Delineate 3D Iron Ore Geology And Resource Models Using The

Delineate 3D Iron Ore Geology and Resource Models Using the Potential Field Method

Authors: Des FitzGerald, Jean-Paul Chilès, Antonio Guillen

Delineate 3D Iron Ore Geology And Resource Models

Using The Potential Field Method

Authors:

Des FitzGerald, Jean-Paul Chilès, Antonio Guillen

For submission to the Iron Ore Conference 27-29 July 2009

Contact Person: Des FitzGerald 2/1 Male Street, Brighton, Victoria 3186 Australia Telephone: +61 03 9593 1077 Fax: +61 03 9592 4142 Email: [email protected]



Delineate 3D Iron Ore Geology And Resource Models

Using The Potential Field Method

Des FitzGerald (MMICA, FAusIMM)

Intrepid Geophysics, 2/1 Male Street, Brighton, Victoria, Australia

[email protected]

Jean-Paul Chilès

Ecole Des Mines de Paris, Fontainebleau, 77305, France

[email protected]

Antonio Guillen

Intrepid Geophysics, 2/1 Male Street, Brighton, Victoria, Australia

[email protected]



ABSTRACT:

Most 3D geological modelling tools were designed for the needs of the oil industry or detailed mine

planning and are not suited to the variety of situations encountered in other application domains. Moreover,

the usual modelling tools are not able to quantify the uncertainty of the geometric models generated. The

potential field method was designed to build 3D geological models from data available in geology and

mineral exploration, namely the geological map and a digital terrain model (DTM), structural data, borehole

data, and interpretations of the geologist. This method considers a geological interface as a particular

isosurface of a scalar field defined in the 3D space, called a potential field. The interpolation of that field,

based on cokriging, provides surfaces that honour all the data. The 3D model and its parts are always

consistent with the observations.

New developments allow the covariance of the potential field to be identified from the structural data. This

makes it possible to associate sensible cokriging standard deviations to the potential field estimates and to

express the uncertainty of the geometric model. It also, for the first time, gives a statistically optimal,

geologically sound way of interpolating geology, other than directly joining the dots as you do with CAD.

Practical implementation issues for producing 3D geological models are presented: how to handle faults,

how to honour borehole ends, how to take relationships between several interfaces into account, how to

model thin beds over many kilometres, how to optimise lithological properties and how to integrate

gravimetric and magnetic data.

We describe all geology surfaces and volumes using implicit functions. These are then rendered onto the

required sections, plans etc. The estimation of ore-body grades and tonnes, using an unbiased and optimal

geostatistical technique, makes use of the stratigraphically bound 3D geology model.

An application to the geological modelling of the Hamersley Iron Ore district, Australia, is briefly presented.



INTRODUCTION The resource evaluation of an Iron Ore deposit is often performed in three steps: (i) delimitation of the boundaries of

the units corresponding to the various geological formations or ore types; (ii) estimating densities; and (iii) estimation

of grades within each unit. In simple cases (e.g., a series of subhorizontal layers), the geometric model can be built

using 2D geostatistical techniques (kriging or cokriging of the elevations or thicknesses of the various horizons) which

also quantify the uncertainty of the model. A recent paper by Osterholt et al. (2009) shows these steps. A lot of effort

has been undertaken to develop 3D modelling tools capable of handling more complex situations (e.g., Mallet, 2003).

Most of them were designed to fulfil the needs of the oil industry, namely for situations where the underground model

can be mostly defined from seismic data. Deterministic methods are also available to interpolate between subparallel

interpreted cross-sections.

When assessing resources, knowledge of the degree of uncertainty of the estimation is as important as the estimate itself.

Uncertainty on the boundaries and volumes of the various units is often a major part of the global uncertainty. When 2D

geostatistical techniques can be used, the quantification of that uncertainty by an estimation variance is a valuable by-

product of the estimation process (Chiles et al., 1999). In contrast usual 3D modelling tools are not able to quantify the

uncertainty attached to the interpolated model, whereas that uncertainty can be quite large.

The potential field method (Calcagno et al., 2008) was designed to build 3D geological models from data available in

geology and mining exploration, namely: (i) a geological map and a digital terrain model (DTM); (ii) structural data

related to the geological interfaces; (iii) borehole data; (iv) gravity data; and (v) interpretations from the geologist. It is

not limited to sedimentary deposits and does not require seismic data (such data would be useful but are seldom

available in geological, mining, and civil engineering applications).

The potential field method defines a geological interface as an implicit surface, namely a particular isosurface of a scalar

field defined in the 3D space—the potential field. The 3D interpolation of that potential field, based on cokriging,

provides isosurfaces that honour all the data. Recent developments allow the covariance to be determined from the

structural data, which makes it possible to associate sensible cokriging standard deviations to potential field estimates

and to translate them into uncertainties on the 3D model.

