The signal/noise ratio concept in underwater optics

The Signal/Noise Ratio Concept in Underwater Optics

H. Hodara and R. J. Marquedant

Underwater ranging capabilities of light are well known to be several orders of magnitude below those ofsound waves. This limitation rules out its use for locating targets. However, the desire on the part ofthe observer to map out a visible two-dimensional display has spurred the applications of light sources tothe problem of target recognition.

1. Theory

Light from a collimated source, whether coherent orincoherent, attenuates in water at an exponential rate.In many cases the system performance is not limited byabsorption but by scattering from particles present inthe beam path. It is convenient to divide the scatteredcomponents into backscattered and forward scatteredcontributions. If the former greatly exceed the re-flected light from the target, contrast is degraded; thesystem is said to be contrast- or backscatter-limited.If forward scattering predominates, the beam angulardivergence may increase over a path long enough tomake the system resolution or forward scatter-limited.As shown in Fig. 1, the scattering properties of waterexhibit much greater directivity in the forward than inthe backward direction. Under these conditions, thecontrast limit is reached at ranges much shorter thanthose where resolution limit occurs. We restrictour discussion to the effects of backscatter on contrastand signal detectability, since it is the first obstacleencountered in trying to extend range for target recog-nition purposes.

Contrast is an ill defined quantity. There are manydefinitions, all of which express a ratio of target-to-backscatter received power. Let P1 and P2 be, respec-tively, the light received from adjacent target elementsdiffering slightly in reflectivity by p = P2 - p1. Themeasured quantities P1 and P2 can be expressed in termsof the target light PT and the backscattered componentPb:

P2 = PT2 + Pb, (la)

P = PTI + Pb- (lb)

The authors are with Tetra Tech, Inc., 630 Rosemead Boule-vard, Pasadena, California 91107.

Received 23 June 1967.

The contrast C is defined as the ratio of the powerdifference to the average total power:

C = (P2 - P,)/2(P2 + P,). (2)

The minimum detectable contrast Cmin is related to theminimum detectable power difference:

APT = PT2 - PTI; (APT << Pb),

through the preceding equations by

Cmin = APT/Pb-

(3)

(4)

Both PT and Pb are directly proportional to the sourcepower PL. If the minimum detectable contrast isreached, further increase in source power does not en-hance it because the ratio [Eq. (4)] remains constantindependently of PL. This has led to certain miscon-ceptions in underwater optics, namely, that underbackscatter-limited conditions, once the minimumdetectable contrast is reached, increasing the sourcepower does not improve visibility. However, visibilityis another ill defined concept and we shall use it spar-ingly. We deal instead with the more exact concept ofsignal taken from communication engineering. Weshow that a more appropriate criterion for the perfor-mance of an underwater electrooptical system is thesignal-to-noise ratio. The signal-to-noise ratio conceptis not new in optics,' but it is important enough for ourapplication that we define precisely what is meant bysignal and noise.

If a laser beam scans a target, the relative intensitychanges in received power from the various targetelements constitute the signal. The diameter of thebeam at the target determines the minimum resolutionelement. The total amount of information is indeeddependent on both the resolution element size and mini-mum detectable intensity level change between adjacentelements. Conceptually, information can be thoughtof in terms of binary digits; we may then consider in thefollowing discussion two resolvable adjacent elements ofthe target to be a white strip and a black strip, corre-

March 1968 / Vol. 7, No. 3 / APPLIED OPTICS 527

100~~~~~0300

40 .40 0

if' , 60 : .. 700H 00

0-10 20 30 4 60 70 46064 3 0lo 0 go

* 00~~~~~~~~~~~I04 -- \ 70,

140°,

I

i~~ ! i50 - I .w-- 009

I 160°8300 1700

Fig. 1. Scattering angular dependence.

larger random fluctuations. If we think of the informa-tion as represented by a time varying electrical current,the signal current varies directly with the received powerP, while the shot noise current varies as (P)1

2. Thesignal can be pulled out of the noise either by increasingthe laser power or by integrating over a time large com-pared with the fluctuations correlation time, by super-posing n displays on a high persistence screen. In eithercase, it amounts to increasing the energy of the receivedsignal; correspondingly the signal-to-noise ratio in-creases as (P)'1 or (n)'11. This is illustrated in Fig.3(b), where the shot noise fluctuations no longer maskthe received power level difference from the two targetelements; the signal has been pulled out of the noise andis now detectable.

