谱反演分辨薄层的基本原理及其简单模型试算 v2.0
DESCRIPTION
Undergraduate ThesisTRANSCRIPT
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09
2013 5 20
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II
ii
1
2
MATLAB cnki MATLAB2012
1.2012.12.15~2013.2.22
2.2013.2.22~2013.4.30 3.2013.4.30~2013.5.15 4.2013.5.15~2013.5.24 ppt
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iii
iii
1 2 3
3 1 2012.9.15~2012.12.15
MATLAB
An Introduction to Seismology, Earthquakes, and Earth Structure MATLAB cnki
2 2013.12.20~2013.3.20 1 2
MATLAB
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IV
iv
3 2013.3.30~2013.4.30
4 2013.5.10
1 2 3 4
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v
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VI
vi
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vii
vii
[]
()(N )
LSQR
MARTLAB
[]MATLAB
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viii
[Abstract]
Spectral Inversion is a method that can, by inversing the seismogram, identify
the thin-bed whose thickness is below the limiting resolution.
Nowadays, oil and gas are of great importance. However, as much of them had
been mined, remains are getting harder to be explored. Technology needs to catch up
with it in order to improve the efficiency of mining. Spectral Inversion is a new method
among many useful ones. Theoretically, it can identify thin-bed with thickness less than
limiting resolution which might largely contains oil and gas.
In the first place, we discussed the basic theory of Spectral Inversion.
Demonstrated the relative theory of Mathematics, Physics and Seismology and derived
formulas from them. They are disciplines of frequency spectral analysis, characteristics
of Fourier Transform, convolution model of seismogram, reflectivity and odd-even
decomposition.
In addition, we derived the objective function of Spectral Inversion. Then we
derived a matrix from the objective function which would be used for programming.
We discussed the ill-posed possibilities of Spectral Inversion and described the LSQR
algorithm used for calculating the objective function.
In the end, based on MATLAB, we practiced the theory and discussed the
influence factors as well as the differences between theories and practical conditions.
[Key words]: Seismic inversion, Spectral Inversion, Reflectivity, Objective Function
MATLAB Simulation
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ix
........................................................................................................................... 1
1.1 ..................................... 1
1.2 ................................... 2
1.3 ..................................... 3
..................................................................................................... 5
2.0 ................................. 5
2.0.1 ................................................ 6
2.0.2 ............................ 9
2.1 .......................................... 12
2.1.1 ............................................. 13
2.1.2 ............................................. 14
2.1.3 ............................................. 17
2.1.4 ..................................... 18
2.2 ...................................... 19
2.3 ............................................ 20
.............................................................................................. 22
3.1 .............................. 22
3.1.1 ............................. 22
3.1.2 ............................. 26
3.2 ............................................ 29
