ATMOSPHERIC PROPAGATION EFFECTS ON HETERODYNE-RECEPTION OPTICAL RADARS by DAVID MICHAEL PAPURT B.S.E.E., B.A., University of Toledo (1977) S.M., Massachusetts Institute of Technology (1979) E.E., Massachusetts Institute of Technology (1980) SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 1982 0 Massachusetts Institute of Technology, 1982 Signature of Author.., Department of Electrical Engineering/and Computer Science May 14, 1982 Certified by........'......... Jeffrey H. Shapiro Tjhsis Supervisor Accepted b.......... Arthur C. Smith Students Graduate on Committee Department Chairman, Archives MA SSACHUSETTS NSitTZTC OF TECHNOLOGY OCT 20 1982 UBRARIES -2- ATMOSPHERIC PROPAGATION EFFECTS ON HETERODYNE-RECEPTION OPTICAL RADARS by DAVID MICHAEL PAPURT Submitted to the Department of Electrical Engineering & Computer Science on May 14, 1982 in partial fulfillment of the requirements for the Degree of Doctor of Philosophy ABSTRACT The development of laser technology offers new alternatives for the problems of target detection and imaging. The performance of such systems, when operated through the earth's atmosphere, may be severely limited by the stochastic nature of atmospheric optical propagation; that is, by turbulence, scattering, and absorption. A mathematical system model for a compact heterodyne-reception laser radar which incorporates the statistical effects of target speckle and glint, local oscillator shot noise, propagation through either turbulent or turbid atmospheric conditions, and beam wander is presented. Using this model, results are developed for the image signal-to-noise ratio and target resolution capability of the radar. Complete statistical characterizations for the radar return are given. An experiment and data processing techniques, aimed at verifying the above statistical models, are described and results given. Notable here is the verification of the lognormal character of the turbulent atmosphere induced fluctuations on the radar return. Target detection performance of the radar is investigated. Regimes of validity for the various target return models are established. Thesis Supervisor: Title: Jeffrey H. Shapiro Associate Professor of Electrical Engineering To my MotheA and FatheA -ot the%'L eoAnt,6, patience, and Zove -4- 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. work significantly. Their many suggestions have improved this 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. -5- 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. -6- TABLE OF CONTENTS Page ABSTRACT...... .................................. 2 ACKNOWLEDGEMEN TS .......................... 4 LIST OF FIGURE S .... .... ................ 8 ...... 14 LIST OF TABLES ............................. 0..... CHAPTER I. CHAPTER II. CHAPTER III. INTRODUCTION..................... A. Laser Radar Configuration.... . .. . . ... ... .. . .. .. .. .. .. . .... 15 . .. 0 17 ATMOSPHERIC PROPAGATION MODELS... 27 A. Free Space Model............ B. Turbulence Model............. 30 C. Turbid Atmosphere Model...... 32 D. Backscatter.................. 34 TARGET INTERACTION MODEL......... A. Planar Reflection Model...... 35 35 31 B. Relationship to Bidirectional Reflectance.... 37 CHAPTER IV. SCANNING-IMAGING RADAR ANALYSIS.. ................ A. Single Pulse SNR............. ................ B. Speckle Target Resolution in the Low Visibility Atmosphere........ ................ C. Identification of Atmospheric Effects ........ 40 41 45 50 D. Turbulence SNR Results....... ................ 52 E. Low Visibility SNR Results... ................ 56 F. Beam Wander Effects.......................... 71 G. Correlation of Simultaneous Speckle Target Returns...................................... 87 H. Backscatter.................................. 90 -7- Page CHAPTER V. THEORY VERIFICATION............................ 96 A. Laser Radar Description.................... 96 B. Data Analysis Techniques................... 99 C. Beam Wander................................ 102 0. Turbulence................................. 113 CHAPTER VI. TARGET DETECTION............................... 135 A. Problem Formulation........................ 135 B. Single Pulse Performance................... 138 C. Multipulse Integration..................... 147 D. Multipulse Performance..................... 150 CHAPTER VII. SYSTEM EXAMPLES................................ 165 A. EXAMPLES................................... 174 CHAPTER VIII SUMMARY........................................ 194 APPENDIX A. DERIVATION OF THE MUTUAL COHERENCE FUNCTION (MCF).......................................... 197 REFERENCES.................................................... 211 BIOGRAPHICAL NOTE............................................. 215 -8- LIST OF FIGURES Page Figure 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 -9- LIST OF FIGURES Figure Page 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 5.7 (5.D.l), a2 = .0045......................................... 114 x 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........................................ 119 x -10- LIST OF FIGURES Pag Figure 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...................................... 120 x 5.11 Normalized histogram of 300 "hot spot" polished sphere returns taken in reduced scanning mode and the lognormal = .0083.................................. 121 x Normalized histogram of 300 "hot spot" polished sphere returns taken in reduced scannina mode and the lognormal 2 = 0.004.................................. 122 PDF (5.D.1), a' x The theoretical curve (4.D.6) and estimates SNRSAT from the data of Figures 5.6,5.9-5.12......................... 124 PDF (5.D.1), 5.12 5.13 5.14 g2 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 -11- 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. 6.5 PF = 10- throuhout.......146 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 -= 102 154 PF 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. 6.9 PF=10-12 throughout. 159 The number of pulses M necessary to achieve PF = 10-12 .99 vs. CNR for a glint target in free-space, two turbulent fluctuation levels, and bad weather.............160 pD= 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 -12- LIST OF FIGURES Paae Figure 6.12 Fifteen pulse detection probility vs. CNR for a glint target in several turbulent atmospheres and a scattering atmosphere. 7.1 PF = 10-12 throughout........................164 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 -13- LIST OF FIGURES Figure 7.11 7.12 7.13 A.1 A.2 Page 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 Field coherence length for a scattering layer near the target and the Nd:YAG system vs. layer thickness......... 190 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 Geometry relating to the definition of specific intensity................... ............................ 198 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 -14- LIST OF TABLES Table Page 7.1 CO2 Laser Radar System Parameters...................... 167 7.2 Nd:YAG Laser Radar System Parameters................... 168 7.3 Atmospheric Parameters at CO2 Laser Wavelength......... 172 7.4 Atmospheric Parameters at Nd:YAG Laser Wavelength. ..... 