Incoherent scatter Faraday rotation measurements on a radar with ...

**on****on** a **radar**

**with** single linear polarizati**on**

Boris G. Shpynev

Institute of Solar-Terrestrial Physics, Irkutsk, Russia

Received 27 August 2001; revised 21 September 2003; accepted 25 March 2004; published 11 May 2004.

[1] This paper outlines an applicati**on** of the incoherent

(Russia) VHF **radar** system, which has an antenna **with** a single, strictly linear polarizati**on**.

Owing to i**on**ospheric **on****on** of wave plane polarizati**on** angle is

essential to obtain electr**on** density. We describe a new technique for electr**on** density

determinati**on** **on** a **radar** **with** single polarizati**on** based **on** the **radar** equati**on**. This

technique does not require additi**on**al external density calibrati**on** using other instruments

such as i**on**os**on**des. INDEX TERMS: 6934 Radio Science: I**on**ospheric propagati**on** (2487); 6952

Radio Science: Radar atmospheric physics; 6964 Radio Science: Radio wave propagati**on**; KEYWORDS:

i**on**ospheric propagati**on**, **radar** atmospheric physics, radio wave propagati**on**

Citati**on**: Shpynev, B. G. (2004), **on****on** a **radar** **with** single linear polarizati**on**,

Radio Sci., 39, RS3001, doi:10.1029/2001RS002523.

1. Introducti**on**

[2] The incoherent

over four decades; a basic review can be found in Evans

[1969, and references therein]. Several IS **radar**s are

currently active worldwide, and all of them are unique

in design, frequency range, and specific signal processing/data

acquisiti**on** techniques. The VHF **radar** system

operated by the Institute of Solar-Terrestrial Physics in

Irkutsk, Russia is no excepti**on**. Since the facility was

designed for military **radar** purposes, a distinctive feature

is its antenna design, which allows for transmissi**on** and

recepti**on** of **on**ly **on**e linear polarizati**on**. This system

characteristic means that the signal at the **radar** receiver

input undergoes quasi-periodic fading, caused by

**on****on** plane **on**

feature can be used for calculati**on** of absolute electr**on**

density, the single polarizati**on** channel leads to distorti**on**

of the IS spectrum or equivalently the correlati**on** functi**on**

of the received signal, and so the analysis is

complicated.

[3] The c**on**cept of using the **on**

of Ne in incoherent

Millman et al. [1961]. In a later work [Millman et al.,

1964] the first **on** plane

angle of an IS signal, received **on** mutually orthog**on**al

linearly polarized antennas, is described. The work of

Farley et al. [1967] and Farley [1969] utilizes for

Copyright 2004 by the American Geophysical Uni**on**.

0048-6604/04/2001RS002523$11.00

RADIO SCIENCE, VOL. 39, RS3001, doi:10.1029/2001RS002523, 2004

**with** opposite circular polarizati**on**s, **with** receivers for

each of the two comp**on**ent circular polarizati**on**s allowing

the measurement of the phase difference between

channels. For the Jicamarca **radar**, **with** a 50 MHz

operating frequency, such a technique has proved optimal.

The work of Pingree [1990] implemented this

technique successfully for regular

the method (**with** subsequent data processing improvements)

is now standard at Jicamarca.

[4]

at a 150 MHz **radar** in Kharkov (Ukraine) [Tkachev and

Rozumenko, 1972]. The wave polarizati**on** plane at the

Kharkov center frequency makes a smaller number of

revoluti**on**s than at Jicamarca. Since the separately

measured polarizati**on** channels at Kharkov are not

completely independent due to crosstalk effects,

Grigorenko [1979] developed a method of Ne definiti**on**

using an analysis of power profile extreme points. The

method works **on**ly at F2 layer heights **with** high electr**on**

density in daytime. If the electr**on** density is low, usually

there are not enough extreme points for restorati**on** of Ne

structure **with** acceptable height resoluti**on**.

