Reflectance and albedo differences between wet and
`dry surfaces
`
`Sean A. Twomey, Craig F. Bohren, and John L. Mergenthaler
`
`It is commonly observed that natural multiple-scattering media such as sand and soils become noticeably
`darker when wet. The primary reason for this is that changing the medium surrounding the particles from air
`to water decreases their relative refractive index, hence increases the average degree of forwardness of
`scattering as determined by the asymmetry parameter (mean cosine of the scattering angle). As a conse(cid:173)
`quence, incident photons have to be scattered more times before reemerging from the medium and are,
`therefore, exposed to a greater probability of being absorbed. A simple theory incorporating this idea yields
`results that are in reasonable agreement with the few measurements available in the literature, although there
`are differences. Our measurements of the reflectance of sand wetted with various liquids are in reasonably
`good agreement with the simple theory. We suggest that the difference between reflectances of wet and dry
`surfaces may have implications for remote sensing.
`
`I.
`
`Introduction
`Everyone is familiar with the fact that sand, clay,
`and similar natural surfaces, as well as many other
`powdered materials, become darker when wet. One of
`the most dramatic modifications of regional reflec(cid:173)
`tance that has been observed and recorded (other than
`snow and cloud cover) was obvious darkening of an
`extensive area of Texas seen in photographs transmit(cid:173)
`ted by Gemini 4 and reproduced in the Bulletin of the
`American Meteorological Society (Ref. 1, Fig. 5; see
`also Ref. 2). We have been unable to find a convincing
`discussion of the physical mechanism responsible for
`this darkening, even though it is so familiar.
`In this paper we describe a mechanism for the dark(cid:173)
`ening of surfaces on wetting and give a simple theoreti(cid:173)
`cal analysis of this mechanism. We also give some
`results of simple experiments. (Albedo as used herein
`is equivalent to irradiance or flux reflectance:
`the
`ratio of reflected irradiance to incident irradiance; in
`general, it depends on the direction of incidence. We
`reserve the term reflectance for what is often called
`bidirectional reflectance-a function of both direc-
`
`Sean A. Twomey is with University of Arizona, Institute of Atmo(cid:173)
`spheric Physics, Tucson, Arizona 85721; C. F. Bohren is with Penn(cid:173)
`sylvania State University, Meteorology Department, University
`Park, Pennsylvania 16802; and J. L. Mergenthaler is with Lockheed
`Palo Alto Research Laboratory, Palo Alto, California 94304.
`Received 16 September 1985.
`0003-6935/86/030431-07$02.00/0.
`© 1986 Optical Society of America.
`
`tions of incidence and of reflection-and adopt a nor(cid:173)
`malization in which a perfect Lambertian reflector has
`a reflectance of unity in all directions.)
`
`II. General Discussion
`When sand or soil is wetted thoroughly, interstitial
`air (refractive index m 0 = 1.0) is replaced by water (m0
`= 1.33). For a given particle with refractive index m
`and diameter d, the optical effective size [ (m -
`m0)/m0]d (see, e.g., Ref. 3, p. 176) is thereby reduced.
`At first sight this would seem to provide an explana(cid:173)
`tion for the observed darkening, but in soils, sand, etc.
`the particles are much larger than the wavelengths of
`visible light, and size, therefore, does not greatly affect
`the scattering efficiency, which for all practical pur(cid:173)
`poses has its asymptotic value of ""2. Thus an expla(cid:173)
`nation based on effective size cannot be entertained.
`Basic physics dictates that discrete particles embed(cid:173)
`ded in a continuous medium must be invisible (optical(cid:173)
`ly undetectable) if the refractive indices of particle and
`medium are exactly equal. Christiansen filters and
`immersion methods for refractive-index determina(cid:173)
`tion are straightforward applications of this principle,
`and they are not restricted to particles of any special
`size or shape.
`Light scattering theory shows that the asymptotic
`value for the radiant power removed by a sphere of
`radius r from an incident beam of irradiance F0 is
`21rr2F0• This value is obtained approximately if rex(cid:173)
`ceeds 10-20 wavelengths (see, e.g., Ref. 4, p. 297); the
`so-called extinction paradox refers to the presence of
`the factor 2, giving 21rr2 rather than just area 1rr2• If
`
`1 February 1986 I Vol. 25, No.3 I APPLIED OPTICS
`
`431
`
`SCOTTS EXHIBIT 1007
`
`

`

`there is no absorption, the power 21rr2F 0 is redistribut(cid:173)
`ed as scattered radiation, and this asymptotic value is
`independent of refractive index. Thus there is an
`apparent conflict between these two fundamental re(cid:173)
`sults, since one implies that scattering by a large parti(cid:173)
`cle does not decrease as its refractive index approaches
`that of the surrounding medium, whereas the other
`dictates that there can be no scattering when the re(cid:173)
`fractive indices are equal.