The above can be put on a simple mathematical basis.We remove the restriction of black and white targetelements and assume two reflectivities, P1, P2 such that

A = P2 - I << P = 2(P + P2). (15)

The sum of the received target and backscatteredpowers PT and Pb of Eq. (1) contribute to the detectedcurrents from the corresponding target elements,

sponding to binary digits 1, 0. There is no loss in gener-ality by doing so rather than taking two strips differingin reflectivity by Ap, corresponding to two gray levels.We discuss, qualitatively, the black and white case andthen derive the pertinent mathematical expressions forthe general gray level case. Figure 2(a) illustrates atypical scan in space x or time t in the absence of back-scatter (no water) and Fig. 2(b) with backscatter. Theeffect of the scattering medium is a reduction in con-trast, as well as in signal APT caused by the addition ofthe backscattered component Pb to the received return.Note that if we were to increase the laser power by afactor of ten, both signal APT and backscatter Pb in-crease by ten, thus keeping the contrast unchanged.

So far we have considered our system and medium tobe noiseless, that is, with no random fluctuations indetected intensity. In the absence of noise, the infor-mation contained in the signal of Fig. 2(b) is as com-plete as the one contained in Fig. 2(a). Indeed, with nonoise present one can always amplify the signal APT ofFig. 2(b) to the level of Fig. 2(a). As is well known in anoiseless system, there is no minimum detectable signaldifference. Note that the abscissa of Fig. 2 can eitherbe identified with distance x or time t. Our signal intime is then made up of an ac wave of amplitude APT SU-perposed onto a dc component of level Pb. What limitsdetectability is noise, introduced by the laser, the me-dium, and the detector. We show in Sec. III that the firsttwo sources of noise are negligible and only detector noiseis significant. At optical frequencies, the predominantsource of fluctuations is shot noise. It is this noise whichplaces a limit on the minimum detectable signal, that is,on the power level change APT between adjacent resolv-able elements. Detectability is impaired when we canno longer distinguish the difference between the twolevels [see Fig. 3(a)], because it is masked by the much

i.)= (es1/hv)P2,

ii = (eo,/hv)P;

(6a)

(6b)

e is the electronic charge, 77 the detector quantum effi-ciency, v the optical frequency, and h is Planck's con-stant. Note that the detector does not separate the

PT+ Pb

ort

Ca)

XOr I

(b)

Fig. 2. Signal return from target: (a) without backscatter, and(b) with backscatter.

IN ALL CASESLEFT SIDE WHITE SIGNALRIGHT SIDE BLACK SIGNAL

PTAPb

SHOT

NOISE SHOT PbNOSE

(a)

SHOT

NOISE I

III SHOTI P

5

NOISE

x o t

(b)

Fig. 3. Signal with noise: (a) signal indistinguishable, and (b)higher energy signal now distinguishable.

528 APPLIED OPTICS / Vol. 7, No. 3 / March 1968

6.

P

PT+ Pb

power contributed by the target elements from thebackscattered component. We can conceptuallyseparate them as long as our final expression for thesignal-to-noise ratio does not involve nonmeasurablequantities. Thus,

i2 = iT2 + ib,

1 = iT1 + ib,

(7a)

(7b)

where

iTI,2 = (e/hv)PT.,,,

ib = (e7/1hv)Pb.

(8a)

(8b)

The detected signal power P expressed in terms ofmeasurable quantities is

P_ - ( - il)2,

which together with Eq. (7) yields

OUTPUT

MAXIMUM -DETECTABLE\ SIGNAL

DETECTABLESIGNAL

(9)

/'tATURATION LEVEL

',

_____ I ___ THRESHOLD LEVEL

I I,, I I/ -, , I III I I

I I

CI I - Pt =E (INPUT)

Fig. 4. General detector response behavior.

