Math GeosciDOI 10.1007/s11004-014-9538-x

SPECIAL ISSUE

Inversion and Geodiversity: Searching Model Spacefor the Answers

Mark Lindsay · Stéphane Perrouty ·Mark Jessell · Laurent Ailleres

Received: 22 May 2013 / Accepted: 11 April 2014© International Association for Mathematical Geosciences 2014

Abstract Geophysical inversion employs various methods to minimize the mis-fit between geophysical datasets and three-dimensional petrophysical distributions.Inversion techniques rely on many subjective inputs to provide a solution to a non-unique underdetermined problem, including the use of a priori model elements (i.e. acontiguous volume of the same litho-stratigraphic package), the a priori input modelitself or inversion constraints. In some cases, inversion may produce a result that per-fectly matches the observed geophysical data, but can still misrepresent the geologicalsystem. A workflow is presented here that offers objective methods to provide inputsto inversion: (1) simulations are performed to create a model suite that contains a rangeof geologically possible models; (2) stratigraphic variability is determined via uncer-tainty analysis to identify low certainty model regions and elements; (3) geodiversityanalysis is then conducted to determine geometrical and geophysical extremes andcommonalities within the model space; (4) geodiversity metrics are simultaneouslyanalysed using principal component analysis to identify the contribution of differ-ent model elements toward overall model suite uncertainty; (5) principal component

M. Lindsay · L. AilleresSchool of Geosciences, Monash University, PO Box 28E, Victoria 3800, Australia

M. Lindsay · S. Perrouty · M. JessellL’Université Toulouse III, Paul Sabatier (OMP), GET, 14 Av. Edouard Belin , 31400 Toulouse, France

M. Lindsay (B) · M. JessellCentre for Exploration Targeting (M006), University of Western Australia, Crawley,WA 6009, Australiae-mail: markdlindsay@gmail.com

S. Perrouty · M. JessellIRD, GET, 31400 Toulouse, France

S. PerroutyDepartment of Earth Sciences, Western University, London, ON N6A 5B7, Canada

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analysis also determines which models exhibit diverse or common geological andgeophysical characteristics which (6) facilitate the selection of models as inputs togeophysical inversion. This workflow is applied to a three-dimensional model of theAshanti Greenstone Belt, southwestern Ghana in West Africa in order to reduce thesubjectivity incurred during decision making, explore the range of geologically pos-sible models and provide geological constraints to the inversion process to producegeologically and geophysically robust suites of models. Results further suggest thatthree-dimensional uncertainty grids can optimize inversion processes and assist infinding geologically reasonable solutions.

Keywords 3D modelling · Uncertainty · Inverse methods · Geodiversity ·Ashanti Greenstone Belt · Geological constraints

1 Introduction

Predicting the outcome of a set of measurements given a known set of parametersis called the forward problem and results in a unique solution. Inverse problems canoffer solutions inferring parameter values describing a system where parameters are notknown; however, inverse problems do not offer unique solutions (Tarantola and Valette1982a,b). Inverse problems are common in geoscience as knowledge of all parametersis rarely known. One common inverse problem is to resolve three-dimensional geo-logical architecture from a geophysical dataset. Successful reconciliation of geologyand geophysics requires explicit knowledge of parameters essential to formulating aforward problem, such as complete descriptions of shape, location and physical natureof relevant geological structures. The challenge is that these types of parameters arevery rarely, if ever, known in geology (Frodeman 1995). An inverse solution is conse-quently required to determine the unknown parameters and define the petrophysicaldistributions observed in the geophysical response.

The least squares inversion method is used in this study to determine an appropri-ate solution to an under-defined geoscientific problem (Tarantola and Valette 1982a;Tarantola 1984). Least squares inversion is a process that conducts iterative forwardmodelling to resolve petrophysical distributions from a measured geophysical field(Oldenburg 1974; Fullagar et al. 2000; Jessell 2001). The a priori input for inversionis a starting model consisting of a selection of petrophysical properties and/or geo-logical surfaces (Oldenburg et al. 1997; Fullagar et al. 2000; Boschetti and Moresi2001; Guillen et al. 2008; Gallardo and Meju 2011; Martin et al. 2013). Estimation ofrock property distributions produces a calculated response that is measured against thenatural, or observed, geophysical response. Mathematical methods, such as residualmisfit and fit-to-data metrics, are used to assess whether the estimated rock propertydistribution adequately reflects the observed geophysical response. Estimation of rockproperty distribution is performed using an objective function, and typically some ana-logue derived from nature is employed to constrain the number of possible solutions(Zelt 1999; Boschetti and Moresi 2001).

Inversion parameters, the structure and composition of the starting model, inversionscheme and what results constitute an adequate answer are all chosen by the operator.

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Finding the optimal set of inversion parameters either requires sensitivity analysisthrough multiple inversion runs or via simultaneous examination of the input dataand the required solution (Zelt 1999; Boschetti and Moresi 2001; Gallardo and Meju2011). Petrophysical constraints not only provide a sample of the natural world toguide the inversion process to provide geophysical contrast between different rockunits (Fullagar et al. 2000; Lane and Guillen 2005; Guillen et al. 2008; Lelièvre et al.2009), but also contain error associated with specimen sampling (Worthington 2002).The standard deviation of the sample is normally used to describe the petrophysicalheterogeneity of the rock type and can also infer the confidence one has with thesampling process. Petrophysical constraints also restrict geological elements frombeing attributed unrealistic values. Minimum and maximum petrophysical values canbe assigned to model elements as lower and upper bounds that the inversion processcannot transgress to improve misfit (Fullagar et al. 2008).

Typical inversion procedures operate with a single a priori model, precludingother geological scenarios from being tested against the observed geophysical field(Boschetti and Moresi 2001), and assume that the input model is the best geologi-cal solution. This assumption is flawed, as geological and geophysical problems areoften as poorly parameterized as each other (Mann 1993; Thore et al. 2002; Jes-sell et al. 2010). The operator is usually aware of how capable the model will be inhonouring both geophysical and geological data, being intimately involved in its con-struction (Royse 2010). The operator’s opinion on model quality is clearly important,but nonetheless subjective, biased and typically not based on any direct quantitativetechniques, making the translation of model quality into inversion input parametersdifficult.

Qualitative review of inversion results is typically performed once mathematicallydefined measures of success have been achieved. A review is necessary as resultssometimes bear little resemblance to what is considered to be a reasonable representa-tion of geological reality. While the algorithm performed according to requirements,the solution is non-unique, and the decision to accept results rests with the operator(Polanyi 1962; Torvela and Bond 2010). Thus, the role geological intuition plays iscritical, as geological idiosyncrasies that have not been retained during inversion maynot be acknowledged as missing from (or added to) the final result (Cooley 2007).Intuition is a fundamentally biased quality and based upon the education, experiencesand background of the operator (Frodeman 1995).

This manuscript presents a process that produces multiple geological realizationsfrom the same input dataset and subjects them to inversion. The process is tested forthe Ashanti Greenstone Belt, southwestern Ghana in West Africa. Model uncertaintyis calculated using techniques from Lindsay et al. (2012) and described by strati-graphic variability. Geodiversity metrics are introduced and used to characterize thegeophysical and geometrical aspects of each model (Lindsay et al. 2013a). Principalcomponent analysis (PCA) is used to simultaneously analyse all geodiversity metricsin order to identify the configuration of model space and which geodiversity metric(or metrics) contribute most to model space variability, providing guidance to inver-sion parameters and reducing the dimensionality of the problem (Bertoncello et al.2013). A priori models are nominated via PCA and selected geological elements aresubjected to gravity inversion. Inversion results are analysed in correlation with model

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uncertainty to assess the viability of the presented technique. The primary aim of theworkflow presented in this manuscript is to produce model elements that honour bothgeological and geophysical data while removing some of the subjectivity from theinversion process.

2 The Ashanti Greenstone Belt, Southwestern Ghana

The Ashanti Greenstone Belt is a gold prospective Palaeoproterozoic granitoid–greenstone belt located in the south of the Leo-Man craton, West Africa (Fig. 1). Fig-ure 2a displays the stratigraphic column adapted for model construction. The AshantiGreenstone Belt comprises a Palaeoproterozoic basement of Birimian Series metavol-canics rocks overlain by metasedimentary rocks (Loh et al. 1999; Adadey et al. 2009;Perrouty et al. 2012) in turn overlain by the Palaeoproterozoic Tarkwaian Series sed-imentary rocks (Kitson 1928; Junner 1940; Pigios et al. 2003; Perrouty et al. 2012).A steep regional gradient seen in the Bouguer gravity anomaly (Fig. 3a), striking north-northeast south-southwest, is interpreted to be a faulted contact between the KumasiGroup to the west and the Tarkwaian Series to the east, indicating the location of theAkropong and Ashanti faults. The Tarkwaian Series comprises a complex and polyde-formed sequence of dolerite sills, phyllites, conglomerates and sandstones (Eisenlohrand Hirdes 1992; Feybesse et al. 2006). Dolerite dykes are common throughout theregion and are noticeably represented in the aeromagnetic datasets (Fig. 3b), thoughhave not been included in the model. Granitoids of typical tonalite–trondhjemite–granodiorite (TTG) composition have intruded the Birimian and Tarkwaian seriesduring the Eoeburnean (2,180–2,150 Ma) and Eburnean (2,130–2,070 Ma) orogenicevents (Perrouty et al. 2012). The south of the region contains a suite of granitoids witha more intermediate composition. Granitoids have been modelled as either TTG-typeEoeburnean granitoids or K-feldspar-rich Late Eburnean grantitoids (Perrouty et al.2012). The youngest geological unit included in the model is a layer of Phanerozoicshales and sandstones, which mostly crop out along coastal regions.