[5] At Irkutsk, the **radar** center frequency is at

154–162 MHz and **radar** can radiate and receive **on**ly

**on**e linear polarizati**on**. However, unlike Kharkov, the

Irkutsk antenna system has a polarizing filter, reducing

the orthog**on**al polarizati**on** level by 30 dB. Therefore it

is feasible to assume that signal polarizati**on** **on** both

transmissi**on** and recepti**on** is strictly linear. This allows

to obtain an exact equati**on** for the received signal and so

RS3001 1of8

RS3001 SHPYNEV: IS MEASUREMENTS USING SINGLE POLARIZATION

give us a soluti**on** which relates it directly to N e.Inthe

present work, we derive this technique **on** the basis of the

**radar** equati**on**.

2.

Equati**on** Accounting for

[6] The **radar** equati**on** can be obtained for the IS

method, accounting for **on** the basis of

the statistical theory of

et al., 1978]. For antennas **with** circular polarizati**on**,

such an equati**on** can be found in Suni et al. [1989] **with**

main techniques developed by Kravtsov and Orlov

[1980]. The derivati**on** of the equati**on**s are complex,

and so we **on**ly specify the basic differences from the

circular polarizati**on** case before carrying out the analysis

of final expressi**on**s. For a derivati**on** of the **radar**

equati**on** we follow an expressi**on** from Suni et al.

[1989]:

UaðÞ¼ t

rffiffiffiffiffi

P

4p

dN t

iðbw0tÞ ree

r

; r a t

c

Z dr

r 2 Dtrað ! lrÞDrecð ! lrÞp

2r

c

e i2k0r

ð1Þ

Here Ua(t) is the complex envelope of the received

signal, P is the peak transmitted power, re is the classical

electr**on** radius, b is the initial transmitted wave phase,

w0 is the transmitter frequency, k0 = w0/c, ~ lr = ~r/r, p =

( ~ lrec [ ~ lr [ ~ lr

~ ltra]]) is the polarizati**on** multiplier,

~ltra,rec are the individual vectors of polarizati**on**, ~Dtra,rec

are the normalized antenna patterns for transmissi**on** and

recepti**on**, dN are the electr**on** density fluctuati**on**s, and

a(t) is the complex envelope of the transmitted signal.

[7] To account for antenna linear polarizati**on**, we

replace the wave functi**on** e i2 0r **with** a sum of geometrical

optic expressi**on**s for ordinary and extraordinary waves:

where

jo;xðÞ¼ r

Z ~r

0

e i2k0r ) 1 =2 e i2k0j o þ e i2k0j x ð2Þ

no;xds ffi

ds ¼ jðÞ r

Z ~r

0

1

2

nds

Z

1

2

~r

0

v ffiffiffi p

u cos a

fðÞ r

ð3Þ

are eik**on**als of ordinary and extraordinary waves

[Kravtsov and Orlov, 1980], no,x is the index of

refracti**on**, v = w2p w2 , u =

0

w2 H

w2 2 Nee

, wp =

0

2

e0me , wH = ej~Bj

. Here

me

Ne is the electr**on** density, e is the electr**on** charge, me is

the electr**on** mass, e0 is the permittivity of free space, ~B is

the magnetic fied, a is the angle between ~B and ~ k0, and

2of8

the integrals are carried out al**on**g the appropriate beams.

Substituting (3) in (2), we have for the wave functi**on**:

e i2k0r i2k0j ðÞ ~r ik0f ðÞ ~r ik0f ðÞ ~r

) e e þ e =2

¼ e i2k0j ðÞ ~r

cos WðÞW r ðÞ¼k0f r ðÞ r

ð4Þ

The variable W(r) is the **on**

polarizati**on** plane from its initial positi**on** **on** the ground

to current range al**on**g the path of wave propagati**on**.

Next we substitute (4) in (1) and c**on**sider the expressi**on**s

for average spectral density S(w, t) = hjU(w, t)j 2 i,

where t is the delay from beginning of emissi**on**.

