The Topology of Geology 1: Topological Analysis
Samuel T. Thielea,*, Mark W. Jessella, Mark Lindsaya, Vitaliy Ogarkoa, J. Florian Wellmannb, Evren
aCentre for Exploration Targeting (M006), School of Earth and Environment, The University of
Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia
bRWTH Aachen University, Graduate School AICES, Schinkelstr. 2, 52062 Aachen, Germany
*Corresponding Author: email@example.com (S. T. Thiele)
Topology has been used to characterise and quantify the properties of complex systems in a diverse
range of scientific domains. This study explores the concept and applications of topological analysis
in geology. We have developed an automatic system for extracting first order 2D topological
information from geological maps, and 3D topological information from models built with the Noddy
kinematic modelling system, and equivalent analyses should be possible for other implicit modelling
systems. A method is presented for describing the spatial and temporal topology of geological
models using a set of adjacency relationships that can be expressed as a topology network, thematic
adjacency matrix or hive diagram. We define three types of spatial topology (cellular, structural and
lithological) that allow us to analyse different aspects of the geology, and then apply them to
investigate the geology of the Hamersley Basin, Western Australia.
Keywords: Topology, Geological Modelling, Connectivity, Model characterisation, Visualisation
Topology – the relationships between discrete elements of a model – is an important constraint for
many geological processes, including deformation and the flow of fluid, heat and electricity. It is
commonly considered, though often not explicitly, when evaluating the economic value of a region
(e.g., Allan, 1989; Knipe, 1997; Pouliot et al., 2008), planning development projects (e.g., Jing and
Stephansson, 1994; Yu et al., 2009) and assessing geohazard risk (e.g., Okubo, 2004).
Geologists have been building 3D geological models of the Earth for over a century. The first models
were physical, built of wood and metal, and were conceived as tools to analyse and communicate
the geometric shapes of subsurface geological features (Anderson, 1884; Barringer, 1892; Cadman,
1927). Physical models were still being designed at the dawn of the digital era (Anstey, 1976). Once
digital 3D geological modelling took over from its physical predecessors, their usefulness expanded
from the still-challenging task of 3D geological visualisation (Tipper, 1976), to providing inputs for
geophysical inversion schemes (Cordell and Henderson, 1968), as the basis for geostatistical analysis
(Pyrcz and Deutsch, 2014), and as the geometric framework for process simulations (Bear and
Cheng, 2010; Bundschuh, 2010).
Topological analysis has also proved useful in diverse scientific domains, including the prediction of
grain-boundary evolution in metals and rocks (Von Neumann, 1952; Glazier, 1993); the evolution of
geomorpological systems (Löwner and Becker, 2013); the nature of the internet (Faloutsos et al.,
1999); comparisons of complex organic molecules (Brohée et al., 2008) and even the form of the
universe (Zeeman, 1964). This paper provides a brief review of topological theory and its
applications in geology, before developing a framework for characterizing, visualising and analysing
the topological properties of geological models and regions and investigating the fundamental
topology of common geological structures.
2. Topological Theory
Topology refers to the properties of space that are maintained under continuous deformation, such
as adjacency, overlap or separation (Crossley, 2006). Egenhofer and Herring (1990) define a set of
eight binary topological relationships that occur between two-dimensional geometries in two-
dimensional space (Fig. 1). These eight also describe the possible relations between three-
dimensional objects in three-dimensional space, although a further 61 relations are possible
between 0, 1, 2 and 3-dimensional objects in three-dimensional space (Zlatanova, 2000).
Figure 1. Eight binary topological relationships between two-dimensional objects A and B, as defined
by Egenhofer and Herring (1990).
In simple three-dimensional geological models, meets (adjacency) is generally the most common
Egenhofer relation, as overlaps and gaps are not normally desired, although obvious exceptions
could be gradual sedimentary facies changes, overprinting alteration haloes or metamorphic zones.