3 Three-Dimensional Modelling Workflow

Figure 4 describes the eight-step workflow used in this manuscript. Detailed descrip-tions the geodiversity metrics methods are supplied in the appendices and are brieflydiscussed in this section. For further information and case studies on steps 1–6(Fig. 4), please refer to Lindsay et al. (2012, 2013a,b).

3.1 Model Construction

The three-dimensional geological model of the Ashanti Greenstone Belt was con-structed using a combination of field, geophysical and remotely sensed data. Field dataconsists of measurements obtained through the outcrop maps of Loh et al. (1999), BHPBilliton, Golden Star and that collected by Perrouty et al. (2012). Field data includepetrophysical measurements, structural observations, lithological descriptions and a

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WassaDamangBogoso

Cape Coast

Sekondi - Takoradi

560000 600000 640000 680000 72000052

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Kawere, Banket and Tarkwa SeriesKumasi GroupSefwi Group volcaniclastic sedimentsSefwi Group lavas Inversion extents

0 10 20 30 405Kilometers

Fig. 1 Location of the Ashanti Greenstone Belt, southwestern Ghana, West Africa. The extents of thegeology in the top figure section correspond to the model extents. Note the inversion extents are defined bythe dashed line. Modified after Perrouty et al. (2012)

revised stratigraphic column also produced by Perrouty et al. (2012). Geophysicalinterpretation of potential field datasets was extensively employed to provide broadergeological understanding over the study area. Geophysical datasets include gravity(Fig. 3a) and aeromagnetic data (Fig. 3b). Gravity data were obtained through theInternational Gravimetric Bureau (BGI, http://bgi.omp.obs-mip.fr/), has a resolutionof approximately 4.6 km per pixel (2.5 arc-minutes) and contains a combination ofGetech ground data (African Gravity Project 1986–1988, http://www.getech.com/history.htm), BGI off and on-shore data and satellite gravity data. Aeromagnetic datawere acquired from the Geological Survey of Ghana and were flown between 1994 and1996. The aeromagnetic surveys were flown at a height of 80 m and with line spacingsof 200 m striking 135◦. The reduced-to-pole (RTP) grid was generated using Inter-national Geomagnetic Reference Field (IGRF) parameters: total field—31,699 nT;inclination—14.5◦ and declination −6.5◦. Geological cross-sections were validated by

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2.55 0.001

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2.75 0.2

2.81 0.05

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2.89 0.04

2.72 0.01

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0.002 0.001

0.002 0.00001

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0.005 0.0001

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0.02 0.0001

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Fig. 2 Model parameters and determination of model uncertainty. a Stratigraphic column and stratigraphicidentifier values (bold numbers). Assigned petrophysical values are listed to the right of each stratigraphicunit. Modified after (Perrouty et al. 2012). b Simplified synthetic example of how stratigraphic variabilityis determined to represent model uncertainty (after Lindsay et al. 2012)

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WassaDamangBogoso

Cape Coast

Sekondi - Takoradi

560000 600000 640000 680000

5200

0055

0000

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0000

6400

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WassaDamangBogoso

Cape Coast

Sekondi - Takoradi

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0055

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6400

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hgiHwoL Geophysical response

0 10 20 30 405Kilometres

(b)(a)

Kilometers Kilometers

Fig. 3 Geophysical grids used to construct the Ashanti Greenstone Belt model. a Bouguer gravity anomalyimage and b reduced to the pole image of the total magnetic intensity. Modified after Perrouty et al. (2012)

Perrouty et al. (2012) through joint forward modelling of magnetic and gravity data viathe Oasis Montaj module GM-SYS (Talwani et al. 1959; Talwani and Heirtzler 1964).Geological cross-sections were validated by (Perrouty et al. (2012)) through joint for-ward modelling of magnetic and gravity data via the Oasis Montaj module GM-SYS(Talwani et al. 1959; Talwani and Heirtzler 1964). Existing geological cross-sectionswere consulted to assist forward modelling in the absence of drill hole logs. Fieldobservations were incorporated into forward modelling where possible, and a geo-logical consistency check with regional interpretation was performed (Perrouty et al.2012).

Input datasets were integrated into a three-dimensional geological modelling sys-tem, 3D Geomodeller (http://www.geomodeller.com/). The modelling system is con-sidered implicit as model elements are calculated using the potential field method(Lajaunie et al. 1997). The location of geological contacts and associated orienta-tion measurements, together with the stratigraphic column, are cokriged to producea scalar potential field (not to be confused with a geophysical potential field). Thethree-dimensional position of geological surfaces can be determined from the scalarpotential field. The boundaries of the geological body are defined by iso-potentialsurfaces and orientation is defined by gradients determined by the orientation mea-surements. The relationship of each geological body to the other is defined by thestratigraphic column. Fault and stratigraphic relationships can be defined to determinewhich faults stop or cross-cut others, and which geological formations are affected byfaults and other formations. These relationships define a model topology represent-

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Model construction

Uncertaintysimulation

Uncertaintydetection

Geodiversityanalysis

End-memberanalysis

Principal ComponentAnalysis

Basement inversionmodelling

Results assessment

stluseRsnoisicedevitcejbuS

Ashanti Greenstone Belt, southwestern

Ghana

Creation ofmodel suite

Obtain stratigraphic variance (L,P)

Geometrical andgeophysical metrics

Determine modelrankings for each metric

Determine influentialmetrics and modelspace boundaries

Property andgeometrical inversion

Correlations withstratigraphic

variability

1.

2.

3.

4.

5.

6.

7.

8.

pertubation

(cell size of voxet)

metrics considered useful

parameters

Type of input data Modelling packages Interpolation algorithms

Magnitude of orientation

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Selection of geodiversity

Assigned inversion

Geophysical dataset

Geological feasibility assessment of inversion

Fig. 4 Workflow diagram depicting the steps taken during the procedure. The subjective decisions madein and results of each step in the workflow are shown

ing the tectonic evolution of the study area. Visualization of geological surfaces isachieved through cokriging of the scalar field, where interfaces are taken from valuesof the field and their orientation from the partial derivatives dv/dx, dv/dy and dv/dz.Extensive case studies and examples are described in Calcagno et al. (2008), Lindsayet al. (2012), Martelet et al. (2004), Maxelon et al. (2009), Schreiber et al. (2010) andVouillamoz et al. (2012).

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3.2 Uncertainty Simulation

Three-dimensional geological models are built using sparse data sets that do not fullydescribe the entire geological system. A number of assumptions and interpolations arethus required to predict geology in regions not described by input data, such as thoseat depth. Knowledge-driven and non-unique interpolation is typically employed togenerate geological relationships between sparse data points. The modelling systemassists in removing some subjectivity associated with knowledge-driven processesthrough interpolation of a repeatable and objective implicit function (Calcagno et al.2008; Lajaunie et al. 1997). Conversely, decisions regarding inclusion of data aremade by the operator, are subjective, and have the potential to significantly affectthe resulting three-dimensional model architecture. Inherent error in input data alsoproduces model uncertainty (Jessell et al. 2010; Wellmann et al. 2010; Lindsay etal. 2012). Whether the data collected adequately represent the architecture of thegeological terrane is another key consideration. Finally, the choice of interpolationalgorithm and their parameters also have significant influence on the resulting model.

Another advantage of implicit modelling techniques is the speed that models canbe constructed. Input of data and setting up the model prior to calculation can betime consuming. Calculation of the model is typically done in seconds once datainput and configuration are complete. Whether this data-driven model is an accuraterepresentation of the geology is decided upon with knowledge-driven assessment bythe operator. If the initial interpolation seems incorrect, subsequent addition of data, orremoval and revision of uncertain data may be performed that fundamentally changesthe geological representation until the result is acceptable. Rapid model calculationallows this form of model editing possible. The difficulty is in capturing the reasoningbehind whether a model seems incorrect or correct. Instead of editing models and onlykeeping the new version, a suite of models is produced increasing the likelihood ofgenerating a model that approaches that seen in nature while providing a basis forquantitative appreciation of model uncertainty (Caumon 2010; Lindsay et al. 2012).

Uncertainty simulation is employed to understand the influence that uncertaindatasets have on the modelling process. The simulation is achieved by perturbingthe orientation measurements that help to shape model elements (i.e. the strike anddip of geological surfaces) via Monte Carlo simulation. Each measurement, eitherdefining geological contacts or fault orientation, is varied to within 5 degrees of theoriginal value using pseudo-random equiprobable perturbation. For example, a mea-surement of 084/62E can be changed to 081/58E or 089/64E and so on, as long as thenew measurement is within 5 degrees of the original. Five degrees was chosen as areasonable estimate of variation that could occur when performing data upscaling ormeasuring surface orientations during geophysical interpretation and field mapping.Varying orientation data attempts to simulate the uncertainty associated with whethera measurement adequately represents a geological structure at different scales or afterdata upscaling. The assumption is that while the input data represent the best possi-ble set of orientation measurements, it is possible that another set exists which betterrepresents nature (Lindsay et al. 2013a).

