Abstract
Cokriging allows predicting coregionalized variables from sampling information, by considering their spatial joint dependence structure. When secondary covariates are available exhaustively, solving the cokriging equations may become prohibitive, which motivates the use of a moving search neighborhood to select a subset of data, based on their closeness to the target location and the screen effect approximation. This paper investigates the efficiency of different strategies for designing a sub-optimal neighborhood wherein the simplification of the cokriging equations is challenging. To do so, five alternatives (single search, multiple search, strictly collocated search, multi-collocated search and isotopic search) are tested and compared with the reference unique neighborhood, through synthetic examples with different data configurations and spatial joint correlation models. The results indicate that the multi-collocated and multiple searches bear the highest resemblance to the reference case under the analyzed spatial structure models, while the single and the isotopic searches, which do not differentiate the primary and secondary sampling designs, yield the poorest results in terms of cokriging error variance.
Similar content being viewed by others
References
Ahmed S, de Marsily G (1987) Comparison of geostatistical methods for estimating transmissivity using data on transmissivity and specific capacity. Water Resour Res 23(9):1717–1737
Almeida AS, Journel AG (1994) Joint simulation of multiple variables with a Markov-type coregionalization model. Math Geol 26(5):565–588
Babak O, Deutsch CV (2009) Collocated cokriging based on merged secondary attributes. Math Geosci 41:921–926
Boezio MNM, Costa JFCL, Koppe JC (2006) Kriging with an external drift versus collocated cokriging for water table mapping. Trans Inst Min Metall Sect B Appl Earth Sci 115(3):103–112
Bohorquez M, Giraldo R, Mateu J (2017) Multivariate functional random fields: prediction and optimal sampling. Stoch Env Res Risk Assess 31(1):53–70
Borkowski AS, Kwiatkowska-Malina J (2017) Geostatistical modelling as an assessment tool of soil pollution based on deposition from atmospheric air. Geosci J 21(4):645–653
Cao R, Ma YZ, Gomez E (2014) Geostatistical applications in petroleum reservoir modelling. J South Afr Inst Min Metall 114(8):625–629
Chilès JP, Delfiner P (2012) Geostatistics: modeling spatial uncertainty. Wiley, New York
Cornah A, Machaka E (2015) Integration of imprecise and biased data into mineral resource estimates. J South Afr Inst Min Metall 115(6):523–530
D’Agostino V, Greene EA, Passarella G, Vurro M (1998) Spatial and temporal study of nitrate concentration in groundwater by means of coregionalization. Environ Geol 36(3–4):285–295
da Silva CZ, Costa JF (2014) Minimum/maximum autocorrelation factors applied to grade estimation. Stoch Env Res Risk Assess 28(8):1929–1938
D’Agostino V, Passarella G, Vurro M (1997) Assessment of the optimal sampling arrangement based on cokriging estimation variance reduction approach. In: Holly FM, Alsoffar A (eds) Water for a changing global community, proceedings of the 27th congress of the international association for hydraulic research forrest. American Society of Civil Engineers, pp 246–252
Dalla Libera N, Fabbri P, Mason L, Piccinini L, Pola M (2017) Geostatistics as a tool to improve the natural background level definition: an application in groundwater. Sci Total Environ 598:330–340
Emery X (2009) The kriging update equations and their application to the selection of neighboring data. Comput Geosci 13(3):269–280
Emery X (2010) Iterative algorithms for fitting a linear model of coregionalization. Comput Geosci 36(9):1150–1160
Emery X (2012) Cokriging random fields with means related by known linear combinations. Comput Geosci 38(1):136–144
Emery X, Peláez M (2011) Assessing the accuracy of sequential Gaussian simulation and cosimulation. Comput Geosci 15(4):673–689
Emery X, Arroyo D, Peláez M (2014) Simulating large Gaussian random vectors subject to inequality constraints by Gibbs sampling. Math Geosci 46(3):265–283
Fabijańczyk P, Zawadzki J, Magiera T, Szuszkiewicz M (2016) A methodology of integration of magnetometric and geochemical soil contamination measurements. Geoderma 277:51–60
Fouedjio F (2018) A fully non-stationary linear coregionalization model for multivariate random fields. Stoch Env Res Risk Assess 32(6):1699–1721
Gálvez I, Emery X (2011) Multivariate resources modelling: which data are relevant for cokriging? In: Beniscelli J, Kuyvenhoven R, Hoal KO (eds) Proceedings of the 2nd international seminar on geology for the mining industry. Gecamin Ltda, Santiago, Chile, pp 10
Gneiting T, Kleiber W, Schlather M (2010) Matérn cross-covariance functions for multivariate random fields. J Am Stat Assoc 105(491):1167–1177
