Ionospheric Tomography I: Fundamentals of tomographic...
50
Ionospheric Tomography I: Ionospheric Tomography I: Fundamentals of tomographic Fundamentals of tomographic imaging of the ionosphere and its imaging of the ionosphere and its applications to radio propagation applications to radio propagation
Transcript of Ionospheric Tomography I: Fundamentals of tomographic...
Iono
sphe
ric T
omog
raph
y I:
Iono
sphe
ric T
omog
raph
y I:
Fund
amen
tals
of t
omog
raph
icFu
ndam
enta
ls o
f tom
ogra
phic
imag
ing
of th
e io
nosp
here
and
its
imag
ing
of th
e io
nosp
here
and
its
appl
icat
ions
to ra
dio
prop
agat
ion
appl
icat
ions
to ra
dio
prop
agat
ion
Intro
duct
ion
to to
mog
raph
yIn
trodu
ctio
n to
tom
ogra
phy
Iono
sphe
ric st
ruct
ure
Iono
sphe
ric st
ruct
ure
Rad
io w
aves
in th
e io
nosp
here
Rad
io w
aves
in th
e io
nosp
here
Mea
surin
g to
tal e
lect
ron
cont
ent
Mea
surin
g to
tal e
lect
ron
cont
ent
Tom
ogra
phic
reco
nstru
ctio
nTo
mog
raph
ic re
cons
truct
ion
Adv
anta
ges,
limita
tions
and
ver
ifica
tion
Adv
anta
ges,
limita
tions
and
ver
ifica
tion
App
licat
ions
to ra
dio
syst
ems
App
licat
ions
to ra
dio
syst
ems
Sum
mar
ySu
mm
ary
Iono
sphe
ric T
omog
raph
yIo
nosp
heric
Tom
ogra
phy
Tom
ogra
phy
Tom
ogra
phy
God
frey
G
odfr
ey H
ouns
field
Hou
nsfie
ldN
obel
Priz
e W
inne
rN
obel
Priz
e W
inne
rEM
I CA
T Sc
anne
rEM
I CA
T Sc
anne
r
X-r
ay G
eom
etry
X-r
ay G
eom
etry
Imag
e of
Bra
inIm
age
of B
rain
Tom
ogra
phic
Imag
ing
Tom
ogra
phic
Imag
ing
••O
btai
n im
age
from
its p
roje
ctio
ns
Obt
ain
imag
e fr
om it
s pro
ject
ions
••
Line
inte
gral
s alo
ng in
ters
ectin
g ra
y pa
ths
Line
inte
gral
s alo
ng in
ters
ectin
g ra
y pa
ths
••M
athe
mat
ical
idea
s lon
g un
ders
tood
Mat
hem
atic
al id
eas l
ong
unde
rsto
od••
Dev
elop
men
t of c
ompu
ters
in 1
960s
Dev
elop
men
t of c
ompu
ters
in 1
960s
••C
AT
scan
ners
CA
T sc
anne
rs••
Succ
esse
s in
med
ical
dia
gnos
tics
Succ
esse
s in
med
ical
dia
gnos
tics
••A
pplic
atio
ns to
geo
phys
ics
App
licat
ions
to g
eoph
ysic
s••
Rad
io to
mog
raph
y of
iono
sphe
reR
adio
tom
ogra
phy
of io
nosp
here
•• Fi
rst d
evel
opm
ents
in U
SA a
nd R
ussi
aFi
rst d
evel
opm
ents
in U
SA a
nd R
ussi
a••
New
exp
erim
enta
l tec
hniq
ueN
ew e
xper
imen
tal t
echn
ique
Iono
sphe
ric T
omog
raph
yIo
nosp
heric
Tom
ogra
phy
•• U
se ra
dio
sign
als f
rom
sate
llite
sU
se ra
dio
sign
als f
rom
sate
llite
s••
Rec
eive
at c
hain
of g
roun
d st
atio
nsR
ecei
ve a
t cha
in o
f gro
und
stat
ions
•• M
easu
re li
ne in
tegr
al o
f ele
ctro
n de
nsity
(TEC
)M
easu
re li
ne in
tegr
al o
f ele
ctro
n de
nsity
(TEC
)••
TEC
alo
ng in
ters
ectin
g ra
y pa
ths
TEC
alo
ng in
ters
ectin
g ra
y pa
ths
••In
vert
data
sets
in re
cons
truct
ion
algo
rithm
Inve
rt da
ta se
ts in
reco
nstru
ctio
n al
gorit
hm••
Obt
ain
2D im
age
of e
lect
ron
dens
ityO
btai
n 2D
imag
e of
ele
ctro
n de
nsity
•• La
rge-
scal
e sp
atia
l stru
ctur
e of
iono
sphe
reLa
rge-
scal
e sp
atia
l stru
ctur
e of
iono
sphe
re
Rad
io T
omog
raph
y us
ing
LEO
Sat
ellit
esR
adio
Tom
ogra
phy
usin
g LE
O S
atel
lites
• N
IMS
– N
avy
Iono
sphe
ric M
onito
ring
Sate
llite
s
• up
to si
x sa
telli
tes,
form
erly
NN
SS
• ci
rcul
ar p
olar
orb
its a
t 110
0 km
alti
tude
• ch
ain
of g
roun
d re
ceiv
ing
stat
ions
• m
easu
re to
tal e
lect
ron
cont
ent a
long
ray
path
s
Rad
io T
omog
raph
y -
Rad
io T
omog
raph
y -
Rec
onst
ruct
ion
Rec
onst
ruct
ion
Tom
ogra
phic
Imag
eTo
mog
raph
ic Im
age
Imag
e of
Spa
tial S
truct
ure
inIm
age
of S
patia
l Stru
ctur
e in
Iono
sphe
reIo
nosp
here
Spat
ial S
truct
ure
in Io
nosp
here
Spat
ial S
truct
ure
in Io
nosp
here
•• W
hy d
evel
op ra
dio
tom
ogra
phy?
Why
dev
elop
radi
o to
mog
raph
y?••
Mos
t exp
erim
enta
l tec
hniq
ues g
ive
Mos
t exp
erim
enta
l tec
hniq
ues g
ive
time
time
serie
s se
ries
••To
mog
raph
y gi
ves
Tom
ogra
phy
give
s sp
atia
lsp
atia
l im
ages
imag
es••
Wid
e-ar
eaW
ide-
area
cove
r fro
m li
mite
d gr
ound
stat
ions
cove
r fro
m li
mite
d gr
ound
stat
ions
•• R
emot
e an
d in
acce
ssib
le re
gion
sR
emot
e an
d in
acce
ssib
le re
gion
s
•• W
hy is
iono
sphe
re st
ruct
ured
?W
hy is
iono
sphe
re st
ruct
ured
?
