## Abstract

Gaussian random field (GRF) conditional simulation is a key ingredient in many spatial statistics problems for computing Monte-Carlo estimators and quantifying uncertainties on non-linear functionals of GRFs conditional on data. Conditional simulations are known to often be computer intensive, especially when appealing to matrix decomposition approaches with a large number of simulation points. This work studies settings where conditioning observations are assimilated batch sequentially, with one point or a batch of points at each stage. Assuming that conditional simulations have been performed at a previous stage, the goal is to take advantage of already available sample paths and by-products to produce updated conditional simulations at minimal cost. Explicit formulae are provided, which allow updating an ensemble of sample paths conditioned on \(n\ge 0\) observations to an ensemble conditioned on \(n+q\) observations, for arbitrary \(q\ge 1\). Compared to direct approaches, the proposed formulae prove to substantially reduce computational complexity. Moreover, these formulae explicitly exhibit how the \(q\) new observations are updating the old sample paths. Detailed complexity calculations highlighting the benefits of this approach with respect to state-of-the-art algorithms are provided and are complemented by numerical experiments.

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## Acknowledgments

Part of this work has been conducted within the frame of the ReDice Consortium, gathering industrial (CEA, EDF, IFPEN, IRSN, Renault) and academic (Ecole des Mines de Saint-Etienne, INRIA, and the University of Bern) partners around advanced methods for Computer Experiments. Clément Chevalier warmly thanks Prof. Julien Bect for fruitful discussions on GRF simulation. David Ginsbourger acknowledges support from the Department of Mathematics and Statistics of the University of Bern and from the Integrated methods for stochastic ensemble aquifer modelling (ENSEMBLE) project funded by the Swiss National Science Foundation under the contract CRSI22_12249/1. The authors are indebted to two anonymous referees, an associate editor, and Yves Deville for a number of remarks having contributed to substantially improve the paper.

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## Appendices

### Appendix A: Universal and Simple Kriging

Let \(Z\) be a \(L^2\) random field defined on a bounded set \(\mathbb {X}\subset \mathbb {R}^d\) with known (and not necessarily stationary) covariance function \(k(\cdot ,\cdot )\) and unknown mean function \(m(\cdot )\) such that \(Z|m \sim \text {GRF}(m,k)\), where \(\text {GRF}(m,k)\) denotes a GRF with mean function \(m\) and covariance function \(k\). A well-known Bayesian approach consists in writing \(m\) as follows

where \(\ell \ge 1\), \(f_1,\ldots ,f_\ell \) are \(\ell \) known basis functions and \(\varvec{\beta }= (\beta _1,\ldots ,\beta _\ell )\) has an improper uniform prior in \(\mathbb {R}^\ell \). In these settings, known as the universal kriging settings, when \(n\) observations \(Z(\mathbf {x}_{1:n})\) are assimilated at points \(\mathbf {x}_{1:n} := (\mathbf {x}_1,\ldots ,\mathbf {x}_n)\in \mathbb {X}^n\), it is known (O’Hagan 1978) that the posterior distribution of \(Z\) is a GRF with posterior (or conditional) mean function \(m^{UK}_n\) and covariance function \(k^{UK}_n\) given by the so-called universal kriging equations

where \(\widehat{\varvec{\beta }} := (\mathbb {F}^\top K^{-1} \mathbb {F})^{-1}\mathbb {F}^\top K^{-1}Z(\mathbf {x}_{1:n})\), \(\mathbf {f}(\mathbf {x}):= (f_1(\mathbf {x}),\ldots ,f_\ell (\mathbf {x}))^\top \), \(\mathbb {F}\in \mathbb {R}^{n\times \ell }\) is the matrix with row \(i\) equal to \(\mathbf {f}(\mathbf {x}_i)^\top \), \(\mathbf {k}(\mathbf {x}):= (k(\mathbf {x},\mathbf {x}_1),\ldots ,k(\mathbf {x},\mathbf {x}_n)) ^\top \), \(K\) is the covariance matrix at the observation points, \(K := (k(\mathbf {x}_i,\mathbf {x}_j))_{1\le i,j\le n}\). The vector \(\varvec{\lambda }^{UK}(\mathbf {x})\) is the vector of \(n\) kriging weights of \(\mathbf {x}_1,\ldots , \mathbf {x}_n\) for the prediction at point \(\mathbf {x}\).

A well-known simpler setting is the case where the non-conditional mean function \(m\) is already known. In that case, the Bayesian approach is no longer necessary and the conditional mean and covariance function of \(Z\) are given by the so-called simple kriging equations, written here in the case where \(m(\cdot )=0\)

If \(m\) is not equal to zero, the simple kriging covariance function \(k_n^\mathrm{SK}\) is unchanged and an application of Eq. (14) to the centred GRF \(Z-m\) yields \(m^\mathrm{SK}_n(\mathbf {x}) = m(\mathbf {x}) + \mathbf {k}(\mathbf {x})^\top K^{-1} ( Z(\mathbf {x}_{1:n}) - m(\mathbf {x}_{1:n}) )\).

### Appendix B: Algorithms

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Chevalier, C., Emery, X. & Ginsbourger, D. Fast Update of Conditional Simulation Ensembles.
*Math Geosci* **47**, 771–789 (2015). https://doi.org/10.1007/s11004-014-9573-7

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DOI: https://doi.org/10.1007/s11004-014-9573-7

### Keywords

- Gaussian random fields
- Residual kriging algorithm
- Batch-sequential strategies
- Kriging update equations