Horizontal visibility and the measurement ofatmospheric optical depth of lidar

Ariel Cohen

In this paper we describe a generalized treatment of the atmospheric visibility distances for a variety of at-mospheric conditions. In the development of the formula for the visibility distance, it is shown how a se-ries of assumptions made in the early work of Koschmieder [Beitr. Phys. Atmos. 12, 33 (1924)] can be re-duced to a single assumption covering more atmospheric conditions. Special attention is paid to the casesin which the extinction coefficient is wavelength-dependent. It is shown that neglecting such a depen-dence may produce errors as large as 20-100%, especially when long visibility distances are considered.The use of a dye-laser radar for the remote sensing of visibility distances is described and discussed.

1. Introduction

The visual range of black objects against the sky iscalculated by use of the following contrast equa-tion-3:

= [L' exp(- f "cYr) dx

- fRA exp (- fXcrdr) dx/ exp (-adr) dx, (1)

where /3 is the scattering coefficient of the atmo-spheric volume seen by the eye at a distance x and ais the extinction coefficient.

The visibility distance R is the distance for whichI cl = 0.02 (this value is not widely accepted; i.e., I =0.05 and 0.055 are used to determine visibility for air-craft operation at airfields in the United States).

Equation (1) can be solved analytically if one as-sumes that /3 and a are constants2 :

R = (3.912/r). (2)

But, except for the two limiting cases (1) the lightconsidered is monochromatic, and (2) a is the sameconstant for all wavelengths, Koschmieder's assump-tion (g) (see Ref. 2, p. 62), that the extinction coeffi-cient a- is a constant for a constant volume scatteringcoefficient /3 (from the observer to the horizon), in-troduces an error.

The assumption that fl is independent of x, whichnormally means that the air is homogeneous, leads tothe following equalities:

The author is with the Hebrew University of Jerusalem, Depart-ment of Atmospheric Sciences, Israel.

Received 12 December 1974.

1 rp (AAxx)I17(A) I (AX, f"X crdr = crL j adr. (3)

For a scattering molecular atmosphere-Rayleighatmosphere-a = f(X) = A- 4 (where X is a wave-length of the scattered light), and for a turbid atmo-sphere that includes aerosol particles, a- = f(X) will bedescribed by the Mie scattering theory that also re-sults in a certain dependence on A. Therefore, inmost atmospheric conditions a, = f(X) #= constant;especially when a includes absorption:

1 Xf+(Xdrf (r)adx, (4)

where kx(r) is the relative intensity of the light foreach wavelength at the distance r.

Hence, if the light, being attenuated, covers thewhole visual range

CAx) * U(x) fo (Ax)i (AX)2 (5)

because the spectral distribution of the light is af-fected by a.

In order to show that the error introduced is notnecessarily negligible, let us consider the horizontallystratified Rayleigh atmosphere. The cone viewed bythe eye is assumed to be illuminated by the directsolar radiation and the diffused light, with a constantradiant density vs distance.

Middleton2 presented such an example [see hisEqs. (8.19) and (8.20)] and derived an approximatesecond-order polynomial in Ax for a&. (Note thathis Eq. (8.21) has an error. It should be

arn = (r) log [ Wj;dx/lf Wy exp((-ar)dX]).

2878 APPLIED OPTICS / Vol. 14, No. 12 / December 1975

He came to the conclusion that the dependence of aon Ax is not marked even in pure air. With hisvalues for the radiant density spectral distributionand the eye spectral response and a corrected equa-tion for oam we found that the value of crX when Ax= 1 km is larger by -20% in comparison with itsvalue calculated for the visibility distance R. Ob-viously, if the scattered radiation spectral distribu-tion flix is different than Middleton's W Y (Ref. 2,Chap. 8) in the sense that the shorter wavelengthsare more pronounced, the error produced by a,, =const. is larger. A similar conclusion can be reachedby introducing absorption, which if present is usuallystrongly dependent upon wavelength. Thus, Mid-dleton's conclusion, based upon the comparison be-tween a and Ax (while it should have properly beenmade between cr and Ax/R-we shall discuss thisstatement further below), may not always be valid.It seems that this fact has been disregarded by sever-al researchers. An example of such neglect can beeasily detected in the Smithsonian MeteorologicalTables.5 With reference made to Duntley3'4 we findthere (page 454 of the Tables): "an atmospherewhich is homogeneous in its lighting and composition(i.e., scattering, absorption, etc.)." This led to thepreparation of Table 160-Bi, where the values of aare given with an accuracy of four digits for visibilitydistances R up to R = 250 km.

