Morris method with improved sampling strategy and Sobol’ variance-based method, as validation tool on numerical model of Richard’s equation

Sunny Goh


Richard’s equation was approximated by finite-difference numerical scheme to model water infiltration profile in variably unsaturated soil. The published data of Philip’s semi-analytical solution was used to validate the simulated results from the numerical scheme. A discrepancy was found between the simulated and the published semi-analytical results. Morris method as a global sensitivity tool was used as an alternative to local sensitivity analysis to assess the results discrepancy. Morris method with different sampling strategies were tested, of which Manhattan distance method have resulted a better sensitivity measures and also a better scan of input space than Euclidean method. Moreover, Morris method at  and Manhattan distance sampling strategy, with only 2 extra simulation runs than local sensitivity analysis, was able to produce reliable sensitivity measures ( , ). The sensitivity analysis results were cross-validated by Sobol’ variance-based method with 150,000 simulation runs. The global sensitivity tool has identified three important parameters, of which spatial discretization size was the sole reason of the discrepancy observed. In addition, a high proportion of total output variance contributed by parameters  and  is suggesting a greater significant digits is required to reduce its input uncertainty range.


Richard’s equation; Morris method; Sobol’s variance-based method; Euclidean distance sampling strategy; Manhattan distance sampling strategy; global sensitivity analysis

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Campolongo F, Cariboni J, Saltelli A. An effective screening design for sensitivity analysis of large models. Environmental Modelling & Software 2007; 22(10): 1509-1518. doi:10.1016/j.envsoft.2006.10.004.

Campolongo F, Saltelli A. Sensitivity analysis of an environmental model: an application of different analysis methods. Reliability Engineering & System Safety 1997; 57(1), 49-69. doi:

Campolongo F, Saltelli A., Cariboni J. From screening to quantitative sensitivity analysis. A unified approach. Computer Physics Communications 2011; 182(4): 978-988. doi:10.1016/j.cpc.2010.12.039.

Campolongo F, Tarantola S, Saltelli A. Tackling quantitatively large dimensionality problems. Computer Physics Communications 1999; 117(1–2): 75-85. doi:10.1016/S0010-4655(98)00165-9.

Caviedes-Voullième D, Garcı´a-Navarro P, Murillo J. Verification, conservation, stability and efficiency of a finite volume method for the 1D Richards equation. Journal of Hydrology 2013; 480(0): 69-84. doi:

Celia MA., Bouloutas ET., Zarba RL. A general mass-conservative numerical solution for the unsaturated flow equation. Water Resources Research 1990; 26(7): 1483-1496. doi:10.1029/WR026i007p01483

Chu-Agor ML., Muñoz-Carpena R, Kiker G, et al. Exploring vulnerability of coastal habitats to sea level rise through global sensitivity and uncertainty analyses. Environmental Modelling & Software 2011; 26(5), 593-604. doi:10.1016/j.envsoft.2010.12.003.

Cohen D, Person M, Daannen R, et al. Groundwater-supported evapotranspiration within glaciated watersheds under conditions of climate change. Journal of Hydrology 2006; 320(3–4), 484-500. doi:

Cukier RI., Fortuin CM., Shuler KE., et al. Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. I Theory. The Journal of Chemical Physics 1973; 59(8): 3873-3878. doi:doi:

Davis A, Kamp S, Fennemore G, et al. Environmental Policy Analysis, Peer Reviewed: A Risk-Based Approach to Soil Remediation Modeling. Environmental Science & Technology 1997; 31(11): 520A-525A (1997). doi:10.1021/es9725662.

De Roo APJ., Offermans RJE. LISEM: a physically-based hydrological and soil erosion model for basin-scale water and sediment management. In: Modelling and Management of Sustainable Basin-scale Water Resource Systems (Proceedings of a Boulder Symposium) 1995. IAHS Publication

Drouet JL., Capian N, Fiorelli JL., et al. Sensitivity analysis for models of greenhouse gas emissions at farm level. Case study of N2O emissions simulated by the CERES-EGC model. Environmental Pollution 159(11), 3156-3161 (2011). doi:10.1016/j.envpol.2011.01.019.

