NBSIR 78-883

PILE CO

NOT RBK

ATMOSPHERIC PROPAGATION EQUATIONS USED IN THE

NBS EARTH TERMINAL MEASUREMENT SYSTEM

W. C. Daywitt

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80303

April 1978

IMBSIR 78-883

ATMOSPHERIC PROPAGATION EQUATIONS USED IN THE

NBS EARTH TERMINAL MEASUREMENT SYSTEM

W. C. Daywitt

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80303

April 1978

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Sidney Harman, Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

CONTENTS

Page

LIST OF FIGURES AND TABLES iv

ABSTRACT 1

1. INTRODUCTION 1

2. AN APPROXIMATE RAY-BENDING EQUATION 2 2.1 Refraction-Angle Error 7 2.1.1 Nonintegral Approximation 7 2.1.2 Nonspherical Symmetry 7 2.1.3 Refractivity Constants 9

2.1.4 Temperature, Pressure, and Water-Vapor

Density Profiles 10

2.1.5 Total Refraction-Angle Error 10

3. ATMOSPHERIC PROFILE MODELS 10

4. AN APPROXIMATE TRANSMISSION-COEFFICIENT EQUATION 11

4.1 Transmission-Coefficient Errors 14

4.1.1 Nonintegral Approximation 15

4.1.2 Nonspherical Symmetry 15

4.1.3 Absorption-Coefficient Constants 15

4.1.4 Temperature, Pressure, and Water-Vapor

Density Profiles 15

4.1.5 Total Transmission-Coefficient Error 16

5. RESULTS AND CONCLUSIONS 16

6. ACKNOWLEDGMENTS 17

7. REFERENCES 18

iii
LIST OF FIGURES AND TABLES

Figure 1. Geometrical idealization of the earth's atmosphere.

Figure 2. World maps of the mean sea-level dry term D and wet term W for

May (from Bean, et al. [7]).

°

Figure 3. Truncated refraction angle and truncation error versus truncation

altitutde at a 3° elevation angle.

Figure 4. Average May refraction angle for Washington, D.C.

Figure 5. Systematic error components for the refraction angle.

Figure 6. Geometrical idealization of the earth's atmosphere showing that

portion (shaded) in the vicinity of the observer of interest in

determining ray bending.

Figure 7. Average May refractivity profile for Washington, D.C., showing

that "area" in which the real profile is assumed to lie.

Figure 8. A portion of the 1962 U.S. standard temperature profile showing

that "area" in which the real profile is assumed to lie.

Figure 9. Total systematic refraction-angle error.

Figure 10. A portion of the 1962 U.S. standard atmospheric temperature

profile showing the three approximations (dashed curves) used in

the text.

Figure 11. A portion of the 1962 standard pressure profile showing the two

approximations (dashed curves) used in the text.

Figure 12. A portion of the 1962 U.S. standard water-vapor density profile

showing the two approximations (dashed curves) used in the text.

Figure 13. Molecular oxygen and water-vapor absorption coefficients versus

frequency (for 293 kelvins, 1013.25 millibars, and 7.75 g/m^).

Figure 14. Zenith attenuation due to molecular oxygen and water vapor versus

frequency (for 300 kelvins, 1013.25 millibars, and 7.75 g/m^).

Figure 15. Truncated zenith attenuation and truncation error versus truncation altitude (at 10 GHz for Washington, D.C., in May).

Figure 16. Systematic error components for the transmission coefficient (at

10 GHz for Washington, D.C., in May).

Figure 17. Total systematic transmission-coefficient error (at 10 GHz for

Washington, D.C., in May).

iv
Refraction-angle errors between various approximations and the

"exact" model (for Washington, D.C., in May).

Approximate refraction-angle equations used in the Earth Terminal

Measurement System.

Approximate transmission^coef f icient equations used in the Earth

Terminal Measurement System.

V

ATMOSPHERIC PROPAGATION EQUATIONS USED

IN THE NBS EARTH TERMINAL MEASUREMENT SYSTEM

W. C. Daywitt

A derivation of equations for approximating the atmospheric

refraction angle and transmission coefficient is outlined. The

approximations apply in the 1 GHz to 10 GHz frequency range and are accompanied by systematic error estimates. They are used

in the NBS Earth Terminal Measurement System for quasi-real-time

calculations concerned with the measurement of earth-terminal

gain-to-noise-temperature ratio (G/T), satellite effective isotropic

radiated power (EIRP), and downlink carrier-to-noise-density ratio

(C/kT).

Key Words: Atmosphere; downlink carrier-to-noise-density ratio (C/kT);

satellite effective isotropic radiated power (EIRP); error analysis;

earth-terminal gain-to-noise-temperature ratio (G/T) ; refraction

angle; satellite communications; transmission coefficient.

1 . INTRODUCTION

the Electromagnetics Division of the National Bureau of Standards has

contracted an Earth Terminal Measurement System (ETMS) designed to determine

earth terminal G/T, downlink C/kT, and satellite EIRP in the 1 GHz to 10 GHz

frequency range using a calibrated radio star [1]. The system operates off the

i-f patch panel of the downlink earth terminal, automatically controlling the

several measurement sequences needed to determine these parameters. In addition

to the programmed sequences, the ETMS contains working equations [2,3] for

calculating G/T, C/kT, and EIRP from measurement data and for calculating the

associated errors on a quasi-real-time basis. The determination of G/T and EIRP

requires that the effects of the atmosphere on the radio-star flux density and on the satellite signal be removed from the measurement data by estimating the

atmospheric transmission coefficient at the time the measurement is performed.

