Estimation of space deformation model for non-stationary random functions

Fouedjio, Francky; Desassis, Nicolas; Romary, Thomas

doi:10.1016/j.spasta.2015.05.001

Cited by 43 publications

(23 citation statements)

References 31 publications

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“…In this work, we consider geologic modeling methods based on interpolation functions (Mallet, 1992(Mallet, , 2004Lajaunie et al, 1997;Carr et al, 2001;Hillier et al, 2014;Fouedjio et al, 2015), here referred to as a function of position ϕ i ð x ! Þ.…”

Section: Introductionmentioning

confidence: 99%

Structural geologic modeling as an inference problem: A Bayesian perspective

Varga

Wellmann

2016

Interpretation

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Structural geologic models are widely used to represent the spatial distribution of relevant geologic features. Several techniques exist to construct these models on the basis of different assumptions and different types of geologic observations. However, two problems are prevalent when constructing models: (1) observations and assumptions, and therefore also the constructed model, are subject to uncertainties and (2) additional information is often available, but it cannot be considered directly in the geologic modeling stepalthough it could be used to reduce model uncertainties. The first problem has been addressed in recent work. Here we develop a conceptual approach to consider the second aspect: We combine uncertain prior information with geologically motivated likelihood functions in a Bayesian inference framework. The result is that we not only reduce uncertainties in the ensemble of generated models, but we also gain the potential to learn additional features about the model parameters. We develop an implementation of this concept in a probabilistic programming framework, in which we extend the functionality of a 3D implicit potential-field interpolation method with geologic likelihood functions. With schematic examples, we show how this combination leads to suites of models with reduced uncertainties and how it provides a deeper insight into parameter correlations. Furthermore, the integration into a hierarchical Bayesian model provides an insight into potential extensions of the method, for example, the interpolation functional itself, and other types of information, such as gravity or magnetic potential-field data. These aspects constitute promising paths for future research.

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Section: Introductionmentioning

confidence: 99%

Structural geologic modeling as an inference problem: A Bayesian perspective

Varga

Wellmann

2016

Interpretation

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“…Remark 5. Note that the notion of χ -isotropy seems to be dependant on the underlying random field X involved in (13). However, after the statement and the proof of Theorem 2, it will be clear that it is in fact not the case.…”

Section: Definition 2 (χ-Isotropic Deformation)mentioning

confidence: 98%

Identification and isotropy characterization of deformed random fields through excursion sets

Fournier

2018

Adv. Appl. Probab.

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A deterministic application θ : R 2 → R 2 deforms bijectively and regularly the plane and allows to build a deformed random field X • θ : R 2 → R from a regular, stationary and isotropic random field X : R 2 → R. The deformed field X • θ is in general not isotropic, however we give an explicit characterization of the deformations θ that preserve the isotropy. Further assuming that X is Gaussian, we introduce a weak form of isotropy of the field X • θ, defined by an invariance property of the mean Euler characteristic of some of its excursion sets. Deformed fields satisfying this property are proved to be strictly isotropic. Besides, assuming that the mean Euler characteristic of excursions sets of X • θ over some basic domains is known, we are able to identify θ.Deformed fields are a class of non-stationary and non-isotropic fields obtained by deforming a fixed stationary and isotropic random field thanks to a deterministic function that transforms bijectively the index set. Deformed fields respond to the need to model spatial and physical phenomena that are in numerous cases not stationary nor isotropic. To give but one example, they are currently widely used in cosmology to model the cosmic microwave background (CMB) deformed anisotropically by the gravitational lensing effect, with mass reconstruction as an objective [14].Our framework is two-dimensional: we set X : R 2 → R the underlying stationary and isotropic field, θ : R 2 → R 2 a bijective deterministic function and X θ = X •θ the deformed field. In fact, most studies on the deformed field model deal with dimension two. The reason for this is that it is the simplest case of multi-dimensionality, the results can be illustrated easily thanks to simulations and it still covers a lot of possible applications, particularly in image analysis. For instance, deformed fields are involved in the "shape from texture" issue, that is, the problem of recovering a 3-dimensional textured surface thanks to a 2-dimensional projection [8].The model of deformed fields was introduced in 1992 in a spatial statistics framework by Sampson and Guttorp in [18], with only a stationarity assumption on X. It is also studied through the covariance function in [16] and in [17].

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“…We use a nonstationary variogram in the exponent function in ( 16) to ensure that (h ij ) is not simply a function of distance. In the context of nonstationary Gaussian processes, Fouedjio et al (2015) propose a semivariogram of the form * (s i , s j ) where * (s i , s j ) = (|| (s j ) − (s j )||), The use of the radial basis function (s) within this semivariogram causes pairs that are closer to o to be more strongly dependent than those pairs that are further away. From ( 16) and ( 17), the Brown-Resnick process with this semivariogram has theoretical (s i , s j ) given by…”

Section: Nonstationary Brown-resnick and Inverted Brown-resnick Processmentioning

confidence: 99%

Spatial deformation for nonstationary extremal dependence

Richards

Wadsworth

2021

Environmetrics

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Modeling the extremal dependence structure of spatial data is considerably easier if that structure is stationary. However, for data observed over large or complicated domains, nonstationarity will often prevail. Current methods for modeling nonstationarity in extremal dependence rely on models that are either computationally difficult to fit or require prior knowledge of covariates.Sampson and Guttorp (1992) proposed a simple technique for handling nonstationarity in spatial dependence by smoothly mapping the sampling locations of the process from the original geographical space to a latent space where stationarity can be reasonably assumed. We present an extension of this method to a spatial extremes framework by considering least squares minimization of pairwise theoretical and empirical extremal dependence measures. Along with some practical advice on applying these deformations, we provide a detailed simulation study in which we propose three spatial processes with varying degrees of nonstationarity in their extremal and central dependence structures. The methodology is applied to Australian summer temperature extremes and UK precipitation to illustrate its efficacy compared with a naive modeling approach.

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Estimation of space deformation model for non-stationary random functions

Cited by 43 publications

References 31 publications

Structural geologic modeling as an inference problem: A Bayesian perspective

Structural geologic modeling as an inference problem: A Bayesian perspective

Identification and isotropy characterization of deformed random fields through excursion sets

Spatial deformation for nonstationary extremal dependence

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