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AD-753 197
RADAR/RADIO TROPOSPHERIC ABSORPTIONAND NOISE TEMPERATURE
Lamont V. Blake
Naval Research Laborator"Washington, D. C.
30 October 1'972
DISTRIBUTED BY:
Pational Technical Information Service0. S. DEPARTMENT OF COMMERCE5285 Port Royal Road, Springfield Va. 22151
NRL Report 7461
N Radar/Radio Tropospheric Absorption
and Noise Temperature
LAMONT V. BLAKE
tJ-3 Radar Geophysics BranchRadar Division
iL
October 30, 1972
\'• NATIONAL TECHNICAL
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'I NAVAL RESEARCH LABORATORYI Washington, D.C.
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RADAR/RADIO TROPOSPHERIC ABSORPTION AND NOISE TEMPERATURE
* 4 OISOC RIPTv( NOTEK ('Ttry Olf po &ld jnylu.,oe deftl)
A final report on one phase of the problem.* *,j V"O.hS) (FL.., 0..*e, Al dfl. In~osl,I leer n~ms)
Lamont V. Blake
., REPORT OA( ?.. TOTAL NO OF PAGRZoOctober 30, 1972 84 I 27
ia CONTRVAAC"T OR GRANV NO RI. OiGiN l&O1*Vl REPORT VNLO.P.RISI
NRL Problem R02-64.2016 -RO•jEC .. U NRL Report 7461
Project SF-11-141-005.15483c. lb. 0 Ti[ 1 NRCPON V NO.I0 (Any Othr• .,bola thot roy b*44#ljI.d
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Department of the NavyNaval Ship Systems CommandWashington, D. C. 20390
Fortran digital-computer routines to calculate absorption loss for radiowave propagation in the tropo-sphere have been developed, and machine plots of radar attenuation as a function of range have been madefor a standard atmosphere, over the frequency range 100 MHz to 100 GHz. The tropospheric noise temper-ature has also been calculated and plotted. This work is an updating of earlier calculations that were lessaccurate, especially in the frequency range above 10 GHz. Separate curves for the oxygen and water-vaporcomponents of the absorption loss are provided in addition to the curves for total loss; this allows the lossfor values of water.vapor density other than the standard 7.5 g/m 3 to be found, by applying a simplemultiplicative factor. The computer routines, listed at the end of the report, have also been used as partof a computer program to calculate the maximum range of a radar system.
D D 147,S/N 0101 -0'0?.6801 Scutitv Closatfication
recutity Claosficotieonf
14 LINK A LwINKl a ýI C_V______ OL. WT SOLE ST Al i.O *
Microwave rtmospheric absorptionMicrowave oxygen abaorptionMicrowave water.vapor absorptionRadio noise "emperatureRadlowave propagation
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DD o.. 14 73 (BACK __ _ __ _ _
(PAGE 2) Secur~iyCliapgi Ica t on
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CONTENTS
A bstract ................. ............ .....................Problem Status .............................................. ii
A uthorization .............................................. i
INTRODUCTION .......................................... 1
ABSORPTION COEFFICIENT FOR OXYGEN .................. 3
ABSORPTION COEFFICIENT FOR WATER VAPOR .............. 5
DISCUSSION OF ABSORPTION.COEFFICIENT FORMULAS ...... 7
MODEL OF THE ATMOSPHERE .............................. 8
Pressure-and-Temperatu'e Model ............................ .Water-Vapor Model ...................................... 9Absorption Coefficients in the Model Atmosphere .............. 10
EQUATIONS FCR CUMULATIVE ABSORPTION ................ 16
TROPOSPHERIC NOISE TEMPERATURE ...................... 17
COMPUTER ROUTINES .................................... 69
REFERENCES ............................................ 78
ABSTRACT
Fortran digital-computer routines to calculate absorption loss for radio.wave propagation in the troposphere have been developed, and machineplots of radar attenuation asia function of range have been made for astandard atmosphere, over thý; frequency range 100 MHz to 100 GHz. Thetropospheric noise temperatuire has also been calculated and plotted. Thiswork is an updating of earlier calculations that were less accurate, especiallyin the frequency range above 10 GHz. Separate curves for the oxygenand water.vapor components of the absorption loss are provided in additionto the curves for total loss; this allows the loss for values of water-vapordensity other than the standard 7.5 g/m 3 to be found, by applying a simplemuliplicative factor. The computer routines, listed at the end of thereport, have also been used as part of a computer program to calculate theinaximum range of a radar system.
PROBLEM STATUS
A final report on one phase of the problem.
AUTHORIZATION
NRL Problem R02-64.Project SF-11-1,il-005-15483
Manuscript submitted July 13, 1972.
RADAR/RADIO TROPOSPHERIC ABSORPTION ANDNOISE TEMPERATURE
INTRODUCTION
The objective of the work reported here was to develop a computer program forcalculating the absorption of radio waves by the oxygen and water vapor of the earth'slower atmosphere - the ,Ionionized region known as the troposphere. Resulting calcula-tions have been used to produce machine-plotted curves of attenuation as a function offrequency and other parameters. The requirement that instigated this work was the needto know the magnitude of the absorption loss in calculating the maximum range of aradar. The computer subrcutines to be described have in fact been incorporated into alarger program that computes radar maximum range (1). However, the results are equallyapplicable to radio systems of any type that use the troposhere as a propagation medium
The frequency range considered is 100 MHz to 100 GHz. Below 100 MHz (andeven somewhat above this frequency) tropospheric absorption is negligible. Above 10r, GIzstrong water-vapor resonance lines result in great absorption. this region is not of muchcurrent interest for radar.
The work described updates earlier work begun in 1958, described in various NRI,reports and other writings (2-8). The original work (2,4,6) resulted in curves plcttcdmanually from manual calculations for frequencies from 100 MHz to 10 GHz. Later(7,8), the NRL NAREC digital computer was used to recalculate the absorption and noisetemperature- more exactly and to extend the frequency range to 100 GHz. However inthat work (as in the earlier work) the "centroid approximation" of Van Vleck (9) wasused to calculate the oxygen absorption; hence the results were not accurate in and nearthe 60-GHz oxygen resonances, that is, in the region from about 50 to 70 GHz. Also,the so-called residual ahsorption below 100 GHz due to strong water-vapor resonances athigher frequencies was approximated by a formula suggested by Van Viick (10), but theconstant factor used has subsequently been shown to give absorption values that are toolow.
In the current work, presented in this report, the oxygen resonances are explicitlyan. individually included in the calculation, so that the re.ults are valid in the regionfrom 50 iu 70 C!!z. The cq,,mti-nn uisd .are thos (-f MW-ký and l~illey (11) as slightlymodified by Reher, Mitchell, and Carte- (12). Improved numerical integration formulasare used for the cumulative-absorption and noise-tempe-ature calculations. An improvedexpression is used for the residual effect of the millimeter-wavelength water.vaporresonances, with a constant based on experimental results near 1 00 GHz reported byStraiton and Tolbert (13). Also a more sophisticated equation for the absorption of the22.235-GItz water-vapor line, due to Liebe (14), is used. Finally an improved model ofthe atmospheric water-vapor content is used. The numerical results of these improvedcalculations differ significantly from the previous results only above about 10 Gltz. Belowthat frequency the differences are small.
2 L.. V. BLAKE
Many authors have made absorption calculations, and it is therefore pertinent to statewhy it is necessary or desirable to publish these of the present report. Among othersthe reports and papers of Bean and Abutt (15), Hogg (16), Schulkin (17), Le Fande (18),and Ulaby and S&raiton (19) may be cited, in addition to those previously mentioned.This list is not exhaustive; in particular, it does not include all of the work published inother countries, some of which is referenced by the -'ithvfI- ,ited. The published resultsby these authors have been either limited to calculation of the absorption coefficient(absorption per unit distance) as a function of frequency rather than calculation of thecumulated absorption along a rry path or limited to calculation of the total absorptionfor paths traversing the entire troposphere. Some results are limited to specific frequencyregions or absorption lines. Moreover the curves presented by most other authors cannotbe read with sufficient accuracy to be useful for practical calculations of radio or radarsystem performance.
For radar detection-range calculations the absorption between two points within the
troposphere is usually needed, rather than total absorption through the entire troposphere.This report., and its predecessors, present curves of tile former tylpe (as well as the latter),with coordinate grids and scales that permit reading valueo to at least two-significant-figure accuracy. The curves of the present report have been plotted using the NRL Gerberplotter (Gerber Model 875 automatic drafting machine) and special computer plottingsubroutines. The coordinate grids as well as the curves are plotted in a single operation,to avoid registration errors. The plotting accuracy is of the order of 0.001 inch on theoriginal plots, which have been photographically reduced to about 1/3 to 1/5 as large forreproduction in this report.
To the author's knowledge results published by others based on computer calculationhave not been accompanied by listings of the computer programs used. The programspreviously written for the NAREC computer, on which the author's previous reportswere based, were written (by Maurice Brinkman, of the NRL Research ComputationCenter) in the NELIAC language, which was never widely used and is no longer muchused at NRL. The newer routines, described in this report, have been written by theauthor in Fortran for the NRL CDC-3800 computer. Complete listings of these routineswill be given in the last section of this report. Although versions of the Fortran languageused with other machines may difier slightly, ttie differences are usually minor, so thatthe routines can be readily adapted to any computer that has a Fortran compiler.
Another feature of the curves presented in this report is the separation of the oxygen
and water-vapor absorption losses, in addition to curves showing the combined absorptionloss. The combined results are for a "standard" atmospheric water-vapor density of 7.5grams per cubic meter. The oxygen component of absorptiii v. relatively stable andinsensitive to day-to-day changes in the atmosphere. For various atmospheric conditionshowever the water-vapor conte:-' may vary between roughly 2 and 20 grams per cubicmeter. The water-vapor absorption is (except for a small an : gnnerr2- :.y ik 'ificant non-linearity) proportional to the water-vapor density, Consequently separate oxygen andwater-vapor curves allow the water-vapor component to be adjusted for any water-vaporcontent other than the standard 7.5 grams per cubic meter by simply multiplying the7.5-gram absorption value by p/7.5, where p is the actual gram--per-cubi c-meter water-vaporcontent. This corrected water-vapor absorption can then be directly add-ed to the oxygenabsorption to obtain the total absorption.
-,-r. -%a"Wn .. s ' ~ l w e~ .. ~4
NRL REPORT 7461 3
The curves presented for cumulative attenuation along a ray path are for thle radarcase of two-way traversal of thle path. Conse-quently the decibel attenuptions are exactlytwice as great as they would be for thle radio-UommunliCation) case of one-way p~ropagationi.
The curves can 1)0 used for one-way systems b~y simp~ly taking half of the decibiel at tevtat ion
values shown for the radar case.
