschosen. This method enabled a close integration of parts of the new code into the existingMOMO structure, which uses the same approach. Furthermore it simplified the definition ofinterfaces with external IDL programs for the spectral and angular integration, for the samereason. The decision, whether to perform the computations azimuthally resolved, can flexiblybe made for every MOMO run. In the following Section, the index m denotes Fourier terms

29

400 500 600 700λ in nm

0.000

0.005

0.010

0.015

J− 0 (µ

=1)

in W

m−

2 m−

1 nm

−1 s

r−1

(a)

0.002 0.003 0.004 0.005 0.006 0.007J+

0 (λ=560nm) in W m−2 m−1 nm−1 sr−1

−100

−80

−60

−40

−20

0

alt i

n m

(b)

Figure 20: (a): Spectral dependency (OLCI channels) of the zeroth order fourier componentof the upward Raman source term of the uppermost water layer in zenith direction. (b):Depth dependency of zeroth order fourier component of the downward (nadir) Raman sourceterm at 560nm for water at a temperature of 288 K and no salinity.

of a Fourier expanded function. So for instance PRm(µ′, µ) is a term of the Fourier expandedphase function. All operations have to be carried out for every necessary Fourier term (threein case of PR) separately. The discretized zenith angles µ′j are calculated following the Gauss-Lobatto scheme. It contains the abscissa value µ = 1 among others and provides the weightscj necessary for the quadrature of all zenith angle dependent discretized functions.

The output of the initial MOMO runs consists of files for each excitation wavelength λ′

containing radiances Lm(µ′j) in Fourier space at the model layer boundaries. To determinethe Raman source term, the spectral integration is carried out in a first step1:

L′m(µ′j , λ)cj =∫

ΛLm(µ′j , λ

′)βRsca(λ′, λ)dλ′. (103)

In the second step, the source term in Fourier space is calculated by replacing the inte-gration in Equation 96 with a sum over the discretized zenith angles µ′j :

JRm(µi, λ) =1

4π

∑j

Pm(µ′j , µi)L′m(µ′j , λ)cj . (104)

The matrix form of this result, with each element of the matrices standing for one obser-vation zenith angle µi and one solar zenith angle µs, is then passed back to MOMO as inputfor the final model run. Since the matrix form of the phase function is split into a forwardscattering matrix P+

m(µ′, µ) and a backward scattering matrix P−m(µ′,−µ) in matrix operatortheory, the source operators are also represented by two matrices j+

m(µs, µ) and j−m(µs,−µ)for the downward and upward contributions to the radiance. From the determined source op-erators at the layer boundaries, the contribution of an elementary layer, for which the single

1The dependence of the functions on the solar zenith angle µs is ignored in this simplified Equation formto give a better overview. In the matrix form of the Equations used in the code, the two matrix indexesstand for the observation- and solar-zenith angles. Furthermore, the contribution of diffuse radiance and thecontribution of direct solar radiance to the source term have to be treated separately due to technical reasonsat this point, but can be summed up after the two integration steps.

30

scattering assumption is appropriate, to the radiances can be calculated. For the source oper-ator of a homogeneous model layer the doubling procedure introduced in Section 2.1 has to beapplied to the elementary matrix operators. Under the assumption that the source operatorsof two adjacent layers within a homogeneous layer are equal, the Equations to determine thesource operators j−20 and j+

02 of a layer after a doubling step have the following form2:

j−20 = j− + t(1− rr)−1(rj+ + j−), (105)

j+02 = j+ + t(1− rr)−1(rj− + j+). (106)

This step is repeated, until the source operator of every homogeneous model layer hasbeen determined, which are then combined by the adding procedure. Finally, the radiancesat every layer boundary, including the first order Raman contribution, can be calculated usingEquations 22.

3 Implementation and Model Description

The radiative transfer model MOMO consists of a number of preprocessors and the radiativetransfer scheme itself. The absorption from gaseous lines is calculated using the correlatedk-distribution approach and the HITRAN 2008 database[49, 50] and has been implementedin IDL[51]. The Fourier expansion of the phase matrices, the atmosphere ocean interface andthe generation of vertical profile files is performed by FORTRAN 77/90 programs like MOMOitself. The general work flow of a MOMO run is shown in Fig. (21). The communication ofthe independent programs and also the MOMO output is realized using human readable (andtherefore writable) text files.

