Geometry of the main discontinuity

$\bullet$ The main discontinuity of the field is located along the line $\eta = \mu (\xi )$if any of the following conditions is fulfilled (see Figure ):

1.

The curve $\mu$ is an envelope of a family of dual lines $\overline{\theta }_{N}$ under $N \in \tau$.

2.

The curve $\tau$ is an envelope of a family of stacking lines $\theta _{M}$ under $M \in \mu$.

The curve $\mu$ satisfies the equations  

$$f(x; \xi ,\eta )=0, \;\; {df(x;\xi ,\eta ) \over dx} =0$$ (122)

where $f(x; \xi ,\eta ) =\theta _{M}(x)-\tau (x)$.

$\bullet$ Each operator ${\bf P}$ defines a one-to-one mapping ${\bf K}(P):\{ \tau \} \rightleftharpoons \{ \mu \}$ of a set $\{ \tau \}$ of curves in the plane (x,y) into the set $\{ \mu \}$ of curves in the plane $(\xi ,\eta)$. This mapping does not depend on the kernel $w(x; \xi ,\eta )$ and is determined only by the family of stacking lines $\{ \theta _{M} \}$.

$\bullet$ There is a one-to-one correspondence between points of curves $\tau$ and $\mu$(Figure ):

$$\omega : \; \; \; M \in \mu \rightleftharpoons N \in \tau$$

$\bullet$ The curve $\theta _{M}$ is a special curve of a mapping ${\bf K}(P)$:

$${\bf K}(P) \theta _{M}=M$$

$\bullet$ The operator $\bar{\bf P}$ , with dual lines as stacking lines, produces the mapping inverse to ${\bf K}(\bar{P})$:

$${\bf K}(P)={({\bf K}(P))}^{-1}$$

that follows immediately from the symmetry of the two conditions in the first statement.

$\bullet$ If the curve $\tau$ ends at the point N₀, then the field $u(\xi , \eta)$has an edge discontinuity that is located along dual line $\eta = \bar{\theta }_{N_{0}}(\xi )$ (Figure ).

$\bullet$ If the aperture X has the end a that does not depend on $\xi , \eta$ and belongs to the interval of determination of the curve $\tau$, then the field $u(\xi , \eta)$ has an edge discontinuity that is located along the dual line $\eta = \bar{\theta }_{N_{0}}(\xi )$ at $n_{0}=(a, \tau (a))$.

$\bullet$ If the value a depends on values $\xi$ and $\eta$, then the field $u(\xi , \eta)$ contains a discontinuity on the line $\lambda$ satisfying the equation

$$\theta [a(\xi , \eta ); \xi , \eta ]- \tau [a(\xi , \eta )]=0 ,$$

(see Figure ).

13: AN APPROACH TO A GENERAL THEORY OF SEISMIC IMAGING

Let equation  

$$s= \psi_1(\lambda),\ \ r= \psi_2(\lambda)$$ (123)

determine an arbitrary ``continuous'' pattern of seismic records and

$$t = \tau(\lambda) = t[ s= \psi_1(\lambda), r=\psi_2(\lambda)]$$

be a traveltime curve of the reflected wave for this pattern and a given reflector. Does an IGO exist that (at given velocities) transforms the discontinuity on the line $t=\tau(\lambda)$ into a discontinuity coinciding with the position of the reflector?

Let us suggest that at the given velocity v(x,t) travel time inversion for the pattern (123) has a unique solution. This suggestion makes sense if it embraces situations such as CSP-pattern $(s=const., r=\lambda)$, CRP-pattern $( s=\lambda, r=const.)$ or zero-offset $( s=r=\lambda)$. We shall show that for homogeneous media a sufficient (but unnecessary!) condition is  

$$\psi_1'(\lambda) \geq 0\ ,\ \psi_2'(\lambda) \geq 0.$$ (124)

