Broadband reflectance and emissivity of specular andrough water surfaces
Miriam Sidran
Complex indices of refraction for water near 251C were obtained from six references, and the most represen-
tative values in the wavelength range of 2000 A to 5 cm were selected. These were used to study the polar-
ized reflectance and emissivity of both specular and wind-roughened surfaces as functions of wavelength and
viewing angle. Applications to remote sensing of sea surface temperature and wave state are discussed, in-
cluding effects of salinity.
1. Introduction
The optical constants of water have been extensivelystudied because of their importance in science andtechnology. Applications include (a) remote sensingof natural water surfaces, (b) radiant energy transfer byatmospheric water droplets, and (c) optical propertiesof diverse materials containing water such as soils,leaves, and aqueous solutions.
In this study, values of the complex index of refrac-tion from six recent articles were averaged by visualinspection of the graphs, and the most representativevalues in the wavelength range of 2000 A to 5 cm weredetermined. These were used to find the directionalpolarized reflectance and emissivity of a specular sur-face and the Brewster or pseudo-Brewster angle asfunctions of wavelength.
The directional polarized reflectance and emissivityof wind-generated water waves were studied using thefacet slope distribution function for a rough sea due toCox and Munk.1 Applications to remote sensing of seasurface temperature and wave state are discussed, in-cluding effects of salinity.
The light upwelling from a water surface is due partlyto surface reflection and thermal emission and partlyto volume scattering by molecules and suspended par-ticles. The volume scattering has been extensivelytreated elsewhere, e.g., by Gordon et al. 2 It is beyondthe scope of this work.
The author is with Baruch College, City University of New York,
Department of Natural Sciences, 17 Lexington Avenue, New York,
If a light beam of intensity Io and wavelength X isincident on a slab of material of thickness x, the inten-sity I of the transmitted beam is given by
I = Io exp(-ax) = Io exp I- (;- ) x] 1
As shown, the Lambert absorption coefficient ac isproportional to the extinction coefficient h. The latteris the imaginary part of the complex index of refractionnc, the real part being the index of refraction n as de-fined by Snell's law. Thus n, = n + ik.
Values of n,k may be obtained by at least fourmethods. Values of k may be found by measuring I inEq. (1). Values of n may then be found by a Kramers-Kronig phase shift (KKPS) analysis of the k spec-trum.3-5 This method is most accurate in spectral re-gions of small k.3 ,6 Alternatively, n,k may be found bya KKPS analysis of the spectrum of reflectance Rmeasured at near-normal incidence.3 5 7 This methodis most accurate in spectral regions of large R. If bothR and k have been accurately determined, n may becomputed directly from Fresnel's equations.5 6 Theseequations may also be used to compute n,k from thereflectances measured at two angles of incidence. 5
In this work, n,k values for a broad spectral rangewere obtained from six recent articles.3 -8 In the 2-2000-,um range, data were available from more than onesource. These data are plotted in Figs. 1 and 2. Thedata sources are discussed below.
Downing and Williams (DW)3 tabulated n,k valuesfor water at 270C in the 2-1000-Atm spectral range.These were derived mostly from earlier studies byWilliams and co-workers, who used several measure-ment techniques. However, their data for X > 500 Axmcame from a study by Ray.9 The DW table givesweighted averages of n,k which favor those values de-rived from measurements of R and a.
Fig. 1. Spectral variation of index of refraction n.
WAVE NUMBER (cm-')
I-
zZ 1.00
z0
0 0.5
zx
10 100 l0WAVELENGTH (,am)
Fig. 2. Spectral variation of extinction coefficient k.
Hale and Querry (HQ)4 tabulated n,k values in the0.2-200-gm range. Their k values, taken from fifty-eight sources, were weighted in favor of those data whichwere for 250C and judged to be most reliable. Theweighted averages for k were plotted against X and asmooth curve drawn manually through the points.Values of n were found by a KKPS analysis of the re-sulting k spectrum.
Zoloratev and Demin (ZD)5 tabulated n,k values forwater at 250C in the 2-50,000-gm range. In the 2.5-50-,m range, they obtained their data by a KKPSanalysis of the measured internal reflectance spectrum.Their data for the other wavelengths were based onvalues from eight references, including HQ and foursources used by DW.
