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Atmospheric Turbulence Parameters fromVisual Resolution

M. L. Wesely and Z. 1. Derzko

Direct visual observations of image blurring by atmospheric turbulence are used to obtain line averages ofthe refractive index structure function coefficient C,,2 along lines of sight in the lower atmosphere. Good

agreement with meteorological point measurements is obtained both above a dry land surface, where tem-perature fluctuations cause most of the image degradation, and above a pond, where humidity fluctuationsare also important. No optical saturation effects are found, and a simplified formula for determining long-

exposure, limiting-resolution angles is verified.

1. Introduction

As optical radiation passes through the atmo-sphere, the amplitude and phase characteristics ofthe radiation can be significantly altered by refrac-tive index variations along the propagation path. Inregions of constant static pressure, these variationsare due to changes in density and composition causedby local fluctuations in temperature and moisturecontent. The atmosphere can have an overwhelm-ingly degrading effect on a total optical system's per-formance. For example, laser beams are stronglydisturbed by thermal turbulence in propagationpaths close to the surface of the earth; a high-fre-quency effect, scintillation, has been studied and re-lated to micrometeorological parameters by many au-thors, among them Livingston et al.,' Kerr and Eiss,2

and Kerr.3 Astronomers are well acquainted with at-mospheric limitations on seeing through cloudlessskies, since observations of star quivering and twin-kling are common. Photographs, taken with suffi-ciently long exposure times, of apparently quiveringlight sources such as stars are blurred by the thermal-ly turbulent atmosphere. 4'5 The extent of imageblurring and laser scintillation is dependent on line-of-sight averages of the refractive index structurefunction coefficient, C,2, which is closely related toits temperature and water vapor pressure counter-parts, CT2 and Ce2, respectively. In the layer of theturbulent atmosphere closest to the earth's surface,

M. L. Wesely is with the Radiological and Environmental Re-search Division, Argonne National Laboratory, Argonne, Illinois60439; Z. I. Derzko is with SUNY, Buffalo, New York 14214.

Received 25 July 1974.

CT and Ce are intimately associated with such mi-crometeorological quantities as the vertical fluxes ofmomentum and latent and sensible heat.6 Also, therelative contributions of CT and Ce to Cn can be ex-pressed in terms of the ratio of the sensible heat fluxto the latent heat flux (commonly referred to as theBowen ratio fi) and the extent of T and e correla-tion. 7

Applied as a means of remotely sensing the envi-ronment, values of C, 2 obtained from image blurringand laser scintillation can be used to help determineline averages of various meteorological variables. Inexperimental studies of propagation phenomenawhere the value of C 2 itself is needed, a simple, reli-able technique of measuring it along a line of sight isan attractive alternative to methods relying on labo-rious point measurements near the line.

In this paper, we investigate the possibility of de-termining C, 2 from measurements of the extent ofblurring of point sources as seen by a human eyeaided with an astronomical telescope. Although theresults presented here might apply to propagationpaths of any length and orientation, the measure-ments in this paper are confined to paths severalhundred meters long and 1-2 m above the surface ofthe earth. A minimum of experimental apparatus isdesired. The diffraction effects of the telescope mustbe considered, but we can neglect the eye's resolutionlimitations because the eye's effective resolving abili-ty is increased nearly proportionally to the amount ofmagnification, which for these studies was greaterthan X100.

II. Exposure Time

For propagation paths a few hundred meters longand close to the earth's surface, the image of a smallobject viewed through an astronomical telescopemight be vertically elongated or compressed, moved

