Acta Geodaetica et Geophysica

, Volume 54, Issue 4, pp 483–497 | Cite as

A heuristic singular spectrum analysis method for suspended sediment concentration time series contaminated with multiplicative noise

  • Fengwei Wang
  • Yunzhong ShenEmail author
  • Qiujie Chen
  • Weiwei Li
Original Study


This paper proposes a heuristic singular spectrum analysis (SSA) approach to extract signals from suspended sediment concentration (SSC) time series contaminated by multiplicative noise, in which multiplicative noise is converted to approximate additive noise by multiplying with the signal estimate of the time series. Therefore both the signal and noise components need to be recursively estimated. Since the converted additive noise is heterogeneous, a weight factor is introduced according to the variance of additive noise. The proposed heuristic SSA approach is employed to process the SSC series in San Francisco Bay compared to the traditional SSA and homomorphic log-transformation SSA approach. By using our heuristic SSA approach, the first 10 principal components derived can capture 96.49% of the total variance with the fitting error of 6.17 mg/L, better than those derived by traditional SSA approach and homomorphic log-transformation SSA approach that catch 88.97% and 87.35% of the total variance with the fitting errors of 14.47 mg/L and 15.03 mg/L, respectively. Therefore, our heuristic SSA approach can extract more signals than traditional SSA and homomorphic log-transformation SSA approach. Furthermore, the results from the simulation cases show that all the mean root mean squared errors and mean absolute errors derived by our heuristic SSA are smaller than the traditional and homomorphic log-transformation SSA, which indicate that the extracted signals by heuristic SSA approach are much closer to the real signals than those by the other two approaches. Therefore it can be conclude that our heuristic SSA approach performs better in extracting signals from SSC time series contaminated with multiplicative noise.


Singular spectrum analysis Suspended sediment concentration Time series Multiplicative noise Missing data 



This work is mainly sponsored by the National Key R&D Program of China (2017YFA0603103) and the Natural Science Foundation of China (Projects: 41731069 and 41274035). We are grateful to the anonymous reviewers for their constructive comments.


