EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 3, 2019, 1052-1068
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Geostatistical analysis with copula-based models of
madograms, correlograms and variograms.

Fabrice Ouoba1, Diakarya Barro2,∗, Hay Yoba Talkibing1

1 LANIBIO, UFR-SEA, Université Ouaga 1, Pr JKZ, Burkina Faso
2 UFR-SEG, Université Ouaga 2, Burkina Faso

Abstract. This paper investigates models of stochastic dependence with geostatistical tools.
Specifically, we use copulas to propose new models of stochastic spatial tools such as variograms,
correlograms and the madograms. Copula versions of covariograms are provided both in second
stationnary and intrinsic frameworks. Moreover, some usual families of models of variograms are
clarified with the corresponding parameters

1. Introduction

Spatial statistics focus on phenomena whose observations is a random process Z =
{Zs, s ∈ S} indexed by a spatial set S = {s1, . . . , sn} while Zs denotes a geographical
space D. Such technics where developed first in geostatistics more specifically from the
for geologists. Geostatistics are applications of probabilistic analysis methods to the study
of phenomena that extends into space and present a structuration. Here, space refers to
be the geographical space, but it may be the temporal axis or more abstract spaces. To
quantify this structure, the geostatistical tools used are mainly the variogram, the correl-
ogram and the madogram depending on the type of sampled data.

While modelling spatial extreme variability of an isotropic and max-stable field, Cooley
et al. (2006) have introduced the F-madogram γF (h) defined by

γF (h) = 1
2E {|F (Z (s))− F (Z (s+ h))|} . (1)

where h is the average value of the separating distance between the two points. This
tool provides a generalization of the so called λ-madogram associated to the distribution
underlying the stochastic process, such as:

γF (h) = 1
2E
{∣∣∣[F (Z (s))]λ − [F (Z (s+ h))]1−λ

∣∣∣} ;λ ∈ ]0, 1[ . (2)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i3.3389

Email addresses: didifab@yahoo.fr (O. Fabrice)
dbarro2@gmail.com (B. Diakarya), talkibingfils@yahoo.fr (H. Y. Talkibing)

http://www.ejpam.com 1052 c© 2019 EJPAM All rights reserved.



O. Fabrice, D. Barro, H. Y. Talkibing / Eur. J. Pure Appl. Math, 12 (3) (2019), 1052-1068 1053

The variogram or the semi-variogram makes it possible to determine whether the distri-
bution or parameters studied have a structure, random or periodic. Its representation has
three characteristic properties: the nugget effect, the range and the sill. The nugget effect
characterizes the variability at the origin. The sill, if it exists, is characterized by the
attainment of a plateau where the semi-variogram become constant with the evolution of
h and the range which characterizes the limiting distance of spatial structuring.

The correlogram function is identical to the linear coefficient between a series of spatial
data. It’s given by:

ρ(s1, s2) = corr(Z(s1), Z(s2)) =
cov(Z(s1), Z(s2))

σs1σs2
. (3)

It is possible to express graphically the correlation between two variables by mean of
their separate distance h. In particular, for spatial extreme or spatial temporal phenomena,
geostatistical tools such as the variogram or correlogram are not appropriate for studying
the spatial structuring of data.

Typically, the madogram or variogram of the first order is used to characterize this
spatial structure of the extreme data. Such as, for all separating distance h,

M(h) =
E(|Z(s+ h)− Z(x)|

2
. (4)

While studying spatial models of extreme values, Barro et al.[2] have considered a set
of locations S = {x1, ..., xs} ⊂ R2, where the process is observed. If Yk,1; ...;Yk,s denote
independent copies from the second-order stationary random field, for k = 1, ..., n, they
poined out that every spatial univariate marginal laws lies in the domain of attraction of
the real-value parametric Generalized extreme value (GEV) distribution, defined spatially
on the subdomain:

Sξ = {xi ∈ S;σi (xi) + ξi (xi) (yi (xi)− µi (xi)) > 0} ⊂ S,

such as:

GEV (yi (xi)) =

 exp

{
−
[
1 + ξi (xi)

(
yi(xi)−µi(xi)

σi(xi)

)] −1
ξi(xi)

}
if ξi (xi) 6= 0

exp
{
− exp

{
−
(
yi(xi)−µi(xi)

σi(xi)

)}}
if ξi (xi) = 0

; (5)

where the parameters {µi(xi) ∈ R}, {σi(xi) > 0} and {ξi(xi) ∈ R} are referred to as
the spatial version of location, the scale and the shape parameters for the site xi respec-
tively.

