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Fast Multidimensional Interpolations

Authors:

Abstract

We have developed a high-performance, flexible scheme for interpolating multidimensional data. The technique can reproduce exactly the results obtained, for example, from Ordinary Kriging and related techniques in 3D, or from Thin-Plate Splines (Briggs' minimum-curvature algorithm) in 2D. Moreover, compared to traditional implementations of these algorithms, our method enjoys large computational-cost savings. The new approach produces an interpolation that obeys a Partial Differential Equation (PDE). The PDE may arise from physically based arguments, but its form can vary widely. It might be specified only implicitly (as through a Model Variogram), or be nonlinear (although a performance penalty could then apply). While a formal equivalence between Kriging and Splines has been known for some time (Matheron (1980)), the present derivation, from radial basis functions, further illuminates this connection. Thus, for example, we can make explicit the PDEs that underlie some of the Model Variograms most often used in Geostatistics. Besides its practical utility, the work thereby acquires a theoretical interest.
ABSTRACT
We have developed a high-performance, flexible
scheme for interpolating multi-dimensional data. The tech-
nique can reproduce exactly the results obtained, for exam-
ple, from Ordinary Kriging and related techniques in 3D, or
from Thin-Plate Splines (Briggs' minimum-curvature algo-
rithm) in 2D. Moreover, compared to traditional implemen-
tations of these algorithms, our method enjoys large
computational-cost savings.
The new approach produces an interpolation that
obeys a Partial Differential Equation (PDE). The PDE may
arise from physically based arguments, but its form can
vary widely. It might be specified only implicitly (as through
a Model Variogram), or be nonlinear (although a perform-
ance penalty could then apply).
While a formal equivalence between Kriging and
Splines has been known for some time (Matheron (1980)),
the present derivation, from radial basis functions, further
illuminates this connection. Thus, for example, we can
make explicit the PDEs that underlie some of the Model
Variograms most often used in Geostatistics. Besides its
practical utility, the work thereby acquires a theoretical
interest.
BACKGROUND
The aim of this paper is to explore the close connec-
tions between several seemingly different topics: Kriging;
radial basis function (RBF) interpolations; Spline interpola-
tions; and the solutions of Partial Differential Equations
(PDEs). We demonstrate that these techniques can pro-
duce families of identical interpolations.
We start with some notation. Throughout this paper, we
denote vectors in a lower case boldface font, and matrices
in an upper case boldface. Let be the position of a
control point at which some scalar value is known. We
denote by an interpolation which passes through the
set of control points .
KRIGING
Simple Kriging
In the traditional formulation of simple Kriging (and its
derivatives; Isaaks and Srivastava (1989)), one estimates
at some interpolation point from a weighted lin-
ear combination of the
, (1)
where is the vector with components and
denotes the inner (dot) product between two vectors. The
weight vector is found by solving the linear system
, (2)
where the are the covariances between
control points in the assumed model, and
gives the covariances between the control points and .
We choose simple Kriging for its mathematical simplic-
ity, and because most (if not all) of the other Kriging tech-
niques can be derived from simple Kriging via additional
constraints and/or via integrals of the resulting interpolation
(e.g. ordinary Kriging, universal Kriging or Kriging with a
trend, block Kriging).
Dual Kriging
In the dual Kriging formulation (Matheron (1982), or
Royer and Vieira (1984), for an English example), one
forms the estimate from a single set of weights ,
. (3)
The unknown is obtained by solving
. (4)
Details are given in Appendix 1, where dual formulation
analogues of techniques like Kriging with a trend are also
outlined.
xin
ui
ux()
Nux
i
() ui1iN≤≤;=
ux0
()
x0
ui
ux0
() w0u,()
uui
ab,()
w0
Cw0b0
=
Cij φxixj
()=
b0φx0xj
()=
x0
ux0
()
f
ux() b0f,()
f
Cfu=
FAST MULTIDIMENSIONAL INTERPOLATIONS
Franklin G. Horowitz
1,2
, Peter Hornby
1,2
, Don Bone
3
, Maurice Craig
4
1
CSIRO Division of Exploration & Mining, Nedlands, WA 6009 Australia
2
Australian Geodynamics Cooperative Research Centre, Nedlands, WA 6009 Australia
3
CSIRO Division of Information Technology, Canberra, ACT 2601 Australia
4
CSIRO Division of Exploration & Mining, Floreat Park, WA 6014 Australia
Reprinted with corrections from Chapter 9 (pp. 53-56) of the 26th Proceedings of the Application of Computers and
Operations Research in the Mineral Industry (APCOM 26), R.V. Ramani (ed.), Soc. Mining, Metall., and Explor. (SME),
Littleton, Colorado, U.S.A., 538 p., 1996.
