estimacion de velocidades

8/20/2019 estimacion de velocidades

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An Optimization Technique for EstimatingVelocities and Fracture Orientation in

Orthorombic Media

by

Debora Cores

José G. Meza

Universidad Simón Boĺıvar

ISPM2000 - ATLANTA

August 2000

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An Optimization Technique for EstimatingVelocities and Fracture Orientation in

Orthorombic Media

by

Debora Cores

José G. Meza


III Jornadas IBO

Intevep, Noviembre 2000

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The Influence of the Seismic ParameterAcquisition on the Optimization Problem for

Estimating Velocities and Fracture Orientation

by

Debora Cores

José G. Meza


Optimization 2001

Aveiro-Portugal, July 2001

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OUTLINE

An orthorombic media (OM)

The Reflection Tomography Problem in OM

Historical Overview

Discretized Problem

Numerical Approach

Numerical Results

Conclusions

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Anisotropy

The velocity does not changes

with the wave propagation direc-

tion.

V

1

6 =

V

2

The velocity changes with the wave propa-

gation direction.

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Anisotropy: An Orthorombic Media (OM)

An orthorombic media is ananisotropic stratified medium with

vertical fractures.

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PROBLEM: Reflection Tomography (SRT) Problem in OM

M i n i m i z e

1

2

k T

r

; T (

V )

k

2

2

T : I R

m

! I R

n

travel time function, T = (

T

1

(

V ) T

2

(

V ) : : : T

n

(

V ) )

where,

T

i

( V ) =

Z

R a y

i

1

V ( x y z )

d l

i

T

r

2 I R

n

real travel time

vector.

V 2 I R

m

is the velocity

vector in OM.

n is the number of layers.

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Historical Overview

ISOTROPIC MEDIA :

Normal Equations:

Gauss Seidel with Successive Over-

relaxation: T. Bishop et al, 1985

Levenberg and Marquardt Method

with SVD descomposition : Lines and

Treitel, 1985: S. Chiu el al, 1986; T.

Zhu and L. Brown, 1987; Farra and

Madariaga, 1998.

Low Storage Opt. Techniques:

Spectral Gradient Method: Castillo,

Cores and Raydan , 2000.

ANISOTROPIC MEDIA:

In 2D elliptical anisotropic medium

Michelena et al. 1994.

In 2D medium with a sinusoidal aproxi-

mation of the velocity: Toshiki et al 1995.

In a 3D Transversally anisotropic

medium : Grechka, 1995.

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DISCRETIZED PROBLEM: Ellipsoidal Aproximation

Contreras et al, 1997

1

V

2

j

=

1

V

2

z

j

c o s

2

(

1

) +

1

V

2

x

j

c o s

2

(

2

) s i n

2

(

1

) +

1

V

2

y

j

s i n

2

(

2

) s i n

2

(

1

)

where j = P S V S H

correspond to the different

wave propagation modes.

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DISCRETIZED PROBLEM: Travel tiem Function

The travel time function for a ray corresponding to the ( i j

) pair reflecting in

the layer k ,

T

i j k

( X Y ) =

P

2 k + 1

h = 2

r

( x

i j k

h

; x

i j k

h ; 1

)

2

v

2

x

h

+

( y

i j k

h

; y

i j k

h ; 1

)

2

v

2

y

h

+

( z

i j k

h

; z

i j k

h ; 1

)

2

v

2

z

h

X = (

x

1

x

2

: : : x

2 n + 1

)

Y = (

y

1

y

2

: : : y

2 n + 1

)

Z = (

z

1

z

2

: : : z

2 n + 1

)

V

x

= (

v

x

1

v

x

2

: : : v

x

2 n + 1

)

V

y

= (

v

y

1

v

y

2

: : : v

y

2 n + 1

)

V

x

= (

v

z

1

v

z

2

: : : v

z

2 n + 1

)

z

i

= f

i

( x

i

y

i

) :

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DISCRETIZED PROBLEM

Consider any symmetry axes, (Group angle 6

=

Ray angle)

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DISCRETIZED PROBLEM

Azimutal Rotation

0

@

c o s ( ) s i n ( ) 0

; s i n ( ) c o s ( ) 0

0 0 1

1

A

Polar Rotation

0

@

c o s ( ) 0 ; s i n ( )

0 1 0

s i n ( ) 0 c o s ( )

1

A

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GENERAL DISCRETIZED PROBLEM

T

i j k

( S ) =

k + 1

X

h = 2

v

u

u

t

i j k

x

h

v

x

h

!

