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A005 LOCAL GEOSTATISTICS
C.L. FARMER
Schlumberger, Abingdon Technology Centre, Abingdon, Oxon, OX14 1UJ, UK,
and OCIAM, Mathematical Institute, University of Oxford, UK.
farmer5@slb.com
1. Introduction
Standard approaches to scattered data interpolation use the theory of Gaussian random functions.
First the correlation function (or equivalently the variogram) is defined, either by direct
assumption based on analogues and conceptual models or sometimes by inference from the data
to be interpolated. Then, a maximum probability argument leads to the various forms of Kriging.
This is justified when the data is plentiful and such that the Voronoi diagram of the data points
has cells that are less than the correlation length in diameter. When data is scant, realisations
conditioned on the scattered data can be drawn from the Gaussian distribution. To many
geostatisticians all is well with this approach, and the view is sometimes expressed that for
practical purposes there is little need for further research on matters of spatial statistics. There
are, in the view of this author, one or two difficulties that undermine this optimistic view.
The first difficulty is when the Gaussian assumption is inadequate (with regard to multi-point
statistics, and not just the one-point distribution). The most widely used non-Gaussian models
require ad hoc devices, such as simulated annealing approaches. Such non-Gaussian methods,
often defined by discrete stochastic models, preclude the use of explicit probability density
functionals and this makes construction of a rigorous Bayesian framework rather difficult. The
second difficulty appears even when the model remains Gaussian – this is the question of
interpolation in the presence of complicated geometric features. Curved layers require the
construction of stratigraphic coordinates. Although this is sometimes possible, in general such
coordinate systems do not exist. This means that in practice, approximate stratigraphic
transformations must be used. The reason for needing stratigraphic coordinates is the non-local
nature of the standard Gaussian random field model.
In this paper a local approach is explored. This is actually a special case of standard Gaussian
theory, but – although not justified here – we believe the analytical results described in the
following indicate the promise of generalisation to any geometry without needing stratigraphic
coordinates. The basis of this local approach is the theory of random fields as studied in
statistical physics.
2. Functional geostatistics
2.1 Introduction
In the following the symbols x and y denote one, two or three dimensional points. Let ϕ = ϕ (x)
be a Gaussian random function of x with mean h = h(x) . This has the probability density
functional (pdf)
π [ϕ ] = C exp{− H [ϕ ]}
Eq. (1)
where H [ϕ ] , called the ‘energy functional’, is given by
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H [ϕ ] = ∫ (ϕ ( x) − h( x)) a( x − y ) (ϕ ( y ) − h( y )) d N x d N y
Eq. (2)
2Ω
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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an integral expression over the N - dimensional reservoir region, Ω , where a is a positive
function such that a( x − y ) = a( y − x) . The non-local nature of the general Gaussian model is
clearly seen in the double integral appearing in equation (2). In equation (1) C is a normalisation
constant such that
Eq. (3)
∫ C exp{− H [ϕ ]}D[ϕ ] = 1
Η
where the function space integral in equation (3) is over some suitable space of functions Η . For
background about functional integration in a style compatible with that adopted here, see [1].
The kernel function a( x − y ) is an input to the model and the definition of the random functions
is such that it can be proved that the solution, g ( x − y ) , of the integral equation
∫ a( x − x′) g ( x′ − y) d
N
x′ = δ ( x − y)
Eq. (4)
Ω
is the correlation function of the random field. The function δ ( x − y ) in equation (4) is the usual
Dirac δ -function. See [1] and [3] for proofs. Thus
g ( x − y ) = ∫ (ϕ ( x) − h( x)) (ϕ ( y ) − h( y )) π [ϕ ] D[ϕ ]
Eq. (5)
Η
The random field, ϕ , can take arbitrarily large, negative or positive values. In general, in
applications, various transforms are applied to ϕ , so that the transformed variable lies in a
required range of values. Note that g is the Green’s function corresponding to the kernel, a.
2.2 Kriging
Consider a situation where ϕ is known without significant measurement error, at a finite number
( M ) of places {x m }, m = 1, M with values {ϕ *m } within the reservoir Ω . One problem is to
estimate a value ϕ (x) at some place x not in the observation set. A possible solution is to seek
the maximum probability interpolant. This follows from minimising the energy functional
subject to constraints that ϕ ( x m ) = ϕ *m , for all m. Depending on the assumptions made about
knowledge of the mean, and the form of the mean, one can recover all the different types of
Kriging. (See [3] for details and references.) It can also be shown that the method of radial basis
functions is the adjoint formulation of the universal Kriging method where the correlation
function provides the radial basis function. The radial basis function formulation requires less
linear algebra than the Kriging formulation and so is more efficient. A generalisation to include
measurement error can be developed in a similar way.
