c 5 5 AZ : H^^ S r | NOAATR NESS 57 NITED STATES ARTMENT OF JMMERCE BUCATION i < TO ' c stems of NOAA. The three principal Offices of NESS are Opera- lions. Systems Engineering, and Research. The NOAA Technical Report NESS series is used by these Offices to facilitate early distri- bution of research results, data handling procedures, systems analyses, and other 'information of interest to NOAA organizations. Publication of a Report in NOAA Technical Report NESS series will not preclude later publication in an expanded or modified form in scientific journals. NOAA Technical Report NESS series is a continuation of the series. ESSA Technical Report, National Environmental Satellite Center (NESC). and of the earlier series. Weather Bureau Meteorological Satellite Laboratory (MSL) Report: numbering follows sequentially. Reports 1 to 37 are listed in earlier publications of this series. Reports 1 to 50 in the series are available from the National Technical Information Service, U.S. Department of Commerce, Sills Bldg.. 5285 Port Royal Road. Springfield, Va. 22151. Price: $3.00 paper copy: $0.95 microfiche. Order by accession number, when given, at end of each entry. Beginning with 51. Reports are available through the Superintendent of Documents, U.S. Government Printing Office. Washington. D.C. 20402. ESSA Technical Reports NESC 38. Angular Distribution of Solar Radiation Reflected NESC 18. from Clouds as Determined from TIROS IV Radiom- eter. I. Ruff. R. Koffler. S. Fritz. J. S. Winston, and P. K. Rao. December 1966. (PB 174 729) NESC 39. Motions in the Upper Troposphere as Revealed by NESC 49. Satellite Observed Cirrus Formation. H. McClure Johnson. December 1966. (PB 173 9%) NESC M). Cloud Measurements Using Aircraft Time-Lapse Photography, L. F. Whitney. Jr.. and E. Paul NESC 50. McClain, April 1967. (PB 174 728) NESC 41. The SINAP Problem: Present Status and Future Prospects. Proceedings of a Conference held at the NESC 51. National Environmental Satellite Center. Suitland. Md.. January 18-20. 1967. E. Paul McClain. October 1967 (PB 176 570) NESC 42. Operational Processing of Low Resolution Infrared (LRIR) Data from ESSA Satellites. Louis Rubin. NESC 52. February 1968. (PB 178 123) NESC 43. Atlas of World Maps of Long-Wave Radiation and Albedo — For Seasons and Months Based on Meas- NESC 53. urements from TIROS IV and TIROS VII. J. S. W inston ami \ . Ray Taylor. September 1967. (PB 176 569) NESC 44. Processing and Display Experiments L sing Digitized NESC 54 ATS-1 Spin Scan Camera Data. M. B. Whitney. R. C. Doolittle. and B. Goddard. April 1968. (PB 178 424) NESC 45. The Nature of Intermediate-Scale Cloud Spirals. NESC 56. Linwood F. Whitney. Jr.. and Leroy D. Herman. May 1968. (AD-673 681) NESC 46. Monthly and Seasonal Mean Global Charts of Brightness From ESSA 3 and ESSA 5 Digitized Pictures. February 1967-Fehruury 1 ( >68. V. Ray Taylor and Jay S. W inston. November 1968. (PB 18(1 717. NESS 55. NESC 17. \ Polynomial Representation of Carbon Dioxide and Water Vapor Transmission. William L. Smith. Vpril 1969. (PB 183 296) Statistical Estimation of the Atmosphere's Geo- potential Height Distribution