r e q u i r e s only 0(N ) m u l t i p l i c a t i o n s Se can see t h i s f a c t
139
TABLE 31
The impedance recovered from nonnoisy response using f a s t Algorithm
L bull - bull j wVJ
i vzOOvK
2( 030000 ) Z( 035000 ) Z( 040000 Ik 045000 ) J( VtOvv
Z( 0^5000 ) Z( 060000 ) Z( 065000 ) Z( 070000 ) L 0gt-VVO T y r bull A bullgt n x
1
Z( - raquo
Zr
Z( 7 f
Z Z(
vo5000
0TOOOO
0gt95000 4 A A r A fl
4 A C- bullbullbull A bull bull bull V
i 4 A n n A
i t i bull
4 n 1 A A r
J ^ - raquo t A
)
)
gt v
j
)
_ ( iivOv -^ ( A A
bull bull
1 bull bull bull ] bull bull
4 - C bull bull ^
150000 155000 160000
f
(
)
)
bull bull bull
- t bull
i bullbull
T l
T 1
L bullbull
Z( - T lt
bull H - r bull i bull
Z( bull ^ _
bull t
A
- 1 4
n
r-
rv
n
-
0
- 1 ^ IS A r
) IVV n 1 A - A
0 bull bull 1
r - c A A ^
Dwv JV
m ^ A A A
raquo fgt A A A
bull V V V -bull v
C A A A
bull A ^ ^ J
J C A A ^
^ A A A -1
1^uvOv - C
1 bull- r r A
1-Ov -raquo r- n V
4 A A A -
i
^
1
)
) )
) )
) 1
j
A J A -T ^ bull - n raquobull r A J raquo^ r bull rraquo
V gt v J T C - mdashTl Z ~ V i iVrcc- bull i T V i I-jr -Ulwu^-tiTVi ir-T ~
wiiJVjZJCLTVi Lrrj -
0i3^25263EI01 ERROR = 0133-253Ef01 ERROR =
A 4 n r - r r ^ raquo I A lt rr-r--np _
V A 7 1wJ ~ mdash T w A UTi
0154D342ET0i ERROR = 0i5774309Ef01 ERROR -
0 1 T C A C C A r I A lt j - n r f n _
A w A A ^ ^ r r - i A rT tn^r i _
A - C T - gt t ^ i A j nF i--nr _ -c-^Jbull i c i T v i c r ur -A lt i T A r T r gt ^ I A lt ^ ^ r n r j _
u - y - u - r i T v i rrrjn -A n c n ^ r A r-^rnr _
OiTJCOiDSCTJi irrur -A ^Airvlt J Tl I A ^bullr^^m^ _
y^yyoiCiicTVi c r -A r A c^^ A T ^ I A T-ri^iH^ mdash
yi-jTbullCLTvi H- ~ Vraquoiiiift30iTyi i rJV -yii27i3LTyi c r uh -
022150GA4E+01 ERROR = 022663934Ei01 ERROR bullbullbull A TTlaquo nnnn^ I A PbullI^l^gt _
023io-3537iT0i trsLT -
Vfovc7ViiTVi 17^jr -yiM230y--iT01 EbullJ -A IC17 bull7TI- I Alt
y i 5 7 0 i 7 J l T y l
^ ^ r- A v ^ raquo A k bull- r r laquo bull T k M
A A ^ ^ T ^
y^O^--A A d A ^ ^ ^ A
y y - v ^ j y A A A ^ laquo laquo ^
y gt y y 7 i raquo
t O v c i O -A A A ^^4 laquo 4
y y y bullltbullbull
00064067 00055357 00046762
A A A bull^^^ bull A
A A A rtTft raquo
y gt y y i i c A A A r AC y y y i j ^ -
e - -^ -1 A rf - f I A J
A j - i r i r ^ bull^C A
A - t A A ^ ^ A J ^ l A y
r rv rk r _
mdash r ^ rt r ^
I - f r ^ r r- -^V bull t laquo
- fT raquo7 t In mdash
^ 1 raquobull lt t f t
rr nnn ^ ^ laquo raquoi i I h M
A A A A lt ^ A
y y y y i i A A A A A ^ r t
- y raquo y y y r i 7 r A A J - A nraquo
- v y y i - v 7 i A A A r t A A n ^
mdash f fc I ) M I ir - n -
A A A T - I A - ^ A
- y v v i - y i y fV A A ^ - laquo - yen laquo
- v y y 7 i i
- bull bull yytycy A A A ^ i T raquo f
V raquo V i J7 A p A r - 4 C ^ ^
- y V v j - j y
O A A w ^ ^ r raquoyyc bull_J
O A A n 4 A raquo y y o i i y
A A A T T A f ^ - y raquo y y 5 y 2 i
O A A m n T o O y 3 i 7 i
A A A ^ ^ ^ r 4
- v v y o o r 7 ^ A A A ri r -bull r
- y y V 7 J C - J i A A J A n ^ ^
- v y i y i r - -A A 4 A n - n 4
- y y i y 7 C 7 i A A 4 laquo r f ^ - lt ^
- j ^ v i i c
C A d ^ laquo 4 ^ ^ y ^ - i i -
A A J ^ ^ T gt
- y y i ^ -
C A i bull^nrrv raquo y i - bull bull bull
A Alt lt C - laquo C
- y y i t i
y i 5 i i
r- - M n
r-1- r n r
A -T A f C A ~ A r A M w bull - bull - - - bull w mdash w 4
A - ^ ^ A n - ^ J - i r - f A j
V gt 0 -vQOC i - T V i
bullraquo-- 1 c i -C T V1 t rr un
f T^ AAf tn A p- 1 A ^raquonnr V t 3 2 6 3 o o v i T U l LrrLTi -
y j c i ^ - L T y l i r - j bull rt -T i -T A bull 1 r r I A 4 r- r r n r y ^ j - ^ y - O w r L T V i i-- bull A - r - i lt - gt n r i A lt i-i- i-rf- I( i f O-^HOUlTvi ZJ bull
A A 4 n ^ rf i
- y y l - 7 A A 4 - ^ l ^ r t ^ r t
__ A A ^ n ^ ^ - - y laquo y i - bull - bull bull
A A 4 ^ J k gt
A 4 ltn ^ -^ n bull
- V I J i 7 Ti-Ci^ A ^ laquo A ^ ^ ^ ^
= - y V i l l J J _ A A rgtr-^ A n
= - y raquo y i ^ - y 7 A A n ^ 4 ^
- j t v i z i c A A - ^ ^ n t ^ C ^
bullV A n ^ ^ ^
TABLE 32
MiO
The impedance recovered from noisy response (0^ = 001) usinq f a s t algorithm
bull 005000 =
Z( 010000 =
Z( 020000 ) = Z( 025000 ) = Z( 030000 ) = Z( 035000 ) = Z( 040000 ) = Z( 045000 ) = Z^ 050000 ) = -bull A C C A A A 4 _
Z( 060000 ) = T A e A A A _
yojyyy - =
Z( 070000 ) = T t A I p A A A _
T A n e A A A y _
L yojyyy --lt ( A n A A A A raquo _
L yt7yyvy --r t A - ^ r - A A A _
L y75yy0 = L iOyyyy bull -
Z(
Z(
L i y j y y y lt -
7 4 4 A A A I _
bull 4 4 T A A ^ V
i gt i - y bull - bull bull - bull -T 4 O A A A A _
i- iiyyyy i -
L i i i j y y y -
Z( 130000 ) = i W bull bull bull V 4 laquoAAy bdquo
4 - e bull bullbull _
iTyy -4 r- A A A A _
i5yyy ) = 4 A A A A
J bull r A- _
i ~ 4 mdash gt A gt A ^ ^
ifvVV bull -
( C - V ~
bull bullbullrAA - _
icjyyl - O A A AA 1 _ bull
itvyyy -iVjyCy ^ = r A A A A A _
Z( 205000 ) = Z( Z-yyy =
StlJJjJ -
^ - raquo A A A V
il-VVJv -
iiyjy -
bullbull -I I- - A A _
A 4 A - ^ n n 4 ) ^ f - lt A 4 fraquorrrM--
vraquoiy077i3hryi irrr -A lt A n laquo T C C C ^ I A4 r-rrnr-
v i y Q i j j j i r v l rwr -y i-r7-i7r7CTvl lJ7i -C 4 4 A A O C raquo I I rt ^ r r r t r _
i i 0 72j-fi^TVi irry -A 4 J7 4 A ^ 4 A r I A 4 r r r r v r _
y 4 i4v j i y tTy l i = 0 4 n m lt bull Tlt I A 4 ^e r (Ar
ii07i-3cTyl trLa =
013400132E+01 ERROR 03S1672Ei0i ERROR =
bull ityoiiiTyi irgtjr = A 4 raquo(nn raquo ^ r A^ I A4 r-l^r^^ _
yiraquoco--ycTyl Lrry = A 4 c -raquoc A -yn bull A 4 ^r r r _ y raquo l i t J - t d T V i n mdash A 4 A nC A- - Tr - ( A 4 ^rrnf y 1 -bullbull-bull i l wI--w T V ^ P A n rraquoraquo A 4 I r- A ^ bull - lt r- I A 4 ^ 4 r n ^
v icjyQr ocTyl c us -A laquo T A lt A n - ^ c ^ 1 A4 ff^-^r- _ A 4 T laquo o n ^ n lt ^ ^ I A 4 rArviA- _ y i 7 7 C 7 i T J l briTiyri -Al 4 A 4 T T 4 - 7 laquo ^ r A 4 r r A ^ t _
ybull 1-iii--iJCTvi _riryr -A bull n c T T I Agt p-rrn^i _ y i C J C ^ J 7 w w T J i _ mdash
yiiu7ijiiTvl LrriU bull A 4 - T n c 4 bull 4 p- I A 4 r^-^--yi
y i j7 j ^o i i Ty i crr -
0 4 n n n lt n n c i I A4 frnr _
A AA 4 ~ltA 4 n f A4 ^ r r n r _
yiyi-4CwltiuTyi iri-yr -VI-C___iTVi Lrruri mdash A n A A T T A - n r A r - r r n t _
yraquoivc y 1CmdashTy i Lririjrraquo bullbull A n4 raquo lt - - ( T L lt r lt - r n n _
