## Thesis etd-09082014-180832 |

Thesis type

Tesi di laurea magistrale

Author

XING, ZHEN

URN

etd-09082014-180832

Thesis title

MPI on sparse Radon transform

Department

SCIENZE DELLA TERRA

Course of study

GEOFISICA DI ESPLORAZIONE E APPLICATA

Supervisors

**correlatore**Prof. Mazzotti, Alfredo

**controrelatore**Prof. Ferrante, Isidoro

**relatore**Prof. Stucchi, Eusebio Maria

Keywords

- Radon transform
- aliasing
- high resolution
- MPI
- ProMAX

Graduation session start date

26/09/2014

Availability

Withheld

Release date

26/09/2084

Summary

In exploration geophysics the Radon transform is applied to seismic data to enhance the signal to noise ratio, to interpolate the data and to attenuate the multiples. Because of its capability in separating the primaries from the multiples, which is a fundamental aspect in the multiple attenuation process, the Radon transform is widely used especially in the processing of seismic marine data.

It can be demonstrated that the ideal linear Radon transform projects seismic reflections into ellipses while the ideal parabolic Radon transform projects the NMO corrected seismic reflections into points. In my thesis I primarily studied the parabolic Radon transform.

Two main problems that arise in the Radon transform computation are the artifacts generated in the transform domain and the long computing time.

The artifacts are generated from two sources. The first is related with a too coarse transform domain sampling rate or a too wide sampling range, the other is due to the limited offset range at near and far offsets. Different sampling criterions are studied to deal with the former source. These criterions have a common statement that a bigger frequency coincides with a limit of smaller sampling rate and range. So if we do a filtering before Radon transform, we can have a larger sampling rate and range in the transform domain. But as a smaller frequency band means a lower resolution, we should only filter out the frequencies between the maximum useful frequency of the signal and the Nyquist frequency. The latter source generates butterfly structures in the parabolic Radon transform domain. The compression of the butterfly structures to points is related with the high signal to noise ratio in time-offset (t-x) domain data. The performance of different methods in the compression of the butterfly structures is studied and compared in the thesis. These methods are the frequency domain sparse Radon transform, the time-frequency domain sparse Radon transform and the iterative shrinkage sparse Radon transform. Moreover, various approaches can be used to solve the system of equations leading to the transform coefficients, namely: the damped least squares, the singular value decomposition and the conjugate gradient method. The singular value decomposition is the most stable method, while the conjugate gradient method can compress the butterfly structures better than the other two. In the thesis the conjugate gradient algorithm is used in the frequency domain sparse and the time-frequency domain sparse Radon transform. The two sparse methods use iterative reweighted least squares to get sparse results. The iterative shrinkage sparse Radon transform can use an any Radon transform method to calculate the initial model and then uses the shrinkage operator to derive a sparser result.

By applying these three sparse methods to a synthetic data, we see that the frequency domain sparse parabolic Radon transform gives the smallest signal to noise ratio and the worst vertical and horizontal resolution in the result, but it has the smallest amplitude attenuation along the offset axis in t-x domain. This method also preserves at best the weak seismic events in transform domain.

Applying the forward and inverse Radon transform by using the time-frequency domain sparse parabolic Radon transform, we get the seismic data with the highest signal to noise ratio and the best vertical and horizontal resolution. But with this method the weak seismic events can be barely seen in the transform domain. Also, it generates the strongest amplitude attenuation along the offset axis in t-x domain.

The iterative shrinkage sparse parabolic Radon transform is able to derive a seismic data with a reasonable signal to noise ratio and at the same time it has the capability to preserve some weak seismic events. Compared with the time-frequency domain sparse parabolic Radon transform, it generates weaker amplitude attenuation along the offset axis in t-x domain.

To deal with the long computing time problem, I used the message passing interface (MPI) to parallelize the iterative shrinkage sparse Radon transform algorithm.

I used the MPI package (Version 1.2.0) from Octave Forge to do the parallel computing. Although sending structures or sending complex numbers are not originally supported by the package, with the help of cell arrays they can be realized.