2

+

i j k

y

h

v

y

h

!

2

+

i j k

z

h

v

z

h

!

2

+

2 n + 1

X

h = 2 n + 2 ; k

v

u

u

t

i j k

x

h

v

x

h

!

2

+

i j k

y

h

v

y

h

!

2

+

i j k

z

h

v

z

h

!

2

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GENERAL DISCRETIZED PROBLEM

i j k

x

h

= D

x

h

c o s (

h

) c o s (

h

) + D

y

h

c o s (

h

) s i n (

h

) ;

D

z

h

s i n (

h

)

i j k

y

h

= ;

D

x

h

s i n (

h

) + D

y

h

c o s (

h

)

i j k

z

h

= D

x

h

s i n (

h

) c o s (

h

) ;

D

y

h

s i n (

h

) s i n (

h

) + D

z

h

c o s (

h

)

D

x

h

= x

i j k

h

;

x

i j k

h ; 1

D

y

h

= y

i j k

h

;

y

i j k

h ; 1

D

z

h

= z

i j k

h

;

z

i j k

h ; 1

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NUMERICAL APPROACH

An optimization scheme for solving

M i n i m i z e k T

r

; T ( S ) k

2

s : t : L S

U

that satisfies the following conditions:

Only function and gradient evaluations (first order information) arerequired

r

f ( S ) =

J

T

f

( S ) (

T ( S ) ;

T

r

)

Global convergence

Fast local convergence

Low computational cost and storage

Box constraints

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NUMERICAL APROACH

Low cost and storage orthorombic ray tracing algorithm (Loreto and

Cores, 1998-1999)

M i n i m i z e T (

X Y Z )

The Spectral Proyected Gradient Method (SPG) (Birgin, Martinez and

Raydan, 1999) to solve,

M i n i m i z e k T

r

;

T ( S ) k

2

s : t : L S

U

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NUMERICAL APPROACH

Spectral Projected Gradient Method (SPG)

Step 1: If k P ( S k

; r f

(

S

k

) )

; S

k

k t o l , then Stop.

Step 2: Nonmonotone Line-search

Step 2.1: Set =

k

Step 2.2: Set S +

=

P (

S

k

; r f (

S

k

) )

Step 2.3: If f (

S

+

)

m a x

0 j k M ; 1

f (

S

k ; j

) +

(

S

+

; S

k

)

T

r f (

S

k

)

then

k

=

, S k + 1

=

S

+

, W k

=

S

k + 1

; S

k

, y k

=

r f (

S

k + 1

) ; r f

(

S

k

)

, go

to Step 3.

else, 2

1

2

]

go to Step 2.2

Step 3: b k

=

W

T

k

y

k

If b k

0 , k + 1

=

m a x

,

else a k

=

W

T

k

W

k

and k + 1

= m i n

f

m a x

m a x

f

m i n

a

k

b

k

g g

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NUMERICAL RESULTS

We consider the following two synthetic models.

−4

−2

0

2

4

−4 −3 −2 −1 0 1 2 3 4

0

1

2

3

4

5

6

MODEL 1

Model 1

−4

−2

0

2

4−4 −3

−2 −10 1

2 34

0

1

2

3

4

5

6

MODEL 2

Model 2

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NUMERICAL RESULTS

The distribution of the sources and recievers was made in:Squared Mesh: a a

b squared Radial Mesh: a circle of radious r .

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

4Squared Mesh

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4Radial Mesh

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NUMERICAL RESULTS

Let the vector ( L U

)

T be the lower and upper bounds of the velocities and

fracture orientation angles respectively.

Unconstrained Case: For i

= 1 : : : 2 n

+ 1

l (

i ) = 0

: 0 0 2

V

x

(

i )

5 0 0 =

u (

i )

l (

i ) = 0

: 0 0 2

V

y

(

i )

5 0 0 =

u (

i )

l ( i ) = 0 : 0 0 2 V

z

( i ) 5 0 0 = u ( i )

l (

i ) = 0

(

i )

9 0 =

u (

i )

l (

i ) = 0

(

i )

9 0 =

u (

i )

Constrained Case: For i = 1 : : : 2

n + 1

l (

i ) = 0

: 2

V

x

(

i )

5 =

u (

i )

l (

i ) = 0

: 2

V

y

(

i )

5 =

u (

i )

l (

i ) = 0

: 2

V

z

(

i )

5 =

u (

i )

l (

i ) = 1 0

(

i )

3 0 =

u (

i )

l ( i ) = 2 ( i ) 9 = u ( i )

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NUMERICAL RESULTS

Initial Iterates:

V

0

x

= V

0

y

= V

0

z

= ( 3 4 5 5 4

3 )

T

0

=

0

= ( 1 0

1 0

1 0

1 0

1 0

1 0 )

T

Note: The velocities are measuared in Km

Angles are measuared in degrees.