April 1975 / Vol. 14, No. 4 / APPLIED OPTICS 847

about rapidly, and blurred. The vertical distortion iscaused by mean refractive index gradients and willnot be discussed in this paper. Image dancing de-rives from rapid changes in the tilt of the mean wave-front at the receiving aperture or, equivalently,changes in the angle of arrival. When a photographwith sufficiently long exposure is taken of the image,the dancing results in apparent blurring. Imageblurring that exists even in instantaneous perceptionis caused by phase distortions along the mean wave-fronts. If image dancing and blurring are inducedentirely by refractive index fluctuations associatedwith the inertial subrange of the turbulence spec-trum, the only atmospheric parameter necessary tocharacterize these effects is C 2.5 Using this as-sumption, Hufnagel and Stanley8 and Fried9 deriveexpressions for atmospheric degradation of images tobe photographed using either short exposures, whichincorporate only the blurring effects caused by phasedistortions along the mean wavefronts, or long expo-sures, which in addition show effects of angle-of-ar-rival fluctuations. The relevant equations show thatfor short exposures the telescope's effective aperturesize has a more variable effect on the performance ofthe total optical system than for long exposures.Thus, although better resolution can be obtainedwith short exposures, long exposures allow less diffi-culty in determining the effects of changing telescopesizes and configurations.

As shown by Clifford et al.,10 the most effectivescale length of refractive index inhomogeneities forproducing phase distortions along a mean wavefront(resulting in blurring) is of the order (X)1'/2, where Xis the wavelength of light and I is the pathlength.Where visible light is emanating from a point sourceacross a typical, near-earth, propagation path, (Xl)1/2

is usually about 2-5 cm. On the other hand, angle-of-arrival fluctuations are associated with scalelengths larger than (XI)1/2 and it follows from consid-erations of geometric optics that the most effectivescale length is of the order of the diameter D of thereceiver aperture. Of course D can be chosen as de-sired, and it will be shown later the D 15 cm is ap-propriate for our purpose here. Other relevant scalelengths we must consider include the inner and outerscales that bound the inertial subrange. Typically,the inner scale 10 in the atmospheric surface layer isless than one centimeter and the outer scale, LO, is'about half the propagation-path height. In summa-ry, < (l)1/ 2 < D < L at heights of 1-2 m. Sincethe scale lengths associated with image blurring arebetween lo and L the use of C 2 in the long expo-sure and short exposure expressions should be valid.

The exposure times relevant in a study of the ef-fects of a particular scale length can be expressed ap-proximately as the length divided by the mean windcomponent VI normal to the line of sight. Tatar-ski5 has shown that the peak of angle-of-arrival fluc-tuations occurs at about the frequency 0.22 VIID,from which we can infer that exposure times of atleast DIVI are needed for obtaining photographs of

images blurred by the full effect of angle-of-arrivalfluctuations. At the same time, however, exposuretimes should be less than L0 /VI. in order to avoid ef-fects of angle-of-arrival fluctuations associated withscale lengths exceeding the inertial subrange.

Experimental data presently available tend to sup-port the contention that, for propagation paths with-in a few meters of the surface, the proper exposuretime r associated with long exposure expressions is anytime span greater than D/VI but less than L 0 /VIFor instance, Livingstonl found no discernible dif-ference in image blurring in photographs with expo-sure times varying from 1 sec to' Y5 sec for his.highwind case when L,/Vi I 1.3 sec and D/VI 0.05sec; for his low wind case when L0 /VJ 4.5 sec andD/VI 0.17 sec, the extent of image blurring re-mained about the same for exposure times rangingfrom 2 sec to Y8 sec. In contrast, in experiments re-ported by Tatarski5 and Andreyev et al.,'2 whereangle-of-arrival fluctuations were directly measured,T 40 sec was successfully employed. This leads usto believe either that L/VI was large, which cannotbe proven here because L, and VI are not given inthe papers, or that refractive index inhomogeneitieslarger than LO did not have a significant effect.

Another set of experimental data concerns the at-mospherically induced spread of the focal spot of alaser beam. As will be shown later, expressions havebeen derived for the long-term-average spread thatinclude identical dependence on path length, C 2,and wavelength as does an empirical expression forimage blurring. In summary, the experimental dataconcerning laser propagation indicate that very longaverages for beam spread or standard deviations ofbeam positions correlate well with C 2 if the propa-gation path is several tens of meters above the sur-face of the earth,13 but for paths within 1 m or 2 mabove the surface this correlation can be poor. 4 15We may conclude that it is crucial in the atmosphericsurface layer that r < L/Vl, but this is easily metat heights of several tens of meters, where Lo >> D.

