modeling of transionospmeric diagrani for consideration of first-order signal statistics, showing...
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Final Quarterly Technical Report
MODELING OF TRANSIONOSPMERIC RADIO PROPAGATION
By: E. J. FREMOUW C. L. RINO
Prepared for:
ROME AIR DEVELOPMENT CENTER GRIFFISS AIR FORCE BASE ROME, NcW YORK 13441
Attentior; MR. RICHARD A. SCHNEIBLE (OCSE)
CONTRACT F30602-74-C-0279
Sponsored by
DEFENSE ADVANCED RESEARCH PROJECTS AGENCY ARPA ORDER NO. 2777
/ A
STANFORD RESEARCH INSTITUTE Menlo Park, California 94025 • U.S.A.
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MODELING OF JRANSIONOSPHKir RADIO PROPAGATION RADI
^ja By: E. J. TREMOUW
C L. RINO
(£>. Augm IB?5
Prepared for:
ROME A|R DCVPLOPMENT CENTER GRIFFISS AIR FORCE BASE ROME, NEW YORK 13441
Attent,0n: {0RCSER),CHARD A- ICHNe,B«
(315) 330-3085
Sponsored by:
I X
^ESsoNDV
BAoNuCuDvA
RpTRCH PR0JECTS A-«V ARLINGTON, VIRGINIA 22209
Attention: MR. JAMES C. GOODWIN
STRATEGIC TECHNOLOGY OFFICE
SRI Project 3416
CONTRAC^ F30602-74-C-0279 /W^ ^ ' - Effective ohn* Contract:' r^ayTg^^ "l Ä^T Contract Expiration Date: 14 May 1975 ^ 9 Amount of Contract: $99 542 Principal Investigator: E. Fremouw
(415) 326-6200, Ext. 2596 ARPA Order No. 2777 Program Code No. 44E20
Copy H0, KsCy
33'X 'TO 0
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CONTENTS
LIST OF ILLUSTRATIONS 111
LIST OF TABLES lv
I REPORT SUMMARY 1
II SCATTERING THEORY EMPLOYED 3
III SCINTILLATION MORPHOLOGY MODELED 23
IV THE CHANNEL-MODEL CODES
V CONCLUSION 41
REFERENCES 47
ii
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ILLUSTRATIONS
mm**
Phasor Diagrani for Consideration of First-Order
Signal Statistics, Showing Elliptical Contour of Equal
Probability Predicted by the Gaussian-Statistics
Hypothesis ^
Scintillation Index S, as a Function of Elongation and
Orientation of the Equal-Probability Ellipse, for Four
Values of Ionospheric Scattering Coefficient, R0 8
Comparison of Algebraic Approximation (Eq. 20) to Values
of the Integral in Eq. (17) 16
Scatter Plot Showing Degree of Correlation Between
Auroral Electroject Index, AE, and Planetary hagnctic
Index, Kp 27
Comparison of Model Calculation (solid curve) with
Diurnal Variation of 136-MHz Intensity-Scintillation
Index Observed (Xs) by Koster (1968) in the Equatorial
Region 29
Comparison of Model Calculation (solid curve) with the
Seasona1. Variation of 136-MHz Intensity-Scintillation
Index ■ bserved (Xs) by Koster (1968) in the Equatorial
Region 29
Comparison of Model Calculation (solid curve) with Diurnal
Variation of 40-MHz Intensity-Scintillation Index Observed
(Xs) by Preddey, Mawdsley, and Ireland (1969) in the Mid-
Latitude Region 30
Comparison of Model Calculations (solid curves) with the
Latitudinal Variation of 400-MHz Intensity-Scintillation
Index, for Three States of Geomagnetic Disturbance,
Observed (symbols) by Evans (1973) Near the Subauroral
Scintillation Boundary 30
Comparison of Calculated (smooth curve) Probability
Density Function for Amplitude with Observations
(histogram) of ATS-5 Transmission at 137 MHz, Made by
Aarons (private communication, 1974) Through the Mid-
Latitude Ionosphere from Near Boston 31
iii
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mmm^Kmmmmi^m^wmm^'i*' «HIM rw^pwil iniaj i
II
10
11
12
13
14
15
16
17
Comparison of Calculated (smooth curve) Cumulative
Probability Function (cpi) for Intensity (or for
amplitude) with Observations (histogram) for Case in
Figure 9 31
Calculated Probability Density Function for Phase for
the Case in Figures 9 and 10 32
Simplified Flow Chart of PROGRAM RFMOD 34
Sample Input Cards for PROGRAM RFMOD 35
Example of Output from Available Version of PROGRAM RFMOD . . 39
Example of Amplitude pdf Output from Available Version
of PROGRAM DIST 42
Example of cdf Output from Available Version of
PROGRAM DIST 43
Example of Phase pdf Output from Available Version
of PROGRAM DIST 44
TABLES
1 Routines in PROGRAM RFMOD 33
2 Data-Card Variables 36
3 Contents of Changing-Paiameter(s) Card 38
4 Routines in PROGRAM DIST 40
iv
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I . REPORT SUMMARY
This is the final quarterly technical report on a one-year contract
to extend and improve an existing empirical model for worldwide behavior
of ionospheii-cally imposed radio-wave sci|ntillation. The objectives of
the project were (1) to improve the accuracy of model-based calculations
of the intersity-sclntillation index and (2) to develop a capability for
full description (from the point of view of engineering t.-plications) of
the first-order, complex-signal statistics that characterize tbn trans-
ionospheric radio communication channel. A follow-on project has been
initiated to extend the channel model to include second-order signal
stauistics in the temporal, spatial, and spectral domains. The first
priority in the follow-on work • ill be to calculate the fluctuation
spectra of relevant signal parameters. In addition, it is intended to
extend validity of the model into the multiple-scatter regime
^ The stated objectives of the just-completed project were accomplished
by starting with the scintillation mod 1 developed by Fremouw and Rino
(1973) , adding a geomagnetic-activity dependence for the behavior of
scintillation-producing ionospheric irregularities, and substituting a
more accurate and general scattering theory for that of Briggs and Parkin
(1963) used in the earlier model. The results of these efforts have
been implemented in two computer programs. The first, entitled RFMOD,
contains the main elements of the scattering theory, the morphological
model for ionospheric irregularity strength and other parameters, and
subroutines for calculating the geometry and other relevant quantities
from user-specified inputs. The main outputs of RFMOD are the intensity-
scintillation index SA, which is the fractional rms fluctuation in re-
ceived signal intensity (Briggs and Parkin, 1963), and an analogous phase
scintillation index $rms (labled PRMS), which is the rms fluctuation of
phase.
■ ■■■
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^ —' — "■'"
:
Three related first-order, signal-statistical parameters also are
output from RFMOD, to be used as inputs to the second program, called
DIST. The latter program permits calculation of the first-order dis-
tribution functions of amplitude and phase for selected values of S4
and/or $rms. There are three types of output from DIST: values of the
probability density functions (pdf) for amplitude and for phase, and
values of the cumulative probability function (cpf) for intensity (or,
equivalently, for amplitude since the argument is output in dB relative
to the undisturbed level). The cpf output constitutes estimation of
fade margin required to combat amplitude scintillation.
