B R D F
RELATIONSHIPS
BETWEEN QUANTITIES WHICH DESCRIBE REFLECTIVE FEATURES OF BOTH LAND AND OCEAN
AREAS
Zbigniew Otremba
Gdynia
Maritime University, 81-225 Gdynia, ul. Morska 83, Poland
zot@am.gdynia.pl
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Goniometric measurement of BRDF
see: subcripts |
Content
1. Bidirectional
reflectance distribution function
2. Remote sensing reflectance and radiance reflectance
4. Remotely sensed reflectance
5. Directional
reflectance factor
Introduction
More than ten
quantities are in use to describe the reflective properties of the earth areas
or any surface. Greatest of reflective quantities
were defined as a consequence of principle of construction of various
reflectance meters adapted for apparent reflective quantities measurements. On
the other hand the inherent reflective quantity is the Bidirectional
Reflectance Distribution Function (BRDF), which is often in use in the land
remote sensing (Liang and Strahler, 1999). Notation introduced to the land
remote sensing, with emphasis on multiangle observations, was put in order
recently (Martonchik et al, 2000), but notation which can be applied simultaneously
for both the land and sea areas has not been well unified yet.
The BRDF cannot
be directly measured, but recently a lot of investigations focused on empirical
and theoretical relationships between multiangle measurements and the BRDF were
derived (Schaaf et al., 2002, Jin et al., 2003). The most significant fact is
that the BRDF allows to derive various apparent reflective quantities - which
is indicated in this paper. Due to the limited volume of this conference-paper
it contains only a cut version of the currently prepared detailed paper on
‘various reflectances’.
1. Bidirectional reflectance distribution
function
The
bidirectional reflectance distribution
function (BRDF) is an inherent quantity and can be treated as a fundament
of all particular ‘reflectances’
using in marine and terrestrial remote sensing. Several reflective quantities
can be derived on the base of the BRDF - as is bellow indicated.
Definition of the BRDF (1.1)
was firstly introduced by Nicodemus et al. (1977).
(1.1)
The above definition dL(qu,ju,l) contains an infinitesimal change of upwelling spectral radiance caused by the infinitesimal change of spectral vector irradiance dEd(l). At the same
time dEd(l) originates from
directional light from strictly one direction qd,jd. Subscripts
‘u’ and ‘d’ relate adequately to words ‘upwelling’ and ‘downwelling’. Whereas
in most cases in land remote sensing ‘r’ is used instead of ‘u’ (reflected) and
‘i’ is used instead of ‘d’ - (incident).
Because the downwelling irradiance relates to
defined direction qd,jd - this information can be included in the
expression 1.1 (Warren, 1982; Perovich, 1994):
(1.2)
It is worth to state that Ed
would not be integrated by angles qd and jd, because in that definition Ed is a
spectrum of irradiance from only one defined direction qd, jd (the rest of
hemisphere is black). In the measure practice one should assume that Lu
is proportional to directional Ed – therefore following simplified
form of definition of the BRDF is
admissible – so called nondifferential form of the average BRDF (Snyder, 1998):
(1.3)
The reflectance - BRDF has only ideal (mathematical) meaning and cannot
be correctly measured when the sky is not black. However, the BRDF has a great
meaning because “shackles” dozen or so reflective quantities (as is shown
bellow).
Definition 1.2 allows
describing the following expressions 1.4 and 1.5 (equivalent one):
(1.4)
In this relation the denominator expresses the infinitesimal irradiance which is
generated by the downwelling radiance
Ld(qd, jd, l). In this way infinitesimal
change of upwelling radiance Lu (measured under direction
qu, ju related to infinitesimal change of solid angle
dWd = 
and related to
infinitesimal change of wavelength dl) can be stated in
the following way:
(1.5)
The two-sided integration of
equation 1.5 yields the expression 1.6. This expression allows the calculation
of the upwelling radiance Lu under defined direction qu, ju.
(1.6)
If the real range of
wavelengths of downwelling light is assumed from ld1 to ld2, then:
(1.7)
In greatest
practical situations, one could assume that no downwelling photon, which
generates more energetic upwelling photon. Then integral 1.7 takes following
form:
(1.8)
Assuming – if the surface of
the land or the surface of the sea is described by both magnitudes: spectral reflectance 
and downwelling
radiation 
-
possibility of calculation the upwelling
radiation comings into being.
