atmospheric propagation
TRANSCRIPT
ATMOSPHERIC PROPAGATION EFFECTS
ON
HETERODYNE-RECEPTION OPTICAL RADARS
by
DAVID MICHAEL PAPURT
B.S.E.E., B.A., University(1977)
S.M., Massachusetts Institute(1979)
E.E., Massachusetts Institute(1980)
SUBMITTEDOF THE
of Toledo
of Technology
of Technology
IN PARTIAL FULFILLMENTREQUIREMENTS FOR THEDEGREE OF
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May 1982
Massachusetts Institute of
of Author..,Department of Electrical
Certified by........'.........
Technology, 1982
Engineering/and Computer ScienceMay 14, 1982
Jeffrey H. ShapiroTjhsis Supervisor
Accepted b..........Arthur C. Smith
Chairman, Department Committee on Graduate Students
ArchivesMA SSACHUSETTS NSitTZTC
OF TECHNOLOGY
OCT 20 1982
UBRARIES
0
Signature
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ATMOSPHERIC PROPAGATION EFFECTS
ON
HETERODYNE-RECEPTION OPTICAL RADARS
by
DAVID MICHAEL PAPURT
Submitted to the Department of Electrical Engineering & Computer Scienceon May 14, 1982 in partial fulfillment of the requirements for the
Degree of Doctor of Philosophy
ABSTRACT
The development of laser technology offers new alternativesfor the problems of target detection and imaging. The performance ofsuch systems, when operated through the earth's atmosphere, may beseverely limited by the stochastic nature of atmospheric opticalpropagation; that is, by turbulence, scattering, and absorption. Amathematical system model for a compact heterodyne-reception laserradar which incorporates the statistical effects of target speckleand glint, local oscillator shot noise, propagation through eitherturbulent or turbid atmospheric conditions, and beam wander ispresented. Using this model, results are developed for the imagesignal-to-noise ratio and target resolution capability of the radar.Complete statistical characterizations for the radar return aregiven. An experiment and data processing techniques, aimed atverifying the above statistical models, are described and resultsgiven. Notable here is the verification of the lognormal characterof the turbulent atmosphere induced fluctuations on the radar return.Target detection performance of the radar is investigated. Regimesof validity for the various target return models are established.
Thesis Supervisor: Jeffrey H. Shapiro
Title: Associate Professor of Electrical Engineering
To my MotheA and FatheA
-ot the%'L eoAnt,6, patience, and Zove
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ACKNOWLEDGEMENTS
It has indeed been an honor and a pleasure to be associated
with my thesis supervisor, Professor J.H. Shapiro. His guidance
throughout the course of my stay at M.I.T. has been invaluable. I
would also like to thank my thesis readers Dr. R.C. Harney and
Professor W.B. Davenport. Their many suggestions have improved this
work significantly. Also, W.B. Davenport, in his role as my
graduate counselor, provided advice, without which, this goal would
never have been realized.
I would like to acknowledge all members of the Optical
Propagation and Communication research group at M.I.T. The many
stimulating discussions, some of a technical nature and many
nontechnical, between myself and members of this group have added
to this work. In particular, I would like to mention Dr. J. Nakai.
I feel fortunate to have made his acquaintance and honored to be his
friend.
Many members of the Opto-Radar Systems group at M.I.T.
Lincoln Laboratory played a role in this research. R.J. Hull,
T.M. Quist and R.J. Keyes deserve special mention for their efforts
in providing me with radar data and the computer facilities to process
this data with.
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Financial support for my doctoral studies has been provided
by the U.S. Army Research Office, Contract DAAG29-80-K-0022. This
support is gratefully acknowledged.
Deborah Lauricella has my appreciation for her pains in
typing this document.
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TABLE OF CONTENTS
Page
ABSTRACT......
ACKNOWLEDGEMEN
LIST OF FIGURE
LIST OF TABLES
CHAPTER I.
CHAPTER II.
CHAPTER III.
CHAPTER IV.
..................................
TS ..........................
S .... .... ................ ......
............................. 0.....
INTRODUCTION.....................
A. Laser Radar Configuration....
ATMOSPHERIC PROPAGATION MODELS...
A. Free Space Model............
B. Turbulence Model.............
C. Turbid Atmosphere Model......
D. Backscatter..................
TARGET INTERACTION MODEL.........
A. Planar Reflection Model......
B. Relationship to Bidirectional
SCANNING-IMAGING RADAR ANALYSIS..
A. Single Pulse SNR.............
B. Speckle Target Resolution inVisibility Atmosphere........
C. Identification of Atmospheric
D. Turbulence SNR Results.......
E. Low Visibility SNR Results...
Reflectance....
................
................
the Low................
Effects ........
................
................
F. Beam Wander Effects..........................
G. Correlation of Simultaneous Speckle TargetReturns......................................
H. Backscatter..................................
2
4
8
14
15
17
27
30
31
32
34
35
35
37
40
41
45
50
52
56
71
87
90
. .. ... . .. .. .. . ..
. . ... .. .. .. . .... 0
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CHAPTER V.
CHAPTER VI.
CHAPTER VII.
CHAPTER VIII
APPENDIX A.
THEORY VERIFICATION............................
A. Laser Radar Description....................
B. Data Analysis Techniques...................
C. Beam Wander................................
0. Turbulence.................................
TARGET DETECTION...............................
A. Problem Formulation........................
B. Single Pulse Performance...................
C. Multipulse Integration.....................
D. Multipulse Performance.....................
SYSTEM EXAMPLES................................
A. EXAMPLES...................................
SUMMARY........................................
DERIVATION OF THE MUTUAL COHERENCE FUNCTION(MCF)..........................................
REFERENCES....................................................
BIOGRAPHICAL NOTE.............................................
Page
96
96
99
102
113
135135
138
147
150
165
174
194
197
211
215
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LIST OF FIGURES
Figure Page
1.1 Laser radar configuration.............................. 18
1.2 Transmitted power waveform............................. 21
1.3 Heterodyne receiver model.............................. 25
2.1 Turbulence field coherence length p0 vs. propagation
path length L for conditions of weak turbulence,
moderate turbulence and strong turbulence.............. 28
3.1 Geometry for defining bidirectional reflectance
p'(x;Tf,T ) '..' '' ' 39
4.1 Reflected spatial modes from resolved and unresolved
speckle targets........................................ 61
4.2 Reflected phasefronts from glint targets............... 64
4.3 Target return PDF for beam wander fluctuations,
Gaussian beam, uniform circular beam center,
|m/rb = 0.0, R/rb = 1.0 ........................... 77
4.4 Saturation SNR due to two types of beam wander
fluctuation............................................ 78
4.5 Target return PDF for beam wander fluctuations;
fan beam, uniformly distributed beam center,
m/rb = .45, R/rb = 1.2................................. 82
4.6 Saturation SNR due to fan beam, uniform beam center
fluctuations........................................... 83
4.7 Target return PDF for beam wander fluctuations;
fan beam, Gaussian distributed beam center, m/rb = 0.8,
a /rb = 1.25........................................... 85
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LIST OF FIGURES
Figure
4.8 Single scattering layer..................................... 91
5.1 Normalized histogram of 100 consecutive retro returns
taken in full scanning mode and theoretical PDF
Eq. (4.F.23), R/rb = 1.3, m/rb = 0.0........................ 105
5.2 Normalized histogram of 200 consecutive returns taken in
full scanning mode and theoretical PDF Eq. (4.F.23),
R/rb = 1.2, m/rb = 0.5...................................... 106
5.3 Normalized histogram of 200 consecutive retro returns taken
in full scanning mode and theoretical PDF (4.F.23),
R/rb = 1.3, m/rb = 0.56..................................... 107
5.4 Normalized histogram of 200 consecutive retro returns from
the "side" pixel of Figure 5.2 taken in full scanning mode
and theoretical PDF (4.F.28), R/rb = 1.2, m/rb = 1.43....... 110
5.5 Normalized histogram of 400 consecutive retro returns
taken in full scanning mode and theoretical PDF (4.F.15),
R/rb = 1.4, m/rb = 0.6...................................... 111
5.6 Normalized histogram of 100 "hot spot" retro returns
taken in reduced scanning mode and the lognormal PDF
(5.D.l), a2 = .0045......................................... 114x
5.7 Time evolution; Row location of hot spot in the l00-60x128
pixel frames used in the example of Figure 5.6.............. 116
5.8 Time evolution; Column location of hot spot in the
l00-60x128 pixel frames used in the example of Figure 5.6... 117
5.9 Normalized histograms of 300 "hot spot" retro returns
taken in reduced scanning mode and the lognormal PDF
(5.D.1), a2 = 0.0138........................................ 119x
Page
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LIST OF FIGURES
Figure Pag
5.10 Normalized histogram of 400 "hot spots" retro returns
taken in reduced scanning mode and the lognormal PDF
(5.D.1), &2 = 0.018...................................... 120x5.11 Normalized histogram of 300 "hot spot" polished sphere
returns taken in reduced scanning mode and the lognormal
PDF (5.D.1), g 2 = .0083.................................. 121x
5.12 Normalized histogram of 300 "hot spot" polished sphere
returns taken in reduced scannina mode and the lognormal
PDF (5.D.1), a'2 = 0.004.................................. 122x
5.13 The theoretical curve (4.D.6) and estimates SNRSAT from
the data of Figures 5.6,5.9-5.12......................... 124
5.14 Normalized histogram of 2000 squared, consecutive
speckle plate returns taken in full scanning mode and
exponential PDF.......................................... 125
5.15 Normalized histogram of 400 consecutive speckle plate
returns taken in full scanning mode and Rayleigh PDF..... 126
5.16 Normalized histooram of 1200 speckle plate returns
taken in reduced scanning mode and Rayleigh PDF.......... 128
5.17 Normalized histograms of 1400 consecutive speckle plate
returns taken in staring mode and Rayleigh PDF........... 129
5.18 Normalized histogram of 300 speckle plate returns taken
in full scanning mode and Rayleigh PDF...................131
5.19 Normalized histogram of the same 300 speckle target
returns as Figure 5.18 and a Rayleigh times lognormal
PDF, a 2 = 0.01........................................... 132
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LIST OF FIGURES
Page
6.1 The probability density function, p (Y) = 2K (2VY).......142
6.2 Single pulse detection probability vs. CNR for a glint
target in bad weather.....................................144
6.3 Single pulse receiver operating characteristics for a
glint target in bad weather...............................145
6.4 Single pulse detection probability vs. CNR for a single
glint target in free-space, three levels of turbulent
fluctuations and bad weather. P F = 10- throuhout.......146
6.5 Likelihood ratio R (Eq. (6.C.3)) and parabola
log R = 3.1 + 0.3441r' vs, matched filter envelope
detector output Ir.......................................149
6.6 Threshold y vs. number of pulses necessary to maintain
-= 102 154PF
6.7 Single-pulse and multipulse detection probability vs. CNR
for a glint target in bad weather. PF = 104 throughout.. 158
6.8 Single-pulse and multipulse detection probability vs.
CNR for a glint target in bad weather. PF=10-12 throughout. 159
6.9 The number of pulses M necessary to achieve PF = 10-12
pD= .99 vs. CNR for a glint target in free-space, two
turbulent fluctuation levels, and bad weather.............160
6.10 The number of pulses M necessary to achieve PD = .99 and
two different false alarm probabilities vs. CNR for a
glint target in low visibility............................161
6.11 Ten pulse detection probability vs. CNR for a glint
target in several turbulent atmospheres and a scattering
atmosphere. PF = 10l2 throughout........................163
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LIST OF FIGURES
Figure Paae
6.12 Fifteen pulse detection probility vs. CNR for a glint
target in several turbulent atmospheres and a scattering
atmosphere. PF = 10-12 throughout........................164
7.1 Geometry for a single scattering layer between radar
and target................................................170
7.2 Geometry for a scattering layer near the target...........171
7.3 Normalized backscatter power from a uniform scattering
profile vs. t................... ......................... 175
7.4 Maximum normalized backscatter power from a scattering
layer L meters from the radar............................177
7.5 Extinguished free-space and MFS resolved speckle target
CNR vs. target range for the CO2 system and a uniform
scattering profile........................................178
7.6 Extinguished free-space and MFS resolved speckle target
CNR vs. tarqet range for the Nd:YAG system and a uniform
scattering profile........................................179
7.7 Atmospheric beamwidth for a single layer scattering
profile vs. layer thickness...............................181
7.8 Extinguished free-space and MFS resolved speckle target
CNR for the CO2 system and a single scattering layer vs.
layer thickness...........................................182
7.9 Extinguished free-space and MFS resolved speckle target
CNR for the Nd:YAG system and a single scattering layer
vs. layer thickness......................................183
7.10 Atmospheric beamwidth for a scattering layer near the
target and the Nd:YAG system vs. layer thickness......... 186
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LIST OF FIGURES
Figure Page
7.11 Extinguished free-space and MFS resolved speckle target
CNR for the Nd:YAG system and a scattering layer near the
target vs. layer thickness............................... 187
7.12 Field coherence length for a scattering layer near the
target and the Nd:YAG system vs. layer thickness......... 190
7.13 Extinguished free-space and MFS unresolved glint target
CNR for the Nd:YAG system and a scattering layer near
the taroet vs. layer thickness........................... 191
A.1 Geometry relating to the definition of specific
intensity................... ............................ 198
A.2 Geometry for relating the specific intensity and the
mutual coherence function................................ 201
A.3 MCF's for plane wave input............................... 207
A.4 Real phase function...................................... 209
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LIST OF TABLES
Table
7.1
7.2
7.3
7.4
CO2 Laser Radar System Parameters...................... 167
Nd:YAG Laser Radar System Parameters................... 168
Atmospheric Parameters at CO2 Laser Wavelength......... 172
Atmospheric Parameters at Nd:YAG Laser Wavelength.
Page
..... 173
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CHAPTER I
INTRODUCTION
The development of laser technology offers new alternatives to
the problems of target detection and imaging. Among the advantages
provided by laser radars over conventional radar systems are increased
angular, range, and velocity resolution with compact equipment. However,
performance of the laser system may be severely limited by the stochastic
nature of atmospheric optical propagation; that is, by turbulence,
absorption, and scattering. Herein, a mathematical model describing a
heterodyne reception optical radar is presented. The model incorporates
the statistical effects of propagation through tubulent and turbid
atmospheric conditions, as well as target speckle and glint, and local
oscillator shot noise.
A convenient theoretical model describing optical propagation
through atmospheric turbulence has been established [1]. In contrast,
optical propagation through bad weather is much more difficult to model
and to date no comprehensive theory exists which characterizes this
propagation regime in complete generality. From the available turbid
atmosphere propagation models the multiple forward scatter (MFS) extended
Huygens-Fresnel formulation [2] is useful in our application. Both the
turbulence model and MFS model are expressed in linear-system form in
which the random nature of the propagation process is represented by a
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stochastic point source response function (Green's function). Such a
linear model leads to a tractable overall radar system model.
In this thesis, we consider the performance of a scanning imaging
system and to a lesser degree that of a target-detection system. By
means of the turbulence model, the primary atmospheric effect limiting
compact* radar system performance has previously been shown to be
scintillation [3-5]. In contrast, beam spread and receiver coherence
loss as well as scintillation will be shown to be important when the MFS
model applies.
To validate the theory measurements made on the compact C02-laser
radar, developed as part of the M.I.T. Lincoln Laboratory Infrared
Airborne Radar (IRAR) project, have been made available. The compact
laser radar system [6] employs a one-dimensional, twelve-element HgCdTe
heterodyne detector, a transmit/receive telescope of 13 cm aperture, and
a 10 W CO2, 10.6 vm laser, which is operated in pulsed mode. For each
pulse, the intermediate frequency (IF) portion of the heterodyne detector
outputs are digitally peak-detected to yield 8 bit range and intensity
values. In the case of a large signal return the intensity value is
essentially the output of a matched filter envelope detector. Hence,
these data can be compared to the theory.
An outline of the topics covered herein is as follows. First, in
this chapter, a description of the laser-radar configuration will be
presented. This will be followed by descriptions of the atmospheric
*The term "compact" indicates a system that can be installed on a vehiclesuch as a truck or airplane.
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propagation and target interaction models, which appear in Chapters II
and III, respectively. The performance of a scanning-imaging radar in
both turbulence and low visibility will be considered in Chapter IV. As
speckle target returns from disjoint diffraction-limited fields-of-view
(FOV) will be shown to be uncorrelated in low-visibility weather, and
each picture element (pixel) of the image is assumed to encompass at
least a diffraction-limited FOV, we need consider only single-pixel
performance. This will be achieved via a signal-to-noise ratio (SNR)
analysis. Also, speckle target resolution in bad weather is considered
here. In Chapter V, theory verification efforts are described.
Single-pulse and multi-pulse target detection is explored in Chapter VI
with emphasis given to low visibility results. In Chapter VII system
examples are presented. Finally, in Chapter VIII, we summarize our
results and present suggestions for future work.
A. Laser Radar Configuration
A model of the laser radar configuration is shown in Figure 1.1.
A series of laser pulses ET(Pt) propagating nominally in the +z
direction are transmitted from the radar located in the z = 0 plane, and
illuminate a target located in the z = L plane (Figure 1.la). A fraction
of the illuminating field Et(PI',t) is reflected in the -z direction back
towards the radar (Figure 1.1b). The nature of this reflected field
E r(',t) clearly depends upon the reflection characteristics of the target,
as does the received field ER(p,t). This recieved field is mixed with a
local-oscillator field E,(P,t) and focussed onto an optical detector. The
W mw
TRANSMITTER
TRANSMITTEDBEAM
E T(-P, t) a) FORWARD PATH
ILLUMINATINGBEAM
Et(-p, t)
0
ATMOSPHERE
RECEIVEDBEAM
ER(p ,t)b) RETURN PATH
REFLECTEDBEAM
Er(pt)
TARGET
z= L
TARGET
Figure 1.1: Laser Radar Configuration
L
RECEIVER
lw w
A TMOSPHERE D
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intermediate frequency (IF) component of the photocurrent comprises
the observed signal.
To as large an extent as possible the analysis has been tailored
to the IRAR project's compact C02-laser radar system. Thus, the
transmitter and receiver are taken to be co-located with common
entrance/exit optics of aperture diameter from 5 to 20 cm. The transmitter
will be assumed to produce a periodic train of rectangular-envelope laser
pulses while the local oscillator operates cw producing an ideal
monochromatic wave offset in frequency by the intermediate frequency
v IF* Targets are assumed to lie along line-of-sight paths a distance L
from the radar where 1 km < L < 10 km. In accordance with the above
conditions and transmitter pulse durations and pulse repetition frequencies
anticipated in real radar scenarios [7,8] we have the following
characterization.
1. The transmitted field has a quasi-monochromatic, linearly
polarized electric field proportional to
ET(Pt) = Re{u (p,t) e 0 } (l.A.1)T
where v is the optical carrier frequency, and P = (x,y) is
a position vector transverse to the direction of propagation.
The complex envelope uT(P,t) is expressed as the product of
a normalized spatial mode F(P) and a time waveform sT(t)
whose magnitude square is the transmitted power PT(t) (Watts).
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uT(P,t) = FT() -T(t) (1.A.2)
PT(t) = Is-T(t)12 (l.A.3)
dPIT( 12= 1 (1 .A.4)
This implies that 1u-T(P,t)I2 is the transmitted power density
(Watts/m 2). The transmitted power waveform PT(t) is assumed
to be as shown in Figure 1.2 where t is the pulse duration
and l/T the pulse repetition frequency.
2. We assume a scalar wave theory to describe the propagation.
We also assume the pulse duration to be short in comparison
to an atmospheric coherence time Tc >> tp and long in
comparison to the reciprocal coherence bandwidth (multipath
spread) of the atmosphere 1/Bcoh < t . The complex envelope
of the illuminating field in the z = L plane can then be
represented by the linear superposition integral
ut(P',t) = dp hLF(PiP) T(p,t - L/c) (l.A.5)
where hLF(P'P) is the stochastic atmospheric Green's function
(point source response) and c is the propagation velocity of
light. The subscripts indicate a path length of L meters in
the +z (or forward) direction. In free space (l.A.5)
becomes the Huygens-Fresnel integral [9]. In the atmosphere
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PT (t)
0 t T p +t 2 , 2 c +Pp p p p p p
Figure 1.2: Transmitted Power Waveform
-22-
hLF(PI',P) is random with statistical characterization
dependent upon weather conditions.
3. A planar target-interaction model is assumed so that for a
stationary target the complex envelope of the reflected field
is
I-r (PI',t) = ut (_',t) T(-') (l.A.6)
where T(P') is the field reflection coefficient at the
point P'. In general T(p') is a random function containing
specular and diffuse components.
4. As before, we can represent the received field as a
superposition integral
u (_gPqt) = d' h LR , ) -r(p',t - L/c) (l.A.7)
In (l.A.7) hLR(P,P') is aoain the stochastic atmospheric
Green's function where the subscript R refers to propagation
in the -z (or return) direction. In turbulence, and in low
visibility when the target range is sufficiently short, the
propagation medium is reciprocal [10,11], i.e.,
hLF(PP') = hLR(', ((1.A.8)
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In low visibility and with a suitably large target range
we take hLF and hLR to be statistically independent. This
is due to the atmosphere decorrelating between pulse
transmission and reception times, and is an appropriate
assumption when the atmospheric coherence time Tc is shorter
than the roundtrip propagation delay 2L/c. We will refer
to the former case as "effectively monostatic" and the latter
case as "effectively bistatic" regardless of the radar
configuration.