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3.3 Uncertainty Detection

The original model, hereafter referred to as the initial model, has been calculated withthe implicit method using datasets comprising unperturbed orientation measurements,that is, the best model according to the operator. The perturbed sets of orientationmeasurements are re-calculated to construct perturbed models, which, together withthe initial model, collectively form a model suite (Lindsay et al. 2012). The next stepis to characterize uncertainty within the model suite. Uncertainty exists in locationsshowing variability between models, such as when a particular geological formationdisplays different geometries or extends to deeper regions from one model to the next.

Characterizing model uncertainty is performed by first discretizing each model intoa voxet, a collection of volumetric pixels (or voxels). Each voxel is attributed with thestratigraphic identifier representing a geological unit within the stratigraphic column(Fig. 2a) to allow easy calculation of stratigraphic range (‘L’) between locations(Fig. 2b). Lindsay et al. (2012) describe uncertainty using two different measurescollectively called stratigraphic variability. The number of stratigraphic units, ‘L’, atthe same X, Y, Z location in each model throughout the model suite comprises the firstcomponent of stratigraphic variability. The second part of stratigraphic variability, ‘P’,describes the deviation away from the most common stratigraphic unit at that X, Y, Zpoint throughout the model suite. The most common stratigraphic unit is determinedfrom the ‘modal model’, a conceptual model that is calculated by determining themodal stratigraphic unit at each point within the voxet. Stratigraphic variability isused to visualize the location and magnitude of uncertainty within a model.

3.3.1 Uncertainty in the Ashanti Greenstone Belt model

Figure 5 shows that levels of uncertainty >4 tend to be associated with the EarlyBirimian surfaces and their contact with tkc (base of the Tarkwaian Group series). Asomewhat surprising result is that only a small amount of uncertainty is associatedwith the modelled faults. The only faults associated with uncertainty dip shallowly,while those with little uncertainty display high dip angles. A shallowly dipping faultis potentially longer and cross-cuts more volume of a model that is tile-shaped (i.e.wide and long but shallow—such as the Ashanti Greenstone Belt model), than a modelvolume that is prism shaped (small on the x and y axes, but deep). A high-angle faultwill be shorter and subsequently cross-cut less of a tile-shaped model volume so, forthe Ashanti Greenstone Belt model, any perturbation of a fault that cross-cuts moreof the model will be associated with more uncertainty.

The number of stratigraphic units assigned to the Early Birimian series is dis-proportionately high when compared with other modelled series. Thus, the levelsof uncertainty associated with the Early Birimian surfaces need to be considered incontext of the complexity of the modelled geology, and uncertainty levels are rel-atively higher. The same, but inverse, consideration is required with respect to theLate Eburnean (granite_k) and Eoeburnean-age granites (granite_ttg) units, as indi-vidual plutons could have been assigned a unique stratigraphic identifier, rather thangrouping multiple intrusions under a single identifier. The apparent disproportion ofuncertainty between the Early Birimian and intrusive units reflects the original pur-

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Str

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View from belowlooking from the SE

View from abovelooking from the NW

tkc surface

Fig. 5 Three-dimensional model of the Ashanti Greenstone Belt and uncertainty grid colour-coded accord-ing to stratigraphic variability—scale at right. Note the uncertainty associated with the base of the Tarkwaian(tkc) and the Early Birimian Series. tkc is bordered in thick grey and the Early Birimian Series unit in black.Uncertainty is represented by points coloured according to stratigraphic variability. Higher magnitudestratigraphic variability has been displayed (>4). The modelled tkc and Early Birimian surfaces have beenobtained from model 92

pose of the model to investigate the interaction between the Birimian and Tarkwaianunits. Other geological units are not represented in as much stratigraphic detail as theyare not considered as important to understanding the interaction between the Birimianand Tarkwaian units.

3.4 Geodiversity Analysis

An uncertain location is one where multiple stratigraphic units are detected acrossdifferent model realizations. It follows that these differences must be due to modelelements’ changing shape or location. Stratigraphic variability describes where and to

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what magnitude uncertainty has affected the model, but not how the model is affectedgeometrically. Quantitative analysis of the varied geometries exhibited by the modelsuite allows models to be compared and then ranked against each other. Geometricalanalyses are performed under a collection of geometrical analyses called geodiversitymetrics (Table 1). Geodiversity metrics include methods that calculate the curvatureof a model surface and find the deepest and shallowest extent of a model element, thevolume of a model element, the surface area of a geological contact and the complexityof a model element. The complexity of a model element is calculated by a six nearest-neighbour calculation and determines the number of different stratigraphic units thatsurround a given point. For example, if the complexity of a point is determined to equalthree, then three different stratigraphic units surround that point, which would representa relatively complex location within the model. For more detailed explanations ofgeometrical geodiversity metrics see Appendix A and Lindsay et al. (2013a).

Average values for each metric from every stratigraphic unit are calculated to pro-vide a representative value for the entire unit, not just individual points. A similarmethod is also employed by Lindsay et al. (2013b) when including geophysical metricsin geodiversity analysis. Geophysical metrics analyse the observed and calculated geo-physical responses, and the difference between the observed and calculated response(the residual). Image analysis and statistical techniques are used to measure differentaspects of model geophysical representation, including root mean square (rms), stan-dard deviation (O’Gorman et al. 2008), entropy (Gonzalez et al. 2003; Wellmann andRegenauer 2011), two-dimensional correlation co-efficient and the Hausdorff distance(dH) (Huttenlocher et al. 1993). Each of these geophysical metrics is described withgreater detail in Appendix B and Lindsay et al. (2013b). The aim of using geodiver-sity metrics is to provide a comprehensive description of geological and geophysicalvariation within the model suite.

3.4.1 End-Member Analysis

End-member analysis is conducted by ranking each model according to each geodi-versity metric. For example, volume of a particular stratigraphic unit or the forwardmodelled gravity misfit can be obtained and results ranked to determine which modelsform the end-member representative for that metric. The end-member representativesfor the Ashanti Greenstone Belt model suite are shown in Table 2. The selection ofstratigraphic units shown was based on which stratigraphic unit best represented modelsuite variability. The end-member values for the depth of tkc show the range of possi-ble depths of contact with the Early Birimian is 1,400 m (between 6,650 and 8,050 m).The contact surface area (CSA) metric measures the surface area of a contact betweentwo stratigraphic units. The most variable units were found within the Early Birimianseries. There is a high range of CSA within the respective end-member representatives,which is reflected in the high degree of uncertainty associated with the Early Birimiansurfaces.

The complexity metrics show similar ranges between each unit. The complexityof the granite_ttg bodies shows similar ranges of complexity to the uncertain EarlyBirimian units bv4 and bvc4. The higher complexity of the Early Birimian can beattributed to the relatively high number of units (11) with which this series has been

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Math Geosci

Tabl

e2

End

-mem

ber

repr

esen

tativ

esfo

rth

eA

shan

tiG

reen

ston

eB

eltm

odel

suite

CSA

bv1

and

bgC

SAbv

c1-b

v2C

SAbg

-bv3

Dee

p.tk

cD

eep.

stk m

bv5-

lbvs

k gbv

5-lb

vsC

omp.

gran

ite_

ttg

Com

p.bv

4C

omp.

bvc4

Vol

.bas

eV

ol.g

rani

te_k

165

(1)

51(1

)4,

586

(89)

−6,6

50(1

7)−9

,450

(25)

−7.0

6×

10−4

(53)

−8.3

4×

1020

(63)

1.46

(79)

1.55

(29)

1.62

(51)

145,

828

(1)

10,5

01(4

4)

4,89

2(7

2)3,

847

(54)

7,79

2(1

)−8

050

(91)

−11,

900

(61)

0.00

13(1

01)

−4.0

7×

10−7

(97)

2.21

(24)

2.30

(1)

2.50

(1)

174,

685

(50)

15,1

78(8

6)

RM

Sgr

avity

RM

Sm

agne

ticd H

grav

.d H

mag

.St

d.gr

avSt

d.m

agE

ntro

pygr

av.

Ent

ropy

mag

.2D

corr

.gra

v2D

corr

.mag

8.86

(78)

55.9

4(7

5)91

1.34

(99)

1,50

1.30

(51)

20.1

3(7

5)29

.83

(75)

0.35

(42)

1.05

(40)

−0.1

6(1

)−0

.19

(87)

9.46

(85)

58.0

5(1

)93

4.03

(32)

1,58

1.46

(36)

21.1

2(1

)33

.22

(1)

0.37

(6)

1.12

(62)

−0.1

7(4

6)−0

.16

(1)

CSA

cont

acts

urfa

cear

ea—

unit

ofm

easu

rem

ent:v

oxel

s,D

eep.

deep

estp

arto

funi

t—m

etre

s,km

mea

ncu

rvat

ure,

kgG

auss

ian

curv

atur

e,C

omp.

com

plex

ity,V

ol.v

olum

e(k

m3),

RM

Sro

ot-m

ean-

squa

re—

mG

als

(gra

vity

),nT

(mag

netic

),d

HH

ausd

orff

dist

ance

,Std

.sta

ndar

dde

viat

ion,

2Dco

rr.2

Dco

rrel

atio

n.T

heC

SAm

etri

cm

easu

rem

entu

niti

sin

‘vox

els’

asan

accu

rate

estim

ate

ofsu

rfac

ear

eaw

asno

tabl

eto

beob

tain

edbe

caus

eth

evo

xels

have

anir

regu

lar

shap

e

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Math Geosci

640000 600000 580000

680000

640000

600000

580000

Str

atig

raph

ic V

aria

bilit

y

4

6

8

12

10

14

2

Ashanti FaultAkropong Fault

Cape Coast Fault

Late Eburneangranitoid

Fig. 6 Overview of the Ashanti Greenstone Belt three-dimensional model showing the lack of high uncer-tainty with the Late Eburnean granites (‘granite_k’). The surface expression of the granites is shown withblack borders. Faults are shown with grey borders. Stratigraphic variability shows low magnitudes (blue)through to extreme (red)—scale at right

modelled. The result is that the units are thin and the potential for complexity increasesin locations where the potential for surrounding units being adjacent increases, suchas where these units are folded. In contrast, the complexity of granite_ttg is due theunit being modelled as an intrusion and subsequently cross-cuts much of the modelledgeology. In addition, some of the granite_ttg bodies outcrop at the surface and extendto the bottom of the model, so the likelihood of complexity is increased with thecorresponding likelihood of cross-cutting other model elements.