2.3 Stochastic sampling
Some applications require the expectation value of some functional of the random field, rather
than an estimate of the field value at a particular place. For example, one might be interested in
the expectation value of the quantity of oil recovered over some time interval. If the functional is
denoted by the symbol F [ϕ ] then this would involve computing functional integrals such as
F = ∫ F [ϕ ]π [ϕ ] D[ϕ ]
Η
Eq. (6)
This is formal, as evaluation of F [ϕ ] will generally involve running a complicated suite of
software including a flow simulator. Nevertheless, it catches the essence of the problem. When
π [ϕ ] is close to a delta-functional, then it makes sense to use Kriging, as evaluation of the
functional at the most probable value leads, in this special case, to a good approximation to the
expectation value. In general however, π [ϕ ] will not be like a delta-functional and one must
resort to other methods. Analytical approaches are not usually practical so this means using
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Monte-Carlo methods. To do this requires discretising the functional so that the functional
integral is approximated by a normal integral (albeit of very high dimension). A topic of current
research interest is to find ways of accelerating the Monte-Carlo method by incorporating prior
knowledge via Bayesian methods. (See [3] for an introduction to relevant literature.)
2.4 Problems with curvilinear and unstructured grids
In general, the reservoir region Ω is not rectangular and the coordinates do not align with the
layers. That is, no one component of the coordinates is constant in a layer. However, in some
cases (as when a simple layering scheme is appropriate) it is possible to introduce a new –
stratigraphic – coordinate system s = S (x) where s represents the stratigraphic coordinate (a
3D point). The compression and stretching of the coordinates then models the actual deformation
of the rocks and provides a natural coordinate system in which to apply Gaussian random fields.
When, however, the deformation of the coordinate system does not mimic a true physical
deformation – as for example the usual coordinate systems that are built with sloping coordinate
lines driven by the geometry of fault positions – there may be small cells at the base of the
reservoir and large cells at the top (or vice versa). In this case then, a second geometric
transformation is required, that maps the actual layers into rectangular layers. In general this is
very difficult to achieve. These difficulties are rather more obvious when an unstructured grid is
required as part of the geological model. As the use of unstructured grids is likely to increase in
the next few years, it is all the more urgent that a satisfactory method for spatial interpolation be
found. Applying geostatistics in a rigorous and transparent fashion, in general geological
geometries is thus an outstanding problem in the geosciences.
If the usual Gaussian random field model is to be used, the correlation functions must generally
be anisotropic with their principal axes aligned with the dip angles and dip normals. If the
stratigraphic coordinates cannot be constructed then the applicability of much geostatistical
theory is in question.
In the remainder of this paper, a local approach to geostatistics is explored. This is a special case
of the usual theory but has potential for generalisations (not developed here) that might alleviate
the difficulties of providing rational techniques for interpolation in general geometries. This is
likely to be achieved by avoiding the need for stratigraphic coordinate transformations.
3 Local density functionals
In the following, energy functionals of the form,
a
H [ϕ ] = ∫ [(∇ 2ϕ ) 2 + b(∇ϕ ) 2 + cϕ 2 ] d N x
Eq. (7)
2Ω
are studied. For simplicity of exposition only the case where h( x) = 0 is considered. In equation
(7) the parameters a, b and c are real. Using standard methods from statistical field theory it can
be shown that the correlation function, g ( x − y ), satisfies the partial differential equation
a∇ 2 (∇ 2 g ) − ab∇ 2 g + acg = δ ( x − y )
Eq. (8)
where suitable boundary conditions (such as the natural boundary conditions for the variational
problem related to the local energy functional or just periodic conditions) will be imposed.
If the energy functional is re-parameterised (using a and b as the symbols again) so that H takes
the apparently more complicated form,
Eq. (9)
H [ϕ ] = a ∫ [(∇ 2ϕ ) 2 + 2b 2 cos(2t )(∇ϕ ) 2 + b 4ϕ 2 ] d N x
Ω
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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then the correlation function satisfies
a∇ 2 (∇ 2 g ) − 2ab 2 cos(2t )∇ 2 g + ab 4 g = δ ( x − z )
Eq. (10)
Using Fourier transform techniques one can show that, in 3-D, where radial symmetry is
exploited in infinite space and r = x − y ,
k sin(kr )
1 1 ∞
dk 4
Eq. (11)
2
∫
2aπ r 0
(k + 2b 2 cos(2t ) + b 4 )
This integral can be evaluated using the calculus of residues or, more easily, by referring to the
tabulated integrals of Gradshteyn & Ryzhik [4]
g (r ) =
ϕ ( x)ϕ ( y ) = g ( x − y ) =
e −| x − y|b cos t sin(| x − y | b sin t )
4πa | x − y | b 2 sin( 2t )
The validity of this result requires that a > 0, b > 0, t <
Eq. (12)
π
.