In the Appendix A, we cover the basic principle of the method, present the inference of the potential field covariance

from the structural data, and explain how the uncertainty of the 3D model can be quantified. In the body of this paper we

examine several practical issues: how to form a covariance matrix, how to handle faults, how to incorporate lithology

property distributions, how to take relationships between several interfaces into account, how to link 3D geometrical

modelling and inverse modelling of gravimetric and magnetic data. We end with a brief presentation of an application to

the geological modelling of the Hamersley Iron Ore district, Australia, and a short discussion.

Importantly, the dual kriging scheme given in Appendix A, gives a mathematical basis for interpolating geological

observations where the observed mapping contacts and dips/strikes are quite sparse. The interpolation scheme for a

geological series is naturally conformable, yielding realistic 3D surfaces that are close to “balanced”, while following



the geological trends. The geostatistical methods used to achieve this end provide unbiased and optimal interpolation

outcomes.

Geology equations To characterise a geological series using the mathematics of potential fields, (a) each observation of a contact adds one

equation to a “global” matrix, (b) each dip vector contributes three equations, (c) the drift for each fault contributes an

equation and (d) the detrending of geology to support universal kriging adds up to 10 equations. The extra covariance

terms of the system are dominated by the structural data. It is this that makes it possible to use a cubic cokriging model

to translate the standard deviations of the dip directions to potential field estimates.

The system of equations for each series forms a square matrix that is positive definite. It has been solved using Gauss

Elimination. Optimisation efforts include use of a Cholesky vector processing and principal component analysis. The

degree of smoothing of the predicted geological surfaces is directly controlled by the range of the variogram for the

series.

From the recent study by the Geological Survey of Victoria, the regional 3D Bendigo model, an Ordovician series was

modelled using 102 structural observations (3 component vector) and 1,582 contacts.

A principal component analysis shows the total domination of the structural data terms in the interpolator as seen in

Figure 1. There are about 310 equations that are important and the rest make only minor contributions. This

demonstrates the principal that “less is more” when it comes to using geological contacts, depending upon the required

smoothness and the scale of your project.

PRACTICAL IMPLEMENTATION ISSUES

The potential field method has been implemented in GeoModeller (www.geomodeller,com), initially developed by

BRGM (the French Geological Survey) and now commercialised by Intrepid Geophysics. Significant support from a

consortium led by Geoscience Australia has also been shown, with the development of an integrated stochastic,

lithologically constrained geophysical inversion module, and more recently, the addition of geothermal simulation

capabilities.

In order to model real-world situations a number of practical implementation issues had to be solved. Apart from

occasional sedimentary examples, a geological body rarely exists throughout a domain. Geological events usually lead

to complex topology where formations cut across or onlap onto each other as a result of deposition, erosion, intrusion or

hiatus.

Such geology can be modelled by combining multiple potential fields and the use of universal kriging principals.

Modelling several interfaces In practical applications when several interfaces are modelled several potential fields are then used. Overturning of the

geology due to extensive folding, faulting and other processes can be accommodated. The method supports modelling of



realistic 3D geometries of intrusives. The important first step for the geologist is to define a stratigraphic column. This

determines how to combine the various potential fields. The column defines the chronological order of the interfaces as

well as their nature, coded as either "erode" or "onlap" .For example, an "erode" potential field is used to mask the

eroded part of the previous formations.

Figure 2 illustrates the rules for modelling complex geology. Different potential field functions are used for different

geological series. These multiple potential fields are managed using Onlap and Erode relations between series. In this

example each series comprises a single formation.

• Interpolated Formation 1 (basement) and data for potential field of Formation 2.

• Formation 2 interpolated using an Onlap relation and data for potential field of formation 3.

• Formation 3 interpolated using an Erode relation

Faults Several methods can be envisaged to handle faults. If they delimit blocks and the potential field is not correlated from

one block to the other, it obviously suffices to process each block separately. Another conventional technique is to

consider faults as screens. The method used in 3D Geological Editor is the method proposed by Maréchal (1984) to

handle faults in the 2D interpolation of the elevation of interfaces, where faults are entered as external drift functions.

This method requires the knowledge of the fault planes and also of the zones of influence of the faults.

Let us start with a very simple example, a normal fault intersecting the whole study zone and dividing it in two subzones

D and D'. That fault induces a discontinuity of the potential field, whose amplitude is not known. Cokriging can

accommodate that discontinuity whatever its amplitude by introducing a drift function complementing the L polynomial

drift functions, for example

f L+1(x) = 1D(x),

or equivalently, in a symmetric form

f L+1(x) = 1D(x) – 1D'(x).

If the polynomial drift functions include the monomial f 1(x) = x (first coordinate) due to the presence of a linear trend of

the potential field, and we have good reasons to suspect not only a discontinuity but also a change of slope of the drift

when crossing the fault, it is advisable to also introduce an additional drift function such as

f L+2(x) = x 1D(x).

