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CHAPTER 5. REFLECTION AT THE SEA SURFACE
R E F L E C T I O N AT T H E SEA S U R F A C E
reflected. This occurs at a water surface (n = +) for i = 539, and
in this case:
1 na-1 a
ps = 2 na+l
Values of the Fresnel reflectance according to the above equations
are tabulated as functions of the solar elevation (Table XVI).
Linearly polarized light which is completely internally reflected
suffers a phase change between the parallel and perpendicular com(1954)
ponents giving rise to elliptically polarized light. WATERMAN
(1958a) have proved that elliptically
polarized light occurs just below the water surface in lines of sight
differing from the vertical by less than the critical angle (see Chapter 6).
T AB L E XVI
REFLECTANCE OF RADIATION AGAINST A CALM SURFACE
Reflectance for dzfuse radiation
The reflection of the diffuse component is more di&cult to express
quantitatively. A first approximation considers diffuse light of equal
radiance from all directions L of the sky. The reflectance p(i) for
the angle of incidence i is taken from the Fresnel equations. It follows
from the definition of reflectance and from eq.5 that :
sin i cos idi
= ]rp(i) sin 2idi
L sin i cos idi
Several workers have evaluated this integral for a smooth water
surface using Fresnel's reflectance. The consistent value is found
to be 6.6 yo (see BURT,1954a).
For the reflectance in the case of a cardional distribution of sky
radiation (eq.24), PREISENDORFER
(1957) has given an exact solution
which yields 5.2 yo.
Reflectance for global radiation
When dealing with the reflectance of global radiation we have to
treat separately the reflectance of direct energy or solar radiation,
ps, and that of sky radiation, pa.
The total reflectance is expressed as the sum:
where E and E, are the incident and reflected irradiance respectively,
and n is the significant ratio of sky radiation to global radiation.
The result of computations of the reflected energy from eq.30
using p,-values from eq.27 and pd = 0.066 and assuming the
incoming radiation to be a given function of the solar elevation, is
illustrated, for example, by the set of curves for several n-values in
1961). The curve for solar radiation
only (s = 0) develops a prominent maximum at 20" as well as a slight
minimum at 50". The intersection point of the curves at 29" represents
the case of p, = pd = 0.066.
The assumption of uniform sky radiation is inadequate, and more
realistic approaches have been made. Cox and MUNK(1956) have
REFLECTION AT T H E SEA SURFACE
n = 0.5
Fig.23. Amount of reflected energy for different portions n o f diffuse radiation
in the global radiation. (After NEUMANN
Solar zenith diStcnet3
Fig.24. Reflected radiance L(i) divided by the sky radiance at zenith for calm
weather and for a Beaufort 4 wind. (After Cox and MUNK,1956.)
also employed a semi-empirical model and considered a clear tropical
sky on the basis of observations at Bocaiuva, Brazil, for the sun at a
zenith angle of GO". They made the simple assumption that the
reflectance is a function of the zenith angle only, ignoring the
increased radiance near the sun (Fig.21). Their results for clear and
overcast sky conditions are presented in Fig.24.
The complete investigation of the interaction of light with the sea
surface confronts us with the additional problem of reflection from
a wind-roughened surface. The following changes in reflection are
immediately realized. With high solar elevations the angle of incidence
will be increased on an average, whereas it will be decreased for low
elevations. The former effect is not important, as the reflectance does
not vary much with the solar elevation ifthis is high. In contrast, the
reflectance for a low sun is drastically reduced by wave action. This
principal feature was first formulated and investigated by LE GRAND
(1939). It appears that the reflectance is independent of the presence
of waves in some intermediate elevation interval. BURT (1954a),
employing a semi-theoretical model, actually found that such independence occurs somewhere between solar elevations of 10" and
The thorough interpretation of the wind effect given by Cox and
MUNK(1956) also takes into account shadowing and multiple reflections for a low sun. Their radiance curves in Fig.25 demonstrate in
essence that the wave action becomes a factor for solar elevations
(yo)OF THE SEA FOR SKYLIGHT
Cox and MUNK
REFLECTION AT T H E SEA S U R F A C E
Solar zenith distance
Fig.25. Reflectance of solar radiation from a flat surface and from a surface
roughened by a Beaufort 4 wind. (After Cox and MUNK,1956.)
below 20”. GRISEENKO
(1959) and HISHIDAand KISHINO(1965)
have demonstrated similar wave effects at low angles. As expected,
the reflection of sky light is less affected by rough sea (Table XVII).
