TELKOM
NIKA
, Vol. 11, No. 9, September 20
13, pp.
5055
~50
6
0
ISSN: 2302-4
046
5055
Re
cei
v
ed Fe
brua
ry 24, 20
13; Re
vised
June 1, 201
3; Acce
pted Jun
e
17, 2013
Inversio
n
of Surface Nuclear Magnetic Resonance by
Regularization with Simulated Atomic Transition
Method
Hong
xin Wu
*
, Guofu Wan
g
, Faquan Z
h
ang, Jincai
Ye, Anqing Sun
Schoo
l of Information a
nd C
o
mmunicati
on E
ngi
neer
in
g, Guilin Un
iversit
y
of
Electronic T
e
chno
log
y
No.1, Jinji R
o
a
d
, Qixi
ng Distri
c
t, 54100
4, Guilin, Chi
n
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
:
w
u
ho
ng
xi
n4
5
@
16
3.com
A
b
st
r
a
ct
Initial w
a
ter
c
ontent
i
m
pacts
the
accuracy
an
d re
s
o
luti
o
n
of th
e i
n
ver
s
ion
in s
u
rfac
e n
u
cle
a
r
ma
gn
etic reso
nanc
e (SNMR
)
. In order to solve this
pro
b
le
m, a n
e
w
meth
od c
a
ll
ed
as regu
lari
z
a
t
i
o
n
combi
ned
w
i
th simul
a
ted
ato
m
ic tra
n
siti
on
meth
od
(
R
SA
T
A
) is prop
os
ed. T
he
inver
s
ion
of SNMR
i
s
transformed
int
o
a
n
u
n
co
nstra
i
ne
d n
o
n
lin
ear
glo
bal
o
p
ti
mi
z
a
tion
prob
le
m,
a
nd
it solv
ed
dir
e
ctly by
RSAT
A
w
i
thout pre-ass
i
gni
ng th
e in
itia
l w
a
te
r content
distributi
on. T
he co
nju
gat
e g
r
adi
ent
li
near iterative alg
o
rith
m
is ad
opte
d
to l
ook for l
o
ca
l
mini
mu
m
an
d g
l
oba
l ex
tre
m
e
v
a
lu
e w
hen
usin
g RSAT
A, an
d
has i
m
prove
d
t
he
inversi
on s
p
e
e
d
. Resu
lts sho
w
that this method
is a v
e
ry
goo
d so
lutio
n
t
o
solv
e the
effect of in
itial w
a
ter
content
and
it
is als
o
b
e
tter than
t
he existin
g
meth
ods on the
o
perati
on efficiency an
d the
acc
u
racy o
f
inversi
on.
Ke
y
w
ords
:
reg
u
lari
z
a
tio
n
, simulate
d ato
m
ic transiti
on al
gorit
hm (SAT
A), inv
e
rsio
n, SNMR
Copy
right
©
2013 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduc
tion
Surface Nu
clear magn
etic
re
son
a
n
c
e (SNM
R) d
e
te
ction te
ch
nol
ogy is
one
a
nd only
dire
ct detecti
on gro
und
wat
e
r method [1]
,
many schol
ars h
a
ve dev
elope
d this theory and ma
de
field test
s o
n
this te
ch
nol
ogy [2-6]. T
h
e inversio
n o
f
wate
r
conte
n
t plays an
i
m
porta
nt rol
e
in
resea
r
ch of S
N
MR.
While
the a
c
cura
cy
and resolutio
n
of SNM
R
in
versio
n a
r
e th
e key fa
ctors
of
inversi
on alg
o
rithm. In order to simplif
y this
proble
m
, we rega
rd the inversi
on pro
c
e
s
s as a
mathemati
c
al
model of sol
v
ing function
al minimu
m. We always calcul
ate this minimum qu
e
s
tion
by linear ite
r
a
t
ion method
s,
su
ch a
s
Steepe
st
De
sce
n
t method [7]
,
Conjug
ate
Gradi
ent met
hod
[8], Newton
method [9], and no
nline
a
r
iterativ
e m
e
thod [10]: Monte Carlo
method [11
]
,
Simulated A
nneali
ng m
e
thod
(SA) [12
], et al.
However, the
line
a
r ite
r
ative
method
s
dep
en
d
much
o
n
the
initial mod
e
l
althoug
h they
have
adv
ant
age
s of
go
od
stability, hi
g
h
comp
utatio
nal
efficien
cy. But it is not easy to get a reasona
bl
e ini
t
ial model in
compl
e
x ele
c
tric conditio
n
s
;
while
the n
o
n
linea
r iterati
v
e method
s based on
ran
dom
search
can o
b
tain th
e global o
p
timal
solutio
n
with
out depe
ndin
g
on the initi
a
l model,
but
it has poo
r stability and sl
ow conve
r
ge
nce
spe
ed.