It should be emphasized that Tables 160-Bi and160-B2 are intended for the determination of R whena is known [see there Eqs. (9) and (10)], normally byits measurement within a shorter path relative to R.As a matter of fact, since the derivation of the visibil-ity distance requires that the light be attenuated to0.02 of its original value (see Sec. III), is practicallyalways measured at distances shorter than R.Hence, the use of Eq. (2) for the determination of Rleads to a systematic error: the real visibility dis-tance is always larger than R predicted in Eq. (2).This result has practical significance whenever largevisibility distances are of interest or when absorptioncannot be neglected. Although this statement seemsto be self-evident, it is worth mentioning, since thiseffect is often neglected. For example, Middleton2

remarked at p. 64 after presenting Koschmieder'stheory: "as absorption is frequently of comparablemagnitude to scattering, its effect is often includedby writing = absorption + Uscattering. But it is notimmediately obvious how this extension is justified"for the derivation of Eq. (2). Probably one couldstate that this extension is justified only whenOrabsorption/uscattering is independent of wavelength,and such a relation is rarely true.

While the error in clear days introduced by thespecial, somewhat narrow filtering of the eye is com-paratively small (20%), the use of a measuring sys-tem that does not contain similar filtering can pro-duce much larger errors. This may explain in partthe 100% and above, practically systematic, devia-tions between the measured extinction coefficients aand the prevailing visibilities when they were over 30km as reported by Muench et al. (p. 29, Fig. 13).6

Horizontal laser-radar backscattering measure-ments can be used for the remote determination of O-A(see Sec. IV). Measurements of this type were takenregularly with our lidar described elsewhere.7 Thosemeasurements indicated that a-\ itself varied signifi-cantly along the laser path in many atmospheric con-ditions.

But the nature of the changes in is different inmany respects as compared with the type of changesin (= 12 oxdX) discussed above. The main differ-ences are: (1) the changes in ax are statistically ran-dom vs distance; (2) the variations of ¢x vs distanceare independent of wavelength and are usuallycaused by fluctuations in time and space of the scat-terers' number density; and, (3) cr\ can vary by ordersof magnitude.

If one neglects the dependence of a vs distance dueto scattering and absorption in a homogeneous atmo-sphere, it still remains to be proved whether (orunder what conditions) the total extinction,

= f 'ud (6)

V o

measured for a known distance can be used in Eq.(2).

Since we now allow fluctuations in the scatterers'number density, the scattering coefficient fi in Eq. (1)would also vary. Therefore, the integral

'OR = fR:(X) exp (- J'udr) dx, (7)

can be easily solved only under certain assumptions.For example, assuming

O(x) = C(x), (8)

where C is a constant independent of x, one immedi-ately obtains

CR'O, R = J Cu(x) exp ( oJ o(x)dr) dx = C( - exp(--R),

(9)and

C = fRa(r)dr.

Unfortunately, while cr is merely a function of thescatterers' properties (number density, size distribu-tion, refractive indices, etc.), is a function of boththose properties and of the radiant density Io.Equation (8) thus includes the assumption that theradiant density does not vary with distance. In realatmospheric conditions, and especially when the skyis partially cloudy, this assumption does not hold.This is probably the reason for the introduction ofassumption (f) (see Ref. 2, p. 62)-a cloudless sky-by Koschmieder in developing his theory.

But it should be mentioned that the cloudless skystill does not necessarily ensure that Io = constant.The earth albedo also plays an important role in thecalculation of 10. Thus, for eliminating such cases,

December 1975 / Vol. 14, No. 12 / APPLIED OPTICS 2879

Koschmieder found it necessary to add the assump-tion (h), stating that the ground is diffusely reflecting(Ref. 2, p. 62).

After all the possible effects are taken into ac-count, Eq. (1), which contains the integrals of thetypes 7, cannot be generally solved analytically, andthe question whether R can be determined by use ofan average extinction coefficient [Eq. (6)] requiresfurther discussion.

11. Derivation of the Horizontal Visibility Distance

The derivation of an expression for the visibilitydistance at a given direction can be based upon thefollowing single assumption: Let w be a small solidangle, say a few milliradians around the direction ofsight, and let Ie be the total luminous flux of thebackground:

1.o = f dwf 10(Xh3(x) exp(f Xadr)dx; (10)

then I.,w = constant, independent of Ro.The visibility distance R in daytime would then be

the pathlength required for attenuating the value ofIcowR to 0.02 I.,wo:

I.,. 0-02 = I,w exp (-:R), (I11)

where a- is the average extinction coefficient along thepath.