From Satellite Radia- tion Measurements. William L. Smith. April 1969. (PB 183 297) Synoptic/Dynamic Diagnosis of a Developing Low- Level Cyclone and Its Satellite-Viewed Cloud Patterns. Harold J. Brodrick and E. Paul McClain. May 1969. (PB 184 612) Estimating Maximum Wind Speed of Tropical Storms from High Resolution Infrared Data, L. F. Hubert. A. Timchalk, and S. Fritz. May 1969. (PB 184 611) Application of Meteorological Satellite Data in Analysis and Forecasting. R. K. Anderson. J. P. Ashman. F. Bittner. G. R. Farr. E. W. Ferguson. V. J. Oliver, and A. H. Smith. September 1969. (AD 697 033) Data Reduction Processes for Spinning Flat-Plate Satellite-Borne Radiometers. Torrence H. MacDonald.July 1970. Archiving and Climatological Applications of Metero- logical Satellite Data. John A. Leese. Arthur L. Booth, and Frederick A. Godshall, July 1970. (COM-71-00076) Estimating Cloud Amount and Height From Satellite Infrared Radiation Data, P. Krishna Ran. July 1970. (PB- 194 685) Time Longitude Sections of Tropical Cloudiness (December 1966-November 1967). J. M. Wallace. July 1970. NOAA Technical Reports The Operational I sc of Satellite-Observed Cloud Patterns in Northern Hemisphere 500-mb Numerical Analysis. Roland E. Nagle and Christopher M. Hayden, April 1971. '' J ? <>, U.S. DEPARTMENT OF COMMERCE Maurice H. Stans, Secretary NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION Robert M. White, Administrator NATIONAL ENVIRONMENTAL SATELLITE SERVICE David S. Johnson, Director a, £ o m o a. o O NOAA Technical Report NESS 57 TABLE OF SCATTERING FUNCTION OF INFRARED RADIATION FOR WATER CLOUDS GIICHI YAMAMOTO, MASAYUKI TANAKA, AND SHOJI ASANO WASHINGTON, D.C. APRIL 1971 UDC 551.501. 721:551.521.32'21. 14:551.576.1:535-1:535.43 535 Physics-radiation 535-1 Infrared radiation 535.43 Scattering of light 551.5 Meteorology .501 Methods of observation .721 Methods of computation of radiation .521 Radiation (meteorology) .14 Albedo of clouds .32 Scattering of radiation (terrestrial and atmospheric) .576.1 Physics of clouds For sale by the Superintendent of Documents, U.S. Government Printing Office. Washington. D.C. 20402- Price $1.00 Contents Abstract 1 1 . Introduction ] 2. Scattering Theory of Light Applied to Clouds 1 2.1 Light Scattering by a Sphere 1 2.2 Light Scattering by Clouds 2 3. Tabulated Functions 2 4. Some Optical Properties of Water Clouds 2 References 8 Phase Function Tables 9 in Digitized by the Internet Archive in 2013 http://archive.org/details/tableofscatterinOOyama Table of Scattering Function of Infrared Radiation for Water Clouds GIICHI YAMAM0T0, MASAYUKI TANAKA, AND SHOJI ASANO Geophysical Institute, Tohoku University, Sendai, Japan 1 ABSTRACT. Scattering of infrared radiation by water clouds has been calculated from the Mie theory. Tables are presented for: the phase function and the degree of polarization for each degree of scattering angle; the volume cross sections for extinction, scattering, and absorption; and the albedo for single scattering. Three models are used, representing altostratus, nimbostratus, and stratocumulus clouds. 