Simpson et al. (SBP)6 determined R and a for waterat 250C in the 56.9-1217-gm range and computed nfrom Fresnel's equations.
Afsar and Hasted (AH)7,8 tabulated n,k values basedon measured reflectances of water at various tempera-tures. Their 19'C data in the 22.22-1733-gm range andtheir 30'C data in the 47 .6-1000-gm range are plottedas squares and circles, respectively, in Figs. 1 and 2.
All data except those of AH are for water near 250 C.Since this is the average temperature for the two AHdata sets, they were included in Figs. 1 and 2. The HQ,DW, and ZD curves in these figures coincide over partsof the spectrum. The most representative values of nin Fig. 1 are given by the HQ curve in the 2-80-gm range
and by the DW curve in the 80-1000-gm range. In Fig.2, the most representative values of k lie along the DWcurve up to 150 Am and then the ZD curve up to 700gim.
The regions of anomalous dispersion in Fig. 1 definefour strong absorption bands, with central wavelengthsat 2.95,6.1, 16.5, and 58 m, corresponding to the peaksof k in Fig. 2. Slightly different shapes, intensities, andlocations are found for these bands depending on themethod used to find n,k. The discrepancies are ex-plained by ZD and DW. Table I shows the peak valuesof k and the extremum n values found in these studiesfor several absorption bands.3-5
Ill. Specular Surface
A. Reflectance, Emissivity, Brewster Angle, andPercent Polarization
The polarized emissivity of a specular surface is givenby equations due to Parker and Abbott10:
2 2(I = ell ~~~~~~~~~~~~~~~(2)(1+Z1) (1+ Z)(S + cos 20
V2 cosO(S + A - sin2O)1/2' (3a)
S + (A2 + B2 ) cos2 0Z11 =X/ cosO[S(A 2 + B2 ) + (A - sin2o)(A2 - B 2 ) + 2AB2]1/2
(3b)
S = (A - sin 2o)2 + B211/2 .
Here is the emissivity for radiation whose electricvector is normal to the plane of viewing, Ell is the emis-sivity for radiation whose electric vector lies in the planeof viewing, and 0 is the viewing angle measured from thenormal to the surface. Quantities A and B are real andimaginary parts, respectively, of the complex permit-tivity, which is n2. Thus A = - k2, and B = 2nk.Equations (2)-(4) may be derived from Fresnel'sequations for the polarized reflectances R 1 and R 11 bymeans of
Table 1. Maximum Values of Extinction Coefficient k and ExtremumValues of Index of Refraction n for Wavelengths in the Absorption Bands
ZD1' HQa DWa
X (M) 2.793 2.800 2.755-2.762n 1.116 1.142 1.138
X (M) 2.941 2.950 2.950-2.959k 0.298 0.298 0.282
(M) 3.125 3.150 3.145-3.155n 1.482 1.483 1.487
(M) 5.882 5.900 5.917-5.952n 1.241 1.248 1.241
X (M) 6.098 6.100 6.098k 0.128 0.131 0.132
X (M) 6.250 6.200 6.250n 1.355 1.363 1.356
A (M) 12.195 12.000 11.905n 1.116 1.111 1.131
a Data from three sources, i.e., references DW,3 HQ,5 and ZD.6
Fig. 4. Spectral variation of reflectance R l polarized normal to theplane of reflection.
WAVE NUMBER (cml )
.,, ,, , , I........ DW
0.4 0.4 D Q
0 0
0.2 0.2
0.1 5'- -- 0.1
.1 1.0 10 100 10, 10
WAVELENGTH tjm)
Fig. 5. Spectral variation of reflectance R polarized parallel to theplane of reflection.
e=1 -R ; e=1-R. (5)
These equations are more convenient to use thanFresnel's equations for finding the polarized emis-sivity and reflectance.
The total emissivity e and total reflectance R of asurface are given by
WAVE NUMBER (cm')
7
a:w4
W
03:
0
al
:It
D 100WAVELENGTH to.m)
0
2.-4a
cr
ra11
r
D
11
I
Fig. 6. Spectral variation of the Brewster or pseudo-Brewster angleOb and of the percent polarization Pb of the radiation reflected at angle
Ob-
UWz4~1
I--J
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
WAVE NUMBER (cm-')10 100 10
0.1 1.0 10 100 103
WAVELENGTH (m1
, , ,,1 01 04 lo,
Fig. 7. Spectral variation of polarized reflectances R 1 and R 11 andof total reflectance R at the Brewster or pseudo-Brewster angle Ob.
