EBS 566/666 notes 2/1/2010
Lecture 8
I.
Exercise. (A) Convert radiant flux () of 1021 quanta s-1 (photons s-1) to J s-1 (watts) for a
wavelength of 550 nm; (B) Convert radiant flux () of 2.5 x 1011 W to quanta s-1.
A. Convert radiant flux () of 1021 quanta s-1 (photons s-1) to J s-1 (watts) for a wavelength
of 550 nm:

energy of a photon is given by = (1989/) x 10-19 J
wavelength is 550 nm
3.62 x 10-10 J/photon * 1021 quanta (or photons)
=3.62 x 1011 J s-1
B. Convert radiant flux () of 2.5 x 1011 W to quanta s-1:
quanta s-1 = 5.03   x 1015
= 5.03 (2.5 x 1011 J s-1 * (550 x 10-9) x 1015
= 6.9 x1020 quanta s-1
II.
Characterizing the light field
(See: http://ceos.cnes.fr:8100/cdrom-00/ceos1/science/baphygb/chap2/chap2.htm)
II.1 Defining a direction in space
The direction of a line through any point on the Earth's surface is defined by 2 angles:
 the zenith angle , between the zenith (point on the celestial sphere located on the
observer's ascending vertical) and the direction observed,
 the azimuth angle  between the North (on the local meridian) and the projection of the
line on the Earth's surface.
The height (altitude or
elevation) is sometimes used
instead of :
h = ( / 2) - ,
 varies along the vertical
plane from 0 to /2 (0° to
90°),
 varies along the horizontal
plane from 0 to 2  (0° to
360°).
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II.2 Definitions
II.2.1 Solid Angle: a solid angle ddelimits
a cone in space: d = dS / r2 (in Steradians, Sr)
where dS is the area cut by the cone over a
sphere of radius r, the center of which is at the
apex of the cone.
II.2.2 Intensity: intensity is the power
emitted by a point source A per solid angle
unit.
IA = dW/d (in W.Sr-1)
If the intensity is the same in all directions, the
source is called isotropic. Whenever a source
does not have the same power in all directions
it is said to be anisotropic.
This notion is rarely used in remote sensing, as
the Earth's surface observed by satellite is not a
point source.
II.2.3 Radiance: radiance (L) is the power
emitted (dW) per unit of the solid angle (d)
and per unit of the projected surface (ds cos)
of an extended widespread source in a given
direction ().
L = d2W / (d. ds. cos) (in W.Sr-1. m-2)
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II.2.4 Irradiance: power received per surface unit from all directions (hemispheric, 2 halves).
E = dW / dS (in Wm-2)
The element of the Earth's surface ds receives an irradiance E from the upper half space and acts
for the sensor as a source of radiance L along a direction .
II.3
Light in water
Velocity of light in a medium = velocity of light in a vacuum divided by the refractive index of
the medium (refractive index of water is 1.33)
Frequency of radiation is the same in water and in a vacuum, but the wavelength diminishes in
proportion to the decrease in velocity in water vs. a vacuum; c and  change in parallel.
Given= c/wheredoes not vary,  must decrease.
III.
Inherent Optical Properties (IOP): IOPs are properties that depend only on the water
and other substances that are dissolved or suspended in it, as distinguished from Apparent
Optical Properties (AOPs). The two fundamental IOPs are absorption (a) and the Volume
Scattering Function (ß). Others commonly derived from these include the total scattering
coefficient (b), backscattering coefficient (bb), and beam attenuation coefficient (c = a + b).
III.1 Absorption (a): The absorption coefficient is the fraction of incident power that is
absorbed within an infinitesimal volume, divided by the distance of propagation through the
volume, expressed in m-1.
III.2 Volume Scattering Function (): The Volume Scattering Function, or VSF, notated
mathematically as ß ("beta") characterizes the intensity of scattering as a function of angle. The
VSF is defined in terms of a beam of light incident on an infinitesimal volume. At each angle
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from 0° (the original angle of the incident light) to 180°, the VSF is the ratio of the intensity of
scattered light (in W/sr) to the incident irradiance (in W/m2), per unit volume (in m3). Therefore
the VSF's units are (W/sr) / (W/m2 m3) = sr-1 m-1.
III.3 Scattering Coefficient (b): Also called the Total Scattering Coefficient: the fraction of a
collimated beam, incident on an infinitesimal volume, that is scattered (in all directions) per
meter of distance. The total scattering coefficient b is calculated by integrating ß over all angles,
and is also equal to the integral of the Volume Scattering Function over all angles.
III.4 Backscattering Coefficient (bb): The backscattering, or backward scattering, coefficient,
in units of m-1. It indicates the attenuation caused by scattering at angles from 90° to 180°. bb is
commonly estimated from measurements of the VSF around a single fixed angle. bf is the
forward-scattering equivalent of bb, indicating scattering in the range of 0° to 90°.
Discussions of scattering often refer to the Scattering Phase Function, which is equal to ß/b.
Because b is the integral of ß with respect to angle, the integral of the phase function with respect
to angle is 1.
III.5 Attenuation Coefficient, Beam (c): The attenuation experienced by a hypothetical
perfectly collimated beam of light. Represented as c, it is equal to a (absorption coefficient) + b
(scattering coefficient). It is measured by a transmissometer, which ideally receives only the
unscattered light that traverses the path from its source to its receiver. Because every practical
transmissometer has imperfect beam collimation and a finite field of view, some scattered light is
collected by the receiver, causing c to be underestimated. For example, HOBI Labs' c-ßeta has
divergence and FOV of less than 0.5° (half-angle), resulting in c values that are slightly higher
(and more accurate) than other transmissometers with wider divergence angles.
IV.
Apparent Optical Properties (AOPs): Properties that depend both on inherent optical
properties (IOPs) and on the light field in which they are measured.
The most widely-used AOP is the diffuse attenuation coefficient Kd. Kd is essentially the
attenuation experienced by sunlight, and its value depends on depth, sun angle, sky conditions,
and shadowing by objects on the surface. However as depth increases, the influence of the
surface illumination characteristics decreases, and Kd eventually reaches an asymptotic value that
is in fact an IOP. Thus the distinction between IOPs and AOPs can sometimes be misleading.
In practice, every optical measurement is dependent on the light field used for the measurement,
but instruments for IOP measurements provide their own controlled light field rather than relying
on ambient light. In contrast, AOPs are typically derived from measurements of ambient light.
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