RFMOD and DIST represent interim versions of evolving transiono-
spheric communication-channel codes. As stated above, capabilities
will be added for calculating second-order statistics. More relevant
in the near term is that there has been no direct testing of calculations
relating to phase scintillation. As was recognized at the outset of the
project, data are not available at present that are directly relevant
to testing phase statistics unambiguously. Useful data are expected
from forthcoming observations of milClfrequency coherent satellite
beacons, and such tests will be made. The forthcoming data may also
result in model changes that will further alter the calculated results
for amplitude scintillation, because testing to date has still necessarily
relied on some observations reported in terms of a partly subjective
scintillation index. In addition, while the assumption of weak, single
scatter contained in the Briggs-and-Parkin theory has been partially
removed, a more complete correction for multiple scatter effects has yet
to be incorporated.
This report represents a summary of the work performed during the
past year and is inteded as a status report on the evolving SRI/ARPA
channel model. Section II describes the scattering theory employed and
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compares some of its salient poit.ts with those of other theories. That
section includes some material reported in earlier quarterly reports,
for the sake of presenting a coherent discussion under one cover, but
there is no attempt to describe the theoretical development in chrono-
logical detail. Section III contains a brief review of the known mor-
phology of scintillation-producing ionospheric irregularities as modeled
in RFMOD. A functional description of the programs RFMOD and DIST is
presented in Section IV, along with instructions for their use. Finally,
in Section V, we present our conclusions on the utility and limitations
of the channel model in its präsent form and some prospects for further
improvement.
II SCATTERING THEORY EMPLOYED
The basic approach to modeling the transionospheric communication
channel is to treat the channel as a time-and-space-varying linear filter
linking the transmitting and receiving antennas of a communication sys-
tem. Slowly varying effects (on a time scale of many minutes or hours),
such as group delay, polarization change, and dispersion of signals
propagating through the smooth ionosphere, can be described by well-
known deterministic methods and are not included in this work. What is
treated are the relatively rapid and random variations (on a time scale
of fractions of a second to a minute or so) In signal parameters that
arise due to scattering by irregularities in ionospheric electron density
and that are referred to as scintillations. Because the channel can be
treated as a linear filter, standard Fourier techniques can be used to
apply the results to modulated signals if a sufficiently general descrip-
tion is provided of the signal statistics resulting when a continuous
(CW) is passed through the channel (Fremouw, 1969). In this section,
we. discuss the first-order statistics of this basic signal.
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Any LW Fignal can be represented by a vector on the complex plane,
such as the phasor E in Figure 1. In this figure, E = ^E^ + IE j is meant
to represent the randomly time-varying complex amplitude of the signal
output from an antenna terminal. In describing the first-order, complex-
signal statistics of E, we are assessing the probability that the tip of
the phasor lies within an elemental area on the complex plain. To begin,
we write E as the resultant of its long-term mean (E) and a zero-mean,
randomly varying component E . We choose to reference all phases to
that of <E>, which is the same as the phase that E would have in the
I
<E> = TJ Re AXIS
LA-3416-12
FIGURE 1 PHASOR DIAGRAM FOR CONSIDERATION OF FIRST-ORDER SIGNAL STATISTICS, SHOWING ELLIPTICAL CONTOUR OF EQUAL PROBABILITY PREDICTED BY THE GAUSSIAN- STATISTICS HYPOTHESIS
absence of channel-imposed perturbations because we have excluded slow
channel variations from the model. (Thus (E ) = 0, and (E > 8 T\ ■ y x
(E>.) The magnitude of (E) (= Tfl is less than the amplitude of the un-
disturbed signal, however, because some of the transmitted energy has been
scattered into the randomly varying component E .
4
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To proceed with the analysts, it is convenient to break E^ into
its own real and imaginary components, S and S , respectively. The x y
convenience stems in part from the fact that the variances of S^ and 0 0 £ *
S a and a , respectively, and their covariance 0 can be calculated y' x y xy
in terms of ionospheric parameters. These are useful calculations to 2 2
perform because, at the very least, the sum of a^ and c^ represents the
powar contained in the scattered signal, which we shall denote as 1^.
That is,
R = <E E*) = c + a . (1) o s s x y
We shall normalize all our calculations to unity total received power,
so R will represent the fraction of flux scattered--!.e., the forward- o ttering coefficient of the ionosphere. A complex quantity analogous
o sea
to the real R that it is useful is the following (Rino and Fremouw, o
1973a):
B = ^ E > = (o2 - o2) + 2i a . (2) oss x y xy
If it should be that the quadrature components, E and E , are
jointly Gaussian random variables, the utility of the three variances
defined above--or alternatively of R and B^-would be very great. Indeed
in this instance the first-order statistics of the received signal E are
totally defined by either of these sets of three parameters (i.e., the
three real variances or the real R and the real and imaginary parts of
B ). Most scintillation observations have been only in terms of the o
magnitude of E (i.e., amplitude or intensity scintillation) because a
sufficiently stable phase reference for measuring its complex value has
seldom been available under scintillation conditions. Thus, there has
been no direct determination of the complex-signal statistics. (This
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data deficiency should be remedied within the next year by means of
careful measurements of signals from the VHF-UHF coherent beacons on
ATS-6 and the Transit satellites and, under a wider range of ionospheric
conditions, by means of observations of the forthcoming DNA-002 multi-
frequency beacon.)
The Gaussian-statistics hypothesis (i.e., that the quadrature com-
ponents are jointly Gaussian variates) has been tested indirectly, however,
by means of amplitude-scintillation data. Rino, Livingston, and Whitney
(1975) assessed this hypothesis and the competing one of log-normal
statistics (according to which the phase and the logarithm of amplitude
are jointly Gaussian variates) by performing chi-square, goodness-of-fit
tests of amplitude histograms to amplitude pdfs calculated from the two
hypotheses. Good fits were found in general for both hypotheses, which
tend toward identical pdfs as the scintillation index de-.reases, but
the Gaussian hypothesis yielded better fits to 11 out of the 12 data
sets tested. This result is fortunate because it permits presentation
of scintillation calculations in a form that will be rather familiar to
communications engineers. In the remainder of this report, we proceed
on the basis of the Gaussian signal-statistics hypothesis.
For Gaussian signal statistics, a contour of equal probability for
the tip of the E-phascr on the complex plane is an ellipse (Beckman and
Spizzichino, 1963), such as the one in Figure 1. The analytical task of
first-order, signal-statistical channel modeling, then, is to relate the
parameters of such an ellipse to parameters of the channel (scattering
irregularities in the ionosphere in the present case). The probability
ellipse is characterized by its size, eccentricity, and orientation on the
complex plane. Clearly. Ro (being the fraction of flux scattered) dic-
tates the size of the ellipse. The eccentricity is dictated jointly by
R and the magnitude of B^ and the orientation is controlled by the phase
angle of B . Spec:fieaiy, the axial ratio of the probability ellipse is
given by
-■' ■LJW' JJ^-—.:-^....^....—..—^.^-J.» ■ ■ - - - - - -~ — ' - -- ^ .^„, .__ ..... .. .