173 -15- 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) Huygens-Fresnel formulation [2] is useful in our application. extended 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 -16- 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 vehicle such as a truck or airplane. -17- 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. analysis. here. This will be achieved via a signal-to-noise ratio (SNR) Also, speckle target resolution in bad weather is considered 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. examples are presented. In Chapter VII system 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 Er(',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 lw mw L TRANSMITTER w A TMOSPHERE D TRANSMITTED BEAM a) FORWARD ET(-P, t) TARGET ILLUMINATING BEAM PATH Et(-p, t) 0 z= L RECEIVER ATMOSPHERE RECEIVED BEAM ER(p ,t) TARGET REFLECTED BEAM b) Figure 1.1: RETURN PATH Laser Radar Configuration Er(pt) -19- 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 T where v 0 } (l.A.1) 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). -20- uT(P,t) PT(t) = = dPIT( -T(t) FT() (1.A.2) Is-T(t)12 12= (l.A.3) 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 becomes the Huygens-Fresnel integral [9]. space (l.A.5) In the atmosphere -21- PT (t) 0 t Pp Figure 1.2: p T +t p p 2, p Transmitted Power Waveform 2 c+ p p -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) -23- In low visibility and with a suitably large target range we take hLF and hLR to be statistically independent. is due to the This decorrelating between pulse atmosphere 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 t u(-,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. -24- This is shown in Figure 1.3 where the pass band filter H(f) has the input with IF complex envelope y(t) = d ~hvJ 0 F*(P) + n(t) -R(t) - (l.A.10) The additive noise n(t) is a zero-mean, white, circulo-complex Gaussian process with <n(t) n*(s)> = tp 6(t - s) where <-> denotes ensemble average, hv0 is the photon energy and q is the detector quantum efficiency. u_,t) (l.A.ll) Substituting for in (l.A.10) and making the definitions P' (l.A.12) d- h LF(PI',P) F T(P) EF(I') = 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 r(t) H(f) Re{y(t) e-} a) IF Model A H(f) U' 2W 2W -I I VIF U VIF b) Normalized IF Filter Frequency Response Figure 1.3: Heterodyne Receiver Model -26- opposed to (l.A.10) which is performed over the receiver From (l.A.10) it can be seen that to maximize signal plane. return we should set F(p) = u (P)/( dpig(p)12) . In to approximate this condition. practice, we set F,(P) = FT) 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 mrij ,2L/c+t~ 2 = l/t J2L/ct 2L/c r(t) (Figure 1.3). r(t) dtf 2 where r(t) is the complex envelope of In imaging, the scene is scanned and the complete image built up from sequential returns Fr| by a diffraction limited FOV or more. 2 separated in space typically 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. -27- 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 = 3/5 2 2 0(1.09 k Cn L)- under clear weather conditions is shown versus path length L for three values of turbulence strength C2 and wavenumber k n corresponding to 10,6 pim wavelength radiation [3]. From -28- 101 X 10.6 p.m 1 2 5 x 101 m-2/3 n C= 5 x 10 m-n" 3 -2 10 2 3 45 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 (C2n = 5 x 10-3 m-2 3 ); 10.6 vim wavelength has been assumed. -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. will take this to be the case. turbid atmosphere We On the other hand, for propagation, the atmospheric coherence 2 length p0 =0 (2/s L 0' F k )2 (see Appendix A), where S' 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 p0). 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. free space propagation is described. turbulence is characterized. First, Next, propagation through 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)] where, via reciprocity, hL = hLF = hLR. In (2.B.1), atmospheric absorption coefficient and x(I',-) ( (I',)) ho(-',-) (2.B.1) a is the 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]. by the central limit theorem [15], x and c Hence, 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 energy x x conservation [16] where the log amplitude perturbation variance is given by -32- 02 x = min(O.124 C2 k7/ n 6 L11 / 6 ,0.5) (2.B.2) 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(',) for either the forward or return path. exp[j (',P)] (2.C.1) h0(P',) 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 Hence, these fields are the sum of a large number of weather. 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 )> = (T1 _-P In (2.C.2), e a exp[-(j 2 + _2))/3pf] ho(TP ,p) IT' - p h*( 2 + , 2 (2.C.2) '(> Ba) is the modified absorption coefficient containing aa wide angle scatter and back scatter contributions and Po= [L e2 k2 (2.C.3) is the atmospheric coherence distance where eF 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. established. The validity of the model is not fully 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. this analysis. We first discuss the planar target model used in 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 and ur(P't), respectively, nominally along the z-axis are noted ut (',t) we have the relation u-(I',t) = t(F',t) r expr-jkjp' - 2 /Rcl 1cf (3.A.l) where r is the intensity reflection coefficient, Rc is the radius of curvature of the mirror and pc 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' - would be our model. (P)I 2 /Rc(p')) (3.A.2) 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 (3.A.3) Ur(I't) = u-t(p',t) T(P') 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') The glint component, T (P'), (3.A.4) 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- -s1 <TS a' <T ( ')> = 0 (3.A.5) <T (() ) Ts(T) )> = 0 (3.A.6) )6~ s 21 _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 ; r = X21AT <- dp' exp1[2f where AT is the target's projected area. plane wave exp(j27 - r) -P '] T(') 12> (3.B.1) If the input field is the 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/m2 sr) in the direction fR to the incident irradiance (W/m2 ) propagating in the direction . W W W REFLECTED 4W MW ww 1W FIELD Ur(U,+) Kiiiiiiii r TARGET-PLANE _ fr FIELD TA RG ET ' INEFFECTIVE z Figure 3.1: W z PLANE OF INTERACTION L Geometry for defining bidirectional reflectance p'(x;Ti,Tr); the target plane field is chosen to be a plane wave of wavelength x propagating in the direction of the unit vector ii (Xfi is the projection of ij on the z = L plane); the radiance of the reflected field is measured in the direction of the unit vector ir (xfr is the projection of ir on the z = L plane). -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 With this kind of diffraction-limited FOV by diffraction-limited FOV. 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 = r(t) dt| 2 J 1l/t (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'ftpP T'~ x = d' T(p') G(p') and the noise n is zero-mean, complex-Gaussian, s(i')i (4A3 lyAen statistically i-ndependent -42- of x, with moments <n2 > <In 0 = 2> = (4.A.4) 1 (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> SNR = - <Ix 12>2 <ln2>)2 (4.A.7) _ Var(I r12) Var( I r12) 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/2 CNR/2 1 SNR SAT where the carrier-to-noise ratio (CNR) (4.A.8) 1 2CNR 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>2 SNRSAT For CNR <I.X12> xi 2) = (4.A.10) AT Var( 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. must calculate <tx! 2 > In order to know these quantities we 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 Ix1 2 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 CNR = T.1P t 2TrP2 0 r ho (4.A.13) L and [3] SNRSAT = 1 For an unresolved glint target, with field reflection coefficient (4.A.14) -45- T (p ) -<s = 2 exp[- L' 12/2r (4. A.15) ] we have CNR = L 4 rrsPTj (4.A.16) More detailed examples of and a saturation SNR equal to infinity. 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') and the, atmosphere is characterized by (2.C.2). (4.B.1) Flp) 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 will be considered. 