The major contribution of this paper is to propose new models of geostatistical depen-
dence tools by using copula functions. Indeed, the variogram, the correlogram and the



O. Fabrice, D. Barro, H. Y. Talkibing / Eur. J. Pure Appl. Math, 12 (3) (2019), 1052-1068 1054

madogram of spatial variable are modeled via the copula underlying their joint distribu-
tion. Specifically, section 2 gives the background tools of stochastic analysis that turn to
be necessary, while section 3 deals with our main results, copulas versions of variogram,
madogram, covariogram and correlogram via copulas.

2. Preliminaries

This section summaries definitions and properties on the copulas of multivariate joint
processes dependence which turn out to be necessary for our approach. For this purpose
the definition of multivariate copula is necessary. Moreover, we provide a survey of the
main geostatistical tools used in this paper.

2.1. Some geostatistical tools in spatial dependence

The covariance of a random field measures the strength of the relationship witch exists
between the random variables witch represents it in the different observation sites. It is
defined on Rd × Rd in R, for all (s1, s2) ∈ Rd × Rd,

c(s1, s2) = Cov[Z(s1), Z(s2)] = E[Z(s1)Z(s2)]−m(s1)m(s2).

Since,

E[Z(s1)Z(s2)] =

∫ +∞

−∞

∫ +∞

−∞
z1z2h(s1, s2)dz1dz2,

the covariance function can still be written such as:

c(s1, s2) =

∫ +∞

−∞

∫ +∞

−∞
z1z2h(s1, s2)dz1dz2 −m(s1)m(s2)

where m(s1) is the mean of Z(s1) and h(s1, s2), the joint density function of Z(s1) and
Z(s2). The Cauchy-Schwarz inequality links the covariance between Z(s1) and Z(s2) to
the variance of Z(s1) and Z(s2),

Cov[Z(s1), Z(s2)]| ≤
√
V ar[Z(s1)]V ar[Z(s2)].

The madogram of a random field, especially used in the extreme case, determines the
strength of the relationship between the random variables that represents it in the different
observation sites. It is set to Rd in R+ by:

∀h ∈ Rd, M(h) =
E(|Z(s1 + h)− Z(s1)|)

2
; ∀s1 ∈ Rd.

2.2. A Survey of Copulas Functions

Copulas functions can be used to describe the dependence of variables or for spatial
interpolation. The copula were introduced by Sklar [10] in order to characterize a vector
Z = (Z1, . . . , Zn) having given marginal laws. According to Sklar’s theorem, all functions



O. Fabrice, D. Barro, H. Y. Talkibing / Eur. J. Pure Appl. Math, 12 (3) (2019), 1052-1068 1055

of continuous multivariate distributions F to marginal F1, . . . , Fd, there is a copula function
C such that

F (s1, . . . , sn) = C[F1(s1), . . . , Fd(sd].

Definition 1. An n-dimensional copula is a distribution function Cn : [0, 1]n −→ [0, 1]
satisfying the following properties.

i) C(u) = 0 if one of the coordinates of u is zero, that is

Cn(u1, ..., ui−1, 0, ui+1, ..., un) = 0; for, all (u1, ..., ui−1, ui+1, ..., un) ∈ [0, 1]n−1.

ii)
Cn(u1, ..., ui−1, 1, ui+1, ..., un) = Cn−1(u1, ..., ui−1, ui+1, ..., un),

, an (n-1) copula for all i.
iii) The volume VB of any rectangle B = [a, b] ⊆ [0, 1]n is positive, that is,

VB([a, b]) = ∆bn
an∆bn−1

an−1
. . .∆b1

a1C(u) =

∫
B
dCn (u1, ..., un) ≥ 0. (6)

where,

∆bk
ak
C(u) = C(u1, . . . , uk−1, bk, uk+1, . . . , un)− C(u1, . . . , uk−1, ak, uk+1, . . . , un) ≥ 0.

