Abstract
Random fields based on energy functionals with local interactions possess flexible covariance functions, lead to computationally efficient algorithms for spatial data processing, and have important applications in Bayesian field theory. In this paper we address the calculation of covariance functions for a family of isotropic localinteraction random fields in two dimensions. We derive explicit expressions for nondifferentiable Spartan covariance functions in \({\mathbb{R}}^2\) that are based on the modified Bessel function of the second kind. We also derive a family of infinitely differentiable, BesselLommel covariance functions that exhibit a hole effect and are valid in \({\mathbb{R}}^{d}\), where d > 2. Finally, we define a generalized spectrum of correlation scales that can be applied to both differentiable and nondifferentiable random fields in contrast with the smoothness microscale.
Similar content being viewed by others
Notes
This notation is ambiguous in \(d=1\) but well defined for \(d \ge 2\).
Henceforward SSRF for simplicity.
There are two additional solutions of opposite sign than \(\kappa _{\pm }\) which are not further considered, since they are either complex or negative real numbers.
References
Abrahamsen P (1997) A Review of Gaussian random fields and correlation functions. Technical Report TR 917. Norwegian Computing Center. Box 114, Blindern, N0314, Oslo, Norway.
Abramowitz M, Stegun IA (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards, Washington
Adler RJ (1981) The geometry of random fields. Wiley, New York
Amit DJ (1984) Field theory, the renormalization group, and critical phenomena, 2nd edn. World Scientific, New York
Bakr AA, Gelhar LW, Gutjahr AL, MacMillan JR (1978) Stochastic analysis of spatial variability in subsurface flows: 1. Comparison of one and threedimensional flows. Water Resour Res 14:263–271
Barker DM, Huang W, Guo YR, Bourgeois AJ, Xiao QN (2004) A threedimensional variational data assimilation system for mm5: implementation and initial results. Mon Weather Rev 132:897–914
Bochner S (1959) Lectures on Fourier integrals. Princeton University Press, Princeton
Caspari E, Gurevich B, Müller TM (2013) Frequencydependent effective hydraulic conductivity of strongly heterogeneous media. Phys Rev E 88:042119
Christakos G (1992) Random field models in earth sciences. Academic Press, San Diego
Cressie N (1993) Spatial statistics. Wiley, New York
Eberhard J (2005) Upscaling for stationary transport in heterogeneous porous media. Multiscale Model Simul 3:957–976
Elogne SN, Hristopulos D, Varouchakis E (2008) An application of Spartan spatial random fields in environmental mapping: focus on automatic mapping capabilities. Stoch Environ Res Risk Assess 22:633–646
Farmer CL (2005) Geological modelling and reservoir simulation. In: Iske A, Randen T (eds) Mathematical methods and modeling in hydrocarbon exploration and production. SpringerVerlag, Heidelberg, pp 119–212
Fiori A, Dagan G, Jankovic I (2011) Upscaling of steady flow in threedimensional highly heterogeneous formations. Multiscale Model Simul 9:1162–1180
Fiori A, Jankovič I, Dagan G (2003) Flow and transport in highly heterogeneous formations: 2. semianalytical results for isotropic media. Water Resour Res 39: 1269.
Gelhar LW, Axness CL (1983) Threedimensional stochastic analysis of macrodispersion in aquifers. Water Resour Res 19:161–180
Genton MG (2002) Classes of kernels for machine learning: a statistics perspective. J Mach Learn Res 2:299–312
Ghanem R, Spanos PD (2003) Stochastic finite elements: a spectral approach. Dover, Mineola
Goldenfeld N (1992) Lectures on phase transitions and the renormalization group. AddisonWesley, Reading
Gradshteyn IS, Ryzhik IM (2007) Table of integrals, series, and products, 7th edn. Academic Press, Boston
Hristopulos D (2003a) Spartan Gibbs random field models for geostatistical applications. SIAM J Sci Comput 24:2125–2162
Hristopulos D (2005a) Erratum: spartan Gibbs random field models for geostatistical applications. SIAM J Sci Comput 26:2176
Hristopulos DT (2003b) Renormalization group methods in subsurface hydrology: overview and applications in hydraulic conductivity upscaling. Adv Water Resour 26:1279–1308
Hristopulos DT (2005b) Spartan Gaussian random fields for geostatistical applications: nonconstrained simulations on square lattices and irregular grids. J Comput Methods Sci Eng 5:149–164
Hristopulos DT, Christakos G (1999) Renormalization group analysis of permeability upscaling. Stoch Environ Res Risk Assess 13:131–161
Hristopulos DT, Elogne S (2007) Analytic properties and covariance functions of a new class of generalized Gibbs random fields. IEEE Trans Inf Theory 53:4667–4679
