# Implementation of migration

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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

A raypath is a course along which wave energy propagates through the earth. In isotropic media, the raypath is perpendicular to the local wavefront. The raypath can be calculated using ray tracing. Let the point ${\displaystyle P{\rm {=}}\left(x_{P},y_{P}\right)}$ be either a source location or a receiver location. The subsurface volume is represented by a ${\displaystyle {\rm {3D\ grid}}\ \left(x,y,z\right)}$ of depth points D. To minimize the amount of ray tracing, we first compute a traveltime table for every surface location, whether the location is a source point or a receiver point. In other words, for each surface location P, we compute the one-way traveltime from P to each depth point D in the ${\displaystyle {\rm {3D}}}$ grid. We put these one-way traveltimes into a ${\displaystyle {\rm {3D}}}$ table labeled by the surface location P.

The traveltime for a primary reflection is the total two-way (i.e., down and up) time for a path originating at source point S, reflected at depth point D, and received at receiver point R. Associated with each trace are two identification numbers, one for the source S and the other for the receiver R. We pull out the respective tables for these two identification numbers. We add the two tables together element by element. The result is a ${\displaystyle {\rm {3D}}}$ table for the two-way traveltimes for that seismic trace.

Let us give a ${\displaystyle {\rm {2D}}}$ example. We assume that the medium has a constant velocity, which we take to be one. Let the subsurface grid for depth points D be given by ${\displaystyle \left(z,x\right)}$ where depth is given by ${\displaystyle z{\rm {=}}{\rm {1,2,...,10}}}$ and horizontal distance is given by ${\displaystyle x{\rm {=}}{\rm {1,2,...,15}}}$. Let the surface locations P be ${\displaystyle \left(z{\rm {=\ 1,}}x\right)}$ where ${\displaystyle x{\rm {=}}{\rm {1,2,...,15}}}$. Suppose the source is ${\displaystyle S=(1,3).}$ We want to construct a table of one-way traveltimes, where depth z denotes the row and horizontal distance x denotes the column. The one-way traveltime from the source (1, 3) to the depth point ${\displaystyle \left(z,x\right)}$ is ${\displaystyle t\left(z,x\right){\rm {=}}{\left[{\left(z-{\rm {\ 1}}\right)}^{\rm {2}}{\rm {+}}{\left(x-{\rm {3}}\right)}^{\rm {2}}\right]}^{\rm {1/2}}}$. For example, the traveltime from source to depth point ${\displaystyle \left(z,x\right){\rm {=}}\left({\rm {4,6}}\right)}$ is

 {\displaystyle {\begin{aligned}t\left({\rm {4,6}}\right){\rm {=}}{\sqrt {{\left({\rm {4}}-{\rm {1}}\right)}^{\rm {2}}{\rm {+}}{\left({\rm {6}}-{\rm {3}}\right)}^{\rm {2}}}}{\rm {=}}{\sqrt {\rm {18}}}{\rm {=4.24}}.\end{aligned}}} (73)

For ease of presentation, we round the computed number 4.24 to 4.2 and enter this rounded number into the slot for the fourth row, sixth column of Table 1. In a similar manner, we compute all of the entries in Table 1.

Next let us compute the traveltime for receiver points. Note that the traveltime from depth point to receiver is the same as the traveltime from receiver to depth point. Suppose the receiver is ${\displaystyle R(1,11)}$. Then the one-way traveltime from the receiver ${\displaystyle (1,11)}$ to the depth point ${\displaystyle \left(z,x\right)}$ is ${\displaystyle t\left(z,x\right){\rm {=}}{\left[{\left(z-{\rm {\ 1}}\right)}^{\rm {2}}{\rm {+}}{\left(x-{\rm {\ 11}}\right)}^{\rm {2}}\right]}^{\rm {1/2}}}$. For example, the traveltime from the receiver (1,11) to depth point ${\displaystyle \left(z,x\right){\rm {=}}\left({\rm {4,6}}\right)}$ is

{\displaystyle {\begin{aligned}t\left({\rm {4,6}}\right)&{\rm {=}}{\sqrt {{\left({\rm {4}}-{\rm {1}}\right)}^{\rm {2}}{\rm {+}}{\left({\rm {6}}-{\rm {11}}\right)}^{\rm {2}}}}{\rm {=5.83.}}\end{aligned}}}

For ease of presentation, we round the computed number 5.83 to 5.8 and enter this rounded number into the slot for the fourth row, sixth column of Table 2. In a similar manner, we compute all of the entries in Table 2.

We obtain Table 3 by adding Tables 1 and 2 entry by entry. Table 3 gives the two-way traveltimes. For example, the traveltime from source (1, 3) to depth point (4, 6) and back to receiver (1, 11) is ${\displaystyle t\left({\rm {4,6}}\right){\rm {=4.24+5.83=}}{\rm {10.07}}}$. This number (rounded) appears in the fourth row, sixth column of Table 3. A contour map of Table 3 would reveal that the contour lines are elliptical curves of constant two-way traveltime for the given source and receiver pair.

