12: OPERATOR OF INTEGRAL GEOMETRY (IGO)

We have connected migration with the idea of wave-field continuation. The great advantage of this approach is the possibility to introduce different ``continuous techniques'' to particular finite difference algorithms. Although this approach was successful in the case of both CSP and zero-offset patterns, it failed in other situations. However, it is not necessary to base migration on wave-field continuation even for the above particular patterns. We have integral transformations  

$$a_{csp}({\bf r})= \int w({\bf r},r) {\bf D}_{-t}^{n} u_{0} (r,t=T(s,{\bf r})+T({\bf r},r))dr$$ (118)

and

$$a_{zop}({\bf r})=\int w({\bf r},r) {\bf D}^{n}_{-t} u_{0}(r,t=2T({\bf r},r)) dr$$

which we can perform in one step.

One may guess that equation (118) is a general form of the transformation of seismic records into the depth (migrated) section. It pushes us to investigate integral transformations of the form:  

$$u(\xi ,\eta )= \int_{X} w(x; \xi , \eta ) u_{0} (x;y= \theta (x; \xi , \eta )) dx \equiv {\bf P} u_{0}(x,y).$$ (119)

This kind of transformation is typical not only in the problem of migration, but also in several other cases:

• slant analysis, when

  $$y=t, \hbox{\hspace{0.5cm}} \theta (x; \xi, \eta ) = \eta + \xi x, \hbox{\hspace{0.5cm}} w \equiv 1$$

• parabolic filtration, when

  $$y=t, \hbox{\hspace{0.5cm}} \theta (x; \xi, \eta ) = \eta + a {(\xi - x)}^{2}, \hbox{\hspace{0.5cm}} w \equiv 1$$

• seismic tomography, when

  u₀ (x,y) = s (x,y) - s₀ (x,y)

  (here s(x,z) is a slowness, $y= \theta (x; \xi , \eta )$ is the equation of a ray with the starting point $\xi$ and end point $\eta$, and $w(x; \xi ,\eta ) = \sqrt{1 + {\vert \theta ^{'}\vert}^{2}}$).

It is also possible to find other examples. Integral transform (119) belongs to the class of integral geometry operators.

In the K-operator case, the method of discontinuities was only one of the possible alternatives; another was the stationary phase technique. For the IGO operator, the method of discontinuities is the only approach that allows us to apply a relatively elementary mathematical concept.

Let us presume that

$$u_0(x,y) \sim A(x) R^{(+)}_q[y-\tau(x)].$$

Again the main idea is that differentiation of the field $u(\xi , \eta)$, with respect to $\eta$, allows us to transform a discontinuity of the field u₀(x,t) into a $\delta$-function. But in this situation it is not so simple because we have essentially used properties of discontinuities of arbitrary order and index. (Some of them were mentioned above.) Now I will add one more: Let t₀ be the only solution of the equation $\phi (t) =0$, then

$${\bf D}^{p}_{(\beta ), t} R ^{\alpha } _{q,\nu } [\phi (t)] ... ...}^{p} R^{(\alpha ')}_{q-p,\nu + p(\alpha ' - \beta )}[\phi (t)]$$

where $\alpha ' = \alpha \; \; {\rm sgn} (\phi_{0}^{'})$, and $\phi_{0}^{'}=\phi^{'}(t_{0})$.After that we get

$$U(\xi, \eta)={\bf D}^{q+1}u(\xi,\eta) \sim \int^{\infty}_{\i... ...heta_{\eta}\vert}^{q+1} \delta [x(x;\xi , \eta ) - \tau (x)] dx$$

Let's summarize the main concepts behind this technique. We shall consider the function $\theta (x; \xi ,\eta )$ as a family of curves $\theta_{M}(x)$ in the plane (x,y) , that depends on the point $M= (\xi , \eta )$ of the plane $(\xi ,\eta)$ (see Figure ).

The curve $y=\theta _{M}(x)$ is called a stacking line. We shall suggest that a given family of stacking lines is regular. It means that for any point N =(x,y) and any value $p \in {\rm P}$ the system of equation  

$$\theta_{M}(x)=y, \; \; {d \theta_M \over dx} = p$$ (120)

with respect to M has one and only one solution (${\rm P}$is a given open set). In other words, through the given point N in the given direction p runs only one stacking line.

If we consider only the first of the equations (120)  

$$\theta_{M}(x)=y$$ (121)

(with fixed x and y) then we have a continuous line of points M satisfying the equation (121). This line is called a dual line (see Figure  ) and denoted as $\eta = \bar{\theta }_{N}(\xi )$.

A discontinuity of wave-field u₀(x,y) located along the line $y=\tau (x)$ influences the appearance of several discontinuities of the field $u(\xi , \eta)$.We shall call the main discontinuity a discontinuity of the field $u(\xi , \eta)$with minimal order q^'.