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`Reflectance and albedo differences between wet and
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`dry surfaces
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`Sean A. Twomey, Craig F. Bohren, and John L. Mergenthaler
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`It is commonly observed that natural multiple-scattering media such as sand and soils become noticeably
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`darker when wet. The primary reason for this is that changing the medium surrounding the particles from air
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`to water decreases their relative refractive index, hence increases the average degree of forwardness of
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`scattering as determined by the asymmetry parameter (mean cosine of the scattering angle). As a conse—
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`quence, incident photons have to be scattered more times before reemerging from the medium and are,
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`therefore, exposed to a greater probability of being absorbed. A simple theory incorporating this idea yields
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`results that are in reasonable agreement with the few measurements available in the literature, although there
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`are differences. Our measurements of the reflectance of sand wetted with various liquids are in reasonably
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`good agreement with the simple theory. We suggest that the difference between reflectances of wet and dry
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`surfaces may have implications for remote sensing.
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`I.
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`Introduction
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`Everyone is familiar with the fact that sand, clay,
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`and similar natural surfaces, as well as many other
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`powdered materials, become darker when wet. One of
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`the most dramatic modifications of regional reflec-
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`tance that has been observed and recorded (other than
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`snow and cloud cover) was obvious darkening of an
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`extensive area of Texas seen in photographs transmit-
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`ted by Gemini 4 and reproduced in the Bulletin of the
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`American Meteorological Society (Ref. 1, Fig. 5; see
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`also Ref. 2). We have been unable to find a convincing
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`discussion of the physical mechanism responsible for
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`this darkening, even though it is so familiar.
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`In this paper we describe a mechanism for the dark-
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`ening of surfaces on wetting and give a simple theoreti-
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`cal analysis of this mechanism. We also give some
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`results of simple experiments.
`(Albedo as used herein
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`the
`is equivalent to irradiance or flux reflectance:
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`ratio of reflected irradiance to incident irradiance; in
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`general, it depends on the direction of incidence. We
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`reserve the term reflectance for what is often called
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`bidirectional reflectance—a function of both direc-
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`Sean A. Twomey is with University of Arizona, Institute of Atmo-
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`spheric Physics, Tucson, Arizona 85721; C. F. Bohren is with Penn-
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`sylvania State University, Meteorology Department, University
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`Park, Pennsylvania 16802; and J. L. Mergenthaler is with Lockheed
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`Palo Alto Research Laboratory, Palo Alto, California 94304.
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`Received 16 September 1985.
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`0003-6935/86/030431-07$02.00/0.
`© 1986 Optical Society of America.
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`tions of incidence and of reflection—and adopt a nor-
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`malization in which a perfect Lambertian reflector has 7
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`a reflectance of unity in all directions.)
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`II. General Discussion
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`When sand or soil is wetted thoroughly, interstitial
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`air (refractive index m0 = 1.0) is replaced by water (mo
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`= 1.33). For a given particle with refractive index m
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`and diameter d, the optical effective size [(m —
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`mo)/mo]d (see, e.g., Ref. 3, p. 176) is thereby reduced.
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`At first sight this would seem to provide an explana-
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`tion for the observed darkening, but in soils, sand, etc.
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`the particles are much larger than the wavelengths of
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`visible light, and size, therefore, does not greatly affect
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`the scattering efficiency, which for all practical pur-
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`poses has its asymptotic value of ~2. Thus an expla-
`nation based on effective size cannot be entertained.
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`Basic physics dictates that discrete particles embed—
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`ded in a continuous medium must be invisible (optical-
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`ly undetectable) if the refractive indices of particle and
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`medium are exactly equal. Christiansen filters and
`immersion methods for refractive-index determina-
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`tion are straightforward applications of this principle,
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`and they are not restricted to particles of any special
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`size or shape.
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`Light scattering theory shows that the asymptotic
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`value for the radiant power removed by a sphere of
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`radius r from an incident beam of irradiance F0 is
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`27rr2F0. This value is obtained approximately if r ex-
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`ceeds 10—20 wavelengths (see, e.g., Ref. 4, p. 297); the
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`so-called extinction paradox refers to the presence of
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`the factor 2, giving 27rr2 rather than just area 7W2.