Stopping Criterium :

k

P ( S

k

; r

f ( S

k

) ) ; S

k

k

2

1 0

; 6

Sun Station Ultra 10

M=8 in the SPG Method

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Model 1 (P-Wave), Unconstrained Case

Squared Mesh, ns=2, nr=15 Radial Mesh, ns=5, nr=16

V

x

V

a

x

V

y

V

a

y

V

z

V

a

z

V

a

x

V

a

y

V

a

z

1.5 1.35 1.7 1.69 1.9 2.12 1.28 1.7 2.23

2 2.97 2.3 2.29 2.5 1.68 2.56 2.3 1.95

3 3 2.8 2.79 3.3 3.3 3.45 2.8 2.86

3 3 2.8 2.8 3.3 3.3 3.45 2.8 2.86

2 2.97 2.3 2.3 2.5 1.68 2.65 2.29 1.94

1.5 1.35 1.7 1.7 1.9 2.1 1.28 1.69 2.23


(Aprox.) (Aprox.) (Aprox.) (Aprox.)

5 34.43 20 20 39.7 19.99

7 48.88 15 14.91 68.52 14.993 4.47 25 23.62 58.25 24.98

3 4.38 25 23.28 58.25 24.98

7 48.84 15 14.91 68.52 14.99

5 35.24 20 20 39.71 19.99

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Model 1 (P-Wave), Constrained Case


V

x

V

a

x

V

y

V

a

y

V

z

V

a

z

V

a

x

V

a

y

V

a

z

1.5 1.49 1.7 1.7 1.9 1.91 1.49 1.69 1.91

2 1.99 2.3 2.29 2.5 2.5 1.99 2.3 2.5

3 3 2.8 2.8 3.3 3.3 2.99 2.8 3.3

3 3 2.8 2.8 3.3 3.3 2.99 2.79 3.3

2 1.99 2.3 2.29 2.5 2.5 1.99 2.29 2.51

1.5 1.49 1.7 1.7 1.9 1.91 1.49 1.7 1.91



5 8.95 20 19.99 8.97 19.99

7 8.68 15 14.91 8.79 14.993 5.81 25 23.42 5.78 24.95

3 5.81 25 23.42 5.78 24.98

7 8.68 15 14.91 8.79 14.99

5 8.95 20 19.99 8.97 19.99

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NUMERICAL RESULTS

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NUMERICAL RESULTS

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Model 1 (P-S Wave), Unconstrained Case


V

x

V

a

x

V

y

V

a

y

V

z

V

a

z

V

a

x

V

a

y

V

a

z

1.5 1.92 1.7 1.66 1.9 1.24 1.14 1.59 2.12

2 1.95 2.3 3.13 2.5 2.37 2.77 2.11 1.58

3 2.85 2.8 2.85 3.3 3.19 3.7 2.85 2.46

2.7 2.83 2.9 2.85 3.1 3.21 3.69 2.84 2.48

1.8 1.81 2 2.17 2.3 2.45 2.76 2.2 1.69

1.3 1.9 1.6 1.64 1.8 1.46 1.28 1.69 2.15



5 71.82 20 19.99 34.49 19.99

7 14.64 15 14.89 53.22 14.993 6.78 25 0 48.34 18.61

3 6.78 25 1.74 46.94 18.55

7 10.44 15 15.03 48.78 14.99

5 64.52 20 19.97 40.57 19.99

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Model 1 (P-S Wave), Constrained Case