`This apparent paradox is resolved if we consider the
`angular distribution of the scattered light: it is more
`and more concentrated around the forward direction
`as the refractive indices tend to equality and exactly
`forward when they are equal (i.e., a nonevent-the
`scattered wave is indistinguishable from the incident
`wave). This effect is shown in a brief table reproduced
`by van de Hulst (Ref. 3, p. 226) from Debye's thesis5
`and is confirmed by Mie computations (see, e.g., Fig. 12
`in Ref. 6). Figure 1 of the present paper shows the
`asymmetry parameter g (i.e., mean cosine of the scat(cid:173)
`tering angle) as a function of the ratio of particle re(cid:173)
`fractive index m to that of the surrounding medium
`m0• The increasing degree of forwardness as the re(cid:173)
`fractive indices of particle and medium move closer
`together illustrates what will be the pivotal point in our
`discussion. ·
`If we turn now to diffuse reflection by multiple(cid:173)
`scattering from soil, sand, powders, etc., it is apparent
`that if such materials could be wetted with a liquid
`having a refractive index exactly equal to that of the
`solid particles (for the moment assumed uniform in
`optical properties and nonabsorbing), they would be
`invisible since no photons could be deflected from
`their original direction of travel. Real solid particles
`are, of course, nonuniform in composition, absorb to
`some extent, and usually possess higher real refractive
`indices than water and most liquids; when they are
`wetted the result is not total darkening, but the mecha(cid:173)
`nism is the same: scattering becomes more forward,
`more scattering events are, therefore, needed to turn a
`photon around, and since each scattering involves a
`finite probability of absorption, fewer photons survive
`the greater number of scattering events so reflection is ·
`diminished.
`As a simple example, consider a hypothetical medi(cid:173)
`um that scatters all photons at 30° only, so that a
`minimum of four scattering events would be needed
`before a normally incident photon could escape. If
`there is one chance in twenty of absorption in each
`event, reflected photons will be less than (0.95),4 or
`81.5%, of the incident stream. Now change the scat(cid:173)
`tering angle to 10°; at least ten scattering events are
`needed, and no more than (0.95),10 or 60%, of the
`incident photons can emerge, which represents appre(cid:173)
`ciable darkening even though the probability of ab(cid:173)
`sorption remained exactly the same for each individual
`scattering event.
`
`Ill. Derivation of Approximate Formulas
`Intuitively, it is obvious that a scattering event
`which deflects the average direction of propagation by,
`
`432
`
`APPLIED OPTICS I Vol. 25, No.3 I 1 February 1986
`
`Refractive Index of Liquid (for particle R.I.1.5)
`1.33
`1.25 12
`1.15
`1.1
`1.5 1.4 1.3
`0.8~,....-,--,-----,-,..---r----r-------,
`
`- -o - Geometric optics
`-----K--- x = 250, m~ =0.0001
`
`---- x=500, mi= 0
`
`Q)
`
`II
`c>
`~ 0.9
`E
`~
`c a.
`
`1.50
`
`Fig. 1. Dependence of the mean cosine of the angle of scattering by
`a sphere (asymmetry parameter) on the refractive index (real part)
`relative to that of the surrounding medium. These calculations
`were made using both Mie theory and geometrical optics. The size
`parameter x is the sphere circumference divided by the wavelength,
`and mi is the imaginary part of its refractive index.
`
`say, 10° must be less effective (for reflection by multi(cid:173)
`ple scattering) than one which deflects it by 30°. If
`photons were scattered in only the forward (0°) and
`backward (180°) directions, the probability of forward
`scattering being f, then 1 - f would be the fraction of
`incident photons turned back in one scattering. In
`defining optical path length r, as customarily used, no
`distinction is made between scattering into small an(cid:173)
`gles and scattering into large angles. Although
`exp( -r) gives the fraction of photons not scattered
`after traversal of an optical path length r, a more
`informative quantity (in this extreme case at least)
`would be exp[-(1 - f)r], which gives the fraction of
`incident photons which are either unscattered or scat(cid:173)
`In reality, scattering distributes
`tered through 0°.
