131_143 adg v5 n01 Mizrahi.pdf


ANNALS OF GEOPHYSICS, VOL. 45, N. 1, February 2002

131

Statistical properties of the deviations
of f0 F2 from monthly medians

Eti Mizrahi (1), Ayşe H. Bilge (1) and Yurdanur Tulunay (2)
(1)  Department of Mathematics, Istanbul Technical University, Maslak, Istanbul, Turkey

(2)  Faculty of Aeronautics and Astronautics, Istanbul Technical University, Maslak, Istanbul, Turkey

Abstract
The deviations of hourly ƒ0F2 from monthly medians for 20 stations in Europe during the period 1958-1998 are
studied. Spectral analysis is used to show that, both for original data (for each hour) and for the deviations from
monthly medians, the deterministic components are the harmonics of 11 years (solar cycle), 1 year and its
harmonics, 27 days and 12 h 50.49 m (2nd harmonic of lunar rotation period L

2
) periodicities. Using histograms

for one year samples, it is shown that the deviations from monthly medians are nearly zero mean (mean < 0.5)
and approximately Gaussian (relative difference range between %10 to %20) and their standard deviations are
larger for daylight hours (in the range 5-7). It is shown that the amplitude distribution of the positive and
negative deviations is nearly symmetrical at night hours, but asymmetrical for day hours. The positive and
negative deviations are then studied separately and it is observed that the positive deviations are nearly independent
of R12 except for high latitudes, but negative deviations are modulated by R12. The 90% confidence interval for
negative deviations for each station and each hour is computed as a linear model in terms of R

12
. After correction

for local time, it is shown that for all hours the confidence intervals increase with latitude but decrease above
60°N. Long-term trend analysis showed that there is an increase in the amplitude of positive deviations from
monthly means irrespective of the solar conditions. Using spectral analysis it is also shown that the seasonal
dependency of negative deviations is more accentuated than the seasonal dependency of positive deviations
especially at low latitudes. In certain stations, it is also observed that the 4th harmonic of 1 year corresponding
to a periodicity of 3 months, which is missing in ƒ0F2 data, appears in the spectra of negative variations.

1. Introduction

The ultimate goal of the work related to the
ionospheric critical frequency ƒ

0
F

2
is prediction

and forecast. However determination of
reliability bounds is also important for planning
in HF communication, radar and navigation

systems. The median values for each hour can
be considered as a first approximation to the
data. The prediction of the median values, based
on standard trigonometric expansions and
parabolic models in terms of the 12 month
smoothed sunspot number R12, proves to be
satisfactory, with the mean square errors in the
range of 3-7% (Baykal, 1998; Bilge and Tulunay,
1998). The estimation of the actual hourly values
of ƒ0F2 is called «forecasting», and it deals with
neural network (Tulunay et al., 2000) and
feedback methods (Bilge and Tulunay, 2000),
in addition to more standard autocorrelation
techniques (Stanislawska et al., 1999). In this
paper, we concentrate on the deviations of ƒ0F2
from monthly medians, denoted by f = ƒ

0
F

2
 –

Mailing address: Dr. Eti Mizrahi, Istanbul Technical
University, Department of Mathematics, 80626 Maslak,
Istanbul, Turkey; e-mail: mizrahi1@itu.edu.tr

Key  words ƒ0F2 – critical frequency – variability



139

Statistical properties of the deviations of f
0
F

2
from monthly medians

that in night hours the negative deviations are
below the positive deviations, while in day time
the roles are reversed. The local time dependency
can also be seen as a shift of the crossing points
as we move longitudinally. We have obtained
similar graphs for each latitude group in table I,

where the similarities of the shapes of the curves
inside each group justify the subdivisions with
respect to latitude.

3.4. Seasonal dependency

As we worked with 1-year samples in the
statistical analysis, we overlooked the seasonal
dependency of the deviations.  We can however
make certain qualitative remarks based on our
time domain plots and our frequency spectra.
The time domain plots of f given in fig. 7a,b
show that at high latitudes, for example
Uppsala, both positive and negative deviations
have seasonal dependency while at lower
latitudes, for example at Rome, only negative
deviations have seasonal dependency. The
amplitudes are higher during equinoxes. This
is consistent with the spectral analysis results
given in fig. 3b, where it can be seen that the
periodicity of 6 months has higher power for
negative deviations.

3.5. Long-term trend analysis

In previous sections, we considered ( f +/ )
as a function of R12. It is also of interest whether
there is a trend irrespective of the solar
conditions. For this purpose, we selected years
with R

12
= 100, R

12
= 50 and R

12
= 20 in rising

and falling phases, obtaining 6 data groups. For
example for R12 = 100 in the falling phase we
have the years (1960, 1970, 1982, 1991). In each
data group we take the difference of the data
for consecutive years that we denote as f +/ .
For example, if there is an increase in f +/ from
1982 to 1991, at a given hour and station, the
corresponding entry in f +/ will be positive.
The relative «strength» of the positive values in

f
+/ will be an indication of an increasing trend.