Pso - (iT2 - i ) 2 (1 0 )The shot noise power through a post detection band-width B is

Pno - 2eB(iT + ib). (11)

We have taken iT to be the average of T2 and i Itdoes not introduce any significant error in the calcula-tion of P,o because of the small difference betweenreflection coefficients as set in Eq. (5). The resultantsignal-to-noise ratio is

P,, X (PT2 - PT) (12)P-o hvB PT + Pb

The final expression in terms of measurable quantities,taking into account the geometry of the experiment, isderived in the Appendix:

P.,o (nPL(a e-lxcz)Pno (hvB ) a2 + 2(-a2

X 1 + b/pa( + 2aa)(z2 + a2)(e+2al _ e-z/a)] (13)

PL is the laser power output, z is the propagation path inwater, b and p are, respectively, the backscatter andreflection coefficient defined in the Appendix, a is thewavelength dependent attenuation factor in water, anda is the collecting aperture radius.

The first parenthesis in Eq. (13) is the minimumdetectable signal-to-noise ratio in the absence of thewater, provided that all the power scattered from thetarget is collected at the receiver (infinite collectingaperture). The second parenthesis is a geometrical fac-tor to account for the finite aperture. The product ofthe first two parentheses is the power received in the ab-sence of water. The third parenthesis accountsforthewater attenuation, and the bracket is the contrast,indeed independent, of the source power.

The preceding equation verifies our premise thatdetectability as described by the signal-to-noise ratioimproves with increased source power. Conversely,

for a given detection threshold, and laser power, therange may be increased until the threshold is met.

In our experimental setup as well as in practical sys-tems

2aa << 1,z > a,

which simplifies Eq. (13) to

p = _-L (_a)_ (-)-P.0 hvB (ca bI ) ( a

(14)

(15)

The fourth power dependence on the number ofattenuation lengths, indicated by the last parenthesis ofEq. (15), reduces the signal at such rapid rate that itcannot be compensated by any reasonable increase insource power. It results from having both source andreceiver colinear, thus presenting the maximum possiblebackscattering volume of water to the detector. Inpractical systems the backscatter is reduced by separat-ing the source and the receiver in order to minimize thevolume of water common to both fields of view. Herelies the laser narrow beam advantage: a small lateralseparation between the source and the detector issufficient to reduce the effective number of attenuationlengths z contributing to backscatter.

Equation (15) is derived assuming the detectionprocess is linear. Figure 4 shows an idealized detectiontransfer curve. There are three different regions:below threshold, linear, and saturation. As long asthe total returned energy, target plus backscatter, fallsbelow the detector saturation level, further increase inpower or integration time will improve detectability.Once the input energy reaches the detector saturationlevel, further energy increase does not improve thesignal-to-noise ratio; it may even degrade it. We nowproceed with the experimental confirmation of theabove.

II. Experimental SetupFigure 5 is a block diagram of the setup. The beam

from an He-Ne (6328-A) Optics Technology laser,


Fig. 5. Experimental setup.

Fig. 6. Photomultiplier output of backscatter and target return.

approximately 1 mW, is directed into a water tank.The laser output is collimated with a variable power(8X to 25X) telescope allowing the diameter of thetransmission beam to be varied. The beam diametervariation is not absolutely necessary but it is an addedconvenience in order to set its value smaller than thediameter of aperture B. Following the collimator isa converging lens and the viewing mirror. Not shownare calibrated attenuators and a chopper that could beplaced between lens and mirror. The viewing mirror islocated at the focal point of the converged laser beam.The laser beam passes through a small hole in the mirror,and the mirror looks directly down the laser beam. Acollecting lens gathers and focuses the backscattered andtarget power intercepted by the viewing mirror. Aper-ture B placed directly in front of the water tank has adiameter slightly larger than the laser beam. Thepurpose of this aperture is to constrain the viewing mir-ror to look directly down the laser beam and not downits side.

The target consists of a wheel about 38 cm in diameterwith alternate white and black spokes. The wheel isrotated in the water at a speed corresponding to a spokefrequency of approximately 0.5 Hz.

Tests were run by changing the distance between thecollecting lens and an aperture C placed in front of the

detector so that in one case aperture C was at the focalpoint of the light coming from the target and in anothercase it was at a focal point of light coming from some-where between the target and glass front of the tank.The position of focus did not effect the results. A 100-Afilter with a center wavelength of 6328 A covers thedetector entrance in order to eliminate extraneouslight. An RCA 7265 photomultiplier operating at1000 V is used to detect the signal. A Tektronix 457oscilloscope displays the photodetected output. Waterturbidity was varied artificially by adding a rubber basecompound (Naugatex) made of small spherical particlesof about 1300 A in diameter.