Integrati**on** of the waves in the far-field range of a

m**on**ostatic transmitting/receiving **radar** antenna give us

the opportunity to express the influence of the antenna

geometry through antenna gain G. This results in the

**radar** equati**on**, which c**on**nects the spectrum of the

received signal **with** the spectrum of plasma fluctuati**on**s

r(r, w). This equati**on** has been derived by many authors

[e.g., Sheffield, 1975]. If we also assume that r(r, w) is

c**on**stant inside the spatial volume occupied by transmitted

pulse, then for spectral density we have

W ¼ 1

ð2pÞ 2

Z

Z

Sðw; tÞ

¼ A

dn ~rðnÞFðwn; tÞ

ð5Þ

Z

dr

Fðw; tÞ

¼ W w; r; t

r2 ð Þ cos2 ðWðÞ r Þ ð6Þ

dt1dt2Oðt1tÞO* ðt2 a t1

ðt2tÞ 2r w n

Þ ei ð Þ t1 ð t2Þ

c

A ¼ PG=k 2 0

2r

c a*

ð7Þ

ð8Þ

In (5)–(7) n, t 1 and t 2 are the integrati**on** variables.

Functi**on** O(t t) determines the delay and the shape of

the signal analysis window, which usually is rectangular

and it length is equal to sounding pulse durati**on**. The

integrati**on** of (5) **on** w gives the total power of the signal,

weight of which is defined by functi**on** (6), which can be

called the modified ambiguity functi**on**. Total

cross secti**on** of plasma for wavelength 2m has the well

known expressi**on** [Buneman, 1962]:

r0 ¼ rðr; t ¼ 0Þ

¼ pr 2 NeðÞ r

e

1 þ TeðÞ=Ti r ðÞ r

RS3001

ð9Þ

So far as we are interested in power from a fixed range

**on**ly, the window O(t t) becomes a delta functi**on** in

RS3001 SHPYNEV: IS MEASUREMENTS USING SINGLE POLARIZATION

time, and by using (9), the power of the incoherent

PðtÞ ¼ pr 2 eA Z

dr

r2 NeðÞcos r 2 2

ðWðÞ r Þ 2r

a t

ð1þTeðÞ=Ti r ðÞ r Þ c

ð10Þ

The equati**on**s (5)–(10) are the required **radar** equati**on**s,

which can be used for derivati**on** of i**on**ospheric plasma

parameters through fitting to the functi**on**s S(w, t) and

P(t). The difference of these equati**on**s from traditi**on**al

**on**es is the presence of **on**

cos 2 (W(r)), in the expressi**on** for power, and in the

modified ambiguity functi**on** (6). When spectral power

density S(w, t) is measured experimentally (Figure 1),

the **on**, which is

visible as two minima near 450 and 575 km. To obtain

necessary spectral power estimates, a l**on**g transmitted

pulse of 750–800 ms is used. At this mode, which we call

as ‘‘l**on**g pulse mode’’, the measured minima of power

always exceed zero level under integrati**on** over the

**on**s near the i**on**osphere

electr**on** density maximum at 300–400 km are not

visible at all. Comparing to traditi**on**al fitting technique

the present **on**e needs the calculati**on** of the ambiguity

functi**on** separately for all heights. In the other detail the

processing of spectra does not differ from traditi**on**al, and

was repeatedly tested at joint Irkutsk - Millest**on**e Hill

observati**on**s.

[8] To take the

experimental spectra and getting electr**on** density itself

we must provide an independent way to derive the value

of W(r). This can be d**on**e by using the expressi**on**s (10),

providing a short transmitted pulse is used (short pulse

mode). If the value of W(r) is obtained, we can simultaneously

fit the IS spectra and determine the Ne(r) profile

by using the well known expressi**on** [Ginzburg, 1967]:

W ðÞ¼ ~r

e3B0 cos a

2e0m2 ew20 c

Z ~r

from which we can obtain Ne(r)

0

NeðÞdz z ð11Þ

dWðÞ r

dr . Here we

assumed that the intensity of magnetic field B0 = j~Bj is

given and c**on**stant inside the range of

Hence, the key task of incoherent

**on** an antenna **with** single linear polarizati**on** is the

solving of the equati**on** (10) relative to Ne(r).

