087_095 adg v–5 n01 Bilge  .pdf


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

87

Variation of the feedback coefficient
with R12 and the geographic latitude

in 1-h ahead forecast of f0 F2

Ayşe H. Bilge (1), Eti Mizrahi (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 «prediction» and «forecast» of the critical frequency of the F

2
 layer (f

0
F

2
) is an important issue for frequency

planning in short wave radio communications. In this context, «prediction» is used for the determination of monthly
median values of f0 F2 for each hour, while «forecast» denotes the determination of hourly values. In a previous
paper we proposed a «sliding window» technique for prediction combined with «feedback» for forecast (Bilge
and Tulunay, 2000). In the present paper we obtain the variation of the feedback coefficient with R12 and geographic
latitude.

1. Introduction

The prediction and forecasting of the
ionospheric critical frequency f0 F2 is crucial in
planning HF communication and for radar and
navigation systems. The monthly median values
of f

0
F

2
 for each hour can be considered as a first

approximation to the data and tabulated values
for these medians provide a good guideline for
frequency planning. However the monthly
medians fail to follow short term irregular
variations and forecasting methods are needed.
The «feedback» method that is widely used in
control engineering and signal processing areas

was applied 1-h ahead of forecast of f
0
F

2
 in a

previous paper (Bilge and Tulunay, 2000). In
the present paper we investigate the dependency
of the feedback coefficient on physical
parameters such as the 12-month smoothed
sunspot number R

12
 and the geographic latitude.

The monthly median values for each hour
depend mainly on solar activity and season.
Various sophisticated models using different
indices for solar activity have been proposed
(Smith and King, 1981; Kane, 1992a,b; Alberca
et al., 1999) but for practical purposes it is
preferable to use R12 as the only physical
parameter, because of its availability, reliability
and predictability (Bradley, 1994). The well-
known hysteresis and saturation effects
characterize the dependency of f0 F2 on R12.
Namely, f0 F2 changes linearly with R12 for low
and medium solar activity and then reaches
saturation. In addition, various periodic
components of the f0 F2 variation are modulated
by R12. The saturation effect is usually dealt with
by using a parabolic fit to the data. More
sophisticated functional fits using square roots

Mailing address: Dr. Ayşe H. Bilge, Department of
Mathematics, Istanbul Technical University, 80626 Maslak,
Istanbul, Turkey; e-mail: bilge@itu.edu.tr

Key words  f
0
F

2
− critical frequency − forecast −

feedback



88

Ayşe H. Bilge, Eti Mizrahi and Yurdanur Tulunay

and higher order polynomials (Bilge and Tulunay,
1998) give better fits but they are not stable for
long-term prediction. The character of the
dependency of f

0
F

2
 on R

12
 is different in rising

and falling phases of a solar cycle and it also
changes from solar cycle to solar cycle. These
differences are important for understanding the
mechanism underlying ionospheric processes,
however for prediction purposes, non-stationarity
of the data can be overlooked by using short term
past data to build the models for prediction.

We have noted that the dependency of f0 F2
on R

12
 is more or less linear over a time span of

2 to 4 years and we developed a «sliding
window» technique to predict the monthly
median f0 F2 using immediately past 48 month
data within 3-4% error, compared with 6-7%
errors based on 20 year models (Baykal, 1998).

Prediction by «sliding windows» and forecast
by «feedback» was proposed in a previous work
(Bilge and Tulunay, 2000), and tested in the
framework of COST 251 action Stanisl/awska
et al., 1999). The feedback coefficient is the key
parameter in single step feedback, and it is
determined by a one-dimensional optimization.
The aim of the present work is to study the
dependency of the optimal feedback coefficient
on R

12
 and geographic latitude.