Ill. Experimental Results

A. Measurements of Reflection From WaterTank Glass Interface

Prior to making signal-to-noise ratio measurements,the reflection from the front of the glass tank was mea-sured to ensure that it was negligible compared with thebackscatter. This was done by focusing the photo-multiplier collecting lens combination on the glass frontsurface, measuring the reflection with the tank emptyand comparing this value with water-Naugatex solu-tion. The glass reflection was measured to be less than10% of the total backscattered power. Specularreflection was avoided by aiming the incident laser beamoff the normal to the wave tank surface so that its re-flection missed the photomultiplier in the experiment.

B. Signal-to-Noise Ratio Measurements

Figure 6 clearly illustrates the backscatter problem.The photomultiplier output is dc coupled to the oscillo-scope. The laser beam is chopped allowing the back-

0

'CL'

0

I-

'C

U)

7 IIDEAL

MEASURED

4_ _ _ _ _ _ _ _ _ _

0 200 400 600 800 10

PHOTO MULTIPLIER OUTPUT, mV00

Fig. 7. Signal-to-noise ratio improvement with increasing illu-mination power.


in Fig. 7 show that, as signal power is increased linearly,the signal-to-noise ratio increases as its square root thusconfirming our theoretical discussion. Figures 8 and 9are oscilloscope traces of the results.

C. Backscatter Spectral Measurements

The detected spectrum of the backscattered signalwas measured with a General Radio analyzer type1910-A, using the experimental setup of Fig. 5. Figure10(a) shows the result. Figure 10(b) is the detectedspectrum of the laser itself. To obtain Fig. 10(b), thelaser output was attenuated and then directly detectedby the photomultiplier. The attenuation was adjustedso that the magnitude of the photomultiplier outputwas identical to that of the backscatter.

The identical spectra in Figs. 10(a) and 10(b) provethere is no backscatter noise at any frequency, such ascould be caused by some preferred particle motion in thewater. Since the shot noise model is valid for thedetected laser signal, it is also valid for the backscattersignal. Note that below 2 kHz, the noise level is higher

(b)

Fig. 8. Target return signal plus noise as a function of illumina-tion power; scale 100 MV/cm. (a) Illumination power 1 P. (b)

Illumination power 0.158 P.

scatter and reflection from the black and white surfacesto be compared. The changes in reflected energy fromthe black and white targets that constitute the signalare much smaller than the backscatter. If the smallpower variations from the target elements are to bedetected, high laser power and a sensitive detector withlarge dynamic range are necessary. The backscatteredenergy, many times larger than the signal energy, tendsto drive the detector into saturation. Indeed, once thedetector is saturated, no further increase in laser powerwill pull the signal out of the noise. In practical sys-tems, backscatter energy must be reduced in order toderive full advantage of the signal-to-noise ratio de-tectability criterion. There are many ways of reducingbackscatter such as source-receiver separation, rangegating,I or polarization discrimination.'

For the purpose of the experiment, backscatter iseasily eliminated by rotating the target. The back-scatter energy contributes a dc component to the photo-current which is eliminated by ac coupling the photo-multiplier to the oscilloscope. The target signal is theac component of the photocurrent.

Signal-to-noise ratio measurements were carried outas a function of laser illumination power. The results

(a)

(b)

Fig. 9. Target return signal plus noise as a function of illumina-tion power; scale 100 uv/cm. (a) Illumination power 0.316 P.

(b) Illumination power 0.5 P.


