Parameter estimation by ensemble Kalman filters with transformed data: Approach and application to hydraulic tomography

Water Resources Research

Volumn 48, Issue 4, 2012, Pages

  Schniger, A ^a,b   Nowak, W ^a   Hendricks Franssen, H J ^c  

^a UNIVERSITY OF STUTTGART   (Germany)

^b BoSS Consult GmbH   (Germany)

^c FORSCHUNGSZENTRUM JÜLICH   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

BAYESIAN UPDATING; COMPUTATIONALLY EFFICIENT; ENSEMBLE KALMAN FILTER; FLOW AND TRANSPORT; GAUSSIANS; MONOTONIC TRANSFORMATION; MULTI VARIATE ANALYSIS; NON LINEAR INVERSION; OCEANIC SCIENCE; PARTICLE FILTER; PSEUDOLINEARIZATION; STATE VARIABLES; SUBSURFACE FLOW;

GROUNDWATER; KALMAN FILTERS; MATHEMATICAL TRANSFORMATIONS; PARAMETER ESTIMATION; TOMOGRAPHY;

GAUSSIAN DISTRIBUTION;

ACCURACY ASSESSMENT; BAYESIAN ANALYSIS; GAUSSIAN METHOD; GROUNDWATER FLOW; HYDRAULIC CONDUCTIVITY; KALMAN FILTER; LINEARITY; MULTIVARIATE ANALYSIS; SUBSURFACE FLOW;

EID: 84859336902     PISSN: 00431397     EISSN: None     Source Type: Journal    
DOI: 10.1029/2011WR010462     Document Type: Article

References (102)