In orde
r to solve the prob
lem of the ini
t
ial value influen
ce an
d re
alize fa
st an
d stabl
e
inversi
on
duri
ng the i
n
version, we
have
pro
p
o
s
ed
a
new
metho
d
whi
c
h i
s
calle
d as combin
ed
regul
ari
z
ation
method
with
simul
a
ted at
omic tran
sitio
n
metho
d
(RSATA) to sol
v
e the optim
al
solutio
n
of SNMR inve
rsi
on. The ide
a
of RSAT
A method i
s
to model the p
r
ocess of ato
m
ic
transitio
n. Th
e obj
ective fu
nction
of
solv
ing the
SNM
R
inve
rsion
i
s
e
quivale
nt to the
en
ergy
of
atomic. And the obje
c
tive functio
n
tendi
ng to a smal
l
e
r value is li
ke the transitio
n from an excited
state with hig
her en
ergy to that of lower ene
rg
y or
grou
nd state.
The prob
abil
i
ty of
transiti
on
from o
ne
en
ergy level
to
anoth
e
r
ca
n
be
cal
c
ul
ate
d
a
c
cordi
ng
to the Bolt
zman di
stri
but
ion
.
Comp
ared wi
th singul
ar va
lue de
comp
o
s
ition (SVD), and SA for the inversi
on of
one layer wa
ter
conte
n
t, the simulation re
sults sh
ow tha
t
the R
SATA
is more stabl
e and fast for efficient SNMR
inversi
on.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 9, September 201
3: 505
5 – 5060
5056
2. For
w
a
rd
Modeling of
SNMR
Assu
ming tha
t
a wire ante
nna is laid o
u
t
on the grou
nd in a circle,
the antenna
is then
energized
by a pul
se
of a
l
ternating
current a
nd an alternatin
g
m
agneti
c
field can
be dete
c
ted
using the same antenna
after the pul
se is termi
nated. Oscillating
with th
e Larm
or frequency, the
SNMR si
gnal
(,
)
E
tq
h
a
s an
ex
pone
ntial e
n
v
elope
and
depe
nd
s o
n
the pul
se
p
a
r
ameter
0
qI
:
*
02
0
0
(
,
)
e
xp(
/
)
c
o
s(
)
Etq
E
t
T
w
t
(1)
W
h
er
e
*
2
T
is the
spi
n
-spin
rel
a
xation time,
and
0
is th
e p
h
a
se
of S
N
MR sig
nal,
0
w
is e
q
ual to
the Larmor freque
ncy
of the proto
n
s
00
wH
, with
0
H
bei
ng the ma
g
n
itude of th
e
geoma
gneti
c
field and
the gyro mag
netic ratio for p
r
o
t
ons. The initi
a
l amplitude
0
()
Eq
can
b
e
cal
c
ulate
d
as
[4]:
00
0
1
1
1
()
s
i
n
(
)
(
)
(
)
2
V
Eq
M
q
w
d
V
rr
(2)
Assu
ming tha
t
stratification
is hori
z
ontal
and ve
rtical d
i
stributio
n of
resi
stivity is known,
()
r
the
sub
s
u
r
f
a
c
e
re
sist
iv
it
y
,
r(
,
)
=x
y
,
z
r
,
()
(
)
z
r
, Equation (2
) ca
n be simplifie
d and written a
s
:
0
0
()
(
,
(
)
,
,
)
(
)
L
Eq
K
q
z
z
w
z
d
z
(3)
Whe
r
e,
00
1
1
,
1
(,
(
)
,
,
)
s
i
n
(
)
2
xy
Kq
zz
M
q
dx
d
y
(4)
We limit integration by
22
(2
)
2
x
yD
and
2
L
D
, where
D
is the antenn
a dia
m
eter.
3
．
Inv
e
rsion of SNMR Data
Assu
me that the gro
und
wat
e
r struct
u
r
e i
s
layer distri
bu
tion, and hen
ce:
()
()
1,
()
0,
jj
j
j
jj
j
wz
w
z
zz
z
z
z
其他
(5)
Whe
r
e
1,
2
,
iM
is the
runni
ng in
dex
of
q
,
1,
2
,
jN
is the
ru
nning i
ndex f
o
r the
wate
r
conte
n
t
w
,
()
j
z
is a set of basi
s
functio
n
s.