The ray bending caused by atmospheric refraction is also estimated for correct

pointing of the earth-terminal antenna during the measurements. The working

equations derived for these estimates and their corresponding errors are the

subject of this report.

A vast amount of effort has been and still is being expanded in the field

of atmospheric studies [4]. Unfortunately, few results of these studies are tailored to the specific needs of the ETMS. For instance, computer programs

like the Longley-Rice program [5] for calculating atmospheric loss are far too

large and time consuming for the ETMS to handle, while some of the simple
ray-bending expressions available in the literature [6] are accompanied by

insufficient information for an evaluation of the systematic errors involved.

The primary ETMS need is for simple expressions, capable of being performed on a quasi-real-time basis, to estimate atmospheric ray bending and transmission and

their associated errors. The transmission equations account for absorption by molecular oxygen and

water vapor and were derived principally for the 1 GHz to 10 GHz frequency

region. Tropospheric scattering loss and the effective loss caused by refractive spreading of the signal wavefront will not be discussed—nor will effects

due to scintillation or multipath fading. Furthermore, since measurement accu- racy is of utmost importance and is maximized by performing measurements in

clear, stable weather only, the effects of inclement weather will be ignored.

The atmospheric models used in deriving the following equations will be

discussed when needed. Generally speaking, however, the models were chosen for

simplicity and amenability to simple calculations and interpretation, while

still producing reasonably accurate estimates. Comparison of the results

derived from these models with more or less sophisticated calculations shows

good agreement.

The approximate equations derived for refractive ray bending will be discussed first along with the assumed atmospheric refractivity model, followed by

a discussion of the reasoning leading to the corresponding error estimates.

Next, the atmospheric model assumed in the transmission-coefficient derivation

will be presented, followed by the corresponding equations and error estimates.

Although many of the calculations are performed using data specific to the

Washington, D.C., area, the results apply to a much wider geographical scale.

2. AN APPROXIMATE RAY-BENDING EQUATION

Earth-terminal measurements using a calibrated radio star as performed by

the ETMS [2] require earth-terminal antenna pointing angles calculated on a quasi-real-time basis with an error of 3 arcminutes or less. Part of this

calculation includes an estimate of the apparent position of the radio star,

whose difference from the true position is caused by refractive bending of the

principal rays from the star as they pass through the atmosphere to the antenna.

A derivation of a simplified equation for this difference, called the refraction

angle, is reviewed in this section. As will be seen, the correction and its

error are both functions of the star's elevation above the horizon.

2
Figure 1 is a cross-sectional view of the earth showing the relationship of an observer at altitude Z above sea level to the refraction angle x. R is the

average sea-level radius of the earth with z representing an arbitrary height

above the observer's altitude, n^ and n represent the atmospheric refractive

index at heights Z and z respectively. The index of refraction is assumed to be

unity above some fictitious height labeled here as the "atmospheric ceiling,"

above which no significant ray bending takes place. 6 is the angle between the

perpendicular-to-the-radius vector from the center of the earth to the point on

the tangent to the ray at altitude z. 9 and 6 represent respectively the

Li J. elevation angles for the apparent and true positions of the radio star or satellite. That is, a principal ray entering the atmosphere at an elevation angle 6^

has an arrival angle at the observer equal to 6 . Thus, the refraction angle t is related to 6 and 6 by the equation

T = e^-e^ (2.1)

For a given angle 6^, x varies with atmospheric conditions, requiring that the

index of refraction be known at each point on the ray path if an accurate calculation of the refraction angle is to be performed. Unfortunately, such com- plete information is never available, and various approximations for x must be

made

By assuming a spherically symmetric or stratified atmosphere, the refraction angle can be calculated from the equation [6]

^=_J^£tne^^ (2.2) n

where ctn0 is the cotangent of the angle 0 appearing in figure 1. The integral

is performed along the ray path, but is too time consuming when performed by the

ETMS to be useful in quasi-real-time. The cotangent can be related to other

parameters appearing in figure 1 through the equation

ctne ( )2 _ 1 -1/2 (2.3)

I ^n2(R+Z)Cose2^ J

'

The index of refraction is related to the atmospheric refractivity N through the

equation

3
n = 1+N/lO^ (2.4)

The assumption of spherical symmetry implies that both n and N are functions of

z only. To complete an atmospheric model for calculating x, a functional form

for N versus altitude must be assumed.

Of the different models that could be chosen for N, the following biexponential model was selected [6,7]

N(z) =

[ D(z)e-(-^>/«l+ W(Z)e-(^-^>/«^ , z < Z

D(Z)e-^t/"l e-(^-^t>/«2 + ^(2)e-(-Z)/Hw

^ ^ ^ ^

^ (2.5)

D(Z) is the refractivity of the "dry" constituents of the atmosphere at the

observer's altitude Z, and W(Z) is the corresponding "wet" refractivity accounting for the presence of water vapor. and H2 are "dry" and "wet" scale

heights applicable respectively below and above the transition altitude Z^

separating tropospheric and stratospheric exponential behaviors. Hw is a "wet"

term scale height. The "dry" and "wet" refractivities are expressible in terms

of measurable atmospheric variables at the observer's altitude through the

eqs [6].

D(Z) =i^4(|P (2.6)

W(Z) , 3.73 X 10^ e(Z) _ r(z)

P(Z) is the total atmospheric nressure in millibars, T(Z) is the atmospheric

temperature in kelvins, and e(Z) is the water-vapor pressure in millibars.

Although there is a considerable amount of average data [7] available for the

parameters appearing in eqs (2.5), there are no equations available for determining Hi, H2, Hw, or Z^ from measurements of ground-level atmospheric variables

themselves. Fortunately, these latter parameters are not needed in some of the

nonintegral approximations of eq (2.2) to be discussed.