2> 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, To facilitate the calculations Gaussian and will be given here. transmitted and local-oscillator spatial modes are assumed FT(P) (~TrP 2 )T T F*(P) exp[- I 2 /2 p2 + j 2 T f-] (4.B.2) (T -exp[- )1f2 P 12 /2 2p R+ ex E-rP/ j2nTr - ] (4.B.3) (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 Vt = 10 km/hr this upper bound on the atmospheric coherence time is p0/Vt is 6.7 -psec at range L 25 psec. = The roundtrip delay time td = 2L/c 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- -'L <_-F( ') -LPF e 2) exp[-p' 2 ALI 2/rbF TrrbF - exp[-p |2 /rcF2 exp[j k (p' - XLfTT) oF c T ]exp[j2pT'S d (4. B.4) where for convenience we have expressed the result in sum and difference coordinates -c = 1 (p - + p ) (4. B.5) (4.B.6) In Equation (4.B.4) the beam radius rbF is rbF = (4.B.7) (AL/pF)2 + (AL/2TrpT) 2 + the coherence distance rcF is 2 rc 2 bF "oF = TT (AL/hrpOF)2 + +(AL/2pT) (4.B.8) + pT + + p2F /4 -49- phase radius of curvature RoF is and the 4p R = + L[T P oF 3pF2 rbF bF + + 3p F The corresponding expressions for <-RC) (4.B.9) T r2 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 0T < X( T)h <' e-2 L ee5 d~p' T (-P') x-- exp[-Ip' - XLfT 2 /r es s 7T2>rh4 (4.B.10) where rbF = rbR r and 1 re r /2 = [(XL/2)rpT 2 + pT] 1 + ~ 2 (AL/2rfTP is the target-plane resolution e~1 spot size. (4.B.11) + PT 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 (pT/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 2 7r 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 Turbulence results will be different atmospheric/target scenarios. 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') ej that behaves like a spherical reflector over the illuminated region T (p') p' -ALfT = F exp(-jkjp'-'|/Rc) T c_ c d (4.D.l) and satisfies Rc << L (XRc is assumed. (4.D.2) 0 (4.D.3) 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 1x_1 2 = CNRgu e 4cr 2 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-amplitude X perturbatiort X( 9,9) . It follows from (4.D.4) that the single pulse image SNR satisfies CNR /2 SNR u g 1 + CNRgu(e gu X - 1)/2 + 1/2 CNRgu (4.D.5) For CNRgu z 5, Equation (4.D.5) takes the standard form (4.A.ll) with -54- SNRSATSAT gu 1 16 a2 e (4.D.6) X If no turbulence is present, we have a2 0 and the saturation SNR X (4.D.6) is infinite as predicted by (4.A.10). For a2 > 0, SNR gu X initially increases with increasing CNR until it reaches the scintillation-limited value (4.D.6). severly limited by turbulence. 2 > 1/16, SNRSAT For cr X -STgu is 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 SNRs CNR /2 16 52 1 + CNR s[1 + 2(e X -1)C]/2 + 1/2 CNRs (4.D.7) where C is the log-amplitude aperture averaging factor [16,41] given by the approximate expression 4 P2 /XL 4T~ 2 (4.D.8) 1 + 4 pT //L in the weak perturbation regime and CNRs is the speckle (denoted "s") target, turbulent atmosphere CNR. If the mean-square reflection does not vary appreciably over the region illuminated coefficient T(p') S -55- by the radar (i.e., the target is resolved) the target return can be expressed as Ix 2 = CNRsr v e2 u (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 6a2 1 - 1 = C(e X 2 e4c - 1) (4.D.10) In (4.D.9), v represents the target speck le and u the scintillation. The probability density function for the unit mean fluctuation w = v e2u is given in [27]. If CNRs z 5, Equation (4.D.7) takes the standard form (4.A.ll) with SNRSAT 1 = 16 5As 1 + 2(e From (4.D.11) note that SNRSAT < (4.D.11) a2 X -_1)c 1 with SNRSAT = 1 corresponding to speckle limited saturation SNR. For more details on turbulence SNR results, including system examples, the reader should see [3-5,26]. -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 CNR = -nP t Pt 0 dpi J dp <T(T ) T*(T)><-F ) L (Pj4 (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 T (p') with The target is pure speckle T(-') P = -S - Ts(P') exp[-I' 2 /r'] (4. E. 2) -57- The mean square reflection coefficient Ts is Gaussian with width rs Although no real target would have the and centered on the z axis. 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 with the speckle target area AT. speckle target if we replace 7rr Assuming (4.E.3) rs<< rbF9rbR i.e., that the target is unresolved by the radar, the carrier-to-noise ratio (4.E.1) becomes CNR su = CNR 0 su 1/3(XL/pOFTO 1/3(XL/p oR7 2 exp[-2'L] a 2 1 + 1 + (XL/2T) 2 + (XL/2rpR p + R LTI 2 *exp (4.E.4) rbF rbR/ rbF + rbR where CNR 0su is the free space unresolved speckle target CNR TIP t = T CNR su hv0 0 r 2 x2 _' [(XL/21pT) 2 + r5s p2][(XL/2pR) 2 + pR] (4.E.5) -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 the PR - 0. The fourth term represents absorption and the last gives 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 at PT = 6.5 cm, the change over from near field to far field occurs 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 (-') square reflection coefficient (4.E.2). with mean Assuming rs >> rbF, rbR so that the target is resolved, the CNR becomes (4.E.6) -59- 0 CNR sr= CNR sr + 1/3(AL/pOF7F) 2 1/3(XL/poRr) 2 (XL/2R)2+p2+( L/2r)2+P exp[-2S;L] exp[-fXLfT 2 (XL/2rpR)2+p2+( L/2 pT) 2 +p2 (4.E.7) /r2 ] where CNR 0 is the free space, resolved, speckle target carrier-to-noise sr ratio CNRP0 sr - Tt p hv0 1 '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 loss. = . Clearly, there can then be no coherence 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 p2 exp[-IK'12 /2r2 ] exp _ T (j')= The intensity reflection coefficient as previously. 2 (4.E.10) T(p)1 2 is Gaussian with width rs We assume that the target is flat so that the phase radius of curvature -61- Beamspread Target RADAR 2rbF Incident power constant over ----...target Free Space Beamwidth Reflected spatial mode is independent of forward-path beamspread a) Unresolved Target .... Target Beamspread RADAR 2r F Free Space Beamwdith Reflected Spati'_'F Mode depends on forward-path beamspread b) Resolved Target Figure 4.1: Reflected Spatial Modes from Resolved and Unresolved Speckle Targets. -62- R g = (4.E.11) O We further assume the target is unresolved (4.E.12) rs << rbF, rbR the radar is in the target's far field r (4.E.13) <<a and the laser is aimed directly at the target (4.E.14) fT = 0 so that the CNR becomes CNR gu = CNR 0 gu -1 1/3(AL/rpoF) 2 (AL/27rpT) 2 + p 1/3(L/TrpoR)2 4r 2 4r 2 1+ (XL/2rpR) 2 + 12R exp[-2 'L] where CNRgu is the free space unresolved flat glint target CNR + 2 rrCF (4.E.15) r 2 rcR -63- CNR 0gu - 4r' nP t tp hv0 4s [(XL/2rrpT) 2 + p{1[(AL/2TrpR) 2 (4.E.16) + 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). loss. The fourth term can be interpreted as return-path beam spread 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 1+ r2 + 2 2 PoF PoR when the target is in the transmitter's far field and PR > PoR' 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 PT > oF. 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 ii 2r Target a) Target smaller than a coherence area, rs < rcF : PoF . Incident Phasefront Reflected Phasefront 2rs arget b) Target larger than a coherence area, rs Figure 4.2: > rcF ~ oF 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" of "unresolved" as in the CNR examples. instead 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 where we have made the definition dP' ! (P') L.( tA ' EF( ' J(4.E.19) (f dp' Ts(-') 1 12 F (4.E.18) -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 (4.E.20) 2 2> = < Assuming that the target is small compared to a coherence area 2 r2 << rF < s cF cR p 2 (4.E.21) PoF (4.E.22) OoR we can then say (4.E.23) I__F (. inside the integral (4.E.18). 2 (4.E.24) 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 The first is that any target satisfying these (4.E.21) and (4.E.22). 