3.3 .................................. 32
.............................................................................................................. 36
4.1 ........................................... 36
4.2 ............................................ 38
4.3 ........................................ 39
4.3.1 LSQR .............................. 39
4.3.2 LSQR .................................. 39
-
X
x
.................................................................................... 41
5.1 ........................................ 41
5.1.1 LSQR ................... 43
5.2 ........................................ 48
5.2.1 ........................... 48
5.2.2 ................. 49
5.3 .................................... 50
5.3.1 ............................... 51
5.3.2 ................................... 53
5.4 .................... 55
.............................................................................................................................................. 56
............................................................................................................................... 57
.............................................................................................................................................. 58
.............................................................................................................................................. 59
A MATLAB.............................. 59
1......................................... 59
2............................... 60
3............................. 60
420 ........................ 64
6................................... 66
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1
1
1.1
1993
2
[1]
30Hz
3000 Widess(1973) 25
25
10
-
2
2
[11]
1.2
80
Widess(1973) Kallweit Wood(1982)[2122]
Partyka et al.(1999), Partyka(2005) Marfurt Kirlin2001
2004Partyka
-
3
3
2006Satinder ChopraJohn P.Castagna
2008Charles I.Puryear JohnP.Castagna
[11]
2009Sanyi Yuan
[19]
2009Satinder.Chopra John P.Castagna
[79]
1.3
1
2
-
4
4
3
LSQR
4
MATLAB
20
-
5
5
-----
2.0
(2.0-1)
Dirichlet
-
6
6
(2.0-1),
Dirichlet[15]:
1
2
3
[2]
2.0.1
1 u(t) = A cos( + ) (2.0.1-1)
A
2(2.0.1-1)
u(t) = A [cos cos sin sin]
= cos + sin (2.0.1-2)
= A cos = A sin
(2.0.1-2)
3 Euler
u(t) = + +
(2.0.1-3)
+ =1
2( +
) , =
1
2(
)
T, =2
u(t)
-
7
7
nu(t)
u(t)
u(t) = cos( + )=0 (2.0.1-4)
= cos() + sin( )=0 (2.0.1-5)
= ()+= (2.0.1-6)
= 2 + 2
= ()
2
2
cos() (n = 0,1,2, ),
= ()
2
2
sin() (n = 0,1,2, ),
= tan1(
),
= ||,
|| = 2 + 2
u(t)
(2.0.1-6)
u(t)(2.0.1-2)
(2.0.1-2) 2006
(2.0.1-2)a||
= tan1(
)
(2.0.1-3)A=440Hz
-
8
8
(2.0.1-3)A=440Hz 2006
= 440()
=2
Dirichlet
u(t) = 1
2()
+
(2.0.1-7)
(2.0.1-6)()
u(t)
() = u(t)+
(2.0.1-8)
u(t) ()(2.0.1-7)(2.0.1-8)
1u(t)
t2
u(t)()()
-
9
9
()
(2.0.1-4) () 2006
(2.0.1-4)1
(1 + ),1, (1 + )
11 + tan1 (1)
(2.0.1-
8)
1
2|(1)| = |(1)| (2.0.1-9)
=
2|(1)|1 =
1
2()[2]
2.0.2
2.0.2.1
()u(t)u(t)
()()u(t)
u(t) ()
-
10
10
2.0.2.2
u(t) ()u(t/a) ()a
t1 =
1 = ,dt1 =
1
u(t/a)+
= u(t1)11
+
= u(t1)111 =
+
(1)
= ()
2.0.2.2-1
:
(2.0.2.2-1)(a) u(t/a)u(t) 2006
(2.0.2.2-1)(b) u(t/a)u(t) 2006
2.0.2.3
u(t) ()
u(t ) () (2.0.2.3-1)
-
11
11
t1 = t
u(t )+
= u(t1)
(1)(1 )+
= u(t1)1(1) =
+
()
2.0.2.4
u1() 1()
u2() 2()
u1()u2(t ) =+
1() 2() (2.0.2.4)
[ u1()u2(t )+
]
+
= [ u2(t )
+
]
+
u1()
(2.0.2.3-1)
u2(t )
+
= 2()
[ u1()u2(t )+
]
+
= 2()
+
u1() =
2() +
u1() = 2() 1()
(2.0.2.4)u1()u2()
u1()u2()
u1() u2() = u1()u2(t ) =+
1() 2()
[2,13]
-
12
12
2.1
(2.1-1)
-----
(1)
A A (t)(t)
(2.1-1)(a)
x (t) 2.1(c)