To eliminate the integral in eq (2.2) and reduce the calculation of x to a simple equation that can be handled by the ETMS in quasi-real-time, a number of

nonintegral approximations to eq (2.2) were examined, the one yielding the least

error when compared to eq (2.2) to be used by the ETMS. The "exact" calculation

4
performed according to eq (2.2) employs the following average data appropriate

to average May weather conditions in Washington, D.C.:

D(o) = 273 , W(o) = 60

Hi = 9.5 km , H2 = 6.2 km

Hw = 2.6 km , Z^. = 11.4 km

Z E 0 km , N = D(o) + W(o)

The first six entries in the table were obtained from global maps [7] of the

parameters involved, two of which are reproduced in figure 2. The circled dot

represents the location of Washington, D.C. Before proceeding to the approximations, it is worthwhile to examine the effect of truncating the upper limit of

the integral in eq (2.2).

In evaluating the integral of eq (2.2), the limits on the integral sign are transformed from "n to n," to "Z to infinity" respectively, raising the question of where to truncate the upper limit. Figure 3 shows the effect of truncating at various altitudes. The upper curve is the refraction angle as calculated by eq (2.2) for an apparent elevation angle of 3° as a function of the

altitude at which the upper limit is truncated. The lower curve gives the

corresponding truncation error, indicating an error less than 0.025 minutes for an altitude of 40 km or greater. Using figure 3 as a guide, an upper limit of

60 km was chosen for the "exact" calculation using eq (2.2), yielding an error

of less than 0.001 minutes. The resulting refraction angle t (as a function of

apparent elevation angle) using the data given in the preceding table is shown

in figure 4. This curve is the "standard" against which the following approximations to eq (2.2) were judged.

The results of six refraction angle approximations were compared to the

"exact" results. These six approximations are summarized in column 1 of table 1 and consist of: (1) a "flat-earth approximation; (2) and (3) zero and firstorder expansions of the eq (2.2) integral in powers of z/ (R+Z) (see fig i and

2

eq (2.3)); (4) a esc 6 fit to the "exact" data at 2° and 20° elevation angles

2 leading to the two coefficients appearing in the fifth equation; (5) a ctn 6

fit at 2" and 20°; and (6) an equation to be discussed later that evolved from

2

the ctn e fit.

The "flat-earth" approximation results from letting R in eq (2.3) increase

without limit (note that r=R+Z+z), yielding an integral in eq (2.2) that is

integrable in closed form. The resulting approximation is found in the first

5
row and column of table 1. The remaining columns in the first row show the

difference between this approximation and the "exact" calculation. For example,

the approximate estimate of x is 5.83' greater than the "exact" estimate at a 3°

elevation angle.

The next two approximations, numbers 2 and 3, are expansions in powers of

z/(R+Z) (after n and n^ are discarded from eqs (2.2) and (2.3)) whose coefficients are functions of the parameters appearing in eq (2.5). The particular

values of these coefficients appearing in the table were calculated from the

preceding data table.

i As previously mentioned, the coefficients of the fourth and fifth approximations in table 1 were determined by fitting these equations with undetermined

coefficients to the "exact" results at 2" and 20° elevation angles. With the

coefficients in the fifth equation determined in this manner for a number of

other locations and months using maps like those appearing in figure 2, the

2 coefficient of the ctn 9^ term was found to be remarkably similar for the various locations and months of the year, the average being -0.0131. With this

fact in mind, an equation of the form given in the outer bracket of the sixth

equation in column 1 was compared to the "exact" calculation for the various

locations and months previously mentioned and was found to be systematically

3.2% below the "exact" calculations. With this 3.2% taken into account, the

sixth equation resulted.

Examination of the errors in table 1 shows the last three approximations to

be considerably superior to the first three at low elevation angles. However,

the fourth and fifth approximations are determined by curve fitting to the

"exact" calculation and are therefore not useful for quasi-real-time calculations. Therefore, the sixth approximation was used to calculate the refraction angle t. The sixth equation of table 1 as it is rewritten in eq (2Tl)* of table 2 is

used by the ETMS to determine the refraction angle x. The ref ractivity

,

N(Z)=D(Z) + W(Z), given by eq (2T2) is derived from eqs (2.6) by using an atmospheric model described later. The saturated water-vapor pressure e^ (eq 2T4)

is a least squares fit to the temperature/ saturated water-vapor pressure curve

[8]. Equation (2T6) will be explained later.

*Equation designations containing the letter T refer to equations found in the

tables at the end of the report.

6
2.1 Refraction-Angle Error

An estimate of the systematic error associated with using the refractionangle eq (2T1) to calculate refraction angles in the real atmosphere will be

reviewed in this section. The total error is comprised of components arising

from several sources; (1) use of the nonintegral approximation eq (2T1) in

place of eq (2.2); (2) assumption of a spherically sjmmietric atmosphere; (3) un- certainties in the constants appearing in eqs (2.6); (4) uncertainties in atmospheric temperature profile; (5) uncertainties in atmospheric pressure profile;

(6) uncertainties in atmospheric water-vapor density profile; and (7) uncertainties in the observer's altitude above sea level.

2.1.1 Nonintegral Approximation

Errors in calculating the refraction angle by using the approximate eq (2T1)

in place of eq (2.2) were obtained in section 2 while deriving eq (2Tl) and are presented in row 6 of table 1. The same process that generated these errors was repeated for a number of other areas around the globe with no radical departures

from those errors given in the table for the Washington, D.C., area. Therefore,

the magnitude of the errors in row 6 was taken as "the" refraction-angle error. Curve number 1 of figure 5 is a graph of this error as a function of apparent

elevation angle. Although not shown on the graph, the error increases quite

rapidly for elevation angles less than 3°; and therefore the range of applicability for eq (2T1) is limited to elevation angles greater than this value. For

most earth terminals this limitation poses no problem.

