(1)
where: J(l,R,r) = the laser induced
spectral radiance at a given l
R = the range
r = the location in the target plane at height
R
dA(R,r) = the element of the laser target area at range
R and location r
p(l,R,r) = the probability
that radiance from R,r will hit the detector
The probability p(l,R,r) is expressed in equation 2.
(2)
where: A0/R^2
= the solid angle over which the receiver optics can see the signal
T(l,R) = the atmospheric transmission factor
of the atmosphere as a function of wavelength and range.
x(l) = the spectral transmission
function of the receiver
x(R,r) = the probability of the radiation
at range R and location r in the target plane of R will reach the receiver,
as a function of the receiving optics.
The spectral transmission function of the receiver, x(l), is a parameter that takes into account any attenuations or amplifications that the receiving stage might do to the raw signal. J(l,R,r) also can be expressed as a product of the laser radiance at R,r and the volume backscattering coefficient b, which is a function of the laser wavelength, the scattered wavelength (which is the same as the laser wavelength in elastic lidar, of which will be discussed later in detail), the range, and the location in the plane of R, r. b is given by equation 3.
(3)
where: Ni(R,r) = the number density of the scattering species
(ds(l)/dW)i s
= the differential scattering cross section with respect to l
Dl
=
the probability that the backscattered radiation is between
l and l+Dl
The time it takes for a photon to make a round trip from the laser back
to the lidar light collecting mechanism is given by equation 4, where c
is the speed of light, and R is the range.
(4)
Plugging equations 2 and 3 into 1 and solving with equation 4 for the upper range limit R yields 5.
(5)
Were J(l,R,r), the target spectral
radiance, has been replaced by the product of b,
the volume backscatter coefficient and 
,
the laser irradiance at position r in the plane R. Also assume that 
and the spectral transmission function 
,
can be thought of as having delta function components centered at l,
assuming that the backscattered wavelengths measured lie well within the
spectral bounds (Dl0) of
the receiving optics. If we assume that the atmosphere is homogeneous
between the laser overlap of the field of view of the telescope and R,
and as a consequence assume that b(l,R,r) is
constant over r, then this term can be factored out of the integral, leaving
equation 6.
(6)
The last three terms on the right can then be considered. Again, if we can approximate the laser radiation is uniform over all r at a range R, and the probability is equally weighted for all r, then the three terms on the right become invariant with r yielding equation 7.
(7)
Where AL(R) is the area at range R that the laser
beam covers. We can then put the limits of integration in terms of
time using equation 4, and using tL as the laser pulse width. The limits
can then be expressed as c(t-tL)/2
and ct/2 respectively, and using the fact that R is much much greater than
the laser pulse length ctL,
it can be shown that power per round-trip time (t = 2R/c) is equation 8.
(8)
I(R) can also be expressed as equation 9.
(9)
where: EL = the output energy of the laser
pulse
T(l,R) = the atmospheric transmission
of the laser pulse
tL = the duration
of the laser pulse
AL(R) = the area at range R that the laser
beam covers
Atmospheric transmission is represented by Beer’s Law (Equation 10),
where k(l,R) is the atmospheric attenuation
coefficient.
(10)
Note that T(l,R) appears twice in equation
8 with equation 9 substituted, so plugging equation 10 into equation 8,
along with equation 9 yields equation 11, the instantaneous power falling
upon the detector.
(11)
If we want to know energy in terms of a detector’s integration time, then we can integrate equation 11 over the limits of 2R/c to 2R/c +td , where td is the detector’s integration time, to obtain equation 12, the scattering form of the basic lidar equation.1
(12)
Here at UMBC the Atmospheric Lidar Group maintains a small
elastic lidar named the Elastic Lidar Facility (ELF) shown below:
The ELF, being a common example of all elastic lidars, shares the same characteristics of all lidar systems. The transmission stage of the ELF consists of a laser and a prism. The laser is a flash lamp pumped, Q-switched Continuum Surelite II Nd:YAG, which is operating at 1064 nm, the fundamental frequency of Nd:YAG, and 532 nm, the first harmonic. The repetition rate for the laser is 10 Hz. After the beam leaves the laser, the beam encounters a steering prism directing the beam skyward.
.
The next figure shows the path of the beam as it exits the prism.
The prism sits upon a stage assembly that has 2 degrees of freedom, enabling
the beam to be aligned and aimed.
The next part of the ELF system is the receiving stage.
A 14” Celestron Schmidt-Cassegrain telescope collects the backscattered
light.