It is commonly accepted that the response time ofthe human eye is of the order of 0.1 sec. Hence, ifD/VI < 0.1 sec < L/VI; visual observations ofimage blurring should conform to the long exposurepredictions of Hufnagel and Stanley8 and Fried.9 Inthe experimental arrangement described later, we useD 15 cm, resulting in DIVI 0.05 sec, and Lo/Vl - 0.17 sec for the typical conditions of V I 3m sec-1, and Lo 0.5 m in the atmospheric surfacelayer. However, visual observations should not beused when 0.1 sec < DVI or 0.1 sec > L 0 /VI; pho-tographs obtained with proper exposure times mightthen be employed successfully.

111. Imaging System

The imaging system consists of a spherical waveleaving the object plane, passing through a number ofelements, and impinging upon an image plane, asshown in Fig. 1. The elements consist of the turbu-lent atmosphere, the telescope, and the human eye,

848 APPLIED OPTICS / Vol. 14, No. 4 / April 1975

OPTICAL SYSTEM

Object plane

Hufnagel and Stanley8 and Fried9 have shown thatfor a long time exposure, the atmospheric MTF for aspherical wave is as follows:

MTF,(i) = exp[-0.5D(,)]. (2a)

The wave-structure function D(g) in the near-field isapproximately equal to the phase-structure function.For a horizontal propagation path of length I alongwhich C 2 is constant, 4

D( ) = 2.91(3/8)K 21D5/3C 2 5/ 3 (2b)

Lens-aperture

In

VIEW OF POINT SOURCES

OO 00 0D

iage plane

0Resolvable Rayleigh Determines Appears as oneas two points criterion d point source

Fig. 1. View of imaging system and point source configurations inthe image plane.

but we can neglect the effects of the human eye be-cause of the high magnification of the telescope. Weshall assume that the telescope consists of a thin, dif-fraction-limited, aberrationless lens. Since the opti-cal transfer function (OTF) of a diffraction-limitedsystem is always real, the OTF of our telescope isidentical to its modulation transfer function, MTFt.We shall neglect absorption and low frequency ormean changes of refractive index along the propaga-tion path. Since we are assuming long exposuretimes, the imaging system's modulation transfer sys-tem, MTFS, is the product of MTFt and the modula-tion transfer function of the atmosphere.9

For a thin, diffraction-limited lens of diameter Dand focal length R,

MTF,(L) = {2/irarc cos(g) - i(1 - 2)1/2] 0 - C 10 ~ ~ ~~~~~i> 1,(1)

where ,u = XKR/D, X is the wavelength of the quasi-monochromatic light source, and K is the spatial fre-quency.'6 Equation (1) is inadequate for the manyastronomical telescopes that are of a reflector or Cas-segrainian type, which have a central obscuration.For brevity in this paper, we shall omit the expres-sions for the MTFt that include a, which is the ratioof the diameter of the obscuration to D. Figure 2shows the behaviors of the MTF(j4's for several as-tronomical telescopes frequently used in terrestrialobservation. For instance, a = 0.3 is common for re-flector telescopes used by amateur astronomers and a= 0.4 for the 12.7-cm-diam Celestron Pacific tele-scope.

However, D(g) is weighted with the 5/3 power of thepathlength. Thus about 50% of the effect of the at-mosphere is attributable to the last 26% of the path,and 75% of the effect to the last 45%.

An example of the MTFa is shown in Fig. 2 forwhat might be considered as moderate thermal tur-bulence and pathlength, since the magnitude of laserscintillation under these conditions is large but notsaturated. The shape of the MTFS appears to bemostly determined by the MTFa.

