This paper is a classic because it has introduced the migration in frequency domain or stolt s migration for seismic data, then adopted also for gpr data. The apex shifted radon transform asrt is an extension of the standard hyperbolic rt, with hyperbolic basis functions located at every point of a data gather. Start with the input wavefield px, z 0, t approximated by the cmp stack, and apply 2d fourier transform to get pk x, z 0. I the fourier transform dnas double helix, the sunspot cycle and the sawtooth signals of electronics can be reduced mathematically to a series of undulating curves. Citeseerx document details isaac councill, lee giles, pradeep teregowda. This process is necessary to overcome the limitations of geophysical methods imposed by areas of complex geology. The transmit delays are determimed from the specified speed % of sound param. An approach to sar imaging by means of nonuniform ffts. The second method in particular appears adaptable to threedimensional migration and migration before stack.
To gain computational speed by using the fast fourier transform fft, the data must be laid on a regular grid by interpolation or other gridding methods, as performed in geosciences or magnetic. Finally jean baptiste joseph fourier 17681830 showed that such an infinite sum, a fourier series, can represent any discontinuous function under general conditions 3. This brings new challenges to seismic data regularization algo. We applied it to the 3 d segeage salt velocity model. Integration property of fourier transform can be used to find the fourier transform of various singals. In this part of the workflow, we will experiment with seismic migration applied to the previously generated dmo stack of the nankai dataset. Efficient stolt migration for large nonuniform single. Stolts fk migration for plane wave ultrasound imaging ncbi. Kirchhoff migration and stolt fk migration with an inverse boundary scattering transform takuya sakamoto, member, ieee. As illustrated by equations 12 and, the stolt s migration includes a nonuniform fourier transform since the k. This is the first of four chapters on the real dft, a version of the discrete fourier transform that uses real numbers. Stolt wave equation migration is known to be simpler method at higher dips and frequencies.
The theoretical basis for prestack migration by equivalent offset gary f. The society of exploration geophysicists the gas research institute. The input to migration is the dmo stacked section figure. Theory of nonstationary linear filtering in the fourier. Migration by fourier transform geophysics geoscienceworld. These algorithms include migration techniques, sar processing, borniterative method, and diffraction tomography. Examples of the migration transform matrices for a single wavenumber are given in figures 4 and 5. Instead of utilizing finitedifference approximations, the twodimensional 2d fourier transform is the fundamental technique of this method. Timemigration velocity analysis by velocity continuation. Fftbased beamforming for ultrasonic imaging has been. An especially efficient algorithm can be derived by means of a coordinate transformation equivalent to that in the stolt frequencywavenumber migration.
As illustrated by equations 12 and, the stolts migration includes a nonuniform fourier transform since the k. This idea underlies a powerful analytical tool to calcu1ate a transform, just lis ten. The simulation results show that this method achieve good results. Blondel and muir, 1993 the sep intime group derived and tested a stolt migration and modeling method that employed the discrete fourier transform dft for nonuniform spacing, the slow fourier transform, to. The level is intended for physics undergraduates in their 2nd or 3rd year of studies. Multiply p by the scale factor which has the interpretation. The frequency wavenumber domain possibility is the poststack stolt migration operator stolt, 1978. The computational count for the fourier transform of the fk migration isonxntlognxlognt, and the scaling will costonxnt. The application of fourier analysis to forecasting the inbound call time series of a call centre bruce g. Seismic migration is the process by which seismic events are geometrically relocated in either space or time to the location the event occurred in the subsurface rather than the location that it was recorded at the surface, thereby creating a more accurate image of the subsurface. Iii stolt migration an example of a low precision usfft is stolt migration 15.
For example, the discrete radon transform 3 involves computation of sums uj xn l1 ul e. Beylkin 1 i introduction the fast fourier transform fft algorithm of cooley and tukey 7 requires sampling on an equally spaced grid which is a signi cant limitation in many applications. Derivation of integration property of fourier transform. After combining the dispersion relation with geometric features of the 2d fourier transform, migration can be performed in the fk domain with a 1d nonstationary shift. Kirchhoff migration the kirchhoff migration technique finds its origin in the field of seismics. Fk stolt migration o this method makes use of the 2d fourier transform to convert the input data from the x,t domain to the k,f domain, where k denotes wavenumber i. More recently, stolt 1978 used fourier transform techniques for migration. Direct fourier migration for vertical velocity variations crewes. The theoretical basis for prestack migration by equivalent. Fourier transforms properties here are the properties of fourier transform.
Frequency domain migration is also based upon a deterministic approach via the wave equation stolt, 1978. How the standard stolts method is adapted for plane wave insonifications is demonstrated in section 2. The ear automatically per forms the calcu1ation, which the. The particular migration algorithm that we will use is stolt migration based on the fourier transform stolt, 1985. A normal 2d inverse fft recovers the migrated data from the. The stolt interpolation is followed by a 1d inverse fourier transform along the same dimension of. Unequally spaced fft and fast radon transform gregory beylkin university of colorado at boulder ipam september 14, 2004. From this early work connecting the wave equation and the fourier transform, much of engineering mathematics of wave motion and transformations has been developed. One of his subsequent major contributions was the extension of his original fk migration algorithm to provide the only comprehensive, inclusive and. We then generalise that discussion to consider the fourier transform. In the examples of the concepts and algorithms presented here, the migration of a. The apex shifted radon transform asrt is an extension of the standard hyperbolic.
