1

Fast Jacobian Mie Library for TerrestrialHydrometeors

Srikumar Sandeep,Student Member, IEEE,and Albin Gasiewski,Fellow, IEEE

Abstract—This paper presents an approach for the fast,accurate computation of several useful Mie-based parametersfor homogenous, spherical, liquid water and ice hydrometeordistributions over a wide range of frequencies, mean hydrom-eteor diameters, and physical temperatures as occur in theterrestrial atmosphere. The absorption coefficient, scatteringcoefficient, backscattering coefficient, and phase asymmetry pa-rameters are cast into functions of three independent vari-ables: frequency, temperature and mean diameter. An expo-nential drop size distribution with a constant fractional vol-ume of 10−6 is used to model polydispersed hydrometeors.The ranges used for frequency, temperature and mean diame-ter are [1, 1000] GHz, [−50,+50]o C and [0.002, 20]mm respec-tively. The functions are then sampled on a logarithmic grid.Trivariate cubic spline interpolation using non uniform B- splinesis then used to efficiently represent these three dimensional func-tions in a compact library. By using this method, four importantcriteria are achieved: 1) fast random computability of any ofthese parameters given the values of frequency, temperature andmean diameter, 2) minimal memory usage by storage of onlyB-spline coefficients, 3) representation of parameters using wellbehaved functional forms amenable to analytical differentiationfor evaluation of Jacobians or alternatively for higher accuracy,B-spline coefficients calculated using true Jacobian values can beused, and 4) negligibly small and bounded error over the entiredomain of the library. These procedures results in considerableacceleration of microwave radiative transfer simulationsacrossa broad frequency spectrum, as demonstrated in calculations forboth scattering and non-scattering atmospheres. The methodsdiscussed can also be applied to other geophysical problemsrequiring rapid calculation of series-based functions of severalindependent variables, where the function evaluation is a timeconsuming process, and maximum error bounds are critical.

Index Terms—Mie, Hydrometeor, Spline, Jacobian, Radiativetransfer, Microwave, Radiance.

I. I NTRODUCTION

Numerical radiative transfer (RT) calculations are essentialfor understanding and assimilating brightness temperaturemeasurements made at various microwave frequencies andgeographical locations. RT model inversions performed usingthese measurements can yield vertical temperature and watervapor density profiles of the atmosphere, which can be usedin numerical weather prediction and climate forecasting [1]–[3]. Alternately, brightness temperature measurements can also

Manuscript received December 4, 2010; revised June 3, 2011;acceptedJuly 3, 2011. This work was supported by award number 1358415from theNational Aeronautics and Space Administration Jet Propulsion Laboratory.

The authors are with the Department of Electrical and Computer Engi-neering, University of Colorado, Boulder, CO 80309 USA (e-mail: sriku-marsandeep@ yahoo.com; al.gasiewski@colorado.edu).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TGRS.2011.2162417

be directly assimilated into numerical weather models [4].Efficient numerical weather forecasting applications requireparticularly fast scattering-based radiative transfer (RT) simu-lations. One of the processes that imparts a high computationalburden for hydrometeor laden atmospheres is the calculationof hydrometeor absorption and scattering coefficients andthe phase asymmetry parameter. The excessive computationaloverhead is a result of the nested summations required inthe calculation of the Mie efficiencies and the subsequentnumerical integration of the Mie efficiencies over the hydrom-eteor drop size distribution. The large number of times thatsuch calculations are required in radiative transfer modelingsuggests that library look-up techniques can provide significantcomputational efficiencies provided that library error canbebounded. Reflectivity modeling for meteorological weatherradar is similarly well understood [5]. In the Rayleigh regime,the backscattering coefficient is related to the reflectivity factorZ, which in turn depends on the precipitation rate. As a partof this work, fast libraries for liquid and ice hydrometeorbackscattering coefficients are also developed.

RT theory describes the interaction of radiation with matterby taking into the consideration the effects of the atmosphericabsorption, emission and scattering due to cloud, fog, snowetcon electromagnetic radiation. It has been thoroughly discussedin numerous references [6]–[8]. One of the first steps inRT modeling is the computation of the absorption vector,extinction matrix and phase matrix at each point within themedium of interest [1]. For plane parallel atmospheric models,this step entails calculations at each of many (typically 50-100)vertical levels of the atmosphere. The atmospheric mediumconsists of gaseous absorbing constituents such as oxygen,water vapor, ozone, and nitrogen. In addition to these gasesthe atmosphere may also contain suspended or falling liquidor frozen water particles in the troposphere and lower strato-sphere. Hydrometeors can absorb, emit, and scatter radiation atmicrowave frequencies. Hydrometeors can be either rain drops,ice crystals, snowflakes, graupel or hail. They are in generalnonspherical in shape. For instance, rain drops are slightlyoblate shaped [9], [10]. Ice crystals are either hexagonal orirregular in shape. However for small hydrometeor dimensions,spherical drop assumption is accurate. In this work, we haveused the assumption of modelling hydrometeors as eitherliquid water or solid ice spheres. Furthermore, in this workonly homogenous liquid water or ice hydrometeors are consid-ered. In reality, many hydrometeors are a mutliphase mixtureof air, ice and water [11], [12]. A future extension of thiswork will deal with the extinction coefficients of nonsphericalhydrometeors. T-matrix method instead of Lorentz-Mie theory

