Bayesian Joint Inversions for the Exploration of Earth Resources

Alistair Reid1, Simon O’Callaghan1, Edwin V. Bonilla1,Lachlan McCalman1, Tim Rawling2 and Fabio Ramos3

1. NICTA, 2. University of Melbourne, 3. University of Sydney{ alistair.reid, simon.ocallaghan, edwin.bonilla, lachlan.mccalman }@nicta.com.au

tim.rawling@unimelb.edu.au, f.ramos@acfr.usyd.edu.au

AbstractWe propose a machine learning approach to geo-physical inversion problems for the exploration ofearth resources. Our approach is based on nonpara-metric Bayesian methods, specifically, Gaussianprocesses, and provides a full distribution over thepredicted geophysical properties whilst enablingthe incorporation of data from different modali-ties. We assess our method both qualitatively andquantitatively using a real dataset from South Aus-tralia containing gravity and drill-hole data andthrough simulated experiments involving gravity,drill-holes and magnetics, with the goal of char-acterizing rock densities. The significance of ourprobabilistic inversion extends to general explo-ration problems with potential to dramatically ben-efit the industry.

1 IntroductionThe discovery of resources in the Earth’s crust is a problemof crucial importance for continued global economic devel-opment, so modern investigation methods make use of vastamounts of geophysical data to improve the efficiency of re-source exploration. In a geological inversion problem, prop-erties such as temperature, conductivity, density, magneticsusceptibility and permeability are inferred from related ob-servations such as gravity, magnetics and seismic reflexion.

Two components are crucial in a geophysical inversion:the assessment of uncertainty over the predicted geophysi-cal properties, and the joint reasoning from multiple sourcesof information. Uncertainty allows for principled decision-theoretic approaches to minimize the risk of acquiring moremeasurements. Joint reasoning through the estimation of sta-tistical dependencies provides a natural mechanism to fusemany of the available data modalities.

In this paper we formulate geophysical inversion as a ma-chine learning problem, and propose an approach based onGaussian processes regression that naturally provides both apredictive distribution over the inverted quantities and a prin-cipled method to fuse different types of observations. Weapply our method to a real dataset from South Australia con-taining gravity and drill-hole data with the goal of charac-

terizing rock densities for geothermal target exploration, andalso to simulated validation data involving gravity, drill-holeand magnetic observations.

2 Related Work

One of the most popular methods to solve inversion problemsin Geophysics is the UBC software developed at the Univer-sity of British Columbia — Geophysical Inversion Facility,based on the work by Li and Oldenburg [1996] and Li andOldenburg [1998]. This software is widely used in the in-dustry and in academia, in particular for gravity inversions,magnetic inversions and mineral exploration [Oldenburg etal., 1998]. These methods have the advantage of being rela-tively easy to understand and quick to obtain an initial solu-tion to an unconstrained inversion problem. However, one ofthe major drawbacks of these techniques is that they do notprovide uncertainty estimates for the properties of interest.

Inverse problems have also been studied in the machinelearning community for general problems[Carreira-Perpinan,2001], and for geophysical problems Yurtsever et al. [2011].From the geostatistics community, a comprehensive overviewof stochastic process priors for inverse problems can be foundin Tarantola [2005]. However, these methods seem to be un-derexploited in the mineral exploration area. Gaussian pro-cesses are, in fact, a Bayesian formulation of the nonparamet-ric priors which have been used in Geostatistics for interpola-tion and regression problems under the name of Kriging, seee.g. Cressie [1993] and Stein [1999]. However, extensions ofstandard regression approaches are required for an inversionproblem because the observations are indirectly related to theproperty of interest.

The importance of the joint analysis of multiple data typesin realistic 3D geological modeling has been studied, for ex-ample, by Guillen et al. [2008] and Fullagar and Pears [2007].Very recent work by Shamsipour et al. [2012] has carried outstochastic inversions of gravity and magnetic data. The mainadvantage of our approach is that we provide a consistentprobabilistic model, where we can use principled marginallikelihood objectives for model selection and leverage all themachinery developed in the machine learning community inrecent years to perform inference with large datasets.