Let us fix a parameter meaning $\lambda,\ t=\tau(\lambda)\ {\rm and}\ \frac{ d \tau}{d \lambda}$. It is evident that the reflection point lies on the the reflection isochron

$$\sqrt{ (x-s)^2 + z^2} + \sqrt{ (x-r)^2 + z^2} = vt.$$

The point we are looking for is the point on the isocron for which  

$$\frac{\partial t}{\partial s}\psi_1'(\lambda) + \frac{\partial t}{\partial r}\psi_2'(\lambda) = \frac{ d t}{d \lambda}.$$ (125)

Let us parameterize the isochron by the angle $\phi$ (see Figure ). Then

$$\frac{ d t}{d r} = f(\psi) \ {\rm and }\ \frac{ d t}{d s} = -f( \pi - \psi)$$

where $f(\psi)$ is a monotonically decreasing function. Then at condition (124) the sum

$$-f( \pi - \psi)\,\psi_1' \ + \ f(\psi)\,\psi_2'$$

is also a monotonic function and it means that equation (125) has a unique solution. The condition (124) is essential as it follows from the example of CMP-pattern $(s=X-\lambda/2, r=X+\lambda/2, X=const.)$, when there are exactly two symmetrical solutions to equation(125). It is easy to notice that in the case of common-offset pattern $( s=\lambda-l/2, r=\lambda+l/2, l=const.)\ ,\ \psi_1'\ =\ \psi_2'\ =\ 1\ \gt\ 0.$

Uniqueness of travel time inversion means that there is a mapping of a set of travel time curves into the set of reflectors (at a given $v({\bf r})$). We pose the following general problem. Let ${\bf K}$ be an arbitrary one-to-one mapping of an arbitrary set of lines $\{ \tau \}$ in the plane (x,z) into a set of lines $\{ \mu \}$ in the plane $(\xi ,\eta)$.Does the IGO ${\bf P}$ exist for which ${\bf K}={\bf K}(P)$? If this operator exists, we shall say that operator ${\bf P}$ realizes the mapping ${\bf K}$.Of course we must determine only the family of stacking lines $\{ \theta _{M} \}$ : the kernal of the operator ${\bf P}$ can be chosen arbitrarily.

It can be proved that there are only three alternatives:

1.

Operator ${\bf P}$ does not exist.

2.

One and only one operator ${\bf P}$ exists.

3.

A continuum of operators ${\bf P}$ exists.

Of course the second alternative is the most important for us.

• We have the second alternative if a common point N of curves $\tau_1$ and $\tau_2$ is mapped into a common point of curves $\mu_1$ and $\mu_2$ when, and only when, $\tau_1$ touches $\tau_2$ at N (see Figure ).
• The family of special curves of mapping ${\bf K}$ originates the family of stacking lines:

  $${\bf K}\hat{\tau} = \hat{M}\,,\ {\rm then }\ \hat{\tau}\, \equiv\, \theta_{\hat{M}}.$$

• If ${\bf K}$ is realized then ${\bf K}^{-1}$ is realized too and $\overline{\theta }_{N}$ is a family of stacking lines.

Let us give an example of the second alternative. We assume that $\tau$ is the traveltime curve for an acceptable pattern and $\mu$ is a reflector on a depth section. It is easy to notice that the ray geometry depends only on the location of the reflector and the slope of the tangent (see Figure ). It immediately follows from this that all conditions for the existence of the second alternative are fulfilled, so there is an operator of integral geometry ( at $\xi=x\ {\rm and}\ \eta=y$ ) that realizes the mapping of traveltime curves into reflector images.

It is evident that special lines of this mapping are traveltime curves for point reflectors.

$$\theta(\lambda;x,z) = T ( s(\lambda); x, z ) + T(x,z;r(\lambda))$$

We shall continue listing some general statements.

• If mappings ${\bf K}_1$ and ${\bf K}_2$ are realized, then the mapping of ${\bf K}= {\bf K}_2\cdot {\bf K}_1$ is realized too.

Example. Let us answer the following question: Can an acceptable pattern (123) be transformed into a time section that is geometrically equivalent to t₀?