Burns (1988) describes an extensive framework for representing the topology of geological models,
using network diagrams (graphs) in which nodes represent geometric elements in different
dimensions (Fig. 2) and arcs represent adjacency relationships (Fig. 3). Under this framework, several
orders of topology are possible, depending on the dimension of the geometric elements for which
adjacency relationships are considered. In three dimensions, for example, it is possible to define
three different orders of topology: a 1st order topology describing adjacent rock volumes (i.e. cells
that share an interface), a 2nd order topology defined by relationships between interfaces (across
edges), and a 3rd order topology representing the adjacency relationships between edges (as defined
by shared vertices). Using this notation, 1st order topology describes geological contact relationships
regardless of the spatial dimensions used to represent them geologically, and higher order
topologies become available as more degrees of freedom are allowed by higher dimensional
representations (Fig. 3).
Figure 2. (a) Geometric elements of a geological model in one, two and three dimensional space, as
defined by Burns (1988). (b) Tabular representation of the same elements. Note that the diagonals
(arrows) show the ‘equivalent’ representation of each element in lower dimensions.
Figure 3. Examples of different orders of topology networks in one (a) and two (b) dimensions. The
order of each topology network depends on the dimension of the geometric element represented by
each node in the network. Edges in this example represent spatial adjacency.
Several authors (Valentini et al., 2007; Sanderson and Nixon, 2015) have used 2nd and 3rd order
topology graphs to describe fracture networks in two and three dimensions. These studies use
graphs where the fractures ('boundaries' in two dimensions and 'interfaces' in three dimensions,
under the terminology of Burns, 1988) are represented as arcs and their intersections (junctions or
edges) by nodes. The number of arcs attached to each node (node degree) can then be used to
characterize the connectivity of the network, and hence potential for fluid communication. Valentini
et al. (2007) applied these methods to fracture networks ranging from the microscale (in a single
olivine grain) to macro-scale (in a ~5 × 10m outcrop), and Sanderson and Nixon (2015) extended the
method to three dimensions.
2nd and 3rd order topologies of geological models have also been used by many authors (e.g., Wu,
2004; Ellul and Haklay, 2006; Ming et al., 2010) presenting methods for maintaining valid surface
geometry (i.e. ensuring models are ‘watertight’ and do not contain self-intersections) during model
construction. This work, while important, relates more to the problem of efficiently and flexibly
representing geometry in a computer, rather than the properties of a geological system, and will not
be discussed further. Analysis of 2D geological models (i.e. maps, Shi and Liu, 2007) and 3D
geological models (Pouliot, et al., 2008) overlaps with GIS analysis, and in particular the use of
buffers to define proximity to geological features in prospectivity analysis (Bonham-Carter, 1994).
Similarly, Wang et al. (2016) present a detailed theoretical framework and data model for
representing geological objects in three-dimensional space based on topological theory. This
contribution is significant as, if implemented, it would allow much easier access to, and hence
analysis of, topological relationships in geological models.
To date, little specific work has been developed for the 1st order topological manifestation of
common geological structures and geological models, even though many types of ore deposits are
controlled by topological relationships such as stratigraphic and intrusive contacts, unconformities
and fault intersections (e.g., Naldrett, 1999; Hildenbrand et al., 2000; Sillitoe, 2010). Burns (1975)
and Burns and Remfry (1976) have used 1st order adjacency relationships derived from map sheets
to automatically determine partially constrained geological histories. Burns et al. (1978) extended
the method to automatically infer possible histories implied by structural and metamorphic
relationships observed in the field. This outcome was achieved by describing temporal relationships
between instantaneous geological events with six binary relations, which we have extended to eight
relationships in Figure 4 to account for events with a finite duration in time. These relations are
conceptually similar to the eight spatial topological relationships of Egenhofer and Herring (1990),
and can be inferred from field observations and then sorted into chronological order using an
algebraic method (cf. Burns, 1975; Potts and Reddy, 1999). The temporal relationships used by these
authors can be represented by a graph in which geological events are nodes and the (temporal)
relationships between them define directed arcs. As geology, and hence a geological model, is
fundamentally determined by its history, usually represented via a combination of the stratigraphic
and structural legends, the temporal relationships between units, structures and events within a
geological model can also be considered as part of its topology.