Hristopulos DT, Elogne SN (2009) Computationally efficient spatial interpolators based on Spartan spatial random fields. IEEE Trans Signal Proc 57:3475–3487
Hristopulos DT, Žukovič M (2011) Relationships between correlation lengths and integral scales for covariance models with more than two parameters. Stoch Env Res Risk Assess 25:11–19
Indelman P, Fiori A, Dagan G (1996) Steady flow toward wells in heterogeneous formations: mean head and equivalent conductivity. Water Resour Res 32:1975–1983
Jankovic I, Fiori A, Dagan G (2003) Effective conductivity of an isotropic heterogeneous medium of lognormal conductivity distribution. Multiscale Model Simul 1:40–56
Jiao Y, Stillinger FH, Torquato S (2007) Modeling heterogeneous materials via twopoint correlation functions: basic principles. Phys Rev E 76:031110
Kramer PR, Kurbanmuradov O, Sabelfeld K (2007) Comparative analysis of multiscale Gaussian random field simulation algorithms. J Comput Phys 226:897–924
Lantuéjoul C (2002) Geostatistical simulation: models and algorithms. Springer, Berlin
Lemm JC (2003) Bayesian field theory. JHU Press, Baltimore
Rasmussen CE, Williams CKI (2006) Gaussian processes for machine learning. MIT Press, Boston
Robin MJL, Gutjahr AL, Sudicky EA, Wilson JL (1993) Crosscorrelated random field generation with the direct fourier transform method. Water Resour Res 29:2385–2397
Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives: theory and applications. Gordon and Breach, Amsterdam
Sarma P, Durlofsky L, Aziz K (2008) Kernel principal component analysis for efficient, differentiable parameterization of multipoint geostatistics. Math Geosci 40:3–32
Schoenberg IJ (1938) Metric spaces and completely monotone functions. Ann Math 39:811–841
Schwartz LM (2008) Mathematics for the physical sciences. Dover, New York
Stein ML (1999) Interpolation of spatial data: some theory for kriging. Springer, New York
Torquato S (2002) Random heterogeneous materials, 1st edn. Springer, New York
VargasGuzmán J, Warrick A, Myers D (2002) Coregionalization by linear combination of nonorthogonal components. Math Geol 34:405–419
Watson GN (1995) A treatise on the theory of bessel functions, 2nd edn. Cambridge University Press, New York
Weaver AT, Mirouze I (2013) On the diffusion equation and its application to isotropic and anisotropic correlation modelling in variational assimilation. Q J R Meteorol Soc 139:242–260
Wendland H (2005) Scattered data approximation, Cambridge monographs on applied and computational mathematics. Cambridge University Press, Cambridge
Yaglom AM (1987) Correlation theory of stationary and related random functions I. Springer Verlag, New York
Yaremchuk M, Sentchev A (2012) Multiscale correlation functions associated with polynomials of the diffusion operator. Q J R Meteorol Soc 138:1948–1953
Žukovič M, Hristopulos DT (2008) Environmental time series interpolation based on Spartan random processes. Atmos Environ 42:7669–7678
Acknowledgments
The author acknowledges funding from the project SPARTA 1591: “Development of SpaceTime Random Fields based on Local Interaction Models and Applications in the Processing of Spatiotemporal Datasets”, which is implemented under the “ARISTEIA” Action of the operational programme “Education and Lifelong Learning” and is cofunded by the European Social Fund and National Resources.
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Below is the link to the electronic supplementary material.
Appendices
Appendix A: Proof of Proposition 1
Proof
Defining dimensionless wavevectors \(u= k\,\xi \) and lag distances \(h=r/\xi \), the spectral integral (9) is simplified as follows:
Equation (27) shows that the only nontrivial parameter is \({\eta _{1}}\); \(\eta _0\) is a multiplicative scale factor, whereas the characteristic length \(\xi \) is absorbed in the nondimensional lag \(h.\) The rational function \(1/\varPi (u)\), where \(\varPi (u)\) is the SSRF characteristic polynomial defined in (7b), admits the following expansion
where \(t_{\pm }^{*}={\left( {\eta _{1}}\pm \Delta \right) }/{2}\) are the roots of \(\varPi (t=u^2)\).
In light of (28), the integral (27) is evaluated using the Hankel–Nicholson formula (11.4.44) in (Abramowitz and Stegun (1972), p. 364):
This equation is valid for \( h>0, {\mathfrak{R}}(z)>0\), and \( 1<{\mathfrak{R}}(\nu )< 2{\mathfrak{R}}(\mu )+\frac{3}{2}\). The above is applied to (27) with (i) \({\eta _{1}}\ne 2\), \(\nu =0\), \(\mu =0\), \(z^{2}_{\pm }=  t_{\pm }^{*}\) and (ii) \({\eta _{1}}= 2\), \(\nu =0\) and \(\mu =1\). In case (ii) we obtain (10b) and in case (i) the following