A n - n n T ( n n r bull A 4 f-r laquo- nr _
y i i 7 7o i i7 tTy i hr-PiLTi -A n n c C A n raquorgt 1 A4 r r i A n -
y2ij6jv^tT01 hrr -O n ^ 4 ^ n - ^ A r I A4 ^ r r n r i _
A nraquonA gt A n r-1 A4 r r n - _
yijoy5vctvl cry -
A n r 4 bullraquoraquobullbullbullbullltbull A 4 f-rr-^p y i i ^ J l - ^ C l D i T y i 2Tl_bull mdash
A bull f 4 n A bullraquo A 1 A 4 ^ rrnr
yijioyjvciTyi r-ryr -A n r n A T n c n ^ l A4 i-rrnr _ y i J 7 y i j j c T y i i r i - y ^ i - ^cnnnn-T-I A4 ^nnnr _ y bullCwO-77-mdashTvi crvrt bull A ^-^^-^vn A - j T r - I A 4 bull n n n A _ bull i 4i bull j i y X y i z r y r -
A A 4 lA^ l A n -
y J y u J 2
A A 4 I r - i c n
V V - - - - I
A A A ^ ^ A l-l
y y y 7 y 7 0 A A A n raquo r laquo t r
ytyyc-i- i j O A 4 j ^ - c ^ ^
y ^ w 7 i
00079127
00073940
00130903 A A A T T - i A
vgtyyij-i-y A A A ^ - 1 4 A A
Oyy iOy A A A A ^ n
^ r -m raquo raquo
_ A A J C ^ -
raquo A A A C - ^
A A A raquo C n il 4
0 A A A A 4 ^ 4 y y u y i i i
A A A ^ ^ n -T
- y y y 7 - A A A ^ 4 n 4
~y yy^f 7i
O A A - - i - i n r y y i - i j
A A A C T 4 A
OOyjjcy
V V - V - W4
0 A4 c n n T y4 7 2 2 i
A Af- 4 irr- y y i i r Z 7 i
C A n n n ^ I A iyxc-7cy
A A n I raquo -raquo
yy-rc-^-A A n C ^ ^
v y i i j i o O A J rvn ^ ^^4
raquo V i C - i i i
v r V b w
A A A ^ A rgt ^ T
vyycci A f^ 4 ^ i 4
yyii--V-A A A T T bull bullraquo
OyyjjoO A A A -^ ^ i ^ -^
A A i
~h
bull W bull gt ^ bull bull - = - r t
A A A r A rf
^ n ^ - raquo~
A j ~ v A ^ A ~ raquo raquo C ^ mdashraquo- r - j^r _
y^Jio^wiTyi zryr -
y2-0057E--01 ERRy- bull-f rJ^^r^r^ raquo t A ^ I A 4 r~r^rr
V raquoi7-7QtOVLTJX LrTV ~ T ^ n c ^ r v c A ( A 4 pr^r^f^ _
bull ^ jdj 7 J V - T V i l ^ L T -
A ^ A n n 4 A - ^ ( A 4 rrr-r- _ bull ^ 7 ^ ^ bull J C T V J C 7- 7 J 7
A raquo lt r r - r ^ bullbull bullbull f - r - n - _ V t w i J Z 7 V J I T i r 7 T i -
A ^ lt -1A - c ^ f r r 1 r bull bull J ^ 7 V O j i J C T V l C7 7 j r ~
A bull raquo n c A n r A r-rraquo--c _
A ^ ^ A A n 4 bull ^ i ~ A 4 ^rrnr _
j f i - - 0 iC 2mTJ i 17 7 J 7
t J laquo Z - 17 i7 J7
i n ^ ^ A ^ mdash ^ gt A 4
A bull ^ 1 j ^ n - mdash r r --i ^ _
A A A7 4 4 C
-y yygti - i -A A A ^ ^ ^ ^
-00035=33 A A 4 - ^ ^ bullmdash ^
mdash I ) i j i cf 7
A A 4 n A -gt r - y 1 i C ^ J O
A A 4 4 A V 4 A 4
- y y i^T^y A A 4 n A T
- y y i c - r i y i A A J c- - raquo c n
- y y i j o o j c A A 4 4 n n A l
mdash i l j l i bull ^ - 4 4
A A A J^ A mdash J
A A A n ^ r ^ ^
141
t i ) Re la t ion t o Robinsonls jfork
Hobinson [ 4 ] developed a dynamic p r e d i c t i v e deconvoshy
l u t i o n scheme to recover the r e f l e c t i o a c o e f f i c i e n t s from a
r e f l e c t e d impulse response for a layered earth system He
der ived the f o l l o w i n g useful recurs ion formulas
^2) = D^^ (z) 4 r^C^^Cz) z C3-85)
with i n i t i a l c o n d i t i o n s
C^(0) = r^ and D (0) = 1 (3-86)
The r e f l e c t i o n impulse response was given by
R()z) = C^(2)
Ontz) (3-87)
where C D^ were c a l l e d the feedforward polynomial
and the feedback polynomial r e s p e c t i v e l y Egs
( 3 - 8 4 ) - ( 3 - 8 7 ) have a s i m i l a r form to egs (3-32) - (3-35) exshy
cept tha t Robinson used the reversed order of i n d i c e s to l a shy
be l the layered sys tem In order to f ind the r e l a t i o n s h i p
the between Robinsons r e s u l t s and those we derived we have
to i n v e r t the order of the i n d i c e s used to l abe l the layered
system Instead of repeat ing the procedure derived by Roshy
b inson we i n v e s t i g a t e C^(2) and D^tz) t o make corresponding
m o d i f i c a t i o n s for egs (3 -84) - ( 3 - 8 7 )
By egs (3-84) - ( 3 - 8 6 ) we have
Do = 1
1 = ^1 S^
142
D = 1 bull r r z
(3-88)
Opon makiag the index change we have
^0= ^0
D= 1 bull r^rz
-J
C = r^ bull ( r bull r jr r )2 bull r z
2 = ^ f i bull J ^ )2 gt r^r^z^ (3-89)
Examing (3-77) and ( 3 - 7 8 ) we f ind the modified recurs ion
formulas
2^0^(12) =r^D^^^(2) bull C ^ ^ T d z ) z (3-90)
^O^^^ = ^no fz) r C (1z) z^ (3-91)
Comparing egs (3-90) and (3-91) with (3-32) we find that h
( T l t n F ( n z ) = D (z) (3-92)
(rrt^ )G(nz) = z C^(1z) (3-93)
Therefore the impulse response R(n^z) in Robinsons model
can be r e l a t e d to F(nz) and G ( n z ) By eqs ( 3 - 9 2 ) (3-93)
and (3-87) we f ind that
143
R(nz) = C^(z)D^(z) = z GCn1z)P(nz) (3-94)
The only distinction between egs (3-94) and (3-48) is that
eg (3-94) has z on the right hand side and eg (3-48) 2^^^
on the right hand side This is due to the fact that
Robinson collected the impulse response right on the surface
and Goupillaud collected it one layer higher than the
surface By taking this fact into account we find that
S^ (Goupillaud) = z RQ (Robinson)
z^^ G(n1z)
F(n2) (3-95)
Eg (3-95) is exactly identical to eg (3-48) In
summary Egs (3-92) (3-93) and (3-95) give the
relationship between the results from the two different
approaches
In this section we have justified that results derived
from the discrete system either those from Berryman and
Greene or those from Robinson can be used to form a fast
algorithm to invert the Gelfand-Levitan matrix which comes
from the continuous inverse problem This fact implies the
eguivalence between the discrete system and the continuous
system
CHAPTER 17
IHALOGI BETWEEN DISCRETE AND COHTIHOOS IBVERSE PROBLEtt
Introduct ion
In t h i s chapter we s h a l l t ry to r e l a t e the a lgor i thms
used for the d i s c r e t e and cont inous problems Me f i r s t d i s -
c r e t i z e the cont inuous earth system by assuming i t has a
number of e g u a l l y - s p a c e d l a y e r s which have corresponding
impedances t o the cont inuous system By using the ML e s t i shy
mation and cepstrum d e t e c t i o n to e s t imate ( r ^ - ) of each
s u b l a y e r we can compute the corresponding impedances from
those c h a r a c t e r i s t i c parameters Secondly by picking an imshy
pu l se response from a 1- layer d i s c r e t e sys tem(f ig-^S) and
using polynomial i n t e r p o l a t i o n to smooth i t wa have a conshy
t inuous impulse response which i s t o be used for the imshy