ABSORPTION COEFFICIENT FOR OXYGEN
As mentioned in the Introduction, the absorption coefficient for oxygen has beencalculated using equations given by Meeks aind Lilley (11), as slightly modified by Reberet al. (12). The theory was originally due to Van Vleck (9), later workers hive refinedsome of thle details. In particular Meeks and Lilley have formulated a refinement of thledependence of the line-breadth constant (which will here be dlenoted Yf) oil altitudle(prc'~sLrv and temperature), and Reber et al. have made a slight further refinement of
this factor.
The Meeks and Lilley equations are a summation of the absorption over maily oxygenresonance lines in thle vicinity of 60 Gli-z. These resonances occur for odd-integral valuesof the rotational quantum number N, that is, for N = 1, 3, 5------. .It has been determined(11) that the contribution of terms above N = 45 is negligible; hence 45 is thle largestvalue included in the summation, For each odd value of N there are two resonancefrequencies, denoted f and fN-- Thiese resonances are listed in Table 14 as given byMeeks and Lilley - The absorption at an arb~itrary frequency f is the sumn of contributionlsfrom each of these collision-broadened resonance lines, p~lus a nonresonant component. .The broadened resonance lines are assumed to have shapes defin(-d by thle Van-Vcevk-Weisskopf formula:
FN± (f) + - -- ) Tlf2 1-f f)2 + (.Af)2 O~N- + f72 +(f)
1
The nonrcsonant contribution is of the form
-ifF0 = (2)f2 + (Aýf) 2
The termis sumnied over odd values of N a-re of tefurm
A-F 2 FN 2 2 -E IkT (3)AN FN M+ F -- N- + FOPNO) V
whereJ2 N(2N +3)(4N+ N +1 (4
2 (N +-1) (2N- I)N - N(5)
4 L. V. BLAKE
Table 1Oxygen Resonance Frequencies
N fN÷ (GHz) fN- (GHz)
1 56.2648 118.75053 58.4466 62.48635 59,5910 60.30617 60.4348 59.16429 61.1506 58.3239
11 61.8002 57.612513 62.4112 56.968215 62.9980 56.363417 63.5685 55.783919 64.1272 55.221-121 64.6779 54.672823 65.2240 54.1294125 65.7626 53.596027 66.2978 53.069529 66.8313 52.545831 67.3627 52.025933 67.8923 51.509135 68.4205 50.994937 68.9478 50.483039 69.4741 49.973041 70.0000 49.4648-13 70.5249 48.9582
45 71.0497 48.4530
2 2(N 2 + N 4 1)(2N + 1) (6)
NNO N(N + 1) (
and
EN!k = 2.06844N (N + 1). (7)
The complete expression for the absorption coefficient is
oz(f, p, T) = CpT- 3 f2 >- AN (8)
N
where D is the atmospheric prosgure and T is the al,.-ooh:tc tempcrature. For f anid L, IIIgigahert4, p in millibars, T in degrees Kelvin, and 0 in decibels pir kolonmeter, C = 2.0058.For a in decibels per nautical mile, C = 3.7148. (One nautical mile is 1852 metersexactly.)
Van Vleck (9) estimated that Af was approximately 0.6 G(1z at 1 atmosphere pressure
and standard sea-level temperature and that it was proportional to pressure and inver.-,ly
NRL REPORT 7461
proportional to the square root of absoluto temperature. Later workers have suý.gested,based on experimental evidence, that the dependence on pressure and temperature is morecomplicated. The Rebet MitchM11-CarL•r (12) modification ur the Mtcks-Lilley (11 ) modelfor this dependence is given in ternis of altitude h above sea level:
Af = g(h) (P/P 0 ) (T 0 /T), (9)
where Po 1013.25 millibars (760 torr), To = 300'K, and
g(h) = 0.640, 0 < h < 8 km, (10)
g(h) = 0.6,10 + 0.04218(h -- 8), 8 < hi < 25, (11)
g(h) = 1.357, h > 26 km. (12)
These are the equations and constants that have been coded in Fortran in ,ubroutine
ALPHA, listed in the last section of this report, for computation of ALPH02, fhe oxygenabsorption coefficient.
ABSORPTION COEFFICIENT FOR WATER VAPOR
The absorption due to water vapor in the frequency region below 100 GH,. can beseparated into two components, one due to the water-vapor resonance at 22.215 GHzand the other due to the residual effect of the many resonance lines above 100 GHz. Theabsorption coefficient due .o the 22.235-GHz line, which will be denoted a22, has beencomputed from equations given by Liebe (14). The following equation is a combinationof his equations 2a, 6, 7, and 8:
C122 = 10 lo(10 e) Sop ; e2 . 4 ( 3 0 T F dBlkm, (13)
where the symbols are defined as
c = 2.998 X 105 km/sec,
SO = 13.92 tiz/torr,*
pw -partial pressure of water vapor (torr),
T - temperature (degrees Kelvin),
F - line-shape factor.
As was done for oxygen, F is taken to he the Van-Vleck-Weisskopf function (here includ-ing the factor f/fr):
"The theoretical value (14) is 14.33 llzltorr, hut the author has been advised by Dr. Ieicbe (privatecommunication dated June 6, 1972) that thi value involves approximations and that the experimentallydetermined value 13.92 it probhably more accurate
6 L. V, BLAKE
! ~~~Fr • A); (-fr--f)2 +(A1f)2 ( f +f)+ (14)
where fr = 22.235 GHz.
The equation given by Liebe for Af is
Af = 17.99 x 10-a 'Pw 1[ 300/T) + 0. 2 0 8 4 6 (Pt, Pw)] (300/T)')63 (15)
where again pw is the water-%apor partial pressure in torr and Pt is the total atmosphericpressure in torr; T is, as before, the temperature in degrees Kelvin. The atmospheric pressurep in millibars is given by
p = 1.33322 Pt, (16)
and the water-vapor partial pressure in torr is given in terms of the water.vat)or density p* in grams per cubic meter (for atmospheric concentrations) byr
P, = pT/288.75. (17)
Combining all of the numerical constants, with f expressed in gigahert/, p in n1W.iars,E nd p, in torr, results in
a = 2.534 x 10-3 Ifl'p (300,'T)' 1 2 e 2 1 444 -1 U, & F] dB,'km (18)
* and
= 17.99 x 10-3 {pw 1(300.T) + 0.20846 (0.75 p -o nw)] (3001T)()it}- (19)
These are the equations and constants that are Fortran c((led in Subroutine ALPHAfor the computation of AT,P1122, the absorption coefficient for th,, 22.235-Gllz water-vapor line, except that th,. constant 2.53.1 in Eq. (18) has been increased to 4.693 togive the absorption in decibels per nautical mile. The water-vapor parameter actuall-employed in the model atmosphere is the water-vapor density p (grams Ier cubic meter)rather than the partfl pressure in torr; it is converted to partial pressure hy Eq. (17).
The residual effect of water-vapor al)sorption lines al)ove. 100 (Glz has heen asSumed,following Ulaby and Straiton (19) and others:*
arcs = CR (P/P0) (P't0) (TO/T)'/ 2 (f,;f 0 12 (20)
where p is the water.vapor density (grams per cubic meter), p is the atmospheric pressuire,T is the temperature, P0, P0, and To are the zero-altitude values of p, p. and T, and f0is a reference value of f. The constant C(R van be taken from oxperimental results at any
SUlaby and Straiton dttributc this exponent For tili depeRndunce on ten'lwratur, tt, V.in Vh,.c', al Atlot hit is not givenl explicit y in Van Vleck's papt-r oin water-valx)r absiorption (10). It is iprobabtly dt'duci dfrom the footnote on page 420 of his paper on oxygen absorption (9).
NRL REPORT 7461 7
suitable frequency f0 . Based on measurements reported by Straiton and Tolbert (13),the value CR = 0.4 dB/km at f0 n 100 GHz has been adopted here. For p in millibars,taking Po = 1023.23 millibars*, To = 288.16'K, p0 = 7.5 grams per cubic meter (thevalue for the standard atmosphere which will be described in a later section), and f0100 GHz, this formula becomes
ores = 7.347 X 10--3 ppT-5/2f2 dB!km. (21)
This is the expression that has been coded in Fortran for the absorption coefficient (named:ALFRES in Subroutine ALPHA) due to the residual effect of the water-vapor lines above7
100 GHz except that the constant 7.347 × 10-3 has been changed to 1.361 X 10-2 togive the absorption in decibels per nautical mile.
DISCUSSION OF ABSORPTION-COEFFICIENTFORMULAS
The total absorption coefficient of the troposphere is the sum of the oxygen andwater-vapor coefficients, and the water-vapor coefficient is in turn the sum of the coefficientsfor the 22.235-GHz line and the residual effect of lines above 100 GHz. Of thesecomponents of the total absorption, the one that has the poorest theoretical basis is theresidual effect of the above-100-GHz (niillimeter-wave and infrared) water-vapor resonances.As has been well known for some time '10,14,18) the application of the theoretical linestrengths and the Van-Vleck-Weisskopf collision-broadening formula, Eq. (1-4), give absorp-tion values that are too low by a factor of at least 5. Other line-shape formulas haveneen proposed, but no fully satisfactory theoretical solution of the problem has beenadvanced, to the author's knowledge. Consequently the empirical formula and constantused here are believed to be as good a solution as can be applied at present.
The use of the Van-Vieck-Weisskopf line-shape formula (20) for oxygen and for the22.23F GHz lines is similarly suspect for absorption prediction at frequencies far removedfrom the resonance free: encies. Hlowever this matter is not so serious for these lines,because at frequencies far removed from the 22.235-GHz line its contribution is dwarfedby the contributions of the oxygen and higher-frequency water-vapor absorption linesand because the oxygen-line contributions at frequencies above about 90 Ghz are similarlydwarfed by the contributions of the millimeter-wave water-vapor lines, except in a verynarrow region about the isolated oxygen line at 118.75 GHz. The oxygen absorption isalso small compared to the water-vapor absorption in the near vicinity of the 22.235-GIlzwater-vapor line. At frequncies well below the z2.Z,35-GHz line the nonresonantcomponent of oxygen absorption is the dominant factor. No criticism of the shape factorfor this component (Eq. (2)) has been found in the literature by the author.
As mentioned in the Introduction, the results for oxygen and for water vapor arepresented in this report separately as well as in combination, so that correction to thelatter can he made for any desired deviation from the standard 7.5-gram-per-cubic-meterwater-vapor content. It was there stated that a simple multiplicative factor p/ip) could beapplied, where f) is the desired water vapor density and p0 - 7.5 is the standard value.AS Eq. (19) shows, this is not quite true; there is a small nonlinear dependence of Af on
*This value or p, is the total prtssure, consisting of the dry-atmosphere standard pressure of 1013.25millibars and the water-vapor partial pressure of 9.98 millibars corresponding to p = .5 g/m
3.l
L. V, BLAKE
p (through p,) in addition to thi linear explicit dependence of a on p, Eq. (18). Asimilar slight nonlinearity would appear in the residual contribution of the higher-frequencylines if an exact expression were used, but again the nonlinearity is slight, and in view ofthe empirical nature of Eq. (20) it was not thought worthwhile to include it.