2.1.2 Nonspherical Symmetry

The assumption of a spherically symmetric or stratified atmosphere was essential to the derivation of eq (2T1). The fact that the real atmosphere

differs from this model leads to an error when using eq (2Tl) to estimate x. With reference to figure 1, this assumption implies that the refractivity N at

the point on the ray path with radius r is identical with the value of N(Z)

given by eq (2.5) directly above the observer's head. With reference to

figure 6, it assumes that the refractivity has equal values at points 1, 2, or

7
3, in fact at any point around the globe with radius R+Z+z. An estimate of the error caused by variation of N between points 1 and 2 of figure 6 is discussed

in this subsection.

The shaded portion of the atmosphere in figure 6 represents that angular

extent *p of the atmosphere responsible for bending the ray shown, and therefore

the area through which the horizontal variation of N should be assessed* For

elevation angles 0 greater than 2°, (J) can be approximated by the equation

where h is the altitude of the "atmospheric ceiling" and R is the average radius

of the earth. Figure 3 can be used to show that (for 6 =3°) 98% of the ray

bending is accounted for with h equal to 25 km.

Using this value for h and 6400 km for R, the <}) calculated from eq (2.7)

represents an upper limit to that angular extent of the atmosphere to be con- sidered in estimating the variation of N for a given 0^. As explained in the

next paragraph, eq (2.7) will be used with the maps in figure 2 to estimate

upper and lower bounds to the average variation of the refractivity along the

ray path shown in figure 6.

To predict the error in using eq (2T1) to estimate x, the first term is

sufficient. Use of this equation implies the functional variation for N(z)

given by eq (2.5) and shown as the solid curve in figure 7. The area between

the dashed curves in this figure is assumed to contain the real refractivity

variation along the ray path shown in figure 6 even though the real variation

may not follow the central curve. The vertical separation between the dashed

curves was determined by using eq (2.7) in the following manner. With 0^ equal

to 2°, eq (2.7) predicts an angle equal to 3.4" which can be translated into a longitude-latitude ellipse around the point shown on the and maps of

figure 2. Using the diameter of these ellipses in the direction of maximum N

variation (perpendicular to the contour lines) in the two maps leads to a value

for N (N =D +W ) of 333±18 (5 . 4%) . The dashed curves in figure 7 were taken to

o o o o be the central curve ±5.4%, and when used to calculate x represent maximum and

minimum limits to the value of x for any real refractivity curve within their

boundaries.

(j) = Cos

-1

1+h/R J Z (2.7)

8
For extraterrestrial radio sources at the observer's zenith, the refraction

angle vanishes no matter what the refractivity profile. Therefore, as 6 approaches 90°, the error in predicting t must also vanish. Correspondingly,

the error in t due to nonspherical symmetry is assumed proportional to ()) of

eq (2.7). The resulting error is shown as curve number 2 in figure 5.

2.1.3 Refractivity Constants

The constants (77.6 and 3.73x10^) appearing in eq (2.6) for D(Z) and W(Z) are uncertain by the amounts 0.2% and 0.9% respectively [6]. The resulting

refraction-angle error calculated from the first term of eq (2T1) is shown as curve number 3 in figure 5.

2.1.4 Temperature, Pressure, and Water-Vapor Density Profiles

The refraction-angle error caused by horizontal gradients in the atmospheric variables leading to the refractivity has already been accounted for in

subsection 2.1.2. Nothing has yet been done, however, to estimate the error caused by uncertainty in the vertical N profile (eq 2.5) itself. The error will

be estimated in this subsection by assuming uncertainties for the temperature,

pressure, and water-vapor density profiles that lead to the N profile predicted

by eqs (2.6).

Atmospheric models assumed for vertical temperature, pressure, and watervapor density variations will be discussed later. However, the first 15 km

portion of the temperature profile model is shown in figure 8 as the solid

curve, shifted in the vertical direction according to the measured gound-level

temperature. The dashed curves correspond to the solid curve ±10 kelvins. The

real temperature profile with the same ground-level temperature as the model is

assumed to lie between the dashed curves. When used to calculate the refraction

angle, the dashed curves then represent lower and upper bounds to the calculation, one-half the difference of which is taken to be the refraction-angle error due to uncertainties in the temperature profile. This resulting error is

plotted as curve number 4 in figure 5.

With reference to the preceding discussion, the refraction-angle error due

to uncertainties in the vertical atmospheric-pressure profile was calculated in

a similar fashion through the use of eq (2Tl), eq (2T2), and a pressure profile

9
model discussed later. It is assumed that the real profile is contained within

bounds established by curves corresponding to the model profile ±10 millibars.

The resulting error as a function of elevation angle is shown as curve number 5

in figure 5.

The water-vapor density is determined at the observer's altitude by measuring the atmospheric temperature and relative humidity, and by using eqs (2T4)

and (2T3) to calculate the water-vapor density. The resulting value is taken as the starting point for a vertical water-vapor density profile similar to the

model profile discussed later. The real profile is assumed bounded by curves corresponding to the model profile ±50%, and leads to the refraction-angle error given by curve number 6 in figure 5.

The refraction-angle error due to uncertainties in the observer's altitude

Z can be easily estimated from eq (2T1). Error curve number 7 is the result

from assuming an uncertainty in Z of ±0.1 km.

