�

Integrating geological uncertainty into combined geological and potential field inversions

Abstract

The reliability of three-dimensional (3D) models are

determined by the combined quality of input data and

supporting intepretations. Uncertainties inherent in

data collection, processing and interpolation result

in 3D models that do not adequately represent the

geological target. Weathered outcrop, unclear structural

and geophysical interpretations or disoriented drill core

are potential sources of uncertainty affecting input

data. In addition, complex geology often has multiple

tectonic interpretations attempting to reconcile the data.

Subsequently, model topology can vary depending on

the tectonic interpretation accepted by the modeller.

We propose a technique that uses implicit modelling

techniques to generate a 3D ‘uncertainty grid’. This grid

is generated by (1) varying structural field measurements

from the input data set; (2) generating models from varied

data sets; and finally (3) assessing the magnitude geological

surfaces differ between each model. The results can be

plotted in a 3D modelling package effectively displaying

areas within the model that display high variation, which

we associate with area of high uncertainty. There is a

fourfold outcomes to collecting this information: (1) the

model can be easily assessed for reliability; (2) further

data collection surveys can be planned and implemented

using the information generated by this technique; (3)

information gathered using this technique can be applied

to geophysical inversion and (4) differing model topologies

can be assessed and compared. Access to this information

should enable more reliable models to be generated.

Introduction

3D modelling is a geoscientific discipline that has shown

dynamic growth in the last decade. Various geoscientific

problems are increasingly being solved with the use of 3D

models, including those applied to minerals exploration,

environmental, tectonic and mining studies (Mallet, 2002).

The quality of a 3D model is indicated by its reliability to

represent real geology. To be reliable, the 3D model needs

to reconcile all available geological and geophysical data

from the study area (Fullagar et al., 2000; Guillen et al.,

2008; Wellman et al., 2010). Further, the ability of the

model to represent geology at the surface (where structural

field observations are usually more abundant) and at

depth (where observations are usually less abundant) is

fundamental for the model to be useful to the practitioner.

Subsequently, modelling deep geological structures usually

requires the use of geophysical surveying techniques,

especially in the absence of drill-hole data (Jessell, 2001;

Martelet et al., 2004). The geophysical inversion is a

powerful data-processing technique that can be used in

conjunction with a 3D model to test and augment the

association of modelled geology with the geophysical

signature (Guillen et al., 2008; Tarantola, 2005). The

combination of geophysical data and inversion is often

used in covered terranes where there are sparse geological

field observations (Betts et al., 2003; McLean and Betts,

2003). A certain amount of control over results is therefore

ceded to computer-based algorithms by the operator.

Subsequently, the success of applying these techniques

to answer a geological problem is heavily dependent on

the quality of data and algorithms. Uncertainty inherent

in the data can therefore be transferred undetected into

the model, adversely affecting reliability.

Mark Lindsay School of Geosciences, Monash University andIRD, Universite Toulouse IIImark.lindsay@monash.edu

Laurent AilleresSchool of Geosciences, Monash Universitylaurent.ailleres@monash.edupeter.betts@monash.edu

Mark JessellIRD, Universite Toulouse IIImark.jessell@lmtg.obs-mip.fr

Eric deKempGeological Survey of Canadaedekemp@.nrcan.gc.ca

Corresponding author: Mark LindsayInstitution: School of Geosciences, Monash Univer-sity / IRD, Universite Toulouse IIIEmail address: mark.lindsay@sci.monash.edu.au

2

Geological uncertainty and geophysical ambiguity

Aspects of geological and geophysical data collection and

processing (including geophysical inversion) need to be

considered when assessing a 3D model. Two of the most

significant sources of model uncertainty are geophysical

ambiguity and geological uncertainty. Concerns associated

with geophysical data ambiguities are well known and have

been acknowledged since Nettleton (1942) began critically

assessing the interpretations of his contemporaries. It

was recognised in his and further studies that a number of

possible outcomes could fit a particular geophysical data

set and render any interpretation meaningless without

the proper geological controls (Clark, 1983; Gunn, 1997).

Endeavours to remove geophysical ambiguity from

geophysical interpretation are critical and are usually

performed by collecting petrophysical data appropriate to

the potential field being utilised (see Joly et al. (2008) ,

Williams et al. (2009) and Stewart & Betts (2009)).

