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## Differential K-operators

The DKO is determined as a solution of the equation

with boundary conditions that supply the uniqueness of the solution.
We have two problems to solve:
I.
The choice of the operator L.
II.
The choice of the boundary conditions.
I.
Let us take the classical eikonal equation

This is a characteristic equation for any hyperbolic equation of the type:
 (80)
at .The classical wave-equation

is the simplest hyperbolic equation of this type. But is it also the best choice?

1. Absence of wave dispersion. Let the coefficient of the equation (80) be constant. After substitution

where and . The equation is reduced to

where .

Looking for the solution in the form

we obtain the dispersion equation

Since in nondispersive media ,only in the case when or

2. Amplitude equivalence. This term means that while discontinuities propagate, amplitudes A change along rays s following the transport equation (see Chapter 4)

which is valid for P and S waves. Using the standard technique of ray theory, it is easy to show that in the case of amplitude equivalence

c=0

and

For homogeneous media the wave equation is the only one that is nondispersive and amplitude-equivalent simultaneously.

3. Nonscattering vertical propagation. If in the amplitude-equivalent equation
 (81)
the following is valid: v=v(z), and , then plane waves propagate in z-direction without scattering. This is easy to show if one inserts

into equation (81).

II. Boundary conditions. There are few types of conditions that can be used for the wave-field continuation with the help of equation (80).

1. Mixed problem with boundary and initial conditions

where t+ = 0 (for forward continuation), t- = T (for reverse continuation), T > tmax, (where tmax is the duration of the seismic record).

This problem has a unique and stable solution (in a restricted domain). It is restricted at (for direct continuation) and (for inverse continuation) if h>0 and (or ).

Representations of the solution:

Finite-difference technique which can be applied for any inhomogeneous media.

Kirchhoff's type integral:

 (82)

where , I+ =(0, t), I(-) =(T-t, T), -Green's function which is the solution of the equation
 (83)
with the zero initial condition at and .

For inhomogeneous media Green's function usually is unknown (although one can calculate ray zero-approximation of ). In the case of wave equation Green's function for homogeneous medium is
 (84)
where .It gives
 (85)
where .

At big values of R the first term in square brackets can be neglected:
 (86)

These expressions are valid only in 3D case. In 2D case Green's function is
 (87)

Inserting this into the equation (82) we obtain for the reverse continuation
 (88)
where .

The operator
 (89)
is by definition the operator of integration of the order 1/2. It is easy to notice that the operator

is the same operator but acting in reverse time. We can denote this constriction as . Now we may also introduce where signs + and - show what time (usual or reverse) is used.

Let the field u0 (x0, t) contain a wave at t=t0. The field u(-) will contain the wave at t = t0 - R/v. Then the main contribution in the integral (88) is made by the values of the field u0 at , that is, .

Taking into account all these considerations and neglecting the term with fast attenuating at , we derive from equation (88):
 (90)

Spectral representation in ()-domain (the wave equation, homogeneous media):

 (91)
where

2. The Cauchy problem with respect to z in the domain for the equation

 (92)
with the conditions,

yields in homogeneous medium to the spectral form

This gives a stable (regular) solution, but the finite-difference solution of equation (92) is stable only in the restricted domain and unstable at .We have the same problem with respect to the Cauchy problem for the wave equation with the conditions:

The natural way to stabilize Cauchy's problem is to cut off all frequencies at k2z < 0.

3. Reconstruction-type wave-field extrapolation. In this case we pose the conditions

and
 (93)
where condition (93) is connected with the source. The solution of this problem gives mixed-type continuation which is nonregular (at ) in both the (r,t) and the domain. The simplest regularization is the same as above.

There is only one reverse type regular continuation into homogeneous medium which satisfies the condition:

 u|z=0 =u0 (94)

Of course we have the whole family of continuation methods if we use instead of equation (94) the condition:

for any .

In inhomogeneous media different conditions produce different solutions. Figure  shows an example of different solutions: (a) is the reversed continuation of a single event at for mixed conditions, and (b) for the Cauchy problem.

In Figure a we use reverse time -t instead of t (we can do it with accordance with relation (51) in Chapter 5) because it promotes better understanding of reflection-refraction pattern at the interface z=d. In reverse time we have the usual process of wave propagation. Dashed lines show some ghost waves.

Next: One-way propagation (PDKO) Up: 8: CLASSIFICATION OF K-OPERATORS Previous: 8: CLASSIFICATION OF K-OPERATORS
Stanford Exploration Project
1/13/1998