2.1.5 Total Refract ion-Angle Error

The total systematic refraction-angle error as a function of elevation

angle is assumed to be the linear sura of the component error curves shown in

figure 5. The total error is shown in figure 9 and is, like the curves of

figure 5, dependent upon atmospheric conditions surrounding the observer's

location. This curve is approximated by eq (2T6), which is used by the ETMS to

predict refraction-angle error. As stated in the introduction, a refraction-angle error of less than

3 arcminutes is sufficient to allow the ETMS to commence its measurement procedure. Figure 9 shows that this condition is met for elevation angles in the

vicinity of 4° or larger.

3. ATMOSPHERIC PROFILE MODELS

Calculations of the earth-space transmission coefficient to be discussed

later are based upon an assumed knowledge of atmospheric temperature, pressure,

and water-vapor density profiles. The model profiles used in this report were derived from the 1962 U.S. Standard Atmosphere [9] and are described in this

section.

10
The solid curved in figure 10 represents the 1962 standard temperature

profile to an altitude of 35 km above sea level. Since very little additional

loss occurs above 25 km, the bend in the standard curve at 32 km is discarded

for simplicity. The resulting curve, consisting of the solid curve below and

the dashed extension 1 above 32 km, was used in the "exact" integral calculation

of the atmospheric transmission coefficient. The curve, consisting of the solid curve below and the dashed extension 2 above 20 km, was used to simplify the

calculation of a nonintegral approximation to the molecular oxygen component of

the transmission coefficient. Dashed curve 3 is an exponential approximation to

the standard curve used to simplify the calculation of the nonintegral approximation to the water-vapor component. All of the resulting curves are uniformly

shifted vertically to make the temperature at Z correspond to local observer

conditions.

The solid curve in figure 11 represents the 1962 standard pressure profile.

The dashed curve 1 removes a slight bend in the solid curve at 20 km and, with

the solid curve below 20 km, is used in the "exact" integral transmissioncoefficient calculation and in the nonintegral approximation to the corresponding oxygen component. Dashed curve 2 is an exponential approximation to the

standard curve used to simplify calculation of the nonintegral approximation to

the water-vapor component. These curves are uniformly shifted vertically to

reflect local conditions.

The solid curve in figure 12 is the midlatitude water-vapor density profile

[10] computed for conditions of the 1962 Standard Atmosphere. Extension 1 which

straightens out the standard curve below 16 km is used with the solid curve below 16 km in the "exact" integral calculation of the water-vapor component of

the transmission coefficient. Dashed curve 2 is an exponential approximation to

the standard curve used to simplify calculation of the nonintegral approximation

to the water-vapor density component. Both curves are shifted to reflect local

conditions.

4. AN APPROXIMATE TRANSMISSION-COEFFICIENT EQUATION

Energy radiated by a satellite or radio star and received by an observer on

the earth's surface suffers an attenuation ^i^^} ^^^^ traverses the atmosphere to the observer. With reference to figure 1, this attenuation represents

11
the accumulated loss from where the ray enters the atmosphere to the observer at

altitude Z, with molecular oxygen and water vapor being the significant contributors to this loss [6]. The transmission coefficient '^(^^^ corresponding to

this attenuation is of primary interest to the observer since it relates the

amount of energy entering the atmosphere to the residual he receives. In terms

of the attenuation

a-(Q^) = 10 . (4.1)

The attenuation itself is calculated from the equation [6]

A(e^) = / (Yl+Y2+Y3)dJl (4.2)

where yi> Y2j Y3 ^.re absorption coefficients (dB/km) , dl is an infinitesimal

arc length along the ray, and the integral is evaluated along the ray from the

observer to the point where the ray enters the atmosphere. yi includes the

nonresonant absorption arising from the zero-frequency molecular oxygen line and

from the absorption due to several natural resonances around 0.5 cm wavelength.

Y2 arises from the water-vapor absorption line at 1.35 cm, and Y3 accounts for

the water-vapor absorption bands above this line. The integral in eq (4.2) is

too time consuming to be performed by the ETMS on a quasi-real-time basis, and

therefore nonintegral approximations were derived. The derivation of these

approximations is the subject of this section.

Expressions for yi> Y2» and yb appearing in the literature [6] contain

line-width factors in their denominators that, due to their dependence on atmospheric variables, complicate the calculation of kiQ^). Discarding these factors

in the denominators causes little error in the 1 GHz to 10 GHz frequency range

of interest in this report. The approximations resulting from discarding these

factors are given by eqs (3T3), (3T6), and (3T9) in table 3 with Z equal to zero. They are plotted in figure 13 for T, P, and p equal respectively to 293

3 kelvins, 1013.25 millibars, and 7.75 g/m . The corresponding zenith attenuation (Q^=90°) as a function of frequency is plotted in figure 14 for 300 kelvins,

1013.25 millibars, and 7.75 g/xa^

.

In general, the ray path in figure 1 must be determined before the integral

in eq (4.2) can be performed. However, for 9^ >_ 3°, this equation can be approximated by

12
A(e ) = (Csc6 ) / (Yi+Y2+Y3)dz (4.3)

^ ^ Z

where the integral in eq (4.3) is performed along the observer's zenith (6 =90°),

giving the zenith attenuation. The error accompanying the approximation embodied

in eq (4.3) will be discussed later.

Before proceeding to a nonintegral approximation for ^{^2} ^^'^ '^i^^}'

of interest to consider the error resulting from truncating the upper integral

limit in eq (4.3). The effect is shown in figure 15 as a function of the truncation altitude. For example, terminating the integral at 8 km results in a 10% error in calculating the zenith attenuation. The graph shows the error to be

insignificant for altitudes of 25 km or greater.

