radiative transfer modeling in the...

CHAPTER3

Radiative transfer modeling inthe cryosphere

Roberto Furfaro, Alberto Previti, Paolo Picca, Jeffrey S. Kargel, and Michael P. Bishop

ABSTRACT

Radiative transfer (RT) modeling plays a key role ininterpreting the radiance measured by multispectralsensors. Glaciers respond to variations in solar irra-diance. At-sensor radiance depends upon glaciersurface material composition and intermixture ofmaterials, solar and sensor geometry, and surfacetopography. To bridge the gap between investiga-tive findings and spectral data, a physically based(i.e., based on first principles) linkage betweenproperties of the observed surface and the measuredelectromagnetic signal should be established. Com-plementing the treatment of related subjects inChapter 2 of this book by Bishop et al., in thischapter we show how RT theory can be adaptedto derive radiative transfer equations (RTEs) thatare commonly employed to properly describe theradiative field within, at the surface of, and aboveglaciers, debris fields, and glacier lakes. RTEs arederived using the basic principle of conservation ofphotons and are simplified to obtain equations thatare more mathematically tractable. Such equationsare numerically solved to compute quantities thatare of interest in remote sensing (e.g., bidirectionalreflectance factor, BRF, and spectral albedo) thatare a function of the optical properties of theobserved surface. Accurate modeling of the opticalproperties of single-material particles (e.g., ice orsnow, water, lithic debris, and carbon soot) is crit-

ical to obtaining meaningful and accurate RTcalculations. The common methods employed todetermine single-scattering albedo and scatteringphase function, for both single-type particles andmixtures, are discussed. In addition, although thebasic conservation of photons holds for bothglaciers and glacier lake water, we have marked aclear distinction between the equation of transferfor glacier surfaces and glacier lake water, as well asbetween the methods employed to describe theiroptical properties. The chapter also providesexamples of RT-based calculations for both BRFand spectral albedo in scenarios typically found inthe cryosphere. Five simulation sets show howremotely measurable quantities depend on the mor-phological and mineralogical properties of the med-ium (e.g., BRF for mixtures of snow and debris;spectral albedo variation for snow and carbon sootwith varying grain size and particle concentration;and spectral variation of glacier lake water reflec-tance as a function of rock ‘‘flour’’ concentration).

3.1 INTRODUCTION

Remote sensing of the Earth’s cryosphere is anactive research area, as glaciological processes areclosely linked to atmospheric, hydrospheric, andlithospheric processes (Bush 2000, Shroder andBishop 2000, Meier and Wahr 2002) and a host

of issues of practical human concern. Global under-standing of cryospheric processes involves analysisof glacier dynamics since they are affected by andcan influence climate change (Kotlyakov et al. 1991,Seltzer 1993, Haeberli and Beniston 1998, Maisch2000). Thus, characterization and estimation ofglacier surface properties, such as ice grain size,rock debris cover, and surface water distribution,become critical to advancing our understanding ofglacier–climate relationships and glacier fluctua-tions (Bishop et al. 2004, Kargel et al. 2005).

Because of the ability of orbiting platforms toprovide global and continuous coverage of vastportions of Earth’s surface, satellite imagery canbe processed to extract information about impor-tant surface properties. Most space-based sensorsoperate passively, measuring the magnitude ofreflected/emitted surface radiance in the visible,infrared, and thermal portions of the spectrum.The Advanced Spaceborne Thermal Emission andreflection Radiometer (ASTER), Moderate Resolu-tion Imaging Spectrometer (MODIS), Polarizationand Directionality of Earth Reflectance instrument(POLDER), and Multi-angle Imaging Spectro-Radiometer (MISR) are examples of currentspace-based instruments that collect such multi-spectral data. Each of these instruments can recordthe directionality and intensity of surface radiance,thereby recording information about scatteringproperties that are a function of wavelength. Foricy surfaces on Earth, we require spatial and tem-poral information about: (1) the amount of clearand dirty ice; (2) snow coverage and rock debris(including both patchy and intimate mixtures);(3) the distribution of ponded or flowing water;(4) the liquid water content of snow and grain sizeof snow and ice; and (5) the extent of partial vegeta-tion cover (Kargel et al. 2005). These compositionaland phase-state parameters are critical to character-izing the current state and recent dynamics ofglaciers and, consequently, they are key to obtain-ing a better understanding of the processes asso-ciated with the cryospheric impacts of climatechange.

Quantitative characterization of surfaceproperties using remotely sensed data requires thedefinition of a functional relationship betweensurface properties and surface radiance. The bi-directional reflectance distribution function(BRDF) and/or the closely related bidirectionalreflectance factor (BRF), as well as the spectralalbedo, are the typical parameters employed todescribe surface reflectance given irradiance, sur-

face morphology, and composition variations.Glacier surfaces are generally comprised of avariety of materials and exhibit a complex reflec-tance distribution depending upon the spatial struc-ture of surface constituents. Spatial and temporalvariations in debris cover and intimate or arealmixtures between coarse-grained glacier ice, snow,liquid water, vegetation, and rock debris contributeto highly variable reflectance as observed by in situand platform-based sensors (Kargel et al. 2005).Modeling plays a central role in investigating therelationships between surface mixtures and reflec-tance, and can assist glacier mapping and charac-terization. BRDF and BRF modeling are importantfunctional components of scientific inquiry becausethey help bridge the gap between investigativefindings and field-based and remote observations.For example, Mishchenko et al. (1999) modeled thedirectional reflectance pattern and its effect onalbedo for four types of soils, each characterizedby a different index of refraction. The BRF patternswere generated for snow using three differentscattering phase functions (hexagonal ice, fractalice, and spherical ice) to examine the effect of icemorphology on reflected radiation.Models can also be used to provide a basis for

surface parameter retrieval. For example, Piatek etal. (2004) use model inversion and laboratory-baseddata to extract the basic optical properties of rego-lith as a function of the chemical composition andgrain size. Radiative transfer modeling was alsoused by Painter et al. (2003) to generate a lookuptable comprising BRF patterns for spherical icewith variable grain size. Each of the elements ofthe lookup table was considered an endmemberof the ice family, and a linear unmixing algorithmwas used to map grain-size distribution over alpineglaciers.Radiative transfer (RT) theory is the logical

quantitative framework and physical basis for mod-eling spectral and directional reflectance as a func-tion of surface composition and granular texture.Its theoretical basis was established by the seminalwork of Chandrasekhar (1960), who derived thebasic radiative transfer equation (RTE) describingthe transport of photons moving in a generic med-ium characterized by specified optical properties.He also derived a set of techniques to determineanalytical and numerical solutions for a largevariety of radiative transfer problems. Whereasthe RTE is capable of describing the radiativeregime within a prescribed host medium, most ofthe efforts involving solutions of the RTE focused

54 Radiative transfer modeling in the cryosphere

on determining the number of photons reflected bya planetary surface as a function of incident radia-tion, viewing geometry, and surface properties. Thegoal was to provide a means for quantitative inter-pretation of the signal collected by airborne and/orspaceborne instruments. For example, Hapke(1981, 1986, 1993, 2002, 2008) devised a semi-empirical RTE-based bidirectional reflectancemodel that, under simplifying assumptions, allowedthe analytical computation of the surface reflec-tance factor and other photometric properties ofinterest. The Hapke model has been successful inmodeling the reflectance factor of a variety ofplanetary surfaces in the solar system (e.g., Helfen-stein et al. 1988, Warell et al. 2009) and it is verypopular in the planetary science community.

More recently, Mishchenko et al. (1999) devisedan efficient and accurate BRF model that computessurface reflectance without the need to evaluate thelight field within the particulate surface. Accuratenumerical models that solve the RTE to determineboth the light field within and reflected by thesurface under investigation are also available. Themost popular model is DISORT (Stamnes et al.1988), which is a generic RT code capable of simu-lating the transport of photons in layered mediawith specified composition and optical thickness.A new multi-layer RT model based on recent theo-retical and numerical advancements in solving theRTE (Siewert 2000) has been developed and testedagainst currently available codes (Previti 2010;Previti et al. 2011; Picca et al. 2007, 2008a, Picca2009). Whereas the new model is general enough tomodel atmospheric RT, the Multi-layer AnalyticDiscrete Ordinate Code (MADOC; Furfaro et al.2014) has been originally conceived to specificallymodel the radiation regime within glacier environ-ments.