V

x

V

a

x

V

y

V

a

y

V

z

V

a

z

V

a

x

V

a

y

V

a

z

1.5 1.42 1.7 1.64 1.9 1.84 1.29 1.59 1.83

2 1.89 2.3 2.14 2.5 2.38 1.98 2.22 2.39

3 2.86 2.8 2.84 3.3 3.17 2.85 2.85 3.19

2.7 2.84 2.9 2.86 3.1 3.23 2.85 2.84 3.21

1.8 1.89 2 2.16 2.3 2.42 1.81 2.09 2.4

1.3 1.38 1.6 1.66 1.8 1.87 1.49 1.7 1.87



5 8.08 20 19.89 5.28 20

7 8.96 15 14.97 8.62 14.98

3 5.48 25 10 5.42 13.02

3 5.45 25 10.25 5.94 14.91

7 8.99 15 15.28 6.62 14.99

5 8.41 20 19.91 8.76 19.99

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NUMERICAL RESULTS

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NUMERICAL RESULTS

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Model 2 (P Wave), Unconstrained Case


V

x

V

a

x

V

y

V

a

y

V

z

V

a

z

V

a

x

V

a

y

V

a

z

1.5 1.9 1.7 1.7 1.9 1.5 1.49 1.7 1.91

2 2.6 2.3 2.08 2.5 2.07 2.01 2.29 2.48

3 3.96 2.8 2.82 3.3 2.62 3.46 2.81 2.96

3 3.97 2.8 2.89 3.3 2.69 3.36 2.8 2.95

2 2.82 2.3 2.49 2.5 1.47 2.01 2.31 2.5

1.5 1.88 1.7 1.69 1.9 1.49 1.49 1.69 1.9



5 90 20 19.8 6.7 19.99

7 38.69 15 0 3.3 12.83

3 48.46 25 45.68 62.77 36.31

3 36.77 25 49.29 57.28 34.31

7 71.88 15 0 3.13 12.88

5 90 20 20.15 6.73 20

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NUMERICAL RESULTS

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NUMERICAL RESULTS

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Model 2 (P-S Wave), Unconstrained Case


V

x

V

a

x

V

y

V

a

y

V

z

V

a

z

V

a

x

V

a

y

V

a

z

1.5 1.92 1.7 1.71 1.9 1.503 1.502 1.69 1.89

2 2.39 2.3 1.87 2.5 1.92 1.76 2.02 2.25

3 3.73 2.8 3.02 3.3 2.29 3.51 2.89 2.96

2.7 3.63 2.9 2.75 3.1 2.71 2.91 2.81 2.76

1.8 2.89 2 2.44 2.3 1.35 1.96 2.28 2.69

1.3 1.77 1.6 1.59 1.8 1.303 1.302 1.59 1.79



5 88.92 20 18.4 0 20

7 32.38 15 0 18.36 22.73

3 66.92 25 22.55 89.16 13.05

3 28.39 25 36.24 86.88 40.32

7 74.75 15 0 14.57 21.16

5 90 20 21.22 1.07 19.98

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Model 2 (P-S Wave), Constrained Case


V

x

V

a

x

V

y

V

a

y

V

z

V

a

z

V

a

x

V

a

y

V

a

z

1.5 1.5 1.7 1.69 1.9 1.86 1.49 1.69 1.903

2 1.91 2.3 2.22 2.5 2.37 1.81 2.06 2.21

3 2.86 2.8 2.85 3.3 3.21 3.01 2.81 3.55

2.7 2.83 2.9 2.85 3.1 3.19 2.73 2.88 2.91

1.8 1.89 2 2.07 2.3 2.42 2.01 2.27 2.59

1.3 1.29 1.6 1.61 1.8 1.84 1.29 1.6 1.81



5 5.84 20 21.49 8.46 20.1

7 7.36 15 15.46 4.93 13.46

3 6 25 20.31 2 25.13

3 5.78 25 19.14 6 20.43

7 7.47 15 14.29 4.47 12

5 5.01 20 18.94 7.64 19.86

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NUMERICAL RESULTS

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NUMERICAL RESULTS

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NUMERICAL RESULTS

0 1000 2000

0

0.02

0.04

0.06

0.08

0.1# rays vs. velocity error

S

R

0 1000 2000

4

4.5

5

5.5

6

6.5

7

7.5# rays vs. polar error

S

R

0 1000 2000

0

1

2

3

4# rays vs. azimutal error

S

R

0 1000 2000

0

20

40

60

80

100# rays vs. cpu−time

S

R

0 1000 2000

500

1000

1500

2000

2500

3000

3500

4000# rays vs. # iteration

S

R

0 1000 2000

0

100

200

300

400

500

600

700# rays vs. # back−tracking

S

R

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CONCLUSIONS

We solve the velocity and fracture orientation inversion problem in OM

using the Spectral Projected Gradient Method (SPG) and an ellipsoidal

approximation of the velocity.