x(t) = 1( 1) + 2( 2) + + ( ) + (2.1-1)
(2.1-1)(2.1-1)
(t)
x(t)(t)
(2)
-
13
13
(2.1-2)(t)
(2.1-2)(a)(2.1-
2)(b)(2.1-2)(c)
[8]
(2.1-2)
2.1.1
,
[5],
, ,,
[4]
[6]
30Hz Ricker( 2.1.1-1)
-
14
14
w(t) = [1 2()2]2()
2
(2.1.1-1)
Ricker
(2.1.1-1)
2.1.2
2.1.2.1
(2.1.2.1-1) Seth Stein,2003
xxy
u 0y
u(x, t)xt
,F = ma,dxdx
-
15
15
yF(x, t) = sin 2 sin 1
(2.1.2.1-1)
F(x, t) = sin 2 sin 1 = 2(,)
2 (2.1.2.1-1)
sin tan ,tan =
(u(x+dx,t)
x(,)
) =
2(,)
2 (2.1.2.1-2)
(u(x,t)
x+2u(x,t)
x2
(,)
) =
2u(x,t)
x2 =
2(,)
2 (2.1.2.1-3)
2u(x,t)
x2=
1
22(,)
2 (2.1.2.1-4)
= (
)
1
2 2.1.2.1-5
u(x, t) = f(x vt)(2.1.2.1-4)
= (
)
1
2
(2.1.2.1-1)
2.1.2.2
(2.1.2.1-4)f(x vt)
u(x, t) = Ae() = Acos( ) + ( ) (2.1.2.2-1)
A(2.1.2.2-1)(2.1.2.1-4)
v = / (2.1.2.2-2)
-
16
16
2.1.2.3
2.1.2.3-1
x = 011,
22x = 0
(2.1.2.3-1) Seth Stein,2003
u1(, ) = (1) + (+1) (2.1.2.3-1)
+xx
-
17
17
+x
u2(, ) = (2) (2.1.2.3-2)
x = 0
x = 0
u1(0, ) = 2(0, ),
+ = (2.1.2.3-3)
(2.1.2.3-3)
+ = 2.1.2.3-4
yx = 0
u1(0,)
x =
u2(0,)
x 2.1.2.3-5
k1( ) = 2 (2.1.2.3-6)
v = (
)1
2, = /v, (2.1.2.3-6)
1v1(A B) = 2v2 (2.1.2.3-7)
2.1.2.3-4(2.1.2.3-7),
12 =B
A= 1v12v2
1v1+2v2 2.1.2.3-8
B
T12 =
A=
21v1
1v1+2v2 2.1.2.3-9
2.1.2.3-82.1.2.3-9
vv[3]
2.1.3
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18
18
s(t) = w(t) r(t) = ()( )+
2.1.3-1
s(t)w(t)r(t)
2.1.3-1 Ricker
2.1.3-1reflection coefficient
Ricker seismogram
2.1.4
2.0.2.4 u1()u2(t ) =+
1() 2()
(s(t)w(t)r(t)s(t) = w(t)
r(t) = ()( )+
)(() = s(t)
+
=2 () = s(t)2
+
)(u(t ) ())
() = s(t)2+
= w(t) r(t)2+
= ()( )+
2
+
-
19
19
= ( )2+
()
+
+
= ()2() +
+
= () 2() +
+
= () () 2.1.4-1
() = () () 2.1.4-2
()s(t)()w(t)()
r(t)
s(t)w(t)r(t)
()()()[9]
[10]()()()
() = [()] + [()] 2.1.4-3
() = [()] + [()] 2.1.4-4
() = [()] + [()] 2.1.4-5
()()()
2.2
f(x)o(x)e(x)
f(x) = o(x) + e(x) 2.2-1
f(x) = o(x) + e(x) = o(x) + e(x) (2.2-2)
{f(x) + f(x) = 2 e(x)
f(x) f(x) = 2 o(x) (2.2-3)
(2.2-3)
f(x) = o(x) + e(x) (2.2-4)
-
20
20
o(x) = f(x) f(x)
2 (2.2-5)
e(x) = f(x)+ f(x)
2 (2.2-6)
r(t)r(t)
r(t)[11]
r(t) = r(t) + r(t)r(t) = r(t)r(t)
2r(t) =
r(t)+r(t)
2 2.2-7
2.2-1
2.2-1 Puryear and Castagna2008
1, 2Even part
Odd part2.2-1
{ + = 1 = 2
2.2-8
2.2-8
{2 = 1+22 = 12
2.2-9
2.3
Puryear and Castagna2008
[11]
2.2-1
-
21
21
+
2.2-1
(2.2-1)(2.2-2)
[()] = [()] [()] [()] [()] (2.2-1)
[()] = [()] [()] + [()] [()] (2.2-2)
=1
-
22
22
3.1
3.1.1
3.1.1-1t1t2
r1r2T
3.1.1-1 Marfurt and Kirlin2001
[12](t):
{ (t)dt+
= 1
(t) = 0, (t 0) (3.1.1-1)
(t)[13]
1() = 0
() (t 0)= (0) (t 0) 3.1.1-2
2() = 0
-
23
23
(t)()dt+
= (0) 3.1.1-3
(t) = 0, (t 0)
(t)()dt+
= (0) (t)dt
+0
0= (0) 3.1.1-4
(t 0)()dt+
= (0) (t)dt
+00
= (0) 3.1.1-5
r(t)[14](t = 0):
g(t)= r1(t 1) + r2(t 2) = r1(t 1) + r2(t 1 )(3.1.1-6)
(3.1.1-6)
R(f) = r1(t 1)+
2 + r2(t 1 )
+
2 (3.1.1-7)
3.1.1-5
R(f) = r1(t 1)+
2 + r2(t 1 )
+
2
= r1 (t 1)+
2 + r2 (t 1 )
+
2
= r121 + r2
2(1+) (3.1.1-8)
(t = 0)
t = 0
g(t) = r1 (t +
2) + r2 (t
2) (3.1.1-9)
(3.1.1-6)(3.1.1-8)
(3.1.1-9)
R(f) = r1 (t +
2)
+
2 + r2 (t
2)
+
2(3.1.1-10)
3.1.1-5
R(f) = r1 (t +
2)
+
2 + r2 (t
2)
+
2
= r1 (t +
2)
+
2 + r2 (t
2)
+
2
= r12(
2) + r2
2(
2) = r1
2
2 + r22
2 (3.1.1-11)
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24
24
(e = cos + sin )(3.1.1-11)
R(f) = r12
2 + r2
22
= r1 (cos(2
2) + sin (2
2)) + r2 (cos(2
2) sin (2
2))
= cos()(r1 + r2) + sin() (r1 r2) (3.1.1-12)
2.2-9{2 = 1+22 = 12
R(f) = 2cos() + 2 sin() (3.1.1-13)
1212
{Re[R(f)] = 2cos()
Im[R(f)] = 2 sin() (3.1.1-14)
t = (t1 +
2)
t = 0
g ((t1 +
2))(3.1.1-
2(a))(3.1.1-2(b))
(3.1.1-2(a))
-
25
25
t(s)t = (t1 +
2)t = 0
g ((t1 +
2))
(3.1.1-2(b))
(3.1.1-2(b)) t(s)t = (t1 +
