Signal-to-Noise Ratio Calculations for Imaging Bi-Static Rayleigh Lidars
Since the invention of the laser in 1961, lidar systems have been developed
to measure a wide variety of atmospheric constituents and parameters. Although
the mono-static configuration is the most prevalent, bi-static techniques have been
proposed for several upper atmospheric studies [e.g. Reagan et al., 1982; Welsh and
Gardner, 1989]. In fact Elterman [1951; 1952; 1953] made the very first Rayleigh
lidar measurements of stratospheric temperature and aerosol profiles sixty years
ago using a bi-static configuration consisting of a vertically pointed searchlight and
an off-axis scanning telescope.
For mono-static systems, the laser and receiving telescope are typically colocated. Range resolution is achieved by transmitting a pulsed laser beam and
range-gating the backscattered signal collected by the receiving telescope. The time
of flight of the laser pulse corresponds to range. For bi-static systems, the laser and
telescope are separated horizontally by up to several km. Range resolution is
achieved by imaging the backscattered beam onto a detector array. The position of
the imaged beam on the detector corresponds to range. Mono-static lidars require
pulsed lasers while bi-static systems can employ either pulsed or cw lasers.
Because of the different viewing geometry and detection strategy, the classic
lidar equations must be modified to properly calculate the backscattered signal
photon count NS and background noise count NB for the bi-static technique. We
assume the laser and telescope are separated by the distance d (see Figure 1). The
laser beam is pointed vertically and the telescope is pointed off-zenith to image the
beam onto the detector array. The signal count is given by
N S (z) 
PLaser
ATele
C
2
C (z) eff
( Laser )z
TeleTAtmos
hc /  Laser
4 (z 2  d 2 )
(1)
Where z is the altitude of the integrated volume (m), PLaser is the average power of
the laser beam (W),  is the integration period (s), hc /  Laser (J ) is the photon energy,
h  6.63 •1034 J / s is Planck’s constant, c  3 •108 m / s is the velocity of light,
 Laser (m) is the optical wavelength of the laser beam, C (z) is the density of the
C
constituent being measured (m-3),  eff
is the effective backscatter cross-section of
the species (m2), ATele is the aperture area of the imaging telescope (m2), Tele is the
optical efficiency of the imaging telescope including the quantum efficiency of the
2
detector and TAtmos
is the 2-way optical transmittance of the atmosphere.
Because of the finite divergence of the laser beam (  Laser ) and the imaging
geometry, the actual vertical resolution of the bi-static lidar (see Figure 1 and Welsh
and Gardner [1989]) is limited to
zBiStat 
 Laser z 2
d
1
.
(2)
Alternatively, to achieve a given vertical resolution, the laser and telescope must be
separated by
d
 Laser z 2
zBiStat
.
(3)
To compute the background noise count we assume the worst-case situation
in which the detector integrates for the complete observation period. If a pulsed
laser is employed and the detector is only activated when the pulse illuminates the
scattering volume, the background noise count could be several orders of magnitude
smaller than the worst-case values. For this worst-case scenario, the background
noise count is given by
NB 
SSky (Laser )ATele 
FOVTele
hc / Laser
(4)
where SSky ( ) is the sky’s spectral radiance (W/m2/nm/sr),  is the optical
bandwidth of the imaging telescope (nm) and  FOV is the solid angle field-of-view of
the telescope. For the bi-static geometry illustrated in Figure 1
 FOV   Laser z 
 Laser dz
z2  d 2
(5)
dz
 z  2
z  d2
and z is the effective vertical resolution of the detector (m), which may be different
from the fundamental resolution of the bi-static configuration, which is given by (2).
If the detector resolution is matched to zBiStat , then
FOV 
z2
2
 2 ;  Laser
2
2 Laser
z d
,
(6)
which is the solid angle field-of-view of a conventional mono-static lidar. In this
case, the signal and background counts and their ratio are given by
PLaser
 A
2
C (z) effC ( Laser ) Laser Tele TeleTAtmos
hc /  Laser
4 d
S ( )A  2
N B ; Sky Laser Tele
 LaserTele
.
hc /  Laser
S ( )
NB
4 d Laser
; Sky Laser
2
N S (z)
PLaser
C (z) effC ( Laser )TAtmos
N S (z) ;
(10)
Notice that the signal count ( N S ) and vertical resolution ( zBiStat ) are largest (see
(10) and (2)) when the separation between the telescope and laser is smallest.
2
The major disadvantage of the bi-static technique compared to a mono-static
lidar is the larger background noise contamination that results when the detector
integrates for the full observation period, instead of just when the scattering volume
is illuminated by the pulsed laser for a mono-static lidar. For example, if a monostatic system is designed to observe Rayleigh scattering to 200 km altitude at 5 km
vertical resolution, then a ranged-gated detector would observe a minimum factor
of 40 = 200/5 reduction in background noise for the maximum possible laser pulse
rate of 750 pps. Larger noise reductions are possible when lower pulse rates are
employed.
Figure 1: Schematic of bistatic and monostatic lidar configurations
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References
L. B. Elterman, “The measurement of the stratospheric density distribution
with the search light technique”, Journal of Geophysical Research, 56, 509-520
(1951)
L. B. Elterman, “A series of stratospheric temperature profiles obtained with
the search light technique”, Journal of Geophysical Research, 58, 519-530 (1953)
L. B. Elterman, “Seasonal trends of temperature, density and pressure to 67.5
km obtained with the search light probing technique”, Journal of Geophysical
Research, 59, 351-358 (1954)
C. S. Gardner, “Performance capabilities of middle-atmosphere lidars:
comparison of Na, Fe, K, Ca, Ca+, and Rayleigh systems”, Applied Optics, 43, 49414956 (2004)
J. A. Reagan, D. M. Byrne, and B. M. Herman, “Bistatic lidar: A tool for
characterizing atmospheric particulates: Part 1-The remote sensing problem”, IEEE
Geoscience and Remote Sensing, GE-20, No. 3 (July 1982)
B. M. Welsh, “Bistatic imaging lidar technique for upper atmospheric studies”,
Applied Optics, 28, 82-88 (1989)
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