1. Aanonsen, S., G. Naevdal, D. Oliver, A. Reynolds, and B. Valles (2009), The ensemble Kalman filter in reservoir engineering-A review, SPE J., 14(3), 393-412.
2. Arulampalam, M., S. Maskell, N. Gordon, and T. Clapp (2002), A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking, IEEE Trans. Signal Process., 50(2), 174-188. (Pubitemid 34291500)
3. Bárdossy, A., and J. Li (2008), Geostatistical interpolation using copulas, Water Resour. Res., 44, W07412, doi:10.1029/2007WR006115.
4. Béal, D., P. Brasseur, J.-M. Brankart, Y. Ourmières, and J. Verron (2010), Characterization of mixing errors in a coupled physical biogeochemical model of the North Atlantic: Implications for nonlinear estimation using Gaussian anamorphosis, Ocean Sci., 6, 247-262.
5. Bellin, A., and D. Tonina (2007), Probability density function of non-reactive solute concentration in heterogeneous porous formations, J. Contam. Hydrol., 94(1-2), 109-125. (Pubitemid 47505376)
6. Bertino, L., G. Evensen, and H. Wackernagel (2003), Sequential data assimilation techniques in oceanography, Int. Stat. Rev., 71(2), 223-241. (Pubitemid 37052980)
7. Bocquet, M., C. A. Pires, and L. Wu (2010), Beyond Gaussian statistical modeling in geophysical data assimilation, Mon. Weather Rev., 138, 2997-3023, doi:10.1175/2010MWR3164.1.
8. Bohling, G. C., X. Zhan, J. Butler, and L. Zheng (2002), Steady shape analysis of tomographic pumping tests for characterization of aquifer heterogeneities, Water Resour. Res., 38(12), 1324, doi:10.1029/2001WR001176.
9. Brauchler, R., J. T. Cheng, P. Dietrich, M. Everett, B. Johnson, R. Liedl, and M. Sauter (2007), An inversion strategy for hydraulic tomography: Coupling travel time and amplitude inversion, J. Hydrol., 345(3-4), 184-198. (Pubitemid 47569213)
10. Burgers, G., P. J. van Leeuwen, and G. Evensen (1998), Analysis scheme in the ensemble Kalman filter, Mon. Weather Rev., 126(6), 1719-1724. (Pubitemid 128597660)
11. Butler, J. J., C. D. McElwee, and G. C. Bohling (1999), Pumping tests in networks of multilevel sampling wells: Motivation and methodology, Water Resour. Res., 35(11), 3553-3560, doi:10.1029/1999WR900231. (Pubitemid 29514627)
12. Cardiff, M., and W. Barrash (2011), 3D transient hydraulic tomography in unconfined aquifers with fast drainage response, Water Resour. Res., 47, W12518, doi:10.1029/2010WR010367.
13. Castagna, M., and A. Bellin (2009), A Bayesian approach for inversion of hydraulic tomographic data, Water Resour. Res., 45, W04410, doi:10.1029/ 2008WR007078.
14. Chen, Y., and D. Zhang (2006), Data assimilation for transient flow in geologic formations via ensemble Kalman filter, Adv. Water Res., 29, 1107-1122, doi:10.1016/j.advwatres.2005.09.007. (Pubitemid 44089284)
15. Cheng, C., and E. Parzen (1997), Unified estimators of smooth quantile and quantile density functions, J. Stat. Plann. Inference, 59(2), 291-307, doi:10.1016/S0378-3758(96)00110-3. (Pubitemid 127162450)
16. Chilès, J. P., and P. Delfiner (1999), Geostatistics : Modeling Spatial Uncertainty, John Wiley, New York.
17. Cirpka, O. A., F. P. J. de Barros, G. Chiogna, and W. Nowak (2011), Probability density function of steady-state concentration in two-dimensional heterogeneous porous media, Water Resour. Res., 47, W11523, doi:10.1029/ 2011WR010750.
18. Clark, M. P., D. E. Rupp, R. A. Woods, X. Zheng, R. P. Ibbitt, A. G. Slater, J. Schmidt, and M. J. Uddstrom (2008), Hydrological data assimilation with the ensemble Kalman filter: Use of streamflow observations to update states in a distributed hydrological model, Adv. Water Res., 31(10), 1309-1324, doi:10.1016/j.advwatres.2008.06.005.
19. Cox, D. R., and N. J. H. Small (1978), Testing multivariate normality, Biometrika, 65(2), 263-272, doi:10.1093/biomet/65.2.263. (Pubitemid 8379487)
20. Crow, W. T., and E. F. Wood (2003), The assimilation of remotely sensed soil brightness temperature imagery into a land surface model using ensemble Kalman filtering: A case study based on ESTAR measurements during SGP97, Adv. Water Res., 26(2), 137-149, doi:10.1016/S0309-1708(02)00088-X. (Pubitemid 36093690)