•""""-•"'• '" " !«■■ "iMiiipiii utmnitmm ""■■""""
1 +
1/2
(3)
and its orientation angle, 6 (shown on Figure 1), is given by
1 TT
(4)
It will be recognized from Eqs. (1) through (4) and the accompanying
discussion that there are several sets of three parameters that can be
used to describe the first-order, complex-signal statistics. For reasons
of convenience, which will become clear in the ensuing discussion, we
choose to work with the parameters R , |B |/R , and 6, As an example of o o o
the relationship of these three parameters to a quantity of direct
applications interest, we have plotted the intensity-scintillation index
S in Figure 2 from the following formula derived by Nakagami (1960): 4
S, = 2R (1 - R ) 4 o o
L^-cos 26 U R2
R 10 1 + (5)
We note that in Briggs' and Parkin's theory, S is calculated
approximately as
cos 26 (6)
I ' .-■..-:- ^..^^^.^l^
—'■■ ■"—' ——^
1.2
1.0
0.8
0.6
Ro = 0.4 \
Ro - 0.36
-0.7 A s ^ ̂
^
1
— •^,
FIGURE 2 SCINTILLATION INDEX S4 AS A FUr.CTION OF ELONGATION AND OP ENTATION OF THE EQUAL-PROBABILITY ELLIPSE, FOR FOUR VALUES OF IONOSPHERIC SCATTERING COEFFICIENT, R0
The following three relationships arise from Eqs. (1) and (2)
2 Ro cos (26) (7)
R [ r. 2 o
0 = y
^ll + ^^cos (26) 2 L Ro (8)
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xy sin (2&) (9)
Th us, Eq. (6) contains o .ly the contribution of the in-phase scattered
nt S to S . It arises from equating fluctuations in E to compone... x 4
fluctuations in the real amplitude A and is accurate only for very weak
scatter [i.e., when the higher-order R terms in Eq. (5) are negligible].
A significant improvement in th« present scintillation model is retention
of all terms in Eq. (5), which accounts for the contribution of fluctua-
tions in the phase-ouadrature component E (=S ) to amplitude scintilla-
tion (as well as for :he influence of the correlation between S and S ). x y
Other differences between the old scattering theory and the new
arise in the manner of calculating R and B . The basic approach used
in the present work is that described by Rino and Fremouw (1973b), in
which the scattering geometry is treated quite rigorously. Briggs and
Parkin simplified the geometry by assuming that the ionospheric scattering
layer effectively can be tilted from the horizontal so that its boundaries
are always normal to the incident radio-wave propagation vector. This
approximation breaks down for large zenith angles but it avoids a great
deal of complication in derivation of the scattering equations. Some of
this complication was avoided by Rino and Fremouw (1973b) by assuming
zero correlation in electron-density fluctuation along the vertical, which
has negligible effect on the scattering equations for nearly vertical
incidence. Again, however, the effect of the assumption becomes un-
acceptable as the incidence angle increases.
During the last quarter of this contract, the theory was recast to
include the scattering effect of structure in all three dimensions of
the ionosphere. An outline of the derivation is given here, together
with the resulting equations for describing first-order signal statistics,
■ - -- ■HHWaMiaiiHViiianmUi «MAMMMMMateMadM
rl mir^mmmmmm^^^ «■liaHW.IIIipjIIIIIBIIIII 11 "I " i"ii«" "Hii .11
which are valid over a greater range of incidence angles than their
counterparts in Quarterly Technical Reports 1 and 2.
The scattering coefficient R represents a special case of the
spatial autocorrelation function of the scattered signal, which is
defined as
R(P) = <E (r) E (r + P)> (10)
namely, the case in which the spatial lag parameter P is zero. A similar
definition, calculated without taking the complex conjugate, can be made
of a quantity B(p), which reduces B in the same special case (i.e.,
zero lag). The approach to evaluating R(P) and B(p) is to express Es
as the scattered field calculated by means of the first Born approxi-
mation, with the scattering medium rer^esented by a spatial spectrum of
ionospheric structure. The field is calculated by integrating the
contribution from different heights within the scattering layer, con-
sidering the diffraction effects arising in propagation to the observing
plane but ignoring effects of multiple scatter. When the complex field
is multiplied by its complex conjugate, the effects cancel.
The result is that R(P) can be expressed as the two-dimensional
Fourier transform of a height-smeared version of the ionosphere's spatial
■pactrun and B(P) can be expressed as the transform of that spectrum
times a propagation factor. Setting the lag parameter to zero is tanta-
mount to integrating over the spectra in the observing plane (as opposed
to calculating the Fourier transforms), with R being an integral over
the scattered field's angular power spectrum and B being an integral
over a complex spectrum that incorporates diffraction effects.
Irobably more important for first-order statistics than the approach
to deriving R and B is the form of the spatial spectrum used in o o
10
± ^^^^^^^^^^^ ------ - -
r mmmimmm!mim*mmmmmmmmmmm*i ■■""■'"" II«IJ*I in
their calculation. In evaluating R and the second factor in Eq. (6), o
Briggs and Parkin assumed a Gaussian-shaped spatial spectrum for the
ionosphere. While convenient because of its Fourier-transfomation
characteristics, a Gaussian spectrum does not appear to represent
ionospheric structure very realistically (Rufenach, 1972; Dyson et al.,
1974) .
A better description is afforded by a power law with a low-frequency
cutoff. When normalized so that its three-dimensional integral is unity 2
times the spatial variance ((AN) ) of electron density, such a spectrum
3 2
[units of (electrons/m ) per unit three-dimensional spatial frequency,
KJ may be written as
3C(tC) = 8rr 3/2 r(v + 1/2) Q3a ((AN)2)
r<v-1) ii + te^2! - -21v + 1/2 (il)
where V is the spectral index defining the power law, a is scale-size
parameter controlling the low-frequency cutoff, a is the ratio of
irregularity scale size along the geomagnetic field to that across the
field, and the three dimensional shape and orientation :>£ the irregulari-
ties (Budden, 1965), are described by
S2(K) = AUO K2 + K2 + B(^2 - 2C(^)K < x y z x z
(12)
where
A(« =
2 2 a + C (t)
»(♦) (13)
2 2 2 B(i|i) = cos 'i + a sin (14)
n
m^t^^^^^m ^.-^M^lac^, . ^..M.
mmm —— imn^nmmnm mmmm* ■w«» ^im'f^mt^mm^mmmfmmm^'l'-'w»" m
c(t) ■ (i - O cos ♦ sin t
and '; = the magnetic dip.
(15)
The coordinate system used is such that K is directed to the geomagnetic
north K to the geomagnetic east, and K downward. ' y z
Using the above spectrum, one obtains the following expression for
R
R =2^r2X2T>- ^ ^JÜ^LW) .
o e r(v - 1) ,, .( r (16)
where r = classical electron radius, e
\ = observing wavelength,
L = thickness of scattering layer,
A' = mLo' + D],
C' = 1/2[D' - D],
2 D' = i + B(\II + 90°) + 2C(t) tan ü cos 0 + B('i) tan 9,
and
D = /LD' - 2(1 + sin2 e)]Z + (B')2 ,
B' = B(^) tan2 6 sin 2C5 + 2C(t) tan 6 sin 0,
9 = incidence angle on scattering layer,
0 = propagation azimuth relative to geomagnetic north at
height of scattering layer.
Eq. (16) was obtained by integrating analytically over the angular
power spectrum of the scattered wave arriving at the observing plane,
as described above. The integral over the complex spectrum associated
with B cannot be performed analytically. However, the real and imaginary o
integrals involved have been solved numerically for a large range of
12
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r ^mmmmm mm -**m~<mmmmiimmß*m* mi^mmmmmmmm*™^*™**!.!" mm. i ■■■
geometrical situations described by the parameters «. *. 6, and 0. Study
of the numerical results has led to approximations suitable for Implemen-
tation In an operating computer code such as RFMOD.
Simplifications have been found by converting the real and Imaginary
integrals to the magnitude and phase angle of Bo and then to the quantities
|B l/R and 6 Uhe latter via Eq. (4)j. The simplification arises from
an^molrlcally noted approximate relationship between the latter two
quantities, as descrlbad In detail In Quarterly Technical Reports 1 and 2.
It permits approximate description of the complete first-order, complex-
signal statistics from solution of the real integral alone. What is
actually calculated is a^, which is related to RelB^ by Eq. (7).
