Geophys. J. Int.

(2006) 164, 670684

doi: 10.1111/j.1365-246X.2005.02729.x

Constrained optimization in seismic reflection tomography:

a GaussNewton augmented Lagrangian approach
F. Delbos,1 J. Ch. Gilbert,2 R. Glowinski3 and D. Sinoquet1
1 Institut

Francais du Petrole, 1 & 4 avenue de Bois-Preau, 92852 Rueil-Malmaison, France. E-mail: delphine.sinoquet@ifp.fr
National de la Recherche en Informatique et en Automatique, BP 105, 78153 Le Chesnay Cedex, France
3 University of Houston, 4800 Calhoun Rd, Houston, TX 77204-3476, USA
2 Institut

GJI Volcanology, geothermics, uids and rocks

Accepted 2005 June 28. Received 2005 April 22; in original form 2004 July 07

SUMMARY
Seismic reflection tomography is a method for determining a subsurface velocity model from
the traveltimes of seismic waves reflecting on geological interfaces. From an optimization viewpoint, the problem consists in minimizing a non-linear least-squares function measuring the
mismatch between observed traveltimes and those calculated by ray tracing in this model. The
introduction of a priori information on the model is crucial to reduce the under-determination.
The contribution of this paper is to introduce a technique able to take into account geological
a priori information in the reflection tomography problem expressed as inequality constraints
in the optimization problem. This technique is based on a GaussNewton (GN) sequential
quadratic programming approach. At each GN step, a solution to a convex quadratic optimization problem subject to linear constraints is computed thanks to an augmented Lagrangian
algorithm. Our choice for this optimization method is motivated and its original aspects are
described. First applications on real data sets are presented to illustrate the potential of the
approach in practical use of reflection tomography.
Key words: augmented Lagrangian, constrained optimization, least-squares approach, ray
tracing, seismic reflection tomography, SQP algorithm.

1 I N T RO D U C T I O N
Geophysical methods for imaging a complex geological subsurface
in petroleum exploration requires the determination of an accurate
wave propagation velocity model. Seismic reflection tomography
turns out to be an efficient method for doing this: it determines the
seismic velocity distribution from the traveltimes associated with
the seismic waves reflecting on geological surfaces. This inverse
problem requires the solution to the specific forward problem, which
consists in computing these traveltimes for a given subsurface model
by a ray tracing method (based on a high-frequency approximation

of the wave equation, see Cerven

y 1989; Jurado et al. 1998). The inverse problem is formulated as the minimization of the least-squares
function that measures the mismatch between traveltimes calculated
by ray tracing and the observed traveltimes.
The main interests of reflection tomography are
(1) its flexibility for handling various types of traveltime data simultaneously (primary reflections but also multiple reflections, traveltimes associated with converted wavesPS data, surface seismic,
well seismic), provided that the ray tracing allows the computation
of such data,
(2) the low computational time of the forward operator, in comparison with the time needed for the calculation of the wave equation
solutions,

670

(3) the reduced number of local minima of the inverse problem,

in comparison with the seismic inversion based on the wave equation
simulation (see for instance Symes 1986 for a study on the choice of
the objective functional in seismic inversion to reduce the number
of local minima), and
(4) its ability to integrate a priori geological information (via a
least-squares formulation).
This method has been successfully applied to numerous real data
sets (Ehinger et al. 2001; Alerini et al. 2003; Broto et al. 2003, among
others). Nevertheless, the underdetermination of the inverse problem generally requires the introduction of additional information to
reduce the number of admissible models. The Bayesian inversion
allows the introduction of different type of data and a priori information, the associated uncertainties being modelled by probability
distributions (see Tarantola 2005, for a detailed description of this
approach). In practice, a classical least-squares formulation (assuming Gaussian probability densities) is used: penalty terms modelling
a priori information are generally added to the seismic terms in the
objective function with a delicate adjustment of the penalty weights
(see Daalen et al. 2004; Krebs et al. 2004, among the most recent
papers on tomography in oil industry).
The standard methodology to invert complex subsurface structures (model composed of several velocity fields and reflectors),
a top-down layer-stripping approach, may be inadequate. This

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Constrained optimization in seismic reflection tomography

approach consists in inverting separately each velocity layers (with
its associated reflectors) starting from the upper layer to the lower
one. To limit bad data fitting for deeper layers, a global inversion
approach, which consists of simultaneously inverting all the velocity layers and interfaces of the subsurface model, is recommended.
But, this method is often discarded due to its convergence troubles:
because of the underlying underdetermination, a global inversion
of complex subsurface structures often leads to a non-admissible
subsurface model on which the ray tracing method fails to compute
the traveltimes. Additional constraints on the model (for instance,
on layer thicknesses to avoid non-physical interface intersections)
are necessary to avoid those non-admissible models. We believe
that the possibility to introduce constraints in the optimization process can overcome part of those difficulties. Equality and inequality
constraints can indeed model many different types of a priori information, especially inequality constraints may help for instance to
match well data with a given uncertainty. An optimization approach
that can face these constraints efficiently will then discharge the final
user of the inversion seismic software from the cumbersome task of
tuning the weights associated with the additional penalty terms in
the objective function.
The goal of the paper is twofold. First, it presents our constrained
nonlinear optimization method and motivates its appropriateness
to constrained reflection tomography. A part of this algorithm is
new and the novelty is presented in the technical Sections 3.3 and
3.4. Second, it illustrates the efficiency of the chosen constrained
optimization method thanks to its application on a 2-D OBC real
data set and then on a 3-D streamer real data set.
We recall the problem of interest and introduce the notation in
Section 2. In Section 3, our Sequential Quadratic Programming
(SQP) augmented Lagrangian approach for constrained reflection
tomography problems is described and motivated. Numerical experiments on real data sets are detailed in Section 4. We conclude with
Section 5.

2 THE SEISMIC REFLECTION

T O M O G R A P H Y P RO B L E M
2.1 The unconstrained problem
Let us first recall the problem of interest and introduce the notation.
The choice of the model representation is crucial for the efficiency of
the methods used to solve the forward and inverse problems. Lailly
& Sinoquet (1996) have discussed the interest of different types of
velocity models. We have chosen here a blocky model, where the velocity distribution is described by slowly varying layer velocities (or
velocity blocks) delimited by interfaces. With this representation,
we introduce explicitly a strong a priori information: the number of
layers. The number of parameters describing the velocity variations
is limited thanks to the explicit introduction of velocity discontinuities (the velocity within a layer varies smoothly). The model is thus
composed of two kinds of parameters: those describing the velocity
variations within the layers and those describing the geometry of the
interfaces delimiting the layers. Parameters describing the velocity
anisotropy can also be included (see Jurado et al. 1998; Stopin 2001,
for more details).
The ith interface is represented by a cubic B-spline function (de
Boor 1978; Inoue 1986)
z i (x, y), whose coefficients define a vector
z i (x and y are the horizontal coordinates). Similarly, the ith velocity field is represented by a cubic B-spline function 
vi (x, y, z) or

vi (x, y) + k z with known scalar k (z is the vertical coordinate);