The brief review above should make it clear that the concept of topology in geological models is
complex and multi-faceted, and that different topological orders can be used for different purposes.
This study focuses on the manifestation of 1st order topologies in geological models (i.e. 3D spatial
relationships between neighbouring cells), although aspects of other types of topology are included
in the discussion, and much of the described methodology can also be applied to lower order
topological relationships. For the rest of this study, cellular topology refers to the 1st order topology
of three-dimensional cells defined by Burns (1988).
Figure 4. (a) Eight binary relations that can be used to describe the relations between geological
events. Relations 1, 4, 5, 6, 7 and 8 were defined by Burns (1975) and used to algebraically constrain
geological history from map relationships and field data. Relations 2 and 3 have been added in order
to complete the set. Field examples of these relations include: (b) an intrusive contact between a
quartz vein and metasediment; (c) a stratigraphic contact between a shale and sandy unit; (d) a
tension vein that is truncated by shear vein, suggesting that the tension vein is either offset by or
coeval with the shear vein; (e) two en-echelon veins, and; (f) an ambiguous contact between two
different intrusive rocks.
3. Automatic calculation of first order topology from geological models
The topology networks for geological models can be calculated automatically by dividing the model
space into topological volumes and then extracting the relationships between these volumes. A
computer method for performing these calculations on voxel models (Fig. 5) has been implemented
as part of the pynoddy project (Wellmann et al., 2015), available at
https://github.com/flohorovicic/pynoddy. This methodology currently only works with Noddy
models, however the method could be implemented in any modelling scheme providing it is possible
to identify the lithology of any point within the model and its location (left, right, inside) with respect
to modelled structures.
We use a similar method to extract the topology of a two-dimensional geological map, represented
by a polygon shapefile describing lithology and a polyline shapefile describing structure (see section
2.4). For simplicity, we ignore structurally and geographically isolated lithology domains in this
instance, as they may maintain connectivity in three-dimensions, and hence only calculate adjacency
relationships between the lithologies. This type of simplified topology based only on lithological
relationships is discussed further in Section 5.1.1.
Figure 5. Methodology used to extract cellular topology networks from voxel models. Voxels are first
given codes according to their location relative to structures in the model (a). These codes are then
aggregated to give a distinct code for each structurally bound rock volume (b), and separate codes
defined representing the lithology of each voxel (c). A ‘flood fill’ algorithm then recursively joins
adjacent voxels with matching codes to identify each discrete volume (cell), coding the voxel
accordingly. Adjacency relationships between these cells can then be identified at the voxel level and
used to construct a topology network (e).
4. Topological Expressions of Geological Structures
Different types of geological structures manifest topologically in different ways, depending on the
physics of the processes creating them and the state of the prior geology. While the specifics of a
structure’s topological manifestation are heavily dependent on its geometry and evolution, some
general characteristics can be observed. In this section, these characteristics are described using
simple synthetic examples, and then applied to identify unconformity structures in a map sheet of
the Mount Bruce area, Western Australia.
4.1. Simple Examples
The simplest non-arbitrary cellular topology is a layer-cake stratigraphy, where each unit only shares
contacts with the strata directly above and below it, and hence its topology can be represented by a
series of linked nodes (a node chain, Fig. 6a). Variations of layer thickness and bedding orientations
will not affect this topology, providing all units are conformable.
Similarly, concordant strata above an unconformity form a node chain, with the lowermost nodes in
this chain linking to units exposed on the basal erosional surface, forming a ‘fan-like’ network (Fig.
6b). The exact configuration of this topological ‘fan’ largely depends on: (1) the topography of the
erosional surface; (2) the geometry of units below the erosional surface; and (3) the thicknesses of
strata deposited above it.
Intrusions (dykes and plugs) have a similar topological expression to unconformities. The intrusive
body can be represented by a single node (as can an unconformity with a single unit deposited
above it) with arcs connecting it to the units it crosscuts, forming another fan-like network.
Intrusions that crosscut an entire model (i.e. continuous dykes) will divide it into separate domains
on either side of the dyke, resulting in a ‘bow-tie’ or ‘star-shaped’ network (Fig. 6c).
