Mod

ern

radi

o te

chni

ques

for

prob

ing

the

iono

sphe

re

Cesi

dio

Bian

chi I

NGV

- R

oma

Ital

y

HF

Dop

pler

sys

tem

HF

Dop

pler

sys

tem

Th

e m

easu

rem

ent

tech

niqu

es o

f th

e D

oppl

er d

rift

in t

he a

dvan

ced

iono

sond

es d

eriv

es f

rom

the

iono

sphe

ric

vert

ical

sou

ndin

g wh

ere

the

retu

rnsi

gnal

is a

naly

sed

in f

requ

ency

dom

ain.

It

can

be

the

iono

sond

e it

self

or

diff

eren

t

inst

rum

ents

abl

e to

per

form

a s

pect

ral a

naly

sis

an

d ca

lcul

ate

the

Dop

pler

fre

quen

cy s

hift

supe

rpos

itio

ned

at t

he r

efle

cted

ech

o si

gnal

radi

al c

ompo

nent

of

the

vect

or

Dop

pler

•H

F D

oppl

er s

yste

m in

par

ticu

lar

cond

itio

nm

easu

res

the

plas

ma

drif

t ve

loci

ty.

•Th

e de

tect

ed r

adia

l com

pone

nt o

f th

e ve

ctor

v is

rela

ted

to t

he

shif

t in

fre

quen

cy b

y:

dtdlcf

f−

=∆

wher

e,

f ∆is

the

fre

quen

cy s

hift

mea

sure

d, c

the

ligh

t ve

loci

ty,

f t

he f

requ

ency

and

( d l/

dt) t

he t

ime

deri

vati

ve o

f th

e p

hase

pat

h l

∫=

0

)(

Sdr

rn

l

Phas

e pa

th

In g

ener

al t

he p

hase

pat

h of

the

wav

e ha

s di

ffer

ent

dire

ctio

n re

spec

t to

the

wav

e ra

y

∫=

s

dsf

nl

0

)co

s()

(α

wher

e α

is t

he a

ngle

bet

ween

the

pro

paga

tion

dire

ctio

n an

d th

e ra

y di

rect

ion.

s 0

s

d

Whe

n t

he r

efra

ctiv

e in

dex

does

not

cha

nge

in t

ime

we h

ave:

vr. cff

−=

∆

wher

e r

the

ray

dire

ctio

n an

d v

the

drif

t v

eloc

ity

of t

hepl

asm

a. T

he v

eloc

ity

v (m

/s) i

s th

e qu

anti

ty t

hat

usua

llywe

wan

t kn

ow.

D1

As

in t

he p

revi

ous

rela

tion

the

ang

ular

pul

sati

on s

hift

∆ω

is:

  wher

e v

is t

he r

adia

l vel

ocit

y of

the

ref

lect

ing

poin

t, c

is

the

light

vel

ocit

y. D

ivid

ing

by

2 π b

oth

side

of

the

abov

eeq

uati

on a

nd r

epla

cing

ω

= 2 π

f w

ith

c k

and

we

obta

in:

    wher

e, f

is t

he f

requ

ency

, ∆f D

is

the

fre

quen

cy s

hift

due

to t

he D

oppl

er e

ffec

t , a

nd k

wav

e ve

ctor

.

cv/

2 ωω

−=

∆

π/kv

f D−

=∆

Dop

pler

inte

rfer

omet

ry

Spec

ial s

pect

ral a

naly

sis

usin

g a

com

plex

FFT

Inpu

t da

ta o

f th

e co

mpl

ex F

FT a

re c

oupl

e of

Nva

lues

or

com

plex

num

ber

(a+j

b)

CFFT

out

put

yiel

ds a

cou

ple

of N

tim

e-in

dipe

nden

t va

lues

from

-N

/2 t

o N

/2 r

epre

sent

ing

ampl

itud

e an

d ph

ase

of e

ach

spec

tral

line

. In

this

con

text

neg

ativ

e sp

ectr

al li

nes

have

aph

ysic

al s

igni

fica

nce.