This is a highly nonlinear problem that has many solutions, so

regularization of the problem is required.

The SPG method obtain good precision for the velocities estimates using

a relative small number of rays and no regularization.

To obtain a good estimate of the azimuthal and polar angle vectors

regularity is essential.

To get a constrained region (regularity ) is not a difficult task in seismic

since the maximum and minimum values of the velocities in the medium

is know a priori.

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A better estimate of the azimuthal angle vector can be obtained if there

are rays in all different azimuths (For example, using Radial Mesh).

None of the Mesh distributions used in this work give enough information

for obtaining a good estimate of the polar angle vector.

The problem in obtaining a better estimate of the polar angle vector is not

the optimization scheme used, but depends on the seismic data

acquisition.

Increasing the number of rays, the error in the velocity vector and in the

azimuthal angle vector can be reduced, obviously this imply an increase

in the cpu-time.

Also, increasing the number of rays, the number of iterations and number

of back-trackings may be reduced.

On the other hand, the error in the polar angle vector increases even if

the number of rays increase, since the seismic data distribution is not the

adequated for estimating the polar angle vector.

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NUMERICAL RESULTS

TEST 1: Corresponding to Model 1

L = ( l

1

: : : l

3 0

)

T

and U = ( u

1

: : : u

3 0

)

T

where l

i

= 0 : 0 0 2 i = 1 : : : 1 8 l

i

= 0 i = 1 9 : : : 3 0

u

i

= 5 0 0 i = 1 : : : 1 8 u

i

= 5 i = 1 9 : : : 3 0

n l s = 3 , n s = 6 , n l r = 4 and n r = 2 8

Initial Velocities Real Velocities Approximated Velocities

V

0

x

V

0

y

V

0

z

V

R

x

V

R

y

V

R

z

V

a

x

V

a

y

V

a

x

2 2 2 1.5 1.7 1.9 1.4999973 1.7019851 1.9019638

3 3 3 2 2.3 2.5 1.9984656 2.3011494 2.5005751

4 4 4 3 2.8 3.3 2.9998747 2.7991592 3.3009248

4 4 4 3 2.8 3.3 2.9997968 2.8008410 3.2994352

3 3 3 2 2.3 2.5 2.0011699 2.2988378 2.4999144

2 2 2 1.5 1.7 1.9 1.5062839 1.6980178 1.8985726

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$

%

NUMERICAL RESULTS


L = ( l

1

: : : l

3 0

)

T

and U = ( u 1

: : : u

3 0

)

T

where

l

i

= 0 : 0 0 2 i = 1 : : : 1 8 l

i

= 0 i = 1 9 : : : 3 0

u

i

= 5 0 0 i = 1 : : : 1 8 u

i

= 5 i = 1 9 : : : 3 0

n l s = 3

, n s = 6

, n l r = 4

and n r = 2 8

Initial Angles Real Angles Approximated Angles

0

0

R

R

a

a

1 1 0 0 1.5322834 0.0033316

1 1 0 0 1.3463286 0.0049373

2 2 0 0 1.4478746 0.0000000

2 2 0 0 1.4356391 0.3929935

2 2 0 0 1.3585243 0.0341731

2 2 0 0 1.5861549 0.0000000

C P U ; t i m e = 2 2 : 8 2 m i n , i t e r = 5 0 7 and l i n e ; s e a r c h e s = 1 2 6

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NUMERICAL RESULTS


L = ( l

1

: : : l

3 0

)

T

and U = ( u

1

: : : u

3 0

)

T

where l

i

= 0 : 0 0 2 i = 1 : : : 1 8 l

i

= 0 i = 1 9 : : : 3 0

u

i

= 6 i = 1 : : : 1 8 u

i

= 4 0 i = 1 9 : : : 3 0

n l s = 3 , n s = 6 , n l r = 4 and n r = 2 8

Initial Velocities Real Velocities Approximated Velocities

V

0

x

V

0

y

V

0

z

V

R

x

V

R

y

V

R

z

V

a

x

V

a

y

V

a

x

3 3 3 1.5 1.7 1.9 1.4867472 1.6086736 1.7609429

3.5 3.5 3.5 2.7 2.5 2.9 2.7073193 2.4508121 2.7925423

4 4 4 3 2.8 3.3 3.0203938 2.7488306 3.1746943

4 4 4 2.9 2.7 3.2 3.0147983 2.7494752 3.1800087

3.5 3.5 3.5 2.6 2.4 2.8 2.6841448 2.4514979 2.8105226

3 3 3 1.4 1.6 1.8 1.5616237 1.6898496 1.7545015

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NUMERICAL RESULTS


L = ( l

1

: : : l

3 0

)