`photons over all directions, and the asymmetry param(cid:173)
`eter g = (cosO), the cosine of the scattering angle
`averaged over many single-scattering events, indicates
`how forward-directed a scattering process is. This
`quantity can be computed for any scattering diagram,
`and for spherical particles can be obtained directly
`from the Mie coefficients. Generalizing from the ex(cid:173)
`treme example of only forward and backward scatter(cid:173)
`ing, one might reasonably expect a scaled optical thick(cid:173)
`ness r' = r(l -g) to be a better indicator of multiple-
`
`

`

`scattering properties than r alone. Numerical tests
`confirm this expectation: for example, computations
`show that two scattering layers have almost the same
`reflection and transmission if r(1 - g) has the same
`value for each even though the separate values of r and
`g are different.
`Up to this point we have neglected absorption.
`Sand, soil, and similar natural finely divided materials
`are, however, far from perfectly white even though
`essentially of infinite optical depth. When absorption
`is present, the fraction of photons surviving an encoun(cid:173)
`ter with a particle is w0 rather than unity, and r is
`composed of a component (1- wo)r due to absorption
`and a scattering component rw0• The single-scatter(cid:173)
`ing albedo w0 for many, but not all, common particu(cid:173)
`late materials is close to unity; for such materials high
`orders of scattering occur and contribute substantially
`to reflection, since after n scatterings a fraction won
`remains unabsorbed [e.g., (0.99) 20 is greater than 0.8,
`and (0.99)1°0 is 0.37].
`The asymmetry parameter clearly has no direct rele(cid:173)
`vance to absorption, so the group (1 - wo)r should
`remain unchanged after scaling. Hence two layers
`, W01
`, g') and ( T, Wo, g) are predicted to
`With properties ( T 1
`be similar in their overall multiple-scattering proper(cid:173)
`ties (e.g., reflection, absorption, transmission) if
`
`(1 - wo')r' = (1- wo)r,
`(1- g')w'0r- (1- g)wor.
`
`(1)
`
`These scaling or similarity relationships, which were
`first used for multiple light scattering by van de Hulst
`and Grossman,7 have been amply validated by numeri(cid:173)
`cal tests (see Ref. 8, p. 398 and elsewhere). It is note(cid:173)
`worthy that, whereas with conservative scattering (no
`absorption) all infinitely deep layers are similar what(cid:173)
`ever the degree Of asymmety ( wo' = 1 and T
`- - co aS T -(cid:173)
`co for g = 1), this is not true in the nonconservative case:
`w0' is a function of asymmetry for wo ~ 1. By division,
`a scaling formula is obtained from Eq. (1) which does
`not contain optical thickness and so can be applied for
`any r; this is further simplified if g' is stipulated to be
`zero meaning that an optically deep layer with actual
`properties (w0, g) will be approximated by an isotropi(cid:173)
`cally scattering deep layer with a single-scattering al(cid:173)
`bedo wo' given by
`
`1
`
`(2)
`
`- 1- g
`-
`wo' = wo---- ·
`1-gw0
`Note that this scaled (or effective) single-scattering
`albedo depends strongly on the asymmetry parameter
`g and goes to zero asg goes to unity. If wo = 0.9 andg =
`0.9, for example, the scaled value for a layer with g' = 0
`is wo' = 0.47, substantially less than 0.9. The depen(cid:173)
`dence of w0' on g is shown in Fig. 2. It is notable that
`for a typical particle refractive index ("'1.5), the
`change from a medium with m0 = 1.0 (air) to m0 = 1.33
`(water) increases the asymmetry parameter from
`around 0.8 to "'0.97 (see Fig. 1), which coincides with
`the region in Fig. 2 where w0' changes rapidly with g.
`Wetting, therefore, reduces the scaled single-scatter(cid:173)
`ing albedo although the actual single-scattering albedo
`is the same or almost the same.
`
`Fig. 2. Scaled single-scattering albedo vs asymmetry parameter.
`Curves are labeled with the actual (unsealed) single-scattering albe(cid:173)
`do.
