Abstract
An automatic, transparent, and regular way to investigate and analyze the spatiotemporal variations in a large, unstructured, and high-dimensional data set is highly desirable in almost every area of knowledge. In light of this, the present study concentrates on a versatile spatiotemporal technique, empirical orthogonal function (EOF), and provides a thorough review of the EOF method with an emphasis on the co-seismic crustal deformation analysis. For this, (i) we provide a mathematical description of the EOF method that decomposes a coherent space–time data set into individual spatial patterns and associated time scales; (ii) we highlight the strength of the EOF method and its several extensions in dealing with correlated data variables, intermittent data gaps, and nonlinear relations among data features; (iii) we discuss prominent applications of the innovative data-summarization EOF method in diverse fields, such as crustal deformation analysis, pattern hunting in climate and atmospheric sciences, reconstruction of gappy data, and ionospheric total electron content (TEC) modeling; and (iv) finally, we implement the EOF method to demonstrate its efficacy in the 3-D co-seismic pattern identification caused by the 2016, \(M_\mathrm{w}\) 7.8, Kaikoura earthquake of New Zealand. As a self-organizing approach, the EOF method not only uncovers the unique dynamic patterns hidden behind the data set, but also is capable of recovering the missing values in a large-volume data set .
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11069-021-04967-4/MediaObjects/11069_2021_4967_Fig1_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11069-021-04967-4/MediaObjects/11069_2021_4967_Fig2_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11069-021-04967-4/MediaObjects/11069_2021_4967_Fig3_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11069-021-04967-4/MediaObjects/11069_2021_4967_Fig4_HTML.png)
Similar content being viewed by others
Data availability
The GPS network data were download from the official website: Ground-Based Earth Observing Network (GeoNet) of New Zealand (https://www.geonet.org.nz). The website was last accessed in May, 2021.
References
Alvera-Azcarate A, Barth A, Rixen M, Beckers JM (2005) Reconstruction of incomplete oceanographic data sets using empirical orthogonal functions: application to the Adriatic Sea surface temperature. Ocean Model 9(4):325–346
Alvera-Azcarate A, Barth A, Beckers JM, Weisberg RH (2007) Multivariate reconstruction of missing data in sea surface temperature, chlorophyll, and wind satellite fields. J Geophys Res Oceans 112(C3)–1:10:1–10
Atasever UH, Kesikoglu MH, Ozkan C (2016) A new artificial intelligence optimization method for PCA based unsupervised change detection of remote sensing image data. Neural Netw World 26(2):141
Beckers JM, Rixen M (2003) EOF calculations and data filling from incomplete oceanographic datasets. J Atmos Oceanic Technol 20(12):1839–1856
Bennett AF (1992) Inverse methods in physical oceanography. Cambridge University Press, Cambridge
Bilitza D (2001) International reference ionosphere 2000. Radio Sci 36(2):261–275
Bjornsson H, Venegas S (1997) A manual for EOF and SVD analyses of climatic data. CCGCR Rep 97(1):112–134
Carroll JB (1953) An analytical solution for approximating simple structure in factor analysis. Psychometrika 18(1):23–38
Chang ETY, Chao BF (2011) Co-seismic surface deformation of the 2011 off the Pacific coast of Tohoku Earthquake: Spatio-temporal EOF analysis of GPS data. Earth, Planets and Space 63(7):649–654
Chang ETY, Chao BF (2014) Analysis of coseismic deformation using EOF method on dense, continuous GPS data in Taiwan. Tectonophysics 637:106–115
Chao BF, Liau JR (2019) Gravity changes due to large earthquakes detected in GRACE satellite data via empirical orthogonal function analysis. J Geophys Res: Solid Earth 124(3):3024–3035