We used those data described in detail in
Mizrahi et al. (2001), provided in the COST 251
CD-ROM, namely data from 48 stations between
1958 and 2000. Eliminating those stations with
less regular data coverage, we based our study
on 13 stations arranged in 3 groups according to
their latitudes. The first group includes the
stations Lycsele, Kiruna, Arkhangelsk, Uppsala
and Leningrad, which lie nearly above 60N line.
The second, mid latitude group includes the
stations in the 50N-57N band, i.e. Slough,
Juliusruh, Moscow, Pruhonice, Kiev. The stations
Tortosa, Rome and Sofia in the 40N-45N band
are arranged in the third group (Mizrahi et al.,
2001).

Our preliminary investigations have shown
that the feedback coefficients for forecast based
on predicted or actual monthly medians differ
in general by about 0.1 and the interrelations
are the same. Thus for simplicity of data
processing we based our forecast on actual
monthly medians.

2.  Forecasting with feedback

The estimation of the actual hourly values of
f

0
F

2
 is called «forecasting», and it is dealt with by

means of neural network (Tulunay et al., 2000)
and «feedback» methods (Bilge and Tulunay,
2000), in addition to more standard auto-
correlation techniques (Stanisl/awska et al., 1999).
The feedback method is a standard tool in control
engineering, for maintaining a constant output or
equivalently for following or tracking a given
signal. At each step, an appropriate multiple or
combination of the measurement error is «fed
back», to modify the controlling signal. Here, the
error is the difference between the measurement
and the prediction of monthly medians. An
appropriate multiple of the error in the i’th step is
added with the reverse sign to correct the
prediction at the i +1’st step. That is, if the
measurement and prediction at the time t

i
 are

respectively f0F2(ti) and f0F2 pred(ti), then the error
at stage t

i
 is

E (t
i
) = f0 F2(ti ) – f0 F2 pred(ti ).

As the predicted value is available at the stage
t

i+1, the forecast at the hour ti+1, denoted by f0 F2
*

is obtained from

f0 F2
*(t

i+1 ) = f0 F2 pred(ti+1 ) – k E(ti )

where k is the feedback coefficient.
This method has been applied to data from

Rome, Poitier and Uppsala ionosonde stations over
1986-1990 (Bilge and Tulunay, 2000). The
comparison of the monthly median (prediction)
data, actual f

0
F

2
 and predicted f

0
F

2
 is shown in

fig. 1, for a typical storm time disturbance followed
by quiet days, Rome, 16-23 April 1958.

In one-step feedback, there is a single parameter
to be adjusted, namely the feedback constant k. The
«best» feedback constant is found by applying
feedback with k ranging in a certain interval and
computing the error with respect to a certain norm.
The value of k corresponding to a local minimum
in the chosen norm of the error is called k*.



89

Variation of the feedback coefficient with R
12

 and the geographic latitude in 1-h ahead forecast of f
0
F

2

We have run the feedback algorithm applied
to actual monthly medians for a feedback
constant in the range 0.2-1.1. We used the l2 norm
of the error as our performance criterion.
However preliminary investigations have shown
that RMS error also leads to same values for
optimal feedback constant.

3.  Results

The optimal feedback coefficient was
computed by minimizing the l2 norm of the error
between the hourly values and monthly medians
for each hour, i.e. constant feedback is used
during a year. A total of 369 samples were
analyzed. The feedback constant ranges all times

between 0.5 and 1.0. The stations are arranged
into three main groups according to their
latitudes.  The percentage of occurrences of the
k* values in each group is given in table I. The
first group includes Lycsele, Kiruna, Ark-
hangelsk, Uppsala and Leningrad; the second
group includes Slough, Juliusruh, Moscow,
Pruhonice, Kiev and the third group includes
Tortosa, Rome and Sofia stations. The number
of samples in these groups is respectively 147,
151 and 71.

Thus the optimal feedback coefficient
ranges mostly in the 0.7-0.9 interval, hence the
value k = 0.8 used in Bilge and Tulunay (2000)
is typical. There is a tendency of k* to be lower
at lower latitudes. The graph of the feedback
constants for all stations versus years is given

Fig.  1.  Comparison of the monthly medians, hourly data and forecast results. Data show a typical storm condition
in Rome 1958 data for 8 days from April 16 to April 23. The forecast by feedback follows well the depression at
the 3rd and 4th days but fails to follow closely the midday and midnight fluctuations.



