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Geostatistics from Elliptic Boundary-Value Problems: Green Operators, Transmission Conditions, and Schur Complements

Juan J. Segura

TL;DR

This work develops an operator-centric framework for Gaussian spatial fields on bounded domains and manifolds with interfaces, embedding boundary and transmission data directly into the covariance structure via the Green operator $G=\mathcal{L}^{-1}$. By tying coercive bilinear forms to a precision–covariance duality, it shows that variograms are derived functionals of the Green operator and depend on domain geometry and boundary conditions; conditioning and kriging follow from standard Gaussian updates in either covariance or precision form. It introduces surface penalties to model defects and interfaces, producing transmission conditions and controlled cross-interface covariance attenuation, and uses Dirichlet-to-Neumann maps and Schur complements for exact domain reduction and boundary-driven upscaling. The paper also demonstrates product-manifold decompositions for nested covariances, non-stationarity via deformation versus metric geometry, and a practical synthesis of dualities with minimal implementation pseudocode for SPDE-based kriging and conditional simulation. Overall, it provides a rigorous operator framework that makes boundary and interface effects explicit in covariance modeling, prediction, and upscaling for geostatistical problems on complex domains.

Abstract

Classical geostatistics encodes spatial dependence by prescribing variograms or covariance kernels on Euclidean domains, whereas the SPDE--GMRF paradigm specifies Gaussian fields through an elliptic precision operator whose inverse is the corresponding Green operator. We develop an operator-based formulation of Gaussian spatial random fields on bounded domains and manifolds with internal interfaces, treating boundary and transmission conditions as explicit components of the statistical model. Starting from coercive quadratic energy functionals, variational theory yields a precise precision--covariance correspondence and shows that variograms are derived quadratic functionals of the Green operator, hence depend on boundary conditions and domain geometry. Conditioning and kriging follow from standard Gaussian update identities in both covariance and precision form, with hard constraints represented equivalently by exact interpolation constraints or by distributional source terms. Interfaces are modelled via surface penalty terms; taking variations produces flux-jump transmission conditions and induces controlled attenuation of cross-interface covariance. Finally, boundary-driven prediction and domain reduction are formulated through Dirichlet-to-Neumann operators and Schur complements, providing an operator language for upscaling, change of support, and subdomain-to-boundary mappings. Throughout, we use tools standard in spatial statistics and elliptic PDE theory to keep boundary and interface effects explicit in covariance modeling and prediction.

Geostatistics from Elliptic Boundary-Value Problems: Green Operators, Transmission Conditions, and Schur Complements

TL;DR

This work develops an operator-centric framework for Gaussian spatial fields on bounded domains and manifolds with interfaces, embedding boundary and transmission data directly into the covariance structure via the Green operator . By tying coercive bilinear forms to a precision–covariance duality, it shows that variograms are derived functionals of the Green operator and depend on domain geometry and boundary conditions; conditioning and kriging follow from standard Gaussian updates in either covariance or precision form. It introduces surface penalties to model defects and interfaces, producing transmission conditions and controlled cross-interface covariance attenuation, and uses Dirichlet-to-Neumann maps and Schur complements for exact domain reduction and boundary-driven upscaling. The paper also demonstrates product-manifold decompositions for nested covariances, non-stationarity via deformation versus metric geometry, and a practical synthesis of dualities with minimal implementation pseudocode for SPDE-based kriging and conditional simulation. Overall, it provides a rigorous operator framework that makes boundary and interface effects explicit in covariance modeling, prediction, and upscaling for geostatistical problems on complex domains.

Abstract

Classical geostatistics encodes spatial dependence by prescribing variograms or covariance kernels on Euclidean domains, whereas the SPDE--GMRF paradigm specifies Gaussian fields through an elliptic precision operator whose inverse is the corresponding Green operator. We develop an operator-based formulation of Gaussian spatial random fields on bounded domains and manifolds with internal interfaces, treating boundary and transmission conditions as explicit components of the statistical model. Starting from coercive quadratic energy functionals, variational theory yields a precise precision--covariance correspondence and shows that variograms are derived quadratic functionals of the Green operator, hence depend on boundary conditions and domain geometry. Conditioning and kriging follow from standard Gaussian update identities in both covariance and precision form, with hard constraints represented equivalently by exact interpolation constraints or by distributional source terms. Interfaces are modelled via surface penalty terms; taking variations produces flux-jump transmission conditions and induces controlled attenuation of cross-interface covariance. Finally, boundary-driven prediction and domain reduction are formulated through Dirichlet-to-Neumann operators and Schur complements, providing an operator language for upscaling, change of support, and subdomain-to-boundary mappings. Throughout, we use tools standard in spatial statistics and elliptic PDE theory to keep boundary and interface effects explicit in covariance modeling and prediction.
Paper Structure (32 sections, 21 theorems, 43 equations, 1 figure)

This paper contains 32 sections, 21 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Operator Backbone: Actions, Green Operators, and Gaussian Fields
  3. Energy spaces and coercive forms
  4. From bilinear form to operator and Green operator
  5. Discrete Gaussian fields and precision--covariance duality
  6. Variograms as derived quadratic functionals
  7. Operator inverse vs. integral kernel (U3)
  8. Polyakov-Type Actions as Variogram Generators
  9. Worked Micro-Examples: Boundaries and Interfaces (U2)
  10. 1D interval: Dirichlet vs Neumann Green kernel and variogram
  11. Implication for empirical variography on bounded domains (U4).
  12. Two-subdomain interface: surface term and derivative jump
  13. Product Manifolds and Nested Covariance Structures
  14. Conditioning, Kriging, and Source Representations
  15. Gaussian conditioning in covariance form
  16. ...and 17 more sections

Key Result

Lemma 2.1

Under Assumption ass:elliptic, there exist $M,\alpha>0$ such that

Figures (1)

  • Figure 1: Schematic domain effects on spatial dependence. Left: the same elliptic operator on a bounded domain yields different Green kernels (hence covariances/variograms) under Dirichlet vs. Neumann boundary conditions, illustrating boundary-induced screening/reflection. Right: an internal interface $S$ with defect penalty $\frac{\alpha}{2}\int_S Z^2\,dS$ (or equivalent transmission condition) reduces cross-interface correlation as $\alpha$ increases, while preserving stronger along-interface continuity.

Theorems & Definitions (45)

  • Definition 2.1: Energy space and bilinear form
  • Lemma 2.1: Boundedness and coercivity
  • proof
  • Theorem 2.1: Lax--Milgram
  • proof
  • Proposition 2.1: Operator induced by a coercive form
  • Definition 2.2: Green operator (variational inverse)
  • Proposition 2.2: Symmetry/positivity
  • Definition 2.3: Gaussian vector with precision
  • Theorem 2.2: Precision--covariance identity
  • ...and 35 more