2)t = 0
2.0.2.3
u(t) ()
u(t ) () (2.0.2.3-1)
g(t t)
g(t t) R(f)2t (3.1.1-15)
(3.1.1-13)
R(f)2t=(2cos() + 2 sin()) (cos(2t) jsin( 2t))
= 2(cos() cos(2t) jcos() sin(2t)) +
2( sin() cos(2t) + sin() sin(2t))
= [2(cos() cos(2t) + 2sin() sin(2t)] +
-
26
26
j[2 sin() cos(2t) 2 cos() sin(2t)] (3.1.1-16)
g(t t)
R(f)2t
Re[R(f)2t]
=2(cos() cos(2t) + 2sin() sin(2t) (3.1.1-17)
Im[R(f)2t]
=2 sin() cos(2t) 2 cos(t) sin(2t) (3.1.1-18)
3.1.2
NN(3.1.2-1)
N + 1NN
(3.1.2-1)
(N + 1)Nr1r2r3r
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27
27
Nt1t2t3t
T1 1N
T2 2 (N 1)
,T
2(
2+ 1)
tt = 0
r1rtr2r(1)t,
g(t)
[14]:
g(t) = r1 (t +T12) + r2 (t +
T22) ++ r
(2)(t +
T(2)
2) + r
(+12 )(t
T(2)
2)
++ r(1) (t 22) + r() (t
12)
(3.1.2-1)
(3.1.2-1)
g(t) = [r (t +T
2) + r(+1) (t
2)]
2=1 (3.1.2-2)
(3.1.2-2)
R(f) = r (t +T
2) 2
+
+ r(+1) (t
T
2) 2
+
2=1
2=1
= r2(
T2) + r(+1)
2(T2)
2=1
2=1
= r2(
T2) + r(+1)
2(T2)
2=1
2=1
= [r2(
T2) + r(+1)
2(T2)]
2=1
(3.1.2-2)
( e = cos +
sin ) (3.1.2-2)
R(f) = {[r[cos (2 (T
2)) + sin(2 (
T
2))] +
2=1
r(+1)[cos (2 (T
2)) sin(2 (
T
2))]}
= [(r + r(+1)) cos(T) + (r r(+1)) sin(T)]
2=1 (3.1.2-3)
-
28
28
{2r(,+1) = r + r(+1)2r(,+1) = r r(+1)
(3.1.2-4)
R(f) = [2r(,+1) cos(T) + 2r(,+1) sin(T)]
2=1 (3.1.2-5)
N
Re[R(f)] = 2r(,+1) cos(T)
2=1 (3.1.2-6)
Im[R(f)] = 2r(,+1) sin(T)
2=1 (3.1.2-7)
t(s)
t = (t +
2)(t = 0)
:
R(f)2t =[2r(,+1) cos(T) + 2r(,+1) sin(T)]
2
=1
[cos(2t) jsin(2t)]
(3.1.2-8)
(3.1.2-8)
R(f)2t =[2r(,+1) cos(T) cos(2t)
2
=1
j2r(,+1) cos(T) sin(2t)
+ 2r(,+1) sin(T) cos(2t)
+ 2r(,+1) sin(T) sin(2t)]
(3.1.2-9)
(3.1.2-9)
-
29
29
R(f)2t ={[2r(,+1) cos(T) cos(2t)
2
=1
+ 2r(,+1) sin(T) sin(2t)]
+ j[2r(,+1) sin(T) cos(2t) 2r(,+1) cos(T)]}
(3.1.2-10)
N
Re[R(f)] =
[2r(,+1) cos(T) cos(2t)+2r(,+1) sin(T) sin(2t)]
2
=1
(3.1.2-11)
Im[R(f)] =
[2r(,+1) sin(T) cos(2t) 2r(,+1) cos(T) sin(2t)]
2
=1
(3.1.2-12)
3.2
2.1.4 (2.1.4-2)() = () ()
(2.1.4-2)
() = ()
() (3.2-1)
Charles I. Puryear John P. Castagna2008
[11]
-
30
30
()rrT
O(r , r , T) = a {Re [()
()] Re[R()]}
2
=1
+ a {Im [()
()] Im[R()]}
2
=1
(3.2-2)
aa
(3.2-2)
s(t),w(t),
r(t),
()()R()
() = ()
()
(3.2-2)()()R()
()()
()
-----
(3.2-2)
a + jb
+ = +
2 + 2+
2 + 2
(3.2-3)
(3.2-3)Re [()
()]Im [
()
()]
Re [()
()] =
[()] [()] + [()] [()]
[()]2 + [()]2
(3.2-4)
-
31
31
Im [()
()] =
[()] [()] [()] [()]
[()]2 + [()]2
(3.2-5)
(3.2-2)
O(r , r , T) = a {[()] [()] + [()] [()]
[()]2 + [()]2
2
=1
Re[R()]}
+ a {[()] [()] [()] [()]
[()]2 + [()]2
2
=1
Im[R()]}
(3.2-6)
(3.2-2)()
[9]
(3.2-1)
() = () () (3.2-7)
(a + jb) (c + jd) = (ac bd) + j(ad + bc) (3.2-8)
(3.2-8)Re[() ()]Im[() ()]
Re[() ()] = [()][()] [()][()] (3.2-9)
Im[() ()] = [()][()] + [()][()] (3.2-10)
O(r , r , T) = a |Re[()] {[()][()] [()][()]}|
2
=1
+a |Im[()] {[()][()] + [()][()]}|
2
=1
(3.2-11)
-
32
32
3.3
(3.2-11) 3.1.2
N
(3.1.2-11)(3.1.2-12)
3.1.2
Re[R()] =
[2r(,+1) cos(T) cos(2t)+2r(,+1) sin(T) sin(2t)]
2
=1
(3.1.2-11)
Im[R()] =
[2r(,+1) sin(T) cos(2t) 2r(,+1) cos(T) sin(2t)]
2
=1
(3.1.2-12)
(3.1.2-11)(3.1.2-12)f
Re[R()]
= [2r(,+1) cos(T) cos(2t)+2r(,+1) sin(T) sin(2t)]
2
=1
2
=1
(3.3-1)
Im[R()]
= [2r(,+1) sin(T) cos(2t)
2
=1
2
=1
2r(,+1) cos(T) sin(2t)]
(3.3-2)
m, = T, n = 2t,(3.3-1)
Re[R()] = [2r(,+1) cos(m,) cos(n)+2r(,+1) sin(m,) sin(n)]
2
=1
2
=1
-
33
33
(3.3-3)
(3.3-2)
Im[R()]
= [2r(,+1) sin(m,) cos(n) 2r(,+1) cos(m,) sin(n)]
2
=1
2
=1
(3.3-4)
(3.3-3)
Re[R()]= {2r(1,) cos(m,1) cos(n)+2r(1,) sin(m,1) sin(n)}2=1
+
{2r(2,1) cos(m,2) cos(n)+2r(2,1) sin(m,2) sin(n)}2=1
+
{2r(
2,
2+1)cos (m
,
2
) cos(n)+2r(2,
2+1)sin (m
,
2
) sin(n)}2=1
(3.3-5)
(3.3-5)
Re[R()]=