5. According to the antenna theorem for heterodyne reception
[12], we can describe the photodetection process as though
it were taking place in the receiver's entrance pupil. Thus,
the detected field is taken to be ER(Pt) + EZ(p,t) where
the cw field E, has complex envelope
j27rv tu(-,t) = P1 F (P) e IF (l.A.9)
In (l.A.9), vIF is the intermediate frequency, PZ the local
oscillator average power and F a normalized spatial mode.
The photocurrent is passed through a rectangular passband
filter of bandwidth 2W centered at vIF. It is straightforward
to show that under the strong local oscillator condition [13]
a normalized (i.e. proportional to the photocurrent) IF
signal r(t) can be expressed as filtered signal plus noise.
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This is shown in Figure 1.3 where the pass band filter H(f)
has the input with IF complex envelope
y(t) = d -R(t) F*(P) + n(t) (l.A.10)~hvJ 0 -
The additive noise n(t) is a zero-mean, white, circulo-complex
Gaussian process with
<n(t) n*(s)> = tp 6(t - s) (l.A.ll)
where <-> denotes ensemble average, hv0 is the photon energy
and q is the detector quantum efficiency. Substituting for
u_,t) in (l.A.10) and making the definitions
EF(I') d- h LF(PI',P) F T(P) (l.A.12)
P' = d h R ,1 )' *( ) ( .A.13)
the IF complex envelope (l.A.10) becomes
y(t) = 2 sT(t-2L/c) dF' T(P') EF R(P') + n(t)
(1.A.14)
The above integral is performed over the target plane as
qw w
Re{y(t) e-} H(f) r(t)
a) IF Model
U'A H(f)
2W 2W
- I
I U
VIF V IF
b) Normalized IF Filter Frequency Response
Figure 1.3: Heterodyne Receiver Model
-26-
opposed to (l.A.10) which is performed over the receiver
plane. From (l.A.10) it can be seen that to maximize signal
return we should set F(p) = u (P)/( dpig(p)12) . In
practice, we set F,(P) = FT) to approximate this condition.
In order not to block an appreciable amount of the signal
return with H(f) we need 14 -1 1/t . To minimize the noise
passed by the filter we set W = 1/t .
In both imaging and target detection applications the initial IF
signal processing is identical. Namely, it is passed through a matched-
filter envelope-detector with output proportional to,2L/c+t~
mrij2 = l/t J 2L/ct r(t) dtf2 where r(t) is the complex envelope of2L/c
r(t) (Figure 1.3). In imaging, the scene is scanned and the complete
image built up from sequential returns Fr| 2 separated in space typically
by a diffraction limited FOV or more. In single-pulse detection ri2
is compared to a threshold where exceeding this threshold indicates
target presence. Optimal processing of returns for multi-pulse detection
also requires knowledge of Fr2 . Before the performance of these systems
can be discussed, however, propagation and targets must be characterized.
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CHAPTER II
ATMOSPHERIC PROPAGATION MODELS
The atmosphere, as an optical propagation medium, differs
markedly from free space. In clear weather, spatio-temporal refractive
index fluctuations are caused by random mixing of air parcels of nonuniform
temperatures. These fluctuations, called atmospheric turbulence, have a
significant effect on optical propagation. In bad weather, scattering
from aerosols, such as haze and fog, and hydrometeors, which include
mist, rain and snow, can also profoundly affect propagation.
We anticipate three atmospheric propagation effects to degrade
performance of the radar. Depending upon the relative sizes of the
phase and amplitude field coherence lengths, transmitter and receiver
apertures, and the target, different effects can become important. On
a qualitative level we have:
1. Beam Spread. In the forward path, if the transmitter
diamter exceeds the atmospheric coherence length p0, in
either turbulent or low visibility weather, random dephasing
of the transmitted field will occur. This results in a
larger target plane illuminating beam than in free space.
In Figure 2.1 the field coherence length p = (1.09 k2 C2 L)-3/50 n
under clear weather conditions is shown versus path length L
for three values of turbulence strength C2 and wavenumber kn
corresponding to 10,6 pim wavelength radiation [3]. From
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101
X 10.6 p.m
1 2 5 x 101 m-2/3
n
C= 5 x 10 m-n" 3
-2
2 3 4510 10 10 10
L (i)
Figure 2.1: Turbulence field coherence length p0 vs. propagation path
length L for conditions of weak turbulence
(C' = 5 x 10-16 m-2/ 3), moderate turbulence
(C2 = 10-14 m-2/3), and strong turbulence
(C2 = 5 x 10-3 m-2 3); 10.6 vim wavelength has been assumed.n
-29-
this digram we see that for typical transmitted beam
diameters of 5 to 20 cm and path lengths of 1 km to 10 km
transmitter beam spread can nearly always be neglected. We
will take this to be the case. On the other hand, for
turbid atmosphere propagation, the atmospheric coherence
length p0 = (2/ s L 0' k2)2 (see Appendix A), where S'0 F
is the effective scattering coefficient and OF the forward
scattering angle, generally is much smaller than typical
beam diameters. Hence, as will be seen in more detail
later, beam spread is an important factor in bad weather,
and must be included in our analysis.
2. Scintillation. The random spatio-temporal amplitude
fluctuations due to constructive and destructive
interference of the randomly lensed light is known as
scintillation. If the target is smaller than the
amplitude coherence length of the atmosphere then the
scintillation modulates the reflected intensity. If the
target is larger than this coherence distance the
scintillation presents itself as a speckling of the
reflected radiation. In both turbulent and turbid
atmospheres this effect is significant, although, we
anticipate that signal return fluctuations will be of
different character in the clear-weather and bad weather
limits.
-30-
3. Coherence loss. At the receiver, surfaces of constant
phase will become wrinkled when the receiver aperture
becomes larger than a phase coherence length (which is a
numerical factor times p 0). When this occurs optimal spatial
mode matching cannot be achieved with F*() = ET(p), resulting
in a performance degradation called coherence loss.
Referring to Figure 2.1, in turbulence, for typical receiver
diameters of 5 to 20 cm and path lengths of 1 to 10 km this
effect can also be neglected. As coherence lengths in low
visibility are normally much smaller than the receiver
aperture this effect is pronounced. More details will be
given later.
The above heuristic descriptions of propagation phenomena are
lacking in mathematical detail. We now provide these details. First,
free space propagation is described. Next, propagation through
turbulence is characterized. Finally, we discuss turbid atmosphere
propagation.
A. Free Space Model
In free space reciprocity (Eq. (l.A.8)) applies with both hLF and
hLR equal to the non-random Green's function [9]
hL(p',P) = jkL exp L - ' (2.A.1)
-31-
where X is the optical wavelength, k = 27/X the wavenumber, and
superscript "o" denotes free space.
B. Turbulence Model
Without loss of generality, the stochastic Green's function for
the clear turbulent atmosphere can be represented as the product of an
absorption term, a random, complex perturbation term and the free
space result [1,31
hL(P1,P) = e- aL/2 exp[x(T',W) + j(',P)] ho(-',-) (2.B.1)
where, via reciprocity, hL = hLF = hLR. In (2.B.1), a is the
atmospheric absorption coefficient and x(I',-) ( (I',)) is the
log-amplitude (phase) perturbation of the field at transverse
coordinate p' in the z = L plane due to a point source excitation at
transverse coordinate P in the z = 0 plane. These perturbations,
before the onset of saturated scintillation, may easily be expressed
as sums of large numbers of independent random variables [1,14]. Hence,
by the central limit theorem [15], x and c are jointly Gaussian random
processes and completely characterized by their means and covariance
functions. Such a complete characterization is provided in reference
[1] but will not be given here. It will suffice to note that the mean
log amplitude perturbation obeys m = -c2 as a result of energyx x
conservation [16] where the log amplitude perturbation variance is given
by
-32-
02 = min(O.124 C2 k7/ 6 L11/ 6 ,0.5) (2.B.2)x n
for Kolmogorov spectrum turbulence with a uniform turbulence strength
C2 profile.
C. Turbid Atmosphere Model
Again, without loss of generality, we can express the stochastic
Green's function as a product form
hL(,) = e -a'L/2 A(',) exp[j (',P)] h0(P',) (2.C.1)
for either the forward or return path. This representation is chosen
to emphasize that the amplitude perturbation A(p',p), and not its
logarithm, is Gaussian. This results from the fact that nearly all
of the light illuminating the target and receiver is scattered in bad
weather. Hence, these fields are the sum of a large number of
independent contributions and, by the central limit theorem [15], can
be taken to be Gaussian.
In Appendix A, the mutual coherence function (correlation
function of the atmospheric Green's function hL) is derived. This is
accomplished by considering scattering from a single particle and
assuming: (1) single scattering is concentrated about the forward
direction, (2) wide-angle scatter and back scatter can be lumped into
absorption so that the real single particle phase function can be
approximated by a Gaussian form, and (3) that L >> 1, where ' is the
-33-
modified atmospheric scattering coefficient, so that the direct
(unscattered) beam contribution to the field is insignificant compared
to the scattered field.
The result we use is [2,17,18]
<hL (Til9 ) h*(p )> = e a exp[-(j 2 + IT' - p 2 +
(T1 _-P _2))/3pf] ho(TP ,p) h*( , 2 (2.C.2)
In (2.C.2), '(> Ba) is the modified absorption coefficient containingaa
wide angle scatter and back scatter contributions and
Po= [L e2 k2 (2.C.3)
is the atmospheric coherence distance where e F is the root-mean-square
forward-scattering angle of the Gaussian single particle phase function.
Furthermore, the variance of the phase perturbation is large enough
to say that hL(P',P) is zero mean and <hL(P',Pl) hL(' P2 )> =
The above model is known as the multiple forward scattering
(MFS) approximation. The validity of the model is not fully
established. It should yield correct results when each scattering
particle is significantly larger than a wavelength X. In this case,
the true single particle phase function would be highly peaked about
the forward direction so that the previous assumptions would apply.
-34-
As discussed in Appendix A the correlation function (2.C.2)
accounts only for scattered light. That is, results derived from
(2.C.2) disregard the unscattered portion of the beam. In order to
take this into account the unscattered beam is taken to be the
free-space result reduced by the extinction (absorption and scattering)
loss. In this thesis, the main theoretical development is aimed at
making use of the scattered light in the context of an optical radar.
Clearly, before use can be made of the scattered power, as calculated
via the MFS theory, it must dominate the extinguished free-space power.
More will be said about this issue in Chapter VII.
D. Backscatter
When the monostatic radar, as described in Chapter I, is used
in inclement weather there may be significant backscatter return from
the hydrometeors and aerosols present in the propagation path. Clearly
this return is undesirable in imaging and target detection applications,
but unavoidable in such weather conditions. To see when such a
return can be significant its power must be compared to those of the MFS
and extinguished free-space target returns. The backscatter
contribution to the radar return will be examined in Chapter IV,
Section H and again in Chapter VII.
-35-
CHAPTER III
TARGET INTERACTION MODEL
Here we discuss how the reflected radiation is related to the
target illumination. We first discuss the planar target model used in
this analysis. This model is then related to the more usual bidirectional
reflectance in the following section.
A. Planar Reflection Model
In scalar paraxial optics reflection of an optical beam from a
spherical mirror is generally represented in terms of a planar
reflection model. If the incident and reflected fields travelling
nominally along the z-axis are noted ut (',t) and ur(P't), respectively,
we have the relation
u- (I',t) = t(F',t) r expr-jkjp' - 1cf2/Rcl (3.A.l)
where r is the intensity reflection coefficient, Rc is the radius of
curvature of the mirror and p c is the transverse location of the center
of curvature. The use of (3.A.1) presupposes -c lies on or near the
z-axis and Rc is much larger than the beamwidth of u-t More generally
we might represent a polished reflecting surface by incorporating into
(3.A.1) spatially varying intensity reflection coefficient, radius
of curvature and center of curvature. That is
-36-
ur(W',t) = ut(',t) r- (') exp(-jk I' - (P)I 2 /Rc(p')) (3.A.2)
would be our model. In accordance with (3.A.2) we shall assume that
for all targets of interest a planar target interaction model (l.A.6)
is applicable
Ur(I't) = u-t(p',t) T(P') (3.A.3)
In general, T(P') will contain two components, the so-called
specular (glint) and diffuse (speckle) reflection components. We
express this as
T(P') = eJO T (p') + Ts(p') (3.A.4)
The glint component, T (P'), is nonrandom and represents the component
of the reflected liqht that is due to the smoothly-varying target shape.
This component may be described by (3.A.2) or (3.A.1). The random
phase e is assumed to be uniformly distributed over [O,2tr] and represents
our uncertainty of target depth on spatial scales on the order of a
wavelength. On the other hand, the speckle component T (_') is random
and represents that part of the reflected light that is due to the
microscopic surface-height fluctuations of the target. This component
may be assumed to be a Gaussian random process with moments [19-21]
-37-
<T ( ')> = 0 (3.A.5)
<T (() ) Ts(T) )> = 0 (3.A.6)
-s 1 s 21 )6~<TS a' _S )> = X Ts( ) 6( - ) (3.A.7)
Use of the above moments is justified by the fact that a purely
diffuse target would turn a perfectly coherent illuminating beam into
a spatially incoherent reflected beam. The quantity Ts (p') can be
interpreted to be the mean-square reflection coefficient at p'.
What is sought in operation of the radar is information about
the target. In terms of the preceding model it is worthwhile to
mention what information we are seeking. We regard T(P') as the target.
For the glint component we are interested in the field reflection
coefficient (P') while for the speckle component we want information
on the mean-square reflection coefficient T(P'). Note that neither
the random phase e nor the exact speckle component field reflection
coefficient Ts(P') is regarded as interesting.
B. Relationship to Bidirectional Reflectance
Let us examine the relationship between the preceeding target
statistical model and bidirectional reflectance, the target signature
quantity that is generally measured [3,8,22]. For the target geometry
-38-
of Figure 3.1, the bidirectional reflectance may be defined as
(X; ; = 1 < dp' exp1[2f - ) -P '] T(') 12> (3.B.1)1 r X2AT - r
where AT is the target's projected area. If the input field is the
plane wave exp(j27 i -p') then the total reflected field is given by
exp(j2r 7 -p') T(p'). The portion of this reflected field that is in
the fr direction is the function-space projection [23] of
exp(j2T i -P') T(I') onto the function exp(j27 Ir -p'), that is, the
Fourier transform of the reflected field. Hence, the bidirectional
reflectance gives the ratio of the average reflected radiance (W/m2sr)
in the direction fR to the incident irradiance (W/m2) propagating in
the direction .
W W W ww
REFLECTED FIELD U r(U,+)r _ TA RG ETKiiiiiiii fr
TARGET-PLANE FIELD ' z
INEFFECTIVE PLANE OF INTERACTION
z L
Figure 3.1: Geometry for defining bidirectional reflectance p'(x;Ti,Tr); the target plane field ischosen to be a plane wave of wavelength x propagating in the direction of the unit vectorii (Xfi is the projection of ij on the z = L plane); the radiance of the reflected fieldis measured in the direction of the unit vector ir (xfr is the projection of ir on thez = L plane).
4W MW 1W W
-40-
CHAPTER IV
SCANNING-IMAGING RADAR ANALYSIS
With atmospheric propagation and target interaction models in
hand we are ready to proceed with the scanning-imaging radar analysis.
This will consist mainly of a signal-to-noise ratio (SNR) analysis.
To a lesser extent resolution and correlation of simultaneous target
returns are also considered.
We assume that an image is built up through scanning a scene
diffraction-limited FOV by diffraction-limited FOV. With this kind of
imaging system successive returns from the same direction are
separated in time typically by tens of milliseconds. Since atmospheric
correlation times in both turbulence and low visibility are considerably
shorter than 10 msec, successive returns can be taken to be independent.
Accordingly, with the SNR definition given below, the N-pulse,
single-pixel SNR is N times the single-pulse SNR. Coupling this with
the fact that speckle target returns from disjoint diffraction limited
FOV's are independent when the MFS model applies, as shown below, the
single-pulse SNR is a reasonable performance measure.
We begin this chapter with a formulation of the single-pulse
SNR problem. In the following section, speckle target resolution in
bad weather is considered. We then discuss how the various atmospheric
degradation effects will be identified in the SNR formulas. Following
-41-
this, SNR results in turbulence and low visibility will be presented.
We then explore the SNR degradation effects of beam wander due to
reset error in the radar aiming mechanism or atmospheric beam steering.
Next, correlation of simultaneous returns from different directions is
considered. Finally, backscatter from particles in the propagation
path is examined.
A. Sinale Pulse SNR
We are interested in the target reflection strength and
accordingly consider
2L/c + tp
Ir12 = 1l/t J r(t) dt| 2 (4.A.1)
- 2L/c
the output of a matched filter envelope detector with input r(t).
Ignoring the passband filter of Figure 1.3 and assuming tp = 1/W (the
tp second integration has approximately the same effect on the IF signal
r(t) as the filter) we have
r12 = Ix + n12 (4.A.2)
where the signal return is given by the target plane integral
j'ft pP T'~ (4A3x = d' T(p') G(p') s(i')i lyAen
and the noise n is zero-mean, complex-Gaussian, statistically i-ndependent
-42-
of x, with moments
<n2> = 0
<In 2> = 1
(4.A.4)
(4.A.5)
Note that the normalization chosen in (4.A.3) leads to the simple
result (4.A.5). The mean of the observation (4.A.2) is
_= 2> + <In12> (4.A.6)
The term <In! 2> is signal independent and due to receiver noise. Hence,
we define the image signal-to-noise ratio to be
(<lr2> - <ln2>)2SNR = _
Var(I r12)
<Ix 12>2
Var( I r12)(4.A. 7)
That is, the ratio of the square of the signal portion of the observation
mean to the observation variance. Since n is zero mean complex-Gaussian
with statistics (4.A.4), (4.A.5) it is straightforward to show
SNR = CNR/2CNR/2 11 SNR SAT 2CNR
(4.A.8)
where the carrier-to-noise ratio (CNR) has been defined to be the ratio
of the signal portion of the observation mean to the noise portion
-43-
CNR (4. A.9)<tn_ 2>
and the saturation SNR is
<|x|2>2SNRSAT = <I.X12> (4.A.10)
AT Var( xi2)
For CNR 5 or CNR 2 (10 SNRSAT we can disregard the last term in
the denominator of (4.A.8) and obtain
SNR = SNRSAT + CNR/2 SNRSAT (4.A.ll)
The maximum value of SNR is, from (4.A.ll), seen to be SNRSAT and is
achieved when CNR >> 2 SNRSAT. Physically, this limiting SNR results
when noise fluctuations become negligible in comparison to signal
fluctuations. From (4.A.8) and (4.A.ll) it can be seen that to complete
the analysis we need to evaluate SNRSAT and CNR for a number of
atmospheric/target situations. In order to know these quantities we
must calculate <tx!2 > and Var(tx1 2 ). Direct calculation of <tx 2 > via
the definition (4.A.3) is not too difficult but use of the same approach
for Var(tx1 2 ), while in principle straightforward, yields results which
quickly become unmanageable. A more reasonable approach is to try to
develop a complete statistical characterization for Ix12 for a number of
atmospheric/target scenarios. This approach, besides being simpler,
has the advantage of providing the information needed for the target
-44-
detection problem. This is the approach to be taken.
Up to now no assumption has been made regarding the statistics
of the signal return x. Therefore, the formulation of the imaging SNR
problem in equations (4.A.1) to (4.A.ll) is equally applicable to free
space, turbulence, and low visibility as well as beam wander induced
fluctuations. To finish this section we give selected free space
SNRSAT and CNR results.
Here we shall assume a Gaussian-beam system described by
FT(P) = F*(P) exp(_IP12/2p2) (4.A.12)(TrpTY
where PT is the radar transmitter and receiver pupil radius.
For a resolved speckle target (i.e. one that is larger than the
illuminating beam size) with intensity reflection coefficient r in free
space it is simple to show that
T.1P t 2TrP2CNR = 0 r (4.A.13)
ho L
and [3]
SNRSAT = 1 (4.A.14)
For an unresolved glint target, with field reflection
coefficient
-45-
T (p ) = exp[- L' 1 2/2r2] (4. A.15)
-<s
we have
CNR = L4rrsPTj (4.A.16)
and a saturation SNR equal to infinity. More detailed examples of
this type will be given later.
B. Speckle Target Resolution in the Low Visibility Atmosphere
We consider the output of the matched filter envelope
detector (4.A.2) when the noise n is negligible so that I 2 T2
where
x(fT hjtp PT d-p' T(p) Eg p') Flp) (4.B.1)
and the, atmosphere is characterized by (2.C.2). In (4.B.1) we have
explicitly noted that the signal return is a function of pulse
propagation direction as defined by TT (-see below). Limiting
ourselves to pure speckle targets, the mean signal return < x(-T 2>
will be considered. In examining this quantity we will be concerned
with what portion of the target contributes to the signal return.
