Radio Science, Volume 8, Number 3, pages 213-222, March 1973
An empirical model for average F-layer scintillation at VHF/UHF
E. I. Fremouw and C. L. Rino
Radio Physics Laboratory, $tan]ord Research Institute, Menlo Park, CaliIornia 94025
(Received May 22, 1972; revised September 18, 1972.)
An empirical approach to modeling the electron-density irregularities in the F layer that
are primarily responsible for amplitude scintillation of VHF/UHF signals has been devised
and tested. An irregularity model was postulated as a function of geomagnetic latitude, local
time of day, season, and sunspot number. The primary parameters of the irregularities that
were postulated were their strength and transverse scale-size. The irregularities were assumed
to be aligned along the geomagnetic field, and their axial ratio was taken as constant, as were
the height and thickness of the irregular layer.
The model was tested by computing the fractional rms fluctuation in received power to be
expected in a given situation, under the weak-scatter assumption, and comparing the. results
against values of this or related quantities reported in the literature. The model then was im-
proved by iteration. The development made use of 12 data sets, and final testing employed
those 12 plus an independent one. Lack of appropriate data precluded testing poleward of
about 70 ø geomagnetic latitude.
The model is offered as a tool for VHF/UHF communication-systems planning, to the
extent that the average value of scintillation in a specified circumstance is of engineering value.
Geophysical application should be limited to such uses as experiment planning, guiding of
intuition, and serving as a basis for more refined modeling.
INTRODUCTION
In an earlier paper [Fremouw and Bates, 1971]
an analytical framework was suggested for sum-
marizing the large amount of data available on radio
scintillation of ionospheric origin. The objectives for
such a summary were to provide a means for pre-
dicting the magnitude of signal fluctuations to be
expected on an arbitrary satellite-to-ground com-
munication path and, hopefully, to contribute some
insight into the production of electron-density ir-
regularities in the F layer.
The procedure envisioned was to model the
scintillation-producing irregularities and to account
for geometrical factors by diffraction-theory calcula-
tions. A tentative model for the rms spatial fluctua-
tion in F-layer electron density, which seemed con-
sistent with salient features of worldwide scintillation
behavior, was postulated as a starting point. Since
most of the data available are for amplitude scintilla-
tions, such a model is inherently limited to irregulari-
ties having a scale small compared with the Fresnel
zone of the observing wavelength at the distance of
the ionosphere. This observational bias has been de-
scribed, for instance, by Ru•enach [ 1971].
Copyright ¸ 1973 by the American Geophysical Union.
213
A first attempt at modeling has now been com-
pleted and is the subject of this paper. The method
is outlined in the next section, and the resulting model
is presented in the third section, along with com-
parisons of results with various observations. The
final section contains an evaluation of the model's
reliability for obtaining scintillation estimates, a dis-
cussion of its limitations, and an assessment of scin-
tillation data.
The model is suitable for estimating the rms
fluctuation in received signal strength (i.e., the
scintillation index) to be expected on a given trans-
ionospheric VHF/UHF (but not SHF) communica-
tion link, under average scintillation conditions. By
average scintillation conditions is meant those to be
expected, on .the average, for a given geomagnetic
latitude, time of day, day of the year, and sunspot
number. Thus the model does not address the ques-
tion of variations in scintillation index from its mean
value for a given set of the above independent vari-
ables.
Such variations are to be expected, for instance,
with changes in geomagnetic activity. At subauroral
and auroral latitudes, scintillation increases during
geomagnetic storms [Little et al., 1962; Aarons et al.,
1964; Aaron& 1970], while near the geomagnetic
214 FREMOUW AND RINO
equator there is .a negative correlation between the
two phenomena at ,solar minimum [Koster and
Wright, 1960] and a slightly positive (or perhaps
zero) correlation near solar maximum [Bandyopad-
hyay and Aarona, 1970].