Spat
ial S
truct
ure
– Io
nosp
heric
Bas
ics
Spat
ial S
truct
ure
– Io
nosp
heric
Bas
ics
•• Ver
tical
pro
files
of e
lect
ron
dens
ity V
ertic
al p
rofil
es o
f ele
ctro
n de
nsity
•• Var
iabl
e ho
rizon
tal s
truct
ure
Var
iabl
e ho
rizon
tal s
truct
ure
•• Int
erac
tion
of b
asic
mec
hani
sms
Inte
ract
ion
of b
asic
mec
hani
sms
Con
tinui
ty e
quat
ion
Con
tinui
ty e
quat
ion
∆∆N
/N
/ ∆∆t t =
q -
L - d
iv(
= q
- L -
div(
NNvv ))
Elec
tron
Elec
tron
dens
ityde
nsity
rate
cha
nge
rate
cha
nge
Prod
uctio
nPr
oduc
tion
Loss
Loss
Tran
spor
tTr
ansp
ort
Iono
sphe
ric B
asic
sIo
nosp
heric
Bas
ics
•• P
rodu
ctio
n (q
)
Pro
duct
ion
(q)
•• s
olar
sola
r euv
eu
v ra
diat
ion
radi
atio
n••
ato
mic
oxy
gen
(O
ato
mic
oxy
gen
(O++ )
pla
sma
in F
2-la
yer
) pla
sma
in F
2-la
yer
•• p
artic
le p
reci
pita
tion
p
artic
le p
reci
pita
tion
•• im
pact
ioni
satio
n im
pact
ioni
satio
n••
hig
h la
titud
es
hig
h la
titud
es••
ion
osph
eric
stor
ms
io
nosp
heric
stor
ms
Iono
sphe
ric B
asic
sIo
nosp
heric
Bas
ics
•• L
oss (
L)
Los
s (L)
•• n
eutra
l atm
osph
ere
chem
istry
ne
utra
l atm
osph
ere
chem
istry
•• m
olec
ular
spec
ies (
N
mol
ecul
ar sp
ecie
s (N
22))••
rea
ctio
n ra
tes
re
actio
n ra
tes
•• te
mpe
ratu
re d
epen
dent
te
mpe
ratu
re d
epen
dent
•• v
eloc
ity d
epen
dent
ve
loci
ty d
epen
dent
•• c
ompo
sitio
n ch
ange
s
com
posi
tion
chan
ges
•• [O
] / [N
[O
] / [N
22] ra
tio] r
atio
•• p
rodu
ctio
n / l
oss p
roce
sses
pr
oduc
tion
/ los
s pro
cess
es••
stor
ms
st
orm
s
Iono
sphe
ric B
asic
sIo
nosp
heric
Bas
ics
•• T
rans
port
div
(
Tran
spor
t di
v(NN
vv ))
••
mot
ion
cons
train
ed b
y m
agne
tic fi
eld
mot
ion
cons
train
ed b
y m
agne
tic fi
eld
•• n
eutra
l win
ds
n
eutra
l win
ds
••
d
iurn
al, s
easo
nal,
stor
m
d
iurn
al, s
easo
nal,
stor
m
•• d
iffus
ion
diff
usio
n
••
O
O
+ + ↔↔
H H+
+
pro
tono
sphe
re
prot
onos
pher
e
••
elec
tric
field
s -el
ectri
c fie
lds -
EExx BB
con
vect
ion
con
vect
ion
••
equ
ator
ial a
nom
aly
eq
uato
rial a
nom
aly
••
hig
h la
titud
es
high
latit
udes
Elec
tron
den
sity
at a
ny p
lace
and
tim
e de
pend
s on
the
bala
nce
Elec
tron
den
sity
at a
ny p
lace
and
tim
e de
pend
s on
the
bala
nce
betw
een
man
y di
ffere
nt p
roce
sses
betw
een
man
y di
ffere
nt p
roce
sses
Iono
sphe
re is
stru
ctur
ed sp
atia
lly o
n m
any
diffe
rent
scal
es
Iono
sphe
re is
stru
ctur
ed sp
atia
lly o
n m
any
diffe
rent
scal
es
Hor
izon
tal S
truct
ure
Hor
izon
tal S
truct
ure
•• B
alan
ceBa
lanc
e be
twee
n th
ese
man
y di
ffer
ent p
roce
sses
bet
wee
n th
ese
man
y di
ffer
ent p
roce
sses
••
Res
ults
in h
oriz
onta
l
Res
ults
in h
oriz
onta
l str
uctu
rest
ruct
ure
in io
nosp
here
in io
nosp
here
••
Nee
d to
und
erst
and
this
stru
ctur
e an
d its
orig
ins
N
eed
to u
nder
stan
d th
is st
ruct
ure
and
its o
rigin
s••
Im
porta
nt fo
r rad
io sy
stem
s
Im
porta
nt fo
r rad
io sy
stem
s ––
hf
hf p
ropa
gatio
n, G
PS c
orre
ctio
nspr
opag
atio
n, G
PS c
orre
ctio
ns
Iono
sphe
ric to
mog
raph
y cr
eate
s im
ages
of s
uch
stru
ctur
eIo
nosp
heric
tom
ogra
phy
crea
tes i
mag
es o
f suc
h st
ruct
ure
••
How
doe
s it w
ork?
H
ow d
oes i
t wor
k?••
H
ow is
radi
o w
ave
affe
cted
by
iono
sphe
re?
H
ow is
radi
o w
ave
affe
cted
by
iono
sphe
re?
Bas
ic M
agne
to-io
nic
Theo
ryB
asic
Mag
neto
-ioni
c Th
eory
How
is ra
dio
wav
e af
fect
ed b
y io
nosp
here
?H
ow is
radi
o w
ave
affe
cted
by
iono
sphe
re?