Efficiency
From the standpoint of the operations count required to
compute an interpolation, a direct implementation of equa-
tions (3) and (4) offers a slight advantage over equations
(1) and (2). Recall that
C
is .
The dual formulation requires solving only the single
linear system (4), which costs operations for practi-
cal algorithms (e.g. Press, et al. (1992) for a readable dis-
cussion). In order to evaluate the interpolation at
different grid points, one needs a further opera-
tions ( evaluations of (3), each of which requires an
dot product between and ). Thus a
direct implementation of dual Kriging requires a grand total
of
(5)
operations.
The simple formulation requires solving the linear sys-
tem (2) at each of the grid points. A direct implementa-
tion requires operations. Evaluation of (1) over
the grid requires a further operations, yielding a
total cost of
(6)
operations. A more sophisticated algorithm might be to
solve (2) by inverting once, costing . Then, over
each of grid points, it would evaluate . Each
matrix multiplication requires operations, resulting
in . The total cost of this algorithm being
, (7)
and it might be more efficient than the direct implementa-
tion (depending on how scales with ). Remember,
sampling is typically pointwise (spot sampling) or linewise
(drill strings, traverses, flight lines), and hence could be
as high as or even , although is
probably more common.
Clearly, the dual formulation is more efficient than
either of the simple formulation algorithms. Equally clearly,
when (common for real world applications),
none
of these algorithms (as stated) is practical on com-
monly available computers. This observation has led to the
development of many localised algorithms, whose primary
computational reason for existence is to force down to
something manageable (like 40 or 50). Of course these
algorithms require large and complex bookkeeping compo-
nents, for such operations as nearest neighbourhood
searches. We will not consider the ramifications of this
modification here.
RBF INTERPOLATION
The theoretical underpinnings of radial basis function
interpolation are described in Powell (1992), and the refer-
ences found therein. Briefly, although the method is not for-
mulated in the language of spatial statistics, the actual
interpolation is computed nearly identically to dual Kriging,
. (8)
Here is as before, and is some polynomial of the
components of . The and the coefficients of the poly-
nomial satisfy an augmented system
. (9)
The only difference between equations (8)-(9) and equa-
tions (3)-(4) is the addition of the polynomial term ,
resulting in the augmentation of the simple Kriging system
of equations (Appendix 1). When we recover
(3), and when , we have the dual Kriging ana-
logue of universal Kriging (also known as Kriging with a
trend).
In RBF parlance, Kriging’s model covariance functions,
, are the radial basis functions upon which the
technique is founded. Just as in Kriging, practitioners of
RBF interpolation choose the form of from a small set of
functions known to be admissible (principally by inducing
the matrix to be invertible). The efficiency of RBF inter-
polation is identical to that of dual Kriging (e.g. described
by expression (5)).
PARTIAL DIFFERENTIAL EQUATIONS
Links to RBFs and Kriging
We now motivate the connection between dual Kriging,
radial basis functions, and linear PDEs by relating the PDE
formalism to the RBF formalism. Suppose that, for a given
, there exists an operator such that
, (10)
where is Dirac’s delta, and
. (11)
NN×
ON
3
()
M
OMN()
M
ON()
f
φx0xi
()
ON
3
()OMN()+
M
OMN
3
()OMN()
OMN
3
()OMN()+
CON
3
()
MC1b0
ON
2
()
OMN
2
()
ON
3
()OMN
2
()+
MN
M
ON
2
() ON
3
() ON()
N10 000,
N
ux0
() fiφx0xi
()
i
Phx()+=
φPhx()
xfi
Phx()
C
ˆf
ˆu
ˆ
=
Phx()
Phx() 0=
Phx() 0
φx0xi
()
φ
C
ˆ
φL
Lφxx
i
()δxx
i
()=
δ
LPhx() 0=
For example, if is a differential operator, is the Green’s
function of , and is a homogeneous solution to
the PDE. Admittedly, (10)-(11) are strong assumptions to
make, however, our aim here is to outline an idea, and we
do not wish to cloud the simplicity of that idea with mathe-
matical trickery dealing with difficult cases.