Hen
c
e,
the kernel ve
ctors are the ele
m
entary
respon
se
s fro
m
the layers
of wate
r,
cha
r
acteri
ze
d by their d
epth
z
an
d thickne
s
s
z
. When we
use a
seri
es
of different pu
lse, equ
ation
(2)
can b
e
discrete
d as:
0
1
()
(
,
)
(
)
ii
j
j
j
j
E
qK
q
z
z
z
w
(6)
In a matrix notation, proje
c
t
ed Equation
(6) ca
n be writ
ten as:
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Inversio
n of Surface Nu
cle
a
r Mag
netic
Re
son
a
n
c
e b
y
Re
gula
r
ization… (Hon
gxi
n
Wu
)
5057
w
=
0
Ae
(7)
Whe
r
e
A
is a re
ctangl
e matri
x
of
M
N
with the element
s
1
,
(,
)
j
j
z
ij
i
z
aK
q
z
d
z
;
T
01
0
2
0
()
,
(
)
,
,
(
)
)
M
Eq
Eq
Eq
0
e
,
0
()
i
Eq
being the set of experimen
tal data;
T
12
(,
,
,
)
jN
ww
w
w
w
，
()
j
j
ww
z
being the vertical distri
buti
on of water
content at spe
c
ific aq
uifer thickne
s
s
。
4. SNMR Inv
e
rsion Op
timization Mo
del
4.1. SNMR In
v
e
rsion based on RSAT
A
Metho
d
Acco
rdi
ng to the cha
r
a
c
teri
stics of the met
hod of SNMR, the aim of SNMR inversi
on is
to get the water conte
n
t and its depth. The und
er
g
r
o
und sp
ace sp
an of 120m is divided into
60
sha
r
e
s
by me
sh g
ene
ration
, and ea
ch o
ne is
an aq
uifer with
2m thi
c
kne
ss. So a
model
with 6
0
equal d
epth l
a
yers i
s
con
s
tru
c
ted. Th
e
water
c
onte
n
t is a col
u
m
n
vector
with
sixty elements
whi
c
h a
r
e all
betwe
en 0
a
nd 1. The i
n
versi
on of o
b
j
e
ctive fun
c
tio
n
is
carrie
d o
u
t according
to
Tikho
nov re
g
u
lari
zation m
e
thod. To find
an
approxim
ate solutio
n
o
f
matrix Equation (8)
2
2
2
()
(
)
m
i
n
L
L
w
dW
w
Aw
(8)
Whe
r
e:
A
is an integ
r
al
of the kernel functio
n
；
d
being the dat
a of experime
n
t
；
W
being the reg
u
lari
zing o
perator.
4.2. The Intr
oduction o
f
TSAT
A Meth
od
Inversio
n of simulation ato
m
ic tra
n
sitio
n
wa
s first p
r
o
posed by Ji
a
y
ing Wa
ng [1
3]. Then
a new n
on-li
near inve
rsio
n method ad
apted to the gene
ral ge
op
hysical inverse proble
m
wa
s
cre
a
ted
by Xuemin
g S
h
i et al.
The
y
analyzed t
he
co
rre
sp
on
den
ce
betwe
en the
atomi
c
transitio
n a
n
d
geop
hysi
cal i
n
versi
on in
T
able 1 [1
4
]. This p
ape
r
will
introd
uce thi
s
meth
od to t
h
e
inversi
on of SNMR, be
sid
e
s
the re
gula
r
i
z
ation
wa
s co
mbined
with this metho
d
.
Table 1. The
Corre
s
p
ondin
g
Relatio
n
be
tween SNMR Detectio
n Inversi
on an
d
Simulated Atomic T
r
an
sition
The inversion of
SNMR
w
a
te
r con
t
ent
Atomic transition
The tar
get level of the SNMR inv
e
rsion
global minimum
Local minimum
Atomic energ
y
le
vel
ground state of
a
t
om
excited state of a
t
om
The ste
p
s of
RSATA meth
od are follo
ws:
1.
Give a random initial model group
12
3
6
0
(,
,
,
)
ww
w
w
T
w
;
2.
Use co
njug
ate gradi
ent lin
ear
iterative method to ge
t tar
get level (local extre
m
um)
and it
s water co
ntent thro
ugh lo
cal
opti
m
ization
inve
rsio
n. If the t
a
rget l
e
vel i
s
less
than the give
n threshold, then go to ste
p
(E);
3.