Another phenomenon with a definite but com-
plicated relationship to scintillation is ionospheric
spread F [Briggs, 1964; Singleton, 1969]. From a
geophysical point of view, comparison between the
scintillation model reported in this paper and a re-
cently completed survey of spread F [Davis, 1972]
may be instructive, but such a comparison has not
yet been performed.
The relation of scintillation to other geophysical
phenomena may be of engineering, as well as sci-
entific, interest. The thrust of this work, however, has
been .to develop a model of mean scintillation trends
as functions of readily accessible parameters such as
latitude and time. In this context, sunspot number is
treated as a measure of epoch for describing long-
term trends in scintillation, a measure which is a
physical variable, to be sure, but one which is rou-
tinely predicted a year in advance.
Clearly, the engineer has more detailed questions
to ask the ionospheric physicist than the model re-
ported here will answer, questions such as the per-
centage of time that a signal may be expected to
fade below a given level. For such questions, the
relation of scintillation index to other geophysical
observables and the statistics of those variables may
be very important. For the specific question above,
a more fundamental need is for the underlying first-
order distribution of the amplitude of a scintillating
signal for a given ionospheric state. A theory relating
this distribution to ionospheric scattering param-
eters and showing that it is not necessarily unique
for a given value of scintillation index (second mo-
ment) will be described in a subsequent paper.
ASSUMPTIONS AND PROCEDURE
The basis for modeling was the theory of diffrac-
tion by a weakly modulating phase screen developed
by Briggs and Parkin [1963]. Accordingly, the fol-
lowing assumptions are inherent in the work: weak,
narrow-angle scatter; a layer that is thick compared
with an irregularity but thin compared with the free-
space propagation distance; and a Gaussian spatial
autocorrelation function.
The above assumptions, for the most part, are ac-
ceptable for a working model of the normal F layer
at VHF/UHF, although two of them have practical
implications for the modeling. The weak scatter as-
sumption represented the most serious limitation of
the theory for our purpose. Checks on the assump-
tion were carried out by calculation, and modeling
was terminated when the necessary condition was
violated. This happened rather often at the common
observing frequencies of 40 and 54 MHz, except in
the midlatitude region.
Arbitrary assumption of a form for the autocor-
relation function limits the frequency range over
which the model will give reliable results; the great-
est accuracy is achieved near .the observing fre-
quencies used in modeling. Since most data available
are from VHF observations, the greatest reliability
may be expected there; it probably extends into the
low UHF spectrum. As will be described in the
next section, the model's reliability does not extend
to the SHF spectrum, at least near the geomagnetic
equator.
The basic calculation in the modeling employed
the following expression, Briggs and Parkin's [1963]
equation 20, for the fractional rms fluctuation in sig-
nal intensity (square of real amplitude) at the ground
as a function of ionospheric and geometrical param-
eters (illustrated in their Figures 1 and 2):
$4 = 2•/2q0o[1 - (cos u• cos u2) •/•' cos (u• -3- u•)/2] •/•
(1)
All ionospheric parameters appear in the factor
00, given by Briggs and Parkin in their equation 13 as
•/4 i)•/•/•/•
qbo = r r,k[(a•o sec ](Ah)•/2(AN) (2)
which is the rms fluctuation in radio-frequency phase
across a plane at the output boundary of the scatter-
ing layer. The primary ionospheric parameters are
the rms fluctuation aN in electron density, the thick-
ness ah of the irregular layer, the transverse irregu-
larity scale-size •0 to the e -x point, and the irregularity
axial ratio a. In addition, 00 depends on the incidence
angle i of the radio wave on the irregular layer and
on the irregularity projection factor •, where • -
sinø- • + cos" ½)x/" and ½ is the angle between the
geomagnetic field and the radio line of sight. The
radio wavelength is given by x, and re is the classical
electron radius.