`If
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`1February 1986 / Vol.25, No.3 / APPLIED OPTICS
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`431
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`SCOTTS EX. 1012
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`SCOTTS EX. 1012
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`there is no absorption, the power 27rr2F0 is redistribut-
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`ed as scattered radiation, and this asymptotic value is
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`independent of refractive index. Thus there is an
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`apparent conflict between these two fundamental re-
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`sults, since one implies that scattering by a large parti-
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`cle does not decrease as its refractive index approaches
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`that of the surrounding medium, whereas the other
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`dictates that there can be no scattering when the re-
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`fractive indices are equal.
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`This apparent paradox is resolved if we consider the
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`angular distribution of the scattered light:
`it is more
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`and more concentrated around the forward direction
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`as the refractive indices tend to equality and exactly
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`forward when they are equal (i.e., a nonevent—the
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`scattered wave is indistinguishable from the incident
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`wave). This effect is shown in a brief table reproduced
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`by van de Hulst (Ref. 3, p. 226) from Debye’s thesis5
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`and is confirmed by Mie computations (see, e.g., Fig. 12
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`in Ref. 6). Figure 1 of the present paper shows the
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`asymmetry parameter g (i.e., mean cosine of the scat—
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`tering angle) as a function of the ratio of particle re—
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`fractive index In to that of the surrounding medium
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`mo. The increasing degree of forwardness as the re-
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`fractive indices of particle and medium move closer
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`together illustrates what will be the pivotal point in our
`discussion. '
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`If we turn now to diffuse reflection by multiple-
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`scattering from soil, sand, powders, etc., it is apparent
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`that if such materials could be wetted with a liquid
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`having a refractive index exactly equal to that of the
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`solid particles (for the moment assumed uniform in
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`optical properties and nonabsorbing), they would be
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`invisible since no photons could be deflected from
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`their original direction of travel. Real solid particles
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`are, of course, nonuniform in composition, absorb to
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`some extent, and usually possess higher real refractive
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`indices than water and most liquids; when they are
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`wetted the result is not total darkening, but the mecha-
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`nism is the same:
`scattering becomes more forward,
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`more scattering events are, therefore, needed to turn a
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`photon around, and since each scattering involves a
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`finite probability of absorption, fewer photons survive
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`the greater number of scattering events so reflection is '
`diminished.
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`As a simple example, consider a hypothetical medi-
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`um that scatters all photons at 30° only, so that a
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`minimum of four scattering events would be needed
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`before a normally incident photon could escape.
`If
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`there is one chance in twenty of absorption in each
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`event, reflected photons will be less than (0.95),4 or
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`81.5%, of the incident stream. Now change the scat-
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`tering angle to 10°; at least ten scattering events are
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`needed, and no more than (0.95),10 or 60%, of the
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`incident photons can emerge, which represents appre-
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`ciable darkening even though the probability of ab-
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`sorption remained exactly the same for each individual
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`scattering event.
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`ill. Derivation of Approximate Formulas
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`Intuitively, it is obvious that a scattering event
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`which deflects the average direction of propagation by,
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`432
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`APPLIED OPTICS / Vol. 25, No. 3 /
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`1 February 1986
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`151.4 1.5
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`0.8
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`Refractive Index of Liquid (for particle R115)
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`1.33
`1.15
`12
`1.25
`1.1
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`—o— Geomeiric optics
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`------.* x=250, mL=O.OOO1
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`.0no
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`=(C058)
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`Asymmetryparameter9
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` 1.0 .
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`1.0
`1.25
`1.50
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`Ra1io of Refractive Indices
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`Fig. 1. Dependence of the mean cosine of the angle of scattering by
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`a sphere (asymmetry parameter) on the refractive index (real part)
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`relative to that of the surrounding medium. These calculations
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`were made using both Mie theory and geometrical optics. The size
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`parameter x is the sphere circumference divided by the wavelength,
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`and mg is the imaginary part of its refractive index.
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`say, 10° must be less effective (for reflection by multi-
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`ple scattering) than one which deflects it by 30°.
`If
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`photons were scattered in only the forward (0°) and
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`backward (180°) directions, the probability of forward
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`scattering being 1‘, then 1 — f would be the fraction of
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`incident photons turned back in one scattering.
`In
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`defining optical path length 7, as customarily used, no
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`distinction is made between scattering into small an-
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`gles and scattering into large angles. Although
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`exp(—r) gives the fraction of photons not scattered
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`after traversal of an optical path length r, a more
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`informative quantity (in this extreme case at least)
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`would be exp[—(1 — f)~r], which gives the fraction of
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`incident photons which are either unscattered or scat-
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`tered through 0°.