Chen Z, Zhang SR, Coster AJ, Fang G (2015) EOF analysis and modeling of GPS TEC climatology over North America. J Geophys Res: Space Phys 120(4):3118–3129
Clark KJ, Nissen EK, Howarth JD, Hamling IJ, Mountjoy JJ, Ries WF, Jones K, Goldstien S, Cochran UA, Villamor P, Hreinsdóttir S (2017) Highly variable coastal deformation in the 2016 Mw7. 8 Kaikōura earthquake reflects rupture complexity along a transpressional plate boundary. Earth Planet Sci Lett 474:334–44
Dabbakuti JRKK, Ratnam DV (2016) Characterization of ionospheric variability in TEC using EOF and wavelets over low-latitude GNSS stations. Adv Space Res 57(12):2427–2443
Dabbakuti JRKK, Ratnam DV (2017) Modeling and analysis of GPS-TEC low latitude climatology during the 24th solar cycle using empirical orthogonal functions. Adv Space Res 60(8):1751–1764
Dawson A (2016) eofs: a library for EOF analysis of meteorological, oceanographic, and climate data. J Open Res Softw 4(1):256
Dittus AJ, Karoly DJ, Donat MG, Lewis SC, Alexander LV (2018) Understanding the role of sea surface temperature-forcing for variability in global temperature and precipitation extremes. Weather Clim Extremes 21:1–9
Dommenget D, Latif M (2002) A cautionary note on the interpretation of EOFs. J Clim 15(2):216–225
Dong D, Fang P, Bock Y, Webb F, Prawirodirdjo L, Kedar S, Jamason P (2006) Spatiotemporal filtering using principal component analysis and Karhunen Loeve expansion approaches for regional GPS network analysis. J Geophys Res: Solid Earth 111:B03405
Feng J, Wang Z, Jiang W, Zhao Z, Zhang B (2016) A new regional total electron content empirical model in northeast China. Adv Space Res 58(7):1155–1167
Forbes JM, Bruinsma S, Lemoine FG (2006) Solar rotation effects on the thermospheres of Mars and Earth. Science 312(5778):1366–1368
Fukuoka A (1951) The central meteorological observatory, a study on 10-day forecast (a synthetic report). Geophys Mag 22(3):177–208
Garcia S, Ramirez-Gallego S, Luengo J, Benitez JM, Herrera F (2016) Big data preprocessing: methods and prospects. Big Data Anal 1(1):1–22
Ghiasi Y, Nafisi V (2016) Strain estimation using ordinary Kriging interpolation. Surv Rev 48(350):361–366
Graham NE, Michaelsen J, Barnett TP (1987) An investigation of the El Niño-Southern Oscillation cycle with statistical models: 1. predictor field characteristics. J Geophys Res: Oceans 92(C13):14251–14270
Greene CA, Thirumalai K, Kearney KA, Delgado JM, Schwanghart W, Wolfenbarger NS, Thyng KM, Gwyther DE, Gardner AS, Blankenship DD (2019) The climate data toolbox for MATLAB. Geochem, Geophys, Geosyst 20(7):3774–81
Gruszczynski M, Klos A, Bogus J (2016) Orthogonal transformation in extracting of common mode errors from continuous GPS networks. Acta Geodyn Geomater 13(3):291–298
Guillaume MAZE (2021) PCAtool, MATLAB Central File Exchange
Hamling IJ, Hreinsdóttir S, Clark K, Elliott J, Liang C, Fielding E, Litchfield N, Villamor P, Wallace L, Wright TJ, D’Anastasio E (2017) Complex multifault rupture during the 2016 Mw 7.8 Kaikōura earthquake, New Zealand. Science. 356(6334)
Hannachi A (2007) Pattern hunting in climate: a new method for finding trends in gridded climate data. Int J Climatol: J R Meteorol Soc 27(1):1–15
Hannachi A, Jolliffe IT, Stephenson DB (2007) Empirical orthogonal functions and related techniques in atmospheric science: a review. Int J Climatol: J R Meteorol Soc 27(9):1119–1152
Hannachi A, Jolliffe IT, Stephenson DB, Trendafilov N (2006) search of simple structures in climate: simplifying EOFs. Int J Climatol, J R Meteorol Soc 26(1):7–28
Hippert-Ferrer A, Yan Y, Bolon P (2019) Gap-filling based on iterative EOF analysis of temporal covariance: application to InSAR displacement time series. in: IGARSS 2019-2019 IEEE International Geoscience and Remote Sensing Symposium, IEEE, P:262–265