Preprocessing Units

VTP SCA PHA ABS

running MOMO preprocessors before running radiative transfer

MOMO – RT model

post processing final result of RT

elementary matrices

doubling and adding

Input and Output

any number of model atmospheres, varying surface albedo, optical thickness,..

INT

Input

Figure 21: MOMO processing and preprocessing scheme.

The radiative transfer model itself is designed for a single core CPU and does not usedifferent cores in a parallel way. We found this to be a small issue since a usual task is togenerate results for more than one atmosphere; i.e. the generation of a look up table. This

2Here, the indexes 0 and 2 denote the boundary positions of the doubled layer. The indexes 01, 01, 12 and21 of the un-doubled layers are needless, due the equality and symmetry of the transmission and reflectionoperators t and r of two adjacent layers. The assumption of a constant excitation radiance within a modellayer requires to adapt the amount of model layers within the water body to achieve the required accuracy ofthe results. If one is interested in the water leaving radiance, especially the thickness of model layers near thewater surface should be low.

31

can be easily parallelized and distributed by running independent instances of MOMO. Theonly constraint for this procedure using any number of CPU cores in a single system is theavailable main memory whichs usage strongly depends on the zenith resolution (quadratic)and Fourier terms (linear).

In the next section we will discuss some of the used numerical techniques used in MOMO.

3.1 Numerical Techniques

3.1.1 Sparse Matrix Multiplication

The Multiplication m = a × b of two square matrices a and b in plain mathematical no-tation may be expressed as mij =

∑k aikbkj . In pseudo code a matrix multiplication can

implemented as:

Doi(Doj(Dok(mij = mij + aikbkj))). (107)

The three do loops can be freely interchanged and either aik or bkj is a constant withrespect to the innermost loop. In the Matrix Operator theory the multiplication with theunity matrix times some scalar are very common so that off diagonal elements may only buildup in later parts of the computation. If the constant of the inner loop is zero the loop mustnot be executed:

DOi(DOk(DOj(IFaik 6=0THEN(mij = mij + aikbkj)ELSE()))). (108)

This simple technique speeds up MOMO calculation with a factor of five to six.

3.1.2 Phase matrix truncation

Input phase matrices with strongly peaked phase functions may need many Fourier expansionterms to be accurately represented. The number of Fourier terms effects the computationtime approximately linear. In reality it is less then linearly since for most real cases for higherFourier terms more zeros populate the matrices which are faster to multiply (see Sec. 3.1.1).By replacing the strongly peaked part of the phase function with a second order polynomial,the number of needed Fourier terms to reach the same accuracy (in terms of deviations of theredeveloped form) is strongly reduced.

0 50 100 150

0.1

1

10

100

1000

104

Scatering Angle

Phas

eFu

nctio

n

original phase function

truncated one

Figure 22: Phase function and truncated phase function. Truncation is done from 15 on.

32

M =

m1 m5 0 0m5 m2 0 00 0 m3 −m6

0 0 m6 m4

:→Mt = δ

M1/δ m5 0 0m5 m2 0 00 0 m3 −m6

0 0 m6 m4

(109)

This alteration inevitable leads to an error in the radiance calculation. We introduce thefactor δ to rescale the other elements of the phase matrix to keep the degree of polarizationin the modified (forward scattered) region. Defining property is, that the rescalation mustkeep the polarization properties of scattered radiation unaltered for all incident polarizationstates ~S[52]:

∀~S V dp(M~S) = dp(Mt~S). (110)

We examine first the degree of circular polarization for this equation:

m4s3 +m6s2

m1s0 +m5s1=m4s3δ +m6s2δ

M1s0 +m5s1δ. (111)

It is easy to see that δ = M1/m1 will solve the equation. It is also easy to verify that thedegree of polarization is also kept using this factor.

3.1.3 Fourier Series Modification for the Conservation of Radiation

The conservation of intensity for non absorbing single scattering is physical reality which maybe violated by the numeric discretezation of the RTE. Incident radiation from ~S(µ′) directionis scattered to all other directions µ described by the phase matrix: ~Sµ

′(µ) = M(µ′, µ)~S(µ′).