Faults crosscutting a model volume also define domains. Unlike intrusions, these domains remain in
contact, and so units on either side of the fault are juxtaposed, forming a ‘ladder-like’ topology (Fig.
Folding and other continuous deformation modes, such as pure and simple shear, have a more
indirect effect as by definition continuous transformations cannot modify topology. They do,
however, change the geometry of geological units and increase geological complexity, meaning that
any later discontinuous events such as faults, unconformities, and intrusions will potentially have
different topological manifestations.
One further consideration is of the scale of discretisation of any digital representation of geology, as
shear zones, for example, although continuous in a mathematical sense, at high resolution in a real
geological setting have discontinuous grain-scale behaviour. Furthermore, if a geological model is
divided into discrete volumes, such as voxels, the effect of a narrow shear zone or a fault may be
identical in terms of placing different lithologies against each other, even if the topological coding of
the two cases would still be different.
Figure 6: Network diagrams of (a) a layer cake stratigraphy; (b) an angular unconformity; (c) an
intrusion cutting an angular unconformity, and (d) a fault cutting an angular unconformity. Network
arcs are coloured according to the nature of the geological interface.
4.2. Mount Bruce Example
To highlight the topological complexity of real geology, and the possibility of using topology analysis
to characterise and identify geological formations and structures, we briefly present a case study
analysing the topological relations between geological units in the Mount Bruce area of Western
Australia, which hosts a sequence of Archean and Paleoproterozoic sediments (Fig. 7a). Specifically,
we investigate the topological relationships formed due to syn-depositional faulting and intra-group
unconformities. While these relationships are not explicitly defined in the geology map, it implies a
set of contact relationships that topology analysis can help resolve.
The stratigraphy in the Mount Bruce area has been divided into six main groups and two basins: The
Ashburton and Hamersley Basins (Martin et al., 2014). The Hamersley Basin is interpreted to be a
late Archean to early Proterozoic (2765-2470 Ma) passive margin adjacent to the Pilbara Craton
(Myers et al., 1996). Three major stratigraphic units are recognized in the basin: the Turee Creek,
Hamersley and Fortescue Groups. Two basement inliers expose unconformable contacts between
the Fortescue Group and granite-greenstone basement of uncertain age. Ashburton Basin sediments
are restricted to the southwest corner of the Mount Bruce map sheet, unconformably overlying
Hamersley Basin rocks. In the mapped area, the Ashburton Basin is subdivided into the Wyloo and
Shingle Creek Groups.
Thorne and Tyler (1997) and Thorne and Trendall (2001) have recognized the occurrence of two
major sets of west-northwest trending faults, the Jeerinah-Sylvania Fault system and the Nanjilgardy
Fault system. These initiated as syn-depositional normal faults and controlled regional variations in
stratigraphy and sedimentation during deposition of the Fortescue Group. Mesoproterozoic
deformation has resulted in a sequence of dome-and-basin refolded fold structures (Thorne and
In this analysis, we use the digital version of the Mount Bruce 1:500,000 scale map and fault network
(Martin et al., 2014), simplified to only include faults longer than 25km (Fig. 7a). Neighbour
relationships of both types of contact (Fig. 7b) were then calculated using the methodology
described in Section 3 and exported to GML format. These relationships were combined with the
minimum age of each formation (arrows point to the older units) to produce a directed graph,
visualised with the yEd Graph Visualisation Library (yWorks, 2016).
The resulting network (Fig. 7c) is complex, but contains a substantial amount of information. The
concentration of faulted contacts within the Hamersley Group, for example, hints at the presence of
synsedimentary faulting in this group, although the smaller faults that are not analysed here are
more common in the Fortescue Group. Faulted contacts between units at the top of the Hamersley
Group (the Boolgeda Iron Formation) and the Fortescue Group (Jerrinah Formation) suggest that
some develop significant offsets.