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the vector v i contains the velocity coefficients. For n v layer velocities and n z interfaces, we collect the coefficients v 1 , . . . , v nv in one
vector v and the coefficients z 1 , . . . , z nz in one vector z. The model
vector m Rn is defined here as m = (v, z). The C 2 smoothness of
the functions describing the model allows the exact computation of
derivatives of the traveltimes with respect to the model, quantities
useful for ray tracing and tomography.
Given a model m and an acquisition survey (locations of the
sources and receivers) a vector of traveltimes T(m) of seismic reflected waves can be computed by ray tracing (see Jurado et al.
1998). The mapping T : Rn Rt : m  T (m) is nonlinear. We assume that it is differentiable. In practice, this assumption may not be
satisfied, in particular, the forward operator may even not be defined
when rays escape from the region of interest or when a layer thickness vanishes (non-physical interface intersections as mentioned in
the introduction).
Reflection traveltime tomography is the corresponding inverse
problem: its purpose is to adjust m so that T (m) best matches a
vector of traveltimes T obs Rt (the observed traveltimes) picked
on seismic data. Since Gauss (1809), it is both classical and natural
to formulate such an inverse problem as a least-squares one:
min

m Rn

1
 T (m) T obs 2 ,
2

(1)

where   denotes the Euclidean norm.

The fact that the problem (1) may be ill-posed has been pointed
out by many authors (see for instance Delprat-Jannaud & Lailly
1993; Bube & Meadows 1999). To ensure well-posedness, a curvature regularization is often introduced (Tikhonov & Arsenin 1977).
We use the sum of the squared L 2 -norms of all the second order
partial derivatives of every velocity 
vi and reflector 
z i (see for instance Delprat-Jannaud & Lailly 1993). Such a regularization term
can be written as m Rm, where R is a symmetric positive semidefinite matrix that only depends on the B-spline basis functions (it is
independent of m). Thus, instead of the problem (1), we consider
the regularized least-squares problem


2
1 
minn f (m) :=  T (m) T obs  + m  R m ,
(2)
mR
2
2
where the regularization weight is positive, and f : Rn R
is called the cost function (or objective function). The choice of
the parameter is a difficult task. In practice, we use the L-curve
method (see Hansen 1992), also called the continuation method
(Bube & Langan 1994): starting from a large regularization weight,
we decrease it regularly to retrieve more and more varying models
until the data are fitted with the expected accuracy. The solution
model is thus the smoothest model that fits the data up to a certain
precision. This methodology allows us to do stable inversions. In
the sequel, when we consider the objective function of the problem
(2), we assume that its regularization weight is fixed.
The unconstrained minimization problem of seismic reflection
tomography, defined by (2), has the following features:
(1) the size of the data space and of the model space can be quite
large (up to 106 traveltimes and 105 unknowns),
(2) the problem is ill-conditioned (Chauvier et al. 2004, have
observed that the condition number of the approximated Hessian
Hk given by (5) below can go up to 109 for a matrix of order
500),
(3) a forward simulation, that is, a computation of T (m), is CPU
time consuming because of the large number of source-receiver
pairs, and

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F. Delbos et al.

(4) the traveltime operator T is nonlinear, revealing the complexity of wave propagation in the subsurface.
To minimize efficiently a function like f in (2), it is highly desirable to have its gradient available. In the present context, thanks to
Fermats principle (Bishop et al. 1985), it is inexpensive to compute
row by row the Jacobian matrix J (m) of T at m. Recall that its (i,
j)th element is the partial derivative
J (m)i j =

Ti
.
m j

(3)

The gradient of the objective is then obtained by the formula

f (m) = J (m) (T (m) T obs ) + Rm. It is also natural to ask
whether one can compute the second derivatives of f . The answer is
however negative. Therefore, using a pure Newton method to solve
(2) is computationally infeasible.
There are at least two classes of methods that can take advantage
of the sole first-order derivatives:
(1) quasi-Newton (QN) methods and
(2) GaussNewton (GN) methods.
Standard QN methods do not use the structure of the least-squares
problems, but have a larger scope of application. They are used for
solving least-squares problems when the computation of the Jacobian matrix J is much more time consuming than the computation
of the residual vector T (m) T obs (see Courtier & Talagrand 1987;
Courtier et al. 1994, for a typical example in meteorology). We have
mentioned above that this is not our case. On the other hand, the GN
methods fully take benefit of the Jacobian matrix J by taking J  J
as an approximation of the Hessian of the first term in f . This algorithm can exhibit slow convergence when the residual is large at the
solution and when T is strongly nonlinear at the solution. This does
not seem to be the case in the problem we consider. Sometimes GN
and QN methods are combined to improve the approximation of the
Hessian of the first part of f by J  J (see Dennis et al. 1981; Yabe
& Yamaki 1995, and the references therein).
The above discussion motivates our choice of a classical linesearch GN method to solve the unconstrained optimization problem
(Chauvier et al. 2000). The kth iteration, k 0, proceeds as follows.
Let mk be the approximate solution known at the beginning of the
iteration. Note
Tk := T (m k )

and

Jk := J (m k ).

2.2 Formulation of the constrained problem

Let us now set down the formulation of the constrained seismic tomography problem. The constraints that can be introduced in the
optimization problem could be nonlinear (for example, we could
force the impact points of some rays on a given interface to be located in a particular area) but, in this study, we limit ourselves to
linear constraints. Even though linearity brings algorithmic simplifications, the optimization problem is difficult to solve because of
the large number (up to 104 ) and the variety of the constraints. These
may be of various types:
(1) constraints of different physical natures: on the velocities, on
the interface depths, or on the derivatives of these quantities,
(2) equality or inequality constraints (examples: fixed value of
the velocity gradient, minimal depth of an interface), and
(3) local or global constraints (examples: local information coming from a well, interface slope in a particular region).
The constrained reflection tomography problem we consider is
therefore formulated as the regularized least-squares problem (2)
subject to linear constraints:

f (m)
mmin
Rn
(6)
m
=
e
C
E

l C m u.
I
In this problem, C E (resp. C I ) is an n E n (resp. n I n) matrix,
e Rn E , and the vectors l, u Rn I satisfy li < u i for all index i.
We note
n C := n E + n I

and

n C := n E + 2n I .

(4)

First, an approximate solution dk to the following tangent quadratic

problem is computed

1 
 Jk d + Tk T obs 2 + (m k + d) R(m k + d).
minn
dR 2
2
This is the quadratic approximation of f about mk , in which the costly
computation of the second derivatives of T has been neglected. By
the choice of the positive semi-definite regularization matrix R, the
Hessian of this quadratic function in d, namely
Hk := Jk Jk + R,

This method is generally able to solve the minimization the problem (2). In some difficult cases, however, the line-search technique fails to force convergence of the sequence {mk } k0 to a
solution. This difficulty may arise when the Hessian of f is very
ill-conditioned and can often be overcome by using trust regions
(see Conn et al. 2000) instead of line-searches. The former method
usually provides more stable and accurate results than the latter
(Delbos et al. 2001; see also Sebudandi & Toint 1993). In any case,
we observe in practice that very few iterations are needed to get
convergence, typically of the order of 10.

