Skip to main content
Log in

Covariance functions motivated by spatial random field models with local interactions

  • Original Paper
  • Published:
Stochastic Environmental Research and Risk Assessment Aims and scope Submit manuscript


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 local-interaction random fields in two dimensions. We derive explicit expressions for non-differentiable 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, Bessel-Lommel 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 non-differentiable random fields in contrast with the smoothness microscale.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others


  1. This notation is ambiguous in \(d=1\) but well defined for \(d \ge 2\).

  2. Henceforward SSRF for simplicity.

  3. 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.


  • Abrahamsen P (1997) A Review of Gaussian random fields and correlation functions. Technical Report TR 917. Norwegian Computing Center. Box 114, Blindern, N-0314, Oslo, Norway.

  • Abramowitz M, Stegun IA (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards, Washington

    Google Scholar 

  • Adler RJ (1981) The geometry of random fields. Wiley, New York

    Google Scholar 

  • Amit DJ (1984) Field theory, the renormalization group, and critical phenomena, 2nd edn. World Scientific, New York

    Google Scholar 

  • Bakr AA, Gelhar LW, Gutjahr AL, MacMillan JR (1978) Stochastic analysis of spatial variability in subsurface flows: 1. Comparison of one- and three-dimensional flows. Water Resour Res 14:263–271

    Article  Google Scholar 

  • Barker DM, Huang W, Guo YR, Bourgeois AJ, Xiao QN (2004) A three-dimensional variational data assimilation system for mm5: implementation and initial results. Mon Weather Rev 132:897–914

    Article  Google Scholar 

  • Bochner S (1959) Lectures on Fourier integrals. Princeton University Press, Princeton

    Google Scholar 

  • Caspari E, Gurevich B, Müller TM (2013) Frequency-dependent effective hydraulic conductivity of strongly heterogeneous media. Phys Rev E 88:042119

    Article  CAS  Google Scholar 

  • Christakos G (1992) Random field models in earth sciences. Academic Press, San Diego

    Google Scholar 

  • Cressie N (1993) Spatial statistics. Wiley, New York

    Google Scholar 

  • Eberhard J (2005) Upscaling for stationary transport in heterogeneous porous media. Multiscale Model Simul 3:957–976

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Farmer CL (2005) Geological modelling and reservoir simulation. In: Iske A, Randen T (eds) Mathematical methods and modeling in hydrocarbon exploration and production. Springer-Verlag, Heidelberg, pp 119–212

    Chapter  Google Scholar 

  • Fiori A, Dagan G, Jankovic I (2011) Upscaling of steady flow in three-dimensional highly heterogeneous formations. Multiscale Model Simul 9:1162–1180

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • Gelhar LW, Axness CL (1983) Three-dimensional stochastic analysis of macrodispersion in aquifers. Water Resour Res 19:161–180

    Article  Google Scholar 

  • Genton MG (2002) Classes of kernels for machine learning: a statistics perspective. J Mach Learn Res 2:299–312

    Google Scholar 

  • Ghanem R, Spanos PD (2003) Stochastic finite elements: a spectral approach. Dover, Mineola

    Google Scholar 

  • Goldenfeld N (1992) Lectures on phase transitions and the renormalization group. Addison-Wesley, Reading

    Google Scholar 

  • Gradshteyn IS, Ryzhik IM (2007) Table of integrals, series, and products, 7th edn. Academic Press, Boston

    Google Scholar 

  • Hristopulos D (2003a) Spartan Gibbs random field models for geostatistical applications. SIAM J Sci Comput 24:2125–2162

    Article  Google Scholar 

  • Hristopulos D (2005a) Erratum: spartan Gibbs random field models for geostatistical applications. SIAM J Sci Comput 26:2176

    Article  Google Scholar 

  • Hristopulos DT (2003b) Renormalization group methods in subsurface hydrology: overview and applications in hydraulic conductivity upscaling. Adv Water Resour 26:1279–1308

    Article  Google Scholar 

  • Hristopulos DT (2005b) Spartan Gaussian random fields for geostatistical applications: non-constrained simulations on square lattices and irregular grids. J Comput Methods Sci Eng 5:149–164

    Google Scholar 

  • Hristopulos DT, Christakos G (1999) Renormalization group analysis of permeability upscaling. Stoch Environ Res Risk Assess 13:131–161

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Hristopulos DT, Elogne SN (2009) Computationally efficient spatial interpolators based on Spartan spatial random fields. IEEE Trans Signal Proc 57:3475–3487

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Jankovic I, Fiori A, Dagan G (2003) Effective conductivity of an isotropic heterogeneous medium of lognormal conductivity distribution. Multiscale Model Simul 1:40–56

    Article  Google Scholar 

  • Jiao Y, Stillinger FH, Torquato S (2007) Modeling heterogeneous materials via two-point correlation functions: basic principles. Phys Rev E 76:031110