1. INTRODUCTION In the paper, "Radiative Transfer in Water Clouds in the Infrared Region" (Yamamoto, Tanaka, and Asano 1970), space did not allow the inclusion of the phase function for the clouds. Requests for these values have prompted us to prepare the tables published here. To satisfy general needs, the phase function is presented not only for the altostratus clouds discussed in our earlier paper, but also for nimbostratus and stratocumulus clouds. Volume cross sections for extinction, scattering, and absorption, and albedos for single scattering are also included. The theoretical basis for computation of the values is described below. 2. SCATTERING THEORY OF LIGHT APPLIED TO CLOUDS 2.1 Light Scattering by a Sphere Scattering of electromagnetic waves by a homo- geneous spherical particle is described by the Mie theory, and is discussed in detail by van de Hulst (1957) and by Kerker (1969). A summary of the results is shown below. The intensity functions ii and t2, which are the components perpendicular and parallel, respec- tively, to the reference plane (i.e., scattering plane), are given as follows: ii(x, m, 6) " * 2n + 1 *i n(n+l) L n=l v ' + b n (x, m)r„(cos 0)} (1) i»(.r, m, d) 51 — ; — —■ rr{b„(x, m)7r„(cos0) -„f? n(n+l) a n (x, m)T„(cos 6)} (2) where a„(x, m) and b n (x, m) are the Mie coeffi- cients, and 7r„(cos#) and t„(cos0) are the angular coefficients as defined by van de Hulst. The effi- ciency factors for extinction and scattering are given by 2 x- Qext(x,m)=—\ (2n + 1 )Re{a„{x, m) + b n (x,m)} (3) Qsca(x, m )—~2j (2l+l){ \On(x, m)\- + \b„(x,m) | 2 } (4) and the efficiency factor for absorption follows by subtraction, (Jabs — Qext Qs (5) Thus, the physical parameters involved are the scattering angle, 6, the complex refractive index of the sphere relative to the surrounding medium, m, and the size parameter 27T/' A (6) 1 Prepared under contract No. E~24^69(N). where r is the radius of scattering particle, and A is the wavelength of light in the surrounding medium. 2.2 Light Scattering by Clouds The total scattering and absorption cross sections per unit volume of the clouds are obtained by inte- grating the Mie single-particle cross section with respect to the appropriate size distribution. The volume cross section. o\ may be given by the integral a ( m . A. ) y-TTr nr-Q (— , m ) n(r)dr (7) where n(r)dr is the number of particles per unit volume with radii between r and r + dr, Q is Q ex t- Qsca. or Q,,bs- and r } and r 2 are the lower and upper limits, respectively, to the radii of the particles. The albedo for single scattering, a>, is defined by (T s (Tabs ) (Tsea (Text (8) where cr sca , cr a & s . and a e xt are the volume cross sections, respectively, for scattering, absorption, and extinction obtained from (7). The elements of the phase function normalized to unity are given by the integrals j-P,(m,0,X) = (~Y — 4 77 \ItT I (d)]. (11) For unpolarized incident radiation, the scattered radiation shows partial linear polarization, with the maximum electric field oscillation being either perpendicular or parallel to the scattering plane. The degree of polarization is given by the ratio PAd)-P 2 (6) PAS) + P 2 (d) with positive and negative values corresponding, respectively, to the two cases mentioned above. 