90

Ayşe H. Bilge, Eti Mizrahi and Yurdanur Tulunay

in fig. 2. We note that the extreme value k = 1
occurs during 1958 and 1959 corresponding to
an extremely high solar activity. On the other
hand, the extreme low value k = 0.5 occurs for
low R

12
 (with an exception of 1969, for Sofia).

We can thus conclude that the value of the
optimal feedback constant increases with R12.

The optimal feedback coefficients and
corresponding forecast errors are shown in figs.

3a-f for selected stations. In these graphs, R
12
 is

drawn to scale, while k* is multiplied by 100 and
the errors are multiplied by 10 for convenient
display. The variation of k* follows the R12
variation quite regularly for Rome, Slough,
Juliusruh, Uppsala, Leningrad and Lycsele
stations, oscillating between 0.6-0.8 for Rome,
and between 0.7-0.9 at higher latitudes. The
variation of k* does not follow R

12
 as regularly at

Table  I.  The percentage of occurrences of the feedback constants for all stations and years.

Feedback constant k* 0.5 0.6 0.7 0.8 0.9 1.0
Latitude:40N-43N % 9.86 15.49 26.76 36.62   9.86 1.41
Latitude:50N-55N % 0.00 2.65 11.26 46.36 39.07 0.66
Latitude:59N-68N % 0.68 0.68 14.29 53.06 30.61 0.68
All stations            % 2.17 4.34 15.45 47.15 30.08 0.81

Fig.  2.  A plot of the feedback constants versus years for all stations.



91

Variation of the feedback coefficient with R
12

 and the geographic latitude in 1-h ahead forecast of f
0
F

2

a

b

Fig.  3a,b.  Values of the feedback constant k (*) versus years for stations Rome (a) and Slough (b). A graph of R12
is shown for comparison. The corresponding errors are shown with (+). The R

12
 is drawn to scale, while k values

ranging between 0.5 and 1.0 are multiplied by 100 and percentage errors ranging between 6%-14% are multiplied
by 10 for convenient display. These figures are typical examples of regular variations with R

12
 at low, mid and high

latitudes.



92

Ayşe H. Bilge, Eti Mizrahi and Yurdanur Tulunay

c

d

Fig.  3c,d.  Values of the feedback constant k (*) versus years for stations Uppsala (c) and Sofia (d). A graph of R12
is shown for comparison. The corresponding errors are shown with (+). The R

12
 is drawn to scale, while k values

ranging between 0.5 and 1.0 are multiplied by 100 and percentage errors ranging between 6%-14% are multiplied
by 10 for convenient display. Figure 3c is a  typical example of regular variations with R

12
 at low, mid and high

latitudes. Figure 3d shows some irregularities.



93

Variation of the feedback coefficient with R
12

 and the geographic latitude in 1-h ahead forecast of f
0
F

2

e

f

Fig.  3e-f.  Values of the feedback constant k (*) versus years for stations Moscow (e) and Kiruna (f). A graph of
R

12
 is shown for comparison. The corresponding errors are shown with (+). The R

12
 is drawn to scale, while k

values ranging between 0.5 and 1.0 are multiplied by 100 and percentage errors ranging between 6%-14% are
multiplied by 10 for convenient display. These figures show some irregularities.



94

Ayşe H. Bilge, Eti Mizrahi and Yurdanur Tulunay

Fig.  4.  A plot of the feedback error versus feedback constant.

Sofia, Pruhonice, Moscow, Arkhangelsk and
Kiruna, while data for Tortosa are scarce. We
have shown the graphs for Rome, Slough,
Uppsala, in figs. 3a-c, and the graphs for Sofia,
Moscow and Kiruna in figs. 3d,f, as typical
examples of regular and irregular behavior in
each latitude band.

An observation of these graphs also reveals
that the errors tend to be higher for low feedback
constant k. A plot of the forecast errors versus
the optimal feedback coefficient k* given in fig.
4 confirms this observation, as the plot has a
negative slope. The errors range mostly between
6% and 12%. Furthermore, we also observed that
the error is not too sensitive to k; in the range
[k* – 1, k* + 1].