plementation of a cont inuous inverse s c a t t e r i n g problem
From cont inous inverse problem to d i s c r e t e inverse problem
We d i s c r e t i z e the cont inuous earth system g iven i n
chapter I I I with egua l ly - spaced l a y e r s whose one-way t r a v e l
t ime i s egual to 0 05 second ( a l s o the sampling time of the
144
145
impulse response R (t) = - 0 5 exp(-0 5t) H (t) ) By impleshy
menting algorithm 1 in chapter I I we can find the r e f l e c t o r
s e r i e s and i t s cepstrum for each layer Examining the
cepstrum (or r e f l e c t o r series) we always find that there
e x i s t s an excess spike between the zero point and the exshy
pected f i r s t spike This implies that there must be an adshy
d i t i o n a l sublayer ex i s t ing between the layers formed by d i s -
cre t i z ing the continuous earth system and indicates the fac t
that the system we probed i s continuous rather than disshy
crete In order to jus t i fy how well t h i s d iscret ized sy s shy
tem simulates the continuous system we pick the expected
f i r s t spike and compute the corresponding re f l ec t ion c o e f f i shy
c ient and one-way t rave l time I t comes out to be wel l -
matched The d i scre t i zed cantinuous model i s shown in
f i g 4 7 According to t h i s model the theoret ica l values of
r s are vJ
1 =
1 - 105
1 bull 105
- = -00243902
- 0 0 5
^ ^ =
105+110
= - 0 0 2 3 2 5 5 8
146
9
0
Af
^^t
llt t
(k-hi)At
Figure 47 The discretized continuous system
147
- 0 0 5
r^ = -002222
110+115
- 0 0 5
r = = - 0 0 2 1 2 7 6 5 4
115+120
The simulation resu l t i s l i s t e d at tab le 33 Although
the computation (simulation) r e s u l t s are pretty c lose to the
t h e o r e t i c a l r e s u l t the former seems to decay a l i t t l e
fas ter than the l a t t e r does This can be improved by
increasing the sampling rate of both the impulse response
and d i s cre t i z ed system By se l ec t ing the sampling time to
be 0005 second we have better re su l t s shown in tab le 34
Theoretical values of r i s in t h i s case are
-0 005
r = = -000249 376
1 +1 005
- 0 0 0 5
I- = = - 0 0 0 2 4 8 1 3 1
1 005+1010
148
- 0 0 0 5
^3 ^ = -0 00246913
1010 + 1015
- 0 0 0 5
^4 = mdash mdash = -0 00245700
1015+1020
149
TABLE 33
E s t i m a t e s of t f o r t h e d i s c r e t i z e d c o n t i n u o u s system with bull A t = 0 05
0- Layer
-00238403
-00221470
-00205727
-00192363
-00179399
-00168264
-00157430
-00148026
-00138858
-00130833
0500000lE-01
05000004E-01
04999999E-01
04999997E-01
05000000E-01
04999999E-01
04999999E-01
04999997E-01
04999997E-01
04999997E-01
1
2
3
4
5
6
7
8
9
10
^so
TABLE 34
E s t i m a t e s of r^ f o r t h e d i s c r e t i z e d c o n t i n u o u s system with -^ A t = 0005 sec
r
-00024882
-00024696
-00024506
-00024332
-00024145
-00023974
-00023793
-00025623
-00023446
-00023281
bull ^ j
04999999E-02
04999999E-02
04999999E-02
05000000E-02
04999999E-02
04999999E-02
04999999E-02
04999999E-02
05000000E-02
04999999E-02
La
1
2
3
4
5
6
7
8
9
10
151
^rom d i s c r e t e inverse problem to continuous
inverse problem
Osing polynomial i n t e r p o l a t i o n to smooth the impulse
response from a 1 - layer earth sytem as shown in f i g 4 8 we
then take t h i s smoothed curve ( f ig 49 ) as an impulse r e s shy
ponse from an unknown cont inuous system to be i d e n t i f i e d
By applying the f i r s t algorithm derived in chapter I I I we
e s t i m a t e the impedance of t h i s unknown system from the genshy
erated continuous impulse response The r e s u l t i s l i s t e d in
t a b l e 35 Examining data in Table 35 we find the c o n t i n u i shy
ty of the impedance v s t r a v e l t ime which impl ies a c o n t i shy
nuous earth system as expected Me a l s o note that the turnshy
ing p o i n t s of the impedance funct ion are located a t the
m u l t i p l e s of 20 which match the turning points on the genershy
ated curve These turning p o i n t s imply that the o r i g i n a l
1 - layer earth system has th ickness 20 A t ( A t - sampling
t ime t h i s i s assumed to be 1 s e c in the o r i g i n a l impulse
re sponse ) The impedance above the surface has been assumed
to be 1 when the algorithm used above was run The r e f l e c shy
t i o n c o e f f i c i e n t on the surface of the 1- layer system i s
0 9 We may then expect the value of the impedance which i s
c l o s e s t t o t h e s u r f a c e i s approximately Z which i s g iven by
Z - 1
= 0 9
Z + 1 (4-1)
152
Figure UBz The impulse response of the 1- layer system in f i g 47
Figure 49 The smoothed curve of fig45 using polynomial interpolation
B^S^MENT
t53
bull-A = 0-9
r -09
Figure 50 The one-layer earth systea
154
TABLE 35
The impedances recovered from the saoothed impulse response (fig 46)
Z( 2( 2( 2( Z( 2( Z( Z( Z( Z(
Zf
(
i I
7 (
7 (
Z( Z( 2( Z( Z( Z ( Z( Z( zlt 7 (
Z( Z( 7 (
Z(
zlt Z( Z( 20
0
0 0 bullJ 0 ( w
0 0 0 0 0 J
KJ
J
0 J
0
4
1
1 1 1 1 1 1 1 i
1 1 1 4
1
1 1 X
1 X
05000 ) 10000 ) 15000 ) 20000 ) 2 5 0 0 0 ) 30000 ) 35000 ) 40000 ) 45000 ) 50000 ) 55000 ) 60000 ) 65000 ) 70000 ) 750 0 0 ) 30000 ) 85000 ) 90000 ) 95000 ) 00000 ) 05000 ) 10000 ) 15000 ) 20000 )
3 0 0 0 0 ) 35000 J 40000 ) 4 5 0 0 0 ) 50000 ) w w y w w
o 0 v v J J O J -gt bull- 0 lt
0 bull-gt v J )
5000 ) 0000 ) 5000 ) 0000 ) 5000 ) 0000 )
0 0
0 K)
0 0 0 0 0 0 0 0 bullJ
0 KJ
J
0 0 0 0 0 0 0
0
0 v
0
KJ
0 0
J
bullJ
0
93603907E+00 86373360E+00 8 0 4 0 0 9 5 5 E T 0 0
75451290E+00 7i346045E+00 67949900E+00 6 5 1 5 S 0 0 0 E T 0 0
6 2 3 3 7 4 5 2 E T 0 0
6107i401E+00 59654780E+00 585911S7E+00 57S40508Ei00 5 7 3 6 7 0 6 9 pound T 0 0
57133143E+00 u 1 Zx 1 7zTjj
57290455E-i-00 5 7 6 1 0 7 7 1 E T 0 0
J o J 5 0 4 c T 0
J O _ O J Q 7 7 L T J J
w 7 1 6 O 7 6 C T U 7
5974512SE4-00 60222962E+00 60578412E+00 60812439E-1-00 609293i2E+00 _ 0 7 O w C 7 O C T w 0
6 v i 1 o 7 c T J J
O 0 C _bull T i 1 i Z T O -J
O V^ 0 1 1 i C T J 0
C bull- V C O -^ 7 OCTJ KJ
w T i 1 i i v c T J
cr - -r -laquo ir -raquo rt c 1 bull_ 7 ^ O w 7 bull_ C T K)
w O 7 i 7 O _ C T J bullbull
5 3 5 5 1 7 S 0 E T 0 0
5 S 1 9 0 2 7 3 E T 0 0
5 7 8 7 2 4 4 4 E T 0 0