Nee
d N
eed
refr
activ
e in
dex
refr
activ
e in
dex
for p
ropa
gatio
n of
radi
o w
ave
in io
nosp
here
for p
ropa
gatio
n of
radi
o w
ave
in io
nosp
here
App
leto
n Eq
uatio
nA
pple
ton
Equa
tion
‘It i
s vir
tual
ly im
poss
ible
for a
n or
dina
ry m
orta
l to
mak
e m
uch
sens
e ‘I
t is v
irtu
ally
impo
ssib
le fo
r an
ordi
nary
mor
tal t
o m
ake
muc
h se
nse
of th
e Ap
plet
on e
quat
ion(
s) in
thei
r ful
l glo
ry.’
of th
e Ap
plet
on e
quat
ion(
s) in
thei
r ful
l glo
ry.’
(( Hun
suck
erH
unsu
cker
and
and
Har
grea
ves
Har
grea
ves ,
200
3), 2
003)
App
leto
n Eq
uatio
nA
pple
ton
Equa
tion
Ref
ract
ive
inde
x (n
) of a
n io
nise
d m
ediu
m w
ith e
lect
ron
dens
ity (N
),R
efra
ctiv
e in
dex
(n) o
f an
ioni
sed
med
ium
with
ele
ctro
n de
nsity
(N),
a m
agne
tic fl
ux (B
) and
ele
ctro
n co
llisi
on fr
eque
ncy
( a
mag
netic
flux
(B) a
nd e
lect
ron
colli
sion
freq
uenc
y ( νν
))
•• X
= (
X
= (
ωωN
N
/ / ωω
) )22
whe
re
whe
re ωω
N
N =
( N
e=
( N
e2 2 // εε
oommee ) )
1/2
1/2
is
is
pla
sma
freq
uenc
ypl
asm
a fr
eque
ncy
X
X
dep
ends
on
d
epen
ds o
n el
ectr
on d
ensi
ty (
N)
ele
ctro
n de
nsity
(N
)
••
Y =
Y
= ωω
B B / / ωω
whe
re
whe
re ωω
B B =
Be
/ m=
Be
/ me
e
isis e
lect
ron
ele
ctro
n gy
rofr
eque
ncy
gyro
freq
uenc
y
YY
de
pend
s on
depe
nds o
n m
agne
tic fi
eld
(B)
mag
netic
fiel
d (B
)
•• Z
=
Z =
νν / / ωω
whe
re
whe
re νν
is
is c
ollis
ion
rate
colli
sion
rate
Tran
s-io
nosp
heric
Pro
paga
tion
Tran
s-io
nosp
heric
Pro
paga
tion
The
radi
o w
ave
freq
uenc
y (ω
) >>
plas
ma
freq
uenc
y (ω
Th
e ra
dio
wav
e fr
eque
ncy
(ω) >
> pl
asm
a fr
eque
ncy
(ω NN
), so
that
X<<
1),
so th
at X
<<1
Neg
lect
ing
the
mag
netic
fiel
d (Y
=0) a
nd c
ollis
ions
(Z=0
) the
N
egle
ctin
g th
e m
agne
tic fi
eld
(Y=0
) and
col
lisio
ns (Z
=0) t
he r
efra
ctiv
ere
frac
tive
inde
xin
dex
now
has
a
now
has
a v
ery
sim
ple
form
very
sim
ple
form
n =
1 –
X /
2 n
= 1
– X
/ 2
ororn
= 1
– N
en
= 1
– N
e22 /
2
/ 2 εε oo
mmee ωω
22
Inse
rting
val
ues a
nd u
sing
In
serti
ng v
alue
s and
usi
ng f f
inst
ead
of
inst
ead
of ωω
for f
requ
ency
for f
requ
ency
n =
1 –
40.
3 N
/ f
n =
1 –
40.
3 N
/ f 2
2
(with
(
with
N
N i
n m
in m
-3
-3 an
dan
d f f
in H
z)in
Hz)
Sinc
e Si
nce
n <
1, p
hase
of w
ave
n <
1, p
hase
of w
ave
in io
nise
d m
ediu
m w
ill in
ioni
sed
med
ium
will
adv
ance
a
dvan
ce w
ith re
spec
tw
ith re
spec
tto
free
-spa
ce p
ropa
gatio
nto
free
-spa
ce p
ropa
gatio
n
Car
rier P
hase
Adv
ance
and
Dop
pler
Shi
ftC
arrie
r Pha
se A
dvan
ce a
nd D
oppl
er S
hift
•• ph
ase
adva
nce
is
pha
se a
dvan
ce is
cum
ulat
ive
cum
ulat
ive
alon
g pa
th a
long
pat
h••
depe
nds o
n th
e d
epen
ds o
n th
e To
tal E
lect
ron
Con
tent
(TEC
) To
tal E
lect
ron
Con
tent
(TEC
) alo
ng sl
ant p
ath
alon
g sl
ant p
ath
NNT T
=
= ∫∫
N d
lN
dl
Num
eric
ally
, N
umer
ical
ly, p
hase
adv
ance
phas
e ad
vanc
e (in
radi
ans)
due
to th
e io
nosp
here
is (i
n ra
dian
s) d
ue to
the
iono
sphe
re is
φφ =
(8.4
5 x
10
= (8
.45
x 10
-7-7) N) N
T T / f
/ f
(w
ith(w
ith N N
TT i
n in mm
-2-2 an
dan
d f f
inin H
z H
z ))
Sin
ce
Sin
ce fr
eque
ncy
freq
uenc
y is
is
rat
e of
cha
nge
of p
hase
rate
of c
hang
e of
pha
se, i
onos
pher
e im
pose
s , i
onos
pher
e im
pose
s D
oppl
erD
oppl
er sh
ift
shift
on
the
wav
eon
the
wav
e
Afte
r tra
velli
ng a
dis
tanc
e A
fter t
rave
lling
a d
ista
nce
dl
dl ( (
ieie d
l/dl/ λλ
wav
elen
gths
) w
avel
engt
hs) p
hase
of t
he w
ave
phas
e of
the
wav
eha
s cha
nged
by
has c
hang
ed b
y 22 ππ d
l/ d
l/ λλ =
=
22ππ
f n d
l / c
f n
dl /
cTh
us o
ver a
pat
h Th
us o
ver a
pat
h l l t
hrou
gh th
e io
nosp
here
the
thro
ugh
the
iono
sphe
re th
e ph
ase
chan
geph
ase
chan
ge w
ill b
e w
ill b
e-(
2-(
2ππ
f / c
) f
/ c ) ∫∫ n
dl /
c =
- n
dl /
c =
- 22 ππ
f l
/ c +
( f
l / c
+ ( 22ππ
x 4
0.3
/ x
40.