E = /2(e + El);
R = /2(RL + R11);
E= 1 -R.
(6a)
(6b)
(6c)
As shown by Eqs. (5) and (6c), the graphs of emissivityof a specular surface vs X may be obtained by reflectingthe corresponding graphs of reflectance (e.g., Figs. 3-5)through the horizontal line for the 0.5 value.
Reflectances R , R 11, and R were computed from Eqs.(2)-(6) using the n,k data of ZD, DW, and HQ. The twosets of AH data were not used here because of theirdifferences in temperature. The SBP data were alsoexcluded because in Figs. 1 and 2 they show excessivescatter.
Figures 3-5 show the variation of reflectance with Xin the 0.2-50,000-gm range for different angles of inci-dence 0. They depict anomalous behavior in the ab-sorption bands, similar to that of n in Fig. 1. In eachfigure, the ZD, DW, and HQ curves for a given 0 coincideover parts of the spectrum. Where they do not coincide,the HQ curve (not shown) lies mostly between the other
Fig. 8. Variation with viewing angle 0 of polarized emissivities ej and ell, of percent polarization Qof percent polarization P of the reflected radiation.
two. The most representative reflectance values are As 0 increases in Fig. 5,those given by the HQ curves up to 100 m and then the some angle 0 = b forDW curves up to 1000,um; beyond this wavelength there maximum value Pb atare only the ZD curves. and Pb with X is showr
The percent polarization P of the reflected radiation value of R 11 and the valuis defined by vs X in Fig. 7. The del
=100R B 1 -R BFigs. 8 for selected waveP = o - . (7) R vs 0 may be visualize
R + i curves of Figs. 8 and Es
z-4
-0
N
-403,
(d)
90
80
70 U
60 Omz
50 0
4
30 N
20 °
10
ANGLE (')
(9)
of the thermally emitted radiation, and
R goes through a minimum ateach wavelength; thus P has athis angle. The variation of b1 by Fig. 6, while the minimum[es of R and R at b are plottedpendence of P on 0 is shown by,lengths. Curves of R, R , andd by referring to the emissivityqs. (5) and (6c).
Fig. 9. Coordinate system for Cox and Munk' facet slope angles 3,-y
and viewing angles 0,0 for wind-roughened water surfaces. Azimuth
angles y and k are measured from the direction of the wind vector.
LU
In the visible and near IR, the minimum R11 is zeroexcept in the absorption bands; the reflected radiationis plane polarized, and Ob is the polarizing or Brewsterangle given by tan- 1 n. This condition is nearly realizedagain at X = 50,000gm (Figs.6 and 7). When R11has anonzero minimum, the reflected radiation is ellipticallypolarized. Then Ob is the pseudo-Brewster angle, givenby a cubic equation in tan Ob in terms of the impedanceand permittivity of water. 12
B. Variation of Emissivity with Angle
The percent polarization Q of the thermally emittedradiation is defined by
(8)E11 + IThe variation of eI, ElI, P, and Q with viewing angle 0 isshown by Fig. 8 for wavelengths spanning the broad,range. All X • 10,um, excluding the absorption bands,are represented by Fig. 8(c). For all X > 10 gim, exceptthose in the absorption bands, the percent polarizationQ at any angle increases with X, i.e., the entire curve ofQ vs 0 rises with increasing X, as shown by Figs. 8(d)-(g).Within the absorption bands, the Q curve behavesanomalously, falling and rising at the wavelengths ofminimum and maximum n. This is shown for the2.95-,um band by Figs. 8(a) and (b). Nearby wave-lengths on either side of this band are represented byFig. 8(c).
For any X, the total emissivity e is constant for all 0<450; this follows from Fig. 8 and Eq. (6a). It is im-portant in remote sensing of thermal radiation fromspace, as discussed later.