The range of possible volumes for the granite_k bodies is 4,677 km3. Model 86displays the highest granite_k volume (15,178 km3) for the entire model suite, withthe range of being 31 % of that value. This high variability (and therefore uncertainty)is not immediately obvious when visually assessing the modelled granite_k bodies’uncertainty (Fig. 6). The granite_k bodies in the eastern section of the model do notappear to be uncertain, aside from where the Cape Coast Fault intersects a body inthe southeastern quadrant. The majority of uncertainty is associated with the gran-ite_k bodies located in the south and southwestern quadrants of the map, where theycross-cut the Early Birimian series units and are cross-cut by the Akropong Fault.This shows that model uncertainty cannot be estimated purely on the measurementsfrom one stratigraphic unit. The interaction with other model elements also needs tobe considered to develop a better understanding of the complex nature of model uncer-tainty. A topological measure of uncertainty (unfortunately beyond the scope of thisstudy) would be desirable to aid this type of analysis. While this model uses a single

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stratigraphic unit to model all the granite_k bodies, it would be more topologicallyaccurate if they were represented with their own individual identifier. The modellingpackage and the geodiversity metrics assume that a single stratigraphic unit is tem-porally contiguous, whereas in reality the Late Eburnean granitoid bodies displaydifferent ages (Oberthür et al. 1998; Loh et al. 1999; Perrouty et al. 2012). Modellingthe Late Eburnean granites individually would reduce the high volume range shownin Table 2, avoid the suite of granites being mislabelled as highly uncertain and wouldfacilitate finding specific bodies displaying high uncertainty.

The rms misfit values calculated by forward modelling are large: between 8.86and 9.46 mGal for gravity data (at most 15 % of the 63.66 mGal dynamic range) andbetween 55.94 and 58.05 nT for magnetic data (at most 12 % of the 479.23 nT dynamicrange). This large misfit is likely due to the lack of geological detail in the TarkwaianSeries units. By including dense (2.87–2.90) and magnetically susceptible (15–78.8× 10-3 SI) (Metelka et al. 2011; Perrouty et al. 2012) doleritic units in the Tarkwaian,large misfits could be avoided. Variation between rms misfit values obtained throughaeromagnetic and gravity forward modelling shows that the calculated geophysicalresponse is affected by uncertainty simulation. The selection of a priori inversionmodels needs to be performed carefully.

Information entropy is a useful technique that reflects the roughness within animage and overall information content (Shannon 1948; Batty 1974; Gonzalez et al.2003; O’Gorman et al. 2008). An entropy metric is applied to the residual grid todiscover whether it is ‘rough’ and contains high variability in values, or smooth andcontains little variability in values (Lindsay et al. 2013b). The residual magnetic andgravity grids have a 1,600 m cell size, and measure 160 km on the x and y axes(25,600 km2 area). The entropy values of the aeromagnetic residual reveal a varietyof results from relatively low entropy values (1.05—model 40) indicating smootherresidual images (i.e. displaying less large or frequent residual anomalies) to higherentropy values (1.12—model 62), which indicate rougher or spikier residual images.The gravity residual images return low entropy values for the residual grids (between0.35 and 0.37), meaning that the gravity residual response fluctuates less than theaeromagnetic residual response. The lesser fluctuation of the gravity signal is due tolonger wavelengths than those exhibited by the magnetic signal. Longer signal wave-lengths reflect how gravity data were collected with larger station spacing (averageof 10 km) than aeromagnetic flight lines (200 m) (Perrouty et al. 2012). In addition,magnetic susceptibility typically displays a larger range of variation, especially in thepresence of magnetically remanent geology (Muxworthy and McClelland 2000). Ahigh entropy value does not indicate a large misfit between observed and calculatedresponses, rather that sections of the residual grid frequently differ. Figure 7 showsthat several high-magnitude residual anomalies fall within the regions of the modelthat contain Tarkwaian series model elements, especially in the aeromagnetic residualimage. Perhaps the model does not adequately represent the level of litho-stratigraphiccomplexity in this region. For example, just two stratigraphic units, td and tkc, repre-sent the Tarkwaian series. Additional units representing the polydeformed sequenceof dolerite sills, phyllites, conglomerates and sandstones may be required to match thehigh-frequency response shown in the observed aeromagnetic grid (Fig. 3b).

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560000 600000 640000 68000052

0000

5500

0058

0000

6100

0064

0000

6700

00

hgiHhgiHResidual

0 20 40 60 8010Kilometres

LowevitisoPevitageN

Coastline

Towns

Tarkwaian Series

560000 600000 640000 680000

5200

0055

0000

5800

0061

0000

6400

0067

0000

hgiHhgiHResidual

0 20 40 60 8010Kilometres

LowevitisoPevitageN

Coastline

Towns

Tarkwaian Series

Bogoso BogosoWassa

Wassa

Damang

Damang

Sekondi - Takoradi Sekondi - Takoradi

Cape CoastCape Coast

(b)(a) scitengamoreAytivarG

Kilometers Kilometers

Fig. 7 Analysis of residual image anomalies and regions showing the highest degree of misfit betweenthe observed and calculated images. The main residual anomalies appear to be associated with the EarlyBirimian and Tarkwaian Series units

The two-dimensional correlation metric results support the dH metric results byshowing that the observed and calculated responses for all models do not match verysuccessfully. The dH results shown in Table 2 are high for both the aeromagneticand gravity calculated responses, supporting the low correlation shown by the two-dimensional correlation metric. Overall, the geophysical metric results suggest that itis very likely the Ashanti Greenstone is not representative of reality, and subsequentlycould be improved through geophysical inversion.

3.5 Principal Component Analysis

Ranking each model according to a geodiversity metric allows easy comparison ofmodels, but only communicates model rankings for that particular metric. A moresophisticated means of analysis is required to determine the model rankings across theentire set of geodiversity metrics. Simultaneous comparison of geodiversity metricsis successfully achieved via principal component analysis (PCA) (Krzanowski 1996;Jolliffe 2002; Scheidt and Caers 2009). There are several reasons why PCA was chosento perform geodiversity metric analysis. Firstly, measurements with different unitscan be compared, for example volume (m3), surface area (m2), rms misfit (mGalsor nT) or depth (m). Secondly, models can be ranked according to Hotelling’s T 2

statistic (Hotelling 1931; Krzanowski 1996) to determine whether they exhibit typicalor diverse characteristics. Finally, variance contained within each metric can be used

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by the PCA to identify which metric contributes most to model suite uncertainty. PCAassumes that the data show linear relationships, which were found in the geodiversityanalysis data. Refer to Appendix C for further descriptions of PCA and the methodexecuted in this manuscript.

3.5.1 Geodiversity Principal Component Analysis

Hotelling’s T 2statistic identified models 3, 38 and 101 as the most diverse examples,in descending order of diversity. Models 92, 33 and 59 were identified as examplessharing the most common characteristics, in descending order of commonality. Iden-tifying the most influential geodiversity metric was aided by three-dimensional PCAvisualization. It was found that just under 50 % of variance could be explained by thefirst three components, with the following 19 components explaining the remainder(Fig. 8). Figure 9a shows a bi-plot, where points and vectors are plotted, each repre-senting different aspects of the PCA. This particular bi-plot is in three-dimensions,with the first three principal components plotted on the x, y and z axes, respectively.The distribution of points represents the relative distance each model has from thecentre of the dataset or model space. The coordinates for each point are scaled withrespect to the influence of each metric, the axis that they are plotted along and thescore obtained from PCA. The locations of these points in the bi-plot are only repre-sentative, making Hotelling’s T 2 statistic a more useful means of identifying common

1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

80

90

100

Principal Component

Var

ianc

e E

xpla

ined

(%

)

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Con

trib

utui

on

Fig. 8 Cumulative distribution diagram showing that just under 50 % of model suite variability is containedwithin the first three components. This requires three-dimensional visualization of the first three componentsshown in Fig. 9

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−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Grav RMS misfit