3. Determinati**on** of Electr**on** Density

[9] To analyze the properties of equati**on** (10) we write

it in the following form:

Z

2

PðtÞ ¼ ~P

2r

ðÞa r t dr; ð12Þ

c

3of8

Figure 1. Example of the altitude distributi**on** of

experimental power spectra measured by l**on**g pulse

T = 750 ms. The variati**on** of spectral power **with** altitude

**on** this plot is caused by

~PðÞ¼pr r

2 eA NeðÞcos r 2ðWðÞ r Þ

r2ð1þTeðÞ=Ti r ðÞ r Þ

ð13Þ

From (12) and (13) we can see that there are two basic

issues for determinati**on** of Ne(r). The first issue is the

c**on**voluti**on** of the experimental power profile **with** the

characteristics of the transmitted pulse. Additi**on**ally,

there is a n**on**linear dependence ~P(r) **on**Ne(r), which

enters in (13) as a multiplier and stands under the integral

sign in expressi**on** (11). Fortunately, the n**on**linear

c**on**necti**on** between ~P(r) and Ne(r) is not the basic problem

of the given technique, since if we replace variables by

using the equati**on**

where

NeðÞ¼ r

1 dW

g dr

g ¼ e3 B0cosa

2e0m 2 e w2 c

ð14Þ

ð15Þ

then we reduce the equati**on** (13) to a coupled differential

equati**on**. Solving this system gives us:

1

A0

Z r

0

~P r 0

ð Þ r 02

1 þ Te r 0

ð Þ=Ti r 0

ð ð ÞÞdr

0

RS3001

sinð2WÞ ¼ W þ þ C ð16Þ

2

where A0 = Apre 2 /2g, and the c**on**stant C =0asW(r =0)

= 0. As the expressi**on** under the integral sign in (16) is

always positive then by measuring the functi**on** ~P(r), and

using known expressi**on**s for A0 and the ratio Te/Ti we can

solve numerically (16) and to determine the value of W(r).

[10] Removing effects of c**on**voluti**on** **with** the transmitted

pulse is a more difficult task. The first problem is

RS3001 SHPYNEV: IS MEASUREMENTS USING SINGLE POLARIZATION

Figure 2. Examples of short pulse power profiles,

corresp**on**ding to different i**on**ospheric c**on**diti**on**s. (top)

Nighttime c**on**diti**on**s **with** low electr**on** density and

200 ms sounding pulse. (middle and bottom) Daytime

c**on**diti**on**s **with** different electr**on** density, 100 ms and

50 ms sounding pulses, respectively.

that for power

same transmitted pulse (750–800 ms) as for spectral

**on** density peak the

effect of c**on**voluti**on** completely hides the

effect. Sec**on**dly, we cannot use very short pulses, the

c**on**voluti**on** **with** which can be neglected, because they

have insufficient energy for

heights. Therefore, for optimal

use pulses which are l**on**g enough to supply necessary

energy, but which are short enough to allow us remove

the c**on**voluti**on** expressed in (12). This ‘‘short pulse

mode’’ goes just after ‘‘l**on**g pulse mode’’ and uses

another carried frequency. The design of transmitter’s

modulators **on** Irkutsk’s **radar** gives us the opportunity to

use the total length of emissi**on** not less then 850 ms, but

i**on**ospheric c**on**diti**on**s sometimes need using such short

sounding pulses at ‘‘short pulse mode’’ as 50 ms. To keep

necessary total pulse length we change for such c**on**diti**on**s

to ‘‘l**on**g pulse’’ durati**on** **with**in (750–800 ms). In

Figure 2,

units versus range are given for different i**on**ospheric

c**on**diti**on**s. The top profile is measured at nighttime **with**

a pulse of T = 200 ms durati**on**, the middle **on**e during day

time **with** a T = 100 ms pulse, and the bottom **on**e during

high electr**on** density c**on**diti**on**s **with** T = 50 ms pulse.

For better understanding of i**on**ospheric c**on**diti**on**s,

corresp**on**ding f0F2 frequencies are presented **on** the

plots. The figure shows that the influence of c**on**voluti**on**

4of8

**on** the transmitted pulse is almost invisible **on** the top and

middle panel of Figure 2 due to the l**on**g period of

**on**s, but c**on**voluti**on** effects clearly appear

**on** the bottom panel as n**on**zero profile minima. As

power profiles can be of different forms, the transmitted

pulse durati**on** must be changed flexibly during the

experiment. Nevertheless, the measured power profiles

dem**on**strate the necessity of dec**on**voluti**on** **with** the

transmitted pulse shape.

