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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 [*]):

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

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

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

$\bullet$ 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}\end{displaymath}

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 ,\end{displaymath}

(see Figure [*]).


Let equation  
s= \psi_1(\lambda),\ \ r= \psi_2(\lambda)\end{displaymath} (123)
determine an arbitrary ``continuous'' pattern of seismic records and

t = \tau(\lambda) = t[ s= \psi_1(\lambda), r=\psi_2(\lambda)]\end{displaymath}

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.\end{displaymath} (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.\end{displaymath}

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}.\end{displaymath} (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)\end{displaymath}

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

-f( \pi - \psi)\,\psi_1' \ + \ f(\psi)\,\psi_2'\end{displaymath}

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:

Operator ${\bf P}$ does not exist.
One and only one operator ${\bf P}$ exists.
A continuum of operators ${\bf P}$ exists.

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

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))\end{displaymath}

We shall continue listing some general statements. 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\end{displaymath}

{\bf K}_2\ :\ \mu \rightarrow t_0\end{displaymath}

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} }\end{displaymath}

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} }.\end{displaymath}

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) ]\end{displaymath}

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.\end{displaymath}

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} }.\end{displaymath}

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 $t={z\over v_{c}}$ (vc - 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).


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Next: Dynamics of the main Up: 12: OPERATOR OF INTEGRAL Previous: 12: OPERATOR OF INTEGRAL
Stanford Exploration Project