With the line factors removed from the denominators of yij Y2> Y3 a-s previously discussed, it is possible to separate the zenith attenuation A into the

following form

A = Yl(T,P,f)£i(Z,T) + Y2(T,P,P,f)^2(Z»T,p) + Y3 (T,P, P , f ) ^^3 (Z, T, p) (4.4)

where Yi> Y2> ^'^^ Y3 are ground-level absorption coefficients corresponding to

atmospheric conditions prevailing at the observer's location. The arguments T,

P, p, and f signify dependence on atmospheric temperature, pressure, water-vapor

density, and frequency respectively at the observer's altitude Z. li, ^3 are scale heights depending upon prevailing conditions. Separation of ^(9^)

into the form given by eq (4.4) leads to the following definitions for £1, and

Z

P(z)

T(z)

11/4

dz (4.5)

Z

P(z)

lT(z)J

3 2.198(— - T(z)' p(z)

dz (4.6)

^3 = /

Z

P(z)

.T(z)J

3/2

P(z) dz (4.7)

T is equal to 293 kelvins, and the factors appearing under the integral signs

ooriginate in the expressions for yij Y2> and Y3 [6]. These factors are not

13
independent because the atmospheric variables are interrelated, allowing for the

simplifications introduced in the next paragraph.

The ideal gas law [11] and curve 2 of figure 10 lead to the following

approximations for z < 11 km:

P(2) ^

P

and for z > 11 km

T(z)

5.26

T

^ (4.8)

P(z) ^ ^-0.158(z-ll) (4.9)

These equations result in curve 1 (including the solid line before 20 km) in

figure 11. When they are used to evaluate ii from eq (4.5), eqs (3T3), (3T4),

and (3T5) in table 3 result.

Reduction of the integrals in £2 ^3 requires that more extensive approximations for T(z), P(z), and p(z) be made. Fortunately, the water-vapor density

falls off rapidly with altitude requiring an approximate fit to the standard

curves for only the first 6 km altitude. The assumed profiles are curve 3 of

figure 10, curve 2 of figure 11, and curve 2 of figure 12. Equations (3T8) and

(3T11) in table 3 for I2 ^^'^ ^3 result from the corresponding reduction of

eqs (4.6) and (4.7).

Table 3, then, is a collection of approximate equations used by the ETMS to

calculate the atmospheric transmission coefficient a(6 ). Errors associated

with these equations are discussed in the following sections.

4.1 Transmission-Coefficient Errors

A derivation of the atmospheric transmission-coefficient equations used by

the ETMS was outlined in the preceding sections. An estimate of the corresponding systematic error will be outlined in this and succeeding subsections.

The error in a(9 ) arises from the following sources: (1) use of the noninteLi gral approximations given in table 3; (2) the nonspherical symmetry of the real

atmosphere; (3) uncertainties in the constants appearing in the expression for

Yl, Y2» ^^'^ ^3 [6]; (4) uncertainty in the atmospheric temperature profile;

(5) uncertainty in the atmospheric pressure profile; (6) uncertainty in the

atmospheric water -vapor density profile; (7) uncertainty in the observer's

altitude; and (8) uncertainty in the observer's viewing angle 9 .

14
4.1.1 Nonintegral Approximation

A computer program was written to perform the "exact" calculation of ^.(9^)

and a(6 ) using eqs (4.1) and (4.2), including the ray-bending equations of

section 2 to predict the "exact" ray path. The "exact" calculations used curve

1 of figures 10, 11, and 12. The results of the "exact" calculation were com- pared with predictions using the equations of table 3. The difference between

the "exact" and approximate calculations was assumed to be the error arising

from the approximate nature of the table 3 equations. Curve number 1 of

figure 16 is the result shown as a function of the angle 9 .

4.1.2 Nonspherical Symmetry

The idealized assumption of spherical atmospheric sjmmietry yielding eqs (4.2)

and (4.3) leads to an error that was estimated in a manner similar to that used

for the nonspherical refraction-angle error of section 2. The percentage

transmission-coefficient error was assumed to be the same as that case leading

to curve 2 of figure 16.

4.1.3 Absorption-Coefficient Constants

The absorption coefficients yi» Y2» ^'^^ Y3 were obtained from the literature [6] and contain constants whose uncertainties are assumed here to be 10%.

The corresponding error in a(9 ) is shown as curve 3 in figure 16.

4.1.4 Temperature, Pressure, and Water-Vapor Density Profiles

Assumed uncertainties in the atmospheric temperature, pressure, and watervapor density profiles have already been discussed in subsection 2.1.4. These

uncertainties were used in a manner similar to those used to establish systematic error bounds on the refraction angle x. The resulting errors are shown as curves 4, 5, and 6 in figure 16.

Transmission-coefficient errors due to uncertainties in the observer altitude Z and elevation angle 9 were calculated from table 3 equations by assuming

Li ±0.1 km and ±0.01° for Z and 9 respectively. Curves 7 and 8 of figure 16 show

C-t these errors.

15
4.1.5 Total Transmission-Coefficient Error

The total systematic transmission-coefficient error is assumed to be a linear sum of the eight component errors shown in figure 16. The result is

shown in figure 17. This curve, like those of figure 16, varies slightly with

atmospheric conditions and frequency. Figures 16 and 17 are appropriate to

average conditions for the month of May in the vicinity of Washington, D.C., and

a frequency of 10 GHz. Equation (3T12) in table 3 is an approximation for

figure 17 that reflects local conditions in general and is used by the ETMS to

calculate the systematic error for a(9 ). This equation implies a 50% systeLa matic error in the atmospheric loss as calculated from the table 3 equations.