IV. Angular Resolution

As shown by Fried, 4 we can define a resolution Rsuch that

R = 87TK0o2S MTF(,) tid,, (3a)

where MTF () is the average MTF of the opticalsystem and Ko = D/(2XR). For the telescope,

Rt = 8K02J MTF,( ,) ,gd, (3b)

and for the total system,

R, = 8rKo2J MTFt(p)MTF,(O),4di. (3c)

Furthermore, Fried explains that R corresponds to a

La.17

IAL

Fig. 2. Computed MTF. The atmospheric MTF is for C, = 1O-7m11/3 , 1 = 700 m, X = 0.55 gm, and D = 15.24 cm. Also, the sys-

tem MTF is the product of MTFa and MTFt at a = 0.3.


d (cm)

Fig. 3. Relationship of minimum resolution length determined byvisual observation of white point sources to C,, when constant over

the entire pathlength.

number of p/m2, and a corresponding resolutionlength is equal to a line width (4R)-'1 2 . However,since we choose to use two point sources of light, theRayleigh criterions may be more applicable. Ac-cording to this criterion, two incoherent point sourcesare just resolved when viewed through a diffraction-limited lens when their separation distance is 1.22XR/D. From Eqs. (1) and (3b) we can find that Rt =7rKO2, and it can be easily shown that Fried's estimateof resolution length is about 0.46 times the estimateusing the Rayleigh criterion. For telescopes having acentral obscuration whose diameter is 0.3 the lens di-ameter, the Rayleigh criterion resolution length is1.975XR/D and, by finding (4Rt)-"/2 for this case, wecan show that Fried's estimate of resolution is about0.42 times the Rayleigh criterion resolution length.

To choose a realistic resolution criterion, we mustconsider the effects of human judgment in decidingwhen point sources are just resolvable. A resolutionlength d can be defined as the separation distancebetween two point sources when they cannot be iden-tified as either one or two apparent sources, but as asingle source scarcely elongated in one direction. Inkeeping with this, we choose a criterion for angularresolution, 0, determined from experimental observa-tions of light sources where we determine a factor Kin (KR)1/2, which is similar to Fried's resolutionlength, that is,

01 = IK8rNKOJ' MTF(L) ud&L]. (4)

To find an appropriate value for K, a 8.89-cm diame-ter Questar telescope with a 0.3 was used in a labo-ratory to determine visually the smallest line sourcespacing that is nearly resolvable at the minimumfocus distance and a magnification of 160X. Sinceno color effects were observed, X 0.55 Am, which isthe wavelength of light the eye detects most efficient-ly. No attempt was made to determine whether Kdepends on X. It was found that K = 2.26, and theminimum resolvable angle predicted by the Rayleigh

criterion was too large by about 56%. Thus, the ap-pearance of point sources in the atmosphere might beas shown in Fig. 1, where the sources are clearly iden-tifiable as two separate sources at the Rayleigh crite-rion distance.

In Fig. 3, the relationships between Cn and d re-sulting from Eq. (4) with K = 2.26 and X = 0.55 gmare shown for different pathlengths. The accuracy ofthese relationships is to be determined in this paper.Also, we shall determine whether the saturation ef-fect that occurs with laser scintillation also occurs inimage blurring detected with long exposures.

An alternative, simplified method for determiningthe limiting resolution of the atmosphere in the near-field is to interpret the effects of the atmosphere asbeing similar to one diffraction-limited lens of diame-ter Do, where 9 ,17

Do = .365X 6/1-'C -6/5

It has been suggested by Gilmartin18 that we mayapply the Rayleigh criterion to result in the followingestimate of angular resolution:

0, = 3 .4X-"/513 5C 61/5, (5)

where the subscript e refers to empirical. Estimatesof de = Oel are shown in Fig. 3. If the value 2.7 isused instead of 3.4 in Eq. (5), Eqs. (4) and (5) give al-most identical results when de is at least two or threetimes larger than the minimum, diffraction-limited,value of d for a particular pathlength.