IV. Wind-Roughened Surface
A. Model of a Rough Surface
To compute the emissivity of wind-generated waterwaves, the surface was assumed to consist of many planefacets having various slopes and large dimensions rel-ative to the wavelength of the radiation. The slope ofa facet is the angle 3 between its surface normal and thevertical direction. The coordinate -y is the azimuth ofthe surface normal measured from the wind vector, asshown in Fig. 9.
C,'I-
C--
t)-4I-
0CZ:
C"
UPWIND
20 10 0 10 20CROSSWIND SLOPE (0)
(a)
0 20 10 0 10 20 30 40-SLOPE () - DOWNWIND
Fig. 10. Facet slope angle j3 distribution probability for a wind-roughened sea, computed for various wind speeds by means of the Coxand Munk' probability function. Curves are normalized to equal
areas: (a) crosswind; (b) upwind-downwind.
When the wind sweeps with constant speed and di-rection across an extended water surface, the waves intime achieve a steady state. Cox and Munk1 derivedthe steady-state distribution function for facet angles
3,'y vs wind speed W. This function is valid for W <14 m/sec, and 1 < 230. Figure 10 depicts the distribu-tion of : values for surface normals lying in the x -z(crosswind) and y-z (upwind-downwind) planes (Fig.9). The maxima occur at # = 0 in the x -z plane, and at
0.6~ UPWIND 0 W 14 m/s ZA_ 0.6 ~ ..... - CROSSWINDJ 60 --I
U' 0.5 50 0
w 0.4 _40 N
0.3 30
0.2 20
0.1 _ :100~~~~~~~~~~~~
0 0 20 30 40 50 60 70 80 90ANGLE (°)
(a)
0.C 100
0.9 - 9
0.8 . 80X: 70/im N Pn0.7 ~ \ 70 :U
> 0.6 - FLAT, NO WIND < , \ 60 mI- -DOWNWINDu) 0.5 --- UPWIND W14m/s50-L' ...... CROSSWINDJ r
O .4 .. 40
0.3 30 S0z0.2 20
0.1 .- 10
0~~~~~~~~~~~~010 20 30 40 50 60 70 8090ANGLE (0)
(b)
Fig. 11. Variation with viewing angle 0 of polarized emissivities eand Ell, of total emissivity , and of percent polarization Q of thethermally emitted radiation for specular and wind-roughened water
surfaces. Wind speed W = 14 m/sec.
small downwind angles -/- 0 in the y-z plane. As Wincreases, the average facet slope increases, i.e., thecurves become broader.
Wave profiles obtained from stereophotographs showthat the average facet size decreases with increasing :or W.1 3 For large values of or W, this size is compa-rable with the lengths of microwaves, which are thendiffracted by the facets. The rough surface analysisbelow is limited to wavelengths for which diffractionmay be neglected.
B. Emissivity
The emissivity of a rough surface in any direction 0,0(Fig. 9) may be found by means of equations due toBasener and McCoyd,14 which add the contributionsmade in this direction by facets of every orientation.However, facets which are hidden from view by the wave
crests do not contribute to the emissivity. This shad-owing becomes more pronounced with increasing 0 orW. The correction factor for the shadowing effect wasfound by means of equations due to Wagner.15
The variation of EI, eli, E, and Q with 0 is shown by Fig.11 for upwind, downwind, and crosswind viewing, thewind speed being 14 m/sec. The solid curves are for aspecular surface. The following conclusions apply toall X for which diffraction may be neglected. Thepolarized emissivities are independent of surfaceroughness for 0 ' 250, while for 0 > 250, the thermalradiation is partly depolarized by the roughness. Itthus contains sea state information for possible detec-tion from earth orbit. However, close spacing of theupwind, downwind, and crosswind curves precludesdetermination of the wind direction from space.
In remote sensing of thermal radiation, the opticalabsorbing path increases with sec 0; therefore, 0 shouldnot exceed 45°. In this 0 range, the total emissivity Eis both constant and independent of surface roughness,which simplifies its use in retrieving sea surface tem-peratures from space.
C. Reflectance
The theory of reflection by a rough surface has beenreviewed by Beckmann and Spizzichino (BS)16 and-treated subsequently by many others. The relevantequations may be solved only for simple surface models.In the model treated by BS which is most relevant tothis work, the surface facets are curved, and their slopedistribution is symmetric and isotropic. A pencil oflight incident on the surface is reflected as a cone ofradiation with its axis in the specular direction.