`In reality, scattering distributes
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`photons over all directions, and the asymmetry param-
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`eter g = (cosfl), the cosine of the scattering angle
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`averaged over many single-scattering events, indicates
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`how forward-directed a scattering process is. This
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`quantity can be computed for any scattering diagram,
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`and for spherical particles can be obtained directly
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`from the Mie coefficients. Generalizing from the ex-
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`treme example of only forward and backward scatter-
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`ing, one might reasonably expect a scaled optical thick-
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`ness 7" = 7(1 - g) to be a better indicator of multiple-
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`scattering properties than T alone. Numerical tests
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`confirm this expectation:
`for example, computations
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`show that two scattering layers have almost the same
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`reflection and transmission if 1(1 — g) has the same
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`value for each even though the separate values of r and
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`g are different.
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`Up to this point we have neglected absorption.
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`Sand, soil, and similar natural finely divided materials
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`are, however, far from perfectly white even though
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`essentially of infinite optical depth. When absorption
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`is present, the fraction of photons surviving an encoun—
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`ter with a particle is wo rather_than unity, and ‘T is
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`composed of a component (1 -_w0)'r due to absorption
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`and a scattering component Two. The single-scatter-
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`ing albedo 0.10 for many, but not all, common particu-
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`late materials is close to unity; for such materials high
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`orders of scattering occur and contribute substantially
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`to reflection, since after n scatterings a fraction coo"
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`remains unabsorbed [e.g., (0.99)20 is greater than 0.8,
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`and (0.99)100 is 0.37].
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`The asymmetry parameter clearly has no_direct rele-
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`vance to absorption, so the group (1 — wo)r should
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`remain unchanged_after scaling; Hence two layers
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`with properties (7’, 000’, g’) and (7, Leo, g) are predicted to
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`be similar in their overall multiple-scattering proper-
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`ties (e.g., reflection, absorption, transmission) if
`(1 — {0007' = (1 — 5w,
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`(l — g’)5)’0r — (1 — (3)5101.
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`(1)
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`These scaling or similarity relationships, which were
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`first used for multiple light scattering by van de Hulst
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`and Grossman,7 have been amply validated by numeri-
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`cal tests (see Ref. 8, p. 398 and elsewhere).
`It is note-
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`worthy that, whereas with conservative scattering (no
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`absorption) all infinitely deep_layers are similar what-
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`ever the degree of asymmety (wo’ = 1 and -r’ -> Go as 7' -+
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`30 for g = 1), this is not true in the ngnconservative case:
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`wo’ is a function of asymmetry for we ¢ 1. By division,
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`a scaling formula is obtained from Eq. (1) which does
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`not contain optical thickness and so can be applied for
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`any 7; this is further simplified if g’ is stipulated to be
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`zero meaning that an optically deep layer with actual
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`properties (mg, g) will be approximated by an isotropi-
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`cally scattering deep layer with a single-scattering al—
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`bedo wo’ given by
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`1 ' f .
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`1 — gwo
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`Note that this scaled (or effective) single-scattering
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`albedo depends strongly on the asymmetry parameter
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`g and goes to zero as g goes to unity.
`If we = 0.9 and g =
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`0.9_, for example, the scaled value for a layer with g’ = 0
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`is wo’ = 0._47, substantially less than 0.9. The depen-
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`dence of wo’ on g is shown in Fig. 2.
`It is notable that
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`for a typical particle refractive index (~1.5), the
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`change from a medium with me = 1.0 (air) to m0 = 1.33
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`(water) increases the asymmetry parameter from
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`around 0.8 to ~09? (see Fig. 1), which coincides with
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`the region in Fig. 2 where 000’ changes rapidly with g.
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`Wetting, therefore, reduces the scaled single-scatter-
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`ing albedo although the actual single-scattering albedo
`is the same or almost the same.
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`’
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`Joe
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`(2)
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`1.0
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`0.99
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`Scaledsingle-scatteringalbedo(:33) 0 (1|
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`0.80
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`1.0
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`10
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`0
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`0.5
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`Asymmetry (9)
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`Fig. 2. Scaled single-scattering albedo vs asymmetry parameter.
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`Curves are labeled with the actual (unscaled) single-scattering albe-
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`do.