Applying to (8) gives
. (12)
This corresponds to an operator equation, where is the
response of a system (described by ) to a set of point
forces of magnitude at points . The coefficients of
determine, or are determined by, the boundary con-
ditions, depending upon your point of view. Equation (12),
together with adequate BCs, renders the problem positive
definite. Apart from the question of BCs, this problem is
equivalent to the Kriging (or RBF) problem. In fact, this is
potentially an interesting route to a proof of the existence of
unique and stable solutions to the latter problems (at least
for certain and ). Thus it would seem plausible
that the Green’s functions of invertible PDE problems are
admissible as RBFs. We have
not
shown that an exists
for arbitrary , and indeed such a statement need not be
true. However, the freedom in choosing boundary condi-
tions for the Green’s function problem, together with the
existence of formal inverses defined in terms of Fourier
methods, can take one a lot further in this direction than
one would initially think possible.
Now suppose in the following that
is
a differential
operator. Then we might consider discretising the system
(12) via a Finite Difference approximation of (more gen-
erally, one could include quadrature for an integro-differen-
tial ). In this discretisation, wherever is known,
is unknown and vice versa. So the discretisation of equa-
tion (12) needs re-arrangement such that all unknowns
appear on the left hand side, and all knowns on the right
hand side.
Next, one imposes boundary conditions (BCs) which
implicitly determine . An example, relevant to ore
grade estimation, might be to require that on a
boundary sufficiently distant from the region of interest.
Numerics
We have just shown that certain simple Kriging, dual
Kriging and RBF interpolations are equivalent to solving a
PDE subject to boundary conditions. We can now employ
one of the fast PDE solvers in order to compute an interpo-
lation.
We solve the discretised version of (12) using the full
multigrid finite difference method. As discussed in (for
example) Press, et al. (1992), this technique solves the
problem on a grid of points in
(13)
operations.
This is independent of the number of control points, and
in fact, the method actually converges faster with increas-
ing density of control points. Compare the scaling behav-
iour (13) with those of the more direct formulations of
Kriging or RBF interpolations: expressions (5), (6), and (7).
As a practical matter, one commonly knows only or
only directly. Determining the missing member of the
pair is straightforward upon consideration of the Fourier
Transform of equation (10) (although the resulting convolu-
tion operator may be unstable).
Our current multigrid implementation has not yet
reached the full potential speed of , due to practical
difficulties involving the representation of control points on
grids of different scales. The scaling behaviour of our cur-
rent implementation is unquestionably better than
and probably better than . We regard this as a
difficulty with our current algorithmic strategy rather than a
fundamental problem with a multigrid approach to interpo-
lation.
Example
As a concrete example of this technique, if ,
is interpreted as displacement, and
as force , then equation (12) would be the PDE
for a two-dimensional thin plate spline (the minimum curva-
ture interpolation of Briggs (1974), common in geophysical
applications). Higher dimensional analogues of thin plate
splines would simply let , and define appro-
priately. Interpolations which are stiffer than thin plate
splines can be found from considering higher powers of
(e.g. the spherical model variogram).
DISCUSSION
This general approach has the obvious capability to put
smooth interpolation onto a physical basis. If an interpola-
tion is required to satisfy (12) where is known from phys-
ical arguments, a direct application of our technique will
yield a physically realistic solution. However, beyond this, it
allows some Kriging interpolations to be calculated quickly
using PDE solution methods.
Kriging methods form the basis of modern conditional
simulation techniques, which seek to characterize the vari-
ability of the predicted quantity in addition to its mean
value. It is interesting to contemplate the relationship
between these conditional simulations, and the PDE formu-
lation. If there is an equivalent PDE formulation, what might
it look like? A Gibbs spatial process perhaps?
Since multigrid is an iterative scheme, given a closed
Lφ
LP
hx()
L
Lu fiδxx
i
()
i
=
u
L
fixi
Phx()
φPhx()
L
φ
L
L
Luxi
() fi
Phx()
ux() 0=
M
OM()
φ
L
OM()
OM
2
()
OM Mlog()
L4
uux()=ffx()=
x2
()
xn
() ∇
4
L
form expression for (and its finite difference representa-
tion) we believe that there is no fundamental reason to
restrict application of this technique to linear . For some
classes of nonlinear PDEs, the only challenge to this gen-
eral scheme should be algorithmic details, and a possibility
of scaling performance that is slightly worse than .