Except for a different numerical factor, Eq. (5) isidentical to equations summarized by Dowling andLivingston' 5 from work by Brown19 and Yura20 forthe long-term-average spreading of the focal spot of alaser beam. Hence, results in this paper should indi-cate the behavior of focal spot size as influenced bythe atmosphere.

V. Experimental ResultsSeveral visual observations of the atmospheric

blurring of point sources have been made. Incoher-ent light sources, usually quartz-halogen lamps ormercury discharge lamps, were placed behind masksthat had various aperture spacings. A practical ar-rangement was to use lamps of several hundred wattsand aperture diameters about 10% of the apertureseparation distances. When significant changes ofbackground light occurred, the light intensity of thelamps were adjusted to give the minimum intensityusable, to keep the eye properly adjusted. In one ex-periment, red filters were placed over the aperturesand a 8.89-cm diam Questar telescope with a 0.3was used at the other end of a 700-m propagationpath 1.5 m above a grass surface. This experimentwas conducted at the Electromagnetic PropagationRange (EMP Range) of the U.S. Army Ballistic Re-search Laboratories at Aberdeen Proving Ground,Maryland. In a second experiment, paths of 400 m,1160 m, and 1500 m were used about 1 m above acooling pond located near Dresden, Illinois; and the


Table I. Summary of Field Data

Resolution MeteorologicalPath

Date Time Height Length Cn Cn

-- 1/3) 107-1/3)(LST) (i) (m) (10 7m

i ii

7-26-72 1930* 1.5 700 3.2 2.47-26-72 2035* 1.5 700 3.2 2.18- 8-72 1100 1.5 700 1.6 3.48- 8-72- 1200 1.5 700 3.8 3.78- 8-72 1400 1.5 700 3.8 3.68-10-72 1005 1.5 700 3.8 3.98-10-72 1305 1.5 700 4.8 5.0 1.78-10-72 1405 1.5 700 4.8 2.5 4.4 1.88-10-72 1805* 1.5 700 2.7 1.4 1.4 -0.708-16-72 1605 1.5 700 1.0 0.7 0.259-19-73 1515 0.93 1500 6.30 6.70 5.90 0.2169-19-73 1605 D.93 1500 6.15 7.24 6.46 0.2449-24-73 1342 0.94 1160 4.10 4.34 3.48 0.1059-24-73 1427 0.94 1160 3.60 3.79 2.99 0.0959-24-73 1506 0.94 1160 3.35 3.50 2.72 0.0869-24-73 1537 0.94 1160 3.30 4.88 3.81 0.08710-3-73 1652 0.96 1500 4.55 5.62 4.74 0.15010-3-73 1821 0.96 1500 6.00 7.52 6.53 0.18910-3-73 1900 0.96 1500 5.40 7.17 6.24 0.19410-3-73 1900 0.80 400 5.35(or 5.03 at 0.96 m)10-3-73 2001 0.96 1500 6.55 7.34 6.39 0.19610-3-73 2001 0.80 400 6.05(or 5.69 at 0.96 m)10-5-73 1747 1.04 1500 5.25 6.70 5.83 0.19310-5-73 1830 1.04 1500 4.40 6.95 6.08 0.20310-5-73 1830 0.80 400 4.90(or 4.49 at 1.04 m)10-9-73 1215 1.04 1500 2.20 3.16 2.52 0.11410-9-73 1310 1.04 1500 2.15 2.26 1.71 0.07310-9-73 1352 1.04 1500 2.10 2.44 1.87 0.07910-9-73 1430 1.04 1500 2.00 2.19 1.64 0.065,10-9-73 1507 1.04 1500 1.75 1.82 1.33 0.05410-9-73 1544 1.04 1500 1.45 1.89 1.42 0.065

*Stable conditionsFor 1972, column "i" is for thermal measure-

ments and column ii" is for gradient measurements. For 1973, the columns

are for linear and vector methods, respectively, of summing temperature and

water vapor contributions to Cn.

telescope was a 15.24-cm diam reflector type (a0.3). Since unfiltered white light sources were usedin the latter experiment and red filters were used inthe former, X was chosen as 0.55 gAm and 0.63 Am, re-spectively. However, since the effect of changes in Xin Eq. (5) is weak, the choice of wavelength for inter-pretation of atmospheric blurring was not critical.This, and the scatter in the data being relativelylarge, indicated that attempts to determine a wave-length dependency from these experiments were fu-tile.

