with boundary conditions that supply the uniqueness of the solution.
This is a characteristic equation for any hyperbolic equation of the type:
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
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
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
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:
where , I+ =(0, t), I(-) =(T-t, T), -Green's function which is the solution of the equation
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
At big values of R the first term in square brackets can be neglected:
These expressions are valid only in 3D case. In 2D case Green's function is
Inserting this into the equation (82) we obtain for the reverse continuation
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):
Spectral representation in ()-domain (the wave equation, homogeneous media):
2. The Cauchy problem with respect to z in the domain for the equation
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
There is only one reverse type regular continuation into homogeneous medium which satisfies 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.