4. Dec**on**voluti**on** of Power Profiles by

Using the Transmitted Pulse Shape

[11] Dec**on**voluti**on** problems are frequently faced in

remote sensing systems. They bel**on**g to a class of

inverse theory problems which in general are difficult

to realize in practice. We can express the c**on**voluti**on**

operati**on** as

Z

gx ðÞ¼

RS3001

fðyÞkx ð yÞdy

ð17Þ

where the real functi**on** f (x) is c**on**volved **with** a kernel

k(x). The most general method of dec**on**voluti**on** is to

perform transiti**on**s to frequency space, to divide the

Fourier-space expressi**on** f (w) by the Fourier-space

expressi**on** k(w), and finally return back to the variable

x [Natterer, 1986]. However, the given approach has an

essential problem in the divergence of the result near the

zeroes of k(w). To solve this c**on**tradicti**on** we could

employ complex regularizati**on** algorithms, which are

often computati**on**ally intensive and very sensitive to

experimental noise. However, for the present problem,

there are features which allow simplificati**on**. For

example, we can assert that the power profile is always

positive definite. Also, we can specify that the

transmitted pulse must have an almost rectangular shape.

Using these features of our measurement technique, we

have developed [Vor**on**ov and Shpynev, 1998] a dec**on**voluti**on**

method which in practice is faster and more

immune to experimental noise compared **with** traditi**on**al

procedures. The method is based **on** the fact that each

positive definite functi**on** can be approximated by the

sum of horiz**on**tal rectangles Ri(x) of identical height and

various lengths (Figure 3b):

fðÞ¼ x

X RiðÞ x

ð18Þ

It is clear that minimal splitting of the original functi**on**

into rectangles must lead to better approximati**on**. We

c**on**sider our kernel k(y) (the transmitted pulse) also to be

rectangular **with** amplitude 1 and **with** durati**on** t0, which

is equal to durati**on** of the transmitted pulse. As the

c**on**voluti**on** of a rectangle Ri(x) **with** a rectangle k(y) isa

trapezoid Ti(x) (Figure 3a) **with** equal slopes at fr**on**t and

RS3001 SHPYNEV: IS MEASUREMENTS USING SINGLE POLARIZATION

Figure 3. Illustrati**on** of approximati**on** method for

dec**on**voluti**on**. (a) C**on**voluti**on** of two rectangles;

(b) approximati**on** of positive definite functi**on**s by

horiz**on**tal rectangles; (c) result of c**on**voluti**on** as a

sum of appropriate trapezoids.

back, **with** durati**on** t0, the approximated equati**on** of

c**on**voluti**on** has the form

gx ðÞ¼ X TiðÞ x

ð19Þ

If we now reverse the task and approximate the c**on**voluti**on**

result by a sum of trapezoids Ti(x) (Figure 3c), and

replace each trapezoid by an appropriate rectangle, we

obtain the required approximati**on** for f (x). Since the

approximati**on** of the positive functi**on** g(x) by a sum of

trapezoids is always possible, the given method is

always stable, and experimental noise will influence

**on**ly the accuracy of approximati**on**. The method

requires a minimal number of calculati**on**s and its

accuracy depends fundamentally **on** the quality of

approximati**on** of the experimental power profile by

trapezoids.

5. Practical Realizati**on** of Ne(r) Derivati**on**

[12] After the dec**on**voluti**on** of P(r) and determinati**on**

of ~P(r), **with** known temperature ratio Te/Ti and c**on**stant

A0 we can express the angle W(r) by using the equati**on**

(16) follows:

1

A0

Z r

0

~P r 0

ð Þ r 02

1 þ Te r 0

ð Þ=Tir 0

ð ð ÞÞdr

0 sinð2WÞ ’ W þ

2

ð20Þ

Here, the symbol ’ in (20) and bey**on**d indicates the use

of the approximate dec**on**voluti**on** algorithm. To determine

in (20) the c**on**stant A0, we assume Te/Ti = 1. This

c**on**diti**on** is almost always true at night, and corresp**on**ds

to the top diagram of Figure 2. In this case, the integral in

5of8

the left part of (20) can be determined for all values of r

from experimental data, and the absolute calibrati**on** of

integral in the left part of (20) between two neighbor

minima is equal to p. Then the c**on**stant A0 can be

determined from the c**on**diti**on**:

A0 ’ 2

Z

p

rjþ1

~PðÞr r

rj

2 dr ð21Þ

where rj are the coordinates of points where complete

fading occurs.

