Parameter estimation of an acoustic
differential equation by optimal control
Usually numerical methods of inversion and
optimization begin with a discretized version of the
Differential equations are often
easier to manipulate directly, so that discretization
can be delayed until last. Here is an example
of optimally perturbing acoustic parameters
to fit recorded acoustic waves. This approach
is seen most often by researchers in optimal control,
such as J.L. Lions 
who also publishes often in the field of geophysics.
This example derives from an explanation I received from
Patrick Lailly in 1984
at the Institut Français du Petrole.
In 1986, I expanded on his example in my lab notebook.
After all these years, I still find it a useful example
of multi-dimensional seismic inversion, so I've finally typed it up.
In a separate paper , I show a detailed
application with the one-dimensional wave-equation.
The methods in this particular paper should be much easier
to generalize to other equations and model parameters.
My preferred methods of optimization (Gauss-Newton)
require linearizing simulation equations and calculating
the adjoint of those linearized equations. Here is how
you can do most of that work analytically.
In the course of this derivation, I will derive what
is called reverse-time migration, with an image directly related
to perturbations of physical parameters.
Let us consider the experiment of a single exploration
seismic field gather with a seismic source places at a surface
and with receivers placed at points
. Each trace is recorded for a finite time
Assume that land experiments record the derivative of pressure with depth.
(Marine experiments record the pressure directly, which is simpler.)
be the acoustic velocity, and
Describe the pressure field
with the following two-dimensional
where the source term
contains body forces.
Use the free surface and causal time boundary conditions:
where is the vertical dimension of .
The surface boundary condition on is not sufficient and
must be extended before is well-determined.
Choose a semi-spherical surface
that cannot be reached by waves
traveling at the maximum velocity for the maximum recorded time :
The radius should be large enough so that no wave reaching this
can be recorded during the time
We can arbitrarily extend the free surface around this closed boundary
without affecting the modeled pressures within the recorded time:
The acoustic differential equation (1)
can also be written in the following variational integral form,
also called the weak form:
for all functions
This integral is true for all (whose second derivatives
are square integrable) if and only if the original equation
(1) also holds.
Conveniently, finite elements  also uses the weak form.
Zero-value boundary conditions (3) and (4)
are implied by the weak form, and are called natural boundary conditions.
Let be the pressure field recorded within the volume
at known receiver positions
for a source located at
Define an objective function to match a recorded pressure field
with a modeled field
in a least-squares sense.
The objective function is a functional of the
three unknown functions of physical parameters , , and .
is a smooth weighting function that we will find convenient.
The optimization problem can be stated as follows: find the
model parameters , , and so that the field
defined by (7), (6),
and (3) minimize the objective function (8).
The objective function has a least-squares form.
The weighting factor compensates for
geometric spreading and absorption to give approximately equal
weight to all parts of the modeled data.
To optimize the model parameters model parameters
, , and according to equations
(7), (6), and (8), one must
discover how a perturbation of the model parameters
will affect the objective function (8).
Let us abbreviate the variational form (7)
of our differential equations in the following form:
is also understood to be a implicit function of
as determined by the acoustic equations.
The exact form of is unimportant for the remainder of this derivation,
except that it must be expressible as a linear functional operator on .
is a nonlinear operator on
By a different choice of model parameters ( , , and ),
one could also linearize with respect to the model , but this
is not necessary.
Similarly, we can rewrite our objective function (8) in
a more general form as
First, let us use the variational equation (9) to relate
perturbations of the model
to perturbations of the pressure
The gradient product can be understood as meaning
and both and are functions of .
In this form the linear operators and
operate on perturbed quantities. Let us replace this form
by one that uses the adjoint operators on .
If and are functions, the adjoint of a linear operator
is defined by
. The brackets
indicate the scalar products in the domains of and .
So we can rewrite our perturbation equation (12) as follows:
If contains only linear differential operators,
then we can obtain their adjoints by integrating by parts, as I will
show in a later section.
This makes explicit the implicit dependence of on .
Next let us relate perturbations in the objective function
to perturbations in the pressure and modeling parameters
used in .
Now for the clever part. The perturbed integral (17)
must be true for all . Choose then so that
(17) equals the first term of the gradient (19)
for all :
For this equation to be true for all , must satisfy
This adjoint system of equations is very similar to the original
set of equations for , but with a new source term -- the
difference between the modeled and the actual data.
If satisfies this equation, then we can combine the variational
equation (17) and the perturbation (19)
of the objective function to remove intermediate dependence on .
We can express this perturbation (22) as
a gradient of the objective function with respect
to model parameters
at a specific location .
The first term of this gradient (23) is the most important
because it provides information about at all positions.
The second term is non-zero only at points
data are recorded.
The simplest possible steepest descent optimization would perturb
a reference value of by a scaled version of its gradient.
One would also sum the perturbations over all source positions
to for a single perturbation of the model.
After one perturbation, the new reference value of
could be used to calculate a new reference wavefield and a new
Let us now leave the general formulation and see
how to handle the acoustic equation in particular.
Let us first perturb the objective function (8)
as the general perturbed form (19):
Next we integrate by parts over .
The boundary term disappears because we require
at the boundary, as does
The next step is to place the variational acoustic equation
(7) as in our general perturbed form (17).
We can integrate the time derivatives in (27)
by parts twice over time.
The divergence and gradient terms of (27)
can be rewritten with the following chain rule and divergence theorem:
is a scalar and is a vector.
is the unit normal
vector to the boundary
of the volume .
So here is the rewritten form of the perturbed variational form
The boundary terms at disappear because must
observe the boundary conditions of . The
the derivative in the direction of the normal to the surface.
Such terms do not disappear. Other integrals that evaluate
on the surface
Now set the right side of the expanded (29)
equal to the first term of of the perturbation
of the objective function (26) for all perturbations
of the wavefield. Then
in the interior. The boundary terms on
must vanish so
These adjoint equations (30) and (31)
require an extrapolation on that is very similar to that on ,
except that extrapolation must now proceed backards in time from .
A single application of this extrapolation is called reverse-time
Errors between the modeled and recorded data act as sources through
the extrapolation at the locations of the receivers.
The first term of in equation (25)
now equals the right side of (29). The boundary
conditions on make the surface
To perturb we take the dot product of the extrapolated
with a filtered version of the previous guess of the wavefield
The perturbation of requires the
application of a differential operator on . The second term of
(from equation ) applies only
at the receiver points and acts as an amplification of the recorded
trace. The perturbation of is simply equal to the adjoint
wavefield evaluated at the source position.
To start the iterations set
and the parameters
and to reasonable guesses. The first perturbation
will attempt to invert the primary reflections and is equivalent to a
so-called reverse-time migration. The second iteration ostensibly would
invert second-order scattering, but the poor choice of
background velocities would make it difficult to predict the second-order
form from the first-order scattering. This inversion, like all inverse
scattering methods, can only invert high-frequency parameters changes
that are likely to create reflections. Smoother changes in
and must be known beforehand.
William S. Harlan.
Separation of signal and noise applied to vertical seismic profiles.
Geophysics, 53(7):932-946, 1988.
Thomas J. R. Hughes.
The Finite Element Method: Linear Static and Dynamic Finite
Dover Publications Inc., 2000.
Optimal Control of Systems Governed by Partial Differential