3. TABULATED FUNCTIONS The optical properties of water clouds are shown in tabular and graphical forms in the wavelength region from 5 to 40 fxm. In the calculation, we used the value of complex refractive index of liquid water evaluated by Ponder and Dechambenoy (1966). The drop size distributions of our cloud models (fig. 1) are those of altostratus (As), nimbostratus (Ns), and stratocumulus (Sc) measured by Diem (1948). The intensity functions (1) and (2) and the effi- ciency factors for extinction (3) and scattering (4) were computed using the computational scheme and recursion criterion of Deirmendjian, et al. (1961). To evaluate the integrals (7) and (9), we replaced each size distribution curve shown in figure 1 by histograms, using radius increments, 8r, of 0.25 (xm (Carrier et al. 1967); particle concentrations for each radius, ?>, are given by n(ri<) Sr^. The validity of our calculation was checked by comparisons with the published results of Deirmendjian (1964), Bauer (1964), and Carrier et al. (1967). The symbols used in the tables are as follows: Volume extinction cross section, -i EXT SCA ABS ALBEDO ANGLE PHASE POL (Text- in cm' Volume scattering cross section, cr sca , incm~'. Volume absorption cross section, cr„6 S , in cm -1 . Albedo for single scattering, co, per unit volume. Scattering angle, 6, in steradians and in degrees measured from the direction of propagation of incident light. Normalized phase function, f(cos 6), per unit volume, defined by (11). Degree of polarization, defined by (12). (12) 4. SOME OPTICAL PROPERTIES OF WATER CLOUDS In the infrared region from 5 to 40/a,m, liquid water has absorption bands (Yamamoto et al. 1970) near 6/jl„, (the v 2 band) and 15/u.m; the latter is a broad, strong band caused by intermolecular vibration, which does not exist in the vapor phase. Figure 2 shows the spectral distribution of volume cross sections for extinction, scattering, and 1 1 1 ■ 1 ! 1 1 1 I 1 l l Concentration ( cm" 3 ) Water content (g rrf 3 ) AS 450 0.28 Sc 350 0.09 NS 330 0.40 10 RADIUS (MM! 15 -1 — ,.l 20 FIGURE 1.- Droplet size distribution of various cloud models. As. Ns, and Sc denote altostratus, nimbostratus, and stratocumulus, respectively. The number density and liquid water content are also shown in the figure. absorption of the altostratus model. The absorption cross section is large in the band regions, whereas the scattering cross section depends mainly on the real part of the complex refractive index. At wave- lengths shorter than 10.5/um, the scattering cross section is larger than the absorption cross section; in the wavelength region beyond 10.5 /xm, the re- lationship is reversed. The total extinction cross section is the sum of the scattering and absorption cross sections. Figure 3 shows the spectral distribution of volume extinction cross sections (solid lines) and of albedos for single scattering (broken lines) for the three cloud models (altostratus, nimbostratus, and stratocumulus). The extinction cross section is largest in nimbostratus, which, as shown in figure 1, contains