The use of copulas in stochastic analysis whas justified by the canonical parametriza-
tion of Sklar, see Joe [9]or Nelsen [12], such that the n-dimensional copula C associated to
a random vector (X1, ..., Xn) with cumulative distribution F and with continuous marginal
F1, ..., Fn is given, for (u1, ..., un) ∈ [0, 1]n by

C(u1, ..., un) = F [F−11 (u1), ..., F
−1
n (un)]. (7)

Differentiating the formula (7 ) shows that the density function of the copula is equal
to the ratio of the joint density f of F to the product of marginal densities hi such as, for
all (u1, ..., un) ∈ [0, 1]n ,

c(u1, ..., un) =
∂nC (u1, ..., un)

∂u1...∂un
=

f
[
F−11 (u1), ..., F

−1
n (un)

]
f1
[
F−11 (u1)

]
× ...× fn

[
F−1n (un)

] . (8)

3. The Main Results of the Study

Let Z be a random field in n sites Z = {Z(s1), . . . , Z(sn)}. Suppose H(si, sj) and
h(si, sj) are the attached distribution and density functions of Z(.) with marginal distri-
butions FZ(si) at the site si.



O. Fabrice, D. Barro, H. Y. Talkibing / Eur. J. Pure Appl. Math, 12 (3) (2019), 1052-1068 1056

3.1. Modeling the Madogram and the F-Madogram via Copulas

The following result provides a relation between the F-madogram and the underlying
copula function.

Theorem 1. Let CF be the copula underlying a stochastic process Z = {Z(s1), . . . , Z(sn)} .
Then, the generalized F-madogram is such that:

γF (h) =

∫ 1

0
udCF,h

(
u

1
λ , u

1
1−λ
)
− 1

2
[C(λ)−Dh(λ)] , (9)

where

C(λ) =
1

2

(
σ2Zs

1 + λσ2Zs

)
,

and

Dh(λ) =
1

2

(
σ2Zs+h

1 + (1− λ)σ2Zs+h

)
,

for λ ∈]0; 1[, h being the average value of the separating distance between the two points.

Proof. Consider a bivariate distribution F, satisfying the key assumption. By noting
that

|a− b| = 2 max(a, b)− (a+ b),

and using this relation in (2), it follows that :

γF (h) =
E
(
2 max

(
[F (Z(s))]λ, [F (Z(s+ h))]1−λ

)
− [F (Z(s))]λ − [F (Z(s+ h))]1−λ

)
2

.

Then,

γF (h) = E
(

max
(

[F (Z(s))]λ, [F (Z(s+ h))]1−λ]
))

− 1
2

{
E
(

[F (Z(s))]λ
)
− E

(
[F (Z(s+ h))]1−λ

)}
(10)

Furthermore,

FZs,Zs+h,λ(u) = P
(

max
(

[F (Z(s))]λ, [F (Z(s+ h))]1−λ
)
≤ u

)
.

Thus,

FZs,Zs+h,λ(u) = P
(

[F (Z(s))]λ ≤ u, [F (Z(s+ h))]1−λ ≤ u
)
,

It yields that

FZs,Zs+h,λ(u) = P
(
F (Z(s)) ≤ u

1
λ , F (Z(s+ h)) ≤ u

1
1−λ
)
,



O. Fabrice, D. Barro, H. Y. Talkibing / Eur. J. Pure Appl. Math, 12 (3) (2019), 1052-1068 1057

Therefore
FZs,Zs+h,λ(u) = CF,h

(
u

1
λ , u

1
1−λ
)

for all λ ∈]0; 1[.

which is equivalent to,

E
(

max
(

[F (Z(s))]λ, [F (Z(s+ h))]1−λ
))

=

∫ 1

0
udFZs,Zs+h,λ(u).

It follows that, for all λ ∈]0; 1[,

E
(

max
(

[F (Z(s))]λ, [F (Z(s+ h))]1−λ
))

=

∫ 1

0
udCF,h

(
u

1
λ , u

1
1−λ
)
. (11)

Furthermore, one have

E
(

max
(

[F (Z(s))]λ
))

=
σ2Zs

1 + λσ2Zs
, (12)

and

E
(

max
(

[F (Z(s+ h))]1−λ
))

=
σ2Zs+h

1 + (1− λ)σ2Zs+h
∀λ ∈]0; 1[. (13)

Using (11), (12) and (13) in (10), it follows that

γF (h) =

∫ 1

0
udCF,h

(
u

1
λ , u

1
1−λ
)
− 1

2

[
σ2Zs

1 + λσ2Zs
+

σ2Zs+h
1 + (1− λ)σ2Zs+h

]

which proves the relation (9) as disserted.