T

and U = ( u 1

: : : u

3 0

)

T

where

l

i

= 0 : 0 0 2 i = 1 : : : 1 8 l

i

= 0 i = 1 9 : : : 3 0

u

i

= 6 i = 1 : : : 1 8 u

i

= 4 0 i = 1 9 : : : 3 0

n l s = 3

, n s = 6

, n l r = 4

and n r = 2 8


0

0

R

R

a

a

0 0 30 10 1.4881100 9.9471689

0 0 30 10 5.3222734 10.2593893

0 0 30 10 0.3525908 9.8410988

0 0 30 10 1.2738337 9.7745561

0 0 30 10 5.3366231 9.8324231

0 0 30 10 4.7458023 10.0201751

C P U ; t i m e = 4 1 : 9 m i n , i t e r = 3 7 9 and l i n e ; s e a r c h e s = 9 3

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NUMERICAL RESULTS


L = ( l

1

: : : l

3 0

)

T

and U = ( u 1

: : : u

3 0

)

T

where

l

i

= 0 : 2 i = 1 : : : 1 8 l

i

= 2 0 i = 1 9 : : : 2 4 l

i

= 5 i = 2 5 : : : 3 0

u

i

= 6 i = 1 : : : 1 8 u

i

= 4 0 i = 1 9 : : : 2 4 u

i

= 2 0 i = 2 5 : : : 3 0

n l s = 4 , n s = 2 0 , n l r = 6 and n r = 6 6

Initial Velocities Real Velocitie s Approximated Velocities

V

0

x

V

0

y

V

0

z

V

R

x

V

R

y

V

R

z

V

a

x

V

a

y

V

a

x

3 3 3 1.5 1.7 1.9 1.4376400 1.7011965 1.9290055

3.5 3.5 3.5 2.7 2.5 2.9 2.6773830 2.4507081 2.8187457

4 4 4 3 2.8 3.3 2.9590742 2.7493184 3.2417599

4 4 4 2.9 2.7 3.2 2.9503655 2.7497884 3.2461665

3.5 3.5 3.5 2.6 2.4 2.8 2.6758592 2.4530501 2.8238717

3 3 3 1.4 1.6 1.8 1.3586700 1.5958847 1.9018278

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NUMERICAL RESULTS


L = ( l

1

: : : l

3 0

)

T

and U = ( u 1

: : : u

3 0

)

T

where

l

i

= 0 : 2 i = 1 : : : 1 8 l

i

= 2 0 i = 1 9 : : : 2 4 l

i

= 5 i = 2 5 : : : 3 0

u

i

= 6 i = 1 : : : 1 8 u

i

= 4 0 i = 1 9 : : : 2 4 u

i

= 2 0 i = 2 5 : : : 3 0

n l s = 4

, n s = 2 0

, n l r = 6

and n r = 6 6


0

0

R

R

a

a

0 0 30 10 37.5290024 9.9980061

0 0 30 10 22.9054416 10.0189242

0 0 30 10 29.4423860 9.9919281

0 0 30 10 28.9710453 9.9914708

0 0 30 10 23.3144158 10.0187249

0 0 30 10 33.8245728 9.9980831

C P U ; t i m e = 1 4 h o u r s , i t e r = 2 6 3 3 and l i n e ; s e a r c h e s = 4 9 1

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NUMERICAL RESULTS

TEST 4: Corresponding to Model 2 L = ( l

1

: : : l

5 0

)

T

and U = ( u 1

: : : u

5 0

)