Holliday JR, Rundle JB, Tiampo KF, Turcotte DL (2006) Using earthquake intensities to forecast earthquake occurrence times. Nonlinear Process Geophys 13(5):585–593
Horel JB (1981) A rotated principal component analysis of the interannual variability of the Northern Hemisphere 500 mb height field. Mon Weather Rev 109(10):2080–2092
Jiang Z, Huang D, Yuan L, Hassan A, Zhang L, Yang Z (2018a) Coseismic and postseismic deformation associated with the 2016 Mw 7.8 Kaikoura earthquake, New Zealand: fault movement investigation and seismic hazard analysis. Earth, Planets and Space 70(1):1–14
Jiang Z, Yuan L, Huang D, Zhang L, Hassan A, Yang Z (2018b) Spatial-temporal evolution of slow slip movements triggered by the 2016 Mw 7.8 Kaikoura earthquake New Zealand. Tectonophysics 744:72–81
Johnson CW, Ben-Zion Y, Meng H, Vernon F (2020) Identifying different classes of seismic noise signals using unsupervised learning. Geophys Res Lett 47(15):e2020GL088353
Jolliffe IT (1986) Principal components in regression analysis. Principal component analysis. Springer, Cham
Jolliffe IT (1990) Principal component analysis: a beginner’s guide-I. introduction and application. Weather 45(10):375–382
Jolliffe IT, Cadima J (2016) Principal component analysis: a review and recent developments. Philos Trans R Soc : Math, Phys Eng Sci 374(2065):20150202
Kaiser HF (1959) Computer program for varimax rotation in factor analysis. Educ Psychol Measurement 19(3):413–420
Kalviainen H (2015) From pattern recognition methods to machine vision applications. advances in independent component analysis and learning machines. Academic Press, Cambridge
Kaplan A, Kushnir Y, Cane MA, Blumenthal MB (1997) Reduced space optimal analysis for historical data sets: 136 years of Atlantic sea surface temperatures. J Geophys Res: Oceans 102(C13):27835–27860
Kim KY, North GR (1997) EOFs of harmonizable cyclostationary processes. J Atmos Sci 54(19):2416–2427
Kim KY, Wu Q (1999) A comparison study of EOF techniques: analysis of nonstationary data with periodic statistics. J Clim 12(1):185–199
Kondrashov D, Ghil M (2006) Spatio-temporal filling of missing points in geophysical data sets. Nonlinear Process Geophys 13(2):151–159
Kositsky AP, Avouac JP (2010) Inverting geodetic time series with a principal component analysis-based inversion method. J Geophys Res: Solid Earth 115:(B03401)
Kumar U, Chao BF, Chang ETY (2020) What causes the common-mode error in array GPS displacement fields: case study for Taiwan in relation to atmospheric mass loading. Earth Space Sci 7(11):e2020EA001159
Lau KM, Chan PH (1986) Aspects of the 40–50 day oscillation during the northern summer as inferred from outgoing longwave radiation. Mon Weather Rev 114(7):1354–1367
Li S, Peng J, Xu W, Qin K (2013) Time series modeling and analysis of trends of daily averaged ionospheric total electron content. Adv Space Res 52(5):801–809
Lin YN, Kositsky AP, Avouac JP (2010) PCAIM joint inversion of InSAR and ground-based geodetic time series: application to monitoring magmatic inflation beneath the Long Valley Caldera. Geophys Res Lett 37:L23301
Liu JY, Chuo YJ, Shan SJ, Tsai YB, Chen YI, Pulinets SA, Yu SB (2004) Pre-earthquake ionospheric anomalies registered by continuous GPS TEC measurements. Annal Geophys 22:1585–1593
Lorenz EN (1956) Empirical orthogonal functions and statistical weather prediction. Science Report 1, Statistical Forecasting Project, Department of Meteorology, MIT. NTIS AD 110268:1–49
Mao T, Wan WX, Liu LB (2005) An EOF based empirical model of TEC over Wuhan. Chin J Geophys 48(4):827–834
Merchant CJ, Minnett PJ, Beggs H, Corlett GK, Gentemann C, Harris AR, Hoyer J, Maturi E (2019) Global sea surface temperature. taking the temperature of the earth. Elsevier, Amsterdam