    Article  CAS  Google Scholar 

  • Kramer PR, Kurbanmuradov O, Sabelfeld K (2007) Comparative analysis of multiscale Gaussian random field simulation algorithms. J Comput Phys 226:897–924

    Article  Google Scholar 

  • Lantuéjoul C (2002) Geostatistical simulation: models and algorithms. Springer, Berlin

    Book  Google Scholar 

  • Lemm JC (2003) Bayesian field theory. JHU Press, Baltimore

    Google Scholar 

  • Rasmussen CE, Williams CKI (2006) Gaussian processes for machine learning. MIT Press, Boston

    Google Scholar 

  • Robin MJL, Gutjahr AL, Sudicky EA, Wilson JL (1993) Cross-correlated random field generation with the direct fourier transform method. Water Resour Res 29:2385–2397

    Article  Google Scholar 

  • Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives: theory and applications. Gordon and Breach, Amsterdam

    Google Scholar 

  • Sarma P, Durlofsky L, Aziz K (2008) Kernel principal component analysis for efficient, differentiable parameterization of multipoint geostatistics. Math Geosci 40:3–32

    Article  Google Scholar 

  • Schoenberg IJ (1938) Metric spaces and completely monotone functions. Ann Math 39:811–841

    Article  Google Scholar 

  • Schwartz LM (2008) Mathematics for the physical sciences. Dover, New York

    Google Scholar 

  • Stein ML (1999) Interpolation of spatial data: some theory for kriging. Springer, New York

    Book  Google Scholar 

  • Torquato S (2002) Random heterogeneous materials, 1st edn. Springer, New York

    Book  Google Scholar 

  • Vargas-Guzmán J, Warrick A, Myers D (2002) Coregionalization by linear combination of nonorthogonal components. Math Geol 34:405–419

    Article  Google Scholar 

  • Watson GN (1995) A treatise on the theory of bessel functions, 2nd edn. Cambridge University Press, New York

    Google Scholar 

  • 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

    Article  Google Scholar 

  • Wendland H (2005) Scattered data approximation, Cambridge monographs on applied and computational mathematics. Cambridge University Press, Cambridge

    Google Scholar 

  • Yaglom AM (1987) Correlation theory of stationary and related random functions I. Springer Verlag, New York

    Google Scholar 

  • Yaremchuk M, Sentchev A (2012) Multi-scale correlation functions associated with polynomials of the diffusion operator. Q J R Meteorol Soc 138:1948–1953

    Article  Google Scholar 

  • Žukovič M, Hristopulos DT (2008) Environmental time series interpolation based on Spartan random processes. Atmos Environ 42:7669–7678

    Article  Google Scholar 

Download references


The author acknowledges funding from the project SPARTA 1591: “Development of Space-Time 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 co-funded by the European Social Fund and National Resources.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Dionissios T. Hristopulos.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 67 KB)


Appendix A: Proof of Proposition 1


Defining dimensionless wavevectors \(u= k\,\xi \) and lag distances \(h=r/\xi \), the spectral integral (9) is simplified as follows:

$$ C_{\mathrm{xx}}(h;{\varvec{\theta }}) = \frac{\eta _0 }{2\pi } \, \int \limits _{0}^{\infty } du \,\frac{ u \, J_{0}(u\,h) }{1+{\eta _{1}}u^2+ u^4}. $$

Equation (27) shows that the only non-trivial parameter is \({\eta _{1}}\); \(\eta _0\) is a multiplicative scale factor, whereas the characteristic length \(\xi \) is absorbed in the non-dimensional lag \(h.\) The rational function \(1/\varPi (u)\), where \(\varPi (u)\) is the SSRF characteristic polynomial defined in (7b), admits the following expansion

$$ \frac{1}{\varPi (u)}= \left\{ \begin{array}{cc} \frac{1}{t_{+}^{*}-t_{-}^{*}} \, \left( \frac{1}{u^2-t_{+}^{*}} - \frac{1}{u^2 -t_{-}^{*}} \right) , &{} {\eta _{1}}\ne 2, \\ \frac{1}{(u^2+1)^2} &{} {\eta _{1}}= 2, \end{array} \right. $$

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):

$$\int _{0}^{\infty } du \frac{u^{\nu +1}J_{\nu }(h\,u)}{(u^2 + z^2)^{\mu +1}} = \frac{h^{\mu }z^{\nu -\mu }}{2^{\mu }\Gamma (\mu +1)} K_{\nu -\mu }(h\,z). $$

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

$$ C_{\mathrm{xx}}(h ;{\varvec{\theta }}) =\frac{\eta _0 \left[ K_{0} (hz_{+}) - K_{0} (hz_{-}) \right] }{2\pi \sqrt{\eta _{1}^{2}-4}} , \quad {\eta _{1}}\ne 2. $$

The coefficients \(z_{\pm }=\sqrt{-t_{\pm }^{*}}\) are plotted versus \({\eta _{1}}\)

Fig. 10
figure 10

Real (a) and imaginary (b) parts of the roots \(z_{+}=\sqrt{-t_{+}^{*}}\) and \(z_{-}=\sqrt{-t_{-}^{*}}\) of the characteristic polynomial Π(u)  = 1 + η 1 u 2 + u 4 according to (11)

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 real-valued.