Proceedings of the Twenty-Third International Joint Conference on Artificial Intelligence

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3 Gaussian Processes for Joint InversionsIn this section we describe our approach to the problem ofgeophysical inversion. We start by introducing Gaussian pro-cess priors in standard regression settings. The formulation isthen extended to inversion problems for which linear forwardmodels exist. We then detail how our modeling approach cannaturally be extended to multi-task settings. Finally, tractableimplementation of this potentially costly model is discussed.

3.1 Gaussian Processes for RegressionGaussian processes are flexible nonparametric priors thathave been used successfully in various machine learning taskssuch as regression and classification (see e.g. Rasmussen andWilliams [2006]). The probability distribution of a func-tion f(x) is a Gaussian process (GP) if for any finite subsetof points x1, . . . ,xN , the function values f(x1), . . . , f(xN )follow a Gaussian distribution. We denote a Gaussian processwith:

f(x) ∼ GP(µ(x), k(x,x�)), (1)µ(x) = E[f(x)], (2)

k(x,x�) = E[(f(x)− µ(x))(f(x�)− µ(x�))], (3)

where µ(x) and k(x,x�) are the mean function and the co-variance function of the GP respectively. A Gaussian pro-cess is completely determined by its mean function and itscovariance function. In a regression setting it is customary toassume a GP prior over the latent functions f(x) and a like-lihood model where these latent functions are corrupted byGaussian noise. In this case, the predictive distribution at atest point x∗ is Gaussian with simple analytical forms for itsmean and variance.

The parameters of the covariance function and the param-eters of the noise process are usually referred to as the hyper-parameters of the GP. These can be learned from data by op-timization of the marginal likelihood. Under the GP priorand the Gaussian likelihood assumptions, it is possible tomarginalize the latent functions analytically, and the result-ing marginal likelihood is also a Gaussian distribution.

3.2 Gaussian Processes for InversionHere we focus on inversion problems for which a linear for-ward model that relates the parameter of interest (i.e. geo-physical property) to the observations can be formulated. Forexplanatory purposes, we take the problem of gravity inver-sion as a running example. In this case, it is well known thatthe density of a body at a particular location and the observedvertical component of the gravitational field are related via anintegral operator (see e.g. Kearey et al. [2002], Ch 6).

Additionally, as in many popular geophysical inversion ap-proaches, we work upon a discretized version of the forwardmodel, where the 3-dimensional region of interest has beenpartitioned into cells, each having a constant value of the pa-rameter of interest. In the case of our gravity example, physi-cal principles imply that the vertical component of the gravi-tational field measured by a sensor is a linear combination ofthe contributions of all the cells as a function of their posi-tions and densities. For details on how these physical modelsover voxel cells are obtained see Li and Oldenburg [1998].

Problem FormulationLet {φ(j)} denote the unknown values of the geophysicalparameter at locations {z(j)}Mj=1, and let {y(i)} be the re-lated observations at the locations {x(i)}Ni=1, with x, z ∈ R

3.Our goal is to reason about the geophysical parameter inthe region of interest ({φ(j)}) given the related observations({y(i)}). In the gravity inversion problem, {y(i)} are mea-surements of the variations of the vertical component of thegravitational field and {φ(j)} are the values of the density ofthe anomalous body responsible for these variations.

Forward ModelAssuming a linear forward model we have that:

f = Gφ, (4)where f is the vector of noiseless observations; φ is the vec-tor of unknown parameters at the 3D locations; G is a knownN × M sensitivity matrix that relates the values of the geo-physical property at different locations to the observations.For a gravity forward model the values of the matrix G canbe determined analytically, assuming simple shapes such asprisms (see e.g. Nagy et al. [2000]). In particular, for a prismdetermined by its start and end coordinates zo and ze, we cancompute the corresponding sensitivity at location x by using:

Gx,z = γg(z)��ze

1

zo1

��ze2

zo2

��ze3

zo3, (5)

where γ is the gravitational constant;g(z) = z1 log(z2+r)+z2 log(z1+r)−z3 arctan

z1z2z3r

; (6)

zo = zo − x; ze = ze − x; and r =�z21 + z22 + z23 .

