Frequency extrapolation through Sparse sums of Lorentzians

Abstract

Sparse sums of Lorentzians can give good approximations to functions consisting of linear combination of piecewise continuous functions. To each Lorentzian, two parameters are assigned: translation and scale. These parameters can be found by using a method for complex frequency detection in the frequency domain. This method is based on an alternating projection scheme between Hankel matrices and finite rank operators, and have the advantage that it can be done in weighted spaces. The weighted spaces can be used to partially revoke the effect of finite band-width filters. Apart from frequency extrapolation the method provides a way of estimating discontinuity locations.

Keywords

sparse sum / Lorentzians / Hankel matrices / finite rank operator / discontinuity location

References

[1] Adamjan V M, Arov D Z, Kreın M G. Infinite Hankel Matrices and Generalized Carathéodory-Fejér and Riesz Problems. Funkcional. Anal. iPriložen., 1968, 2(1): 1-19.

[2] Andersson F, Carlsson M. Alternating Projections on Non-Tangential Manifolds. Constructive Approximation, 2013, 38(3): 489-525.

[3] Andersson F, Carlsson M, de Hoop M V. Sparse Approximation of Functions Using Sums of Exponentials and Aak Theory. Journal of Approximation Theory, 2011, 163(2): 213-248.

[4] Andersson F, Carlsson M. A Fast Alternating Projection Method for Complex Frequency Estimation. Proceedings of the GMIG, 2011, 11: 21-36.

[5] Beylkin G, Monzón L. Nonlinear Inversion of a Band-Limited Fourier Transform. Applied and Computational Harmonic Analysis, 2009, 27(3): 351-366.

[6] Beylkin G, Monzón L. On Approximation of Functions by Exponential Sums. Applied and Computational Harmonic Analysis, 2005, 19(1): 17-48.

[7] Chartrand R, Sidky E Y, Pan X. Frequency Extrapolation by Nonconvex Compressive Sensing. Biomedical Imaging: From Nano to Macro, 2011. IEEE International Symposium on IEEE, 2011, 1056-1060.

[8] Claerbout J. Earth Soundings Analysis Processing Versus Inversion, 1992 Cambridge: Blackwell Scientific Publications

[9] Claerbout J. Multidimensional Recursive Filters Via a Helix. Geophysics, 1998, 63(5): 1532-1541.

[10] Clayton R W, Wiggins R A. Source Shape Estimation and Deconvolution of Teleseismic Bodywaves. Geophysical Journal of the Royal Astronomical Society, 1976, 47(1): 151-177.

[11] Dasgupta S, Nowack R L. Frequency Extrapolation to Enhance the Deconvolution of Transmitted Seismic Waves. Journal of Geophysics and Engineering, 2008, 5(1): 118-127.

[12] Eckart C, Young G. The Approximation of One Matrix by Another of Lower Rank. Psychometrika, 1936, 1(3): 211-218.

[13] Escalante C, Gu Y J, Sacchi M. Simultaneous Iterative Time-Domain Sparse Deconvolution to Teleseismic Receiver Functions. Geophysical Journal International, 2007, 171(1): 316-325.

[14] Horn R A, Johnson C R. Topics in Matrix Analysis, 1994 Cambridge: Cambridge University Press

[15] Ligorria J P, Ammon C J. Iterative Deconvolution and Receiver-Function Estimation. Bulletin of the Seismological Society of America, 1999, 89(5): 1395-1400.

[16] Papoulis A. A New Algorithm in Spectral Analysis and Band-Limited Extrapolation. Circuits and Systems, IEEE Transactionson, 1975, 22(9): 735-742.