A finite fault (limited extent) is modelled with a drift function with a bounded support. The fault vanishes on the support

boundaries; inside that support, the function takes on positive values on one side of the fault plane, with a maximum at

the centre of the fault, and negative values on the other side. Figure 3 illustrates how that method takes faults into

account. In real-world applications a fault plane is not a planar surface. It is often only known by some points on its

surface and unit vectors orthogonal to it. Its geometry is also modelled by a potential field.



Boreholes The primary use of boreholes in this method is to provide observations of the contacts between different lithologies.

This requires a mapping from the detailed downhole logs to the scale at which you wish to work. Recent work has been

directed at making this much easier for the geologist. Figure 4 shows a borehole log and a corresponding borehole

section through a 3D project and a demonstration where the misfit is less than 0.2% overall. Sometimes a fault may

cross the borehole. An ability to re-interpret the lithological log interactively and add a fault contact can be an

important means of getting the 3D geology interpretation to work.

CHALLENGES FOR MODELLING IRON ORE DEPOSITS As with all thin bed, stratigraphically controlled geological units, the challenges for the project geologist that must be

addressed in the modelling are:

1. Develop a concept for spatial distribution of lithology.

2. Transverse isotropic interpolation of the beds:

An anisotropic covariance is used to model thin beds less than 1 metre thick over a lateral extent covering many

kilometres.

3. Vertical exaggeration during visualisation:

This is important to enable fine tuning of the economic horizons in the context of a large lateral extent.

4. Limited faults:

Local limited faults can be modelled easily and modified to gauge their influence.

5. Forward modelling of the gravitational response of the geology.

Independently observed geophysical datasets are commonly available. They provide a very important means of checking

the model. This includes an ability to model real topography and high rock density units in limited surface relief. One

aim here is to simulate what would be observed from a low flying aircraft with a next generation gravity gradiometer on

board. The other aim is to use ground gravity as an independent tool to check the fit of the model to the ‘reality’.

LITHOLOGICAL PROPERTY ESTIMATION AND MODELLING Both a drilling database and detailed observations of topography and gravity should be used in estimating the density of

the ore-body in the model. We are involved in an on-going study with Rio Tinto Exploration to demonstrate how

sensitive an airborne gravity gradiometer (AGG) needs to be to compete with the accuracy and usability of ground based

gravity acquisition. Existing systems, FALCON, Bell Geospace and Arkex are generally thought to resolve to no better

than 8 Eö/√Hz or in layman’s terms a difference in the gravitational acceleration locally of eight parts in 109 is lost in

the noise.

The setting for these tests is the Pilbara where there is considerable topographic relief (>100m) associated with an

unweathered near-surface iron ore deposit. This buried deposit has a large volume and has a higher density than the

surrounding host rocks.

The airborne gravity survey has an average drape clearance of 80 metres with up to +/- 50 metres near the cliff top. A

detail digital terrain model with a spatial resolution of better than 25 metres and a vertical resolution to +/- 5 centimetres



was used in this study. The aim is to test how sensitive a next generation gravity gradiometer instrument, being flown in

a conventional survey aircraft, needs to be to find and delineate iron ore resources quickly and efficiently. There are

three well advanced teams working on next generation instruments namely, Rio Tinto, Gedex and Arkex. Our work

shows an instrument with an error of around 1 Eö/√Hz would deliver a powerful exploration tool with significantly

improved capability of resolving near-surface density anomalies.

Figure 5a shows the acquired gravity gradient signal before any attempt is made to remove the terrain effects. The data

is processed to continue the signal to a smoothed drape surface that is a good approximation to the average clearance.

This removes all flight line based biases.

A classical “Hammer” method terrain correction is then applied to remove the terrain effects, assuming the background

rock density is 2.67g/cc. Figure 5b then shows the remaining density anomaly map. In this case, the “target” ore-body

is the one shown in the cliff face. Other types of buried targets are present, but are not of interest to the subject of this

paper.

Without doubt, one of the greatest weaknesses in creating 3D geological models to use in both exploration mapping and

resource estimation, is the assigning of realistic lithological properties to the model. Geophysical surveys of gravity

gradiometry has an important part to play here. The integration of density and lithology to produce a detail forward

model of the predicted 3D gravity response of the mapped area is an important check that the model is reflecting

independently observed gravity datasets to an acceptable level. Importantly you do not have to assume homogeneity of

properties as you also have a 3D geology model to help interpret your data.

Estimating tonnes and grade Once the various lithological units have been delimited, we have to tackle the estimation of grade and tonnage. This can

be done with geostatistical techniques, namely kriging, or better with cokriging in order to simultaneously and

consistently estimate the iron grade and the grade of by-products and penalty substances.