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The limiting case of equation (12) when t = 0 is of interest in which case the Green’s function
reduces to
e −| x − y|b
ϕ ( x)ϕ ( y ) = g ( x − y ) =
Eq. (13)
4πab
4 Some further examples of local geostatistics
4.1 Biharmonic-Laplace functional
An interesting example is provided by the case, where the correlation function is not derivable
from the first example,
H [ϕ ] = a ∫ [(∇ 2ϕ ) 2 + b 2 (∇ϕ ) 2 ] d N x
Eq. (14)
Ω
This leads to the correlation function, in 3D,
1
− x− y b
(1 − e
)
Eq. (15)
2
4πa x − y b
Taking the limit as b goes to zero one finds, γ (r ) = g (0) − g (r ) ( the variogram, which is the
ϕ ( x)ϕ ( y ) = g ( x − y ) =
average of 0.5 (ϕ ( x) − ϕ ( y )) 2 ) is such that the standard linear variogram is recovered;
x− y
γ ( x − y) =
Eq. (16)
8πa
This is also the Green’s function for the 3D biharmonic equation, and a widely used radial basis
function in 3D. In this case the correlation function does not exist in the limit b goes to zero.
4.2 Helmholtz functional
This functional is given as an example in several texts on statistical field theory.
H [ϕ ] = a ∫ [(∇ϕ ) 2 + b 2ϕ 2 ] d N x
Eq. (17)
Ω
and has the corresponding correlation function satisfying the equation
− a∇ 2 g + ab 2 g = δ ( x − z )
Eq. (18)
which is the equation for the Green’s function for Helmholtz equation. In 3D this has the
solution
− x− y b
e
Eq. (19)
g ( x − y) =
4πa x − y
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5 Kriging and Briggs’ method
Consider the problem, as stated earlier, where the value of the field ϕ is known at a finite
number ( M ) of places {x m }, m = 1, M to have the corresponding values {ϕ *m } .
Introducing Lagrange multipliers, {λ m }, m = 1, M seek the minimum of the functional
H [ϕ ] + ∑ λ m (ϕ ( x m ) − ϕ *m )
Eq. (20)
m
Using the calculus of variations one finds the interpolant satisfies the equation
a∇ 2 (∇ 2ϕ ) − 2ab 2 cos 2t∇ 2ϕ + ab 4ϕ = ∑ λ mδ ( x − x m )
Eq. (21)
This has the solution
ϕ ( x) = ∑ λ m g ( x − x m )
Eq. (22)
m
m
where the coefficients, λ m , are to be determined by solving the linear system
ϕ *n = ∑ λ m g ( x n − x m )
Eq. (23)
m
which is a standard form of radial basis function interpolant. This derivation is for the case
without the affine contribution, which is needed for the case with non-zero mean. See reference
[3] for a discussion of this generalisation.
The equation just written is the generalisation of Briggs’ [2] method and closely related to the
‘discrete smooth interpolation’ (DSI) method of Mallet [4]. One sees that traditional
‘deterministic’ interpolation methods are closely related to Kriging, and are maximum
probability interpolants for the case of local geostatistics.
7 Conclusion
We have seen that a local model for the functional pdf, with ‘biharmonic’ terms included leads to
correlation functions of considerable generality. It remains to be shown in future work that the
method is useful and a starting point for curvilinear grid generalisations. However, the simple
results that have been derived indicate this to be a fruitful line of enquiry.
Acknowledgement
I thank the Royal Society of the United Kingdom and Schlumberger for research support.
References
1. Binney, J.J., N.J. Dowrick, A.J. Fisher and M.E.J. Newman, The Theory of Critical
Phenomena, Oxford University Press, Oxford, 1992.
2. Briggs, I.C. Machine contouring using minimum curvature. Geophysics, 39, 39-48, 1974.
3. Farmer, C. L. Geological Modelling and Reservoir Simulation. To be published in
‘Mathematical Methods and Modeling in Hydrocarbon Exploration and Production’. Editors:
Armin Iske and Trygve Randen, Springer-Verlag, Heidelberg, 2004.
4. Gradshteyn, I.S. and I.M. Ryzhik. Table of Integrals, Series, and Products. Academic Press,
San Diego, 2000.
5. Mallet, J.L. Discrete smooth interpolation. ACM Transactions on Graphics, 8(2), 121-144,
1989.
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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