3 / c
f c
f ) ) ∫∫ N
dl
N d
l
Cha
nge
in p
hase
of a
wav
e C
hang
e in
pha
se o
f a w
ave
trave
lling
at t
he sp
eed
of li
ght
trave
lling
at t
he sp
eed
of li
ght
Phas
ePh
ase
adva
nce
adv
ance
du
e to
the
due
to th
e m
ediu
mm
ediu
m
Mea
sure
men
t of T
ECM
easu
rem
ent o
f TEC
by D
iffer
entia
l Car
rier P
hase
Met
hod
by D
iffer
entia
l Car
rier P
hase
Met
hod
•• In
pra
ctic
e, m
easu
rem
ent o
f
In p
ract
ice,
mea
sure
men
t of p
hase
phas
e re
quire
s a
requ
ires a
ref
eren
cere
fere
nce
sign
al si
gnal
•• O
ne w
ay in
whi
ch th
is c
an b
e ac
hiev
ed is
for t
he so
urce
to tr
ansm
it
One
way
in w
hich
this
can
be
achi
eved
is fo
r the
sour
ce to
tran
smit
two
cohe
rent
freq
uenc
ies
two
cohe
rent
freq
uenc
ies
deriv
ed fr
om a
com
mon
osc
illat
or d
eriv
ed fr
om a
com
mon
osc
illat
or
•• F
orm
s the
bas
is o
f the
Form
s the
bas
is o
f the
Diff
eren
tial C
arri
er P
hase
Diff
eren
tial C
arri
er P
hase
or
or D
iffer
entia
l D
iffer
entia
l
D
oppl
er
D
oppl
er te
chni
que
used
for
tech
niqu
e us
ed fo
r tom
ogap
hy
tom
ogap
hy
Diff
eren
tial C
arrie
r Pha
se (D
iffer
entia
l Dop
pler
)D
iffer
entia
l Car
rier P
hase
(Diff
eren
tial D
oppl
er)
Tech
niqu
eTe
chni
que
Sate
llite
tran
smits
two
Sate
llite
tran
smits
two
cohe
rent
co
here
nt fr
eque
ncie
s fr
eque
ncie
s f f a
nd
and
pf,
pf, w
here
p is
con
stan
tw
here
p is
con
stan
t(N
IMS
sate
llite
s use
d fo
r tom
ogra
phy
trans
mit
on 1
50M
Hz
and
400M
Hz
(NIM
S sa
telli
tes u
sed
for t
omog
raph
y tra
nsm
it on
150
MH
z an
d 40
0MH
zso
that
so
that
p =
8/3
p =
8/3
))C
ompa
re
Com
pare
rec
eive
d ph
ase
of lo
wer
freq
uenc
yre
ceiv
ed p
hase
of l
ower
freq
uenc
y w
ith th
at o
f the
w
ith th
at o
f the
hig
her
high
erfr
eque
ncy
divi
ded
by p
freq
uenc
y di
vide
d by
p∆φ∆φ
=
= φφ
f f - - φφ
pf
pf =
{-2
= {-
2 ππ f
l/ c
+ 2
f l/
c +
2ππ
x40.
3 N
x40
.3 N
T T / f
c}
/ f c
} ––
{(1/
p)({
-2{(
1/p)
({-2ππ
p f l
/ c +
2 p
f l/
c +
2ππ
x40.
3 N
x40
.3 N
T T / p
f c}
/ pf c
}Th
e Th
e fir
st a
nd th
ird
term
s can
cel
first
and
thir
d te
rms c
ance
l bec
ause
they
repr
esen
t the
pha
se b
ecau
se th
ey re
pres
ent t
he p
hase
chan
ges f
or
chan
ges f
or fr
ee sp
ace
prop
agat
ion
free
spac
e pr
opag
atio
n of
the
two
sign
als a
long
the
path
of t
he tw
o si
gnal
s alo
ng th
e pa
th
Thus
the
Thus
the
diffe
rent
ial p
hase
shift
di
ffere
ntia
l pha
se sh
ift d
ue to
the
iono
sphe
ric T
EC is
due
to th
e io
nosp
heric
TEC
is∆φ∆φ
= {1
= {1
–– 1
/ p
1 /
p22) (
8.45
x 1
0 ) (
8.45
x 1
0 -7-7
) N) NT T
/ f/ f
Can
mea
sure
C
an m
easu
re r
elat
ive
phas
e ac
cura
tely
rela
tive
phas
e ac
cura
tely
but
still
hav
e 2
but
still
hav
e 2 ππ
am
bigu
ity to
am
bigu
ity to
solv
e to
get
so
lve
to g
et a
bsol
ute
TEC
abso
lute
TEC
Abs
olut
e TE
C fr
om D
iffer
entia
l Car
rier
Abs
olut
e TE
C fr
om D
iffer
entia
l Car
rier
Phas
e M
easu
rem
ents
Phas
e M
easu
rem
ents
•• W
ith tw
o or
mor
e st
atio
ns c
an m
atch
the
obse
rvat
ion
in th
e
With
two
or m
ore
stat
ions
can
mat
ch th
e ob
serv
atio
n in
the
regi
on o
f ove
rlap
to g
ive
the
sam
e ve
rtica
l TEC
regi
on o
f ove
rlap
to g
ive
the
sam
e ve
rtica
l TEC
••
Hen
ce o
btai
n
Hen
ce o
btai
n ab
solu
te
abso
lute
TEC
mea
sure
men
ts v
ersu
s lat
itude
TEC
mea
sure
men
ts v
ersu
s lat
itude
Equi
vale
nt
Equi
vale
nt
Ver
tical
TEC
Ver
tical
TEC
Latit
ude
Latit
ude
Slan
t TEC
and
Ver
tical
TEC
Slan
t TEC
and
Ver
tical
TEC
•• S
lant
TEC
Sla
nt T
EC
NNT T
=
= ∫∫
N d
l, N
dl,
whe
re
whe
re l l
is a
is
a s
lant
slan
t ray
pat
h fr
om sa
telli
te to
gro
und
ray
path
from
sate
llite
to g
roun
d
•• V
ertic
al T
ECVe
rtic
al T
EC (
need
ed fo
r com
paris
ons a
nd m
odel
s) (
need
ed fo
r com
paris
ons a
nd m
odel
s)
NNT T
=
= ∫∫
N d
h,
N d
h, w
here
w
here
h h is
a
is a
ver
tical
vert
ical
pat
h th
roug
h io
nosp
here
pat
h th
roug
h io
nosp
here
•• E
quiv
alen
t Ver
tical
TEC
Equi
vale
nt V
ertic
al T
EC
In p
ract
ice,
mos
t mea
sure
men
ts a
re sl
ant T
EC b
ut a
re c
onve
rted
In p
ract
ice,
mos
t mea
sure
men
ts a
re sl
ant T
EC b
ut a
re c
onve
rted
to th
e ve
rtica
l usi
ng a
n as
sum
ed
to th
e ve
rtica
l usi
ng a
n as
sum
ed th
in-s
hell
iono
sphe
reth
in-s
hell
iono
sphe
re a
t a c
hose
n a
t a c
hose
nm
ean
heig
ht (
mea
n he
ight
( dl
= d
h dl
= d
h se
cse
c χχ) g
ivin
g ) g
ivin
g eq
uiva
lent
ver
tical
TEC
equi
vale
nt v
ertic
al T
EC..