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 mean-square sense requires that all second-order 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 second-order spectral moment

$$ \varLambda _{d}^{(2)}:=\int _{{\mathbb{R}}^d} d{\mathbf{k}}\, k^2 \, \widetilde{C_{\mathrm{xx}}}({\mathbf{k}},{\varvec{\theta }}).$$

For the SSRF spectral density in \(d=2\) the above becomes

$$ \varLambda _{2}^{(2)} \propto \lim _{{k_{c}}\rightarrow \infty } \int _{0}^{{k_{c}}} dk\, \frac{k^3 }{1 + {\eta _{1}}(k\xi )^2 + (k\xi )^4}. $$

This integral develops a logarithmic divergence as \({k_{c}}\rightarrow \infty \). Hence, the SSRF is mean-square non-differentiable. \(\square \)

Appendix B: Proof of Proposition 2


Let \(J_{\nu }(\cdot )\) be the Bessel function of the first kind of order \(\nu \), and define

$$ {\mathcal{A}}_{\mu ,\nu }(z)=\int _{0}^{1} dx \, x^{\mu } J_{\nu }(z\,x), $$

where \(z=u_{c}\, h\), \(\nu =d/2-1\), 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

$$ {\mathcal{A}}_{\mu ,\nu }(z) =\frac{2^{\mu } \Gamma \left( \frac{\nu +\mu +1}{2}\right) }{z^{\mu +1} \, \Gamma \left( \frac{\nu -\mu +1}{2}\right) } + \frac{(\mu + \nu -1) J_{\nu }(z) S_{\mu -1,\nu -1}(z)}{z^{\mu }} - \frac{J_{\nu -1}(z) S_{\mu ,\nu }(z)}{z^{\mu }}.$$

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

$$ C_{\mathrm{xx}}^{BL}(h;{\varvec{\theta }}) =\frac{ u_c^{1+d/2}\, h^{1-d/2}}{(2\pi )^{d/2} \eta _0\, \xi ^{2d} } \int _{0}^{1} d x \, {x^{d / 2} J_{d/2-1}(x h u_c)} \cdot \left[ 1+{\eta _{1}}(x u_c)^2+ (x u_c)^4 \right]. $$

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

$$ C_{\mathrm{xx}}^{BL}(z;{\varvec{\theta }}) =\frac{g_{0}({\varvec{\theta }})}{ z^{\nu }} \left[ {\mathcal{A}}_{\nu +1,\nu }(z) + {\eta _{1}}u_{c}^2 {\mathcal{A}}_{\nu +3,\nu }(z) \right. \left. + \, u_{c}^4 {\mathcal{A}}_{\nu +5,\nu }(z)\right] ,$$
$$ g_{0}({\varvec{\theta }}) = \frac{ {k_{c}}^{d}}{(2\pi )^{d/2} \eta _0\, \xi ^{d} }. $$

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

$$ \nu - \mu +1 = - 2\,l, \;\text{ where } \; l=0,1,2. $$

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 non-negative 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 Bessel-Lommel SRF in the mean-square 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

$$ \varLambda _{d}^{(2n)} = \int _{0}^{{k_{c}}} dk\, {k^{n+d-1} }{\, \left[ 1 + {\eta _{1}}(k\xi )^2 + (k\xi )^4 \right] } $$

for \({k_{c}}\in {\mathbb{R}}\). \(\square \)

Appendix C: Proof of Proposition 3


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:

$$ \tilde{\kappa }_{1,2} = \sqrt{\frac{ \pm \sqrt{{\eta _{1}}^2 \, (1 - \alpha )^2 - 4 \alpha (\alpha -2)} - {\eta _{1}}\, (1 - \alpha )}{2(2 - \alpha ) \xi ^2}}. $$

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

$${\mathcal{S}}_d \, \eta _{0} \, \xi ^{1-2\alpha } \, \int _{0}^{\infty } dx \,\phi _{\alpha }(x) $$
$$ \text{ where } \quad \phi _{\alpha }(x) =\frac{x^{1+2\alpha }}{ 1 + {\eta _{1}}\, \xi ^2 \, x^2 + \xi ^4 \, x^4 }. $$

