Constrained gravity 3D litho-inversion applied to Broken Hill Antonio Guillen *
Gabriel Courrioux
Ph.Calcagno
BRGM France
[email protected]
BRGM France
[email protected]
BRGM France
[email protected]
Richard Lane
Terry Lees
Philip McInerney
Geoscience Australia
[email protected]
Monash University, Australia
[email protected]
Intrepid Geophysics, Australia
[email protected]
SUMMARY We present an example of a constrained inversion method that uses a categorical property, lithology, as the primary model parameter. A 3D geological model is supplied as the starting model. The topology of this model is used as prior information, together with parameters that define the probability density function of a secondary property, density, and observations of the gravity field. The 3D geological model is a prediction of the geometry of geological interfaces given points where these interfaces have been observed, structural observations of the orientation of these interfaces and a table describing the relationships between the various geological units. The interfaces are modelled as equipotential surfaces, and cokriging is used to interpolate between the supplied points. The inversion method returns a number of models which are consistent with the supplied prior information. This ensemble of acceptable models can be analysed statistically to derive conclusions. A 3D geological model of Broken Hill was constructed from a range of geological inputs. Constrained inversion of ground gravity was then carried out. Prompted by the results of initial inversions, a number of adjustments were made to both the geological model and estimates of the density for each of the units. The final inversion was used to demonstrate a high degree of internal consistency amongst these amended forms of the prior information. The use of airborne gravity gradiometry (AGG) data was deferred pending modification of the forward modelling algorithm to incorporate the band-limited character of these observations. Key words: 3D inversion, Monte Carlo, 3D geological modelling, geostatistics, gravity.
INTRODUCTION To better understand the geological processes that have operated in the past, geologists make observations on both the surface and in the subsurface, at points where outcrop, drillholes and other sub-surface exposures allow. They are then faced with the challenge of organising these observations into logical groupings and projecting the boundaries of these units such that the 3D volume of interest is completely filled th
and there is a single unit present at any point. This interpretive process is uncertain and ambiguous. The geologist can make use of various remote sensing geophysical methods to provide feedback on the interpreted extensions of geological objects. Inversions of gravity and magnetic data can be used for this purpose, but unfortunately the solutions are non-unique. Despite this, an inversion procedure can be used to analyse a geological model for consistency with potential field observations, and to provide insight into possible modifications to the geological model that would improve this consistency. One further complication is that the perception of consistency is dependent on spatial scale, as observations at more detailed scales will undoubtedly reveal increasing levels of inconsistency. We present a method to obtain a 3D probabilistic description of geological objects that utilises available data such as geological maps, borehole data, physical properties, structural data and geophysical measurements such as gravity and/or magnetic potential fields. The degree of uncertainty in the inversion output is investigated by generating and analysing an ensemble of models that are suitably consistent with the input observations. The method of constructing a geological model and testing this model with geophysical inversion has been implemented in software, 3DWEG. We illustrate this implementation by constructing a geological model of the Broken Hill district and performing constrained inversion of a ground gravity dataset.
METHOD Construction of a Geological Model A geological model in the context of this paper is a representation of the sub-surface subdivided into a number of mappable rock units, i.e. the 3D analogy of a geological map. Geological observations from surface mapping, drilling and/or underground workings are used to produce interpreted surface geological maps and cross-sections. On the basis of this information and a set of modelling assumptions, a complete and continuous model for geological objects at all points in the subsurface is created. The method implemented in 3DWEG makes use of observations that define the location of interfaces between rock units, structural observations that provide the orientation of these interfaces, and a table that defines the geometrical relationships between the units. A unique solution for the 3D geometry of the interfaces between rock unit is obtained by assuming that the observed points for each interface lie on a potential field surface. Structural observations define the
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Extended Abstracts
Constrained gravity 3D litho-inversion applied to Broken Hill