Cox and Munk point out that with an absolutely flat sea the horizon
would not be visible. Actually the sea always contrasts with the sky
and becomes darker when the wind increases. The shadowing is also
examined by LAUSCHER
(1955), who computed data which amply
illustrate the irregular reflection pattern from a wave of maximal
steepness (Michell wave).
Glitter of the sea
Glitter is a phenomenon bearing upon reflection of solar radiation.
This special aspect of reflection has given rise to much speculation.
The glitter pattern being open to everyone’s observation, it has in
popular speech been given poetic names such as “the road to
1941) or “the golden bridge” (STELENAU,
1961). The glitter arises when a flat surface is roughened by wind and
the image of the sun formed by specular reflection explodes into
glittering points. This is because water facets occur with an orientation so as to reflect sunlight to the observer. Increasing roughness
will enlarge the width of the glittering band. The phenomenon is
most spectacular at solar elevations of 30-35” (cf. Fig.23). The
pattern becomes narrower when the sun sets.
CONCEPT O F ALBEDO
The distribution of radiance of glitter has been the subject of
theoretical treatment with a view to estimating the slopes of the sea
surface. The interested reader may consult the original papers by
(1952), Cox and MUNK(1956), SCHOOLEY
(1961), and MULLAMAA
The latter has also considered the polarization of the reflected
glitter, which is shown to be dependent both on solar elevation and
on the direction of reflection. The polarization at the maximum
radiance is near zero at i = 0" and increases with zenith distance to
100 yo at i = 35-45", subsequently decreasing to 20 % at i = 90".
CONCEPT O F ALBEDO
In order to scrutinize reflection events and secure adequate definitions, the following symbols are pertinent:
Ead= downward irradiance in air.
Eau= upward irradiance in air.
Ewd= downward irradiance in water.
Ew = upward irradiance in water.
Since reflection against the sea surface takes place from above as
well as from below, two kinds of reflectance are distinguished: ( 7 )
p a = reflectance in air; and (2)pw = reflectance in water.
The albedo A of the sea is defined as the ratio of the energy leaving
the sea to that falling on it:
The above definitions yield the following identities :
Ead-Eau = Ewd-Ewu
Eau = paEad+Ew-pwEw
From eq.31 the albedo may therefore be written:
Some confusion in reflection studies has resulted from failure to
distinguish the reflection (at a surface) in a strict sense, namely p a ,
and the albedo which is the sum of p a and the percentage of light
REFLECTION AT T H E SEA SURFACE
back-scattered from the sea. Furthermore, the factor pw must be
accounted for. Because of the relatively small variation of the scattering function in the back-scatter field, we can assume in a first approximation that the upwelling light is completely diffuse and unpolarized.
Because Fresnel reflection beneath the surface involves total reflection
in the angle interval 48.6-90", integration of eq.29 yields pw = 48%.
Hence it is not permissible to neglect this factor as most workers
EXPERIMENTAL VALUES OF REFLECTANCE
A selection among the multitude of reflection observations is called
for. The systematic studies of albedo as a function of various parameters made by ANDERSON
(1954) and FORS(1954) will be our
primary references. The Lake Hefner results suggest that the albedo
under clear skies is only weakly dependent on wind speed and air
mass turbidity, including the effects of clouds. We may infer that the
albedo is primarily a function of solar altitude. Even with a low
stratus cloud cover, the dependence on the position of the sun
persists. This is consistent with the findings of NEIBURGER
but conflicts with earlier statements (Chapter 4).