21. de Barros, F. P. J., and W. Nowak (2010), On the link between contaminant source release conditions and plume prediction uncertainty, J. Contam. Hydrol., 116(1-4), 24-34, doi:10.1016/j.jconhyd.2010.05.004.
22. Dechant, C., and H. Moradkhani (2011), Radiance data assimilation for operational snow and streamflow forecasting, Adv. Water Res., 34, 351-364, doi:10.1016/j.advwatres.2010.12.009.
23. Deutsch, C. V., and A. G. Journel (1998), GSLIB: Geostatistical Software Library and User's Guide, 2nd ed., Oxford Univ. Press, New York.
24. Efron, B. (1982), The Jackknife, the Bootstrap and Other Resampling Plans, vol. 1, 92 pp., Soc. for Ind. and Appl. Math., Philadelphia, Pa.
25. Evensen, G. (1994), Sequential data assimilation with a nonlinear quasigeostrophic model using Monte Carlo methods to forecast error statistics, J. Geophys. Res., 99, 10,143-10,162, doi:10.1029/94JC00572.
26. Evensen, G. (2009), The ensemble Kalman filter for combined state and parameter estimation, IEEE Control Syst. Mag., 29(3), 83-104.
27. Fiorotto, V., and E. Caroni (2002), Solute concentration statistics in heterogeneous aquifers for finite Péclet values, Transp. Porous Media, 48, 331-351, doi:10.1023/A:1015744421033. (Pubitemid 34735973)
28. Fritz, J., I. Neuweiler, and W. Nowak (2009), Application of FFT-based algorithms for large-scale universal kriging problems, Math. Geosci., 41(5), 509-533, doi:10.1007/s11004-009-9220-x.
29. Gordon, N. J., D. J. Salmond, and A. F. M. Smith (1993), Novel approach to nonlinear/non-Gaussian Bayesian state estimation, IEE Proc., Part F, Radar Signal Process., 140(2), 107-113.
30. Gottlieb, J., and P. Dietrich (1995), Identification of the permeability distribution in soil by hydraulic tomography, Inverse Probl., 11, 353-360.
31. Gu, Y., and D. Oliver (2006), The ensemble Kalman filter for continuous updating of reservoir simulation models, J. Energy Res. Technol., 128, 79-88.
32. Gu, Y., and D. Oliver (2007), An iterative ensemble Kalman filter for multiphase fluid flow data assimilation, SPE J., 12(4), 438-446, doi:10.2118/108438-PA.
33. Han, X., and X. Li (2008), An evaluation of the nonlinear/non-Gaussian filters for the sequential data assimilation, Remote Sens. Environ., 112(4), 1434-1449, doi:10.1016/j.rse.2007.07.008. (Pubitemid 351375237)
34. Hendricks Franssen, H.-J., and J. J. Gomez-Hernandez (2002), 3D inverse modelling of groundwater flow at a fractured site using a stochastic continuum model with multiple statistical populations, Stochastic Environ. Res. Risk Assess., 16(2), 155-174. (Pubitemid 34488733)
35. Hendricks Franssen, H.-J., and W. Kinzelbach (2008), Real-time groundwater flow modeling with the ensemble Kalman filter: Joint estimation of states and parameters and the filter inbreeding problem, Water Resour. Res., 44, W09408, doi:10.1029/2007WR006505.
36. Hendricks Franssen, H.-J., and W. Kinzelbach (2009), Ensemble Kalman filtering versus sequential self-calibration for inverse modelling of dynamic groundwater flow systems, J. Hydrol., 365(3-4), 261-274, doi:10.1016/j.jhydrol. 2008.11.033.
37. Hendricks Franssen, H.-J., A. Alcolea, M. Riva, M. Bakr, N. van der Wiel, F. Stauffer, and A. Guadagnini (2009), A comparison of seven methods for the inverse modelling of groundwater flow. Application to the characterisation of well catchments, Adv. Water Res., 32(6), 851-872, doi:10.1016/j.advwatres.2009. 02.011.
38. Illman, W. A., J. Zhu, A. J. Craig, and D. Yin (2010), Comparison of aquifer characterization approaches through steady state groundwater model validation: A controlled laboratory sandbox study, Water Resour. Res., 46, W04502, doi:10.1029/2009WR007745.
39. Johnson, R. A., and D. W. Wichern (1988), Applied Multivariate Statistical Analysis, Prentice-Hall, Englewood Cliffs, N. J.
40. Kitagawa, G. (1996), Monte Carlo filter and smoother for non-Gaussian nonlinear state space models, J. Comput. Graphical Stat., 5(1), 1-25.
41. Kitanidis, P. K. (1986), Parameter uncertainty in estimation of spatial functions: Bayesian analysis, Water Resour. Res., 22(4), 499-507, doi:10.1029/WR022i004p00499.
