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