orientation of a local tangential plane to the potential field. The supplied relationships determine priority when interfaces intersect. Cokriging is used to interpolate between the supplied observation points. Physical Properties It is assumed that the probability density function (pdf) of density and/or magnetisation is known for each geological unit. To perform the inversion, we supply the shape of the distribution for each relevant physical property of each unit (e.g. Gaussian, lognormal, etc), together with parameters such as the mean and standard deviation that define the pdf. Constrained Potential Field Inversion Overview The geological model is discretised into a 3D matrix of cells, termed voxels, to produce an initial rock unit (‘lithology’) model. Lithology, a categorical variable (i.e. one that acts as a label rather than as a numerical value), is the primary model parameter. The present implementation of 3DWEG holds the lithology of surface voxels fixed throughout the inversion. The lithology associated with subsurface voxels information is free to vary, subject to the condition that the topology of the initial model remains unchanged. The inversion explores variations to the initial model which reproduce the supplied gravity or magnetic data to within a desired tolerance level. The adopted strategy is to randomly walk around the space of possible models for a given set of a priori information. This approach was proposed by Mosegaard and Tarantola (1995) and developed in 2D by Bosch et al. (2001). Many transient models are derived from the initial model using an iterative procedure. At each iteration, we make one of two possible changes. The physical property (density or magnetic susceptibility) for a randomly selected voxel that is separated from the boundary of that unit may be modified. Alternately, the lithology of a voxel that lies on the interface between two or more units may be modified and a new physical property assigned to that voxel according to a random selection from the pdf of the relevant physical property distribution for the new lithology. The voxels to which the latter operation can be applied is restricted by the constraint that the topology of the model is not altered. The change in the misfit between the observed potential field data and the potential field response calculated for the modified model is determined. This change is examined in a probabilistic framework to determine whether the modification to the model is accepted. Detailed Outline of the Inversion Algorithm The inversion algorithm can be defined using 11 steps.
1. Build the a priori geological model To obtain the starting model (m0) we build a geological model constrained by a set of data (p0) that cannot be modified and by a set of hypothetical data (ph) that can be modified as long as the change does not alter the topology of the model. The model is completely filled by regions representing geological units. To minimise edge effects during the inversion, the th
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geological model is padded on all sides using various reflection transformations of the supplied geological model. 2. Define the a priori physical property laws Parameters that define the pdf of density or magnetisation for each rock unit must be defined. For example, the density pdf could be defined by a Gaussian distribution with a supplied mean and standard deviation. 3. Discretise this model A method of creating weak random disturbances of the model is required for the litho-inversion method to work effectively. We subdivide the study zone into a 3D matrix of cells and assign to each of the cells the properties of the appropriate lithological unit. The choice of the cell-size controls the magnitude of the disturbance. 4. Compute topology The spatial network relationships (topology) defining connections between homogeneous rock unit regions are determined. The topology must remain unchanged throughout the inversion 5. Make a list of the boundary or frontier cells The boundary or frontier cells need to be identified and updated during the inversion. The lithology parameter of a frontier cell may be changed during the inversion provided this does not alter the topology of the model. This allows the boundaries to migrate during the inversion. 6. Compute the gravity or magnetic effect for each cell A forward model calculation for each voxel using a unit physical property value is carried out. 7. Initialise the density or magnetisation values Values of physical property for each voxel are randomly selected according to the unit that the voxel belongs to and the appropriate pdf defined in Step 2. The likelihood of the model (current likelihood), is defined by the following expression
L(m 0 ) = k exp(−S(m 0 ) / σ 2 ) where
S (m0 )
,
represents the misfit with geophysical data, eg
(
)
N 2 S(m 0 ) = 12 ∑ g l (m 0 ) − data l l=1
where
g l (m0 )
represents the predicted potential field
response of the model at the point of observation l,
data l represents the measurement of the field at 2 point, and σ represents the variance of the data.
the same
8. Compute the geophysical effect of the model The geophysical effect of the model is computed by weighted summation of the unit response functions computed in Step 6. The weights are the present physical properties of the voxels. This model is called the current model, m cur . 9. Disturb the model The model is disturbed according to one of two possible schemes; a modification of the physical parameters for a voxel internal to a unit, or a modification of the geometry (i.e. re-
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Extended Abstracts
Constrained gravity 3D litho-inversion applied to Broken Hill
allocation of a point of data to another lithological unit) and re-initialisation of the physical property according to the pdf of the new lithological unit. The scheme employed is chosen randomly.
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possibility of major structures within the stratigraphy (Noble, 2000; Gibson and Nutman, 2004).