This chapter focuses on a fundamental treatmentof RT principles and the modeling of the radiationreflected by glacier surfaces over the spectral rangebetween 0.4 and 2.5 mm (VIS/NIR). We emphasizethe processes required to compute surface reflec-tance (e.g., BRF and BRDF) and spectral albedo,which are the most meaningful parameters in opti-cal remote sensing of Earth and planetary surfaces.In Section 3.2, the principles of RT are introducedand explained with special emphasis on a modelingapproach to adapt the generic RTE to properlydescribe the radiative regime for glacier surfacesand glacier lakes. In Section 3.3, we provide athorough description of the optical properties ofmaterials comprising a glacier surface (snow, ice,

rock debris) and, separately, the optical propertiesof glacier water. Methods currently employed tomodel the optical properties of mixtures are alsoreported. Section 3.4 briefly reviews the variety ofnumerical methods available to solve the RTE anddiscusses some of the RT codes available in theliterature. We then show some examples of simula-tions of albedo and BRF for a variety of scenariosthat are typically found on glacier surfaces.

3.2 RADIATIVE TRANSFER MODELING

OF GLACIER SURFACES

Modeling the radiative regime of glacier surfacesrequires a quantitative understanding of howphotons interact with complex surface composi-tions. RT theory provides the physical basis formodeling spectral and directional reflectance as afunction of composition and granular texture. TheRT problem and its mathematical machinery wereinitially developed by Chandrasekhar (1960), whoapplied the model to the transfer of light in atmo-spheres, oceans, and the interstellar medium. Here,we are interested in detailing a modeling approachthat yields RT equations capable of quantifyingthe light reflected and transmitted by glaciers andglacier lakes.We begin by considering the general problem.

Sunlight is comprised of photons traveling throughthe atmosphere. Ultimately, they reach the Earth’ssurface and interact with media such as that of thesurface of a glacier. The host medium is usuallycharacterized by an ensemble of particles with prop-erties that are a function of their composition andspatial arrangement. Commonly, such ensembles ofparticles include ice ranging from multicentimeterglacier ice to fine-grained snow and frost, liquidwater, suspended particles in the water, and rockdebris or soot resting on the glacier or intermixed inthe optical zone of the ice and snow. The ensemblesare modeled here as dense particulate media thatare illuminated by both direct and diffuse beams ofphotons and whose optical properties can be indi-vidually described by a finite number of parameters.Such parameters are intended to describe how themedium interacts with light particles. Generally,photons are either absorbed or scattered. A singlephoton can undergo multiple scattering before it iseither absorbed by, or exits from the medium backinto the atmosphere and space. Modeling radiative

Radiative transfer modeling of glacier surfaces 55

transport in dense particulate media is commonlydone either stochastically or deterministically.Here, we will utilize a deterministic modelingapproach that consists in formulating an integro-differential equation derived from a fundamentallaw of physics (i.e., the balance of photons in theappropriate phase space). In classical RT theory,the medium is assumed to be a collection of dimen-sionless scattering and absorbing centers uniformlydistributed in a differential volume. The photons’behavior is determined by the probability of scatter-ing and/or absorption within the host medium,assuming that streaming is possible between inter-actions. If the conservation of photons is applied insix-dimensional phase space (i.e., position andvelocity), the following equation is utilized:

1

c

@I�ðr;I; tÞ@t

þIErI�ðr;I; tÞ þ StotðrÞI�ðr;I; tÞ

¼ 1

4�

ZdI 0 Ssðr;I0 ! IÞI�ðr;I0; tÞ ð3:1Þ

where I�ðr;I; tÞ is the spectral radiance (energy/m2

sr s) of photons at location r traveling in thedirection I ¼ ð�; ’Þ within the cone dI. Spectralradiance is the (unknown) physical quantity thatdescribes the light distribution within, entering,and exiting the host medium. Importantly, � and’ are cosines of the inclination angle and the azi-muth angle, respectively, which are used to describethe photons’ direction of motion. The two terms onthe left-hand side of eq. (3.1) represent net energyloss of photons streaming out of the phase spacewhich is balanced by energy loss due to the scatter-ing and absorption (third term) and the inscatteringof photons in the phase space (right-hand side). Theparticipating medium is described by the absorptionand scattering coefficients. Here, StotðrÞ is the totalinteraction coefficient, defined as the sum of theabsorption and scattering coefficient (StotðrÞ ¼SabsðrÞ þ SscaðrÞ). In addition, Ssðr;I 0 ! IÞ isthe differential scattering coefficient (also calleddifferential scattering cross section or inscatteringcoefficient) which describes the probability thatphotons traveling in the I

0 direction are scatteredin the dI about I direction. In conventional RTtheory, the host medium is assumed to be ‘‘rota-tionally invariant’’ (i.e., the differential scatteringcoefficient depends only on the angle between I

andI0). With the latter position, and assuming that

the transport of photons is a function of only onespatial variable (depth, �), it is customary to writethe differential scattering as SsðI 0 ! IÞ ¼

SsðI 0EIÞ ¼ !�ð�Þp�ð�; cosYÞ where !�ð�Þ andp�ð�; cosYÞ are the single-scattering albedo andthe scattering phase function, respectively, andY ¼ cos�1ðI 0EIÞ is the angle between the twodirections (see Section 3.2.2 for a more precisedefinition of both !�ð�Þ and p�ð�; cosYÞ).To complete the mathematical description of the

balance of photons interacting with the host med-ium, proper boundary conditions that account forthe radiative flux of photons entering the mediummust be provided. Importantly, for the case of ice,snow, debris, and their mixtures, the scattering pro-cess does not alter photons’ energy (i.e., interactionwith the host medium does not shift wavelength).Therefore, photons maintain their energy whilechanging direction. Eq. (3.1) can then be solvedindependently at each wavelength to determinethe radiative regime of the observed surface. Insupraglacial lakes, Raman scattering requires thatphotons change energy during their interaction pro-cess. Consequently, eq. (3.1) must be modified withan additional wavelength-dependent source termthat accounts for such effects on overall photonbalance (see Section 3.2.3). Bioluminescence is nor-mally absent or negligible and is ignored for glaciallakes.Eq. (3.1) is an integro-differential whose solution

has challenged researchers for decades. In the radia-tive transfer and particle transport communities, itis known as the linearized Boltzmann equation(LBE) and generally does not have any analytical,closed-form solution. Although eq. (3.1) is the basisfor our modeling, several assumptions must beenforced to derive a more manageable form. Oneimmediate simplification comes from the observa-tion that the 1=c term factoring the time derivativeof I�ðr;I; tÞ is smaller than the intensity flux timerate (Davis and Knyazikhin 2005). The latterimplies that a steady state is reached almost instan-taneously. Indeed, radiative transfer in passiveremote-sensing applications is generally modeledas a stationary phenomenon; time-dependent RTproblems are considered only when modeling theresponse of the surface to active pulsed illuminationby remote-sensing instruments (e.g., LiDAR).

3.2.1 RT modeling approach for

glacier surfaces

Glacier surfaces and supraglacial lakes generallyexhibit a complex spatial arrangement of different


materials. Consequently, any RT model should beflexible enough to permit the incorporation ofcompositional and spatial structure scenarios thatare typically found in such environments (e.g., atwo-layer arrangement that includes an upperlayer of mixed snow and debris overlaying abottom layer of pure bubbly ice). The naturalvariability of surface composition and lake sus-pended debris, however, makes it extremely difficultto develop a comprehensive RT model that is sim-ple (i.e., mathematically tractable), accurate, andcomputationally inexpensive. Clearly, if a compre-hensive description of the optical properties appear-ing in the LBE is available, one may attempt todirectly solve eq. (3.1). While this is in principlepossible, the dimensionality of the problem (threepositions and two angular variables) coupled withthe integro-differential nature of the LBE makeit virtually impossible to find accurate and fastsolutions. Consequently, the RTE must besimplified.