5. RESULTS AND CONCLUSIONS

An outline for the derivation of equations used by the ETMS to calculate

atmospheric refraction angle and transmission coefficient and their corresponding systematic errors has been presented in sections 2, 3, and 4, leading to the

approximations found in tables 2 and 3. The error components for the refraction

angle and transmission coefficient are presented in figures 5 and 16 respectively, with the corresponding total errors (linear sums of the component errors)

presented in figures 9 and 17. The equations in both tables apply only to

stable, clear weather conditions.

Figure 9 shows that the equations in table 2 can be used to calculate the

refraction angle to an error of 3 minutes or less for elevation angles greater

than 4°. This accuracy is sufficient for the ETMS to commence its earth-terminal

measurement procedures in most geographical locations now in operation.

Equation (3T12) is an approximation for the transmission-coefficient error shown in figure 17, revealing how the error varies with weather conditions

through the loss factor "l-aCG^)." This equation implies that the error in the

loss itself is approximately 50%, independent of elevation angle, weather

(stable) conditions, and frequency (in the 1 GHz to 10 GHz range).

16
6 . ACKNOWLEDGMENTS

The work reviewed in this report represents a small part of the total

effort expanded to design and construct an earth-terminal-measurement system for

G/T, C/kT, and EIRP measurements. The system was built for the U.S. Army Communications Command at Fort Huachuca, Arizona, whose support is gratefully

acknowledged

17
7 . REFERENCES

[I] Wait, D. F., Satellite earth terminal G/T measurements, Microwave Journal,

20, No. 4 (April 1977).

[2] Daywitt, W. C, Error equations used in the NBS precision G/T measurement

system, NBSIR 76-842 (Sept. 1976).

[3] Daywitt, W. C., Error equations used in the NBS earth terminal measurement

system, NBSIR 78-869 (Dec. 1977).

[4] Hill, D. A., A survey of earth-to-satellite propagation factors between 2.5

and 275 GHz, Published in an Office of Telecommunications Report, OT

Report, 74-43 (July 1974).

[5] Longley, A. G., Rice, P. L. , Prediction of tropospheric radio transmission

loss over irregular terrain: A computer method-1968, ESSA Research Laboratories Technical Report, ERL 79-ITS 67 (July 1968).

[6] Bean, B. R., Dutton, E. J., Radio meteorology, NBS Monograph 92 (March 1966).

[7] Bean, B. R. , et al., A world atlas of atmospheric radio refractivity, ESSA

Monograph 1 (1966).

[8] Handbook of Chemistry and Physics, Ed. C. D. Hodgman (Chemical

Rubber Publishing Co., Cleveland, OH, Feb. 1960).

[9] U.S. Standard Atmosphere, 1962, Prepared under sponsorship of the National

Aeronautics and Space Administration, U.S. Air Force and the U.S. Weather

Bureau (1962).

[10] Handbook of Geophysics and Space Environments, Ed. S. L. Valley (Air

Force Cambridge Research Laboratories, Office of Aerospace Research, U.S.

Air Force, 1965)

.

[II] Mcintosh, D. H., and Thom, A. S., Essentials of Meteorology (SpringerVerlag, New York, Inc., 1972).

18
APPARENT DIRECTION

OBSERVER'S

^^^^^^ ZENITH /

CENTER OF

EARTH

Figure 1. Geometrical idealization of the earth's atmosphere.
20
10 20 30 40

TRUNCATION ALTITUDE (Ioti)

Figure 3. Truncated refraction angle and truncation error versus truncation

altitutde at a 3° elevation angle.

21
22
ELEVATION ANGLE (DEGREES)

Figure 5. Systematic error components for the refraction angle.

23
Figure 6. Geometrical idealization of the earth's atmosphere showing that

portion (shaded) in the vicinity of the observer of interest in

determining ray bending.

24
25
26
4

^

3

toLU

ucii:

oe: o

on

on

1

10 20

ELEVATION ANGLE (DEGREES)

30

Figure 9. Total systematic refraction-angle error.

27
300!-

00

Ixl

on

LiJ 2:

200LJ_l

SOLID CURVE - 1962 U.S. STANDARD ATMOSPHERE

EXTENSION (1) - "EXACT" MODEL

EXTENSION (2) - APPROXIMATE MODEL (DRY)

EXTENSION (3) - APPROXIMATE MODEL (WET)

15 20

ALTITUDE (km)

35

Figure 10. A portion of the 1962 U.S. standard atmospheric temperature

profile showing the three approximations (dashed curves) used in

the text.

28
10 I I I ' ' ' ^ ' » I I I I I ,1 0 4 8 12 16 20 24 28

ALTITUDE (km) .

Figure 11. A portion of the 1962 standard pressure profile showing the two

approximations (dashed curves) used in the text.

29
ALTITUDE (km)

Figure 12. A portion of the 1962 U.S. standard water-vapor density profile

showing the two approximations (dashed curves) used in the text.

30
31
FREQUENCY (GHz)

Figure 14. Zenith attenuation due to molecular oxygen and water vapor versus

frequency (for 300 kelvins, 1013.25 millibars, and 7.75 g /m )

.

32
.06

TRUNCATION ALTITUDE (km)

Figure 15. Truncated zenith attenuation and truncation error versus truncation altitude (at 10 GHz for Washington, D.C., in May).