If a modification of the geometry is required, choose a cell from the list of frontier cells using an equi-probable random sampling method. For the selected cell, ci, identify all the formations present on the various faces. Then randomly sample from this list to choose a formation. If Fj is the selected formation, assign this formation to the cell. At the same time, sample the physical property according to the pdf defined in Step 2. If a simple modification of the physical parameters is required, choose a voxel from the list of cells that are completely internal to a unit using an equi-probable random sampling method. Re-sample the physical property value according to the relevant pdf defined in Step 2. Increment the iteration number, n, for the new model. 10. Compute the geophysical effect of the disturbed model The geophysical effect of the disturbed model is calculated by adding the weighted change in response for the voxel that has been disturbed to the current model response. The weight in this case is the change in physical property brought about by the disturbance. Note that this is a relatively small computation when compared with the normalised forward model calculation in Step 6 or with the full weighted summation performed in Step 8. 11. Compute the likelihood of the disturbed model The likelihood of the disturbed model is expressed as
L(m dis ) = k exp( −S(m dis ) / σ 2 ) . If
L(m dis ) > L(m cur ) then keep the disturbed model. If
L(m dis ) ≤ L(m cur ) ,then
keep
m dis
with
random
sampling and a probability equal to L( m dis ) / L(m cur ) . If we decide to keep m dis , then assign the disturbed model to the current model, m cur = m dis . During the initial part of the inversion, the data misfit of the current model follows a generally decreasing trend (Figure 5). At some point, the data misfit reaches an asymptotic value and we begin to store the models. These stored models are an exploration of the probability space of acceptable models. After completing Step 11, the inversion returns to Step 9 and continues to iterate around this loop. Typically, the first model with acceptable misfit is achieved within 1 million iterations. An ensemble of models that can satisfactorily explain the geophysical signature might be explored by continuing for a further 10 million iterations.
RESULTS FOR THE BROKEN HILL DISTRICT
Figure 1. Plan view of the Broken Hill geological model. The colours correspond to the geological units shown in Table 1. The presence of a fault is indicated by a white line. The location of Section N6 (Figure 2) and the seismic line (Figure 3) are shown as black lines. The project covers an area 20 by 20km, with the coordinates for the top-left corner being 535000E 6470000N (GDA94,MGA54). The following geological questions concerning the geometry of the units were posed at the beginning of this study; Do the major units flatten at depth? What is the regional extent of each unit? The objective was therefore to use the geological modelling tool to construct various geological hypotheses and to use inversion modelling to test the geological models for consistency with the gravity data. 3DWEG allows the rapid construction and editing of 3D geological models that are based on input observations, supplemented by various hypothetical observations. The 3D volumetric model proposed by Pasminco (Archibald et al., 2000) was used as a starting point. This model was modified using the method outlined in this paper to ascertain whether this topological and geometrical model could explain the observed gravity anomaly, and then to explore the probability space of models that can also meet this objective.
Lithology The units that are present in the model are listed in Table 1. The published geological map (after Willis, 1989) was a primary input along with 5 regional-scale geological crosssections. The final geological model is shown as a geological map in Figure 1, and on Section N6 in Figure 2.
Geological context We present the results of litho-inversion applied to geological, density and gravity from Broken Hill. The project area was a 20 by 20km area (Figure 1) extending to a depth of 5km. The rock units and their relationships are based on the GSNSW synthesis (Willis, 1989), however it is noted that there is the th
Figure 2. Section N6 through the 3D model. The colours of
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Extended Abstracts
Constrained gravity 3D litho-inversion applied to Broken Hill
the units are as shown in Table 1. (Length: 20km; V/H=1) The Alma Gneiss is generally considered an intrusive unit of the same age as the Thackaringa Group. To be properly represented in 3DWEG as an intrusive unit, it needed to be placed at the top of the sequence and given an erosional relationship with respect to the rest of the sequence. Structure Management of faults is a key issue in constructing a realistic 3D geological model. The number of faults introduced into the model was kept to a minimum. A satisfactory geological model at this broad scale could be constructed with just 2 faults. Other more extensive models with up to 10 faults are in preparation. Complex structural effects of the Broken Hill terrane that are not represented in the model include high temperature shears, boudinage and transposition. Geophysical Data The Rift unit was derived from an interpretation of a crustal seismic section (Gibson et al., 1998). ‘Observations’ for the top of this unit were digitised using the section as a backdrop.