[13] If we begin

when Te/Ti 6¼ 1, we can modify our technique to c**on**sider

the Te/Ti ratio as a c**on**stant inside an interval between

neighbor minima, as can be seen in Figure 2 (middle and

bottom plots). For this case, to obtain absolute calibrati**on**

we define the set of normalizing c**on**stants A0 j

separately for every j-th interval. In this case we obtain

A0 j ’ 1

Z

p

rjþ1

~PðÞr r

rj

2 dr ð22Þ

Of course, the c**on**stants A0 j will differ from the actual

c**on**stant A 0 at least by a factor of 2, but this does not

affect our result since we are **on**ly interested in N e(r).

[14] If the c**on**stant A 0 is already determined, and the

power profile has a sufficient number of polarizati**on**induced

minima (as in the middle and bottom diagrams

of Figure 2), from (20) we can obtain an estimati**on** of

the T e/T i ratio itself. If we take into c**on**siderati**on** the

variati**on**s of the **radar** system c**on**stant versus the time,

and fix the ratio T e/T i c**on**stant inside the intervals

between neighbor minima of the profile ~P(r), then for the

j-th interval we have

ðTe=TiÞj ’

rjþ1 R

rj

pA0

~PðÞ r r2dr 1 ð23Þ

This equati**on** can be used both for processing of the IS

spectra and for a more exact definiti**on** of Ne(r). We

should note that everywhere at Ne(r) processing we need

the Te/Ti ratio (not Te and Ti separately).

[15] During numerical processing of the equati**on** (20)

the soluti**on** becomes unstable in regi**on**s close to zeroes

of a power profile. This happens both because the signal

fades to the point where it is comparable to noise, and

because of the n**on**linear behavior of

sinð2WÞ 2

RS3001

near the

minima when small changes of integral in the left part of

(20) corresp**on**d to large variati**on**s of the angle W(r). To

avoid this instability, we instead use an approximati**on** of

the profiles near minima locati**on**s before calculati**on** of

W(r).

RS3001 SHPYNEV: IS MEASUREMENTS USING SINGLE POLARIZATION

Figure 4. Processing of short pulse T =70ms power

profiles (thin solid line). The first stage excludes

c**on**voluti**on** **with** the transmitted pulse and accounts for

the factor r 2 (dashed line). The sec**on**d stage solves the

equati**on** (20) and determines Ne(h) (thick line). The

shape of Ne(h) is not proporti**on**al to envelope of ~P (r)r 2

due to different value of Te/Ti ratio.

[16] The soluti**on** of the equati**on** (10) can also be

carried out by direct fitting of a specified profile shape

to the entire profile P(r), accounting for c**on**voluti**on** **with**

the transmitted pulse and for the

However, this kind of algorithm requires much more

computing resources and is in many cases unacceptable

for regular

compromise algorithm in which the first approximati**on**

of W(r) is determined by solving the equati**on** (20). Once

this has been obtained, least squares fitting is used **with**

the direct formula (10). This is very helpful at high

altitudes, where owing to the r 2 factor the influence of

noise is much more significant.

[17] Unfortunately, direct **on**

density in Irkutsk at heights lower than 160 kilometers

are currently impossible owing to ground clutter. However,

indirect informati**on** about E regi**on** c**on**diti**on**s can

be obtained **on** the basis of the value of total electr**on**

c**on**tent, from the Earth’s surface up to a height of

160 kilometers. These estimates are made possible since

the height of the first polarizati**on** minimum in practice is

always higher than 160 km. This informati**on** may be

useful for analysis of E regi**on** variati**on**s during major

geomagnetic storms.