33
1. NON-INTEGRAL APPROXIMATION

2. NON-SPHERICAL SYMMETRY

0 10 20 30 40 50 60

ELEVATION ANGLE (DEGREES)

Figure 16. Systematic error components for the transmission coefficient (at

10 GHz for Washington, D.C., In May).

34
ol \ \ I \ \ i \ i 0 10 20 30 40 50 60 70 80

ELEVATION ANGLE (DEGREES)

Fi gure 17. Total systematic transmission—coefficient error (at 10 GHz for

Washington, D.C., in May).

35
Q

QQ

dd

M

CD

Q CO Q QQ Q

*H

Q

<D r—1 w

i_l

CT) p,p,

<^ o on

•U

CO o d w

QJ

j_) 00 00

lO

r*V-i * 'Pi

COo CO

y^ 00 CO

l_r CO w in

CO

cu u3C

•HEo

<d

cn

C3o

•i-i ue

•HOU

Cua

COoo

CNN c

CO 4J

CsJ ooo• oI ooooI

cndI ooIo

CO o

N3

OI ooooooI oI ooI HM oodod

00ooI oI

COoHM

Oo

CMO

00o

COooI ood

CNJ

COod

SI

03

CDd

CO

CO

.—I oo o

CO o

CMCO

CO

CMo

in

1—1 COo

CDC

4-) I I Nl N) oXo

00

CD

C! 4J O

in

CO

CD

•u u o

vOo

00

CM

COo

CO

CO

CO

II

36
H

CMH

H

CM

cnaI

N!

Nl

O

ON

I—

CMO• O+

CM

a CM u SO o

in

m CO

1—1

H o o

p-i 0 o H /~\ 1 o N

^-^ Nl Ph in

CD 1—

d CM

4-> CM u O NN ' +

cn1 or-lX o

00

in • E-i CO 00so

II <N

0)

SO

CN4

Cl

IIQ.

to

H

CM

mH

CM

H

soI O1—

X

CO . CO

CMCO

CM•

COHmm

r~COmOoI CM

EhCOo•—1 vO

00

CMO

soo

00

in

CMCM

CM

in00

so

CM

Csj

s£> H

CM

fOaI

CD

CJ u

CM

U)W

d)u

<x

Jja

CO>o

CM

PCTJ

0) u3 e

4-1 f3 •H

II

•H

>^

4J •H

COaQ

}-l oa

CO >

COd

•H>

•H

N!H

II

•H

T3•HE3

PC

cu>

•Hu

CO iH

QJ Pi

II JC

J2o

II

CM

IIo

aCOa

•H>

CD

CO

CM e

in

CMCO

37 o O
H

CO

H

CO

fOH

CO

H

CO

mH

CO

vOH

CO CO

00H

CO

0) T33

4-1 •H

in

u

01>

CO O

CDO

CO u

<:I o

Nl

CD

23

T3

00

CM

Csl

>-

II <

P-iO oH

is: in

r—I

cso

(4-1

r-4 >-

II

'.

/"^ I WG , (U .) H •

CM CM

vO

CN in y-i CO+

inCO1 o.—1 •-4 •—

in

CM

l4-t

t—t

CM

r-t I—1

CM a\

a^ CM o +

<f COOI—1

c—(

>-

Cv3 CMO

in

o\oo

in

CM+

in

oo

NII CMOm1—1 CM

CMO• OI

in

in

vOOom

0^

HI

00

CM

COo

/-

N

CvJ >- oH

Nlm1—1 CMO

CO

CTi o

II

Ht /—s

14-1 CM CM CM• m

CM

OH

CMo+o

CM

Csl

Nl J

PQC

•H

CO

COC

•rl>

•HNo

•H

«i a

mmH

OHM

CM

CO

CMoH

NBS-1 14A (REV. 7-73)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUIUJCATION OR REPORT NO.

NBSIR 78-833

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLK AND SUBTITLE

Atmospheric Propagation Equations Used in the NBS Earth

Terminal Measurement System

5. Publication Date April 1978

6. Performing Organization Code

276.05

7. AUTHOR(S)

W. C. Daywitt

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

UCrAK 1 MCN 1 Ur CUMMcKCC

Washington, D.C. 20234

10. Project/Task/Work Unit No. 2765266

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Same as #9.

13. Type of Report & Period

Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-wotd or less (actual summary of most si^ificant information. If document includes a significant

bibliography or literature survey, mention it here.)

A derivation of equations for approximating the atmospheric

refraction angle and transmission coefficient is outlined. The

approximations apply in the 1 GHz to 10 GHz frequency range and are accompanied by systematic error estimates. They are used

in the NBS Earth Terminal Measurement System for quasi-real-time

calculations concerned with the measurement of earth-terminal

gain-to-noise-temperature ratio (G/T), satellite effective isotropic

radiated power (EIRP), and downlink carrier-to-noise-density ratio

(C/kT).

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only tfie first letter of the first key word unless a proper name; separated by semicolons) Atmosphere; downlink carrier- to-noise-density ratio (C/kT);

satellite effective isotropic radiated power (EIRP); error analysis;

earth-terminal gain-to-noise-temperature ratio (G/T); refraction

angle; satellite communications; transmission coefficient.

18. AVAILABILITY^Unlimited

1 ! For Official Distribution. Do Not Release to NTIS

1 1 Order From Sup. of Doc, U.S. Government Printing Office Washineton. D.C. 20402. SD Cat. No. CI

3

1 1 Order From National Technical Information Service (NTIS)

Springfield, Virginia 22151

19. SECURITY CLASS

(THIS REPORT)

UNCL ASSIFIED

21. NO. OF PAGES

45

20. SECURITY CLASS

(THIS PAGE)

UNCLASSIFIED

22. Price

$4.50

USCOMM-DC E9042-P74