Deviations from the ‘chain-like’ network expected for a conformable stratigraphy (Fig. 6a) suggest
the presence of lateral facies transitions or unconformities. The Boolgeeda Iron Formation, for
example, shows typical offlap-unconformity topology (c.f. Fig. 6b), overlying and in contact with
many older formations (Fig. 7c). Conversely, the Bunjina Formation shows topological relationships
suggestive of an onlap topology, contacting many units that are younger than it (Fig. 7d).
Figure 7: Topological analysis of the Map Bruce map sheet. (a) Extract of the 1:500,000 digital map
of Western Australia showing the area equivalent to the 1:250,000 Mt Bruce map sheet. (b)
Automatically generated formation-level topology visualised as a network diagram. Dashed lines
represent faulted contacts, solid lines represent stratigraphic contacts, and dotted lines represent
relationships which are in part faulted and in part stratigraphic. Line width is a function of total
contact length. Arrows point to the unit with the older minimum age, and where the two units have
the same minimum age they are drawn with an arrow on both ends. Shaded boxes show the
different stratigraphic groups. (c) Subset of the topology network showing relationships for the
Booldega Iron Formation, which is in contact with seven older units via both fault and stratigraphic
contacts. The latter suggests an off-lap relationship. (d) Subset of network showing relationships for
the Bunjinah Formation that is in contact seven younger units, with both fault and stratigraphic
contacts. The latter suggests an on-lap relationship.
5. Topology Analysis and Visualisation
5.1. Visualisation Techniques
Visualising complex networks is notoriously difficult (Becker et al., 1995), as the previous example
highlights. Network diagrams (Fig. 5, 8a) can be useful for simple geological models, but quickly
become complicated and difficult to interpret in realistic scenarios (e.g., Fig. 7b). Instead, adjacency
matrices (Fig. 8b) and hive diagrams (Fig. 9) can be more useful.
A graph containing n nodes, which is equivalent to a geological model with n contiguous volumes,
can be represented as an n×n adjacency matrix (Godsil and Royle, 2013), in which each node is
represented by a row and a column, and each element Exy of the matrix is scored if the xth and yth
node are adjacent (share an arc). Types of adjacency relationship (e.g., faulted, intrusive) can be
represented symbolically, as in Fig. 8b. Diagrams of this type can still be complex and intimidating,
however specific relationships and general patterns can easily be identified.
Hive diagrams (Krzywinski et al., 2012) can also be used to visualise complex networks (Fig 9), and
are useful for comparing different networks with respect to defined network properties. Hive
diagrams contain three axes, along which nodes are ranked according to a property of interest (e.g.,
stratigraphic age, degree, volume), and arcs added between adjacent nodes on each axis. To
incorporate arc attributes such as surface area in the diagram, the nodes on some axes of the hive
diagram represent arcs in the original network.
Take, for example, two units of different ages that share a common contact (Fig. 9a). A node
representing each unit will appear on the “stratigraphic age” axis, linked to a single node on the
“structural age” axis that is positioned according to the age of the contact between the two units,
and to another single node on the “surface area” axis that is positioned according to the surface area
of the common contact. In this example, only the surface-area axis is a scaled axis, whereas the
other two represent a simple ordering.
Figure 8. Model topology displayed as a 3D network (a) and adjacency matrix (b). A render of the 3D
model is also included for reference (top).
Figure 9. Schematic diagram illustrating the use of a hive diagram (see section 5.1 for definition) to
visualise topology networks (a). An example hive diagram (b) created from the topology network
shown in Fig. 8 has also been included. Hive diagrams are useful for conveying general or emergent
properties of topology networks. This hive shows, for example, that: (1) the stratigraphic contacts
tended to have the highest surface area; (2) all but the oldest of the pre-unconformity units were
exposed on the unconformity surface, and; (3) fault offset is small compared to the general thickness
of the stratigraphic units, as faulted contacts tend only to juxtapose units close to each other in the
5.1.1. Structural and lithological topology
To reduce the complexity of cellular topology, and the ensuing interpretation challenges, two
different generalised forms of model topology are introduced here, each highlighting different
topological properties. These simplifications are hereafter referred to as structural and lithological
A model’s structural topology is derived by collapsing all conformable stratigraphic relationships
within a structurally contiguous volume (i.e. a volume defined by discontinuities such as faults,
intrusions and unconformities), and represents the topology of structurally bound rock volumes (Fig.