Fox GA., Muñoz-Carpena R, Sabbagh GJ. Influence of flow concentration on parameter importance and prediction uncertainty of pesticide trapping by vegetative filter strips. Journal of Hydrology 384(1–2), 164-173 (2010). doi:10.1016/j.jhydrol.2010.01.020

Goh EG., Noborio K. Sensitivity Analysis on the Infiltration of Water into Unsaturated Soil. In: Proceedings of Soil Moisture Workshop, Hiroshima University Tokyo Office in Campus Innovation Center 2013, pp. 66-68.

Gosling SN., Arnell NW. Simulating current global river runoff with a global hydrological model: model revisions, validation, and sensitivity analysis. Hydrological Processes 2011; 25(7): 1129-1145. doi:10.1002/hyp.7727

Haverkamp R., Vauclin M, Touma J, et al. A Comparison of Numerical Simulation Models For One-Dimensional Infiltration. Soil Science Society of America Journal 1977; 41(2), 285-294. doi:10.2136/sssaj1977.03615995004100020024x

Istok J. Step 4: Solve System of Equations. In: Groundwater Modeling by the Finite Element Method. pp. 176-225. American Geophysical Union, 2013.

Jhorar RK., Bastiaanssen WGM., Feddes RA., et al. Inversely estimating soil hydraulic functions using evapotranspiration fluxes. Journal of Hydrology 2002; 258(1–4): 198-213 (2002). doi:

Kabala ZJ., Milly PCD. Sensitivity analysis of flow in unsaturated heterogeneous porous media: Theory, numerical model, and its verification. Water Resources Research 1990; 26(4): 593-610. doi:10.1029/WR026i004p00593.

Lagerwall G, Kiker G, Muñoz-Carpena R, et al. Global uncertainty and sensitivity analysis of a spatially distributed ecological model. Ecological Modelling 2014; 275(0): 22-30. doi:

Ma Y, Feng S, Su D, et al. Modeling water infiltration in a large layered soil column with a modified Green–Ampt model and HYDRUS-1D. Computers and Electronics in Agriculture 2010; 71, Supplement 1(0): S40-S47. doi:

Min CH., He YL., Liu XL., et al. Parameter sensitivity examination and discussion of PEM fuel cell simulation model validation: Part II: Results of sensitivity analysis and validation of the model. Journal of Power Sources 2006; 160(1): 374-385. doi:10.1016/j.jpowsour.2006.01.080.

Moreau P, Viaud V, Parnaudeau V, et al. An approach for global sensitivity analysis of a complex environmental model to spatial inputs and parameters: A case study of an agro-hydrological model. Environmental Modelling & Software 2013; 47(0), 74-87. doi:

Morris MD. Factorial sampling plans for preliminary computational experiments. Technometrics 1991; 33(2): 161-174 (1991). doi:10.2307/1269043.

Namin MM., Boroomand MR. A time splitting algorithm for numerical solution of Richard’s equation. Journal of Hydrology 2012; 444–445(0): 10-21. doi:

Nathan R, Safriel UN., Noy-Meir I. Field Validation and Sensitivity Analysis of a Mechanistic Model for Tree Seed Dispersal by Wind. Ecology 2001; 82(2): 374-388. doi:10.1890/0012-9658(2001)082[0374:fvasao];2.

Nossent J, Elsen P, Bauwens W. Sobol’ sensitivity analysis of a complex environmental model. Environmental Modelling & Software 2011; 26(12): 1515-1525. doi:

Pannell DJ. Sensitivity analysis of normative economic models: theoretical framework and practical strategies. Agricultural Economics 1997; 16(2), 139-152. doi:10.1016/S0169-5150(96)01217-0.

Pappenberger F, Beven KJ., Ratto M, et al. Multi-method global sensitivity analysis of flood inundation models. Advances in Water Resources 2008; 31(1): 1-14. doi:

Poeter EP., Hill MC. Documentation of UCODE: A computer code for universal inverse modeling, vol. 98. vol. 4080. DIANE Publishing, 1998.