A closer look at the actual material and particle-size arrangements typically found on glaciersprovides the insight that simplified geometries cansupport rigorous but simplified RTEs to describethe BRDF and spectral albedo. A vertically layeredapproach implies that radiant intensity dependsonly on one spatial (optical depth) and two angularvariables. The layering assumption is justified byreal case examples found in natural glacier environ-ments. For example, Fig. 3.1 depicts monomineralicand monolithologic materials, intimate mixtures,and areal mixtures of materials on the Root andKennicott Glaciers in the Wrangell Mountains,Alaska. Whether a patch of surface material is effec-tively monomineralic or monolithologic depends onthe scale and the situation. To explain areal mix-tures, consider Figs. 3.1(A, B). At the ASTER pixelfootprint scale (15 m/pixel for VNIR), the majorglaciological features (especially medial morainesand intermoraine lanes of ice, snow, and firn) wouldbe totally resolved and are each optically thick;these materials may be approximated as either purerock debris or pure medium-grained ice (firn). AMODIS pixel (250 m) would not resolve the indi-vidual medial moraines and ice patches; in this case,linear areal mixing models could adequately simu-late reflectance. So long as photons do not penetratethrough grains far enough to interact with differenttypes of minerals, the reflectance signature ofsuch patch-wise areal mixtures is simply the area-averaged reflectance spectra of the mineral compo-nents making up the surface.

Figs. 3.1(C–J) show that more complex situa-tions, not well modeled as areal mixtures, are alsocommon on glaciers. Most glaciers possess broadareas of impure ice stained by small amounts ofsurficial or intermixed rock grit. As detailed close-up images show (Figs. 3.1(I, J); J being an extremecolor stretch of I), the rock grit is so fine that it isoptically thin and to some extent is intermixed withice in the upper few millimeters; this slight debris-laden ice has a structure of massive, clean, coarsecrystalline ice overlain by a zone just a couple ofmillimeters thick where ice is effectively intermixedwith fine silicate rock particles. We refer to a homo-geneous or random intermixing of two (or more)phases, with one phase having high photon trans-mittance relative to the other phase, like an intimatemixture. Thus, intimate mixtures permit photons tointeract with two or more materials of differingoptical properties (e.g., ice with intermixed rockflour). As we demonstrate in the simulation section,such intimately mixed surfaces exhibit unique spec-tral properties and unique bidirectional reflectancecharacteristics different from pure ice, pure rock, orareal mixtures of rock and ice.The glacier surface also has centimeter-scale grit-

free ‘‘windows’’ that expose the coarse crystalline‘‘blue ice’’, and optically thick centimeter-size rockparticles. Photons are admitted through the grit-free windows into coarse-grained massive ice belowthe gritty ice. The grittier areas admit and emitfewer photons per square millimeter, but they dom-inate the area fraction of this surface. AlthoughFigs. 3.1(I, J) point out the real world complexityof such surfaces from a radiation transfer perspec-tive, we can approximate the geometry as a deeplayer of clean coarse crystalline ice overlain by athin layer of an intimate mixture of ice with finerock detritus. The more complex real world geom-etry can then be simulated by patches of intimatelymixed dirty ice with patches of pure clean ice andpatches of pure debris. Complex, realistic scenarioscan therefore be generated in a stepwise fashion.The layering assumption reduces the dimension

of the RTE yet allows meaningful modeling of awide range of spatially complex material structures,including multi-layers of intimate mixtures. In thisframework, two major configurations for materialarrangement are possible. In the first configuration,intimate mixtures of ice and sediment can be mod-eled as a homogeneous single layer (optically thick),where the optical properties of the layer are derivedby a weighted average of the optical properties ofthe pure components (see Section 3.3.4). In the sec-


ond configuration, the surface is assumed to exhibitan idealized multi-layer structure. Each layer ismodeled as homogeneous, with a defined opticalthickness, and its composition may include eithera pure material or an intimate mixture of differentelements (ice, snow, debris). In addition, RT modelsshould be able to describe scenarios where, in asingle-layer configuration, the surface exhibitspatches of homogeneous material. Within the limitsof its spatial resolution, any remote-sensing instru-ment measures the spectrally average electromag-netic radiation reflected by the area correspondingto one pixel. Assuming an independent pixelapproximation (IPA), RT models can be devisedto compute the flux intensity of heterogeneouspatched areas by calculating the average flux inten-sity of homogeneous subpixel components weightedwith their area coverage.

3.2.2 Radiative transfer equation in

layered mixtures of snow, ice, and

debris

The major assumption that reduces the problemdimension is to set the medium to be infinite inthe horizontal direction so that photon transportwithin the medium is one dimensional (the depthvariable). If a single-layer configuration is consid-ered, a further simplification is obtained by workingwith a spatially homogeneous medium. Verticalheterogeneity can be described by assuming amulti-layer configuration where the medium is sub-divided in many layers, each having different opticalproperties. Under these conditions, the LBE can besimplified. The spectral radiant intensity for a ver-tically heterogeneous medium is governed by thefollowing 1D, two-angle RTE (Chandrasekhar


Figure 3.1. (A–J) Examples of monomineralic and monolithologic materials, intimate mixtures, and areal mixtures

of materials on the Root and Kennicott Glaciers in theWrangell Mountains, Alaska. On the right side of the figure are

illustrated a set of typical scenarios that can bemodeled usingmulti-layer 1D, two-angle radiative transfer equations

(eq. 3.3). The geometrical configurations include single-layer and multi-layer arrangements of mixtures of ice and

rock debris. Figure can also be viewed as Online Supplement 3.1.

1960, Siewert 1978, 2000):

�@

@�I�ð�; �; ’Þ þ I�ð�; �; ’Þ

¼ !�ð�Þ4�

Z 2�

0

d’0Z 1

�1d� 0 p�ð�; cosYÞI�ð�; � 0; ’0Þ

ð3:2Þwhere, for a given wavelength �, I�ð�; �; ’Þ is thespectral radiant intensity, � ¼

Rx0 �tot dx

0 is theoptical thickness (varying in the range � 2 ½0;D�),� 2 ½�1; 1� is the cosine of the polar angle (meas-ured from the positive �-axis), ’ 2 ½0; 2�� is theazimuth angle, !�ð�Þ ¼ �scað�; �Þ=�absð�; �Þ is thespatially dependent single-scattering albedo,defined as the ratio between scattering coefficient�scað�; �Þ and total interaction (extinction) coeffi-cient (�totð�; �Þ ¼ �scað�; �Þ þ �absð�; �Þ (i.e., sumof absorption and scattering coefficient), andp�ð�; cosYÞ is the spatially dependent scatteringphase function.

In the case of a multi-layer approximation, themedium is subdivided into N layers each havingdifferent optical properties. For each layer, thesingle-scattering albedo and the scattering phasefunction are defined either for a single material orfor a mixture, respectively as !s

�, s ¼ 1; . . . ;N andps�ðcosYÞ, s ¼ 1; . . . ;N. For each of the N layers,we write the following equation (s ¼ 1; . . . ;N):

�@

@�IðsÞ� ð�; �; ’Þ þ I

ðsÞ� ð�; �; ’Þ

¼ !ðsÞ�

4�

Z 2�

0

d’0Z 2�

0

d�0 pðsÞ� ðcosYÞI

s�ð�; � 0; ’0Þ

ð3:3ÞThe RTE must be equipped with boundary con-ditions describing radiant intensity at the top andbottom layers. Moreover, the continuity of radiancebetween layers must be enforced. For s ¼ 1 (topboundary condition):

Ið1Þ� ð0; �; ’Þ ¼ S0�ð�� 0Þ�ð’� ’0Þ þ f ð�; ’Þ ð3:4Þ

At the interface between slabs:

IðsÞ� ð�

ðsÞ; �; ’Þ ¼ Iðs�1Þ� ð� ðsÞ; �; ’Þ

IðsÞ� ð�

ðsþ1Þ; �; ’Þ ¼ Iðsþ1Þ� ð� ðsþ1Þ; �; ’Þ

9=; ð3:5Þ

For s ¼ N (bottom-boundary condition):

IðNÞ� ð0;��; ’Þ ¼ 0 ð3:6Þ

where � 2 ½0; 1�. Eq. (3.4) represents the incidentdirectional field (first term on the RHS) and diffuse

radiation field (second term at the RHS). Eq. (3.5)ensures that the continuity of radiant intensity atthe interface between layers is respected. Eq. (3.6)imposes a non-reentrant boundary condition at thebottom layer. Typically, the bottom layer isassumed to be optically thick (i.e., semi-infinite).