CSA bv1−bg

CSA bg−bv3

Deep st

Comp. granite−LateEb

Vol granite−LateEb

dH−grav

2D corr grav

Entropy grav

dH mag

Std dev mag

Component 1

Com

pone

nt 2

2D bi−plot view: Component 1 and 2

−0.5 0 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Grav RMS misfit

Mag RMS misfit

CSA bv1−bg

CSA bvc1−bv2

CSA bg−bv3

Deep stb

Comp. bv4

Comp. bvc4

Vol base

dH−grav

Std dev grav

2D corr grav

dH mag

Std dev mag

2D corr mag

Component 2

Com

pone

nt 3

2D bi−plot view: Component 2 and 3

−0.5 0 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Grav RMS misfit

Mag RMS misfit

CSA bv1−bg

CSA bg−bv3

Comp. bv4Comp. bvc4

Vol base

dH−grav

v2D corr grav

Entropy grav

dH mag

Std dev mag

2D corr mag

Component 1

Com

pone

nt 3

2D bi−plot view: Component 1 and 3

−0.5

0

0.5

−0.5

0

0.5−0.5

0

0.5

Component 1

Comp. bv4

3D bi−plot view

dH−grav

Component 2

Com

pone

nt 3

Comp. granite−LateE

Vol granite−LateEb

Std dev gra

CSA bvc1−bv2

Mag RMS misfit

Std dev grav

CSA bvc1−bv2

(a) (b)

(c) (d)

b)c)d)

Key

Comp. granite−LateEb

Std dev mag

Std dev grav

CSA bv1−bgCSA bvc1−bv2

Comp. bvc4

Vol base

2D corr mag

2D corr grav

Vol granite−LateEb

Fig. 9 Results of combined principal component analysis on the Ashanti Greenstone Belt model suite.a Three-dimensional bi-plot of the first three principal components. Component 1 is plotted on the x-axis,Component 2 on the y-axis and Component 3 on the z-axis. To aid visualization each coloured section in arepresents the corresponding border of each two-dimensional plot in b–d

and diverse models. While it may be tempting to do so, it is not appropriate to assignlikelihood to a model based on its T 2 ranking. While a model close to the centre ofmodel space shares similar geodiversity characteristics as a high proportion of othermodels , this does not indicate a higher likelihood of that model representing nature.Likelihood could only be assigned if nature was almost or completely represented bythe data. The situation in all regional geological studies is that of an underdeterminedproblem and requires us to assume that any solution is as likely as another, within thebounds of geological feasibility.

Figure 9b–d shows sections through the three-dimensional plot to assist in visualiz-ing the somewhat complex arrangement of coefficient vectors, each labelled accordingto the geodiversity metric it represents. The longer vectors represent metrics that con-tain relatively higher model suite variability. Labels belonging to the shorter vectorshave been removed to simplify the diagrams. The distance a vector plots from a par-ticular component axis represents the level of variance explained by that vector in thatcomponent. The closer a vector plots to a component axis, the closer the association.Vectors that plot on the right or upper side of the diagram have positive associationsand vice versa for those that plot to the left or lower side. The largest model suitevariance is contained within Component 1 (∼20 %); therefore, any long vector with a

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close association with Component 1 (plotted on the x-axes in Fig. 9a, d) is of interest asit can be used to identify possible sources of model suite variability. Close associationsto Component 1 are split between standard deviation of gravity; the CSA of bvc1–bv2and bv1–bg; the aeromagnetic forward model RMS misfit and the aeromagnetic stan-dard deviation. A large angle between these metrics and the x-axis (Component 1) isshown in Fig. 9a. This suggests that these metrics may have a closer association withother components.

Figure 9b highlights the association of geophysical metrics (dH gravity, two-dimensional correlation gravity) and the volume and complexity of the Late Eburneangranites with Component 2. Figure 9c and d supports the association that geophysicalmetrics have with model suite variability, along with the CSA within the Early Birim-ian Series. Similarly oriented vectors can reveal the interaction between geometricaland geophysical metrics within the model suite. There appears to be some covariancebetween CSA (CSA bg-bv3, CSA bvc1–bv2) and gravity geophysical metrics (standarddeviation and dH—Fig. 9b), among the gravity geophysical metrics (two-dimensionalcorrelation and dH), the volume and complexity of the Late Eburnean granites, andthe CSA within the Early Birimian (Fig. 9c). The covariance between these particularmetrics confirms that the calculated gravitational response of the model suite is sen-sitive to geometrical variation within the models suite and that gravity data could bea useful input for inversion.

3.6 Basement Inversion

Gravity inversion of the tkc surface was performed to determine the depth and shape ofthe base of the Tarkwaian Series. The observations made during end-member analysisand PCA suggest that basement-style inversion of the tkc surfaces calculated frommodels 92, 33, 59 (the typical model representatives) and models 3, 38 and 101 (thediverse model representatives) may be successful. (1) tkc forms the basement of theTarkwaian series and the interface between the Tarkwaian and the Early Birimianunits. (2) tkc is associated with uncertainty. (3) End-member analysis identified thatthe depth of tkc is highly variable. (4) The depth of tkc affects the underlying EarlyBirimian units. The CSA of the Early Birimian units was identified by the PCA asinfluential in terms of overall model suite variability; therefore, the depth of tkc islikely to be a primary cause of this variability. (5) Geophysical forward modelling ofboth aeromagnetic and gravity data has shown that the largest variation in residual isassociated within a region defined by the boundaries of tkc. Inversion can be used withincreased confidence to resolve some of the uncertainties in the model suite as inputswere chosen using an integrated analysis of geological factors.

Commercial inversion packages now offer methods that jointly change both geom-etry and distribution of petrophysical properties (Fullagar et al. 2008; Guillen et al.2008). VPmg software (Fullagar et al. 2008) provides a means of defining the con-tribution of problematic geometrical and property elements in each iteration of theinversion. This approach is useful as it does not assume that the geometry of the modelelement is correct or the assigned petrophysical properties homogenous. The inversionprocedure iteratively optimizes the geometry, and then property of the basement unit,independent of each other.

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Petrophysical character can be useful to differentiate lithologies (Clark 1997;Manger 1963). For example, different rock types exhibit different density ranges,especially between mafic (3.0 gm/cm3‘td’) and sedimentary rocks (2.55 gm/cm3 ‘st’)Fig. 2a. However, density value overlap between some rock types, such as betweenMpohor mafic rocks (2.75 gm/cm3and σ = 0.2) and the Late Eburnean granite rocksand granitoid intrusions, can also be observed (Fig. 2a, 2.74 gm/cm3). In cases whereoverlap is present, a geometrical boundary can be used to differentiate rocks of sim-ilar petrophysical character. The VPmg (Vertical Prism magnetic gravity) inversionalgorithm is designed to use magnetic or gravity potential field data supplied as agrid. Inversion is performed on a discretized version of the three-dimensional modelcreated by subdividing the model into 1 km by 1 km vertical prisms extending fromthe Earth’s surface to the base of the model. Each prism is further subdivided into twolayers, one layer representing basement and the other representing cover. Geometricalperturbation is achieved by moving the subdivision boundary between cover and base-ment representatives along the z-axis in each prism. Density property perturbation isachieved by changing the assigned property value for each prism within the basementonly. Upper and lower density bounds were provided so that the inversion algorithmwas restricted from using values outside of sampling density ranges (Fig. 2a).

The solution of each phase of the iteration was accepted according to the reductionof the forward response misfit and by the degree of fit to the data (the Chi squared datanorm ‘L2’) (Fullagar 2009). The objective of the inversion iteration is to halve the Chisquared data misfit while using the smallest possible modification. The inversion willterminate if successive iterations do not result in reduced misfit, if a predeterminedmisfit threshold is obtained or if the maximum number of iterations is completed. Theinversion is considered a success if all these conditions are met.

The use of a uniform probability density function to create heterogeneity in theproperty distribution in the basement to the Tarkwaian series is justified in that no dataexist to suggest that this is not a reasonable assumption. A valid, although imprac-tical (access to outcrop and deep rocks) and cost-prohibitive (stratigraphic drillingis expensive) approach would be to accumulate enough petrophysical data from thebasement to the Tarkwaian series until the distribution of petrophysical values can bemore accurately defined. Examples of non-uniform distributions include anisotropictrends such as density variably increasing with depth and along regional fold axes,or magnetic susceptibility increasing with proximity to contacts or faults. This is acritical issue facing those employing inversion, especially in metamorphic terranessuch as that described in this manuscript. One could argue that inversion should notbe employed in these regions, as petrophysical values are undersampled, and het-erogeneity is presumed to be uniform. Undersampling of all data is ubiquitous ingeology and will remain so. However, uncertainty analysis can guide geoscientists towhere sampling should be focused, so the physical system can be defined with morecertainty.

3.6.1 Inversion with Magnetic Data

Magnetic data as a constraint for inversion were investigated as the finer spatial reso-lution could provide important local information (Aitken and Betts 2009; Caratori

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Tontini et al. 2009; Williams et al. 2009). Also, using two independent potentialfield datasets can reduce ambiguity associated with using a geophysical datasets toconstrain geology (Fullagar et al. 2004; Saltus and Blakely 2011). The aeromag-netic data available were of sufficient resolution, however were ill-suited for inver-sion purposes due to the pervasive presence of dolerite dykes (Fig. 10a). After fil-tering of the dykes was performed (Fig. 10b), their signal in the magnetic datasetwas still present as seen in the residual grid produced by inversion (Fig. 10c). Thestrong geophysical presence of the dykes after filtering combined with their exclu-sion from the a priori model confirmed that the magnetic data were inappropriate forinversion.