(5)

is usually positive definite. This property makes it possible to minimize the above quadratic function by a preconditioned conjugate
gradient algorithm. The next model estimation is then obtained by
the formula
m k+1 = m k + k dk ,
where k > 0 is a step-size computed by a line-search technique
ensuring a sufficient decrease of f at each iteration.

It is said that an inequality constraint is active at m if it is satisfied

with equality for this m. The inequality constraints of (6) can be
active/inactive in 3nI ways, since each of them can be either inactive or active at its lower or upper bound (three possibilities). This
exponential amount is sometime referred to as the combinatorial
aspect of an inequality constrained optimization problem. Determining which of the inequality constraints are active at a solution
turns out to be a major difficulty for the algorithms.
2.3 First-order optimality conditions
 be a local solution to (6). Since f is assumed to be differLet m
entiable and the constraints are linear (thus qualified), there exist

E Rn E , 
l Rn I , and 
u Rn I such that the following Karush,
Kuhn and Tucker conditions (KKT) hold

(a) f (
m ) + C E
E + C I (
u 
l ) = 0

 = e, l C I m
u
(b) C E m
(7)
(c) (
l , 
u ) 0

 l) = 0, 
 u) = 0.

(d) 
l (C I m
u (C I m

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Constrained optimization in seismic reflection tomography

These equations are fundamental in optimization and are the basis of many algorithmic approaches to solve (6). Figuring out their
meaning is easier when there are only equality constraints. Then
the first equation simply expresses the fact that the gradient f (
m)
 must lie in the range space of C E , which is
of the objective at m
perpendicular to the constraint manifold. This geometrical view of
optimality looks quite natural. In the presence of inequality con must satisfy the
straints the meaning of (7) is that, to be optimal, m
constraints (see (b)) and the gradient f (
m ) must be in the dual
; admittedly a
cone of the tangent cone to the feasible set of (6) at m
more complex property. We refer the reader to the book of Fletcher
(1987) or the review paper by Rockafellar (1993) to get more insight
on (7).
The vectors 
E , 
l , and 
u in (7) are called the Lagrange or
KKT multipliers, and are associated with the equality and inequality
constraints of (6). From (a) and (d) in (7), we see that they are used to
decompose f (
m ) on the gradients of the active constraints (some
of the C i s). It can be shown that 
i tells us how the optimal value
f (
m ) varies when the ith constraint is perturbed. Condition (a) can
also be written
m (
m, 
) = 0,


where 
:= (
E , 
l , 
u ) and the function  : Rn RnC  R,
called the Lagrangian of problem (6), is defined by
(m, ) = f (m) + E (C E m e)
l (C I m l) + 
u (C I m u).

(8)

l 
u .
We note 
I := 
Equations (d) in (7) are known as the complementarity conditions.
They express the fact that a multiplier 
i associated with an inactive
inequality constraint vanishes. For some problems, the converse is
also true: active inequality constraints have positive multipliers. It
is then said that strict complementarity holds at the solution:
 (
li < C i m
l )i = 0
 < u i (
u )i = 0.
Ci m
3 S O LV I N G T H E C O N S T R A I N E D
SEISMIC REFLECTION TOMOGRAPHY
P RO B L E M
In this section we motivate and describe the optimization method
used to solve (6) or its optimality conditions (7). A more detailed
description is given by Delbos (2004). The operating diagram of the
overall algorithm is presented in Fig. 1 and can help the reader to
follow the different levels of the approach.
3.1 Motivation for the chosen algorithmic approach
Presently, numerical methods to solve a nonlinear optimization
problem like (6) can by gathered into two classes:
(1) the class of penalty methods, which includes the augmented
Lagrangian approaches and the interior point (IP) approaches and
(2) the class of direct Newtonian methods, which is mainly
formed of the SQP approach.
Often, actual algorithms combine elements of the two classes, but
their main features make them belonging to one of them.
In penalty methods, one minimizes a sequence of nonlinear functions, obtained by adding to the cost function in (6) terms penalizing

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Figure 1. Operating diagram of the constrained optimization method.

more or less strongly the equality and/or inequality constraints as

the iterations progress. For example, in the IP approaches, the inequality constraints are penalized in order to get rid of their combinatorial aspect, while the equality constraints are maintained, since
they are easier to handle (see for instance Byrd et al. 2000, and the
references therein). The iterations minimizing approximately each
penalty function are usually based on the Newton iteration, which
requires finding the solution to a linear system. Therefore, the overall work of the optimization routine can be viewed as the one of
solving a sequence of sequences of linear systems. This simplified presentation is mainly valid far from a solution, since close to
a solution satisfying some regularity assumptions, a single Newton
step is often enough to minimize sufficiently the current penalty
function (see Gould et al. 2000, for example). Now, each time a step
is computed as a solution to a linear system, the nonlinear functions
defining the optimization problem have to be evaluated in order to
estimate the quality of the step. It is sensible to define an iteration
of the penalty approach as formed of a step computation and an
evaluation of the nonlinear functions.
On the other hand, an SQP algorithm is a Newtonian method
applied to the optimality conditions, (7) in our case (see part III in
Bonnans et al. (2003) e.g.). Therefore, there is no sequence of nonlinear optimization problems to solve approximately, like in penalty
methods. As a result, it is likely that such an approach will need less
iterations to achieve convergence. Since, here also, the nonlinear
functions defining the optimization problem need to be computed
at each iteration to validate the step, it is likely that less nonlinear
function evaluations are required with an SQP approach, in comparison with a penalty approach. Nevertheless, each SQP iteration is
more complex, since it requires solving a quadratic program (QP),
which is an optimization problem, with a quadratic cost function
and linear equality and inequality constraints.
The discussion above shows that the choice of the class of algorithms strongly depends on the features of the optimization problem
to solve. The key issue is to balance the time spent in the simulator (to evaluate the functions defining the nonlinear optimization
problem) and in the optimization procedure (to solve the linear systems or the QP). In the unconstrained seismic reflection tomography
problem, we have said in Section 2.1 that most of the CPU time is
spent in the evaluation of the traveltimes (simulation) and that the
GN algorithm converges in very few iterations (around 10). When

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choosing the algorithmic approach for the constrained version of the

problem, we anticipated that the number of iterations will also be
small with a Newton-like algorithm and that, in the current state of
their development, IP algorithms will be unlikely to converge in so
few iterations. This is our main motivation for developing an SQPlike algorithm: to keep as small as possible the number of nonlinear
function evaluations. This strategy could be questioned with a ray
tracing using massive parallelization; we leave this topic for future
investigations.
3.2 A GaussNewton sequential quadratic programming
approach
We have already mentioned that the SQP algorithm is a Newtonlike method applied to the optimality conditions of the nonlinear
optimization problem under consideration, (7) in our case. In its
standard form, it then benefits from a local quadratic convergence.
Let us specify this algorithm for the constrained seismic reflection
tomography problem.
The main work at the kth iteration of an SQP algorithm consists
in solving the following tangent QP in d (see Chapter 13 and (13.4)
in Bonnans et al. 2003), in order to find the perturbation dk to be
given to mk :



