Changes

==How migration is accomplished==

==How migration is accomplished==

One senses the massive scale of the computer intensive two-dimensional mapping involved in the transformation from unmigrated to migrated data in Figure 4. While these schematic sections depict ''what'' migration aims to accomplish, they say little about ''how'' it is done. All of the many methods of doing migration are founded on solutions to the ''scalar wave equation'', a partial differential equation that models how waves propagate in the earth. A simple form of the wave equation is as follows:

One senses the massive scale of the computer intensive two-dimensional mapping involved in the transformation from unmigrated to migrated data in [[:file:seismic-migration_fig4.png|Figure 4]]. While these schematic sections depict ''what'' migration aims to accomplish, they say little about ''how'' it is done. All of the many methods of doing migration are founded on solutions to the ''scalar wave equation'', a partial differential equation that models how waves propagate in the earth. A simple form of the wave equation is as follows:

:<math>\frac{\partial^{2}P}{\partial z^{2}} + \frac{\partial^{2}P}{\partial x^{2}} = \frac{1}{V^{2}}\frac{\partial^{2}P}{\partial t^{2}}</math>

:<math>\frac{\partial^{2}P}{\partial z^{2}} + \frac{\partial^{2}P}{\partial x^{2}} = \frac{1}{V^{2}}\frac{\partial^{2}P}{\partial t^{2}}</math>

where ''P''(''x, z, t'') is the seismic amplitude as a function of reflection time ''t'' at any position (''x, z'') in the subsurface, and ''V'' is the seismic wave velocity in the subsurface, a function of both ''x'' and ''z''. Disturbances initiated by a seismic energy source are assumed to propagate in accordance with solutions to the wave equation. Migration, then, involves a running of the wave equation ''backward in time'', starting with the measured waves at the earth's surface ''P''(''x, z'' = 0, ''t''), in effect pushing the waves backward and downward to their reflecting locations.

where ''P''(''x, z, t'') is the seismic amplitude as a function of reflection time ''t'' at any position (''x, z'') in the subsurface, and ''V'' is the seismic wave velocity in the subsurface, a function of both ''x'' and ''z''. Disturbances initiated by a seismic energy source are assumed to propagate in accordance with solutions to the wave equation. Migration, then, involves a running of the wave equation ''backward in time'', starting with the measured waves at the earth's surface ''P''(''x, z'' = 0, ''t''), in effect pushing the waves backward and downward to their reflecting locations.

All current computer-based approaches to migration involve this backward solution to the wave equation. The earliest form, ''Kirchhoff summation migration'', intuitively follows the situation depicted in Figure 1. In essence, recorded amplitudes on CMP traces are summed along the diffraction trajectories dictated by the assumed subsurface velocity distribution, and the sums are placed at the apexes of the curves, one curve for each sample point in the output migrated section.

All current computer-based approaches to migration involve this backward solution to the wave equation. The earliest form, ''Kirchhoff summation migration'', intuitively follows the situation depicted in [[:file:seismic-migration_fig1.png|Figure 1]]. In essence, recorded amplitudes on CMP traces are summed along the diffraction trajectories dictated by the assumed subsurface velocity distribution, and the sums are placed at the apexes of the curves, one curve for each sample point in the output migrated section.

''Finite difference migration'' numerically integrates the wave equation by the method of finite differences to push seismic waves backward into the subsurface. A third category of migration approaches is ''f-k migration'', which operates via Fourier transforms in the frequency wavenumber (f-k) domain. In general, Fourier transform methods provide elegant means of solving partial differential equations. When applied to migration, this elegance is often complemented by high computational efficiency.

''Finite difference migration'' numerically integrates the wave equation by the method of finite differences to push seismic waves backward into the subsurface. A third category of migration approaches is ''f-k migration'', which operates via Fourier transforms in the frequency wavenumber (f-k) domain. In general, Fourier transform methods provide elegant means of solving partial differential equations. When applied to migration, this elegance is often complemented by high computational efficiency.