Let Z =
{
Z(x), x ∈ Rd

}
be a regular random field defined on Rd. It is a well known

that the madogram associated to the random field Z is the function M , mapping Rd to
R+ such as:

∀h ∈ Rd,M(h) =
E(|Z(x+ h)− Z(x)|)

2
, x ∈ Rd. (14)

Proposition 1. ( Copula-based madogram )
If Z(.) is a stationary random field of order two then, the relation between its madogram
and the copula function bivariate is given by:

M(h) =

∫ 1

0
F−1Z (u)dCh(u, u)− µ,

where µ = E(Z(x + h)) = E(Z(x)), Ch(., .) being the jointed copula function which de-
scribes the dependence structure between two remote sites of h.

The following figure provide a representation of the joint copula of these two variables.



O. Fabrice, D. Barro, H. Y. Talkibing / Eur. J. Pure Appl. Math, 12 (3) (2019), 1052-1068 1058

Figure 1: Graph of a joint distribution of copula

Proof. Recall that |a− b| = 2 max(a, b)− (a+ b). Using this relation in (14), it’s comes
that for all h, x ∈ Rd

M(h) =
E (2 max[Z(x+ h), Z(x)]− Z(x+ h)− Z(x))

2
. (15)

Since E(.) is a linear application, the relation (15) gives

M(h) =
E (2 max[Z(x+ h), Z(x)])− E (Z(x+ h))− E (Z(x))

2
.

Thus,
M(h) = E (max[Z(x+ h), Z(x)])− µ. (16)

Furthermore,

P (max[Z(x+ h), Z(x)] ≤ z) = P (Z(x+ h) ≤ z, Z(x) ≤ z) ,

which is equivalent to:

P (max[Z(x+ h), Z(x)] ≤ z) = Ch (P (Z(x+ h) ≤ z) , P (Z(x) ≤ z))

Thus,
P (max[Z(x+ h), Z(x)] ≤ z) = Ch (FZ(z), FZ(z)) ,

so,

E (max[Z(x+ h), Z(x)]) =

∫ 1

0
F−1Z (u)dCh(u, u).

Substituting this expression in the equation (16) and taking into account that z = F−1Z (u),
we obtain the result:

M(h) =

∫ 1

0
F−1Z (u)dCh(u, u)− µ.

Which proves the assertion.



O. Fabrice, D. Barro, H. Y. Talkibing / Eur. J. Pure Appl. Math, 12 (3) (2019), 1052-1068 1059

3.2. Modeling the covariogram via copulas

The variogram allows to measure the linear dependence between the random variables
of a field. For two given sites, the associated variogram is given by

ϑ(si, sj) = V ar(Z(si)− Z(sj)) = V ar(Z(si)) + V ar(Z(sj))− 2ĉ(si, sj). (17)

These tools do not take into account the extreme data observed in the different observation
sites. However, the copula function makes it possible to model the extreme data and to
detect any nonlinear link between different observation sites. So, it is necessary to express
the variogram and the covariogram via the copula to allow the model to take into account
the spatial structure even in case of extremes data. The model could also be able to detect
the presence of some nonlinear dependence.

Theorem 2. Let Z = {Z(s1), . . . , Z(sn)} be a stochastic process with variogram given by
(14). Then, the covariogram ĉ(si, sj) and the copula function are linked by the relation.

ϑ(si, sj) = σ2Z(si) + σ2Z(sj)− 2ĉ(si, sj) (18)

where

ĉ(si, sj) =

∫ 1

0

∫ 1

0
F−1Z (u)F−1Z (v)c(u, v)dudv −mimj .

The quantity mi being the mean of Z(si); c(u, v) the copula density function attached to
Z(si) and Z(sj).