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 ss 2>2 Var( xf 2 ) (4.E.26) 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) r2 >> r2 (4.E.28) cF s 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 In fact, the target can be stringent limitation on the target size. either resolved or unresolved and still satisfy them. The saturation SNR is now given by SNRSAT = where "sZ" denotes large speckle target. resolved glint targets are now presented. 1 (4.E.30) The cases of small and -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 PT x_= h T eJe T (T') d' E-RP) -F(E.)3 (4. E.31 ) If we assume the target is small, i.e. a s cF r2 << rcR and Lg(U), (4.E.33) ') in the above integral by then we can replace g(P') and (4.E.32) respectively, as in the small speckle target case. ( It then follows that fx 2 = CNR v w (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 SNR SAT 1 g (4.E.35) -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>> r 2 > rbbF 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) SNRSAT gr = 1 (4.E.39) so that 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 SAT ==Var(w) Var(v) + Var(w) + Var(v) (4.F.2) The expression reduces to SNRSAT = SNRSATbw (4.F.3) -72- when SNRSATta >> SNRSATbw (4.F.4) and SNRSATta > 1 (4.F.5) where SNRSATta Var(w) (4.F.6) is the saturation SNR when only target/atmospheric fluctuations are present and SNRSATbw Var(v) (4.F.7) 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 (14.F. 8) -73- or equivalently IxI = <lxI> z (4.F.9) The random variable z is the unit mean fluctuation on lxi and is related to v by V We develop statistical models for z here. (4.F.10) 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,), t 2= Tr exp(- p' (4.F.11) Trb where the beam center (x,y) is random and circularly distributed (4.F.12) -74- (X-m )2+(y -Mmy)2 < R' 7TR2 p (4.F.13) (XY) elsewhere 0 An unresolved target of area AT is assumed to be located at P' = so that the reflected power is AP P AT ut() t t exp(- 2 r2 Trb 2 m (4.F.14) /r2) Since pm is random it is clear that the reflected power is also. lijil = < R of this on (m + m2) = For <IxI>z is given by the probability density 2 r 2~ b 1 Tr R2 Z 2 PZ (Z) = 2 0 Cos -2rb ln(Z/Zmx) 21r1 rb/ - R2 + nZ/Zmax I I Zmin ZZ mid Zmid <Z <Zmax el sewhere (4.F.15) where -75- m12 /2r2 e Zmax (4.F.16) 1 du u exp - 2 u2 .0 Zmid R02 b exp 1 b - R)2 (4. F.17) Zmax b R )2 ( ... =exp Zm (4.F.18) max b In (4.F.16), Io(-) is the zero-order modified Bessel function of the first kind. For Jml > R the PDF of z becomes 2 2 2 b 1 cos T R2 Z 1 2 r ln(Z/Zmx ) R2 + |i2 2m1 rb/- 2 ln(Z/Zmax) Z*nF.9)<_< Zmid (m4i pZ (L)= (4. F.19) . 0 el sewhere -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 (4.F.20) SNRSAT 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 1/128 th 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) where Py has units (W/m) and the beam center x0 is random with distribution (4.F.21) W W W w qW 01 1W MW I I I I W I 2- N -4 -4 N 0~ 1 0.6 0.7 08 0.9 1.0 1.1 1.2 1.3 z Figure 4.3: Target return PDF for beam wander fluctuations; Gaussian beam, uniform circular beam center, tII/rb = 0.0, R/rb =l.. 116597-N low qw W I 16(b I I 50 --- 40 2 I I I I I I GAUSSIAN BEAM, CIRCULARLY DISTRIBUTED BEAM CENTER FAN BEAM, UNIFORMLY DISTRIBUTED BEAM CENTER 1-N I 30 00 Z 20 10 - - 0 0 Figure 4.4 1 2 3 4 5 6 7 8 9 Saturation SNR due to two types of beam wander fluctuation. 10 -79- 1 m - R < X < m+ R (4. F.22) p (X)= 0 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. an unresolved target located at p' = Given (4.F.21), (4.F.22) and 0, the probability density for z becomes for 0 < m < R rb/ 2 R Z <_Z <_Zmid Z[-2 ln(Z/Zmax rb/ R PZ (Z) = Zmid . z[-2 ln(Z/Z -Z < Zmax )]2 ,max 0 el sewhere (4.F.23) where -80- 1 Zmax -r 2 (4.F.24) Q m+ R R 0 I r b b_ 1 (m - R)2 Zmid = exp Zmax (4.F.25) Zmax (4.F.26) r2 rb 1 (m + R)2 min = rb and (4.F.27) exp(-x 2 /2) dx Q(y) = y /i is related to the complementary error function by Q(y) = erfc(y/42)/2. For m > R the pdf for z is rb/2R Zmin Z[-2 ln(Z/Z max Z < Zmid 2 (4.F.28) pZ (Z)= 0 elsewhere -81- where Zmax 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 7 rb [Q{v2 (m-R)/rb} - Q{W2 (m+R)/rb1 -1 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 SATbb is shown as a function of R2 /rl for m/rb = 0 in Figure 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 0 ( = exp[-(X - m)2 /2cy ] 1 / (4.F.30) 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]. unresolved target located at P' = For the case of an 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 Figure 4.5: 0.75 z 1.0 1.25 Target return PDF for beam wander fluctuations; Fan beam, uniformly distributed beam center, m/rb = .45, R/rb = 1.2. 116600-N 60 1 m/rb 50- 1 I 1 1 1 1 1 I 1 1 I I 0 40- ~in Z ~30 20 10- 0.8 . 0 - -10 1 0 12 3 4 5 R/rb 116594-N Figure 4.6: Saturation SNR due to fan beam, uniform beam center fluctuations. -84- 2- 7r b/aX rb x [-2 ln(Z/Zmax)] exp(-m 2 /2a ) PZ(ZZma cosh maxF Z az IFx ZJ mrb[-2 ln(Z/Zmax b C -2 0< Z < 0 max elsewhere (4.F.31) where Zmax is given by m 2/2r2 max = (1 + a/r x )2 + CF2/r2 A sketch of (4.F.31) is shown in Figure 4.7 for m/rb a /rb = 1.25. (4.F.32) b exp = .8 and 4 III Ij III 111111 1111111 liii 0.25 0.5 liii 111111 liii N N 0 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 distributed beam center, m/rb = 0.8, aX/rb = 1.25. 116598-N -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 2 xy2'rr exp{{(X-m )2 + (Y-m )2]/2a2} X Y (4.F.33) 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 r2 rb expV _ i Zmax c 2 1 2 rb pz(Z) =- max 2 L Ia0 bn /- / m o 0 <7Z<7Z - -- max elsewhere (4.F.34) where -87- m12 /r Zm zmax = [1 + 2 a2 /b]exP2 /r ] exp 1b 1 +-a2/ r2 1 ~ (4.F.35) 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 twelveelement 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( x 1 )I2 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(fT 2 ) are zero-mean, complex Gaussian random variables, as is the case for a large speckle target, and < x(l we can say <h L(P' 1) hL Cov(Ix(Y P ) 2 )> = 2I(T T2 )> = 0, which is true when ) 0, then Tl 2 T2 )> so that we now need only the correlation function <x x*>. 2 (4.G.1) 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 <X( 0 { dp' Ts(p') - 2 'L eTTa exp[-Ip' - X2_ 2 exp [-IXLd 12 /Por Trrb Lfc 2 2 /r] exp j2r r S rres d-(' - XLTcj (4.G.2) -89- where fd fTl ~ T2 (4. G.3) T +T (4. G.4) f and the correlation distance pcor is given by r P2c o r + 2-+_ 2 bT (PT/Po) (4.G.5) 2 From (4.G.2), (4.G.3) it is clear that c We find that ec = XS cor/L NT in the two limiting cases PT (4.G.6) o and It can be concluded p o under far-field propagation conditions. PT 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 ut(',z) Tb(p',z) (4.H.1) We assume that T (',z) is a random quantity Ur(I',z) = as in equation (3.A.3). and can be statistically characterized as thought it target were a speckle w ww qw .9!t IZI) ') Az) RADAR !