A Simple Substitute for Conditional Expectation : The Disjunctive Kriging

Matheron, G.

doi:10.1007/978-94-010-1470-0_14

Cited by 228 publications

(75 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…, u * n (x 0 ), additional structural assumptions on Z are needed, and depending on these assumptions one distinguishes simple, ordinary, universal and intrinsic kriging. There are still quite some other forms (such as complex kriging [44], indicator kriging [40] or disjunctive kriging [51,53]) but we shall only discuss the aforementioned ones due to their close connection to kernel interpolation.…”

Section: Krigingmentioning

confidence: 99%

Interpolation of spatial data – A stochastic or a deterministic problem?

2013

View full text Add to dashboard Cite

Interpolation of spatial data is a very general mathematical problem with various applications. In geostatistics, it is assumed that the underlying structure of the data is a stochastic process which leads to an interpolation procedure known as kriging. This method is mathematically equivalent to kernel interpolation, a method used in numerical analysis for the same problem, but derived under completely different modelling assumptions. In this article we present the two approaches and discuss their modelling assumptions, notions of optimality and the different concepts to quantify the interpolation accuracy. Their relation is much closer than has been appreciated so far, and even results on convergence rates of kernel interpolants can be translated to the geostatistical framework. We sketch the different answers obtained in the two fields concerning the issue of kernel misspecification, present some methods for kernel selection, and discuss the scope of these methods with a data example from the computer experiments literature.

show abstract

Section: Krigingmentioning

confidence: 99%

Interpolation of spatial data – A stochastic or a deterministic problem?

2013

View full text Add to dashboard Cite

show abstract

“…The tenants of Geostatistics immediately fought back, and introduced discontinuous Geostatistical models. The concept of disjunctive Kriging was introduced by Matheron as early as 1973 (Matheron 1973(Matheron , 1976, but not commonly used; Journel (1983), Journel and Isaaks (1984), Journel and Alabert (1990) and Journel and Gomez-Hernandez (1993) developed the Indicator Kriging approach, where an indicator can take (at any given point in space) the value zero or one, depending on whether the point is inside or outside a given facies. Matheron and the Heresim Group at the French Institute of Petroleum developed the Gaussian Threshold model, where a continuous Gaussian random function in space generates a given facies if the function value at a point in space falls between two successive prescribed thresholds (Matheron et al 1987(Matheron et al , 1988see also Rivoirard 1994;Chiles and Delfiner 1999;Armstrong et al 2003, for a review of such stochastic models).…”

Section: Geostatistics Fights Back: Discontinuous Facies Modelsmentioning

confidence: 99%

Dealing with spatial heterogeneity

et al. 2005

View full text Add to dashboard Cite

Heterogeneity can be dealt with by defining homogeneous equivalent properties, known as averaging, or by trying to describe the spatial variability of the rock properties from geologic observations and local measurements. The techniques available for these descriptions are mostly continuous Geostatistical models, or discontinuous facies models such as the Boolean, Indicator or Gaussian-Threshold models and the Markov chain model. These facies models are better suited to treating issues of rock strata connectivity, e.g. buried high permeability channels or low permeability barriers, which greatly affect flow and, above all, transport in aquifers. Genetic models provide new ways to incorporate more geology into the facies description, an approach that has been well developed in the oil industry, but not enough in hydrogeology. The conclusion is that future work should be focused on improving the facies models, comparing them, and designing new in situ testing procedures (including geophysics) that would help identify the facies geometry and properties. A world-wide catalog of aquifer facies geometry and properties, which could combine site genesis and description with methods used to assess the system, would be of great value for practical applications.RØsumØ On peut aborder le problme de l'hØtØrogØnØitØ en s'efforçant de dØfinir une permØabilitØ Øquivalente homogne, par prise de moyenne, ou au contraire en dØ-crivant la variation dans l'espace des propriØtØs des roches à partir des observations gØologiques et des mesures locales. Les techniques disponibles pour une telle description sont soit continues, comme l'approche GØosta-tistique, soit discontinues, comme les modles de facis, BoolØens, ou bien par Indicatrices ou Gaussiennes SeuillØes, ou enfin Markoviens. Ces modles de facis sont mieux capables de prendre en compte la connectivitØ des strates gØologiques, telles que les chenaux enfouis à forte permØabilitØ, ou au contraire les facis fins de barrires de permØabilitØ, qui ont une influence importante sur les Øcoulement, et, plus encore, sur le transport. Les modles gØnØtiques rØcemment apparus ont la capacitØ de mieux incorporer dans les modles de facis les observations gØologiques, chose courante dans l'industrie pØ-trolire, mais insuffisamment dØveloppØe en hydrogØo-logie. On conclut que les travaux de recherche ultØrieurs devraient s'attacher à dØvelopper les modles de facis, à les comparer entre eux, et à mettre au point de nouvelles mØthodes d'essais in situ, comprenant les mØthodes gØo-physiques, capables de reconnaître la gØomØtrie et les propriØtØs des facis. La constitution d'un catalogue mondial de la gØomØtrie et des propriØtØs des facis aquifres, ainsi que des mØthodes de reconnaissance utilisØes pour arriver à la dØtermination de ces systmes, serait d'une grande importance pratique pour les applications.Resumen La heterogeneidad se puede manejar por medio de la definición de características homogØneas equivalentes, conocidas como promediar o tratando de describir la variabilidad espacia...

show abstract

“…These include an over-smoothing and non-applicability for certain problems such as the estimation of probabilities. At least two extensions have been proposed, disjunctive kriging (Matheron, 1976) and indicator kriging (Journel, 1985). The disjunctive kriging estimator can be thought of as a generalization of (8) …”

Section: Non-linear Extensionsmentioning

confidence: 99%

Spatial interpolation: an overview

Myers

1994

Geoderma

165

View full text Add to dashboard Cite

The interpolation of spatial data has been considered in many different forms. The various forms of kriging are among the best known in the earth sciences although techniques such as inverse distance weighting were and are in use for spatially located data. In the numerical analysis literature various forms of splines and more recently radial basis functions have been developed and used. Because these techniques have been developed in very different contexts the relationship between them has not always been apparent. Various forms of kriging are considered as well as kernel estimators, splines and radial basis functions. By using the dual form of kriging and the positive definiteness property of the variogram connections are shown between splines, kriging and radial basis functions. One of the distinctions between kriging and other interpolators is the incorporation of the support of the samples and explicit estimation of linear functionals such as spatial integrals.

show abstract

A Simple Substitute for Conditional Expectation : The Disjunctive Kriging

Cited by 228 publications

References 0 publications

Interpolation of spatial data – A stochastic or a deterministic problem?

Interpolation of spatial data – A stochastic or a deterministic problem?

Dealing with spatial heterogeneity

Spatial interpolation: an overview

Contact Info

Product

Resources

About