57A15632E+00 574370^3E+00 5 7 3 5 4 0 2 1 E T 0 0
83729E+00
155
By solving (5-1) we obtain Z = 19 Examining the result
obtained in table 35 we find Z = 093608907 The deviation
of the computed Z from the expected Z is not surprising beshy
cause we did not take care of the scaling problem caused by
the sampling time Recalling that the sampling time used in
the algorithm for the inverse scattering problem is 005 sec
instead of 1 sec we thus have to rescale Z by multiplying
it by lAt and obtain Z = Z^t = 187217814 which is pretty
close to the expected value 19
By selecting two simple examples discussed above to ilshy
lustrate the anology between the algorithms used for the
discrete and continuous earth system we may infer that the
continuous inverse problem can be discretized and solved by
the algorithms used in the discrete inverse problem and
the impulse response from the discrete system can be
smoothed and identified by the algorithms used in the contishy
nuous scattering problem
CHAPTER f
CORCLOSIOH
The seismic inverse problem has been investigated for
the discrete and continuous earth systems and the simulashy
tion as well as its result ifere discussed in the previous
chapters As described before ML estimation and cepstrum
detection work fine to estimate the characteristic parameshy
ters (r-0^) as long as noise is not too serious If the
Input of the system is not given we may use the cepstrum
(algorithm 3) to find it but with the limitation of the
shortpass filter we were not able to find the input other
tlian the unit spike The problem may be solved by using a
so-called comb filter to filter out the spikes and restorshy
ing the cepstrum of the input with some sort of smoothing
scheme [ 14 ] This is left for future study since the inshy
put is usually assumed to be known for the inverse problem
For a continuous earth system the technigue developed for
the inverse scattering problem helped to solve the inverse
problem The mast exciting result is that the estimated
Impedance is extremely insensitive to noise and that reasoshy
nable estimates for impedlnnce can be obtained up to noise
level 0^= 001 The essential step in solving the inverse
156
157
scattering problem lies in solving the Gelfand-Levitan inshy
tegral eguation which was solved by three integral approxishy
mation rules in this thesis The Gelfand-Levitan integral
eguation may be solved by some other numerical methods and
this may be a good topic for future study The major disadshy
vantage of the technigue used to solve the inverse scattershy
ing problem is that it reguires the impedance of the earth
system to be continuous This reguirement limits the applishy
cation of the inverse scattering technigue to the real case
This may be another problem to be solved in the future
BIBLIOGBAPHT
1 A T Oppenheim and B W S c h a f e r D i g i t a l S igna l P r o c e s s i n g Englewood C l i f f s NJ P r e n t i c e - H a l l 1975
2 P Eykhoff System I d e n t i f i c a t i o n John Wiley Sons New York 1960
3 J H Mendel N E Nahi and M Chan S y n t h e t i c Seismograms u s i n g the s t a t e - s p a c e approach Geophys i c s Vo l 4 4 Ho 5 p p 8 3 0 - 8 9 5 May 1979
4 E A Robinson Dynamic P r e d i c t i v e D e c o n v o l u t i o n G e o p h y s i c s Vol- 2 3 pp 7 8 0 - 7 9 8 Dec 1975
5 N A A n s t e y S e i s m i c P r o s p e c t i n g I n s t r u m e n t s S i g n a l C h a r a c t e r i s t i c s and Instrument S p e c i f i c a t i o n s Gebruder B o r n t r a e g e r B e r l i n 1970
6 F H a b i b i - A s h r a f i Es t imat ion of Parameters in L o s s l e s s Layered Media S y s t e m s Ph D d i s s e r t a t i o n U n i v e r s i t y Southern C a l i f o r n i a Nov 1978
7- N E Nahi Est imat ion Theory and A p p l i c a t i o n s New York K r i e g e r 1976
8 H L T r e e s D e t e c t i o n Es t imat ion and Modulation Theo
ry-- p a r t I Hew York Academic 1970
9 T Y Young A R e c u r s i v e Method for S i g n a l R e s o l u t i o n l E E s T r a n s Aerospace E l e c t r o n S y s t Vol AES-5 pp 4 6 - 5 1 Jan 1969
10 J A Stuffer Generalized Liklihood Signal Resolution IEEE TransInform Theory Vol IT-21 pp 276-282 Hay 1975
11 B- G Lichtenstein and T I Young The Resolution of Closely Spaced Signals IEEE Trans Inform Theory Vol IT-14 pp288-293 Mar 1968
12 D G Childers D P Skinner and R C Kemerait The Cepstrum A Guide to Processing Proceedings of
159
160
IEEE V o l 65 No 10 p p 1 4 2 8 - 1 4 4 3 Oct 1977
1 3 A V Oppenheim ( E d ) A p p l i c a t i o n s of D i g i t a l S i g n a l P r o c e s s i n g Englewood C l i f f s NJ P r e n t i c e - H a l l 1978
14 P L S t o f f a P L Buhl and G tt Bryan The A p p l i c a t i o n of Homomorphic D e c o n v o l u t i o n t o Shal low-Water Marine S e i s m o l o g y Geophys i c s V o l 3 9 pp 4 0 1 - 4 1 6 Aug 1974
1 5 See Re ference 1 2 pp 1 4 3 1 - 1 4 3 2
16 R W S c h a f e r Echo Removal by D i s c r e t e Genera l i zed Linear F i l t e r i n g Ph D d i s s e r t a t i o n M I T Cambridge MA 1968
17 S e e R e f e r e n c e 1 2 p 1442
1 8 See Reference 14 pp 4 1 3 - 4 1 5
19 T J U l r y c h A p p l i c a t i o n of Homomorphic Deconvo lu t ion t o S e i s m o l o g y Geophys i c s Vol 36 pp 6 5 0 - 6 6 0 Aug 1971
2 0 R G N e w t o n S c a t t e r i n g Theory of Saves and P a r t i c l e s McGraw-Hill New York 1966
2 1 L D Faddeev The I n v e r s e Problem in the Quantum Theort o f S c a t t e r i n g J- Math P h y s i c s Vo l 4 p p 7 2 - 1 0 4 1963
2 2 H E Moses and C H deRidder P r o p e r t i e s of D i e l e c t r i c s from R e f l e c t i o n C o e f f i c i e n t s i n One-Dimension HI T- L i n c o l n Lab Tech Rep No 3 2 2 pp 1 -47 1963
2 3 I Kay The I n v e r s e S c a t t e r i n g Problem I n s t i t u t e of Math S c i e n c e Div of E l e c t r o - M a g n e t i c Research New York U n i v e r s i t y Efl-74 pp 1 -29 1955
2 4 J A Ware and K Aki Continuous and D i s c r e t e I n v e r s e Problems In A S t r a i t i f i e d E l a s t i c Medium Part I J Acoust - S o c Am V 4 5 pp 9 1 1 - 9 2 1 1969