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`A one-to—one relationship exists between reflectance
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`and single-scattering albedo for an infinitely deep iso-
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`tropic layer, and fairly simple formulas have been de-
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`rived for that case by ChandraSekhar.9 His treatment
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`shows that a radiant flux (irradiance) F0 incident at
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`zenith angle cos‘1 #0 gives rise to a (multiply scattered)
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`reflected intensity (radiance)
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`(3)
`up) = 217; if; Hummer...
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`where ,u is the cosine of the direction of the reflected
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`intensity. The H functions are smooth and monoton-
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`ic, being unity at argument zero for all coo and reaching
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`a maximum between 1_and 3 at argument unity de-
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`pending on the value of too; values of H(u) are tabulated
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`in Ref. 9 (p. 125), and they can be computed readily by
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`iteration on any small computer. The albedo (irradi-
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`ance or flux reflectance) is obtained by integrating I(M)
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`over all p. and can be shown to be (see, e.g., Ref. 10)
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`(4)
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`R = 1 — \/1 — (30190.0).
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`Equations (3) and (4) give reflectance and albedo as
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`functions of single-scattering albedo for an infinitely
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`deep layer of isotropic scatterers. Real particles do
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`not scatter isotropically, so Eqs. (3) and (4) are not
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`directly applicable, but by scaling we can find an iso-
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`tropic layer Which is similar to (i.e., closely approxi-
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`mates) the anisotropically scattering layer of interest;
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`hence we can apply Eqs. (3) and (4) (as approxima-
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`tions) to_ anisotropic scatterers but must use wo’ in
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`place of we in these equations.
`_
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`Although the single-scattering albedo coo of particles
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`in natural scattering layers is not known, the scaled
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`single-scattering albedo wo’ is by Eq. (4) directly infer—
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`able from the reflectance. We have plotted in Fig. 3
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`the relationship between scaled single-scattering albe-
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`do and both zenith reflectance [Eq. (3)] and albedo
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`[Eq. (4)] for incident illumination at 41.4° from the
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`zenith (#0 = 0.75).
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`1 February 1986 / Vol. 25, No.3 / APPLIED OPTICS
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`433
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`

`

`Reflectance
`
`D03
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`.0as
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`1.0
`
`Reflectance towards zenith
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`0 8
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`----- -<>-- Albedo (llux reflectance)
`
`0.2
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`Fig. 3. Dependence of albedo and zenith reflectance (for incident
`light at 41.4" from zenith) on scaled single-scattering albedo wo’.
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` 0
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`“a
`0.
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`My)VWet(well
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`105,
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`(a) Scaled single-scattering albedos for an infinitely deep
`Fig. 4.
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`scattering layer wet by a liquid with refractive index mg. The
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`particles are much larger than the wavelength and have a refractive
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`index of 1.5.
`(b) Inference of the reflectance produced by wetting.
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`Quantitative tests of this were made by wetting vari-
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`ous surfaces (sand, soil, concrete) with liquids of differ-
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`ent refractive index—glycerol, benzene, carbon disul-
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`fide, sugar solutions—in sunlit conditions and
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`observing the change in the reading of a photometer
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`viewing the test surfaces.
`(In most experiments, a
`
`(5)
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`_
`r "
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`’
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`'
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`Wetting a finely divided material increases the
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`asymmetry parameter g and hence, according to Eq.
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`(2), decreases the scaled single-scattering albedo wo’.
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`For particles that are large compared with the wave-
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`length, the asymmetry parameter is almost indepen-
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`dent of size, being determined primarily by the ratio of
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`particle refractive index to that of the surrounding
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`medium (Fig. 1). Given the reflectance or the albedo
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`of the dry surface, the value of wo’ is obtained from Fig.
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`3. The value of wo’ when the surface is wet can then be
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`obtained from the expression
`-;
`I
`_
`—
`mod
`1'y
`”Owet _
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`erdI-y + 1 — dery
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`which results from applying Eq. (2) for both wet and
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`dry conditions. For weakly absorbing particles larger
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`than the wavelength, the actual single-scattering albe-
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`do wo is not substantially changed by wetting, whereas
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`the scaled value 010’ is.
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`As a numerical example, consider a surface which
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`when dry has a reflectance of 0.3 for the illumination
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`envisaged in Fig. 3; this figure shows the corresponding
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`value of (90’ to be 0.825.
