The method of time fields (Common Source Pattern)

During World War II the Soviet geophysicist J. Riznichenko proposed the following method for reconstruction of a reflector in a medium with a given velocity $v({\bf r})$.Let the source be located at point ${\bf s}$ and let $z = h({{\bf r}})$ be a surface that shows the location of a reflector R. In a medium D between surfaces $\Sigma$ and R two eikonals can be introduced:

• eikonal of the direct wave: $\tau_d({\bf r})$
• eikonal of the wave reflected from R: $\tau_r({\bf r})$.

Both eikonals satisfy the eikonal equation  

$$\vert\nabla \tau \vert^2 = {1\over{v^2({\bf r})}}.$$ (53)

The initial condition for $\tau_d$ is  

$$\tau_d({\bf s}) = 0 ;\qquad \qquad\tau_d({\bf r}_0) \ne 0 , \quad {\bf r}_0 \ne {\bf s};$$ (54)

initial condition for $\tau_r$ is  

$$\tau_r({\bf r})\vert _{R} = \tau_1({\bf r}_{R}),$$ (55)

where  

$$\tau _1({\bf r}_{R}) = \tau_d({\bf r})\vert _{R}$$ (56)

${\bf r}_{R}$ are curvilinear coordinates on the surface R. $\tau_d$ and $\tau_r$ are determined as the direct eikonal's continuations.

It is by definition that eikonals $\tau_d$ and $\tau_r$ have the same values on the reflector R:  

$$\tau_d \vert _{R} = \tau_r \vert _{R}.$$ (57)

In fact the surface R is a geometrical collection of points ${\bf r}$ that satisfies equation  

$$\tau_d({\bf r}) = \tau_r({\bf r}).$$ (58)

We suggest that travel-times $\tau_0({r}_0)$ for a common source are given on the surface $\Sigma$. Since the velocity-function $v({\bf r})$ is proposed to be known, we can determine the reverse eikonal continuation $\tau^{(-)}({\bf r})$.Both functions $\tau^{(-)}({\bf r})$ and $\tau_r({\bf r})$:

• satisfy the same equation (53)
• coincide with $\tau_0({\bf r}_0)$ on $\Sigma$
• approach the surface $\Sigma$ from below.

This means that  

$$\tau^{(-)}({\bf r}) = \tau_r({\bf r}), {\bf r} \in D$$ (59)

and the reflector's location is determined by  

$$\tau^{(-)}({\bf r}) = \tau_d({\bf r}).$$ (60)

In fact, Riznichenko proposed not only this theoretical scheme but also a graphical scheme. Huygens principle founded method of reconstructing $\tau^{(-)}$ and $\tau_d$ in the 2D case. Nowadays this second scheme has lost its significance since we have a great deal of different computer-flexible techniques to solve equation (53). Due to the principle of reciprocity, absolutely the same scheme works when, instead of $\tau_0({\bf r}_0)$, we have $\tau_0(s,r=const)$: travel-times for common receiver patterns with source location s.

It is easy to extend this technique to converted waves PS or SP: in this case we must use different velocities for reconstructing $\tau_d$ and $\tau^{(-)}$.