MODEL OF THE ATMOSPHERE
The absorption coefficients, for which formulas have been given in the precedingsections, describe the rate at which signals traversing an atmospheric path will be attenuated,as a function of the pressure, temperature, and water-vapor density. To find the altitudedependence of the absorption coefficient, it is necessary to have a model of the altitudedependence of pressure, temperature, and water vapor in the atmosphere.
Pressure-and-Temperature Model
The mode! adopted here for the pressure p and temperature 'r of the standardatmosphere is the U.S. extension to the ICAO Standard Atmosphere (21); this is, forthe altitudes considered here (up to 100,000 feet or 30.48 kilometers), the same as the
ARDC model atmosphere.
The equations describing the temperature (degrees Kelvin) and pressure (millibars)
of this atmosphere in terms of the geopotential altitude hg (meters) are*
T = 288.16 - 0.0065 h " < 0j hg <• 11000, i22)
p = 1013.25 (T1288.16)IJ
T = 216.66 ( 11000 < hg < 25000, (23)
p (226.32,'T) exp [-3(hg - 11000)]f
T = 216.66 + 0.003 (hg -25000 )J25000<250 h• < 47000. (24)
p = 24.886 (216.66/T)'y
where
a = 5.2561222,
S= 0.034164794
y = 11.388265.
The relationship between geopotential altitude hg and geometric altitude h, is
hg = rha/(r + ha), (25)
*Reference 21, pp. 6 and 7, Eqs. (13) and (Vi).
NRL REPORT 7461 9
where r is the radius of the earth, taken to 356,766 meters. This relationship, whichis Eq. (10) oi Ref. 21, is valid for the altif nge nonsidered here.
The absorption losses were computed in tecrms of akitude in feet and decihels pernautical mile, to conform to the measurement units employed by Navy radars. The con-version from feet to meters is 1 foot = 0.3048 meter exactly; the conversion from nauticalmiles is 1 nautical mile = 1852 meters exactly.
This model of the atmosphere as incorporated into the absorption-coefficient computersubroutine ALPHA, as a table of values at 75 specified altitudes, in the form of FortranDATA statements. The pressure is given by the dimensioned identifier PP and the tempera-ture by TT. The 75 altitudes (hf) range from 0 to 100,000 feet in steps Ah that aregraduated according to the following scheme:
Ah = 100 ft, hf < 2000 ft,
Ah = 1000 ft, 2000 < hf < 30000 ft,
Ah = 2000 ft, 30000 < hf < 70000 ft,
Ah = 5000 ft, 70. 1,0, < hf < 10000 ft.
This graduation provides more closely spaced points at low altitudes, where absorption isgreater and therefore where more accurate interpolation of the tabulated values is dgsirable.The interpolation is provided by the numerical integration subroutines used in computingthe cumulative absorption. As will be discussed in detail later, the integration is done bya modification of Simpson's rule, so that the interpolation is more accurate than would beobtained by using the trapezoidal rule (linear interpolation).
Water-Vapor Model
No accepted standard profile of water vapor has been universally adopted. The oneused here is taken from a report by Sissenwine et al. (22). Their values for the water-vapor density up to 32 km (104,987 ft) ar,, given in Table 2. As shown, the vapor densityis specified at 2-kmn intervals, and the surface value is 5.947 g/m3 . The values correspond-ing to the 75 altitudes specified in Subroutine ALPHA were found by interpolating. Thesevalue,- were then multiplied by the factor 7.5/5.947 = 1.26114, to convert them to thesurface water-vapor value of 7.5 g/m 3 that seems to have been adopted as a standard byworkers in the field of water-vapor absorption. The interpolation was done by use of aspecial computer subroutine by a method equivalent to using a French curve on a plotof the data points. This method is described (in terms of curve plotting) in an NRLMemorandum Report (23). The resulting values of water-vapor density are incorporatedin Subroutine ALPHA in the form of a DATA statement for the dimensioned identifier RR.
Subroutine ALPHA also has a COMMON statement containing a variable namedRHOFAC. This quantity is set equal to 1 in a DATA statement, but may be set to othervalues by use of the COMMON statement in the main program or in another subroutine.All the values of water-vapor density in the model atmosphere are multiplied by RIIOFACin Subroutine ALPHA. Consequently if for example calculations of the attenuation andnoise temperature for a surface water-vapor density of 15 g/m 3 are wanted, the desiredrps•ilt. will he obtained by setting RHOFAC = 2.
10 L. V. BLAKE
Table 2Midlatitude Mean Water-Vapor Densities*
Altitude (kin) Density (g/m 3 )
0 5.947 X 1002 2.946 X 1004 1.074 x 1006 3.779 x 10-18 1.172x 10-1
10 1.834 x 10-212 3.708 X 10-314 8.413 X 10-416 6.138 X 10--418 4.449 X 10-420 4.449 X 10-422 5.230 x 10-i24 6.138x 10-426 7.191 x 10-428 5.230 X 10-430 3.778 x 10--432 2.710 X 10-4
"From Ref. 22, Table 3.
This procedure may result in vapor densities that exceed the saturation values forthe temperatures assumed. However this is probably not a serious violation of realism.It means that the temperature values should also be adjusted when the water-vapor densityis increased significantly. But since the water-vapor attenuation depends somewhat weaklyon the temperature, the inaccuracy that results from failure to readjust the tropospherictemperatures to values compatible with the increased vapor densities is probably slight.
It is noteworthy that p (h) does not decrease monotonically in this mode!. It doesdecrease up to 18 kin, remains constant to 20 kin, and then actually increases from 20 to26 kin; thereafter it again decreases. However, this nonmonotonic behavior is probablynot too significant from the absorption viewpoint, because it occurs at a vapor-densitylevel that is too small to contribute appreciably to the absorption.
Absorption Coefficients in the Model Atmosphere
The variation of the absorption coefficients at zero altitude in this model atmospherefor oxygen, the 22-GHz water-vapor line, the residual effect of higher-frequency waier-vapor lines, total water vapor, and combined oxygen and water-vapor absorptions areplotted for the frequency range 100 MHz to 100 GHz in Figs. 1 through 5. Figure 6illustrates the behavior of the absorption coefficients with altitude at several frequenciesin the most-used part of the radar frequency spectrum, from 100 MHz to slightly above30 GHz. These curves have been machine plotted from computations made with computersubroutine ALPHA, which is listed in the last section of this report. The plots have beenmade in terms of decibels per kilometer to facilitate comparison with other similar curves,although the curve, of cumulative absorption loss. shown later, are plotted to nautical-milescales.
NRL REPORT 7461 11
1 0 2 , , , .. . .
10' -COL
too
CL
-r 10-2*10-3
I-.
100 1000 10000 100000
FREOUENCTY MHZ
Fig. 1 - Absorption coefficient for oxygen at standard sea-level pressure and temperature
102 , ,.
101
100
CLi
C-
cc 10-2
10-3
10 -4 - .. ___toI .- --- I L LLI _________I___
100 1000 10000 1001300FREOUENCTr MHZ
Fig. 2 - Absorption coefficient for the 22.235-GHz-resonance component of water vaporat standard sea-level pressure and temperature
12 L. V. BLAKE
102
lot
1O0
3C o
W
A10'1
10-3:=-
10-24
1O0 1000 10000 100000
FREQUENCYP MHZ
Fig. 3 - Absorption coefficient for the residual effect of higher-frequency water-vaporresonance lines at standard sea-level pressure and temperature
102
10' _ _ _ _ _ _ _ _
100
r
SID " (n -CO
I0-3
0 10' -4__________
100 1000 10000 100000
FREFUENCTr M'IZ
Fig. 4 - Total absorption coefficient for water vapor at standard sea-level pressureand temperature
NRL REPORT 7461 13
102
to,
too
UJC-
0* 10"10
to-,
(J,4Lnor 10 "2
10'
100 1000 10000 100000
FREUNC r* M-HZ
Fig. 5 - Absorption coefficient for oxygen and water vapor combined at standardsea-level temperature and pressure
14 L. V. BLAKE
32500 MHZ
10~ 150000 tIz
300W
0 100-0
RLTITUOE' KM
(a) oxygen component
to-' -------- -- 15000 MHZ
10000 MHZ3000 MHZ
C3
Cr
(0 10 20 30
PrTITLJE. KM
(b) water-vapor component
S
NRL REPORT 7461 15
1001
1 0 1 *-.- - 3 2 5 0 C" '0- ) - .
SI -' -
T A X ICC HI
1 0 -2 1.. . .. .102
a: 10--
io
o 0 20 30
AL71TToE., KM
(c) oxygen plus water vapor
Fig. 6 - Variation of the total absorption coefficient in the standardatmosphere as a function of altitude for several frequencies
AM-
16 L. V. BLAKE
EQUATIONS FOR CUMULATIVE ABSORPTION
To find the total attenuation along a ray path in the atmosphere, it is necessary tointegrate the absorption coefficient along the path. This path is formally defined by thefollowing relation between range R (as would be measured by a radar) and height h:
hiC dR '6R(hl) = dhd. (26)
To calculate cumulative absorption along a ray path from the earth's surface to analtitude h, the bending of the ray path caused by refraction should be taken into account.This is a function of the ray elevation angle as well as the refractive index profile. Therefraction is significant only at relatively low angles. In principle the refractive indexshould be calculated from the water-vapor/temperature profile of the atmosphere. However,since the refraction does not have a major effect on the absorption, it was decided to takeit into account simply by assuming a standard exponential model of the refractive index,the so-called CRPL Exponential Atmosphere (24) for surface refractivity Ns = 313,
I given by
n(h) = 1 + 0.000313 e-kh, (27)
L where n is the refractive index and k = 0.00004385 for h in feet.
Taking the refraction into account, from Snell's law the derivative in Eq. (26) isgiven bb,
dR n(h) (28)dh ]// n0 c os 0Q -12'
V' In(l - h/ro)J
in which n(h) is given by Eq. (27), no = 0.000313, 00 is the initial elevation angle of theray (angle at h = 0 relative to the horizontal), and ro is the earth's radius. (The earth'sradius for the CRPL Exponential Atmosphere is taken to be 6370 kin, :r 2.0899 X 107 ft.)
The cumulative attenuation along this ray path is given by
A(R 1 ) 2 (1 u(h) dh, (29)0J
in which R 1 = R(h 1 ), a(h) is the absorption coefficient at height h, and
s . 1 dR. (30)
dh n dh
That is, ds/dh is given by Eqj. (28) with 1 substituted for n(h) in the numerator; R is thedistance as measutcd by a radar, along the ray path, and s is the corresponding geometricdistance. The factor 2 in Eq. (29) is used to obtain the radar attenuation, for twowaytraversal of the path.
___ I_ _ _ _ _ _ _
NRL REPORT 7461 17
The cumulative absorption is calculated in a Fortran subroutine named ATLOSS, whichis listed in the last section of the report. The derivatives ds/dh and dR/dh are computed intne short subroutine named DDH; the derivatives are therein named DSDH3 and I)RDII3.