The

y m

eans

mot

ion

of t

he s

ourc

e aw

ayfr

om t

he o

bser

vati

on s

ite.

A r

adio

wav

e o

f an

gula

r fr

eque

ncy

ω i

s se

nt t

owar

ds t

heio

nosp

here

whe

re it

is r

efle

cted

. In

the

freq

uenc

y ra

nge

of 2

-15

MH

z th

e so

lid b

eam

ang

le o

f th

e em

ploy

ed a

nten

na is

ver

ywi

de a

nd t

he g

ain

is a

s lo

w as

1-3

dB

depe

ndin

g on

the

freq

uenc

y, s

o th

at a

n ar

ea o

f hu

ndre

ds o

f sq

uare

km

is

illum

inat

ed.

Bec

ause

of

the

rip

pled

iono

sphe

ric

surf

aces

and

the

vo

lum

ein

hom

ogen

eiti

es,

the

sig

nal

is r

efle

cted

bac

k fr

om v

ario

uspo

int

sour

ces

that

sat

isfy

the

ref

lect

ion

law.

The

se p

oint

sour

ces

if m

ovin

g ar

e co

nsid

ered

as

Dop

pler

sou

rces

tha

tsu

perp

osit

ion

a D

oppl

er s

hift

(∆ω

1 ∆ω

2 ...

....)

to t

he s

igna

l.

D2

The

mov

ing

sour

ce (

s )

adds

a s

hift

in f

requ

ency

ac

cord

ing

toth

e ab

ove

equa

tion

and

in t

he a

nten

na w

e re

ceiv

e a

com

posi

tesi

gnal

wit

h di

ffer

ent ∆ω

s th

at a

re t

he c

ontr

ibut

ion

of t

he s

sign

ific

ant

sour

ces.

In c

ase

of

a si

ngle

rec

eivi

ng a

nten

na

the

spec

tral

ana

lysi

sfu

rnis

hes

all t

he s

pect

ral c

ompo

nent

s an

d th

e re

late

d ph

ases

and

cann

ot

disc

rim

inat

e th

e sp

atia

l di

stri

buti

on

of

the

diff

eren

t so

urce

s.

In

case

of

3 or

m

ore

ante

nnas

it

is

po

ssib

le

with

an

inte

rfer

omet

ric

Dop

pler

te

chni

que

to

solv

e th

e sp

atia

ldi

stri

buti

on o

f th

e si

gnif

ican

t so

urce

s. 

Inte

rfer

omet

ric

anal

ysis

2D-

geom

etry

Onl

y on

e so

urce

(ref

lect

or p

oint

)

No

com

posi

te e

cho

sign

al

Dop

pler

sou

rce

Dop

pler

syst

em

mor

e th

an 9

0 km

Spac

ed a

nten

na a

rray

ab

out

50 m

Ant

enna

1

Ant

enna

a

l

rece

ived

sig

nal

A1

= A

01 c

os [(ω

+ ∆ω

)t +

φ1 ]

A a

= A

0a c

os [(ω

+ ∆ω

)t +

φa

]

Phas

e re

lati

on

φ1

= k

r1 + δ

φa =

k ra

+ δ

= k r1

+ k⋅l

a + δ

Phas

e at

the

ant

enna

1

Phas

e at

the

ant

enna

a

∆ φ

= φ1

- φa

= k⋅ l

a ⋅

= k

l a c

os ( θ

) = 2π

l cos

( θ)/

λ

Phas

e di

ffer

ence

bet

ween

ant

enna

1 an

d a

D7

Each

ant

enna

a, n

egle

ctin

g th

e en

viro

nmen

tal e

lect

rom

agne

tic

nois

e, w

ill r

ecei

ve a

sig

nal o

f am

plit

ude

A (t

) giv

en b

y:  