`
`(3)
`
`A one-to-one relationship exists between reflectance
`and single-scattering albedo for an infinitely deep iso(cid:173)
`tropic layer, and fairly simple formulas have been de(cid:173)
`rived for that case by Chandrasekhar.9 His treatment
`shows that a radiant flux (irradiance) F0 incident at
`zenith angle cos-1 p,0 gives rise to a (multiply scattered)
`reflected intensity (radiance)
`-
`1 WoJ..Lo
`l(J..L) = -
`-+- H(J..L)H(J..L0)F0,
`4
`J..L
`J..Lo
`1l"
`where p, is the cosine of the direction of the reflected
`intensity. The H functions are smooth and monoton(cid:173)
`ic, being unity at argument zero for all w0 and reaching
`a maximum between 1 and 3 at argument unity de(cid:173)
`pending on the value of w0 ; values of H(p,) are tabulated
`in Ref. 9 (p. 125), and they can be computed readily by
`iteration on any small computer. The albedo (irradi(cid:173)
`ance or flux reflectance) is obtained by integrating l(p,)
`over all p, and can be shown to be (see, e.g., Ref. 10)
`R = 1- v1- woH(J..L0).
`Equations (3) and (4) give reflectance and albedo as
`functions of single-scattering albedo for an infinitely
`deep layer of isotropic scatterers. Real particles do
`not scatter isotropically, so Eqs. (3) and (4) are not
`directly applicable, but by scaling we can find an iso(cid:173)
`tropic layer which is similar to (i.e., closely approxi(cid:173)
`mates) the anisotropically scattering layer of interest;
`hence we can apply Eqs. (3) and (4) (as approxima(cid:173)
`tions) to anisotropic scatterers but must use wo' in
`place of w0 in these equations.
`Although the single-scattering albedo w0 of particles
`in natural scattering layers is not known, the scaled
`single-scattering albedo w0' is by Eq. (4) directly infer(cid:173)
`able from the reflectance. We have plotted in Fig. 3
`the relationship between scaled single-scattering albe(cid:173)
`do and both zenith reflectance [Eq. (3)] and albedo
`[Eq. (4)] for incident illumination at 41.4° from the
`zenith (p,o = 0.75).
`
`(4)
`
`1 February 1986 I Vol. 25, No.3 I APPLIED OPTICS
`
`433
`
`

`

`Wetting a finely divided material increases the
`asymmetry parameter g and hence, according to Eq.
`(2), decreases the scaled single-scattering albedo wo'·
`For particles that are large compared with the wave(cid:173)
`length, the asymmetry parameter is almost indepen(cid:173)
`dent of size, being determined primarily by the ratio of
`particle refractive index to that of the surrounding
`medium (Fig. 1). Given the reflectance or the albedo
`of the dry surface, the value of w0' is obtained from Fig.
`3. The value of wo' when the surface is wet can then be
`obtained from the expression
`
`'
`
`(5)
`
`_
`_,
`Wowet-
`
`1- gwet
`rwodry
`r= - - - ·
`'
`_,
`rwodry + 1 - Wodry
`1- gdry
`which results from applying Eq. (2) for both wet and
`dry conditions. For weakly absorbing particles larger
`than the wavelength, the actual single-scattering albe(cid:173)
`do wo is not substantially changed by wetting, whereas
`the scaled value wo' is.
`As a numerical example, consider a surface which
`when dry has a reflectance of 0.3 for the illumination
`envisaged in Fig. 3; this figure shows the corresponding
`value of wo' to be 0.825. If a particle with real refrac(cid:173)
`tive index 1.5 (typical of sand and many common min(cid:173)
`erals at visible wavelengths) is surrounded by water
`(m0 = 1.33) instead of air, the relative refractive index
`is reduced to 1.13, and from Fig. 1, the asymmetry
`parameter increases from "'0.83 to 0.96, giving for the
`ratio r in Eq. (5) the value 0.23. Thus wetting reduces
`w0' from 0.825 to 0.52. According to Fig. 3, the corre(cid:173)
`sponding reflectance is 0.12, less than half of the origi(cid:173)
`nal reflectance. The procedure for determining wet
`from dry reflectances is outlined schematically in Fig.
`4.
`
`IV. Predictions of the Effect of Wetting on Albedo and
`Reflectance
`A. Wetting of Sand and Soil
`Natural surfaces are wetted on a large scale only by
`water; for a typical particle refractive index-few nat(cid:173)
`urally occurring common materials differ markedly
`from 1.5-wet albedo can be predicted from dry albedo
`in a manner similar to that described in the previous
`section. Few data on albedos (or reflectances) of natu(cid:173)
`ral surfaces in both wet and dry states could be found
`in the literature; what we have ·been able to find is
`shown in Fig. 5.
