# Lagrangian analysis by clustering

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- 22 Citations

## Abstract

We propose a new method for obtaining average velocities and eddy diffusivities from Lagrangian data. Rather than grouping the drifter-derived velocities in geographical bins, we group them by nearest-neighbor distance using a clustering algorithm. This yields sets with approximately the same number of observations, covering unequal areas. A major advantage is that, because the number of observations is the same for the clusters, the statistical accuracy is more uniform than with geographical bins. We illustrate the technique using synthetic data from a stochastic model, employing a realistic mean flow. The latter represents the surface currents in the Nordic Seas and is strongly inhomogeneous in space. We use the clustering algorithm to extract the mean velocities and diffusivities and compare the results with the corresponding quantities from the stochastic model. We perform a similar comparison with the means and diffusivities obtained with geographical bins. Clustering is more successful at capturing the mean flow and improves convergence in the eddy diffusivity estimates. We discuss both the advantages and shortcomings of the new method.

## Keywords

Lagrangian analysis Eddy diffusivity Binning Clustering## Notes

### Acknowledgements

The work is part of the Poleward project, funded by by the Norwegian Research Council Norklima program (grant number 178559/S30). Details are found on http://www.iaoos.no/ and http://folk.uio.no/ingako/my_files/POLEWARD_WEBPAGE_MAIN.html.. Harald Engedahl provided the MIPOM velocities. We appreciate useful comments from two anonymous reviewers.