-46-
Before giving the expression for the mean signal return it is
worthwhile to develop some preliminary results. Namely, the field
spatial correlation functions <F(Cj) gp)> and qR
should be found. For the resolution issues discussed in this section
it is necessary to know these correlation functions only for p= p.
The more general result ( 5 is useful in the sequel, though,
and will be given here. To facilitate the calculations Gaussian
transmitted and local-oscillator spatial modes are assumed
FT(P) exp[- I 2/2 p2 + j 2 T f-] (4.B.2)T (~TrP 2 )T
(T
F*(P) -exp[- P 1 2 /2 p + j2nTr - ] (4.B.3)ex E-rP/ 2 )1f2 R(RR
In the above equations PT and PR correspond, respectively, to the
transmitter pupil radius and receiver pupil radius and fT determines
the direction of propagation as this pulse will illuminate a circular
region in the z = L plane centered on the transverse coordinates
P = ALfT.
The atmospheric Green's functions hLF(P',P) and hLR(PP') are
taken to be statistically independent. This, as stated earlier, is
the "effectively bistatic" assumption. In order for this assumption to
be reasonable when the radar is in a monostatic configuration (i.e. when
the transmitter and receiver are colocated) the roundtrip delay for
-47-
pulse propagation to the target and back to the radar must be longer
than the atmospheric coherence time Tc. An upperbound on this coherence
time [24,25] can be taken to be the time for a frozen atmosphere moving
at the transverse wind velocity Vt to move a coherence distance p ,
i.e. Tc < P0/Vt. This expression ignores random motion of the air
molecules which could decorrelate the atmosphere much more quickly than
the preceding expression would suggest. For T' = s'L = 10,
forward scattering angle eF = 10 mrad, 10.6 ym radiation and transverse
wind velocity V t = 10 km/hr this upper bound on the atmospheric
coherence time is p0/V t 25 psec. The roundtrip delay time td = 2L/c
is 6.7 -psec at range L = 1 km so that delay is equal to the upper
bound when range is approximately 3 km. Even with the maximum value
for coherence time the "effectively bistatic" assumption can be made
for reasonable target- ranges.
The spatial modes (4.B.2), (4.B.3) and the definitions (l.A.12),
(l.A.13) in combination with the moment (2.C.2) are used to develop
expressions for the field spatial correlation functions <L 4$>, <L 4>.
To facilitate interpreting the results of later sections, the
atmospheric coherence distance p0 is taken to be different for the
forward and return paths. Specifically, poF replaces p0 in (2.C.2) for
hLF(P',p) while poR replaces p0 for hLR(p',). Physically, such an
assumption is unreasonable for a monostatic radar and is made only to
aid interpretation. The correlation functions are then given by
-48-
-'Le 2 exp[-p'TrrbF
ALI 2/rbF
- exp[-p |2 /rcF2 exp[j k (p' - XLfTT) ]exp[j2pT'SoF c T d
(4. B. 4)
where for convenience we have expressed the result in sum and difference
coordinates
(4. B.5)
(4.B.6)
In Equation (4.B.4) the beam radius rbF is
rbF = (AL/pF)2 + (AL/2TrpT) 2 +
the coherence distance r cF is
(4.B.7)
2 2
bF "oF
TT (AL/hrpOF)2 + +(AL/2pT) + pT +
rc = (4.B.8)
<_-F( ') -LPF ) 2
-c = 1 (p - + p )
+ p2F /4
-49-
and the phase radius of curvature RoF is
4p + 3pF 2 rbFR = L[T + bF (4.B.9)oF P + 3p F r 2 T
The corresponding expressions for <-RC) L*(p)> are found from
(4.B.4) - (4.B.9) by replacing pT by PR and PoF by PoR everywhere so
that r F becomes rbR, rcF becomes r2R, and ROF becomes ROR.
Assuming the radar is "effectively bistatic," that it uses
the Gaussian spatial modes (4.B.2), (4.B.3) with PT = PR and poR= oF Po
and that the target is pure speckle with moments (3.A.5) - (3.A.7),
the mean signal return is
< X( T)h T e-2 L x-- d~p' T (-P') exp[-Ip' - XLfT 2 /r es<' 0 7T 2>rh4 s ee5
(4.B.10)
where rbF = rbR r and
1 ~
re r /2 = [(XL/2)rpT 2 + pT] 1 + 2 (4.B.11)(AL/2rfTP + PT
is the target-plane resolution e~1 spot size. Equation (4.B.10) can be
interpreted as saying the signal return is an average over a spot of
-50-
area r es centered at Lf If the target is in the radar's far field
this becomes
rres 2 (XL/2mpT) 2 [1 + ( T (4.B.12)
which can be interpreted as saying that the signal return is an average
taken over approximately 1 + 4/3 (p T/P)2 diffraction-limited fields of
view.
C. Identification of Atmospheric Effects
Although we discussed earlier a number of atmospheric effects
expected to degrade the performance of an optical radar, no indication
was given as to how these effects might be identified in our performance
analysis. Here we discuss how this identification will be made in the
SNR and CNR formulas to follow.
1. Forward-path beam spread loss. If PT is the transmitted
beam radius and p0 the atmospheric coherence distance, any
degradation due to beam spread in the forward path can be
eliminated by letting PT become small in comparison to p0 .
Under this condition diffraction would dominate any
atmospherically induced beam spread and this loss mechanism
should be eliminated from the CNR and SNR formulas. Note
that this approach will be useful only in interpreting
low-visibility results as beam spread was already shown to
be unimportant in turbulence.
-51-
2. Return path beam spread loss. After reflection of the
illuminating beam the phase fronts of the return beam will
undergo additional wrinkling and hence additional beam spread
will be incurred. This source of beam spread loss is not
as easily identifed in the equations as the forward-path
loss. But it should not be present when the target is pure
speckle as the reflected light is effectively radiating
into the 27r steradians solid angle in front of the target,
nor should it be present when the target is pure glint and
smaller than the coherence distance p as diffraction then
dominates beamspread.
3. Receiver coherence loss. As this loss mechanism is due to
a wrinkled phase front in the received field it can be
eliminated by decreasing the receiver pupil radius PR to
satisfy PR < Po. In this case, the received phase front
will be flat over the receiver aperture and the spatial
modes of the received and local oscillator fields will
match. Coherence loss is not important in turbulence, as
discussed earlier, so this approach is useful only in low
visibility.
4. Scintillation. The effects of scintillation will be
difficult to see in the CNR and SNRSAT formulas. These
effects will become apparent when complete statistical
characterizations of target returns are presented.,
-52-
We now proceed with a series of SNR, CNR examples corresponding to
different atmospheric/target scenarios. Turbulence results will be
presented first, followed by low visibility results.
D. Turbulence SNR Results
The results cited in this section have previously appeared
in references [3-5,26]. They assume the Gaussian-beam system
(4.B.2), (4.B.3) with perfect transmitter, local oscillator mode
matching Fp) = F*(p), PT = PR. We present these results as a series
of examples.
Case 1. Turbulence, Unresolved Glint Target
Here a pure glint target T(p') = T (p') e j that behaves like
a spherical reflector over the illuminated region
T (p') = F exp(-jkjp'-'|/Rc) p' -ALfT (4.D.l)c_ c T d
and satisfies
Rc << L (4.D.2)
(XRc 0 (4.D.3)
is assumed. Equation (4.D.2) should often if not always hold. It can
-53-
be shown that the effective radiating region of the target (4.D.1)
(i.e. the region within the illuminated portion of the target that
makes an appreciable contribution to the target return) has nominal
diameter (R c)P. Hence (4.D.3) amounts to saying that the effective
radiating region is smaller than an atmospheric coherence area. A
target satisfying (4.D.l)-(4.D.3) is called a "single glint" target.
Furthermore, assuming that the target lies entirely within the
illuminating beam, i.e. it is unresolved, we find that the signal
return can be expressed as
- 4cr 2
1x_12 = CNRgu e X exp[4X(p ,0)] (4.D.4)
where p' is the glint reflection point of (4.D.1) and exp[4X(',U)]
is a lognormal random variable. In (4.D.4) CNRgu is the unresolved
(denoted "u") glint (denoted "g") target, tubulent atmosphere
carrier-to-noise ratio and a2 is the variance of the log-amplitudeX
perturbatiort X( 9,9) .
It follows from (4.D.4) that the single pulse image SNR
satisfies
CNR /2SNR u gu (4.D.5)
g 1 + CNR gu(e X - 1)/2 + 1/2 CNRgu
For CNRgu z 5, Equation (4.D.5) takes the standard form (4.A.ll) with
-54-
SNRSAT 1 (4.D.6)SAT 16 a2gu e X
If no turbulence is present, we have a2 0 and the saturation SNRX
(4.D.6) is infinite as predicted by (4.A.10). For a2 > 0, SNRX gu
initially increases with increasing CNR until it reaches the
scintillation-limited value (4.D.6). For cr2 > 1/16, SNRSAT isX -STgu
severly limited by turbulence. Multiframe averaging is required to
overcome this limit.
Case 2. Turbulence, Speckle Target
When the target is assumed to be pure speckle, T(p') = T '
the single pulse image SNR satisfies
CNR /2
SNRs 16 52 (4.D.7)
1 + CNR s[1 + 2(e X -1)C]/2 + 1/2 CNRs
where C is the log-amplitude aperture averaging factor [16,41] given by
the approximate expression
4 P2 /XL4T~ 2 (4.D.8)1 + 4 p T //L
in the weak perturbation regime and CNRs is the speckle (denoted "s")
target, turbulent atmosphere CNR. If the mean-square reflection
coefficient T(p') does not vary appreciably over the region illuminatedS
-55-
by the radar (i.e., the target is resolved) the target return can be
expressed as
Ix 2 = CNRsr v e2u (4.D.9)
where v is a unit mean exponential random variable and u is a Gaussian
random variable, statistically independent of v, with mean -a2 and
variance a 2 satisfying
2 1e4c - 1 = C(e
In (4.D.9), v represents the target speck
The probability density function for the
w = v e2u is given in [27]. If CNRs z 5,
standard form (4.A.ll) with
SNRSAT =5As
6a2X - 1)
(4.D.10)
le and u the scintillation.
unit mean fluctuation
Equation (4.D.7) takes the
1
16 a21 + 2(e X -_1)c
(4.D.11)
From (4.D.11) note that SNRSAT < 1 with SNRSAT = 1
speckle limited saturation SNR.
For more details on turbulence SNR results,
examples, the reader should see [3-5,26].
corresponding to
including system
-56-
E. Low Visibility SNR Results
In this section first CNR and then SNR results will be
developed for bad weather radar operation. Again, the Gaussian
transmitted and local oscillator spatial modes (4.B.2), (4.B.3) as
well as an atmosphere characterized by (2.C.2) are assumed. The
correlation function (4.B.4) then applies to this case where we will
take the coherence distances for the forward and return paths to be
different poR PoF* This will aid in interpreting the results of this
section.
Assuming that the atmospheric coherence time Tc is short
enough to justify saying the radar is "effectively bistatic" the CNR
(4.A.9) becomes
-nP tCNR = Pt dpi dp <T(T ) T*(T)><-F ) L (Pj4
0 J (4.E.l)
Our task is now reduced to evaluating the integral (4.E.1). This is
done as a series of examples, each corresponding to a different target.
We will interpret the results in terms of the previously mentioned
atmospheric degradation mechanisms.
Case 1. Low Visibility, Unresolved Speckle Target
The target is pure speckle T(-') = T (p') with-P -S
Ts(P') exp[-I' 2/r' ] (4. E. 2)
-57-
The mean square reflection coefficient Ts is Gaussian with width rs
and centered on the z axis. Although no real target would have the
form (4.E.2) it is chosen to allow closed form evaluation of (4.E.1).
Also the CNR results below would be applicable to any unresolved
speckle target if we replace 7rr with the speckle target area AT.
Assuming
rs<< rbF9rbR (4.E.3)
i.e., that the target is unresolved by the radar, the carrier-to-noise
ratio (4.E.1) becomes
CNR = CNR 0 exp[-2'L]su su 1/3(XL/pOFTO 2 1/3(XL/p oR7 2 a
1 + 1 +
(XL/2T) 2 + p (XL/2rpR + R
*exp LTI 2 (4.E.4)
rbF rbR/ rbF + rbR
where CNR 0 is the free space unresolved speckle target CNRsu
TIP t x2 r 2
CNR 0 = T _ s r5 (4.E.5)su hv 0 ' [(XL/21pT) 2 + p2][(XL/2pR)2 + pR]
-58-
The second multiplicative term in (4.E.4) is the forward-path beam
spread loss since it approaches 1 as PT +* 0. Similarly the third term
can be identifed as the coherence-loss term as it approaches 1 as
PR - 0. The fourth term represents absorption and the last gives the
loss due to the misalignment of the radar, i.e., that the center of
the illuminating field is IXLfT1 from the target center. Note that no
return-path beam-spread loss is evidenced by (4.E.4), as predicated
earlier. It should also be noted that the total free space beamwidth
(XL/2pT) 2 +p is used instead of the far field approximation
(XL/27rrpT) 2. At C02 wavelength 10.6 ypm and transmitter pupil radius
PT = 6.5 cm, the change over from near field to far field occurs at
approximately 2.6 km, midway through the expected useful range of the
radar.
Case 2. Low Visibility, Resolved Speckle Target
The target is again pure speckle T(P') = T (-') with mean
square reflection coefficient (4.E.2). Assuming
rs >> rbF, rbR (4.E.6)
so that the target is resolved, the CNR becomes
-59-
0CNR sr= CNR sr
+ 1/3(XL/poRr) 2 1/3(AL/pOF7F)2
(XL/2R)2+p2+( L/2r)2+P (XL/2rpR)2+p2+( L/2 pT) 2+p2
exp[-2S;L] exp[-fXLfT 2 /r2 ] (4.E.7)
where CNR0 is the free space, resolved, speckle target carrier-to-noisesr
ratio
CNRP0 - Tt p 1sr hv0 'T (XL/2TpT) 2 + p + (AL/2pR)2 + PR
(4.E.8)
The final two terms in (4.E.7) are identifed as absorption and
misalignment losses as before. The forward-path beam spread and
coherence loss are given by the second term. The second term in the
denominator of the quantity
1+ 1/3(XL/poRr) 2
(XL/2TrpR) 2 + p + (AL/2rp) 2 + p2
+ 1/3(XL/poF')
(AL/27rpR) 2 + p2+ (AL/27rpT) 2 + 2
(4.E.9)
gives the coherence loss while the last term gives the forward-path
beam spread loss. We see that forward-path beam spread loss in the
-60-
resolved case is affected by both PT and PR but affected only by PT
in the unresolved case equation (4.E.4). In order to understand the
difference in these two cases, consider the situation when the return
medium is free space p = . Clearly, there can then be no coherence
loss. From Figure 4.la it can be seen that in the unresolved case only
the on-target (essentially constant) incident power density is affected
by beam spread. The spatial mode of the reflected light depends on
target shape and not on forward path beam spread. Hence, changing PR
cannot reduce forward-path beam spread loss as the spatial mode of
the reflected light is independent of the beam spread. In Figure 4.lb
(resolved case) we have the situation in which the reflected light has
a spatial mode that depends upon forward-path beam spread and all of
the incident power is reflected. By choosing PR large enough, any
loss associated with this beam spread can be eliminated. That coherence
loss is affected by PT in (4.E.7) can be similarly explained.
Case 3. Low Visibility, Unresolved Flat Glint Target
The target is pure glint T(p') = ejeT (p') where
T (j')= p2 exp[-IK'12 /2r2] exp _ 2 (4.E.10)
The intensity reflection coefficient T(p)1 2 is Gaussian with width rs
as previously. We assume that the target is flat so that the phase
radius of curvature
-61-
Beamspread
Target
RADAR 2rbF
Incident powerconstant over
----...target
Free Space Beamwidth
Reflected spatial mode is independentof forward-path beamspread
a) Unresolved Target
.... Target
Beamspread
RADAR 2r F
Free SpaceBeamwdith
Reflected Spati'_'FMode depends onforward-path beamspread
b) Resolved Target
Figure 4.1: Reflected Spatial Modes from Resolved and Unresolved SpeckleTargets.
-62-
Rg
= O
We further assume the target is unresolved
rs << rbF, rbR
the radar is in the target's far field
r << a
and the laser is aimed directly at the
fT = 0
(4.E.13)
target
(4.E.14)
so that the CNR becomes
CNR = CNR 0gu gu1/3(AL/rpoF)2
(AL/27rpT)2 + p
1/3(L/TrpoR )2
(XL/2rpR) 2 + 12R
-1
4r 2 4r2
1 + +r 2 r 2rCF rcR
exp[-2 'L] (4.E.15)
is the free space unresolved flat glint target CNRwhere CNRgu
(4.E.11)
(4.E.12)
-63-
nP t 4r'CNR0 - tp 4s (4.E.16)
gu hv0 [(XL/2rrpT) 2 + p{1[(AL/2TrpR)2 + PI]
Since the effective radiating region of a spherical glint target has
nominal diameter (AR ) the above results can be applied to the case
of finite R by replacing 2rs with (AR )2. In (4.E.15), the second,
third and fifth terms are recognized as forward-path beam spread loss,
receiver coherence loss and absorption loss, respectively, as in
(4.E.4). The fourth term can be interpreted as return-path beam spread
loss. Justification of this last identification follows from the
fact that this term approaches 1 as rs -+ 0. To see more clearly what
is transpiring, note that this term is given by
(4.E.17)r2 r2
1+ +2 2
PoF PoR
when the target is in the transmitter's far field and PR > PoR'
PT > oF. Since a coherent radiator in the target plane of size rs
would have beam size 1/3(XL/p oR)2 + (XL/27r s)2 in the far field, it
is clear that beam spread would affect beam size only when rs >oR'
which is exactly when the third term in the denominator of (4.E.17)
becomes significant. To see how low visibility in the forward path
would affect return-path beam spread consider Figure 4.2. In Figure
4.2a an essentially flat phase front is reflected from the target
-64-
Incident Phasefront
Reflected Phasefront
ii2r
Target
a) Target smaller than a coherence area, rs < rcF : PoF
Incident Phasefront
Reflected
2rs
Phasefront
arget
b) Target larger than a coherence area, rs > rcF ~ oF
Figure 4.2: Reflected Phasefronts from Glint Targets
.
-65-
while in Figure 4.2b a wrinkled phase front is reflected. The
wrinkled phase front would clearly cause beam spreading after
reflection. This beam spreading becomes significant when rs >oF;
this is exactly when the second term in (4.E.17) becomes significant.
We now turn our attention to developing complete statistical
characterizations of the single-pulse target return. These
characterizations will be of the form (4.D.4), (4.D.9) wherein the
signal return |x 2 is represented as a product of constants and
independent random variables. This is most easily accomplished via a
series of examples.
Case 1. Low Visibility, Small Speckle Target
Note that the target has been designated as "small" instead
of "unresolved" as in the CNR examples. The reason for this is the
requirement on the size of the target is different here than previously,
as will become apparent. Describing the target T( ') = T (P) by
(4.E.2) we express the signal return (4.A.3) as
x_= p)T dp' Ts P 2P F (4.E.18)
where we have made the definition
dP' ! (P') L.( ' EF( 'tA J(4.E.19)
(f dp' Ts(-') 1 12 F
-66-
Assuming T(p') is a Gaussian random process it is a simple matter to
show, by consideration of the conditional moments of c given (R and F
that c is a zero mean complex Gaussian random variable with independent
identically distributed real and imaginary parts and variance
< 2> = 2 (4.E.20)
Assuming that the target is small compared to a coherence area
(4.E.21)r2 << rF2 p 2
< cF PoF
s cR OoR (4.E.22)
we can then say
(4.E.23)
__F I (. 2 (4.E.24)
inside the integral (4.E.18). It then follows that the signal return
can be expressed as
x 2 = CNR u v w (4. E. 25)
-67-
where u, v, and w are independent, identically distributed, unit mean,
exponential random variables. In (4.E.25), u represents target
fluctuations and v, w the forward and return path atmospheric
fluctuations. These atmospheric fluctuations represent the randomness
imposed by the terms (4.E.23), (4.E.24).
Two points need to be mentioned relating to the assumptions
(4.E.21) and (4.E.22). The first is that any target satisfying these
conditions is necessarily unresolved as rcF < rbF and rcR < rbR'
Second, is that this condition is an extremely stringent one as poF and
poR may easily be less than a millimeter.
The saturation SNR for this case is now easily calculated
from (4.E.25)
SNRSAT <X 2>2 (4.E.26)ss Var( xf2 ) 7
where "ss" denotes small speckle target. A physically more realistic
situation is now presented.
Case 2. Low Visibility, Large Speckle Target
Again, the target T(p') = T (') is described by (4.E.2) so
that (4.E.18) applies where E is a complex Gaussian random variable.
Assuming that the target is large compared to a coherence area
-68-
r2 >> r2 (4.E.27)s cF
r2 >> r2 (4.E.28)s cR
we can then argue there is sufficient target plane aperture averaging
so that the integral (4.E.18) becomes nonrandom and can be replaced by
its mean. Taking this to be the case we then have
jx_| = CNR u (4.E.29)
where u is a unit mean exponential random variable representing target
fluctuations.