T

where

l

i

= 0 : 2 i = 1 : : : 3 0 l

i

= 0 i = 3 1 : : : 5 0

u

i

= 6 i = 1 : : : 3 0 u

i

= 4 0 i = 3 1 : : : 4 0 u

i

= 5 i = 4 1 : : : 5 0

n l s = 1 , n s = 2 , n l r = 2 and n r = 1 5

V

0

x

V

0

y

V

0

z

V

R

x

V

R

y

V

R

z

V

a

x

V

a

y

V

a

x

2.5 2.6 3 1.5 1.7 1.9 1.5161705 1.7044085 1.8036717

3.5 3.2 3.6 2.7 2.5 2.9 2.6627301 2.4650854 2.8372655

4 3.8 4.1 3 2.8 3.3 2.9680257 2.7636329 3.2310795

4.1 4.3 4.5 3.3 3.5 3.6 3.1759822 3.4724621 3.6971851

4.3 4.5 4.5 3.5 3.6 3.8 3.7108556 3.7538805 3.5634800

4.3 4.5 4.5 3.4 3.5 3.7 3.8196233 3.5490186 3.5673048

4.1 4.3 4.5 3.2 3.4 3.5 3.1410118 3.4148884 3.5903418

4 3.8 4.1 2.9 2.7 3.2 2.9575074 2.7362643 3.2367196

3.5 3.2 3.6 2.6 2.4 2.8 2.6577356 2.4378346 2.8389060

2.5 2.6 3 1.4 1.6 1.8 1.4447071 1.5913248 1.8147302

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NUMERICAL RESULTS


1

: : : l

5 0

)

T

and U = ( u 1

: : : u

5 0

)

T

where

l

i

= 0 : 2 i = 1 : : : 3 0 l

i

= 0 i = 3 1 : : : 5 0

u

i

= 6 i = 1 : : : 3 0 u

i

= 4 0 i = 3 1 : : : 4 0 u

i

= 5 i = 4 1 : : : 5 0

n l s = 1 , n s = 2 , n l r = 2 and n r = 1 5

0

0

R

R

a

a

23 3 30 0 26.9826548 0.0023920

23 3 30 0 27.8955924 0.0056462

23 3 30 0 28.1072325 0.0169888

23 3 30 0 32.2204377 0.9686877

23 3 30 0 36.8106643 3.5928424

23 3 30 0 6.5430180 0.9112516

23 3 30 0 38.4896379 1.0487732

23 3 30 0 27.2661348 0.0000001

23 3 30 0 28.0762195 0.0000000

23 3 30 0 22.3459052 0.0065288

C P U ; t i m e = 2 : 1 7 h o u r s , i t e r = 2 2 6 4 and l i n e ; s e a r c h e s = 5 2 9

50

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NUMERICAL RESULTS


1

: : : l

5 0

)

T

and U = ( u 1

: : : u

5 0

)

T

where

l

i

= 0 : 2 i = 1 : : : 3 0 l

i

= 0 i = 3 1 : : : 5 0

u

i

= 6 i = 1 : : : 3 0 u

i

= 4 0 i = 3 1 : : : 4 0 u

i

= 5 i = 4 1 : : : 5 0

n l s = 2 , n s = 6 , n l r = 4 and n r = 2 8

V

0

x

V

0

y

V

0

z

V

R

x

V

R

y

V

R

z

V

a

x

V

a

y

V

a

x

2.5 2.6 3 1.5 1.7 1.9 1.5146239 1.6834034 1.8664965

3.5 3.2 3.6 2.7 2.5 2.9 2.6520520 2.4716170 2.8821383

4 3.8 4.1 3 2.8 3.3 2.9540754 2.7565136 3.2766908

4.1 4.3 4.5 3.3 3.5 3.6 3.2934095 3.5050076 3.6028292

4.3 4.5 4.5 3.5 3.6 3.8 3.7115094 3.6230909 3.5623813

4.3 4.5 4.5 3.4 3.5 3.7 3.7822892 3.5025558 3.5610915

4.1 4.3 4.5 3.2 3.4 3.5 3.2050763 3.3926163 3.4969652

4 3.8 4.1 2.9 2.7 3.2 2.9172908 2.7421486 3.2543635

3.5 3.2 3.6 2.6 2.4 2.8 2.6143580 2.4293056 2.8551777

2.5 2.6 3 1.4 1.6 1.8 1.3901383 1.6163345 1.8183832

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NUMERICAL RESULTS


1

: : : l

5 0

)

T

and U = ( u 1

: : : u

5 0

)

T

where

l

i

= 0 : 2 i = 1 : : : 3 0 l

i

= 0 i = 3 1 : : : 5 0

u

i

= 6 i = 1 : : : 3 0 u

i

= 4 0 i = 3 1 : : : 4 0 u

i

= 5 i = 4 1 : : : 5 0

n l s = 2 , n s = 6 , n l r = 4 and n r = 2 8

0

0

R

R

a

a

23 3 30 0 32.3391389 0.0000489

23 3 30 0 34.1106580 0.0000030

23 3 30 0 32.3994191 0.0060426

23 3 30 0 28.2734365 0.2313614

23 3 30 0 33.7757315 1.8283798

23 3 30 0 14.0161269 1.4759419

23 3 30 0 31.3560557 0.2281124

23 3 30 0 31.2219247 0.0000007

23 3 30 0 32.5670208 0.0000002

23 3 30 0 26.4797392 0.0000259

C P U ; t i m e = 1 6 h o u r s , i t e r = 5 6 6 1 and l i n e ; s e a r c h e s = 1 2 4 7