2r(1,) sin(m,1) sin(n) +
2
=1
+ 2r(2,2+1)
sin (m,2) sin(n)
+ 2r(1,) cos(m,1) cos(n) + + 2r(2,2+1)
cos(m,2) cos(n)
=1
=1
(3.3-6)
(3.3-6)
Re[R()] = [2 sin(m,1) sin(n)2 sin (m,2) sin(n)]12
[
r(1,)
r(2,2+1)
]
21
2
=1
+ [2cos(m,1) cos(n)2cos(m,2) cos(n)]12
[
r(1,)
r(2,2+1)
]
21
2
=1
(3.3-7)
(3.3-7),Re[R()]
-
34
34
Re[R()]
=
[ 2 sin(m1,1) sin(n1) 2 sin (m1,2
) sin(n1)
2 sin(m2,1) sin (n
2) 2 sin (m
2,2) sin (n
2)]
22
[
r(1,)
r(2,2+1)
]
21
+
[ 2cos(m1,1) cos(n1) 2cos(m1,2
) cos(n1)
2cos(m2,1) cos (n
2) 2cos(m
2,2) cos (n
2)]
22
[
r(1,)
r(2,2+1)
]
21
(3.3-8)
Im[R()]
=
[ 2 sin(m1,1) cos(n1) 2 sin (m1,2
) cos(n1)
2 sin(m2,1) cos (n
2) 2 sin (m
2,2) cos (n
2)]
22
[
r(1,)
r(2,2+1)
]
21
[
2cos(m1,1) sin(n1) 2cos(m1,2) sin(n1)
2cos(m
2,1) sin(n
2) 2cos(m
2,2) sin(n
2)]
22
[
r(1,)
r(2,2+1)
]
21
(3.3-9)
(3.2-11)
O(r , r , T) = a |Re[()] {[()][()] [()][()]}|
2
=1
+a |Im[()] {[()][()] + [()][()]}|
2
=1
(3.2-11)
[
]
[]1
= [[()]
[()]]1
(3.2-12)
= {2[()] sin(m,) sin(n) 2[()] sin(m,) cos(n)},(1,2)
-
35
35
= {2[()] 2cos(m,) cos(n) + 2[()] cos(m,) sin(n)},(1,2)
= {2[()] sin(m,) cos(n) + 2[()] sin(m,) sin(n)},(1,2)
= {2[()] 2cos(m,) cos(n) 2[()] cos(m,) sin(n)},(1,2)
= [
r(1,)
r(2,2+1)
]
21
= [
r(1,)
r(2,2+1)
]
21
[()] = [
Re[(1)]
Re [ (2)]]
21
[()] = [
Im[(1)]
Im [ (2)]]
21
(3.2-12)
= (3.2-13)
-
36
36
4.1
Moho
(4.1-1)
(4.1-1) 2003
Herglotz-Wiechert
-
37
37
Herglotz-Wiechert
Backus and Gilbert (19671968)
md m~,
,
=+(4.1-2)
(4.1-2) 2003
[16]
-
38
38
4.2
Hadamard(1923),
:l)2)3)
, [17]
4.1
(2009)[19]
-
39
39
4.3
(LSQR)
LSQR
LSQR[9]
4.3.1 LSQR
Paige,Saunders(1982) Golub Kahan LSQR
=
(Conjugate Gradients)
[20]
Paige,Saunders(1982)
LSQR(CG)LSQR
LSQR
4.3.2 LSQR
(3.2-13)
1 = 1 (4.3.2-1)
1
1
LSQR
(1)LSQR
= , 0
0 = 02 = 2
=1
0 = 0
0 =00
-
40
40
0 = 0
0 = 02 = 2
=1
0 = 0
0 =00
0 = 0
0 =00
(2) LSQR
+1 =
+1 = +12
+1 =+1+1
+1 = +1 +1
+1 = +12
+1 =+1+1
(3) LSQR
= 2 + +12
+ = +2
+1 = +1 +1+1+12
+1 = +1
+1 =+1
LSQR MATLAB
-
41
41
(5.0-1)
(5.0-1)
5.1
(5.1-1):
-
42
42
(5.1-1)
30Hz(5.1-2)
(5.1-2)
(5.1-3)
(5.1-3)
(5.1-3)
widess
8
[21]
(5.1-4)
(5.1-4)
5.1-5)
-
43
43
5.1-5)
0.0035 103 LSQR
1.2129e-16 0
MATLAB LSQR
5.1.1 LSQR
103 1.2129e-16
1251020 50
60 70 80 103 5.1.1-
15.1.1-10
5.1.1-1 1
-
44
44
5.1.1-2 2
5.1.1-3 5
5.1.1-4 10
-
45
45
5.1.1-5 20
5.1.1-6 50
5.1.1-7 60
-
46
46
5.1.1-8 70
5.1.1-9 80
5.1.1-10 103
1
70 0.0035
70 10-3 10-8
7080103 50
-
47
47
(5.1.1-11) 0 20
1 103
1 0.3728 0.4778
2 0.1835 0.3214
5 0.0697 0.2443
10 0.0299 0.1675
20 0.0126 0.1058
50 0.0023 0.0310
60 0.0010 0.0157
70 2.2893e-08 0.0035
80 1.7590e-11 0.0035
103 1.2129e-16 0.0035
(5.1.1-1)
(5.1.1-11)
0
0.1
0.2
0.3
0.4
0.5
0.6
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0
-
48
48
5.2
()
5.2.1
(5.2.1-1)
(5.2.1-1)
0.8 0.6 20
120 2(ms) 2(ms)
30Hz
(5.2.1-2)
(5.2.1-2)
Kallweit and Wood(1982)[22]
2=
1
2.6 (5.2.1-1)
-
49
49
()
,
2
( 30Hz)
12.82(ms)(5.2.1-2) 7 8
((5.2.1-2)(
))
(5.2.1-3)
(5.2.1-3)
5.2.2
(5.2.2-1)
-
50
50
(5.2.2-1)
0.8 0.6-0.6
20 120 2(ms)
2(ms)
(5.2.2-2)
30Hz
(5.2.2-2)
15
(5.2.2-3)
(
)
5.3
-
51
51
5.3.1
15Hz20Hz25Hz30Hz35Hz40Hz45Hz50Hz
55Hz ((5.3.1-1))
1(ms)300 45
(5.3.1-1) 30Hz
(5.3.1-2)15Hz 55Hz
30Hz
15Hz
-
52
52
(5.3.1-2) 15Hz 55Hz( 5Hz)
15Hz(5.3.1-3)
(5.3.1-3) 20Hz 55Hz( 5Hz)
(5.3.1-3) 20Hz 20Hz 55Hz
30Hz
30Hz 25Hz
30Hz,
-
53
53
5.3.2
(5.1-2)
1%5%10%
(5.1-2)
(5.3.2-1)(5.3.2-3) 1%5%10%
(5.3.2-1)(a) (5.1-2) 1%
(5.3.2-1)(b) (5.1-2) 1%
(5.3.2-2)(a) (5.1-2) 5%
-
54
54
(5.3.2-2)(b) (5.1-2) 5%
(5.3.2-3)(a) (5.1-2) 10%
(5.3.2-3)(b) (5.1-2) 10%
1%
5% 10%
-
55
55
5.4
(
)
[5]
1)
2)
-
56
56
----
-
57
57
[1].[D].(),2009.