Richards LA. Capillary Conduction of Liquids through Porous Mediums. Journal of Applied Physics 1931; 1(5): 318-333. doi:10.1063/1.1745010.

Saltelli A, Annoni P. How to avoid a perfunctory sensitivity analysis. Environmental Modelling & Software 2010; 25(12): 1508-1517. doi:10.1016/j.envsoft.2010.04.012.

Saltelli A, Annoni P, Azzini I, et al. Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index. Computer Physics Communications 2010; 181(2), 259-270. doi:

Saltelli A, Chan K, Scott EM. Sensitivity analysis Wiley series in probability and statistics. Willey, New York, 2000.

Saltelli A, Ratto M, Andres T, et al. Elementary Effects Method. In: Global Sensitivity Analysis. The Primer. pp. 109-154. John Wiley & Sons, Ltd, 2008a.

Saltelli A, Ratto M, Andres T, et al. Variance-Based Methods. In: Global Sensitivity Analysis. The Primer. pp. 155-182. John Wiley & Sons, Ltd, 2008b.

Saltelli A., Tarantola S, Campolongo F, et al. The Screening Exercise. In: Sensitivity Analysis in Practice. pp. 91-108. John Wiley & Sons, Ltd, 2004.

Saltelli A, Tarantola S, Chan KPS. A Quantitative Model-Independent Method for Global Sensitivity Analysis of Model Output. Technometrics 1999; 41(1): 39-56. doi:10.1080/00401706.1999.10485594

Schaibly, J.H., Shuler, K.E.: Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. II Applications. The Journal of Chemical Physics 1973; 59(8), 3879-3888 (1973). doi:doi:

Sepúlveda FD., Cisternas LA., Gálvez ED. The use of global sensitivity analysis for improving processes: Applications to mineral processing. Computers & Chemical Engineering 2014; 66(0), 221-232. doi:10.1016/j.compchemeng.2014.01.008.

Sobol’ IM. On sensitivity estimates for nonlinear mathematical models. Matematicheskoe Modelirovanie 1990, 112–118.

Stange F, Butterbach-Bahl K, Papen H, et al. A process-oriented model of N2O and NO emissions from forest soils: 2. Sensitivity analysis and validation. Journal of Geophysical Research: Atmospheres 2000; 105(D4), 4385-4398. doi:10.1029/1999jd900948

Tang Y, Reed P, Wagener T, et al. Comparing sensitivity analysis methods to advance lumped watershed model identification and evaluation. Hydrol. Earth Syst. Sci. 2007; 11(2); 793-817. doi:10.5194/hess-11-793-2007.

Tarantola S, Giglioli N, Jesinghaus J, et al. Can global sensitivity analysis steer the implementation of models for environmental assessments and decision-making? Stochastic Environmental Research and Risk Assessment 2002; 16(1), 63-76. doi:10.1007/s00477-001-0085-x.

Tu J, Yeoh GH., Liu C. Computational fluid dynamics: a practical approach. Butterworth-Heinemann, 2007.

Vazquez-Cruz MA., Guzman-Cruz R, Lopez-Cruz IL., et al. Global sensitivity analysis by means of EFAST and Sobol’ methods and calibration of reduced state-variable TOMGRO model using genetic algorithms. Computers and Electronics in Agriculture 2014; 100(0), 1-12. doi:10.1016/j.compag.2013.10.006

Vereecken H, Maes J, Feyen J. Estimating unsaturated hdraulic conductivity from easily measured soil properties. Soil Science 1990; 149(1), 1-12.

Wagener T, van Werkhoven K, Reed P, et al. Multiobjective sensitivity analysis to understand the information content in streamflow observations for distributed watershed modeling. Water Resources Research 2009; 45(2), W02501 (2009). doi:10.1029/2008WR007347.

Yang J. Convergence and uncertainty analyses in Monte-Carlo based sensitivity analysis. Environmental Modelling & Software 2011; 26(4), 444-457. doi:

Zheng C, Bennett GD. Applied Contaminant Transport Modeling, 2 ed. John Wiley & Sons, New York, 2002.



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