Prior and Likelihood ModelsIn this work we assume a Gaussian process prior over the rockproperties of interest φ and an isotropic likelihood model:

φ(z) ∼ GP(0,κφ(z, z�)), (7)

y = f + η with (8)

η ∼ N (η|0,σ2I), (9)where κφ(·, ·) is the covariance function in the parameterspace and σ2 is the noise variance. The GP prior over thefunctions φ translates into a Gaussian prior over the geophys-ical quantities at the locations of interest:

φ ∼ N (φ|0,Kφφ), (10)where Kφφ is the M×M covariance matrix obtained by eval-uating the kernel κφ(·, ·) at the locations of interest {z(j)}Mj=1.

InversionInversion within a Bayesian framework is straightforward asit corresponds to computing the posterior distribution of theparameters of interest φ given our observations y. This iseasily obtained by conditioning, where we need to computethe covariance structures:

Kyφ = GKφφ, (11)

Kyy = GKφφGT + σ2I. (12)

Hence we obtain that the predictive distribution is given by:φ|G,y ∼ N (φ|µφ|y,Σφ|y) with: (13)

µφ|y = KTyφK

−1yy y (14)

Σφ|y = Kφφ −KTyφK

−1yy Kyφ. (15)

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3.3 Multi-task settingsOur ultimate goal in addressing geophysical inversion prob-lems is to fuse different data sources within a single prob-abilistic framework. As an initial step, we have adopted amulti-task learning approach based on the model of Bonilla etal. [2008]. This model assumes that the covariance betweentwo different geophysical quantities (tasks) at two distinct lo-cations decomposes as:

κ(χ(x),φ(x�)) = kχφκ(x,x�), (16)

where χ and φ denote distinct geophysical quantities, suchas density and magnetic susceptibility, and kχφ is the covari-ance between these quantities. This means that instead ofhaving uncorrelated priors with distinct κφ(·, ·) and κχ(·, ·)as in Equation (7), we share “statistical strength” by usingthe correlated prior in Equation (16). Despite the simplicity,our experiments show that, under certain assumptions on thedependency of the quantities of interest, this model performswell, and observing data from an additional source does im-prove the quality of the inversion results, especially when theobservations are very sparse.

3.4 Hyper-parameter LearningLet the set of hyper-parameters θ include the parameters ofthe covariance function κφ(·, ·) and the noise variance σ2.We learn these hyper-parameters by maximization of the logmarginal likelihood. As in the regression case, we can inte-grate out φ analytically and obtain the marginal likelihood:

y|G,θ ∼ N (y|0,GKφφGT + σ2I), (17)

where, for notational simplicity, we have omitted the implicitdependency of Kφφ on the kernel hyper-parameters.

3.5 Computational ConsiderationsThe flexibility of our approach comes at the expense of a highcomputational cost in space and time. While computing theinverse covariance matrix is expensive at O(N3), our frame-work is more sensitive to the resolution of the output gridbecause computing Kyy = GKφφG� + σ2I requires largematrix multiplications at a cost of O(NM2 +MN2). In ourproblem M � N because the output grid spans three dimen-sions, while the observations span two dimensions on the sur-face or one dimension along drillholes. This section outlinesan approximation of Kyy to reduce the cost of using a largegrid to O(M logM) and an additional O(N2) to populate theelements of the covariance matrix from a lookup table.

Let K = GKφφGT be the noiseless version of Kyy .Hence, the (i, j)th entry of K is given by:

Kij =M�

a=1

M�

b=1

G(xi, za)κφ(za, zb)G(xj , zb), (18)

Where G(x, z) is the sensitivity at observation location x cor-responding to the cell at location z as defined in Equation (5).If we assume that the covariance function κφ(·, ·) is station-ary, and the cell locations {z(i)}Mi=1 lie in a regular spatialgrid, the number of distinct entries in Kφφ is reduced from

O(M2) to O(M). Similarly, given that our sensitivity func-tion G(x, z) is stationary, if we assume that our observationlocations {x(i)}Ni=1 are placed on a regular grid and lie abovethe cell columns, the total number of distinct entries in G isreduced from NM to M .