6. Spatial Resoluti**on** and Measurement

Accuracy

[18] Since during data processing we carry out dec**on**voluti**on**,

formally (mathematically, in view of accuracy

of dec**on**voluti**on** algorithm) the spatial resoluti**on** of ~P(r)

**on**ly by the sampling frequency.

However, owing to noise and to the previously

6of8

menti**on**ed problem of algorithm instability near the

profile zeroes, the same resoluti**on** for Ne(r) cannot

usually be achieved. It may be seen that there is no signal

near the minima and there are no data here. However, in

practical use, the positi**on**s of minima points are the data

itself and, fortunately, the variati**on** of Ne(r) at heights of

zeroes (above 200 km) are slow, and these data gaps are

not critical if some approximati**on** is used. As a best

approximati**on**, we select the spatial resoluti**on** which

results using a transmitted pulse of durati**on** 50–200 ms.

Of cause, resoluti**on** will be better when Ne(r) islarger

and the transmitted pulse is shorter.

[19] Since the noise level also defines the accuracy of

Ne(r) derivati**on**, for estimati**on** of measurement error the

following algorithm is used. The profile Ne(r), obtained

as a result of calculati**on**s, is substituted in expressi**on**

(10), and the resulting power profile P 0 (r) is compared

**with** the experimental profile P(r). Using a sliding

window **with** a durati**on** T, an average square-law

deviati**on** s 2 (rj) = PjþT

(P 0 (ri) P(ri)) 2 is calculated at

i¼j

RS3001

every height. The value of s 2 , expressed in percentage

terms, is then used as the relative error of

In this way, calculati**on** errors appear in the value of

measurement error, including errors near minima, from

noise and other causes.

[20] An example of calculati**on** results is shown in

Figure 4, where the thin c**on**tinuous line is an experimental

power profile, the dotted line is the profile ~P(r)

after dec**on**voluti**on** and correcti**on** for the factor r 2 , and

the thick line is the restored shape of Ne(r) after the

algorithm has been applied. Curves are presented in

relative units. The shape of Ne(h) is not proporti**on**al to

Figure 5. Comparis**on** of critical frequencies derived by

the described technique and by i**on**os**on**de. C**on**tinuous

lines corresp**on**d to **radar** data **with** 6 minute resoluti**on**.

Circles are i**on**os**on**de data **with** 15 minute resoluti**on**.

RS3001 SHPYNEV: IS MEASUREMENTS USING SINGLE POLARIZATION

Figure 6. Example of data processing **on** April 14,

2001. (bottom) Normalized power profile, plotted as a

functi**on** of time. (top) Ne(h) profile. The black curves **on**

the bottom panel are the **on**

be used for analysis of traveling i**on**ospheric disturbances

(TID). TIDs are visible as wavelike distorti**on** of

isopleths in the time interval 14–22 LT. TID height

distributi**on** and dynamics are also visible in variati**on**s of

Ne(h) (top panel).

envelope of ~P(r)r 2 due to different value of Te/Ti ratio at

different altitudes. The greater value of Te/Ti, the less the

area under power profile curve.

7. Testing and Applying the Method

[21] The described method of Ne(r) calculati**on** has

been implemented for processing of regular experimental

data measured at the Irkutsk incoherent **radar**. For

testing of the method, we have compared the critical

frequency of the F2 i**on**osphere maximum, calculated

from Ne(h) profile, **with** f0F2 data of an i**on**os**on**de

located at Irkutsk which has resumed operati**on**s this

year. The comparis**on** is presented in Figure 5, where

circles corresp**on**d to the i**on**os**on**de data **with** 15 minute

time resoluti**on**, and c**on**tinuous lines represent the **radar**

data **with** 6 minutes resoluti**on**. As **on**e can see from this

plot, there is good agreement between the measured

values, even for the daytime interval **on** October 3, when

wavelike variati**on**s of f0F2 are present. Some differences

in the plots may be caused both by the different time

resoluti**on** of methods and the slightly different separati**on**s

of **radar** and i**on**os**on**de (approximately 120 km).

7of8

[22] To show the capabilities of our technique, in

Figure 6 we present two c**on**tours, received **on** April 14,

2001. One of them (bottom panel) is the normalized

power profile, plotted as a functi**on** of time, and another

**on**e (top panel) is the N e(h) profile result of our algorithm,

plotted in the same manner. The black curves **on** the

bottom panel are the **on**

used for analysis of traveling i**on**ospheric disturbances

(TID) as in the work of Goodman [1971] for investigati**on**

of neutral internal and surface gravity waves. TIDs

are visible in Figure 6 as wavelike distorti**on**s of

isopleths in the time interval 14–22 LT. Time variati**on**s

of N e(h) resulting from our algorithm (top panel of

Figure 6) show not **on**ly changing electr**on** density **with**

time, but also indicate the heights where these variati**on**s

appear and their dynamic behavior **with** altitude.