(a)

40dB

30dB

2odB

0dB

0

40dB

30dE

2Od1

= | z -; . ,_ . . 1 1 1 1 I - I I I I = Ii- -L+'i -- :'i - - m y - --~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-

4 i -'-I-- -

___- " LE I~~~~~~~~~~~~~ 2_

FREQUENCY IN Hz

(a)

_; _ . _ E _~ I-- l G - :_ 1-<~~~~~~~~~~ - _L.- -_,._ ,a.IC ,5GL_

l t -- F- t jt= [1T - r-

-I

10dB ' O S

~~~ i - ~ ~ ~ L t _ _ 3 ~~~~ L E V E L0QE E iE 20 1 2

FREQUENCY IN Hz

(b)

Fig. 10. (a) I)etected spectrum of backscatter plus laser.Detected spectrum of laser.

20dB

10dB i ir

I

(b) O

a:

I

D. Water Attenuation Length Measurements

The attenuation length of the water used to obtain- the signal-to-noise ratio measurements was 19 cm.

Figure 12 shows the change of the photomultiplier out-put vs light transmission distance through the water.Figure 13 shows attenuation vs distance for ordinarytap water (44 cm).

Figure 14 is a block diagram of the experimentalsetup used to obtain attenuation length measurements.

-- Since the laser beam remains collimated as it passesthrough the water and the detecting surface of thephotomultiplier is considerably larger than the diameterof the laser beam, it is not necessary to consider any1/r2 ( = range) dependence. Transmission attenua-tion is directly proportional to ear. In Fig. 14, r is

LI¶:l P ¶ W NOSE- - - ' _ -LLEVEL

2FREQUENCY IN kH. -

Fig. 12. Attenmation of water-Naugatex solution.

3

Fig. II. 1)etected spectrum of photomultiplier.

than shot noise. This additional noise was found to beE,-

IL

00111=1

2

5?

in the photomultiplier; it could be argued that if therewere any backscatter noise it could have been masked bythe photomultiplier noise below 2kHz. However, thisis ruled out since there is no significant difference inthe amplitudes of Figs. 10(a) and (b). Furthermore,the signal-to-noise ratio would not increase as the squareroot of the input power, thus contradicting the results ofFig. 7.

Figure 11 shows the spectrum of the photomultiplieroutput with the laser off. This output is caused byextraneous light while the measurement was made.Comparison of Fig 11 with Fig. 10 shows that the dataof Fig. 10 are well above any extraneous signal.


a

PATH LENGTH IN TAP WATER, cm

Fig. 13. Attenuation of tap water.

I 1�i la-libliI111 �:" 1,6 !�

_I

r- I I I -

1

3

I

11

varied by simply moving the front surfaced mirrors toincrease the transmission path length; the photo-multiplier output is recorded as a function of pathlength.

IV. Summary

We have shown that under contrast-limited condi-tions, the minimum distinguishable signal is independentof backscatter and determined solely by detector shotnoise. The backscattered light masks the signal in twoways: (1) through its shot noise contribution and(2) by driving the detector into saturation. In princi-ple, the viewing range can be extended independently ofthe backscatter by increasing the source power orintegration time. In practical systems the trade-offbetween power and contrast-limited range is a fewadditional attenuation lengths because of the exponen-tial decay of light in water and the limited dynamicrange of the detectors.

AppendixThe experimental setup of Fig. 5 is equivalent to thesimplified geometry sketched in Fig. 15, where thesource and receiver are collinear. The scattering me-dium (water) extends between = 0 and the targetlocated at = z. Through appropriate optics theincident beam is made to diverge from S. An apertureof radius a placed at P = 0 reduces the backscatter.

Calculation of Backscatter ReturnThe laser output power is PL. If multiscattering is

neglected (a reasonable approximation), the radiationattenuates in the water exponentially reaching a scat-tering volume element dV at with the value PLe-a.The volume scattering function of Fig. 1 (6) yieldsfor the scattered power at within the cone angle sub-tending by the iris,

dPb(v) = PL2r ff e-ayrf(O) sin~dOd¢. (Al)

a is a function of both and the incidence angle 6i withrespect to the scattering volume. Since we are onlydealing with O = 0 in the backscattered case, we candrop this symbol altogether.

