Structure
Functions of Temporal Phase Variations in Vertical Sounding as a
Diagnostic Tool for Ionospheric Small-Scale Irregularities
N.A.
Zabotin, Rostov State University, Rostov-on-Don, 344090 Russia
J. W.
Wright, 1915 Spruce Avenue, Longmont, Colorado, 80501 USA
ABSTRACT
We present a new method of the
ionospheric irregularity investigation based on using of temporal
structure function of the phase of the vertical sounding signal.
Modern digital ionosondes (e.g. the dynasonde) measure the echo
phase with very high resolution and precision at closely spaced
antennas, frequencies and times. A ‘stringing’
procedure gives continuous and unambiguous phase variation data
for time intervals of any desired length. Quasi-periods of tens
of seconds up through several minutes are caused by large scale
movements of the ionospheric plasma, while short variations are
due to interaction of sounding signal with small-scale
irregularities. That is why statistical properties of short phase
variations are interconnected with statistical characteristics of
the ionospheric small-scale irregularities.
The most convenient tool for
statistical description of small-scale structures in both
temporal (phase variations) and spatial (electron density
irregularities) domains is the structure function (SF).
Theoretical relation between structure functions of the phase
variations and the irregularities can be obtained in the
framework of a simple model of frozen irregularity horizontal
drift. This relation allows one to solve both direct and inverse
problems.
We show that some modification
of this approach (which we call rudimentary structure function,
or RSF concept) can be applied to analysis of dynasonde
recordings obtained in its standard B- and K-modes. Our results
suggest a significant diagnostic potential of the RSF method.
What is the ‘structure
function’?
SF is one of common statistical
characteristics of any random function (random field). It is
defined as the mean square of difference of the
function values taken at two different points. When
random function is statistically
homogeneous, SF depends only on
vector connecting those two points. Thus it can be defined for temporal
phase variations as 
and for spatial electron density
irregularities 
as 
where averaging over
realization ensemble is supposed. In the case of field aligned irregularities the function 
depends only on the
absolute value of the transverse component of vector 
(that is on 
). Unlike
the correlation function or dispersion,
large scale variations (
) do not influence SF. Moreover, structure
function is the only natural mathematical analog for the inexact physical
definition 'variation (disturbance) at a certain scale
length' with relation to random functions.
How is phase SF connected
with irregularity SF?
The following consideration is somewhat
simplified but simplifications are not of principle character.
Let us consider the layer of isotropic plasma with small-scale
stationary random irregularities infinitely stretched along the
slanting 'magnetic' lines. We shall assume that no multibeam
interference exists. But
horizontal drift of the layer and the 'frozen' irregularity field
with velocity 
will be taken into account (See Fig. 1). Hence, the
ray path of the sounding signal for two different times passes
through different irregularities. This is considered the primary
reason of the short
period phase variations observed at
the fixed point 
on the ground surface.
Let 
be the reflection level, 
is the level of the ionospheric layer
beginning, 
, 
is
regular part of the layer profile (
), 
is irregular part of 
connected with
electron density irregularity field by the relation 
. Random
phase variation due to irregularities is given by well known
expression with sufficiently broad applicability field:
.
Using definition of structure function, one
can obtain for the case of vertical sounding of the
linear plasma layer (
) the following relation:
where
=
and representation 
has been used.
The above expression can be generalized for
the case when reflection occurs along oblique
ray path with polar angle 
and azimuth angle 
:
where 
.
What are SF features
predicted by the theory?
The irregularity structure
function 
can be easily calculated if the irregularity spatial spectrum is known. For example, if the spatial spectrum has
the following, quite typical for the ionospheric conditions form:
,
where 
and 
are components of vector 
orthogonal and parallel to the geomagnetic field
vector and delta-function allows for infinite stretching of the
irregularities, the corresponding structure function is
.
Here 
is the Gamma-function, 
is the Macdonald function and the
normalizing constant 
should be determined from the condition 
(
has the meaning of
averaged 
distortion value in the scale length 
).