The cumulative integration is performed by a modified Simpson's rule which is
described in an NRL Memorandum Report (25). However instead of the actual subroutinedescribed in that report an equivalent procedure is programmed in Subroutine ATLOSS.The method allows the cumulative absorption to be found at each of the 75 ranges (heights)at which a is specified, rather than only at every other one (half as many), as wouldresult if the usual form of Simpson's rule were employed.
This integration cannot be performed however at h = 0 when 00 = 0, because thenthe integrand becomes infinite. A special technique is used to approximate the integralin a small region about h = 0 where 00 = 0 (actually, in this case, from h = 0 to h = 200feet); this technique was described in a paper by the author (26).
Subroutine ALPHA is called by Subroutine ATLOSS for a specified frequency, andthe resulting 75 absorption coefficients o(h) are used to calculate the absorption (usingEq. (29)) at 75 corresponding ranges (found from Eq. (26)) along the ray path for thespecified elevation angle. These points allow a curve of attenuation in decibels to beplotted as a function of the range in nautical miles, with frequency and elevation angleas parameters. Such curves are given for 38 frequencies in the range 100 MHz to 100 GHzfor several elevation angles, as Figs. 7 through 98. Below 1500 MHz only the oxygenattenuation is plotted (Figs. 7 through 17), since water-vapor attenuation is negligible.At 1500 MHz and above plots are given for oxygen and water vapor separately, and for thetotal attenuation. These curves can bc used for estimating tropospheric attenuation forearth-based radars, for targets in the troposphere. They can also be used for estimatingattenuation for poit-to-point radio communication when one terminal is earth-based andthe other is airborne (ground-to-air, ship-to-air, or vice versa), by taking half of the plotteddecibel values.
The water-vapor and total-attenuation curves are plotted for the standard water-vapordensity (at zero altitude) of p. = 7.5 g!m 3 . The total attenuation for any other valueof Ps can be obtained by multiplying the decibel value read from the water-vanor curveby p,/7.5 and then adding the result to the decibel value read from the correspondingoxygen curve.
Curves of the absorption through the entire troposphere, for oxygen and water vaporseparately and for the total, have also been plotted, in Figs. 99 through 101.
TROPOSPHERIC NOISE TEMPERATURE
The troposphere radiates thermal noise, ind Lhe tropospheric noise temperature asseen by an earth-based radar or radio antenna is given by the following equation (27):
Tn = 0.2303 a(R) Tt(R) e- 0. 2 3 0 3 1O a(r)dr dR, (31)
18 L. V. BLAKE
0.3s
0.20 .. 0
0.5
I..
r0 10 S 20 20 30 5
als gioe the tota absrpio loss.0
10.0
0.0 6__
0 50 30 IS 200 250 30 35
R~flA-TDTRItT O5TRNE IIJ~iAL MLES
Fi.8-LdNasrto osdet oye o a ah ntcsadr0.mapti . 00M, 2.I
NRL REPORT 7461 19
2.0
5.
00.5
~0.s -j -0..
0p T so 00 ISO 20 2 300 - -
Fig. 9 -Radar absorption loss due to oxygen for ray paths in the
0.01.6 *
10.0
0 SO 00 IO 200 2SO 0023 00-OTE105ICE(A8~.MLS
Fig.10 adarabsrptin lss de t~ '~~ygn fo ra pats S thstnadatopee.t4C4?
20 L. v. BLAKE
3.0
2.5 II A
- .___ __________ ,.0
.0.0
0 so to0 ISO 200 2SO 300 350PflRA-T0-TROGET OISTRNCL INAUTICtt MILESI
Fig. 11 - Radar absorption loss due to oxygen for ray paths in thestandard atmosphere at 500 MHz
3.0f
S2.5 0.0
S1................. .
2.0 -. 0
5,0
10.0
0.00 50 100 I 200 250 300 35t,
RAOAP-T0-1RRGET OISTPNCE (iUTICAL MILES)
Fig. 12 - Radar absorption loss due to oxygen fee ray paths in thestandard atmosphere at 600 Mliz
NRL REPORT 7461 21
3.0
0.0
2.51
IN12.0a1. Iz
10.0
00 s 100 ISO 200 250 300 350-0~TO.TAR0T DISTANCE UTC.MIS
Fig. 13 - Radar absorption -os. duc to oxygen for ray paths in thestandard atrnosphtrc at 700 MHz
0.0
5.0
2.00C"
0 .
0 so 100 ISO ?00 250 300 350
RARP~-TOT-WET DISTANCE fN1UTICP4 MILES)
Fig. 13 - Radar absorption loss due to oxygen for ray paths in thestandard atmosphere at 800 MHz
22 L. V.BLAKE
0.0
1.0
2.0
10.
0 s 100 ISO 200 250 300 350
AAOAAf-70-TAWZT 0I31JKt (hUJTIrCIL MILES)
Fig. 15 -Radar absorption toss due to oxygen for ray paths in thestandard atmosphere at 900 MIHz
4 F
0.0
2.0
10.0
0 so 100 ISO 200 2501 300 350
-TO-TAACCT OISTAICE IN4lIJIICAL MILES)
Fig. 16 -Radar abs~orption toss due to oxygen for ray paths in 'he~standard atmosphere at 1000 MW.z
NRL REPORT 7461 23
0. 0
S.0
10.0
0 50 too ISO 200 250 300 3SO
FWR-TO-TAAET DISTANCE 1INFJT1CAL MILES)
Fig. 17 -- Radar absorption loss due to oxygen for ray paths in the£ standard atmosphere at 1200 MHz
I 2.0
r)r
50 100 150 200 "so 300 350
*Z.. FI0 TWIGET DISTANCE ,,J~.TICAL MILES:
[,;k. 18- Oxga c.,nipuii.-iit of therla& i.da b~upt*o. c n iaý p~i~fi, i.,o
standard atmosphere at 1500 NMHz.
24 L. V. BLAKE
' .10 ... . .... .
,0 1g.,IN 00.0
S0.04 0 .0
i i0.5 G. w 1.0
12.0.0O
0.00 10.00 50 100 IS 200 250 300 350
AROA-O-T-1i•T OISTNCt 'NAJTICAL MILES)
Fig. 19 - Watcr-Vapor componcnt of the radar absorption lo- for ray pathsin the standard atmosphere at 1500 MHz
'4 0.0
3 D , 0, 5
0 -------0 5) ino ISO 200 250 300 350
Pqg-T0-TARGX. I 01fNCE (NAUTICAL. MILES)
Fig- 20 - Total radar absorption loss for ray pathF in thestandard atmosphere at 1500 Nf[.
NRL REPORT 7461 25
0.0
2.0
50.0
0 so C100 150 200 250 31E 350
* OPIRTO.TART 0ISTANCE (NAUTICAL NilES'
Fig- 21 -Oxygen component of the radar absorption loss for ray paths in thestanrdard atmosphere at 2000 NlI1z
0. 10
0. 08
0 .0 -
0.060
0 50 00 10 20 250 300.50
0-.T0T05T( P4TC MLS
th stndr 1t00pcr aSO 2000 300 35
AROWTO-TRGETOISWE IAUTIAL MLES
F
26 L. V. BLAKE
* °r
0.0:3
.0.
¶ ~2.0 j
r0 so 100 ISO 200 250 300 350
VIA-TO-TAXAT DISTANCE (NAUTICAL MILES)
'Fig. 2.3 - Total radar absorption loss for ray palths, Iin the
standard atmosphere at 2000 MI-z0.
3 ks
2.0
I. C _ _ __"_ I i •
0 0.
0 s I :so 200 250 300 350
SI•PO-TO-TIARGT DISTAN'CE (N4AUTICAL MILES)
Fig. 24 - Oxygen component of the radar absorption loss for ray paths inthe standard atmosphere at 2500 MNlz
I,
NRL REPORT 7461 27
0.161
0.080
5.0
0.0.0
~-TO-TAGT 0ISTANCE (NAUTICAL. MILESI
Fig~. 25 -Water-vapor componeat of the radar absorption loss for ray pathsin the standard atmosphere at 2500 MHz
I 0.0
10.0
'I2 u
00A-O-..E D1YW M1AT~ ILESI 0
Fig. 26 - To tal radar absorption loss for ray paths in the.~.aindard, atrosphcrc at 2500 MIIz
28 L. V. BLAKE
.0 14
0.0 -
10.01
0 i
! ' , i , 2.0
0 So 100 150 200 250 300 3S0AR!-T0 -Td T•d). DISTANCE INA:UTICAL MILESI
Fig. 27 - Oxygen component of the radar absorption loss for ray paths inthe standard atmosphere at 3000 MIi4
0. 20
0.16 . .. 0.0 A
S0.12
o •0. 5 .
€ 0.08 1.0
0,0,{• . .2.0
• - S.0
10.010.00
0 50 D00 ISO 200 250 300 350
RODAR-TO-TARGET DISTANCE INAUTICAL MILESI
Fig. 28 - Water-vapor component of the radar ahrorption I-RR fnr ray p3!hFin the standard atmosphere at 3000 MHz
NRL REPORT 7461 29
f " •0.0
-0.5
1.0 ra A-2.0 t
- - S.0I
10.0
0410
0 S 100 ISO 200 250 300 350
AA-T0-TCi• DISTANCt INAUTICAL MILES
Fig. 29 - Total radar absorption loss for ray paths in thestandard atmosphere at 3000 Ml.
10.0
3 _ ý
42.0
10.0
0 Sc0 100 150 200 250 300 3S0
MAD-TO-TARRCt DIST
ANCE INAUiTICQL MILES,
Fig. 50 - Oxygen component of thec radar absorption loss lor ray paths inthe standard atmosphere at 3500 Mliz
30 L. V BLAKE
0.30 1.0
t 2.0
77 .100.00
0 so 100 ISO 200 250 3S0MALIA-TO-TARGET 0IST~tiCE (NaMUT;CRL MiLESI
Fig. 31 -Water-vapor component of the radar absorption loss for ray pathsin the stantdard atmosphere at 3500 NIHz
0.0
7.s.0
10.01
0 so 100 150 N00 250 300 3C~n
RA0ARk_!O.TAR~t 0ISTANC_ fN9.jTI1AL MILESI
atmosphere at 3500co hHz
INRL REPORT 7461 31
I _____________________________________________ _________I S _ _ _ _ _ _ _ _ _ _ _0.0_
I4
22
5.0
0 so 100 ISO 200 250 S00 IS0PAPA~-TO-TAET DISTANCWE (NAUTICAL ((ILESI
Fig. 33 -Oxygen component of the radar abs'rptinn Ino,, for ray paths inthe standard atmosphere at 4000 hMHz
0.30I- .
-a 11.0-- 4 _
~O.I0 -. -2.0
0.0 - 0 0
--.000.