Aa (t

) =

A0 a

cos

[(ω

+∆ω

)t +

φa ]

Aft

er t

he q

uadr

atur

e sa

mpl

ing

the

sign

al o

f th

e tw

o fo

llowi

ngdi

scre

te-t

ime

sequ

ence

s Xa

and

Ya

will

be o

btai

ned:

Xa

(τ n

) =

A0a

(s)

cos(

∆ω

(τ n

) + φ

a )

Ya (τ n

) =

A0a

(s)

sin

( ∆ω

(τ n

) + φ

as)

wher

e τn

is

the

sam

plin

g ti

me

inte

rval

. If

the

sam

plin

g is

perf

orm

ed a

t ex

actl

y th

e ti

me

peri

od τ

n of

the

car

rier

wav

eω

, the

sam

plin

g it

self

ac

ts li

ke a

filt

er r

ejec

ting

the

car

rier

ω a

nd

the

two

sequ

ence

s wi

ll co

ntai

n on

ly t

he D

oppl

er s

hift

∆ω.

D8

Am

plit

ude

A(τ1

) an

d ph

ase φ (τ1

) o

f th

e si

gnal

at a

giv

en t

ime τ1

are

:

φ =a

rcta

n [

Ya (τ 1

) /

Xa

(τ1

)]

AX

Ya

a=

+(

)(

)τ

τ1

12

2

D9

The

two

disc

rete

-tim

e se

quen

ces

Xa

and

Ya

are

the

inpu

t of

the

algo

rith

m t

he p

erfo

rms

a co

mpl

ex F

ast

Four

ier

Tran

sfor

m,

(CFF

T).

The

FF

T of

the

N s

ampl

es (w

here

N is

a p

ower

of

2)ca

n be

wri

tten

as:

    wher

e n

is a

n in

dex

that

run

s fr

om –

N/2

to

N/2

,

d i

s a

dum

my

inde

x us

ed t

o pe

rfor

m t

he o

pera

tion

, an

d f

a (n

) is:

 fa

(n) =

Xa

( τn)

+ iY

a ( τ

n)    It

is w

orth

to

appl

y a

tape

ring

fun

ctio

n (

Han

ning

or

othe

rs)

to t

he d

iscr

ete

and

fini

te t

ime

sequ

ence

s to

avo

id t

he r

ingi

ngsi

x (x

)/x

aft

er t

he s

pect

ral a

naly

sis.

Fd

fn

ea

nN

Na

iN

dn(

)(

)/

/=

=−−

−Σ

22

12π

∆ φ

=2 π

l co

s(θ)

/ λ

3D-

geom

etry

3D 2

The

mov

ing

sour

ce (

s ),

that

are

are

as o

f ap

prop

riat

e RC

Sad

ds a

shi

ft i

n fr

eque

ncy

to t

he c

arri

er w

ave

and

in

the

ante

nna

arra

y re

ceiv

es a

com

posi

te s

igna

l wit

h di

ffer

ent ∆ω

sth

at a

re t

he c

ontr

ibut

ion

of t

he s

sou

rces

.

In c

ase

of

4 a

nten

nas

it is

pos

sibl

e wi

th a

n in

terf

erom

etri

cD

oppl

er t

echn

ique

to

obta

in

the

spat

ial

dist

ribu

tion

of

the

mai

n so

urce

s. 

3D3

Pr

inciple

of D

oppler

int

erfe

romet

ry    In

pr

inci

ple

the

int

erfe

rom

etri

c D

oppl

er t

echn

ique

rel

y on

the

freq

uenc

y an

d ph

ase

anal

ysis

of

the

sign

al p

icke

d-up

fro

msp

aced

ant

enna

s ly

ing

in a

pla

ne a

t an

app

ropr

iate

dis

tanc

e. I

nou

r de

scri

ptio

n we

re

fer

to

4

ante

nnas

pl

aced

in

th

ege

omet

ric

bary

cent

re a

nd i

n ve

rtex

es o

f eq

uila

tera

l tr

iang

leas

in

figu

re.