Make
the
si
mulated
ato
m
ic tran
sited
acco
rdin
g to the ta
rget
level. The transitio
n
prob
ability of target level are deci
ded by
Equation (9);
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 9, September 201
3: 505
5 – 5060
5058
1,
ex
p
(
)
/
,
ij
ji
i
j
EE
p
EE
k
T
E
E
(9)
4.
update
the model pa
ram
e
ter,
**
w=
w
+
δ
w
, where
*
w
is
“s
teady”
solution,
δ
w
is
rand
om num
ber, then go t
o
step (B
);
5.
Output targ
et level and pa
rameter of the
inversio
n.
5. In
v
e
rsion
Resul
t
s and
Analy
s
is
In orde
r to demon
strate t
he perfo
rma
n
c
e of RSATA
method, we
used a
singl
e layer
model
con
s
i
s
ting of a 10-m-thick ho
ri
zontal, hom
o
g
eneo
us in fre
e
spa
c
e
at depth of 20m
and
three
-
layer m
odel con
s
istin
g
of two 10-m-thick
ho
rizontal, homog
eneo
us in fre
e
spa
c
e at d
epth
of 20m and 1
00m and a 8
-
m-thick ho
rizontal, hom
og
eneo
us in fre
e
spa
c
e at de
pth of 60m.
In the
cal
c
ula
t
ion of thi
s
se
ction
we
first
gi
ve a
wate
r
conte
n
t of the
model,
and
calcul
ate
the strength
of the
signal
i
e
a
t
different p
u
l
s
e m
o
me
nts,
then
certai
n signal-to
-
n
o
ise
ratio (SNR)
of the Gaussi
an white n
o
ise is add
ed to
i
e
,so we get
e
,signal-to
-
noi
se i
s
define
d
by (10):
2
10
2
2
0*l
o
g
i
i
e
SN
R
ee
(10)
Assu
me that
the whole
section i
s
comp
os
ed o
f
s
i
xty
2
-
m-
th
ick
a
c
qu
ir
es
, th
e
geoma
gneti
c
field is 500
00
nT, the anten
na is laid
out
on the gro
u
n
d
in a circle
with a diam
eter
of 100m, and
it is excited b
y
sixty curren
t
pulse
s, the large
s
t pul
se i
s
10000
A
ms
.
It can be se
en from the i
n
versi
on results that RS
ATA method
is effective for sin
g
le
aquifer mo
de
l, especi
a
lly whe
n
we ad
d the noise
o
f
20 dB signal-to-noi
se ra
tio, the inversion
result is still con
s
i
s
tent wi
th the model. With
the co
mplexity of the model co
mplex, the water
conte
n
t of three layer
aqui
fer mod
e
l at depth of
1
0
0
m
wa
s not a
c
curate
eno
u
gh in Fig
u
re
3,
whi
c
h i
s
le
ss
than mod
e
l, let alone th
e
water co
ntent
with ad
ding
noise in Fig
u
re 4. At the sa
me
time we
can
see th
ere
are
three m
o
re l
a
yers
mod
e
l
in Figu
re 5
a
nd Figu
re
6, Ho
wever, fro
m
Figure 5 and
Figure 6
we
found that th
e amplitude
wh
i
c
h we cal
c
ulate
d
by inversio
n re
sult
s in
Figure 3
an
d
Figure 4
is
co
nsi
s
tent
with t
he a
m
p
litude
whi
c
h
we
cal
c
ulate
d
by i
n
versio
n m
ode
l .
It is
th
e
mu
lti-
s
o
lu
tion
s
pr
ob
le
m in
g
e
oph
ys
ics
th
a
t
we
a
l
s
o
en
co
un
te
r
e
d
ea
r
t
h
in
SN
MR
.