A summary of the experimental field data is shownin Table I. The estimates of C, from resolution werecalculated using Eq. (4) with K = 2.26. Most mea-surements were taken during unstable conditions,when sensible heat flux was directed away from thesurface and the mean refractive index increased withheight. The data for 1972 were collected at the EMPRange where the wind direction was usually J90' ofa line drawn from the light sources to the telescope'slocation. As a result, the fetch across the level, gras-sy surface was greatest near the telescope where the


influence of thermal turbulence on the blurring wasmost heavily weighted. Within 50 in of the telescopeand along the propagation path, meteorological mea-surements were obtained. Two closely-spaced resis-tance thermometers21 were used to obtain 5-15 minaverages of CT, from which an estimate of Cn at 1.5-m height was calculated. Fifteen-minute averagegradients of temperature and winds close to theground provided the estimates of Cn, via the Rich-ardson number method as summarized by Weselyand Alcaraz.6 The effect of humidity fluctuationswere neglected, as is frequently appropriate over landsurfaces.

At the Dresden cooling pond, the wind directionswere +450 from the light sources to the telescope, ex-cept for the 400-m path that was used when the windwas approximately normal to the chosen line of sight.For the long paths, the fetches were inadequate nearthe light source but were sufficient at the important,near-telescope part of the propagation path. Mea-surements of Bowen ratio, wind speed, and thosevariables necessary for low-level bulk aerodynamicestimates of sensible and latent heat fluxes22 wereobtained over water, usually within 400 m of the tele-scope. The friction velocity was calculated as 0.045times the wind speed at 1 m, which corresponds to adrag coefficient of about 0.0013 at 10 m. From theseparameters, usually obtained as 15-30 minute aver-ages, estimates of Cn were obtained that included theeffects of both temperature and humidity fluctua-tions.

If temperature and water vapor pressure fluctua-tions were perfectly correlated at Dresden, a linearsum of their two influences on refractive index fluc-tuations would have been correct. That is, after CTwas calculated as detailed by Wesely and Alcaraz6

the following could be used:

Cn = CTAp(T)-2(l + 003/8) (6)where A, = 78.7 X 10-6 K mb-1, p is the atmosphericstatic pressure, and (T) is the mean absolute tem-perature. This equation differs slightly from Weselyand Alcaraz's Eq. (17) because they used an errone-ous value for their A2, which should be equal toabout 66.3 X 10-6 K mb'1. As explained earlier, theBowen ratio ,3 is the ratio of the vertical fluxes of sen-sible to latent heat, and is proportional to CT/Ce.This ratio, frequently measured by micrometeorolog-ists, was obtained at Dresden from water surfacetemperatures and wet and dry temperatures at 50cm. If the temperature and water vapor pressurefluctuations were not correlated, a vector sum wouldhave been appropriated Then

C = CAIp(T)-2[1 + (0.03/)2]1/2.

were taken at the land site to show whether the ther-mal or the gradient method show better agreementwith the resolution data. The purpose here is toshow that the resolution method of measuring C isnot rigged to work only for a restricted set of experi-mental conditions. As is apparent in Fig. 4, theredoes not appear to be a saturation effect, whichwould result in underestimates of Cn from resolutionfor long paths or strong refractive index fluctuations.The values of Cn from optical measurements at theDresden cooling pond have a correlation coefficientof 0.956 with the linear estimates and 0.957 with thevector estimates. The average slope for the linear es-timate is 0.77 ad for the vector estimate is 0.82.This, and inspection of Table I, reveal that the opti-cal measurements of Cn at the Dresden cooling pondfor both long and short paths appear to be systemati-cally low by a few percent. This may have resultedfrom an increased mechanical turbulence within asignificant portion of the propagation path, especial-ly near the light sources where there was inadequatefetch away from rough land surfaces and obstacles.Also, subsequent investigation of water surface tem-peratures near the propagation paths indicate thatpoor site conditions occasionally result in a morerapid decrease of C with height than that predictedby theory and used to calculate the meteorological es-timates of C at Dresden.