00 so 00 ISO 200 250 300 350
RADAR-70-T~GaT 0ISTANE ItAUTIC4L MILE51
Fig- 34 -Walcr-vapor component of the radar absorption loss for ray pathsin the standard atmosphere at 4000 MHz
32 L. V. BLAKE
142I
2 2.0
10.0
0 -) 100 ISO 200 250 300 350
RAO0R-T0.?AAGET DISTANCE INACTICAt. 'ILESI
F~ig. ~35 -Total radar absorption loss foi ray paths in the standardatmosphere at 4000 MHz
S
- . 0.04 Ir2 .0
0 so 100 ISo 200 250 300 350
PA0PAT0-TAACET DISTANCE INPAJTlCPA PIILE51
Fig. 36 - Oxygen component of the radar absorption loss for ray paths in
the standard atmosphere at 5000 MHzI
NRL REPORT 7461 33
LS~0. 0.---
1.0
0.20
___7_7570
10.0
0 so 100 I5O 200 250 300 350
PAO0-TO-TAKET DISTANCE (NAUTICAL MILMS
Fig. 37 -Water-vapor component of the radar absorption loss for ray path&in the sAandard atmosphere at 6000 MHz
S 0.0
------- -- -.
I I1 I
01~ --- '-
atophr at 500.M
34 L. V. BLAKE
0.0
S.0
00 so 100 ISO 200 250 300 350
FF(-70-YARGET 01!TAlNCE (NAUTICAL MILES]
Fig. 39 Oxygen comnnonent of the radar absorption loss for ray paths in thestandard atmosphere at 7000 1vHz
4 I4 r
0.6~is. .-
1 _______2.0
6O 5nI0so 200 2SO 300 350R~AOA-TO-TPAGET DISTANE INAUIJ~CRA MILES)
Fig. 40 -Water-vapor component of the radar absorption loss for ray paths inthe standard atmobphere at 7000 Milz
NRL REPORT 7461 35
6 I -I, , . ~ I -j
I..I11t- j* 0.4
i- i I .--- " ,
"'-4 - / •-- • ' . - 4.. . li
"j--/--- - '2.0
I :I // , i i _, i - --
. '- -- _ . _ 5.02 t
-10.0IV iI ,: I 3
0 so Ito ISO 2 M0 250 300 3S0
FWCI-T0 -T AG .T OISTANCE lNiF TIC:L MILES)
Fig, 41 - Total radar absorpticn. loss for ray paths in the standard
atmosphere at 7000 MHz
0.0
r- i - ' - ... -• -•--- •~ --- 1 --1 --1t .
I ' ' --- - "' I. - I 8 .
PA R-T0-T RG.T .ISTA-.t I.J• T .ICA _ MILES)
Fi.42 - Oxgncomponent ofteradar Pbaorption loss for ray paths in
the staindard atmosphere at 10 6Hiz* --I.- --"*. - ---- -r.. . . ... o. o
36 L. V. BLAKE
3.0 r-- m-11 ~ ~ - --1T -- ~-*--
I ' I i I
2.5 .
0.0 9_g 2.0 . . . -- ! i i o o
1.00
I2.
0 10.0
RA~DAI-T0.TAP.CET DISTANCE [NAUTICAL MILES)IFig. 43 Wtrvprcomponent of the radar absorption loss for ray paths
I. . ._ _ . . " ! -T - i o
rI0.0
6-
____ ____ _ 0.0!
5.0
0 50 100 ISO 200 250 300 350
RIADAR-TO-TPRRLT 0i5TANCE !NAUTICAL MILESI
Fig. 44 - Total radar absorption loss for ray paths in thestandard, atmosphere at 10 GlHz
I T-- .
NRL REPORT 7461 37
6I I
S..... -----
--I*I I ; 1 ,
-11.0
0 50 10 0) 150 200 250 300 350
P0ADRRTO-T6#Aj 0I5¶ANCf (NRUTIC&L MILES[
Fig. 45 - Oxygen component of the radar absorption loss for ray paths inthe standard atmosphere at 15 GH0
i *--- .,
I : 1-'I -
i I ; rI . IS I . '. 1I
-J ... I
', . .. .. I - - 0.o
. , : , I:I,j/ i I •- , 1 0 g3 r
I# - I . ,I -'-- .- ,~
-2 2.-----r-
I -__ _ __ _ - - L ___ 5.0. ....... 10.0
0w10 so DOo ISO 200 250 30), 350
FVOW-TO-TAtET DIStI :E INlUTIC.L IILES)
Fig. 46 - Water-vapor component of the radar Absorption loss for ray pathsin the standard atmosphere at 15 -Hz
I138 , V. BLAKE
1 .0
L I I
. i.
I 1 . ""S-------------; 4.. .. ... .. .. - 1- , ... , '1° •
- -L
, I .'I ,, 'I
Il~
**-'-*! .; * ;--- -
0.
2 .
If - , !I I . .I•
0. -1 ..- , - -
o so 1O0 ISO 200 250 300 3SO
PADA.TO-IAFQ DISTANCE (NAUTICAL MILES)
Fig. 48 -- Oxygen component of the radar Raoor[ptinn oth for ray psai inthe standard atmosphere at 20 GHz
NRL REPORT 7461 39
- "
I20 .-4 I •
_____ I,_ _ _ _ _ _ _ _ _ _. 5.0 -
_ _"_ _ _I _I _ _ _, Itn t s rp 2.0
. i 0
1---- -10.0
Si Io 1: • I ---- * ---------:-- .
0 L.0
50 ioo 150 200 250 300 350RADAR-TO-T1RC.T OISTCKE INqUTICkL MILESI
FilF. 49 -g W'ater-vapor component of the radar absorption los for ray pathsin Lhe standard atmosphere at 20 GHz
'__0.0
- - - -- -- - - -I- - - - w.
2.%0
I 1C.O
0 SC 1o0 ISO 200 250 300 350
AFFO):W-TO-TAfla-1 01517AKEF INAIUTICRJL M'ILE.S)
Fig. 'it T-- olI• rprl•r Fihurplinn In€., fnr r~iy palhk in thostandard atmosphere at 20 GHz
40 L. V. BLAKE
7 -
-~ <- -- ~-~ -~-0.0
- . -4- - - 4 - --
2.0
10.0
00 50 100 ISO 200 250, 300 350
AAAAT0-TA~kGE DISTANCE INAUTICAL MILES,
Fig. 51 - Oxygen component of the radar absorption loss for ray paths inthe standard atmosphere at 22.235 GHz
70 0.0
60 00-420 0 5
so t. TOT5ET05.CtIUIRL5L5
40 .5 ~.e ao opoeto h aa bopto o~ o a ahatmosh"'~ t 22.31.0H
r
NRL REPORT 7461 41
so [ ,. . ........ . I
0.0
S- : ' --4 --
6 . .. . ... 4 .. . . . . -." I I I60 41 ~ >50 -
-1.0K . .. . . : , ! I
,~. : 7277 __•___ / .0
I . I
0SO 200 250 300 350
(WOI.T0-TI:CFT OS01dTt ICE RJtCRI. MIL.ESIFi!g. 5,3 -- Totail radar absorption loss for ray ptith ii ;.lie
standard atmosphere at 22.235 GHz
* i , i - . - .- .* .
2.0
101
-J _ . / ,, . : .. . . :
10.0
0 5.
0 so 00 ISO 200C 250 300 3S0
nRAOI-T0-TARRGT OISTANCtE (NUTICAL MILES)
Fig. 54 - Oxygen component of the radar absorption loss for ray paths inthe standard atmosphere at 22.3 G0Hz
L7
!6
I\ - i • I =- Ir : -
iI
42 L. V. BLAKE
SII
~~~~S T° - !.. ---Y ------ -- . .. - - - t - -I / I ! i - iI- -00
"-'/ ' - , ' 0.L9 ..- - ... --7_ _ ___ _ _ _ _ _
g' Ij Fi I 1I ,oI / -.-
;' ,. II I I ,
I . I-TO-TA T DISTAKE INAUICI. MILFS
Fig. 55 - Water-vapor component of the radar absorption loss for ray pathsin the 6tandard utmo•phere at 25 GHz
601 I I -
--- t -- -.. .- -0 ..... 0.'. ', • ' Z2Y _.---- I -- ,io I
I -~".5
K4 ----4
21) ----
-T -
-. 1 17 . ,7 , . . . ,I-
4I .I .
0 so I0 ISO 200 250 300 350FRW-T--TAGET DIST- . IA - --. -N MILES)
Fi/ . 5C - Tota I radar absarption to= for ray path;n thstandard atmosphere at 25 GI-z
P_.
NRL REPORT 7461 43
10I IT
0.01
,________,,I
4 2.0
2 5.0
0 J..0 50 10 ISO 200 250 300 3SO
RmMOA-TO-TRWCT DISTANCE (NAUIlCL. MILES] iFig. 57 - Oxygen component of the radar absorption loss for ray paths in I
the standard atmosphere at 30 GHz
30' I I I
101
_ _._-___: . .. . .. . . . _-- - -_.. . ,_ . ._ _ . I1 - . . . . . - 1 . . .
+2._I.. . . -- _ - -, - -I I I, -
SI
I./ I/ I
. -.0
0 so 100 150 200 250 300 350
D-t0-tmrLT DISTACE (i/rJTIC, MILES,
Fig. 58 -- Water-vapor component of the radar absorption loss, for r" pathsin the standard atmosphere at 30 'HI
44 L. V. BLAKE
,0 ' I 4 j' ' I I
30-- .I i Ii. / I :
5 -- - .. - .. .... .- --rJ -4 -4 _,_22.
S.0S• t • 1 0.0
0 so 100 50 200 250 300 150M O A•R •- T0 -TA R .CT O IS TA K E (N AU T IC A L .M IL E S I
Fig. 59 - Total radar absorption loss for ray paths in thestandard atmosphere at 30 GHz
'4
I
--.. . ; -.. . . . . ... . . ... ... . .. . . I....- .. . .. ... ...I - - - -•-
16.......... . ..... .. •. . . .. . . . .:
. . .- ~. -. •- - - - - - - - - - - - - . 0.0
/ _ __-- ___ ____ - -- ;-- - -- - -- - ,0.00 .. . .. •. .L ;. 25 .0 35.
0 0-I_
_0
o0 so 100 150 2w 250 300 3so -
, -�,r�i3,�A34'�,r;�-�~tM z t C ShC:(-
Fig. 60 - Oxygen component of the radar absorption loss for ray paths in
standsrd atmosphere at 32.5 GHz
NRL REPORT 7461 45
30 F-___
0.0
-. V 0.5
Io -.-.T :,-1
110.,- -- I - ..- ' .
• [ E /I - . . . .... .•... - . ; - a ' - , . .- --..:,
-".- " -2 -' '
0-0 50 100 150 2w 250 300 3S0
A:I0A-T0OTRtGET DISTANCE NflJTlCAL MILES!
Fig. 61 - Water-vapor component of the radar absorption loss for ray pathsin the standard atmosphere at 32.5 0Hz
4I0 f - - 7- --T r " -
i0 0
30~ .
____ ____ 0.5
a a-
- - - - - -- ~ 10.0
F•N-t0-TY•.,T OISTANCE INA'Jt1CAL MILESI
Fig. 62 -Total radar absorption loss for ray paths in the standardat-nosphere at 32.5 GHz
46 L. V. BLAKE
16 J
~~i7 0.0 -
-< H 0.