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Mean values of density for the various units were obtained from an analysis presented in Lane et al. (2003) (Table 1). The initial values excluded any contribution from amphibolite. An adjustment in mean density for some units was made during the interpretation exercise, factoring in the contribution of a percentage of amphibolite (mean density of 3.05g/cm3). Prior to inversion, a user-supplied reference density value is subtracted from each of the means to produce relative density. The standard deviation values used in the inversion were chosen empirically. A reduction in variance of density with increasing sample volume from hand specimen scale, relevant for the results reported in Lane et al. (2003), to the size of the voxels used in the inversion would be expected. However, there was insufficient information available to calculate this scaling relationship. Results and Analysis The 3D geological model encompasses a parallelepiped 20km long, 20km wide and 5km deep (Figure 4). For inversion, the model was discretised using a grid of 80*80*21 voxels, each with dimensions 250m x 250m x 250m.
Figure 3. Broken Hill seismic section. The interpreted ‘rift’ unit is shown in the light hatching. (Length: 16km; V/H=1) The aeromagnetic data have only been briefly considered in this project. An ongoing debate on the relative degree of structural or stratigraphic control on magnetite concentrations is noted. Investigation of different hypotheses via constrained magnetic litho-inversion is being considered for the future. Ground based vertical gravity data with variable quality and spacing from 25m to 2km (Lane and Peljo, 2004) were used to produce a grid of simple Bouguer anomalies (terrain density of 2.75g/cm3) with a cell size of 250m. This grid was trimmed to the horizontal extent of the geological model. Prior to inversion, a constant value of 9mGal was subtracted from the data. Vertical gravity data are also available from a FALCON AGG survey (Lane et al., 2003). The nature of the acquisition and processing of these data limit the spatial wavelengths to a band between approximately 400m and 5-10km. This band limitation along with the terrain clearance requires different forward modelling parameters to be used when inverting the ground and airborne gravity datasets. Inversion of the AGG data will be carried out when the forward modelling routine has been modified to produce band-limited output consistent with the characteristics of the AGG data. An alternate (partial) solution would be to augment the missing long wavelengths in the AGG data with suitably upward continued and low passed ground gravity data. Physical Property Data
th
Figure 4. Perspective view from the northeast of the 3D geological model. The colours of the units are as shown in Table 1. Initial inversion results produced systematic data misfits that correlated with the plan-view location of the Thorndale Gneiss and the Clevedale Migmatite. A second inversion was carried out after adjusting the mean density for these units to 2.80 and 2.73g/cm3 respectively (Table 1). This is not unreasonable as geological mapping indicates that 10-15% amphibolite is present in these units. A further conclusion from the initial inversion exercise was that the Clevedale Migmatite unit (with the original density of 2.69g/cm3) is more restricted in extent than originally modelled. Figure 5 shows the evolution of the misfit between the observed and calculated ground gravity data. The average misfit per observation is less than 0.1mGal after 1 million iterations, and there is no obvious signal from a geological unit reporting in the misfit plot in the bottom right of the figure. Assuming that the discretisation of the inversion model and the constraints imposed during the inversion do not significantly impact the misfit that is achieved by the inversion, this average misfit can be interpreted as an estimate
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Extended Abstracts
Constrained gravity 3D litho-inversion applied to Broken Hill
of the precision level for the ground gravity grid. An independent assessment of these data suggested that this would not be an unreasonable uncertainty value for the ground gravity observations used in the inversion. Following inversion, the set of acceptable models was examined. The probability of finding any particular unit in 3D space was calculated and viewed on sections. The middle panel in Figure 6 is a section showing the likelihood of finding the Broken Hill Group in the context of where it was initially modelled, as indicated by lines tracing the boundaries of the interfaces in the initial geological model. Other parameters that can be derived for each voxel and displayed in a similar fashion include the most probable unit, the most probable unit where the probability exceeds 0.95 (i.e. those parts of the model where the unit is well defined and consistent with the gravity data), the average density and the standard deviation of density.
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Gneiss, Clevedale Migmatite and Rift Series. The distribution of amphibolites both spatially and stratigraphically plays a major role in the density and therefore the gravity response. Further resolution of the geology through inversion or modelling of gravity data will require a more detailed evaluation of the distribution of amphibolites.
ACKNOWLEDGEMENTS 3DWEG inversion software was developed by BRGM. The Broken Hill geological modelling work was funded by an Innovation Access Programme grant under the Australian Government’s innovation statement, Backing Australia's Ability, together with support from three sponsors: Geoscience Australia, the NSW Department of Mineral Resources and the Predictive Mineral Discovery CRC. Their support is gratefully acknowledged. The AGG data were acquired and made available by these same sponsors. The ground gravity data were obtained from the National Gravity Database maintained by Geoscience Australia. One of the authors (Lane) publishes with the permission of the CEO, Geoscience Australia.