The Fresnel-distance parameters u• and u2 in equa-
tion 1 are defined as
u• = tan -• (2Xz/•r•o")
u,. = tan -• (2Xz/r/5?•/o ')
(3)
(4)
F-LAYER SCINTILLATION MODEL 215
where
z = ZlZ•/(z• + z•) (5)
where Zl is the distance from the receiver to the
center of the scattering region and z2 is that from the
region center to the transmitter. The geometry is fur-
ther specified in Briggs and Parkin's equations 1, 2,
and 3.
Equations 1 through 5 were coded, along with a
number of auxiliary expressions, to permit calcula-
tion of the scintillation index S4 as a function of the
F-layer model being developed and of various satel-
lite and radio-star observing conditions. The main
modeling endeavor was to provide proper parameter
values for use in calculating the rms phase fluctua-
tion, 00. By far the greatest effort was put into select-
ing the appropriate behavior of rms electron-density
fluctuation, zxN.
Before describing the AN modeling, we shall dis-
cuss selection of the other geophysical quantities in-
volved in the calculations. The simplest to handle was
the layer thickness Ah which was easily treated as a
constant. Doing so means that model testing was
actually of the product AN(Ah)•/•; separating the
effects of the two variables would be impossible,
given the published scintillation data. Nonetheless, in
order to model AN as accurately as possible, a value
was taken for Ah from measurements reported in the
literature, namely, 100 km [Liszka, 1964; ¾eh and
Swenson, 1964; Kent and Koster, 1966]. While it is
possible that, from time to time, the center height h
varies through much of the F layer, it too was taken
as constant. Observations published in the above lit-
erature suggest an average value of 350 km without
systematic trends, and this value was used.
For the axial ratio a the constant value 10 was
used, based on observations performed under a
variety of conditions [Jones, 1960; Liszka, 1963;
Koster, 1963]. More recent observations of Kent and
Koster [1966] and especially of Koster et al. [1966]
show that the irregularities can be much more
elongated in the equatorial region. In this region th•
field-aligned irregularities are nearly horizontal, how-
ever; thus, they are usually viewed from a quasi-
transverse aspect, and the value of a then has little
effect on the scintillation index.
The remaining irregularity parameter to be con-
sidered is the transverse scale size •.0. At the outset of
the work it was planned to treat it as a constant also.
During the course of the modeling, this idealization
was found unacceptable for treating scintillation fre-
quency dependence. Therefore, a rudimentary model
for •o as a latitudinal variable was introduced into the
work in addition to the more complete one, involving
latitude, time of day, season, and sunspot number,
for AN.
The essence of the procedure was to postulate
models for AN and $0, to insert the model values in
equation 2 along with the other parameters needed,
and then to employ equations 2 and 1 to calculate
the value of $4 expected for a given set of pub-
lished observations. In this manner, the model was
tested and improved, using 12 data .sets from a
variety of observational circumstances. A thirteenth
set, not used in model development, was included
in final testing.
The procedure was designed to account for dis-
similar experimental circumstances and data-reduc-
tion procedures. For each data set, the transmitter
and receiver locations used in calculation were chosen
to be representative of the actual ones, and the mag-
netic-field geometry was accounted for on the basis
of an earth-centered, but axially tipped, dipole model.
After the scintillation index was calculated, averages
were performed in a manner similar to those per-
formed by the observer. The final result then was
compared with the reduced data presented in the
literature.
The index first calculated was $4. The program also
converted to S1, S2, or Sa, on demand. The conver-
sions made use of the simple proportionality between
the four indices suggested by Briggs and Parkin
[1963] on the basis of the Rayleigh distribution and
verified by Bischo# and Chytil [1969] for conditions
under which the Nakagami approximate distribution
may be employed. We note that the latter conditions
have not been established clearly and suggest this as
a fruitful topic for theoretical investigation.
The papers used gave scintillation magnitude either
as one of the above four indices or as some other
index calibrated in terms of one of the above. In the
latter case, the quoted index was converted to one of
the above for comparison with the calculations.