In the Cox and Munk model of a rough sea, the slopedistribution is anisotropic and asymmetric (Fig. 10).Each facet reflects light in the locally specular direction.Its reflectance is here assumed to be a function of X andlocal angle of incidence given by Eqs. (2)-(6). Due tothe anisotropic slope distribution, the axis of the re-flection cone need not be in the specular direction oreven in the plane of incidence.
Light normally incident on a rough sea is reflectedvertically upward by facets of zero slope. The reflectionintensity is proportional to the total area of such facets,which decreases with W, as shown by Fig. 10. Thus thereflection intensity provides a measure of surfaceroughness.
Figures 12(a)-(e) show the angular distribution of thepolarized reflectances in the plane of incidence forvarious angles of incidence i. The reflection cones arerepresented as lobes, whose sizes and angles depend onX and W and also on the azimuth of the incident ray. Inthese figures, X = 170 i, W = 14 m/sec, and the inci-dent ray points downwind.
For small i id 0, the axes of the R 1 and R 11lobes lie onopposite sides of the angle for specular reflection. Asi increases, the R lobe grows larger, and the R 11lobeshrinks. Before it vanishes, however, a second R11 lobeappears with its axis pointing in the direction /27r. Thetwo R 11lobes coexist within a narrow range of i encom-passing i = b for each wavelength. As i approaches
Fig. 12. Directional polarized reflectances R 1 and RBI of wind-roughened water surfaces for wind speed W = 14 m/sec, optical
wavelength X = 170 ,im, and various angles of incidence i.
1/2-7r, the second R11 lobe becomes comparable in size withthe R lobe. At grazing incidence, both lobes degen-erate into rays, so that the surface reflects specularly.
V. Sea Surface Temperature Sensing in the 8-14-,um Band
In remote sensing of sea surface temperature, thevertical thermal radiance Bt received by a satellitesensor is given by Eqs. (10), which are derived else-where17 :
Here -r is the atmospheric transmittance, e is the normalemissivity of water, and B8 is the radiance of a black-body at the sea surface temperature. Thus B1 is thereceived surface radiance. The components B 2 and B 3both refer to received atmospheric radiance; Ba and Brare defined elsewhere.1 7 Equations (10) are solved forB8, all other quantities being either measured or cal-culated, and T, is found by inverting the Planck func-tion.17
The 8-14-gim spectral band is chosen for discussionhere because (a) it is used in remote sensing and (b) theatmospheric transmittance -r in this band is a fairlywell-known function of atmospheric moisture con-tent.17 18 Water vapor is the chief radiation absorberin this band.
In Eqs. (2)-(4), n and k (and therefore A and B) arefunctions of salinity.1 9 However, the E value computedfor pure water differs from that of seawater by <0.5%.When used in Eqs. (10), it causes an error of <0.20C inretrieved T8 .
Since E in this band lies between 0.96 and 0.995, ap-proximation E = 1 is routinely used in sea surface tem-perature retrieval. However, this has been shown tocause an error of -0.5 to -1 0C for very dry atmo-spheres.1 7 For very moist atmospheres, the error is only'-0.2 0C.17
Term B3 in Eqs. (10) gives the downward atmosphericradiance after it has been reflected at the sea surfaceand attenuated in its upward path to the sensor. Al-though B3 is very small, i.e., <0.004Bt, it cannot be ne-glected without causing an error of 0.1-0.3 0 C in re-trieved T. In deriving B 3, the surface was assumedspecular with only normally incident and reflected ra-diation.17 This approximation is justified because itcauses a negligible error in T, amounting to<0.050C.VI. Summary and Conclusions
Values of n,k for water near 250C, taken from sixrecent articles, were compared over a broad spectrum(Figs. 1 and 2). The largest discrepancies occur in theabsorption bands, due to anomalous behavior in thesebands, and the variety of observational methods. Thedata sources agree well for 2-15-gum wavelengths, afterwhich discrepancies increase with wavelength. Themost representative values of n,k were selected by visualinspection of the graphs.