8. C**on**clusi**on**

[23] We have described a method of Ne(r) derivati**on**,

**with** account for

an antenna **with** a single linear polarizati**on** for i**on**ospheric

research. For realizati**on** of incoherent

**on** such systems, we must carry out

simultaneously two types of **on**al

**on** functi**on**s)

using a l**on**g pulse, and additi**on**al

power using a short pulse. These allow us to take into

account i**on**ospheric

and to obtain the absolute values of electr**on** density

**with**out additi**on**al external calibrati**on**.

[24] Acknowledgments. The author is grateful to A. P.

Potekhin, I. I. Orlov, O. I. Berngardt, A. L. Vor**on**ov, and P. J.

Ericks**on** for useful discussi**on** and help **with** preparing the

paper. This work was supported by grants of RFBR (Russian

Foundati**on** for Basic Research) NSh-272.2003.5 and 01-05-

65374.

References

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Buneman, O. (1962), Scattering of radiati**on** by the fluctuati**on**s

in a n**on**equilibrium plasma, J. Geophys. Res., 67, 2050–

2053.

Evans, J. V. (1969), Theory and practice of i**on**osphere study by

Thoms**on** **radar**, Proc. IEEE, 57, 496–530.

Farley, D. T. (1969), **on**

incoherent

Farley, D. T., J. P. McClure, D. L. Sterling, and J. G. Green

(1967), Temperature and compositi**on** of the equatorial i**on**osphere,

J. Geophys. Res., 72, 5837.

Ginzburg, V. L. (1967), Propagati**on** of Electromagnetic Waves

in a Plasma, Nauka, Moscow.

Goodman, J. M. (1971), Traveling i**on**ospheric disturbances

observed through use of a Thoms**on**

RS3001 SHPYNEV: IS MEASUREMENTS USING SINGLE POLARIZATION

employing **on**

Phys., 33, 1763–1777.

Grigorenko, E. B. (1979), I**on**os. Issled., 27, 60–73.

Kravtsov, Y. A., and Y. I. Orlov (1980), The Geometric Optics

of N**on**homogeneous Media, Nauka, Moscow.

Millman, G. H., et al. (1961), Effect of **on****on**

incoherent back**on**s, J. Geophys. Res., 66,

1564–1568.

Millman, G. H., V. C. Pineo, and D. P. Hynek (1964), I**on**ospheric

investigati**on**s by the **on**

back

Natterer, F. (1986), The Mathematics of Computerized Tomography,

John Wiley, Hoboken, N. J.

Pingree, J. E. (1990), Ph.D. thesis, Cornell Univ., Ithaca, N. Y.

Rytov, S. M., Y. A. Kravtsov, and V. I. Tatarsky (1978),

Introducti**on** to Statistical Radio Physics, Nauka, Moscow.

8of8

RS3001

Sheffield, J. (1975), Plasma Scattering of Electromagnetic

Radiati**on**, Academic, San Diego, Calif.

Suni, A. L., V. D. Tereshchenko, E. D. Tereshchenko, and B. Z.

Khuduk**on** (1989),

High-Latitude I**on**osphere, 182 pp., AN SSSR, Apatity.

Tatarsky, V. I. (1967), Wave Propagati**on** in a Turbulent

Atmosphere, 548 pp., Nauka, Moscow.

Tkachev, G. N., and V. T. Rozumenko (1972), Geomagn.

Aer**on**, 12(4), 657–661.

Vor**on**ov, A. L., and B. G. Shpynev (1998), Excluding of c**on**voluti**on**

**with** sounding impulse in experimental

Scatter power profile, Proc. SPIE, 3583, 414–418.

B. G. Shpynev, Institute of Solar-Terrestrial Physics, P.O.

Box 4026, 664033 Irkutsk, Russia. (shpynev@iszf.irk.ru)