Therefore, Ki,j can be expressed using functions that op-erate on the discrete displacements between observation loca-tions and/or cell centers. Let G and κφ be the stationary func-tions (that operate on discrete displacements) correspondingto G and κφ respectively. Hence we have that:

Ki,j =M�

a=1

M�

b=1

G(xi − za)κφ(za − zb)G(xj − zb). (19)

In order to further improve the efficiency of our algorithm,we introduce the following approximation. We sum over thesame template grid of sensor-cell displacements regardless ofeach sensor’s position over the output grid. This means thatwe can characterize Equation (19) by using only displace-ment variables.

Let K(·) be the stationary function corresponding to K andhx ∈ Hx denote the displacement between two observationlocations (x and x�). Similarly, let ha,hb ∈ Hz denote dis-placements between an observation location (x) and a loca-tion in the 3D volume of interest (z). Hx and Hz are sets ofdisplacements defined over three dimensions, although in ourproblem setup the displacement between sensors has a heightcomponent of zero. Hence we have that:

K(hx) =�

hb∈Hz

G(hb)�

ha∈Hz

G(ha)κφ(hx + hb − ha),

(20)where we note that the cardinality of the discrete displace-ment sets Hx and Hz is not much greater than the number ofcells M . For example, if we consider a cube with M cellsas the volume of interest, the number of distinct discrete dis-placements is |Hz| = 4M . We can now recognize that Equa-tion (20) computes an element of a nested discrete convolu-tion. We first convolve G with κφ and then convolve the out-put of this operation with (a flipped version of) G. This pro-cess is depicted in Figure 1, which shows the grids used fora squared exponential covariance function, and a gravity sen-sitivity model. Discrete convolution is applied to these gridsto compute a covariance-vs-sensor-displacement lookup ta-ble (where we are only interested in the slice with no verticaldisplacement).K is most efficiently computed using zero padded, dis-

crete frequency space convolutions based on the fast Fouriertransform.The convolution theorem states that the cost of thisapproach is O(M logM) time, while the covariance matrixover observations is populated by N2 table lookups.

The change in behavior introduced by our approximationis is illustrated in Figure 2. We have assumed that the gridaround the sensors extends uniformly regardless of position- which actually removes the edge effects from the covari-ance matrix, causing the function to behave as if the integra-tion had padding cells. The emergent difference between thecovariance functions is shown in the bottom row, where the

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Figure 1: Computation of an approximate covariance func-tion using discrete convolutions (denoted with

�). The

matrices Kφφ and G are evaluated over regular displace-ment grids, and convolved to obtain a covariance-vs-sensor-displacement lookup table K.

Figure 2: Top: The common dependence of two nearby sen-sors using matrix multiplication (left) and a stationary dis-placement template (right) is illustrated for observations nearthe grid boundaries. Bottom: The corresponding covariancematrices computed over a grid of sensors.

envelope induced from the finite grid is visible in the multipli-cation case (left image) but not the convolution result (right).

4 Outline of ExperimentsOur experiments focus on the 3D inversion problem for de-termining the density contrast of rocks below the Earth’s sur-face. This is crucial in many applications, including the char-acterization of hot dry rocks up to 5km deep when explor-ing geothermal energy targets (see e.g. Huenges and Ledru[2010], Ch 2). However, gravity alone provides a very poordepth resolution so it is necessary to fuse additional sourcesof information into the inversion models in a principled way.

Our experiments investigate gravity inversion on both areal dataset and a simulated scenario. On the simulateddataset, we perform joint inversions with gravity and drill-hole observations on a realistic geological structure and also

(a) (b)

Figure 3: (a) The density of the dipping body along with thelocation of the two simulated drill-holes. (b) Simulated grav-ity observations corresponding to the dipping body.

investigate the inclusion of magnetic susceptibility observa-tions. On the real dataset, we study a region in the CooperBasin in South Australia, fusing ground-based gravity obser-vations with drill-hole core samples that provide very sparsedirect observations of the density.

4.1 Simulated DataThe simulated structure is a dipping body, where a slab ofdense, magnetically susceptible material is present. It is in-clined from the surface and has properties typical of igneousrock that may have intruded into the crust. The geometry ofthis scenario is shown in Figure 3a where the density of thedipping body is shown, along with two simulated drill-holes— one drill-hole passes through the body, while the nearerone misses the body.