In order to quantify the dependency of k* on
the latitude and on R12, we grouped the yearly

samples from all stations, according to the
latitudes of the stations as in table I, and the R

12

value of the corresponding years as given in
table II.

The average value of k* corresponding to
each group of latitude and R12 range is given in
table III.

We note that, except for the second column,
the average k* in the highest latitude range is
slightly lower than the value in the mid latitude
range. This result is similar to the behavior of
the variation of the upper deciles of the negative
deviations with R12, presented in Mizrahi et al.
(2001) and the «On the day-to-day variation of
MUF over Europe» reported in Kouris et al.
(2000). These parameters increase with latitude
but then decrease slightly after 60N. As the
feedback compensates for the disturbances, it is



95

Variation of the feedback coefficient with R
12

 and the geographic latitude in 1-h ahead forecast of f
0
F

2

plausible that we need higher feedback constants
when the fluctuations in the data have larger
amplitudes, which also explains the correlated
latitude dependency of k* and the upper deciles.

REFERENCES

ALBERCA, L.F., G. JUCHNIKOWSKI, S.S. KOURIS, A.V.
MIKHAILOV, V.V. MIKHAILOV, G. MIRO, B. DE LA
MORENA, P. MUHTAROV, D. PANCHEVA, G.J. SOLE, I.
STANISL/ AWSKA and T. XENOS (1999): Comparison of
various f0 F2 single-station models for European area,
Acta Geophys. Pol., 47, 42-56.

BAYKAL, Ş. 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 modeling of f

0
F

2
 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
the ionospheric critical frequency f0 F2 1-hour ahead,
Geophys. Res. Lett., 27, 1383-1386.

BRADLEY, P.A. (1994): Further study of f
0
F

2
 and M(3000)F

2

in different solar cycles, Ann. Geofis., 37 (2), 201-208.
KANE, R.P. (1992a): Solar cycle variation of f

0
F

2
, J. Atmos.

Terr. Phys., 54, 1201-1205.
KANE, R.P. (1992b): Sunspots, solar radio noise, solar EUV

and ionospheric f0 F2, J. Atmos. Terr. Phys., 54, 463-466.
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.

MIZRAHI, E., A.H. BILGE and Y. TULUNAY (2001): Statistical
properties of the deviations of f0 F2 from monthly
medians, Poster presented at EGS XXVI Symposium,
Nice, France, March 2001.

SMITH, P.A. and J.W. KING (1981): Long term relationship
between sunspots, solar faculae and the ionosphere,
J. Atmos. Terr. Phys., 43, 1057-1063.

STANISL/ AWSKA, I., LJ.R.CANDER, E. TULUNAY, Y. TULUNAY,
A.H.BILGE, G.JUCHNIKOWSKI, M. KADIOĞLU, C.
ÖZKAPTAN and Z. ZBYSZYNSKI (1999): Instantaneous
mapping of 1-h ahead f0 F2 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 f0 F2 values up to
twenty four hours in advance, Phys. Chem. Earth (C),
25, 281-285.

Table  III.  Average values of the optimal feedback coefficient in different latitude groups and R
12

 ranges.

R
12
: 0-29 R

12
: 30-59 R

12
: 60-109 R

12
: 140-189

Latitude: 40N-43N    % 6.588 6.824 7.680 7.917
Latitude: 50N-55N    % 7.750 8.111 8.380 8.862
Latitude: 59N-68N    % 7.543 8.162 8.275 8.708

Table  II.  The breakdown of years 1958-1998 into R
12

 ranges.

  R
12

                                            Years
0-29 (1963,1964,1965,1975,1976,1985,1986,1995,1996,1997)

30-59 (1961,1962,1966,1973,1974,1977,1984,1987,1993,1994)
60-109 (1960,1967,1968,1969,1970,1971,1972,1978,1982,1983,1988,1992,1998)

140-189 (1958,1959,1979,1980,1981,1989,1990,1991)