F-K Demigration In Homogeneous Media: Implementation And Application

The seismic imaging is the finally aim of the seismic exploration and also the kernel of reflect seismology . Nowadays, the seismic migration imaging is the most important seismic imaging, the migration which contain time migration and depth migration is that time domain is transformed to migrated section . At the same time there is an inverse process to migration that we call demigration. Under a demigration ,the migrated image is transformed from the depth domain( time domain )back to the time domain. Similar to migration, demigration also assumes a basic role in the complete imaging theory. Demigration is the inverse process to seismic migration, provided the macro-velocity model, the measurement configuration, and the ray code are kept the same, In other words, when changing macro-velocity measurement configuration and ray code, the demigration will have a lot of implementation, for example configuration transform remigration ray-code transformation forward modeling and enhancing signal-to-noise. At present there are three methods for migration that are Kirchhoff migration F-K migration and finite difference migration. Similar to migration, demigration also have different approaches for its implementation. F-K demigration is processed in the frequency-wavenumber domain. Using the same ideas of F-K migration in constant, layer and complex velocity media model, F-K demigration can also be completed. The reason of emphasizing the constant velocity F-K demigration is that its exact solution could become a reference standard for other approximate solution and it is the base of the F-K demigration in complex velocity media. Migration image spectrum mapping to demigration section spectrum is the key of the constant velocity F-K demigration.Constant velocity F-K demigration implementation: through the migrated section to be two-dimensional Fourier transform migrated section spectrum, from the dispersion of the relationship between migrated section mapped to the observation section ,then time observation section to do anti-two-dimensional Fourier transform the time observation. Due to discrete calculation, the mapping relates to interpolation and the original single mapping relation appears two interpolation strategy. One strategy is that equal interval points interpolate unequal interval points, another strategy that unequal interval points interpolate equal interval points is inverse to front strategy. To achieve the two interpolation, the strategy is feasible, but the strategy one involved subtraction, two similar number will affect the accuracy. In order to reserve the high frequency of the migration section ,the angular frequency interval of demigration section must assumeΔω=Δω' . If small interval the high frequency components must be lost ,but when the migrated section some high frequency spectrum is zero, reduce angular frequency interval appropriate will not result in the loss frequency components. The object of interpolation is complex, so this is a complex interpolation in real domain, hence we put forward three methods which respective are real and image respective interpolation, module and phase respective interpolation, Frequency domain direct interpolation. By analyzing impulse demigration model, the precision of real and image respective interpolation is restricted by the shake factor, so when removing it ,the precision have a great improvement, but there also have limitations. First, this method is relevant with reflective of the location, in other words must first know the specific location of the reflector. Second, when the reflection of the level the impact of shake factor can be completely removed, but the layers are tilted reflection, the impact can not be completely removed. As a result of phase is a discontinuity function, precision of module and phase respective interpolation is lower. Frequency domain direct interpolation is a undistorted interpolation in theory. However we usually need truncation for calculation efficiency, this will introduce error. On the singularity point the function value will become infinitude for which the integral can be unstable, the approximate substitution can solute the problem, in this paper we adopt Tayler expandedness and pada expandedness.Demigrating the impulse migration image ,the demigration section is accord with ray theory on the travel time. Same as the migration there is period effect in the demigration, on account of Fourier transform. Add zeros used in this article and add attenuation trace are good methods for eliminating the period effect ,but double the length of the original data, horizontal and vertical direction need to fill zero, which caused a threefold increase in the amount of data, increase the amount of computation, this is also the current frequency - calculation of the wave number domain Bottlenecks. There are lots of applications with demigration. Forward modeling is one of them. Demiration and forward modeling is the biggest difference between the different starting point, the demigration from the migrated section to get observation section ,and forward modeling from the geological structure to observation section. Forward modeling by demigration ,first through geological structure to construct migration section then demigrate it. Calculation shows that forward modeling by demigration is feasible and have a certain prospect. Finally, different models are implemented by Kirchhoff demigration and F-K demigration ,both the result are basically the same .