3.2.3 Radiative transfer equation in

glacier lake waters

RT modeling for glacier-related surface water canbe based on using a plane-parallel approximation,where the water column is assumed to be layered,and surface irradiance is assumed to be uniformover the surface. Because of small horizontal varia-tion in water optical properties, the RT problemcan be treated spatially using the depth variable.Using notation that is customary in the ocean opticscommunity, radiance is governed by the followingintegro-differential equation:

�@

@zLðz; �; ’; �Þ þ cðz; �ÞLðz; �; ’; �Þ

¼ bðz; �ÞZ 2�

0

d’0Z 1

�1d�0 �ðz; �0 ! �; ’ 0 ! ’; �Þ

�Lðz; �0; ’0; �Þ þ Sðz; �Þ; ð3:7Þ

where z is downward distance from the lake surface,� is the cosine of the polar angle, and ’ is theazimuth angle in the horizontal plane. The sum ofthe scattering and absorption coefficient is calledthe beam attenuation (extinction) coefficientcðz; �Þ ¼ aðz; �Þ þ bðz; �Þ, whereas �ðz; �0 ! �;’0 ! ’; �Þ is the scattering phase function. Proper-ties aðz; �Þ, bðz; �Þ, and cðz; �Þ are commonlyknown as inherent optical properties (IOPs). IOPsare critical parameters that describe the bulk opticalbehavior of lake water and are of general interest ininverse remote sensing. Sðz; �Þ is a source functionthat accounts for the Raman scattering phenom-enon. Such a term complicates the nature of theequation, and finding accurate numerical solutionsbecomes more challenging. This is due to the factthat in such cases the problem cannot be solved‘‘one wavelength at a time’’ as in the conventionalcase (see Section 3.2.2). The latter implies that in eq.(3.7), the wavelength variable is coupled with bothspatial and angular variables. Nevertheless, a sim-plified version of eq. (3.7) which does not depend onSðz; �Þ may be sufficient to describe photons’behavior because bioluminescence sources are verysmall during the daytime, and both fluorescenceand Raman scattering become negligible.


Glacier lake water is generally assumed to beeither homogenous (well mixed) or verticallystratified which, as in the case for glacier surfaces,enables the use of multi-layer configuration models.Eq. (3.7) is also equipped with boundary conditionsthat are similar to the one described in the previoussection. The top layer is illuminated by an incidentand diffuse source (see eq. 3.4). If lake depth islarge, the medium is optically thick and non-reentrant conditions at the bottom layer (eq. 3.6)are appropriate. If the lake is shallow, reflectiveboundary conditions, which depend on lake bottomcomposition, shall be imposed at the bottom layer,although we do not provide simulations of suchinstances.

3.3 OPTICAL PROPERTIES OF SNOW,

ICE, DEBRIS, MIXTURES, AND

GLACIER LAKE WATER

The RTE can be solved as a function of surface/water properties (e.g., mineralogical composition,grain size distribution) and solar/sensor geometry.The RTE describes the multiple scattering ofphotons through the host medium and criticallydepends on single-particle optical behavior. Fromthe photon–medium interaction point of view, suchbehavior is regulated by the ability of the hostmedium to absorb and scatter light in any direction.Indeed, a single particle is characterized by absorp-tion and scattering efficiencies that are related tothe probability that a photon will be scattered orabsorbed by a single particle of defined shape andsize. Scattering and absorption processes areaccounted for via two optical property parameters:(1) single-scattering albedo, which represents theprobability of a single photon to be scattered inany direction; and (2) the scattering phase functionp�ðcosYÞ (or �ðz; �0 ! �; ’0 ! ’; �Þ for glacierlake water), with Y being the angle between theincident and scattering direction. The phase func-tion specifically describes the probability that aphoton coming from a direction I

0 ¼ ð�0; ’ 0Þ isscattered by a particle in direction I ¼ ð�; ’Þ. Par-ticulates that typically comprise surfaces in glacierenvironments are rotationally invariant (i.e., theprobability of scattering depends only on the rela-tive angle between incident and scattered radiation).While single-scattering albedo is a scalar parameterranging between 0 and 1, the phase function is

customarily described by a series of coefficientsrepresenting the projection of p�ðcosYÞ onLegendre polynomials. Generally, snow, ice, soil,andwater exhibit a strongly forward-peaked scatter-ing function (i.e., relatively large single particles tendto scatter radiation primarily in the forward direc-tion). The latter implies that a large number ofcoefficients (of the order of thousands) are requiredto accurately characterize the single-particle phasefunction. The asymmetry parameter g, which isdefined as the cosine of the average of p�ðcosYÞ(also as the degree of forward scattering of themedium), may be conveniently employed to reducethe number of parameters required to describe theoptical properties of a single particle. The popularHeyney–Greenstein model is therefore widely usedto approximate the phase function for the particlesof interest:

pðcosYÞ ¼ 1� g2

ð1� 2g cosYþ g2Þ3=2; g 2 ½�1; 1�

ð3:8Þ

The scattering phase function can be expanded inLegendre polynomials PlðcosYÞ with coefficientsthat are directly related to the asymmetry parameterin the following manner:

pðcosYÞ ¼XLl¼1

�lPlðcosYÞ ð3:9Þ

�l ¼ ð2j þ 1Þg l ð3:10Þ

Therefore, single-particle optical properties can be,at minimum, characterized by knowledge of single-scattering albedo and asymmetry parameter. Alter-natively, a number of scattering coefficients arerequired to describe the particle phase function.Such parameters depend on the size of the particle,wavelength, index of refraction, and the shape ofthe particle. The most common approach to deter-mine optical properties is to assume that the particleis a perfect sphere. In this case, optical propertiescan be computed in closed form by directly solvingMaxwell equations. Mie theory (Wiscombe 1980)has been widely used to model the optical propertiesof snow, ice, and soil. More involved approaches,such as the T-matrix method (Watermann 1971,Mackowski and Mishchenko 1996) and the ray-tracing method (Bohren and Barkstrom 1974,Macke et al. 1996), have been developed to accountfor arbitrary and irregular shapes. Such methods,however, tend to be computationally expensive.


3.3.1 Snow

From the radiative transfer perspective, snow canbe viewed as a collection of ice particles immersed inair. A single particle of ice has variable shape andsize and an exact description of single-scatteringalbedo and phase function involves using methodsof geometric optics. Yang and Liou (1998) appliedMonte Carlo–driven ray-tracing algorithms to com-pute the optical properties of ice crystals for a sub-stantial variety of shapes including plates, hollowcolumns, bullet rosettes, and ice aggregates.Whereas a database of properties has been gener-ated, accurate description of the phase function forsuch complex shapes implies that one needs RTcode capable of handling thousands of phase func-tion coefficients to compute the radiative regime inmedia with irregular particles, which results inimpractical computationally expensive calculations.A more popular approach employed to describesnow particle optical properties makes extensiveuse of Mie-based code. Founded on Mie theory(Wiscombe 1980, 1996), such code can effectivelycompute efficiencies, single-scattering albedo, andasymmetry parameters as a function of grain size.However, calculations based on Mie theory may beseverely limited by the fact that the theory worksrigorously only if the particles are assumed to beperfect spheres. Whereas single snow particles arenot spheres, an ensemble of snow particles is pos-tulated to behave as ‘‘optically equivalent spheres’’(i.e., a collection of snow grains having the samevolume/surface ratio; Dozier et al. 1987). Mugnaiand Wiscombe (1980) demonstrated that a collec-tion of nonoriented spheroids have the same scat-tering behavior as an ensemble of spherical particlesof equivalent size. Mie-based code requires knowl-edge of the complex index of refraction n ¼ nr þ ni.Generally, the real part is interpreted as phasevelocity and the imaginary part describes absorp-tion loss for an electromagnetic wave movingthrough the particle. For the case of ice, theimaginary part is linked to volume absorption ofice (Wiscombe and Warren 1980). Fig. 3.2 showsboth real and imaginary parts of the index of refrac-tion for pure ice (Wiscombe 1980, Warren andBrandt 2008). Importantly, the magnitude of theimaginary part varies across the visible and thenear-infrared (NIR) by many orders of magnitudeimplying that: (1) ice is transparent in the visibleregion (small imaginary index of refraction); and(2) ice is moderately absorptive in the NIR(imaginary index of refraction is larger and