3.6.2 Gravity Inversion

The tkc surfaces from models 92, 33, 59, 3, 38 and 101 were each subjected to inver-sion. Figure 11 shows the differences in geometry between the selected a priori tkcsurfaces. Constraints were placed upon the edges of the tkc surfaces to prevent geomet-rical modification during inversion as the basal contact of the Tarkwaian is relativelywell-defined by field mapping, as reflected by the relatively low uncertainty associatedwith the zero elevation part of tkc (Fig. 5—highlighted in grey). Density constraintsare applied to both the cover layer (2.55 gm/cm3)—taken from the Phanerozoic coverunit ‘st’ and kept static through inversion) and tkc (2.7 gm/cm3, Perrouty et al. 2012;Perrouty et al. 2012) to provide a starting property. Each inversion run was set to exe-cute 100 iterations, with each iteration comprising one geometrical and one property(petrophysical) perturbation phase. The inversions were deemed successful as (1) bothor one of the data norms (L1 and L2) ≤1; (2) each modification saw a reduction inRMS misfit and Chi squared data misfit and (3) inversion convergence resulted in anRMS misfit that approximately equaled the standard deviation of the residual betweenthe calculated and observed responses. All final models produced RMS errors of 0.20mGal, except for model 101 which had an RMS error of 0.19 mGal. The inverted tkcmodel surfaces did not exceed the possible depth extents outlined in the end-memberresults, providing additional confidence that inversion results are feasible.

3.6.3 Inversion Differences

Variability maps were generated by calculating standard deviation maps (Fig. 12),following the technique shown in Aitken et al. (2013) from the six models generatedafter inversion. Values outside of the boundaries of tkc were masked to remove valuesoutside the tkc region of interest. Figure 13a shows that the largest density differencesbetween inversion results are located in the central west area and around Damang.Intermediate levels of difference are located in (1) the far northeastern corner, (2)5 km of Bogoso and (3) south-southwest of Damang. Figure 13b shows that thelargest geometrical differences are seen near (1) Bogoso, where a very high anomalyexists in the same region as density difference anomaly; (2) south-southwest of Bogosoand (3) east of Damang. These results suggest that the inversion algorithm found itdifficult to resolve these anomalous regions and/or that the initial selected modelsgeometries played a significant role in the final result. It is possible that after small

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6000

0064

0000580000610000640000

hgiH

woLG

eoph

ysic

al r

espo

nse

010

2030

405

Kilo

met

res

6000

0064

0000

580000610000640000

hgiH

woLG

eoph

ysic

al r

espo

nse

010

2030

405

Kilo

met

res

6000

0064

0000

580000610000640000

Was

saD

aman

gB

ogos

oW

assa

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ang

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oso

Was

saD

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gB

ogos

o

hgiH

hgiH

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idua

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010

2030

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met

res

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evitisoP

evitageN

(c)

(b)

(a)

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ers

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Inte

rpre

ted

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loca

tions

sho

wn

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hite

Fig

.10

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omag

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data

and

the

influ

ence

ofdo

leri

tedy

kes.

aPr

e-m

aske

dT

MI

aero

mag

netic

grid

show

ing

the

pres

ence

ofN

NE

-an

dW

SW-t

rend

ing

dyke

s.b

TM

Iae

rom

agne

ticgr

idaf

ter

mas

king

and

fillin

g.c

Res

idua

lgr

idof

the

mag

netic

base

men

tin

vers

ion

show

ing

the

mag

netic

sign

atur

eof

the

dyke

sre

mai

nev

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ter

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ring

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Fig. 11 Colour-coded elevationresidual between tkc surfacesfrom Model 3 and Model 92, theextent of which is outlined inblack. Red contour lines havebeen added to emphasizeresidual geometry

-800

-600

-400

-200

0

200

400

600

Residualdepthcontours(100minterval)

Res

idua

l dep

th (

m)

DensityModel A

DensityModel B

DensityModel C

2.75

2.70

2.76

2.57

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2.77

2.61

2.58

0.02

0.11

0.11

0.08

0.05

0.06

0.04

0.01

0.03

Fig. 12 Comparison technique employed to detect variation between different inverted models. The exam-ple shown here uses density values (g/cm3). Models A, B and C are fictitious inverted models and thecells shown are a sample of a larger grid. The ‘σ ’ grid at right displays the standard deviation from thecorresponding cell in each model, i.e. the top-left cell (0.02) is the standard deviation of the top-left cellvalue from Model A (2.75), Model B (2.76) and Model C (2.72)

number of iterations (up to ten) the input model may still retain its original propertiesand influence the inversion result. However, it is unlikely that after 100 iterationsthe initial model would still be influential. Analysis of all changes performed by theinversion algorithm also revealed that virtually the same locations were perturbed forboth geometry and property across the set of six inverted models (Fig. 14). The almostperfect correlation coefficients shown in Table 3 emphasizes that there was almost nodifference in location for geometry and property changes during inversion. Differencesbetween the inverted models cannot then be attributed to possible variations betweenperturbed locations in each inversion run. Another reason why the inversion algorithmhas difficultly resolving these anomalous areas needs to be uncovered.

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600000 640000

5500

0058

0000

6100

0064

0000

Residual

0 6 12 18 243KilometresCoastline

Towns

Tarkwaian Series

600000 640000

5500

0058

0000

6100

0064

0000

Residual

0 6 12 18 243Kilometres

Coastline

Towns

Tarkwaian Series

Kilometers

(b)(a) yrtemoeGytisneD

Bogoso

Wassa

DamangDamang Bogoso

WassaDamang

Kilometers Kilometers

Fig. 13 Variability map showing the differences between inversion results for a density and b geometryacross inversion model results. Property values are used to calculate the density variability in a and depthvalues to calculate the geometrical differences in b. The variability maps are generated using standarddeviation value for each x, y point (see Fig. 12)

Table 3 Correlation coefficient matrices for (a) density and (b) geometry modification variability maps

(a) 3 33 38 59 92 101

3 1.0000

33 0.9980 1.0000

38 0.9992 0.9979 1.0000

59 0.9982 0.9992 0.9974 1.0000

92 0.9981 0.9992 0.9986 0.9982 1.0000

101 0.9989 0.9960 0.9976 0.9962 0.9959 1.0000

(b) 3 33 38 59 92 101

3 1.0000

33 0.9965 1.0000

38 0.9980 0.9965 1.0000

59 0.9971 0.9990 0.9964 1.0000

92 0.9972 0.9983 0.9981 0.9973 1.0000

101 0.9982 0.9938 0.9955 0.9943 0.9941 1.0000

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Den

sity

Geo

met

ry3 33 38 59 92 101

Fig. 14 Variability maps showing that the location and magnitude of modifications the inversion algorithmhas performed to basement (density) and tkc surface (geometry). Each map displays the differences betweenthe first and last iteration for both density and geometry, with blue = low, green = moderate and red =high. Models 3, 33, 38, 59, 92 and 101 are shown in that order from left to right. Density comparisons areshown in the top row, geometry comparison in the bottom row

Correlation with uncertainty assisted in addressing why some areas were more dif-ficult than others to solve during inversion. The three-dimensional uncertainty grid(Sect. 3.3) was converted into a two-dimensional representation by projecting thethree-dimensional grid onto a plane located at the topographic surface. The meanvalue of stratigraphic variability was used to incorporate the values in each column.The two-dimensional stratigraphic variability grid shown in Fig. 15a was masked toremove values outside the region defined by the borders of tkc. The correlation coeffi-cient for the differences in density (Fig. 15b) between the inverted models and strati-graphic variability is 0.77. The correlation coefficient for the differences in geometry(Fig. 15c) between the inverted models and stratigraphic variability is 0.67. Thesehigh coefficient values indicate that there is a link between locations identified asgeologically uncertain, and locations geophysical inversion finds difficult to reconcileagainst the observed geophysical response. Thus, the inversion algorithm could beguided by stratigraphic variability to focus on areas of high uncertainty to achievehigher confidence in inversion results.