2. Remote sensing reflectance and radiance reflectance
Remote
sensing reflectance RSR is an apparent quantity and is in use
when reflective properties are determined just above the earth surface.
Primarily RSR were called the remotely
sensed reflectance (Zaneveld, 1982), currently - the term remotely sensed reflectance is applied
for another quantity (see section 4).
If RSR is measured or modeled towards definite light condition
then becomes inherent-type one. The quantity RSR is very often used
by oceanographers to describe light leaving sea-surface. Measurement can be
carried out from satellite (Tozzi et al., 2002), aircraft (Gatebe et al., 2003)
or vessel (Arst and Haltrin, 2002). Independently on position of RSR
meter the quantity is referred to position just above the earth/sea/ocean
surface. The magnitude RSR measured on define height h is called radiance reflectance RL. Radiance reflectance can also be measured in the bulk of water
(Chang, 2002).
Definition of remote sensing reflectance RSR takes following form:
(2.1)
as well as radiance reflectance RL:
(2.2)
where: Lu(l) and respectively Lu(l,h) represent upwelling spectral radiance under direction qu = 0
Ed(ld) and respectively
Ed(l,h) represent
downwelling spectral planar irradiance just above the earth surface expressed
by relation 2.3 (one could compare with 1.9) and downwelling spectral planar
irradiance at defined height h.
(2.3)
Taking into consideration
definition of the reflectance-BRDF (1.16) and definition of the remote sensing reflectance (2.3) one can
describe relationship between both magnitudes (2.4), but under condition, that
downwelling radiance Ld(qd,jd) is known.
(2.4)
3. Irradiance reflectance
The irradiance reflectance RE
is an apparent-type magnitude defined by relation
(3.1)
where: Es(l) is an upwelling vector irradiance just above the sea/land surface
Eo(l) is a downwelling
vector irradiance just above the sea/land surface
Relationship between irradiance reflectance and BRDF:
(3.2)
4. Remotely sensed reflectance
The remotely sensed reflectance
is usually defined similarly to the irradiance
reflectance, but measured directly above the sea surface. This magnitude is
sometimes understood as an apparent, sometimes as an inherent type.
· If inherent – then
downward irradiance must be artificially produced by the laser beam.
· If apparent - then the magnitude is
measured by irradiance reflectance meter situated just above the sea/land
surface.
In addition, if apparent type
of remotely sensed reflectance is
considered, the rule of measurement is the same like for irradiance reflectance. Whereas if inherent - only measurements of
relative values are possible. Value of measurement of remotely sensed reflectance depends on apparatus features and
geometry of the measurement.
Relationship between the remotely
sensed reflectance and the reflectance
- BRDF:
· If apparent – then the relationship is described by expression 3.1.
· If inherent – then the relationship between the remotely sensed reflectance and the reflectance - BRDF is as follows:
(4.1)
5. Directional reflectance factor
The directional reflectance factor
is the same as the radiance reflectance
but multiplied by p:
(5.1)
The idea of using the value p is connected with
the fact that if the sky radiance at all points of the hemisphere is
independent of angle (and is equal to 
), then
(5.2)
In such situation the value p in expression 5.1
disappeared and downwelling irradiance Ed(l) is replaced by downwelling radiance 
:
(5.3)
For this reason r(l) is dimensionless.
Relationship between the directional
reflectance factor and the reflectance
- BRDF:
(5.4)
where: Ld(qd,jd,l) is downwelling radiance just above the sea surface
6. Ocean color reflectance
The ocean color reflectance RC
is an apparent magnitude
(6.1)
where Ed expresses solar irradiance integrated by full
wavelength range.
Relationship between ocean color reflectance and BRDF:
(6.2)
7. Discussion
The above listed definitions
represent a wide set of quantities, which were established to describe
reflective properties of sea areas especially. Practical application of any one
of them depends on several factors. The first one is a kind of device using for
measurement of reflective features. For example - if one uses only the
reversible (up-down) irradiance meter then only the radiance reflectance can be measured. But despite of the radiance reflectance depends on angular
reflective features of the surface those features are not evident. Otherwise
that reflective features included in the reflectance
can be recalculated to the radiance
reflectance, but reverse process is very difficult. The direction of
relationships between various ‘reflectances’ is showed bellow.