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need to be used for this purpose [13]. The problem of applyingan iterative numerical radiative transfer model to hydrometeor- laden atmospheres has been addressed in [1], within whichhydrometeors are modeled as dielectric spheres. The analyticalsolution to the absorption and scattering of electromagneticwaves by a sphere of arbitrary radius, permittivity and per-meability was provided by Mie [14]–[16], from whose theorythe absorption efficiency, scattering efficiency, backscatteringefficiency, and phase function asymmetry of a single dielec-tric sphere can be calculated. However, expressions for theabove parameters are in the form of infinite summations,where each term contains spherical Bessel functions requiringsoftware evaluation. Accordingly, doubly nested summationsare required to evaluate either the cross-sectional efficienciesor the phase function asymmetry for a single hydrometeor.Owing to hydrometeors of varying radii, clouds are modeledby a polydispersed distribution of hydrometeors. The hydrom-eteor coefficients of a polydispersed cloud is obtained bythe integration of the corresponding efficiencies multiplied bythe sphere cross-sectional area over the entire hydrometeordistribution. Numerical quadrature is used to perform thisintegration, resulting in a third level of nesting. Due to theneed for accuracy, the resulting numerical integration oftenrequires many tens of thousands of elementary calculationsforeach atmospheric spatial point. Depending on the hydrometeormean electrical size, this computational burden is comparableto or greater than that of the vertical RT quadrature itself.

This paper presents an efficient approach based on cubicB-spline interpolation for fast, accurate calculation of theMie hydrometeor coefficients and phase asymmetry parameterfor exponential size distributions of terrestrial hydrometeors.B-splines are standard for representing curves and surfacesin computer graphics and computer aided design [17], [18].Recently, they have been used for volume reconstruction ofarbitrarily dimensioned data. For the present Mie application,lookup tables (LUT) for the absorption coefficient, scatteringcoefficient, backscattering coefficient, and phase matrix asym-metry parameter for both liquid and ice hydrometeor distribu-tions are generated. These LUTs are three dimensional arrayscoding the values of these products as a function of the threeindependent variables of temperature, average hydrometeorsize, and frequency over a wide range of these parameters. Intotal, eight LUTs corresponding to four different productsfortwo types of hydrometeors (liquid and frozen) are generated.Spline interpolation is applied on any of these eight LUTsto evaluate the corresponding products. The method resultsinconsiderable acceleration of microwave radiative transfer sim-ulations across a broad frequency spectrum and fast calculationof radar reflectivity for weather radar data assimilation. Wedemonstrate the effectiveness of this approach by comparingradiative transfer simulations using the B-spline librarytothe conventional series approach within scattering and non-scattering atmospheres. These intercomparisons confirm ac-ceptable reconstruction accuracy for most relevant problemsin terrestrial radiative transfer and radar studies.

II. M IE THEORY

The electromagnetic absorption and scattering propertiesofa single sphere are described by the absorption efficiencyηa,scattering efficiencyηs and backscattering efficiencyηb [16],[19]. Let Si (Wm-2) be the average power density of theelectromagnetic wave incident on the sphere, andPa (W), Ps

(W) andPb (W) be the absorbed, scattered and backscatteredpower respectively.Pa, Ps andPb are related toSi throughthe absorption cross sectionσa (m2), scattering cross sectionσs (m2) and backscattering cross sectionσb (m2). They aregiven by the following relations,

σa =Pa

Si

; σs =Ps

Si

; σb =Pb

Si

(1)

The absorption efficiencyηa, scattering efficiencyηs andbackscattering efficientyηb are defined as the ratio of thecorresponding electromagnetic cross-sections to the physicalcross-section of a sphere of radiusa,

ηa =σa

πa2; ηs =

σs

πa2; ηb =

σb

πa2(2)

The extinction cross-sectionσe and extinction efficiencyηeare similarly given by the sum of the corresponding absorptionand scattering quantities,

σe = σa + σs ; ηe = ηa + ηs (3)

The ηe, ηs and ηb for a homogenous dielectric sphere ofarbitrary radius and refractive index is given by Mie theoryas,

ηe (x,m) =2

x2

∞∑

n=1

(2n+ 1)Re {an + bn} (4)

ηs (x,m) =2

x2

∞∑

n=1

(2n+ 1){

|an|2 + |bn|

2}

(5)

ηb (x,m) =1

x2

∣

∣

∣

∣

∣

∞∑

n=1

(−1)n(2n+ 1) (an − bn)

∣

∣

∣

∣

∣

(6)

In (4-6),x is the size parameter given byx ≡ 2πaλ

andan, bnare the Mie coefficients, which are generally complex andcalculated as follows,

an =m2jn (mx) [xjn (x)]

′

− jn (x) [mxjn (mx)]′

m2jn (mx) [xh1n (x)]

′

− h1n (x) [mxjn (mx)]

′ (7)

bn =jn (mx) [xjn (x)]

′

− jn (x) [mxjn (mx)]′

jn (mx) [xh1n (x)]

′

− h1n (x) [mxjn (mx)]

