Application of POCS algorithm for the reconstruction of three-component seismic data in quaternion Fourier domain

Document Type : Research Article


1 M.Sc. Graduated, Department of Geophysics, Research and Science Branch, Islamic Azad University, Tehran, Iran

2 Professor, Department of Earth Physics, Institute of Geophysics, University of Tehran, Tehran, Iran


Three-component (3C) seismic data acquisition method samples seismic wave field at each station along three Cartesian coordinates, simultaneously. Many reservoirs have been discovered and determined by the generation and recording of P waves only, but the P wave alone cannot provide a comprehensive description of the reservoir characteristics. In some studies, S-wave information is required in addition to P-wave information to get a correct estimation from reservoir properties. By the three component seismic acquisition, P and S waves’ information can be recorded simultaneously. More often in seismic surveys, one cannot sample seismic wave field uniformly the along spatial direction due to environment limitations or instrument malfunctions; inevitably we have to use interpolation methods for reconstruction of missing traces. Reconstruction of missing or noisy traces is done using the projection onto convex sets (POCS). The POCS algorithm is a simple algorithm which is suitable for reconstruction of irregularly lost traces in a regular grid using multiple repetitive Fourier transforms. Conventional methods for reconstruction of missing traces in three component acquisition is usually done by implementation of POCS on each component separately, which could damage any subtle features in the record. This research introduces a method to reconstruct all three components at once using the quaternion Fourier transform and Projection onto Convex Sets (QPOCS).Quaternions in mathematics are a commutative numbers system that extend the complex numbers system. As the ordinary complex numbers can be displayed on two dimensions, these numbers can also be displayed on four dimensions. Quaternions were first introduced by William Rowan Hamilton when looking for a way to extend complex numbers to three dimensions. He knew how to sum and multiply three-dimensional numbers, but he was looking for a way to divide these numbers into each other. In 1843, Hamilton discovered that the division of quaternions requires a fourth dimension. Quaternion Algebra is often shown with H (in honor of Hamilton). The two-component data vector representation in the frequency domain can be obtained by putting the real and imaginary parts of each component in the arguments of a quaternion. This method allows operators to apply both components simultaneously. Quaternions are converted to Frequency-wavenumber domain by Quaternion Fourier Transform (QFT) and a single domain spectrum for both components is defined using the polar representation of the Quaternions. Quaternions have other applications in seismic data processing such as computing spectral attributes, multi-component velocity analysis and multi-component deconvolution. The advantage of this method is because of the spectral overlapping of the components in the frequency-wavenumber domain, thus the perpendicularity of input components is preserved (signals are not interconnected) and similarities between components are maintained that helps improve the quality of reconstruction. The coding of this method has been done in MATLAB environment and results of applying the proposed method on 3-component synthetic and real seismic data are compared to that of the POCS algorithm when applied on each component separately. The results of reconstruction using QPOCS algorithm indicate a better quality for reconstructed seismic data and in the output data, the percentage of produced artifacts is lower than that of the POCS algorithm on each component alone.


Main Subjects

Abma, R. and Kabir, N., 2006, 3D interpolation of irregular data with a POCS algorithm, Geophysics, 71, 91–96.
Bihan, N. L. and Mars, J. I., 2001, New 2D complex and hypercomplex seismic attributes, Presented at the 71st Conference of the Society of Exploration Geophysicists, SEG.
Ell, T., 1992, Hypercomplex spectral transformations, PhD thesis, University of Minnesota.
Ell, T. A., 1993, Quaternion-fourier transforms for analysis of two-dimensional linear time-invariant partial-differential systems, 32nd IEEE Conf. Decision and Control, 1830–1841.
Ell, T. A. and Sangwine, S. J., 2007, Hypercomplex fourier transforms of color images, IEEE Trans, Image Process, 16, 22–35.
Galloway, E. and Sacchi, M. D., 2007, POCS method for seismic data reconstruction of irregularly sampled data: CSPG CSEG Convention, 555.
Gao, J. J., Chen, X. H., Li, J. Y. and Liu, G.C., 2010, Irregular seismic data reconstruction based on exponential threshold model of pocs method, Applied Geophysics, 7, 229–238.
Gao, J. and Sacchi, M. D., 2011, Convergence improvement and noise attenuation considerations for POCS reconstruction, Presented at the 73rd EAGE conference and exhibition.
Gao, J., Stanton, A., Naghizadeh, M., Sacchi, M. and Chen, X., 2013, Convergence improvement and noise attenuation considerations for beyond alias projection onto convex sets reconstruction, Geophysical Prospecting, 61, 138–151.
Gerchberg, R. W. and Saxton, W. O., 1972, A practical algorithm for the determination of phase from image and diffraction plane pictures, Optik 35, 227–246.
Grandi, A., Mazzotti, A. and Stucchi, E., 2007, Multicomponent velocity analysis with quaternions, Geophysical Prospecting, 55, 761–777.
Hamilton, W., 1866, Elements of quaternions, Longmans Green.
Jiang, T., Gong, B., Qiao, F., Jiang, Y., Chen, A., Hren, D. and Meng, Y., 2017, Compressive seismic reconstruction with extended POCS for arbitrary irregular acquisition, SEG Technical Program Expanded Abstracts, 4272-4277, doi:10.1190/segam2017-17632472.1.
Krieger, L. and Grigoli, F., 2015, Optimal reorientation of geophysical sensors, A quaternion based analytical solution, Geophysics, 80(2), 19-30.
Menanno, G. M. and Mazzotti A., 2012, Deconvolution of multicomponent seismic data by means of quaternions: Theory and preliminary results, Geophysical Prospecting, 60(2), 217-238.
Pinilla, J., Etcheverlepo, A. and Ojeda, G., 2016, A piecewise linear threshold model for five-dimensional interpolation of seismic data using POCS method, SEG Technical Program Expanded Abstracts, 4134-4138, doi:10.1190/
Sangwine, S. and Bihan, N., 2005, Quaternion Toolbox for Matlab, Software Library, Available at
Sangwine, S. J. and Ell, T. A., 2000, The discrete fourier transform of a colour image: Proc. Image Processing II Mathematical Methods, Algorithms and Applications, 430–441.
Stanton, A. and Sacchi, M., 2011, Multicomponent seismic data reconstruction using the quaternion Fourier transform and POCS. SEG Technical Program Expanded Abstracts: pp. 1267-1272.
Stanton, A. and Sacchi, M., 2013, Vector reconstruction of multicomponent seismic data, Geophysics, 78(4), 131-145.
Wang, S. Q., Xing, G. and Yao, Z. X., 2010, Accelerating POCS interpolation of 3d irregular seismic data with graphics processing units: Computers & Geosciences, 36, 1292–1300.
Witten, B. and Shragge, J., 2006, Quaternion-based signal processing, Presented at the 76th Conference of the Society of Exploration Geophysicists, SEG.