--r(I',Z) SCATTERING SLAB HAS APRA A ANDir IS 14 Figure 4.8: z Single scattering layer; backscattered field. JKAz CENTERED AT - (P',z) is incident field and ur(P',z) is D' = o 116601-N -92- <T b(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 4 7r 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 2 |t( 0 ,z)| 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 Tb A Z2 -93- Combining (4.H.3), (4.H.4) we have =ab(z) p(z) Tb4r 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( ER(p',z) 9',z) (4.H.6) If we let PT(t) T0 = |ST(t)|2 = E 6(t) where E (4.H.7) = 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) hv0 4-7r at(s) p(s) ds) Az exp(-2 -0 { dp'IE0(-1',z)12 IO(5,z)I2 (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 tP dz 6(t - 2z/c) h1 <ly(t) 12> = 0 J 2ab(z) p(z) dp'l (p',z)2j F(p ,z)|2 0 -z -exp(-2 (4.H.9) at(s) p(s) ds) -O Convolving (4.H.9) with the waveform 0 < t < t T t (4.H.10) P T(t) = 0 elsewhere gives the mean-square output of H(f) (Figure 1.3) c 2 t < x~)2> t pP T = dz (t-t hv0 2 I' 4T b(z) p(z) d'1 R(p',z)|2|((p',z)2 ) 2 p z exp(-2 J -0 at(s) p(s) ds) (4.H.l1) -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 p PT2 < 1nt t1 2 b <|x (tb2 h o exp(-2atpz) eld dz 22 )l+ [27rPT 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. treats only scattered light. The MFS theory 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: scanning mode, (2) reduced scanning mode, and (3) (1) full 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. separated by 200 Each detector is 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. limit of a large target return (i.e. CNR In the 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>2 SAT :Var( x1 2 ) (5.B.1) can be estimated by m2 SN SAT (5.B.2) = where m is the sample mean of the squared data N m = and y2 is the sample variance xi |i2 (5.B.3) -100- a2 N Ni=1 (jx2 - - m)2 (5.B.4) 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) = 2) Var(X1 N (5.B.6) = N SNR (5.B.7) The ratio SAT Var(m) 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 probabilities k of occurrence p1p2 k .Z pi = 1. In a chi-square goodness of i=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 2 (f. - np.)z np (5.B.10) i=l where f. is the number of occurrences of the ith outcome and k Sf. = n i=l (5.B.11) 1 Denoting the calculated value of X2 as X we see that X 00 = 0 indicates perfect correspondence with theory while X1 large tends to discredit the hypothesis. A quantitative measure of the validity of the -102- is provided by the level of 'Pk theoretical distribution P1 SP2 '3'' significance a a = Prob(x 2 > X2 0 ) (5.B.12) 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.28) and (4.F.29) via five examples. (4.F.19), (4.F.23), 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 In fact, not a single example of are dominated by random aiming error. 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 C2n and log-amplitude perturbation variance a2 at two wavelengths. For the first four examples of this section the X scintillation measurements indicate that a2 was approximately 0.0005. X (4.F.6) From (4.D.6), SNRSAT STgu = 125 SNR SAT ta (5.C.1) -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 2 = 0.0026 so that time the data for the fifth example was taken cy X SNRSAT = (5.C.2) ta gu Again ~: 24 SNRSAT T was much smaller than this value and we conclude that SAT 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. minimized, calculated value of chi-squared for this data is X with 8 degrees of freedom. This = 4.10 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), 1.4 dB. I'SAT = 6.6 dB, a difference of only 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 ruiui~v u v p i ; 6.0 5.. 4.0 Z(Z) I 3.0 0 a.0 I.* . i .II 0.0 #.So 0.60 0.70 0.30 0.90 1.06 1.10 1.10 I. 1.30 I I 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. t I I IF I I -F- I I I I I I I I I I I I I I I I I S. 9 I- __j I pz(Z) a.s L. 4 e.g I II I 0.50 I -. I i I - lA 1.90 0.75 - m~~ md ~ I - LL..L 1.25 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. r Fi-l P1111 El 1111 1I 1 IF I, 6.0 5.. 4.0 I 3.0 4 I -- e.9 i.e ,.. 0. S.as E e.5. 0.75 .I I I I." 1.2s I -I I M 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. -108- taken over a period of approximately 14 seconds and begun approximately 47 seconds after the finish of the data of Figure 5.1. calculated to be X2 0 = Chi-squared was 14.54 with 12 degrees of freedom corresponding 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 with 0 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 1w ~t-I I a I I I I I I I I I I I I I I I I I I T a.w0 1.s@ 1.00 CD I Pz(Z) 0.50 0.S 'Ii I I - 11111 6.60 II 1 11 I 1.00 I 1.50 I ~I I - - Ii2.50 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. III 111111 I I III I lit I I II I I I *.75 *.s*9 -J Pz ) 0.2S .U£ L I 0.25 S.69 i Iliad 0.75 . . . . .A . A I 1.S0 I . . I i . I.25 I I III .. 1.6s III . I II . - . 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. -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 Iw 3.,1 a.. -J Pz ) 1.0 0.0 L S.o I 0.4 I I I 0.3 1.3 1.6 I I U I 3.0 1.4 I 3.3 I U I U I 2.3 3.1 I 4.0 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 -115- p (Z) z = /2 1 2a Z exp- X with g 2 X = 0.0045. 18G2 L (ln Z + 2u 2 ) X U(Z) (5.D.l) X Simultaneous scintillation measurements indicate that a 2 = 0.0027, but, the value of 0.0045 was chosen to minimize the X 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 equal, ~ 25 prads. Since these two numbers are very nearly 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 .W 1W ( 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- LU i ; i . I I - I .I . I . . 111MkIT1MU1, I -- ----31 17 Brad 29 IW I F-r-T--r-T I 0B tThtYt-t-t-t-99999+999@~@-i--I-f** 27 25 I FFVrFrrTIYrrmTYY1-l-Y-tIY1-t~tw-.- 0 20 -t+9. l-. H- 1 60 40 4-01 1 -I-44 . I40@+-4 O4* 14@4-14 [-14-1-14 &4-&-464 14-44--1 1-1 1 1 & 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 . ,. . ,. . .. . . .. Mw w 1w I I I 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 I I I I I I I I T 1 1 1 1 1 1 1 1 i -" - . . . . 1 1 1 1 1 1 1 1 1 1 1 L--L 84 . . . . .. . . . . . . . I 7.8 prad 82 80 . I . . I . I . . I . ; i +-V4-t-4 -+-4-4-4-4-4M4-4 I 14 LYITI----HT±.---------TF~TTfl~l7MftTitu1 Ufi llLLTIUJJ tl1LtIJ~l~I1~ -4 73 C, I 75 -- - - - - - C 111111 If I I I I I I I I I I I I I I I I I I I I I I1- 20 40 60 80 100 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. -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) Simultaneous scintillation measurements with a2 = 0.0138 to minimize X2. X indicate that a2 = .0189. Chi-squared was calculated to be X2 = 12.78 0 X 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) to minimize X = 12.85 with 15 degrees of freedom. 2 = 0.018 where a' X 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 X2 X 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 2 a = .004 minimizes X2 = 11.93 with 11 degrees of freedom indicating X a level of significance a between .30 and .50 and, again, excellent w w qw SIII 1111111113i 1 i1 11 iI -1 3.S@ 1.50 pZ(Z) i.4s .mE.'- Mo, *.0 0.4 0.3 l.a 1.6 3.0 3.4 8.S 3.1 3.6 4.4 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 -1I 1.50 7 ( 7 1.86 P (Z) 0.60 I 6.8 I III - O.6 0 - I I * ~ I I M) U 1 1.016 I I II I .d. I & 4 iii 1. 5 j~~l 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 MW RW -IL 0. I I I I pz (Z) tse pzz 0.0SAS5 1.00 1.50 3* 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 II ii gI li i i li i i I i i Il 3., 2.9 pZ(Z) 1. . S.. 1 i. 0.. .i . . ..~ 0.S0 ~.I _. .I1 . 1 . .1. . I I I I I I . i I '.se I.49 L I- 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 x = 0.004. -123- agreement between theory and data. 2 X The scintillation measurements give .0026 for this figure. 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 minimizes X 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. example) is squared. The data in this example (and only this 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. X The calculated value of = 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- 10 100 2a - X 100 2 8"10 10' 102= o -points plotted against minimizin1 10 4 1I3 1O6 2 ol f pints lotted aga ins minimizig 22 xX Figure 5.13: The theoretical curve (4.D.6) and estimates 'SNASAT from 5.9-5.12. data of Figures 5.6, the 00040 I . I I I I I 1 1 1 1 1 1 1 I 1 0 . 0030 0.0020 N, Pz (Z) 0.001 0 0 - 0 0. I --- 500. 