`The greatest relative change in albedo on wetting
`occurs for dry albedos around 0.3-0.6, which decrease
`by "'0.1. (An albedo of zero implies total absorption,
`whereas an albedo of unity implies no absorption
`whatsoever, so neither of these extreme values is
`changed by wetting.)
`
`B. Experiments Using Liquids of Different Refractive
`Indices
`Figure 6 shows a photograph 11 of sand that was wet
`with water and benzene. It is readily apparent that
`the liquid of higher refractive index (benzene) pro(cid:173)
`duced a darker surface than the liquid of lower refrac(cid:173)
`tive index (water), as argued in the preceding section.
`
`434
`
`APPLIED OPTICS I Vol. 25, No.3 I 1 February 1986
`
`- - - Reflectance !awards zenith
`------o-· Albedo (fluxrefleclance l
`
`tO
`
`0.8
`
`~
`~ 0.6
`
`~ c::
`0.4
`
`0.2
`
`Fig. 3. Dependence of albedo and zenith reflectance (for incident
`light at 41.4° from zenith) on scaled single-scattering albedo wo'·
`
`(a)
`
`to
`
`h
`~0.5
`
`Dry w'
`
`(a) Scaled single-scattering albedos for an infinitely deep
`Fig. 4.
`scattering layer wet by a liquid with refractive index m0• The
`particles are much larger than the wavelength and have a refractive
`index of 1.5. (b) Inference of the reflectance produced by wetting.
`
`Quantitative tests of this were made by wetting vari(cid:173)
`ous surfaces (sand, soil, concrete) with liquids of differ(cid:173)
`ent refractive index-glycerol, benzene, carbon disul(cid:173)
`fide, sugar solutions-in sunlit conditions and
`observing the change in the reading of a photometer
`(In most experiments, a
`viewing the test surfaces.
`
`

`

`• Kondro1yev
`
`+Sellers
`
`1.0
`
`0.8
`
`0.6
`
`0 -c
`Q)
`.c
`<!
`~ 0.4
`
`0.2
`
`0.6
`0.4
`Dry Albedo
`
`0.8
`
`1.0
`
`Fig. 5. Wet vs dry albedos obtained by the method indicated in Fig.
`4. The wetting liquid is water (m0 = 1.33). The crosses are experi(cid:173)
`mental data given by Sellers13 for natural surfaces; the circles are
`similar experimental data reported by Kondratyev. 15
`
`c
`Q) u
`~OA
`..2
`~
`
`Refractive Index of Medium
`Fig. 7. Computed curves of zenith reflectance for a range of values
`of (unsealed) single-scattering albedo. The dark circle shows the
`measured value for Ottawa sand before wetting. The crosses are
`experimental results for wetting by sugar solutions with refractive
`index ranging from 1.33 to 1.48. The open circles are for wetting by
`benzene and glycerol.
`
`the reflectance and liquid refractive index are the de(cid:173)
`pendent and independent variables, respectively, di(cid:173)
`rectly pertaining to the experiment. The sets of
`curves in this figure are for different values of the
`absorption coefficient of the particles, which is the
`most important parameter in calculations; w0 was cal(cid:173)
`culated using Mie theory, strictly applicable only to
`spheres of uniform composition, which assuredly the
`experimental particles were not. The data points lie
`close to one of the theoretical curves but do not coin(cid:173)
`cide exactly with any of them. Considering the non(cid:173)
`uniformity of real materials, the degree of agreement
`seems satisfactory for sugar solutions, somewhat less
`than satisfactory for benzene and glycerol.
`
`C. Angular Distribution of Reflectance
`From the theory developed previously the angular
`distribution of reflectance and the dependence of albe(cid:173)
`do on the direction of illumination are given by H
`functions [Eqs. (3) and (4)]. When theoretical predic(cid:173)
`tions are compared to some data from the literature
`the com(cid:173)
`(e.g., Ref. 13), the disagreement is serious:
`puted variation of albedo with direction of incident
`illumination is much less than that given in this refer(cid:173)
`ence. However, measurements obtained by hemi(cid:173)
`spherical omnidirectional sensors are subject to con(cid:173)
`siderable errors; albedos obtained by Kuhn and
`Suomi14 by integration of directional data showed only
`a slight variation with solar angle compared with a very
`
`1 February 1986 I Vol. 25, No.3 I APPLIED OPTICS
`
`435
`
`Fig. 6. Sand wet by water and benzene (from Ref. 11).