Theoretical dependence of
the phase structure function on some parameters has been illustrated
in Fig. 2a. Parameters 
= 5 MHz and 
= 50 km were used here. The magnetic field
inclination angle 
was
set equal to 0.2 rad what simulates situation in TromcЬ. For the
irregularity spatial spectrum parameters 
at the scale length 
km, 
and 
= 10 km were used.
The drift velocity direction was determined by angle 
=0.5 rad. The polar
angle of the reflected signal was set equal to 0.2 rad, that is
the case of 
was
chosen. It was done to demonstrate dependence of the effect on
relative orientation of the ray and of the magnetic field line
passing through the sounder location. Thus dependence of the
structure function on two parameters is shown: on the azimuth
angle 
and drift
velocity 
. Color
in the figure marks different values of 
and line thickness corresponds to the
drift velocity value 
.
One can see that the SF
amplitude depends on the ray orientation and on the . It is evident also
that it must depend on the . The SF amplitude is especially large when ray
direction is close to the magnetic line direction. It is of
importance that SF slope is practically independent on 
, 
and . At the same time,
the SF slope depends on the spectrum index 
(see Fig. 2b). Therefore, the SF slope can serve as good
indicator of the irregularity spectrum index .
How is the phase SF measured?
Generally speaking, phase of a wave is
determined ambiguously with an accuracy to integer number of 
. But, if the phase is
measured frequently enough (as with modern digital ionosondes
under vertical sounding), special procedure called stringing
allows one to obtain continuous series of the phase values
determined unambiguously even if they differ more than by .
Long-period (tens of
seconds) variations of the phase are caused by large-scale
movements of the ionospheric plasma. But we are interested now in
small-scale structures. That is why strung phase data must be detrended
before calculation of the SF. Both preliminary stages of the data
processing (stringing and detrending) are illustrated in Fig. 3 in
terms of the TromsЬ dynasonde recording 26 Oct 1985 starting at
1130UT. This recording uses only minimal options of the system:
4-pulse sets at 50 pulses/sec, plus a .04 sec pause between sets;
single fixed
frequency, a time series of 1000
pulse sets in 160 seconds; sampling 50 complex amplitudes at 10 msec
intervals between TOAs of 228.29 and 301.74 km.
Resulting phase structure
functions for this recording are presented in Fig. 4.
There are two large groups of the SFs, which have been marked by
colors. Lilac, red and green constitute the first group. Lilac
and red curves are from the wings of the TOA interval, where
signal/noise ratio is small. Green ones are from the TOA bands
where glint interference takes place. Both cases are not
described by the theory which does not take into account
multibeam interference effects. Another group (black and blue
curves) are from the TOA bands where single glint dominates by
amplitude and therefore multibeam interference is not essential.
As one can expect, features of the SFs belonging to the second
group are in full agreement with the theory stated above. The SF shape in the band of
large lags can be elaborated by filtering of
long-periodic harmonics from the strung detrended phase data (see
Fig. 5). This operation allows one to estimate the
dispersion of phase variations caused by small-scale
irregularities (it is about 1 rad2 in
the presented case). Filtering does not influence the SF behavior
in the small-lag band and its most remarkable property — log-log-linear character. This property implies possibility of
realization of the rudimentary structure function
concept that significantly extends the SF method
applicability.
What is rudimentary
structure function?
The starting band of the full-scale structure
function in the log-log coordinates is well
approximated by the following linear dependence:
log(/rad2) = 
+
·log(
/sec)
.
Structure indices and that have been
introduced here depend upon all parameters which SF itself
depends on. Their merit is possibility of their determination
using only few
smallest lags of the full-scale SF.
It allows one to use for the purpose of the irregularity
diagnostics large volume of the information obtained in standard
B and K dynasonde modes.
Almost all dynasonde ionograms
today are made using B-mode. In a typical B-mode ionogram, one
uses a variant of the following time/frequency pattern of pulse
sets. 'x' is one pulse set:
Radio frequencyxxxxxxxxxxxxxxxxxxxx xxetc.