  1. Adarsh S, Reddy MJ (2015) Multiscale analysis of suspended sediment concentration data from natural channels using the Hilbert–Huang transform. Aquat Procedia 4:780–788CrossRefGoogle Scholar
  2. Adarsh S, Reddy MJ (2016) Multiscale characterization of streamflow and suspended sediment concentration data using Hilbert–Huang transform and time dependent intrinsic correlation analysis. Model Earth Syst Environ 2(4):199. CrossRefGoogle Scholar
  3. Adelmann H (1998) Butterworth equations for homomorphic filtering of images. Comput Biol Med 28(2):169–181CrossRefGoogle Scholar
  4. Ambrosino F, Thinová L, Briestenský M et al (2019) Anomalies identification of Earth’s rotation rate time series (2012–2017) for possible correlation with strong earthquakes occurrence. Geod Geodyn. CrossRefGoogle Scholar
  5. Aubert G, Aujol J (2008) A variational approach to removing multiplicative noise. SIAM J Appl Math 68(4):925–946CrossRefGoogle Scholar
  6. Barrios-Muriel J, Romero F, Alonso FJ et al (2016) A simple SSA-based de-noising technique to remove ECG interference in EMG signals. Biomed Signal Proc Cont 30:117–126CrossRefGoogle Scholar
  7. Broomhead D, King G (1986) Extracting qualitative dynamics from experimental data. Phys D 20(2–3):217–236CrossRefGoogle Scholar
  8. Buchanan PA, Schoellhamer DH (1998) Summary of suspended-solids concentration data, San Francisco Bay, california, water year 1996. Cent Integr Data Anal Wis Sci Cent 94(5):33–35Google Scholar
  9. Buchanan PA, Ruhl CA (2000) Summary of suspended-solids concentration data, San Francisco Bay, California, water year 1998, open file report 99-189, U.S. Geological Survey, 41 ppGoogle Scholar
  10. Buchanan PA, Schoellhamer DH (1999) Summary of suspended solids concentration data, San Francisco Bay, California, water year 1997, open file report 00-88, U.S. Geological Survey, 52 pp.
  11. Cheng RT, Gartner JW (1985) Harmonic analysis of tides and tidal currents in south San Francisco Bay, california. Estuar Coast Shelf Sci 21(1):57–74CrossRefGoogle Scholar
  12. Clifford NJ, Richards KS, Brown RA, Lane SN (1995) Scales of variation of suspended sediment concentration and turbidity in a glacial meltwater stream. Geogr Ann 77(1/2):45–65CrossRefGoogle Scholar
  13. Cloern JE (1987) Turbidity as a control on phytoplankton biomass and productivity in estuaries. Cont Shelf Res 7(11):1367–1381CrossRefGoogle Scholar
  14. Cole BE, Cloern JE (1987) An empirical model for estimating phytoplankton productivity in estuaries. Mar Ecol Prog 36(3):299–305CrossRefGoogle Scholar
  15. Dettinger M, Ghil M, Strong C, Weibel W, Yiou P (1995) Software expedites singular-spectrum analysis of noisy time series. EOS Trans Am Geophys Union 76(2):12–21Google Scholar
  16. Durand S, Fadili J, Nikolova M (2010) Multiplicative noise removal using L1 fidelity on frame coefficients. J Math Imaging Vis 36(3):201–226CrossRefGoogle Scholar
  17. French JR, Burningham H, Benson T (2008) Tidal and meteorological forcing of suspended sediment flux in a muddy mesotidal estuary. Estuaries Coasts 31(5):843–859CrossRefGoogle Scholar
  18. Golyandina N (2010) On the choice of parameters in singular spectrum analysis and related subspace-based methods. Stat Interface 3(3):259–279CrossRefGoogle Scholar
  19. Golyandina N, Korobeynikov A, Zhigljavsky A (2018) Singular spectrum analysis for time series. Springer, Berlin, pp 1–272. CrossRefGoogle Scholar
  20. Golyandina N, Zhigljavsky A (2013) Singular spectrum analysis for time series. Springer, Berlin, pp 1–118. CrossRefGoogle Scholar
  21. Gupta S, Chauhan R, Saxena S (2005) Robust non-homomorphic approach for speckle reduction in medical ultrasound images. Med Biol Eng Comput 43(2):189–195CrossRefGoogle Scholar
  22. Harris TJ, Yuan H (2010) Filtering and frequency interpretations of singular spectrum analysis. Phys D 239(20):1958–1967CrossRefGoogle Scholar
  23. Hassani H, Mahmoudvand R, Zokaei M et al (2012) On the separability between signal and noise in singular spectrum analysis. Fluct Noise Lett 11(2):1–11CrossRefGoogle Scholar
  24. Hassani H, Mahmoudvand R (2013) Multivariate singular spectrum analysis: a general view and new vector forecasting approach. Int J Energy Stat 1(1):55–83CrossRefGoogle Scholar
  25. Hoang A (2012) Resuspension of bottom sediment on inner shelf-a case study of north-western coast of Taiwan.
  26. Jiao W, Jiang Y, Lin S (2015) Modified signal de-noising approach for multiplicative noise based on empirical mode decomposition. J Mech Eng 51(24):1–8CrossRefGoogle Scholar