1

minn Fk (d) := gk d + d  Hk d

dR
2
(9)
(QPk )

d
=
e
C

E
k

lk C I d u k .
The cost function of this problem has a hybrid nature. Its linear part
is obtained by using the gradient of the cost function of (6) at mk ,
which is


gk = Jk Tk T obs + R m k ,
where we used the notation in (4). Its quadratic part should use,
ideally, the Hessian of the Lagrangian  defined in (8), in order to get (quadratic) convergence. Since the constraints in (6)
are linear, only the Hessian of f plays a role. Like for the unconstrained problem, we select the GN approximation Hk (see
(5)) of this Hessian in order to avoid the computation of the expensive second derivatives of T. The constraints of (9) are obtained
by the linearization of the constraints of (6) at mk . Writing C E (mk
+ d) = e and l C I (mk + d) u leads to the constraints of eq. (9)
with
e k := e C E m k , lk := l C I m k , and u k := u C I m k .
Let dk be a solution to (QP k ). Near a solution, the SQP algorithm
updates the primal variables mk by
m k+1 := m k + dk .

(10)

The SQP algorithm is actually a primal-dual method, since it also


generates a sequence of multipliers {k } RnC , which aims at
approximating the optimal multiplier associated with the constraints
of (6). These multipliers are updated by
k+1 = QP
k ,
QP
where k is the triple formed
QP
and (k ) u , associated with the

(11)
QP
(k ) E ,

QP
(k ) l ,

of the multipliers
equality and inequality constraints
of (QP k ). Because of the linearity of the constraints, these multipliers
do not intervene in the tangent QP, but they are useful for testing
optimality and for the globalization of the algorithm (Section 3.5).

The following property of the SQP algorithm deserves being

quoted for the discussion in Section 3.3. When strict complementarity holds at a non-degenerate primaldual solution (
m, 
) to (6) (see
Section 2.3) and when the current iterate (mk , k ) is close enough
to (
m, 
), the active constraints of the tangent QP are those that are
 (see Theorem 13.2 in Bonnans et al. 2003).
active at the solution m
Therefore, the difficult task of determining which constraints are
,
active at the solution to (QP k ) disappears once mk is close to m
since the constraint activity is unchanged from one tangent QP to
the next one.
The technique used to solve the tangent QP is essential for the
efficiency of the SQP approach. In particular, because of the property
mentioned in the previous paragraph, it should take benefit of an a
priori knowledge of the active constraints. In the next two sections,
we concentrate on this topic, which is represented by the right-hand
side blocks in Fig. 1. These sections have a strong algorithmic nature;
the non-interested reader can skip them without loosing the leading
strand of the algorithm. We come back to the globalization of SQP
(the part of the algorithm that is depicted by the bottom block in the
left-hand side of Fig. 1) in Section 3.5.

3.3 Solving the tangential quadratic problem by an

augmented Lagrangian method
Because Hk is positive semi-definite (and usually positive definite),
the tangent QP problem (9) is convex. Such a problem has been
the subject of many algorithmic studies; we mention the following
techniques:
(1) active set (AS),
(2) augmented Lagrangian (AL), and
(3) interior points (IP).
Let us now motivate our choice of developing an AL algorithm
to solve the QP in (9). The AS approach is often used to solve the
QPs in the SQP algorithm. It has the advantage of being well defined, even when the problem is non-convex, and of being able to take
advantage of an a priori knowledge of the active constraints at the solution. However, since this algorithm updates the active set one constraint at a time, it suffers from being rather slow when the active
set is not correctly initialized and when there are many inequality
constraints. For large problems, this can be a serious drawback and
we have discarded this method for that reason. The IP algorithms are
very efficient to solve convex QPs but, presently, they have some
difficulty in taking benefit of a good knowledge of the active constraints as this is often the case after a few iterations of the SQP
algorithm. On the other hand, the inherent ill-conditioning of the
linear systems they generate and the necessity to use here iterative
methods to solve them have appeared to us as deterrent factors.
The AL approach that we have implemented goes back to
Hestenes (1969) and Powell (1969). This is a well-established
methodology, designed to solve nonlinear optimization problems,
although its properties for minimizing a convex QP does not seem
to have been fully explored (see Delbos & Gilbert 2005). It is adapted
to large problems, since it can be implemented in such a way that it
does not need any matrix factorization (Fortin & Glowinski 1983;
Glowinski & Le Tallec 1989). In the context of seismic tomography problems, a version of the AL algorithm has been proposed by
Glowinski & Tran (1993) to solve the tangent QP of the SQP method.
The present contribution takes inspiration from that paper and goes
further by improving the efficiency of its augmented Lagrangian QP
solver. Let us detail the approach.

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The QP (9) is first written in an equivalent form, using an auxiliary
variable y Rn I (we drop the index k for simplicity):

F(d)
min

(d,y)Rn Rn I

(QP )
C E d = e , C I d = y

l y u.
Next, the AL associated with the equality constraints of that problem
is considered. This is the function Lr : Rn Rn I RnC  R defined
at d Rn , y Rn I , and = ( E , I ) RnC by
Lr (d, y, ) = F(d) + E (C E d e ) + r2  C E d e 2
+ I (C I d y) + r2  C I d y 2 .
The scalar r > 0 is known as the augmentation parameter and can
be viewed as a weight scaling the penalization of the constraints of
(QP ) in Lr . A deeper (and more complex) interpretation of these
parameters is given below, allowing the algorithm to change them
regularly.
Algorithm 1. An AL algorithm to solve (9).
data : 0 RnC and r 0 > 0;
begin
for j = 0, 1, 2, . . . do
if (C E d j  e ) & (C I d j  y j ) then return;
by Algorithm 3, find a solution (d j+1 , y j+1 ) to

min(d,y) Lr j (d, y, j )

l y u;
if (12) is solved then
 j+1


j
E := E + r j C E d j+1 e


j+1
I := Ij + r j C I d j+1 y j+1
else
j+1 := j ;
end
choose a new augmentation parameter r j+1 > 0 by
Algorithm 2;
end
end

(12)

(13)

The precise statement of our version of the AL algorithm to solve

(9) or (QP ) can now be given: see Algorithm 1. This method, in
particular the update of the multipliers by (13), has a nice interpretation in terms of the proximal point algorithm in the dual space
(see Rockafellar 1973). It is not essential to give this interpretation
here, but this one is very useful for proving the properties of the
method, including its convergence. In this theory, the augmentation
parameter r in the definition of Lr is viewed as a step-size, damping
the modification of the multipliers in (13). The factor of r = r j in
this formula is indeed a negative subgradient of the dual function at
j+1 . Note that these step-sizes r j can now change at each iteration
of Algorithm 1, while the penalty approach behind the definition
of Lr makes the possibility of such a change less natural. On the
other hand, it can be shown that the algorithm converges if (9) has a
solution and if the sequence {r j } j0 is chosen bounded away from
zero. If, in addition, r j is chosen larger than some positive Lipschitz
constant L (usually unknown, unfortunately), the norm of the equality constraints converges globally linearly to zero: this is inequality
(14) below, to which we will come back. Actually, the larger are the