4 Geological Significance

Comparison of the inverted models indicates that the inversion algorithm responded togeological structures existing in both the geological model and geophysical datasets.Cross-validation with the depth bounds determined through end-member analysis(Table 2—maximum depth of tkc = 8,050 m) was performed and no part of the tkcsurface exceeded the depth limit of 8,050 m, precluding the need to geologically justifythe presence of anomalously deep sections of the Tarkwaian Series. The geometry ofthe tkc surface overall has been modified to be shallower. The greatest geometricalchange, as identified in the variability maps, is located in the central western area closeto the Ashanti Fault. A region of thick Tarkwaian Series of sedimentary rocks seen

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600000 64000055

0000

5800

0061

0000

6400

00

Stratigraphic variability

0 6 12 18 243Kilometres

Coastline

Towns

Tarkwaian Series

600000 640000

5500

0058

0000

6100

0064

0000

Residual

0 6 12 18 243KilometresCoastline

! Towns

Tarkwaian Series

600000 640000

5500

0058

0000

6100

0064

0000

Bogoso

Wassa

Damang

Kilometers

Density

Geometry

(a) (b)

(c)

Fig. 15 Comparison of stratigraphic variability with inversion differences maps. a Image showing the pro-jection of uncertainty (stratigraphic variability) onto a zero-elevation, horizontal two-dimensional surface.This image was used to correlate model uncertainty with b density and c geometry inversion modellingshown. b and c are reproduced from Fig. 13a, b

Fig. 16 Inversion resultsshowing geometrical changesnear the Ashanti Fault (labelled).The input model (a) andinversion surfaces (b) have beenpainted according to their depthin metres (scale at right). Notefeature 1 from the input modelhas not been retained duringinversion modelling, whilefeature 2 has been enlarged. Theoverall depth of the tkc surface isshallower after inversion

0

1000

2000

3000

4000

5000

6000

7000

Met

ers

Ashanti Fault

Inverted tkc

Input model tkc

1

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in the a priori models was not retained in the inverted models (Fig. 16—feature ‘1’),though a region of thicker sedimentary rocks 23 km northeast along the strike of theAshanti Fault near Bogoso has been thickened correspondingly (Fig. 16—feature ‘2’).At the location of feature ‘2’, three faults interact with the base of the Tarkwaian: theAshanti Fault, striking northeast–southwest; an unnamed fault interpreted by Perroutyet al. (2012), striking west-southwest–east-northeast; and the faulted Tarkwaian/EarlyBirimian contact. The protrusion of the tkc surface through the faulted Tarkwaian/EarlyBirmian contact is not a reasonable geological proposition. The result was likelyproduced as the inversion algorithm is unaware of these faults and their topology, asfaults are removed during discretization of the model. Another possible solution canbe offered geophysically. The inversion has modified the tkc surface to account for alower magnitude gravity anomaly in this location (Fig. 3a). A number of geologicalreasons can be made to account for this anomaly, including a thick layer of relativelyunconsolidated basin infill (unlikely, given the age and metamorphic history of theregion) or (more likely) the presence of Late Eburnean granitoid near the unnamedfault, a conduit which facilitated magma transport and emplacement.

The density distribution displayed in the tkc surface does not present too many obvi-ous geological dilemmas. Possible Early Birimian structures beneath the tkc surfaceare reflected in the density distribution and also correlate to the Early Birimian modelelements (Fig. 17). Folded surfaces can be interpreted from the density response. Thelarge high-magnitude density anomaly between Wassa and Damang is spatially linkedto a large-scale isoclinal fold in the model. The density anomaly may be due to eithera higher volume of higher density stratigraphic units due to the presence of the fold orthat isoclinal folding produced an accumulation of higher density rocks (for example,amphibolites) in the thickened hinge zone.

The next course of action would be to make the necessary adjustments to the geo-logical model and petrophysical distribution and to resubmit to the workflow proposedhere. The typical three-dimensional modelling workflow uses iterations of geometricalmodelling, inversion, revision, re-modelling, inversion and so on, until an acceptableresult is reached. The workflow here differs from current methods by removing theneed to make small model adjustments and re-invert as these small adjustments are sim-ulated by exposing a range of geological possibilities to inversion. The model adjust-ment phase of the typical three-dimensional modelling workflow is now significantlyoptimized in proportion to the insights gained from the proposed analyses. Multipleeffectual adjustments can be made in one adjustment phase, rather than in many.

5 The Future of Geophysical Inversion

This section assesses using three-dimensional uncertainty grids as guides for geophys-ical inversion. It has been shown in the previous section that geologically uncertainregions (Fig. 15) are correlated to regions receiving heavy modification via geophysicalinversion (Fig. 13a, b). The significance of this correlation is that a three-dimensionaluncertainty grid attributed with stratigraphic variability values can assist inversion. Bytargeting inversion on uncertain regions, the inversion algorithm can focus on locationsthat require the heaviest modification, rather than relying on least squares or stochas-

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600000 640000

640000

600000

560000

620000

580000

000046000026 000046000026

640000600000

600000

2.82.752.72.652.6

Density (gm/cm )3

tkc boundary

(a) Plan view

mottob morf weiV (d)mottob morf weiV(c)

(b) Oblique view from southeast

The oblique view has been sliced to expose the coincidenceof the density anomaly and the fold

Location of slicein oblique view

nwohs era stinu naimiriB ylraEnwohs ton era stinu naimiriB ylraE

Density property contours are shown in b), c) and d).

0.05 gm/cm intervals:

0.01 gm/cm intervals:

The density anomaly discussed is circled in white.

3

3

Fig. 17 Model 92 inverted model density distribution. A large density anomaly is circled in white in allviews. b The model surfaces have been ‘sliced’ from the east to expose the centre east of the model. Thelocation of the slice is shown in a with a red line

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Base of theTarkwaian Basin

Pro

babi

lity

(%)

Stratigraphic ID Stratigraphic ID

Probability (%

)

1

2

(a)

(b)

-5000 -4000 -3000 -2000 -1000 0MetersDepth

contours(500minterval)

Stratigraphic Variability

4 6 8 1210 14

0 2 4 6 8 10 120

2

4

6

8

10

12

14

16

18

10 11 13 150

5

10

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base

bvc1

bv1

bvc2

bv2

bg bv3

bvc3

bvs

bv4

bvc4

bv5

bv4

bvc4

Mpo

hor

tkc

Fig. 18 Example of how a three-dimensional stratigraphic variability grid guides inversion. a View fromthe west of a surface representing the base of the Tarkwaian Basin (tkc) overlain by the points representingstratigraphic variability. Note the depth and stratigraphic variability scales at the top-right. The red boxindicates the location of b). Two locations shown in b) are highly uncertain (1) and less uncertain (2). Thehistograms display the stratigraphic identifier value of each different unit detected at these locations on thex-axis) (the corresponding unit name is labelled) and the probability of their occurrence on the y-axis

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tic methods to search for where and how modifications should be applied. Additionalinversion constraints are offered by a three-dimensional uncertainty grid that suppliesa frequency histogram of geological units to the inversion algorithm by defining whichstratigraphic units may exist at any given point, followed by the likelihood of each unitbeing present. The inversion could perform modifications according to the frequencyhistogram (Fig. 18), rather than relying on global constraints such as perturbation periteration or petrophysical ranges. The integration of an uncertainty grid into inversionwould increase the likelihood of finding a solution that honours both geological andgeophysical data.

6 Conclusions

The workflow described in this manuscript reduces subjectivity in the geophysicalinversion process by (1) producing multiple geological realizations from input dataand (2) guiding the choice of geophysical input data through geodiversity analysis.Geological models now have more use than just a starting point for inversion or acontainer of constraints for the distribution of petrophysical properties. The opera-tor is now informed of the geological and geophysical possibilities contained withinthe model. Geodiversity end-member analysis and PCA help to guide the choice ofinversion parameters and assist in assessing results by reducing the dimensionality ofthe problem. Complex interactions within the Ashanti Greenstone Belt model suitewere identified with the aid geodiversity analysis and PCA. Notably, the relationshipbetween gravity data misfit and the CSA of the Early Biriman Series supported thechoice of gravity data for inversion. PCA of geodiversity metrics selected models 92,33, 59, 3, 38 and 101 and provided multiple inversion starting points. Visualization ofstratigraphic variability highlighted sources of uncertainty, such as modelling wholecollections of granite bodies as one geological unit, rather than as individual bod-ies. Perhaps most importantly, the three-dimensional stratigraphic variability grid hasbeen recognized as a new inversion constraint that can incorporate additional geologi-cal information into a dominantly geophysical process. Although subjective decisionsare still required by the operator, the techniques used in this workflow remove somesubjectivity while simultaneously increasing the role of geological input. Finally, theinversion result provided a geologically reasonable model of the Ashanti GreenstoneBelt, southwestern Ghana.

Acknowledgments Thanks go to Intrepid Geophysics for technical assistance and access to the 3DGeomodeller API (Application Programming Interface). Special thanks to Philip Chan of the MonasheResearch Centre for his patience in providing technical assistance with Monash Sun Grid computing. Manythanks to Eric de Kemp and his skills with GoCAD. We are grateful for the thorough and constructive reviewsfrom one anonymous reviewer and Li Zhen Cheng, Université du Québec en Abitibi-Témiscamingue.Finally, we thank Guillaume Caumon and Pauline Collon-Drouaillet for their efforts in compiling theSpecial Publication on 3D Structural Modelling.

Appendix: Description of Geodiversity Metrics

The following summary of geodiversity metrics is a compilation of the methods pre-sented in Lindsay et al. (2013a,b).

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Dep

th (

m)

Dep

th (

m)

0 Shallowest (200 m) Deepest (1100m)

Dep

th m

etric

calc

ulat

ion

Vol

ume

met

ricca

lcul

atio

n

Model X Model Y

721 = slleC231 = slleC

Largest volume Smallest volume

Fig. 19 Formation depth and volume calculations. The geological bodies are shown as a set of voxelsrather than surfaces, according to the described method. Formation depth analysis detects the location ofthe deepest or shallowest voxel attributed to a particular geological unit. Volume analysis is performed witha voxel count of the geological unit. Reprinted from Tectonophysics, 594, Lindsay et al. (2013a), Copyright2013 with permission from Elsevier

A1. Geometrical Geodiversity Metrics

A1.1. Formation Depth and Volume

The shallowest extent of a stratigraphic unit is calculated by determining the depth ofthe shallowest voxel in the formation under study. The deepest extent is determinedfrom the deepest voxel of a given formation (Fig. 19). Volumes are determined bymultiplying the count of stratigraphic unit voxels with the voxel volume, or by simplydisplaying the voxel count (Fig. 19).