## References

- Bauer S, Swenson MS, Griffa A, Mariano AJ, Owens K (1998) Eddy mean flow decomposition and eddy diffusivity estimates in the tropical Pacific Ocean. J Geophys Res 103(C13):30855–30871CrossRefGoogle Scholar
- Bauer S, Swenson MS, Griffa A (2002) Eddy mean flow decomposition and eddy diffusivity estimates in the tropical Pacific Ocean: 2. Results. J Geophys Res 107(C10):3154CrossRefGoogle Scholar
- Brink KH, Breadsley RC, Paduan J, Limeburner R, Caruso M, Sires JG (2000) A view of the 1993–1994 California Current based on surface drifters, floats, and remotely sensed data. J Geophys Res 105(C4):8575–8604CrossRefGoogle Scholar
- Colin de Verdiere A (1983) Lagrangian eddy statistics from surface drifters in the eastern North Atlantic. J Mar Res 41:375–398CrossRefGoogle Scholar
- Davis RE (1991) Observing the general circulation with floats. Deep-Sea Res Suppl 38:S531–S571CrossRefGoogle Scholar
- Davis RE (1998) Preliminary results from directly measuring mid-depth circulation in the Tropical and South Pacific. J Geophys Res 103:24619–24639CrossRefGoogle Scholar
- Falco P, Griffa A, Poulain P-M, Zambianchi E (2000) Transport properties in the Adriatic Sea as deduced from drifter data. J Phys Oceanogr 30:2055–2071CrossRefGoogle Scholar
- Fratantoni DM (2001) North Atlantic surface circulation during the 1990’s observed with satellite-tracked drifters. J Geophys Res 106(C10):22067–22093CrossRefGoogle Scholar
- Garraffo Z, Griffa A, Mariano AJ, Chassignet EP (2001) Lagrangian data in a high-resolution numerical simulation of the North Atlantic II. On the pseudo-Eulerian averaging of Lagrangian data. J Mar Syst 29:177–200CrossRefGoogle Scholar
- Griffa A (1996) Applications of stochastic particle models to oceanographical problems. In: Adler R, Muller P, Rozovskii B (eds) Stochastic modelling in physical oceanography. Birkhauser, Boston, pp 114–140Google Scholar
- Jakobsen PK, Ribergaard MH, Quadfasel D, Schmith T, Hughes CW (2003) Near-surface circulation in the northern North Atlantic as inferred from Lagrangian drifters: variability from the mesoscale to interannual. J Geophys Res 108(C5):3251CrossRefGoogle Scholar
- Kanungo T, Mount DM, Netanyahu NS, Piatko CD, Silverman R, Wu AY (2002) An efficient k-means clustering algorithm: analysis and implementation. IEEE Trans Pattern Anal Mach Intell 24(7):881–892CrossRefGoogle Scholar
- Koszalka I, LaCasce JH, Orvik KA (2009) Relative dispersion in the Nordic Seas. J Mar Res 67:411–433CrossRefGoogle Scholar
- LaCasce J (2005) Statistics of low frequency currents over the western Norwegian shelf and slope I: current meters. Ocean Model 55:213–221Google Scholar
- LaCasce J (2008) Statistics from Lagrangian observations. Prog Oceanogr 77(1):1–29CrossRefGoogle Scholar
- LaCasce J, Engedahl H (2005) Statistics of low frequency currents over the western Norwegian shelf and slope II: model. Ocean Model 55:222–237Google Scholar
- LaCasce JH (2000) Floats and f/H. J Mar Res 58:61–95CrossRefGoogle Scholar
- Lloyd SP (1982) Least squares quantization in PCM. IEEE Trans Inf Theory 28(2):129–137CrossRefGoogle Scholar
- Lumpkin R (2003) Decomposition of surface drifter observations in the Atlantic Ocean. Geophys Res Lett 30(14):1753CrossRefGoogle Scholar
- Lumpkin R, Flament P (2001) Lagrangian statistics in the central North Pacific. J Mar Syst 29:141–155CrossRefGoogle Scholar
- Lumpkin R, Garraffo Z (2005) Evaluating the decomposition of Tropical Atlantic drifter observations. J Phys Oceanogr 22:1403–1415Google Scholar
- Lumpkin R, Treguier A-M, Speer K (2002) Lagrangian eddy scales in the Northern Atlantic Ocean. J Phys Oceanogr 32:2425–2440Google Scholar
- MacKay DJC (2003) Information theory, inference, and learning algorithms. Cambridge University Press, CambridgeGoogle Scholar
- Mariano A, Ryan E (2007) Lagrangian analysis and prediction of coastal and ocean dynamics (LAPCOD review). In Griffa A, Kirwan AD, Mariano AJ, Ozgokmen T, Rossby T (eds) Lagrangian analysis and prediction of coastal and ocean dynamics, Chapter 13. Cambridge University Press, Cambridge, pp 423–467CrossRefGoogle Scholar
- Orvik KA, Niiler P (2002) Major pathways of Atlantic Water in the northern North Atlantic and Nordic Seas towards Arctic. Geophys Res Lett 29(19):1896CrossRefGoogle Scholar
- Owens WB (1991) A statistical description of the mean circulation and eddy variability in the northwestern North Atlantic using SOFAR floats. Prog Oceanogr 28:257–303CrossRefGoogle Scholar
- Poulain P-M (2001) Adriatic Sea surface circulation as derived from drifter data between 1990 and 1999. J Mar Syst 29:3–32CrossRefGoogle Scholar
- Poulain P-M, Warn-Varnas A, Niiler PP (1996) Near-surface circulation of the Nordic Seas as measured by Lagrangian drifters. J Geophys Res 101:18237–18258CrossRefGoogle Scholar
- Rossby HT, Riser SC, Mariano AJ (1983) The western North Atlantic—a Lagrangian viewpoint. In: Robinson AR (ed) Eddies in marine science. Springer, Heidelberg, pp 66–91Google Scholar
- Rupolo V (2007) Observing turbulence regimes and Lagrangian dispersal properties in the oceans. In Griffa A, Kirwan AD, Mariano AJ, Ozgokmen T, Rossby T (eds) Lagrangian analysis and prediction of coastal and ocean dynamics, Chapter 9. Cambridge University Press, Cambridge, pp 231–274CrossRefGoogle Scholar
- Saetre R (1999) Features of the central Norwegian shelf circulation. Cont Shelf Res 19:1809–1831CrossRefGoogle Scholar
- Sallee JB, Speer K, Morrow R, Lumpkin R (2008) An estimate of Lagrangian eddy statistics and diffusion in the mixed layer of the Southern Ocean. J Mar Res 66:441–463CrossRefGoogle Scholar
- Skagseth Ø, Orvik KA (2002) Identifying fluctuations in the Norwegian Atlantic Slope Current by means of empirical orthogonal functions. Cont Shelf Res 22:547–563CrossRefGoogle Scholar
- Swenson MS, Niiler PP (1996) Statistical analysis of the surface circulation of the California Current. J Geophys Res 101(C10):22631–22645CrossRefGoogle Scholar
- Taylor GI (1921) Diffusion by continuous movements. Proc Lond Math Soc 20:196–212CrossRefGoogle Scholar
- Thompson A, Heywood KJ, Thorpe SE, Renner AH, Trasvina A (2009) Surface circulation at the tip of the Antarctic Peninsula from drifters. J Phys Oceanogr 39:3–25CrossRefGoogle Scholar
- Veneziani M, Griffa A, Reynolds AM, Mariano AJ (2004) Oceanic turbulence and stochastic models from subsurface Lagrangian data for the Northwest Atlantic Ocean. J Phys Oceanogr 34:1884–1906CrossRefGoogle Scholar
- Zhurbas V, Oh IS (2003) Lateral diffusivity and Lagrangian scales in the Pacific Ocean as derived from drifter data. J Geophys Res 108(C5):3141CrossRefGoogle Scholar