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augmentation parameters r j , the faster is the convergence. The only

limitation on a large value for r j comes from the ill-conditioning
that such a value induces in the AL and the resulting difficulty or
impossibility to solve (12). This is why a test for updating the multipliers by (13) has been introduced. For ensuring convergence, the
test prevents the multipliers from being updated when (12) is not
correctly solved (a heuristic less restrictive than this test is used by
Delbos 2004).
Algorithm 1 is a dual method, since it essentially monitors the dual
sequence { j } j0 ; the augmentation parameters r j and the primal
variables (d j+1 , y j+1 ) are viewed as auxiliary quantities. The choice
of the initial multiplier 0 depends on the outer SQP iteration index.
When k = 0, Algorithm 1 simply takes 0 = 0, unless an estimate of
the optimal multiplier is provided. When k > 0, Algorithm 1 takes
for 0 the dual solution to the previous QP. A similar strategy is used
for setting r 0 : when k = 0, r 0 is set to an arbitrary value (the effect
of taking r 0 = 1 or r 0 = 104 is tested for the 3-D real data set in Section 4.2) and, when k > 0, r 0 is set to the value of r j at the end of the
previous QP.
Algorithm 1 can be viewed as transforming (9) into a sequence
of bound constrained convex quadratic subproblems of the form
(12). These subproblems have a solution, as soon as (9) has a solution (see Proposition 3.3 in Delbos & Gilbert 2005, for a weaker
condition). Clearly, the major part of the CPU time required by Algorithm 1 is spent in solving the bound constrained subproblems
(12). We describe an algorithm for doing this efficiently in Section
3.4: Algorithm 3. Two facts contribute to the observed success of
this method. First, a bound constrained QP is much easier to solve
than (9), which has general linear constraints (see More & Toraldo
1991, and the references therein). Second, because of its dual and
constraint convergence, the AL algorithm usually identifies the active constraints of (9) in a finite number of iterations. Since often
these active constraints are stable when mk is in some neighborhood
of a solution, the combinatorial aspect of the bound constrained
QPs rapidly decreases in intensity as the convergence progresses
(and usually disappears after very few AL iterations).
We have already made it clear that the choice of the augmentation
parameters r j is crucial for the efficiency of Algorithm 1. Two pitfalls
have to be avoided: a too small value slows down the convergence,
a too large value makes it difficult to find a solution to (12). It is
usually not easy to determine an a priori appropriate value for the
augmentation parameter, so that updating r j in the course of the AL
iterations by observing the behaviour of the algorithm looks better.
In our implementation of Algorithm 1, the update of r j at iteration
j 1 is done by a heuristic close to the one given in Algorithm 2.
This one deserves some explanations.
Algorithm 2. A heuristics for updating r j in Algorithm 1.
data : r j1 , r j , j , des , j1 , and j ;
begin
if (12) is solved then
r j+1 := r j ;
1
if j > des then r j+1 := r j j /des ;
else
2
r j+1 := r j /10;
if (r j < r j1 )&( j > j1 ) then
3
stop [failure of Algorithm]
end
end
end

676

F. Delbos et al.

A too small value of r j can be deduced from the observation of j

:= j+1 / j , where j := (C E d j e , C I d j y j ) is the Euclidean
norm of the equality constraints of (QP ). It is indeed known from
Theorem 4.5 of Delbos & Gilbert (2005) that there is a constant
L > 0 such that, for all j 1, there holds


L
j min 1, j .
(14)
r
This estimate is valid provided (12) is solved exactly. If such is the
case and if des (0, 1) is a given desired convergence rate for the
constraint norm, a value j > des is then an incitement to increase
the augmentation parameter. This update of r j is done in statement
1 of Algorithm 2.
On the other hand, a difficulty to solve (12) can be detected by
the impossibility to satisfy the optimality conditions of that problem
at a given precision in a given number of preconditioned conjugate
gradient (CG) iterations (the algorithm to solve (12), based on CG
iterations, is explained in Section 3.4). The heuristics has also to
decide whether a failure to solve (12) is due to a too large value
of r j , in which case decreasing r j is appropriate (statement 2), or
to an intrinsic excessive ill-conditioning of the problem, in which
case a definitive failure is declared (statement 3). For this, it uses
the sensitivity to r j of an estimate j of the condition number of the
preconditioned Hessian
1/2

Mr j := Pr j

1/2

Q r j Pr j

where Q r j : = H +r j (C E C E + C I C I ) is the Hessian with respect

to d of the criterion Lr j of (12) and P r j (Q r j ) is a preconditioner
for Q rj . The estimate j makes use of the Rayleigh quotients of M rj
computed during the preconditioned CG iterations. According to
Proposition 2.3 of Fortin & Glowinski (1983), the condition number
of Q rj grows asymptotically linearly with r j . The same law does not
hold for M rj , since hopefully r j intervenes in P rj . Nevertheless, it
is important to have an estimate of the condition number of M rj ,
not of Q rj , since it is M rj that governs the performance of the CG
algorithm. In view of these comments, it seems reasonable to say
that, if a preceding decrease of the augmentation parameter, r j <
r j1 , has resulted in an increase of the condition number estimate,
j > j1 , it is likely that a new decrease of r j will not improve the
conditioning of problem (12). In that case and if it is not possible
to solve (12) with r j , a decision to stop is taken in statement 3 of
Algorithm 2; the problem is declared to be too hard to solve.
The actual heuristics for updating r j in our implementation of
the AL algorithm has other safeguards, detailed by Delbos (2004),
but the logic is essentially the one presented in Algorithm 2. Experiments with two different initial values r 0 of the augmentation
parameter are shown in Section 4.2, illustrating the behaviour of the
heuristics adapting r j .
To conclude the description of Algorithm 1, we still need to say
a word on its stopping criterion and to specify the value of the
QP
multiplier k used in (11). There are many ways of showing that
the stopping criterion makes sense. The shortest one here is probably
to observe that a solution (d j+1 , y j+1 ) to (12) satisfying C E d j+1 = e
and C I d j+1 = y j+1 is actually a solution to (QP ); d j+1 is then a
solution to (9). Finally, the optimality conditions of problems (9)
and (QP ) show that one can take

QP 
j+1
k E = E ,

QP 
j+1 
j+1 
,
k l = max 0, I , and QP
k u = max 0, I
is the value of the multiplier on return from Algorithm
where
1 at the kth iteration of the SQP algorithm.
j+1