S.0
- ~ I 10.0
0 so 100 ISO 200 2;0 30 350AAAR0-TO-TAAGET OISTANCE (NAUTICAL. MILES)
Fig. 63 -Oxygen component of the radar absorption loss for ray paths inthe standard atmosphere at 35 0Hz
20--- -- ---- ---
-- 0.0
__ _ _ _ __ _ _ 1.0
0 s0 t0o ISO 200 250) 300 350OAn0?1?aAIFT VWfllT0!5CF INQ.I- ira HOF-.)
Fig. 64 - Water-vapor component of the radar absorption 1toss for ray pithsin the standard atmosphere at 35 0Hz
NRL REPORT 7461 47
I 0.0
20------------- 1 .
102.
5.0
10.0
0 so too 1$0 200 250 300 350
Fig. 65 Total radar absorption loss for ray paths in the standard
atmosphere at 35 GHz
I 30 -25
4- 20.
*1 0I _ _2.0
10.0
I0 so 100 I-SC 200 250 300 350FRqADAR-0-TRAGET OISTANCE (NA~UTICAL MILES)
Fig. 66 - Oxygen component of the radar absorption loss for ray paths inthe standard atmosphere at 40 GHz
48 L.V. BLAKE
- - .- . . ... . .. .o0.0
2!.
S---- i-15. .
- I ______ --- --- ~ -- .0
0 '-i i I0.0
0 50 100 ISO 2)3 253 30C 350
S~nO•AR-!0 - TRACEYT 0|sr?<E lNcALJTICA- M'IlLES]
Fig. 67 - Water-vapor component of the radar absorption loss for ray pathsi,a the standard atmosphere at 40 GHz
60
47so., 0.0
2 1
2. 0-5 .0
0 50 00 ISO 200 250 300 350
RAI0-10 7ARCT DISTANCE (NAUTICAL MIL.ES,
F~ig. bb - Total radar absorption loss for ray paths in the standardatmosphere at 40 GHz
NRL REPORT 7461 49
0.0
I . .. . . .. ..... ..- - - - -. " -- . . . . -,. . .. " -
II, ,ji : !40
:0 L -0- 0 5 0 3
0 S_, __o ISO , _00 250 300 350
AR.O•-TO-TAIGEIT OISTW . INA:JTICAL MILES)
Fig. 69 -- Oxygen component of the radar abso•rptiwn logs for ray path.% inthe standard atmosphere at 45 GHz
20 - 4,• ' '
10 ' / ..0
0) 50 100 M5 200o 2--0 33ý 350
R13DA9-TO-1A9G(T OlS'Cl(Cf (NQIJTICI•L 'IML[1
Fig. 70 -- WaLer-vapor component oi the radar absorption loss tor ray pathsin the standard atmosphere at V/, 0117
50 L.V BLAKE
80--0.
t i 0.5
70. 1 --
30 2 .0
-- 1.0
0 s0 I00 ISO 200 250 300 350
RfPOZA-TO.1AGET OISTRNC( (NAUTICAL MILES)
Fig. 71 -Total radar absorption loss for ray~ paths in the standardatmosphere at 45 GHz
2011 - ----.--- ----- --------
0.0
120Ž3 0
Fig.- 1 .i~1ii7 .Ii~~i 0.5
the stnadamsheea 06h 5.
NRL REPORT 7461 51
"40 _T _ T . - -
:2Ch----------...-... -' . - - -* -- . . . ...,
20.
IS. D
10.0
0 s0 ID I S IO 250 3"0 350
PAA-TO-TARGT OIST~ACE (Nk-7CAL MILES)
Fig. 73 -Water-vapor compoltenl of the radar absorption loss fwr ray pathsin the standard atmosphere at 50 GHz
I I ' 0.0' : " •
s o -
I2.
|. . ...... • - 5
'p I•
I.0 so 100 1 50 200 250 100 350
• N•0P•RA -T0-TARCE.{ 015TAW•E INP,'TICAL M41L£SI
I•. ~~ ~Fig. 7 4 Toa \•'t r-apo o lK•e~ ft•dar absorption loss• for ray pathsintesadr
a pa-
i i
0- --
F---4- oalrda b-rp -n-os-o' rypah n ---- -tandard
" ~atmosphere at 50 (~lzl
52 L. V. BLAKE
500. ..... . .. .
u, . -- -- ---T0 IFIO0 0
g .. .. --! '-. '' °'
0-j0.0
I - - -2.0
100.
0 s0 IOo ISO 5 00 250 300 3S0
t I-TO-TRFIGET lI5 NCL IqlUTIICAL MILiS)
Fig. 7 -5 Oxygen component of the r adar absorption loss for ray paths inthe standard atmosphere at 55 GHz
SO
* ~~0 ...
- - ----------------- i2.0
- 5.0
0 s0 100 IsO 200 250 300 350qANAe.10-rARGFT DISIANPCE INALJTICAL MILES1
Fig. 6 - Water-vapor component of the radar absorption loss for ray pathsin the st~sndard atmosphere at 5.5 0iz
NRL REPORT 7461 53
3GO -- -- T_ o.,
10001 .. -
0 50 10 0 150 200 250 300 350
RADAR- TO-TAFICET DISTRNC[ INAU7ICkL MILESI
Fig. 77 -- Total radar absorption loss for ray paths iii the standard
atmosphere at 55 GHz
I . ..' -
16000 . .. . . - - - - - - - -...
-j ,
SI / • . .. .... .
I u ~
:iSI ooo. -
1 000 O SO Ž0 20 0 5
LP A~ OT R E 0I.0EIN UI A M L S
4000[ .
S. .. ._ o. I
0so.!00.. . . . - --... - 4-. oo
1 00 0 0 1 020'. . . . . . . . . . . . . . . . . . . . . . . . . . . -5 S 5
Fig. 79 -- Oxygen component of the radar abso~rption loss for ray p~aths in
the, standard atmo)sphoro a- 60 GHz
0 • r.........................-... - . v ,I
54 L. V. BLAKE
60.
50
1I 4
-0 : - . . .. . ..- - . 11. 0
.S.0
I-• • - .I - ~
' , 20 .. / ". - -.. . -
.2.0
5 1 - " '- -- -
0 50 ICO 1SO 200 250 300 350
RA'•]R-T0-TAR.ET DISTANCE INAUT!Ck. MILESI
Fig. 79 - Water-vapor component of the radar absorption loss for ray pathsin the standard atmosphere at 60 GHz
S~~20000 I
, I16000 r1 1
I
I 2.0
0 50 100 )so 200 250 300 350
RA0AR-T0-TR•GET OIS'ANCE IN•4OTICL 'ILES)
Fig. 80 - Total radar absorption loss for ray paths in the standardatmosphere at 60 GHz
• • = ,= • = • -',r- . • , '•-• •=7..•^, .. --..--' - . -- - - • •. •- =.-" • -.. • • ,• . ... . ..
NRL REPORT 7461 5
30007 O.S -
2500- .. -
1 0.0
1000------ . -- - -- - 2.0
I 0. 0
0 5a0 so 50IS 200 21,0 300 350
t Fi-TO-TARCT M)STANCE ftWJTICA, MILES)
Fig. 81 -- Oxygen component of the radar absorption loss for ray paths in
Pr the standard atmosphere at 65 GHz
70 ~ ~ ~ ~ ~ -....... . .. -.. ........ r- . 2 ........ . ........
40--
30 -- 1.0-J
I~AI2CC 50 30 3 t
20T-~E 0S~C NUICL4~~
Fig.82 - Wtervapo coponet o theradr aborpion ossfor ay2a.hin te sandad amosperea! 6 G4
56 L. V. BLAKE j
2$°°r -- - -
; oo ....... ...... .. * . . . .. *_ _ .---- ------*---
10.0
1O0000 2.0
f SO0 rr
SC 5G 1 O0 ! so 200 2S0 300 350
AAOAA-T3-TAACET 01STANCE 1ANUTICAL MILES1
Fig. 83 - Total radar absorption loqA for ray paths in the
standard atmosphere at 65 GHz
300 F--- -.. . . -- T... - . .. . . . .--
.. . . ."
' I2000
-J (P150 L 1.01
1I00 .. . . ... 2.O
o/so, I- e Z ..!0.0
0 so 100 ISO 200 250 30C !50-
R.ADAR- T0-TA F T DISITNCE rNAUTICnL MILESI
Fig. 84 - oxygen component of the radar absorption loss for rpy paths inthe standard atmosphere at 70 Gliz
I
iNRL REPORTr 7461 57A
70F7 7 7 7 7 7 7 0.0-... • . . . .. . -4 .. • . . ., . --
-- 41
-19 - .I
i :
I
30 : -4
:..0
0 s0 100 ISO 200 2SO 300 350
IAOAA-TO-TRRAET CISTANCE (NAUTICAL M1L Sl
I
Fig. 85 - Wster-vapor compolnent of the radar absorption loss for ray paths A
in the standard atmosphere at 70 GHz
400 -i . I
0.0
I ,o .. . . /t I- .
---. .. 0. 5,0
0.!00~
0 5D 100 ISO 200 250 300 350
RA .- :O-TPtC.T 0ISTA4:E :%•jtiCqL MILESI
Fig. 86 - Total radar absorption loss for ray path% in thestandard atmospheic at 70 Glz Ij
58 L. V. BLAKE
20C3 -- - ---. . : .. . . . ..
; i
S~160.
I(. -'
--- 40.5
_2.0'40 _ ... -
t - 10.0
I c00 5C 03 1C 2CC 25C 3CC 350
PQD•r-T0-'nrtGE' rISTANCE 'NAUT!CPL MILES1Fig. 97 - Oxygen eomponent of the radar absorption loss tor ray paths in
the standard atmosphere at 75 Gliz
907~I7.7 7 7
70° " - - - .
SO - Z .
40"
I -4
i3 0j
201 .
100 e.O
- 1o.0
so I GO i SO 200 250 301 5
ACIOAR-T0-YARGET, OIST•INC_ INAUTICni- MILES)
Fig. 88 - Water-vapor component of the radar absorption loss for ray paths
in the standard atmosphere at 75 GHz
NRL REPORT 7461 59
% 160
1201 -- - - - ---. 1.4
so2.0 t
So_.... ____ _ _ . .- . ... . .I
0 so 100 ISO 200 250 300 350S~RADRR-T0-- ýFIGET DISTANCE INAUTICAL MILESI-
Fig. 89 - Total radar absorption loss for ray paths in thestandard atmosphere at 75 GHz
.I ' 0.0•0.. .. ....... . . " ....... ........ . -. . ... * - --
- 0.5 ~
2 -
Ž0- - . . . . . . 5.0
to. .. 10.0I. . -
I0
C 50 IO0 150 200 250 3CC 3S:
r3A(q!O.Q-I0 QRG T DISTANCE iNI•UTICIL MI1E51
Fig. 90 - Oxygen component of the radar absorption loss for ray paths inthe standard atmosphert at 80 Gliz
60 L. V. BLAKE
'00 7 7.