Table 1. One-way traveltimes from source point (1, 3) to the depth point indicated by the row and column.
Horizontal distance, x
Depth, z 2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0
2.2 1.4 1.0 1.4 2.2 3.2 4.1 5.1 6.1 7.1 8.1 9.1 10.0 11.0 12.0
2.8 2.2 2.0 2.2 2.8 3.6 4.5 5.4 6.3 7.3 8.2 9.2 10.2 11.2 12.2
3.6 3.2 3.0 3.2 3.6 4.2 5.0 5.8 6.7 7.6 8.5 9.5 10.4 11.4 12.4
4.5 4.1 4.0 4.1 4.5 5.0 5.7 6.4 7.2 8.1 8.9 9.8 10.8 11.7 12.6
5.4 5.1 5.0 5.1 5.4 5.8 6.4 7.1 7.8 8.6 9.4 10.3 11.2 12.1 13.0
6.3 6.1 6.0 6.1 6.3 6.7 7.2 7.8 8.5 9.2 10.0 10.8 11.7 12.5 13.4
7.3 7.1 7.0 7.1 7.3 7.6 8.1 8.6 9.2 9.9 10.6 11.4 12.2 13.0 13.9
8.2 8.1 8.0 8.1 8.2 8.5 8.9 9.4 10.0 10.6 11.3 12.0 12.8 13.6 14.4
9.2 9.1 9.0 9.1 9.2 9.5 9.8 10.3 10.8 11.4 12.0 12.7 13.5 14.2 15.0

Table 4 gives the amplitude values of a deconvolved trace (i.e., the trace with primary reflections only). There are only three nonzero amplitude values on this trace. In other words, we have three lumps of butter, namely –1, 2, and 3, to be spread onto three slices of bread, namely 10, 13, and 18.

Table 2. One-way traveltimes from receiver point (1, 11) to the depth point indicated by the row and column.
Horizontal distance, x
Depth, z 10.0 9.0 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0 1.0 2.0 3.0 4.0
10.0 9.1 8.1 7.1 6.1 5.1 4.1 3.2 2.2 1.4 1.0 1.4 2.2 3.2 4.1
10.2 9.2 8.2 7.3 6.3 5.4 4.5 3.6 2.8 2.2 2.0 2.2 2.8 3.6 4.5
10.4 9.5 8.5 7.6 6.7 5.8 5.0 4.2 3.6 3.2 3.0 3.2 3.6 4.2 5.0
10.8 9.8 8.9 8.1 7.2 6.4 5.7 5.0 4.5 4.1 4.0 4.1 4.5 5.0 5.7
11.2 10.3 9.4 8.6 7.8 7.1 6.4 5.8 5.4 5.1 5.0 5.1 5.4 5.8 6.4
11.7 10.8 10.0 9.2 8.5 7.8 7.2 6.7 6.3 6.1 6.0 6.1 6.3 6.7 7.2
12.2 11.4 10.6 9.9 9.2 8.6 8.1 7.6 7.3 7.1 7.0 7.1 7.3 7.6 8.1
12.8 12.0 11.3 10.6 10.0 9.4 8.9 8.5 8.2 8.1 8.0 8.1 8.2 8.5 8.9
13.5 12.7 12.0 11.4 10.8 10.3 9.8 9.5 9.2 9.1 9.0 9.1 9.2 9.5 9.8
Table 3. Two-way traveltimes obtained by adding Tables 1 and 2 entry by entry. The source point is (1, 3), the receiver point is (1, 11), and the depth points are indicated by the row and column.
Horizontal distance, x
Depth, z 12 10 8 8 8 8 8 8 8 8 8 10 12 14 16
12 10 9 8 8 8 8 8 8 8 9 10 12 14 16
13 11 10 10 9 9 9 9 9 10 10 11 13 15 17
14 13 12 11 10 10 10 10 10 11 12 13 14 16 17
15 14 13 12 12 11 11 11 12 12 13 14 15 17 18
17 15 14 14 13 13 13 13 13 14 14 15 17 18 19
18 17 16 15 15 15 14 15 15 15 16 17 18 19 21
19 18 18 17 16 16 16 16 16 17 18 18 19 21 22
21 20 19 19 18 18 18 18 18 19 19 20 21 22 23
23 22 21 20 20 20 20 20 20 20 21 22 23 24 25
Table 4. Amplitude values at the indicated times for a deconvolved seismic trace.
Deconvolved seismic trace
Time 0 1 2 3 4 5 6 7 8 9
Amplitude value 0 0 0 0 0 0 0 0 0 0
Time 10 11 12 13 14 15 16 17 18 19
Amplitude value –1 0 0 2 0 0 0 0 3 0
Table 5. Amplitude values spread out onto ellipses. To see the ellipses, connect the numbers with the same value.
Horizontal distance, x
Depth, z 0 –1 0 0 0 0 0 0 0 0 0 0 0 0 0
0 –1 0 0 0 0 0 0 0 0 0 –1 0 0 0
2 0 –1 –1 0 0 0 0 0 –1 –1 0 2 0 0
0 0 2 0 0 –1 –1 –1 –1 –1 0 0 2 0 0
0 0 0 2 0 0 0 0 0 0 0 2 0 0 0
3 0 0 0 0 2 2 2 2 2 0 0 0 0 3
0 3 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 3 3 0 0 0 0 0 0 0 3 3 0 0
0 0 0 0 0 3 3 3 3 3 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

We construct Table 5 from Table 3 as follows. In Table 3, replace all entries with zero except the entries with the values 10, 13, or 18. Replace all entries 10 with the lump –1. Replace all entries 13 with the lump 2. Replace all entries 18 with the lump 3. The result is Table 5. Table 5 represents the pieces of bread (i.e., ellipses) from one trace. This operation must be repeated for all traces in the survey, and all the resulting tables must be added together. The final image appears by the constructive and destructive interference among the individual trace contributions.

The above-described procedure for a constant velocity is the same as the procedure with a variable velocity except that with a variable velocity the traveltimes have to be computed according to the velocity function ${\displaystyle v\left(x,y,z\right)}$. The eikonal equation, as developed in this chapter, can be used to achieve that end.