42. Kitanidis, P. K. (1995), Quasi-linear geostatistical theory for inversing, Water Resour. Res., 31(10), 2411-2420, doi:10.1029/95WR01945. (Pubitemid 26475164)
43. Kitanidis, P. K. (1997), Introduction to Geostatistics: Applications to Hydrogeology, Cambridge Univ. Press, Cambridge, U. K.
44. Kong, A., J. Liu, and W. Wong (1994), Sequential imputations and Bayesian missing data problems, J. Am. Stat. Assoc., 89(425), 278-288.
45. Leisenring, M., and H. Moradkhani (2010), Snow water equivalent prediction using Bayesian data assimilation methods, Stochastic Environ. Res. Risk Asses., 25(2), 1-18, doi:10.1007/s00477-010-0445-5.
46. Li, W., W. Nowak, and O. A. Cirpka (2005), Geostatistical inverse modeling of transient pumping tests using temporal moments of drawdown, Water Resour. Res., 41, W08403, doi:10.1029/2004WR003874. (Pubitemid 41393333)
47. Li, W., A. Englert, O. A. Cirpka, J. Vanderborght, and H. Vereecken (2007), Two-dimensional characterization of hydraulic heterogeneity by multiple pumping tests, Water Resour. Res., 43, W04433, doi:10.1029/2006WR005333. (Pubitemid 46799539)
48. Liu, G., Y. Chen, and D. Zhang (2008), Investigation of flow and transport processes at the MADE site using ensemble Kalman filter, Adv. Water Res., 31(7), 975-986, doi:10.1016/j.advwatres.2008.03.006.
49. Liu, J. S., and R. Chen (1998), Sequential Monte Carlo methods for dynamic systems, J. Am. Stat. Assoc., 93(443), 1032-1044.
50. Liu, N., and D. S. Oliver (2005), Ensemble Kalman filter for automatic history matching of geologic facies, J. Pet. Sci. Eng., 47(3-4), 147-161, doi:10.1016/j.petrol.2005.03.006. (Pubitemid 40829359)
51. Lorentzen, R., G. Naevdal, B. Valles, A. Berg, and A. A. Grimstad (2005), Analysis of the ensemble Kalman filter for estimation of permeability and porosity in reservoir models, paper presented at SPE Annual Technical Conference and Exhibition, Soc. of Pet. Eng., Dallas, Tex.
52. Margulis, S. A., D. McLaughlin, D. Entekhabi, and S. Dunne (2002), Land data assimilation and estimation of soil moisture using measurements from the Southern Great Plains 1997 Field Experiment, Water Resour. Res., 38(12), 1299, doi:10.1029/2001WR001114.
53. Matheron, G. (1973), Le krigeage disjonctif, Intern. Note N-360, 40 pp., Centre de Géostatistique, Ecole des Mines de Paris, Paris, France.
54. Montzka, C., H. Moradkhani, L. Weihermüller, H.-J. Hendricks Franssen, M. Canty, and H. Vereecken (2011), Hydraulic parameter estimation by remotely-sensed top soil moisture observations with the particle filter, J. Hydrol., 399, 410-421.
55. Moradkhani, H., K. L. Hsu, H. Gupta, and S. Sorooshian (2005a), Uncertainty assessment of hydrologic model states and parameters : Sequential data assimilation using the particle filter, Water Resour. Res., 41, W05012, doi:10.1029/2004WR003604. (Pubitemid 40856462)
56. Moradkhani, H., S. Sorooshian, H. Gupta, and P. Houser (2005b), Dual state-parameter estimation of hydrological models using ensemble Kalman filter, Adv. Water Res., 28(2), 135-147. (Pubitemid 40102390)
57. Naevdal, G., L. V. Johnsen, S. I. Aanonsen, and E. H. Vefring (2003), Reservoir monitoring and continuous model updating using ensemble Kalman filter, Soc. Pet. Eng. J., 10(1), 66-74, doi:10.2118/84372-PA.
58. Nakano, S., G. Ueno, and T. Higuchi (2007), Merging particle filter for sequential data assimilation, Nonlinear Proc. Geophys., 14(4), 395-408. (Pubitemid 47116504)
59. Ng, G. H. C., D. McLaughlin, D. Entekhabi, and B. Scanlon (2009), Using data assimilation to identify diffuse recharge mechanisms from chemical and physical data in the unsaturated zone, Water Resour. Res., 45, W09409, doi:10.1029/2009WR007831.
60. Nowak, W. (2009), Best unbiased ensemble linearization and the quasilinear Kalman ensemble generator, Water Resour. Res., 45, W04431, doi:10.1029/2008WR007328.
61. Nowak, W. (2010), Measures of parameter uncertainty in geostatistical estimation and geostatistical optimal design, Math. Geosci., 42(2), 199-221
62. Nowak, W., S. Tenkleve, and O. Cirpka (2003), Efficient computation of linearized cross-covariance and auto-covariance matrices of interdependent quantities, Math. Geol., 35(1), 53-66. (Pubitemid 38149279)