To simplify notation we define

$$ I_{\alpha }(\phi ):=\int _{0}^{\infty } dx \phi _{\alpha }(x). $$

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 real-valued function \(\forall x \in {\mathbb{R}}\) which is integrable \(\forall \alpha \in [0, 1]\). If there is a real-valued 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

$$ I_{\alpha }(\phi )= \int _{0}^{\infty } dx \lim _{n \rightarrow \infty } g_{n}(x) \phi _{\alpha }(x) = \lim _{n \rightarrow \infty } \int _{0}^{\infty } dx g_{n}(x) \phi _{\alpha }(x). $$

We define the following auxiliary function

$$ g_{n}(x) : = \frac{ 2^{\alpha }\, \Gamma (\alpha +1) \, J_{\alpha }(x/n)}{ \left( \frac{x}{n}\right) ^\alpha }. $$

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. 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. 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. 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. 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. 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

$$ I_{\alpha }(\phi ) = \lim _{n \rightarrow \infty } (2 n)^{\alpha } \Gamma (\alpha +1) \,\tilde{I}_{\alpha }(\phi ) $$


$$ \tilde{I}_{\alpha }(\phi ) = \int _{0}^{\infty } dx \, \frac{J_{\alpha }(x/n) x^{1+\alpha }}{ 1 + {\eta _{1}}\xi ^2 x^2 + \xi ^4 x^4 }. $$

The integral \(\tilde{I}_{\alpha }(\phi )\) is evaluated by means of the Hankel-Nicholson formula (29) (\(\nu =\alpha ,\) \(\mu =0\) for \({\eta _{1}}\ne 2, \, \mu =1\) for \({\eta _{1}}= 2\)) which leads to

$$ \tilde{I}_{\alpha }(\phi ) = \left\{ \begin{array}{cc} \frac{z_{+}^{\alpha } \, K_{\alpha }(z_{+}/n) - z_{-}^{\alpha } \,K_{\alpha }(z_{-}/n)}{\sqrt{{\eta _{1}}^2-4}} &{} {\eta _{1}}\ne 2\\ \frac{K_{\alpha -1}(1/n)}{2n} = \frac{K_{1-\alpha }(1/n)}{2n} &{} {\eta _{1}}= 2. \end{array} \right.$$

To evaluate \(\lim _{n \rightarrow \infty }\tilde{I}_{\alpha }(\phi )\) for \(1> p >0\) we use the series expansion (Schwartz 2008) of the K-Bessel function

$$ K_{p}(x) =\frac{1}{2} \left[ \Gamma (p) \left( \frac{2}{x}\right) ^{p} \left( 1+ O(x^2) \right) + \Gamma (-p) \left( \frac{x}{2}\right) ^{p} \right. \left. \quad \left( 1+ O(x^2) \right) \right].$$

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 \(O(x^{2-p})\) 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 K-Bessel function, (40), (41), and (42) we obtain the following equation (where \(\Delta = \sqrt{{\eta _{1}}^2-4}\)):

$$ I_{\alpha }(\phi ) =\left\{ \begin{array}{cc} \frac{\Gamma (1-\alpha )\Gamma (1+\alpha ) \, \left[ \left( {\eta _{1}}+ \Delta \right) ^{\alpha } - \left( {\eta _{1}}- \Delta \right) ^{\alpha }\right] }{2^{\alpha +1}\alpha \Delta } &{} {\eta _{1}}\ne 2\\ \frac{\Gamma (1-\alpha ) \, \Gamma (1+\alpha ) }{2} &{} {\eta _{1}}= 2. \end{array} \right. $$

Finally, based on (43), the definition (38) and (36), (24a) is proved. \(\square \)

Appendix D: Proof of Proposition 4


Based on the spectral density (13) it follows that the denominator in (23) is given by

$$\int _{{\mathbb{R}}^d} d{\mathbf{k}}\,k^{2\alpha } {\widetilde{C_{\rm{x}\rm{x}}}}^{BL}(k;{\varvec{\theta }}) = \frac{{\mathcal{S}}_{d} {k_{c}}^{d+2\alpha }}{\eta _{0} \xi ^d} \times \left( \frac{1}{d+2\alpha } + \frac{{\eta _{1}}{k_{c}}^2 \xi ^2}{d+2\alpha +2} + \frac{ {k_{c}}^4 \xi ^4}{d+2\alpha +4} \right). $$

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 Non-negative \({\eta _{1}}\)

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. 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).

  1. 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. 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. 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 \({k_{c}}.\)

    1. (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 \(\kappa _{-} < {k_{c}}\).

    2. (b)

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

      1. i.

        If k c  < κ +, then κ* = κ and λ (α) c is given by (25b).

      2. ii.

        If k c > κ +, then κ*= arg max(φ(κ ),φ(k c)) and λ (α) c is given by (25c).

\(\square \)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

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).

Download citation

  • Published:

  • Issue Date:

  • DOI: