## Abstract

The regularisation on a segment of a stationary or intrinsic variogram model is usually expressed as an integral. By introducing an auxiliary function, it is shown that this integral is a linear combination of elementary terms. This paper shows how these terms are arranged and provides a simple and safe procedure to calculate them. An important application of this procedure is the deregularisation problem, that is, the conversion of a regularised variogram to a point variogram. This ill-posed problem is tackled using an iterative optimisation algorithm designed to produce a family of point variograms that best fit the initial regularised variogram. Subsequently, each of these point variograms can be regularised on any segment, which results in a fully coherent family of simple and cross regularised variograms.

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## Notes

If a nugget effect resulting from a microstructure is observed at support \(\lambda \), then it can be written as \(\nu / \lambda \), where \(\nu \) is the so-called nugget constant. Then the nugget effect at any other support \(\mu \) is \(\nu / \mu \).

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## Acknowledgements

The authors thank Pr. Ute Mueller and an anonymous reviewer for their careful reading of the first version of this paper and their constructive comments that contributed to removing a number of mistakes and inconsistencies.

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## Appendices

### Appendix 1: Proofs in the Stationary Case

Let us recall that *C* is a function of positive type. If it is continuous at the origin (no nugget effect), then a theorem by Bochner (1955) states that *C* is the Fourier transform of a positive measure *dF* (called the spectral measure of *C*)

There exists a relation between the auxiliary function and the spectral measure. To obtain it, we start with definition (1) and replace *C* by its Bochner’s expression. After changing the order of integration, we obtain

The inner integral can be calculated as

and since the spectral measure is even, the *ius* term cancels out, and we get

which shows that \(\vartheta \) is a variogram with spectral measure *F* (Chilès and Delfiner 1999). More explicitly, \(\vartheta \) is nothing but the variogram of the intrinsic process

We now turn to the calculation of \(Cov \{ Z_\lambda (x) , Z_\mu (y) \} \). Using the spectral representation of *C*, we have

Of course, there is no inconvenience in inserting one value \(+1\) or \(-1\) next to each exponential term. It then suffices to apply formula (16) to complete the proof of formula (2).

### Appendix 2: Proofs in the Intrinsic Case

Exactly as in the stationary case, the auxiliary function

is non-negative and even, increasing and convex.

In what follows, the variogram is supposed without a drift. Then it admits a spectral representation of the form

where *F* is a positive measure without an atom at the origin and satisfying

Replacing \(\gamma \) by its spectral representation, and permuting integrals, we obtain

Moreover, we have

from which we derive

which is the spectral representation of a generalised covariance of order 1.

Regarding the proof of formula (11), we have

Moreover, we have

which completes the proof.

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Lantuéjoul, C., Bush, D.E., Stiefenhofer, J. *et al.* Regularisation and Deregularisation of Covariances and Variograms for Univariate Random Functions in One-dimensional Space.
*Math Geosci* **55**, 609–624 (2023). https://doi.org/10.1007/s11004-022-10043-9

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DOI: https://doi.org/10.1007/s11004-022-10043-9