Lidar Measurements: Atmospheric Constituents ...
219
? 2 2;2 222222~V2V22222222m
,
:t Ov_g~~ncUonT-=:;Z:
Distancex -
Figure 10.1: Schematic representation of the overlap integral O(x) of the laser beam with the
receiver field of view, as a function of distance x.
10.2.1 Mie lidar
In the lower troposphere where aerosols such as dust, haze and fog are the dominant scatterer,
most of the lidar backscatter radiation comes from particles that are comparable in size with the
wavelength of the scattered light. The theory of this process was first described by Gustav Mie
(1868-1957). Although Mie's theory rigorously applies to spherical and homogeneous particles
only, it has become customary to call devices Mie lidars that register elastically backscattered
laser radiation and analyze the return signals for the distribution and properties of the scattering
aerosol.
In Eq. (10.3) the transmittance T decreases monotonically with distance x according to
(10.4)
where a is the extinction coefficient: clearly the local extinction coefficient at point e must be
integrated out to distance x to obtain the transmission between the lidar and the point x where
the backscattering occurs; the power 2 to which T is raised reflects extinction of the primary
as well as the backscattered light.
Introducing Eq. (10.4) into Eq. (10.3) leads to an equation with one measured quantity (P)
and two unknowns (a and (3). A way out of this dilemma is what is today known as Klett's
inversion method (Klett, 1981; Fernald, 1984). It consists in the postulate of a relation
(3 = Fa~
(10.5)
between the unknowns, F and r;, being two constants. The lidar equation then turns into an
ordinary differential equation of the Riccati type, the solution of which is
(10.6)
with the abbreviation
S(x) = In[x 2 p(x)]
(10.7)
and a set of boundary values
000 = a(xo), So = S(xo)
(10.8)
at some given distance Xo for which the extinction coefficient a is sufficiently well known. This
can, e.g., be the case at the beginning of the lidar range at which an in-situ device is available
219
? 2 2;2 222222~V2V22222222m
,
:t Ov_g~~ncUonT-=:;Z:
Distancex -
Figure 10.1: Schematic representation of the overlap integral O(x) of the laser beam with the
receiver field of view, as a function of distance x.
10.2.1 Mie lidar
In the lower troposphere where aerosols such as dust, haze and fog are the dominant scatterer,
most of the lidar backscatter radiation comes from particles that are comparable in size with the
wavelength of the scattered light. The theory of this process was first described by Gustav Mie
(1868-1957). Although Mie's theory rigorously applies to spherical and homogeneous particles
only, it has become customary to call devices Mie lidars that register elastically backscattered
laser radiation and analyze the return signals for the distribution and properties of the scattering
aerosol.
In Eq. (10.3) the transmittance T decreases monotonically with distance x according to
(10.4)
where a is the extinction coefficient: clearly the local extinction coefficient at point e must be
integrated out to distance x to obtain the transmission between the lidar and the point x where
the backscattering occurs; the power 2 to which T is raised reflects extinction of the primary
as well as the backscattered light.
Introducing Eq. (10.4) into Eq. (10.3) leads to an equation with one measured quantity (P)
and two unknowns (a and (3). A way out of this dilemma is what is today known as Klett's
inversion method (Klett, 1981; Fernald, 1984). It consists in the postulate of a relation
(3 = Fa~
(10.5)
between the unknowns, F and r;, being two constants. The lidar equation then turns into an
ordinary differential equation of the Riccati type, the solution of which is
(10.6)
with the abbreviation
S(x) = In[x 2 p(x)]
(10.7)
and a set of boundary values
000 = a(xo), So = S(xo)
(10.8)
at some given distance Xo for which the extinction coefficient a is sufficiently well known. This
can, e.g., be the case at the beginning of the lidar range at which an in-situ device is available