•• M
ost r
efer
ence
s to
TEC
are
in fa
ct to
Mos
t ref
eren
ces t
o TE
C a
re in
fact
to e
quiv
alen
t ver
tical
TEC
equi
vale
nt v
ertic
al T
EC..
•• M
easu
re T
EC in
TEC
uni
ts, w
here
Mea
sure
TEC
in T
EC u
nits
, whe
re 1
TEC
U
1 TE
CU
≡≡ 1
01016
16
mm-2-2
Iono
sphe
ric T
omog
raph
yIo
nosp
heric
Tom
ogra
phy
Bas
ics o
f ion
osph
eric
tom
ogra
phy
- pix
els
Bas
ics o
f ion
osph
eric
tom
ogra
phy
- pix
els
For a
ll ra
y pa
ths b
etw
een
sate
llite
and
gro
und
stat
ions
For a
ll ra
y pa
ths b
etw
een
sate
llite
and
gro
und
stat
ions
M
M s
uch
sim
ulta
neou
s equ
atio
nssu
ch si
mul
tane
ous e
quat
ions
In m
atrix
not
atio
n:In
mat
rix n
otat
ion:
y =
y =
Ax
Ax
y
-
y
-sl
ant t
otal
ele
ctro
n co
nten
t mea
sure
men
tssl
ant t
otal
ele
ctro
n co
nten
t mea
sure
men
tsxx
-
-
unkn
own
elec
tron
den
sitie
sun
know
n el
ectr
on d
ensi
ties
The
prob
lem
of i
onos
pher
ic to
mog
raph
y is
to so
lve
this
equ
atio
n to
find
The
prob
lem
of i
onos
pher
ic to
mog
raph
y is
to so
lve
this
equ
atio
n to
find
the
elec
tron
den
sitie
sth
e el
ectr
on d
ensi
ties
AAijij
leng
th o
fle
ngth
of i i
thth ra
y pa
th in
ray
path
in j j
thth p
ixel
pix
elxx j j
elec
tron
dens
ity in
elec
tron
dens
ity in
j j thth
pix
el p
ixel
yy ii el
ectro
n co
nten
t alo
ng
elec
tron
cont
ent a
long
ii th
th
ra
y pa
th is
app
roxi
mat
ed b
yra
y pa
th is
app
roxi
mat
ed b
y y y
ii =
= ∑∑
AAijij
xx jj , su
mm
ed fr
om
, sum
med
from
j =
1 to
Nj =
1 to
N
Tom
ogra
phic
Alg
orith
ms -
Iter
ativ
eTo
mog
raph
ic A
lgor
ithm
s - It
erat
ive
•• E
arly
alg
orith
ms u
sed
Ea
rly a
lgor
ithm
s use
d Ro
w A
ctio
n M
etho
dsRo
w A
ctio
n M
etho
ds••
Itera
tive
Itera
tive
solu
tions
solu
tions
•• Su
cces
sive
cor
rect
ions
to a
n as
sum
ed in
itial
Succ
essi
ve c
orre
ctio
ns to
an
assu
med
initi
alBa
ckgr
ound
Iono
sphe
reBa
ckgr
ound
Iono
sphe
re••
Nee
d ba
ckgr
ound
bec
ause
of i
ncom
plet
e in
form
atio
nN
eed
back
grou
nd b
ecau
se o
f inc
ompl
ete
info
rmat
ion
••N
o ho
rizon
tal r
ay p
aths
No
horiz
onta
l ray
pat
hs
•• AR
T (A
lgeb
raic
Rec
onst
ruct
ion
Tech
niqu
esAR
T (A
lgeb
raic
Rec
onst
ruct
ion
Tech
niqu
es
Afte
r k it
erat
ions
the
set o
f ele
ctro
n de
nsiti
es is
giv
en b
y
A
fter k
iter
atio
ns th
e se
t of e
lect
ron
dens
ities
is g
iven
by
whe
rew
here
with
w
ith aa
i i rep
rese
ntin
g th
ere
pres
entin
g th
e i i t
h th ro
w o
f ro
w o
f A
A a
nd
and λλ k
k a
rela
xatio
n co
nsta
nt <
1a
rela
xatio
n co
nsta
nt <
1
Tom
ogra
phic
Alg
orith
ms –
Dire
ct In
vers
ion
Tom
ogra
phic
Alg
orith
ms –
Dire
ct In
vers
ion
Dis
cret
e In
vers
e Th
eory
(DIT
)D
iscr
ete
Inve
rse
Theo
ry (D
IT)
••
Man
y di
ffer
ent m
athe
mat
ical
tech
niqu
es h
ave
been
use
d by
M
any
diff
eren
t mat
hem
atic
al te
chni
ques
hav
e be
en u
sed
by
diff
eren
t wor
kers
diff
eren
t wor
kers
••
Fin
ding
an
appr
opria
te
Find
ing
an a
ppro
pria
te b
ackg
roun
d io
nosp
here
back
grou
nd io
nosp
here
to in
itial
ise
any
to in
itial
ise
any
algo
rithm
is th
e al
gorit
hm is
the
key
to io
nosp
heri
c to
mog
raph
yke
y to
iono
sphe
ric
tom
ogra
phy
••
Use
ext
ensi
on o
f DIT
met
hod
of
Use
ext
ensi
on o
f DIT
met
hod
of F
rem
ouw
Fre
mou
w e
t al.