`
`small fiber-optic pickup was used.) Before and after
`introduction of the test surface, a reference surface
`(white paper or an NBS-calibrated standard diffuser)
`was brought into the sensor field of view, enabling both
`relative and absolute reflectances to be inferred. No
`attempt was made to infer albedos since this requires
`integration over all directions. The theoretical ex(cid:173)
`pressions given earlier had the single-scattering albedo
`w0 as a variable, which is not controllable but rather an
`externally prescribed unknown. (Absorption in these
`surfaces is a result of traces of impurities rather than
`an intrinsic property of either the bulk material or the
`Ideally, w0 would be varied
`surrounding medium.)
`and dry and wet reflectances measured for different
`wetting liquids, but this is not possible in practice.
`One can only infer w0' from one measurement (dry
`reflectance) and then compare the wet reflectance to
`the theoretical prediction.
`All our experiments gave similar results; rather than
`present all of them, we restrict ourselves to those ob(cid:173)
`tained with Ottawa sand wetted by aqueous sugar solu(cid:173)
`tions, the refractive index of which can be varied by
`changing the concentration, 12 and a few results for
`benzene and glycerol. The results are plotted in Fig. 7;
`
`

`

`physical reason for this is that refraction causes an
`incident bundle of rays to occupy a larger solid angle on
`transmission; although the amount of radiant energy
`has not changed, its disposition has. Thus, in proper
`circumstances, the radiance of an object under water
`may be less than above water by the factor 1/m0
`2•
`This can be demonstrated easily enough with a few
`pans of water and a white plastic spoon. What is
`observed depends on the nature of the pan (white or
`dark) and even which side of the spoon one faces. If its
`bowl is partially submerged in water in a dark pan, the
`submerged part is noticeably darker than the part
`above water provided that one faces the convex side of
`the bowl one faces. But now turn the spoon over so
`that one faces the concave side one faces. In this case
`the submerged part will not be so noticeably darker
`than the part above water. Now repeat this experi(cid:173)
`ment using a white pan filled with water. The darken(cid:173)
`ing is hardly noticeable at all, regardless of the orienta(cid:173)
`tion of the spoon.
`This geometrical mechanism for darkening of ob(cid:173)
`jects under water, which does not entail a change in
`their reflecting properties, depends on the extent to
`which multiple reflections are important. A perfectly
`white object, infinite in lateral extent, will be no less
`bright under water than above water because under
`water it is illuminated not only directly but indirectly
`as the result of many multiple reflections between it
`and the water-air interface. Only in the limiting case
`of an object not illuminated by any multiply reflected
`light (either because it or its surroundings are black)
`2 when it is under
`will its reflectance be reduced by 1/m0
`water. All other objects will suffer radiance reduc(cid:173)
`tions lying between these two extremes. A white
`spoon, for example, suffers different amounts of
`brightness reduction when submerged depending on
`the extent to which it is illuminated indirectly as well
`as directly, which in turn depends on its orientation as
`well as the nature of its surroundings under water.
`What we have called the geometrical mechanism for
`the reduction of the albedo of ground on wetting was
`put forward by Angstrom. 20 He recognized that the
`magnitude of the reduction depends on the albedo of
`the ground when dry, and he even gave an explicit
`expression for the dependence of wet albedo on dry
`albedo and the refractive index of the wetting liquid.
`But he also stated emphatically that the "diffuse re(cid:173)
`flection power of the surface . . . is assumed to be
`unaltered through the presence of the liquid." We
`have argued the contrary: wetting changes the diffuse
`reflection power by making the scattering more for(cid:173)
`ward. To determine which mechanism is dominant,
`recourse must be had to measurements.