larger cloud droplets than the other clouds and is optically the most dense (highest liquid water content); it is smallest in the optically thin (lowest liquid water content) stratocumulus. In the far infrared region beyond about 10.5/xm, albedos of the three cloud types are less than 0.5; that is, the contribution of absorption to the total extinction is larger than that of scattering. Figure 4 shows the values of the phase function, P(cos 6), for the altostratus model at wavelengths of 5, 10, 20, and 40/xm as a function of scattering angle, 0. The predominance of the forward scatter- ing, which is a characteristic property of Mie scattering, is clearly exhibited at all wavelengths. However, it is more marked at shorter wavelengths, for which the size parameter x (i.e., particle size relative to wavelength) becomes larger for a given size distribution of cloud droplets. The pattern of the phase function would become flatter and smoother in a cloud composed of droplets with smaller size parameters. These features also are evident in the comparative phase functions of altostratus, nimbostratus, and stratocumulus at a wavelength of 10/xm (fig. 5). The forward scattering is most marked in nimbostratus, which is composed of droplets with the largest size parameters among the three model clouds. 0.15 0.10- X UJ b < b < o (S) b 0.05 20 WAVELENGTH (/"M) 30 40 FIGURE 2. — Volume extinction (ovj-,), scattering (ov„), and absorption (o-„<„,) cross sections of altostratus as a function of wavelength. 4 0.15 - CO CO CO o or o o »- o z r- X Ixl Ul => § 0.10 0.05 0.0 o o CD 20 WAVELENGTH {/M 30 40 FIGURE 3. -Volume extinction cross section (solid lines) and the albedo for single scattering (broken lines) for nimbostratus (Ns), alto- stratus (As), and stratocumulus (Sc) as a function of wavelength. 2 r 0'- (f) O CJ 0' - -2 - 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i i _ :— \ \ As — — \ \ \. X = 40 A Lm — ■"■ \ — - \\ \v 20/im -A- - >v SfjLfn V - \ 10/im - 1 1 1 1 1 1 i i 1 i i 1 i i 1 I 1 30 60 90 120 9 (DEGREES) Figure 4. -Phase function P(cos 0) for altostratus at wavelengths of 5, 10, 20, and 40/um 6 50 180 30 60 90 120 9 (DEGREES) 150 180 Figure 5. - Phase function P(cos 6) for stratocumulus (Sc), altostratus (As), and nimbostratus (Ns) at wavelength of 10 /uvn. REFERENCES Bauer. E.. '"The Scattering of Infrared Radiation From Clouds," Applied Optics, Vol. 3. No. 2. Feb. 1964. pp. 197-202. Carrier. L. W., Cato. G. A., and von Essen, K. J., "The Back- scattering: and Extinction of Visible and Infrared Radiation by Selected Major Cloud Models," Applied Optics, Vol. 6, No. 7. July 1967. pp. 1209-1216. Deirmendjian. D.. Clasen, R., and Viezee, W., "Mie Scattering With Complex Index of Refraction, Journal of the Optical Society of America, Vol. 51, No. 6, June 1961, pp. 620-633. Deirmendjian. D.. "Scattering and Polarization Properties of Water Clouds and Hazes in the Visible and Infrared," Applied Optics, Vol. 3. No. 2, February 1964, pp. 187-196. Diem. M.. "Messungen der Crosse von Wolkenelementen II," Meteorologische Rundschau, Vol. 1, Nos. 9-10, March- April 1948, pp. 261-273. Kerker, M., The Scattering