Figure 1. The
Inversion
Re
sults fo
r Singl
e
Layer with
out
Noise
Figure 2. The
Inversion
Re
sults fo
r Singl
e
Layer with
No
ise of s/n
=
20
dB
0
5
10
15
20
25
30
35
40
-12
0
-10
0
-80
-60
-40
-20
0
I
n
v
e
rs
i
o
n
res
u
l
t
s
w
a
te
r
c
o
nte
n
t/%
de
pt
h
/
m
i
n
v
e
rs
i
o
n
res
u
l
t
M
ode
l
0
5
10
15
20
25
30
35
40
-1
2
0
-1
0
0
-8
0
-6
0
-4
0
-2
0
0
I
n
v
e
rs
i
o
n
re
s
u
l
t
s
w
a
te
r
c
ont
e
n
t/
%
de
pt
h/
m
in
v
e
r
s
io
n
r
e
s
u
lt
M
odel
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TELKOM
NIKA
ISSN:
2302-4
046
Inversio
n of Surface Nu
cle
a
r Mag
netic
Re
son
a
n
c
e b
y
Re
gula
r
ization… (Hon
gxi
n
Wu
)
5059
Figure 3. The
Inversion
Re
sults fo
r Thre
e
Layers witho
u
t Noise
Figure 4. The
Inversion
Re
sults fo
r Thre
e
Layers with Noise of s/n
=
2
0dB
Figure 5. The
SNMR Fo
rward of 3-l
a
yer
Aquifers and
the Reforwa
r
d of RSATA without
Noi
s
e Inversi
on Re
sult
s
Figure 6. The
SNMR Fo
rward of 3-l
a
yer
Aquifers and
the Reforwa
r
d of RSATA with
Noi
s
e of s/n
=
20dB Inversio
n Re
sults
Table 2. The
Compa
r
i
s
on
of Different Kinds of Inversi
on Method
s
signal-to-noise
rati
o
Inversion method
s
RSATA
SVD
SA
accurac
y
/%
time/s
accurac
y
/%
time/s
accurac
y
/%
time/s
40 90.3
2.73
60.9
50.2
88.2
8.91
30 86.1
2.45
50.4
52.6
80.3
9.32
20 83.2
2.87
40.7
59.2
74.3
10.33
10 81.7
3.12
30.1
60.3
70.5
11.78
The a
c
curacy
of inversi
on
and its
rate o
f
conv
ergen
ce at different
noise for
sing
le layer
water mod
e
l
by RSATA, SVD, and S
A
algorith
m
are
given in
Table
2. It shows that
RSATA
algorith
m
ha
s obtai
ned
hi
gher
accu
ra
cy and big
ger
rate of
conv
erge
nce with
lowe
r SNR
for
observation d
a
ta, while SVD nee
ds a hi
gher S
NR at
l
east 40
dB to compl
e
te the inversi
on.
6. Conclusio
n
Reg
u
lari
zatio
n
combi
ned
with simul
a
tio
n
atomic tran
sition algo
rith
m is prop
ose
d
to the
inversi
on
of SNMR in thi
s
p
aper.
Fro
m
th
e sim
u
lation
experim
ents, we ca
n see
i
n
versi
on re
su
lts
never be effe
cted by the initial water
content
and th
is method h
a
s
better p
e
rf
orma
nce on the
inversi
on
of
singl
e-laye
r
and m
u
lti-lay
e
r a
quifer m
odel
s. Moreo
v
er, the u
s
e
of the conju
g
a
te
0
5
10
15
20
25
30
35
40
45
50
-
120
-
100
-80
-60
-40
-20
0
I
n
v
e
rs
i
o
n
res
u
l
t
s
w
a
te
r
c
onte
n
t/%
d
e
pth/m
in
v
e
r
s
io
n
r
e
s
u
l
t
Mo
d
e
l
0
5
10
15
20
25
30
35
40
45
50
-
120
-
100
-8
0
-6
0
-4
0
-2
0
0
I
n
v
e
rs
i
o
n
res
u
l
t
s
w
a
te
r
c
onte
n
t/%
d
e
pth/m
in
v
e
r
s
io
n
r
e
s
u
lt
M
odel
0
1
2
3
4
5
6
7
8
9
10
0
50
10
0
15
0
20
0
25
0
30
0
35
0
40
0
45
0
50
0
q/
A
*
s
A
m
pl
i
t
ud
e
/
n
V
M
o
del
F
o
rw
ard D
a
t
a
In
v
e
r
s
i
o
n
R
e
fo
r
w
a
r
d
D
a
t
a
0
1
2
3
4
5
6
7
8
9
10
0
10
0
20
0
30
0
40
0
50
0
60
0
q/
A
*
s
A
m
pl
i
t
ud
e
/
n
V
M
ode
l
F
o
r
w
ar
d
D
a
t
a
In
v
e
r
s
i
o
n
R
e
fo
r
w
o
r
d
D
a
t
a
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 9, September 201
3: 505
5 – 5060
5060
gradi
ent line
a
r iterative o
p
timization
algorithm
to fi
nd local and
globe extre
m
um obviou
s
ly
increa
sed th
e
spee
d of cal
c
ulatio
n du
rin
g
the
inversi
on. Com
pare
d
with SVD a
nd SA, we can
see it is
more
fit to the inversio
n of lo
w SNR
sign
als. S
o
more accu
rate unde
rg
ro
und info
rmati
o
n
can
be d
e
tected by this al
gorithm
and i
t
also p
r
ovide
s
a
n
e
w way of
solving oth
e
r
p
r
obl
ems of
low SN
R.
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