Two approximations are involved in formulating this part of the
problem. The first-neglect of a diffraction effect related to layer
thickness-appears numerically to be valid essentially anywhere outside
the scattering layer. The second holds so long as the spectrum of
irregularities contributing to Ex fluctuations is cut off at the low-
frequency end by diffraction effects rather than by an Inherent cutoff
of the ionospheric spatial spectrum. Both in-situ (Dyson et al., 1974)
and scintillation observations (Rufenach, 1572; Rlno, Livingston, and
Whitney, 1975) support this view for all frequencies of interest. We
have, therefore, ^oceeded on the basis of this "near-zone" approximation,
but we have Incorporated a test In RFHOD to safeguard against violation
of the assumption underlying it. On the basis of these two approximations,
2 one may write a /R as follows:
x = 2(v - 1/2) V A ' AZ sec
i vl 2 (C ) VA" a
1/2
13
L. in MBiMin^^iaMh^M »m^^^^M^^mmM ■ -- -^ ...^ J^_^-M—^.,
■Pi— "" '■"■ ^^mmmmmmmiif^mK>nmi>m»imi^n^m^^mm^^^n . i. IMI*. •'—- ——~ ^-^—
oj 2TT
sin2 [g2 7(9,0^j
r „ , 2 2 2 v + 1/2 [(ß ) cos 9 + sin cpj
dcp dq (17)
where
and
VXz = Fresnel-zone radius at scattering layer
2 2 7(0,0') = [l + tan e cos (0 - «(*)]
ß' =NA^7
;' = i i -i — tan 2 D' - 2(1 + sin 0)
(18)
(19)
The main diffraction effect dictating the first-order statistics of
a scattered signal is controlled by the Fresnel-distance parameter
appearing in parentheses in front of the integral in Eq. (17). The
integral itself contains only geometric factors. Although the integral
cannot be solved analytically, it has been possible to find a useful
algebraic approximation to it. The approach was to numerically eva'Mate
the integral, which was named F(ß',6, -,), for selected points in a
three-space of ß', 9, and 0', and then to fit judiciously chosen
algebraic functions to the points. The resulting functiors then were
incorporated into RFMOD. The procedure was as follows.
First, the integral was solved numerically for P =1,2, 5, 10, and
20, with 6 ranging from 0° through 80° (which exceeds the maximum in-
cidence angle of a transionospheric communication link on the F layer,
by virture of the ionosphere's curvature), and with 0 stepped between
0° and 90° (since the t' dependence is symmetrical about 90 ). After
the results were plotted, inspection of the curves suggested a behr/ior
of the following form:
F(ß', 6, $') = ^a(ß', 6) + b(ß', 6) + (b - a) cos 20', (20)
14
— - ■ '■ ■ ' • •■■■^ ^ -^ - ■ --■
. . —ini^ ■
. ... ...
"I«*™« mmmmmmm^^^**!""" ' "" ■' r-^-^mmmmmm
where a = the valuo of the integral when 0' = W2, aid b = the value
when 0 = 0.
T'he values of a and b were then plotted as functions of 6 for given
values of ß'. Again the plots suggested analytic fits of the form
aCp'jP) = c(|') sec 6 mce') (21)
b(ß',6) = c(ß') sec .n(ß') (22)
where c(ß') = the value of the function when 9 = (J ' = 0 and where the
exponents m(ß') and n(ß') were chosen by forcing Eqs. (21) and (22) to
match the numerical value of the integral when 6 = 70°, which is about
its largest value for penetration of the F layer. The values of c, m,
and n were then employed in a one-dimensional, least-squares fitting
routine to obtain best-fit polynomial exoansio;." of c(ß ), m(ß ), and
n(ß').
The results of the above procedure were combined into an algebraic
approximation to the integral in Eq. (17), and the approximation was
then used to calculate curves of F(ß', 6, 0') over the ranges of 6 and
0' and for the values of ß' originally employed in the numerical in-
tegration. Finally, similar curves were calculated for other (both
intermediate and more extreme) values of ß', and numerical integrations
were performed for these new cases. The entire collections of numerical-
integration and algebraic-approximation results were then compared to
determine the efficacy of the approximate expression. An example of the
comparison is presented in Figure 3 for a value of ß' not involved in
development of the approximation; the smooth curves represent the
algebraic fit and the individual points are results of the numerical
integration. The fit is considered quite adequate for application to
scintillation modeling. Thus, RFMOD contains the following expression:
15
- - - — - ■ ■
mmtm^mmmmm «■piMnpiwwp ^IUWM^PIPBIBPIUPPHI«»! wm*w^^mmmmmmimmmimm*m »HUPP«.» UP
-- 2C. - ih 2 (C)
A Az sec 9
I
2 F(ß', 9, 0') ,
4TTQ
(23)
where FCß', 9, 0') is calculated from Eq. (20)
1.8 -
1 r 1 r
LA-anKi-s
FIGURE 3 COMPARISON i F ALGt^.AIC APPROXIMATION (Eq. 20) TO VALUES OF THE
INTEGRAL IN EC (17)
It is clear from Eqs. (7), (8), and (9), that the first-order
statistical problem would be completely solved at this point if one 2
could find an approximation to o in some manner analogous to that 7 7 Xy 2
above for a iO being simply R - o ). However, no such approximate x y ox
16
* M^jm^ni, i ^. ^ ■._ WjIMHIiyiiriHMMiHUII
mwmmm mimmmmmmm***m mmm^^^>r^"'vmjm'>^^mmmm\i» u. iwm^mmmmi^^™**^*'^*
solution to the imaginary part of the B integral has been found, and
numerical integration each time RFKOD is run seems unacceptably inefficient,
Fortunately, as mentioned above, synthesis of the numerical integration
results that have been obtained has pointed to another simple procedure.
Specifically, when |B \/i and 6 were formed from the real and imaginary o o
integrals for a number of geometrical situations, it was found that t'ie
relationship between Iß I/R and 6 was not very dependent on geometry oo
and was well approximated by the following simple formula:
6 = - (1 - B /R ) 4 o o
(24)
Not only is the relationship between 6 and |B 1/R rather insensitive ' o o
to geometry, but also the first-order signal statistics are not very
sensitive to that relationship. For first-order statistics, the main
effect of diffraction is a simu''t aneous rotation (increasing 6) and
circularization (decreasing JB |/R ) of the correlation ellipse illus- o o
trated in Figure 1, as the scattered wave propagates with increasing Z in
Eq. (23). Wha. is important is the rate at which this approach to Rice
statistics (IB |/R = 0 and ^ = 45°) takes place rather than the precise o o
relation between ellipse elongation and eccentricity. Thus, we have
proceeded on the basis of Eq. (24), which, when combined with Eq. (7),
yields the following transcendental equation relating lB0l/R0 to a
x^R0:
B o TT
cos -
(■■ Ml R
0 4 1 - 2 (25)
Eq. (25) is fit very voll, especially in the rear and intermediate
zones, by the following approximation:
2 l!oi /
R =o-98(i - i-55r o \ o
17
(26)
„Mg*. d^iaiMAfti^H — - - .