1000. I 1500. .L iJ I I I I 2000. Z Figure 5.14: Normalized histogram of 2000 squared, consecutive speckle plate returns taken in full scanning mode and exponential PDF. w w Aw 7K1 =I11 , 1 1 1 1 1 1 1 1 1 11 0.75 0.50 a.) pz ) 0.25 0.as 11 I I 0 .0, 0.4 I 0.8 1I 11I I 1.8 1 I1 1.6 I9 1- 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. -127- Pz(Z) = (5.D.2) Z exp[-TrZ 2 /4] u(Z) 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 x20 = 14.92 with 16 degrees of freedom indicating a is between 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 X20 = 23.44 with 19 degrees of freedom indicating at is between 0.20 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 a 2 = 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 X0 = 285 w Vw low I I I U I A I I I iI I I @.76 0.50 N) Pz 0, ) .as 0.0 0.0 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. w w w - I I I I I I I I I I I I I I I I I A I I I 1.00 T 0.5 N pz(Z) 0.26 i-i III 0. * 0.4 0.6 III 1.3 I LI 1.6 11111111 _ a.$ 3.4 i.E 3.8 _ _ 3.6 _ _ 4.0 _ j 4.4 Z Figure 5.17: Normalized histogram of 1400 consecutive speckle plate returns taken in staring mode and Rayleigh PDF. -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. were to view a "rough" target through free space in starina If we 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. of X20 = The value 13.51 with 20 degrees of freedom in Figure 5.18 indicating a MW 1.80 0.75 0 * 50 Pz CA) ) OS 0 .0.4 --, 1.8 I I 8,.$ I ~ - I-A L 3~ -.8~ 3.9 dL. 4.0 414 Z Figure 5.18: Normalized histogram of 300 speckle plate returns taken in full scanning mode and Rayleigh PDF. qw MW 0.76 - -' *aEO N) pZ(Z) 0.26 kil Ss.6 0.0 0.4 -L 0.1 1.a 1.8 3.0 3.4 eA. 3.8 . .8 4.0 .. I.. 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. -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= 0.01 0 to x' = 13.20 with 19 degrees of freedom in Figure 0 5.19 indicating a is again between 0.80 and 0.90 and excellent agreement between the PDF and the data. a 2 = 0.15. For both figures, scintillation measurement gives 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 a 2 is more than an order of magnitude away from the measured value. case is not as yet clear. But the scintillation measurements indicate that the atmospheric coherence distance p beam size. Exactly why this is the is smaller than the radar 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. should be directed towards understanding this. More effort -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 {:?:ng(t) n (t) r.(t) H0 = , (6.A.1) 1 -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 x = hv) T .i df' T(p') ERip' EFip' = 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 r1 L (6.A.4) H0 The likelihood ratio L is given by M L = CX2 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) (6.A.6) dtj 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 < H0 y (6.A.7) -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 > yjH 0 ) (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) ln P (6.B.4) so that (6.A.7) becomes H Ir 2 - 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. If rx! 2 jx 2 Nonrandom = 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 D =12 2 + 2!xI)_|2)I 0 (z LxI) (-2lnP F) = Q(/2-- (6.B.5) 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 ) satisfying (4.D.l)-(4.D.3) the detection probability is [3,5] 4A PD where dx p (x) Q[(2 CNR xA gu e 2 X) e2 X,(-21nPF) ] (6.B.6) -140- p(x) = X V2Tra 2 exp[-(x + a2)2/2&21 X X (6.B.7) X is the pdf for the log-amplitude perturbation X. Case 3. Ix1 2 Exponential Here we consider the case IX2 (6.B.8) = CNR v 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 D = p(CNR + 1)l F (6.B.9) 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- (6.B.10) 2 U +l)_l P (CNRsre dU pu(U) F u PD where Pu (U) exp[-(U = 2 + u 2 )2 /2a (6.B.11) )] v/2Trcf2 is the pdf of the aperture-averaged log amplitude perturbation u. Case 5. Ix1 2 A Product of Two Exponential Random Variables The case of a small glint target in low visibility is considered here Ixi 2 (6.B.12) = CNR v w where v, w are IID unit mean exponential random variables. The probability density for the product y = vw is given by (6.B.13) py (Y) = 2K (2vY) u(Y) where K (-) is the modified Bessel function of the second kind of order zero, and u(-) is the unit step function. (6.B.13) is shown in Figure 6.1. case is given by the integral A sketch of The detection probability in this -142- 2.0 1.6 ... 1.2 0.8 0.4 0.0 0 Figure 6.1: 1 2 3 The Probability Density Function, p (Y) yi 4 = 2 K (2Y) 0 -143- ( f P = in P 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 In Figure 6.2 the detection probability PD is plotted atmospheres. 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. plotted vs. CNR for |X 2 In Figure 6.4, PD is 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 SINGLE PULSE IXI2 =IX2>VW 99.9 99 95 90 2 70 P =10 F 50 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-- 99.99 99.9 I ~_ SINGLE PULSE I X|2 I I 2> VW 99 CNR 40 dB 95 90 C 70 C., a, 0. 0 C- 50 dB _CNR=20 30 10 5 0.1 CNR = OdB 0.01 1010 -6 106 -4 -2 PF Figure 6.3: Single pulse receiver operating characteristics for a glint target in bad weather. .l46- 99.99 1 1 1 1 / I 1 1 SINGLE PULSE 99.9 PF F 99 P 0-7 2= 95- .2 = 0.01 90- x/x 2 = 0.1 2 =0.5 70CL 50 a. 30- , - |_ 2 2 =< X( >VW 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 observe X 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 if X 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. about this in the next chapter. in the Ix1 2 = More will be said Again, as in Figure 6.2, the bumps <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- = CNR, For CNR i = 1,2,... ,M this is not too difficult. Consider the likelihood ratio (6.A.5) M L = i R i=1 (6.C.1) where R. = For jx.1 2 = dX p (X) e-X IX{2X } (6.C.2) CNR vi wi, (6.C.2) becomes R. = dX 2K 2 S10 jdX/CKo N -NR j e I{2X Ir 1} o -I (6.C.3) This integral has been evaluated numerically using cautious adaptive Romberg extrapolation [29]. where R is plotted vs. The results are shown in Figure 6.5 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 101I CNR = OdB 10 CNR 20 dB 10 8 106 105 4 10 R 103 10 log R = 3.1 + 0.344 Irl2 CNR zOdB CNR 10 10-2 20 dB - CNR= 40dB - - 10~ 3.1 + 0.344 log R -3 r|2 -4 101 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. -150- log(R 1 ) = A(CNR) + B(CNR) Ir 1 2 (6.C.5) or R. = eA(CNR) eB(CNR) r*iJ2 (6.C.6) where we have explicitely noted that A and B are functions of CNR. Using (6.C.6) in place of (6.C.3) in the likelihood ratio (6.C.1) gives, for equal CNR, the Neyman Pearson test H Mi Ir 2 (6.C.7) Y 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 H Z= r (6.D.1) Y H0 for several target/atmospheric scenarios. The use of (6.D.1) in the 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) can -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 F r(MMY) (M- 1)!'' (6.D.2) a- (6.D.3) where F(a,x) = e-u du 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. accurate approximate results. Hence it is worthwhile developing 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. PF Hence the results that follow only apply in the regime 0.40, PD > 0.60. Defining two conditional semi-invariant moment generating functions 10(s) = M ln<exp(slr|j2 )H> s > 00 (6. D.4) -152- s < 0 'Pl(s) = M ln<exp(slr12) IH 1> (6.D.5) where s is a real variable, approximate expressions for the false alarm probability PF and miss probability PM PFF = x 1 - PD are oi(s 00 (S 0 )/2] Q[/vlgso) so] 0 0 ) - s0 0 (s 0 ) + s20 (6.D.6) QE-vil(s, ) sl (6.D.7) P = exp[il(sl) for M = - sl S + l(Sl )S 2 1 5. In (6.D.6), (6.D.7) dot denotes differentiation with respect to s, Q(-) is the complemented error function .co (6.D.8) exp(-x 2 /2) dx Q(y) = /- r and sl' s1 are solutions to the equations respectively. My = O(s) (6.D.9) my = l (S) (6.D.10) The approximation used to derive (6.D.6), (6.D.7) was an application of the central limit theorem [15]. Specifically, -153- M j tn the sum 2 is taken to be a Gaussian random variable even i=l though the Ir.! i = 1,2,...,M are not