2 5 See Reference 21 pp 7 2 - 8 0
2 6 J G Berryman and R R Greene D i s c r e t e I n v e r s e Methods f o r E l a s t i c Waves i n Layered Media G e o p h y s i c s V o l 4 5 No 2 pp 2 1 3 - 2 3 3 Feb 1980
2 7 P L G o u p i l l a u d An Approach t o I n v e r s e F i l t e r i n g of Near -Surface Layer E f f e c t from S e i s m i c R e c o r d s
161
Geophysics V26 PP 754-760 1961
^^ llJln^rsltf^^V Synthesis of A Layered Medium from I t s Acoust ic Transmission Response Geophysics V 3 3 pp 264-269 1968 f^i^^^^
APPENDIX List of FORTRAN programs to implement algorithms
(1) MLCEPFOR
(2) PMLDECFOR
(3) INVHTFOR
(4) INVHT13F0R
(5) INHT138F0R
(6) NOISEFOR
( 7 ) FSTINVFOR
162
163^
(1) The FORTRAN program MLCEPFOR for algoritrm 1 in Chapter II
bull
c n r
Usin^ MeMiiTiuiTi Liklihood EstiiTistion snd Hoffiofiiopphic Decorivolution
to iden t i fy the e3rtn Isjirjerfid system fr-oc the noi-=s bull=eipoundi0srsT
The r-sr-BJiieters to be iden t i f i ed ere Reflect ion Coeff icient
Bnd One-wey Travel Time of esch subls=er
Bdditive noise has been assuifsed to be white ^sussisn snd i t C3n be jSenersted b^ running 3 pro^rsiTi NOISEtEXE which i- wr i t ten to c rea te 3 noisy seismoarstTi with non-noisy seisiTiQSrsni 3S input
K E H i_ -J j bulllaquobull M 0 y i gt J J 4 0 n ( 0 J - 0 H j J M J u lt D v raquo J i J -J bullbull-bull
t-
7 bull= Ncisy Response fpoffs Isyered eer th systeTi U = Lp^oin^- s t s t e waveforiTi of sub 1 ayer 0 -- Downsioini^ s t o l e wBveforiJi of sublsyer H - Input source to Isyered esT-th systyis bull 1 bullraquobull r-mdash r bullbull
I f rE bull CNicK Tnc n h r L C L - luc u u E r r i L i c r ^ u r Cr _M =bullgt - r- r- r r bullbull rt
iviir i f r l iPL f ENTEK rir SHnrLiNu ij-nt OF Tnsi iNrUi rcL-jE Lc
Set i n i t i a l v3lue of loop psrsfTieter H=0
Siven the seisiTio^r-sn to be processed a no tne -(bullbullgt kha i 1 pJ t S i poundbull f 31 bull-
IL l u =^i JiiJ-to ri i - laquo r- -f bull 7 7 bullbull T
-rHL--- 11 L 1 bull-bullbwiigt-Jf fW J
bull ^ ^ bull bull U l l c i t ipiioins snd downsoini wa^efcrn f^cfi noisy sei^c2 usinii niiiui Likelihood Estinisti
LlJ ^ij i - J l - - I I bull T 1 - X bullbull I bullbull Lpound - bull ^ bull bull 4 ^ ^
bull 1 r ^ ~ ~ T bullbull M bullbullbull _ 11 bullbullbull 1 1 i T i i f f i i z i t bull bull fraquo I r V bull bull r 1 1 j laquo bull - bull bull 1 J laquo I laquo 1 bullbull bull
_ 1 bullbull bullbullbull ] V pound L e _bull euro P S- I- P J M -1 L i_f- r 5
-rt f r w _J ^
-bullbullbulld l c c = t
164
c
r
c
Print the reflector series and its cepstrum for liyer 7
IF(NNraquoEQ7) THEM DO 75 I=ij5040 l)RITE717) A d )
^^ yRITE(72r7) CEPId) 75 CONTINUE
END IF
After examine the reflector series and obt3inin^ ih= bullbullo-stinr of the first spikeraquo find the smpiitude of the first spike and compute the reflection coefficient end one-w3y trevei tii e
CALL REFONE(AjDELTjREFONEfRyTIGH)
Print the reflection coefficient 5nd one-w3y trsvei tiine
VRITE(705) RJTJNN 5 FORHATdOXREFLECTION COEF ^ yFiOw 3XHHE-yAY TRAVEL T-ME 1 = E17raquo3r FOR LAYER 12)
Use recursive reletionship to find the next state
CALL RECURSIVE(UjDflGHjR) ONE(NN)=T
Increese loop PsrBfiieter
n - r i T l Ir(HLEraquo6) GO TO 77
y F0RMAT(Ei7raquo8 2 F0R1AT(F107)
o r END
L Subroutine to compute reflection coefficient snd one-w-s trsvei tinse
ouoftuu JAz hhrJiyh M itL I rcr ^ br^c hhr Luc ri-v i inz ijr bull DIMENSION Alt5040)REF(50)fONE(50)
Need to input the 3ocstion of the first sPike before proceedi this subroutinet The location of the first spike can bs deterniined from COHTAL screenv f w f r- bullbull -bull r bullbullbull bull T raquo t -- I f I -v f i T raquo r I (bull r bullbull bull Li Cr (mdash T r- rk f -bull ii T -bull r- I b bullbullbullbull - bull n r i t L - J L M iL N r u 1 i v-r PL r i r C I OV L--Z i 1mdash r - --i bullbull- bullbull J I -r
ri L - - i (bullbull-E-^v
bull r i 1H K J o ^1 l O L i f J - J4v 1 -^ b 1 y ^ u I i bull-bull ri ~ ^ v ^bull A bull mdash 1 gt-
ft r ft
bullI Jmdash J - J - ^ bullgtbull - - bull J ft ft - s ~ f raquo i r - i i
J bullbull- I J iz gt i bullbullbull t bull nci^i
H i bullmdash bulli T i bull - - bull v _ f l -I bullgt bull- bullbull J - r - | - T
bull t bull bull _ ~ t gtbull ^ - 1 bull - - bull ^ L- bullbull bull I mdash r mdash - r
r T - r - mdash - r - I
c r
1 6 5
SUH=00 DO 50 1=150 IF(ONpound(I)tNE0raquo0) THEN SUii SUM-VONEd) cLoc GO TO 60 END IF
50 CONTINUE 60 TRAVTIME=0 5(DELAY-3UH)
IGH=JNINT(TRAVTIMEDELT) RETURN END
Subroutine to find the next s tate usin^ recursive relationship
SUBROUTINE RECURSIVE(UDdGH7R) DIMENSION U(5040)raquoD(5040)
N=5040--IQH DO 30 1=1fH L=I-1 D bull 5040-L) ==D (5040-L-IGH)
30 CONTINUE DG 35 1=1 dGH
35 DXI)=00 DO 40 1=1 J5040 II==I-I-I6H IFdIraquoGTgt5040)THEH Ud)=-RMid) d O-R) GO TO 40 CNJ i i -d)^(Ud-f-I6H)-R^Dd))d0~R) n r T v= 1 O + R ) f D (I) -RU (I)
40 CONTINUE gt- Imdash V ( ( - I
I-r jr-V
iub rout ins for CePstruiTs analysis to reconver reflected rsries
ri 1 raquogt rt i-v 11T T M p p lt-1- 1 i lt w r n i-i r- gt- rlt rgt i- n bullbullbull
Duijrbb i viz nLzr U r M rraquo h P Q Lcr ri CcF x )
U bull= Up^oin5 s t a t e estiniated by HL estiniation A = The real par t of r e f l ec to r s e r i e s B - The itiiaainary ^sft of r e f l ec to r s e r i e s H = ihe input sisiTiol to the layered ear th systei P ~ The real r s r t cf the cepstrum of the make-up coTPosite
stltte n ~ the iwiasiinary par t of the cepstruirs of the ^bove s t a t e CEPR - The re51 par t of the cepstruiii of the re^lector s e r i e s CEPI - Tiie iia-2inary par t of tiie cepstruir of tiie r e f l ec to r
s e r i e s I- n- A I A r - ft 4 gt f imdash ft A ft rlaquo r - ft ft - bull r ^ bull ft - r - t - J i ^ - i bull - r - r -r raquo r ^ ft A ft
-Jiiri H u J bull ^ i ^J^bull bull r v bull w- j J ^ - j bull Llaquo r r- j ^ ^ j - L z ~ --bull^ r r- A I ( r- ft i ftlt T gt 1 -r nr ft raquo -bull bullbull y Z ^ bulli bull bull
- r ^ i r- bullbull f raquo c ft laquo ft M bullbull T bullbull A bull 1 J c ft i
