A fast finite-element software for gravity anomaly calculation in complex geologic regions Yongen...
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Transcript of A fast finite-element software for gravity anomaly calculation in complex geologic regions Yongen...
A fast finite-element software for gravity anomaly calculation
in complex geologic regions
Yongen Cai Department of Geophysics Peking University,
Beijing, 100871
Chi-yuen WangDepartment of Earth and Planetary ScienceUniversity of California, Berkeley, CA 94720
Introduction
• For geologically complex regions, forward computation of the gravity anomaly of a density model may be computationally demanding and the bottle-neck in gravity inversion.
• We present a fast finite-element software for solving this problem.
V
dddzyx
zGzyxg
2/3222 )()()(),,(),,(
P(x,y,z)
dv
GBOX( R.J.Blakely,1995)2 2 2
i j1ijk k i ijk j
i 1 j=1 k=1 k ijk
j ijk i
2 2 2 i j kijk i i i ijk i
x yg [z tg x log(R y )
z R
y log(R x )
R x y z , ( 1) ( 1) ( 1) x )
x
y
z
P(0,0,0)
R
Boundary value problem
),,,(42 zyxG
,),,(),,(11
zyxzyx SS
),,,(),,(22
zyxgzyxg SS
g (x, y, z) = -
Boundary condition
22
31
rM
ICBA
r
GM
2 2
4
3 31
2
3
GM A B C Ig
r M r
A B C IG
r r r
( Jeffreys, 1962)
FEM formulation
s
gdSdVGdVzyx
F
42
1)(
222
0)( F
v
eee
m
e SS
gdSdVGdVzzyyxx1
04
p
i ii 1
i i i i
( , , ) h
h (1 )(1 )(1 ) /8 for p=8
FKΦ
Accuracy verificationDensity model for verifying
(c= 0.001 kg/m4 )30
20
(z)( z)c
3p_0
21 0
1
p ( )k kc
GBOX( average density)
FFEM( distributed density)p
i ii 1
( , , ) h
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.0 1.0 2.0 3.0 4.0 5.0 6.0Distance (km)
g(m
Gal
)Exact solution
GBOX
FFEM
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.0 1.0 2.0 3.0 4.0 5.0 6.0Distance (km)
g(m
Gal
)
Exact solution
GBOX
FFEM
0.00
1.00
2.00
3.00
4.00
5.00
0.0 1.0 2.0 3.0 4.0 5.0 6.0Distance (km)
g(m
Gal
)
Exact solution
GBOX
FFEM
Application to Taiwan
Source elements: 76,500
Source nodes: 83,448
Calculated gravity points
GBOX:4636 points only at ground surface
FEM: 285488 at all nodal points
Computer: PC with 2.3 GHz CPUs
Comparison between FFEM and GBOXmGal
FFEM: used cpu time : 280 s GBOX: used cpu time : 6780 s
Application to Sirrea Nevada (Cai, Zhang and Wang, 2006)
Calculated Bouguer anomaliesby FFEM
Calculated Bouguer anomaliesby classical method
Conclusion
• A software FFEM is provided which is more accurate and much faster than the classical integration method, if density in the material body is highly heterogeneous.
• The computational efficiency for the FFEM method is more pronounced in regions with greater heterogeneities.
Density model
The density distribution can be obtained from the velocity from seismic tomograph.
p
i ii 1
i i i i
( , , ) h
h (1 )(1 )(1 ) /8 for p=8