The coefficients \(z_{\pm }=\sqrt{t_{\pm }^{*}}\) are plotted versus \({\eta _{1}}\)
in Fig. 10. For \({\eta _{1}}>2\) both \(z_{+}\) and \(z_{}\) are real numbers, hence proving (10a). For \(2< {\eta _{1}}<2\) \({\mathfrak{R}}(z_{+})={\mathfrak{R}}(z_{})\), whereas \({\mathfrak{I}}(z_{+})={\mathfrak{I}}(z_{})\), i.e., \(z_{} =\overline{z_{+}}\). The analytic continuation property \(K_{0}(\overline{z})=\overline{K_{0}(z)}\) ((Abramowitz and Stegun 1972, p. 377)) leads to (10c) which is explicitly realvalued.
1.1 Continuity
A stationary SRF is mean square continuous \(\forall {\mathbf{s}}\in {\mathbb{R}}^d\) if and only if its covariance function is continuous at zero lag (Adler 1981; Abrahamsen 1997). This condition is satisfied for the SSRF covariance.
1.2 Differentiability
Differentiability of the SRF \(X({\mathbf{s}},\omega )\) in the meansquare sense requires that all secondorder partial derivatives of the covariance function at \(\Vert {\mathbf{r}}\Vert =0\) exist ((Adler 1981, p. 27)). This requirement is equivalent to the convergence of the secondorder spectral moment
For the SSRF spectral density in \(d=2\) the above becomes
This integral develops a logarithmic divergence as \({k_{c}}\rightarrow \infty \). Hence, the SSRF is meansquare nondifferentiable. \(\square \)
Appendix B: Proof of Proposition 2
Proof
Let \(J_{\nu }(\cdot )\) be the Bessel function of the first kind of order \(\nu \), and define
where \(z=u_{c}\, h\), \(\nu =d/21\), and \(\mu > (\nu +1)\). Then, \({\mathcal{A}}_{\mu ,\nu }(z)\) is evaluated using (Gradshteyn and Ryzhik (2007), eq. (6.561.13), p. 676) as follows
Further, we use the normalizing variable transformations \(x=k/{k_{c}}\), \(h=r/\xi ,\) and \(u_{c}={k_{c}}\xi \). In view of the dimensionless variables \(x, h, u_c\), the integral (12) becomes
In light of (30), (32) and using \(z =u_{c} h = {k_{c}}r\) as the dimensionless distance, the function \(C_{\mathrm{xx}}^{BL}({\mathbf{r}};{\varvec{\theta }}) \) defined by (12) is given by
For the three terms \({\mathcal{A}}_{\mu ,\nu }(z)\) \((\mu =\nu +1, \nu +3, \nu +5)\) included in \(C_{\mathrm{xx}}^{BL}(z;{\varvec{\theta }})\), the parameters \(\mu , \nu \) satisfy the relation
Equations (16) follow directly from (33) which expresses \(C_{\mathrm{xx}}^{BL}(z;{\varvec{\theta }})\) in terms of \({\mathcal{A}}_{\mu ,\nu }(z)\), and from (31) which expresses the integrals \({\mathcal{A}}_{\mu ,\nu }(z)\) in terms of Lommel functions. In view of (35), the Gamma function contributions to \({\mathcal{A}}_{\nu +2l+1,\nu }(z)\) in (31) vanish due to the poles of \(\Gamma (n)\) at \(n \in \mathbb {Z}_{0,}\).
1.1 Permissibility
The nonnegative definiteness of \(C_{\mathrm{xx}}^{BL}(z;{\varvec{\theta }})\) is based on Bochner’s theorem and the fact that, according to (13), \(\widetilde{C_{\mathrm{xx}}^{BL}}(k;{\varvec{\theta }}) \ge 0\) for \({\eta _{1}}>2\).
1.2 Differentiability
The existence of the \(n\)th order partial derivatives of the BesselLommel SRF in the meansquare sense requires that all the partial derivatives of order \(2n\) of \(C_{\mathrm{xx}}^{BL}(z;{\varvec{\theta }})\) exist at \(z=0\). This condition is ensured by the convergence of the \(2n\)th order spectral moment, i.e., of the integral
for \({k_{c}}\in {\mathbb{R}}\). \(\square \)
Appendix C: Proof of Proposition 3
Proof
To find the supremum of \(f(k):=k^{2\alpha } \, \widetilde{C_{\mathrm{xx}}}(k;{\varvec{\theta }})\) we consider the extremum condition \(\mathrm{d} f(k)/\mathrm{d} k=0\), which admits the following two roots:
For \( 0 \le \alpha < 1\) only \(\tilde{\kappa }_{1} \in {\mathbb{R}}\) and \(\sup f(k) = f(\tilde{\kappa }_{1})\).
According to (7) the denominator of (23) becomes
To simplify notation we define
In order to calculate the integral (36) we use Lebesgue's dominated convergence theorem (Schwartz 2008) expressed as follows:
Theorem 2
Let \(\phi _{\alpha }(x)\) be a realvalued function \(\forall x \in {\mathbb{R}}\) which is integrable \(\forall \alpha \in [0, 1]\). If there is a realvalued function \(g_{n}(x)\) such that (i) \(\lim _{n \rightarrow \infty } \phi _{\alpha }(x) \, g_{n}(x) = \phi _{\alpha }(x), \, \forall x \in {\mathbb{R}}\) and (ii) \( \phi _{\alpha }(x) \, g_{n}(x) \le \phi ^{*}(x), \forall x \in {\mathbb{R}}\), where \(\phi ^{*}(x)\) is an integrable function, then
We define the following auxiliary function
Condition (i) of Theorem 2 is satisfied because \(\lim _{n \rightarrow \infty } g_{n}(x)=1\) based on the infinite series expansion of the Bessel function of the first kind around zero ((Watson 1995, p. 40)).
To prove the condition (ii) we apply the following steps.