Goovaerts P (1997) Geostatistics for natural resources evaluation. Oxford University Press, New York
Goulard M, Voltz M (1992) Linear coregionalization model: tools for estimation and choice of cross-variogram matrix. Math Geol 24(3):269–286
Hohn ME (1999) Geostatistics and petroleum geology, 2nd edn. Kluwer Academic, Dordrecht
Jalalalhosseini SM, Ali H, Mostafazadeh M (2014) Predicting porosity by using seismic multi-attributes and well data and combining these available data by geostatistical methods in a South Iranian oil field. Pet Sci Technol 32(1):29–37
Journel AG (1999) Markov models for cross-covariances. Math Geol 31(8):955–964
Journel AG, Huijbregts CJ (1978) Mining geostatistics. Academic Press, London
Kitanidis PK (1997) Introduction to geostatistics: applications to hydrology. Cambridge University Press, London
Lark RM, Ander EL, Cave MR, Knights KV, Glennon MM, Scanlon RP (2014) Mapping trace element deficiency by cokriging from regional geochemical soil data: a case study on cobalt for grazing sheep in Ireland. Geoderma 226–227(1):64–78
Masihi A, Zarei M (2010) Permeability modeling using ANN and collocated cokriging. In: 72nd European association of geoscientists and engineers conference and exhibition 2010: a new spring for geoscience. Incorporating SPE EUROPEC 2010. European Association of Geoscientists and Engineers (EAGE), vol 5, pp 3939–3943
Minnitt RCA, Deutsch CV (2014) Cokriging for optimal mineral resource estimates in mining operations. J South Afr Inst Min Metall 114(3):189–203
Myers DE (1982) Matrix formulation of cokriging. Math Geol 14(3):249–257
Olea RA, Raju NJ, Egozcue JJ, Pawlowsky-Glahn V, Singh S (2018) Advancements in hydrochemistry mapping: methods and application to groundwater arsenic and iron concentrations in Varanasi, Uttar Pradesh, India. Stoch Env Res Risk Assess 32(1):241–259
Pan G, Gaard D, Moss K, Heiner T (1993) A comparison between cokriging and ordinary kriging: case study with a polymetallic deposit. Math Geol 25(3):377–398
Pawlowsky-Glahn V, Egozcue JJ, Olea RA, Pardo-Igúzquiza E (2015) Cokriging of compositional balances including a dimension reduction and retrieval of original units. J South Afr Inst Min Metall 115(1):59–72
Rivoirard J (2001) Which models for collocated cokriging? Math Geol 33(2):117–131
Rivoirard J (2004) On some simplifications of the cokriging neighborhood. Math Geol 36(8):899–915
Roberts BL, McKenna SA (2009) The use of secondary information in geostatistical target area identification. Stoch Env Res Risk Assess 23(2):227–236
Schwab AM, Buckner S, Bramald JA, Cass J (2011) Improving reservoir modeling through integration of seismic data in eocene turbidites for West Brae field, central North Sea, United Kingdom. AAPG Mem 96:107–119
Stein A, Van Dooremolen W, Bouma J, Bregt AK (1988) Cokriging point data on moisture deficit. Soil Sci Soc Am J 52(5):1418–1423
Subramanyam A, Pandalai HS (2004) On the equivalence of the cokriging and kriging systems. Math Geol 36(4):507–523
Subramanyam A, Pandalai HS (2008) Data configurations and the cokriging system: simplification by screen effects. Math Geosci 40(4):425–443
Tolosana-Delgado R, van den Boogaart KG (2013) Joint consistent mapping of high-dimensional geochemical surveys. Math Geosci 45(8):983–1004
Uygucgil H, Konuk A (2015) Reserve estimation in multivariate mineral deposits using geostatistics and GIS. J Min Sci 51(5):993–1000
Vargas-Guzmán J, Jim Yeh TC (1999) Sequential kriging and cokriging: two powerful geostatistical approaches. Stoch Env Res Risk Assess 13(6):416–435
Wackernagel H (1988) Geostatistical techniques for interpreting multivariate spatial information. In: Chung CF, Fabbri AG, Sinding-Larsen R (eds) Quantitative analysis of mineral and energy resources. Reidel, Dordrecht, pp 393–409
Wackernagel H (2003) Multivariate Geostatistics: an introduction with applications. Springer, Berlin
Xu W, Tran TT, Srivastava RM, Journel AG (1992) Integrating seismic data in reservoir modeling: the collocated cokriging alternative. In: 67th SPE annual technical conference and exhibition. Society of Petroleum Engineers, SPE paper 24742, pp 833–842
Yates SR, Warrick AW (1987) Estimating soil water content using cokriging. Soil Sci Soc Am J 51:23–30
Acknowledgements
The first author acknowledges the Nazarbayev University for funding this work via “Faculty development competitive research Grants for 2018–2020” under Contract No. 090118FD5336. The second author acknowledges the Chilean Commission for Scientific and Technological Research (CONICYT), through Grant CONICYT PIA Anillo ACT1407.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Madani, N., Emery, X. A comparison of search strategies to design the cokriging neighborhood for predicting coregionalized variables. Stoch Environ Res Risk Assess 33, 183–199 (2019). https://doi.org/10.1007/s00477-018-1578-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00477-018-1578-1