ARMA order model detection using minimum of Kurtosis: application on seismic data

  • Hocine BellahseneEmail author
  • Abdelmalik Taleb-Ahmed
Review Paper


The objective of this study is to find the order and coefficients of non-low-phase causal filters for ARMA (auto regressive moving average) filter model, using the Kurtosis minimization criterion. This method is based on the Kurtosis calculation of the treated sample at the input level and its identification at the output of the ARMA model. For this purpose, the order and coefficients of the AR (auto regressive) part are identified using the Yule-Walker algorithm at order two and then extended to order four. To obtain the MA (moving average) part, the AR components are calculated at first from the ARMA filter by deconvolution. Then, spectrally equivalent and minimum phase (SEMP) MA filter is identified using the Durbin algorithm at second and fourth order. Finally, the correct filter is found when the Kurtosis value of the output ARMA filter reconstituted is the closest to the Kurtosis of introduced signal. The proposed method is then tested on simulated processes and applied to real seismic data to perform blind deconvolution and obtain the reflectivity coefficients of subsoil studied.


Fourth-order cumulants ARMA filters Kurtosis Blind identification Seismic traces Reflectivity coefficients 



The first author would like to express his sincere gratitude to Professor Jean Michel Rouvaen, his co-supervisor in doctoral studies, for his supervision and advice in the IEMN laboratory. He also thanks his former teacher, Dr. Chahed Idir.


  1. Akaike H (1973) Information theory and an extension of the maximum likelihood principle. In: Proceed 2nd Internat’l Symp on Inform Theory, Budapest (Hung), pp 267–281Google Scholar
  2. Bellahsene H, Chahed I, Djeddi M (1999) Identification de filtres ARMA à phase non minimale par maximum de kurtosis. In: Proceedings of the 2nd Internat. Conf on Electron, Signals, Systems and Automatics, Algeria, pp 107–111Google Scholar
  3. Bellahsene H, Fatani IFE (2011) Utilisation des HOS pour le calcul du filtre optimal et de la rduction du temps de convergence en DMT, ICGST, ACSE journal on computer and engineering, pp 69–74, Istanbul, TurkeyGoogle Scholar
  4. Boumahdi M, Lacoume JL (1994) Blind identification of FIR systems using the kurtosis. In: Proceedings of the 7th workshop on statistical signal and array processing, Québec (Can), pp 191–194Google Scholar
  5. Boumahdi M, Lacoume JL (1994) Blind identification of non-minimum phase filters by maximising the kurtosis, EUSIPCO 1994 - Special HOS session, Edinburgh, UKGoogle Scholar
  6. Broersen PMT (1998) The Quality of Models for ARMA Processes. IEEE Trans Signal Process 46:1749–1752CrossRefGoogle Scholar
  7. Chi CY, Chen C-Y, Chen C-H, Feng C-C (2003) Batch processing, Algorithms for blind equalization using higher order statistics. IEEE Signal Process Mag 20:25–49CrossRefGoogle Scholar
  8. Comon P (1992) MA identification using fourth order cumulants. Signal Process 26:381–388CrossRefGoogle Scholar
  9. Haseyama M, Kitajima H (2000) An ARMA order selection method with fuzzy reasoning. Signal Process 81:1331–1337CrossRefGoogle Scholar
  10. Mendel JM (1991) Tutorial on higher order statistics (spectra). Proc IEEE 79:278–305CrossRefGoogle Scholar
  11. Moddermeijer R (1999) Testing composite hypotheses applied to AR order estimation; the Akaike criterion revisited, Signal Proc., Symp Proceed, Leuven (Belgium), pp 135–138Google Scholar
  12. Sayed Ali (2008) Adaptive filters. Wiley, HobokenCrossRefGoogle Scholar
  13. Shuichi O, Hideaki S, Hironori Y (1999) Adaptive blind equalization of multichannel FIR systems. Proc IEEE, Internat’l Symp Circuits Systems 3:66–73Google Scholar
  14. Tugnait JK (1985) ARMA spectral estimation. IEEE Trans Acoust Speech Signal Proc ASSP–23:160–163CrossRefGoogle Scholar
  15. Wang L, Libert GA (1994) Combining pattern recognition techniques with Akaike’s information criterion for identifying ARMA models. IEEE Trans Signal Proc 42:1388–1394CrossRefGoogle Scholar

Copyright information

© Saudi Society for Geosciences 2018

Authors and Affiliations

  1. 1.L.Z.AFSNV Université de BejaiaBejaiaAlgeria
  2. 2.I.E.M.N., D.O.A.E., (U.M.R., C.N.R.S.8520 U.P.H.F.)Université de ValenciennesValenciennes CedexFrance

Personalised recommendations