The conditions (4.E.27), (4.E.28) do not impose a very
stringent limitation on the target size. In fact, the target can be
either resolved or unresolved and still satisfy them. The saturation
SNR is now given by
SNR SAT = 1 (4.E.30)
where "sZ" denotes large speckle target. The cases of small and
resolved glint targets are now presented.
-69-
Case 3. Low Visibility, Small Glint Target
The target T(p') = e a T (P') is given by (4.E.10) so that
the signal return is
[nt P T
x_= h TeJe d' T (T') E-RP) -F(E.)3
If we assume the target is small, i.e.
s a cF
r2 << rcR
(4.E.32)
(4.E.33)
then we can replace g(P') and ') in the above integral by (
and Lg(U), respectively, as in the small speckle target case.
follows that
fx 2 = CNR v w
It then
(4.E.34)
where v and w are independent identically distributed unit mean
exponential random variables representing forward and return path
fluctuations, respectively. The saturation SNR for this case is
1SNR SAT g (4.E.35)
(4. E.31 )
-70-
The same comments regarding the implications of (4.E.32), (4.E.33)
apply here as in case 1.
Case 4. Low Visibility, Resolved Glint Target
For a glint target T(P') = ej T (P') specified by (4.E.10)
that satisfies
r 2>> rb 2r > bF
s bR
(4.E.36)
(4. E.37)
it can be easily argued, by considering an unfolded geometry for the
radar configuration Figure 1.1, that the received field u (5,t) is
Gaussian. It then follows from Eq. (l.A.10) that x is Gaussian and
|x2 exponential
1x1 2 = CNR v (4.E.38)
so that
SNRSAT gr= 1 (4.E.39)
This complete the single pulse signal-to-noise ratio analysis
for the "effectively bistatic" radar. In the next section, the effects
-71-
of beam wander on unresolved target returns is considered.
F. Beam Wander Effects
The power reflected from an unresolved target will fluctuate
due to beam wander effects from radar pulse to radar pulse. The
source of the beam wander may be actual reset error in the aiming
mechanism of the radar or beam steering effects of turbulent atmosphere
propagation [1]. To account for these fluctuations in our model the
expressions for the matched-filter envelope-detector output IX| 2
(i.e. (4.D.4), etc.) are multiplied by another unit mean random
variable. If we let the unit mean random variable w represent the
collective effects of target/atmospheric fluctuations (excluding beam
steering) and the unit mean random variable v represent beam wander
induced fluctuations then we can say in general
IX 2 = CNR w v (4.F.1)
Taking w and v to be independent the saturation SNR is
SNRSAT = (4. F.2)SAT =Var(w) Var(v) + Var(w) + Var(v)
The expression reduces to
SNRSAT = SNRSATbw (4.F.3)
-72-
when
SNRSATta >> SNRSATbw
SNRSATta
SNRSATta
(4.F.4)
(4.F.5)> 1
Var(w) (4.F.6)
is the saturation SNR when only target/atmospheric fluctuations are
present and
(4.F.7)SNRSATbw Var(v)
is the saturation SNR when only beam wander fluctuations are present.
By extension, when the conditions (4.F.4), (4.F.5) are satisfied we
can say
JxJ 2 = CNR v
and
where
(14.F. 8)
-73-
or equivalently
IxI = <lxI> z
The random variable z is the unit mean fluctuation on
related to v by
lxi and is
V (4.F.10)
We develop statistical models for z here. This is most conveniently
accomplished as a series of examples.
Case 1. Circular BeamUniform Circular Beam Center
Say that the illuminating intensity is Gaussian
ut(-P,), 2=t exp(- p'
TrTrb(4. F. 11)
where the beam center
(x,y) (4.F.12)
is random and circularly distributed
(4.F.9)
-74-
(X-m )2+(y -M my)2 < R'7TR2
p (XY)
0
(4.F.13)
elsewhere
An unresolved target of area AT is assumed to be located at P' =
so that the reflected power is
A PP AT ut() 2 t t exp(- 2/r2)
r2 mTrb
(4.F.14)
Since pm is random it is clear that the reflected power is also. For
lijil = (m + m2) < R of this on = <IxI>z is given by the probability
density
r 2~2 b 1 CosTr R2 Z
-2rb ln(Z/Zmx) - R2 + I
21r1 rb/ nZ/Zmax IZmin ZZ mid
Zmid <Z <Zmax
2
2
0 el sewhere
(4.F.15)
where
PZ (Z) =
-75-
e1
2 du u exp - u2 R0 2
.0 b b
Zmid exp
Zm =exp
1 - R)2
b
... R ( )2
b
Zmax
max
In (4.F.16),
first kind.
Io(-) is the zero-order modified Bessel function of the
For Jml > R the PDF of z becomes
2 2 1 2 r ln(Z/Zmx R2 + |i2
2 b 1 cos )T R 2 Z 2m1 rb/- 2 ln(Z/Zmax)
pZ (L) = .
Z* <_< Zmid(m4i nF.9)
(4. F. 19)
el sewhere
Zmax (4.F.16)
(4. F. 17)
(4.F.18)
m12 /2r2
0
-76-
where Zmax' Zmid' and Zmn are again given by (4.F.16)-(4.F.18). A
sketch of this density is shown in Figure 4.3 for lin/rb = 0 and
R/rb = 1. The saturation SNR for this fluctuation with -mi/rb = 0 is
is given by
SNRSAT (4.F.20)bw R2 1 - exp(-2R 2/r )
2r (1 - exp(-R 2 /r ))2
and is sketched versus R2 /r in Figure 4.4.
Case 2. Fan Beam, Uniformly Distributed Beam Center
The M.I.T. Lincoln Laboratory mobile infrared radar is
sometimes operated with a beam shape that is not circular as in case
1, but is instead expanded in the vertical direction to produce a fan
beam [6]. Furthermore, when in scanning mode the vertical scan is
required to move only 1/128th as fast as the horizontal scan and
therefore it appears that the vertical aiming error is negligible.
With this in mind the target illuminating intensity is assumed to be
P- 2 exp(-(x' - x)2/r2) (4.F.21)
where Py has units (W/m) and the beam center x0 is random with
distribution
W W W w
N
N0~
0.6 0.7 08 0.9 1.0 1.1 1.2 1.3z
Figure 4.3: Target return PDF for beam wander fluctuations; Gaussian beam,
uniform circular beam center, tII/rb = 0.0, R/rb =l..
01 I I I I I
2-
1
-4-4
116597-N
qW MW 1W W
qwW
I 16(b 1-N
I I I I I I I I I
50 GAUSSIAN BEAM,CIRCULARLY DISTRIBUTED BEAM CENTER
40 --- FAN BEAM,UNIFORMLY DISTRIBUTED BEAM CENTER
2 3000
Z 20
10 - -
0
0 1 2 3 4 5 6 7 8 9 10
Saturation SNR due to two types of beam wander fluctuation.
low
Figure 4.4
-79-
1
p (X) =
0
m - R < X < m+ R
(4. F.22)
el sewhere
such a uniform distribution is reasonable if one considers the
friction in the bearings that hold the beam aiming mirrors as the
dominant influence on the aiming error. Given (4.F.21), (4.F.22) and
an unresolved target located at p' = 0, the probability density for z
becomes for 0 < m < R
rb/2R
Z[-2 ln(Z/Zmax
rb/ R
z[-2 ln(Z/Z )]2
,max
0
Z <_ Z <_ Zmid
Zmid -Z < Zmax
el sewhere
(4.F.23)
where
PZ (Z) = .
-80-
1
-r
2 R 0 rI b_
(4.F.24)
Q m+ R
b
Zmid = exp
min =
Q(y) =
y /i
and
1
1
(m - R)2r2rb
(m + R)2
rb
Zmax
Zmax
exp(-x 2 /2) dx
is related to the complementary error function by Q(y) = erfc(y/42)/2.
For m > R the pdf for z is
r b /2R
Z[-2 ln(Z/Z max 2
pZ (Z) =
Zmin Z < Zmid
(4.F.28)
elsewhere
Zmax
(4.F.25)
(4.F.26)
(4.F.27)
0
-81-
where Z max Zmid, Z are again given by (4.F.24)-(4.F.26). The
saturation SNR for this case is given by
SNRSA = R1(4.F.29)SSATbw r Q{2(m-R)/rb} - Q{2(m+R)/rb -1
7 rb [Q{v2 (m-R)/rb} - Q{W2 (m+R)/rb1 2
A sketch of pz (Z) is shown in Figure 4.5 for m/rb = .45 and R/rb = 1.2.
It is interesting to note that the step in pz (Z) is due to the fact
that the reflected power depends not on the random variable x0 , but on
|x0 |. SNR is shown as a function of R2/rl for m/rb = 0 in FigureSATbb
4.4 and as a function of R/rb for various values of m/rb in Figure 4.6.
Case 3. Fan Beam, Gaussian Distributed Beam Center
Here we again assume the fan beam (4.F.21) but assume that
the beam center x0 is Gaussian distributed
p ( = 1 exp[-(X - m)2/2cy ] (4.F.30)0 / 2T2
x
Such a distribution for x0 is reasonable if the dominant influence on
beam wander is a thermal noise voltage in the radar aiming mechanism
or tubulent atmosphere beam steering [1]. For the case of an
unresolved target located at P' = 0 the probability density for z is
T.5 I I I j I I I I I I I I I I I I I -
5.0-
2.5-
0.5 0.75 1.0 1.25z
Figure 4.5: Target return PDF for beam wander fluctuations; Fan beam, uniformly
distributed beam center, m/rb = .45, R/rb = 1.2. 116600-N
60 1 1 I 1 1 1 1
50- m/rb 0
40-
~in~30
Z 20
10- 0.8
0 - .
-10 1 1 1 1 I I I0 12 3 4 5
R/rb
116594-NFigure 4.6: Saturation SNR due to fan beam, uniform beam center
fluctuations.
-84-
-2rb x Z7r b/aX az IFx
[-2 ln(Z/Zmax)] ZJ
exp(-m 2 /2a ) mrb[-2 ln(Z/Zmax -2
PZ(ZZma cosh b CmaxF
0< Z < max
0 elsewhere
(4.F.31)
where Zmax is given by
m 2/2r2+ = (1 + a/r )2 exp b (4.F.32)
max x + CF2/r2
A sketch of (4.F.31) is shown in Figure 4.7 for m/rb = .8 and
a /rb = 1.25.
4 III Ij III 111111 1111111 liii liii 111111 liii
N
N
0 0.25 0.5 0.75 1.0 1.25 1.5 1.75 2.0
z
Figure 4.7: Target return PDF for beam wander fluctuations; Fan beam, Gaussian 116598-Ndistributed beam center, m/rb = 0.8, a X/rb = 1.25.
-86-
Case 4. Circular Beam, Gaussian Circular Beam Center
Here we again assume that Gaussian intensity (4.F.ll) but
assume that the beam center P = (x,y) is Gaussian distributed
p (XY) - 2 exp{{(X-m )2 + (Y-m )2]/2a2} (4.F.33)xy2'rr 2 X Y
This distribution on pm is reasonable if beam wander is dominated by
thermal noise in the radar aiming mechanism or turbulent beam
steering. For an unresolved target located at P' = 0 the probability
density for z is
r21 rb expV _ i
Zmax c 2 2
rb
pz(Z) =- Ia 2 /- bn /max 0 L m
o <7Z<7Z- -- max
0 elsewhere
(4.F.34)
where
-87-
m12 /r 2
Zm = [1 + a2 /r ] exp 1b (4.F.35)zmax /b]exP2 1 + -a2/ r21 ~ bj
It is clear that beam wander can severely limit SNRSAT. It
is of interest to note (from Figures 4.4 and 4.6) that the limitation
is not too severe so long as the beam center varies within the area
of a diffraction limited spot (i.e. R/rb < 1) but becomes severe when
the opposite is true (R/rb > 1).
G. Correlation of Simultaneous Speckle Target Returns
In the introduction it was mentioned that the Lincoln
Laboratory compact CO2 laser radar employs a one-dimensional twelve-
element detector array. The transmitted energy is matched to this
array in that it is a fan beam, as mentioned earlier, compressed in
one transverse direction and expanded in the other. This situation
could be thought of as the detection of twelve simultaneously
transmitted, coherent, laser pulses. In order to make use of these
simultaneous measurements it is of interest to consider the
correlation between them. In the regime where the noise n is
negligible we have from (4.A.2) that the output of any matched-filter
envelope-detector is Ir 12 =XT) 2 where x(T) is given in (4.B.1).
-88-
Hence, we would like to determine Cov( x1 )I 2 1 T2)[), the
covariance between the simultaneous target returns from two directions.
This is a difficult calculation to make directly. But, if x( T '
x(fT2) are zero-mean, complex Gaussian random variables, as is the case
for a large speckle target, and < x(l ) T2 )> = 0, which is true when
we can say <h L(P' 1 ) hL P 2)> = 0, then
Cov(Ix(Y ) 2I(T 2 Tl T2)> 2 (4.G.1)
so that we now need only the correlation function <x x*>. From this
quantity, the correlation angle, ec, will be found, where
<x(Tl) X*T2)> 0 for the arrival angle difference I fTl - fT2I Z 'c'
For the Gaussian spatial modes (4.B.2) and (4.B.3) with
PT = PR, the "effectively bistatic" assumption, the MFS propagation
model with poR = PoF = po and a speckle target, it is a tedious but
straightforward calculation to show that
- ntpPT - 2 'L X2_
<X( eTTa 2 exp [-IXLd 12/Por0 Trrb
{ dp' Ts(p') exp[-Ip' - Lfc 2/r]2 exp j2r r d-(' - XLTcj
S rres
(4.G.2)
-89-
where
fd fTl ~ T2 (4. G. 3)
f T +T (4. G. 4)
and the correlation distance pcor is given by
r 2
P2 b (4.G.5)c o r + T2-+_ (PT/Po) 2
From (4.G.2), (4.G.3) it is clear that
c cor/L (4.G.6)
We find that ec = XS NT in the two limiting cases PT o and
PT p o under far-field propagation conditions. It can be concluded
that the correlation angle eC is essentially independent of
atmospheric conditions and is given by the field of view of a single
detector. This implies that matched-filter envelope-detector outputs
from adjacent pixels can be averaged for improved SNR if the target
is speckle and larger than a diffraction-limited FOV. Also, for the
multipulse detection problem discussed in Chapter VI, adjacent pixels
of large speckle targets can be taken to be independent observations.
-90-
H. Backscatter
As discussed earlier, when the monostatic radar is used in
inclement weather there may be significant backscatter return from
the hydrometeors and aerosols present in the propagation path. In
this section, the backscatter contribution to the matched-filter
envelope-detector output is found by considering first order multiple
scatter [28] as the propagation mechanism. The development begins by
characterizing a thin layer of scatterers as a planar target. The IF
signal due to a transmitted impulse of energy and this single-scattering
layer is integrated to give the impulse response. This is then
convolved with a rectangular power waveform (Figure 1.2) resulting in
the backscatter contribution to the matched-filter envelope-detector
output.
Consider Figure 4.8. The field ut(P',z) is incident on a thin
layer of scatterers of cross sectional area A centered at p' = p0.
The backscattered field u(p',z) is given by
Ur(I',z) = ut(',z) Tb(p',z) (4.H.1)
as in equation (3.A.3). We assume that T (',z) is a random quantity
and can be statistically characterized as thought it were a speckle
target
w w qw
.9!t IZI)
!--r(I',Z)
14 z JKAz
Single scattering layer; -
backscattered field.
(P',z) is incident field and ur(P',z) is
RADAR
Figure 4.8:
) ' Az)
SCATTERING SLAB HASAPRA A ANDir IS
CENTERED AT D' = o
116601-N
w
-92-
<Tb(p, Z) j(3 ,z)> = A2 Tb(i ,z) 6(_ - T (4.H.2)
Since each scattering particle is at a random position within this
layer (4.H.2) is reasonable. We seek to express Tb in terms of the
backscattering cross section ab which is normally considered [28].
The quantity ab associated with a given particle is the area
intercepting the incident radiation which, when scattered isotropically
into 47r steradians solid angle, produces an echo at the radar equal
to that from the particle. If the region between the scattering layer
and the radar is free-space and the radar is in the far field of the
scatterers
I = 4bTZ 2 |t( 0,z)|2 pA Az (4.H.3)
is the power density (j;/m 2 ) of the backscattered field on the z = 0
plane where p is the particle number density (m- 3). In terms of the
target model (4.H.1) this same density is given by
I = <J dp' ho(p,p') _t(p',Z) Tb (',z)2>(
(4.H.4)
u-t( P 0,Z)12 T b A
Z2
-93-
Combining (4.H.3), (4.H.4) we have
T =ab(z) p(z)b4r Az(4.H.5)
where ab and p have been allowed to be functions of z.
Consider again Figure 4.8. The contribution to the IF complex
envelope from the scattering layer is from (l.A.14)
y(t) = 2 s-T(t - 2z/c) dp' Tb(p',z) -F( 9',z) ER(p',z)
(4.H.6)
If we let
PT(t) = |ST(t)|2 = E 6(t)T0 (4.H.7)
where E = 1, then by the theory of the first-order multiple scatter [28]
the mean square contribution to the IF complex envelope becomes
<1y(t)12> = 6(t - 2z/c) - ab(Z) p(z) dp'IE0(-1',z)12 IO(5,z)I2hv 0 47r {exp(-2 at(s) p(s) ds) Az
-0
(4. H. 8)
-94-
where use has been made of (4.H.5), R F are free-space field
patterns and at is the single particle total cross section. Integrating
(4.H.8) over z, the mean-square IF impulse response is
<ly(t) 12> = dz 6(t - 2z/c) h1 tP 2ab(z) p(z) J dp'l (p',z)2j F(p ,z)|2
0 0
-z
-exp(-2 a t(s) p(s) ds)
-O(4.H.9)
Convolving (4.H.9) with the waveform
t T
P T(t) =
0
0 < t < t
elsewhere
gives the mean-square output of H(f) (Figure 1.3)
c2 t
< x~)2> =
(t-t )2 p
t pP T 2 I'dz hv0 4T b(z) p(z) d'1 R(p',z)|2|((p',z)2
z
exp(-2 J at(s) p(s) ds)
-0(4.H.l1)
(4.H.10)
-95-
which is essentially the output of the matched-filter envelope-detector.
For the Gaussian spatial modes (4,B.2), (4.B.3) with PT =PR and a
uniform scattering profile ab(z) = ab' at(z) = at, p(z) = p (4.H.11)
becomes
ct
< 1 t1 nt p P T2 2 exp(-2atpz) dz<|x (tb2 h b eld
o 22)l+
[27rP T Z
(4.H.12)
Equations (4.H.11), (4.H.12) give the backscatter power incident on
the radar receiver normalized by the local oscillator shot noise
power and, as such, should be compared to the CNR. When this
normalized backscatter power is smaller than the CNR it can be
neglected. But, when it is larger it can dominate target return.
This will be considered in more detail in Chapter VII.
-96-
CHAPTER V
THEORY VERIFICATION
The IRAR project's compact CO2 laser radar provides a good
opportunity for verfication of the preceding theory. The MFS theory
treats only scattered light. However, at the CO2 wavelength of 10.6 im
the albedo of a single scattering particle is rarely larger than a
half [37], so the extinguished free-space portion of the beam will
dominate the scattered portion in inclement weather. This implies that
verification of the turbid atmosphere theory is difficult, if not
impossible, with the IRAR system. Hence, our efforts were concentrated
on verifying the turbulence models, of Chapter IV, Section D and the
beam wander models of Chapter IV, Section F. In this chapter we begin
with a brief description of the CO2 laser radar used to make the
measurements. This is followed by a description of the techniques used
in the data analysis. In the final two sections the beam wander and
turbulence models are examined in terms of measured data.
A. Laser Radar Description
The compact laser radar system [6] employs a one-dimensional,
twelve-element HgCdTe detector, a transmit/receive telescope of 13 cm
aperture, and a 10 W, 10.6 -pm, CO2 laser, which is operated in pulsed
mode. The radar system can be operated in three modes: (1) full
scanning mode, (2) reduced scanning mode, and (3) staring mode. When
-97-
the radar is operated in full scanning mode there are two frame
rate/imaging options. In the first option the full, vertically stacked,
twelve element detector array is employed along with a fan shaped
illuminating beam, that has been compressed horizontally and expanded
vertically to match the detector array shape. Each detector is
separated by 200 rads from its neighbors directly above and below in the
direction it views. A 60 x 128 pixel image is constructed by sweeping
the fan beam horizontally through 128 - 100 prad steps five times. This
image, which takes 1/15th seconds to produce, then fills a 12 mrad x 12
mrad FOV.
In the second full scan, frame rate/imaging option a single
detector is employed along with a circular-symmetry Gaussian shaped
illuminating beam. A 60 x 128 pixel image is constructed by sweeping
the circular beam through 128 - 100 prad steps 60 times where each of
the 60 horizontal sweeps is separated by 200 prads in direction from
the one immediately preceding it. This image which takes 12/15th
seconds to produce, again fills a 12 mrad x 12 mrad FOV.