I exploited two for loops in the iterative shrinkage sparse Radon transform algorithm to do the parallelization. In the parallel computing code the master node divided the input into pieces. At the beginning it sent each slave node one piece. When a slave node had finished calculating and had sent the results back to the master node, the latter would send a new piece to it if there were still pieces of input left.

When we used the damped least squares method to calculate the initial model of the iterative shrinkage sparse Radon transform, the parallel part occupied an extremely small part of the whole calculation. In this case the parallel computing code could not be faster than the direct calculation code according to Amdahl's law.

When we used the frequency domain sparse Radon transform to calculate the initial model in the iterative shrinkage sparse Radon transform, the acceleration of the computing by using MPI was evident. With 6 processors we achieved a 3.17x speedup when we set the iteration in the frequency domain sparse Radon transform part as 200 and split the frequencies into 25 parts.

Although in the thesis the parallel computing by using MPI is only applied to the iterative shrinkage sparse Radon transform algorithms, it can be applied to other Radon transform algorithms in the same manner.

The selective use of the zero padding during the use of the fast Fourier transform algorithm, and the exploitation of the property of the Toeplitz matrix and the circulant matrix are also important for the acceleration of the Radon transform algorithms.

The demultiple abilities of the frequency domain sparse parabolic Radon transform, the time-frequency domain sparse parabolic Radon transform, the iterative shrinkage sparse parabolic Radon transform from my codes, and the frequency domain sparse parabolic Radon transform, the standard parabolic Radon transform from ProMAX were compared by using a synthetic marine seismic data. Although the time-frequency domain sparse parabolic Radon transform and the iterative shrinkage sparse parabolic Radon transform from my codes were able to give much better horizontal and vertical resolution than the other Radon transform methods, their incapabilities in showing weak seismic events and the producing of the strong attenuation of the amplitudes along the offset axis in t-x domain made them nearly unusable in attenuating multiples in the marine seismic data. Neither could the frequency domain sparse parabolic Radon transform from ProMAX give a satisfactory demultiple result. The standard parabolic Radon transform algorithm from ProMAX and my frequency domain sparse parabolic Radon transform algorithm could attenuate the multiples efficiently. The former was more efficient in attenuating far offset multiples than the latter, but the former generated more artifacts at near offset. Both of the two algorithms were less efficient at the end point when there was an abrupt end of a multiple in seismogram.

I tested whether the spherical divergence correction should be done before or after Radon transform. The results showed that the latter case was not better than the former.

Both my frequency domain sparse parabolic Radon transform code and the standard parabolic Radon transform algorithm from ProMAX were applied to a real marine data to attenuate the multiples. The multiples were attenuated effectively.

It can be demonstrated that the ideal linear Radon transform projects seismic reflections into ellipses while the ideal parabolic Radon transform projects the NMO corrected seismic reflections into points. In my thesis I primarily studied the parabolic Radon transform.

Two main problems that arise in the Radon transform computation are the artifacts generated in the transform domain and the long computing time.

The artifacts are generated from two sources. The first is related with a too coarse transform domain sampling rate or a too wide sampling range, the other is due to the limited offset range at near and far offsets. Different sampling criterions are studied to deal with the former source. These criterions have a common statement that a bigger frequency coincides with a limit of smaller sampling rate and range. So if we do a filtering before Radon transform, we can have a larger sampling rate and range in the transform domain. But as a smaller frequency band means a lower resolution, we should only filter out the frequencies between the maximum useful frequency of the signal and the Nyquist frequency. The latter source generates butterfly structures in the parabolic Radon transform domain. The compression of the butterfly structures to points is related with the high signal to noise ratio in time-offset (t-x) domain data. The performance of different methods in the compression of the butterfly structures is studied and compared in the thesis. These methods are the frequency domain sparse Radon transform, the time-frequency domain sparse Radon transform and the iterative shrinkage sparse Radon transform. Moreover, various approaches can be used to solve the system of equations leading to the transform coefficients, namely: the damped least squares, the singular value decomposition and the conjugate gradient method. The singular value decomposition is the most stable method, while the conjugate gradient method can compress the butterfly structures better than the other two. In the thesis the conjugate gradient algorithm is used in the frequency domain sparse and the time-frequency domain sparse Radon transform. The two sparse methods use iterative reweighted least squares to get sparse results. The iterative shrinkage sparse Radon transform can use an any Radon transform method to calculate the initial model and then uses the shrinkage operator to derive a sparser result.

