兰伯特余弦定理在微波漫反射的应用
微波与地物相互作用的过程会在地物表面产生镜面或者漫反射。
一般情况下，如果地物表面是光滑的，反射电磁波服从斯涅耳定律。但是，当地物表面是粗糙表面，漫反射电磁波散射服从的是兰伯特余弦定理。
具体特点如下：
Lambert's
cosine law
From
Wikipedia, the free encyclopedia
See also: Lambertian reflectance
In optics, Lambert's cosine law says
that the radiant
intensity observed from a "Lambertian" surface
is directly proportional to the cosine of the angle θ between the
observer's line of sight and the surface normal.
The law is also known as the cosine emission law or Lambert's
emission law. It is named after Johann Heinrich Lambert, from his Photometria,
published in 1760.
An important
consequence of Lambert's cosine law is that when such a surface is viewed from
any angle, it has the same apparent radiance. This
means, for example, that to the human eye it has the same apparent brightness
(or luminance). It
has the same radiance because, although the emitted power from a given area
element is reduced by the cosine of the emission angle, the size of the
observed area is decreased by a corresponding amount. Therefore, its radiance
(power per unit solid angle per unit projected source area) is the same. For
example, in the visible
spectrum, the Sun is
not a Lambertian radiator; its brightness is a maximum at the center of the
solar disk, an example of limb darkening. Ablack body is
a perfect Lambertian radiator.
Lambertian
scatterers
When an area
element is radiating as a result of being illuminated by an external source,
the irradiance (energy
or photons/time/area) landing on that area element will be proportional to the
cosine of the angle between the illuminating source and the normal. A
Lambertian scatterer will then scatter this light according to the same cosine
law as a Lambertian emitter. This means that although the radiance of the
surface depends on the angle from the normal to the illuminating source, it
will not depend on the angle from the normal to the observer. For example, if
the moon were a Lambertian scatterer, one would expect to see its scattered
brightness appreciably diminish towards the terminator due
to the increased angle at which sunlight hit the surface. The fact that it does
not diminish illustrates that the moon is not a Lambertian scatterer, and in
fact tends to scatter more light into the oblique angles than
would a Lambertian scatterer.
Details of
equal brightness effect
|
Figure 1: Emission rate (photons/s) in a normal and
off-normal direction. The number of photons/sec directed into any wedge is
proportional to the area of the wedge.  |
|
Figure 2: Observed intensity (photons/(s·cm2·sr))
for a normal and off-normal observer; dA0 is the area
of the observing aperture and dΩ is the solid angle subtended by the
aperture from the viewpoint of the emitting area element.
|
The situation
for a Lambertian surface (emitting or scattering) is illustrated in Figures 1
and 2. For conceptual clarity we will think in terms of photons rather than energy or luminous energy.
The wedges in the circle each
represent an equal angle d Ω and, for a Lambertian surface,
the number of photons per second emitted into each wedge is proportional to the
area of the wedge.
It can be seen
that the length of each wedge is the product of the diameter of
the circle and cos(θ). It can also be seen that the maximum rate of
photon emission per unit solid
angle is along the normal and diminishes to zero for θ =
90°. In mathematical terms, the radiance along
the normal is I photons/(s·cm2·sr) and the number
of photons per second emitted into the vertical wedge is I dΩdA.
The number of photons per second emitted into the wedge at angle θ is I cos(θ) dΩ dA.
Figure 2
represents what an observer sees. The observer directly above the area element
will be seeing the scene through an aperture of area dA0 and
the area element dA will subtend a (solid) angle of dΩ0.
We can assume without loss of generality that the aperture happens to subtend
solid angle dΩ when "viewed" from the emitting area
element. This normal observer will then be recording I dΩ dAphotons
per second and so will be measuring a radiance of
photons/(s·cm2·sr).
The observer
at angle θ to the normal will be seeing the scene through the
same aperture of area dA0 and the area element dA will
subtend a (solid) angle of dΩ0 cos(θ). This
observer will be recording I cos(θ) dΩ dA photons
per second, and so will be measuring a radiance of
photons/(s·cm2·sr),
which is the
same as the normal observer.
See also
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此外，在兰伯特反射的镜面反射部分中一部分保持相干特征，称为相干散射分量，其余方向的散射分量则称为非相干分量。而且它们的极化方式也会一部分改变。