[2].[M]..:,2006:1-9,61-84.
[3]Seth Stein, Michael Wysession. An Introduction to Seismology, Earthquakes, and Earth Stru
cture[M].Blackwell Publishing Ltd,2003:29-34.
[4].[M].,1996,189.
[5].[J].,2008,43(1):123-128.
[6].[J].:,2000,24(1):77-84.
[7].[D].:,2012.
[8].[D].(),2011.
[9]. LSQR [D].(),2012.
[10].[M].:,2005:1-19,149-179.
[11]Charles I. Puryear1 and John P. Castagna1. Layer-thickness determination and stratigraph
ic interpretation using spectral inversion: Theory and application[J].Geophysics,2008, VOL.7
3,NO.2,P.R37R48.
[12] []. MATLAB [M].:,2002:101-228.
[13] [].[M].,2007:1-12,112-231,296-337.
[14]K. J. Marfurt, R. L. Kirlin.Narrow-band spectral analysis and thin-bed tuning[J].Geophys
ics,2001,VOL. 66, NO. 4, P.12741283.
[15] [].[M].:,2007:1-34.
[16].[M].:,2003:1-12.
[17] Mario Bertero, Tomaso A. Pogcio, Vincent Torre. Ill-Posed Problems in Early Vision[J].
PROCEEDINGS OF THE IEEE,1988,VOL.76,NO.8.
~~[18]Jacques Hadamard. Lectures on Cauchy's Problem: In Linear Partial Differential Equations
[DB/OL]. books.google.com.2003.
[19]Yuan S. Y.et al.Ill-posed analysis for spectral inversion[J].Expanded Abstracts of 79th A
nnual International SEG Meeting,2009,2447-2451.
[20]Christopher C. Paige, Michael A. Saunders. LSQR: An Algorithm for Sparse Linear Equation
s and Sparse Least Squares.[J].ACM Transactions on Mathematical Software,1982, Vol8,No.1,43-
71.
[21]M. B. Widess. How thin is a Thin Bed?[J].Geophysics,1973,vol.38,No.6.1176-1180.
[22]Kallweit and Wood. The limits of resolution of zero-phase wavelets. Geophysics,1982,vol.
47,No.7.1035-1046.
-
58
58
word
-
59
59
A MATLAB
1
f1=sym('sin(pi/2*t)')
subplot(5,2,3)
ezplot(f1,[-4,4])
hold
f2=sym('sin(pi/2*t*2)')
subplot(5,2,4)
ezplot(f2,[-4,4])
hold
f3=sym('sin(pi/2*t*3)')
subplot(5,2,5)
ezplot(f3,[-4,4])
hold
f4=sym('sin(pi/2*t*4)')
subplot(5,2,6)
ezplot(f4,[-4,4])
hold
f5=sym('sin(pi/2*t*5)')
subplot(5,2,7)
ezplot(f5,[-4,4])
hold
f6=sym('sin(pi/2*t*6)')
subplot(5,2,8)
ezplot(f6,[-4,4])
hold
f7=sym('sin(pi/2*t*7)')
subplot(5,2,9)
ezplot(f7,[-4,4])
hold
f8=sym('sin(pi/2*t*8)')
subplot(5,2,10)
ezplot(f8,[-4,4])
hold
f12345678=f1+f2+f3+f4+f5+f6+f7+f8
subplot(5,1,1)
ezplot(f12345678,[-4,4])
axis([-4 4 -8 8])
-
60
60
2
fm=30;%
dt=0.001;%
number=100;%
t=-number/2+1:number/2;
a=(1-2*(pi*fm*t*0.001).^2).*exp(-(pi*fm*t*0.001).^2);
subplot(3,1,1);
plot(t,a);
title('Ricker-');
xlabel(' tms');
ylabel(' A');
for i=1:100
f(i)=10*(i-1);%
end
Y=abs(fft(a));%fourier
subplot(3,1,2)
plot(f,Y);
title('Ricker ');
axis([0 100 0 16])
xlabel(' fhz');
ylabel('');
% 0.001s 100 0.1s
10hz
3
clear
dt=0.004;
nt=120;%nt 1
flag=1;
for j=1:nt
r(j)=rand()*(-1)^j*0.1;
r(j)=0;
end
example=2;
if(example==1)
r(20)=1; r(19)=-0.5;r(21)=-0.5;
r(30)=-0.5;r(31)=0.5;
-
61
61
r(50)=0.8;r(51)=-0.6;r(52)=-0.5;
r(80)=0.7;r(79)=-0.6;r(81)=-0.6;
r(100)=0.68;r(101)=-0.5;r(99)=-0.5;
end
if(example==2)
r(20)=0.71;r(22)=0.71;
r(40)=-0.5;r(42)=-0.5;
r(60)=0.8;r(63)=0.8;
r(80)=-0.7;r(82)=0.7;
r(100)=-0.65;r(101)=-0.65;
end
[w,tw] = ricker_chg(nt,dt,30);%
seis=conv(r,w);%
%noise=randn(1,239)*0.04;
%seis=seis+noise;
mm=round((max(size(seis))-nt)/2);seis=seis([mm:mm+nt-1]);
nf=nt;df=1/(nt)/dt;
t1=[1:nt]*dt;
if(flag==1)
figure;
subplot(411);plot(t1,r);axis tight;title('');xlabel(' t(s)');ylabel('');