^""■■■"■•■■, ' ' ■ u" ■' ■ ■ wmmmmmmmrmm^mm
Inserting Eqs. (7) and (26) into Eq. (5), we obtain the following very
useful formula for scintillation index:
S4- "■Afhtffv R°
1/2
(27)
While containing some approximations in its third terra, Eq. (27) is a
considerable improvement over the formula of Briggs and Parkin for S
[Eq. (6)3, which totally neglected both the second and third terms. The
improvement increases as the scattering becomes more intense (increasing
R )• o
One reason for separating the first two terms in Eq. (27) into two
factors is that R is amenable to a simple multiple-scatter correction, o
The theory discussed so far is limited to the regime of weak, single
scatter for the entire scattering region (i.e., R « I)• However, an
important effect of multiple scatter, namely extinction of the non-
scattered signal component (represented by (E) in Figure 1), can be
accounted for simply by requirü g conservation of total power flux while
applying the first Born approximation to elemental layers within the
scattering region. From the work of Fejer (1953), Bramley (1955),
Fremouw (1968), and Rino (1975), it follows that extinction can be
accounted for by replacing the single-scattering coefficient calculated
from Eq. (16) by a multiple-scattering coefficient defined by
R = 1 - e c
(28)
It is R' that is used (in place of R ) in Eq. (27) for calculation o 0
of scintillation index, which is output as S4, in RFMOD; this multiple-
scattering coefficient (R') is itself output fro RFMOD as RZRO. An
18
,... . . . . . ^-.-. .....*■. >.., —-|,|,t , n ,
«WiW> ji ■"•"■'■"■", '
important task for the second contract year is to find an analogous
multiple-scatter correction for a /R and the related quantities UJ/R r x o o o
and 6. Presently, the latter two parameters are calculated from Eqs.
(26) and (24), respectively, on the basis of Eq. (23). They are output
from RFMOD as BMAGR and DELTA, respectively, for use as inputs to the
second channel-model code, DIST. A single-scatter approximation warni-g
is printed if R Ä !• o
Once R', |B j/R , and 6 are calculated, then application of the o o o
Gaussian-statistics hypothesis permits calculation of any desired first-
order, signal-statistical parameter. The calculation of the intensity-
scintillation index.
A ((A2 - (A2))2)1/2
<A2> (29)
as described above, is an example. An analogous phase-scintillation
index may be defined as
rms i <(l - <$))2>1/2 (30)
= «$2) - <$)
2)1/?- (31)
Given the phase reference chosen in Figure 1, it is well known (e.g.,
Beckman and Spizzichino, 1963) that under the postulate of Gaussian
statistics, the joint probability density function (pdf) for the real,
E , and imaginary, E , pa x y
E, is given by
rts of the received signal's complex amplitude.
P (E ,E ) = xy x y
2TTO O x y
2 2x/r : Vl - p
exp 1
2D
VE - T1)E E -2 ^H
a a x y
2(1 - P )
2
(32)
19
-■^i— ■■ * . ...„..^^—_,. -.-^ ^ggmm^|M^^^^^^^ _ M ^^„jto^^^^^^^^ _ J
r —-—-— *m^*i
where p = a /o a and where 1) may be calculated from Tl = vl - R and xy x y u
the a's fromEqs. (7), (8), and (9). The individual pdf? for amplitude
and for phase can be calculated by changing variables in tq. (32) from
component to polar form and then integrating over the other variable.
That is.
and
2r
P(A) = 1 A P (A cos I, A sin i) dl xy
p($) =1 A P (A cos I, A sin i) dA Jo xy
(33)
(34)
Analytic solutions to the integrals in Eqs. (33) and (34) have been
known for some time in special cases (e.g., for Rice statistics, in which
a =0 and a = a ) , but not so for the generalized Gaus.-ian statistics xy x y needed to describe scintillation. Recently, however, Hatfield (1975)
has found the following general analytic solution for Eq. (34):
TT 2
a exp x
p(l) = ■:(i - o )
2TTO a A - P2 f*(l) x y 1
(1 - p ) exp fy)
2(1 - D )
. V2n(l - P ) + f2(*) ^ *■ e^c
f2(*)
/2(1 - P2) (35)
where / 3 2 '
f (») = Vcos I/O - 2P cos $ sin $/(>T a ) + sin I/O x y
2 x, 2
y (36)
2 and f ($) = "H Leos I/o - D sin 4/(0 0 ) J/£ (t)
2 x x x y i (37)
2n
. UHttHUWiUdAb^. MttHm^biu
mfF^mmmmmm mmmmwmmmmmmrmmm* <•"•' '' -m iKiiPimin
U>fortunarely, there is no known analytic solution for the moments of
the distribution in Eq. (35). Therefore, the distribution itself has
been incorporated in RFNOD, together with a numerical integration routine
for calculating its first and second momeu.s, as follows:
and
2TT
<*) -I i o (|) di
2r
<$2) = f $2 p (») d$
(38)
(39)
The results of Eqs. (38) and (39) are then inserted into Eq. (31) for
calculation of the phase-scintillation index, which is output as PRMS.
As described above, the signal-statistical outputs of RFMOD are
B' IB l/R 6 S , and $ . The last two outputs constitute an intensity o o o' ' 4' rms scintillation index and a phase-scintillation index, respectively, and
represent our judgment as to the first-order statistical parameters most
likely to be soupht by a user for synoptic purpoeefi. The other three
outputs are provided for use in posing more detailed questions about
first-order signal statisticr in particular cases, by means of the
DIST code
The DIST code is essentially a device for calculating and outputing
the amplitude and phase pdfs from inputs of R^, IBJ/RJ, and 6. For
phase, it is simply a matter of evaluating Eq. (35) for the given input
parameters and outputing the result as P(PHI) . Unfortunately, there is
no known analytical solution to Eq. (33) analogous to Hatfield's phase
distribution. Thus, it is necessary to perform the phase integration
over the joint distribution (32) numerically. This integration is per-
formed in DIST, and the result is output as P(A) versus A (the latter in
21
. ■ „■-■^^-*^—.,.■-,,,—^..,. . . -
""MnmmiHipppMP«* 1 ■ ■ ■ "^^mmmmmmm
units of undisturbed amplitude). In addition, the following integration
also is performed numerically:
P(A < V) ■f P(A) dA (40)
This resulting cumulative probability function (cpf) of amplitude is
then output as CPF versus 20 log V (labeled dB). Since the independent
variable is output in decibels, CPF may be thought of as the cumulative 2
distribution of intensity (I = A ) as well as of amplitude (A); it
represents the link margin needed to mitigate scintillation fading to a
desired reliability threshold.
22
1^. -^^— — ■ ■ —
'■ " •'>"■-•" '" IN« "I "-i ■•ll-lll.l ■"■■■■ll-- mpirs«p»iwiiwi*>p»ip"«w«wiw
III SCINTILLATION MORPHOLOGY MODELED
The major work performed on this contract was development and
implementation of the generalized scattering theory described in Section
II. The other important aspect of modeling the transionospheric scintil-
lation channel is description of the ionosphere's scattering structure as
a function of several variables. The structural characteristics that
must be described pertain to the strength, size, and shape of the radio-
wave-scattering irregularities and the thickness of the scattering layer.
All the relevant parameters appear in Eq. (16). The shape of the irregu-
larities is described by the ionospheric spatial spectrum LEqs. (11) and
(UyJ, which is characterized mainly by the power-law spectral index, v.
and by the ratio, a, of scale-size parameter along the geomagnetic field
to that across it. The size of the irregularities is scaled by the
cross-field scale-size parameter, a. The strength of the irregularities
is described by the spatial variance of electron density ((AN) ), and
the thickness of the scattering region is L.