(199
2 an
d 19
94) t
o e
t al.
(199
2 an
d 19
94) t
oob
tain
ob
tain
bac
kgro
und
iono
sphe
reba
ckgr
ound
iono
sphe
re
••
Des
crib
ed b
y lin
ear c
ombi
natio
n of
a n
umbe
r of
D
escr
ibed
by
linea
r com
bina
tion
of a
num
ber o
f bas
is fu
nctio
nsba
sis f
unct
ions
with
diff
eren
t w
ith d
iffer
ent w
eigh
tings
wei
ghtin
gs
Firs
t Sta
ge o
f Rec
onst
ruct
ion
Firs
t Sta
ge o
f Rec
onst
ruct
ion
To o
btai
n To
obt
ain
basi
s fun
ctio
nsba
sis f
unct
ions
•• v
ertic
al fo
rm:
Cha
pman
pro
files
of E
and
F re
gion
sve
rtica
l for
m:
Cha
pman
pro
files
of E
and
F re
gion
s•• s
pann
ing
com
plet
e ra
nge
foun
d in
dat
a se
t of
span
ning
com
plet
e ra
nge
foun
d in
dat
a se
t of
•• pea
k he
ight
s, p
eak
heig
hts,
•• sca
le h
eigh
ts a
nd sc
ale
heig
hts a
nd•• s
cale
-hei
ght g
radi
ents
in to
psid
e sc
ale-
heig
ht g
radi
ents
in to
psid
e••
hor
izon
tal s
truct
ure:
Fou
rier s
erie
s with
pow
er-la
w ta
per
h
oriz
onta
l stru
ctur
e: F
ourie
r ser
ies w
ith p
ower
-law
tape
r
•• U
se
U
se tr
unca
ted
Sing
ular
Val
uetr
unca
ted
Sing
ular
Val
ueD
ecom
posi
tion
(SVD
)D
ecom
posi
tion
(SVD
) to
obta
in se
t of
to o
btai
n se
t of
Empi
rica
l Em
piri
cal O
rtho
norm
alO
rtho
norm
al F
unct
ions
Fun
ctio
ns(( E
OFs
EOFs
) to
form
) t
o fo
rm b
asis
.ba
sis.
•• B
ackg
roun
d io
nosp
here
Back
grou
nd io
nosp
here
can
be
desc
ribed
can
be
desc
ribed
by
by li
near
com
bina
tion
linea
r com
bina
tion
of th
ese
with
of t
hese
with
diff
eren
t di
ffer
ent w
eigh
tings
wei
ghtin
gs
Tom
ogra
phy
Alg
orith
ms –
Dis
cret
e In
vers
e Th
eory
Tom
ogra
phy
Alg
orith
ms –
Dis
cret
e In
vers
e Th
eory
Back
grou
nd io
nosp
here
Back
grou
nd io
nosp
here
des
crib
ed b
y d
escr
ibed
by
linea
r com
bina
tion
linea
r com
bina
tion
of a
num
ber o
f o
f a n
umbe
r of
basi
s fun
ctio
nsba
sis f
unct
ions
with
diff
eren
t w
ith d
iffer
ent w
eigh
tings
wei
ghtin
gs
b =
B x
b =
B x
b -
b
-
the
the
elec
tron
den
sity
elec
tron
den
sity
val
ues i
n pi
xels
, v
alue
s in
pixe
ls,
B
- B
-
basi
s fun
ctio
nsba
sis f
unct
ions
repr
esen
ting
iono
sphe
re re
pres
entin
g io
nosp
here
x -
x
-
wei
ght
wei
ght g
iven
to e
ach
basi
s fun
ctio
n g
iven
to e
ach
basi
s fun
ctio
n
Usi
ng
Usi
ng H
-
H
- ge
omet
ryge
omet
ry m
atrix
of p
ath/
pixe
l int
erse
ctio
ns m
atrix
of p
ath/
pixe
l int
erse
ctio
ns
y
-
y
-m
easu
red
TEC
mea
sure
d TE
C ≈≈
TEC
thro
ugh
back
grou
nd (
TEC
thro
ugh
back
grou
nd ( H
BxH
Bx))
Thus
Th
us
y =
H B
x
y =
H B
x
or
or
y =
A x
y =
A x
whe
re n
ow
whe
re n
ow A
= H
BA
= H
B
Tom
ogra
phy
Alg
orith
ms –
Dis
cret
e In
vers
e Th
eory
Tom
ogra
phy
Alg
orith
ms –
Dis
cret
e In
vers
e Th
eory
Nee
d to
solv
eN
eed
to so
lve
y =
A x
y =
A x
•• R
efor
mul
ate
to a
void
nee
d fo
r
Ref
orm
ulat
e to
avo
id n
eed
for a
bsol
ute
calib
ratio
nab
solu
te c
alib
ratio
n of
mea
sure
d TE
C o
f m
easu
red
TEC
•• U
se
U
se y
y as
as
diff
eren
ces
diffe
renc
es in
TEC
bet
wee
n su
cces
sive
ray
path
s, th
at is
, use
in T
EC b
etw
een
succ
essi
ve ra
y pa
ths,
that
is, u
sere
lativ
e TE
Cs
rela
tive
TEC
s
•• S
olve
to fi
nd
So
lve
to fi
nd x
-
x -
the
wei
ght g
iven
to e
ach
basi
s fun
ctio
n in
the
linea
rth
e w
eigh
t giv
en to
eac
h ba
sis f
unct
ion
in th
e lin
ear
com
bina
tion
that
bes
t des
crib
es th
e ba
ckgr
ound
iono
sphe
reco
mbi
natio
n th
at b
est d
escr
ibes
the
back
grou
nd io
nosp
here
,,co
nsis
tent
with
the
mea
sure
d TE
Cco
nsis
tent
with
the
mea
sure
d TE
C
2.