increases with wavelength). Wiscombe and Warren(1980) studied the behavior of pure ice single-particle optical parameters using Mie theory andshowed that the extinction coefficient and asym-metry parameter are relatively insensitive to wave-length (typical value for g ranges between 0.88 and1), and that !� (or the co-albedo 1� !�) is mainlyresponsible for the spectral variation of snowalbedo. Generally, !� is very close to 1 in the opticalregion (highly scattering snow medium across thevisible) and decreases monotonically reaching aminimum value of 0.5 in the NIR. Increasing grainsize is shown to decrease !�. Generally, values of 50mm are assumed for fresh snow, whereas 1 mm isassumed for grain clusters or wet snow. Water inthe snowpack is not usually modeled, as the indexof refraction of water is very close to that of ice.

3.3.2 Glacier ice

Whereas snow and ice are the major material com-ponents, the physical state and the optical proper-

Optical properties of snow, ice, debris, mixtures, and glacier lake 61

Figure 3.2. Spectral behavior of the real part (A) and

imaginary part (B) of the complex index of refraction for

pure ice. Figure can also be viewed as Online Supple-

ment 3.2.

ties of various glacier surface characteristics underinvestigation vary dramatically. Snowfall is trans-formed to ice through a variety of mechanismsincluding: (1) mechanical settling; (2) sintering;(3) refreezing of meltwater; and (4) refreezing ofsublimating ice. The grain size of surface ice varieswidely. In the accumulation zone, fine snow maydominate. In the firn zone (annealed snow), grainsize may effectively be in the range of several milli-meters. In the ablation zone, bubbly ice may havean effective grain size of a few millimeters, butdense, well-crystallized ice may have grain sizes of1–10 cm. In some cases, glaciers may be effectivelylayered in the optical zone, with a dusting of snowor bubbly ice overlying denser, coarser ice. Multiplescattering of such complex structures requiresproper modeling of the properties of single com-ponents which are usually mixed with other com-ponents (snow, bubbles, rock debris, and soot, seeSection 3.3.4) and possibly arranged in multi-layerconfigurations. However, glacier clean ice may bemodeled as a collection of bubbles trapped within amatrix of transparent ice.

The firn, which is snow material after the trans-formation process has begun, is initially porous andcontains interconnected air channels. As the densityincreases above 880 kg/m3, the channels close offresulting in a mixture of ice and bubbles trappedwithin the glacier body. Mullen and Warren (1988)provided a framework for modeling the volumetricscattering of such bubbly ice. Indeed, in a pure icesample containing only air bubbles, the physics ofinteraction between the photons and the host med-ium is such that absorption occurs in the ice matrixand scattering occurs at ice bubble boundaries.Thus, the absorption process is generally separatedfrom the scattering process. Scattering is dominatedby the size and distribution of the air bubbles withinthe ice. If bubbles are assumed to be spheres, Mietheory can be employed to compute the scatteringefficiency (and subsequently the scattering coeffi-cient) as well as the asymmetry parameter as afunction of bubble size. In this case, the Mie-basedcalculation follows the same process as in the caseof snow, but with the imaginary part of the complexof refraction set to zero (n ¼ nr). Marston et al.(1982) found good agreement with the scatteringof a single air bubble in water, which has an indexof refraction similar to ice. Conversely, absorptionis assumed to be exclusively a function of theamount of ice per unit volume. For example,Bohren (1983) assumed that the absorption coeffi-cient of an air–ice mixture is that of pure ice multi-

plied by the volume fraction of ice in the sample.Once the scattering and absorption coefficients areavailable, both the extinction coefficient and single-scattering albedo can be computed to complete theoptical characterization of the volumetric scatteringof clean ice.Microtopography is important on some glacier

surfaces and can introduce complex scatteringand microshadowing effects. We do not address thisimportant topic in this chapter.

3.3.3 Rock debris

Debris-covered glaciers include varying amounts,grain sizes, and spatial arrangements of rock debris.Moraines may consist mainly of boulders and coverthe ice completely. Debris patches may be scatteredamongst clean ice exposures. Fine rock flour may beintimately mixed with ice. The optical characteriza-tion of single-particle absorption and scattering forsoil/sediment is, however, very difficult. Generally,soil particle distributions can vary in size, shape,and mineralogy. Whereas Mie theory has beenfairly successful in describing the optical propertiesof snow and ice, it has not been able to accuratelyreproduce the optical properties of soil particles.Indeed most materials found on Earth and planet-ary surfaces exhibit a nonuniform structure andcomposition.Two decades ago, Hapke (1993) presented a gen-

eralized model for the scattering efficiency of largeirregular particles. The model is based on an exten-sion/generalization of the equivalent slab model forirregular particles. The model is approximate butcan be used to compute scattering efficiency inclosed analytical form and has been mostlyemployed in the planetary science community(e.g., Warell et al. 2009). More recently, T-matrixcode has been made available to describe the opticalproperties of particles that are large and irregular.Based on the linearity of Maxwell’s equations(Watermann 1971), the method has some computa-tional advantage where the particle under consid-eration is axial symmetric, because the matrix iseasily subdivided into independent matrices, yield-ing more efficient and faster calculations. T-matrixmethods are also available to compute the opticalproperties of particle clusters with definedorientation (Mackowski and Mishchenko 1996).Generally, such algorithms are computationallyexpensive, and recently some of the available codehas been redesigned to run on parallel clusters ofmachines (Mackowski and Mishchenko 2011).


3.3.4 Mixtures

The optical properties of single particles can beemployed to determine the optical behavior ofmulticomponent mixtures. Computing single-scattering albedo and the asymmetry parameterfor a mixture is fairly straightforward. Let usconsider the case of two pure components eachcharacterized by spherical particles of radius r1and r2 and with scattering coefficient �sca;1ð2Þ ¼n1ð2Þ�r

21ð2ÞQsca;1ð2Þ and total interaction (extinction)

coefficient �tot;1ð2Þ ¼ n1ð2Þ�r21ð2ÞQtot;1ð2Þ where

Qsca;1�2 and Qtot;1�2 are the scattering and extinc-tion efficiencies for type 1(2) particles, and n1ð2Þ isthe number of particles per unit volume of type 1(2)particles. The combined single-scattering albedo ofa mixture of two types of particles can be deter-mined as follows:

!mix ¼�sca;1 þ �sca;2�tot;1 þ �tot;2

¼n1�r

21Qsca;1 þ n2�r

22Qsca;2

n1�r21Qtot;1 þ n2�r

22Qtot;2

ð3:11Þ

If particles are not spherical, �r2 shall be replacedby relative particle cross-sectional area. If particlesize is modeled as a size distribution, then comput-ing an integral with the weighted size distributionis required. Similarly, combined phase functionmoments are computed by averaging the momentof each component and weighting it by its scatteringcoefficient. For example, the asymmetry parameterof a two-component mixture is computed as fol-lows:

gmix ¼g1�sca;1 þ g2�sca;2�sca;1 þ �sca;2

¼g1n1�r

21Qsca;1 þ g2n2�r22Qsca;2

n1�r21Qsca;1 þ n2�r

22Qsca;2

ð3:12Þ

3.3.5 Glacier lake water

Glacier water includes lakes, streams, and smallpuddles and rivulets. In this section we will notconsider thin films (such as wetted rocks andwater-saturated snow) or optically thin pondswhere substrate effects are important. The opticalproperties of optically thick glacier water aredescribed by explicitly characterizing the absorp-tion, scattering, and total beam attenuation coeffi-cients as a function of the various componentspresent in the water body. Optical properties areresponsible for the large variation in the supra-glacial lake colors that can be observed in alpine