In terms of the frequencywavenumber fk spectrum mapping functions and 2d fast fourier transform fft, the stolt based ashrt is much faster than ashrt implemented in the time domain trad et al2012, ibrahim and sacchi 2015, gong et al2016a. Center for wave phenomena colorado school of mines golden, co 80401, usa past support was received from. The splitstep fourier method is developed and applied to migrating stacked seismic data in two and three dimensions. Yarovoy, senior member, ieee abstractin this paper, we propose a fast and accurate radarimaging algorithm that combines kirchhoff migration with stolts. Kirchhoff and stolt migration signal and image centre. Although the kirchhoff migration has been developed for the backpropagation of scalar pressure wavefields, it is often applied. It is important to understand that this approach is an approximation and may. This fast algorithm, valid for q constant with depth, can be extended to accommodate depthvariable q by. A brief introduction to the fourier transform this document is an introduction to the fourier transform. The second in principle when the horizontal coordinate or coor scheme effects a fourier transform in both space and dinates are replaced by their fourier conjugates. Papoulis systems and transforms with applications in optics new york. Kirchhoff migration and stolt fk migration with an inverse boundary scattering transform takuya sakamoto, member, ieee, toru sato, member, ieee, pascal j. The second scheme effects a fourier transform in both space and time. History of the wave equation and transforms in engineering.
Stolt migration vaillant and fomel, 1999, and stolt residual migration sava, 1999b,a. The method forms common scatter point csp gathers for each migrated trace and then images those gathers with a migration algorithm. If we compute anglegathersin conjunction with stolt residual migration, this method becomes even more attractive since the seismic images are already transformed to the fourierdomain,which makes the true cost of the transformation insigni. Migration with fourier transforms was also studied by claerbout 1977 and lynn 1977.
In this formulation, prestack stolt residual migration is a constantvelocity ratio method, not a constant velocity method. Prestack residual migration in the frequency domain. Jump to content jump to main navigation jump to main navigation. Vz fk migration crewes research report volume 10 1998 361 direct fourier migration for vertical velocity variations gary f. Directional ltering of seismograms using slant stack radon transform. Application of poststack migration to seismic data. Margrave abstract the stolt fk migration algorithm is a direct i. In these studies, finite fourier transforms are employed for obtaining a direct solu tion of the wave equation. Antileakage fourier transform for seismic data regularization. A general linear theory describes the extension of the convolutional method to nonstationary processes. Robert stolt was awarded the reginald fessenden award for his pioneering landmark contribution. Interpolate p onto a new mesh so that it is a function of k x and k z. Inverse q filtering by fourier transform geophysics. On applications of unequally spaced fast fourier transforms g.
By applying the nonuniform fast fourier transform fft to the acquisition of the frequencywavenumber spectrum fws, the efficiency of stolt migration for nonuniform surveys is improved. The filter application equations and the expressions to move the filter. This theory can apply any linear, nonstationary filter, with arbitrary time and frequency variation, in the time, fourier, or mixed domains. Deblending using an improved apexshifted hyperbolic radon. These algorithms do not explicitly model multiple reflections, converted waves, surface waves, or noise. Any such energy present in data input to migration is treated as primary reflections. The mathematical description of such an operator is similar to the kinematic post stack time migration equation, with the horizontal coordinate being not midpoint but offset. Stolt s algorithm for constant velocity thus involves the following steps.
Even though ground penetrating radar has been well studied and applied by many researchers for the last couple of decades, the focusing problem in the measured gpr images is still a challenging task. On the fast fourier transform of functions singularities, 1995. The fk migration for pwi is inspired by the original fourier migration introduced by stolt for seismic imaging 10, 11. I introduction the fast fourier transform fft algorithm of cooley and tukey 7 requires sampling on an equally spaced grid which is a signi cant limitation in many applications. In this paper, nonuniform fast fourier transform nufft algorithm is introduced to realize the fast computing of integral formula in image reconstruction directly. Although there are many methods offered by different scientists, there is not any complete migration focusing method that works perfectly for all scenarios. High frequency random noise attenuation on shallow seismic. Introduction stolt migration is one of the useful gpr imaging algorithms which has the advantage of fast.
Wave equation migration is known to be simpler method at higher dips and. The fk migration for pwi has been adapted from the stolt migration for. Migration algorithms can be classified under three main categories. Although the kirchhoff migration has been developed for the backpropagation of scalar pressure wavefields, it is often applied with success to electromagnetic waves. This migration method, which is implemented in both the frequencywavenumber. The multivelocity version proposed by ibrahim also employs this stolt based operator in each. The theoretical basis for prestack migration by equivalent offset. Ultrawideband radar imaging using a hybrid of kirchhoff. Stolts fk migration for plane wave ultrasound imaging. Full text views reflects the number of pdf downloads. The stolt migration is used and is introduced in section iv. Subsurface imaging using groundpenetrating radar measurements.
Combining fk filter with minimum entropy stolt migration. Direct fourier migration for vertical velocity variations. The fourier dual algorithm to eom, called equivalent wavenumber. Denoising and migration techniques for target identi.
The advantage is apparent for cascaded migration or migration with multiple velocity models. Mar 11, 2017 integration property of fourier transform is discussed in this video. Applying a 2d fourier transform with respect to the spatial distance x and the time t to spatial frequency kx, the result is an unfocused wavenumber. Bellc answ police assistance line, tuggerah, nsw 2259, email. Samples of stolt migration of impulses are shown in figure 9. Pdf stolts fk migration for plane wave ultrasound imaging. Introduction groundpenetrating radar has many applications ranging from detection of buried pipes to nondestructive probing for contaminants to mapping of the subsurface layers. Stolt, migration by fourier transform, geophysics, vol. Fkmig does not use the transmit time delays as input % parameters. In these cases, the cost of stolt migration increases in direct proportion to the number of velocity models, while the cost of velocity continuation stays the same. The residual migration transforms using a scaled version of the original velocity. The application of fourier analysis to forecasting the. Cambridge core solid earth geophysics seismic imaging and inversion by robert h. Siam journal on scientific computing society for industrial.
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