Geological uncertainty is generally associated with cross

section interpretations, structural field mapping and drill-

core logging (Groshong, 2006; Thore et al., 2002). Some

geologists may not agree on the location and orientation

of a fault or stratigraphic contact or an inaccurately

calculated drill-hole angle can adversely affect geological

interpretation (Groshong, 2006). Sparse measurements,

under-sampling or a lack of clear observational data

require interpretation with a degree of uncertainty to

develop a hypothesis (Gallerini and De Donatis, 2009;

Wellman et al., 2010). These data ambiguities can have a

compounding negative effect on model reliability.

Further uncertainty can be introduced through

interpolation of the input data. Implicit modelling

applications rely on algorithms to determine model

architecture, but may not make geologic sense even

though the data is honoured exactly. This is especially true

when only sparse data is available. Whether the geology

shown in the generated model is reasonable is usually not

assessed in 3D modelling workflows, except as an esthetic

decision by an informed observer.

Calculating geological uncertainty

Our cognitive system, after a period of time, accepts

a 3D geological solution as the ‘truth’. We tend not to

consider that some or all of the geological architecture

represented in the model is a ‘best-guess’ effort and

other geometries may be possible (Tarantola, 2006). The

3D solution being offered may honour the geological and

geophysical data input, but may be one of many possible

solutions that also honour the data. Understanding

where geological uncertainty lies within a 3D model can

help mitigate geological misrepresentations. Further,

quantified geological uncertainty could be used as

input to guide geophysical inversion, effectively adding

a geological constraint into the inversion process. We

propose a technique that explores the 3D model space

with a given geological data set. This technique uses

information gathered from the model space to assess

uncertainty within the 3D model volume.

We use an implicit modelling application, 3D

Geomodeller, to create a reference model with the

geological data set (Fig. 1). This includes ‘strike-and-dip’

orientation measurements that define geologic surfaces.

Orientation measurements are then varied in both strike

Figure 1. (a) Reference model shown in 3D Geomodeller in plan view and (b) as interpolated in three dimensions. This model shows an refolded antiform cross-cut by two thrust fault gen-erations and a normal fault.

(a)

(b)

3

and dip within ± five degrees of the original measurements

to produce a large number of models, each different from

the reference model (Fig. 2). Measurements are varied in

this manner to produce different model interpretations,

simulating the variation in geological interpretation that

may be observed over a geological terrane.

Areas of uncertainty can be identified as those that

show a high degree of lithology variation across the

suite of models after the original orientation data has

been perturbed. This is done by calculating the standard

deviation of sampled lithology index integers at a discrete

point within a regular sampling grid for each model. The

standard deviation of grid points are collated to form an

uncertainty map, and when draped over the reference

model geologic surfaces, emphasize regions of high

uncertainty (Fig. 3a). Uncertainty grid points can be

converted to a voxet and calculated similar to a resource

block model to determine the volume of uncertainty

within a model (Fig. 3b). This uncertainty information can

then assist identification of areas that require more data

to constrain modelled geology.

We anticipate that uncertainty maps will also provide

a measure to assess geology in knowledge-based

interpretations. Regions rich in data density (i.e. cross

sections) and those with sparse measurements (i.e.

regions between cross sections) can be interrogated

without bias for regions of high or low uncertainty after

implicit interpolation.

Potential Field Inversion

The purpose of potential field inversion is to reduce the

misfit between observed and forward calculated potential

Figure 2. Model A is the reference model (shown in Fig. 1), models 1 and 10 were calculated from varied input data. The red circles shown areas of significant difference (and high vari-ation) between each model

Figure 3. (a) Uncertainty map of model shown in Fig 1. Colour scale represents uncertainty (standard deviation) magnitude. (b) Uncertainty volume showing the top 25th percentile of val-ues representing 1.97% of the overall volume. Both methods of visualisation show high uncertainty associated with fault-ing

(a)

(b)

Figure 4. Conceptual diagram describing the proposed method of investigating model space. Yellow circles represent different models generated by varying orientation measurements, black lines represent topological boundaries. Only models that sat-isfy both geological and geophysical data (the blue region) are accepted by the process (shown by dashed lines).

4

field data. This is performed by perturbing the geometry

or physical property distribution of a reference model

iteratively until the misfit falls below a given threshold.

The process is ambiguous and does not include a check to

verify whether the results honour the original geological

input data.

