Advertisement

Simulation of intrinsic random fields of order k with a continuous spectral algorithm

Original Paper
  • 42 Downloads

Abstract

Intrinsic random fields of order k, defined as random fields whose high-order increments (generalized increments of order k) are second-order stationary, are used in spatial statistics to model regionalized variables exhibiting spatial trends, a feature that is common in earth and environmental sciences applications. A continuous spectral algorithm is proposed to simulate such random fields in a d-dimensional Euclidean space, with given generalized covariance structure and with Gaussian generalized increments of order k. The only condition needed to run the algorithm is to know the spectral measure associated with the generalized covariance function (case of a scalar random field) or with the matrix of generalized direct and cross-covariances (case of a vector random field). The algorithm is applied to synthetic examples to simulate intrinsic random fields with power generalized direct and cross-covariances, as well as an intrinsic random field with power and spline generalized direct covariances and Matérn generalized cross-covariance.

Keywords

Non-stationary random fields Generalized direct and cross-covariances Generalized increments of order k Spectral density 

Notes

Acknowledgements

The authors acknowledge the funding by the Chilean Commission for Scientific and Technological Research (CONICYT), through Projects CONICYT/ FONDECYT/ INICIACIÓN EN INVESTIGACIÓN/ N\(^{\circ }\)11170529 (Daisy Arroyo) and CONICYT PIA Anillo ACT1407 (Xavier Emery). Two anonymous reviewers are also acknowledged for their constructive comments.

References

  1. Apanasovich TV, Genton MG, Sun Y (2012) A valid Matérn class of cross-covariance functions for multivariate random fields with any number of components. J Am Stat Assoc 107(497):180–193CrossRefGoogle Scholar
  2. Arroyo D, Emery X (2015) Simulation of intrinsic random fields of order \(k\) with Gaussian generalized increments by Gibbs sampling. Math Geosci 47(8):955–974CrossRefGoogle Scholar
  3. Arroyo D, Emery X (2017) Spectral simulation of vector random fields with stationary Gaussian increments in \(d\)-dimensional Euclidean spaces. Stoch Environ Res Risk Assess 31(7):1583–1592CrossRefGoogle Scholar
  4. Bochner S (1933) Monotone Funktionen Stieljessche Integrale and Harmonische Analyse. Math Ann 108:378–410CrossRefGoogle Scholar
  5. Buttafuoco G, Castrignano A (2005) Study of the spatio-temporal variation of soil moisture under forest using intrinsic random functions of order \(k\). Geoderma 128(3–4):208–220CrossRefGoogle Scholar
  6. Cassiani G, Christakos G (1998) Analysis and estimation of natural processes with nonhomogeneous spatial variation using secondary information. Math Geol 30(1):57–76CrossRefGoogle Scholar
  7. Chilès JP, Delfiner P (2012) Geostatistics: modeling spatial uncertainty, 2nd edn. Wiley, New YorkCrossRefGoogle Scholar
  8. Chilès JP, Gable R (1984) Three-dimensional modelling of a geothermal field. In: Verly G, David M, Journel AG, Maréchal A (eds) Geostatistics for natural resources characterization. Reidel, Dordrecht, pp 587–598CrossRefGoogle Scholar
  9. Christakos G (1992) Random field models in earth sciences. Academic, San DiegoGoogle Scholar
  10. Christakos G, Thesing GA (1993) The intrinsic random-field model in the study of sulfate deposition processes. Atmos Environ Part A Gen Top 27(10):1521–1540CrossRefGoogle Scholar
  11. de Fouquet C (1994) Reminders on the conditioning kriging. In: Armstrong M, Dowd PA (eds) Geostatistical simulations. Kluwer, Dordrecht, pp 131–145CrossRefGoogle Scholar
  12. Dimitrakopoulos R (1990) Conditional simulation of intrinsic random functions of order \(k\). Math Geol 22(3):361–380CrossRefGoogle Scholar
  13. Dong A, Ahmed S, Marsily G de (1990) Development of geostatistical methods dealing with the boundary conditions problem encountered in fluid mechanics of porous media. In: Guérillot D, Guillon O (eds) 2nd European conference on the mathematics of oil recovery. Edition Technip, Paris, pp 21–30Google Scholar
  14. Emery X (2010) Multi-Gaussian kriging and simulation in the presence of an uncertain mean value. Stoch Environ Res Risk Assess 24(2):211–219CrossRefGoogle Scholar
  15. Emery X, Arroyo D, Porcu E (2016) An improved spectral turning-bands algorithm for simulating stationary vector Gaussian random fields. Stoch Environ Res Risk Assess 30(7):1863–1873CrossRefGoogle Scholar
  16. Emery X, Lantuéjoul C (2006) TBSIM: a computer program for conditional simulation of three-dimensional Gaussian random fields via the turning bands method. Comput Geosci 32(10):1615–1628CrossRefGoogle Scholar
  17. Emery X, Lantuéjoul C (2008) A spectral approach to simulating intrinsic random fields with power and spline generalized covariances. Comput Geosci 12(1):121–132CrossRefGoogle Scholar
  18. Gneiting T, Kleiber W, Schlather M (2010) Matérn cross-covariance functions for multivariate random fields. J Am Stat Assoc 105(491):1167–1177CrossRefGoogle Scholar
  19. Haas A, Jousselin C (1976) Geostatistics in petroleum industry. In: Guarascio M, David M, Huijbregts C (eds) Advanced geostatistics in the mining industry. Springer, Dordrecht, pp 333–347Google Scholar
  20. Huang C, Yao Y, Cressie N, Hsing T (2009) Multivariate intrinsic random functions for cokriging. Math Geosci 41(8):887–904CrossRefGoogle Scholar
  21. Kitanidis PK (1999) Generalized covariance functions associated with the Laplace equation and their use in interpolation and inverse problems. Water Resour Res 35(5):1361–1367CrossRefGoogle Scholar
  22. Lantuéjoul C (2002) Geostatistical simulation: models and algorithms. Springer, BerlinCrossRefGoogle Scholar
  23. Madani N, Emery X (2017) Plurigaussian modeling of geological domains based on the truncation of non-stationary Gaussian random fields. Stoch Environ Res Risk Assess 31:893–913CrossRefGoogle Scholar
  24. Maleki M, Emery X (2017) Joint simulation of stationary grade and non-stationary rock type for quantifying geological uncertainty in a copper deposit. Comput Geosci 109:258–267CrossRefGoogle Scholar
  25. Matheron G (1973) The intrinsic random functions and their applications. Adv Appl Probab 5:439–468CrossRefGoogle Scholar
  26. Pardo-Igúzquiza E, Dowd PA (2003) IRFK2D: a computer program for simulating intrinsic random functions of order \(k\). Comput Geosci 29(6):753–759CrossRefGoogle Scholar
  27. Suárez Arriaga MC, Samaniego F (1998) Intrinsic random functions of high order and their application to the modeling of non-stationary geothermal parameters. In: Twenty-third workshop on geothermal reservoir engineering. Technical report SGP-TR-158, Stanford University, Stanford, pp 169–175Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of ConcepciónConcepciónChile
  2. 2.Department of Mining EngineeringUniversity of ChileSantiagoChile
  3. 3.Advanced Mining Technology CenterUniversity of ChileSantiagoChile

Personalised recommendations