  27. Kalteh AM (2016) Improving forecasting accuracy of streamflow time series using least squares support vector machine coupled with data-preprocessing techniques. Water Resour Manag 30(2):747–766CrossRefGoogle Scholar
  28. Kisi O, Shiri J (2012) River suspended sediment estimation by climatic variables implication: comparative study among soft computing techniques. Comput Geosci 43:73–82CrossRefGoogle Scholar
  29. Kumar KS, Rajesh R, Tiwari RK, (2018) Regional and residual gravity anomaly separation using the singular spectrum analysis-based low pass filtering: a case study from Nagpur, Maharashtra, India. Explor Geophys 49(3):398–408CrossRefGoogle Scholar
  30. Li W, Shen Y, Li B (2015) Weighted spatiotemporal filtering using principal component analysis for analyzing regional GNSS position time series. Acta Geod Geophys 50(4):419–436CrossRefGoogle Scholar
  31. Liu J (2003) Estimation of suspended sediment concentration in rivers utilizing principal components analysis with ETM+ data. In: Proceedings of SPIE—the international society for optical engineering, 4897Google Scholar
  32. Mohamoud YM (2014) Time series separation and reconstruction technique to estimate daily suspended sediment concentrations. J Hydrol Eng 19(2):328–338CrossRefGoogle Scholar
  33. Partal T, Cigizoglu HK (2008) Estimation and forecasting of daily suspended sediment data using wavelet–neural networks. J Hydrol 358(3–4):317–331CrossRefGoogle Scholar
  34. Ran J, Tangdamrongsub N, Shi J et al (2019) GRACE observed mass loss in the middle and lower Yangtze basin. Geod Geodyn 10(02):69–74Google Scholar
  35. Rajaee T (2011) Wavelet and ANN combination model for prediction of daily suspended sediment load in rivers. Sci Total Environ 409(15):2917–2928CrossRefGoogle Scholar
  36. Sadeghpour Haji M, Mirbagheri SA, Javid AH, Khezri M, Najafpour GD (2014) Suspended sediment modelling by SVM and wavelet. Građevinar 66(3):211–223Google Scholar
  37. Schoellhamer DH (1996) Factors affecting suspended-solids concentrations in south San Francisco bay, california. J Geophys Res Oceans 101(C5):12087–12095CrossRefGoogle Scholar
  38. Schoellhamer DH (2001) Singular spectrum analysis for time series with missing data. Geophys Res Lett 28(16):3187–3190CrossRefGoogle Scholar
  39. Shen Y, Peng F, Li B (2015) Improved singular spectrum analysis for time series with missing data. Nonlinear Process Geophys 22:371–376CrossRefGoogle Scholar
  40. Sofowote UM, McCarry BE, Marvin CH (2008) Source apportionment of PAH in Hamilton Harbour suspended sediments: comparison of two factor analysis methods. Environ Sci Technol 42(16):6007–6014CrossRefGoogle Scholar
  41. Tyagi V, Wellekens C (2006) Fepstrum and carrier signal decomposition of speech signals through homomorphic filtering. In: IEEE international conference on acoustics, speech and signal processing, 2006. ICASSP 2006 proceedings.
  42. U.S. Environmental Protection Agency (1992) State of the estuary: dredging and waterway modification. U.S. Environmental Protection Agency San Francisco Project, Chapter 8, pp 191–215Google Scholar
  43. Vautard R, Yiou P, Ghil M (1992) Singular-spectrum analysis: a toolkit for short, noisy chaotic signals. In: Conference proceedings on interpretation of time series from nonlinear mechanical systems, vol 58, issue 1. Elsevier North-Holland, Inc., pp 95–126Google Scholar
  44. Walling DE (1977) Limitations of the rating curve technique for estimating suspended sediment loads, with reference to British rivers. In: Erosion & solute matter transport in inland waters, vol 122. International Association of Hydrological Sciences Publication, Wallingford, pp 34–38Google Scholar
  45. Wang F, Shen Y, Li W, Chen Q (2018) Singular spectrum analysis for heterogeneous time series by taking its formal errors into account. Acta Geodyn Geomater 4(192):395–403CrossRefGoogle Scholar
  46. Watson PJ (2016) Identifying the best performing time series analytics for sea level research. In: Rojas I, Pomares H (eds) Time Series Analysis and Forecasting. Springer, Cham, pp 261–278CrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó 2019

Authors and Affiliations

  • Fengwei Wang
    • 1
  • Yunzhong Shen
    • 1
    Email author
  • Qiujie Chen
    • 1
  • Weiwei Li
    • 2
  1. 1.College of Surveying and Geo-informaticsTongji UniversityShanghaiPeople’s Republic of China
  2. 2.College of GeomaticsShandong University of Science and TechnologyQingdaoPeople’s Republic of China

Personalised recommendations