On the focal plane the telescope, there is an optics package that
is comprised of removable neutral density filters, a dichroic filter, and
two narrow band pass filters. Since Mie scattering has such a large
backscattering cross section, the signal is often large enough to saturate
the detectors. Neutral density filters in the optical path are used to
alleviate this, and can be replaced with higher or lower values in order
to obtain the best signal under the changing atmospheric conditions.
The spectral transmission factor of the receiver x(l) is a function
of the filters, mirrors, and detectors used in this stage.
This is a schematic showing the ray tracing of light inside the telescope
and optics package. Light, after being collected by the telescope’s main
mirror, reflects off the secondary mirror, and is then focused to the back
of the telescope. The light then passes though the first stage of
neutral density and to a dichroic filter, which is used to split the light
into its 532 and 1064 nm components. From this filter, the 1064 nm
wavelengths are allowed to pass straight through, going through another
neutral density filter, through a 1064 narrow band pass filter, and finally
into the APD. The 532 nm beam reflects 90 degrees off the dichroic,
through another neutral density filter, and then finally through a 532
narrow band pass filter before it reaches the PMT. The ELF system
uses a Hamamatsu H6780 PMT for the 532 nm channel, and an Analog Modules
710-427 APD for the 1064 nm channel. From the detectors, the signals
are then routed via coaxial cables to a Gage 12100 series digitizer card
installed in a Gateway P550 computer.
A screen shot of the ELF software. The basic function of
the software is to average profiles, save raw data, and display current
raw and background subtracted range square corrected data; the latter of
which will be explained in detail later. The program enables the
user to change sampling rates of the Gage cards, as well as invert the
negative going signal of the PMT to positive voltages. The ELF software
triggers off a Q-switch pulse signal from the laser, and since the laser
operates at 10 Hz, raw lidar profiles are received at 10 times a second.
User specified averaging is used to smooth noise out of the raw data signal.
Typically, one minute averages are used; meaning that 600 shots are averaged
together to produce one profile.
An example of a typical one minute averaged profile can be seen below.
The green and red signals are from the PMT and APD respectively. The green curve shows a characteristic raw lidar profile. The peak of profile corresponds to crossover, where the laser beam is fully inside the field of view of the telescope. The graph plots percent full scale voltage versus height. Equation 4 is used to change round trip flight time into R, or height. By setting t equal to the sample rate of the digitizer card and solving for R, the “bin width” of that sampling rate can be obtained; meaning that each point along the profile can now be correlated to a distance or height, and not just a round trip flight time.
In order to extract more meaning from a raw lidar signal, several corrections
must be made. First, the entire signal is background subtracted,
to negate all ambient background light effects. The average of the
last few (50) points of the profile are taken as the average ambient background
light since the signal is very low at this point. The average of
these points is then subtracted from every point in the profile.
A better way to background subtract would be to pre-trigger; meaning that
an average background value could be taken microseconds before the laser
fires and that value be used as the background. This is often the
background subtraction technique of choice.
Secondly the lidar profile must be R square corrected.
Since each lidar signal falls off as 1/R^2 like any electo-magnetic wave,
this effect must be negated in order to see any signal structure at the
higher altitudes and atmospheric attenuation of the signal. By multiplying
each data point by it’s numerical location in the data array and the bin
width, quantity squared, then the 1/R^2 dependence of the signal is effectively
nulled, and nearly equal weighting is obtained over the whole profile.
This shows a raw lidar signal that’s been background subtracted and R^2 corrected. Notice that much more structure can be seen at the higher bins, and the signal lacks the range dependant decay of the raw signal. If several profiles of one minute background subtracted range square corrected data are placed on a color plot, figure 10 shows the time-height display for a selected channel. The colors represent relative signal strength, and are variable by the ELF software user. Clouds show up as the whitish and reddish areas due to their strong signal return, and things such as the planetary boundary layer show as yellows and blues.
The future of the software lies in its ability to accurately calibrate the system, and use that calibration to achieve concrete results. In the lidar equation, parameters such as telescope area, laser power, detector integration time, and the receiver spectral response function can all be lumped into a single “system constant” term. This can be achieved by taking profiles on unusually clear days, and assuming only Rayleigh scattering. After the system constant is obtained, quantities such as Aerosol backscattering ratio can be derived. It’s hoped that the software will be developed to this point by early summer.
The primary function of the ELF, along with many elastic lidar systems,
is to study aerosols and clouds. Since Mie and Raleigh scattering
can be observed from the laser probing of these atmospheric constituents,
lidar makes a perfect instrument perform detailed studies. Many successful
aerosol and cloud studies have been done with elastic lidar.1
1) Raymond M. Measures, Laser Remote Sensing: Fundamentals and Applications 2nd Ed. (Krieger, Florida, 1992)