-
Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
Delineate 3D Iron Ore Geology And Resource Models
Using The Potential Field Method
Authors:
Des FitzGerald, Jean-Paul Chils, 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]
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Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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 Chils
Ecole Des Mines de Paris, Fontainebleau, 77305, France
[email protected]
Antonio Guillen
Intrepid Geophysics, 2/1 Male Street, Brighton, Victoria,
Australia
[email protected]
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Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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.
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Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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 spacethe 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
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Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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
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Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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 Marchal (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.
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Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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 E/Hz or in laymans 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
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Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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 E/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
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Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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.
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Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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.
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Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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 cole des
Mines de Paris was funded by BRGM.
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Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
References Aug C, (2004) Modlisation gologique 3D et
caractrisation des incertitudes par la mthode du champ de
potentiel,
PhD thesis, cole des Mines de Paris, 198 p.
Calcagno P, Chils 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
Chils 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).
Marchal 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 Marchal), Part 1:
271294 (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
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Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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(x) T(x), = 1, , m, all valued to 0. Two
classical choices for x consist in taking either
the point x0 whatever , or the point x1 (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 x. 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(x) / u denote any partial derivative at x and N
denote the total number of such data (in practice N is a
multiple of 3 and the x form triplets of common points). Let us
recall that the x do not necessarily coincide with the x
(the latter are located on the interfaces whereas the former can
be located anywhere).
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Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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
( )* * 01 1
( ) ( ) ( ) ( ) ( )M N T
T T T Tu = =
= +
x x x x x
where the weights and , 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
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 xlll
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 (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 Chils 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 (x) / u with the same
unknown coefficients b 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.
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Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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).
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Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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 CK(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
< =
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)]/ CK 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 CK(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.
-
Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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.
-
Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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.
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Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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.
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Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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.
-
Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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.
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Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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.
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Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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.
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Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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
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Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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.
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Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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.
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Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
Appendix Figures:
Figure A1. Principle of the potential-field method. Top: surface
datapoints at interfaces and structural data; bottom: vertical
cross-section through the 3D model.
-
Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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. X// and
X 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
Y
Z
Y// X//
X
-
Delineate 3D Iron Ore Geology and Resource Models Using the
Potential Field Method
Authors: Des FitzGerald, Jean-Paul Chils, Antonio Guillen
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).