We close with the following remark: since some Kriging
problems can clearly be solved by solving PDEs, we sus-
pect that there are many potential synergies to be found in
applications (such as flow field simulation in petroleum
engineering) where the output from an interpolation speci-
fies the geometry for further physical simulation. Both
aspects of the problem could now be solved from within the
same software base.
APPENDIX 1
RBF Equivalent of Ordinary Kriging
The linear system for simple Kriging arises from a mini-
mization problem. The conditions that the variance is mini-
mized with respect to the unknown parameters , are
expressed as equation (2). For ordinary Kriging, to impose
an unbiased solution, equation (2) must be solved with the
constraint
,(A1)
which also ensures that a constant function is interpolated
exactly. Together, equations (2) and (A1) are solved via the
method of Lagrange multipliers, yielding a new augmented
system (in partitioned matrix format)
. (A2)
We rewrite equation (A2) as
. (A3)
Now, in the Kriging formulation, the value of the interpo-
lation at a point is expressed as
. (A4)
Here, the inner (dot) product between dimensional vec-
tors is denoted by . The augmented vector
must be constructed such that the equality between the left
two terms of equation (A4) is preserved. The only augmen-
tation that preserves that equality is
. (A5)
Given this, we now derive a radial basis function equiv-
alent to the ordinary Kriging system. Start with the ordinary
Kriging expression for the value at some interpolation point
. (A6)
Since we have assumed is invertible, and given that it is
also symmetric, it follows that there exists an such that
. (A7)
These are the radial basis function equations augmented
by a constant offset. This corresponds to the case of Equa-
tion (8) with . This augmented RBF repre-
sentation can be evaluated at to give
. (A8)
But by (A3) we know that this can be rewritten as
. (A9)
Rewriting (A9) as a matrix manipulation, regrouping, and
transposing, we find
. (A10)
Substituting from (A7) it is found
. (A11)
But this is exactly the ordinary Kriging solution for .
Therefore, if there is a solution to the augmented RBF
defined by (A7) then it is equivalent to the ordinary Kriging
solution and vice versa.
Extension for detrending with constrained higher
order moment
Consider an interpolation of a scalar function of .
The augmented RBF system which includes linear func-
tions of the coordinates in , will interpolate any lin-
ear function exactly. The equations determining the and
the three coefficients of are
L
L
OM()
wi
wi
1=
C1
10
w0
µ
b0
1
=
C
ˆw
ˆ0b
ˆ0
=
x0
ux0
() w0u,()
Nw
ˆ0u
ˆ
,()
N1+
==
N
αβ,
αβ(,)
N
u
ˆ
u
ˆu
0
=
ux0
() w
ˆ0u
ˆ
,()
N1+
=
C
ˆ
f
ˆ
C
ˆf
ˆu
ˆ
=
Phx() const=
x0
ux0
() b
ˆ0f
ˆ
,()
N1+
=
ux0
() C
ˆw
ˆ0f
ˆ
,()
N1+
=
ux0
() w
ˆ0C
ˆf
ˆ
,()
N1+
=
ux0
() w
ˆ0u
ˆ
,()
N1+
=
x0
2
Phx()
fi
Phx()
, (A12)
where is a vector of the th coordinates of the set of
control points, . (Here, denotes the th
coordinate at sample point .) Equation (A12) adds further
constraints to the first order moments of the “forces” ,
these being and . This is
like linearly detrending the data and kriging simultaneously.
The resulting interpolation is
, (A13)
where . Clearly, higher orders may
be handled similarly, so we have the dual Kriging/RBF ana-
logue to universal Kriging/Kriging with a trend.
ACKNOWLEDGEMENT
This paper is published with the permission of the
Director of the Australian Geodynamics Cooperative
Research Centre.
REFERENCES
Briggs, I.C., 1974 “Machine Contouring Using Minimum
Curvature,” Geophysics, v. 39, no. 1, pp. 39-48.
Isaaks, E.H. and Srivastava, R.M., 1989 “An Introduction to
Applied Geostatistics,” Oxford University Press, New
York.