The technique proposed here can be used to improve

the role geology has in the potential field inversion process

(Fig. 4). Firstly, the varied models can be used as a suite

of reference models subjected to inversion. Secondly,

uncertainty information can be used to guide the potential

field inversion by identifying which model regions should

be perturbed and, based on the magnitude of uncertainty,

to what degree. Finally, geophysically acceptable models

that do not honour geological data are rejected based

on a series of, as yet, undefined geological objective

functions. The result will be a suite of both geologically

and geophysically acceptable models associated with

a 3D uncertainty volume. Different topologies can also

be examined by changing fault relationships or polarity.

By examining the model space more thoroughly,

accommodating inherent input data uncertainty and

honouring the input data, a more reliable 3D product can

be produced.

ReferencesBetts, P. G., Valenta, R. K., and Finlay, J., 2003, Evolution of the Mount Woods

Inlier, northern Gawler Craton, Southern Australia: an integrated structural and aeromagnetic analysis: Tectonophysics, v. 366, p. 83-111.

Clark, D. A., 1983, Comments on magnetic petrophysics: Bull. Austr. Soc. Explor. Geophys, v. 14, p. 49–62.

Fullagar, P. K., Hughes, N. A., and Paine, J., 2000, Drilling-constrained 3D gravity interpretation: Exploration Geophysics, v. 31, p. 017-023.

Gallerini, G., and De Donatis, M., 2009, 3D modeling using geognostic data: The case of the low valley of Foglia river (Italy): Computers & Geosciences, v. 35, p. 146-164.

Groshong, R. H., 2006, Elements of map-scale structure, 3-D structural geology: A practical guide to quantitative surface and subsurface map interpretation: New York, Springer, p. 1-32.

Guillen, A., Calcagno, P., Courrioux, G., Joly, A., and Ledru, P., 2008, Geological modelling from field data and geological knowledge: Part II. Modelling validation using gravity and magnetic data inversion: Physics of the Earth and Planetary Interiors, v. 171, p. 158-169.

Gunn, P. J., 1997, Quantitative methods for interpreting aeromagnetic data: a subjective review: AGSO Journal of Australian Geology and Geophysics, v. 17, p. 105-114.

Jessell, M., 2001, Three-dimensional geological modelling of potential-field data: Computers & Geosciences, v. 27, p. 455-465.

Joly, A., Martelet, G., Chen, Y., and Faure, M., 2008, A multidisciplinary study of a syntectonic pluton close to a major lithospheric-scale fault - Relationships between the Montmarault granitic massif and the Sillon Houiller Fault in the Variscan French Massif Central: 2. Gravity, aeromagnetic investigations, and 3-D geologic modeling: J. Geophys. Res., v. 113.

Mallet, J.-L., 2002, Geomodeling: New York, Oxford University Press.Martelet, G., Calcagno, P., Gumiaux, C., Truffert, C., Bitri, A., Gapais, D., and

Brun, J. P., 2004, Integrated 3D geophysical and geological modelling of the Hercynian Suture Zone in the Champtoceaux area (south Brittany, France): Tectonophysics, v. 382, p. 117-128.

McLean, M. A., and Betts, P. G., 2003, Geophysical constraints of shear zones and geometry of the Hiltaba Suite granites in the western Gawler

Craton, Australia: Australian Journal of Earth Sciences, v. 50, p. 525 - 541.

Nettleton, L. L., 1942, Gravity and magnetic calculations: Geophysics, v. 7, p. 293-310.

Stewart, J. R., and Betts, P. G., 2009, Late Paleo-Mesoproterozoic plate margin deformation in the southern Gawler Craton: Insights from structural and aeromagnetic analysis: Precambrian Research.

Tarantola, A., 2005, Inverse problem theory and methods for model parameter estimation, Society for Industrial and Applied Mathematics.

Tarantola, A., 2006, Popper, Bayes and the inverse problem: Nat Phys, v. 2, p. 492-494.

Thore, P., Shtuka, A., Lecour, M., Ait-Ettajer, T., and Cognot, R., 2002, Structural uncertainties: Determination, management, and applications: Geophysics, v. 67, p. 840-852.

Wellman, F. J., Horowitz, F. G., Schill, E., and Regenauer-Lieb, K., 2010, Towards incorporating uncertainty of structural data in 3D geological inversion: Tectonophysics.

Williams, H. A., Betts, P. G., and Ailleres, L., 2009, Constrained 3D modeling of the Mesoproterozoic Benagerie Volcanics, Australia: Physics of the Earth and Planetary Interiors, v. 173, p. 233-253.