The initial model postulated contained the follow-
ing parameters' Ah -- 100 km, h - 350 km, a - 10,
•.0 = 1 km, and
AN ---- A N•(R, D, t, X) -[-- A N,•(t, X) -[-- ANn(R, t, X)
(6)
where the independent variables are the following:
mean sunspot number R, day of the year D, time of
day t, and geomagnetic latitude X. The three terms
216 FREMOUW AND RINO
specifying AN were, respectively, equatorial, midlati-
tude, and high-latitude contributions to the rms
fluctuation of electron density, as described mathe-
matically in the third section of the paper by
Fremouw and Bates [1971].
In the initial model, equation 6 was defined
quantitatively by 14 numerical constants, to be
evaluated by comparison of model-based calculations
of scintillation index against observed values. For
the most part, the changes in the initial model that
came about through iterative testing were in the
nature of evaluating the constants. Some changes in
form were made, however, most notably the addition
of a fourth term to account for autorally associated
scintillation. The result is presented in the next sec-
tion; the reader concerned with calculational details
may find a complete description of the model's
evolution in report form [Fremouw and Rino, 1971].
THE RESULTING MODEL AND ITS
LIMITATIONS
As a result of the procedure described in the pre-
ceding section, the following empirical model for
scintillation-producing irregularities in the F layer is
put forth: center height of the irregular layer = 350
km, thickness of the irregular layer -- 100 km, ratio
of the scale size along the geomagnetic field to that
transverse = 10, transverse scale size (to e -x spatial
autocorrelation) = •o, and rms fluctuation of elec-
tron density - AN. Mathematical expressions are
given below for •o and AN in equations 7 and 8, re-
spectively, in terms of the following independent
variables: h = geomagnetic latitude in degrees (•0 is
treated as a function of )t only), t = local time of
day in hours, D - day of year out of 365, R -- sun-
spot number.
The model for •o is as follows:
•0 = 300 q- 600{1 q- erf [CA - 12)/31}
- 450{ 1 q- erf [(X- 62)/3]}
200{ 1 q- erf [CA- 69)/31} rn (7)
It consists essentially of steps at particular geomag-
netic latitudes; in order to avoid discontinuities, steps
are described by error functions, the widths of which
are about 6 ø . This model is very rudimentary as
compared with that for AN, but it is a considerable
improvement over assuming a constant value for
scale-size, especially as regards the frequency de-
pendence of scintillation.
The model for AN consists of four additive terms,
the influence of e. ach being dominant in different
regimes of geomagnetic latitude, as follows:
AN = A N•q(R, D, t, X) q- AN•ia(t, X)
-[' • Nhi (R, t, X) q- A Naur(R, t, X) (8)
where
AN•q = (5.5 X 10ø)(1 -}- O.05R)
[1-- 0.4 •r(/• q- 10•1
ß cos i 72• '/3
ß {exp [--(5)21} el/m a (9)
(
AN,•ia = (6.0 X 108 ) 1 q- 0.4 cos
10
{ - -t).l}
•v• = (2.? x •o ø) 1 + err .•x• t) J
•N• = (5.0 X 10*)R
0.03R
where
3,b = 79 -- 0.13R -- (5 q- 0.04R)
ß cos Oft/12) degrees (13)
Equation 9 describes the well-known peaking
of equatorial scintillation in the midnight hours and
the decay of activity through the early morning
hours, a simple harmonic seasonal dependence with
peaks at the equinoxes, a linear dependence on sun-
spot number, and a Gaussian latitudinal dependence
which drops to e -x 12 ø on either side of the geomag-
netic equator. Equation 10 describes the simple
diurnal and latitudinal variations of scintillation at
middle latitudes that were suggested by Fremouw and
Bates [1971].
