cations. For example, it has been used in chemistry to determine molecular structures from nuclear mag-
Ž .
netic resonance data Glunt et al., 1993 ; in statistics and cartography to solve multidimensional scaling
Ž .
problems Luengo et al., 1998 ; in the solution of partial differential equations, where the dimensions
Ž of the linear systems are very large Molina and
. Raydan, 1996 ; and very recently in seismic explo-
ration to estimate velocities by reflection tomogra- phy, which brings enormous computational advan-
Ž .
tages Castillo et al., 2000 . This work is presented in the following manner.
In Section 2 we pose the ray tracing problem as a nonlinear optimization problem and we also present
the model parametrization. A description of the SG method when it is used for solving the ray tracing
problem is presented in Section 3. In Section 4 some convergence properties of the SG method and of the
proposed optimization problem are presented. In Sec- tion 5 we present some numerical results and we also
illustrate the computational advantages of the method when compared to other optimization techniques.
Finally, in Section 6 we present some conclusions.
2. Optimization problem
In this section we formulate the problem of trac- ing rays from a fixed source to a fixed receiver as
the solution of an optimization problem. A ray traveling from a source X g
R
3
to a
s
receiver X g R
3
and crossing different reflectors of
r
the subsurface satisfies the Fermat’s principle, i.e., the ray follows the trajectory of minimum travel time
between X and X . For that reason, for each pair of
s r
Ž .
source receptor X , X
crossing n layers, where
s r
each layer is defined by the region delimited be- tween two consecutive interfaces of the subsurface,
the problem is a nonlinear unconstrained optimiza- tion problem
d l
X
r
X
r
Minimize T x , y, z s
1
Ž .
Ž .
H
X
s
Õ x , y, z
Ž .
X
s
Ž .
where Õ x, y, z is the velocity of the medium and, d l is the differential length of the ray.
It is important to mention that the proposed scheme consists in solving m nonlinear optimization
Ž . problems of the form
1 where, m is the total
number of rays. In this work, we assume that the segment of a ray between two consecutive interfaces,
Ž .
say the i y 1 th and ith interfaces, is a straight line Ž
. Ž
segment l , as Mao and Stuart 1997 proposed see
i
. Fig. 1 ,
2 2
2
l s x y x
q y y y q z y z
Ž .
Ž .
Ž .
i i
iy1 i
iy1 i
iy1
2
Ž .
Ž .
where x , y , z
are coordinates of the ray at the
i i
i
interfaces, i s 1, . . . ,2 n q 1. Under the assumption that the segment of a ray
between any two consecutive interfaces is a straight line segment, we have that the trajectory of a ray for
Ž .
a pair of source–receptor X , X
is a piecewise
s r
linear path crossing n q 1 interfaces. So, the opti- Ž .
mization problem 1 can be written as follows:
2 nq1
l X ,Y , Z
Ž .
i X
r
Minimize T
X ,Y , Z s ,
Ž .
Ý
X
s
3= Ž2 ny1.
Õ X ,Y , Z
Ž .
Ž .
X ,Y , Z g R
i
is2
3
Ž .
where
T
X s x , y , z ,
Ž .
s 1
1 1
T
X s x , y
, z ,
Ž .
r 2 nq1
2 nq1 2 nq1
T
X s x , . . . , x ,
Ž .
2 2 n
T
Y s y , . . . , y ,
Ž .
2 2 n
T
Z s z , . . . , z ,
4
Ž .
Ž .
2 2 n
Õ is the velocity in the midpoint of the ray path
i
Ž .
segment delimited by the i y 1 th and ith inter-
faces, and it satisfies Õ s Õ for i s 2, . . . ,
i 3q2 nyi
Fig. 1. Parametrization of a general 3D model.
n q 1 in the case of non-converted waves; otherwise, the down going velocities Õ , i s 2, . . . , n q 1, are
i
different from the up going velocities Õ , i s n q
i
Ž .
2, . . . , 2 n q 1 see Fig. 1 . The z coordinates of the ray trajectory, for i s
i
Ž . Ž .
1, . . . , 2 n q 1 in Eqs. 2 and 4 , depend on the ray Ž
. trajectory coordinates x , y
in the following way
i i
z s f x , y ,
for i s 1, . . . ,2 n q 1, 5
Ž .
Ž .
i i
i i
where the function f defines the ith interface, i s
i
1, . . . , n q 1 and, for i s n q 2, . . . , 2 n q 1, f s
i
Ž .
f . Then, it is clear that the n q 1 th interface,
2 nyiq2
Ž f
, corresponds to the reflecting interface see Fig.
nq 1
. 1 . Moreover, the interface functions f
and the
i
velocities Õ are obtained by biharmonic splines
i
Ž .
Sandwell, 1987 . Therefore, the optimization prob- Ž .
Ž lem
3 can be expressed as a function of
X, .
4 ny2
Y g R
as follows
2 nq1
l X ,Y
Ž .
i X
r
Minimize T X ,Y s
. 6
Ž .
Ž .
Ý
X
s
Õ X ,Y
Ž .
i
is2
It is well known that a necessary condition for a Ž
. ray trajectory X, Y
to be a minimizer of problem Ž .
6 is that the gradient of the travel time functional,
X
r
Ž .
=T X, Y , is equal to zero. Here, the gradient
X
s
vector is given by = T
X
r
X ,Y
Ž .
X
s
T X
X
r r
ET X ,Y
ET X ,Y
Ž .
Ž .