REFERENCES Archibald, N. J., Holden, D., Mason, R. and Green, T., 2000, A 3D Geological Model of the Broken Hill ‘Line of Lode’ and Regional area: Unpublished report, Pasminco Exploration. Bosch M., Guillen A. and Ledru P., 2001, Lithologic tomography: an application to geophysical data from the Cadomian belt of northern Brittany, France: Tectonophysics, 331, 197-228. Gibson, G., Drummond, B., Fomin, T., Owen, A., Maidment, D., Gibson, D., Peljo, M. and Wake-Dyster, K., 1998, Reevaluation of Crustal Structure of the Broken Hill Inlier through Structural Mapping and Seismic Profiling: AGSO Record 1998/11.
Figure 5. Evolution of the average misfit per grid cell for 8 000 000 iterations. The bottom 4 panels provide feedback on the data and data misfit.
CONCLUSIONS A constrained gravity litho-inversion method was applied to a region surrounding Broken Hill. This enabled us to investigate the probability space of 3D lithological models that are compatible with the geological hypothesis (i.e. the starting model), estimates of density for each geological unit and the supplied gravity data. It was found that the geometry of the interfaces in the initial geological model and estimates of density were for the most part compatible with ground gravity data. Modifications to the extent of the Clevedale Migmatite and to the mean density of the Thorndale Gneiss and the Clevedale Migmatite in a revised model improved the compatibility with the ground gravity data. Density contrasts for the units modelled dictate that discrimination via inversion of gravity is most effective between high density Broken Hill Group, intermediate density Sundown and Thackaringa Groups and low density Alma th
Gibson, G. M. and Nutman A. P., 2004, Detachment faulting and bimodal magmatism in the Palaeoproterozoic Willyama Supergroup, south-central Australia: keys to recognition of a multiply deformed Precambrian metamorphic core complex: J. Geol. Soc. Lond., 161, 55-66. Lane, R., Milligan, P. and Robson, D., 2003, An Airborne Gravity Gradiometer Survey of Broken Hill: in Peljo, M. (comp.) Broken Hill Exploration Initiative, Abstracts from the July 2003 conference, Geoscience Australia Record 2003/13, 89-92. Lane, R., and Peljo, M., 2004, Estimating the pre-mining gravity and gravity gradient response of the Broken Hill AgPb-Zn Deposit: Extended abstract, ASEG 17th Geophysical Conference and Exhibition, Sydney. Mosegaard K., and Tarantola A., 1995, Monte Carlo sampling of solutions to inverse problems: J. Geophys. Res., 100, No. B7, 124321-12447. Noble M. P., 2000, The Geology of the Broken Hill Synform, N.S.W. Australia: MSc Thesis, Monash University. Willis, I. L., 1989, Broken Hill Stratigraphic Map: New South Wales Geological Survey, Sydney.
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Extended Abstracts
Figure 6. NW-SE sections showing various products derived from the ensemble of acceptable models: the most probable lithology; the most probable lithology with areas where this probability is less than 0.95 shown in black; probability of finding Broken Hill Group (black is zero probability whilst white is a probability of one); mean relative density; standard deviation of density. The lines shown on each section are the geological boundaries of the starting model. (Length: 28km; V/H=1) Table 1. Geological units, relationships and density parameters used in the inversions. Map symbol
Geological unit
th
Relationship
Density distribution
Initial mean density (g/cm3)
Final mean density (g/cm3)
Standard deviation (g/cm3)
Alma Gneiss
Intrusive (erosional)
Gaussian
2.69
2.69
0.05
Paragon Group
Unconformable (onlap)
N/A
N/A
N/A
N/A
Sundown Group
Unconformable (onlap)
Gaussian
2.81
2.81
0.05
Broken Hill Group
Unconformable (onlap)
Gaussian
2.84
2.84
0.05
Thackaringa Group
Unconformable (onlap)
Gaussian
2.80
2.80
0.05
Thorndale Gneiss
Unconformable (onlap)
Gaussian
2.73
2.80
0.05
Clevedale Migmatite
Unconformable (onlap)
Gaussian
2.69
2.73
0.05
Rift Series
Unconformable (onlap)
Gaussian
2.72
2.72
0.05
ASEG 17 Geophysical Conference and Exhibition, Sydney 2004.
Extended Abstracts