of Light and Other Electromagnetic Radiation, Academic Press, New York, 1969, 666 pp. Pontier, L., and Dechambenoy, C, "Determination des Constants Optiques de 1'Eau Liquide Entre 1 et 40p. Application au Calcul de son Pouvoir Reflecteur et de son Emissivite," Annates de Geophysique, Vol. 22, No. 4, October-December I960, pp. 633-641. Van de Hulst. H. C, Light Scattering by Small Particles, John Wiley and Sons, New York, 1957, 470 pp. Yamamoto, C, Tanaka, M., and Asano, S.. "Radiative Transfer in Water Clouds in the Infrared Region," Journal of the Atmospheric Sciences, Vol. 27, No. 2, March 1970, pp. 282-292. 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CLTT (<) rl J >0 Kl O O roOOCvJOf^ T CM rl •© rO CM O T LOCOfOO iTlfO rHr^T TLr\r^OLriOLf\rlCOLf\rMOCOOLn rOAJrdO r\^^lTi«I'tt1 , W v)(M t\) oj dH Hrl H OC0^if\ , T , 0C\JAIddHdO[0> l Criini'o o O oo 00 o 00000000000000000000000003 o 00 OO oO 3 Od (\JPO 0I>oCMT , 0X>O oj ^^^^tH^H^w v-41-t cm cm cm cm cm ro ro f-ororoTT ^T^iiMTuri^^vO ,© o o r*. r^ pvp^rv co CDlTi dO dNCVIMO Xt^OO TO-LT* Oifir-t ^)dN ND O 3>0 d dM WfO ^ ^^<0>OM!t)>ood(\l CNJ'OT T-T*-0 O OJ O t > O D ^ ^ d O oiM t D^) D ^J N d-O 0>D^ 0^in^'O^Ho>DNOri!X) lv -'0 I> ^ "O \ JO^ Q-CV9 N^T (MO r-xTrH>OT rl>0 «-d !> rO'JO ^ » T -D OBfOt>--OrvAt^r\|r^r\l!N k rsjr^^j' N, *rHOr-tO T-lOrH-O joH^r od om'O ri N > =>-\jt «\> d"Ott)Hr\ o\jn >\o>o o3-ONotNdtcdnc\^ >\o > Z OOoO 3 O tH tH H drl H tMAjCV C>J M M 1*3 *3 ^) "O * T TTVlTi r*0 OO'^ M 33 D > > > OOorHrH ^1^^ -\j-0-0"0 ■<••••••• •• ••• ...... ...... ... .••• ...••*•• .•• • OO0O0OOOO OO 0O0O OOO oOoOOo^OOoOoOOOOOOOOOrlddrlddrlddrldrl PHASE FUNCTION TABLES -Continued HHHHHHHrJr400G0003000000 0000000aODOOOOaoO^OOO030000CD0C3OO0O0:^Z3O0O000O^>00 000<»" I I I II I I I !♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ + ♦♦♦*♦♦♦♦♦♦♦ I I I I I I I I I ajiiijcuajajLij ULUULUdjijuajJJdjajLUijjj'iJ-LJjiUiiiiiLUiJiJLUdJUdju^ij'iJ'UU^u.i) u u uajuuudj OOOOHlfiO'OlD03)fMtn^)lf\'OHN'0!DON^r\)fl - rvfOOOOiO'rtlv|v.fOlOIDIA'OrsH(D(00t>Vi"UC\]'rivo W»>OfO»*Dt\IOOOOHH(\ir\lf0^t^OMOOO(\l h OlAOMDOOOOO'Nto^HVO!)"0 0^(\ll>o iTi^>ci^r^(Eci)0-<>w^^^w^»H^«HTH^^^WTHC\j(\jrvjtv(vc\jC\jrvjror^)roc\j(\j(\j(\iTHTHr^c\ir- <> O "O => J^ ^" O -O f*^ ^CM^t-l>>*H^X*»H>> W -O 3D f TO ^ ^ > I3H»tt33»01\3JM) XX>X)rH.O«H^'00>0'OTHS>Oif\fO."MTH=»ODl^-r»«OOOOOOI^^DSJOO>0.'\J»rV'OX\t\l»'00>On^'0o=^Ai.M"0**xi^^i*-ooa)t>oaTHrt(\i , 0'0« - ^x\'0^i s »ooo>»c>TH'-(.'M"0'0> ^ ri ri H H ri rl d rl rl H rl H H ri H ri rl ,\ ,M .\ .\ M \ M M M M .M .M ,M -.M CVi \l \l r\l .\J M .M AJ .'J M .M .M ,\i .M . 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HI IX' 'XI IX' X X' Jr-f < JCMMft cr .p A: = = t- n -c * n e= in cr ii I— o + + + + + + + + + + t/> Q HJUJUJUJlDUJ IXtLUUJLlilXI 3 XI !rt»"Oin3>MiOc«-(\lif\ i— an «tc\j*r>cct)*-iroipoi^r s * <* oor^i^f^^r^30 a r! rl rl rl rl iH rl rl rl t-l IC » IT IT «NN CO O O Ul I - i«5 CM iH => O CO r^ O !T1 _J r^ CV N CV 1^ 1-1 C rIO r-! trcvoo B * C K N c » ZCCIIaOccc^ t- < • CV CV (V CM CM K! tO m W M £ I 1 C ) GO HI >*. w IT CPOPCC\JC\Ji\JC\iC\J(\J CMHH ,-i*-tOOOOOc:OOOOOOoOOOOO 00<=>oOoOOOOOOOOOOt-ttH ^■oooooooo ooc oooo ooooo oo ooo oooc oo oooooooooooooooooo I I I I I I I I I III II ++++ + +++++ + +++++++++++++++++ + + + +++ I I *-l • OJ UJLLJLDUJUJllJUJlUmiLJUJWLUWLUm <=> <=> -joojiftoor*.