′ (8)

where jn (x) , h1n (x) are spherical Bessel and Hankel func-

tions of first kind and ordern respectively. In the aboveequationsm (f, T ) ≡ ns(f,T )

nb(f,T ) is the relative complex refractiveindex of the material of the sphere with respect to the back-ground medium,ns is the refractive index of the sphere andnb

is the refractive index of the background medium (usually air).The refractive indicesns, nb are both functions of frequencyf and temperatureT . In these general expressions both thesphere and background are assumed to be non-magnetic. Inpractice the summations in (7-8) are truncated at a finitenumber of terms. The maximum number of terms is commonly

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taken to be the next integer closest tox+ 4x13 + 2 [16]. The

phase function asymmetry parameter G defines the relativeproportion of energy scattered in the forward versus backwarddirections [1]. The Mie phase function asymmetry for a singlespherical scatterer is,

G =4

x2ηs (x,m)

∞∑

n=1

[

n (n+ 2)

n+ 1Re {a∗nan+1 + b∗nbn+1}

+2n+ 1

n (n+ 1)Re {a∗nbn}

]

(9)

For G = 1, there is only scattering in the forward directionand G > 0 denotes a preferential scattering in the forwardversus backward direction. Similarly, forG = −1 there isscattering only in the backward direction andG < 0 denotesa preferential scattering backwards. In (4-6,9), the valueofthe size parameterx determines the number of significantterms in the series, and as a result the scattering regime ofthe particle. For electrically small spheres withx << 1, asingle term is usually sufficient. This is the Rayleigh scat-tering approximation [20]. For electrically large sphereswithx >> 1 geometrical optics, physical optics, or ray tracing,approximations can often be used effectively. Nonetheless, inthis work the truncated Mie summations are exclusively usedwithout further approximation.

III. L OOK-UP TABLE CALCULATION

The radius of hydrometeors varies from∼ 1µm for haze,fog, and small cloud particles up to about∼ 10 mm for largefrozen particles such as graupel, snow and small hail. Thestatistical characterization of the hydrometeor particlesize fora polydispersed cloud is given by a drop size distribution,n(D) [1], [19]. A general functional form ofn(D) (mm-1m-3)is the modified gamma distribution given by,

n (D) = N0 (ΛD)Pe−(ΛD)Q (10)

wheren (D) dD is the number of particles per unit volumewith diameters in the range[D,D + dD] (mm) and N0

(mm-1m-3), Λ (mm-1), P andQ are distribution parameters. Byconsideringn(D) given by (10) as a scaled probability densityfunction, closed form expressions for important statisticalquantities such as mean diameter, diameter variance, particlenumber density, and fractional volume can be derived [1],[19]. For example, the mean diameter〈D〉 (mm) and fractionalvolume of waterfv are given by,

〈D〉 =1

Λ

Γ(

P+2Q

)

Γ(

P+1Q

) (11)

fv =10−9πN0

6Λ4QΓ

(

P + 4

Q

)

(12)

whereΓ (.) is the gamma function. The aggregate absorption,scattering and backscattering coefficients in (Npm-1) can becomputed as,

κα (f, T ) =π

4

∞∫

0

D2n (D) ηα

(

πD

λ,m (f, T )

)

dD (13)

where α = a, s, b. Similarly, the aggregate phase functionasymmetryg (f, T ) for a polydispersion of spherical hydrom-eteors is given by,

g (f, T ) =

∫∞

0 D2n (D) ηs(

πDλ,m

)

G(

πDλ,m

)

dD∫∞

0D2n (D) ηs

(

πDλ,m

)

dD(14)

Most of the commonly used drop size distributions forprecipitation such as Marshall - Palmer (MP), Laws andParsons, and Sekhon - Srivastava (SS) are of the exponen-tial form [1], [19]. Accordingly, to avoid the problem ofcataloguing a four-fold infinity of size distributions we limitthis study to exponential distributions for whichP = 0 andQ = 1. Exponential distribution can be used to model majorhydrometeor particles such as rain, snow, graupel and hail[21]. In this case〈D〉 = 1

Λ . Although the upper limit ofintegration in (13,14) is infinite, the practical upper limit onthe drop diameter is about6 mm [22]. However, we have useda conservative upper limit of about 15 mean particle diametersdue to the exponential decay factor. Byn = 15, it is notedthat the relative error in the integral becomes bounded by9.6× 10−4 times the real value, viz.:

∫ n〈D〉

0D2e−

D〈D〉 dD

∫∞

0 D2e−D

〈D〉 dD= 1− e−n

[

0.5n2 + n+ 1]

(15)

A simple empirical sensitivity analysis was performed toconfirm the above mentioned upper limit. For 1000 randomvalues of(f, 〈D〉 , T ), κb was calculated for liquid hydrom-eteors for values ofn ranging from 5 to 25 in steps of5. The relative error percentage was calculated for eachincrement ofn. The averages of these error percentages tendto converge to a very small value with each increment. Theaverage percentage errors for the increments 5-10,10-15,15-20,20-25 are14.5%, 0.31%, 0.023% and0.021% respectively.This confirms that forn > 15, there is hardly any contributionto the coefficient value. In this work, we have used meandiameters ranging from2 µm to 20 mm to cover the rangeof terrestrial hydrometeors encountered in realistic precipita-tion models. The constantN0 for an exponential drop sizedistribution is a function of〈D〉, and can be calculated usingequations (11,12). The average fractional volume of liquidor frozen precipitation rarely exceeds5 × 10−6. SinceN0

is directly proportional to fractional volume,fv, a constantvalue fv = 10−6, corresponding to1 gm-3 of liquid waterwas used in this work. In this case,N0 = 1000