In reduced scan mode, a single detector is employed along
with a circular-symmetry, Gaussian shaped illuminating beam. A
60 x 128 pixel image is constructed in the same way as in the second
full scan, frame rate/imaging option except that the horizontal and
vertical steps are one twelfth as large. The image, which takes
12/15th seconds to produce,fills a 1 mrad x 1 mrad FOV.
When the radar is operated in staring mode a single detector
is employed along with a circular-symmetry, Gaussian shaped illuminating
-98-
beam. Radar returns are measured from a single diffraction limited
FOV of approximately 50 virads (diameter of the e~ contour).
Successive laser illuminating pulses are separated in time by
104 bpsecs.
In all modes of operation the IF portion of the heterodyne
detected photocurrents are linearly passband filtered and then video
detected. The output of the video detector is proportional to the
magnitude of the envelope of the input (making it proportional to the
square root of the target return power). The video detector output is
then digitally peak-detected to yield 8 bit range and intensity values
which are stored on magnetic tape for off-line processing. In the
limit of a large target return (i.e. CNR 10) a stored intensity value
is essentially the output of a matched filter envelope detector and,
in terms of our previous radar model, is proportional to lxi (Equation
(4.A.3)).
All the data that has been examined comprises intensity returns
from three targets: (1) A retroreflector of approximately 2 cm diameter
which is well modeled as a pure glint target; (2) a polished sphere of
approximately 10 cm diameter, which, again, can be modeled as a glint
target; and (3) a 1 m x 1 m flame-sprayed aluminum plate which can
be taken to be a pure speckle target. All data discussed in this
chapter was recorded under the condition of high CNR. We have used
this data to verify the models of Chapter IV, Sections D and F.
Specifically, efforts have been aimed at statistically verifying
-99-
equations (4.D.4), (4.D.6), (4.D.9), (4.D.11), (4.F.15), (4.F.19),
(4.F.23), (4.F.28), and (4.F.29). In the next section, the data
analysis techniques used for this will be described.
B. Data Analysis Techniques
Two principal types of calculations were used in the data
analysis. The first is an estimate of the saturation signal-to-noise
ratio, and the second is a chi-square goodness of fit test against
theoretical probability distributions.
If the stored intensity values, denoted 1xil, i = i,...,N,
are independent and identically distributed the saturation SNR which
is defined in (4.A.10) as
SNR< _X12>2SAT :Var( x1 2 )
can be estimated by
m2
SN SAT =
where m is the sample mean of the squared data
Nm = xi |i2
(5.B.1)
(5.B.2)
(5.B.3)
and y 2 is the sample variance
-100-
Na2 (jx2 - m)2 (5.B.4)Ni=1 -
The performance of (5.B.2) in estimating SNRSAT is considered to be
approximated by the performance of (5.B.3) in estimating m. The mean
and variance of (5.B.3) are
<M> =< (5.B.5)
Var(i) = Var(X12) (5.B.6)N
The ratio
= N SNR (5.B.7)Var(m) SAT
indicates that the standard deviation of the estimate m is l/(N SNRSAT)P
of its mean so that if we required, for example
(Var(m))2 < 100 (5.B.8)
for a ±1% RMS error we must have enough samples to satisfy
N SNRSAT > 1002 (5.B.9)
For the case of a speckle target, SNRSAT < 1 from which it follows
that a minimum of 104 samples are required for a ±1% RMS error. For
-101-
a speckle plate it is not too difficult to record this many measurements.
Unfortunately, over the period of time required to record this many
samples the characteristics of the radar itself may change. For example,
when observing an unresolved target the mean beam center m (see
Chapter IV, Section F) in fact drifts in time so that, as will be seen,
only a maximum of a few hundred points are recorded under the same
conditions. More will be said about this stationarity issue later.
Suppose that we have k mutually exclusive, collectively
exhaustive outcomes for some experiment with theoretical probabilitiesk
of occurrence p1p2 k .Z pi = 1. In a chi-square goodness ofi =1
f-it test [38,39] the hypothesis that the outcomes of the experiment
are governed by this distribution is tested against the hypothesis
that they are not. If n independent trials of the experiment are run
we calculate
k (f. - np.)z2 np (5.B.10)
i=l
where f. is the number of occurrences of the ith outcome and
kSf. = n (5.B.11)
i=l 1
Denoting the calculated value of X2 as X we see that X = 0 indicates00
perfect correspondence with theory while X1 large tends to discredit
the hypothesis. A quantitative measure of the validity of the
-102-
theoretical distribution P1 SP2 '3'' 'Pk is provided by the level of
significance a
a = Prob(x 2 > X 2 ) (5.B.12)0
where the above probability is calculated assuming the theoretical
distribution is correct. It can be shown [38] that the random variable
X2 (5.B.10) is approximately a chi-square random variable of k - 1 - m
degrees of freedom so long as npi > 5 for all i = 1,2,...,k and m is
the number of parameters in the theoretical distribution that are
estimated from the data. Generally a value of a areater than or equal
to .05 is regarded as verifying the theoretical distribution.
In the applications of this test that are found here, the
underlying probability distribution is that of a continuous random
variable. In this case, the pi, i = 1,2,...,k are calculated as the
probabilities of the outcome falling into one of k contiguous intervals.
C. Beam Wander
In this section, we compare radar data from the retroreflector
target at 1 km range to the predictions of (4.F.15), (4.F.19), (4.F.23),
(4.F.28) and (4.F.29) via five examples. The data for the first four
of these examples was taken while the radar was operating in full
scanning mode with the first frame rate/image option so that the fan
beam, uniformly distributed beam center model of (4.F.21), (4.F.22)
-103-
is appropriate. The data for the last example was taken while the
radar was operating in full scanning mode but with the second frame
rate/image option so that the circular beam, uniform circular beam
center model (4.F.ll), (4.F.13) applies.
Typically, when the IRAR system observes an unresolved glint
target in full scanning mode, the fluctuations on the target return
are dominated by random aiming error. In fact, not a single example of
atmospheric fluctuations dominating aiming error fluctuations could be
found in all of the full scanning mode data processed. The examples
of this section then represent typical samples of full scanning mode,
unresolved glint target data.
At the same time as the IRAR data was being collected one-way
scintillation measurements, whose purpose was to provide an accurate
estimate of the state of the atmosphere, were also being made. This
setup consisted of a GaAs laser and CO2 laser located approximately 10 m
to the side of the targets, and sensors corresponding to these lasers
located 10 m to the side of the radar. This equipment then provided
values of turbulence strength C2 and log-amplitude perturbation variancen
a2 at two wavelengths. For the first four examples of this section theX
scintillation measurements indicate that a2 was approximately 0.0005.X
From (4.D.6), (4.F.6)
SNRSAT = SNR 125 (5.C.1)STgu SAT ta
-104-
The estimates of the saturation SNR, IEhSAT, for these examples were
much smaller than (5.C.1) so that we conclude (4.F.4), (4.F.5) are
satisfied and the beam wander fluctuations dominate target/atmospheric
fluctuations. The scintillation measurements indicate that at the
time the data for the fifth example was taken cy2 = 0.0026 so thatX
SNRSAT = SNRSAT ~: 24 (5.C.2)
gu ta
Again T was much smaller than this value and we conclude thatSAT
beam wander fluctuations dominate target/atmospheric fluctuations.
Figures 5.1-5.5 summarize the results of these five examples.
Figure 5.1 shows the theoretical distribution (4.F.23) with R/rb = 1.3,
m/rb = 0.0 along with a normalized histogram of 100 consecutive data
points taken over a period of approximately 7 seconds. The values
of R/rb and m/rb given above for the theoretical PDF were chosen to
minimize the calculated value of chi-squared as are all parameters of
theoretical PDF's in all examples of this type in this chapter. This
minimized, calculated value of chi-squared for this data is X = 4.10
with 8 degrees of freedom. This corresponds to a level of significance
a. between .8 and .9 indicating excellent agreement between theory and
data. For R/rb = 1.3, m/rb = 0, equation (4.F.29) gives SNRSAT = 8 dB
while from the data and (5.B.2), I'SAT = 6.6 dB, a difference of only
1.4 dB. Figure 5.2 shows the distribution (4.F.23) with R/rb = 1.2,
m/rb = 0.50, and a normalized histogram of 200 consecutive data points
Z (Z)
6.0
5..
4.0
3.0
a.0
I.*
0.0#.So
ruiui~v u v ; i
I I
p
I I
0.60 0.70 0.30 0.90 1.06 1.10 1.10 1.30 1.40
Z
Figure 5.1: Normalized histogram of 100 consecutive retro returns taken in full scanning
mode and theoretical PDF Eq. (4.F.23), R/rb = 1.3, m/rb = 0.0.
I
. .I .
0
i
t I I IF I I -F- I I I I I I I I I I I I I I
S. 9 I-
pz(Z)a.s L.
e.g I I II I I I i lA
0.50
4
I-. md - - m~~ ~ -
0.75 1.90
I I I
LL..L1.25
Z
Figure 5.2: Normalized histogram of 200 consecutive retro returns taken in full scanning
mode and theoretical PDF Eq. (4.F.23), R/rb = 1.2, m/rb = 0.5.
__j
I
6.0
5..
4.0
3.0
e.9
i.e
0.
r Fi-l P El 1111 1111 1I IF 1
,.. E
S.as e.5. 0.75
I I II."
--
1.2s
I,
I 4
1.s0
Z
Figure 5,3: Normalized histogram of 200 consecutive retro returns taken in full scanning
mode and theoretical PDF (4.F.23), R/rb = 1,3, m/rb = 0.56.
M. - -
I
II I
-108-
taken over a period of approximately 14 seconds and begun approximately
47 seconds after the finish of the data of Figure 5.1. Chi-squared was
calculated to be X2 = 14.54 with 12 degrees of freedom corresponding0
to a level of significance a between .25 and .30, still an excellent fit.
For R/rb = 1.2, m/rb = 0.50, equation (4.F.29) gives SNRSAT = 5 dB while
(5.B.2) gives $ITSAT = 5.9 dB. In Figure 5.3, R/rb = 1.3 and m/rb = .56.
The histogram is of 200 data points taken over 14 seconds beginning
20 seconds after the finish of the data of Figure 5.2. Chi-squared was
calculated to be X = 19.17 with 13 degrees of freedom corresponding to
a level of significance a between .10 and .20 which is still quite
acceptable. For R/rb = 1.3 and m/rb = .56, SNRSAT = 4.5 dB, while
SNRSAT = 4.7 dB.
For the above three examples m/rb changed from 0 to .50,
47 seconds later, and then to .56 another 20 seconds later. This gross
aiming error change of approximately one half a diffraction limited
FOV could be due to many causes. Movement of the people inside the
mobile radar vehicle or a change in the wind velocity against the side
of this vehicle could easily be responsible. This difficulty in
dealing with drifts and non-stationary effects is typical of highly
quantitative atmospheric propagation experiments.
The radar return corresponding to the pixel immediately to
the side (right or left, depending on which side of the retro the beam
center is) of the pixel in which the retro return is strongest should
also be dominated by the retro return. If the beam wander model
-109-
(4.F.21), (4.F.22) correctly describes the situation then the target
return corresponding to this "side" pixel should be distributed
according to (4.F.28) with the same R/rb as the neighboring "hot"
pixel. Further, since there are two diffraction limited F0V's (e
points) between beam centers (which are separated by 100 yirads), m/rb
from the "hot" pixel plus m/rb from the "side" pixel should sum to 2.
In Figure 5.4 is shown a normalized histogram of 200 consecutive
points from such a "side" pixel along with the PDF (4.F.28) with
R/rb = 1.2 and m/rb = 1.43. The 200 data points of this figure
correspond to the "side" pixel of the 200 data points of Figure 5.2.
The value of R/rb which best fits the data is 1.2 in both cases and
the sum of the m/rb' s corresponding to these two pixels is 1.93.
Chi-squared was calculated, for Figure 5.4, to be X2 = 21.14 with0
12 degrees of freedom corresponding to a level of significance a
between .025 and .05, indicating fair agreement between theory and
data. For R/rb = 1.2 and m/rb = 1.43, SNRSAT = 0 dB, while
SAT = -1.4 dB. Further examples of this type were difficult to
obtain in our data set. Presumably this is because the beam is not
well described by a Gaussian form beyond the e-2 points.
Figure 5.5 shows the final example of this section. This time,
the data was taken while the radar was operating in full scanning mode
with the second frame rate/imaging option so that the beam wander
model (4.F.11), (4.F.13) applies. The figure shows the. theoretical
PDF (4.F.15) with R/rb = 1.4 and m/rb = 0.6 and a normalized histogram
a.w0
1.s@
1.00
1.00
I ~I I I1.50
Z
Figure 5.4: Normalized histogram of 200 consecutive retro returns from the "side" pixel of
Figure 5.2 taken in full scanning mode and theoretical PDF (4.F.28), R/rb = 1.2,
m/rb = 1.43.
T
I I I
Pz(Z)
0.50
0.S 'Ii I6.60
Ii
2.50I 1 11 - - --
1w
~t - I I a I I I I I I I I I I I I I I I I I I
11111
CDI
111111 III I I I lit I I III II I I I
.U£ L
S.69
I i Iliad A I I I i I I III I I. . A . . . . . . . .. I . . - .0.75 1.S0 I.25 1.6s
III
-J
1.75
Z
Figure 5.5: Normalized histogram of 400 consecutive retro returns taken in full scanning mode and
theoretical PDF (4.F.15), R/rb = 1.4, m/rb = 0.6.
*.75
*.s*9
Pz )
0.2S
0.25
-112-
of 400 consecutive target returns taken over a 5 minute period. The
calculated value of chi-squared was X = 26.08 with 8 degrees of
freedom indicating a level of significance a of approximately 0.001.
This small value of a and lack of agreement between the theoretical
PDF and histogram shapes tend to discredit the model (4.F.ll), (4.F.13).
But before any conclusions are drawn it should be noted that the data
in this example was taken over a period of 5 minutes as opposed to a
maximum of 14 seconds in the case of the first four examples of this
section. This data collection interval difference was due to the factor
of 12 difference in frame rate between the two frame rate/imaging full
scanning mode options. There is then a much larger opportunity for
non-stationary and unidentified effects to cause radar return
fluctuations in the final example than in the first four. To test
the model (4.F.11), (4.F.13) it seems necessary to acquire data in a
much shorter period than has been done.
What these figures and discussion indicate is that even in the
absence of target and atmospheric fluctuations, random aiming errors
can cause fluctuations to be impressed upon the target return. These
fluctuations are severe when the aiming error standard deviation is
comparable to a diffraction limited field of view R/rb = 1. In
Figures 5.1-5.4, R/rb was consistently equal to 1.2 or 1.3. As this
data was taken over a period of 90 seconds, in scanning mode, the aiming
error of the IRAR system is a fact and quite severe.
The above remarks apply to unresolved targets. For
resolved targets the implications of beam wander are not as severe.
-113-
Namely, the beam wander provides a mechanism for spatially averaging
several diffraction limited spots in a single pixel. As the target is
already resolved this is not too limiting and, as will be seen in the
next section, the wander increases the appropriateness of the speckle
target statistical model.
D. Turbulence
The data processing examples given in this section can be
broken into two groups. The first group involves radar returns from
the retro and sphere (glint targets) which are expected to be
distributed according to (4.D.4), while the second group involves radar
returns from the flame sprayed plate (speckle target) and should follow
(4.D.9).
All data in the first (glint target) group was collected while
the radar was operating in reduced scanning mode as this data could be
processed to eliminate the effects of beam wander, whether due to
atmospheric beam steering or radar aimpoint jitter. This was
accomplished by selecting only the single, maximum intensity value,
data point from each 60 x 128 pixel frame. This maximum point, termed
the "hot spot," should then be lognormally distributed from frame to
frame. In Figure 5.6, a normalized histogram of 100 consecutive. retro
reflector returns selected according to the above procedure is shown
along with the lognormal distribution
3.,1
a..
Pz )
1.0
0.0SL I I I I I U I I I I U I U I I.o 0.4 0.3 1.3 1.6 3.0 1.4 3.3 2.3 3.1 4.0
-J
4.4
Z
Figure 5.6: Normalized histogram of 100 "hot spot" retro returns taken in reduced scanning mode
and the lognormal PDF (5.D.1), g 2 = .0045.x
Iw
-115-
p (Z) = 1 exp- (ln Z + 2u 2) U(Z) (5.D.l)z /2 2a Z 18G2 X
X L X
with g 2 = 0.0045. Simultaneous scintillation measurements indicateX
that a2 = 0.0027, but, the value of 0.0045 was chosen to minimize theX
calculated value of chi-squared. This calculated value was X2 = 12.70
with 7 degrees of freedom indicating a level of significance a between
0.05 and 0.10 and fair agreement between theory and data.
In Figures 5.7 and 5.8, the time evolution of the location of
the hot spot within the 100 - 60 x 128 pixel images used for the
example of Figure 5.6 is shown. Specifically, Figure 5.7 shows the
row location with separation between rows corresponding to an angle
of 17 Vrads, while Figure 5.8 shows the column location with
separation between columns corresponding to an angle of 7.8 Prads.
These figures indicate that the RMS angle error of the radar return is
on the order of 15 prads. The independent scintillation measurements
indicate that the RMS angle error due solely to atmospheric beam wander
effects is X/p ~ 25 prads. Since these two numbers are very nearly
equal, and any aimpoint jitter would cause the RMS angle error of the
radar return to increase, we can conclude that the beam wander and
non-constant behavior of the hot spot location is due to atmospheric
beam steering effects. We can also conclude that while in reduced
scanning mode the frame-to-frame RMS aiming error is less than
LU
3117 Brad
29
tThtYt-t-t-t-99999+999@~@-i--I-f** I40@+-4 O4* 14@4-14 [-14-1-14
0B 27
25
I FFVrFrrTIYrrmTYY1-l-Y-tIY1-t~tw-.- -t+9. l-. H- 1 4-01 1 -I-44 . &4-&-464 14-44--1 1-1 1 1 &
0 20 40 60 80 100
FRAME NUMBER
Figure 5.7: Time evolution; Row location of hot spot in the 100 - 60 x 128 pixel frames used in
the example of Figure 5.6.
( I I I I I I I I I I I I f I I I I I I I I I I I I i I I I I I I I I I I I I I I I r- . I - I . . I I . . I . , . . ,. . . . . . . . ..
-- --- --
I W I F-r-T--r-T
.W 1W
i ; i
I
I
I111MkIT1MU1,
Mw w 1w
I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I T
84
7.8 prad82
80
C,
LYITI----HT±.---- ---- ----
TF~TTfl~l7MftTitu1 Ufi llLLTIUJJ tl1LtIJ~l~I1~itIU73
75C
11111120 40 80 10060
FRAME NUMBER
Figure 5.8: Time evolution; Column location of hot spot in the 100 - 60 x 128 pixel frames used in
the example of Figure 5.6.
-4
f I I I . i i i i 1 1 1 1 1 1 1 1 i
-" - . . . . 1 1 1 1 1 1 1 1 1 1 1 L--L
. . . . .. . . . . . . . I . I . . I . I . . I . ; i +-V4-t-4 -+-4-4-4-4-4M4-4 I 1 4
- - -- - - - - If I I I I I I I I I I I I I I I I I I I I I I1-
I
-118-
25 yprads. This is in contrast to full scanning mode in which we saw
this RMS aiming error was approximately R/v'3 rb = .7 diffraction limited
F0V's or 35 yirads.
In Figure 5.9, a normalized histogram of 300 consecutive
"hot spot," retro reflector returns is shown along with the PDF (5.D.1)
with a2 = 0.0138 to minimize X2. Simultaneous scintillation measurementsX
indicate that a2 = .0189. Chi-squared was calculated to be X2 = 12.78X 0
with 9 degrees of freedom for a level of significance a between 0.10
and 0.20 indicating very good agreement between theory and data.
Figure 5.10 shows a normalized histogram of 400 consecutive
"hot spot" retro returns along with the PDF (5.D.1) where a'2 = 0.018X
to minimize X = 12.85 with 15 degrees of freedom. The level of
significance a for this example is between 0.50 and 0.70 indicating
excellent agreement between the data and theory. The scintillation
measurements give a2= 0.055, a factor of three larger than the X2X
minimizing value, though.
In Figures 5.11 and 5.12, histograms of 300 consecutive "hot
spot" polished sphere returns are shown along with the PDF (5.D.1).