subplot(412);plot(tw,w);title('ricker ');axis tight;xlabel(' t(s)');ylabel('');
subplot(413); plot(t1,seis);title('');axis tight;xlabel(' t(s)');ylabel('');
subplot(414);plot(t1,r,t1,seis);title(' ');axis tight;xlabel('
t(s)');ylabel('');
end
cr=fft(r);creal=real(cr);cimage=imag(cr);cabs=abs(cr);cangle=angle(cr);
nn=ceil(max(size(cr))/2);n=0:nn-1;f=n*df;
if(flag==1)
figure;
subplot(411);plot(f,creal(1:nn));title('');axis tight;
subplot(412);plot(f,cimage(1:nn));title('');axis tight;
subplot(413);plot(f,cabs(1:nn));title('');axis tight;
subplot(414);plot(f,cangle(1:nn));title('');axis tight;
end
cw=fft(w);cwreal=real(cw);cwimage=imag(cw);cwabs=abs(cw);cwang=angle(cw);
nb=ceil(max(size(cw))/2);n=0:nb-1; df1=1/(max(size(w)))/dt;fa=n*df1;
if(flag==1)
-
62
62
figure;
subplot(411);plot(fa,cwreal(1:nb));title('');
subplot(412);plot(fa,cwimage(1:nb));title('');
subplot(413);plot(fa,cwabs(1:nb));title('');
subplot(414);plot(fa,cwang(1:nb));title('');
end
cseis=fft(seis);csreal=real(cseis);csimage=imag(cseis);csabs=abs(cseis);csang=angle(cseis);
nse=ceil(max(size(cseis))/2);n=0:nse-1;fse=n*df;
if(flag==1)
figure;
subplot(411);plot(fse,csreal(1:nse));title('');axis tight;
subplot(412);plot(fse,csimage(1:nse));title('');axis tight;
subplot(413);plot(fse,csabs(1:nse));title('');axis tight;
subplot(414);plot(fse,csang(1:nse));title('');axis tight;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% size(cw),size(cseis)
%ir=ifft(cw);
%figure;plot(tw,ir);title('');
%cxr=cseis./cw;xr=ifft(cxr);figure;plot(t1,abs(xr));title('');
nb=ceil(max(size(cw))/2);n=0:nb-1; df1=1/(max(size(cw))-1)/dt;fa=n*df1;
nt2=nt/2;
b=[csreal(1:nt2)';csimage(1:nt2)'];
dw=(nt-1)*dt/2;
for i1=1:nt2
fi=df*(i1-1);
for j=1:nt2
tj=(nt-1-2*(j-1))*dt;
mij=pi*fi*tj;
ni=2*pi*fi*dw;
sinm=sin(mij); cosm=cos(mij); cosn=cos(ni); sinn=sin(ni);
sin_sin=sinm*sinn; sin_cos=sinm*cosn;
cos_cos=cosm*cosn; cos_sin=cosm*sinn;
a11(i1,j)=cwreal(i1)*sin_sin-cwimage(i1)*sin_cos;
a12(i1,j)=cwreal(i1)*cos_cos+cwimage(i1)*cos_sin;
a21(i1,j)=cwreal(i1)*sin_cos+cwimage(i1)*sin_sin;
a22(i1,j)=cwimage(i1)*cos_cos-cwreal(i1)*cos_sin;
end
end
a=[a11,a12;a21,a22]*2;
%figure;imshow(a);colorbar;pause;
-
63
63
%cond(a,inf),cond(a,1),cond(a,2),cond(a,'fro'),pause;
tol=1e-19;
maxit=105;
m1=[];m2=[];
[x,sign0,residual,iter]=lsqr(a,b,tol,maxit,m1,m2);%lsqr
%[x,sign0]=lsqr(a,b,tol,maxit);
residual,iter,
if(sign0==0)
disp(' LSQR converged to the desired tolerance TOL within MAXIT iterations.');
end
if(sign0==1)
disp('LSQR iterated MAXIT times but did not converge.');
end
if(sign0==2)
disp(' preconditioner M was ill-conditioned.');
end
if(sign0==3)
disp('LSQR stagnated (two consecutive iterates were the same).');
end
if(sign0==4)
disp('one of the scalar quantities calculated during LSQR became too small or too large to
continue computing.');
end
ro=x(1:nt/2);re=x(nt/2+1:nt);
r1=(ro+re);r2=flipud((re-ro));
%figure;plot(t1(1:nt/2),r1);
%figure;plot(t1(nt/2+1:nt),r2);
rt=[r2',r1']';
r=r(1,1:119);
r=[0 r];
diff=r-rt';
figure;
subplot(311);plot(t1,r);axis tight;title('');xlabel(' t(s)');ylabel('');
subplot(312);plot(t1,rt);title('');xlabel(' t(s)');ylabel('');axis tight; %set(gca,
'YTick',[0,0.24,0.48] );