L_ir Lc J jJ-iJ bull 7 n jj-^J r jJ^-J bullraquobull J 1 - _ - _ - 4 L _ _ 1 4 I ^ J -bull bull bull I - I J 3 bull y bull-bull e bullbullbull bullJVir^Jz 1 5 z-3bullbull V z r J j l i l - l f i I~J _
- i bull- ft T _ 4 IJ ft ^ bull V i ~L f y-jj bull ^_ I I T bull J T - I Imdasht - A - r I bull I
i 1- f i - bullgt
166
Ad)=Ad)Ur(FLOATJd-l)) Bd)=0raquo0
C C To create s time sectuence whose fourier transforn is p- uivalent C to the derivstive of the spectruni of the composite state
INTd)=-lraquo0-FLOATJd-i)Ad) 20 CONTINUE
f-
C To conipute the spectrufii of the composite state
99 CALL FFTY(AyBd) DO 30 1=1f5040 Fd)=Ad)^d00raquo0)-fBd)f00draquo0 IF(CAB3(Fd))raquoLTd0E-20) Fd) = 10E-20 Fd)=10Fd) Ad)=REALFd)) Bd)=AIMA6(Fd))
30 CONTINUE CALL FFTY(AjBj-i) DO 35 1=2521^5040 Ad)=00 Bd)=00
35 CONTINUE CALL FFTYCArBd) bull00 36 1=1 5040
36 Fd)=Ad)d050raquo0)IBd)^(00draquo0 fS
bullbullute the derivative of the composite s ta te spectrusi T_
-bullJ
io compute th
ri d ) =V bull J
Bd) = INTd) CONTINUE-L M L L rr t MHC i
To compute the division cf the derivative of the spectrus7 and the spectrum
UU JJ X-l7JltJ^gtJ b l=HKljkKl ^ gtJ v bull V ) rr ( i I- ( V y bull i bull 0 H I bull =G (I ir bull I Ad)=REAL(Hd)) Ba)^^AIHAG(Hd)) vU ivhjt
To t ak e i nve rse f ou r i e r t r ars f o ri o f bullbull esJ 11 c oifPU t ed a t previous stai^e and find the cepstruni
bdquo H L L r- I f t -i r- J
iu pass filterins to obtain the cepstrjfi toy ire reTect c M r t c c
-bullbull w t - f - l r - l y i mdash n -^Ugtmdash -i bullraquobull A r - f T -r 1-1-V bullr -i I bullraquobull Mmdash -f bull i r- 1 ^ i i- i- I bull c ^ c i tr^ i h e ~ i ur i i r v j ru J r t b r r _bull_bullbullbull 4 -- I - 1 - -r ^ cr T r
bull--cr =4 i r - t bullbull ft T _ J T -^ bullbull V mdash i deg i -L A t r bullbull _^ -ft JJ I mdash bullbull A
V - bull bull ^ - bull C ft -ft
16 r
^ io recover the reflector series from its cepstrum
DO 90 1=25040 Ad)=-iraquo0Bd)(FLOATJd-l)) Bd)=00
90 CONTINUE CALL FFTY(ABd) DO 95 1=1f5040 QR=EXF(Ad)) Ad)=QR|tCOS(Bd)) Bd)=ORSINltBd))
95 CONTINUE CALL rFTY(AjBT-l)
Unweighting
DO 10 1=1r5040 Diy=W(FLGATJd-l)) IF(DIVraquoEQraquo0raquo0) DIV=lraquo0E-20
10 Ad)=FLOATJ(K)Ad)Diy 45 FORMAT(13)
RETURN END
168
(2) The FORTRAN program PMLDECFOR for algorithm 2 in Chapter II
The only difference between (1) and (2) is the subroutine
listed below
c
BUBROUTINTE TO DECGNVOLV TC OBTAIN THE REFLECTOR nr- ri T r-1-
jcric3
Rn
bull - )
UBROUTINE DECQNVvUjHrRjRIHAG) EAL UC EAL R( OMPLEX n 10 I I)=U(
K J 1-0
10 4 J) 1 5 v 4 JraquoH ( 5 0 4 bull) B bull 5 -J 4 gt7) L- ( J -J 4 Kgt J J i- K J J O J 4 0 raquo K i M A b K i u 4 0
r 15bullbullbull4 f b ( l u40 -bullraquoH K 5vHU ^ =15040 I) 0
Hi n=Hf gt bull A i raquo
- gt I r T J111
laquo t I M_i_
1 p
I - f -t
r ^ b gt bullr r ir
B(
1 = 1 = I) = T y
r I LL
i-i-r-r-r r
V _ 4 i I laquo 1 r ^ J
M I I
Cd) Fd) REAL A T i i A
HUE r r
TYArB) TY(CDd)
v4u- J bull J ft ft ft I raquoi ^ T V |- f t ft J ft V I bull i U U J T D bull i bull bullbull V V t bull-bull i raquo )fedO00)-fDd)) |c(00l0) d) )raquoEQraquo0raquo0) G d ) = iOE i n i
( H d ) ) G ( H d ) )
A B - I )
I- bullbull I 1 I ^ I 1
iO T t i mdash laquo i I t laquo i J
r- T A T bull
^- I pound 1 = f H gt j _ I
bullbull i i i - i u J bull i-ft n bull T T w 1 gt gtbull-
KpoundTUR^^
169
(3) The FORTRAN program INVHTFOR to solve inverse-scattering problem using the Trapezoid rule to approximate Gelfand-Levitan integral equation
b Inverse-Scattering Rroblem Technique to detpr-ir-inp the i iplt=denlt= Qt earth system
IMPLICIT INTEGER2 l-gti) IMPLICIT REALrS (A-HPO-Z^
DIMENSION R(2043)yYd00)72d00) DELT=0raquo05 DO 10 I=1204S READlt11J7) R d )
10 CONTINUE DO 20 1=12047 L=I-1 R(2048~L)=R(2047-L)
20 CONTINUE nd^=0raquou
To Calculate YCn^m)
vihL c^ii xr ) FORMATdOXUsin^ Trapezoid rule to appro-iiiiate inteij
and usin^ Householder fornrula to inverse niat J- i lOX ra t ion f r-e-i^i
value of Zbull) = 1 T - jLu 4u 1 = 1raquo50
1-MLL 1 iNV -l K T J1C- i
bull f i
To calculate impedance by assumins unit impedance ~-00 bulle surfi
HL-C=UraquoV
i i j JJ J = i f f - t - i
cr--ijn-( J i bull r T T tmdash raquo T -
jar-I c- i^f - ft j - r bullbullbull gt ( 1 mdash
runriM bull bull c l r- -1 f l gtlt - r bullbull -gt V
1mdash T bull^ 1
t-1
ft
s - 1
n 1
rhr
bullJ
-r
- V
i J
1 Am bull
bull^ f
i V
bull
(
gt
) T
i
n
i
1
J-i
7
r o
) bull
bull M
r-rt f It mt t
1trade
T- P - I
i Cl
f p r- r^r J
r^r- r bullbullbull raquobull
OLCrJJ I 1 i 1( r i Tl i l-CL -bull
T - bull -1 T - r T f f bullj-ft -bull gt V ^
i_iU J I hrMU-c i-i-r u^ bullgtbullbull -bull l fft T - gt I V bull J ft ft
1 -ncf-M-1 u^- bull Lbull-bullbull 5
t r i gt li ^t i^ V f
bull ^ ft ft 1 - - I - I - I a t I I raquo
1 mdash 1
A ^ 4 -V ft J -
M 1 ^ i v j Ij
170
10 CONTINUE DO 30 J=1I H=I-J+1 DO 40 L=iM
40 A(L+J-lfM)=R(L-M)-DELT 30 CONTINUE
DO 45 J=ld 45 A(JraquoJ) = lraquo0+AUyJ)
CALL^GANDIdjHrAfAl) DO 50 j = l d
50 Y(J )=Al ( Jd ) RM=(Yd)-i^0)DELT RM=RMdgt0t0v55|ltDpoundLTRH) Diy=10-0raquo5DELTRM DO 60 J = l d - 1
60 Y(J)=Y(J)tDIVDELT Y d ) = (Yd5-iraquo0)^DIvDELT RETURN r- i I r
ri-iu
eUrhbu iNc 0 LJcNcrtmc f-Mrxlt MNB iNvt^^Sc iT
ffi--iti- or nn r i o cc ir-ivcRrcL
IS - flti-i fin ur DLULIX i r ivimbci nHir ucNcPiAs ci H i - i P i r - i T rigt ifft bullftT-Aftlt~
rncviuu--- o i HJc bull--DrcwiHl frt br nMihi^ o r c ihVchsci
IHPLICIT INTEGERI^2 d-N) IMPLICIT REALMS (A-Hn-2) 1 f r h n i n f-J n t 1 bull I ij laquo i J I lt lt bullraquo r bull 1 J U f J A H i Jraquo i bull-bull J 7 r b i N V K X J r 1 bull- U DIMENSION AdOOf 100)^0(100) DO 10 I=1N1 QRd)=QdfNl) CONTINUE
0 INvEKbc GcNcrgtHEI nMir iA
J-^ru gtcQ gt2- pcN
Rr^-Hydi)=GR(2)DTM RGINy(l72)=-QRd)DTM rv-jlfV 2 i 1 =rtblNv bull 1 i raquot r T bull 11 1 - mdash1 1 ft Tt T lt
r_i_gtc
-- r bull r bull I r
A ^
(bullbullbull 1 -J - ~ H M i J
L U i 1 bull_ - 4 1 mdash - 11 I lt bull gt I I bull- [ -bullbull bull bullgt i I V
-bull bull r V T bull bull X f H L r -3 a i
T bdquo 4 i
i - J i i
bullr I i N V i i J
c
1 7 1
C SUBROUTINE TO INVERSE MATRIX USING HOUSEHOLDER FORMULA