environments ranging from exquisite turquoise andother pastels, to brilliant green, cobalt blue, andgray. The colors of optically deep lakes pertainmainly to their suspended sediment loads, especiallyclastic sediment in glacial lakes and substrate effectsfor optically shallow lakes.Suspended clastics in glacier meltwater have two

major components: (1) rock flour—mainly clay tofine silt size—produced by glacier grinding at thebed; (2) silt and sand introduced by nonglacialstreams and incorporated into glacier waters frommass-wasting deposits or other sources that havenot been pulverized at the glacier bed. These com-ponents are usually distinct in their grain-rounding,faceting, and particle size–frequency distributions.In heavily glacierized basins (where glacial sedimentinput is much greater than nonglacial input) innonpolar climates (where glaciers have thawedbeds), glacier rock flour overwhelmingly dominatesmost lakes. Glacial grinding produces extremelyfine particles due to rock being forced to slide overother rocks, whereas in ordinary streams, particlesare generated as chips that are shed from largerclasts during water-driven impacts. The energyand momentum relations are such that particles afew microns in radius become ineffective abradingagents in normal streams. Hence, glacial rock flourincludes particles sizes from hundreds of microns toa tenth of a micron or smaller, whereas non-glacialstreams normally carry suspended load particlesizes from a few hundred microns down to a fewmicrons. Although particle sizes on the scale of lightwavelengths can total up to a small fraction of thetotal mass of suspended load, those size ranges candominate scattering due to the large specific surfacearea of those fine grains. Such particle sizes canbe virtually absent from nonglacial waters, thusaccounting for some of the difference between thecolors of typical nonglacial lakes versus glaciallakes.From the optical perspective, turbid nonglacial

lake waters, which are generally classified as ‘‘Case2’’ waters (Mobley 1994), can be difficult tocharacterize because they often consist of waterplus dissolved organic and inorganic matter. Mostglacial lakes and streams have negligible organiccomponents, but some isolated ponds can developa biota and suspended and dissolved organic mat-ter. These lakes are strongly influenced by coloreddissolved organic matter (CDOM also known asGelbstoff, yellow substance, or gilvin), detritus,mineral particles, bubbles, and other substancesincluding phytoplankton. Such particles are not

Optical properties of snow, ice, debris, mixtures, and glacier lake 63

very well characterized in terms of type and loca-tion, and they do not covary with phytoplanktonconcentration as in Case 1 waters. Generally, theoptical properties of such waters are assumed to beadditive, where the absorption and scattering coeffi-cient can be written as a sum of the modelsemployed for absorption and scattering of the indi-vidual components:

að�Þ ¼ aW ð�Þ þ aPð�Þ þ aXð�Þ þ aYð�Þ ð3:13Þ

bð�Þ ¼ bW ð�Þ þ bPð�Þ þ bXð�Þ þ bYð�Þ ð3:14Þ

where aW ð�Þ; aPð�Þ; aXð�Þ; aY ð�Þ (bW ð�Þ; bPð�Þ;bXð�Þ; bY ð�Þ) are the absorption (scattering) coeffi-cients for pure water, phytoplankton, mineral par-ticles/detritus, and CDOM, respectively. Modelsand/or experimental data for individual compo-nents are more or less available. For pure water,scattering and absorption coefficients have beenmeasured. Fig. 3.3 shows the wavelength-dependentscattering and absorption coefficient of pure wateras measured by Pope and Fry (1997). Opticalmodels for phytoplankton and CDOM are avail-able from the literature (Prieur and Sathyendranath1981). Generally, CDOM absorption is assumed todecay exponentially with wavelength, whereas scat-tering due to dissolved microscopic pigments isassumed to be negligible. Phytoplankton absorp-tion is a complex function of its concentrationand wavelength, and requires two empirical wave-length-dependent functions. Phytoplankton scatter-ing is proportional to the inverse of the wavelength

and directly proportional to its concentration to thepower of 0.6. Suspended particle optical propertiesin glacier lakes are not well known and require insitu measurement. Suspended sediments tend to bemostly scatterers although iron-rich sediments mayexhibit strong absorption features. Heege (2000)measured the optical behavior of suspended par-ticles in Lake Constance and found that the par-ticles had no influence on the overall absorptioncoefficient (aXð�Þ ¼ 0) whereas they heavily influ-enced the scattering properties of the lake waters. Itwas found that the backscattering coefficient wasindependent of wavelength and proportional tothe particle concentration.Scattering in glacier and oceanic environments is

strongly forward peaked with peak ratio betweenforward and backward single-scattering ratios ofthe order of 105. Various models are available todescribe such a type of scattering behavior althoughtwo types of phase functions are generally used forwater including the Petzold phase function (Petzold1972) and the Fournier–Fourand model (Fournierand Fourand 1994). Mobley et al. (2002) examinedthe effect of various phase functions on the com-puted radiance and concluded that: (1) knowledgeof the appropriate phase function is as importantas knowledge of the absorption and scatteringcoefficient; and (2) accurate knowledge of the shapeof the phase function is not required as long asthe backscattering fraction is correct. It was alsoshown that the Fournier–Fourand model worksvery well.

3.4 NUMERICAL SOLUTION OF

THE RTE

Computing the radiative regime within glaciers, aswell as the amount of radiation reflected by glaciersurfaces and/or glacier lakes, requires solving theRTEs formally presented in the previous sections.Due to its mathematical complexity, an analyticaldescription of the light field is virtually impossible.Over the past few years, many approximate meth-ods have been developed to provide analyticalexpressions for the multiple scattering of photonsin snow, ice, and soil. One of the most popularanalytical methods is based on approximating theradiance as one upwelling and one downwellingstream of photons as they are constrained to moveonly in two directions (two-stream approximation).


Figure 3.3. Spectral behavior of absorption and scat-

tering coefficients for pure water. Figure can also be

viewed as Online Supplement 3.3.

Such an assumption allows the determination ofanalytical formulas that quickly compute upwellingand downwelling diffuse fields (see Wiscombe andWarren 1980 for an example of a two-stream calcu-lation applied to the problem of determining snowalbedo). In an attempt to provide a quantitativetool for the analysis and interpretation of the lightreflected by planetary surfaces, Hapke (1981) deter-mined an analytical expression for the reflectancefactor of an optically thick surface. He decomposedthe radiative field into two major components con-sisting of a scattered component and a multiple-scattered component. An approximate solutionwas obtained by: (1) determining an exact expres-sion for the first scattered component; and (2) deter-mining an approximate analytical expression forthe multiple-scattered component, assuming thatthe host medium is isotropic from the second col-lision on.

Whereas approximate methods provide theanalyst with analytical expressions that are easyto implement, they do not provide accurate solu-tions of the RTE. Indeed, accurate solutions can befound only by solving the RTE numerically. Overthe past two decades, the widespread availability ofhigh-speed digital computers permitted researchersto develop and test more efficient and faster algo-rithms to compute radiance and the reflectance fac-tor. All the available numerical algorithms havebeen based on some form of discretization of thespatial and angular variable, thus resulting in RTsoftware that can be used to simulate the lightreflected by various surfaces.

One of the most efficient ways to solve the RTE isto discretize the angular variable in a set of finitedirections, and then solve the resulting differentialequations. Cast as an eigenvalue problem, the set ofequations can be solved using conventional numer-ical methods (e.g., relaxation method) to find ananalytical expression for radiance along fixed direc-tions. The DISORT code (Stamnes et al. 1988) isthe most popular and is based on discretizing theangular variable. It has been widely used by thecommunity to compute the reflectance factor andspectral albedo of snow as a function of grain size(e.g., Aoki et al. 2000, Painter et al. 2003).