VI. Conclusions

Field experiments indicate that measurements ofQ, can be taken by visual observations of atmospher-ic blurring of two point sources of light. For lines ofsight at 1.0 m and 1.5 in above level surfaces, thehuman eye's response is slow enough for use of longexposure estimates of image degradation, but fastenough to detect and ignore image dancing caused byspatial inhomogeneities larger than the outer scale ofisotropic turbulence. Image blurring for long expo-

I:0

E

0

C.)-j

C.)

a-0

(7)

The results from both methods of summing areshown in Table I; as argued by Wesely and Alcaraz(1973), the actual value of Cn; might be closest tothat obtained from linear summing.

It is evident from Table I that measurements of C,from visually determined resolution are valid under alarge variety of situations. Clearly, insufficient data

Fig. 4. Summary

2.0 3.0 4.0 5.0 6.C

METEOROLOGICAL Cn(10 7 m"1/3 )

of field data from long propagation paths, asgiven in Table I.


sures is not subject to a strong saturation effect likethat which affects laser beam scintillation, but giventhat short exposure blurring can comprise less thanone-third of long exposure blurring,9 we cannot ruleout the possible occurrence of significant saturationeffects for short exposures. Also noteworthy is that,although a wind direction being precisely along theline of sight (VI = 0) would theoretically result inthe slowing of angle-of-arrival fluctuations to the ex-tent that visual resolution would improve towardthat appropriate for short exposures, no systematicvariation of observed blurring with wind directionwas seen. A possible explanation may be associatedwith the large amount of mechanical turbulence pres-ent in the atmospheric surface layer; even with VInear zero when considered as an average all along thepath, local values of VI found instantaneously atpoints along the path can be large.

The equipment necessary for estimates of C islimited to a telescope and light sources, and no fieldcalibrations are necessary. If the atmosphere's limit-ing resolution angle is at least 4X/D for a typical por-table, astronomical telescope with Cassegrainian op-tics, the effects of the telescope can be ignored and asimple calculation of Cn from de, 1, and X with Eq. (5)and a coefficient of 2.7 can be made. Conversely,when the telescope's diffraction limitations becomeimportant in the total optical system, the MTF of thetelescope should be theoretically or experimentallydetermined and Eq. (3c) used. When very large di-ameter telescopes are used or where propagationpaths are several tens of meters above the surface,long exposure measurements require averaging timesof several seconds, and photographs, instead of visualobservations, should be taken.

During field experiments, particular attentionmust be paid to the path weighting effects of Cn onthe observed resolution angle. The value of Cn mea-sured will be most representative of the path nearestthe telescope, whereas according to Lawrence23 esti-mates of laser scintillation weights the center of thepath most heavily. To obtain estimates of Cn fromresolution that is representative of surface conditionsand a path with enough fetch all along its length can-not be found, it is desirable that the telescope be lo-cated in the downwind position and that the chosenline of sight be as close to the surface as possible.

The initial work for this paper was carried outwhile both authors were on military duty at the U.S.Army Ballistic Research Laboratories, AberdeenProving Ground, Maryland. The authors thank T. J.Gilmartin of the M.I.T. Lincoln Laboratories for hissuggestions that led to the use of visual observations.Also, appreciation is expressed to E. C. Alcaraz at theU.S. Army Ballistic Research Laboratories for his ad-vice and guidance. Meteorological measurements atthe Dresden cooling pond were carried out in cooper-ation with P. Frenzen and B. B. Hicks of ArgonneNational Laboratory.

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