Proof. Let ui = F (Z(si)) = FZ(si). It’s follow that:

zi = F−1Z (ui) =⇒ dui = fZ(zi)dzi,

where fZ(si) is the density function of the variable Z(si).
So, it comes that

dzidzj =
1

fZ(zi)fZ(zj)
duiduj .

Similarly, according to Sklar’s theorem [10], it follow that, for all couple of sites
(si, sj) ∈ S2

H(si, sj) = C(FZ(si), FZ(sj)).

Moreover,

c(u, v) =
f
[
H−11 (u), H−1n (un)

]
f1
[
F−11 (u1)

]
× ...× Fn

[
F−1n (un)

] . (19)

The relation between the joint density function h(si, sj) and the joint density function of
the copula c(u, v) is given by:

h(si, sj) = c(u, v)fZ(zi)fZ(zj).



O. Fabrice, D. Barro, H. Y. Talkibing / Eur. J. Pure Appl. Math, 12 (3) (2019), 1052-1068 1060

Using these expressions in the covariogram expression, it follows that

ĉ(si, sj) =

∫ 1

0

∫ 1

0
F−1Z (u)F−1Z (v)c(u, v)dudv −mimj .

By using the last relation in the variogram expression, it follows that

ϑ(si, sj) = σ2Z(si) + σ2Z(sj)− 2

∫ 1

0

∫ 1

0
F−1Z (u)F−1Z (v)c(u, v)dudv − 2mimj .

The following figure provide a representation of some theoretical variogram

Figure 2: Graph of some theoretical variograms

3.3. Modeling the correlogram via copulas

The correlogram measures the spatial dependence between two sites si and sj for all
i and j. The following result gives a relation between the correlogram and the copula
function.

Theorem 3. Let Z = {Z(si), . . . , Z(sj)} be a stochastic process on a geostatistical domain
S. For two sites si, sj ∈ S, the correlogram is given, via the associated copula by

ρ(si, sj) =

∫ 1

0

∫ 1

0

F−1Z (u)

σZ(si)

F−1Z (v)

σZ(sj)
c(u, v)dudv − mi

σZ(si)

mj

σZ(sj)
; (20)

where mi denotes the mean of Z(si), c(u, v) = ∂2C(u,v)
∂u∂v being the density function of the

copula C and σZ(si) and the standard error of Z(si) and Z(sj).



O. Fabrice, D. Barro, H. Y. Talkibing / Eur. J. Pure Appl. Math, 12 (3) (2019), 1052-1068 1061

Proof. By definition, the correlogram of a random field in two observation sites is given
by:

ρ(si, sj) =
ĉ(si, sj)

σZ(si)σZ(sj)
=

1

σZ(si)σZ(sj)
× ĉ(si, sj).

Using the result of precedent theorem (theorem 3), we get (20).

3.4. Stationnary framework for covariance modeling

In spatial context, the stationarity describes in a way, a form of spatial homogeneity of
regionalization. From a mathematical point of view, stationarity hypothesies consists in
assuming that the probabilistic properties of a set of values do not depend on the absolute
position of the associated sites, but only on their separation.

Under the assumption of the second order stationarity of the random field Z(.), the
mean function deviates a constant and the covariance depends only on the distance sepa-
rating the sites. So,

E(Z(si)) = µ ∀i = 1, n and ĉ(si, sj) = ĉ(si − sj) = ĉ(hij).

Previous relationships can be written differently. The following result gives a rela-
tionship between the covariogram and the copula function in second-order stationarity
case.

Corollary 1. In a second order stationarity framework the covariogram is given by

ĉ(si, sj) = ĉ(hij) =

∫ 1

0

∫ 1

0
F−1Z (u)F−1Z (v)chij (u, v)dudv − µ2, (21)

where hij = |si − sj | and chij (u, v) the jointed copula density function of the two localized
variables at two remote sites of hij.

Proof. Under the assumption of two order stationarity, the result of theorem (Theorem
3) gives the relation (21).