2. F
ind
Find
sm
alle
r -sc
ale
stru
ctur
esm
alle
r -sc
ale
stru
ctur
e ::Ite
rativ
eIte
rativ
e se
cond
stag
e us
es
seco
nd st
age
uses
bac
kgro
und
iono
sphe
reba
ckgr
ound
iono
sphe
re
gene
rate
d by
the
first
stag
ege
nera
ted
by th
e fir
st st
age
as st
artin
g co
nditi
on fo
r a
s sta
rting
con
ditio
n fo
ral
gebr
aic
reco
nstr
uctio
n te
chni
que
(ART
)al
gebr
aic
reco
nstr
uctio
n te
chni
que
(ART
)al
gorit
hm.
algo
rithm
.
•• Im
age
Grid
:Im
age
Grid
:A
ltitu
de:
25 k
m
Lat
itude
: 0.
25 d
egre
eA
ltitu
de:
25 k
m
Lat
itude
: 0.
25 d
egre
e
•• C
an u
se m
etho
d to
inco
rpor
ate
othe
r typ
es o
f ion
osph
eric
C
an u
se m
etho
d to
inco
rpor
ate
othe
r typ
es o
f ion
osph
eric
m
easu
rem
ents
- fo
r exa
mpl
e, io
noso
nde
data
mea
sure
men
ts -
for e
xam
ple,
iono
sond
e da
ta
Seco
nd S
tage
of R
econ
stru
ctio
nSe
cond
Sta
ge o
f Rec
onst
ruct
ion
Rad
io T
omog
raph
y:R
adio
Tom
ogra
phy:
Adv
anta
ges a
nd L
imita
tions
Adv
anta
ges a
nd L
imita
tions
Adv
anta
ges
Adv
anta
ges
•• N
ew e
xper
imen
tal t
echn
ique
New
exp
erim
enta
l tec
hniq
ue••
Spat
ial i
mag
es o
f lar
ge-s
cale
den
sity
stru
ctur
esSp
atia
l im
ages
of l
arge
-sca
le d
ensi
ty st
ruct
ures
•• W
ide
cove
rage
from
lim
ited
grou
nd st
atio
ns
Wid
e co
vera
ge fr
om li
mite
d gr
ound
stat
ions
••
Com
plem
enta
ry ro
le to
oth
er in
stru
men
tsC
ompl
emen
tary
role
to o
ther
inst
rum
ents
Lim
itatio
ns
Lim
itatio
ns
••U
nder
stoo
d at
ear
ly st
age
Und
erst
ood
at e
arly
stag
e••
Lim
ited-
angl
e te
chni
que,
no
horiz
onta
l ray
pat
hsLi
mite
d-an
gle
tech
niqu
e, n
o ho
rizon
tal r
ay p
aths
••In
com
plet
e in
form
atio
n on
ver
tical
stru
ctur
eIn
com
plet
e in
form
atio
n on
ver
tical
stru
ctur
e••
Tem
pora
l cov
erag
e de
pend
ent o
n sa
telli
te o
rbits
Tem
pora
l cov
erag
e de
pend
ent o
n sa
telli
te o
rbits
Doe
s it w
ork?
Doe
s it w
ork?
Expe
rimen
tal s
tatio
n ch
ains
use
d by
UW
A g
roup
Expe
rimen
tal s
tatio
n ch
ains
use
d by
UW
A g
roup
UK
UK
Sval
bard
Sval
bardSc
andi
navi
aSc
andi
navi
a
Doe
s it w
ork?
Doe
s it w
ork?
Ver
ifica
tion
usin
g EI
SCA
T ra
dar
Ver
ifica
tion
usin
g EI
SCA
T ra
dar
Diff
eren
ces i
n La
yer H
eigh
tD
iffer
ence
s in
Laye
r Hei
ght
Ker
sley
et a
l. (1
997)
Rad
io T
omog
raph
ic Im
agin
g:R
adio
Tom
ogra
phic
Imag
ing:
App
licat
ions
to p
ract
ical
radi
o sy
stem
sA
pplic
atio
ns to
pra
ctic
al ra
dio
syst
ems
•• C
ompl
emen
tary
to io
noso
nde
mea
sure
men
ts C
ompl
emen
tary
to io
noso
nde
mea
sure
men
ts••
Val
idat
ion
of io
nosp
heric
mod
els
Val
idat
ion
of io
nosp
heric
mod
els
•• M
appi
ng o
f ion
osph
eric
par
amet
ers
Map
ping
of i
onos
pher
ic p
aram
eter
s••
Obl
ique
soun
ding
Obl
ique
soun
ding
•• H
F ra
y tra
cing
HF
ray
traci
ng••
HF
dire
ctio
n fin
ding
HF
dire
ctio
n fin
ding
•• Sp
ace
wea
ther
Spa
ce w
eath
er
Rad
io T
omog
raph
y an
d Io
noso
ndes
Rad
io T
omog
raph
y an
d Io
noso
ndes
Slou
gh51
.5N
Lann
ion
48.8
N
S
L
S S
L L
0250
UT
0320
UT
0439
UT
Tom
ogra
phy
and
Test
ing
of E
mpi
rical
or
Tom
ogra
phy
and
Test
ing
of E
mpi
rical
or
Para
met
eris
ed M
odel
sPa
ram
eter
ised
Mod
els
Tom
ogra
phy
UK
IRI-
95
PIM
CO
ST23
8
Dab
as a
nd K
ersl
ey (2
003)
Tom
ogra
phy
and
Cou
pled
The
rmos
pher
eTo
mog
raph
y an
d C
oupl
ed T
herm
osph
ere
Iono
sphe
reIo
nosp
here
Pla
smas
pher
e P
lasm
asph
ere
Mod
el (S
UC
TIP)
Mod
el (S
UC
TIP)
Iden
den
et a
l.(1
999)
a) I
RI-
95 a
lone
a) IR
I-95
alo
ne
b) I
RI-
95 p
lus
b
) IR
I-95
plu
s to
mog
raph
yto
mog
raph
y
Rad
io T
omog
raph
ic Im
agin
g:R
adio
Tom
ogra
phic
Imag
ing:
App
lied
to
App
lied
to m
appi
ng o
f ion
osph
eric
par
amet
ers
map
ping
of i
onos
pher
ic p
aram
eter
s
Map
s of p
eak
elec
tron
dens
ity (N
mF2
) ove
r Eur
ope
Map
s of p
eak
elec
tron
dens
ity (N
mF2
) ove
r Eur
ope
Dab
as a
nd K
ersl
ey (2
003)
Val
idat
ion
of fo
F2 M
aps
Val
idat
ion