Gaussian. M Hence, the requirement 5. For the PF calculation H tn.!22 , a r2= is true so that unit mean exponential random variable, hence s< 1 y1(s) = -M ln(l - s) (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 P M(3s /2 - s M (1- s )~ exp F o (L- s0) s Q 0 L (6.D.13) ~" 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. as a series of examples. We present our results -154- 100 90- 80- 70-12 PF 10 60- M 50- 40- 30- 20- 10- 0 5 10 20 15 25 30 35 Y 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 - s ln(-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 Ix1 2 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 mean exponential random variable. applies to the three scenarios: = CNR v where v is a unit As before, this characterization -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. (6.D.16) s < CNR + 1 Resolved Speckle Target in Turbulence For a target with radar return described by (4.D.9) yi(s) becomes [31] ii(s) = dU p(U)[1 - (1+ M ln { CNRsr e2 U)s]l} s < 0 (6.0.17) with Pu (u) 2 exp[-(U + a 2 ) 2 /2C ] = (6.D.18) 2 /2 ar Case 5. Ix 2 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 - ey- dt ln (-sCNR) + ln Ss - 1 sCNR s < 0 (6.D.19) -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 Ix1 2 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 a 2 , and in low X 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 atmospheric X 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- 99.99 I I I I I I 99.9 99 F- M =30 95 -/ 90 - M =10- 70 CL C. 4) . 50 30 2 VW |X|2 10 -P ~ 5 4 1=M1 F 01 0.1 10 U 20 30 40 CNR (dB) Figure 6.7: 2Single-pulse and multipulse detection probability vs. CNR for a glint target in bad weather. PF= 104 throughout. -159- v 99.99 I 99.9 99 K- = 30 M 95 90 M 1 70 4, 'a 50 < PI 30- jVW 10 12 10 5 S>V Iw 0.1 0.01 I -20 I -10 0 10 20 CNR (dB) Figure 6.8: Single-pulse and multipulse detection probability vs PF CNR for a glint target in bad weather. 1 12 throughout. -160- 100 90 P M = 0.01 P= D 0.99 PF = 10-12 70 - GLINT TARGET LOW VISIBILITY bu |>12 M 2> VW 50 40- 30- GLINT TARGET 20 -X 2= 0.05 GLINT TARGET, FREE SPACE GLINT TARGET 10- 0 10 20 30 40 50 .2 = 0.5 x 60 70 CNR(dB) Figure 6.9: The number of pulses M necessary to achieve PF = 116596-N -12 pD = .99 vs. CNR for a glint target in free-space, two turbulent fluctuation levels, and bad weather. -161- rv-~I~i I I 90 1 *1 I 0.01 P =0.99 D 10-7 M F so - I 70k- p --- = 0.01 P = 0.99 10-12 M PF I 60K GLINT TARGET, LOW VISIBILITY 2 1 lxi2 <x > vw M 50 40 30 . 20 10 0 10 20 30 40 50 60 70 CNR (dB) 116595-N Figure 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 visbility. -162- PF 12. -10 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- 99.99 S=0.01 -x. 99.9 )V W I - =0.5 A -2 99 1XI - -x 95 90 - 70 I C a, C.) 50 a) 0. 0 30 - a- 10 5 PF = 10- 12 M =10 1 0.1 0.01 0 10 i I 20 I 30 1I 40 CNR (dB) Figure 6.11: Ten pulse detection probability vs. CNR for a glinti target in several turbulent atmospheres and a scattering atmosphere. PF = 10-12 throughout. 99.99 2'2 I2 II >vw 99.9 - X .1 2 x = 0.5 X 99 ,/ - 95 I -- 90 70 C U 50 0. 0 0~ 30- K 10 1012 - 5 F F M = 15 1 01 I01 0.01 L I Ui 1u I 20 I I 30 40 CNR (dB) Figure 6.12: Fifteen pulse detection probability vs. CNR for a glint target in several turbulent atmospheres and a scattering atmosphere. PF = 0-12 throughout. -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 CNR 0 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 scatteredlight 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 -167- WAVELENGTH X 10.6 -pm PHOTON ENERGY hv0 1.87 x 10-20 PEAK POWER PT 10 kW PULSE DURATION tp 100 ns BEAM DIAMETER 2PT 13 cm DETECTOR QUANTUM EFFICIENCY r 0.9 46 Table 7.1: CO2 Laser Radar System Parameters -168- WAVELENGTH A 1.06 pm PHOTON ENERGY hv 0 1.87 x 10 PEAK POWER PT 10 kW PULSE DURATION t 100 ns BEAM DIAMETER 2PT 0.5 cm DETECTOR QUANTUM EFFICIENCY rn 0.9 Table 7.2: Nd:YAG Laser Radar System Parameters 9 1 -169- cover. Two layered profiles are considered in this chapter. consists of a layer of scatterers of thickness The first 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. in Figure 7.2. This second situation is depicted The formulas of Chapter IV, Section E do not apply without modification to the layered scattering scenarios. necessary modifications are not too extensive. Fortunately, 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. RADAR FREE-SPACE L SCATTERING LAYER I FREE-SPACE s L L Figure 7.1: Geometry for a single scattering layer between radar and target. TARGET RADAR FREE-SPACE SCATTERING LAYER TARGET -4 Ls L. L Figure 7.2: Geometry for a scattering layer near the target. -172- Extinction Coefficient St (km1 ) Modified Scattering Coefficient ' (km~ ) Backscattering Cross Section Per Unit Haze Cloud .005 10 .0005 4 .0005 .2 50 50 - Volume (km1 ) RMS Forward Scattering Angle Table 7.3: 6F (mrad) Atmospheric Parameters at CO2 Laser Wavelength -173- Haze Extinction Coefficient t (km~ 1 ) Modified Scattering Coefficient s' (kmi Cloud .07 20 .05 17 .01 5 ) Backscattering Cross Section Per Unit Volume (km 1 ) RMS Forward Scattering Angle 10 1 eF (mrad) Table 7.4: Atmospheric Parameters at Nd:YAG Laser Wavelength -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 Since the integration interval in (4.H.12) is ct p/2 = 15 m, systems. 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 L1 meters from the radar < 1Sbi) 2> <h, x-(L 'nIt 0PT h2o t 2 cobP P22 T -L. 2 dz c- t 2p 1 + 2 27T 2 z2 (7.A.1) 70 50 fi1jf~z~f~~ -- 30 A C%4 - V-__ CO 2 System 10 V ~1 - -10. t Nd:YAG System -7 LL-~-- ----------- -30 0 20 40 60 t(iisec) Figure 7.3: Normalized backscatter power from a uniform scattering profile vs. t. -176- This maximum normalized backscatter return results when the pulse of length ct p/2 (meters) has just entered the scattering layer. case, extinction can be neglected as exp(-2 t ct p/2) duration considered. In this 1 for the pulse 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 Ls) 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 ~K ~ CO2 System 30 ^M -77- Nd:YAG System A :f C\J ><P 10 v ~Iit -~ 10 -~ iL::yiiI$:T7!1i7:7f:::.: :7: ::Z7 ~ - :- -~:c:::r:V -30 0 2 4 6 Li (km) Figure 7.4: Maximum normalized backscatter power from a scattering layer L. meters from the radar. -178- 100 80 CNR0 et- 2 tL L-:rr_ _= 60 L N 40 CNR L 4 20 -- T ____- ~---- -- '77 0 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. -179-, 80 60 -= o -2 tL CN R et Ci 40 -o ~I-v I~ LAL -~ -~ 20 CNR §1272:. 0 77_ 77 7:: 7 L= 1 ----- -20 0 - z__ L'=7 8 4 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. -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 in place of (2.C.3). 2 0 3 kol(7.A.2) = 2 o ' Ls O k2 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 11 80 CO 2 System P~7iZEZ2Z27 60 ~Th~- E - 4k I V7t~z~ S.- 40 20 Nd:YAG System ~z~zc~z~Z2V77 0 400 200 600 Ls (m) Figure 7.7: Atmospheric beamwidth for a single layer scattering profile vs. layer thickness. -18280 - _7 60 CNR 0 e-2tLs 40 tR\7 t-Z 20-- CNR --7::_ 0 ' -20 C -- 400 200 600 Ls (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. 70 50 30 - i __ 10 - * CNR -10 - t CNR0 - 7 ---- --- s ---- -30 200 C 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. -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 layer thickness Ls. = 2 km, L0 = 1 km vs. scattering 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), appropriate if in place of (2.C.3) we use (4.E.8) are again -185- p 2 0 'Ls e2k 2 (7.A.3) ' 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 gate properly we can receive the true MFS target return. we range 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. -186- 1.0 0.8 0.6 ------------z S.