Differential K-operators

The DKO is determined as a solution of the equation

$${\bf L} u = \rho {\partial^{2} u \over \partial t^{2}}$$

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

$$\vert\nabla \tau \vert^2 = {1\over{v^2({\bf r})}}.$$

This is a characteristic equation for any hyperbolic equation of the type:  

$$a \Delta u + \sum {b_i} {\partial u \over \partial x_{i}}+c{... ...artial t} + h u = \rho {\partial ^{2} u \over \partial t^{2}} ,$$ (80)

at $v=\sqrt{\frac{a}{\rho}}$.The classical wave-equation

$$\Delta u = \frac{1}{v^2({\bf r})} \frac{\partial^2 {u}}{\partial {t^2}}$$

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

$$u = {\overline u}\exp \left(-{1\over 2} \sum {\beta_i x_i} + {1\over 2}\alpha t \right)$$

where $\beta_i =b_i/a$ and $\alpha =c/a$. The equation is reduced to

$$\triangle{\overline u} + {\overline h}{\overline u} - {1\over v^2}{\partial^2 {\overline u}\over\partial t^2} = 0$$

where ${\overline h} =h/a + v^2\alpha^2/4 + \sum {\beta_i^2/4}$.

Looking for the solution in the form

$$\exp[i (\omega t - \sum_{i=1}^{3}k_i x_i)],$$

we obtain the dispersion equation

$$\omega_{1,2} =\pm v\sqrt{\sum k_i^2 + {\overline h^2}}$$

Since in nondispersive media $\sum k_i^2 = v^2 \omega^2$,only in the case when ${\overline h} =0$ or

$$h= -{ (c v^{2} + \sum {b_{i}^{2}\over a}) \over 4}.$$

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

$$2 {dA\over{ds}} + A\left[ v \Delta \tau + {d\over{ds}} \ln(\rho v^2) \right] = 0$$

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

$$b_{i}={\partial a \over \partial x_{i}}.$$

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  

$$\sum {\partial \over \partial x_{i}} (\rho v^{2} {\partial u... ... \partial x_{i}}) =\rho {\partial^{2} u \over \partial t^{2}},$$ (81)

the following is valid: v=v(z), $\rho = \rho (z)$ and $v(z) \rho (z) = const.$, then plane waves propagate in z-direction without scattering. This is easy to show if one inserts

$$u = A f \left( t - \int_{0}^{z} {d \xi \over v(\xi)} \right)$$

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

$$\begin{array} {ll} u\vert _{z=0} = u_{0}{\:}{\:}(or {\:}{\:}... ... {\partial u \over \partial t}\vert _{t=t^{\pm}} = 0\end{array}$$

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

This problem has a unique and stable solution (in a restricted domain). It is restricted at $t \rightarrow + \infty$ (for direct continuation) and $t \rightarrow - \infty$ (for inverse continuation) if h>0 and $+c \geq 0$ (or $-c \geq 0$).

Representations of the solution:

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

$\bullet$ Kirchhoff's type integral:

 

$$u^{(\pm)}({\bf r}, t)= \int\limits_{I_\pm}d\tau\int\limits_... ...(\pm)}({\bf r}, {\bf r}_0, t-\tau)\over \partial z} d{\bf r}_0$$ (82)

where $\sum : z=0$, I₊ =(0, t), I_(-) =(T-t, T), $g^{(\pm)}({\bf r}, {\bf r}_0, t)$-Green's function which is the solution of the equation  

$${\bf L}u - \rho {\partial^2 u\over\partial t^2} =\delta ({\bf r}-{\bf r}_0)\delta (t)$$ (83)

with the zero initial condition at $t=t^\pm$ and $u\vert_{z=0} =0$.

For inhomogeneous media Green's function usually is unknown (although one can calculate ray zero-approximation of $g^{(\pm)}({\bf r}, {\bf r}_0, t)$). In the case of wave equation Green's function for homogeneous medium is  

$$g^{(\pm)}({\bf r}, {\bf r}_0, t) = {\delta\left( t\mp {R_+\... ... R_+} - {\delta\left( t\mp {R_-\over v}\right) \over 4\pi R_-}$$ (84)

where $R_\pm =\sqrt{(x-x_0)^2 + (y-y_0)^2 + (z\mp z_0)^2}$.It gives  

$$\begin{array} {cl} u^{(\pm)}({\bf r}, t) = & {1\over 2\pi}\... ...bf D} u({\bf r}_0, t\mp{R\over v}) \Big ] d{\bf r}_0\end{array}$$ (85)

where $R=\mid {\bf r} - {\bf r_0}\mid, \cos \theta =z/R$.