In Figure 5.11, a2 = .0083 minimizes X = 10.19 with 12 degrees of
freedom indicating a level of significance a between 0.50 and 0.70 and
excellent agreement between theory and data. The scintillation
measurements give a2 = .0245 for this figure. In Figure 5.12,X
a2 = .004 minimizes X2 = 11.93 with 11 degrees of freedom indicatingXa level of significance a between .30 and .50 and, again, excellent
w w
*.0 0.4 0.3 l.a 1.6 3.0 3.4 8.S 3.1 3.6 4.4
Z
Figure 5.9: Normalized histogram of 300 "hot spot" retro returns taken in reduced scanning mode
and the lognormal PDF (5.D.1), a2 = 0.0138.x
3.S@
1.50
i.4s
Mo,
pZ(Z)
SIII iI i 11 1 1 1111111113i -1
.mE.'-
4.4
qw
77(
I I I I- - I * ~ I I U 1 I .d. I & 4
1.016
iii j~~l
1. 5 11.4
Z
Figure 5.10: Normalized histogram of 400 "hot spots" retro returns taken in reduced scanning mode
and the lognormal PDF (5.D.1), a2 = 0.018.x
-1I
1.50
1.86
P (Z)
0.60
I I III
6.8 O.6 0
M)
I
MW RW
pz (Z)
0. -IL I I I I
tse
0.0SAS5 1.00 1.50 3*
pzz
Z
Figure 5.11: Normalized histogram of 300 "hot spot" polished sphere returns taken in reduced
scanning mode and the lognormal PDF (5.D.1), a = .0083.
w Iw
3.,
2.9
p (Z)Z
1. .
S.. 10..
gI II li ii i i i li i i i I Il
I I I I I I . i I L I-i i ~ ~
0.S0 I.49 '.se a."
Z
Figure 5.12: Normalized histogram of 300 "hot spot" polished sphere returns taken in reduced
scanning mode and the lognormal PDF (5.D.1), cV2 = 0.004.x
. . . . .. .I _ . .I . . . . .1 1 1
-123-
agreement between theory and data. The scintillation measurements give
2 .0026 for this figure.X
The single glint SNRSAT curve (4.D.6) is shown in Figure 5.13
along with the estimates fN SAT calculated from the data of Figures
5.6, 5.9-5.12. Each of the five values of SONSAT is plotted twice;
once vs. measured a2 values (from the scintillation measurements),X
indicated by squares, and second vs. the value of a2 that minimizesX
X", indicated by circles. Here we see very good agreement between the
predictions of (4.D.6) and the data, at least within the limited range
of available a2 values.X
The data in the examples of the second (speckle target) group
was collected in all three modes of operation. In Figure 5.14, a
normalized histogram of 2000 consecutive speckle plate, squared,
intensity returns taken by the radar operating in full scanning mode
with the first (fan beam) frame rate/imaging option is plotted along
with an exponential PDF. The data in this example (and only this
example) is squared. As simultaneous scintillation measurements give
a2 = .02, Equations (4.D.11) and (4.D.9) indicate that this squared
data should fit an exponential distribution. The calculated value of
X = 17.84 with 11 degrees of freedom indicating a is between 0.05
and 0.10 and reasonably good agreement between theory and data.
The data of Figure 5.15, which shows a normalized histogram of
400 speckle plate returns and a unit mean Rayleigh PDF
-124-
10
2a
100
- X
102= o -points plotted against minimizin
2
1O6 2 ol pints lotted aga ins minimizig
10' 8"10 1
22
xX
100
10 4 1I3 f
Figure 5.13: The theoretical curve (4.D.6) and estimates 'SNASAT fromthe data of Figures 5.6, 5.9-5.12.
I I I I I I 1 1 1 1 1 1 1 1 I0 . 0040
0 . 0030
0.0020
Pz (Z)
0.001 0
0 0- --- I I .L iJ I I I I
0. 500. 1000. 1500. 2000.
Z
Figure 5.14: Normalized histogram of 2000 squared, consecutive speckle plate returns taken
in full scanning mode and exponential PDF.
N,
w Aw w
0.75
0.50
pz )
0.25
0.as
0 .0, 0.4 0.8 1.8 1.6 8.0 P.4 2.8 3.2 3.6 4.0 4.4
Z
Figure 5.15: Normalized histogram of 400 consecutive speckle plate returns taken in full
scanning mode and Rayleigh PDF.
K 1 7 =I11 , 11 1 1 1 1 1 1 1 1 1
1 1 I I I I I I 1I 11 I9 1 1 1-
a.)
-127-
Pz(Z) = Z exp[-TrZ 2 /4] u(Z) (5.D.2)
was also collected in full scanning mode, as in the previous example,
but with the second (circular beam) frame rate/imaging option.
Scintillation measurements give a2 = .004 so that by (4.D.ll), (4.D.9)
the data should be distributed according to (5.D.2). The calculated
value of x2 = 14.92 with 16 degrees of freedom indicating a is between0
0.50 and 0.70 and excellent agreement between theory and data.
In Figure 5.16 a normalized histogram of 1200 data points is
shown along with the Rayleigh PDF (5.D.2). The 1200 data points were
taken from 48 consecutive 60 x 128 pixel reduced scan images. The
25 data points from each frame were arranged in a 5 x 5 matrix across
the target, each separated in angle by approximately 200 lirads from its
nearest neighbors. Scintillation measurements give a2 = 0.004 so that,
again, the data should be Rayleigh distributed. The calculated value
of X2 = 23.44 with 19 degrees of freedom indicating at is between 0.200
and 0.25 and very good agreement between theory and data.
In Figure 5.17, we have a normalized histogram of 1400 speckle
plate returns taken in staring mode along with the Rayleigh PDF
(5.D.2). Scintillation measurements give a2 = 0.004 at the time
these measurements were made so that the data, according to (4.D.9),
(4.D.11), should be Rayleigh distributed. But contrary to this the
data has a significantly narrower distribution than the theoretical
prediction. Further, the calculated value of chi-squared is X = 2850
w
@.76
0.50
Pz )
.as
0.00.0 0.4 0.3 1.a 1.6 2.6 3.4 3.3 3.8 3.6 4.0 4.4
Z
Figure 5.16: Normalized histogram of 1200 speckle plate returns taken in reduced scanning mode
and Rayleigh PDF.
I I I U I A I I I iI I I
N)0,
low Vw
w w w
N
i-i III I LI
T
11111111* 0.4 0.6 1.3 1.6 a.$ 3.4 i.E 3.8 3.6 4.0 4.4
Z
Figure 5.17: Normalized histogram of 1400 consecutive speckle plate returns taken in staring
mode and Rayleigh PDF.
1.00
0.5
pz(Z)
0.26
0.
IIIj
- I I I I I I I I I I I I I I I I A I I I I
_ _ _ _ _ _
-130-
with 18 degrees of freedom indicating extremely poor correspondence
between theory and data. To understand this disparity between theory
and experiment we must consider the nature of a speckle target. If we
were to view a "rough" target through free space in starina mode and
both the target and radar were held perfectly fixed in space, i.e., all
movement and vibrations had been eliminated, then it is clear that all
target returns would be the same. That is, under the above conditions,
the target return probability distribution would be an impulse. It is
only when, due to atmospheric beam steering effects, radar aiming
errors, and target movement, we begin to sample several diffraction
limited spots on the target that we begin to see the Rayleigh
statistics. In Figure 5.17, we are at an intermediate stage between
looking at a single diffraction limited spot, resulting in an impulsive
data distribution, and sampling many spots, resulting in Rayleigh or
exponential statistics as in Figures 5.14-5.16. So that we conclude
that the speckle nature of a target depends on more factors than just
the roughness of the target surface and is closely related to how many
diffraction limited spots on the target are sampled by the radar.
The final example of this section is summarized in Figures
5.18 and 5.19. The same normalized histogram of 300 speckle returns
taken in full scanning mode with the second frame rate/image (circular
beam) option is shown along with a Rayleigh PDF in 5.18 and a Rayleigh
times lognormal PDF (see Equation (4,D.9) and [27]) in 5.19. The value
of X2 = 13.51 with 20 degrees of freedom in Figure 5.18 indicating a0
1.80
0.75
0 * 50
Pz )
OS 0 0.4.- , -- I I I - I-A L dL .
Z
Figure 5.18: Normalized histogram of 300 speckle plate returns taken in full scanning mode
and Rayleigh PDF.
CA)
MW
1.8 8,.$ 3~ -. 8~ ~ 3.9 4.0 414
qw MW
0.76
*aEO
p (Z)Z
0.26
Ss.60.0 0.4 0.1 1.a 1.8 3.0 3.4 eA. 3.8 .8 4.0 4.4 4A1
Z
Figure 5.19: Normalized histogram of the same 300 speckle target returns as Figure 5.18
and a Rayleigh times lognormal PDF, a2 = 0.01.
- -'
kil -L . .. I..
N)
-133-
is between 0.80 and 0.90 and excellent agreement between the PDF and
the data. The calculated value of X2 is minimized by choosing 92= 0.010
to x' = 13.20 with 19 degrees of freedom in Figure 5.19 indicating a0
is again between 0.80 and 0.90 and excellent agreement between the
PDF and the data. For both figures, scintillation measurement gives
a2 = 0.15. At this level of turbulent fluctuations, a marked departure
from Rayleigh statistics is expected. But contrary to this, the data
fits a Rayleigh distribution very well and when fit to the Rayleigh
times lognormal PDF the X minimizing value of a2 is more than an order
of magnitude away from the measured value. Exactly why this is the
case is not as yet clear. But the scintillation measurements indicate
that the atmospheric coherence distance p is smaller than the radar
beam size. Hence, the model (4.D.9) does not strictly apply. It is
suspected that this fact plays a significant role in explaining the
divergence between theory and experiment here.
In this chapter, we have given many examples of our theory
verification efforts. The model for aimina error, at least in the case
of a fan illuminating beam, seems to be correct. Also, the lognormal
character of the atmospherically induced fluctuations, at least from
a glint target, appears to have been verified. What is less certain
is the Rayleigh times lognormal character of the speckle plate return
in clear weather. Indeed, the essentially free space result of
speckle plate Rayleigh statistics was verified. But in heavier
turbulence it was difficult to find examples which clearly followed
-134-
(4.D.9). Exactly why this was the case is not clear. More effort
should be directed towards understanding this.
-135-
CHAPTER VI
TARGET DETECTION
Here we discuss multipulse detection of targets at known
range by the radar. That is, we address the problem of optimally
deciding between the hypothesis that there is a target present at
range L and the hypothesis that there is no target present. This
decision is based upon the returns from M transmitted laser pulses.
The detection problem just posed is most relevant to objects viewed
from a ground-based radar against a nonreflecting background
(i.e. the sky) as background clutter is ignored. In the first
section of this chapter we formulate the problem mathematically.
Following this, expressions for single pulse performance are given.
Multipulse integration is discussed in the next section. Finally,
linear integration multipulse performance is presented.
A. Problem Formulation
Consider transmitting M laser pulses from the radar and
observing the IF returns ri(t), i = 1,2,...,M. These pulses can
occur sequentially as in Figure 1.2, concurrently but mutually
separated in angle by at least a diffraction limited FOV, or both.
If no target is present the IF signals are pure noise while if a
target is present the IF signals are return plus noise. Mathematically,
-136-
if we call target absent hypothesis H0 and target present H1 the IF
complex envelopes corresponding to the M transmitted pulses satisfy
n (t) H 0
r.(t) = , (6.A.1)
-1 { :?:ng(t) 1
i = 1,2,... ,M
where r.(t) is observed from 2L/c seconds to 2L/c + t seconds after
transmission of pulse i. The xi are given by
'rtpP '2 r .i
x = hv) T df' T(p') ERip' E Fip' = 2ij e
(6.A.2)
where the 8. can be taken to be mutually independent, uniformly
distributed random variables over [0,27r]. The n (t) are mutually
independent, circulo-complex, white, zero mean Gaussian processes with
<n (t) nt(s)> = tp 6 6(t - s) (6.A.3)
As a scanning radar with frame rates slower than 20 Hz is assumed, the
IF returns r.(t), i = 1,2,...,M can be taken to be independent under
either hypothesis. The optimal Neyman-Pearson likelihood ratio test
(LRT) [23] for target detection is then
-137-
H
L r1 (6.A.4)
H0
The likelihood ratio L is given by
M CX2L = II dX p e- I {2X (6.A.5)
where p, iI(X) is the probability density function for the magnitude
of the signal return x and
Ir.i = (1/tp) {r(t) dtj (6.A.6)
is the output of a matched filter envelope detector. The integral
(6.A.6) is taken over the t second interval beginning 2L/c seconds
after transmission of the ith pulse. In (6.A.5) I {-} is the
zeroth-order modified Bessel function.
As no assumption has been made regarding the statistics of
the xi this formulation applies to low visibility as well as turbulence.
In particular, if we consider the case of a single transmitted pulse,
M = 1, the LRT (6.A.4) reduces to the threshold test
H
2 < y (6.A.7)H0
-138-
The optimality of the above test applies regardless of the target
scenario and atmospheric conditions. The performance of the test, as
we shall see in the next section, unfortunately is not similarly
independent of these details.
B. Single Pulse Performance
Here we discuss the performance of the Neyman-Pearson test
(6.A.7). This evaluation involves the determination of the
probability of detection PD and the false alarm probability PF
PD = Prob(1r_1 2 > yjH 1 ) (6.B.1)
PF = Prob( r12 > yjH0 ) (6.B.2)
where the subscript on r has been dropped.
The false alarm probability is independent of the
atmospheric/target scenario and depends only on the LRT threshold.
P = exp(-y) (6.B.3)
so that (6.A.7) becomes
H
Ir 2 - ln P (6.B.4)
H0
-139-
In contrast, the detection probability depends on the atmospheric/target
scenario as well as threshold. We present our results as a series of
examples.
Case 1. jx2 Nonrandom
If rx! 2 = CNR is known, as in the case of a glint target in
free space, the detection probability is given by [3,23]
dz z exp(-(z 2 + 2!xI)_|2)I 0 (z LxI)D =12(-2lnP F) (6.B.5)
= Q(/2-- W, (-2 ln PF
Equation (6.B.5) is Marcum's Q-function.
Case 2. Single-Glint Target in Turbulence
For a single-glint target T(p') = eje T ('-_g(4.D.l)-(4.D.3) the detection probability is
PDdx p (x) Q[(2 CNRxA gue
) satisfying
[3,5]
X) e2X,(-21nPF) ]
where
(6.B.6)4A 2
-140-
p(x) = exp[-(x + a2)2/2&21 (6.B.7)X X X
V2Tra 2
X
is the pdf for the log-amplitude perturbation X.
Case 3. Ix1 2 Exponential
Here we consider the case
IX2 = CNR v (6.B.8)
where v is a unit mean exponential random variable. Equation (6.B.8)
applies to the three scenarios:
1. Free space, speckle target.
2. Low visibility, large speckle target.
3. Low visibility, resolved glint target.
The detection probability is [23]
p = p(CNR + 1)l (6.B.9)D F
Case 4. Resolved Speckle Target in Turbulence
For a target whose radar return is described by (4.D.9) the
detection probability is [3,5]
-141-
dU pu(U) P (CNRsre2U +l)_lu F
Pu (U) = exp[-(U + u2 )2 /2a2 )]
v/2Trcf2
is the pdf of the aperture-averaged log amplitude perturbation u.
Case 5. Ix12 A Product of Two Exponential Random Variables
The case of a small glint target in low visibility is
considered here
Ixi 2 = CNR v w (6.B.12)
where v, w are IID unit mean exponential random variables. The
probability density for the product y = vw is given by
py (Y) = 2K (2vY) u(Y)
where K (-) is the modified Bessel function of the second kind
order zero, and u(-) is the unit step function. A sketch of
(6.B.13) is shown in Figure 6.1. The detection probability in
case is given by the integral
(6.B.13)
of
this
PD
where
(6.B.10)
(6.B.11)
-142-
2.0
1.6
... 1.2
0.8
0.4
0.0
0 1 2 3 4
Figure 6.1: The Probability Density Function, p (Y) = 2 K (2Y)yi 0
-143-
( f in PP = dt exp -t + CNR t1 (6.B.13)
We are most interested in case 5 here as it represents new
work. Also, it is worthwhile, in a practical sense, to contrast this
case with case 2 as both (6.B.6) and (6.B.13) are detection
probabilities for small, single-glint targets in two limiting
atmospheres. In Figure 6.2 the detection probability PD is plotted
vs. CNR for Ix1 2 a product of two exponentials and various false
alarm probabilities. In Figure 6.3 we have the receiver operating
characteristics, PD vs. PF as the threshold y is varied, for several
values of CNR. From these figures we see that PD is a monotonically
increasing function of both CNR and PF. We also observe, from
Figure 6.2, that if just two or three dB of CNR can be obtained over,
say, the PF = 10~ case, the false alarm probability can be improved
by four orders of magnitude to PF = 1011 while maintaining constant
detection probability PD. The bumps found in the curves of Figure
6.2, at high CNR, are thought to be due to the numerical integration
of (6.B.13) and are not real oscillations. In Figure 6.4, PD is
plotted vs. CNR for |X 2 a product of two exponentials and a single
glint target in turbulence (Equation (6.B.6)) for PF = 10 7. Each
curve for a single glint target in turbulence corresponds to a
--141-
99.99
99.9 SINGLE PULSE IXI2 =IX2>VW
99
95
90
2 70
P =1050 F
CL
30-
10
5 -pF =10-4-
P F P107
S1-11
0.1 F
0.01-20 0 20 40 60
CNR (dB)
Figure 6.2: Single pulse detection probability vs. CNR
for a glint target in bad weather.
-145--
~_ SINGLE PULSE
1010
I I
X|2 2> VW
99.99
99.9
99
95
90
-6 106
Figure 6.3: Single pulse receiver operating characteristics for a glint
target in bad weather.
CNR 40 dB
_CNR=20 dB
CNR = OdB
70
50
30
C
C.,
a,0.
0C-
10
5
0.1
0.01-4 -2
I I
PF
.l46-
99.99 1 1 1 1 / I 1 1
99.9 SINGLE PULSEP 0-7
F PF
99 2=
95- .2 = 0.01
x/x90-
2 = 0.1 2 =0.5
70-
2 2CL 50 - , |_ =< X( >VW
a. 30-
10
5-
1I
0.1 /0.01/
-20 0 20 40 60
CNR (dB)
Figure 6.4: Single pulse detection probability vs. CNR for a single
glint target in free-space, three levels of turbulent
fluctuations and bad weather. PF = 10~7 throughout.
-147-
different value of a2 with a2 = 0 indicating free space. We observeX X
improved performance of the jxj 2 random curves over the jx1 2 nonrandom
curve (a2 = 0) in the region CNR 5 9 dB. This can be understood ifX
one considers the likelihood of a fluctuation bringing the matched
filter envelope detector output above threshold in this low CNR
regime. It is also evident from this figure that significant CNR
increases will be needed to maintain high PD values on a glint target
in the presence of strong turbulence or scattering. This figure
indicates that for equal CNR a glint target in the turbid atmosphere
is easier to detect than the same target for saturated scintillation
C2 = .5. The problem is that in turbulence the CNR is essentially the
free space CNR whereas, from (4.E.15), the CNR in bad weather can be
significantly reduced from the free space value. More will be said
about this in the next chapter. Again, as in Figure 6.2, the bumps
in the Ix1 2 = <tx1 2 > vw case are thought to be due to the numerical
techniques used to integrate (6.B.13).
C. Multipulse Integration
As should be evident from Figures 6.2-6.4 adequate detection
performance cannot be maintained at lower CNR for a single radar pulse.
In order to improve upon this situation we find it necessary to use
several pulses in combination. In this section, we address the
problem of approximating the multipulse decision rule (6.A.4) when
the signal return jx! 2 is a product of two exponential random variables.
-148-
For CNR = CNR, i = 1,2,... ,M this is not too difficult. Consider
the likelihood ratio (6.A.5)
ML = i R (6.C.1)
i=1
where
R. = dX p (X) e-X IX{2X } (6.C.2)
For jx.1 2 = CNR v i wi, (6.C.2) becomes
R. = dX 2K 2 I{2X Ir 1} (6.C.3)S10 N jdX/CKo -NR j e o -I
This integral has been evaluated numerically using cautious adaptive
Romberg extrapolation [29]. The results are shown in Figure 6.5
where R is plotted vs. Ir[ for several values of CNR along with the
curve
log(R) = -3.10 + 0.344 Ir12 (6.C.4)
It is evident from comparing the plot of (6.C.4) and the CNR = 40 dB
plot of (6.C.3) that (6.C.3) is well approximated by a parabola
-149-,
116765-M
0 1 2 3 4 5 6
1'r
Figure 6.5: Likelihood ratio R (Eq. (6.C.3)) and parabola
log R = 3.1 + 0.3441r|2 vs. matched filter
envelope detector output Irl.
10 8
106
105
10 4
R 103
10
10
10-2
10~
-3
-4101
101I
10
CNR = OdB
CNR 20 dB
log R = 3.1
+ 0.344 Irl2CNR zOdB
CNR 20 dB -
- CNR= 40dB -
log R 3.1 + 0.344 r|2
-150-
log(R1 ) = A(CNR) + B(CNR) Ir 12
R. = eA(CNR) eB(CNR) r*iJ2
where we have explicitely
Using (6.C.6) in place of
gives, for equal CNR, the
Mi
noted that A and B are functions of CNR.
(6.C.3) in the likelihood ratio (6.C.1)
Neyman Pearson test
H
Ir 2 Y (6.C.7)
HO
This test is then very nearly optimal. The performance of (6.C.7) is
investigated in the next section.
D. Multipulse Performance
Here we consider the PD F behavior of the threshold test
Z= r
H
H0
Y
for several target/atmospheric scenarios. The use of (6.D.1) in
case of a small glint target in bad weather was justified in the
last section. If lxf 2 is an exponential random variable (6.D.1)
(6.D.1)
the
can
or
(6.C.5)
(6.C.6)
-151-
be shown to be optimal for CNR = CNR, i = 1,2,...,M. At any rate,
use of (6.D.1) can be considered to be arbitrary if it is not optimal.