At the local scale (e.g. a core or a small block) the ore tonnage is the product of ore volume by ore density; similarly, the

metal tonnage is the product of ore tonnage by ore grade (expressed in weight percentage). In the case of an iron ore

deposit there is a high correlation between ore grade and ore density, which shall be taken into account. If the

measurements include ore density and ore grade for all the samples, it suffices to work with the volumetric grade,

namely the product of ore density by ore grade in %. Otherwise, the correlation between density and grade shall be

studied. In both cases we have to model ore density in order to estimate the ore tonnage (See Figures 6 and 7). This shall

be done on the basis of the data available at different scales (geophysical interpretation of gravity data, analysis of bulk

samples or core samples, etc.). Multivariate geostatistics provides tools for that integration (support change modelling,

cokriging, external drift kriging, etc.).

An important issue for the sound application of geostatistics is the correct modelling of spatial correlations. In sub

horizontal deposits, the lateral grade variations are usually much smoother than the vertical ones, and the variogram

analysis considers the horizontal variogram and the vertical variogram. In more complex layered deposits, the analysis



of spatial correlations shall consider the variations along the layers and orthogonal to them. This is done by

'horizontalizing' the data. The fact that the geological model has been built with the potential field approach provides a

consistent means to perform that step. For example, if a layer is defined by two potential values of a common potential

field, the value of the potential at any point in the layer can be used as a new vertical coordinate. In the system defined

by the original horizontal coordinates and the new vertical coordinate, the main anisotropy directions are the horizontal

and vertical directions, so that the analysis of spatial correlations can be carried out in the usual way. Kriging can be

done in that system and then exported in the original physical coordinate system.

APPLICATION TO THE HAMERSLEY DISTRICT The geological scale and purpose of the model can vary enormously. GeoModeller has been used for regional scale

geological modelling of the Alps and the Massif Central in Europe. For example, Maxelon (2004), and Maxelon and

Mancktelow (2004), used it to model foliation fields and a juxtaposition of nappes with a strong folding in the Lepontine

Alps.

Australian regional cases include the Gawler Craton, Bendigo, Burdekin 3D studies.

At the mining scale, the Broken Hill, Guillen et al.(2004), Bendigo Gold Mine, San Nicholas, Lane (2008) and Peruvian

Andes studies indicate the diversity and complexity of the geological environments. This method has been applied to the

Hamersley region by various groups. The recent paper by Osterholt et al. (2009) details how BHP Billiton in

association with SRK are routinely using the method to report exploration target size and type with ranges of uncertainty

compliant with the Australasian Code for Reporting Mineral Resources and Ore Reserves (the JORC code). Figure 8 is

reproduced with permission from this paper. Data during early evaluation work is usually sparse and historically not

sufficient to support public reporting of resources. They give a methodology to address the uncertainty using an holistic

view to develop:

1) geology and grade scenarios,

2) 3D geology modelling to create the volumes and

3) grade modelling.

For this paper, we report on some work done in the Hamersley to build a 3D model using these methods, using the

Stratigraphic Pile shown in Figure 9. A study area 5 km by 2 km by 1 km was chosen. The iron ore bearing formations

are folded and faulted and then overlain by colluvium or recent sediments as shown in section, Figure 10. The beds are

extensive laterally. The desire to model thin beds over an extensive area was one of the study objectives. Vertical

exaggeration of up to 3 to 1 assists in this task.

Many similar sections are created and interpreted, as well as the geology at the topographic surface. Borehole lithology

data is also used to constrain the third dimension, as each of the formations is modelled. The 3D model is realised by

calculating each of the series independently and then applying the onlap/erode rules to resolve the final layout.

Figure 11 shows a 3D perspective view of the geological model. The colours correspond to the geological units shown

in Figure 10. The presence of three longitudinal faults is clearly seen.



Figure 12 shows the near surface gravity response that would be expected from the model.

This is a very useful independent check that the model and observed gravity are in close agreement

There is a desire to extend the sensitivity of geophysical instruments to enable a better realisation of density anomalies

and the geometries of ore-bodies, Figure 13 shows what might be expected if a Full Tensor gravity gradiometer was

used in this area. The signature of the faults in the gravity is weak. This is where a full tensor magnetic gradiometer

system would help.

Recent extensions to the GeoModeller technology include

• An integrated borehole, conventional geostatistical capability as described above. This initiative is being under

taken in association with Geovariance.

• Speed and detail enhancements to the prediction of the geophysical responses by using 3D Fast Fourier Transform

technology.

• Batch scripting for the high fidelity rendering of geological contacts and faults.

• Predicting the temperature gradients based upon thermal conductivity properties and heat production rates.

POTENTIAL FIELD METHOD SUMMARY The method presented here is designed for 3D geological models of ore deposits built from interface points and

polarised orientation data. The methodology is designed for cases where the geology is known at sparse locations, e.g.

when data are available on the surface but not at depth. The orientation data, i.e. dip measurements, are not necessarily

located on the geological interfaces. They can represent stratifications or foliations related to the contacts. Data are

interpolated through a potential field implicit function continuously defined in the entire 3D domain. Thus, the model

predicts the geological formation at any 3D point.