`If measurements were made using only water as the
`wetting liquid, it would be very easy to conclude that
`the geometrical mechanism proposed by Angstrom is
`indeed responsible for the observed darkening. It is
`only when liquids with different refractive indices are
`used that the issue can be settled. If, for example, the
`geometrical darkening mechanism were dominant, the
`reduction in reflection in going from a wetting liquid
`
`0
`"C
`Cl.l
`.0
`
`~
`
`0
`
`0
`
`• Sellers, dry
`o Sellers, wet
`A Kondrotyev, stony-dry
`t:. Kondratyev, grey-green soil
`
`p.--
`
`Fig. 8. Calculated dependence of albedo (for representative single(cid:173)
`scattering albedos) on the direction of incident illumination [Eq. (4)]
`compared with data from Sellers13 and from Kondratyev.15
`
`strong variation obtained when a hemispherical in(cid:173)
`strument was used. Albedos tabulated by Kondra(cid:173)
`tyev,15 in marked contrast with those presented by
`Sellers,13 varied only "'10% over all solar angles. The
`curves in Fig. 8 show computed directional depen(cid:173)
`dence for several values of wo' together with data from
`Refs.13·(p. 30) and 15 (Table 4.10). Equation (4) gives
`a dependence similar to that of the Kondratyev data;
`the data of Sellers, however, are very different, so no
`conclusive judgment can be made concerning the ade(cid:173)
`quacy of the theory.
`Although for convenience we have cited Sellers as
`the source of the albedo data shown in Fig. 8, they are
`not his measurements. He obtained them from anoth(cid:173)
`er secondary source16 in which the measurements
`made originally by Buttner and Sutter17 are presented.
`These authors did not measure albedos for solar eleva(cid:173)
`tions of less than "-'20°; albedos for elevations less than
`this were obtained by extrapolation. So the sharp
`increase in albedo with decreasing elevation may be
`what the authors expected it to be rather than what it
`really was. If one looks at sand (both wet and dry) in
`various directions, it is difficult to accept that its albe(cid:173)
`do is almost 100% at near-glancing incidence, unless, of
`course, the sand is completely covered by water.
`
`V. Other Darkening Mechanisms
`Although we have argued that the mechanism for
`darkening of sand on wetting is increased forward scat(cid:173)
`tering by the sand grains, there are other possible
`mechanisms which must be addressed. For example,
`in going from one homogeneous (nonabsorbing) medi(cid:173)
`um to a different one, radiance is not conserved (even if
`the transmittance is taken to be unity) but rather the
`product of radiance and refractive index squared.
`This result has been derived by Milne (Ref. 18, p. 7 4),
`for example, although it was known to Planck (Ref. 19,
`p. 35) and even earlier to Helmholtz (Ref. 20, p. 233).
`Consider, for example, light transmitted from water
`to air. The transmitted radiance is (ignoring trans(cid:173)
`mission losses) less than that incident by the factor
`1/mo2, where m0 is the refractive index of water. The
`
`436
`
`APPLIED OPTICS I Vol. 25, No.3 I 1 February 1986
`
`

`

`with refractive index 1.35 to one with refractive index
`1.4 could be at most row7%, whereas the data of Fig. 7
`show a reduction of row40%. we, therefore, conclude
`that at least for the case we have investigated geomet(cid:173)
`ric darkening is of considerably less magnitude than
`that wrought by increasing the forwardness of scatter(cid:173)
`ing by grains.
`There is yet another mechanism for darkening,
`which at first glance may seem different from the one
`we have discussed. Suppose that instead of a medium
`consisting of more or less distinguishable grains, we
`have a homogeneous medium with a rough surface.
`Frosted glass is a simple example. When such glass is
`wet by water it becomes noticeably darker. We argue
`that this is merely a variation on a theme: wetting the
`glass makes the scattering more forward (i.e., more
`light is transmitted into the glass rather than being
`diffusely reflected by it).
`
`VI. Concluding Remarks
`Apart from offering a straightforward physical ex(cid:173)
`planation of a very common observation, the theory
`developed here would appear to be relevant to remote
`it suggests inclusion of wet, as well as dry,
`sensing:
`spectral reflectance for characterizing natural surfaces
`(other than dense vegetation). Since the refractive
`index of the wetting liquid is accurately known, one
`can characterize the complex refractive index (or its
`effective value for the real nonuniform world) much
`better by two measurements than by a single measure(cid:173)
`ment. We also note that the extent to which a surface
`darkens on wetting decreases with increasing (real)
`refractive index of the surface material, which may
`have implications for remote sensing.
`
`The work presented in this paper was supported in
`part by a grant from the Office of Naval Research.
`While preparing the final manuscript, the second au(cid:173)
`thor (CFB) was spending the summer in the Geophysi(cid:173)
`cal Sciences Branch of the Cold Regions Research and
`Engineering Laboratory and in the Department of
`Physics and Astronomy at Dartmouth College. Also
`he is grateful to William Doyle and Alistair Fraser for
`helpful discussions.
`
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