In Fourier space the zeroth matrix M0 describes the mean of the angular distribution. InFourier space we can therefore describe the energy conservation:

Iµ′ =∫dµ~Sµ

′

0 (µ) =∫dµ(M0(µ′, µ)~S(µ′))0. (112)

Considering the unpolarized incident solar radiation the matrix product simplify’s andthe conservation of intensity can be written as:

∀µ′ → 1−∫dµM11(µ′, µ)0 = 0→ εµ′ . (113)

Due to the limited number of zenith angles and the limited number of usable Fourier termsthe left hand side of equation (113) may not vanish and is set to ε. Increasing of zenith anglesand Fourier terms will diminish this residual but may not be possible due to constrains inavailable computation memory or time resources. Hence we modify the phase matrix:

Mm11(µ′, µ′)0 = M11(µ′, µ′)0 −

εµ′

gµ′(114)

As described in section 3.1.2 the phase matrix is then modified to keep the polarizationstate of the scattered radiation unaltered and the rescalation factor δ becomes:

δµ = 1− εµgµM11(µ, µ)

. (115)

33

The effect of these phase function modification is emphasized in Fig. (23). The highrescase is using 180 zenith Gaussian points and the unaltered phase matrix. The original andmodified cases are using only 60 zenith Gaussian points but the original and the modifiedphase matrix. The scattering function is a cloud test case with strongly peaked phase matrixwith strong features due to the narrow size droplet distribution used. The ground is black andthe optical thickness is 5 which represent a case dominated by multi scattering. The results ofthe highres is seen as truth. The modified version agrees very well with the highres results butneeded significantly less computation effort. The usage of the original (unmodified) phasematrix is leading to errors in the intensity and degree of polarization. The case is chosen sothat the effect of the discussed procedure becomes visible; for usual standard cases the effectwouldn’t be as pronounced since real droplet size distributions and Rayleigh scattering tendto dilute the pronounced features of such special cases.

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5.00

10.00

Μ in deg

I

up downOriginal à à

Modified ô ô

Highres

(a) Comparison of upward and downward azimuthal averaged intensity.

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0 20 40 60 800.00

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0.20

Μ in deg

dop

up downOriginal à à

Modified ô ô

Highres

(b) Comparison of upward and downward azimuthal averaged degree of polarization.

Figure 23: Effect of diagonal balancing of residuals to better conserve the total intensity ofthe system. Comparing intensity and degree of polarization for the original and balancedcase using a cloud model with optical thickness 5 and a black underlying surface. Azimuthalaveraged calculations where done using 180 (high res) and 60 zenith angles. For the lowerresolved cases the original and the modified cases where calculated and the high resolutioncase is seen as truth.

3.1.4 Geometric Series of Reflection Matrices

The term (1 − R1R2)−1, with the Ri being reflection matrices, have to be calculated in thedoubling and adding scheme and of the most computational expensive terms in the model

34

since it involves the inversion of a matrix with dimensions zenith×npol×zenith×nnpol. Wherezenith is the number of Gaussian points for the zenith and npol is the number of polarizationparameters used in the program. For scalar calculations nnpol is one and for calculations withpolarization it is 3 (I,Q,U) or 4 (I,Q,U,V). This indicates that calculations with polarizationmay be approximately a factor 32 = 9 or 42 = 16 more expensive than scalar computations.That makes the effective treatment of the calculations very important. The term (1−R1R2)−1

can be interpreted as limit of the geometric series∑∞

k=0(R1R2)k since it describes the multiplereflection in between layer boundaries.

(1− a)−1 = 1 + a+ a2 + a3 + a4 + a5 + ... (116)

Setting a as the matrix product of the reflection matrices Eq. (116) can be used to computean approximate value of the expression. The result is an approximation since the series canonly evaluated up to a finite number of terms. We use a procedure to calculate higher ordersof this series with less computational effort. In Eq. (117) we show that by setting bracketsin the right way and factoring out we can rewrite this expression into a recursive series:

(1− a)−1 = 1 + a︸ ︷︷ ︸=:s0

+ a2 + a3︸ ︷︷ ︸(1+a)a2︸ ︷︷ ︸

s0+s0a2=:s1

+a4 + a5 + ...

=s1 + (1 + a+ ...)︸ ︷︷ ︸s1

a4 + ...