63. Nowak, W., R. L. Schwede, O. A. Cirpka, and I. Neuweiler (2008), Probability density functions of hydraulic head and velocity in three-dimensional heterogeneous porous media, Water Resour. Res., 44, W12433, doi:10.1029/2007WR006668.
64. Nowak, W., F. P. J. de Barros, and Y. Rubin (2010), Bayesian geostatistical design: Task-driven optimal site investigation when the geostatistical model is uncertain, Water Resour. Res., 46, W03535, doi:10.1029/2009WR008312.
65. Osnes, H. (1995), Stochastic analysis of head spatial variability in bounded rectangular heterogeneous aquifers, Water Resour. Res., 31(12), 2981-2990, doi:10.1029/95WR01956. (Pubitemid 26444459)
66. Pan, M., E. F. Wood, R. Wójcik, and M. F. McCabe (2008), Estimation of regional terrestrial water cycle using multi-sensor remote sensing observations and data assimilation, Remote Sens. Environ., 112(4), 1282-1294, doi:10.1016/j.rse.2007.02.039. (Pubitemid 351375233)
67. Qin, J., S. Liang, K. Yang, I. Kaihotsu, R. Liu, and T. Koike (2009), Simultaneous estimation of both soil moisture and model parameters using particle filtering method through the assimilation of microwave signal, J. Geophys. Res., 114, D15103, doi:10.1029/2008JD011358.
68. Reichle, R. H., D. B. McLaughlin, and D. Entekhabi (2002a), Hydrologic data assimilation with the ensemble Kalman filter, Mon. Weather Rev., 130, 103-114. (Pubitemid 34838242)
69. Reichle, R. H., J. P. Walker, R. D. Koster, and P. R. Houser (2002b), Extended versus ensemble Kalman filtering for land data assimilation, J. Hydrometeorol., 3, 728-740. (Pubitemid 36122408)
70. Rings, J., J. Huisman, and H. Vereecken (2010), Coupled hydrogeophysical parameter estimation using a sequential Bayesian approach, Hydrol. Earth Syst. Sci., 14(3), 545-556, doi:10.5194/hess-14-545-2010.
71. Rivoirard, J. (1994), Introduction to Disjunctive Kriging and Non-linear Geostatistics, Oxford Univ. Press, New York.
72. Rubin, Y. (2003), Applied Stochastic Hydrogeology, Oxford Univ. Press, New York.
73. Rubin, Y., and G. Dagan (1988), Stochastic analysis of boundaries effects on head spatial variability in heterogeneous aquifers: 1. Constant head boundary, Water Resour. Res., 24(10), 1689-1697, doi:10.1029/WR024i010p01689. (Pubitemid 19299585)
74. Rubin, Y., and G. Dagan (1989), Stochastic analysis of boundaries effects on head spatial variability in heterogeneous aquifers: 2. Impervious boundary, Water Resour. Res., 25(4), 707-712, doi:10.1029/WR025i004p00707. (Pubitemid 19305680)
75. Schwede, R., O. Cirpka, W. Nowak, and I. Neuweiler (2008), Impact of sampling volume on the probability density function of steady state concentration, Water Resour. Res., 44,W12433, doi:10.1029/2007WR006668.
76. Simon, E., and L. Bertino (2009), Application of the Gaussian anamorphosis to assimilation in a 3-D coupled physical-ecosystem model of the North Atlantic with the EnKF: A twin experiment, Ocean Sci., 5, 495-510.
77. Skjervheim, J.-A., G. Evensen, S. I. Aanonsen, B. O. Ruud, and T. A. Johansen (2007), Incorporating 4D seismic data in reservoir simulation models using ensemble Kalman filter, SPE J., 12(3), 95789, doi:10.2118/95789-PA.
78. Smith, A. F. M., and A. E. Gelfand (1992), Bayesian statistics without tears: A sampling-resampling perspective, Am. Stat., 46(2), 84-88.
79. Smith, L., and F. W. Schwartz (1981), Mass transport: 2. Analysis of uncertainty in prediction, Water Resour. Res., 17(2), 351-369, doi:10.1029/WR017i002p00351.
80. Smith, P. J., K. J. Beven, and J. A. Tawn (2008), Detection of structural inadequacy in process-based hydrological models: A particle-filtering approach, Water Resour. Res., 44,W01410, doi :10.1029/2006WR005205.
81. Smith, S. P., and A. K. Jain (1988), A test to determine the multivariate normality of a data set, IEEE Trans. Pattern Anal. Mach. Intell., 10(5), 757-761, doi:10.1109/34.6789.
82. Snyder, C., T. Bengtsson, P. Bickel, and J. Anderson (2008), Obstacles to high-dimensional particle filtering, Mon. Weather Rev., 136, 4629-4640, doi:10.1175/2008MWR2529.1.