SUBROUTINE INVH(NfA^C^B) C C C NmdashDIM OF INMERSED MATRIX TO BE GENERATED C AmdashINPUT BLOCK INVERSED MATRIX C 3mdashOUTPUT INVERSED MATRIX TO BE GENERATED C CmdashN-DIM ARRAY OF NEU ENTRIES WHICH EXTEND BLOCK MATRIX TO C THE MATRIX OF INTEREST
IMPLICIT INTEBER-2 d-N) IMPLICIT REAL3 (A-H0-Z) DI MENS I ON A (100 d 00) B (100 100) C (100) B1 (100 100 bull B2 bull 100 100) DIMENSION Cld00)rC2d00) C(N)=(C(N)i0)2raquo0 DO 10 I=2N-1 DO 15 J=2yN-l
15 BldJ)=Ad-lyJ-l) 10 CONTINUE
DO U J=iyN B l d j J ) = 0 raquo 0 B l ( J d ) = 0 0 B I N J J ) = O gt O B1(JfN)=0raquo0
ir CONTINUE B l d d ) = 1 0 Bl(NyN)=lraquo0 bCALHK= 1 0 M bull vTL- bull N J ) Cld)=Cd)SCALAR Ci(N)=C(N)^SCALAR DO 20 I=2N-i SUM=0raquo0
25 SUM=3UM-fC(J)^Bl(Jd)SCALAR Cld)=SUM
20 CONTINUE i 11J bull J 1 bull N
ZO B ( N d ) = B i ( N d ) - C l d ) SUM=00 tiU ZZ i - i bullbullbull
j bUn=bUnT tgt X bull A gt X f-b X oLnLMn-1 bull bull-gt i raquoVTjUn P T J bull = ( 1 J
C 2 ( N ) = S U M DC 40 1=2N-1
m I w _ i oun- vJ
TM 50 J=2-N-l T^-^ 3UM=SUM4-Bldf J ) C ( J
t
LONTIr-Lii I b WW i mdash - bull - - - f t t bull V t
1 - mdash-v V V t r i ^ gt bull ^ - t O r raquo A bull
yj bull J) bull=b2 d - - -Bl s N J foL-ii-rir - - I -ft I J ( t C
I raquo _ v J bullbull T I raquo r i ^ r
172
(4) The FORTRAN program INVHT13F0R to solve the inverse-scattering
problem using both the trapezoid rule and Simpsons 13 rule to
approximete the Gelfand-Levitan integral equation
t I
C Inverse-Scatterins Problem Technieue to deter-iiine the i-riplt=dance b of earth system
IMPLICIT INTEGER-- 2 d-N) IMPLICIT REAL)iS (A-H0-2) DIMENSION R(2048)J Y d00) Z d00 DELT=005 DO 10 I=l2048 READdi7) R d )
10 bull CONTINUE DO 20 I=lr2047 L=I-1 R(2043-L)=R(2047-L) CONTINUE Rd)=0gt0
To Calculate Y-nyfii)
FORMATdOX dnverse-Scatterin^ iTiethod to determine the bull 1 diiipedance of ea r th systemISX^with inipulse response t 2i3XR(T) = - 0raquo5ieltp(-0^5t)1^H(t)d3Xwhere H(t) i s a u n i t 3 s tep funct ion )
DO 40 I=lraquo40 N=2-tI CALL INV(NyRYDELT)
lU LnLbuuhic inrciHr-fc r i MCDuniir^ ui^i i1^clbullbullbulliM_c Mruvc rurrMLc
A p -1 1 _ w I 1 I r ft
A bull -- A
HL-L- i = V laquo V U J O J - i bull - i
mdashmdash ft - r fl u ft ft
- i I bull bull - rt b_- i f-raquobull-- 4 f t mdash - ^ bdquo ft ft hLlo--J tJ T- r- A _ ft-- M ft n
i -
H _bull bullbull bull ~ H b _bull O T I bull ~^ bull i 1^ r^ J I A r^ r - ( A - ft -V bull l-L -bullbullHi-L- i T iHL- - j mdash H U L - j -y - _ J ft I A ~ - 4 rbull-1 I f J - ft bullbull-r L- - - i f - bullbullbullrub-bullbullbull iC_ -tvi V V bullbull (bull J
R^==ii005 Imdash r r -- ii _ 1 - 1 I- I- -T T y f 4 ft r i r
K b mdash v 1 bull bull-bullTrbullbull~i bull i bull i raquo J T r - r I If- T-imdash - n s r- -r T bull (mdash-iTl bullbullft
~ i 1 c ^ i i i r w rrM ii i crvKUix
173
c c
bull 4 bullbullbull
DO 10 J=1T DO 20 L=id A(JfL)=0raquo0 CONTINUE CONTINUE DO 30 J=ld M=I-J-M DO 40 L=1M A(L+J-1M)=R(L+1))|DELT CONTINUE DO 11 J=2d2
11 AJd)=5raquo0^A(Jd)6raquo0 DO 12 J=ld-i2 DO 13 L=2df2 Ir(A(LjJ)raquoNEraquo00) THEN A(L-J)=40A(LfJ)3n END IF
13 CONTINUE 12 CONTINUE
DO 16 J = 2 d - 2 r 2 DO 17 L=2dr2 1 F ( A ( L J J ) N E raquo 0 0 ) THEN A(LfJ)=2tOA(LJ)3laquo0 END IF
17 CONTINUE Id CONTINUE
DO 45 J = l d -^ A-JyJ) = lraquo04-A( J f J )
JLb O l J = l raquo l
6- B(J)=00 DO 62 J=2f l -2y2
62 B(J)=DELTR(J-d)(-6raquo0) Bd)=DELT^Rd + l ) ( -6raquo0 ) i l raquoO CALL INVH2(AdyX) DO 63 J=1I
3UM=0raquo0 V bull C -i L ~ 1 i
0 o u I -J ~ bullgt U n -- K J L ) I L
Y bullj=SUr C--J LUI-i i i i v b c
RH=(Yd ) - l gt0 ) DELT RH=RMdOTO5DELTRM) T-TU=i gt0-0gt=ifDEi TRH Tl bull- f t I _ ^ T J UU OV bull_- i - i
bull bullbull^ bull bullbull I ^ y bull 4 ^ -(f Tlaquo V I I ff r f -V
c i J = gt J J f i i i v [ lEL t T - -bullbull Jl 1 (bull bullbull L J V L C L
v C f bull j C -J - mdash -
bullraquobull I Imdash - M - ) -
laquo
174
u
c c c r
A N X
SUBROUTINE INVH2(AfNraquoX)
INPUT MATRIX TO BE INVERSED DIM OF THE MATRIX N MUST BE EVEN NUMBER OUTPUT INVERSED MATRIX
C
r
tr J
10
13
IMPLICIT INTEGER-2 d-N) IMPLICIT REALMS (A-H0-2) DIMENSION AClOOdOO) jXdOOdOO) JBI dOOdOO) B2d00 100)
INITIALIZATION
DO 5 1=1N DO 6 J=lrN XdyJ)=0raquo0 CONTINUE
TO INVERSE 2 BY 2 CENTRAL BLOCK MATRIX AT FIRST STEP
H=N2 DTM=AMyM)5J(A(M+ljM+l)-AM7M+i)A(MTljH) X(MM)=A(M+iMll)DTM XM--ljM-fl)=A(MfM)DTM X(MfM-M)=-A(MdyM)DTM X ( MTI M) =--A (M J MTI ) DTM IF(HraquoEQ1) GO TO 45 TO EXTEND CENTRAL INVERSED BLOCK TO A N-DIMgt MATRIX WITH UNITY DIAGONAL ELEMENT EXCEPT THE INVERSED BLOCK
DO 10 1=1M-1 XdfM-fld+H-M) = 10 X(n-IM-D = l0
CONTINUE
TO SET UP LOOP PARAMETERS FOR EXTENDING INVERSION TO THE WHOLE MATRIX
K=0 L - L r i K=K^1 M1=H-K M2=M-KfL-l L H L L H I H A j i M u i N j n i i i bull r ^ j bullJ b A -J i = n i AI
lt U l i d gt = hjL 7 li-t-X K U 7 i -
DO o j j = n j n 2 J u ^Ki bullJ=ni n- T 1 bull T I f j t T I V ^i 1 J bull- - bull A bull- bull AA J bull -mJ bull
iLC
J- r - -rraquo t i 11
r i o u I- r J bull J i i bull ill
bull bull lt ( T J P I O T T bull 1 bull- I bullbull bull bull -t bullbullbullbull bullbull-lt bull-lt bull - - ( 1 f r i t ( I j I | r ^ ( f -J ^ | p v bull y j i bullr bull- bull r t p -- T - T ( - r bull -bull rmdash n Jmdash -r bull t bullbull f bull J t bull I J f i - bull trade f i r - rt a i t mdash f j bull ^ mdash 1 ^- i ( J U raquo U bull T- - ^ b - bullf r I V bullbull T V i f bull 1 - bull I I - -T raquo
t t B -
1 7 5
6 BldraquoJ)=0raquo0 5 CONTINUE
DO 10 I=MiyH^-l 10 Cd)=A(M2raquor)
C(M2) = (AltM2yM2)-U0)20 3CALAR=10d0+C(M2)) Bl(M2rMi)=CMl)SCALAR Bl(M2yM2)=CM2)SCALAR DC 20 I=Ml-fiM2~l
SUM=0raquo0 DO 30 J=MllljM2-i
SUM=SUMiC(J)--^Xdd) 30 CONTINUE
Bl(M2d)=SUM-SCALAR 20 CONTINUE
RETURN END
SUBROUTINE H2
SUBROUTINE H2(AXrB2jLyKjNrM7MlJM2) IMPLICIT INTEGER5IC2 d-N) IMPLICIT REALMS (A-H^O-Z) DIMENSION XdO0dOO)AdO0dOO)B2dOOd0O)CldOO)C2dvO) DIMENSION CCdOO) DATA C11000raquo0C21000raquo0 DO 5 1=1N DO 6 J=l7N
o Ox I J ) =v y rr
XJ
CONTINUE DO 10 I=MiM2 Cld)=X(M2d) C2d)=AdM2) CONTINUE C2(M2)=(C2(M2)-10)2raquo0 nUM=0raquo0 DO 20 I=Ml7M2 OUM=3LltMTL-1 bull i ^Li i SCALAR=i0d0T3uM) CC(M1)=C2(M1) f t - v ( bull^ bull bull bullbull ^ M
bullbull-bullbull MA - c u r l uO bull-bullJ j = n i T i n A - j