In o

rder

to

bett

er d

iscr

imin

ate

the

pha

seco

ntri

buti

ons

it i

s im

port

ant

to m

axim

ize

the

tria

ngle

siz

e(m

ore

than

50

m),

anyw

ay t

o el

imin

ate

the

phas

e am

bigu

ity,

the

dist

ance

s be

twee

n th

e an

tenn

as f

rom

the

cen

tre

of t

hetr

iang

le m

ustn

’t b

e m

ore

than

the

wav

elen

gth λ

empl

oyed

in

the

soun

ding

sys

tem

.

D4

Now

, lik

e in

the

iono

sphe

ric

vert

ical

sou

ndin

g, s

uppo

se t

o se

nda

radi

o fr

eque

ncy

(RF)

pul

se t

hat

illum

inat

e a

larg

e ar

ea,

the

vari

ous

iono

sphe

ric

mov

ing

isod

ensi

ty s

urfa

ces

refl

ect

the

RFsi

gnal

wit

h a

supe

rpos

itio

n sh

ift

Dop

pler

. A s

ingl

e re

flec

tor

sco

ntri

bute

s wi

th a

n ec

ho s

igna

l tha

t in

the

rec

eivi

ng a

nten

na 1

has

an a

mpl

itud

e A

giv

en b

y: 

A 1

= A

01 c

os [(ω

+ ∆ω

s )t

+ φ

1s ]

  wher

e, A

01 a

nd φ

1s

are

resp

ecti

vely

the

max

imum

am

plit

ude

and

the

phas

e at

the

rec

eivi

ng a

nten

na 1

. Fo

r th

e ge

neri

can

tenn

a a

the

tim

e va

ryin

g am

plit

ude

will

be:

 A a

= A

0a c

os [(ω

+ ∆ω

s )t

+ φ

as ].

 

D5

 

The

phas

e te

rm d

ue t

o th

e so

urce

s in

the

ant

enna

1 is

: 

φ1s =

ks

⋅ r1

s + δ

  w

here

k t

he w

ave

vect

or,

r1s

is t

he o

rien

ted

vect

orfr

om t

he a

nten

na 1

and

the

sour

ce s

and

δ i

s th

e ph

ase

valu

e at

the

leve

l of

the

sou

rce

s.

I

n th

e ne

xt a

naly

sis

we c

an n

egle

ct δ

bec

ause

it is

aco

nsta

nt v

alue

tha

t di

sapp

ear

in t

he o

pera

tion

of

subt

ract

ion.

D6

If w

e as

sum

e th

e an

tenn

a 1

as r

efer

ence

poi

nt,

the

phas

efo

r th

e ge

neri

c an

tenn

a a

is:

  φ

as

= k

s ⋅ r

1s + δ

+ k

s ⋅ l

a = φ

1s +

ks ⋅ l

a ⋅  wh

ere

la is

the

ori

ente

d ve

ctor

fro

m t

he a

nten

na 1

and

the

ante

nna

a. It

mea

ns t

hat

the

phas

e di

ffer

ence

s be

twee

n th

ean

tenn

a 1

and

the

gene

ric

ante

nna

a is ks

⋅ l a

.It

mus

t al

so b

e no

ted

that

bec

ause

of

the

huge

dis

tanc

e of

the

r1s

com

pare

d wi

th t

he s

mal

l di

stan

ce b

etwe

en t

hean

tenn

as la

, the

vec

tor

ks

has

the

orie

ntat

ion

in a

ll th

e 4

ante

nnas

.