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

$$u^\pm ({\bf r}, t)\simeq {1\over 2\pi} \int\limits_\infty^{... ...s\theta\over vR}{\bf D}u({\bf r}_0, t\mp{R\over v}) d{\bf r}_0$$ (86)

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

$$g^+({\bf r}, {\bf r}_0, t) = {1\over 2\pi}\left[{H(t-{R_+\ov... ... {H(t-{R_-\over v})\over\sqrt{t^2 - ({R_-\over v})^2}} \right]$$ (87)

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

$$\begin{array} {cl} U^{(-)}({\bf r}, t) = & {1\over \pi v} ... ...r v})\over \sqrt{\tau-t+{2R\over v}} } \Big ]dx_0\end{array}$$ (88)

where $\dot{u} ={\bf D}u$.

The operator  

$$I^{1\over 2}f(t) = \int\limits_{-\infty}^t {H(t-\tau) \over\sqrt{\pi(\tau - t)}} f(t)\, d\tau$$ (89)

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

$$\int\limits_t^\infty {H(\tau-t) \over\sqrt{\pi(\tau - t)}} f(\tau)\, d\tau$$

is the same operator but acting in reverse time. We can denote this constriction as ${\bf I}_{(-)}^{1/2}$. Now we may also introduce ${\bf D}_{(\pm)}^{1/2} = {\bf D}_{(\pm)} {\bf I}_{(\pm)}^{1/2}$ 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 $\tau +R/v \simeq t_0$, that is, $\tau -t + 2R/v \simeq t_0 - R/v - t_0 + R/v + 2R/v \simeq 2R/v$.

Taking into account all these considerations and neglecting the term with fast attenuating at $R\rightarrow\infty$, we derive from equation (88):  

$$U^{(-)}({\bf r}, t) \simeq {1\over \sqrt{2\pi}}\int {\cos\t... ...vR}}{\bf D}_{(-)}^{1/2} u_0\left(x_0, t+{R\over v}\right) dx_0$$ (90)

$\bullet$ Spectral representation in ($\omega, k$)-domain (the wave equation, homogeneous media):

 

$$u^{(-)}({\bf r},t) = {\bf F}^{-1}_{x,y,t} \left\{ {\tilde {u} _{0} e ^{ik_{z}^{\pm} z}} \right\}$$ (91)

where

$$k_{z}^{\pm} = \left\{ \begin{array} {ll} \mp \vert k_{z}\ver... ...\geq 0 \\ i \vert k_{z}\vert & k_{z} ^{2} < 0\end{array}\right.$$

$$k^2_z={\omega^2 \over v^2}-k^2_x-k^2_y$$

2. The Cauchy problem with respect to z in the $(\omega,{\bf k})$ domain for the equation

 

$${d^{2} \tilde{u} \over dz^{2}} + k_{z}^{2} \tilde{u} = 0$$ (92)

with the conditions,

$$\begin{array} {ll} \tilde{u} (\omega ,k_{x}, k_{y}; z=0) = ... ...\over \partial z})_{z=0}=ik_{z}^{\pm } \tilde{u}_{0}\end{array}$$

yields in homogeneous medium to the spectral form

$$\tilde{u}^{(\pm )} = \tilde{u}_{0}e^{ik_{z}^{\pm}z} .$$

This gives a stable (regular) solution, but the finite-difference solution of equation (92) is stable only in the restricted domain $z <z^{\ast }$ and unstable at $z \rightarrow \infty$.We have the same problem with respect to the Cauchy problem for the wave equation with the conditions:

$$\begin{array} {ll} u\vert _{z=0} = u_{0} \\ \\ {\partial u ... ... {\bf F}^{-1} \left\{ {ik_{z}^{\pm } u_{0}} \right\}\end{array}$$

The natural way to stabilize Cauchy's problem is to cut off all frequencies at k²_z < 0.

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

$$u\vert _{z=0} = \tilde u_{0}$$

and  

$${\partial u \over \partial z} \vert _{z=0} = \sigma _{0}$$ (93)

where condition (93) is connected with the source. The solution of this problem gives mixed-type continuation which is nonregular (at $z \rightarrow \infty$) in both the (r,t) and the $(\omega,{\bf k})$ 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:

$$\left[ \alpha u + (1- \alpha ) {\partial u \over \partial z} \right] _{z=0} = u_{0}$$

for any $\alpha , 0< \alpha <1$.

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 $t=\tau _{0}$ 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.