-^ I gt ^i _ ft ft
vu 4v -MiTi f n-c- -ft I ( rft I I v i I ft ftgt I gt 4 v bull bb i =SUnTL-A J f A K X J (bullft ft T i _^ ift I I i l j _ i ) mdash JlljfI
rnNl INUc 0 50 i=MlrM2 bullsect2 ( I J--CC d)-^C 1( J)+3CALAR f t 1-1 bullbull T ^ A
_-U- 1 bullbullJ c r 7 l i r i - I bullc t or-I
176
(5) The FORTRAN program to solve the inverse-scattering
problem using the trapezoid rule Simpsons 13 and
Simpsons 38 rule to approximate the Gelfand-Levitan
Integral equation The only difference from (4) is the
subroutine listed below
ftgt T T l r bull bull bull Vgt T V11 gt T r raquo T-bullmdash f V
Dubpub ir tc iNV bull u f r bull) vc_ s IMPLICIT INTEGER2 d-N) IMPLICIT REAL5i=S (A-H0-Z) DIMENSION YdOO)yAdOOdOO)fAAdOOdOO)Xd00100) DIMENSION R(2043)Bd00)BBd00 DO 10 J = i r l DO 20 L = l d A(J L)=0raquo0 AA(JL)=0raquo0
20 CONTINUE 10 CONTINUE
DO 30 J=lfl ^ M=I-J+1 DO 0 L=1raquoM A^LTj-ljM)=R(L-fl)^DELT AA(L-fJ-lM)=A(LiJ-lyM)
MO CONTINUE 0 CONTINUE
DO 11 J = 2 d f 2 A 4 A 1 T mdash C j - i l A f T 1 ft
b 1A J = i i - i o i IF(AL J)bullNEbull0raquo0) THEN HrLfJ)=4raquo0-A(LyJ) 30 END IF
13 CONTINUE 12 CONTINUE
DO 16 J=2d-22 _b i L-A7ii
I - ( A - L J ) N E raquo 0 0 ) THEM bull A L7J =20^A(LJ ) 3raquo0 END IF
r^ CONTINUE J f t i tT i f (r~ iO bUl iMOC
Vl i i _ lt V
gtbull j - b v J V f I y _ ftj ft
- - I l l - 1 - bullII I I -
Z J L U N I XriJZ -- -ft -t I _ (-i T bullft ft
UJ O- - - -A i - A
r -r V _ r i r - T | i - T i
i r- T V ^ J
r -gt i - Vi J - i i
1 i - bullbullraquo
I T A A I
i mdash i u i-
177
^^i^^Uld-L)raquoNEraquo00) THEN IF(LraquoNEraquoL1) THEN A(Jld-L)=9gt0AA(Jld-L)3raquo0
ELSC
A(JldL)=30AA(Jld-L)4raquo0 Li=LiTgt5
END IF ELSE
GO TO 41 END IF
42 CONTINUE 41 CONTINUE
DO 45 J = l d 45 AltJrJ)=10+A(JyJ)
CALL INVH2(AdfX) DO 63 J=ld SUM=00
DO 64 L=ld 64 SUM=SUMiX(JL)fB(L)
YJ)=SUM 63 CONTINUE
RM=(Yd)-l0)DELT RM=RM ( 1 OiOraquo5DELricRM) DIV=10-0raquo5-4DELTRM DO 60 J = l d - 1
60 Y(J)=Y(J)^DIVDELT Yd) = (Yd)-l0)WIVDELT RETURN
178
(6) The FORTRAN program NOISEFOR to generate a noisy seismogram
or impulse response
C GENERATION SYNTHETIC SEISMOGRAM CORRUPTTED BY GAUSSIAN C WHITE NOISE C
REAL V(2048)7A(2048)OUTNOISE(2043) TYPE ENTER VARIANCE ACCEPT 7JVAR P I = 3 d 4 1 5 9 11=351251319 12=532151319 Lu 1 0 j = l yiv-^io
i~rMbullbull Li
0 vTlV^=SQRT(-2raquo0-^AL0Q(Xl))^C0S(201^rI1fX2)^S0RT(vAR) f 1 n c T mdash bulllt laquo^ ft A bullgt
yb _bull- i mdashiibio JO RcHDviU7 H i)
n i-ft ft T _ raquo - i ft A ft
n ( lt - r M f t T f t r - r raquo _ A T ( bull bull bull bull raquo _b NOiDCbullgt I bullbull - M M TV X wRiTc 45 f ObTNuxcc K X gt
o-j I-b IM i r-i b e rbmiMi lt c i tc f t - r - r i o i br Imdash y IT C IL
179
(7) FSTINVFOR T A fast algorithm to solve the inverse Problem with multiplications O(N^)
u To inve r t Gelfand-Levitan iTiatri usin^ a fas te r al^orithiTi
with 0(N2)
bullr
L-
1
J V
20
4J
^r bull 1 bull bull i ft bull i C bullbull ft bull J ft A p I 4 (I n 11 lt 4 I n laquo Ibull J ft ft -raquo r gt
nci-iL n K1 Ov i gt i vO y ij bullbull i vvjraquojx x Jj J JVA VO ft bull i jb i (5b bullbull Lraquou iy i i = l y l o u
ncHihll Rgtii rUrsnH i ci7togt
r r-1 T ~ ft e
i i C L - V v V J
rN iraquo=r i ^LiCL DO 5 M^lfSO
I=2fM
To coiTiPute F G and re f l ec t ion coef f ic ien t RN bull A raquonV^r ft 1 fi T r-r -
L-HL-L r bnN r ^jyr-r rN j I bcL raquo Usin^ F G and RM to coiiiPute FN and V which i s re la ted t
PN=lraquo0-l-RNd) DO 20 I I = l d - i
pN=PNdO-RNdI)^RNdI)) 1 mdash bullbull ft I f I M_ I i I ^ bull X f J Vift CMT bullraquo T ^ bull I b Ai-J X i - ^ X
I 1 -r T 1 r- T T ft -r T V V r- 4 raquo Z bulllt V i J mdash f X X J mdashuJ i i i bull K r bull X gt mdashO bull 1 )
Xltr f t T T _ 4 T i_ t b i i mdash i i
l T T i i i T - r v r - v | - bull i i mdash V i i i ST bull
I- 4 T T )bull J T J ft r r - f T
I 4 _ L- lt 4 ft I Tr -T-J ft irIgt 4
DO 50 11=2I i~ r T raquo ~ l ~ ~ - ^ bullbull -u- bull t- r- t T l f t C-tf- i bullbull T i T T
K J 1 bull = K i bull i i f U - i C L A b raquo J bullbullbullIS J bull U C l
To coiiipute impedance ba Kernel intes-ration ft ft _ raquo ft
r -ft A T T _ bull T
Iiu ob i i - ^ y i -raquo V mdash A ft ft gt 4 ft cr -i- raquobull r-1 T- 11 4 raquo L bull M 1 =Alaquo_ OT i (bull b T V i- bull_bullbull LCL i f iS i
F bull bull r- 0 r 0 0 ff p u t a t i Q n
to Kernel
r-r vi)raquo-r- -r
i-r- r lt-r 1 7 7 v bull rr-_- gt U^ i fti^IU
bull i r b n- bull= i i bull b T r- r i v n bull- bullbull i gt v r 7 bullbull
rin tout 1 rr -r 1 r-bull-bull I--bull-bull j ri f1 9 -bull 1 I I 1 raquo r 7 7^ M f
I t- IT - r - i ^^ r ^
gtmdash - S - i A IT i -i U f I- i O A
r- r I I -r 1 raquo1 bull I r-bull 1 iM 1 i^tJXX I -r -v -bull D L T Tl
bullrv cr
1 r - - I bull - bull bull
r I 1 T It ~ I I T bullraquo i I bull f _ ^ M M I J bull bullbull ^ - Imdash A I T - J - - - - raquo-bull t _ IT V ^ _ =
r A 1 r - r - 4 - f
J bull ^ 1 - -1 ^
y r- I l ^ -
180
L=N-1 END IF
U
c c
To coTlaquoPute RN = S(N)0(N-i)
25 CALL SNdjFSRyDELT) CALL QNLldjRNyQ) R N d ) = S d ) Q d - l )
To conipute F and 6 usin^ recursion forjlas
40
p l - K N i i bull _ s X -Ki-t i bullbull - 1 -K-- i jhri i xr K irZijtpoundj inhie
00 TO 26 END IF r o ZKJ ix-^x-x
r i i 1 - r r -d i rrN ( i -f i0 d I - 1 D xi) ~bu v i 1 - i -r jv 11) -- F ( i X
bullif bullt ft T T _ 4 -r
FFdI )=Fd) GGdI)=Gd)
i r V i + LTN inEf T _ T 1 4 X-XTi
nn TO ^i END IF RETURN
Subroutine to conpute S(N)
Tl r ^ I I T - T gt - bull raquo
rnuu I ir-C Cl 7 1 bull J r c L l Imdash 4 ^ A bull ^ 4 ^ A ^ r- 4 l
HL r i 00 r 3 d 0 0 ft bull x bb gt i C bull
f lC_ mdash I i r - - i
i ncfx
i J I V ^
bull bull- i b 1 t i raquobull[ n -1 T t _ 1 -r J W J U ^ J m 4 4
CUf = 2b T r i i i T K i T J mdashi i ) I iCL 1 - gt 1 1
Cf- -our
bullTMi
- i_j Q bullbull J (J f 1 f e bull -bull 0 c 01i p u -bull s i mdash i
r j r h Jb i ifc wNi i ^ i r- ij r A J n c bull ft ft bull ^ f 4 ft ft
1 rgt gt- I I p -r _ 4 ^ r i 4 gt - i - i-i i 4 gt
JF bull 1 t b i i - ^ r e -
J I V - I~4 i X r r r- T- I - -^ ri--k r I I-V-T 4 bull I bdquo mdash r r - b i _ _ bull bull i ^
i- i bull 4 ^ r lr -gt raquo ^ bull ibull- l - r u r _
c