Munekane H (2012) Coseismic and early postseismic slips associated with the 2011 off the Pacific coast of Tohoku Earthquake sequence: EOF analysis of GPS kinematic time series. Earth, Planets and Space 64(12):1077–1091
Nantasenamat C, Isarankura-Na-Ayudhya C, Naenna T, Prachayasittikul V (2009) A practical overview of quantitative structure-activity relationship. Exper Clin Sci 8:74–88
Navarra A, Simoncini V (2010) A guide to empirical orthogonal functions for climate data analysis. Springer Science & Business Media, New York
Nguyen C (2019) Development of Geodetic Imaging Techniques and Machine Learning for Marsh Observation (Doctoral dissertation, Texas A & M University-Corpus Christi)
North GR (1984) Empirical orthogonal functions and normal modes. J Atmos Sci 41(5):879–887
Obukhov AM (1947) Statistically homogeneous fields on a sphere. Uspekhi Fizicheskikh Nauk 2(2):196–198
Obukhov AM (1960) The statistically orthogonal expansion of empirical functions. bulletin of the academy of sciences of the USSR. Geophys Ser 1:288–291
Orellana M, Cedillo P (2019) Outlier detection with data mining techniques and statistical methods. In: 2019 International Conference on Information Systems and Computer Science (INCISCOS), IEEE, pp:51–56
PandaSK Gedam SS, ** S (2015) Ionospheric TEC variations at low latitude Indian region, Satellite Positioning-Methods, Models and Applications. Tech-Publisher. Rijeka, Croatia:149–174
Pappas C, Papalexiou SM, Koutsoyiannis D (2014) A quick gap filling of missing hydrometeorological data. J Geophys Res: Atmos 119(15):9290–9300
Pedregosa F, Varoquaux G, Gramfort A, Michel V, Thirion B, Grisel O, Blondel M, Prettenhofer P, Weiss R, Dubourg V, Vanderplas J (2011) Scikit-learn: machine learning in Python. J Mach Learn Res 12:2825–2830
Radiguet M, Perfettini H, Cotte N, Gualandi A, Valette B, Kostoglodov V, Lhomme T, Walpersdorf A, Cano EC, Campillo M (2016) Triggering of the 2014 Mw 7.3 Papanoa earthquake by a slow slip event in Guerrero, Mexico. Nature Geosci 9(11):829–833
Raphael MN (2013) The NCAR Command Language (Version 6.1.2) [Software]. UCAR/NCAR/CISL/VETS, Boulder, Colorado
Reynolds RW, Smith TM (1994) Improved global sea surface temperature analyses using optimum interpolation. J Clim 7(6):929–948
Richman MB (1986) Rotation of principal components. J Climatol 6(3):293–335
Rishbeth H, Muller-Wodarg ICF, Zou L, Fuller Rowell TJ, Millward GH, Moffett RJ, Idenden DW, Aylward AD (2000) Annual and semiannual variations in the ionospheric F2-layer: II physical discussion. Annal Geophys 18:945–956
Rundle JB, Donnellan A, Fox G, Crutchfield JP, Granat R (2021) Nowcasting Earthquakes: Imaging the Earthquake Cycle in California with Machine Learning. Earth and Space Science
Segall P, Matthews M (1997) Time dependent inversion of geodetic data. J Geophys Res: Solid Earth 102(B10):22391–22409
Sharma Y, Pasari S, Dikshit O, Ching K E (2018) GPS-based monitoring of crustal deformation in Garhwal-Kumaun Himalaya, ISPRS-International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, XLII-5, pp 451–454
Sharma Y, Pasari S, Ching KE, Dikshit O, Kato T, Malik JN, Chang CP, Yen JY (2020) Spatial distribution of earthquake potential along the Himalayan arc. Tectonophysics 791:228556
Sharma Y, Pasari S, Neha (2021) Indian plate motion revealed by GPS observations: preliminary results. In: Kulshrestha R, Shekhar C, Jain M, Chakravarthy SR (eds) Mathematical modeling and computation of real time problems: an interdisciplinary approach, pp 203–213 (CRC Press)
Shi X, Tapponnier P, Wang T, Wei S, Wang Y, Wang X, Jiao L (2019) Triple junction kinematics accounts for the 2016 Mw 7.8 Kaikoura earthquake rupture complexity. Proc Nat Acad Sci 116(52):26367–75
Shrivastava MN, Gonzalez G, Moreno M, Chlieh M, Salazar P, Reddy C, Baez JC, Yanez G, Gonzalez J, Llera JC (2016) Coseismic slip and afterslip of the 2015 Mw 8.3 Illapel (Chile) earthquake determined from continuous GPS data. Geophys Res Lett 43(20):10710–10719