Similarly, under the assumption of second-order stationaity, the correlogram and the
variogram are expressed as a function af the copula by the relation,

Proposition 2. Assuming that hij = si−sj is the average distance and using the relation
(21), then, the correlogram is such as

ρ(hij) =

∫ 1

0

∫ 1

0

F−1Z (u)F−1Z (v)

ĉ(0Rd)
chij (u, v)dudv − µ2

ĉ(0Rd)
. (22)

and

ϑ(hij) = 2

[
ĉ(0Rd) + µ2 −

∫ 1

0

∫ 1

0
F−1Z (u)F−1Z (v)chij (u, v)dudv

]
. (23)



O. Fabrice, D. Barro, H. Y. Talkibing / Eur. J. Pure Appl. Math, 12 (3) (2019), 1052-1068 1062

Proof. Under the hypothesis of two order stationary and using the relation (21), the
result of theorem (Theorem 3) gives the result (22) and (23).

Proposition 3. The relationship between the variogram and the covariogram is such
than,

ĉ(hij) = ĉ(0Rd)−
1

2
ϑ(hij).

Proof. Let us consider the relations (21) and (23). It is known that:

ĉ(si, sj) = ĉ(hij) =

∫ 1

0

∫ 1

0
F−1Z (u)F−1Z (v)chij (u, v)dudv − µ2,

and

ϑ(hij) = 2

[
ĉ(0Rd) + µ2 −

∫ 1

0

∫ 1

0
F−1Z (u)F−1Z (v)chij (u, v)dudv

]
.

Adding the two relations above, we get:

2ĉ(hij) + ϑ(hij) = 2ĉ(0Rd).

So

ĉ(hij) = ĉ(0Rd)−
1

2
ϑ(hij).

3.5. Covariogram in Stationnary intrinsic framework

Consider an intrinsic random field Z(.) without drift, that is, the average of the incre-
ments is zero and the variance of the increments is the variogram. The intrinsic hypothesis
is written:

E[Z(x+ h)− Z(x)] = 0,

and
V ar[Z(x+ h)− Z(x)] = E

{
[Z(x+ h)− Z(x)]2

}
= 2γ(h). (24)

By integrating the intrinsic hypothesis, we obtain the following result.

Theorem 4. Let Z(.) An intrinsic random field without drift. It follows that the covari-
ance is such that

ĉ(si, sj) = ĉ(hij) =

∫ 1

0

∫ 1

0
F−1Z (u)F−1Z (v)chij (u, v)dudv − µ2, (25)

while the correlogram is:

ρ(hij) =

∫ 1

0

∫ 1

0

F−1Z (u)F−1Z (v)

ĉ(0Rd)
chij (u, v)dudv − µ2

ĉ(0Rd)
.



O. Fabrice, D. Barro, H. Y. Talkibing / Eur. J. Pure Appl. Math, 12 (3) (2019), 1052-1068 1063

And the variogram

ϑ(hij) = 2

[
ĉ(0Rd) + µ2 −

∫ 1

0

∫ 1

0
F−1Z (u)F−1Z (v)chij (u, v)dudv

]
.

Where µ denote the mean, ĉ(0Rd) the variance and chij (u, v) the density copula function.

Proof. By considering again the relation (24) we obtain:

γ(h) =
1

2
E
{

[Z(x+ h)− Z(x)]2
}
. (26)

Now, it comes that

E
{

[Z(x+ h)− Z(x)]2
}

= E([Z(x+ h)]2)− 2E(Z(x+ h)Z(x)) + E([Z(x)]2).

So, by taking into account the density of the copula,

E
{

[Z(x+ h)− Z(x)]2
}

= ĉ(0Rd) + µ2 − 2

∫ 1

0

∫ 1

0
F−1Z (u)F−1Z (v)chij (u, v)dudv + ĉ(0Rd) + µ2.

Furthermore, we have,

E
{

[Z(x+ h)− Z(x)]2
}

= 2ĉ(0Rd) + 2µ2 +−2

∫ 1

0

∫ 1

0
F−1Z (u)F−1Z (v)chij (u, v)dudv.

Using this last relation in the equation (26), we get finaly,

ϑ(h) = 2

[
ĉ(0Rd) + µ2 −

∫ 1

0

∫ 1

0
F−1Z (u)F−1Z (v)chij (u, v)dudv

]
. (27)

The variogram expression is similar in the intrinsic and second-order case, we deduce that
the covariance and correlogram expressions remain unchanged.