`If a particle with real refrac-
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`tive index 1.5 (typical of sand and many common min-
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`erals at visible wavelengths) is surrounded by water
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`(m0 = 1.33) instead of air, the relative refractive index
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`is reduced to 1.13, and from Fig. 1, the asymmetry
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`parameter increases from ~0.83 to 0.96, giving for the
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`ratio r in Eq. (5) the value 0.23. Thus wetting reduces
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`wo’ from 0.825 to 0.52. According to Fig. 3, the corre—
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`sponding reflectance is 0.12, less than half of the origi-
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`nal reflectance. The procedure for determining wet
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`from dry reflectances is outlined schematically in Fig.
`4.
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`Predictions of the Effect of Wetting on Albedo and
`IV.
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`Reflectance
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`A. Wetting of Sand and Soil
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`Natural surfaces are wetted on a large scale only by
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`water; for a typical particle refractive index—few nat-
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`urally occurring common materials differ markedly
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`from 1.5—wet albedo can be predicted from dry albedo
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`in a manner similar to that described in the previous
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`section. Few data on albedos (or reflectances) of natu-
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`ral surfaces in both wet and dry states could be found
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`in the literature; what we have been able to find is
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`shown in Fig. 5.
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`The greatest relative change in albedo on wetting
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`occurs for dry albedos around 0.3—0.6, which decrease
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`by ~0.1.
`(An albedo of zero implies total absorption,
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`whereas an albedo of unity implies no absorption
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`whatsoever, so neither of these extreme values is
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`changed by wetting.)
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`B. Experiments Using Liquids of Different Refractive
`Indices
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`Figure 6 shows a photograph“ of sand that was wet
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`with water and benzene.
`It is readily apparent that
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`the liquid of higher refractive index (benzene) pro-
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`duced a darker surface than the liquid of lower refrac-
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`tive index (water), as argued in the preceding section.
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`434
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`APPLIED OPTICS / Vol. 25, No. 3 /
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`1 February 1986
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`

`

`We1Albedo
`
`
`I Knndrmyev
`
`{-1 Sellers
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`0
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`0.2
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`0.3
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`1.0
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`0.6
`0.4
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`Dry Albedo
`
`Fig. 5. Wet vs dry albedos obtained by the method indicated in Fig.
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`4. The wetting liquid is water (m0 = 1.33). The crosses are experi-
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`mental data given by Sellers13 for natural surfaces; the circles are
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`similar experimental data reported by Kondratyev.15
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`Fig. 6. Sand wet by water and benzene (from Ref. 11).
`
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`small fiber-optic pickup was used.) Before and after
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`introduction of the test surface, a reference surface
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`(white paper or an NBS-calibrated standard diffuser)
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`was brought into the sensor field of View, enabling both
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`relative and absolute reflectances to be inferred. No
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`attempt was made to infer albedos since this requires
`
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`integration over all directions. The theoretical ex-
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`pressions given earlier had the single-scattering albedo
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`too as a variable, which is not controllable but rather an
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`externally prescribed unknown.
`(Absorption in these
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`surfaces is a result of traces of impurities rather than
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`an intrinsic property of either the bulk material or the
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`surrounding medium.)
`Ideally,
`(.00 would be varied
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`and dry and wet reflectances measured for different
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`wetting liquids, but _this is not possible in practice.
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`One can only infer wo’ from one measurement (dry
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`reflectance) and then compare the wet reflectance to
`
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`the theoretical prediction.
`.
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`All our experiments gave similar results; rather than
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`present all of them, we restrict ourselves to those ob-
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`tained with Ottawa sand wetted by aqueous sugar solu—
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`tions, the refractive index of which can be varied by
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`changing the concentration,12 and a few results for
`
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`benzene and glycerol. The results are plotted in Fig. 7;
`
`1.0
`
`0.8
`
`
`
`Reflecmnce(:1lug
`
`.04:
`
`0.2
`
`0.75
`
`
`1.6
`
`1.0
`
`1.2
`1.4
`
`Refractive Index of Medium
`
`
`Fig. ’7. Computed curves of zenith reflectance for a range of values
`
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`of (unsealed) single-scattering albedo. The dark circle shows the
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`measured value for Ottawa sand before wetting. The crosses are
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`experimental results for wetting by sugar solutions with refractive
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`index ranging from 1.33 to 1.48. The open circles are for wetting by
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`benzene and glycerol.