π〈D〉4, where〈D〉

is in mm. The coefficients for a different value of fractionalvolume can be obtained by scaling the results according to theactual density of hydrometeors. Therefore, equations (13,14)reduce to,

κα (f, 〈D〉 , T ) ≈ π10−6

4

15〈D〉∫

0

D2N0 (〈D〉) e− D

〈D〉 ηα(

πDλ, m

)

dD

(16)

g (f, 〈D〉 , T ) ≈

15〈D〉∫

0

D2N0(〈D〉)e− D

〈D〉 ηs(πDλ

,m)G(πDλ

,m)dD

15〈D〉∫

0

D2N0(〈D〉)e− D

〈D〉 ηs(πDλ

,m)dD(17)

where D is in mm andκα (f, 〈D〉 , T ) is in Npm-1. Theabove products are functions of three independent variables:

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frequency, temperature, and the mean diameter of the poly-dispersion. Since the Mie efficiencies are a function of therefractive index of the sphere, the complex dielectric constantsof liquid water and ice as a function of frequency andtemperature are also implicitly required.

Each product LUT is a three dimensional array that storesthe values ofκa, κs, κb, g at discrete values of the inde-pendent variable triplet(f,< D >, T ). The ranges used foreach independent variable are :f ∈ [1, 1000] GHz, 〈D〉 ∈[0.002, 20] mm andT ∈ [−50,+50]oC. For ice hydrometeors,T ∈ [−50, 0]oC is the temperature range used. Due to thelarge range of mean diameter and frequency, a logarithmicsampling grid was used for these two variables. Ray’s modelwas used for the dielectric constant of pure water at microwavefrequencies [23]. The dielectric constant of ice at microwavefrequencies was obtained from Warren [24]. Linear interpo-lation was used to obtain values of the dielectric constant atfrequencies that are not listed in [24]. The Mie efficienciesarecalculated using routines provided by Maetzler [25]. Theseroutines based on Mie formulations in [16], but stable forsize parameters up to∼ 1000. Once the Mie efficiencies(ηa, ηs, ηb, G) and N0 were calculated, adaptive Simpson’squadrature was used for the numerical evaluation of thedistribution integrals (16,17). The surface plots of hydrometeorabsorption, scattering, backscattering coefficients and phaseasymmetry parameter for both liquid and frozen hydrometeorsis shown in Fig.1. These plots can be considered as slices fromthe product LUTs at a constant temperature.

IV. B- SPLINE INTERPOLATION

Spline functions are piecewise polynomials on subintervalsthat are joined together with prescribed continuity conditions[17]. A spline function of orderk consists of piecewisepolynomials of maximum degreek− 1. Therefore a spline oforder 1 is composed of piecewise constants, a spline of order2 consists of piecewise linear polynomials, etc. A univariatespline functionS(x) can be represented in the piecewisepolynomial form as,

S (x) = Pi (x) ;x ∈ [ξi, ξi+1) (18)

where{ξi}l+1i=1 are a strictly increasing sequence ofl+1 break

(or tie) points ofS(x). A spline function of orderk has thefollowing continuity conditions at its breaks.

Pi (ξi+1) = Pi+1 (ξi+1) ; i ∈ [1, l− 1]

P(n)i (ξi+1) = P

(n)i+1 (ξi+1) ; i ∈ [1, l− 1]

;n = 1, 2, . . . k − 2

(19)

where P(n)i (x) stands for thenth derivative of Pi (x).

The spline interpolation problem can be stated as follows:given the values of a function at break points,{g (ξi)}

l+1i=1,

the spline functionS(x) needs to be determined such thatS (ξi) = g (ξi) ; i ∈ [1, l+ 1]. Splines of order 4 (cubicsplines) are twice continuously differentiable at the breakpoints and most commonly used in practice. For a cubicspline, each polynomial section would be of degree 3, i.e.Pi (x) = ci3(x− ξi)

3+ci2(x− ξi)

2+ci1(x− ξi)+ci0. Given

the values of the functiong at the break points, the continuityconditions, and boundary conditions (usually the values ofthe first and second derivatives ofS(x) at the first and lastbreak points), the coefficients of each piecewise polynomialcan be uniquely determined. Simple tridiagonal linear systemsof equations are solved to obtain{cij ; i ∈ [1, l], j ∈ [0, 3]}.

An alternate representation of the spline function is knownas the B-form representation [17], [18], [26]. This represen-tation is based on the Curry-Schoenberg theorem, which isstated as follows: All the spline functions of orderk andbreak sequence{ξi}

l+1i=1 form a linear space denoted byΠk,ξ.