WATER TANK

Fig. 14. Experimental setup for attenuation length measure-ments.

Fig. 15. Simplified geometry of the experimental setup used inFig. 5.

The backscatter coefficient b is defined as

b = 2 7r | (O) sinodo;7rT/2

(A2)

a-(0) as seen from Fig. 1 is approximately constant in theback direction, thus yielding

o(O) = b/27r, r/2 < 0 < r. (A3)

Substitution of Eq. (A3) in Eq. (Al) gives after integra-tion

(A4)dPb(r) = dPLe-at(l - osO)d¢.

From the sketch above,

cos0 = /(W' + a)6. (A5)

The power reaching the aperture at = 0 from theelement dV at is

dPb = dPb(r) er- (A6)

Substituting Eqs. (AG) and (A5) in Eq. (A4) and inte-grating gives for the total backscattered power at thedetector,

Pb = bPLJ' e- 2ar {1 - [/(P2 + a)1 }d. (A7)

In order to integrate the second term of Eq. (A7), weapproximate part of the integrand:

+/(r' + a')1 1 - e-/a. (A8)

Such approximation never introduces errors in theintegrand larger than 15%. Substitution of Eq. (A8)in Eq. (A7) yields after integration,

Pb = abPL 1 - exp[-(2aa + 1)lz/a(2aa + 1)

For our experimental setup

2aa << 1,

thus yielding

Pb = abPL(l -eZ/a).

Calculation of Target ReturnThe target is taken to be Lambertian.

according to the law:

p'(0) = (p/7) cos ',

(A9)

(A10)

(All)

It reflects

(A12)


which is consistent with the definition of the reflectioncoefficient:

rf/2p = 2r fo p'(0) sin ' do'. (A13)

Here 6' is measured with respect to the target normal asshown in Fig. 15. The light reaching the target ele-ment is PLe `; the reflected light that reaches back tothe aperture within an incremented solid angle is then

dPT = PLe-aZ27rp'(0) sin O' do'. (A14)

Substituting Eq. (A12) in Eq. (A14) yields the integral

PT = PPLe 2az f 2coso' sin0'do'.

OT is the total cone angle subtending theaperture

sinOT = a/(z' + a2)l.

Integrating Eq. (A15) and substitutingintegration in Eq. (A16) yields

PT = pPLe - 2az[a2/(2 + a,)].

(A15)

target at the

Calculations of Contrast and Signal-to-NoiseRatio

From Eq. (A2) we get the contrast equation

C = (PT2 - PT)/(PT + Pb),

which gives, via Eqs. (A9) and (AlS),

AP/PI + [blpa( + 2aea)] (2 + a2) (e2az - -z/a)

The signal-to-noise ratio is obtained from Eq. (A12):

P.0 71PPL 2z a

P.. huB a' + Z2

A -

(A18)

1 + [b/pa(l + 2aa)] (Z2 + a2) (e2 az - e z/- (A20)

We would like to acknowledge useful discussionswith Robert T. Hodgson and Paul Heckman of NUWC,Pasadena and Michael Shumate of the Jet Propul-sion Laboratory who suggested the noise measurement.

(A16) References

the limit of 1. Peter Elias, J. Opt. Soc. Am. 43,229 (1953).2. P. J. Heckman, Jr., and R. T. Hodgson, IEEE J. Quantum

Electron. QE-3, 445 (1967).(A17) 3. G. D. Gilbert and John C. Pernicka, Appl. Opt. 6, 741 (1967).

Summer School for Nonspecialists in Optics

Applied Optics Section, Imperial College, London

10-21 June 1968

This introductory optics course of about thirty-six lectures, designed for scientists and

engineers who have little or no specialized training in applied optics, emphasizes

basic ideas rather than the detailed development of formulas. The course will be

given by staff members of the Applied Optics Section W. D. Wright, M. J. Kidger, K.

H. Ruddock, A. Thetford, W. T. Welford, and C. G. Wynne. Lectures will treat Gaus-

sian optics and aberrations (5 lectures), interferometry and the testing of optical in-

struments (3 lectures), sources and detectors (5 lectures), thin films (3 lectures),

polarization (3 lectures), physical theory of image formation, with applications (5

lectures), image forming systems (5 lectures), visual optics and photometry (4 lec-

tures), design and setting up of complete optical system (3 lectures); they will be

supplemented with laboratory demonstrations. Lecture notes will be circulated in

advance. Accommodation is limited to 100 persons. The course fee is $60.00. Applic-

ation forms may be obtained from The Registrar, Imperial College, London, S.W.7, U.K.


The signal/noise ratio concept in underwater optics

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