** tioa oro r*)ir» , J'f s ' no ^ 'J■ , o , T^o^^^oo(D^^lT^co , 0(OlO'J■(\lOT^ -C » ro dO(D*<0O(VO| , 0CD , *Wtfi H CVMT »0«>^CMf(N0COrttm\iWW^'lflN0*{Vl«CW O ■ OrHNo(CO'(\iCDlfi>Crl'ON'JO^OfOO' , TO i rO , COr(C\irOr^C\)HO , Ofn OlTi ^HOtD'C , f(\|O(00^C\JHir\O *o w o o^^wtHrv^^r^o^cv^r^wHwcvcvfor^sr^^iftXiiftiftiftift^^^i^ro^^ *-t ir> • ^" • It OOOOQOOOC3 COC OOOOOCOOOOOOCOOOOOOO oOCoOOOOOOOCCCOOOO + + + + + + + + + + + + ++ ++ + + + ++ ++ + + + + ++ +++ +++ ++++ +++ + + +++ + + + UJUJLJJUJLLLLIUJUJLL LULL'LU LL" UJLULULL1UJUJ LLIUJUJ LUUJLLJUJUJUJUJLIJLLJUJ LLJ UJ LU UJ UJ LL LL LU UJ UJ LLi LLi UJ Lb U-> LU UJ UJ U- (/)N^ :\J T O i-H > 0*0*0 OrO ^ O -T fO P*- DfO "O O TT» > t-t -O ft O'JTi^ r^^ o HOO ■O^HfO X> ; T\rOt-*^)rO'0<0 , J\ "O O Xi tft roi-i oioO ri ao (C o qri odro id ^ w o o x-t i t-too r-* <3" o tft o ro -o icon r^^rroao i-iV3jr» ur\ o >* ao ro*o o>x>^ ift o x> uo counrocv oo d r^ >o Lft ift ift «ft ^ft jt\ □_ *-t tH O O O O O O CO C-*. >OH ^ D HT N C\J N fO O C\J sf i-l C\J 1ft O iftCM O CO f^- Lft "J- "3" rO rO fO rO rO CVJ C\iC\lC\JC\JfMC\JC\JCNjOJ r^rvpN-r^r^f^.NO«0>0-0 >OOinT1'IOCV(\lHHH!DOirTK'rOC\J(\IC\IHH HHrlrlHrtHrlrlHrlHrlHHrlHH O OO OO O 003 T> O O 00000000000000000300 CDOOoOOOOOOOOOOOOOO uj YHtHt^cNjroro^^ift^oor^r^X'XO^o^ooHTHc^jcvrorororo^^^ O rHW*HfHWWiHr-tr-lr-)T-t,-lt--i T H«-*tHr-lT-lt-ltHtHrH o ft *h -OT-irvrvji^ro D ro o r*- r> o r*. ro o >0 *0 o >o ro o ^ro o -CO o-oro aO'O ro 'T lft ift •or-* rs coo O o -r-t «-t c\J "O uoto r>D Ajrv *-i o oi\t) o'oocoT-i^r-ooojTr*- o oj ift x) o >o o rn ^r o ;> dno'IM-iomhoo^ :o ^ o ift _Jof s ^.^-c\j>r^^-c\j>r*. 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PHASE FUNCTION TABLES -Continued ^ ^ in OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOwf(\lT-IW*-l»-fC\IO o o oo oooooozozdozsoooooocdococdcoo oooooooooooooooooooooooo^- I I I I I I I + + + ♦♦+♦♦++♦♦♦+ + ♦+++♦+♦+♦♦♦♦♦+ + + ♦+♦♦1 I I ( I I I I jujjij.Lijjaj.jjajjjjj.ij'jjajjjjjiiJjjijjLUJjjJw D4-O"0OAIOOHif\r\J^"cd ^ o if\ to oooddc\i(\i")i , ov ^" irviA^>r^ cd o tH c\j ^1- ia n o ocnjpo ^"ir\ir\^"C\JO toricco •© if\ cm ro o o ■ON DiDO^OdHddddddddddddd «d C\J CM CM CVJ CM CM CM rOfOrO f) rO PO K> f*3 CM CM ddlTiOM^^WOO O 03 C0> OOOOOSOSOSOO^OaOCaOOC^OOOOJCSOOOOOOO 03O33O03'CDC3'QOO0303OOOOOOOO I I I I I <-H rd td W rd HHd Hddddddddddddd dd dd dd oooooooaoooooooooooooooooooooooooooooooooooooooooo I LI I I I I I I I I I I I 1 I I I I I I I L I I I f 1 I I I 1 I I I I I I I I I 1 I I f I I t I t JJJJJJJJJJJJJJJJ-lJJJ-UdJlJJ'JJWlUWWWJJJJW /ico>\ v )>3s^« , d>3?ij3ND\o , oofi>ry'0(03irinr\ , TiS'j\^oofl'«oddOdry)Bii>^)03 , ow OOC\J^^^^dX»fOd>Od^C\J'0>OdO^C00 4 '\JC\idlfi>tOBCO I o.m o o:\j>jf^ir*:o-3-c\jsrr^rooco:ooir»o:r\o cm r**- r-v o ^- u*n -3- cm 00 d(\ik oco >dOOdMOtfOOdiOd 3- i f\M>>-XN OHo^rvH'OC\JC>fON l ,| , Ofl'in X> "N. **n OO^X^A^J-^ ^"^"^"-OO-OOO'O-O'OO'O'OOO oo-o o o o o o y) O ^O ^) O J ^ J 'O -O M \ \1 ,\ ,\l uX'DBDOooo^ocDzjoooddHddcMCNjcMcAjCMrofOi'Oforo 'j q- .r*-r^r^ 13 HHHrldrldHddHrlHrlHddHrldrlHHddHHHrlHHdrlHHHdHdHd (^ 3D X)> 3 O d CM M O t ^ if\ OOf^ X> D > OO d *d CM nO rO 0 ,M H 3 O D rv O Pi T !