To represent a function using B-splines we choose a strictlynon-decreasing sequence of points{τi}

l+k+1i=1 called the knot

sequence. The knot sequences need not coincide with the breakpoints. In the B-form representation, the spline function isexpressed as a weighted superposition of basis functions calledB-splines (or, basis splines). For example, the B-spline oforder1 is given by,

B1i (x) =

{

1 ; x ∈ [τi, τi+1)0 ; otherwise

(20)

B-splines of higher order can be determined from lower orderB-splines using the Cox-de Boor recurrence relation [17],

Bki (x) =

(

x− τiτi+k−1 − τi

)

Bk−1i (x)

+

(

τi+k − x

τi+k − τi+1

)

Bk−1i+1 (x)

(21)

Using the Cox - de Boor relation, the B-spline of order 4(or, cubic B-spline) can be obtained and is given by (25).The B-splinesBk

1 , Bk2 , . . . , B

kl+1 defined on the knot sequence

{τi}l+k+1i=1 form the basis for the spline space,Πk,ξ. It can be

shown that the B-form representation of the spline functionS (x) of orderk is given by,

S (x) =

l+1∑

i=1

αiBki (x) ;x ∈ [ξi, ξi+1) (22)

where{αi}l+1i=1 are the B-spline coefficients. Importantly, B-

splines have local support. Hence theith B-spline of orderk,Bk

i (x) = 0;x /∈ [τi, τi+k). For example, cubic B-splines havesupport between five successive knots. The local support ofB-splines results in computational advantages for the B-formrepresentation over the piecewise polynomial representationof a spline function. A good choice of knot sequence{τi}

l+5i=1

for cubic B-spline interpolation can be determined from thebreaks{ξi}

l+1i=1 as follows,

τ1 = τ2 = τ3 = τ4 = ξ1

τl+5 = τl+4 = τl+3 = τl+2 = ξl+1

τi =ξi−3 + ξi−2 + ξi−1

3; 5 ≤ i ≤ l + 1 (23)

Extension of spline function theory to multiple variables canbe obtained by tensor product construction of univariate B-splines. In this case a trivariate cubic spline functionF (x, y, z)is represented in B-form as,

F (x, y, z) =

Nx∑

i=1

Ny∑

j=1

Nz∑

m=1

αijmB4i (x)B

4j (y)B

4m (z) (24)

5

B4i (x) =

(x−τi)3

(τi+3−τi)(τi+2−τi)(τi+1−τi); x ∈ [τi, τi+1)

(x−τi)(x−τi)(τi+2−x)(τi+3−τi)(τi+2−τi)(τi+2−τi+1)

+ (x−τi)(τi+3−x)(x−τi+1)(τi+3−τi)(τi+3−τi+1)(τi+2−τi+1)

· · ·(τi+4−x)(x−τi+1)(x−τi+1)

(τi+4−τi+1)(τi+3−τi+1)(τi+2−τi+1); x ∈ [τi+1, τi+2)

(τi+4−x)(x−τi+1)(τi+3−x)(τi+4−τi+1)(τi+3−τi+1)(τi+3−τi+2)

+ (τi+4−x)(τi+4−x)(x−τi+2)(τi+4−τi+1)(τi+4−τi+2)(τi+3−τi+2)

· · ·(x−τi)(x−τi+3)(x−τi+3)

(τi+3−τi)(τi+3−τi+1)(τi+3−τi+2); x ∈ [τi+2, τi+3)

−(x−τi+4)3

(τi+4−τi+1)(τi+4−τi+2)(τi+4−τi+3); x ∈ [τi+3, τi+4)

(25)

In this work, the three dimensional functionF can be any ofthe following eight functions: absorption coefficient, scatteringcoefficient, backscattering coefficient, or phase asymmetryfor liquid water and ice hydrometeors, with the independentvariables being frequency, temperature and mean diameter,

F (f, T, 〈D〉) =

Nf∑

i=1

NT∑

j=1

N〈D〉∑

m=1

αijmB4i (f)B

4j (T )B

4m (〈D〉)

(26)The αijm in (26) are the stored Mie B-spline coefficients,and MATLAB Spline toolbox was used for the calculation ofthese coefficients. The generation of B - spline coefficientsisa one time process that does not need to be repeated. Thecode for reconstructing the Mie products is a straightforwardimplementation of (26).

V. RESULTS AND DISCUSSION

In this section we discuss the five main aspects of this work- memory and overhead reduction, radiation Jacobian, recon-struction error, computational savings, and radiative transfersimulations. The simulations and related software codes werewritten in the MATLAB environment.

A. Memory and Computational Overhead Reduction

To generate the Mie LUTs the mean diameter and frequencyranges were logarithmically gridded with 200 and 60 points,respectively, and the temperature range was gridded linearlywith an interval of2.5oC. The eight normalized Mie productswere sampled at these grid points and stored as eight LUTs.The total memory usage of these LUTs is∼ 10.6 MB. Theproduct LUTs are subsequently used for spline interpolation(i.e., piecewise polynomial spline interpolation). The eightproduct LUTs were converted to eight corresponding B-splineLUTs which contain the B-spline coefficients. From (26), itcan be inferred that the size of B-spline LUTs is nearly thesame as the size of corresponding product LUTs. The disad-vantage of using piecewise polynomial spline interpolation isthat the values stored in the product LUTs need to be convertedto the piecewise spline polynomial coefficients. This procedureis not localized to a small subset of intervals, thus resultingin the required calculation of all the polynomial coefficientsfor each interpolation. Hence there is a fixed overhead timerequired for the evaluation of all the piecewise polynomialcoefficients irrespective of the number of product values thatneed to be interpolated. In the trivariate case, if the number ofvalues in an individual product LUT isN , the total number

of piecewise polynomial coefficients will be43N = 64N .Therefore in our case, assuming one product LUT is∼ 1.25MB in size, a RAM usage of80 MB will be required whenperforming the piecewise polynomial spline interpolation. Thishigh memory usage also limits the possibility of storingthe piecewise polynomial coefficients as LUTs. The fixedoverhead and high memory usage thus obviates many of theadvantages of conventional splines.