In-situ measurements of the ionospheric spatial spectrum (Dyson et
al., 1974) show a remarkably consistent spectral index, corresponding
very nearly to the Kolmorov value of 4/3 for V, under most ionospheric
conditions; we have adopted this value as a constant in the model. The
irregularity axial ratio, a, also has been taken as constant, with a
value of 10, as a result of the data review performed by Fremouw and
Rino (1973). There is no direct measurement of the outer scale, a; it is
simply known to be large compared with a km or so (Rufenach, 1972; Dyson
et al., 1974; Sagalyn et al., 1974). We have found that a = 3 km pro-
duces a good fit to the amplitude-scintillation data analyzed by Rino,
Livingston, and Whitney (1975), and this value is currently employed as
a constant in the model.
23
mm**r*mmmim <•>" •' PWW-P^I—- ,"",,u
The remaining two ionospheric parameters, ((AN)2) and L, appear in
the scintillation calculations only as a product. We have set L = 100 km
as a representative value and have expressed the behavior of scintillation
in terms of the following empirical model for the rms electron-density
variation, ((AN)2)1'2:
((AN)2)1/2 = AN (X, T,D,R) + AN (X,T) + AN.(X,T,K ) e m n p
+ AN (X.T.K ) el/m' a p (41)
where '• = geomagnetic latitude in degrees
T = time of day in hours,
D = day of the year out of 365,
R = sunspot number,
K = planetary magnetic index. P
The terms in Eq. (41) are
AN = (1.2 X 10 )(1 + 0.05R) e
1 - 0.25 cos TT D 4- 10
91.25 exp
fT + 1.5
+ exp T - 22.5
F
exp _ M2
\l2/ -
3 if T s 22.5 6 if T > 22.5 (42)
AN 8 N / TTT \
= (4.4 X 10 )(1 + 0.33 cos —) exn fX - 32.5^ i 15 ,
AN, = (2.0 X 10 ) h
1 + erf X - 68.5 + 1.82 Kp -f (5 + 0.55 Kp) cos (TTT/12)
18.7 - 0.36 KD - (1 + 0.11 Kp) cos (TTT/12)
24
(43)
(44)
1 ._! ■ M ...... -.L.. .-- ■ .
...l*.,^!....^. -■. ..-.
- --- -- ' - -■ ■ - -^ "iimi i in ■" ■— —-"-^ — - — — ^„J—^
.•—p—i —— HHRP111 •w«i n»1' ■PVHPMPMII I H
and AN = (1.1 a
x 108) K exp P
f - 77 + 1.82 Kp + 4 cos (TTT/12)^
0.32 K«
(45)
Eqs. (42) and (43) describe, respectively, the equatorial and mid-
latitude behavior of scintillation, and the high-latitude behavior is
described by Eqs. (44) and (45). All four terms of Eq. (41) contain the
well-established nighttime maximum of scintillation. The equatorial term
also includes a seasonal variation, displaying equinoctial maxima. The
most complicated behavior occurs at high latitudes, above the "scintil-
lation boundary" (Aarons et al., 1969). Equation (44) essentially
describes the behavior of the boundary location as a function of time of
day and geomagnetic activity, and Eq. (45) accounts for sein*illation
that is directly related to auroral activity.
In the Fremouw-and-Rino 1973 model, the relationship of scintilla-
tion to geomagnetic activity was not included. The only solar-terrestrial
observable employed in the earlier model was the sunspot number, which
was intended as a measure of epoch during a solar cycle. However, it is
clear that the scintillation boundary migrates equatorward, along with
the auroral oval, with increasing planetary magnetic activity (Aarons
et ll., 1963; Evans, 1974). This was known at the time the 1973 model
was developed, but the dependence was excluded to avoid double accounting
for other aspects of scintillation behavior, such as seasonal and solar-
cycle variation.
Pope (1974) modified our high-latitude term to include a K de-
pendence without checking on the possibility of seasonal double accounting.
We have had five years of K values plotted and have found no troublesome P
seasonal dependence. We have, therefore, adopted Pope's basic revision
of replacing the sunspot-number dependence in our old high-latitude term
with a K dependence. We have not used his exact formulation, however, P
which deleted our earlier description of the variation in statistical
25
■^L ■-"■* ■ «MMMfiiktfdia IIIIüI—iMMii. nimünagiM,., .„^utamaannHB,,.
——' P?pi""P»IIW"I^IWP»"IHii»W?»W»«B»'W"^'W«
width of the scintillation boundary with its location. We find that there
is such a variable-width behavior in the in-situ data of Sagalyn et al.
(1974) ; it is described by the denominator of the error-function argument
in Eq. (44).
Having incorporated a K dependence in Eq. (44), it was a simple
extension to include the K -dependent migration of the auroral oval in P
Eq. (45). A more accurate index to use might be the auroral-electrojet
index, AE, but we have elected to use only K for simplicity. The two
magnetic indices, K and AE, are rather highly correlated, as shown in
Figure 4. To avoid the necessity for a user to specify two magnetic
indices rather than just one, we have elected to exclude AE from the
model unless later analysis of scintillation data shows that including
it would improve the model's accuracy significantly. In a similar vein,
we i.ave avoided including a geomagnetic dependence in the equatorial
term, retaining instead the original sunspot-number dependence. Further
experimental work is necessary before a reliable geomagnetic dependence
can be incorporated in the equatorial term; a likely candidate for a
magnetic index is D , the seasonal and solar-cycle behaviors of which st
are similar to those of scintillation.
In addition to the above modifications to the 1973 model, some of the
constants employed have been changed. These changes stem partly from
improved calibration of some data sets used in development of the original
model. The data sets in question (Aarons et al., 1963; Koster, 1968;
Joint Satellite Studies Group, 1968) were presented in terms of a hand-
scaled scintillation index, SI (Whitney and Malik, 1968), which only
recently (Whitney, 1974) has been calibrated reasonably accurately to S^.
It is still to be noted that the relation between SI and S^ must be
scintillation-rate dependent, and data characterized directly by S^
are strongly preferred for modeling.
26
■- " - ' -—■■ - - ■ -"■- ■■-- .--.■ ^■.—... lataWMiä
r mmmmmmmmmmmmmmm IIWJI««"
"
t
— o E o) o •-
5} < c
ffi
ii
_ 8
x
z
I —1 o re
-J
8 i z
8 i
8 S
8
—^ 8 CM
• . •
L^ L iliU Ol+o>| + »l+^|+^o|■,■'"l+*, IN I ♦ •■ I + O
cr a oc ^ O - O X
LLI
oi Hi Lil U ■E P a u S z D u
if ll
ai l! < - to <
D a
27
■ ■ -^ nmiM _^
WM«l IWW^^^W^^^t wmmi'uim m j IIIWIII|«II - IIWWW«W«W^WP^™
Figures 5 through 8 show the degree of fit between the model
now in RFMOD and several data sets that were presented by the observers
either in terms of S, or in terms of an empirical index such as SI, 4
which we then converted to P,. Figures 9 and 10 compare the amplitude 4
pdf and cdf calculated by means of DIST with corresponding histograms
developed from AT? 5 data provided by Aarons (private communication,
1974). The corresponding pdf for phase is shown in Figure 11; no phase
data exist at this time for comparisoi .
IV THE CHANNEL-MODEL CODES
Of the two computer programs, RFMOD and DIST, the former is by
far the more complicated. It is a modification of the program BPMOD,
described by de la Beaujardiere and McNiel (1971), and consists of the
routines listed in Table 1. Its very simplified flow chart is shown in
Figure 12. Using RFMOD consists essentially of preparing input cards,
of which there are three types: (1) a title card, (2) namelist cards,
and (3) a changing-parameter(s) card. Three sample sets of input cards
are shown in Figure 13, and each type of card is described below.