Smith CM, Faulds JE, Brown S, Coolbaugh M, Lindsey CR, Treitel S, Ayling B, Fehler M, Gu C, Mlawsky E (2021) Characterizing Signatures of Geothermal Exploration Data with Machine Learning Techniques: An Application to the Nevada Play Fairway Analysis, \(46^th\) Workshop on Geothermal Reservoir Engineering Stanford University. California, Stanford
Su X, Meng G, Su L, Wu W, Liu T (2020) Coseismic and Early Postseismic Deformation of the 2016 M w 7.8 Kaikōura Earthquake, New Zealand, from Continuous GPS Observations. Pure Appl Geophys 177(1):285–303
Taylor MH, Losch M, Wenzel M, Schroter J (2013) On the sensitivity of field reconstruction and prediction using empirical orthogonal functions derived from gappy data. J Clim 26(22):9194–9205
Thomson RE, Emery WJ (2014) Data Analysis Methods in Physical Oceanography. Newnes
Tiampo KF, Rundle JB, Klein W, Ben-Zion Y, McGinnis S (2004) Using eigenpattern analysis to constrain seasonal signals in Southern California. Computational Earthquake Science Part I. Springer pp1991–2003
Tiampo KF, Rundle JB, McGinnis SA, Klein W (2002a) Pattern dynamics and forecast methods in seismically active regions. Earthquake Processes: Physical Modelling, Numerical Simulation and Data Analysis Part II. Birkhäuser, Basel, pp 2429–2467
Tiampo KF, Rundle JB, McGinnis S, Gross SJ, Klein W (2002b) Eigenpatterns in southern California seismicity. J Geophys Res: Solid Earth 107(B12):ESE8-13
Uwamahoro JC, Habarulema JB, Buresova D (2019) Highlights about the performances of storm-time TEC modelling techniques for low/equatorial and mid-latitude locations. Adv Space Res 63(10):3102–3118
Vautard R, Yiou P, Ghil M (1992) Singular-spectrum analysis: a toolkit for short, noisy chaotic signals. Phys D: Nonlinear Phenom 58(1–4):95–126
Wallace LM, Barnes P, Beavan J, Van Dissen R, Litchfield N, Mountjoy J, Langridge R, Lamarche G, Pondard N (2012) The kinematics of a transition from subduction to strike-slip: an example from the central New Zealand plate boundary. J Geophys Res: Solid Earth 117(B2):879
Wallace JM, Dickinson RE (1972) Empirical orthogonal representation of time series in the frequency domain. part I: theoretical considerations. J Appl Meteorol Climatol 11(6):887–892
Wan W, Ding F, Ren Z, Zhang M, Liu L, Ning B (2012) Modeling the global ionospheric total electron content with empirical orthogonal function analysis. Sci China Technol Sci 55(5):1161–1168
Weare BC, Nasstrom JS (1982) Examples of extended empirical orthogonal function analyses. Mon Weather Rev 110(6):481–485
Xu C (2016) Reconstruction of gappy GPS coordinate time series using empirical orthogonal functions. J Geophys Res: Solid Earth 121(12):9020–9033
Zhang DH, **ao Z, Hao YQ, Ridley AJ, Moldwin M (2011) Modeling ionospheric foF2 by using empirical orthogonal function analysis. In: Annales Geophysicae, Vol. 29, Copernicus GmbH, pp:1501–1515
Zhang D, Ridley AJ, **ao Z, Hao Y (2012) A global model: empirical orthogonal function analysis of total electron content 1999–2009 data. J Geophys Res 117:A03328
Acknowledgements
Some of the figures were prepared using MATLAB and GMT. Constructive comments and useful suggestions of two anonymous reviewers are greatly appreciated. The first author [Neha] thankfully acknowledges the financial support from the CSIR-UGC-NET (Ref. No: 1197/CSIR-UGC NET JUNE 2017).
Funding
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Rights and permissions
About this article
Cite this article
Neha, Pasari, S. A review of empirical orthogonal function (EOF) with an emphasis on the co-seismic crustal deformation analysis. Nat Hazards 110, 29–56 (2022). https://doi.org/10.1007/s11069-021-04967-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11069-021-04967-4