Let us define ${\bf K}_1$ and ${\bf K}_2$ such that

$${\bf K}_1\ :\ \tau(\lambda) \rightarrow \mu$$

$${\bf K}_2\ :\ \mu \rightarrow t_0$$

where $\mu$ is the position of a reflector on the depth section. We have shown above that ${\bf K}_1$ is realized. The possibility of realizing the transformation of depth section into the time section t₀(x) is obvious so the operator ${\bf K}$ is realized. Stacking lines of the corresponding operator ${\bf P}$ are travel time curves from the circular reflector with center at x and radius $\rho = vt_0$, because this reflector gives a point on the zero offset time (non-migrated) section, and consequently originates a special curve of mapping ${\bf K}$.Another example: cascaded migration. Let ${\bf P}_1$ be determined by the system of stacking lines:

$$\theta_1 = \sqrt{ \eta_1^2 + \frac{ 4(x-\xi_1)^2}{v_1^2} }$$

and ${\bf P}_2$ by the system of stacking lines:

$$\theta_2 = \sqrt{ \eta_2^2 + \frac{ 4(\xi_1-\xi)^2}{v_2^2} }.$$

The system of stacking lines for the operator ${\bf P} = {\bf P}_1 \cdot {\bf P}_2$ in the general case is determined by the formula

$$y= \theta(x; \xi,\eta) = \theta_1[ x; \xi_1, \theta_2(\xi_1;\xi,\eta) ]$$

at $\xi_1 = \Psi(x,\xi,\eta)$ where the function $\Psi$ is derived from the equation

$$\frac{ d\theta_1}{d\xi_1} : \frac{\partial \theta_1}{\partial \eta_1} + \frac{ \partial \theta_2}{\partial \xi_1} = 0.$$

Application of all these equations to cascaded migration gives the system of stacking lines:

$$\theta(x;\xi_1,\eta) = \sqrt{ \eta^2 + \frac{ 4 (\xi -x )^2}{ v_1^2 + v_2^2} }.$$

Now we shall give an example of the first alternative: the case of head waves (Figure  ). It is easy to show that in the case of the CSP-pattern there is a unique solution of travel time inversion for head waves ( for a given v₁ and v₂ ). But touching of refractors does not guarantee touching of traveltime curves. So the mapping ${\bf K}$is not generally realized.

Nevertheless, it can be done for some special cases. For example, when $\{ \mu \}$is a set of planar refractors, this produces an operator ${\bf P}$ which approximately realizes the mapping ${\bf K}$ for refractors with small curvature.

In Chapter 7 we have testified that image ray technique does not guarantee exact time-to-depth migration if time migration was performed in a simpler model of medium. Does the correct time-to-depth migration algorithm in this case exist? The answer is positive. The mapping ${\bf K}$ we are looking for is a product of three mappings: scaling operator ${\bf S}$ that, with help of substitution $t={z\over v_{c}}$ (v_c - continuation velocity for time migration), connects reflector images $\{\mu _{t}\}$in time-sections with reflector images $\{ \mu _{z}\}$ in depth section; mapping ${\bf K}_1$ that connects $\{ \mu _{z}\}$with zero-offset travel-time curves $\{ t_{0}\}$ in a simpler model of medium; and mapping ${\bf K}_2$ that connects to reflectors $\mu$ in a true model. So ${\bf K}= {\bf K}_{2}{\bf K}_{1}{\bf S}$. Mappings ${\bf K}_1$ and ${\bf K}_2$ are realizable, so their product is realizable too.

Finally, an example of the third alternative: CMP stacking. Traveltime $\tau(l)$is always an even function so any family of even curves $\{ \theta(l;\tau_0,v_s(\tau_0)) \}$ guarantees touching at $l=0\ {\rm and}\ t=\tau_0$ (Figure ). This means that CMP stacking always gives a section that is geometrically equivalent to t₀(x).

14: DYNAMIC ASPECTS OF IGO AND THEIR APPLICATION TO TOMOGRAPHY