# Migration in the case of constant velocity

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Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 2 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

The two-way traveltime for a primary reflection is the time it takes for the seismic energy to travel down from the source ${\displaystyle S{\rm {=}}\left(x_{S},y_{S}\right)}$ to depth point ${\displaystyle D{\rm {=}}\left(x_{D},y_{D},z_{D}\right)}$ and then back up to the receiver ${\displaystyle R{\rm {=}}\left(x_{R},y_{R}\right)}$. Deconvolution and other multiple-removing methods ideally produce a trace that consists of primary reflections only. Such deconvolved traces can be used for imaging. The deconvolved trace ${\displaystyle {\textit {f}}\left(S,R,t\right)}$ gives the amplitude of the reflected signal as a function of two-way traveltime t, which is given in milliseconds from the time that the source is activated. We know S, we know R, we know t, and we know ${\displaystyle f\left(S,R,t\right)}$. The problem is to find D, which is the depth point at which the reflection occurred.

In an isotropic medium, the physical properties at a point are the same in all directions. In particular, the wave velocity at a point is the same in all directions. In an isotropic medium, the rays are orthogonal trajectories of the wavefronts. In other words, the rays are normal to the wavefronts. However, in an anisotropic medium, the rays need not be orthogonal trajectories of the wavefronts. In a homogeneous medium, the physical properties are the same throughout the medium.

Here we consider the case of a homogeneous isotropic medium. Within a homogeneous isotropic material, the velocity v has the same value at all points and in all directions. The rays are straight lines because, by symmetry, there is no preferred direction along which they can deviate from the straight path. The two-way traveltime t is the elapsed time for a seismic wave to travel from its source to a given depth point and return to a receiver at the earth’s surface. Thus, the two-way traveltime t is equal to the one-way traveltime ${\displaystyle t_{\rm {1}}}$ from the source point S to the depth point D plus the one-way traveltime ${\displaystyle t_{\rm {2}}}$ from the depth point D to the receiver point R. Note that the traveltime from D to R is the same as the traveltime from R to D. We can write ${\displaystyle t{\rm {=}}t_{\rm {l}}\ {\rm {+}}t_{\rm {2}}}$, which in terms of distance is

 {\displaystyle {\begin{aligned}vt&{\rm {=}}{\sqrt {{\left(x_{D}-x_{S}\right)}^{\rm {2}}{\rm {+}}{\left(y_{D}-y_{S}\right)}^{\rm {2}}{\rm {+}}{\left(z_{D}-z_{S}\right)}^{\rm {2}}}}{\rm {+}}{\sqrt {{\left(x_{D}-x_{R}\right)}^{\rm {2}}{\rm {+}}{\left(y_{D}-y_{R}\right)}^{\rm {2}}{\rm {+}}{\left(z_{D}-z_{R}\right)}^{\rm {2}}}}.\end{aligned}}} (72)

We recall that an ellipse can be drawn with two pins, a loop of string, and a pencil. The pins are placed at the foci, and the ends of the string are attached to the pins. The pencil is placed on the paper inside the string, so the string is taut. The string will form a triangle. If the pencil is moved around so that the string stays taut, the sum of the distances from the pencil to the pins will remain constant, and the curve traced out by the pencil is an ellipse. Thus, if vt is the length of the string, then any point on the ellipse could be the depth point D that produced the reflection for that source S, that receiver R, and that traveltime t. We therefore take that event and move it out to each point on the ellipse.

Suppose we have two traces with only one event on each trace. Suppose both events come from the same reflecting surface. Figure 6b depicts the two ellipses. In the spirit of Huygens’ construction, the reflector must be the common tangent to the ellipses. This example shows how migration works.

What do we have? We have a set of traces in which each trace is associated with a given source point and a given receiver point. On each trace is a fixed number of amplitude values in which each value is associated with a discrete time index. Out of all these data, we just want to pick one amplitude value and then determine the set of depth points in the earth that could give rise to this value. Under the simplifying assumption of constant velocity, all such depth points lie on an ellipsoid. In other words, we know the location of all such depth points for the amplitude value in question. The next step is to assign a value to each point on the ellipsoid in question. We assign the same value as the amplitude value in question. For example, if the amplitude value is the number 3, then we assign the number 3 to each point on the ellipsoid in question. The terminology used is: Take the amplitude at each digital time instant t on the trace and spread out the amplitude onto the constructed ellipsoid. In other words, we have taken one amplitude value on a trace and, like butter on bread, we have spread the lump out onto an entire ellipsoidal surface. Then we repeat this operation for every amplitude value on every trace. In our mind’s eye, every amplitude value is spread out onto its own ellipsoid. Each lump of butter has its own piece of bread.

The next step is superposition. We superimpose all the pieces of bread. In other words, all the numerical ellipsoids are added together. (In practice, as soon as an ellipsoid is computed, it would be added cumulatively to the running sum.) Interference tends to destroy the noise, and we are left with the desired 3D image of the earth. This process is an elaborate numerical version of what J. Clarence Karcher did graphically in 1921 (Figure 5). The authority for this procedure is the time-honored Huygens’ construction along with the interference principle of Fresnel.