Geological interfaces in the model are particular isosurfaces extracted from the potential field. They may have any kind

of 3D geometry: multilayer type, recumbent folds, complex intrusions, etc.

The geometry of faults is computed by applying the same method. Faults can be infinite within the 3D domain,

interrelated in a fault network, or finite.

The throw of the faults are predicted from the other field observations and do not need to be modelled in detail.

Anisotropic interpolation of thin beds allows the geologist to control the geological sequence over many kilometres with

sparse data observations. Inequality constraints such as a borehole finishing within granite are also handled using a

“Gibbs” iterative solver.

Geological rules are defined to model complex geology where formations onlap onto or erode another. These rules are

also used to automatically assign the right geological interface between two consecutive formations. This methodology

automatically provides the intersections between geological units, enables fast modelling and allows the geologist to

focus on geological interpretation.

As the geological pile defines the topology of the model, one can modify it without changing the basic data to produce

alternative interpretations and geometries. This capability makes it possible to progressively update the model when new

data or interpretation is available. It is this ability to quickly realize several scenarios that has found favour with the

BHP Iron Ore group.



Future Work GeoModeller has a very active development program. In a short time frame it is expected that (1) inferred apparent dip

of structures from a seismic section will be supported, (2) thin bodies similar to the current fault modeling will be

supported, (3) faults will displace faults, predicting their throw, (4) simulation of geological and geothermal

uncertainties will be formalized, (5) data rich portions of the project show higher fidelity and (6) geostatistics for

property, tonnes and grade can be made via a direct ISATIS plug-in.

Another future possible extension of the fundamental approach outlined here concerns the geological gradient. The

gradient of a random function is rarely a unit vector. GeoModeller treats the structural data as a unit vector ignoring the

“strength” of the trend. The ideal would be to sample both a structural direction and a structural intensity, but this is

possible only in very specific cases. Aug (2004) has shown on simulations of actual situations that replacing actual

gradients by unit vectors usually has a minor impact on the determination of the covariance and the cokriging. A useful

improvement of the method may be to extend the interpolation to support an optional “strength or intensity” value.

CONCLUSION Both geology and geophysics practice needs re-engineering to simplify the identification of buried economically

significant resources. The new geoscience framework includes:

1. Quantitative and repeatable geology in 3D. The decisions are ‘what scale’ and ‘what purpose’.

2. Airborne systems that deliver gravity and magnetic signatures of rocks 10 times more precisely than 1980

technology. The key here is driving noise from instruments towards 1 Eotvos or 100 pico Tesla per meter (pT/m).

3. Appropriately built 3D geophysical simulation models from the geology to help create the ‘right’ interpretations.

This sensible joining of the disciplines of structural geology interpretation, resource estimation and computational

geophysics provides a novel method for increasing the productivity of senior geoscientists leading to faster and better

3D modelling of orebodies. The integration of gravity and gradiometry provides independent checking for the model

and helps to constrain the economic geology.

The rapid delineation of the iron ore resources, using an implicit lithology model based upon all mapping and sparse

drilling provides estimates that are much closer to the JORC (Joint Ore Reserves Committee) code spirit than just using

polygonal based estimates.

Acknowledgements Theo Aravanis of Rio Tinto Exploration initiated work on sensitivity studies for detecting buried iron ore deposits using

gravity gradiometry.

BHP Iron Ore kindly allowed the inclusion of their exploration resource model.

Geological Survey of Victoria has allowed the mention of the Bendigo 3D geological model discussion.

The Australian Government has funded Intrepid Geophysics via a Commercial Ready Grant.

The 3D FFT work was stimulated by Jeff Phillips of the USGS.

The geostatistical research work carried out at the École des Mines de Paris was funded by BRGM.



References Aug C, (2004) Modélisation géologique 3D et caractérisation des incertitudes par la méthode du champ de potentiel,

PhD thesis, École des Mines de Paris, 198 p.

Calcagno P, Chilès JP, Courrioux G, Guillen A, (2008) Geological modelling from field data and geological knowledge.

Part I. Modelling method coupling 3D potential field interpolation and geological rules, Physics of the Earth and

Planetary Interiors

Chilès J P, Delfiner P (1999) Geostatistics: Modelling Spatial Uncertainty. 695 p (Wiley: New York)

Lane R, McInerney P, Seikel R (2009) Using a 3D geological mapping framework to integrate AEM, gravity and

magnetic modelling – San Nicolas case history in Proceedings ASEG 18th Geophysical Conference and Exhibition.

McInerney P, Golberg A, Holand D (2007) Using airborne gravity data to better define the 3D limestone distribution at

the Bwata Gas Field, Papua New Guinea, Proceedings ASEG 18th Geophysical Conference and Exhibition, Perth.