10a), characterising model architecture without the complexity that results from the inclusion of
lithological units. Conversely, the lithological topology of a model retains information about the
adjacency relationships between individual lithologies within a stratigraphic series, but discards
information regarding relative spatial location (Fig. 10b). This reduction is done by simplifying the
cellular topology to a set of pairwise relationships between lithologies equal to the number of
lithostratigraphic units defined in the legend. General structural information (i.e. A is in faulted
contact with B) can be preserved, but the specific context (i.e. different structurally bound volumes
of A and B) is lost, and hence the nodes in a lithological topology network have no spatial
Figure 10. Examples of structural and lithological topologies. A topology network (a) and associated
adjacency matrix (b) can be simplified by: (c) collapsing stratigraphic contacts to produce a structural
topology network, or; (d) ignoring topological domains and using lithological contact relationships to
produce a lithological topology matrix.
6. Discussion: A Framework for Geological Topology
Geological events have an inherent topological nature (Perrin et al., 2005), with successive
geological discontinuities (structures) each modifying a pre-existing topology. This section suggests
a broad framework for topology in geology that can include both spatial and temporal aspects, and
discusses some future applications of topology in geological modelling and uncertainty analysis.
6.1. Topology in Geology
The geological framework defined by Burns (1988) is robust for situations in which spatial adjacency
is the topological relationship of interest, but it is unclear how this relates to other topological
properties. As such, a broader framework is proposed, which can include temporal aspects such as
crosscutting and overprinting relationships and broader spatial relations. This framework is
essentially a synthesis of the ideas presented by Burns (1975), Burns et al. (1978), Burns (1981, 1988)
and Egenhofer (1989).
Geological topology can be defined by two broad classes: spatial topology and temporal topology.
Spatial topology relates to the topology of geological geometries, and is defined by Egenhofer
relationships (or if necessary the relations defined by Zlatanova, 2000) between geometric elements
(Fig. 2) of different dimensions. Generally, the “meets” relationship is most useful as geological
models tend not to allow gaps or overlaps, although when modelling some properties (e.g.,
alteration halos, mineral distributions or structural fabrics) topologies that include overlap are
possible. As in the framework of Burns (1988), multiple orders of topology are possible, depending
on the dimensions of the geometric elements of interest.
Lithological and structural topology represent simplifications of a 1st order spatial topology. Similarly,
it may be possible to derive multi-scale topology graphs to better capture the multi-scale nature of
geology. For example, the topology of a basin might be defined at a broad scale by its structural
topology, then nodes in this topology assigned information regarding the topology at a formation
level, nodes representing individual formations assigned information regarding the topology of
individual units, and so on.
Temporal topologies are similar to one-dimensional spatial topologies (i.e. relationships between
intervals), although they are fundamentally distinguished by the directionality of time. This
distinction has two important implications. Firstly, relationships that could not be considered
topological in a spatial context can be included: The older than and younger than relationships
defined by Burns (1975) do not have analogues in a spatial topology (the equivalent statement in 1D
space would be ‘left’ vs ‘right’ or ‘upper’ vs ‘lower’, which are not maintained after negative scaling),
but can be considered topological in a temporal sense as negative scaling is impossible. A geological
example of this behaviour is an overturned stratigraphy, where spatial relations (superpositions)
have been reversed, while temporal relations remain unchanged.