of fo
F2 M
aps
•• Tom
ogra
phy
from
UK
To
mog
raph
y fr
om U
K
s
tatio
ns +
Chi
lton
s
tatio
ns +
Chi
lton
i
onos
onde
i
onos
onde
•• Val
idat
ion
usin
g V
alid
atio
n us
ing
iono
sond
es n
ear
iono
sond
es n
ear
troug
h an
d at
mid
-
tro
ugh
and
at m
id-
latit
udes
latit
udes
Val
idat
ion
of M
aps u
sing
Iono
sond
esV
alid
atio
n of
Map
s usi
ng Io
noso
ndes
•• U
se o
f tom
ogra
phic
imag
e gi
ves b
ette
r agr
eem
ent w
ith
Use
of t
omog
raph
ic im
age
give
s bet
ter a
gree
men
t with
io
noso
nde
foF2
than
any
of t
he m
odel
s alo
ne
io
noso
nde
foF2
than
any
of t
he m
odel
s alo
ne••
Pot
entia
l for
Pot
entia
l for
now
cast
ing
now
cast
ing
Tom
ogra
phy
and
Obl
ique
Tom
ogra
phy
and
Obl
ique
Iono
gram
s Io
nogr
ams
Tom
ogra
phy
Cha
inTo
mog
raph
y C
hain
IRIS
Obl
ique
IR
IS O
bliq
ue
Soun
der
Net
wor
kSo
unde
r N
etw
ork
Hea
ton
et a
l. (2
001)
•• P
rofil
es a
t mid
-poi
nt sh
ow g
ood
agre
emen
t in
F-la
yer
Pr
ofile
s at m
id-p
oint
show
goo
d ag
reem
ent i
n F-
laye
r••
Tom
ogra
phic
imag
es m
ay h
elp
in a
sses
smen
t of
To
mog
raph
ic im
ages
may
hel
p in
ass
essm
ent o
f
ass
umpt
ions
nee
ded
for o
bliq
ue
a
ssum
ptio
ns n
eede
d fo
r obl
ique
iono
gram
io
nogr
am re
duct
ion
redu
ctio
n
Tom
ogra
phy
and
HF
Ray
Tra
cing
Tom
ogra
phy
and
HF
Ray
Tra
cing
Estim
atio
n of
Estim
atio
n of
Max
imum
Usa
ble
Max
imum
Usa
ble
Freq
uenc
y (M
UF)
Freq
uenc
y (M
UF)
Rog
ers e
t al.
(200
1)
•• T
omog
raph
y w
ith io
noso
nde
inpu
t gav
e sm
alle
st e
rror
s
Tom
ogra
phy
with
iono
sond
e in
put g
ave
smal
lest
err
ors
in e
stim
atio
n of
MU
F(F2
)in
est
imat
ion
of M
UF(
F2)
•• B
ette
r tha
n FA
IM, P
IM a
nd IR
I-95
mod
els
B
ette
r tha
n FA
IM, P
IM a
nd IR
I-95
mod
els
•• C
limat
olog
ical
mod
el b
ette
r for
E-la
yer M
UF
C
limat
olog
ical
mod
el b
ette
r for
E-la
yer M
UF
Tom
ogra
phy
and
HF
Dire
ctio
n Fi
ndin
gTo
mog
raph
y an
d H
F D
irect
ion
Find
ing
War
ringt
on e
t al.
(200
2)
Rad
io T
omog
raph
ic Im
agin
g: a
pplie
d to
HF
DF
Rad
io T
omog
raph
ic Im
agin
g: a
pplie
d to
HF
DF
TEC
plo
ts
TEC
plo
ts v
s vs la
titud
e fr
om U
Kla
titud
e fr
om U
Kto
mog
raph
y ch
ain
tom
ogra
phy
chai
n
Latit
ude
0526
UT
0402
UT
0213
UT
2319
UT
1904
UT
0107
UT
0713
UT
TEC
•• Tr
ough
wal
l sup
porti
ng H
F T
roug
h w
all s
uppo
rting
HF
prop
agat
ion
prop
agat
ion
Tom
ogra
phy
and
Ver
tical
TEC
Tom
ogra
phy
and
Ver
tical
TEC
VTE
CV
TEC
Equi
v Eq
uiv
Ver
t V
ert T
ECTEC
Oth
er fo
rms o
f ion
osph
eric
‘tom
ogra
phy’
Oth
er fo
rms o
f ion
osph
eric
‘tom
ogra
phy’
•• St
atis
tical
imag
ing
of io
nosp
heric
irre
gula
ritie
sSt
atis
tical
imag
ing
of io
nosp
heric
irre
gula
ritie
s
•• G
PS im
agin
gG
PS im
agin
g
TEC
map
TEC
map
Elec
tron
dens
ity im
age
Elec
tron
dens
ity im
age
•• F
ollo
w te
mpo
ral c
hang
es
Follo
w te
mpo
ral c
hang
es••
Lim
ited
heig
ht re
solu
tion
Li
mite
d he
ight
reso
lutio
n
Con
clus
ions
Con
clus
ions
••ve
rsat
ile n
ew e
xper
imen
tal t
echn
ique
vers
atile
new
exp
erim
enta
l tec
hniq
ue
••la
rge-
scal
e sp
atia
l stru
ctur
ela
rge-
scal
e sp
atia
l stru
ctur
e
••w
ide
area
cov
erag
e fr
om fe
w g
roun
d st
atio
nsw
ide
area
cov
erag
e fr
om fe
w g
roun
d st
atio
ns
••ap
plic
atio
ns to
app
lied
radi
o sc
ienc
eap
plic
atio
ns to
app
lied
radi
o sc
ienc
e
••ap
plic
atio
ns to
geo
phys
ical
rese
arch
appl
icat
ions
to g
eoph
ysic
al re
sear
ch
••fo
otpr
ints
of s
pace
-wea
ther
pro
cess
esfo
otpr
ints
of s
pace
-wea
ther
pro
cess
es
Rad
io T
omog
raph
ic Im
agin
gR
adio
Tom
ogra
phic
Imag
ing
![Gamma-Ray Emission Computed Tomographic Image ... · disposal [1]. The combination of Transmission Computed Tomography (TCT) with Emission Computed Tomography (ECT) is used in non-destructive](https://static.fdocuments.in/doc/165x107/5f3bcd0219f7ef6e106afdb8/gamma-ray-emission-computed-tomographic-image-disposal-1-the-combination.jpg)