Mallet JL (2003) Geomodelling pp 599 (Oxford: Oxford).

Maréchal A (1984) Kriging seismic data in presence of faults. in: Geostatistics for Natural Resources Characterization

(Eds: G Verly, M David, A G Journel and A Maréchal), Part 1: 271–294 (Reidel: Dordrecht).

Osterholt V, Herod O, Arvidson H (2009) Regional Three-Dimensional Modelling of Iron Ore Exploration Targets,

Proceedings AusIMM Orebody Modelling and Strategic Mine Planning 2009



APPENDIX A

BASIC PRINCIPLE OF THE POTENTIAL FIELD METHOD The basic method is designed to model a geological interface or a series of subparallel interfaces Ik, k = 1, 2, …

(Calcagno et al., 2008). The principle is to represent the geology by a potential field, namely a scalar function T(x) of

any point x = (x, y, z) in 3D space, designed so that the interface Ik corresponds to an isopotential surface, i.e. the set of

points x that satisfies T(x) = tk for some unknown value tk of the potential field. Equivalently, the geological formation

encompassed between two successive interfaces Ik and Ik' is defined by all the points x whose potential field value lies in

the interval defined by tk and tk'. In figurative terms, in the case of sedimentary deposits T could be seen as the time of

deposition of the grain located at x, or at least as a monotonous function of that geological time, and an interface as an

isochron surface.

Data types T(x) is modelled with two kinds of data, as shown in Figure A1:

(i) Points known to belong to the interfaces I1, I2, …, typically 3D points discretizing geological contours on the

geological map and intersections of boreholes with these interfaces; and

(ii) Structural data: in the case of sedimentary rocks the stratification is parallel to the geological horizons. We measure

a unit vector normal to the stratification. They can also be unit vectors orthogonal to foliation planes for metamorphic

rocks. Measurements are made on outcrops or in boreholes, either on the interfaces or anywhere within a formation.

For the interpolation of the potential field, these data are coded as follows:

(i) Since the potential value at m + 1 points x0, x1, …, xn sampled on the same interface is not known, these data are

taken as m increments T(xα) – T(x’α), α = 1, …, m, all valued to 0. Two classical choices for x’α consist in taking either

the point x0 whatever α, or the point xα–1 (the choice has no impact on the result; other choices are possible provided

that the increments are linearly independent). Since the sampled data can be located on several interfaces, let M

represent the total number of increments (it is equal to the total number of data points on the interfaces minus the

number of interfaces).

(ii) The unit vector normal to each structural plane is considered as the gradient of the potential field, or equivalently as

a set of three partial derivatives ∂T(x) / ∂u, ∂T(x) / ∂v, ∂T(x) / ∂w at some point xβ. The coordinates u, v, w are defined in

an orthonormal system; this system can be the same for all the points or a specific system can be attached to each point

(the result does not depend on the choice provided that the three partial derivatives are taken in consideration). In the

sequel let ∂T(xβ) / ∂uβ denote any partial derivative at xβ and N denote the total number of such data (in practice N is a

multiple of 3 and the xβ form triplets of common points). Let us recall that the xβ do not necessarily coincide with the xα

(the latter are located on the interfaces whereas the former can be located anywhere).



Interpolation of the potential field The potential field is then only known by discrete or infinitesimal increments. It is thus defined up to an arbitrary

constant. So an arbitrary origin x0 is fixed and at any point x the potential increment T(x) – T(x0) is kriged. The

estimator is in fact a cokriging of the form

( )* *0

1 1

( ) ( ) ( ) ( ) ( )M N T

T T T Tuα α α β β

α= β= β

∂′− = µ − + ν

∂∑ ∑x x x x x

where the weights µα and νβ, solution of the cokriging system, are in fact functions of x (and x0). One may wonder why

the potential increments are introduced in that estimator since their contribution is nil. The key reason is the weights νβ

are different from weights based on the gradient data alone. Conversely, the gradient data also play a key role, because

in their absence the estimator would be zero for any x.

Cokriging is performed in the framework of a random function model. T(.) is assumed to be a random function with a

polynomial drift

0

( ) ( )L

m b f=

=∑x xl

ll

and a stationary covariance K(h). Since the vertical usually plays a special role, the degree of the polynomial drift can be

higher vertically than horizontally and the covariance can be anisotropic. For example, if we model several subparallel

and subhorizontal interfaces, it makes sense to assume a vertical linear drift of the form m(x) = b0 + b1 z, i.e. with two

basic drift functions f 0(x) ≡ 1 and f 1(x) = z. A geological body with the shape of an ellipsoid would correspond to a

quadratic drift, i.e. to the 10 basic polynomial coefficients with degree less than or equal to 2.