In contrast, B-form interpolation is both localized to a smallsubset of intervals, requires a large fixed overhead time to cal-culate theαijm coefficients, but does not require the temporarystorage of large number of spline coefficients. However, whensufficiently large numbers of products needs to be evaluatedby interpolation, the spline interpolation has an advantage overB-spline method. This computational advantage is due to thefact that for the spline interpolation, once all the piecewisepolynomial coefficients are calculated, the evaluation of thesepiecewise functions is extremely rapid compared to the threenested summations required for B-splines, as given by (26).

B. Radiation Jacobian

The radiation Jacobian is defined as the derivative of theradiance field with respect to any electromagnetic parameterof the atmosphere [27]. Therefore, the derivative of anycoefficient with respect to (for example) the mean diametercan be evaluated by differentiation of (16), (17) using Leibnitzrule for differentiation under integral sign. Surface plots ofthe radiation Jacobian of different Mie products for ice andliquid hydrometeors calculated using analytical differentiationare shown in Fig. 2. One possible way to evaluate Jacobian isfacilitated by analytical differentiation of the cubic B-spline.Therefore, the derivative of any coefficient with respect to(for example) the mean diameter can possibly be evaluatedas follows,

dF (f, T, 〈D〉)

d〈D〉=

Nf∑

i=1

NT∑

j=1

N〈D〉∑

m=1

αijmB4i (f)B

4j (T )

dB4m (〈D〉)

d〈D〉

(27)However, this procedure cannot produce accurate estimatesconsistently. This is because the B-spline coefficients werecalculated using the product coefficients. A better approachis to generate B-spline coefficients using analytically calcu-lated Jacobian values in the(f,< D >, T ) grid and then usethese B-spline coefficients,βijm in (26). Error analysis wasperformed by comparing the B-spline evaluated coefficientvalues with analytically obtained values for 20000 random(f,< D >, T ) points. This was done forκa, κs for both liquid

6

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Fig. 1. Polydispersed hydrometeor absorption coefficient(κa), scattering coefficient(κs),backscattering coefficient(κb) and phase asymmetry parameter(g) versus(f, 〈D〉). κa, κs, κb

are in dB/km. (a) Liquidκa. (b) Ice κa. (c) Liquid κs. (d) Ice κs. (e) Liquid κb. (f) Ice κb.(g) Liquid g. (h) Ice g

7

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Fig. 2. Polydispersed hydrometeor absorption coefficient(κa), scattering coefficient(κs),backscattering coefficient(κb) and phase asymmetry parameter(g) Jacobians versus(f, 〈D〉)

at T = 0oC : (a) Liquid dκad〈D〉

. (b) Ice dκad〈D〉

. (c) Liquid dκsd〈D〉

. (d) Ice dκsd〈D〉

. (e) Liquid dκbd〈D〉

.

(f) Ice dκbd〈D〉

. (g) Liquid dgd〈D〉

. (h) Ice dgd〈D〉

8

and ice hydrometeors. The mean fractional error in all the fourcases were of the order of10−4. The maximum fractional errorwas less than 0.1 for all the cases except liquidκa for whichit was 1.8. However, the method given by (27), can be usedin a 2D case, where we are only concerned about radiometricfrequencies, thereby reducing the 3D LUT to a 2D LUT andhence allowing for much finer discretization along the〈D〉and T axes.

C. Computational Savings

In this section, the improvement in computational speedusing the fast Mie library is discussed. In Fig.3, the loga-rithmic ratio of the B-spline interpolation time to the exactcalculation time using the full Mie series is illustrated forliquid hydrometeor absorption atT = 0oC. This time ratiowill be nearly the same for all other products and at anytemperature. From the Fig. 3, it can be seen that there isconsiderable speedup when the hydrometeors are large indiameter compared to the wavelength. In this case, the timeconsuming step of integration over the drop-size distribution isobviated. Usually radiative transfer simulations are performedat the channel frequencies of the remote sensing instrument. Insuch a case, the B-spline coefficients for only the instrument’schannel frequencies need be stored. This reduced storagenecessitates only two dimensional interpolation, which isfasterthan the three dimensional interpolation. Equivalently, for thesame amount of memory, in the two dimensional case a finergridding can be used to provide extreme accuracy.

Fig. 3. Logarithmic ratio of the calculation time for B-spline interpolationto that for exact Mie series calculation of polydispersed liquid hydrometeorabsorption coefficients. An exponential drop-size distribution, temperatureT = 0oC and fractional volumef = 10−6 are assumed

D. Reconstruction Error Analysis

In this section, we derive an expression for the maximumerror that can be allowed for the reconstructed values ofMie products, and use this to determine the overall libraryaccuracy. Any error in the absorption or scattering coefficient

will result in an associated error in the computed brightnesstemperature. By using the condition, that the maximum al-lowed brightness temperature error is bounded by a prescribedvalue we can derive expressions for the maximum error boundon the coefficients. We use the approach in [28] to derive theabsorption coefficient incremental weighting function (IWF)for downwelling radiation. The IWF describes the relationshipbetween infinitesimal variations in any atmospheric parameterand the downwelling brightness temperature. The downwellingbrightness temperature observed at heighth and zenith angleθ for a non-scattering plane-parallel atmosphere is,