Recently, Siewert (2000) proposed a new formu-lation of the discrete ordinate approach for a singlelayer with specified optical properties and thickness.The method, called the analytical discrete ordinatesmethod (ADO), is a novel semi-analytic approachin which the angular variable is discretized to deter-mine a set of ordinary differential equations that

have unknown radiant intensities along discretedirections. The set of equations are solved by:(1) numerically solving the resulting eigenvalueproblem to compute the homogeneous solution;and (2) use of a modified Green’s function formula-tion to compute the particular solution of the set ofdifferential equations arising from angular dis-cretization. More recently, the method has beenextended to include multiple layers of optical prop-erties with special routines that give the method theability to quickly and efficiently handle thousandsof layers (Picca et al. 2007, 2008a, Picca 2009, Pre-viti 2010, Previti et al. 2011). The principles behindthe ADO have been implemented in a novel RTcode called Multi-layer Analytical Discrete Ordi-nates Code (MADOC, Furfaro et al. 2014). Devel-oped in a MATLAB environment, MADOC will besoon made available to the larger scientific com-munity. Importantly, MADOC was used to providethe simulation examples in the next section. Dis-crete ordinate methods have also been used to gen-erate code capable of computing matter/energyinteractions in water. Indeed, the Coupled OceanAtmosphere Radiative Transfer code (COART; Jinet al. 2006) extended the DISORT numerical plat-form to handle the change in photon directionbetween the water and atmosphere interface dueto differences in the index of refraction.In principle, both angular and spatial variables

can be discretized to transform the RTE into a setof differential equations that can be implementedusing a digital computer. The most elementaryapproach to what is called ‘‘vanilla discretization’’is the SN1 method (Lewis and Miller 1984). Onemajor drawback of the method is that accuratesolutions can only be obtained if the discretizationmesh is sufficiently fine. For optically thick mediaand settings with forward-peaked phase functions,the method becomes computationally expensive.More recently, it has been shown that accuratesolutions of the RT equation can be obtained bysolving a sequence of RT problems using SN fromcoarser to finer grids and coupled with accelerationtechniques to find the mesh-independent limit(Ganapol and Furfaro 2008).Two other alternative methods are suitable for

RT calculations involving particulate media. The‘‘adding–doubling method’’ (Hansen and Travis1974) is based on the idea that the medium is slicedin thin layers with light entering the top layer. Theangular reflectance (and transmittance) of the com-

Numerical solution of the RTE 65

1 SN is the codeword for discrete ordinates method.

bined layers is obtained by superposing the reflec-tance and transmittance of the individual layers.The adding–doubling model has been implementedby de Hann et al. (1987). Another efficient RTmodel devised to compute the bidirectional reflec-tance factor of an optically thick surface of parti-culate media (including snow, ice, debris, andmixtures) was implemented by Mishchenko et al.(1999). The method is based on iteratively solvinga nonlinear integral equation (an Ambartsumianequation) derived from photon conservation. Themethod is efficient because it does not need to solvefor light inside the medium and may be ideal formodeling reflected radiance.

Finally, the light field in water can be computedusing a commercially available program calledHYDROLIGHT (Mobley 1994). Written in For-tran, the code uses the invariant-embeddingapproach to solve eq. (3.7) for each of the azimuthalcomponents. It has an extensive library of opticalproperties, and may be purchased for a fee.

3.5 GLACIER RADIATIVE TRANSFER

SIMULATION EXAMPLES

Proper modeling of the BRF and spectral albedobased upon surface properties requires solving theRTE as presented in eqs. (3.3) and (3.7). Here, werecall their definitions (see Chapter 2 of this bookby Bishop et al.). The bidirectional reflectance fac-tor (BRF) represents the ratio between the radianceemitted in any particular direction by the surface,and the radiance that would be reflected into thesame direction given an ideal Lambertian surface,illuminated by the same incident geometry. Con-versely, the spectral albedo of a surface is a dimen-sionless ratio of the radiant energy scattered awayfrom the surface to that received by the surface at aspecified wavelength. In this section, a set of numer-ical examples that show how RT theory can beemployed to model BRF and spectral albedo fora variety of configurations typically found in alpineglaciers and glacier lakes are presented. The


Figure 3.4. RT-based model of spectral albedo for a layer of optically thick pure snow as a function of wavelength

(0.4–2.5 7m) and grain size (50, 200, and 1,000 7m). The solar zenith angle is 30�. Comparewith Figure 2.8 (Bishop

et al.’s chapter). Figure can also be viewed as Online Supplement 3.4.

MADOC platform (Previti 2010, Previti et al. 2011,Furfaro et al. 2014) has been employed in all simu-lations involving glacier surfaces. Conversely, para-meterization of HYDROLIGHT, as presented byAlbert and Mobley (2003), has been employed tomodel reflectance factors for glacier lake scenarios.

In the first set of simulations, RT theory has beenemployed to model the spectral albedo of a layer ofoptically thick pure snow as a function of wave-length (0.4–2.5 mm) and grain size (Fig. 3.4). Thesolar zenith angle was set at 30�. The single-particleoptical properties of single-scattering albedo andasymmetry parameter, were computed using theMATLAB-based Mie code provided by Matzler(2004). The ice complex index of refraction wasused as input and grain sizes of 50 mm (fresh finesnow), 200 mm (fresh coarse snow) and 1 mm(annealed snow; i.e., firn) were considered.

As shown in Fig. 3.4, albedo is very sensitive tograin size and decreases as the radius of the snowparticle increases. From the physical point view,

photons have higher chances to be scattered atthe boundary between fine snow grains and air.Incrementally larger grain size has the effect ofincreasing the mean free path, giving photons ahigher chance to travel through the ice, and asmaller chance to be scattered and exit the snow-pack. Whereas larger snow particles are both moreabsorptive and more forward scattering, it can beshown that the decrease in albedo is mainly due tothe fall of !� in the NIR regime, where the asym-metry parameter increases only slightly (Wiscombeand Warren 1980). Importantly, in the visibleregion of the spectrum, snow particles are highlyscattering (!� very close to one), which explainswhy snow has generally such a high albedo, inde-pendent of grain size.In the second set of simulations, a RT model was

employed to compute the spectral albedo of a layerof optically thick pure snow as a function of wave-length (0.4–2.5 mm) and solar zenith angle (Fig. 3.5).Grain size was fixed and assumed to be 50 mm. Mie

Glacier radiative transfer simulation examples 67

Figure 3.5. RT-based model of spectral albedo for a layer of optically thick pure snow as a function of wavelength

and solar zenith angle. The particle grain size is fixed and assumed to be 50 7m. Figure can also be viewed as Online

Supplement 3.5.

theory is used to compute single-particle opticalproperties. As expected, the angle of incidence ofthe incoming radiation illuminating the snowpackhas a large effect on albedo. As shown in Fig. 3.5,albedo increases with increasing solar zenith angle.Wiscombe and Warren (1980) explained this phe-nomenon by postulating that, because of the highinclination with respect to the zenith, photonsentering the medium travel close to the upper sur-face of the snowpack and therefore scatteringevents give light particles a higher probability ofexiting the snowpack surface.

In the third set of simulations, a RT model wasemployed to compute the spectral albedo of a layerof an optically thick mixture of pure snow andcarbon soot as a function of grain size (50, 200,and 1,000 mm; see Fig. 3.6). The optical propertiesof the two constituents have been independentlycomputed using Mie theory. The optical propertiesof carbon soot were computed (assuming the sootparticles are modeled as spheres) by setting the soot

radius equal to 0.1 mm and particle density equiva-lent to 0.3 ppmw (parts per million weight). Thesoot complex index of refraction is assumed to beconstant across the spectral range of interest andequal to 1:95þ 0:79i. Fig. 3.6 shows the resultingspectral albedo as a function of wavelength alongwith the radius of the snow particles. Simulationsshow that a small amount of carbon soot is suffi-cient to lower albedo in the visible region of thespectrum where ice has the lowest absorption(highly scattering media with single-scatteringalbedo very close to one). As expected, albedoreduction is more marked for larger snow particles.Relatively high scattering between 0.4 mm and 0.7mm increases the probability that photons are morelikely to experience multiple scattering, thereforeincreasing the probability of encountering a carbonparticle and being absorbed. As discussed above,increasing snow particle grain size increases averagefree mean path, further increasing the probability ofencountering a carbon particle and being absorbed.