0.0
I . ,. I
205.
. .- . . ..... -... . ... . • 10.0
C 5s I0= 159 200 2SC 30i 350
RPDlR-Y0-MaUT 0lS!V NCE INMATICAL MILt•S
Fig. 31 - Water-vapor component of the radar absorption loss for ray path•;in the standard atmosphere at 80 GHz
200 1- : r- --
,Io - - - . -....-. .-
I.
s I 30
standard atmosphere at 80 GFIz
NRL REPORT 7461- 61
So---- iI
0.0
/ 1.0- I...
/ I - g
•,/ I -/ .e
.a•20: .... •, - -
0 . . . . . .. . . .. .. -] 5.
0 0.
10 - . . _ _______ . .
0 so-'iS 200 .. .25
C so 1 oo 2 sc 300 350R50A9-T0-TARGC T •ISTANCE (NAUTICAL MILESI
Fig. 93 - Oxygen component of the radar absorption loss for ray paths in
the standard atmosphere at 90 GHz
160 iIIS "2 -. . r
4 0
- _____ - _____ -0.0
o . ....-.. . . . . . _ _ oIo * .-_2 .- "/_ ".-7. _-_ .-.. i. ;- -. . :; -LL i; L 2J ''
L0 so IO0 ;so 2-0 250 300 350
K-H -Wbii U-IM4ML, UI•i'ý144. IhNWJIIL-'tL "•It'..)
Fig. 94 - Water-vapor component of the radar absorption loss for ray pathsin the standard atmosphere at 90 Gl0zIT
rirr~'_
62 L. V. BLAKE
t2
2 -0 ----.--..- - - -
160 t . . . .0-
,r : . :~120
81.*2.0
- h I
, _ _ _
0~o
0 50 100 [so 200 250 300 350
RI0FR-T0-TAR0ET OISTANCC ItlAjTICAý MILES'
Fig. 95 - Total radur absorption loss for ray paths in thestandard atmosphere at 90 GH2
40 .. .
-___0.5
2.0
15.0
S0.0
0 SC I ,l ISO 20K 250 300 350
M4HUM-IU-IHwLat2 Ulý)IHNLL 0N4M,11101 MýLt•I
Fig. 96 - Oxygen compcnent of the radar absorption loss for ray path=; inthe standard atmosphere at 100 GHz
IT
NRL REPORT 7461 63
200 .- . "
160 - -- -t -
K 0~.0 *
a -i
. I
10- , - . ... . ..-- - -
. ... .. . .....-. ....... ....... . . .. ..._ . ....___ 0.0
2'•0 -. --- " - - - -- _'
.5.0;0.0
0so I o0 IS0 200 2SO 3, u 350WA- 70•-T-'WiGEt LUISIANCEf. N,.UT::.k MILES)
Fig. 97 - Water-vapor component of the radar absorptimnio ioob for ray pathsin the standard atmosphere at 100 (uHz
160 .
-- - --- c .
-- - 5.C
0 50 100 !,.C0 200 2.0 300" -
WhjA4 TO 01,5- T l A~hC.4 lJ ol',q !A. IltES"
Fig. 96 -- TotAi riadir absorption ioss for ray paths in ' nestandard atmosphere at 100 (Gl~z
64 L. V. BLAKE
S33WO3 6319NU NOIJUA313
0u~'0 C 0 0 00
00 C)
j 0g.
It~
Fr:r
IJ
r 0
CiCDL 11111L .. .. LC
CD D C
NRL REPORT 7461 65
o~~ CnO 0en0 00
I Nil
.r -
C-
-I M
o cc
- 0
tW~JOJJ~-IddU Y1N O1LNU1j-M O NIdl-1
II
C~-jIS119130 3WHJ;JY -UI JO 11SUU -a----- -O -~lJA
66 L. V. BLAKE
S33HO30 0 0 630N 00PJ3
0 0- : CD C) C)O 0)(0 M
_: 0 L
-C
CDCD
-9-
C 0
CD M
- C -
CD ~ ~ jj3~dOOi U1~J 1NY BMOI~JNIBl~1
a .2
C'Sd
ra
NRL REPORT 7461 67
where dR is measured along the ray path (Tn, in degrees Kelvin, is a function of the ray "elevation angle), Tt is the thermal temperature of the troposphere, defined by the standardatmosphere of Subroutine ALPHA, and or is the absorption coefficient. (The factors0.2303 are needed in this equation because a is here assumed to be in decibels per unit
distance.) ]
The previously described modification of Simpson's rule cannot be used for this Iintegration, because it is an integration with respect to range (R) rather than height (h).The height intervals are (within each of the four height regions defined in SubroutineALPHA) uniformly spaced, as required by Simpson's rule; but the corresponding rangeintervals are not uniformly spaced, because of the nonlinear relationship between rangeand height in the space above a spherical earth and in a refracting atmosphere (for allelevation angles except 90 degrees). Therefore another special modification of Simpson'srule was devised to fit this situation; it is Fortran-eoded in Subroutine INTGRT.
When the absorption loss is great, as it is at the oxygen resonance frequency forexample, most of the tropospheric noise seen by a ground-based antenna comes from the Ilowest layer of the atmosphere. that is, the integrand of Eq. (31) is large at R = 0 and -decreases steeply beyond R = 0, going quickly to virtually zero. Because of the extremedeparture of the integrand from a parabolic curve in this circumstance, the Simpson'srule integration is not accurate. An analytic approximation to the integral in this regionis numerically more accurate than numerical integration of the "exact" expression.Therefore such an approximation is used for the integration over the first two height(range) intervals, corresponding to the first 200 feet of atmospheric altitude. The approxima-tion is derived by assuming that both a and Tt, Eq. (31), are constants within thoseintervals. If R1 and R 2 are the ranges at h = 100 and h = 200 feet, then Eq. (31) forthese intervals may be written
=2 ~R1 ke-klkR dR + 2 dR, (32)lTn T I k 2 e 2
where k1 = 0.2303571, k 2 = 0.2303U2, Tt, is the average tropospheric temperature in therange interval 0 to R1 , Ttz is the average temperature in the interval R1 to R 2 , and •1and & 2 are the corresponding average ahsorption coefficients. Integrating this expressiongives
ATn = TtI(I - e-k, R11 + TL2 (e-kRl _ e-k 2 R 2 ). (33)
This is the approximation employed in Subroutine ATLOSS for the first two range intervals.(The approximation is used whether the attenuation is large or not, because it is equallyaccurate when the attenuation is not large.)
The noise temperature is calculated in Subh,,utine ATLOSS, since it is efficient tocalculate it concomitantly with the absorption. The result is reported as the outputparameter ATMP. A p!gt of the resulting values for several elevation angles in the frequencyrange 100 NIHz to 100 Gli.-. is given in Fig. 102. This quantity is not computed andIplotted separately for the o):ygen and water-vapor contributions, because they are notlinearly additive as are the absorption values.
68 L. V. BLAKE
C)
T i Il I -T r ] C
I IA 1 9kl0
I ii' l~i I I
ujICD
ICD
CD 00-
*030 'J3uflUY~dW31 IS0N
NRL REPORT 7461 69
COMPUTER ROUTINES
The Fortran subroutines ATLOSS, ALPHA, DDH, and INTGRT, which have beendiscussed in the preceding sections of this report, are listed on the following pages. Theircalling sequences, pertinent COMMON blocks, and definitions of the parameters are asfollows-
SUBROUTINE ATLOSS (FMHZ, ELEV, ATMP)
COMMON/RGA/RG(75), ATTN(3,75)
FMHZ - radio frequency, megahertz (input)
ELEV - initial ray elevation angle, degrees (input)
ATMP - tropospheric noise temperature, degrees Kelvin (output)
RG -- 75 monotonically increasing values of range, nautical miles, alongthe ray path from h = 0 to h = 100,000 ft (output)
ATWN - 75 corresponding decibel radar attenuation values (output)for oxygen (1), water vapor (2), and oxygen plus water vapor (3)
SUBROUTINE ALPHA (FMHZ)
COMMON/PTR/PP(75), TT(75), RR(75), ALPH(3,75)
COMMON/H20/RHOFAC
FMHZ - frequency, megahertz (input)
PP, TT, RR - standard atmosphere values, specified in DATA statements(available as output via the COMMON block if desired)
RHOFAC - A numerical factor by which the water-vapor-density values ofthe standard atmosphere are multiplied. If RHOFAC is not setto some other value by the user, the value RHOFAC = 1, set bya DATA statement, will apply.
ALPH(I,J) - the 3-by-75 output array of absorption coefficients. The I subscriptshave the meanings I = 1, oxygen; I = 2, water vapor; I = 3, oxygenplus water vapor. The subscripts J = 1 through 75 correspond tothe 75 altitudes to which the RG values of Subroutine ATLOSS alsocorrespond.
SUBROUTINE DDII(H)
COMMON/RRG/REFO, RAD, GRAD, U
COMMON/DRS/DSDH3, DRDH3, AN
Aij
I
70 L. V. BLAKE
:H - altitude, feet (input)
FEFO, RAD, GRAD, U - refractive-index profile parameters transmitted from IATLOSS
DSDH3, DRDH3 - ds/dh and dR/dh corresponding to H (output) I
AN - refractive index n(h) at H (output)
SSUBROUTINE INTGRT (HI, H2, Y1, Y2, Y3, AREA)
If X1, X2, and X3 are successive points on the x axis of a cartesian coordinatesystem and Y1, Y2 and Y3 are the corresponding y values, then
HI = X2 - X1 and H2 = X3 - X2. AREA (output parameter) is the value of Ithe integral:
X3
ý y(x) dx, (34)Il I
where y(x) is the second-degree polynomial that passes through the points Y1,Y2, and Y3. It is not necessary that H1 = H2.