1.
For condition (ii) it suffices to show that \( \phi _{\alpha }(x) \, g_{n}(x) \le \phi _{\alpha }(x) \), because \(\phi _{\alpha }(x)\) is integrable. Given that \(\phi _{\alpha }(x) >0\), the latter is equivalent to \( g_{n}(x)  \le 1\).

2.
We use the integral representation of \(J_{\alpha }(z)\) given by (Gradshteyn and Ryzhik (2007), 8.411.4), where \(z \in {\mathbb{R}}\):
$$ J_{\alpha }(z) = \frac{2\left( \frac{z}{2} \right) ^{\alpha }}{\Gamma (\alpha +1/2)\Gamma (1/2)} \, \int _{0}^{\pi /2} d\theta \sin ^{2\alpha }\theta \cos \left( z \cos \theta \right) $$ 
3.
Since \(\sin ^{2\alpha }(\theta ) \cos \left( z \cos (\theta )\right) \le 1\) and \(\Gamma (1/2) = \sqrt{\pi }\) it follows from the above that \(J_{\alpha }(z) \le \frac{\left( \frac{z}{2} \right) ^{\alpha } \sqrt{\pi }}{\Gamma (\alpha +1/2)}.\)

4.
In light of this inequality and (39), proving that \(g_{n}(x) \le 1\) is equivalent to showing that \( \mu _{\alpha }:= \Gamma (\alpha +1) / \Gamma (\alpha +1/2) \le \sqrt{\pi }.\)