Figure 3.6. RT-based model of spectral albedo for a layer of an optically thick mixture of pure snow and carbon

soot as a function of wavelength and snow grain size (50, 200, and 1,000 7m). Soot particle concentration is

assumed to be 3 ppmw. The optical properties of the two constituents have been independently computed using

Mie theory. Figure can also be viewed as Online Supplement 3.6.

In the NIR region (above 0.9 mm) the influence ofcarbon particles on albedo is limited, as its reduc-tion is dominated by the stronger absorption of ice.This simulation shows that carbon soot and gener-ally other impurities (e.g., dust; Wiscombe andWarren 1980) may have a large impact on the over-all energy budget of glaciers.

The fourth set of simulations shows an exampleof a BRF for an optically thick mixture of soil andice as a function of the percentage volume of thetwo pure components. The mixture BRF was com-puted at a specified wavelength in the visible regionof the spectrum (0.63 mm). Input parameters,including the complex index of refraction, grainsize, and grain size distribution for both soil andsnow are reported in Table 3.1 (Warren 1984, Mish-chenko et al. 1999). A total of 641 coefficients of theLegendre expansion were used to properly describethe phase function. Fig. 3.7 demonstrates the reflec-tance variability associated with intimate mixturesof ice and sediment/soil. Clearly, the magnitude ofreflectance decreases as the percentage of sediment/soil increases. The anisotropic nature of reflectanceis also noteworthy, as pure ice exhibits a highlyvariable pattern that is directionally dependent.The anisotropic pattern changes with increasingsediment, such that azimuthal reflectance variations

become more spatially homogeneous. The forward-scattering component also decreases in extent andmagnitude, while the backscatter componentdecreases in extent, and exhibits an increase in mag-nitude associated with increasing debris.In the fifth set of simulations, the reflectance

patterns of typical glacier lake water were computedas a function of particle and phytoplankton con-centration, wind speed, and sensor geometry. Fig.3.8 depicts spectral curves from 390 to 710 nm.Simulations were implemented using the analyticalfit of HYDROLIGHT for Case 2 (turbid) waters(Albert and Mobley 2003). Clearly, increasingthe suspended matter content tends to decreasereflectance variability (flatten spectral curves) andincrease the magnitude of reflectance from the lake

Glacier radiative transfer simulation examples 69

Table 3.1. Input optical parameters employed for

MADOC BRF simulations of intimate and areal

mixtures of ice and soil.

Parameters Ice Soil

Wavelength (mm) 0.63 0.63

Particle diameter (mm) 30 10

Variance 0.2 0.1

Distribution Log Modified

normal gamma

Index of refraction (real) 1.31 1.55

Index of refraction (imag) 0 0.001

Max radius 35 N/A

Min radius 25 N/A

Single-scattering albedo 1 0.85413

Max scattering coefficients 641 641

Type of simulation Intimate Intimate

mixture mixture

Figure 3.7. BRF simulations for intimate mixtures (in

volume percentage) of ice and sediment/soil. The over-

all magnitude (i.e., albedo) decreases as the percentage

of soil increases. BRF patterns are also a function of the

mixture percentage. This set of simulations shows that

BRF patterns can be potentially used to discriminate

between various surface materials and conditions in

glacier environments. Figure can also be viewed as

Online Supplement 3.7.

water surface. In fact, as the particle concentrationincreases, scattering dominates over absorption. Inaddition, measurements over Lake Constance water(Gege 1998, Heege 2000) showed that scattering(and backscattering) is fairly insensitive to wave-length. In most cases, glacier lake turbidity is sohigh that large optical depths effectively precludetransmittance of light that has reflected off the lakebottom, except in uncommon cases of extremelyshallow lakes and areas fringing the edges of deeplakes. The more usual bathymetry of glacier lakescauses very narrow edge zones where the bottom issensed, so to a good approximation one can com-pletely ignore bottom reflectance and assume acondition of infinite optical thickness. However,by modifying bottom boundary conditions (i.e.,assuming that the bottom of the lake is reflecting),RT code for water can be employed to compute thespectral reflectance of shallow lakes. Importantly,such simulations require guestimating the spectralreflectance properties of the lake bottom.

3.6 CONCLUSIONS

Radiative transfer theory is a powerful tool that canbe employed to quantify the amount of radiationreflected by glacier surfaces and glacier lake water.The equations of radiative transfer, which describethe balance of photons that are being absorbed andscattered in their interaction with the host medium,are mathematically complex. Nevertheless, over thepast few decades, advancements in numerical mod-eling of RTEs have provided the scientific commun-ity with RT software that permits the computationof physical quantities of interest in remote sensing(e.g., BRF and spectral albedo) both efficiently andaccurately. Such RTE-based radiance calculations,however, are only meaningful if knowledge of themedium’s optical properties is available. SuchRTEs require the input of explicit models forsingle-scattering albedo and scattering phase func-tion, which in turn depend upon the use of ab-sorption and scattering coefficients (as well as


Figure 3.8. Simulated spectral reflectance curves from 390 nm to 710 nm as a function of suspended particles and

phytoplankton concentrations, wind speed, and Sun angle. Simulations are based on an analytical fit of the

HYDROLIGHT radiative transfer model adapted to simulate Case 2 water. (A) Effect of suspended matter on

deep-lake water reflectance. As concentration increases, spectral variability decreases, although the magnitude of

reflectance increases across the wavelength range. (B) Effect of phytoplankton, causing increased reflectance with

a peak in the green region of the spectrum. (C) Effect of wind speed (u) on spectral reflectance. A decrease in wind

speed results in higher reflectance. (D) Effect of solar zenith angle on reflectance. The range of angles causes a small

change in reflectance but no significant change in absorption. Figure can also be viewed as Online Supplement 3.8.

efficiencies), the asymmetry parameter (or Legendrecoefficients describing series expansion of the phasefunction), particle concentrations, wavelength ofinterest, as well as size and shape of the single-typeparticles/mixture of many-type particles describingthe medium. Mie theory is the most popular way todetermine such properties.

Based on the assumption that a single particle isspherical, Mie-based code has been designed tocompute the aforementioned parameters as a func-tion of the complex index of refraction and particlesize. The spherical particle assumption has beenshown not to be limiting for snow and ice mixtures,but does not accurately model soil/sediment opticalproperties. More advanced techniques based on theT-matrix and ray-tracing methods have beendesigned to model the optical properties for par-ticles of irregular shape. Nevertheless, they tendto be computationally expensive.

The optical properties of glacier lake water havebeen modeled by experimentally determining thescattering and absorption coefficient of the variouscomponents comprising the host medium (purewater, phytoplankton, particle concentration,CDOM). Models for a scattering phase functionthat is suitable to describe the in-scattering of Case2 waters has also been proposed.

The RT simulation examples have demonstratedthat modeling can be effectively employed as aninvestigative tool to assess the information contentin satellite imagery. Furthermore they show howtypical glacier surface conditions and glacial lakesoptically respond as a function of their morpholo-gical and mineralogical composition. Consequently,the coupling of RTmodeling and multispectral digi-tal image analysis can be used to assess importantbiophysical parameters of glacier environments,such that the assimilation of sensor data can pro-duce spatiotemporal quantitative informationregarding the observed medium including ice grainsize, percentage of carbon soot in snow, concentra-tion of glacier flour in lakes, pixel debris coverpercentages. Such attempts or algorithms areusually called ‘‘physically based inversion algo-rithms’’, and they are essentially models based onfirst principles which are employed to describe thefunctional inverse relationship between a sensor-collected signal and the properties of the observedsurface/medium (Furfaro 2004, Furfaro et al. 2007,Picca et al. 2008b). It is important to note, however,that such RT and coupling issues can be difficultproblems to solve. Indeed, it is well known thatinverse problems are ill posed in that they do not

depend continuously on the data. Recent mathe-matical advancements in the field of statisticsapplied to machine learning (e.g., neural networks,Gaussian processes, regularized sliced inverseregression), however, have made model inversionmore manageable. Examples of such approachescan be found in the literature for a variety of appli-cations in Earth and planetary science (e.g., Furfaroet al. 2007, Morris et al. 2008, Bernard-Michel et al.2009). The latter reinforce the significance of RTmodeling for studying and understanding theEarth’s cryosphere.

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