Once the basic functions f ℓ(x) of the drift and the covariance K(h) of T(.) are known, we have all the ingredients to

perform a cokriging in the presence of gradient data, as shown in Chilès and Delfiner (1999, section 5.5.2). Indeed, the

drift of ∂T(x) / ∂u is simply ∂m(x) / ∂u, i.e. a linear combination of the partial derivatives ∂f ℓ(x) / ∂u with the same

unknown coefficients bℓ as for m(x), the covariances of partial derivatives are second-order partial derivatives of K(.),

and the cross-covariances of the potential field and partial derivatives are partial derivatives of K(.).

Implementation of the cokriging algorithm Since the potential increment data in fact do not contribute to the final cokriging estimate, the estimator can be seen as

an integration of the gradient data. To preserve the spatial continuity of the cokriging estimates it is wise to work in a

unique neighbourhood, namely to effectively include all the data in the cokriging of T(x) for every x. If we are not

interested in the cokriging variance, cokriging can be implemented in its dual form, which has two advantages: (i) the

cokriging system is solved once, (ii) that form is especially suited when cokriging is considered as an interpolator,

because it allows an easy estimation of T(x) – T(x0) at any new point x. The latter property is very useful to display 3D

views of the geological model with an algorithm such as the marching cube, which starts from the estimation of T(x) –

T(x0) at the nodes of a coarse regular grid and then requires intermediate points to be predicted to track the desired

isopotential surface.



INFERENCE OF THE COVARIANCE OF THE POTENTIAL FIELD In usual geostatistical applications, the covariance or variogram of the variable under study is modelled from the sample

variogram of the data. In the present case, we have few measurements of the potential T(x), and the potential increments

used for the interpolation cannot be used for the inference of K since they all have a zero value. The choice of the model

followed from these considerations:

(i) At the scale considered, geological interfaces are smooth rather than fractal surfaces which implies that the

covariance is twice differentiable. A cubic model was considered a good compromise among the various possible

models, because it has the necessary regularity at the origin, and a scale parameter (the range) which can accommodate

various situations.

(ii) The scale parameter a and sill C of the covariance K(h) determine the sill of the variogram of the partial derivatives:

it is equal to 14 C / a2 in the case of an isotropic cubic covariance considered here. When there is no drift and the

geological body is isotropic (e.g., a granitic intrusion), the unit gradient vector can have any direction so that its variance

is equal to 1. The variance of each partial derivative is then equal to 1/3. A consistent choice for C once the scale

parameter a has been chosen is thus C = a2 / 42. That value shall be considered as an upper bound for C when the

potential field has a drift, because in that case the mean of the potential gradient is not equal to zero so that its variance

is shorter than 1 (its quadratic mean is 1 by definition).

(iii) Sensible measurement variances can also be defined (nugget effects).

The assumption of an isotropic covariance model is too restrictive and can be relaxed. In practice the covariance K(h) is

supposed to be the sum of several cubic components Kp(h), each one possibly displaying a zonal or geometric

anisotropy. To avoid too much complexity, the main anisotropy axes u, v, w, are common to all the components of a

series.

Thanks to these formulae the covariance parameters of K (nugget effect, scale parameter of each covariance component

in the three main directions, sill of each component) are chosen so as to lead to a satisfactory global fit of the directional

sample variograms of the three components of the gradient. An automatic fitting procedure based on the Levenberg-

Marquardt method has been developed to facilitate that task (Aug, 2004).

Figure A2 shows an example of such a fitting. One thousand, four hundred and eighty-five structural data were sampled

in an area of about 70 × 70 km2 in the Limousin (Massif Central, France). The main (u, v, w) coordinates here coincide

with the geographical (x, y, z) coordinates. Since the structural data are all located on the topographic surface, the

variograms have been computed in the horizontal plane only. Note that the sill of the variogram of the vertical

component is much lower than that of the horizontal components. This is due to the fact that the layers are subhorizontal

so that the vertical component of the gradient displays limited variations around its non-zero mean. The model K

includes three components, the second of which only depends on the horizontal component of h and the third one on the

N-S component (zonal anisotropies).



UNCERTAINTY ON THE 3D MODEL Case studies have shown that the use of a sound covariance model improves the model in comparison with the use of a

conventional model. An additional interest in using a covariance fitted from the data is the possibility of obtaining

sensible cokriging standard deviations.

When the "true" covariance of the potential field is known, a meaningful cokriging standard deviation σCK(x) can be

associated with the cokriging of T(x) – T(x0). The calculation of that standard deviation requires the use of the standard

form of the cokriging system, which calls for more computing time than its dual form (this is the price to pay for

knowing the uncertainty attached to the geological model). Let us suppose that some geological formation is defined by

the set of points x such that T(x) – T(x0) is comprised between two values t and t'. Assuming that the potential field is a

Gaussian random function, an assumption which seems reasonable in the present context, the probability that a given

point x belongs to that formation is

{ }* * * *

0 00

CK CK

' ( ( ) ( )) ( ( ) ( ))Pr ( ) ( ) '

( ) ( )

t T T t T Tt T T t G G

σ σ − − − −

≤ − < = −

x x x xx x

x x

where G is the standard normal cumulative distribution

function.