Secondly, relationships in a temporal topology are directed – if A is younger than B then B is not
younger than A – whereas the common Egenhofer relations are not directed (if A meets B, then B
also meets A). Hence, graphical representations of temporal topology require directed arcs, while
graphs of spatial topology do not except where more obscure topological relations such as ‘contains’
are used. In some instances, such as the Mt Bruce example (Section 4.1; Fig. 7b), aspects of both the
spatial and temporal topologies can be represented in a single graph. In summary, spatial topologies
are defined by Egenhofer relationships between geological volumes, surfaces or lines, while
temporal topologies are defined by temporal relations (cf. Burns, 1975, and Fig. 4) between events
In a geological context, spatial boundaries represent natural processes (e.g., intrusion, change in
depositional environment or fracturing), and hence, spatial topology can be used to infer temporal
topology and vice versa. The key to this transition is the concept of a process model (Burns, 1981),
which allows spatial relationships to be translated into temporal ones. For example, we know that
some intrusions form through the injection of magma into a fracture, and hence we might infer that
the spatial adjacency between a sediment and a dyke indicates that the dyke is younger than the
sediment. It is important to note that if the wrong process model is invoked, then the inferred
temporal relation may be incorrect. For example, the sediment in the previous example may
unconformably overlie an older intrusion.
It is worth mentioning that a model’s spatial topology is a time-evolving property (Michalak, 2005),
as it is for geomorphological systems (Löwner et al., 2013), and we only observe the final product.
Topological relationships critical to the formation of an ore deposit, for example, may no longer exist
as they may have been modified by later events. Faults, in particular, may result in rapidly evolving
topologies over the time period that they are active.
Overall, geology has its own ‘flavour’ of topology, as all of the topological relationships we observe
result from a limited set of physical processes (e.g., deposition, intrusion, alteration), each with
distinctive topological signatures (Fig. 6). The extensive history of most geological regions means
that the cumulative topological manifestation of these processes can be exceedingly complicated
(e.g., Fig. 7). However, as a result, the analysis and ‘disentanglement’ of a region’s topology can help
elucidate its history and the processes that have shaped it.
Figure 11. Framework for geological topology outlined in this study. A geological model has two
broad classes of topological relationships: spatial and temporal. These are linked by a process model,
which can be used to infer age relations from spatial relations or vice versa (provided the correct
process is implied and relationships are not ambiguous).
6.2. Limited-extent faults and splays
The methodology for extracting cellular topologies from geological models presented in this work is
generally robust, although it has some limitations. Most significantly, the method assumes that any
point within the model volume can be located relative to (e.g., left or right of) each fault, and hence
requires faults to crosscut the entire model volume at the time of their formation (Fig. 12a). Non-
continuous faults will not do this, and under the current method limited-extent faults (Fig. 12b) and
fault splays (Fig. 12c) cannot be processed unless only the lithological topology is being calculated, as
in the Mt Bruce example.
One possible solution would be to implement a null or ‘not-affected’ option in the topology codes,
which could be used to define an artificial topological boundary around the region of influence
surrounding a non-continuous structure (Fig. 12b). As the tips of fault splays will fall on the edge of a
structure’s region of influence, the splays will divide this artificial topological domain into further
regions (Fig 12c). Another possible solution in some situations would be to use a 2nd order topology
instead, representing contacts, fault segments and splays directly as nodes and their intersections
(edges) as arcs, similar to the method for describing fracture networks presented by Valentini et al.
(2007) and Sanderson and Nixon (2015).
Faults that transition to shear zones at depth are also problematic, because while the structure may
fully divide the model, the discontinuity effecting cellular topology disappears as the deformation
becomes ductile. This change could be managed by dividing the structure into an abutting non-
continuous fault and non-continuous shear zone.
Figure 12. The method of calculating cellular topology described in this work requires faults to divide
the model volume, as in (a), at the time of their formation. Hence the method currently cannot
resolve non-continuous faults (b), or fault splays (c). One possible method for calculating the
topology of such structures is to create an artificial topological boundary surrounding the ‘region of
influence’ of non-continuous structures, shown by black dashed lines in (b) and (c). Voxels within this
domain are classified according to their position relative to the structure (e.g., hanging wall,
footwall) and voxels outside the domain are given a null code for that structure. (c) If the region of
influence is expanded or shrunk such that its boundary includes the tips of fault splays, they also
define unique topological domains, as shown by unique numbers here.
6.3. Future Directions
The framework for ‘geological topology’ presented has a variety of applications including: automatic
history interpretation and validation (c.f. Burns, 1975; Burns and Remfry, 1976); as additional model
characteristics to be fed into geodiversity analysis (c.f. Lindsay et al., 2013; Lindsay et al., 2014), and
as a tool for quantifying differences between geological models in uncertainty analyses.