>flO> , OA"3'fO,\ldOODN^c>^>^>^>TB^co^Boco^Bror^Ai\^ji^v\iNC\irvryNHOdOd ^rd^*dtdTHrHTH^id^THrHTdtdiHrHC^C\JC\JC^<>Jt\jC>JC\JC\i(^ O^r0?0^^CMfM^rMr»J^C\irMddddddddddddddddddddd(M{\)W(0W^f0ddddddddd ^OO CO-OOOOCDOCO a OOGOCDCDOOOOOOOOOOCDCD OOOOCDOCDOOO 0000)0000000 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I UJJJJJUUJJUJJJ'JJJJJJ'JJ'JJLUlJJilJLUW JOBO^Ol'OSN >NaO^O(>NJ ,, T^N"00 , OC03't O CM ^0%OC><>:Dtdj'\td^COCM , .r\CO"0 , Tr-v I\ r\] >0 =5 O \£\ CO oo-ocororocvjincodTin>ooirin d d r, o o d t oc>t>in^Hiriror^rvtc(D<)0'{>-Ht-4ror\jc\jr^ doinNWOO tr\ro o o to m ir\ oj>aioo*o(\,iao o dfo ^ o cd tv 7 ^ i*) .*) t ir\ -© r*- o r*} ^" oo*r-.mro^"cocoLr\ ocMoroOi-tr^-mcM orotdr><)rdwc\jt%iroin<3r^co^WTHtdWiHTd(^jc\jC>j(^c\j(>jrvj{Aj^ O 03 O z: O C3 O OOOOS50OO9OS3 LOO o 03 00 OOOOOOOOOOOOOOODOOOOODOOOO I I I I I I I 1 I I I I I I I I I I I I I I I I I 1 I I I 1 I I I I I I I I I 000c ooocooooooooocooooaoo + +++ + + + + + + + + + + ++ + + + + + + + + + + + + +++ + + + ++ + + +++ + + + ++ + + + I I UJLULUUJmUJLlJLUUJUJWLJJLUlULULUmUJLUWW Si ^ x, o \i x; 3- r\ -e ,-* 3* u r 00: oririO^.\jPriT^3' <>NCdtPi.M^>Oti)OONd\OOnD^n\Pfi^^Mx3\0=;ddriOtO^^O'ON^ri^^>ON Ix»d>'OBIl OOrH"M^J>r>r\"\JTT AO t Jl i-H^r'3- ^*dr>>OOrHOX) , 00 03 0TdO:0:Or''v x <, \, o. ^■-•c-c>\,'C~r'3rvc-;\icoo-^"voo^ = u> m © ^ ,m d r; o > B m n *m > \N^di>(vo<'.'Od2^H BDDXNO^^q-v3OM.\ddddC0NO^t'0'0M^dddd>NOfliN^T'0'0.\.'\J.'NJdddddddCM> OOO 00030000300000000000 000 O O O OO OO OO OO O O O O OOOOOOOOOOO cjrr. c\^^ir , ci^aDocd(\ii*)^ if\'Or*-cco>oc\j^j- , caG ccv^ ( CcDc(\f>ccoc(\itoccca i ^ocDcDf\iococ a c p, t- -C r-: (v ry rv ;/; CD v) o ^- o if\ 3 ^ ,-; o h r*^ 1^ CO O O =-<- tH'.MrOi v 5^^"ir\'OOI s 'CDCOOC3HC\l f \l. v 20*0 LLio^^.-ocEirgr^-rH-ooiri^^rtDfOcDtxirv Tdvooocor^oO'J^ cDi^^iri^FOOjir\jHot>c\j ^^i^^rvc>r^^rNJChr^^tdc>'0^^o^>o^Tdc>^co^x>rox).^co. v 2cDror^ e30rd^ir*>o:Go(MroLi^r^c>ocM^<>r^c>rHro^cc^ir\cocMLr\<>CM^ Z o cc ~.r. r;t-HT- , ^t-rHiH v cmvAio.>cvcv ^ro^ro^^^Lf*Lriir»o^^r^r^ao<3Dcoc>c^ < COOCOOOOOOOOOOOOOOOOOOOOOOODOOOOOOOOOOOdddddddddndd PHASE FUNCTION TABLES -Continued cz vr^^trc^^torntr^r o> o in o « o cm M- 10 4) «om»» ir>^« »h « r~ f~ mo a. no »riMO oin o u\ C* nMom or^« (Mri(\iino nir no a>oHNNfOv*iriot^^orvrv(DoooTHC\iro^^N(DoHi*)^"ir»'Or^DO(DN'OTo^cc(\j o^o^odroNOO^ >o o o ro m ro t-t o x> ro o Or-troao < rvfo no c>>o cm^ non ro oo n ^o© Na o^N(\in>oooo>oinnLr\ «ocMMnif>odna)rooiDNo>o , o lo o lc>tH<>rOTHinro^cCLriLr\r^c\ir^iriijnr^<>c>j^r^ovOsor^r^-0(\j<>r^'CLAj^ a. i^ cm o cm x ^ d©toroH(>(D'OLhfl'nrvjc\jdoo(>o>0'0 , (>o>noood NN«*c>CM>ooooodddddr>;N(M!M(Mn«n«io»Tirvif, tr* ir\ irv >o o oo-oi^i^i^r^i^ao O, rlddddddddddddddddddddddddddddrlddddddddddd NCOItlOOOd(VIWIO»itir« lu idooxxi^vOun^rocM t-i o o x n o if\ o'icf^»«ch«>^»^x^xroXfOXWxr r ;r^rvjr~(\Ji^r\it^r\jr^fvit^^o»dCT-i is roooKr^oq-r^T-i^x dire dircw iro > '(Vi>co»to>cofor^ovr-»-. *ec^-iirec(Miro'Cv>cO'rcooi^r» o ^r Z WVtllMniCiO ONNNC(C»0'OOOr>T4rlK(Vnnn<)'V4'IMlMn'0 0<:iNNNOCOrl<-! 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