5.
Based on the inequality \( \mu _{\alpha } < \sqrt{\alpha +1/2}\) (valid for \(\alpha > 1/4\)) the maximum upper bound of \( \mu _{\alpha }\) for \(0 \le \alpha \le 1\) is \(\sqrt{3/2} < \sqrt{\pi }\). Hence, in light of the previous step \(g_{n}(x) \le 1\). This concludes the proof of condition (ii).
In light of the above, we can use dominated convergence to calculate \(I_{\alpha }(\phi )\) as follows
where
The integral \(\tilde{I}_{\alpha }(\phi )\) is evaluated by means of the HankelNicholson formula (29) (\(\nu =\alpha ,\) \(\mu =0\) for \({\eta _{1}}\ne 2, \, \mu =1\) for \({\eta _{1}}= 2\)) which leads to
To evaluate \(\lim _{n \rightarrow \infty }\tilde{I}_{\alpha }(\phi )\) for \(1> p >0\) we use the series expansion (Schwartz 2008) of the KBessel function
For \({\eta _{1}}=2\), \(p = 1  \alpha \), and \(x=1/n\) the dominant contribution at \(n \rightarrow \infty \) comes from the \(O(x^{p})\) term of the first series on the right hand side, which gives \( K_{1\alpha }(1/n) \approx \frac{1}{2} \Gamma (1  \alpha ) (2n)^{1  \alpha }\).
For \({\eta _{1}}\ne 2\) the \(O(x^{p})\) term of the first series cancels out due to the difference between the two Bessel functions, whereas the \(O(x^{2p})\) terms vanish at the limit \(n \rightarrow \infty \). A finite contribution comes from the \(O(x^p)\) term of the second series on the right hand side, i.e., \(z_{+}^{\alpha } \, K_{\alpha }(z_{+}/n)  z_{}^{\alpha } \,K_{\alpha }(z_{}/n) \sim \frac{\Gamma (\alpha )}{2(2n)^{\alpha }}\left( z_{+}^{2\alpha }  z_{}^{2\alpha } \right) \).
Thus, based on the above asymptotic analysis of the KBessel function, (40), (41), and (42) we obtain the following equation (where \(\Delta = \sqrt{{\eta _{1}}^24}\)):
Finally, based on (43), the definition (38) and (36), (24a) is proved. \(\square \)
Appendix D: Proof of Proposition 4
Proof
Based on the spectral density (13) it follows that the denominator in (23) is given by
Let us define the function \(\phi (k) = k^{2\alpha } \,\widetilde{C_{\mathrm{xx}}}^{BL}(k;{\varvec{\theta }})\). The numerator in (23) is then given by \(\sup _{{\mathbf{k}}\in {\mathbb{R}}^d} \, \phi (k) = \phi (\kappa ^{*})\) where \(\kappa ^{*} = \underset{{k} \in [0, {k_{c}}]}{\arg \max } \phi (k)\).
1.1 Nonnegative \({\eta _{1}}\)
For \({\eta _{1}}\ge 0\), \(\phi (k)\) is a monotonically increasing function of \(k\); thus, \(\kappa ^{*}= {k_{c}}\) and \( \phi (\kappa ^{*}) = \frac{{k_{c}}^{2\alpha }}{\eta _{0} \xi ^d}\, \left( 1 + {\eta _{1}}\, {k_{c}}^2 \, \xi ^2 + {k_{c}}^4 \xi ^4 \right) \). In light of (44), this leads to (25a).
1.2 Negative \({\eta _{1}}\)
For \({\eta _{1}}<0\), \(\phi (k)\) develops local extrema at the wavenumbers that solve the equation \(d\phi (k)/dk=0\), i.e., at the \(\kappa _{\pm }\) given by (26c)^{Footnote 3}.

1.
Complex \(\kappa _{\pm }\)
For \({\eta _{1}}^2 < 4 \alpha (\alpha +2) / (\alpha +1)^2\), the \(\kappa _{\pm }\) are complex numbers and thus \(\phi (k)\) does not develop local extrema for \(k\in {\mathbb{R}}\). Hence, \(\kappa ^{*} = {k_{c}}\) and \(\lambda _{c}^{(\alpha )}\) is given by (25a).

2.
Real \(\kappa _{\pm }\)
For \(4 > {\eta _{1}}^2 \ge 4 \alpha (\alpha +2) / (\alpha +1)^2\), the \(\kappa _{\pm }\) are real numbers, one corresponding to the position of a local minimum and the other to a local maximum.

1.
If \(\alpha >0\), \(d\phi (k)/dk\propto 2\alpha \, (k\xi )^{2\alpha 1}\) for \(k\ll 1\), and thus \(\phi (k) \) increases monotonically. The maximum of \(\phi (k)\) thus occurs at \(\kappa _{} < \kappa _{+}\).

2.
If \(\alpha =0\), then \(d\phi (k)/dk\propto 2{\eta _{1}}\, (k\xi )\) and thus \(\phi (k) \) decreases monotonically for \(k\ll 1\). In this case also \(\phi (k)\) becomes maximum at \(\kappa _{}=0,\) whereas the minimum occurs at \(\kappa _{+} = \sqrt{{\eta _{1}}/2} / ({k_{c}}\xi )\). Again, we distinguish between two cases depending on the relation between \(\kappa _{}\) and \({k_{c}}.\)

(a)
If \(\kappa _{} > {k_{c}}\), then \(\kappa ^{*} = {k_{c}}\) and \(\lambda _{c}^{(\alpha )}\) is given by (25a). For \(\alpha =0\) it holds that \(\kappa _{}=0\) and thus \(\kappa _{} < {k_{c}}\).

(b)
If \({k_{c}}> \kappa _{}\), we further distinguish the following cases

(a)
\(\square \)
Rights and permissions
About this article
Cite this article
Hristopulos, D.T. Covariance functions motivated by spatial random field models with local interactions. Stoch Environ Res Risk Assess 29, 739–754 (2015). https://doi.org/10.1007/s0047701409330
Published:
Issue Date:
DOI: https://doi.org/10.1007/s0047701409330