- 0.4 0.2 -------- - 0 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 -CNR T0 CN RO e -10 7 7_ F7e 2: - t -30 C 400 200 600 Ls (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 These are in a uniform scattering profile apply. 2 CNR = CNR 0 e 1/3(XL/Trpin )2 (7.A.3) r2 1 + (XL/2rpT P 2 +T +c where 4r 4 rlP tp CNR 0 = T h\)0 s p (7.A.4) [(XL/27r T2 + pZ] as before and rb rb = ( ( L/7Tpin) 2 22+(L/21rpT) + XLw 2 (AL/7T) (7.A.5) +p +A2rP rb (7.A.6) r2 c r [ (AL/Tpout )2 + p2] - [p - (ALrpmid 2 -189- 2 p out 2 k2 ' L2 e2 L3 - (7.A.7) L s F1 p in 2 k2 e2 sF P2mid 6' k22 L (7.A.8) L3 s 82 s F L2 3LL 2L3 - s (7.A.9) 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. difficult to "see" for this target. Hence the MFS power is 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 dominant. = < 2> vw case) requires that the MFS return be It also requires that the target radius rs be less than the field coherence length rc. In Figure this case vs. scattering layer thickness. 7.12, rc is plotted for We see that for Ls > 300 m, where the MFS return dominates the condition rsS approximate equality. rc holds with -190-. 1.0 I 4~ z 0 .Q1008 0.0 006 i7I 7 E C-) 5- 0.0 004 ---- ---- 0.0002 77 7. - 0.0 0 -7--.7 -- ------ 400 200 600 LS(in) Figure 7.12: Field coherence length for a scattering layer near the target and the Nd:YAG system vs. layer thickness. -19170 -Lr -- - - - T- 50 T__-- - - - - Ju t. - -' - -~ - ,7, 7 1-:t H7 ,7- 7 + 7_ =tE= .. ... .. ... . .. ::i 7 --CIN R 7- 1171-~-x\ I11~1 j~~J - X .-m-- M .. ....CINR 30x.b - 0 <K.~jL2V2ee2BtLs 400 200 j1Z2Ft1" K e -26tLs 600 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. -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 heterodynereception 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 the radar in a scanning-imaging application, first to analyze 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) 0 / e / 1 ... 00 d/ IIb da Figure A.: Geometry relating to the definition of specific intensity. J01 -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 In (A.2), t(s) t I~r~ 5 4wf dZ' p( T,Q) I(r,Q') = 0 (A.2) 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) 4 7r 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 where - [Ij2) ~ (il) (A.4) -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 P = jd r Jdp' ~r d 2 p P 2,!L) -circ(2152 where r(P, 2 ,z) = exp(-j k y -p'-(p o/d) -p 2 ))circ(2 1- --p /dR cir(21p' - pDI/d) (A.7) <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 INDEPEND ENT SPATIALLY VARYING PROPAGATION MEDIUM I- DIAMETER dR FOCAL LENGTH f CENTERED AT 0 DETECTOR AREA A = ird CENTERED AT u~,L) .... D I C3 §0 Z=O Figure A.2: 4 =f i Z=L Z= L+f Geometry for relating the specific intensity and the mutual coherence function. -202- the (z = L + f) detector plane and circ(2[pl/d) is a circular pupil function defined by I 1PI < d/2 circ(2I5I/d) = (A.8) 0 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 P dp C =I dpPd A C) ) C VL) , exp(--jk 's Pd )cic_1cT -o 2- circ(2I-pc + 2 -P0 /dR) 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 '(c' d,L) = is reasonable to assume that ,pdL) -'(p 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 pd > Pa (A.11) where pa << dR/ 2 , equation (A.10) becomes A P 7rd2 =R dd (A.12) Fo'd,L) exp(-jk s-pd) If we write this same detected power in terms of the specific intensity we have rd2 P = I(P o,s ,L) R 0 4 A D (A.13) 2 Combining (A.12), (A.13), we have that r' and I are related by a Fourier transformation I(f oL) T' (PC') d, L) =i = drdfr dL) exp(-jk - d s I(-Pc'5s0,L) exp(+jk so -Pd) (A.14) (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) cPdt Jk '7d -7cr(c~dz 9 - + ,P ,z) cpd~ (A.16) s(Td) F'Cpc' d,z) = 0 where P(Pd) is the Fourier transform of the phase function p(s) jks *d P(Pd jds 0p(so) e 0 (A.17) d Equation (A.16) has been solved [18,28] and '(pc' dZ) = j Jc' '(', dpd r dp -z - exp - ',0)e(Xz) rr 2 ep j Pd rz S) dS - I a.o s(s) C + 1 - P zpl-]+pd s c d 1 ds (O (A. 18) which implies <hL (,p -) h*(-2,p )> = ho(p L -exp - ,5 ) ho(, { ( a(z) dz - 2 '2I) L BS (Z) 1 - P [1 .0 .0 + zp dz (A. 19) -205- 5 For a plane wave input r'(p', ,0) JU1 = 2 the transmitter plane integrals (A.18) can be performed and r'(pc'PdL) = JR12 exp[- L ' rL exp -[1-P(Pd a(z)dz 0 (z)dz] (A.20) If a Gaussian form for the phase function is assumed exp(- Is12 /2 2) (A.21) F then P(pd) is P(d)= exp(-k 2 e .Id 2 /2) (A.22) 1 - k 2 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 S(z) = S' a(z) = then (A.19) becomes <hL (1,p ) h*(p 2 ,p )> = ho(phpi) h (p2' -e-aL expf Ed 2 2 + d d+ Id 3p 2 (A.23) -206- where p2 0 (A.24) 2 asL k2 e which for plane wave input implies -1 - L r'(PC'Pd,L) = JU 2 e 12 p (A.25) e Equations (A.20) and (A.25) have been plotted in Figure A.3. clear that if It is 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' 1 2 Jul 2 2 I yI e-pa L L - 2 -e3s L (I- P(p ) e~Ra L -I 2-P4 2 e-(lea + G) L 0 d Figure A.3: MCF's for plane wave input. -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 S= In place of (A.26) @ ; = Na + s(1 s (A.27) a, s in (A.23), (A.24) where a+ a + s s t (A.28) and -e =2Tr p(8) sin e de -10 (A.29) -209- 4 p (e) GE 8 90 8 (deg) Figure A.4: Real phase function. 180 -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 (A.30) 0 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- REFERENCES 1. J.H. 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Van Trees, Detection Estimation and Modulation Theory, Vol. 1, J. Wiley and Sons, New York, 1968. 24. J.H. Shapiro and C. Warde, "Optical Communication through Low Visibility Weather," Optical Engineering, January, February 1981, Vol. 20, No. 1, pp. 76-83. 25. A. 26. D.M. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2, Academic, New York, pp. 296-301, 1978. Papurt, J.H. Shapiro, and R.C. Harney, "Atmospheric Propagation Effects on Coherent Laser Radars," to appear in Proceedings of the SPIE, Vol. 300. -213- 27. J.H. Shapiro and S.T. Lau, "Turbulence Effects on Coherent Laser Radar Target Statistics," to appear in Applied Optics. 28. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1, Academic, New York, 1978. 29. C. de Boor, "CADRE: An Algorithm for Numerical Quadrature," Mathematical Software, (John R. Rice, ed.) Academic Press, New York, 1971, Chapter 7. 30. I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products, Academic Press, New York, 1980. 31. J.H. Shapiro, "MIT Target Detection Analysis," Internal Memo, Lincoln Laboratories, Lexington, MA, December 29, 1978. 32. H.L. Van Trees, Detection, Estimation and Modulation Theory, Vol. 1, J. Wiley and Sons, New York, 1968, Sect. 2.7. 33. S.J. Halme, B.K. Levitt and R.S. Orr, "Bounds and Approximations for Some Integral Expressions Involving Lognormal Statistics," Res. Lab. Electron. Quart. Prog. Rept. 93, 163-175, MIT (1969). 34. E.V. Hoversten, R.O. Harger and S.J. Halme, "Communication Theory for the Turbulent Atmosphere," Proc. IEEE 58, 1626-1650 (1970). 35. S.T. Lau, "Pulse-Integration Detection Performance," Internal Memo, Lincoln Laboratories, Lexington, MA, July 23, 1981. 36. G.C. Mooradian, M. Geller, L.B. Stotts, D.H. Stephens, and R.A. Krautwald, "Blue-Green Pulsed Propagation Through Fog," Applied Optics, Vol. 18, No. 4, pp. 429-441, February 1979. 37. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions, American Elsevier, 1969. 38. R.L. Anderson and T.A. Bancroft, Statistical Theory in Research, McGraw-Hill, 1952. 39. C. Chatfield, Statistics for Technology, Halsted Press, 1970. 40. T.S. Chu and D.C. Hogg, "Effects of Preciptation on Propagation at .63, 3.5 and 10.6 Microns," Bell System Technical Journal, May-June 1968, pp. 723-759. -214- 41. R.S. Lawrence and J.W. Strohbehn, "A Survey of Clear-Air Propagation Effects Relevent to Optical Communications," Proc. IEEE 58, 1523-1545, 1970. -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.