subplot(313); plot(t1,diff);title('');axis([0 0.48 -0.7 0.7]);xlabel(' t(s)');ylabel('');
-
64
64
420
clear
dt=0.002;
nt=120;%nt 1
flag=0;
nx=20;
r=zeros(nt,nx);
example=2;
if(example==1)
for ix=1:nx
r(20,ix)=0.8;
r(20+ix,ix)=0.6;
end
end
if(example==2)
for ix=1:nx
r(30,ix)=0.8;
r(30+ix,ix)=0.8
r(30+ix*2,ix)=-0.6;
end
end
ft=0;
[w,tw] = ricker_chg(nt,dt,20);%
for ix=1:nx
seis(:,ix)=conv(r(:,ix),w);%
mm=round((max(size(seis(:,ix)))-nt)/2);
seis(1:nt,ix)=seis([mm:mm+nt-1],ix);
end
seis=seis(1:nt,:);
nf=nt;df=1/(nt)/dt;
t1=[1:nt]*dt;
if(flag==1)
figure1;
subplot(311);wigb(r,1,[1:1:nx],[ft:1:ft+nt-1]*dt);title('');
subplot(312);plot(tw,w);title('ricker ');axis tight;
subplot(313);wigb(seis,1,[1:1:nx],[ft:1:ft+nt-1]*dt);title('');
end
cr=fft(r);creal=real(cr);cimage=imag(cr);cabs=abs(cr);cangle=angle(cr);
-
65
65
nn=ceil(max(size(cr))/2);n=0:nn-1;f=n*df;
cw=fft(w);cwreal=real(cw);cwimage=imag(cw);cwabs=abs(cw);cwang=angle(cw);
nb=ceil(max(size(cw))/2);n=0:nb-1; df1=1/(max(size(w)))/dt;fa=n*df1;
if(flag==1)
figure;
subplot(411);plot(fa,cwreal(1:nb));title('');axis([0 125 -4 4]);
subplot(412);plot(fa,cwimage(1:nb));title(''); axis([0 125 -4 4]);
subplot(413);plot(fa,cwabs(1:nb));title('');axis([0 125 -4 4]);
subplot(414);plot(fa,cwang(1:nb));title('');axis([0 125 -4 4]);
end
cseis=fft(seis);csreal=real(cseis);csimage=imag(cseis);csabs=abs(cseis);csang=angle(cseis);
nse=ceil(max(size(cseis))/2);n=0:nse-1;fse=n*df;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
nb=ceil(max(size(cw))/2);n=0:nb-1; df1=1/(max(size(cw))-1)/dt;fa=n*df1;
nt2=nt/2;dw=(nt-1)*dt/2;
for i1=1:nt2
fi=df*(i1-1);
for j=1:nt2
tj=(nt-1-2*(j-1))*dt;
mij=pi*fi*tj;
ni=2*pi*fi*dw;
sinm=sin(mij); cosm=cos(mij); cosn=cos(ni); sinn=sin(ni);
sin_sin=sinm*sinn; sin_cos=sinm*cosn;
cos_cos=cosm*cosn; cos_sin=cosm*sinn;
a11(i1,j)=cwreal(i1)*sin_sin-cwimage(i1)*sin_cos;
a12(i1,j)=cwreal(i1)*cos_cos+cwimage(i1)*cos_sin;
a21(i1,j)=cwreal(i1)*sin_cos+cwimage(i1)*sin_sin;
a22(i1,j)=cwimage(i1)*cos_cos-cwreal(i1)*cos_sin;
end
end
a=[a11,a12;a21,a22]*2;
%figure;imshow(a);colorbar;pause;
%cond(a,inf),cond(a,1),cond(a,2),cond(a,'fro'),pause;
tol=1e-19;
maxit=5000;
m1=[];m2=[];
-
66
66
for ix=1:nx
b=[csreal(1:nt2,ix);csimage(1:nt2,ix)];
[x,sign0,residual,iter]=lsqr(a,b,tol,maxit,m1,m2);%lsqr
% residual,iter,
if(flag==1)
if(sign0==0)
disp(' LSQR converged to the desired tolerance TOL within MAXIT iterations.');
end
if(sign0==1)
disp('LSQR iterated MAXIT times but did not converge.');
end
if(sign0==2)
disp(' preconditioner M was ill-conditioned.');
end
if(sign0==3)
disp('LSQR stagnated (two consecutive iterates were the same).');
end
if(sign0==4)
disp('one of the scalar quantities calculated during LSQR became too small or too
large to continue computing.');
end
end
ro=x(1:nt/2);re=x(nt/2+1:nt);
r1=(ro+re);r2=flipud((re-ro));
%figure;plot(t1(1:nt/2),r1);
%figure;plot(t1(nt/2+1:nt),r2);
rt(:,ix)=[r2',r1']';
end
figure;
subplot(311);wigb(r,1,[1:1:nx],[ft:1:ft+nt-1]*dt);title('');axis tight;xlabel('
');ylabel(' t(s)');
subplot(312);wigb(rt,1,[1:1:nx],[ft:1:ft+nt-1]*dt);title('');axis tight; xlabel('
');ylabel(' t(s)');
subplot(313);wigb(seis,1,[1:1:nx],[ft:1:ft+nt-1]*dt);title('');axis tight;xlabel('
');ylabel(' t(s)');
5
noise=randn(1,239)*0.04;%0.04
seis=seis+noise;
4 29