TDB (h, θ, ν) = T bB (TCB, ν) exp

[

−

∫ ∞

h

κoa (z) sec θdz

]

+

∫ ∞

h

T bB (T (z) , ν) sec θκo

a (z)

exp

[

−

∫ z

h

κoa

(

z′)

sec θdz′

]

dz

(28)

In (28), ν denotes the frequency andTCB, κoa (z) , T (z) and

T bB (.) represents the cosmic background temperature, absorp-

tion coefficient as a function of altitude, physical temperatureas a function of altitude and blackbody brightness temperaturefunction respectively. The downwelling brightness temperatureis due to the cosmic background radiation and the thermalemission, both attenuated by the atmospheric absorption. Ifδκa (z) is the variation in the absorption coefficient profile, thecorresponding variation in the observed downwelling bright-ness temperature,TDB (h, θ, ν) denoted byδTDB is given by,

δTDB =

∫ ∞

h

W ↓κa

(z, θ, ν) δκa (z)dz (29)

where W ↓κa

(z, θ, ν) is the absorption IWF for down-welling brightness temperature. An approximate expression forW ↓

κa(z, θ, ν) is derived by substitutingκo

a (z)+δκa (z) in (28)and subtracting the orginal expression forTDB (h, θ, ν),

W ↓κa

(z, θ, ν) = −T bB (TCB, ν) sec θ exp

−

∞∫

h

κoa (z) sec θdz

+ T bB (T (z) , ν) sec θ exp

−

z∫

h

κoa (z) sec θdz

− sec2 θ

∞∫

z

T bB

(

T(

z′)

, ν)

κoa

(

z′)

dz′

exp

−

z′

∫

h

κoa

(

z′′)

sec θdz′′

(30)

By applying the following assumptions to (30), the IWF canbe simplified: 1) the observation point is assumed to be on theground i.e.,h = 0. 2) κa (z) = 0; z > H , whereH = 8 kmis the atmospheric scale height. 3)κo

a (z) = κoa i.e. a constant

atmospheric absorption coefficient profile is assumed, and 4)

9

the thermodynamic temperature,T (z) = T is constant. Thesimplified IWF then becomes,

W ↓κa

(z, θ, ν) = sec θ exp [−κoa sec θH ]

(

T bB (T, ν)− T b

B (TCB, ν)) (31)

The simplified expression forδTDB is obtained by substituting(31) in (29),

δTDB = δκaH sec θ exp [−κoa sec θH ]

(

T bB (T, ν)− T b

B (TCB, ν)) (32)

Since we are concerned about the maximum possible valueof the brightness temperature deviation,|δTDB|max, (32) ismaximized by choosing an extreme atmospheric temperatureof T = 325K. In this case,|δTDB|max becomes,

|δTDB|max = 2.56× 106 δκa sec θ exp [−κoa sec θH ] (33)

Constraining the maximum brightness temperature error by 10mK yields,

|δTDB|max ≤ 0.01K ⇒δκo

κoa

≤3.9× 10−9

κoa sec θ exp [−κo

a sec θH ](34)

For a constantκoa, the expression on the right side of (34) can

be minimized with respect toθ, thus resulting in an expressionfor maximum allowable fractional error in the reconstructedvalue of the absorption coefficient,

|δκa

κoa

|max =

{

8.48× 10−5 ; κoa < 1

H3.9×10−9 expκo

aH

κoa

; κoa > 1

H

(35)

It should be noted that the assumptions we have used to derivethis expression are conservative, and not normally encounteredin practice.

The model in (35) can be used to evaluate the accuracyof the spline reconstructed absorption coefficient values.InFig. 4, the fractional error in reconstructed values of liquidand ice hydrometeor absorption are plotted against the corre-sponding absorption coefficients for 2000 random values of

0 0.5 1 1.5 2 2.5 310

−10

10−8

10−6

10−4

10−2

100

102

104

κa (Np/km)

κ a frac

tiona

l err

or (

dκ a/κ

a)

Liquid κ

a

Frozen κa

Error bound ( ∆TB = 0.01 K))

Error bound ( ∆TB = 0.1 K)

Fig. 4. Fractional error in reconstructed values of liquid and ice hydrometeorabsorption coefficient vs. liquid and ice hydrometeor absorption values. Themaximum allowable error bounds forTDB = 0.01K and 0.1 K are shown.

(f,< D >, T ) and fractional volume. The maximum allow-able fractional error in the absorption coefficient for brightnesstemperature errors less than 0.01 K and 0.1 K are alsoplotted. It can be seen that for the prescribed level of librarydiscretization the liquid absorption coefficient fractional erroris always well below the 0.01 K error bound. However theice absorption coefficient fractional error is higher than thewater absorption coefficient, but still smaller than∼ 0.1K.This error is attributed to the ice dielectric model that wasusedin this work [24], which has discontinuities in its derivatives.The reconstruction error for ice hydrometeors can be reducedfurther by either finer sampling or by smoothing the icedielectric constant model before calculating the ice extinctioncoefficients.