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`the reflectance and liquid refractive index are the de-
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`pendent and independent variables, respectively, di-
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`rectly pertaining to the experiment. The sets of
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`curves in this figure are for different values of the
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`absorption coefficient of the particles, wh_ich is the
`
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`most important parameter in calculations; we was cal-
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`culated using Mie theory, strictly applicable only to
`
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`spheres of uniform composition, which assuredly the
`
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`experimental particles were not. The data points lie
`close to one of the theoretical curves but do not coin—
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`cide exactly with any of them. Considering the non-
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`uniformity of real materials, the degree of agreement
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`seems satisfactory for sugar solutions, somewhat less
`
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`than satisfactory for benzene and glycerol.
`
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`
`C. Angular Distribution of Reflectance
`
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`From the theory developed previously the angular
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`distribution of reflectance and the dependence of albe-
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`do on the direction of illumination are given by H
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`functions [Eqs (3) and (4)]. When theoretical predic-
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`tions are compared to some data from the literature
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`(e.g., Ref. 13), the disagreement is serious:
`the com-
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`puted variation of albedo with direction of incident
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`illumination is much less than that given in this refer—
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`ence. However, measurements obtained by hemi-
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`spherical omnidirectional sensors are subject to con-
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`siderable errors; albedos obtained by Kuhn and
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`Suomi14 by integration of directional data showed only
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`a slight variation with solar angle compared with a very
`
`1 February 1986 / Vol. 25, No.3 / APPLIED OPTICS
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`physical reason for this is that refraction causes an
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`incident bundle of rays to occupy a larger solid angle on
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`transmission; although the amount of radiant energy
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`has not changed, its disposition has. Thus, in proper
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`circumstances, the radiance of an object under water
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`may be less than above water by the factor 1/m02.
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`This can be demonstrated easily enough with a few
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`pans of water and a white plastic spoon. What is
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`observed depends on the nature of the pan (white or
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`dark) and even which side of the spoon one faces.
`If its
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`bowl is partially submerged in water in a dark pan, the
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`submerged part is noticeably darker than the part
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`above water provided that one faces the convex side of
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`the bowl one faces. But now turn the spoon over so
`that one faces the concave side one faces.
`In this case
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`the submerged part will not be so noticeably darker
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`than the part above water. Now repeat this experi-
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`ment using a white pan filled with water. The darken-
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`ing is hardly noticeable at all, regardless of the orienta-
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`tion of the spoon.
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`This geometrical mechanism for darkening of ob-
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`jects under water, which does not entail a change in
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`their reflecting properties, depends on the extent to
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`which multiple reflections are important. A perfectly
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`white object, infinite in lateral extent, will be no less
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`bright under water than above water because under
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`water it is illuminated not only directly but indirectly
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`as the result of many multiple reflections between it
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`and the water—air interface. Only in the limiting case
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`of an object not illuminated by any multiply reflected
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`light (either because it or its surroundings are black)
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`will its reflectance be reduced by 1/me2 when it is under
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`water. All other objects will suffer radiance reduc-
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`tions lying between these two extremes. A white
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`spoon, for example, suffers different amounts of
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`brightness reduction when submerged depending on
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`the extent to which it is illuminated indirectly as well
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`as directly, which in turn depends on its orientation as
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`well as the nature of its surroundings under water.
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`What we have called the geometrical mechanism for
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`the reduction of the albedo of ground on wetting was
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`put forward by Angstrom.20 He recognized that the
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`magnitude of the reduction depends on the albedo of
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`the ground when dry, and he even gave an explicit
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`expression for the dependence of wet albedo on dry
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`albedo and the refractive index of the wetting liquid.
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`But he also stated emphatically that the “diffuse re—
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`flection power of the surface
`is assumed to be
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`unaltered through the presence of the liquid.” We
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`have argued the contrary: wetting changes the diffuse
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`reflection power by making the scattering more for-
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`ward. To determine which mechanism is dominant,
`recourse must be had to measurements.
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`If measurements were made using only water as the
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`wetting liquid, it would be very easy to conclude that
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`the geometrical mechanism proposed by Angstrom is
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`indeed responsible for the observed darkening.
`It is
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`only when liquids with different refractive indices are
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`used that the issue can be settled.
`If, for example, the
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`geometrical darkening mechanism were dominant, the
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`reduction in reflection in going from a wetting liquid
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`P 1
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`1-Albedo
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`0
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`' Sellers, dry
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`oSellers, wet
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`A Kondratyev. stony-dry
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`A Kondratyev. grey-green soil
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`Fig. 8. Calculated dependence of albedo (for representative single-
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`scattering albedos) on the direction of incident illumination [Eq. (4)]
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`compared with data from Sellers13 and from Kondratyev.15
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`strong variation obtained