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NUMERICAL RESULTS


1

: : : l

5 0

)

T

and U = ( u 1

: : : u

5 0

)

T

where

l

i

= 0 : 2 i = 1 : : : 3 0 l

i

= 0 i = 3 1 : : : 5 0

u

i

= 6 i = 1 : : : 3 0 u

i

= 4 0 i = 3 1 : : : 4 0 u

i

= 5 i = 4 1 : : : 5 0

n l s = 3 , n s = 1 2 , n l r = 5 and n r = 4 5

V

0

x

V

0

y

V

0

z

V

R

x

V

R

y

V

R

z

V

a

x

V

a

y

V

a

x

2.5 2.6 3 1.5 1.7 1.9 1.5015321 1.6815115 1.8446112

3.5 3.2 3.6 2.7 2.5 2.9 2.6330562 2.4550646 2.8898359

4 3.8 4.1 3 2.8 3.3 2.9731147 2.7550248 3.2525118

4.1 4.3 4.5 3.3 3.5 3.6 3.3050384 3.453214 3.6168122

4.3 4.5 4.5 3.5 3.6 3.8 3.9870890 3.8343516 3.0960452

4.3 4.5 4.5 3.4 3.5 3.7 3.9948479 3.7634267 3.1293398

4.1 4.3 4.5 3.2 3.4 3.5 3.2320777 3.3985567 3.5229264

4 3.8 4.1 2.9 2.7 3.2 2.9321892 2.7441214 3.2395075

3.5 3.2 3.6 2.6 2.4 2.8 2.6083586 2.4465557 2.8750930

2.5 2.6 3 1.4 1.6 1.8 1.3929838 1.6172687 1.8505478

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NUMERICAL RESULTS


1

: : : l

5 0

)

T

and U = ( u 1

: : : u

5 0

)

T

where

l

i

= 0 : 2 i = 1 : : : 3 0 l

i

= 0 i = 3 1 : : : 5 0

u

i

= 6 i = 1 : : : 3 0 u

i

= 4 0 i = 3 1 : : : 4 0 u

i

= 5 i = 4 1 : : : 5 0

n l s = 3 , n s = 1 2 , n l r = 5 and n r = 4 5

0

0

R

R

a

a

23 3 30 0 34.3991447 0.0043882

23 3 30 0 35.4810917 0.0013095

23 3 30 0 30.9784433 0.0881668

23 3 30 0 33.9740601 0.0000000

23 3 30 0 33.5488468 3.7197016

23 3 30 0 37.6389450 0.3137009

23 3 30 0 32.450822 0.0000000

23 3 30 0 28.059081 0.087736

23 3 30 0 34.2884020 0.0047640

23 3 30 0 25.9546206 0.0020617

C P U ; t i m e = 6 0 h o u r s , i t e r = 3 9 7 4 and l i n e ; s e a r c h e s = 9 6 2

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NUMERICAL RESULTS

150 rays 840 rays 2700 rays

k

S

r

;

S

k

k

2

1.749 0.622 1.27

k

V

r

;

V

k

k

2

0.631 0.54 1.25

k (

r

r

) ; (

k

k

) k

2

1.63 0.317 0.24

Iterations 2264 5661 3974

Line-searches 529 1247 962

CPU-time 2 16 60

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CONCLUSIONS

We solve the velocity and fracture orientation inversion problem in OM

using the Spectral Projected Gradient Method (SPG) and an ellipsoidal

approximation of the velocity.

This is a highly nonlinear problem that has many solutions, so

regularization of the problem is required..

The SPG method obtain good precision for the velocities estimates using

a relative small number of rays and no regularization.

To estimate de azimuthal and polar angles regularity is essential.

A better estimate of the azimuthal angle can be obtained if there are rays

in all different azimuths (For example using Radial Mesh).

The ray tracing takes most CPU time required for the inversion, so a

parallel low cost ray tracing will reduce the CPU time.

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estimacion de velocidades

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