In order to quantify these trends, we separated
the positive and negative parts of f +/ and
computed their norms, denoted as n

p
and n

m
. If

the ratio n
p
/n

m
is larger than 1 there is a trend to

increase from one year to the next in the same
group, while if n

p
/n

m
is less than 1 there is a

trend to decrease. We found that for positive
deviations from monthly medians the ratio n

p
/n

m

Fig.  5. Comparison of the histogram plots of the
deviations from monthly medians for Rome at 24 UT
and 9 UT.

Fig.  6. Positive (solid line) and negative (dotted
line) deviations for Slough, Juliusruh, Kaliningrad,
Moscow (in the 50°N-57°N latitude band). Plots are
shifted each by 2 units.



140

Eti Mizrahi, Ayşe H. Bilge and Yurdanur Tulunay

is larger than one for 15 out of 17 times, while
for negative deviations from monthly medians,
it is larger than one only for 10 out of 17 times.
Thus we can conclude that for positive variations
there is a trend to increase with time irrespective
of the solar conditions. In the literature, linear
trends in ƒ

0
F

2
are tied to the lowering of the F

2

layer caused by the increasing greenhouse effect
(Bremer, 1992) and the increase in magnetic
storm activity (Clilverd et al., 1998; Danilov,
2001).

4.  Modelling positive and negative variations
     separately

For each station year and hour, we
computed the 90% confidence interval, i.e. the
value ( f +/ )c such that ( f

+/ ) is less than ( f +/ )c
with a probability of 0.9, for the given station,
hour and year. We have seen that these upper
deciles for positive deviations are virtually
independent of R12 and we decided to in-
vestigate the negative deviations. In general
( f )

c
for a given hour (UT) is expected to

depend on R12, the latitude and longitude of
the station and the month of the year. As we
worked with yearly samples of data, we lost
the information related, for example, to the
semi-annual and seasonal dependencies. On
the other hand, the time domain plots in fig. 6

suggest that the longitude dependency reflects
in a shift of the curves, hence using a local
time correction we may assume that ( f )c is
a function of the latitude, local time and R12.
We shall look for a linear model with respect
to R

12
.

As an example, in fig. 8 we show ( f )
c
for

Sverdlovsk for each hour as a function of R12 .
The data are quite scattered, but we can still
look for a linear trend using least squares
approximation: For each station and each hour
we obtained the slope and the intercept of an
approximating line, by minimizing the least
squares error between the data and the linear
model. For selected representative stations and
hours these errors range between 9.9% and
21.3%, which indicates the existence of some
linear trend. As a result, we obtain a model where
we express the 90% confidence interval in terms
of R

12
 as below.

( f )c (latitude, local time, R12) =

= A (latitude, local time) R
12

 +

+ B (latitude, local time).

Where the «coefficients» A and B depend on the
latitude and local time. In order to analyse the
dependency on the latitude, we evaluate ( f )c

(4.1)

Fig.  7a,b.  Yearly variation of the positive and negative deviations for Uppsala (high latitude) and Rome (low
latitude).

a b



141

Statistical properties of the deviations of f
0
F

2
from monthly medians

as given by eq. (4.1) for a representative value
of R

12
, say the median, corresponding to (R

12
)

median

= 62.63. This gives a representative value of ( f )
that we denote as ( f )m given below.

( f )
m
 (latitude, local time)  =

= A (latitude, local time)(R12) median +

+ B (latitude, local time).

Note that ( f )m is a function of two variables
only. For each station, we plot ( f )m with
respect to local time and present the results in
fig. 9a-d grouped with respect to their
latitudes. It can be seen that the graphs for
stations between 60°N-70°N are less coherent,
the maxima are located at 12-14 LT, and the
amplitudes are less than 14 MHz. For the
stations in the 50°N-57°N band, the variations

Fig.  8.  Upper deciles of f- for Sverdlovsk, with respect to R
12

 for all hours.

are extremely coherent, there is a distinct peak
at 12 LT, and the amplitudes reach 18 MHz.
Down to 45°N-51°N band, all graphs have a
peak again around noon, but their amplitudes
seem to have a shift, their respective maxima
ranging from 10 to 16 MHz. In the lowest
latitude band, 40°N-43°N, the peaks are
located around 10-11 LT, the maxima range
from 7 to 14 MHz, but the lowest value of
7 MHz, for Tortosa is questionable, because
the data for Tortosa has large gaps. The
incoherency of the graphs of equal latitude/
different longitude stations shows that the
local time correction at high latitudes may not
compensate for spatial dependency. On the
other hand, graphs for lower latitude station
groups are coherent; hence we may say that
local time correction is successful at least
below 60°N latitude. We also note that the
variation curves shift up with latitude up to



143

Statistical properties of the deviations of f
0
F

2
from monthly medians

deviations have larger amplitude at 12 UT and
the histograms are wider at daytime. The negative
deviations are modulated by R12 at all latitudes,
but positive deviations have a R12 dependency
only at high latitudes. Also from the frequency
spectrum we observed that the 27 days periodicity
disappears at night time (24 UT). At high latitudes
both positive and negative deviations have
seasonal dependency while at low latitudes
positive deviations are nearly independent of
season. The longitude dependency reflects as a
shift with respect to local time.

