Asymptotic pseudo-unitary operators as orthonormal bases

It is interesting to note that a wide class of integral operators, routinely used in seismic data processing, have the form of operator (22) with the ``Green'' function  

$$G (t,\bold{y};z,\bold{x}) = \left\vert\frac{\partial}{\part... ...ld{y}) \delta \left(z - \theta(\bold{x};t,\bold{y}) \right)\;.$$ (24)

where we have split the variable x into the one-dimensional component z (typically depth or time) and the m-dimensional component $\bold{x}$ (typically a lateral coordinate with m equal 1 or 2.) Similarly, the variable y is split into t and $\bold{y}$. The function $\theta$ represents the summation path , which captures the kinematic properties of the operator, and A is the amplitude function.

Impulse response (27) is typical for different forms of Kirchhoff migration and datuming as well as for velocity transform, integral offset continuation, DMO, and AMO. Integral operators of that class rarely satisfy the unitarity condition, with Radon transform (slant stack) being a notable exception. In an earlier paper Fomel (1996), I have shown that it is possible to define the amplitude function A for each kinematic path $\theta$ so that the operator becomes asymptotic pseudo-unitary . This means that the adjoint operator coincides with the inverse in the high-frequency (stationary-phase) approximation. Consequently, equation (25) is satisfied to the same asymptotic order.

Using asymptotic pseudo-unitary operators, we can apply formula (26) to find an explicit analytic form of the interpolation function W, as follows:

$$\begin{eqnarray} W (t, \bold{y}; t_n, \bold{y}_n) = \int\!\!\int G (t, \bold{y... ...{y} ) - \theta(\bold{x};t_n,\bold{y}_n) \right) \, d \bold{x}\;.\end{eqnarray}$$ (25)

Here the amplitude function A is defined according to the general theory of asymptotic pseudo-inverse operators as  

$$A = \frac{1}{\left(2\,\pi\right)^{m/2}} \, \left\vert F\,\w... ...\vert\frac{\partial \theta}{\partial t}\right\vert^{(m+2)/4}\;,$$ (26)

where

$$\begin{eqnarray} F & = & \frac{\partial \theta}{\partial t}\, \frac{\partial^2 \... ...ac{\partial^2 \widehat{\theta}}{\partial \bold{y}\, \partial z}\;,\end{eqnarray}$$ (27) (28)

and $\widehat{\theta} (\bold{x};t,\bold{y})$ is the dual summation path, obtained by solving equation $z=\theta(x;t,y)$ for t (assuming that an explicit solution is possible).

For a simple example, let us consider the case of zero-offset time migration with a constant velocity v. The summation path $\theta$in this case is an ellipse  

$$\theta(\bold{x};t,\bold{y}) = \sqrt{t^2 - \frac{(\bold{x}-\bold{y})^2}{v^2}}\;,$$ (29)

and the dual summation path $\widehat{\theta}$ is a hyperbola  

$$\widehat{\theta}(\bold{y};z,\bold{x}) = \sqrt{z^2 + \frac{(\bold{x}-\bold{y})^2}{v^2}}\;.$$ (30)

The corresponding pseudo-unitary amplitude function is found from formula (29) to be Fomel (1996)  

$$A = \frac{1}{\left(2\,\pi\right)^{m/2}} \, \frac{\sqrt{t/z}}{v^m z^{m/2}}\;.$$ (31)

Substituting formula (34) into (28), we derive the corresponding interpolation function  

$$W (t, \bold{y}; t_n, \bold{y}_n) = \frac{1}{\left(2\,\pi\r... ...rt{t\,t_n}}{v^{2m} z^{m+1}}\, \delta (z - z_n) \,d \bold{x}\;,$$ (32)

where $z = \theta(\bold{x};t,\bold{y})$, and $z_n = \theta(\bold{x};t_n,\bold{y}_n)$. For m=1 (the two-dimensional case), we can apply the known properties of the delta function to simplify formula (35) further to the form  

$$W = \frac{v}{\pi} \, \left\vert\frac{\partial}{\partial t... ... \left[v^2 (t + t_n)^2 - (\bold{y}-\bold{y}_n)^2\right] }}\;.$$ (33)

The result is an interpolator for zero-offset seismic sections. Like the sinc interpolator in formula (19) that is based on decomposing the signal into sinusoids, interpolation (36) is based on decomposing the zero-offset section into hyperbolas.