3.4 Solving the Lagrange problem by the GPASCG

algorithm
Problem (12) is solved in our software by a combination of the
gradient projection (GP) algorithm, the active set (AS) method, and
conjugate gradient (CG) iterations. This GPASCG algorithm is
a classical and efficient method for minimizing a large scale bound
constrained convex quadratic function, see More. & Toraldo (1991),
Friedlander & Martnez (1994), Nocedal & Wright (1999), and the
references therein. We adapt it below to the special structure of
problem (12), in which the variables d and y intervene differently in
the objective and only y is constrained.
A brief description of the three ingredients of the GPASCG
algorithm is necessary for the understanding of the discussion below.
The AS method solves (12) by maintaining fixed a varying choice
of variables yi to their bounds, while minimizing the objective with
respect to the other variables, which are supposed to satisfy the
constraints. Each time the minimization would lead to a violation
of some bounds, a displacement to the boundary of the feasible set
is done and some new variables yi are fixed to their bounds. The
minimization is pursued in this way up to complete minimization
with respect to the remained free variables or (in our case) up to
the realization of a Rosen-like stopping criterion. The GP algorithm
intervenes at this point to inactivate a bunch of erroneously fixed
variables and, possibly, to activate others. This GPAS algorithm
proceeds up to finding a solution to (12). Finally, CG iterations are
used to minimize the objective on the faces of the feasible set that
are activated by the GPAS algorithm.
Minimizing the objective of (12) in (d, y) jointly can be inefficient, in particular when there are many inequality constraints in
the original problem (6), since then the presence of the auxiliary
variable y increases significantly the number of unknowns. Our first
adaptation of the GPASCG algorithm consists in setting up a
minimization in d only, while y is adapted to follow the change in
d. Let us clarify this. Suppose that W I is the working set at
a given stage of the algorithm, that is the set of indices i of the
variables y i that are fixed at one of the bounds li or u i . We note
C := (C E C I ) , V := I \W, W := E W , and denote by C V
(resp. C W ) the matrix obtained from C I (resp. from C) by selecting
its rows with index in V (resp. in W ). For the current working set W ,
the algorithm has to solve (we drop the index j of the AL algorithm)
min Lr (d, y, ).

(d,yV )

with implicit bound constraints on yV [lV , u V ] and with

yW fixed. The optimality conditions of this problem can be

Figure 2. A 2-D view of one iteration of the gradient projection (GP)

algorithm: the ellipses are the level curves of the objective function, the
vector G is its negative gradient at the current iterate, and the shadow box
represents the feasible set.

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written


(H + rC C)d

rC V yV

= g C +

rC V d + r yV = V .

rC E e


rC W
yW ,

(15)

Substituting the value of yV given by (15) into the first equation

gives






+ rC W
H + rC W
yW .
(16)
d = g C W
+ rC E e
CW
W
This is the equation that is solved by the preconditioned CG algorithm. During this process, yV is updated so that (15) is continually
verified. It can be shown that the directions modifying (d, y) are conjugate in Rn Rn I with respect to the Hessian matrix of Lr (, , ),
so that this method can be viewed as a conjugate direction algorithm
that maintains the iterates in the affine subspace defined by equation
(15).
In this process, as soon as yV hits a new bound, the working set W
is enlarged. Then a new CG process is started on the updated equations (16) and (15), from the (d, y) obtained so far. Note that the
updated (15) is verified at the beginning of this new process, since
it is obtained by deleting equations from (15) and since (15) was
verified at the end of the previous CG process. We will see that
(15) is also verified at the end of a GP phase of the algorithm, so
that this equation makes no difficulty to be maintained all along the
GPASCG algorithm, provided this one starts by a GP phase.
The goal of the gradient projection (GP) phase of the algorithm
is to change drastically the working set if this one is felt to be very
different from the set of the active constraints at the solution. As
explained below, the GP phase is actually an adapted version of
a single iteration of the GP algorithm (see Bertsekas 1995 e.g.);
more iterations would be useless in our case, as we will see. Its
property mentioned above, which is observed in practice, is then
also supported by the fact that the GP algorithm identifies the active
constraints at the solution in a finite number of iterations when strict
complementarity holds.
An iteration of the standard GP algorithm forces the decrease
of the objective of (12) along the piecewise linear path obtained
by projecting the steepest descent path on the feasible set (see
in Rn I , the
Fig. 2). If P[l, u]
stands for the projection operator on [l, u]
projected steepest descent path emanating from the current iterate
(d, y) is the mapping


d gd
p : ( > 0) 
,
P[l, u]
(y g y )
where gd (resp. gy ) is the gradient of Lr with respect to d (resp. y).
There holds
g y = I r (C I d y).
In the standard GP algorithm, a local minimum or a step-size ensuring a sufficient decrease of the complex piecewise quadratic function
 Lr ( p(), ) is usually taken. Because of the particularly simple structure of Lr in y, we prefer maintaining dfixed and minimizing
completely


 Lr d, P[l, u]
(y g y ), .
This is our second adaptation of the standard GPASCG algorithm.
It has the following interesting property. Because the Hessian of Lr
with respect to y is a multiple of the identity matrix, the new y is the
u]
of the unconstrained minimizer of Lr (d, , ):
projection on [l,


I
.
(17)
y := P[l, u]
d
+
C
I

r

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Observe that, if W is now set to {i : yi = li or u i }, equation (15) is

satisfied, as claimed above.
Algorithm 3 summarizes our version of the GPASCG algorithm. The use of Rosens stopping test for the CG iterations and
other algorithmic details are not mentioned. See Delbos (2004) for
further information on the implementation.
Algorithm 3. The GPASCG algorithm to solve (12).
data : r , , d := 0;
begin
GP := true;
while optimality of (12) is not satisfied do
if GP then
compute y by (17);
update W := {i : yi = li or u i } and V := I \W ;
GP := false;
else
while lV < yV < u V do
use CG iterations to solve (16) in d, while
updating yV to satisfy (15);
end
update the index sets W and V ;
if W has not been enlarged then GP := true;
end
end
end

3.5 Globalization by line-search

It is now time to remember that the constrained minimization problem (6) we want to solve is nonlinear and that, just as in unconstrained optimization, the update of the model mk by (10), where dk
is the computed solution to the quadratic problem (QP k ), is unlikely
to yield convergence if the initial estimate m 0 is not close enough to
a solution. For forcing convergence from a remote starting model,
we follow the standard globalization technique presented in Chapter
15 of Bonnans et al. (2003), which uses an exact penalty merit function. Using a filter method would have been an alternative, but we
did not try it. We have also implemented a line-search technique,
since the combination of trust regions with a QP approximately
solved by an augmented Lagrangian algorithm is a technique that
does not seem to have been explored. Due to its usefulness to solve
the unconstrained tomography problem, we plan to investigate this
possibility in a future research.
We use the exact penalty function
: Rn R defined by

(m) = f (m) +  (m),

where
 (m) =

i | Ci m ei |

iE

i max(li Ci m, 0, Ci m u i ),

iI

in which the i s are penalty weights. Exactness of the penalization

means that a solution to (6) is also a minimizer of
. To get that
important property, the i s need to be large enough, although finite.
It is usual to update them at some iteration, so that they always satisfy


 

i  QP
for i E
k i + ,

QP    

QP   
(18)
, 
 + , for i I,
i max 
k

l i

u i

678

F. Delbos et al.
QP

where k is defined after (11) and > 0 is a small security threshold.

