Implementation of migration
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| Series | Geophysical References Series |
|---|---|
| Title | Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing |
| Author | Enders A. Robinson and Sven Treitel |
| Chapter | 2 |
| DOI | http://dx.doi.org/10.1190/1.9781560801610 |
| ISBN | 9781560801481 |
| Store | 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): P{\rm =}\left(x_P,y_P\right) be either a source location or a receiver location. The subsurface volume is represented by a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\rm 3D} grid. We put these one-way traveltimes into a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\rm 3D} table for the two-way traveltimes for that seismic trace.
Let us give a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(z,x\right) where depth is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z{\rm =} {\rm 1,2,...,10} and horizontal distance is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x{\rm =} {\rm 1,2,...,15} . Let the surface locations P be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(z{\rm =\ 1,}x\right) where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x{\rm =} {\rm 1,2,...,15} . Suppose the source is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(z,x\right) is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(z,x\right){\rm =}\left({\rm 4,6}\right) is
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} 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{align} ()
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R(1,11) . Then the one-way traveltime from the receiver Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): (1,11) to the depth point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(z,x\right) is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(z,x\right){\rm =}\left({\rm 4,6}\right) is
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} 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{align}
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 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.
| 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.
| 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 | |
| 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 | |
| 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 |
| 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): v\left(x,y,z\right) . The eikonal equation, as developed in this chapter, can be used to achieve that end.
Continue reading
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|---|---|
| Migration in the case of constant velocity | Appendix B: Exercises |
| Previous chapter | Next chapter |
| Wave Motion | Visualization |
Also in this chapter
- Reflection seismology
- Digital processing
- Signal enhancement
- Migration
- Interpretation
- Rays
- The unit tangent vector
- Traveltime
- The gradient
- The directional derivative
- The principle of least time
- The eikonal equation
- Snell’s law
- Ray equation
- Ray equation for velocity linear with depth
- Raypath for velocity linear with depth
- Traveltime for velocity linear with depth
- Point of maximum depth
- Wavefront for velocity linear with depth
- Two orthogonal sets of circles
- Migration in the case of constant velocity
- Appendix B: Exercises
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