This definition of the visibility distance is in com-plete accordance with the requirement of the con-trast Eq. (1) [see also Eq. (4.1) in Ref. 2], since ob-Vi0US1Y o,- - IO,R = R,o, where I is defined in Eq.(7).

The apparent brightness B of a black object seenat a given distance r would therefore be

B = exp (-'Ur), (12)

or, when absorption and/or spectral scattering are ofinterest:

B = I,, f 2tw(X) exp (-axr)dX.I."~L

(13)

The main differences between this definition ofvisibility and Koschmieder's derivation are as fol-lows:

(1) Koschmieder's assumptions are a special some-what limited case of atmospheric conditions thatobey the general assumption mentioned above.

(2) The atmosphere producing a practically con-stant L,, at a given direction can contain scatteredclouds, i.e., sections along the light path illuminatedby both diffused and direct solar light or by diffusedlight alone. The assumption only signifies that in astatistical sense the atmosphere is approximately thesame along and around x. This is also true for thereflecting ground.

(3) The scattering and attenuating atmospherewithin the cone described by the solid angle w canfluctuate randomly (for example, due to winds and

turbulence). The theory will hold for all atmospher-ic conditions with scales of fluctuations smaller thanthe visibility distance. In practice, the scale of fluc-tuations should be less than the atmospheric dis-tances used for measuring 5- or A - s.

(4) The measurement of the visibility distance isnow based upon the measurement of the backgroundilluminance (and its spectral distribution) in additionto a single measurement (or a series of measure-ments) from which a (orax - s) is calculated.

We shall describe below one method of measure-ment based upon Eq. (13).

111. Lidar Measurement of Visibility

Figure 1 reproduces a scope display of a laser-radarecho from a horizontal path. The correspondinglaser-radar equation is given by7:

I(x) = 0*3(1r, x) exp (-2f rodr) /x2, (14)

where I is the vertical display of the scope recordedat a time t = 2xlCo (C0 is the velocity of light) afterthe laser shot, A is a constant of the laser-radar-scope system, and o is the laser output energy. Theindex X is used to denote the fact that , and a are thecorresponding scattering and extinction values forthe laser wavelength, while r denotes the backwardscattering.

Since the average extinction value ax is requiredfor the horizontal visibility distance calculation, thefollowing analysis method is suggested.

We first divide the range R1 x < R3 (see Fig. 1)into two equal ranges R1R2 = R2R. In most practi-cal cases the scatterers' density would randomly fluc-tuate along the laser path. For such cases, if RiR2 islarge enough compared to the spatial length of thefluctuations, one can assume that the average scat-tering coefficient f(lxr,x) is the same in R1R 2 andR2R3:

X, 1 fR2 , x)dx = J f3:(r, x)dx ,AR '=1 ARR 2

where AR = R2 - R, = R3- R2.

(15)

HZ;0

pi h2 k3 >

TIME (Distance) t 2XCO

Fig. 1. Reproduction of laser-radar return for the measurementof extinction. 2- R = R3 -R2, R i chosen to exceed theminimal distance at which the laser beam isj totally within the lidar

telescope field of view,

2880 APPLIED OPTICS / Vol. 14, No. 12 / December 1975

For any point x in R1 R2 and a corresponding pointx2 = x1 + AR in R2R 3, one can write

I(xO1 /I(x2 ) = PxOT, x0)/j,9x(w, X2 ).X 22/'X

2 exp(2aCrLv.AR), (16)

or- 1 r 2 a -s7-- ]U,1 1 AdiR +

1 f R2A LA R (> +

(IF x)x2)AR)(x + R)2)dx

+ A /7 in 3(7r,x)dx - A in i(v, x + R)dxj.

(17)

By making use of Eq. (15), we get

= 2AR [nIRIR 2

-In 'R 2R + (2R2 n R2 - RI n R - R3 n R3)], (18)

and at, can be calculated from the recorded values ofI.

This procedure of measurement, if ax = f (), canbe repeated for various wavelengths by use of a dyelaser radar (which produces monochromatic light inany desired wavelength within the visible range) anda set of interference filters.

In addition to the measurement of ax, one has tomeasure each time what is considered the back-ground noise in lidar work. This noise is measurable(in daytime) and its value is simply proportional toI_,W0 (Xi), where w0 is the lidar telescope field of view.If L-xiand IW(Xg) are measured in such a way thatthe main characteristics of the spectral dependenceof I,, 0(X) can be determined, the following expres-sion would then be used for the visibility distance R:

0.O2Zl)xi4,wj{>i) = L4~J. 0 ,~0 (x)exp HCxjR) (19)

where fr is the eye spectral response. No calibrationof the proportionality factor connecting tX, ) andthe lidar background noise is nee ded, provided that itis the same, or within a known ratio, for all wave-lengths.