E. Radiative Transfer Simulations

In this section, the results of RT simulations performedto verify the accuracy of the developed library is presented.These results confirm the applicability of the library for RTcomputations. The atmosphere used for the RT simulationswas the 1976 version of U.S. Standard Atmosphere, withwater vapor density in each level corresponding to50% of thesaturation vapor density at the level pressure and temperature.Surface reflectivity is computed at a particular frequency byassuming an ocean surface with salinity3.5% and using theRay model [23] and Klein and Swift model [29] for dielectricconstant and conductivity respectively. The base atmospherewas modified by inserting liquid and ice hydrometeors over1 − 10 km [1]. The simulations were carried out in thefrequency range[1, 1000] GHz in steps of1 GHz.

In the first simulation, multiple scattering was ignored bysetting the extinction coefficient of each level to the sum ofabsorption coefficient and scattering coefficient. The relativedifference in the top of the atmosphere brightness temperaturefor a downward looking radiometer was obtained as a func-tion of frequency. The relative difference corresponds to thedifference between the cases when real Mie series value andspline interpolated values of extinction coefficients wereused.The maximum and the mean value for this relative differenceare1.15× 10−3% and3× 10−5% respectively. In the secondRT simulation, multiple scattering was taken into account byusing Henyey-Greenstein phase function and a scattering basedRT model [1]. The maximum and mean value of the relativedifference are9.72× 10−2% and7.21× 10−3% respectively.

VI. CONCLUSION

In this work, the absorption coefficient, scattering coeffi-cient, backscattering coefficient, and phase asymmetry param-eter of both liquid and ice spherical, homogenous hydromete-ors as a function of frequency, temperature and mean diameterof the hydrometeor distribution is represented in a piecewisefunctional form using trivariate cubic B-splines. By usingthismethod, it was possible to achieve significant computationtime reduction for calculating the extinction parameters andphase asymmetry parameter, especially for large hydrometeorsand at high microwave frequencies. Furthermore, the recon-struction error that is caused by the spline interpolation is neg-ligible enough to preclude any adverse impact on the accuracy

10

of radiative transfer simulations for most relevant terrestrialapplications. The memory requirement for this fast libraryisaround 10.6 MB for all eight products. The library further sup-ports evaluation of the radiation Jacobian by either the rapidanalytical differentiation of the B-spline basis functions or forhigher accuracy, B-spline coefficients can be calculated byusing true Jacobian values. The reconstruction time, memoryusage can be further decreased with improvement in accuracyby storing the B-spline coefficients only at frequencies ofinterest where radiative transfer simulation is required.Thecubic B-spline based approximation method can be appliedto other geophysical problems, where function evaluation is atime consuming process and reconstruction accuracy is critical[30]. Future work will include applying these techniques tononspherical or even multi-phase hydrometeors.

ACKNOWLEDGMENT

The authors are grateful to Prof. Gregory Beylkin of Univer-sity of Colorado, Boulder for valuable suggestions regardingB-splines and Prof. Christian Maetzler for providing the codeto calculate Mie efficiencies. The authors would also liketo thank Dr. Bjorn Lambrigsten of NASA Jet PropulsionLaboratory for his support of this study. This work was fundedby award number 1358415 from the NASA Jet PropulsionLaboratory.

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by Small Particles. New York: Wiley - Interscience, 1983.[17] C. de Boor,A Practical Guide to Splines. Springer - Verilag, 2001.[18] D. F. Rogers,An Introduction to NURBS : with historical perspective.

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Srikumar Sandeep received the B.E degree fromUniversity of Auckland in 2006. He worked as asoftware engineer in Trimble navigation for a year.He is currently a graduate student at University ofColorado, Boulder. His technical interests includeelectromagnetics, applied mathematics, remote sens-ing and software development. He was awardedthird prize in the 2011 USNC/URSI National RadioScience Meeting paper contest.

Albin J. Gasiewski (StM‘81, M‘88, SM‘95, F‘02) isProfessor of Electrical and Computer Engineering atthe University of Colorado at Boulder and Directorof the CU Center for Environmental Technology. Hereceived the Ph.D. degree in electrical engineeringand computer science from the Massachusetts Insti-tute of Technology in 1989. Previously, he receivedthe M.S. and B.S. degrees in electrical engineeringand the B.S. degree in mathematics from Case West-ern Reserve University in 1983. From 1997 through2005 he was with the U.S. National Oceanic and

Atmospheric Administration’s (NOAA) Environmental Technology Labora-tory in Boulder, Colorado, USA, where he was Chief of ETL’s MicrowaveSystems Development Division. From 1989 to 1997 he was a faculty memberat the Georgia Institute of Technology. He has developed andtaught courseson electromagnetics, remote sensing, instrumentation, and wave propagationtheory.

Prof. Gasiewski is a Fellow of the IEEE, Past President (2004-2005) ofthe IEEE Geoscience and Remote Sensing Society, and founding memberof the IEEE Committee on Earth Observation (ICEO). He is a member ofthe American Meteorological Society, the American Geophysical Union, theInternational Union of Radio Scientists (URSI), Tau Beta Pi, and Sigma Xi.He currently serves as Chair of USNC/URSI Commission F. He served on theU.S. National Research Council’s Committee on Radio Frequencies (CORF)from 1989-1995. He was the General Co-chair of IGARSS 2006, in Denver,Colorado, and a recipient of the 2006 Outstanding Service Award from theGRSS.