Geostatistical interpolation using copulas

Abstract

[1] In many applications of geostatistical methods, the dependence structure of the investigated parameter is described solely with the variogram or covariance functions, which are susceptible to measurement anomalies and implies the assumption of Gaussian dependence. Moreover the kriging variance respects only observation density, data geometry and the variogram model. To address these problems, we borrow the idea from copulas, to depict the dependence structure without the influence of the marginal distribution. The methodology and basic hypotheses for application of copulas as geostatistical methods are discussed and the Gaussian copula as well as a non-Gaussian copula are used in this paper. Copula parameters are estimated using a division of the observations into multipoint subsets and a subsequent maximization of the corresponding likelihood function. The interpolation is carried out with two different copulas, where the expected and median values are calculated from the copulas conditioned with the nearby observations. The full conditional copulas provide the estimation distributions for the unobserved locations and can be used to define confidence intervals which depend on both the observation geometry and values. Observations of a large scale groundwater quality measurement network in Baden-Württemberg are used to demonstrate the methodology. Five groundwater quality parameters: chloride, nitrate, pH, sulfate and dissolved oxygen are investigated. All five parameters show non-Gaussian dependence. The copula-based interpolation results of the five parameters are compared to the results of conventional ordinary and indicator kriging. Different statistical measures including mean squared error, relative differences and probability scores are used to compare cross validation and split sampling results of the interpolation methods. The non-Gaussian copulas give better results than the geostatistical interpolations. Validation of the confidence intervals shows that they are more realistic than the estimation variances obtained by ordinary kriging.

Upper tail dependence in rainfall extremes: would we know it if we saw it?

“…The fast growth of multivariate frequency analysis (thanks partly to the introduction of apparently more manageable statistical tools such as copulas) has led to an extensive application of multivariate models to a variety of hydrological analyses going from the the study of the relationships between the characteristics of objects such as drought events and hydrographs (e.g., Serinaldi et al 2009;Volpi and Fiori 2012) to the study of the occurrence of extreme events at multiple sites (e.g., Ghizzoni et al 2010) to spatial interpolation and simulation problems (e.g., Bárdossy 2006;Bárdossy and Li 2008;Bárdossy and Pegram 2013). This intense activity resulted in a large body of literature that was almost unavoidably focused on showing the potential application rather than on the actual nature of the variables at hand, the possible shortcomings of the methods used, and the reliability of multivariate methods applied to the usually very short hydrological time series.…”

Cópulas en geoestadística o lo que se puede hacer con coordenadas y estructuras de dependencia

“…La primera fue usada en principios de los cincuenta por Danie G. Krige, en Sudáfrica, para ampliar técnicas estadísticas para la estimación de las reservas de minerales (Bárdossy & Li 2008). En los años sesenta el trabajo de Krige fue formalizado por el matemático Georges Matheron, desde entonces ha sido ampliamente utilizado enáreas como la minería, la industria petrolera, hidrología, meteorología, oceanografía, el control del medio ambiente, la ecología del paisaje y la agricultura.…”

“…A pesar de estos desarrollos, el modelado espacial a menudo se basa en hipótesis gaussianas, que muchas veces no se consideran realistas para los tipos de datos y se reportan datos atípicos que causan problemas en las investigaciones (Bárdossy & Li 2008). …”

Random Mixing: An Approach to Inverse Modeling for Groundwater Flow and Transport Problems

“…For further information, the reader is referred to Nelson (1999). The applicability of copulas in spatial problems is extensively described in Bárdossy and Li (2008), Haslauer et al (2012), and Guthke (2013). Using a Gaussian copula, the transmissivity field W (x) has to be transformed to a multinormal field Z (x).…”