The behavior of high-latitude scintillation other
than that directly associated with auroral disturbance
is described in equation 11. This behavior is attrib-
uted to diurnal and solar-cycle mi•ations of the
scintillation boundary, as described in equation 13.
The basis for the error-function form of equation 11
was developed in the appendix of Fremouw and
Bates' [1971] paper. Equation 12 describes what is
believed to be aurorally associated scintillation arising
F-LAYER SCINTILLATION MODEL 217
in a region, near the auroral oval, of which the lati-
tudinal extent is proportional to sunspot number, as
is the strength of the irregularities it contains.
Comparisons of the scintillation index calculated
from the above model with the observations used in
iterative evaluation of the model are shown in Fig-
ures 1 through 6. This is followed, in Figure 7, by
comparison of calculated values with a set of ob-
served values not employed in the development of
the model. The calculated curves are solid where the
assumption of weak scatter is satisfied (00 < 0.7),
and dashed where the assumption is questionable
(0.7 < 00 _< 1.0). Where the assumption is invalid
(00 > 1.0), no calculated value is given. These
somewhat arbitrary numerical choices were made by
inspection of Briggs and Parkin's [1963] Figure 3.
Comparison of results with the observations of
Koster [1968] appear in Figure 1; the fits are rea-
sonably close where the weak-scatter assumption
holds. The rise of the observed values in the evening
hours which is more abrupt than those calculated
could be accounted for by a change in form of the
equatorial term of the /xN model, and parameter
adjustments could reduce other discrepancies. This
hardly seems justified, however, in the light of two
more serious limitations of the model at equatorial
latitudes.
The first limitation stems from lack of an opportu-
nity to test the predicted sunspot-number dependence
of scintillation. There appear to be no long-term
equatorial data available in terms of quantitative
indices, although there remains the possibility of
calibrating some earlier observational results in such
terms (J. R. Koster, personal communication, 1971).
The equatorial term of the scintillation model may
be considered relatively reliable at VHF/UHF under
average ionospheric conditions for sunspot numbers
on the order of 100 (typical of solar maximum). For
other sunspot numbers, however, it is only an un-
tested estimate, and more experimental work is
needed.
The second limitation may be inherent in the
average nature of the model but is of some practical
concern and a good deal of scientific interest. In the
past few years, instances of significant scintillation on
surprisingly high frequencies (as high as 6 GHz)
have been reported by equatorial observers [Chris-
tiansen, 1971; Skinner et al., 1971; Crait and Wester-
lund, 1972]. The model developed in this work
would not have predicted this turn of events.
It may be that the observed SHF scintillations are
1.0
z 0.5
!
• X X X•X
• X
t t X
_ X
X
/
5 I0 15 20
T•ME (hr)
x
25
1.0
Fig. 1.
I i I , I , I , I , I , 1
50 I00 150 200 250 300 350
DAY
Comparison of model calculations with geosta-
tionary-satellite observations from Ghana [Koster, 1968].
The top is diurnal variation' frequency -- 136 MHz, sun-
spot number _-- 107, and day number -- 31. The bottom
is seasonal variation: frequency -- 136 MHz, sunspot num-
ber -- 97, and time is 0200. As in all figures, the observa-
tions are shown as discrete points, and the calculations as
a curve. The curve is solid where the weak-scatter assump-
tion is valid and dashed where it is questionable. Where
it is invalid, no calculated results are given.
not a manifestation of average ionospheric conditions,
as the term is meant herein. On the other hand, this
inadequacy of the model may stem from the assump-
tion of an unproven spatial autocorrelation function
(Gaussian) in the diffraction calculations, as de-
scribed in the preceding section. For reliable ex-
trapolation over a wide frequency range, it is neces-
sary to have a realistic description of the spatial
spectrum of the F-layer structure. Work has begun
toward this end for middle latitudes [Ruienach,
1971], but the need is more pressing at equatorial,
and probably auroral, latitudes from the modeling
viewpoint.
At middle latitudes the model produced quite ac-