(117)

Defining the recursive series sn+1 := sn + sna2n with s0 := 1 + a the expression may be

approximated by choosing an sufficiently large n for the series sn. We stop this series whenthe trace of the higher order series element is sufficiently small.

3.2 Computation Time Versus Accuracy

The execution time of a model run is determined by the zenith resolution (number of Gaus-sian quadrature points) and the number of used Fourier terms that determine the azimuthalaccuracy of the model calculations. In this section we briefly discuss the effects of reducingthe resolution to the accuracy of the calculations.

As a test case we choose the cloud test case from[53]. This test case involves a nonabsorbing cloud of scatterers of optical thickness of 5 over a black surface. The phase matrixis strongly peaked and can serve as a good and rather difficult to compute test case. In theupper two sub figures of Fig. (24) we show the differences of MOMO computations with theassumed truth results from SCIATRAN[53] for different zenith resolution and Fourier terms.In the lower part of the figure we show similar results but with truncated phase matrices.

The results can be reproduced with high accuracy when using high resolution both inzenith and azimuth (since the result is very strongly peaked). Decreasing the resolution isleading to higher deviations in the intensity and in the Q,U and V components. In the lowerpart of the figure we show similar results but where using modified (truncated) phase matrices.In panel (c) one can see that the model can as expected not reproduce the strong peak in thediffuse forward scattered radiation in 15 around the position of the sun. Away from thatregion the lower resolved results agree with the given truth. In panel (d) we show the results

35

0 20 40 60 80

0.001

0.01

0.1

1

10

Μ in °

mean difference of Sciatran and MOMO for the I components

nft nΜ Cloud30 3040 4050 5060 60300 128

(a) Intensity differences with logarithmic scale fororiginal phase matrices.

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10-4

0.001

0.01

0.1

Μ in °

mean difference of Sciatran and MOMO for Q,U,V components

(b) Q, U and V component differences with loga-rithmic scale for original phase matrices.

0 20 40 60 80

0.001

0.01

0.1

1

Μ in °

mean difference of Sciatran and MOMO for the I components

nft nΜ Cloudmod

30 3040 4050 5060 60

(c) Intensity differences with logarithmic scale formodified (quadratic replacement) phase matrices.

0 20 40 60 802 ´ 10-5

5 ´ 10-5

1 ´ 10-4

2 ´ 10-4

5 ´ 10-4

0.001

0.002

Μ in °

mean difference of Sciatran and MOMO for the I components

nft nΜ Cloudmod

30 3040 4050 5060 60

(d) Q, U and V component differences with loga-rithmic scale for modified (quadratic replacement)phase matrices.

Figure 24: Comparison of Momo results for increasing model (zenith and Fourier components)resolution and the assumed truth.

for Q, U, and V that agree with the truth over the complete zenith range. This shows thatthe rescalation of the truncated phase matrices (see. Sec. (3.1.2)) is performing well for thepolarization properties .

Absolute values for this comparison for the intensity are shown in Fig. (25). The black linerepresents the high resolution computation (the truth) and the lines with circle and squaresresults for lower resolution computations. The higher angular resolved of the two cases agreesbetter with the SCIATRAN results than the lower resolved case. A case with truncated phasematrix with the same resolution as the lowest is shown with stars and represents the truthaway from the forward scattering direction very well. The forward peak is of course notmatched.

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0 20 40 60 80

1

2

5

10

20

50

100

200

Μ in °

I¯

comparison of downward directed radiances for original and truncated phase matrices Hcloud modelL

nft nΜ Cloudmod

300 128 original ¬ truthæ 30 30 originalà 60 60 original* 30 30 modified

Figure 25: Comparison of Momo results for increasing model (zenith and Fourier components)resolution for original phase matrices and a modified (quadratic replacement) case.

36

Concluding we can state that as long as the radiance in forward scattering direction isnot important for the application in question (but the computation time is) the phase matrixtruncation and rescalation can and should be applied to the problem. The polarization in theforward scattering direction is a lot less affected by this procedure. The deviations shown heredo not represent the deviations for real (usual) cases since they have been selected to show theeffects the the truncation and rescalation as clear as possible and do not necessary representthe deviations for real atmospheres. For applications where these effects are unknown theycan be easily tested before comprehensive studies are being computed.

37

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