IV. First-Order Approximation to Visibility Distance

When absorption and scattering are wavelength-dependent, the following equation can be used for thederivation of R:

0. 02 =f dX fj 3x (x) exp[f Cax(r)dr]dx/

fX 2 fdo A (xexpf 9x(r)drldx. (20)

In order to solve this equation we shall limit our-selves to the constant radiant density assumption,and Eq. (8), where C may be a function of X but notof x.

Equation (20) thus becomes:

L 0.02 = [2 O exp Rc ax(r)drjdxf~', Gxl fx= f1 "'0; exp (-crxR)dx, (21)

where

L = f " EA.

Let us now expand exp(-OAR) = 0.02 exp(3.912 -aXR) in the form:

exp (-uR) = 0.02(1 + 3.912 - arR + . . . ). (22)

Substitution of Eq. (22) into (21) results in:

1. = J (4.912 - xR + . . . )dx.xj Ux

Neglecting terms of the form (3.912 - xR)n/n! forn Ž 2 results in

R = (3.912 L0/Th, (23)

where

1 = dr and d = [ /3xdX,.R OXEquation (23) has the advantage that when the at-

mosphere is homogeneous, I and are constantsalong the path. When a is wavelength-independent,Eq. (23) can be simplified to Eq. (2).

One way of measuring can be based upon themeasurement of the background noise in the (lidar)telescope directed to a distant black object coveringits field of view. (A smaller black object would re-quire a further calibration.) Let IR = a be thatbackground noise; then R is simply:

R In 0.02 RR=in (1 - a)I (24)

In the derivation of Eq. (24) we neglected againhigh orders of

(In 1 - axRa)

a neglect justified for R1 < R when Eq. (22) holds.Note that this approximation should really applymainly in a rather narrow band of wavelengths (com-pared to the whole visible range) due to the filteringof the eye. This derivation also shows clearly thatone has to compare the values of axR vs aR, while thedependence of a on R alone may be misleading.

V. Conclusion

We may summarize as follows:(1) The increase of air pollution (and hence ab-

sorption) on the one hand and the growing interest inlong visibility distances on the other are both situa-tions where ax is strongly X-dependent. Therefore,

December 1975 / Vol. 14, No. 12 / APPLIED OPTICS 2881

the problem of measuring horizontal visibility shouldnot be restricted to objective systems requiring a con-stant average ax over different distances. The use ofa dye-laser radar is an example for an objective sys-tem that can deal accurately with horizontal visibili-ties by remote sensing.

(2) It seems that there is no simple derivation forthe visibility distance when I cannot be assumedconstant vs distance. But if a constant IO is accept-able, the ever-changing (in a short scale) atmosphereshould not be treated point by point, but rather bythe use of average-integral values. The advantage ofsuch a treatment is clearly seen in the derivation ofthe simple visibility distance formula for all atmo-spheric conditions (including various amounts ofcloudiness and different earth albedo along the path)when ID is agreed to be constant. This requirementis the sum of all possible effects and does not have tobe treated in integrals of the type in Eq. (6). Suchintegrals do require for their analytical solution thewhole set of assumptions suggested by Koschmieder.It appears that the values of the radiant density mul-tiplied by the corresponding scattering coefficient ateach distance from the observer are irrelevant to the

problem of the simplified (meaning a constant aver-age &) visibility distance in the atmosphere.

(3) Simplified definition and measuring proce-dures of visibilities can be developed for a first-orderabsorption and scattering X dependence that can eas-ily be expanded to higher orders of X dependence.

The author wishes to thank E. Doron of the De-partment of Atmospheric Sciences, Hebrew Universi-ty of Jerusalem, for helpful discussions.

References1. H. Koschmieder, Beitr. Phys. Atmos. 12, 33 (1924).2. W. E. K. Middleton, Vision Through the Atmosphere (Univer-

sity of Toronto Press, Toronto, 1952).3. S. Q. Duntley, J. Opt. Soc. Am. 38,179 (1948).4. S. Q. Duntley, J. Opt. Soc. Am. 38, 237 (1948).5. R. J. List, Smithsonian Meteorological Tables (Smithsonian

Institution, Wash., D.C., 1958); see also Ref. 6.6. H. S. Muench, E. Y. Moroz, and L. P. Jacobs, Development and

Calibration of the Forward Scatter Visibility Meter (Air ForceCambridge Research Laboratories, Technical Report-74-0145,1974).

7. A. Cohen and M. Graber, J. Appl. Meteor. 14,400 (1975).

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