The title card is used to assign any desired title of up to 80
characters to a calculation case or series of cases. It is read by
the driver program, RFMOD, into the array LBL with a 20A4 format, and
printed in the output by the subroutine IMPRIME; it ha^ no effect what-
soever on the rest of the program. Any number of cases can be run
under a common title, which will be printed in the output for each case
The namelist cards are used to input the fixed parameters for a
calculation case. They are read by subroutine READIN, by means of fie
namelist INPAR. The user gives the name and desired value for each
quantity listed above the dashed line in Table 2 that is to be fixed.
The number under the heading K in Table 2 is an index used in the
28
■^^"■■^^^■~"" ' " ' PWll i. •»■.in- IIIII»III -»vn^qpB^^B
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29
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30
Z) o -J > z < < - O OC z
UJ ^ |_
9 _i < 5 Z d a r ^- o2? ^ Q O
O T J - t H <5i ft -^ ^ ^ a- LU
O o —
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< CO
O OC a
O.S 1.0 I.S 2.0 2.6 3.0 3.5 *MPUTUOCA)NDQTU«SED «MPUTUOE
«.0 4.9 S.O
FIGURE 9 COMPARISON OF CALCULATED (smooth curve) PROBABILITY DENSITY FUNCTION FOR AMPLITUDE WITH OBSERVATIONS (histogram) OF ATS-5 TRANSMISSION AT 137 MHz, MADE BY AARONS (private communication, 1974) THROUGH THE MID-LATITUDE IONOSPHERE FROM NEAR BOSTON
^ > >
*
-.-i"»
i
0.001 ■30 26 •20 -16 -10 -5 0
SIGNAL STMEN0TH--0B
FIGURE 10 COMPARISON OF CALCULATED (smooth curve) CUMULATIVE PROBABILITY FUNCTION (cpf) FOR INTENSITY (or for amplitude) WITH OBSERVATIONS (histogram) FOR CASE IN FIGURE 9. (Calculations of cpf are intended for use in assessing fade-margin requirements.)
31
— - -- ■■■----- ...^--.-J- ^ —.. -^ ■^-.. — —
r — ■■■ ■'
——— wmm. — -^-^ -^»
z LU □
< CQ
o Q.
0.020
0.0«
0.0)6
O.OK
0.012
0.010
0.008 -
o.ooe
0.004
0.002
-I—■"
-200
-i—'—i—•—r
S- - 0.282 4
-180 -100 -SO 0 BO PHASE «NOU-DCOnCO
_. 100 IBO 200
FIGURE 11 CALCULATED PROBABILITY DENSITY FUNCTION FOR PHASE FOR THE CASE IN FIGURES 9 AND 10. (No observed phase üata exist for comparison.)
32
■t. -—- - ■"■—"- ■ -— I llll J
■ ■ ' 1 ■■ I 1 «"I"! IWP-^—
Table L
ROUTINES IN PROGRAM RFMOD
Name
BLOCK DATA
SUBROUTINE READIN,
with ENTRY INC
SUBROUTINE SCINT
SUBROUTINE SIGS
FUNCTION RMSDN
FUNCTION RFBPT
SUBROUTINE FINDZ
SUBROUTINE MGFLD
SUBROUTINE AZNSIDE
SUBROUTINE COORD
FUNCTION ERF
SUBROUTINE CPRMS
FUNCTION PHAS,
with ENTRY PRT
FUNCTION FL, with
ENTRY F2
SUBROUTINE SIMPA
SUBROUTINE IMPRIME
Function
Initializes common parameters.
Performs program inputting, and increments the
changing parameter(s).
Controls the basic scintillation calculation and
provides quantities needed for SUBROUTINE SIGS.
Performs the bjisic scintillation calculations.
Contains the worldwide model for strength of
F-layer irregularities used in the scintillation
calculation.
Evaluations the diffraction integral by an
algebraic approximation.
Computes the reduced distance from scattering
layer to receiver for use in evaluating Fresnel-
zone radius.
Contains an earth-centered, axially tapped,
dipole model of the geomagnetic field and cal-
culates the geomagnetic coordinates of, and the
field components at, a specified point.
Given the latitude and longitude of two points,
A and B, computes the azimuth of B as seen from
A and the great-circle arc AB.
Calcul?tes coordinates of a desired point on a
specified great-circle arc.
Computes the Error Function.
Controls calculation of rms phase.
Evaluates the Hatfield phase distribution of
a scintillating signal.
Provides ir.tegrand functions for evaluating
moments of phase distribution.
Adaptively performs numerical integration
using Simpson's rule.
Performs program outputting.
3 3
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READ THE FIXED PARAMETERS.
READ THE NAME OF THE CHANGING PARAMETER AND ITS START, END. AND INCREMENT VALUES.
CHANGING PARAMETER = INITIAL VALUE - INCREMEN
I INCREMENT THE CHANGING PARAMETER.
I n COMMUTE INTENSITY AND PHASE
SCINTILLATION INDICES AND PROBABILITY-ELLIPSE PARAMETERS
I STORE ALL PERTINENT VALUES.
DISPLAY THE VARIOUS PARAMETERS FOR THIS CASE, AND THE VALUES OF THE SCINTILLATION INDICES
AND THE PROBABILITY-ELLIPSE PARAMETERS
J LA-1079-28R
FIGURE 12 SIMPLIFIED FLOW CHART OF PROGRAM P.FMOD
34
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35
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Table 2
DATA-CARD VARIABLES
K Variable Name
1 FREO
2 SSN
J DAY
4 TIME
s RLAT
6 RLON
7 HK
8 TLAT
9 TLON
LO HT
11 FKP
II
Unit
Megahertz
None (floating point)
None (floating point)
Decimal hours
Decimal degrees
Decimal degrees
Meters
Decimal degrees
Decimal degrees
Meters
Decimal value 0.0 through 9.0
None (value 1 or 0)
Definition
Frequency
Sunspot number
Day of year
Meridian time at iono-
spheric penetration point
Receiver latitude
Receiver longitude
Receiver altitude
Transmitter latitude
Transmitter longitude
Transmitter altitude
Planetary magnetic
index, Kp
Data flag (see text)
12
13
RCRD
TCRD
Decimal degrees
Decimal degrees
Receiver coordinates
TranFmitucr coordinates
common VARPAR to identify the parameters. The user may list the fixed
parameters in any order on the namelist cards.
A parameter input on a namelist card will keep the specified value
until changed by means of a new namelist card or until input on a changing-
parameter(s) card. The first colmtt) of each namelist card must be blank,
and the first characters on the first namelist card must be &1NPAR. The
last namelist card in a sequence must terminate with the characters &END.
}6
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The changing-parameter(s) card is employed by the user to select the
independent variable(s) for a given calculation case. He may choose
either of two program modes. In Mode 1, he chooses any one of the param-
eters listed in Table 2 for which K is between 1 and 11, inclusive, and
inputs its name on the changing-parameter card. In this mode, the first-
order signal-statistical parameters, including S^ and ^rmg. are calculated
as functions of the single changing parameter. In Figure 13, the cards
identified as Set 1 are for two sequential cases run in Mode 1, under a
common title.
Mode 2 permits the user to increment the position of either the re-
ceiver or the transmitter along an arc of a great circle. In this mode,
both the latitude and longitude of the receiver or of the nransmitter be-
come changing parameters, and the signal-statistical paramete-s are cal-
culated for equidistant points along the arc. Mode 2 is selected by naming
RCRD or TCRD on the changing-parameters card (see Table 2). In Figure 13,
the cards identified as Set 2 are for a run in this mode.