Similarly, if we are interested in the interface passing through the point x0, namely in the set of points x such that T(x) –

T(x0) = 0, the variable R(x) = [T*(x) – T*(x0)]°/ σCK measures the likelihood that x belongs to the interface. Indeed,

writing the obvious relation

T(x) – T(x0) = T*(x) – T*(x0) + cokriging error

we see that x belongs to the interface if and only if T*(x) – T*(x0) is equal to minus the cokriging error, or equivalently if

R(x) is equal to minus the standardised cokriging error (the ratio of the error by σCK(x)). The value of that error is not

known but it is a variable with zero mean and unit variance.

For example, assuming again that the potential field is Gaussian, the area defined by |R(x)| < 2 includes about 95% of

the actual interface. Figure A3 displays R(x) for the top of the lower gneiss unit in the Limousin.

The black line corresponds to R(x) = 0, i.e. to the isovalue surface of the cokriged potential field passing by the data

points sampled on that interface. The true interface is likely to be found in the light-coloured area, whereas the darkest

area can be considered as a forbidden area. This capability is not routinely made available within GeoModeller.



Figures

Figure 1. A principal components analysis of the dual kriging equation system used to interpolate the Ordovician Units in the 3D Bendigo Model. The first 310 components are derived from structural observations and the rest are the geological contacts.



Figure 2. Complex geology is modelled using different potential-field functions for different geological series. These multiple potential fields are managed using Onlap and Erode relations between series. In this example each series comprises a single formation. (a) Interpolated Formation 1 (basement) and data for potential field of Formation 2. (b) Formation 2 interpolated using an Onlap relation and data for potential field of formation 3. (c) Formation 3 interpolated using an Erode.



Figure 3. Handling faults. Top: data points located on two interfaces and structural data; middle: model built without introducing any fault; bottom: model taking faults into account.



Figure 4. A reconciliation of the borehole lithology log against the 3D model. The dual kriging technology knows the data to better than 0.2%, whilst also accommodating all surface mapping etc.



Figure 5a. Simulation of the cross line gravity gradient terrain response of DTM from LIDAR data, 10m cell size.

Figure 5b. Terrain corrected gravity shows the same small escarpment now with an embedded high density “iron ore” deposit.



Figure 6 Experimental Variogram is derived directly from the 3D point data. The model variogram is then used to interpolate in 3D, the estimated quantity.

Figure 7. Cut-off grade can be imposed by selecting via a histogram, the portion of the population on interest.



Figure 8. Geological section of Brockman iron-formation hosted orebody (from unpublished internal BHP Billiton report). This together with sparse drillhole dta is used to capture geological uncertaintly in grade-tonnage estimates, using the Potential Method.

Figure 9. Representative section of a 3D model created to model interaction of folding and faulting on the Brockman formation. Vertical exaggeration is set to 2:1.



Figure 10. Geological units and the relationships, showing the onlap or erosional relationship between different series.

Figure 11. Plan view of the geological model. The colours correspond to the geological units shown in Figure 10. The presence of longitudinal faults are clearly seen. The project covers an area 5 km by 2 km by 1 km thick



Figure 12. Forward model of the vertical gravity component (Gz) at a fixed elevation above the plan view of the Iron Ore geological model. This is normally what is collected on the ground. Units are mGals.



Figure 13. Forward model of the gravity gradient Gzz (top)and Gyz (bottom) at a fixed elevation above the plan view of the Iron Ore geological model. Units are Eotvos. This is the vertical and north gradients of the usual gravity measurement. There is not a clear expression of the faults in these images. The gravity gradient data indicates the folded nature of the higher density rocks.



Appendix Figures:

Figure A1. Principle of the potential-field method. Top: surface data—points at interfaces and structural data; bottom: vertical cross-section through the 3D model.



Figure A2. Example of fitting of the covariance of the potential field from the sample variograms of the partial derivatives of the potential field. Limousin dataset, Massif Central, France. γX// and γX┴ denote the variogram of the partial derivative ∂T/∂x respectively along and orthogonally to direction x.

0 5000 10000 15000 20000 25000

0.0

0.1

0.2

0.3

Variogramme

Distance

γγγγY┴

γγγγZ┴

γγγγY//

γγγγX//

γγγγX┴



Figure A3. Representation of the uncertainty of the top of a geological unit by the variable R(x) (upper gneiss unit, Limousin). The data (geological map and structural data) are all located on the topography. Top: map of a zone of 65 km × 65 km in the horizontal plane with elevation 500 m ; bottom: vertical E-W cross-section with 62 km extension and 34 km depth. The black curve represents the kriged interface. The true interface is in fact in the coloured zones, with a smaller probability as the zone is darker. The darkest zones can be considered as exclusion zones. After Aug (2004).

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