Remark 1. The variogram is related to the covariogram by the relation:

ĉ(si, sj) = ĉ(hij) =

∫ 1

0

∫ 1

0
F−1Z (u)F−1Z (v)chij (u, v)dudv − µ2. (28)

and

ϑ(hij) = 2

[
ĉ(0Rd) + µ2 −

∫ 1

0

∫ 1

0
F−1Z (u)F−1Z (v)chij (u, v)dudv

]
. (29)

Adding the two relations above, we get finaly,

2ĉ(hij) + ϑ(hij) = 2ĉ(0Rd)

and

ĉ(hij) = ĉ(0Rd)−
1

2
ϑ(hij).



O. Fabrice, D. Barro, H. Y. Talkibing / Eur. J. Pure Appl. Math, 12 (3) (2019), 1052-1068 1064

3.6. Mains families of models of variograms

In this section we summary some of the most usefull models of variagrams in spatial
modeling

a) Families with bearing or transition models

Families or Modeles of variograms Parameter

1

· Pure nugget effect model

γ(hij) =

{
0 for i = j
ĉ ∀i 6= j

· Meaning: reflects an absence of spatial structuring, due at
the presence of an undetectable micro-structure experimentally.

2

· Spherical model of parameter range a and sill ĉ (valid in Rd, d 6 3)

γ(‖ hij ‖) =

{
ĉ(7
‖hij‖2
a2
− 35

4
‖hij‖3
a3

+
7

2
‖hij‖5
a5
− 3

4
‖hij‖7
a7

) pour 0 6‖ hij ‖6 a

ĉ pour ‖ hij ‖> a
· Meaning: the presence of an undetectable micro-structure experimentally.

a

3

Gaussian model of parameter a and sill ĉ

γ(‖ hij ‖) = ĉ(1− exp(−‖ hij ‖
2

a2
))

· Meaning: the sill is reached asymptotically and the practical range

can be taken equal to a
√

3

a
√

3

b) Bessel and Polynomial models of variogram

Models includes mainly two models: the k modified model characterized



O. Fabrice, D. Barro, H. Y. Talkibing / Eur. J. Pure Appl. Math, 12 (3) (2019), 1052-1068 1065

1

· Bessel modified model

γ(‖ hij ‖) = ĉ

{
1− 1

2α−1Γ(α)
(
‖ hij ‖
a

)αKα(
‖hij‖
a )

}
characterized by sill ĉ, scale factor a and parameter α, and
Γ being the function of Euler interpolating the factorial K(α) .

Kα(u) =

π

(
Σ∞k=0

1

k!Γ(−α+ k + 1)
(
u

2
)2k−α − Σ∞k=0

1

k!Γ(α+ k + 1)
(
u

2
)2k+α

)
2 sin(απ)

with α > 0

2

· The Bessel J model

γ(‖ hij ‖) = ĉ

{
1− (

‖ hij ‖
2a

)−αΓ(α+ 1)Jα(
‖ hij ‖
a

)

}
;

characterized by sill ĉ, scale factor a and parameter α, and
Jα being the Bessel function of the first kind of order α,

Jα(u) = (
u

2
)αΣ∞k=0

(−1)k

k!Γ(α+ k + 1)
(
u

2
)2k..

· α = −1/2 for cosine model
· α = 1/2 for sinus model

3

· Polynomial Models with a scope and threshold ĉ

γ(‖ hij ‖)=

 ĉ

(
35
12
‖hij‖
a − 35

8
‖hij‖3
a3

+
7

2

‖ hij ‖5

a5
− 25

24
‖hij‖7
a7

)
pour 0 6‖ hij ‖6 a

ĉ pour ‖ hij ‖> a

on Rd for d 6 3

4

· An other Polynomial Models with a scope and threshold ĉ

γ(‖ hij ‖) =

 ĉ

(
5

2

‖ hij ‖
a
− 5

2

‖ hij ‖3

a3
+
‖ hij ‖5

a5

)
pour 0 6‖ hij ‖6 a

ĉ pour ‖ hij ‖> a

Figure 3: Graph of the extremal coefficient



REFERENCES 1066

4. Conclusion

The results of the study provides important characterizations of the variogram, the
correlogram in a copula framework. Especially, they show on one hand that these tools are
limited when data includes extremes values, in an other hand, that they have a copulawise
extension, allowing the copula to model the data in the spatial context. Moreover, the
study provides tools to analyze data and perform a comparative study with existing tools.

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