The false alarm probability PF of (6.D.1) is independent of
target/atmospheric scenario and depends only on number of pulses M and
threshold y. An exact expression is
p r(MMY) (6.D.2)F (M- 1)!''
where
F(a,x) = a- e-u du (6.D.3)
is the incomplete Gamma function [30]. Evidently exact performance
results are difficult to obtain for the test (6.D.1), and, once
obtained are generally cumbersome. Hence it is worthwhile developing
accurate approximate results. Towards this end, we use a modified
Chernoff bound procedure [31,32] to derive these results. This technique
is useful for estimating the area underneath the tails of a probability
distribution. Hence the results that follow only apply in the regime
PF 0.40, PD > 0.60.
Defining two conditional semi-invariant moment generating
functions
10(s) = M ln<exp(slr|j2)H> 0s > 0 (6. D.4)
-152-
'Pl(s) = M ln<exp(slr12) IH1 >s < 0 (6.D.5)
where s is a real variable, approximate expressions for the false
alarm probability PF and miss probability PM = 1 - PD are
PF = x oi(s ) - s (s ) + s20 (S )/2] Q[/vlgso) so]F 0 0 0 0 0 00 0
P = exp[il(sl) - sl S + l (Sl )S 21 QE-vil(s, ) sl
(6.D.6)
(6.D.7)
for M 5. In (6.D.6), (6.D.7) dot denotes differentiation with
respect to s, Q(-) is the complemented error function
.co
Q(y) = exp(-x2 /2)
/- r
dx (6.D.8)
and sl' s1 are solutions to the equations
My = O(s)
my = l (S)
(6.D.9)
(6.D.10)
respectively. The approximation used to derive (6.D.6), (6.D.7) was
an application of the central limit theorem [15]. Specifically,
-153-
Mthe sum j tn 2 is taken to be a Gaussian random variable even
i=lthough the Ir.! i = 1,2,...,M are not Gaussian. Hence, the requirement
M 5.
For the PF calculation H is true so that r2= tn.!22 , a
unit mean exponential random variable, hence
y1(s) = -M ln(l - s) s < 1 (6.D.ll)
Equation (6.D.9) is easily solved to give the threshold
1 (6.D.12)1s 0
with approximate false alarm probability
M M(3s /2 - s sP (1 - s )~ exp Q 0 (6.D.13)
F o (L - s0) L ~"
Equation (6.D.13) can be inverted numerically to give y as a function
of M and PF* In Figure 6.6, we show M vs. y for PF = 1012. Clearly
a modest amount of pulse integration leads to a drastic reduction in
threshold.
The function v1l(s), and hence the detection probability PP'
depends on the target/atmospheric scenario. We present our results
as a series of examples.
-154-
100
90-
80-
70-
10 -12PF
60-
M 50-
40-
30-
20-
10-
0 5 10 15 20 25 30 35Y
116599-N
Figure 6.6: Threshold y vs. number of pulses necessary to
maintain PF 0-12
-155-
Case 1. jxj 2 Nonrandom
If 1x2 = CNR is known the moment generating function is
given by [31]
y (s) = M CNR T - ln(-s) s < 1 (6.D.14)
This result is useful in the jxf 2 random cases as we can average
exp(vil(s)/M) with respect to the statistics of Ix12 to find p1 (s).
Case 2. Single-Glint Target in Turbulence
For a single glint target T(p') = e a T (P') satisfying
(4.D.1) - (4.D.3) we find that [31]
pl (s) = M ln s Fr{- s CNR, 0; 2r s < 0 (6.D.15)
where Fr(a,0;c) is the lognormal density frustration function [1,33,34].
Saddle-point integration techniques for Fr(a,0;c) that are in the
literature [27,33] permit rapid accurate numerical evaluation of -pl(s)
and, in turn, of PD.
Case 3. |x12 Exponential
Here we consider the case of Jxj 2 = CNR v where v is a unit
mean exponential random variable. As before, this characterization
applies to the three scenarios:
-156-
.1. Free space, speckle target.
2. Low visibility, large speckle target.
3. Low visibility, resolved glint target.
For this case, we have [31]
Pi(s) = -M ln[l - s(l + CNR)]
Case 4. Resolved
For a target with
becomes [31]
Speckle Target in Turbulence
radar return described by (4.D.9) yi(s)
ii(s) = M ln { dU p(U)[1 - (1+ CNRsr e2U)s]l} s < 0 (6.0.17)
with
(6.D.18)Pu (u) = exp[-(U + a 2 )2 /2 C 2]
/2 ar2
Case 5. Ix2 A Product of Two Exponential Random Variables
The case of a small glint target in low visibility is considered
here, jx2 = CNR vw, where v, w are IID unit mean exponential random
variables. We have that
(S) = M CNR - ln (-sCNR) + lnSs - 1sCNR
ey- dt s < 0 (6.D.19)
s < CNR + 1(6.D.16)
-157-
As in the single pulse M = 1 case we are most interested in
case 5 above and in contrasting it with case 2 as they both deal with
small, single glint targets. Numerical work on case 2 has appeared
in [35] and will be used for comparison.
In Figure 6.7, PD is plotted vs. CNR for Ix12 a product of
two exponentials, PF = 10~4, and several values of M. In Figure 6.8,
is a similar set of curves for PF = 12. These two figures indicate
the performance improvement obtained through pulse integration.
In Figure 6.9, the number of pulses M necessary to achieve a
performance of PF = 10-12 ,D = 0.99 is plotted vs. CNR for a glint
target in free space, in turbulence for two values of a2 , and in lowX
visibility. This figure indicates that, for equal CNR a glint target
seen through scattering conditions can be detected more easily than
when seen through saturated scintillation a2 = 0.5. Also atmosphericX
pulse-integration performance for large M is very near that for free
space as only 1 or 2 more dB of CNR is necessary to achieve
PF = 10-12 PD = 0.99 when M 20 in the atmosphere except in heavy
turbulence.
In Figure 6.10, we again plot M vs. CNR, for a glint target
in bad weather, PD = 0.99, and two values of false alarm probability
-15-
I I
M =30
-/- M =10-
|X|2 2 VW
- ~ P 1=M1 4
F
10 20 30
CNR (dB)
Figure 6.7: Single-pulse and multipulse detection probability 2-
vs. CNR for a glint target in bad weather. PF= 104
throughout.
99.99
99.9
99 F-
95
90
70
50
30
10
5
CL
C.
4).
01
0.1
U 40
I II I
-159-
M = 30
jVW12
S>V Iw
100
CNR (dB)
Figure 6.8: Single-pulse and multipulse detection probability
vs
PF
CNR for a glint target in bad weather.
1 12 throughout.
99.99
99.9
I
99 K-
95
90
70
4,'a 50
30- PI <
10
10
5
0.1
0.01
M 1
20-20 -10
v
I I
-160-
100
90P = 0.01 P= 0.99
M D
PF = 10-12
70 -
bu GLINT TARGETLOW VISIBILITY|>12 2> VW
M 50
40-
30-
GLINT TARGET 2= 0.0520 -X
GLINT TARGET, FREE SPACE
GLINT TARGET .2 = 0.510- x
0 10 20 30 40 50 60 70CNR(dB) 116596-N
Figure 6.9: The number of pulses M necessary to achieve PF = -12
pD = .99 vs. CNR for a glint target in free-space, two
turbulent fluctuation levels, and bad weather.
-161-
M
F
--- p =
MP F
0.0110-7
GLINT TARGET,
1 lxi2 <x
.
P =0.99D
0.01 P = 0.9910-12
LOW VISIBILITY2
> vw
10 20 30 40 50 60 70CNR (dB)
116595-NFigure 6.10: The number of pulses M necessary to
achieve PD = .99 and two different false
alarm probabilities vs.
in low visbility.
CNR for a glint target
rv-~I~i I
90
I1*1I
II
60K
M 50
40
30
20
10
0
so -
70k-
-162-
PF -10 12. From this figure it is clear that for just 1 or 2
extra dB of CNR a 5 order of magnitude improvement in false alarm
probability can be achieved without sacrificing detection sensitivity.
In Figures 6.11, 6.12, PD is plotted vs. CNR for M = 10, 15
respectively, PF = 10-12, a glint target in several turbulent
atmospheres, and the same target in a turbid atmosphere. Again we
see that for equal CNR the target in the turbid atmosphere is easier
to detect than for saturated scintillation a' = 0.5.X
The implications of the above results cannot be properly
accessed without considering resolution and CNR in bad weather. From
(4.B.11), (4.E.4), etc., we have that in bad weather, both resolution
and CNR are degraded from their corresponding free-space values while [5]
in turbulence these same quantities remain essentially unchanged as
compared to free-space. What this means is that targets which would
be resolved in turbulence (and free-space) may not be in bad weather.
Further, the CNR available to the radar for the same target, viewed
through the two limiting cases of clear and scattering atmospheres,
will be much smaller in the latter case as compared to the former.
Hence, the difficulty in detecting a target in bad weather is expected
to increase as compared to in turbulence. In the next chapter, CNR
and resolution curves are presented for typical systems and various
targets in bad weather. These curves, coupled with the results just
presented, can be used to access target detectability in inclement
weather.
-163-
S=0.01-x.-2 A
-
-x
- I
1XI )V W -I
30 -
PF = 10- 12
M =10
10 20
CNR (dB)
Figure 6.11: Ten pulse detection probability vs.
target in several turbulent atmospheres and a
glintiscattering
atmosphere. PF = 10-12 throughout.
99.99
99.9
99
=0.5
95
90
70
50
Ca,C.)a)0.
0a-
10
5
1
0.1
0.010 30 40
CNR for a
I I1i I
99.99
99.9
99
2'2- X .1
- ,/
-- I
I2II >vw
2x = 0.5X
30-
10 KF
-1012
M = 15
01 I-
Ui 1u
CNR (dB)
Figure 6.12: Fifteen pulse detection probability vs.
a glint target in several turbulent atmospheres
and a scattering atmosphere. PF = 0-12 throughout.
95
90
70
50
C
U
0.
00~
5 F
1
0.0120 30 40
CNR for
I I I
01
L I
-165-
CHAPTER VII
SYSTEM EXAMPLES
The preceding chapters have pointed out three primary
contributions to the optical radar return:
(1) the carrier-to-noise ratio CNR due to MFS propagation,
which as discussed in Chapter II and Appendix A, disregards
the unscattered portion of the light beam;
(2) the extinguished free-space carrier-to-noise ratio
CNR0 e-2tL, which is due to the unscattered light; and
(3) the normalized backscatter power <1x%(t)1 2 > from scatterers
in the propagation path between radar and target which, in
the context of an imaging or detection application is
undesirable.
The main theoretical development of this thesis has been the scattered-
light MFS propagation theory as applied to an optical radar. In this
chapter, we introduce two representative hypothetical radar systems
and examine, for a variety of target/atmospheric scenarios, the
relative strengths of the above mentioned radar-return contributions.
Most significant here are examples which indicate where the MFS return
is dominant, for these establish regions of applicability for our
theoretical work.
-166-
The two hypothetical radar systems we employ in this chapter
are a CO2 laser radar, and a Nd:YAG laser radar. Their essential
parameters are summarized in Tables 7.1 and 7.2, respectively. The CO2
laser radar parameters are chosen to closely match a typical existing
system of that type, such as the mobile, Lincoln Laboratory infrared
radar mentioned previously. The Nd:YAG laser radar parameters are
chosen purposely to give an example of a system which can exploit
scattered light. Indeed, the system as described here is on the edge
of current technology and hence, at this time, may be unrealizable. It
provides, however, a context in which MFS propagation is often the
dominant target return mechanism.
The CNR results of Chapter IV, Section E all assume a uniform
scattering profile between radar and target. The examples in this
chapter will indicate that neither the C02 system nor the Nd:YAG
system will be able to make use of scattered light in this situation
because of the dominance of the extinguished free-space power. It
turns out, however, that there are several interesting situations in
which the MFS power is dominant, at least for the Nd:YAG system. These
situations arise when the propagation path does not possess a uniform
scattering profile between radar and target, but rather the scattering
is concentrated within a layer comrpising a small to modest fraction
of the total path length. Such situations. occur in the context of an
airborne radar searching the ground through a cloud or fog cover, or a
ground-based radar searching the sky, again through a cloud or fog
WAVELENGTH X
PHOTON ENERGY hv0
PEAK POWER PT
PULSE DURATION tp
BEAM DIAMETER 2PT
DETECTOR QUANTUM EFFICIENCY r
1.87 x 10-20
10 kW
100 ns
13 cm
0.9
46
Table 7.1: CO2 Laser Radar System Parameters
-167-
10.6 -pm
WAVELENGTH A
PHOTON ENERGY hv 0
PEAK POWER PT
PULSE DURATION t
BEAM DIAMETER 2PT
DETECTOR QUANTUM EFFICIENCY rn
1.87 x 10 9 1
10 kW
100 ns
0.5 cm
0.9
Table 7.2: Nd:YAG Laser Radar System Parameters
-168-
1.06 pm
-169-
cover.
Two layered profiles are considered in this chapter. The first
consists of a layer of scatterers of thickness Ls sandwiched between
two free-space layers of thickness Li and Lo, where the distance from
the radar to scattering layer is L the distance from the scattering
layer to the target is LO and the total target range is L = Li+ Ls+ L .
This situation is depicted in Figure 7.1. The second layer geometry
consists of scatterers of thickness. L placed near the target with L.
meters of free-space between the radar and the scatterers, so that the
total target range is L = Li + Ls. This second situation is depicted
in Figure 7.2. The formulas of Chapter IV, Section E do not apply
without modification to the layered scattering scenarios. Fortunately,
necessary modifications are not too extensive. Hence, we will indicate
what the appropriate modifications are as the examples are discussed.
Finally, before delving into our calculations we must indicate
values for the atmospheric parameters. The numbers we use are
summarized in Tables 7.3 and 7.4, and are typical [18,37,40]. The
numbers cited for haze are used in all examples in which a uniform
scattering profile is assumed. These examples are intended to predict
how the radar systems would perform in a terrestrial situation. The
numbers cited for a cloud are used in all examples in which a layered
scattering profile is assumed. These examples are intended to predict
radar performance in air to ground or ground to air scenarios.
FREE-SPACE
L
SCATTERINGLAYER
L
L
FREE-SPACE
Figure 7.1: Geometry for a single scattering layer between radar and target.
RADAR TARGET
Is
FREE-SPACE
L.
L
SCATTERING TARGETLAYER
Ls
Figure 7.2: Geometry for a scattering layer near the target.
RADAR
-4
-172-
Extinction Coefficient St (km1 )
Modified Scattering Coefficient ' (km~ )
Backscattering Cross Section Per Unit
Volume (km1 )
RMS Forward Scattering Angle 6F (mrad)
Atmospheric Parameters at CO2 Laser Wavelength
Haze
.005
.0005
.0005
50
Cloud
10
4
.2
50 -
Table 7.3:
-173-
Extinction Coefficient t (km~1 )
Modified Scattering Coefficient
s' (kmi )
Backscattering Cross Section
Per Unit Volume (km 1 )
RMS Forward Scattering Angle
eF (mrad)
Table 7.4: Atmospheric Parameters at Nd:YAG Laser Wavelength
Haze
.07
.05
.01
10
Cloud
20
17
5
1
-174-
A. Examples
Figure 7.3 gives plots of the normalized backscatter power from
a haze vs. delay t, Equation (4.H.12), for both the CO2 and Nd:YAG
systems. Since the integration interval in (4.H.12) is ct p/2 = 15 m,
we can regard <1 xb(t)1 2> as the backscatter return from range L = ct/2
so that, for example, t = 10 psec corresponds to L = 1.5 km and
t = 70 ypsec corresponds to 10.5 km. Hence, Figure 7.3 covers the entire
target range of 1 km < L < 10 km considered. What we see from this
figure is that in a haze <!xg(t)(2 > is always less than 20 dB for the
CO2 system, and below 0 dB for the Nd:YAG system beyond t = 10 ypsec
(L = 1.5 km). We will see later that the target return power for a
resolved speckle target will dominate this backscattered power for both
systems, so that < x-(t)1 2 > can be disregarded for uniform haze
profiles.
Before presenting the above mentioned target return result,
we shall give our second and last backscatter example. Following a
procedure similar to that of Chapter IV, Section H, we can use single
scatter theory [28] to derive the following expression for the maximum
normalized backscatter power from a scattering layer L meters from the1
radar
< 1Sbi) 2> 'nI t 0P T 2 cobP<h, x-(L P22 h2 dz (7. A.1)
o t T -L. c- t 1 + 2 z22 2p 27T 2
70
50
30
- V-__
-10.
t -7
CO 2 System
Nd:YAG System
~1
LL-~--
20 40 60
t(iisec)
Figure 7.3: Normalized backscatter power from a uniform scattering
profile vs. t.
-- fi1jf~z~f~~
A
C%4
V
10
-300
-----------
-
-176-
This maximum normalized backscatter return results when the pulse of
length ct p/2 (meters) has just entered the scattering layer. In this
case, extinction can be neglected as exp(-2 t ct p/2) 1 1 for the pulse
duration considered. Plots of the maximum backscatter are shown in
Figure 7.4 for both radar systems, where the atmospheric parameters
assumed are those of a cloud. As one might expect this backscatter
return is significantly larger than for a haze. However, if the radar
uses a range gate that effectively closes the receiver aperture until
the desired moment, this backscatter power can be combatted. As the
laser pulse propagates through the cloud the backscatter will decrease
because of extinction. A good approximation to the backscatter power
from a distance Ls into the scattering layer is (7.A.1) multiplied by
exp(-2 tLs). If this quantity is smaller than either CNR or
CNR 0 exp(-2t L s) then the backscattered power can be excluded via range
gating. Later calculations will show this to be the case.
The case of a resolved speckle target seen through a haze
(uniform scattering profile) is summarized in Figures 7.5 and 7.6. Figure
7.5 gives the CO2 system extinguished free-space and MFS CNR, where the
target mean square reflection coefficient r= 0.5. Figure 7.6 gives the
same quantities for the Nd:YAG system with the same value for r. In both
cases the extinguished free-space return clearly dominates the MFS return
so that the target return is best described by (4.D.9) and resolution is
given by the free-space result. Also by comparing the extinguished free-
space curves with the backscatter curves of Figure 7.3 we see that
-177-
70
50
30
-77-
~Iit -~
10
-~ ~ -iL::yiiI$:T7!1i7:7f:::.: :7: ::Z7 :-
-~:c:::r:V
0 2
CO2 System
Nd:YAG System
4
Li (km)
Figure 7.4: Maximum normalized backscatter power from a scattering layer
L. meters from the radar.
~K ~
^M
A
C\J
><P 10v
6
:f
-30
-178-
100
80
60
40
20
0
L N
CNR
____- -- ~---- -- T
0 4 8 12
L(km)
Figure 7.5: Extinguished free-space and MFS resolved speckle target CNR
vs. target range for the C02 system and a uniform scattering
profile.
L 4
L-:rr_ _=
'77
0 - 2 tLCNR et
0
-179-,
80
60
40
-o
20
0
-20
-= o -2 tLCi CN R et
~I-v I ~ -~ -~
§1272:.
CNR
4 8 12
L ( km)
Figure 7.6: Extinguished free-space and MFS resolved speckle target CNR
vs. target range for the Nd:YAG system and a uniform
scattering profile.
LA L
77_
7 7 7:: L=
7 1 ----- -
L'=7 z__
-180-
backscatter is insignificant in a haze for a resolved speckle target.
The dominance of the extinguished free-space return can be understood
if one considers that for either radar system, even at target range
L = 10 km, the beam has not propagated through a single optical thickness.
That is, StL < 1.
For a resolved speckle target with a thin layer of scatterers
between it and the radar we might expect, because the optical thickness
is much larger than in the previous example and many more scatterers
are encountered, that the MFS return would dominate. For the Nd:YAG
system, since the single scatter albedo is nearly unity at this wavelength,
we shall see that this is in fact the case. If the speckle target is
as in Figure 7.1 with L. >> Ls, L0 >> Ls, it is a simple matter to show
that the appropriate results of Chapter IV (i.e., (4.B.7)-(4.B.9),
(4.E.7), (4.E.8)) all hold with p0 given by
P = 2 2 0 3 kol(7.A.2)o ' Ls O k2
in place of (2.C.3). With this in mind, the case of a resolved speckle
target of mean square reflection strength. r = 0.5 and a thin scattering
layer between radar and target is summarized in Figures 7.7-7.9.
Figure 7.8 shows the CO2 system extinguished free-space and MFS CNR
for this scenario with L. = 2 km, L0 = 1 km vs. scattering layer
thickness Ls. These same quantities are shown for the same conditions
100
80
60
~Th~-- 4k V7t~z~
CO System2
Nd:YAG System~z~zc~z~Z2V77
0 200 400
SLffi~~F I
600
L s (m)
Figure 7.7: Atmospheric beamwidth for a single layer scattering
profile vs. layer thickness.