^ xxxxxxxxxxxxxxxxxxxxxxxx
| xxxxxxxxxxxxxxxxxxxxxxx
| xxxxxxxxxxxxxxxxxxxxxx
| xxxxxxxxxxxxxxxxxxxxx
| xxxxxxxxxxxx
| xxxxxxxxxxx
| xxxxxxxxxx
| xxxxxxxxx
+--------------------------------------------> time
In this example, we have a
"B-mode ionogram of 3 ramps of 4 frequencies per
block". Simple ionograms (I-mode) can be described as 1 ramp
of (say) 1000 frequencies, or 1000 ramps of 1 frequency per
block. K-mode could be described as (say) a B-mode of 1000 ramps
of (say) 4 frequencies, but this would not be the most flexible
way to achieve K-mode.
This is design of a single
8-pulse pulse set:
Pulse xxxxxxxxxx#1 #2 #3 #4 #5 #6 #7 #8 xxxUnits
____________________________________________________________
Time xxxxxxxxxx-7x -5 -3x -1x 1x 3x 5x 7xxxx0.005 sec
Frequency xxxxxx1 x-3x 3x -1x -1x 3 -3x 1 xxx1 KHz
Ant,Receiver1 xxxe xN xS xe xxe xS xN xe
Ant,Receiver2 xxxW xs xsx E xxE xs xs xW
Pulse set supplies four lags for the
structure function: 0.01, 0.03, 0.05, and 0.07 sec. Obviously,
ramp repetitions of pulse sets introduces another series of
temporal lags. The relation of these lags to the shorter lags in
the constituent pulse sets depends on the number of pulses/set
and the number of frequencies per ramp.
| Rudimentary
structure function (RSF)
is the structure function that contains only few initial
lags sufficient to determine structure indices and by fitting. |
In case of dynasonde B-mode sufficient
statistics of the realizations of the phase differences needed to
calculate RSF is achieved by using of counts relating to
different frequencies. Statistical
homogeneity of the realization
ensemble is provided in this case by introducing the factor f
-4 compensating for the frequency
dependence of the .
How does the RSF
method works?
We cite here for
illustration the
results of processing of data of two dynasonde campaigns. The
first one — 'UKDYNA' — lasted 30 hours and used only
B-mode. The time series of the structure indices for
this campaign is shown in Fig.
6. Analogous time series for
another campaign are shown in Fig. 7. Their number
is twice as much than in the previous case because distinctive
feature of this campaign is using of two dynasonde modes, B and
K, interlacing. Its duration is also different, about 72 hours.
Besides the structure indices, owing to
standard POLAN, UNIPHASE and VFITr algorithms, we have time
series for the whole set of other parameters: echolocations,
drift velocity, ionospheric layer thickness etc. All these
parameters serve as input information for special algorithm of
the inverse problem solving based on the above theoretical
expressions for the phase structure function . The only additional
simplification we make on this stage is assumption that 
value is equal to 7.5
km. But SF weakly depends on the . Output information for the inverse
problem algorithm is two parameters of the irregularity spectrum model cited above: spectrum index and irregularity
level in the
scale length 1 km.
Time series for the
irregularity spectrum parameters are shown in Figs. 8 and 9
for two dynasonde campaigns correspondingly. Lack of some data
points is caused mainly by two reasons: by absence of the F layer
reflections due to the ES and by
occurrence of the >2
cases. Such cases are not described by the present variant of the
theory. They are more characteristic of F layer reflections and
are probably caused by multibeam interference. Despite this
shortcoming, the method gives extensive and unique information
about the temporal
dynamics of the small-scale
irregularity spatial spectrum. Note clear diurnal variation of
the spectrum parameters during the UKDYNA campaign which is in
qualitative agreement with the results of other, more rough,
methods.
CONCLUSION
Our results suggest a significant diagnostic
potential of the phase SF/RSF method with relation to small-scale
ionospheric irregularities.
Figure 1
Figure 2a
Figure 2b
Figure 3
Figure 4
Figure 5
Figure 6.
Figure 7
Figure 8
Figure 9