Directional polarized reflectances, Brewster andPseudo-Brewster angles, also behave anomalously in the
absorption bands (Figs. 3-7). The polarized emissivi-ties of wind-roughened surfaces are independent ofroughness for viewing angles up to 250C, after which thethermal radiation is depolarized by roughness (Fig. 11).Thus the polarized thermal radiation contains wavestate information for possible detection from earthorbit.
For practical viewing angles in remote sensing, thetotal emissivity is both constant and independent ofwave state. Therefore, in retrieving sea surface tem-peratures, knowledge of the wave state is not required;the normal emissivity of a specular surface may be usedin Eqs. (10).
The effect of salinity on n,k in Eqs. (2)-(6) may alterthe reflectance by 10% in the 8-14-gm band.19
Therefore, the n,k values for pure water are not appli-cable to Mie scattering by fog particles of high salinityfound near the sea.19 The effect of salinity on isusually not significant. However, in remote sensing ofT for very saline waters in very dry climates, such asthe Dead Sea and Caspian Sea, small errors may resultfrom the use of the pure water values of e in Eqs. (10).More accurate n,k values may be needed in theseareas.1 9
Unpolarized light incident on a rough surface isscattered into overlapping cones of polarized radiation(Fig. 12). The intensity in the specular direction de-creases with increasing roughness. Thus in remotesensing with lasers or radar, the reflected beam containswave state information, which is being retrieved in stateof the art measurements.
The author is grateful to Rose Halwer for extensivehelp in searching the literature and to Grum de Hen-seler for programming assistance. Thanks are also dueto the Educational Computing Center of Baruch College
for the use of its facility. This work was supported bya grant from the City University of New York Com-puting Facility.References1. C. Cox and W. Munk, J. Opt. Soc. Am. 44, 838 (1954).2. H. R. Gordon, 0. B. Brown, and M. M. Jacobs, Appl. Opt. 14,417
(1975).3. H. D. Downing and D. Williams, J. Geophys. Res. 80, 1656
(1975).4. G. M. Hale and M. R. Querry, Appl. Opt. 12, 555 (1973).5. V. M. Zoloratev and A. V. Demin, Opt. Spectrosc. 43, 157
(1977).6. 0. A. Simpson, B. L. Bean, and S. Perkowitz, J. Opt. Soc. Am. 69,
1723 (1979).7. M. N. Afsar and J. B. Hasted, Infrared Phys. 18, 835 (1978).8. M. N. Afsar and J. B. Hasted, J. Opt. Soc. Am. 67, 902 (1977).9. P. S. Ray, Appl. Opt. 11, 1836 (1972).
10. W. J. Parker and G. L. Abbott, in Symposium on Thermal Ra-diation of Solids, S. Katzoff, Ed., NASA SP 55 (U.S. GPO,Washington, D.C., 1964), pp. 11-28.
11. F. F. Hall, Jr., Appl. Opt. 3, 781 (1964).12. G. P. Ohman, IEEE Trans. Antennas Propag. AP-25, 903
(1977).13. W. Marks, in Oceanography from Space (Woods Hole Oceano-
graphic Institute, 1965), p. 386.14. R. F. Basener and G. C. McCoyd, in Grumman Research De-
15. R. J. Wagner, J. Acoust. Soc. Am. 41, 138 (1966).16. P. Beckmann and A. Spizzichino, The Scattering of Electro-
magnetic Waves from Rough Surfaces (Macmillan, New York,1963), Chap. 7.
17. M. Sidran, Remote Sensing Environ. 10, 101 (1980).18. J. E. A. Selby, E. P. Shettle, and R. A. McClatchey, Supplements
LOWTRAN 3B AFCRL-TR-76-0258, Environmental ResearchPaper 587, Air Force Geophysics Laboratory, Optical DivisionProject 7670, Hanscom AFB, Mass. (1976), pp. 12-16.
19. D. E. Hobson, Jr., and D. Williams, Appl. Opt. 10, 2372 (1971).
1981November 9-10 An Overview of Droplet Sizing MethodsCourse Director: William D. BachaloTopics include: A review of light scattering and imagingof spherical particles, characteristics of availableinstruments, their applications, and limitations.Fee: $300. Sponsored by Spectron Development Laboratories,Inc., 3303 Harbor Blvd., Costa Mesa, CA 92626. For moredetails contact Dr. Bachalo at (714) 549-8477.