P~7iZEZ2Z27
E
S.-
40
20
11
-182-
80
60
40
tR\7
t-Z
0
CNR 0 e-2tLs
20--
CNR
-20 ' --C 200 400 600
L s (m)
Figure 7.8: Extinguished free-space and MFS resolved speckle target
CNR for the CO2 system and a single scattering layer
vs. layer thickness.
- _7
--7::_
200 400 600
Ls (m)
Figure 7.9: Extinguished free-space and MFS resolved speckle
target CNR for the Nd:YAG system and a single
scattering layer vs. layer thickness.
70
50
30
10
-10
-30C
- i __
- * CNR
CNR0 - t s
- 7 --- --- --- ----
-184-
and the Nd:YAG system in Figure 7.9. For the CO2 system we see
that the extinguished free-space beam dominates. But for a
scattering layer thicker than 150 m the MFS power dominates so
that it is appropriate to describe the target return by (4.E.29).
The backscatter can be disregarded as (considering the geometry
of Figure 7.1) it can be easily range gated out.
Besides power return we must also consider resolution, i.e.
whether or not the beam lies entirely on the target, as assumed
for this example. Figure 7.7 shows the atmospheric (MFS) beamwidth
for both radar systems, and L = 2 km, L0 = 1 km vs. scattering
layer thickness Ls. Since the CO2 radar return is given by the
extinguished free-space result, beam size in this case is given by
the free-space result and the upper curve can be disregarded. But
since for the Nd:YAG system the MFS return dominates for Ls > 150 m
the beamwidth we must consider is the atmospheric limited result
shown. Note that if the target is 10 m or larger in diameter it
is resolved and the curves, Figure 7.9, apply.
For a resolved speckle target with a scattering layer near
the target (Figure 7.2) we expect that forward-path beam spread
loss and receiver coherence loss should be less severe than in the
previous examples thus increasing the MFS return relative to the
extinguished free-space return. For this case it can be easily
shown that Equations (4.B.7)-(4.B.9), (4.E.7), (4.E.8) are again
appropriate if in place of (2.C.3) we use
-185-
p 2 ' (7.A.3)0 'Ls e2k 2 Lsl
An example of the use of the Nd:YAG system in this situation is
summarized in Figures 7.10 and 7.11 where we have assumed L. = 1 km.
Figure 7.11 shows both the extinguished free-space and MFS CNR for
this situation. As the MFS return dominates we need to consider the
atmospheric beamwidth shown in Figure 7.10. For a scattering layer
thickness of less than 400 m a target of 1 m radius or more is
resolved so that the target return is described by (4.E.29). The
location of the scattering layer, in this case, is such that the
backscatter power must be carefully considered. From Figure 7.4,
the maximum normalized backscattered power is 31 dB. In Figure 7.11,
we plot the product of this maximum backscattered power and the
extinction factor exp(.-2t Ls ) vs. Ls. It is clear that if we range
gate properly we can receive the true MFS target return. For example,
if our target is 300 meters inside the scattering layer then
CNR = 23 dB. From Figure 7.11, the backscatter return falls below
this level at Ls = 50 m so that if we open the range gate
2 x 250 m/c = 1.67 iisec or less before the target return arrives we
will receive the target return and not the backscatter power. This
should not be too difficult if we have apriori knowledge of where the
cloud begins.
0
-186-
1.0
0.8
0.6
-zS.- 0.4
0.2
0 200 400 600
Ls (m)
Figure 7.10: Atmospheric beamwidth for a scattering layer near the
target and the Nd:YAG system vs. layer thickness.
------------
-- --------
-187-
70
50
30
T0
-10
-CNR
7 7_
CN RO e
2: - tF7e
-30C 200 400 600
L s (m)
Figure 7.11: Extinguished free-space and MFS resolved speckle target CNR
for the Nd:YAG system and a scattering layer near the target
vs. layer thickness.
-188-
For an unresolved glint target in the situation depicted
in Figure 7.2, CNR equations similar to those for the same target
in a uniform scattering profile apply. These are
CNR = CNR 0
1/3(XL/Trpin )21 +
(XL/2rpT 2P +T
2
er2
+ c
rlP tpCNR 0 = T p
h\) 0
4r 4s
[(XL/27r T2 + pZ]
as before and
rb = ( XLw 2+(L/21rpT)2 +prb ( L/7Tpin) 2 + +A2rP
(AL/7T) 2 rb
r [ (AL/Tpout )2 + p2] - [p - (ALrpmid 2
where
(7.A.3)
(7.A.4)
r2c
(7.A.5)
(7.A.6)
-189-
p 2 2 L 2 (7.A.7)out ' k2 e2 L3 - Ls F1
p 2 L (7.A.8)in k2 e2 L3
sF s
P2 2 L2 (7.A.9)mid 6' k2 82 3LL - 2L3s F s s
An example of the use of the Nd:YAG system in this situation is
summarized in Figures 7.12, 7.13 for L. = 1 km. Figure 7.13
shows extinguished free-space and MFS CNR for a glint target of
radius rs = 0.25 mm and mean square reflection coefficient r = .5.
For scattering layer thickness L > 300 m the MFS power dominates.
But at this point, CR0 e s = CNR = 6 dB. Hence the MFS power is
difficult to "see" for this target. The maximum backscatter power
is again 31 dB and just as in the last example can be range gated
out.
The small-glint detection theory analysis in Chapter VI
(i.e. the Jxj 2 = < 2> vw case) requires that the MFS return be
dominant. It also requires that the target radius rs be less than
the field coherence length rc. In Figure 7.12, rc is plotted for
this case vs. scattering layer thickness. We see that for Ls > 300 m,
where the MFS return dominates the condition rs S rc holds with
approximate equality.
-190-.
i7I 7
---- ----
I 4~
7. -7--.7- -- ------
0 200 400 600
L S(in)
Figure 7.12: Field coherence length for a scattering layer near the
target and the Nd:YAG system vs. layer thickness.
1.0
0 .Q1008
0.0 006
E
C-)5-
0.0 004
0.0002
0.0
z
77
-191-
-- - - - T- -L r
T__-- - - - -
t. -' - -~ --
H7 1-:t 7
,7, 7 ,7- + 7_
=tE=....... ......
::i 7
--- CIN R
1171 -~-x\ - X j1Z2Ft1"
.-m -- M K -26tLs.. ....CINR e
400 600
<K.~jL2V e2BtLs30 x 2e.b
- 0 200
L s(M)
Figure 7.13: Extinguished free-space and MFS unresolved glint
target CNR for the Nd:YAG system and a scattering
layer near the target vs. layer thickness.
70
50
Ju
7-
11~1 I-- j~~J
-192-
The above examples indicate the regions of applicability
of the various target return models considered in this thesis in
terms of two laser radar systems. We observed that for resolved
speckle targets and the Nd:YAG system, the MFS power can easily
domainte all other radar returns. For the CO2 system, we found that
it is appropriate to treat low visibility weather as a purely
extinction phenomenon. Hence, resolution of the Nd:YAG system can
be degraded in bad weather, from the free-space result while this
degradation does not exist for the CO2 system. The reason for this
is simply the higher single scatter albedo at the shorter wavelength.
That is, nearly all light incident on a scattering particle at the
Nd:YAG wavelength is scattered while at the CO2 wavelength about
half the power is absorbed.
It is clear that in an imaging application any radar system
that makes use of scattered light is going to have degraded
resolution as compared to free-space. This leads one to the
conclusion that scattered light is most useful in a detection
application. At this point, though, it is not clear whether one
would want to design a system specifically to use scattered light.
That is, the question of whether one detects targets more easily
with a system designed, as in the Nd:YAG system above, to have
the MFS return dominate or, as in the CO2 system, to have the
extinguished free-space return dominate has not been answered. In
order to answer this question the implications of the CNR results,
-193-
as in the examples of this chapter, on the false alarm/detection
probabilities needs to be investigated and the tradeoffs evaluated.
-194-
CHAPTER VIII
SUMMARY
In this thesis, we have examined the use of a heterodyne-
reception optical radar in both imaging and target detection
applications. As noted early in this work, such systems may be
severly limited by the stochastic nature of atmospheric optical
propagation; that is, by turbulence, absorption, and scattering. We
began the thesis by presenting a mathematical system model which
incorporates not only the statistical effects of propagation through
either turbulent or turbid atmospheric conditions, but also target
speckle and glint and local oscillator shot noise. We later
augmented this model to account for beam wander induced fluctuations,
whether due to radar aimpoint jitter or turbulent atmosphere beam
steering.
Once this model was established we used it first to analyze
the radar in a scanning-imaging application, Most important here
were the issues of resolution and signal-to-noise ratio. In
previous work turbulent atmosphere resolution had been shown to be
the same as free-space. We showed, as was expected, that turbid
atmosphere resolution is degraded from the free-space value. We
found, due to the statistical nature of the local oscillator shot
noise, that the signal-to-noise ratio depended on two quantities:
-195-
SNRSAT and CNR. The former being the SNR limit set by target return
fluctuations, the latter being the average target return to shot
noise power ratio. In turbulence, these quantities had previously
been evaluated for a number of interesting targets. We proceeded to
evaluate these same quantities for inclement weather. In the course
of doing so, complete statistical characterizations of the radar
return were developed for several bad weather situations, as had
been done previously in turbulence. In both turbulence and bad
weather, we endeavored to interpret CNR and SNRSAT results in terms
of intuitively pleasing descriptions of target interaction and
atmospheric propagation. These interpretations greatly enhance our
understanding of the mechanisms that degrade optical radar
performance.
The second major use we made of our system model was in
analyzing the detection capability of the radar. For each complete
statistical description of the radar return developed earlier, a
separate detection performance analysis was required. We concentrated
on the case of a small glint target in bad weather, as this
represented new work.
We were also concerned with verifying the theoretical
statistical characterizations of the radar return. Due to the nature
of the available experimental setup we were limited to verifying our
beam wander and turbulent atmosphere models. We found very good
correspondence between theory and data in many cases. Of particular
-196-
interest here was the verification of the lognormal character of
the atmospherically induced fluctuations.
Our last task involved the establishment of regimes of
validity for the various target return models. This was done in
terms of two hypothetical optical radar systems and involved comparing
relative values of three quantities: The extinguished free-space
radar return; the MFS radar return; and the atmospheric backscatter
return. It was shown that some regime existed wherein every target
return model was valid.
In the future, several topics which extend and adjoin this
work might be investigated. These include: continuation of theory
verification work; a similar type of analysis, as found herein,
performed on a direct detection system which might make better use
of scattered light; generalization of this work to include doppler
shift-moving target indication; use of an array of detectors in
place of the single detector assumed in this work to make better
use of scattered light; and an investigation with other values for
system parameters than in this thesis (for example, unequal transmitter
and receiver aperture diameters). All of these topics are important
for a more thorough understanding of optical radar potential, and
should all be investigated
-197-
APPENDIX A
DERIVATION OF THE MUTUAL COHERENCE FUNCTION (MCF)
In this appendix, the details of the derivation of the
multiple forward scattering (MFS) propagation model, Eq. (2.C.2) are
given. We begin by defining the specific intensity I(r,f) and give its
governing equation, the equation of transfer, which is essentially a
statement of energy conservation. This equation is specialized to the
case of a collimated beam and the specific intensity is then shown to
be related to the MCF by a Fourier transformation. The specialized
equation of transfer is then Fourier transformed, and its solution,
the MCF, given. We then apply this result to real scattering
atmospheres and derive the multiple forward scatter (MFS) propagation
theory.
Consider a flow of wave energy at some point r in a random
medium. For a given direction defined by the unit vector U we can find
the average power flux density per unit solid angle. This quantity
I(F,!) is called the specific intensity [28] and has units W m-2 sr~ .
Operationally, we can say that the amount of power dP flowing within a
solid angle d0 and through an elementary area da oriented in the
direction of the unit vector ? is (Figure A.1)
/ ...
e / d/
Figure A.: Geometry relating to the definition of specific intensity.
0
00
1
J01
I I b
da
-199-
dP = I(r,2) cos(6)da do (A.1)
The equation that governs the evolution of the specific intensity is
the equation of transfer [28] and for a sourceless medium is
Q*V I ,2 t I~r~ 4wf5 dZ' p( T,Q) I(r,Q') = 0 (A.2)
In (A.2), t(s) is the extinction (scattering) coefficient (m~1 ) and
p(f,f') is the single particle phase function normalized to satisfy
dff p(2,Q') = 1 (A.3)47r
Equation (A.2) can be interpreted as saying that the change in power
flowing in the 0 direction due to propagation in a random scattering
medium is equal to the power scattered into that direction from all other
directions less the power absorbed and, hence, is a statement of
energy conservation.
If we now assume that all light of interest is propagating
nominally in the +z direction, so that we can say
S= (, /l - [Ij2) ~ (il) (A.4)
where
-200-
S = (s ,s ) (A.5)
then (A.2) becomes
s v- I(P, ,z) + Z I(Fiz) + pt(,isz) -- TS ds' p(S-') I(P,' ,z) = 0
(A.6)
In (A.6), r = (P,z) where P = (x,y) is the coordinate vector transverse
to the direction of propagation, Vp = x 3/ x + y 3/3y is the two
dimensional del operator and the phase function is assumed to be a
function only of the difference in the output direction and the direction
of the incident wave. Also p(s) is assumed to be sufficiently narrow
that the limits on the integral in (A.6) can be extended to infinity.
To relate the specific intensity to the mutual coherence
function consider Figure A.2. The power incident on the detector is
easily shown to be
r ~r p P 2,!L) k -P = jd d 2 Jdp' exp(-j y p'-(p -p 2 ))circ(2 1- -- p /dR
-circ(2152 o/d) cir(21p' - pDI/d) (A.7)
where r( P, 2,z) = <u(Pi,z) u*(P2,z)> is the MCF, the p, and P2
integrals are over the (z = L) receiver plane, the p' integral is over
RANDOM,TIME INDEPENDENT SPATIALLY VARYINGPROPAGATION MEDIUM
u~,L) ....
Z=O
I- DIAMETER dRFOCAL LENGTH fCENTERED AT
Z=L
0
DETECTORAREA A = ird 4CENTERED AT =f i
D
§0
Z= L+f
Figure A.2: Geometry for relating the specific intensity and the mutual
coherence function.
C3I
-202-
the (z = L + f) detector plane and circ(2[pl/d) is a circular pupil
function defined by
circ(2I5I/d) =
I
0
1PI < d/2
(A.8)
elsewhere
If we assume the detector and AD is equal to a diffraction limited
field of view
AD = (Xf/dR) 2 (A.9)
the detector plane integral can be approximated so that (A.7) becomes
=I A C)P dp C dp Pd , ) C VL) exp(--jk 's Pd )cic_1cT -o R
2- -circ(2I-pc + 2 -P 0 /d R) (A.10)
where r' is the MCF in terms of the sum and difference receiver plane
coordinates pc = + 2)/2 and Pd = l - P2. As the field u(P,z)
has propagated through L meters of the random, time independent
spatially-varying propagation medium it is reasonable to assume that
'(c' d,L) = -'(p ,pdL) over the receiver aperture. Further assuming
that r'(. ,Pd,L) is narrower than the receiver aperture in Pd' i.e.,
-203-
'(PoPd, L) Z 0 (A.11)pd > Pa
where pa << dR/ 2, equation (A.10) becomes
A 7rd2 -P =R d d F o'd,L) exp(-jk s-pd) (A.12)
If we write this same detected power in terms of the specific intensity
we have
rd2 AP = I(P ,s ,L) R D
o 0 4 2(A.13)
Combining (A.12), (A.13), we have that r' and I are related by a
Fourier transformation
I(f oL) =i drdfr dL) exp(-jk - (A.14)
T' (PC') d, L) = s d I(-Pc' 5s0,L) exp(+jk so -Pd) (A.15)
If we Fourier transform (A.6), we then have the differential
equation that the MCF must satisfy
-204-
1 d v I,z +) + 5 -'(ic'Pd,z) + ,P ,z)Jk '7 d -7cr(c~dz 9 cPdt cpd~
- s(Td) F'Cpc' d,z) = 0
where P(Pd) is the Fourier transform of the phase function p(s)
jks *dP(Pd jds 0p(so) e 0 d
Equation (A.16) has been solved [18,28] and
'(pc' dZ) = j dpd r '(', ,0)e jdp Jc' ' (Xz) 2 e p Pd C c
rr-z rz + s d- exp - S) dS - I s(s) 1 - P zpl-]+pd ds1
(O a.o
(A. 18)
which implies
= ho(p ,5 ) ho(, 2 '2I)
{ L ( L
-exp - a(z) dz - BS (Z) 1 - P [ 1 - + zp dz
.0 .0 (A. 19)
(A.16)
(A.17)
<hL (,p -) h*(-2,p )>
-205-
For a plane wave input r'(p', 5 ,0) = JU1 2 the transmitter plane
integrals (A.18) can be performed and
rL ' L
r'(pc'PdL) = JR12 exp [- a(z)dz exp -[1-P(Pd 0 (z)dz]
If a Gaussian form for the phase function is assumed
exp(- Is 12 /2 2)
F
(A.20)
(A.21)
then P(pd) is
P(d)= exp(-k 2e .Id 2 /2)
(A.22)
1 1 - k2 e2 d 2 / 2
where a two-term Taylor series expansion of P has been indicated. If
we use this two-term Taylor series and further assume a uniform
scattering profile along the propagation path a(z) = S(z) = S'
then (A.19) becomes
<hL (1,p ) h*(p 2 ,p )> = ho(phpi) h (p2' 2
-e-aL expf Ed 2 + d d+ Id 2
3p(A.23)
-206-
where
p2 2 (A.24)0 asL k2 e
which for plane wave input implies
- L -1 12 p
r'(PC'Pd,L) = JU 2 e e (A.25)
Equations (A.20) and (A.25) have been plotted in Figure A.3. It is
clear that if sL >> 1, then (A.25) is a good approximation to (A.20).
By extension we say that under the same condition (A.23) is a good
approximation to (A.19). In order to see what is neglected by using
(A.23) in place of (A.19) consider again Figure A.3. The correlation
that remains between the field sampled at two highly separated points
in the upper curve (for large IPd1, ' =u2 exp(-(a + S)L)) can
only be due to unscattered light as scattered light should, by
physical reasoning, become uncorrelated at large 1-d'. Therefore we
conclude that use of (A.25) in place of (A.20) and, by extension (A.23)
in place of (A.19), amounts to treating only the scattered light and
disregards the unscattered beam. Hence, the unscattered power needs to
be considered separately from the scattered power.
To conclude our development of the MFS propagation model
consider the sketch of a real phase function in Figure A.4. Besides
being highly peaked in the forward direction there is significant
1'
Jul 2 e-pa L
1 2 - L
2 e~
-e3s
Ra L 2-P4 2
0
Figure A.3: MCF's for plane wave input.
2Iy I
-I
e-(lea + G) L
d
L (I- P(p )
-208-
wide angle and back scatter. Our aim here is to apply the previous
theoretical development to such a real world situation. The procedure
we follow has previously been used by Ross et al. [18] and Mooradian
et al. [36]. Reasonably good, but inconclusive experimental
verification of this prcoedure has been reported by both groups of
researchers. First, we truncate the real phase function at eE, as
shown in Figure A.4, chosen to contain the forward scatter peak and
lump all scattering at angles wider than 8E into a modified absorption
coefficient. Hence we would use
; = Na + s(1 @ (A.26)
S = s (A.27)
In place of a, s in (A.23), (A.24) where
a+ s a + s t (A.28)
and
-e=2Tr p(8) sin e de (A.29)
-10
-209-
4
90
8 (deg)
Real phase function.
p (e)
1808 GE
Figure A.4:
-210-
is the forward scattering efficiency. Second, the forward scatter
peak is approximated by the Gaussian phase function (A.21) where 6F3
the effective rms forward scatter angle, is given by [17]
6 2-rr 0 (A.30)F 27rp(p)J
Clearly the forward scatter efficiency must satisfy 0 < 0 < 1. To
simplify our calculations in Chapter VII, we choose D = 0.57 as a
reasonable value [17] in place of (A.29). Equation (2.C.2) is therefore
(A.23) with parameters chosen according to the above procedure.
-211-
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-215-
BIOGRAPHICAL NOTE
David Papurt was born in Toledo, Ohio on July 18, 1954.
He graduated from Thomas A. DeVilbiss High School in June 1972. Dr.
Papurt then entered the University of Toledo where he received the
B.S. degree in Electrical Engineering in June 1977 and the B.A.
degree in Music in August 1977.
In September 1977, Dr. Papurt became a graduate student in
the Department of Electrical Engineering and Computer Science at
the Massachusetts Institute of Technology where he specialized in
communication theory. He received the S.M. degree in Electrical
Engineering and Computer Science and the E.E. degree from MIT in
September 1979 and June 1980, respectively. He is a member of the
Society of Photo-Optical Instrumentation Engineers.
Dr. Papurt is currently Assistant Professor of Electrical
Engineering at Northeastern University, Boston, Massachusetts. He
has co-authored a paper on "Atmospheric Propagation Effects on
Coherent Laser Radars," that will appear in the Proceedings of the
Society of Photo-Optical Instrumentation Engineers.