X X
s s
[ . . . , , . . . ,
, . . . ,
ž
E x E y
i i
i s 2, . . . ,2 n 7
Ž .
where the partial derivatives are:
X
r
Ž .
ET X ,Y
X
s
E x
i
Ž .
Ž .
El X ,Y
EÕ X ,Y
i i
Ž .
Ž .
Õ X ,Y y
l X ,Y
i i
E x E x
i i
s
2
Ž .
Õ X ,Y
i
Ž .
Ž .
El X ,Y
EÕ X ,Y
iq 1 iq1
Ž .
Ž .
Õ X ,Y y
l X ,Y
iq 1 iq1
E x E x
i i
q ,
2
Ž .
Õ X ,Y
iq 1
8
Ž .
X
r
Ž .
ET X ,Y
X
s
E y
i
Ž .
Ž .
El X ,Y
EÕ X ,Y
i i
Ž .
Ž .
Õ X ,Y y
l X ,Y
i i
E y E y
i i
s
2
Ž .
Õ X ,Y
i
Ž .
Ž .
El X ,Y
EÕ X ,Y
iq 1 iq1
Ž .
Ž .
Õ X ,Y y
l X ,Y
iq 1 iq1
E y E y
i i
q ,
2
Ž .
Õ X ,Y
iq 1
9
Ž .
for i s 2, . . . , 2 n. Now, for the general 3D ray tracing we write the
Snell’s law as a nonlinear system of equations as follows:
sin a sin a
Ž .
Ž .
i iq1
s ,
i s 2, . . . ,2 n 10
Ž .
Õ Õ
i iq1
where for each i, a is the angle of incidence and
i
a is the refraction angle at the ith interface. It is
iq 1
clear that the number of Snell’s nonlinear equations Ž
. is 2 n y 1, one for each point
x , y , z in the
i i
i
trajectory of the ray except for the source and the receiver. Using the definition of the sine and cosine
of an angle we have that
x y x s l sin a cos f ,
Ž .
Ž .
i iy1
i i
i
y y y s l sin a cos f ,
Ž .
Ž .
i iy1
i i
i
z y z s l cos a ,
Ž .
i iy1
i i
Fig. 2. Parametrization used in Snell’s Law to obtain a nonlinear system of equations.
where a is the angle that the segment between
i
Ž .
Ž .
x , y
, z and
x , y , z forms with the
iy 1 iy1
iy1 i
i i
vertical axes and the angle f is the corresponding
i
azimuthal angle, as Fig. 2 illustrates. Moreover, these angles satisfy,
y y y
i iy1
sin f s ,
Ž .
i
ˆ
l
i
x y x
i iy1
cos f s ,
Ž .
i
ˆ
l
i 2
2
ˆ
l s l sin a s
x y x q y y y
.
Ž .
Ž .
Ž .
i i
i i
iy1 i
iy1
11
Ž .
Therefore, from these equations we can write the Snell’s equations as a non-linear system of equations
of the form,
F X ,Y s 0, 12
Ž .
Ž .
where,
ˆ ˆ
l l
i iq1
F X ,Y s y
, ,i s 2, . . . ,2 n.
13
Ž .
Ž .
l Õ l
Õ
i i iq1 iq1
ˆ
Ž .
Since in a 2D homogeneous medium, l rl s
i i
Ž .
Ž .
El rE x and EÕ rE x s 0, we can establish from
i i
i i
Ž .
Ž .
the nonlinear system of Eqs. 12 and 13 the fol- lowing result.
Theorem 2.1. In a 2D homogeneous isotropic
medium solÕing the Snell’s non-linear system 12 and 13 is equiÕalent to forcing the gradient of the
traÕel time function giÕen by Eqs. 7 – 9
to be equal to zero.
Therefore, the first order necessary conditions for Ž .
the travel time function given by Eq. 6 are identical Ž
. to Snell’s nonlinear system of Eq. 12 . Thus, New-
Ž . ton’s method applied to Eq. 6 will generate identi-
cal iterates to the ones generated by the Newton’s method applied to the Snell’s nonlinear system of
Ž .
Eq. 12 . Moreover, in a 2D homogeneous medium finding a ray path using Snell’s equations to obtain
Ž . an initial iterate for solving problem
6 , by an optimization scheme, seems to be redundant. There-
fore, an optimization technique for solving problem Ž .
6 that does not require a close initial ray path to converge and, that requires very inexpensive compu-
tations, will avoid the redundancy on the use of different approaches for solving the problem and
also will guarantee fast convergence. In the 3D homogeneous medium, we can only establish that the
first order necessary conditions associated to the
Ž . travel time function 6 imply that the Snell’s nonlin-
ear system of equations are satisfied. Based on the conclusions made on the above
paragraph, we propose to solve the unconstrained Ž .
nonlinear optimization problem 6 with the global Ž
. SG method Raydan, 1993, 1997 . This low storage
optimization technique has been used recently in Ž
. inversion tomography, Castillo et al. 2000 , obtain-
ing many computational advantages. This low stor- age technique presents four advantages: requires low
computational storage and few floating point opera- tions when compared with other optimization tech-
Ž niques, second order information
Hessian of the .
travel time function is not needed and, it is a global technique, in the sense that it converges from any
Ž .
initial ray trajectory X , Y , which is in sharp
contrast with the method proposed by Mao and Ž
. Stuart 1997 . A brief but concise description of the
Ž . SG method when applied to problem 6
can be found in the next section.
3. Global spectral gradient method