The subroutine listings follow. (Slightly different but essentially the same subroutines
are used in a computer program that calculates the maximum range of a radar; this programis described Ref. 1.) The lengths of these subroutines (number of memory locations 1required) are as follows:
Subroutine Name Octal Length Decimal Length
ATLOSS 756 494
ALPHA 520 336DDH 211 137INTGRT 112 74
Totals 2021 1041
I4
!I
LNRL REPORT 7461 71
A6/23/72
flLmEJSI(Th 166(4), 11:1H4)
Cli~).N/RS/ ,fl3.7)nHI. AN
DlATA (R F s O 0 1 ) ( AI!?)$9 g l l ) ( R n~ n (41 5
flATA (1LAS9aQ..14FlAS7:IU(,),(C~fN..1T3.2f25Un.5)
iý (FMHZ .t.FLAST.,A'qr' LEV.E'.FLAST) kFTURN
FLASI 9 LF
TmFTAzE-LPV/57.295/795S~i:SIN( IHEIA)Ch=COj( THLTA)S ý:SiNGSNq
UzQ~*t~)*2- ?,RF - EoF0
U. (FMHZ.kO, ýLASI) (aj Tr-)55CALL ALP1¶A(FmHZIFL AST"'M011
A YTE N 1 AT71E .N z0,
CALL DDH(HI%'P)fH9lz'IfiRDm3
TtMP a 0
FArjziJELHCj)/(3.e6')7E.j55~)FA2-.w ACL
IMAE: 166(J)
CAL LIDH(.')
nm'l-2zp0SY'i3
H: '-IC L H:.#j)
YezALe(1g,(41)*DSDHi
72 ýL. V. BLAKE
06/23/12
V.J:ALRN( 1,K+2)aDS~w4YJ72AýPH(2,K#ý)ODlb¶.3
Al N 3x A voI ý (ELEV ALT. 1. ,AND. m .01. 201.) 5. 6
SCt:CS.CS*(. (/kAD-s4EF0.iWAU/RPI.)CLI I. / (CCQ60/'6,jji55)PH00=ie~. CC4Hj
r FVILL(I)NING 1S APPRCYIbIATIE1N REQ,.iIQED NEAR TWETAmO AND Hoo FO!R RANOEc CALCULATIO~N, QANGF Ih LALCULATEIM THU'S FOR H 100 ANDJ He 20nl WHEN
c EtLFVAlHN ANCnLF 16 L~h TH4AN I. DEGREF,RiNzC,:C(APROD/( SQRTrC Pp.'4oSS) *SNJMfli aRNC
e7 Aý-RqIXIMAIE ATTENUjATIONJ IS RAN'IE (TWf-wAY) TIMES AVEPA5E VALUE erC AL.PHA IN THE RANGE INTCHVAL,
AITEN12RNGO(ALPH(l.1) * ALPP4C1.2) )Al TENd:RNj0(ALPH(W.1) + ALPH(2,2))
C RADAR RANGE IS GF uETP(h; RANGE TIMES AVERAGE REFRAtTVE INDEX,-4r,=R.Gs(Hp1 # 1. & Rtrc,.:XP(.GRADsH1)) * 5
GO~ To 76 vz,:qG1(DRCH3)
R'.C:RNG. DS* Ct:.zL) S/((ANI.AN2)-,5)
AT 'ENisATTENi .AT1 (Y
AITENW:ATIEN? # AIT3.(Y3e)7 R6(K+(J.)zRNG
AýT-N(j,K.1)t ATTEN1AlT7 N(d~, K. 3,1 AT 7 ENL'AlT.N(J,K+1IzAlTENj * ATIEVN?GO TO (10#11) ISIG
11 Pn~TD=2..CC*HRN Ti:C~l. 1*PROD/( SQ R Tr ( P .~n0*S S *'S NDz-?=RNG- 0S1AlTEN3.SRNU@(ALP'4(3.,1 # ALPH(1,J) )AlTEN2zRNGd(ALPW(2d,1) & ALPH(2,3) )RNGzRNGi.(RPJ # . PtrU.EWP(-GRAV*H o to
Gv~ T--l 12
AITEP~j1 A7TENJ # ATT2(Y.S)Y1:Yli
AiTEN~e: ATTEN2 * ATT2(Y3?)
AlTN (1. K+2 ):SA Is
AtTN(J,K.k):ArTENl + ATTEN21
NUL REPORT 7461 73
zýl (RO ( K*1 -R,(K))/ ( ANj AN')o. 5)I
C, APPROXIMAIION EiM0J.AYEU IN Pj.ACF ') FHkST P'T!'uQA T IfN STPC I~tED TG GIVE VALIrl Pt:ULTs IN H4TGhWAITENUAT~vN rA",:. ANALYTICC APPROAIMATIeN STARTS AT STAIFM7NT 70.
/0 Ct~:O.5.CeNST.(ALPH(3.u.*ALPHC1,2))
A6'-)S13EXPF (-Ct-X$DSiAL0S2cEXPF (-Cf:Y*DI)S)
DIFElP:(o.ýICONST. ((TI (l).TT(ie)).(1 ,-ALeSl) *( TT(7)*TT(l))*tALý,S?I, -A6.C1S3))I
GO) Tn. 7 2/t ALlSSa1O.#*(-ATTNJ(3,w.1)*. *l)
TW?:ALPH(3,K.I)oTT(K*2A.AL7'$SSCALL INTGRT(DS1,nS2,TiJ1 TP2,TP3,l)TeMF)
12 Tt:P r TFMP.D)IEMP
Tb-I a TP3
Y12=YJ2
ANI 2 A.%3
01 Cv)NT INUE60~ CO~NTINUE
Al MP3TEMPvCoNbT E~I
74 L. V. BLAKE
CE(mN/w.20/RwerACrIMEN5I9N FTPP(23),PIRP C.31r1mENI(D~N rELIIc4)&j* 1A1(4)
ICATA(CONSTmJ,7143) *(COKST2:*,206mi)
2A7,3A27,67,8923,6a.42c5,68,94701,A.4741 *7C,UDOC.7U.52GQ. ?1.0d97)
25i,25 ,5 .591 5U.9ý4,5C.4A~-4v93 ,(.4,fb93 ,4 . 4 )S!ATA (PWZ
le.12047,E2.2/e91..,2 4,.49.,,64267J7.2
15.63~19E-2.,96,.,,5E2,. I
rATA (cT--
P; .776 L*2# .8 566P 2,2. #7,36 E#2, 2 71f ý+2
l?,8 971 *2, ,66 7ýF4 t2 P~c~cý+2,2 84ý 7/r ?,2 A617 E02 i .SY~F+P
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NRL REPORT 7461 77
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78 L. V. BLAKE
REFERENCES
1. Blake, L.V., "A Fortran Computer Program to Calculate the Range of a Pulse Radar,"NRL Report 7448, August 1972.
2. Blake, L. V., "Radar Attenuation by Atmospheric Oxygei,," papeor presented al. URSICommission 2 meeting of May 6, 1959, Washington, D.C.
3. Blake, L.V.. "Interim Rcpo)rt on Basic Pulse-Radar Maximuzn.Range Calculation,"NRL Memorandum Report 1106, Nov. 1960.
4. Blake, L.V., "Curves of At inospheric-Absorpt ion Loss for Use in Radai Range Calcula-tin, NRL Report 5601, Mar. 1961.
5. Blaice, L.V., "Recent Advancements in Basic Radar Range Calculation Technique,"IflLE Trans. AiIL-5(No. 2), 154-164 (Apr. 1961).
6. Blake, L.V., "Tropospheric Absorption Loss and Noise Temperature," Trans. IREAP-1O(No. 1), 101-102 (JTan. 1962).
7. Blake, L.V.. "A Guide t,) Basic Pulse-Radar Maxinium-Range Calculation, Part I-Equations, IDefinitions, and Aid's to Calculation," NRL, Report 5868, Dec. 1962, andSecond Edition, NRL Report 6930, Dec. 1969; anid "Part 2?- Derivations of Equa-tions, Bases of Graphs, and Additional Explanations," NRL Report 7010, Dec. 1969.
8. Blake, L.V. "Prediction of Radar Rang'"," Ch;ýpter 2 of Rodar Hindbook, M.I.SkolniL, editor, McGraw-Hill, New York, 1970.
9. Van X'leck, J,H., "Lie Ab-;orp*t.on of Mlicrow~aves iby Oxyg(-n," Plys. Rev. 71 (No. 7),413-424 (Apr. 1, 1947).
10. X an Vleck, J.H., "The Absorption of Microwave's by Uncondensed Water Vapor,"r'hys. Ruy. 71 (No. 7), 425-433 (Apr. 1, 1947).
11. Meeks, M.L., and 1.i'ley, A.E, "The Microwave Spec-trurn of Oxygen in the Earth'5,Atmosphere," j. Geophy.;_ Res. 68 (No. 6): 16S.3-1703 (Mar. 1. 5, 1963).
12. Reber, T' E., Mitchell. R.L., arid Carter. C.~,_ -Attenuation of the 5-Mmn WavelengthBand in a Variable Atmosphers,.' IEEE Trans. -118(o.1, 472-479 (July 1970).
13. Straiton, A.W., and Tolbert, C.W.- "Anomalies in the Absorption of Radio Waves byAtmospheric Gases," Proc. TRE .18(No. 5), 898-903 fMay, 1960)1.
14. Liebe, H.J,, "Calculated Tropospheric Dispersion and Absorption Due to the 22-GtlLWater Vapor Line," IEEE Trans. AP-1 7(No. 5), 621-62-7 ~Scpt. 1969).
15. iiean, B.R., and Abbott, R., 'Oxygen and Water Vapor Absorption of Radio Wavesii, the Atmosphere," (3eofisica Pura E Applicata (Milatno) .37, 127-144 (1957).
16. ;iogg, D.C., "Effective Antenna Temperaturps Due to Oxygen ano Water Vaplor irthe Atmosphere," J. AppI. Phys. 30, 1417-1419 (Sept. 1959).
17. Schulkin, M., "Determination oi Microwave Atmosptieric Absorption Using Extra-terrestrial Sources," NRL Report 3843, Oct. 1951.
16. Le Fande, R.A., "Attenuation of Microwave Radiation for Paths Throuvh thvAtmospriere," NF~L Report 6766, Nov. 1968.
NRL REPORT 7461 79 A
* I
19. Ulaby, F.T., and Straiton, A.W., "Atmospheric Absorption of Radio Waves Between150 and 350 GHz," IEEE Trans. AP-!8(No. 4), 479-485 (July 1970).
20. Van Vleck, J.H., and Weisskopf, V.F., "On the Shape of Collision-Broadened Lines,"Rev. Mod. Phys. 17 (Nos. 2 and 3), 227-236 (Apr.-July 1945).
21. Minzner, R.A., Ripley, W.S., and Condron, T.P., "U.S. Extension to the ICAO Standard
Atmosphere," Dept. of Commerce Weather Bureau and USAF ARDC CambridgeReseaxch Center, Geophysics Research Directorate, Government Printing Office,Washington, D.C., 1958.
22. Sissenwine, N., Grantham, D.D., and Salmela, H.A., "Humidity up to the Mesopause,"Air Force Cambridge Research Laboratories Report AFCRL-68-0550 (Air ForceSurveys in Geophysics No. 206). Oct. 1968.
23. Blake, L.V., "A 'French Curve' Computer Plottihg Subroutine," NRL MemorandumReport 2335, Sept. 1971.
24. Bean, B.R., and Dutton, E.J., "Radio Meteorology," National Bureau of StandardsMonograph 92, Government Printing Office, Mar. 1966, pp. 65-73.
25. Blake, L.V., "A Modified Simpson's Ru!e and Fortran Subroutine for CumulativeNumerical Integration of a Function Defined by Data Points," NRL MemorandumReport 2231, Apr. 1971.
24i. Blake, L.V., "Ray Height Computation for a Continuous Nonlinear AtmosphericRefractive-Index Profile," Radio Science 3 (New Series) (No. 1), 85.92 (Jan. 1968).
27. Dicke, R.H., et al., "Atmospheric Absorption Measurements with a Microwave Radiom-eter," Pbys. Rev. 70, 340.348 (Sept. 19.. 6).
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