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Generally utilized magnitudes for describing of
reflective features of both land and ocean surfaces,
as well as interrelate directions
suggesting possibility of deriving the reflectance-BRDF
to other quantities.
The BRDF-reflectance and the bidirectional
reflectance factor (BRF) are analogues quantities (differ only by the
number p, BRF=pBRDF). The BRF is unit-less quantity (whereas unit of BRDF is 1/rad).
Another reflective quantity -
the irradiance reflectance is usually
called by oceanographers: the volume
reflectance or the subsurface
irradiance reflectance, when they want to describe vertical light balance
in the bulk of water.
Inherent reflective
magnitudes hold greatest amount information about the earth surface or the
earth surface waters, but they are very difficult to measure. On the other hand
a primitive factor such as the irradiance
reflectance may be not sufficient for representing land or water
properties. Apparent magnitudes are useful for quick and rough monitoring or estimation
of current or long-term nature processes. Fortunately there are a lot of cases
when mathematical relationships between reflective apparent factor and
environmental features can be settled. That mathematical relationships are
usually experimentally settled formulas, so they are called “engineering
model”. Just if the model comes from fundamental physical considerations then
becomes a “physical model”. Models are very often mixed – engineering-physical.
Examples of strictly physical models are products obtained by Monte Carlo (MC) methods. The MC methods
need detailed physical input data. If marine environment is considered a lot of
parameters describe features of the sea-water components, which takes dissolved
or suspended form. Dissolved substances are characterized in the input of model
by spectra of absorption coefficient and by concentration. Suspended substances
are characterized by spectra of scattering and absorption coefficients and by
volume (or phase) angular scattering functions. Unfortunately one can indicate
a shortage of knowledge about the above listed optical factors of sea-water
components. Something comfortable is that there are still some investigations
carried out on plankton cells [Krol et al., 2001], oil suspension (Otremba, 2002),
air bubbles (Stramski and Têgowski, 2001), suspended sand grains (Haltrin et
al., 2001). Stressing attention to ocean optics one could note, that current
research directions are dispersed and is independently planned something
overmuch. One of indicators of such situation is – for example – using more
than 10 various magnitudes describing reflective features, which cannot be tied
by mathematical relationships. On the other hand operating by only one
magnitude (the reflectance) would not be possible due to the survey
difficulties. Current methods allow for measurements of only limited components
of the reflectance (BRDF). In my researches the reflectance-BRDF of sea areas
contaminated various form of oil is modeled using the Monte Carlo ray tracing method. This paper indicates that the obtained reflectance-BRDF can be recalculated to
various ‘reflectances’ which are in use in the remote sensing.
In order to
roughly describe the reflective features of land or ocean surfaces besides of
above listed quantities also further quantities are in use very often. For
example the spectral albedo, which is
the ratio of the power returning from a unit surface to the power incident upon
that unit surface. Unfortunately that definition is not unequivocal and can be
interpreted in many different ways – it is a separate arrangement worthy
problem. All magnitudes used for describing of reflective features can also be
related to the time or to define time-period relation. In
my opinion the harmonization of nomenclature used by the remote sensing
community for describing the land-ocean regions would be useful.
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Annex
One can find in the literature or net other quantities which can be
described by the BRDF – for example:
Bidirectional
Reflectance Factor BRF is the ratio of the radiance reflected from the target in the
direction, to the
radiance reflected from a Lambertian reflectance-reference panel in the same
direction measured under identical illumination conditions.
In
satellite remote sensing BRF is defined as follows:
where: L – radiance, 
- solar irradiances,
D – Sun-Earth distance (in astronomical units), q - the solar zenith angle
The
anisotropy factor ANIF of a surface in a particular view direction is defined as the
ratio of the reflectance in that direction to the nadir. Since ANIF is a ratio
of two reflectance factors it is also a unit-less quantity. The ANIF values for
an ideal Lambertian surface would be unity in all view directions.
Anisotropic
Index ANIX is
the ratio of the maximum and the minimum reflectance factors observed in a
given azimuth plane for a given spectral band is known as the Anisotropy Index.
Since this quantity is also a ratio of two BRFs it is a unit-less quantity.