By applying these three sparse methods to a synthetic data, we see that the frequency domain sparse parabolic Radon transform gives the smallest signal to noise ratio and the worst vertical and horizontal resolution in the result, but it has the smallest amplitude attenuation along the offset axis in t-x domain. This method also preserves at best the weak seismic events in transform domain.

Applying the forward and inverse Radon transform by using the time-frequency domain sparse parabolic Radon transform, we get the seismic data with the highest signal to noise ratio and the best vertical and horizontal resolution. But with this method the weak seismic events can be barely seen in the transform domain. Also, it generates the strongest amplitude attenuation along the offset axis in t-x domain.

The iterative shrinkage sparse parabolic Radon transform is able to derive a seismic data with a reasonable signal to noise ratio and at the same time it has the capability to preserve some weak seismic events. Compared with the time-frequency domain sparse parabolic Radon transform, it generates weaker amplitude attenuation along the offset axis in t-x domain.

To deal with the long computing time problem, I used the message passing interface (MPI) to parallelize the iterative shrinkage sparse Radon transform algorithm.

I used the MPI package (Version 1.2.0) from Octave Forge to do the parallel computing. Although sending structures or sending complex numbers are not originally supported by the package, with the help of cell arrays they can be realized.

I exploited two for loops in the iterative shrinkage sparse Radon transform algorithm to do the parallelization. In the parallel computing code the master node divided the input into pieces. At the beginning it sent each slave node one piece. When a slave node had finished calculating and had sent the results back to the master node, the latter would send a new piece to it if there were still pieces of input left.

When we used the damped least squares method to calculate the initial model of the iterative shrinkage sparse Radon transform, the parallel part occupied an extremely small part of the whole calculation. In this case the parallel computing code could not be faster than the direct calculation code according to Amdahl's law.

When we used the frequency domain sparse Radon transform to calculate the initial model in the iterative shrinkage sparse Radon transform, the acceleration of the computing by using MPI was evident. With 6 processors we achieved a 3.17x speedup when we set the iteration in the frequency domain sparse Radon transform part as 200 and split the frequencies into 25 parts.

Although in the thesis the parallel computing by using MPI is only applied to the iterative shrinkage sparse Radon transform algorithms, it can be applied to other Radon transform algorithms in the same manner.

The selective use of the zero padding during the use of the fast Fourier transform algorithm, and the exploitation of the property of the Toeplitz matrix and the circulant matrix are also important for the acceleration of the Radon transform algorithms.

The demultiple abilities of the frequency domain sparse parabolic Radon transform, the time-frequency domain sparse parabolic Radon transform, the iterative shrinkage sparse parabolic Radon transform from my codes, and the frequency domain sparse parabolic Radon transform, the standard parabolic Radon transform from ProMAX were compared by using a synthetic marine seismic data. Although the time-frequency domain sparse parabolic Radon transform and the iterative shrinkage sparse parabolic Radon transform from my codes were able to give much better horizontal and vertical resolution than the other Radon transform methods, their incapabilities in showing weak seismic events and the producing of the strong attenuation of the amplitudes along the offset axis in t-x domain made them nearly unusable in attenuating multiples in the marine seismic data. Neither could the frequency domain sparse parabolic Radon transform from ProMAX give a satisfactory demultiple result. The standard parabolic Radon transform algorithm from ProMAX and my frequency domain sparse parabolic Radon transform algorithm could attenuate the multiples efficiently. The former was more efficient in attenuating far offset multiples than the latter, but the former generated more artifacts at near offset. Both of the two algorithms were less efficient at the end point when there was an abrupt end of a multiple in seismogram.

I tested whether the spherical divergence correction should be done before or after Radon transform. The results showed that the latter case was not better than the former.

Both my frequency domain sparse parabolic Radon transform code and the standard parabolic Radon transform algorithm from ProMAX were applied to a real marine data to attenuate the multiples. The multiples were attenuated effectively.

File

Nome file | Dimensione |
---|---|

There are some hidden files because of the review of the procedures of theses' publication. |