Table XVIIl shows the true reflectance for a flat level surface and
clear sky. These values have been drawn chiefly from Anderson's
results with due correction for the existing low shortwave backscattering from the water of Lake Hefner.
In rough weather it is difficult to distinguish between the true
reflectance and the shortwave light back-scattered from the sea. White
caps and air bubbles in the surface layer contribute greatly to the
albedo. Experiments have verified the reduction of reflectance at
low solar elevations owing to wave action. The present author found
TABLE X V I I I
REFLECTANCE OF UNPOLARIZED RADIANT ENERGY (SUN+ SKY) FROM A HORIZONTAL
DISPERSION O F REFLECTION
Fig.26. Reflectance (yo) of global radiation for different solar elevations as a
function of wavelength. Corrected for cosine error of the collector.
that a flat sea surface is a barrier to the light from a distant lighthouse
tower, whereas the reflection of this light from a rough surface is
DISPERSION OF REFLECTION
It is established by observations that for solar elevation below 30"
a dependence of reflection of irradiance on wavelength is developed
and RUTTNER(1941) have given the right explanation of this effect which - since the dispersion of light reflection is
only 6% -must be associated with the amount of dif€use light in the
global light. This is 90 % in the violet and 22 yoin the red for a solar
elevation of 10"(Fig.22). The average angle of incidence is thus much
less for the violet than for the red, which accounts for the stated
difference in reflection.
R E F R A C T I O N A T THE S E A S U R F A C E
The interface between air and sea is a boundary between two media
of different optical density. An electromagnetic wave falling on the
surface decomposes into two waves ; one is refracted and proceeds
into the sea, the other is reflected and propagates back into the air
The law of refraction is:
a relation known as Snell's law. The angles i a n d j are defined so as to
describe a refracted wave as being in the same plane as the incident
wave. For definiteness we may take the refractive index n of sea water
relative to air to be 8. From a geophysical point of view there is no
need to account for the variation of the index with temperature,
salinity and wavelength (Chapter 2, p.25).
For the case of grazing incidence (i = 90') a limiting angle of
refractionj = 48.5" is obtained. As seen from the sea, the sky dome
is compressed into a cone of half-angle 48.5'. If they form an angle of
more than 48.5" with the vertical, upward travelling rays are totally
reflected at the surface (Fig.27).
Fig.27. Refraction and total internal reflection at the sea surface.
EFFECT OF WAVES O N REFRACTION
Refraction is associated with polarization according to Fresnel's
formulae. Direct sunlight penetrating into the water becomes partly
polarized. Calculations by MULLAMAA
(1964a) testify that the degree
of polarization increases with the sun's zenith distance and reaches
magnitudes as great as 27 yo during sunset.
The linearity of skylight polarization remains unchanged by refraction though there is a rotation of the plane of oscillation.
C H A N G E OF RADIANCE A N D I R R A D I A N C E AT T H E SURFACE
It follows from simple geometrical considerations that the radiance
L, below water of refractive index n is na times the radiance Lain air
reduced by reflection losses at the surface (GERSHUN,
L, = #a( 1-&La
The downward irradiance in air and in water are respectively:
d E a = La cos i d o = Lacos i sin id+&
dE, = L, cosj d w = L, cosj sinjdr$dj
From eq.33 and 34 it follows that:
dE, = (1 -p)dEa
which verifies the obvious fact that the interface between air and
water changes irradiance on a plane parallel t o the interface by
EFFECT OF WAVES O N REFRACTION
Refraction according to Fig.27 is described by Snell's law only in
the ideal case of a flat surface. Waves cause fluctuations of the direction of refracted rays. Deviations of the direction of maximum
radiance can amount to as much as f 1 5 yo according to theoretical
deductions of MULLAMAA
(1964a). Because of wave action, the image
of the sun as seen from beneath the surface disintegrates into a glitter
pattern with features different from those of the reflected glitter. The
refracted glitter subtends a smaller angle and is of the order of 1,000
times more intense than the reflected glitter. Due to lens action of