More broadly, geological topology could provide a novel method for characterising and comparing
geological features (Deutsch, 1998; Hilaga et al., 2001; Fufa, 2004; Li and Yang, 2009; Bazaikin et al.,
2013). The topology analysis of the Mt Bruce map sheet provides an example of some of these
possible applications. The unconformable relationships described in section 4.2 (Fig. 7) are not
obvious on the map sheet and would be difficult to otherwise identify. Hence, direct analysis of the
topology highlights implicitly defined but otherwise hidden relationships. Obviously, this analysis is
brief and geographically limited, given that other relationships may exist on adjacent map sheets,
however this is one of the powers of the approach. It is possible that variations in the topological
relationships on a regional scale could be used to highlight underlying variations in distinct sub-
basins. Graph-theory-based methods for comparing topology graphs or sub-graphs might then assist
in the identification and quantification of such variations.
Similarly, ore deposits or tectonic terranes could be characterised and compared at a simple level
using their topology. While direct topological comparison in terms of specific lithologies may not be
possible, the overall structure, connectivity or complexity of the topological network, or specific
aspects such as the fault topology, may be revealing. For example, comparisons of the topological
relationships between alteration zones, vein generations and intrusive bodies in porphyry copper
deposits might highlight some genetic requirements of the mineral system.
In addition the idea of “fuzzy topology” (Shi and Liu, 2007) may aid in the comparison between
different regions. More advanced comparisons could potentially include relative temporal or
structural relations, although equivalence between geological events or structures in different
locations may be difficult to define.
It might also be worthwhile to produce a set of ‘topological rules’ that must be met for a model to
be ‘geologically reasonable’, derived from physical restrictions implied by the process model or the
directionality of time. For example, self-contradictory cycles (loops of connected nodes) could easily
be identified in a temporal topology. Automatic identification of the processes responsible for
specific contact relationships given their broader context might also be possible, as highlighted by
the identification of unconformable relationships in the Mt Bruce map sheet.
Finally, the set of temporal relations defined by Burns (1975) assumes that events are essentially
instantaneous, and hence models for temporal topology currently cannot describe diachronous,
coeval or overlapping events. This limitation will cause problems for geological structures such as
syn-sedimentary and reactivated faults, coeval intrusive and extrusive volcanic suites or
contemporaneous metamorphism and deformation. While we have been able to define further
temporal relationships to cope with these structures, inferring them from spatial topology using the
methodology of Burns (1975) is not trivial.
The topology of three-dimensional geological models can be described for water-tight
models purely in terms of adjacency and temporal relationships, whereas a more general
scheme that considers overlapping volumes and time relationships is necessary for
alteration halos, microstructural overprinting and diachronous event sequences. These
relationships are a direct result of the physical processes that have shaped the geology and
The topology of a geological model can be represented and interpreted as: two- and three-
dimensional network diagrams; adjacency matrices, and; hive diagrams. Each one of these
complementary techniques highlights different properties of geological topology.
We have developed an automatic system for extracting 1st order 3D topological information
from models built with the Noddy kinematic modelling system, and equivalent analyses
should be possible for other implicit modelling systems.
Topological analysis is fertile ground for further study, with under-explored applications
including: (1) use as a tool for comparing and characterising different geological possibilities
in uncertainty and sensitivity analyses; (2) automatically inferring geological history from
geometry via a process model, and; (3) characterizing and comparing lithological
associations in ore deposits, tectonic domains or sedimentary basins.
This work derives from an MSc thesis by STT. It was supported by resources provided by the Pawsey
Supercomputing Centre with funding from the Australian Government and the Government of
Western Australia. STT acknowledges the support of a University Postgraduate Award and Top-Up
Scholarship. MWJ was supported by a Western Australian Fellowship and the Geological Survey of
Western Australia. MDL was supported by the Geological Survey of Western Australia and the
Exploration Incentive Scheme. Eric de Kemp and David Sanderson are thanked for their useful
feedback during the review process.
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