Matheron, G., 1980 “Splines and Kriging - Their Formal
Equivalence,” in Down-to-Earth Statistics - Solutions
looking for geological problems, ed. D.F. Merriam, pp.
77-95, Syracuse University Geology Contributions.
Matheron, G., 1982 “Pour une analyse krigeante des don-
nées régionalisées,” Note interne, n°732, CGMM, Fon-
tainebleau, 20 p.
Powell, M.J.D., 1992 “The Theory of Radial Basis Function
Approximation in 1990”, in Advances in Numerical Anal-
ysis, Vol. 2: Wavelets, Subdivision Algorithms, and Ra-
dial Basis Functions, ed. W.A. Light, pp. 105-210, Oxford
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Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flan-
nery, B.P., 1992 “Numerical Recipes in C; The Art of Sci-
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Royer, J.J., and Vieira, P.C., 1984 “Dual Formalism of Krig-
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Co., Dordrecht.
Cc
1c21
c1000
c2000
1000
f
σ1
σ2
µ
u
0
0
0
=
cj
j
xij()
{}
i1N=xij()
j
i
fi
fixi1()
i1=
N
0= fixi2()
i1=
N
0=
ux0
() b
ˆ0f
ˆ
,()
N1+
=
b
ˆ0b0x01()x02()1
=
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The main mathematical techniques used in building geological models for input to fluid flow simulation are reviewed. The subject matter concerns the entire geological and reservoir simulation modelling workflow relating to the subsurface. To provide a realistic illustration of a complete fluid flow model, a short outline of two-phase incompressible flow through porous media is given. The mathematics of model building is discussed in a context of seismic acquisition, processing and interpretation, well logging and geology. Grid generation, geometric modelling and spatial statistics are covered in considerable detail. A few new results in the area of geostatistics are proved. In particular the equivalence of radial basis functions, general forms of kriging and minimum curvature methods is shown. A Bayesian formulation of uncertainty assessment is outlined. The theory of inverse problems is discussed in a general way, from both deterministic and statistical points of view. There is a brief discussion of upscaling. A case for multiscale geological modelling is made and the outstanding research problems to be solved in building multiscale models from many types of data are discussed.
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Problems of scattered data interpolation are investigated as problems in Bayesian statistics. When data are sparse and only available on length scales greater than the correlation length, a statistical approach is preferable to one of numerical analysis. However, when data are sparse, but available on length scales below the correlation length it should be possible to recover techniques motivated by more numerical considerations. A statistical framework, using functional integration methods from statistical physics, is constructed for the problem of scattered data interpolation. The theory is applicable to (i) the problem of scattered data interpolation (ii) the regularisation of inverse problems and (iii) the simulation of natural textures. The approaches of kriging, radial basis functions and least curvature interpolation are related to a method of ‘maximum probability interpolation’. The method of radial basis functions is known to be adjoint to the universal kriging method. The correlation functions corresponding to various forms of Tikhonov regularisation are derived and methods for computing some samples from the corresponding probability density functionals are discussed.
Article
Machine contouring must not introduce information which is not present in the data. The one-dimensional spline fit has well defined smoothness properties. These are duplicated for two-dimensional interpolation in this paper, by solving the corresponding differential equation. Finite difference equations are deduced from a principle of minimum total curvature, and an iterative method of solution is outlined. Observations do not have to lie on a regular grid. Gravity and aeromagnetic surveys provide examples which compare favorably with the work of draftsmen.
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In Applied Geostatistics the authors demonstrate how simple statistical methods can be used to analyse earth science data. In clear language, they explain how various forms of the estimation method called kriging can be employed for specific problems. A case study of a simulated deposit is the focus for the book. This model helps the student develop an understanding of how statistical tools work, serving as a tutorial to guide readers through their first independent geostatistical study.
Dual Formalism of Kriging
  • J J Royer
  • P C G Vieira
  • M Verly
  • A G David
  • A Journel
  • Nato Marechal
  • C Asi Series
Royer, J.J., and Vieira, P.C., 1984 " Dual Formalism of Kriging, " in Geostatistics for Natural Resources Characterization, Part 2, ed. G. Verly, M.David, A.G. Journel, and A. Marechal, NATO ASI Series C: Mathematical and Physical Sciences, Vol. 122, pp 691-702, D. Reidel Pub. Co., Dordrecht.