In our case, the convexity of the QP (9) defining dk and the inequalities (18) ensure that
decreases along dk . A line-search
along this direction is then possible. A step-size k > 0, typically
determined by backtracking, can be found such that for some small
constant (0, 1/2):

(m k + k dk )
(m k ) + k k ,

(19)

where  k := f (mk ) dk  (mk ) can be shown to be negative

when mk is not stationary. Now the model and the multiplier are
updated by

m k+1 = m k + k dk


(20)
k+1 = k + k QP
k k .
Algorithm 4 summarizes the overall algorithm to solve
problem (6).
Algorithm 4. The overall algorithm to solve (6).
data : m 0 , 0 ;
begin
for k = 0, 1, 2, . . . do
if optimality of (6) holds at (m k , k ) then stop;
compute (dk , QP
k ) a primal-dual solution to (12) by
Algorithm 1;
update to satisfy (18);
determine a step-size k > 0 by backtracking,
in order to satisfy (19);
update (m k+1 , k+1 ) by (20);
end
end

4 A P P L I C AT I O N S O F T H E
CONSTRAINED REFLECTION
T O M O G R A P H Y O N R E A L D AT A S E T S
4.1 2-D PP/PS data set

inverse problem does not allow the joint inversion of the velocities,
interfaces and anisotropy parameters ( parameter is particularly undetermined, Stopin 2001). This applied methodology is obviously
not optimal. Indeed, the manual tuning of the anisotropy parameters
requires a lot of time: an important numbers of anisotropic inversion with different pairs (, ) have to be performed before getting
a satisfying result. Secondly, it is very hard to make this method accurate: we note a discrepancy of 150 meters for the reflector depth
h5 compared to the depth given by the well logs data. Finally, it
turned out impossible to determine the anisotropy parameter so
that both the reflector depths of h4 and h5 given by the well logs are
reasonably matched.
The solution we propose here is to compute a model using our
constrained inversion method in order to fit the reflector depths given
by the well, while considering (, ) as additional unknowns to determine. This consists in the inversion of P- and S-velocity variations
(described by functions v(x, y, z)), of the geometries of interfaces
h4 and h5 and of the two anisotropy parameters, i.e. inversion of
1024 parameters from 32 468 picked traveltimes (associated with
reflections of PP waves and PS waves on h4 and h5). The estimated
picking errors are the following: 5 ms for the PP traveltimes associated with reflections on h4 and h5 and 8 ms for the PS traveltimes
associated with reflections on h5 (see Figs 3 and 4). Those uncertainties are taken into account in the cost function (The Euclidian
norm is weighted by the inverse of the square of the uncertainties).
In Tables 1 and 2, we have respectively summed up the results
of the final models obtained with the unconstrained and constrained
inversions. The final model (Fig. 5 right) of the constrained inversion
matches the traveltime data with the same accuracy (Fig. 6) than the
result obtained by the unconstrained inversion (Fig. 5 left), and it
strictly verifies the reflector depths given by the well logs. For this
test, we have introduced equality constraints to match the reflector
depths at well location, however inequality constraints could have
been used to take into account the uncertainty on the well data. This
solution has been obtained after only nine nonlinear SQP iterations.
We see that, the introduction of constraints at wells for the two
reflectors h4 and h5 reduces the underdetermination and allows for
a joint inversion of velocities, interfaces, and anisotropy parameters. The resulting model matches the traveltime (see Fig. 6) and
well data. The values of its anisotropy parameters are very different
from those obtained with the unconstrained inversion (Tables 1 and

In this section, we present an application of constrained reflection

tomography to one 2-D line of a 3-D 4C OBC (Ocean Bottom Cable) survey with PP and PS data from bp.1 Broto et al. (2003) have
already interpreted and studied this data set using an unconstrained
inversion method. The velocity model is described by four velocity
layers and five interfaces (cf Fig. 5 left). The isotropic assumption
was satisfying until the last layer (layer which contains the last two
interfaces h4 and h5). By applying the anisotropic inversion methodology of Stopin (2001) on the last layer, they obtained a model that
fits the traveltimes better than any of the previous isotropic models
and that, in addition, has more reliable velocity variations. Two parameters (, ) describe the velocity anisotropy: can be seen as
a measure of the an-ellipticity, whereas controls the near vertical
velocity propagation of the P waves (Thomsen 1986; Stopin 2001).
The value of the anisotropy parameter has been obtained by
a trial and error approach in order to match approximately the h5
depth given by well logs. Actually, the underdetermination of the
1 4C refers to the 4 components, 1 hydrophone and 3 orthogonal geophones,
that allow both compressional (P) and shear wave data (S) to be recorded.

Figure 3. Picked traveltimes associated with reflections on h4 versus offset

and receiver location.

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Figure 4. Picked traveltimes associated with reflections on h5 versus offset and receiver location. (PP data: left, PS data: right)

Figure 5. The velocity models (Vp and Vs) on the left hand side are computed with the unconstrained inversion method; those on the right hand side are
computed with the constrained inversion method. The white crosses locate reflector depths measured from the deviated well logs, which are imposed as
constraints. These additional constraints allows the software to determine the anisotropy parameters in the same run.


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F. Delbos et al.

Figure 6. Distributions of traveltime misfits associated with all reflectors for model obtained with unconstrained inversion (left) and for model obtained with
constrained inversion (right).
Table 1. Inversion results of the unconstrained inversion.
Layer 4

RMS of
traveltime
misfits

Depth mismatch
at well location

h4-PP
h5-PP
h5-PS

3.6 ms
4.6 ms
9.8 ms

190 m
150 m

6.2 per cent

2 per cent

2) and we observe velocity variations that were not present in the

model obtained by reflection tomography. The depth migration of
the data with the obtained velocity models would have been useful
to compare the two results. In this paper, we limit ourselves to the
application of the constrained optimization method.

4.2 3-D PP data set

Table 2. Inversion results of the constrained inversion.
Layer 4

RMS of
traveltime
misfits

Depth mismatch
at well location

h4-PP
h5-PP
h5-PS

6.3 ms
3.9 ms
8.1 ms

0m
0m

8.2 per cent

15.9 per cent

During the European KIMASI project, reflection tomography was

applied on a 3D North Sea data set from bp (Ehinger et al. 2001). A
P-velocity model was obtained thanks to a top-down layer-stripping
approach where lateral and vertical velocity variations within
Tertiary, Paleocene and Cretaceous units (this last layer being divided in two velocity layers) have been determined sequentially. A
strong velocity underdetermination in the upper Tertiary layer was
detected during the inversion process due to the large layer thickness

Figure 7. Velocity model (slices along x (left) and along y directions at one of the five well locations) obtained with a layer-stripping approach using the
unconstrained reflection tomography. This model corresponds to the v(x, y, z) velocity parameterization (see Table 3). The RMS value of the traveltime misfits
is 6.1ms.