We quantified the R12 dependency of the
positive and negative deviations as a linear model
for the %90 confidence interval for each station.
We found that the positive deviations were rather
independent of R12, and we computed repre-
sentative values for the upper deciles of negative
variations for each station and local time. On the
other hand, we found an increase in the 90%
confidence intervals of positive deviations after
elimination of R12 effects, which were not
apparent in the negative deviations.

For stations below 60°N, the longitude
dependency was eliminated by a local time shift
and the upper deciles have a coherent behaviour
for stations grouped according to latitudes. The
location of the peaks moves from 11 LT to 14 LT
from south to north, and the peak amplitudes start
from 10 MHz, increase to 18 MHz and then
decrease to 16 MHz above 60°N. This latitude
dependency agrees with recent investigations
where Kouris et al. (2000) also observed an
increase in MUF with latitude followed by a
decrease above 60°N.

As we worked with 1 year samples of data
for the study of the deviations from monthly
medians, we overlooked any seasonal depen-
dencies in the quantitative study of the deviations.
Seasonal variations appeared only in frequency
spectra. Semi-annual (Rishbeth et al., 2000)
variations as well as the seasonal variations of
the hysteresis effects (Buresova and Lastovicka,
2000) are discussed in the literature.

Acknowledgements

This work is partially supported by the
COST271 EEEAG-TUBITAK project. The

authors are indebted to the comments by the
referees; Section 3.5 was added on referee’s
comments.

REFERENCES

BAYKAL, S.A. (1998): A model study of ionospheric
critical frequencies for telecommunication purposes,
M. Sc. Thesis, Middle East Technical University,
Ankara.

BILGE, A.H. and Y. TULUNAY (1998): Semi empirical
single station modelling of f0F2 variations: spectral
analysis, Poster presented at the 23rd General
Assembly of European Geophysical Society, Nice,
France.

BILGE, A.H. and Y. TULUNAY (2000): A novel on-line
method for single station prediction and forecasting
of ionospheric critical frequency f

0
F

2
1-hour ahead,

Geophys. Res. Lett., 27, 1383-1386.
BREMER, J. (1992): Ionospheric trends in mid-latitudes

as a possible indicator of the atmospheric greenhouse
effects, J. Atmos. Terr. Phys., 54, 1505-1511.

BURESOVA, D. and J. LASTOVICKA (2000): Hysteresis
of f 0 F2 a t European middle latitudes, Ann.
Geophysicae, 18, 987-991.

CLILVERD, M.A., T.D.G. CLARK, E. CLARKE and H.
RISHBETH (1998): Increased magnetic storm activity
from 1868 to 1995, J. Atmos. Sol.-Terr. Phys., 60,
1047-1056.

DANILOV, A.D. (2001): F2-region response to geo-
magnetic disturbances, J. Sol.-Terr. Phys., 63,
441-449.

KOURIS, S.S., D.N. FOTIADIS and B. ZOLESI (1999):
Specifications of the F-region variations for quiet
and disturbed conditions, Phys. Chem. Earth (C),
24 (4), 321-327.

KOURIS, S.S., D.N. FOTIADIS and R. HANBABA (2000):
On the day-to-day variation of the MUF over Europe,
Phys. Chem. Earth (C), 25 (4), 319-325.

RISHBETH, H., I.C.F. MÜLLER-WODARG, L. ZOU, T.J.
FULLER-ROWELL, G.H. MILLWARD, R.J. MOFFETT,
D.W. IDENDEN and A.D. AYLWARD (2000): Annual
and semiannual variations in the ionospheric F

2

layer:II. Physical discussion, Ann. Geophysicae, 18,
945-956.

STANISL/ AWSKA, I., LJ. R. CANDER,  E. TULUNAY, Y.
TULUNAY, A.H. BILGE, G. JUCHNIKOWSKI, M.
KADIOGLU, C. ÖZKAPTAN and  Z. ZBYSZYNSKI
(1999): Instantaneous mapping of 1-hour ahead f0F2
forecasting values over Europe, in COST 251 4th
Workshop, Madeira, Portugal, 22-26 March 1999,
235-241.

TULUNAY, E., C. ÖZKAPTAN and Y. TULUNAY (2000):
Temporal and spatial forecasting of the f

0
F

2
values

up to twenty four hours in advance, Phys. Chem.
Earth (C), 25, 281-285.