The required contents of the changing-parameter(s) card are listed
in Table 3. Note that in Mode 1, if the final value of a changing param-
eter is smaller than the initial value, the increnent value must be speci-
fied as negative. Absence of the required negative sign in this situation
will cause the run to terminate, and an error message will be printed.
After reading a set of cards, the program will keep the same title
and read only namelist and c nging-parameter(s) cards, so long as II = 1.
After encountering II = 0, it will read a new title card. The program
will terminate if the name on the changing-parameter(s) card does not
correspond to one of those numbered 1 through 13 in Table 2. Thus, to
terminate a run, the user may prepare a changing-parameter card contain-
ing an unfamiliar word, such as the FINI appearing in Figure 13.
37
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Table 3
CONTENTS OF CHANGiNG-PARAMETER(S) CARD
Columns
I through 4
9 through 16
17 through 24
25 through 32
33 through 40
41 through 42
Mode-1 Content
Name
Initial value
Final value
Increment value
Mode-2 Content
Name
Initial latitude
Initial longitude
Final latitude
Final longitude
Number of increments
Symbol
in Program
Name
Al
\2
A3
A4
NA5
For model development and testing, we employed a plotting routine
in conjunction with RFMOD to produce graphs such as those in Figures 5
through 8. The available version of RFMOD provides an array that the user
may employ with his own computer peripherals. In the absence of user
modification, the array will be output in tabular form along with identify-
ing information, as shown for a test case in Figure 14. In the case shown,
the changing parameter was the latitude of the transmitter, which was
taken to be in a nolar-orbiting satellite passing over a receiving station
near Boston (namely, the Sagamore Hill site of the AFCRL group headed by
Aarons). The changing parameter is followed in each row by the azimuth
and elevation of the satellite, the three probability-ellipse parameters,
and then by the calculated scintillation indices. The input cards for
this run are identified as Set 3 in Figure 13.
It is envisioned that a user will inspect the values of S4 and/or
PRMS predicted for his communication link by RFMOD and then decide if he
wants more detailed" signal-statistical information for a particular
situation. In the example of Figure 14, for instance, he might decide
Hby i
38
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that he wants the full first-order distributions of intensity and phase
for the case in which S4 on his channel is .538 and PRMS is 32.835 degrees
To obtain the distributions he would employ DIST, inputting RZRO = .371,
BMAGR = .924, and DELTA = 3.419 degrees, on a single card in F10.3 format.
The DIST code consists of the routines listed in Table 4.
Table 4
Name
SUBROUTINE PDFI
SUBROUTINE PARRAY
SUBROUTINE GAUP
FUNCTION PR
FUNCTION XINTGR
SUBROUTINE PDFP
SUBROUTINE PHAS
FUNCTION ERF
SUBROUTINE CPFI
ROUTINES IN PROGRAM DIST
Funct ion
Controls calculation and output of probability density function for amplitude, by means of one- dimensional integration over two-dimensional Gaussian probability density function.
Initializes amplitude array for SUBROUTINE PDFI.
Controls calculation of Gaussian probability density function.
Computes two-dimensional Gauspian probability density function.
Performs integration by Simpson's rule.
Controls computation and output of probability density function for phase, calling SUBROUTINE
PHA3.
Evaluates Hatfield phase distribution.
Computes the Error Function.
Calculates and outputs the cumulative probability density function for amplitude or intensity (expressed in dB) by performing Simpson's rule integration over probability density function for amplitude.
40
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Again, for modeling we employ a plotting routine in conjunction with
DIST to produce graphs such as those in 9, 10, and 11. The available
version of DIST simply outputs the array from which such graphs are made,
in the line-printer formats shown in Figures 15, 16, and 17. As with
RFMOD, a user may wish to employ DIST in conjunction with his own plotting
routine or in some other output form. Alternatively, he may wish to
reformat the array listing, perhaps outputting fewer values and possibly
presenting CPF in terms of percent or decimal link reliability (the one's
complement of CPF) rather than directly as in Figure 16.
V CONCLUSION
It has been our attempt to develop a channel model that may be used
conveniently for calculating first-order signal statistics, expressed in
a format relevant to systems application. The resulting codes, RFMOD
and DIST, represent the first-year output of a broader effort to
definitively model the transionospheric communication channel. These
codes are to be augmented by routines to permit calculation of second-
order statistics, such as fluctuation spectra of amplitude and phase, as
well as signal correlation at spaced receivers and across bands of
frequency. Moreover, the existing version of RFMOD is likely to evolve
over the next year in two important ways.
First, it is intended to modify SUBROUTINE SIGS in RFMOD to fully
account for multiple scatter. Theoretical work now underway is hinting
at a possible form for the modification. The single-scatter theory
described in Section II of this report describes the simultaneous rota-
tion (increasing 6) and circularization (decreasing |B0|/R0) of the
first-order probability ellipse for the received signal's complex ampli-
tude (illustrated in Figure 1), as the wave propagates away from the
scattered medium. In the single-scatter limit, the ellipse emerges from
41
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the scattering layer highly elongated (|B0|/R0 W 1) and oriented nearly
vertically (6 « 0) on the complex plane.
Multiple-scatter calculations for the special case of normal inci-
dence on the scattering layer show that some circularization (|B0j/R0
M e"^), but no rotation (6 M 0), is produced Dy the reseat.erings. This
is an intuitively understandable result and, if corroborated by similar
calculations for off-normal incidence, is expected to lead to a multiple-
scatter correction for |ß0|/R0 analogous to Eq. (28) for R0. It will
presumably also lead tc a modification of Eq. (24) for relating the
ellipse orientation to its elongation at the receiver. The result is
certain to be a reduction in the third term of Eq. (5) for intensity-
scintillation index as the fraction of flux scattered (Ro) increases.
It will presumably limit values of S4 to something near unity and improve
agreement between calculated and observed values for moderate to strong
scintillation.
The other important area in which RFMOD is expected to evolve is
that of scintillation morphology, which will entail updating of FUNCTION
RMSDN. As more definitive scintillation data become available, the model
of electron-density irregularity strength contained in RMSDN presumably
will be improved by a process that may be viewed as calibration [i.e.,
the main changes should be adjustment of the coefficients in Eqs. (42)
through (45)]. This will be particularly so in regions where the model
is currently calibrated in terms of ad hoc indices such as SI instead of
in terms of S,. One such region is the important one centered on the 4
geomagnetic equator. Useful equatorial data are expected from observa-
tions of the NOAA ionospheric beacon on board NASA's ATS-6, particularly
measurements of S, at 140 and 360 MHz. During periods of weak to 4
moderate scintillation, estimates should also be obtained of 0^
at 140 MHz. The ATS-6 observations also will provide histograms
4.
-- - ^- -^r^.L^^L-.^.. . ... -..- . ........ ....■■
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of amplitude (under all conditions)) and of phase (under conditions of
weak to moderate scintillation) for testing the predictions made by the
DIST code.
Pending the improvements described above, we offer RFMOD and DIST
as the best way to estimate the first-order signal statistics to be
expected at the output of a user's transionospheric communication link,
short of performing a series of observations over that particular link.
It must be remembered that the estimated values are those expected for
average ionospheric conditions in the observing situation (frequency,
transmitter and receiver location, time of day, K . etc.) specified by
the user. There is presumably some statistical distribution of each
parameter estimated about its calculated value, and the present model
does not account for such statistical variation. Possibly of equal
importance, the model has yet to be subjected to testing against observa-
tions above 400 MHz (e.g., in the SHF band).
4b
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REFERENCES
Aarons, J., J. P. Mullen, and S, Basu, "Geomagnetic Control of Satellite
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