83. Straface, S., T. C. J. Yeh, J. Zhu, S. Troisi, and C. H. Lee (2007), Sequential aquifer tests at a well field, Montalto Uffugo Scalo, Italy, Water Resour. Res., 43, W07432, doi:10.1029/2006WR005287. (Pubitemid 350194484)
84. Sun, A. Y., A. P. Morris, and S. Mohanty (2009), Sequential updating of multimodal hydrogeologic parameter fields using localization and clustering techniques, Water Resour. Res., 45, W07424, doi:10.1029/2008WR007443.
85. Tong, J., B. X. Hu, and J. Yang (2010), Using data assimilation method to calibrate a heterogeneous conductivity field conditioning on transient flow test data, Stochastic Environ. Res. Risk Assess., 24(8), 1211-1223, doi:10.1007/s00477-010-0392-1.
86. Van Delft, G., G. Y. El Serafy, and A. W. Heemink (2009), The ensemble particle filter (EnPF) in rainfall-runoff models, Stochastic Environ. Res. Risk Assess., 23(8), 1203-1211, doi:10.1007/s00477-008-0301-z.
87. Van Der Merwe, R., A. Doucet, N. De Freitas, and E. Wan (2001), The unscented particle filter, Adv. Neural Inf. Process. Syst., 96, 584-590.
88. Van Leeuwen, P. J. (2009), Particle filtering in geophysical systems, Mon. Weather Rev., 137, 4098-4114, doi:10.1175/2009MWR2835.1.
89. van Leeuwen, P. J. (2010), Nonlinear data assimilation in geosciences: an extremely efficient particle filter, Q. J. R. Meteorol. Soc., 136(653), 1991-1999.
90. Vasco, D. W., and K. Karasaki (2006), Interpretation and inversion of lowfrequency head observations, Water Resour. Res., 42, W05408, doi:10.1029/2005WR004445. (Pubitemid 43955908)
91. Vesselinov, V. V., S. P. Neuman, and W. A. Illman (2001), Threedimensional numerical inversion of pneumatic cross-hole tests in unsaturated fractured tuff: 2. Equivalent parameters, high-resolution stochastic imaging and scale effects, Water Resour. Res., 37(12), 3019-3041. (Pubitemid 34108262)
92. Vrugt, J., C. Diks, H. Gupta, W. Bouten, and J. Verstraten (2005), Improved treatment of uncertainty in hydrologic modeling: Combining the strengths of global optimization and data assimilation, Water Resour. Res., 41, W01017, doi:10.1029/2004WR003059. (Pubitemid 40359297)
93. Weerts, A. H., and G. Y. H. El Serafy (2006), Particle filtering and ensemble Kalman filtering for state updating with hydrological conceptual rainfall-runoff models, Water Resour. Res., 42, W09403, doi:10.1029/ 2005WR004093. (Pubitemid 44619764)
94. Wen, X. H., and W. H. Chen (2005), Real-time reservoir model updating using ensemble Kalman filter, paper presented at Reservoir Simulation Symposium, Soc. of Pet. Eng., Houston, Texas.
95. Wen, X. H., and W. H. Chen (2006), Real-time reservoir model updating using ensemble Kalman filter with confirming option, SPE J., 11(4), 431-442, doi:10.2118/92991-PA. (Pubitemid 46099492)
96. Wen, X. H., and W. H. Chen (2007), Some practical issues on real-time reservoir model updating using ensemble Kalman filter, SPE J., 12(2), 156-166, doi:10.118/111571-PA. (Pubitemid 47046334)
97. Xiong, X., I. M. Navon, and B. Uzunoglu (2006), A note on the particle filter with posterior Gaussian resampling, Tellus, Ser. A, 58(4), 456-460, doi:10.1111/j.1600-0870.2006.00185.x. (Pubitemid 44070045)
98. Yeh, T. C. J., and S. Liu (2000), Hydraulic tomography: Development of a new aquifer test method, Water Resour. Res., 36(8), 2095-2105, doi:10.1029/2000WR900114. (Pubitemid 30610900)
99. Zafari, M., and A. Reynolds (2005), Assessing the uncertainty in reservoir description and performance predictions with the ensemble Kalman filter, paper presented at Annual Technical Conference and Exhibition, Soc. of Pet. Eng., Dallas, Texas.
100. Zhou, H., J. Gomez-Hernandez, H.-J. Hendricks Franssen, and L. Li (2011), An approach to handling non-Gaussianity of parameters and state variables in ensemble Kalman filtering, Adv. Water Res., 34(7), 844-864, doi:10.1016/j. advwatres.2011.04.014.
101. Zhu, J., and T.-C. J. Yeh (2005), Characterization of aquifer heterogeneity using transient hydraulic tomography, Water Resour. Res., 41, W07028, doi:10.1029/2004WR003790. (Pubitemid 41197307)
102. Zhu, J., and T.-C. J. Yeh (2006), Analysis of hydraulic tomography using temporal moments of drawdown recovery data, Water Resour. Res., 42, W02403, doi:10.1029/2005WR004309.