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Table 3. The different velocity parametrizations tested for the inversion of the Tertiary layer (in the layer-stripping approach applied by Ehinger
et al. 2001) for the unconstrained inversion and in a global constrained approach (last line of the table).
Tertiary velocity parameterization
Velocity type
V 0 (x, y)
V 0 (x, y) + 0.2z
V (x, y, z) without constraints
V (x, y, z) with constraints

RMS of traveltime misfits

Vertical velocity gradient

Layer 1

Top of Paleocene

fixed to 0/s
fixed to 0.2/s
inverted 0.01/s
inverted 0.18/s

4.3 ms
3.4 ms
4.1 ms
3.9 ms

4.2 ms
4.6 ms
4.4 ms
4.4 ms

Mean depth mismatch at the

five well locations
96 m
300 m
100 m
0m

Table 4. Description of the equality and inequality constraints introduced in the inversion. We succeed to obtain a model that matches the data
and the 2300 constraints.
Constraints
Model obtained with the
Model obtained with the
UNCONSTRAINED inversion
CONSTRAINED inversion
Mean depth
mismatch at the 5
well locations

tpal
tchalk
bchalk

96 m
132 m
140 m

0m
0m
0m

0.1 < k < 0.3/s

k = 0/s

k 0.18/s

2.5 < vpal < 4 km s1

3.5 < vichalk < 5.7 km s1
4.2 < vchalk < 5.8 km s1

ok
ok
ok

ok
ok
ok

Vertical velocity
gradient in Tertiary
Velocity range

(2.5 km) and to the very poor ray aperture (despite the introduction of the intermediary reflector named layer 1). Several velocity
parametrizations (Table 3) were inverted and led to solution models
that match traveltime data with the same accuracy. These different
tests are time consuming (each test is a whole inversion) and the
reflector depths given by well data are not well retrieved, this information being not explicitly introduced in the inversion process. One
of the resulting model is presented in Fig. 7.
To obtain a model consistent with well data, we propose to apply our developed constrained tomography inversion. The interface
depths are constrained at five well locations and we constrain the
range of variations of the vertical velocity gradient in the Tertiary
layer thanks to well measurements (Table 4). As for the application
on the first data set, the equality constraints on interface depths at
well locations could have been relaxed by inequality constraints to
model the uncertainty on the well data. A global inversion of the
four velocity layers is preferred to the layer-stripping approach: it
avoids bad data fitting for deeper layers due to errors in shallow
layers. The simultaneous inversion of all the layers is guided by
constraints on layer thicknesses to avoid any non-physical interface
intersections: Fig. 8 shows an example of non-admissible model obtained by global unconstrained inversion (it presents non-physical
interface intersections that make layers vanish in the pointed region
and thus a large number of rays are discarded).
The experiment then consists in a global inversion of 127 569
traveltimes (with offsets ranging from 0 to 1.3 km for layer 1 interface and from 0 to 3 km for the other interfaces and with a picking
error of 5 ms for the shallowest reflectors and of 7 ms for ichalk and
bchalk interfaces) for 5960 unknowns describing 4 velocity layers
and 5 interfaces, subject to 2300 constraints (Table 4). The constraints on the velocity variations (resp. on the layer thicknesses)
are applied on a grid of 10 10 10 (resp. 20 20 points). The
results are presented in Fig. 10: the obtained model matches the data
with the same accuracy (Fig. 9) as the models obtained by Ehinger
et al. (2001) and verifies all the introduced constraints (see Tables

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Figure 8. Interfaces (slice along x) obtained with a global inversion using the unconstrained reflection tomography. Difficulties are encountered
during the GN iterations. Some iterates lead to non-admissible models in
which the forward operator is not defined. For instance, the model found at
iteration 7 presents non-physical intersections that make layers vanish in the
pointed region and thus a large number of rays are discarded. The resulting
discontinuity of the cost function leads to convergence troubles of the GN
method.

4 and 3). The model obtained by unconstrained inversion (Fig. 7)

and the model obtained by constrained inversion (Fig. 10) are very
different: by the geometry of the interfaces and by the velocity variations within the layers. The introduction of constraints leads to local
velocity variations at well locations that may perhaps be attenuated
by a stronger regularization. As for the first application, we limit
ourselves to the validation of the described optimization algorithm
to handle numerous equality and inequality constraints. A depth
migration step should help to analyse the obtained result.

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Figure 9. Distributions of traveltime misfits associated with all reflectors for model obtained with unconstrained inversion (left) and for model obtained with
constrained inversion (right).

Figure 10. Velocity model (slices along x (left) and along y directions at one of the five well locations) obtained with a global inversion using the constrained
reflection tomography (constraints described in Table 4). The RMS value of the traveltime misfits is 6.5 ms.

The total number of conjugate gradient iterations for each GN

step (the total number of conjugate gradient iterations takes into
account all the iterations of augmented Lagrangian method) is less
than 104 (less than twice the number of unknowns), which looks
like a very good result for a problem with 2300 constraints. In this
experiment, only six nonlinear SQP iterations are necessary to reach
convergence.
The automatic adaptation of the augmentation parameter r j (see
Algorithm 2 in Section 3.3) is illustrated in Fig. 11. The initial value
r 0 can be chosen in a large interval. Algorithm 2 modifies r j in
order to obtain a good convergence rate of the multipliers in the AL
algorithm (Algorithm 1), without deterioration of the conditioning
of the bound constrained problem (12).

5 C O N C LU S I O N
Reflection tomography often requires the introduction of additional
a priori information on the model in order to reduce the underdetermination of the inverse problem. A nonlinear optimization method
that can deal with inequality linear constraints on the model has
been developed. This dedicated method has proved its efficiency
on tomographic inversions of many synthetic examples with various types of constraints and of some real examples, including those
related to the two real data sets presented in this paper. The number
of GN iterations has the same order of magnitude than the number
of GN iterations necessary in the unconstrained case (10 iterations). This and the fact that solving the forward problem is time

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Constrained optimization in seismic reflection tomography

683

Figure 11. Automatic adaptation of augmentation parameter r j : two experiments are performed, with an initial augmentation parameter either set to r 0 = 1
(left) or to r 0 = 104 (right). Algorithm 2 adapts r j during the AL iterations. It may decrease its value to improve the conditioning of problem (12) or increase
r j to speed up the convergence of the constraint norm to zero (the desired convergence rate des is fixed to 103 ).

consuming were preponderant factors in favour of a SQP approach,

instead of an interior point method. The chosen combination of the
augmented Lagrangian relaxation and the active set method for solving the quadratic optimization subproblems is efficient even for a
large number of constraints. An extension of the method to nonlinear
constraints is currently tested. At last, the algorithm developed for
updating the augmentation parameter discharges the user of the software from such a technical concern and allows an adequate choice
of its value in terms of the convergence rate of the augmented Lagrangian algorithm.

AC K N OW L E D G M E N T S
We would like to thank Guy Chavent, Patrik Lailly and Roland
Masson for helpful discussions and advises at various stages of the
elaboration of this work. We acknowledge Karine Broto and Anne
Jardin for their precious help for the applications on real data sets.
The data sets studied in this paper were kindly provided by bp. We
would like to thank the editor, Andrew Curtis, the referee Gilles
Lambare and the anonymous second referee for their constructive
remarks on this paper.

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