A1.2. Average Mean Curvature

Curvature is determined by finding the orientation of a normal section rotated aroundthe surface normal (

−→N ) at P, a given point on a folded surface (Lisle and Toimil

2007). The maximum magnitude of curvature along the intersection of the foldedsurface is principal curvature k1. Curvature of the intersection of the surface and anormal section perpendicular to k1 is principal curvature k2 so that k1 > k2. Positiveprincipal curvature indicates convex-upward and negative principal curvature indicatesconcave-upward. Mean curvature (M) is the arithmetic mean of k1 and k2

M = k1 + k2

2. (1)

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Fig. 20 Short-distanceneighbour calculationconstrained to locating the sixclosest neighbours in a regulargrid made of cells withequidistant axes. The samplelocation assigned with astratigraphic ID of ‘1’ issurrounded by cells assignedwith IDs of ‘1’, ‘2’ and ‘3’resulting in a neighbourhoodrelationship metric result of ‘3’.Modified from Lindsay et al.(2013a)

1

1

1

2

2

Metric result = 3

313

When M > 0 an antiformal (convex) surface is observed, when M < 0 a synformal(concave) surface is observed. M = 0 indicates the surface is a flat plane or a surfacewhere k1 = −k2 known as a ‘perfect saddle’ (Lisle and Toimil 2007). Positive Gaussiancurvature (G)

G = k1 · k2. (2)

indicates that both principal curvatures k1 and k2 have the same sign and the surfaceresembles a dome or a basin, if inverted. Negative Gaussian curvature indicates thatprincipal curvatures have different signs and the surface resembles an antiformal orsynformal saddle (Fig. 20) (Lisle and Toimil 2007; Mynatt et al. 2007). Please referto Besl and Jain (1986), Lisle and Robinson (1995) and Lisle and Toimil (2007) forfurther details and case studies.

A1.3. Neighbourhood Relationships and Contact Surface Area

Neighbourhood relationships are calculated with a k-nearest neighbour algorithm (k-NN). The locations of k nearest neighbours around a point of reference are found, andfrom that their Euclidean distance can be calculated (Bremner et al. 2005; Friedmanet al. 1977). The Euclidean distance is used to constrain which voxels are counted asneighbours and the shortest distances measured along eastings, northings and depthaxes resulting in a six-neighbour relationship (Fig. 20). The surface area of the contactbetween stratigraphic units is identified together with the proportion of overall contactrelationships within the three-dimensional volume.

A2. Geophysical Geodiversity Metrics

A2.1. Root Mean Square

A root-mean-squared value, or ‘rms’, is

xrms =√

1

n(x2

1 + x22 + · · · + x2

n ), (3)

where x1, x2 . . . xn equals the difference between one value and another in a grid.

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A2.2. Standard deviation

The standard deviation (s) of an image is taken as a measure of value spread over agrid

s =(

1

n − 1

n∑i=1

(xi − x)2

) 12

. (4)

A2.3. Entropy

Information entropy (E) is used to measure the average bits per pixel over an entireimage, representing its global information content (Batty 1974; O’Gorman et al. 2008;Shannon 1948)

E =N∑i

pi log pi , (5)

where E is the sum of all products of p (probability) of each possible outcome (i) outof N total possible outcomes. E = 0 indicates that the image is dominated by largeregions of the same value. If in a 1-bit system of two integer values 0 and 1, Emax = 1,

the image is made of equal proportions of possible values in this case. E = 1 reflectsthat it is equally likely to find a ‘0’ or a ‘1’ in a given image.

A2.4. Two-Dimensional Correlation Coefficient

Two-dimensional correlation coefficients are typically calculated in geophysical andengineering applications to track changes in two- and three-dimensional objects. Thetwo-dimensional correlation coefficient r is calculated using

r =∑

m∑

n(Amn − A)(Bmn − B)√∑m

∑n((Amn − A)2)(

∑m

∑n(Bmn − B)2)

, (6)

where A is the global mean of set one and B is the global mean of set two. The purposeof this technique is to identify when patterns in two different sets resemble one another.

A2.3. Hausdorff Distance

The Hausdorff distance measures how far points in two different subsets are fromeach other. The distance can then be used to understand the level of resemblancetwo superimposed objects have to each other (Huttenlocher et al. 1993; Olson andHuttenlocher 1997; Rucklidge 1997; Sim et al. 1999; Wang and Suter 2007). TheHausdorff distance, dH, between two sets is

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dH(X, Y ) = max

{supx ∈ X

infy ∈ Y

d(x, y),supy ∈ Y

infx ∈ X

d(x, y)

}, (7)

where X and Y are two non-empty subsets of a metric space (M, d), ‘sup’ is thesupremum, ‘inf ’ is the infimum and x and y are points within sets X and Y , respectively.The supremum defines the upper bounds of subset Y within set X , whereas the infimumdefines the lower bounds of subset Y within set X . First, determine the shortest distancefrom point x1 to any point in Y, then the shortest distance from point x2 to any pointin Y, point xn to any point in Y. The Hausdorff distance is the largest distance of thosemeasured from x1, x2, . . ., xn .

6.1 A3. Two-Stage Principal Component Analysis Procedure

The following summary of PCA is a compilation of the methods presented in Lind-say et al. (2013a,b). Common sources of model suite variability can be identifiedby combining metric variability into principal component analyses. Thus, PCA hasbeen chosen to understand the multidimensional problem presented in this manu-script. PCA is a multivariate exploratory data technique that allows complex datainteractions to be displayed (Jolliffe 2002). PCA transforms the data orthogonallyand highlights relevance by re-organization of the data with respect to the attributeunder analysis. Central to PCA is conversion of the potentially correlated originalvariables, i.e. the geodiversity metrics, into uncorrelated principal components. Dataconversion is performed so the first principal component displays the greatest variance,with each subsequent component displaying progressively lower degrees of variance.Each component contains a contribution of variability from all the metrics submit-ted to the PCA, while each following component contains the next highest possibledegree of remaining variance, with the proviso that it is uncorrelated to the precedingcomponents (Jolliffe 2002).

The MATLAB princomp function (http://www.mathworks.com.au/help/toolbox/stats/princomp.html) was used to calculate the coefficients, or loadings, of metriclinear combinations. Principal components are calculated in the following manner:

1. Statistical calculations such as the mean, subtraction of deviations from the meanand covariance matrix calculation.

2. Sort the eigenvalues and eigenvectors of the covariance matrix in descending order.3. Determine eigenvector contribution to eigenvalues.4. Basis vector determination.5. Project z-score-converted original dataset onto the basis vectors.

Loading vectors show the amount of variability any metric has toward a principalcomponent (see Fig. 21). The length of a vector and the angle it has with the principalcomponent axis indicate the strength of the relationship. The longer the vector, themore variability contained within the metric it represents. The smaller the incidenceangle of the loading vector to the principal component axis, the more related the vectoris to that principal component. Stage one analyses individual stratigraphic units withina metric, to find those which best describe variability within each metric. The first stageis critical to identifying: (1) redundant metrics, and filtering them from analysis and;

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Fig. 21 Two-stage PCA methodused in this contribution shownas a flowchart. Stage onegeodiversity analyses of thevolume and depth of formations‘A’, ‘B’, ‘C’ and ‘D’ are abovethe solid black line. Noteformations are subscripted‘V’(volume) or ‘D’(depth) toindicate the relevant geodiversitymetric. Stage two results arebelow the black line and arewhere all the metrics (volumeand depth, in this example) aresimultaneously examined withPCA. Note that any metrics, notjust depth and volume, can beincluded in stage two. Thevolume of formations B and C(circled) and the depth offormations A and D are shown instage one to contribute most tomodel suite variability. The mostinfluential metric for model suitevariability, the depth offormation A, is revealed in stagetwo. Reprinted fromTectonophysics, 594, Lindsay etal. (2013a), Copyright 2013 withpermission from Elsevier

Barycentre Outliers

Component 1

Com

pone

nt 2

0

0

0

Component 1

Com

pone

nt 2

0

0

Component 1

Com

pone

nt 2

0

Model

Loading vector

C

B

D

A

D

A

B

C

B

D

CD

ADV

V

V

V

V

V

D

D

D

D

CVBV selected

for stage two PCA

Model space boundaries

AD

PCA - Stage oneFormation volumemetric

PCA - Stage oneFormation depth metric

PCA - Stage twoCombinationanalysis

and

DDAD selectedfor stage two PCA

and

Most influentialmetric overall is

(2) retaining those metrics that do adequately describe variability in the model suitefor analysis in stage two. The filtering of redundant metrics is only performed forthose metrics that analyse each stratigraphic unit (depth, volume and short-distanceneighbourhood relationship). Metrics that represent the model with a single value,such as the geophysical metrics (refer Sect. A2), are input directly into stage two.

Stage two repeats the PCA process, with all the metrics described in Sects. A2and A3. A simplified example of the two-stage PCA process is shown in Fig. 21.

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Stratigraphic units B and C contribute most to the first principal component and secondcomponent variability respectively for the volume metric. Units A and D contributemost variability to the first and second principal components respectively for the depthgeodiversity metric. Units B and C represent the volume metrics, and units A and Drepresent the depth metrics. Figure 21 shows that in stage two, where the volumeand depth metrics are combined, depth of unit A has been determined to be the mostinfluential in terms of model suite variability.

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