D6

Refe

rrin

g to

th

e pr

evio

us

figu

re

the

phas

e di

ffer

ence

betw

een

the

ante

nna

1 and

the

gen

eric

ant

enna

a is

: 

φ 1as

= φ1

as -φ1

as =

ks ⋅

l a⋅

= k

s la

cos

( θ)

φ 1as

=2π

/ λ

l cos

( θa)

wher

e θa

is

the

angl

e be

twee

n ks

and

la⋅

Th

e

phas

edi

ffer

ence

bet

ween

the

ant

enna

1 a

nd t

he g

ener

ic a

nten

na a

,ac

cord

ing

to t

he a

bove

equ

atio

n, is

a f

unct

ion

of

the

angl

e θ

betw

een

the

vect

or k

and

l.

a

I

t is

wor

th t

o re

mar

k th

at d

iffe

rent

Dop

pler

sou

rces

are

dist

ingu

isha

ble

by

diff

eren

t ∆ω

s val

ues

(whe

re s

is t

he n

umbe

r of

sou

rces

).

For

a gi

ven

Dop

pler

sou

rce ∆ω

, kno

wing

the

ran

ge r

,th

e wa

vele

ngth

λ,

th

e di

stan

ce

l of

se

para

tion

betw

een

the

cons

ider

ed a

nten

nas

and

mea

suri

ng t

heph

ase

diff

eren

ces φ1

as, i

t is

pos

sibl

e to

cal

cula

te t

hean

gle θ

and

con

sequ

entl

y th

e ho

rizo

ntal

pos

itio

n of

the

sour

ce in

the

iono

sphe

re.

D7

Each

ant

enna

a, n

egle

ctin

g th

e en

viro

nmen

tal e

lect

rom

agne

tic

nois

e, w

ill r

ecei

ve a

com

posi

te s

igna

l of

ampl

itud

e A

give

n by

:  Aa

(t)

= ∑

s A

0 a(s

) co

s [(ω

+∆ω

s )t

+ φ

as]

wher

e s

indi

cate

s al

l the

pos

sibl

e so

urce

. Aft

er t

he q

uadr

atur

esa

mpl

ing

the

sign

al

of

the

two

follo

wing

di

scre

te-t

ime

sequ

ence

s Xa

and

Ya

will

be o

btai

ned:

Xa

(τ n

) =

∑s

A0a

(s)

cos(

∆ω

s (τ

n ) +

φas

)

Ya (τ n

) =

∑s

A0a

(s)

sin

( ∆ω

s (τ

n ) +

φas

) wh

ere

τn i

s th

e sa

mpl

ing

tim

e in

terv

al.

If t

he s

ampl

ing

ispe

rfor

med

at

exac

tly

the

tim

e pe

riod

τn o

f th

e ca

rrie

r wa

ve ω

,th

e sa

mpl

ing

itse

lf

acts

like

a f

ilter

rej

ecti

ng t

he c

arri

er ω

and

the

two

seq

uenc

es w

ill c

onta

in o

nly

the

Dop

pler

shi

ft∆ω

s.

4 ta

ble

are

the

outp

ut o

f th

e FF

T

Dri

ft v

eloc

ity

Calc

ulat

ion

err2

Σsw

sV

rs.W

s.

2

Σsw

ser

r2Σs

ws

Vrs

Ws

Σsw

s

Radi

al v

eloc

ity

Ws ca

lcul

ated

by

mea

ns o

f la

st-s

quar

e fi

t an

alys

is

Ws =

- ∆ω

s ·c/2

ωwh

ere

s is

the

sour

ce in

dex

and

ws a

wei

ghti

ngfa

ctor

rel

ated

to

the

num

ber

of s

ourc

es

1

By s

etti

ng t

he p

arti

al d

eriv

ativ

e of

resp

ect

to V

x, V

y an

d Vz

∂

/ ∂

Vx,

∂

/ ∂

Vy

and

∂

/ ∂

Vz,

to z

ero

thre

e si

mul

tane

ous

equa

tion

are

obt

aine

dfr

om w

hich

Vx,

Vy

and

Vz a

re c

alcu

late

d.

err2

err2

err2

err2