The background material in this volume has been assignedthe average density of the Earth’s crust (2.67g/cm3). Thedipping body was assigned a density of 2.9g/cm3 and a mag-netic susceptibility of 2× 10−4 in SI (which is used in a laterscenario). A voxel grid of 50×50×25 cells was used. Grav-ity observations (in milliGals) were forward simulated ontoan array of 50× 50 observations centered above each columnof voxel cells. These observations are computed as anomalies(the difference between the gravity observed and the grav-ity from a grid filled with the mean density of 2.67). IIDGaussian noise with a standard deviation of 1% of the meananomaly observations was applied to these observations, pro-ducing the observations shown in Figure 3b. Note that thesensitivity of gravity readings drops off quickly with depth,so the deep dipping body produces only a subtle smear-likesignature to the right of the body’s anomaly.

4.2 Real Dataset: Cooper BasinThe real dataset contains Bouger anomaly gravity measure-ments and core-sample densities from the Cooper Basin for-mation in South Australia. This data has been provided by theSouth Australian Department of Manufacturing, Innovation,Trade, Resources and Energy (DMITRE). The Cooper Basinformation is comprised of a basin-shaped basement, over-lain with multiple sedimentary layers. The basement rocksare metamorphic, ranging from high grade gneisses to lowergrade schist, and contain numerous intruded granite bodies.The large granite structures induce moderate to strong lows

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(a) (b)

Figure 4: Outputs of the Gaussian process inversion algo-rithm using only gravity observations of the dipping body.

in the regional gravity datasets due to their relatively lowerdensities. The sedimentary layers overlaying the basement,particularly the Eromanga basin, contain significant hydro-carbon accumulations, so significant drilling and geophysicaldata collection has been conducted in the region.

Data PreparationFor the Cooper Basin data we used the Bouguer anomaly,which accounts for the Earth’s total mass, variations in ra-dius with latitude, the effect of elevation on the measurementsand the mass of the local topography such as mountains andvalleys. In addition, raw drill-hole information provided thedepth of samples and corresponding densities (drawn fromanalysis of the target formation). The drill samples tend tooccur on the formation boundaries, although there are manyexceptions. To mitigate the influence of this clustering aroundboundaries, density measurements over the extent of each tar-get formation were averaged - averages over lines can be nat-urally incorporated into the inversion algorithm.

5 Empirical Results and AnalysisThis section presents empirical results of our inversion algo-rithms on the simulated dataset (dipping body) and the realdataset (Cooper basin).

5.1 Dipping BodyGravity observations were inverted in isolation by applyingthe GP inversion algorithm to learn the hyperparameters ofan anisotropic polynomial covariance function. Because theaccuracy of gravity surveys is well understood, appropriatesmall noise variances were included in the model.

The inversion produced the mean shown in Figure 4a. TheGP formulation also provides uncertainty, which is shown inFigure 4b. This inversion is ill posed; we cannot infer whethera gravity signature corresponds to a large mass deep down, ora small mass on the surface, so there is little inherent depthresolution. The uncertainty in this case appears featurelessbecause of the uniform sampling pattern. The mean is moreinteresting, as it shows the body extending down from the sur-face, although the inclination of the body cannot be resolved.Forward simulation of gravity observations on the predictivemean confirms that this is indeed one of the infinite numberof valid solutions to the problem, selected through the priorover density structure.

Fusing drill-hole measurements with the gravity observa-tions can further constrain particular features, because the

(a) (b)

Figure 5: Outputs of the Gaussian process inversion algo-rithm after fusing gravity and drill observations of the dippingbody.

Metric GravityOnly

Gravity &Drill Holes

UIQ 0.480 0.589Corr. 0.538 0.620

(a)

Method RMSE(g/cm3)

GP 1.06UBC 2.06

(b)

Table 1: Quantitative evaluation of inversion results on thedipping body. (a) UIQ index [Wang and Bovik, 2002] andcorrelation between the predicted density and the true densityfor GP inversion. (b) Mean square error of GP vs UBC.

drill samples the density directly at specific locations. In ourscenario one of the simulated drill-holes passes through thebody. Using the multi-task model, updated mean and variancevolumes are obtained: Figure 5a and Figure 5b. The problemremains ill posed, but (as the variance shows), the model isnow confident at depth where drill observations are provided,adding two columns of low-variance. The drill-holes can onlyimprove the model in the vicinity where they are correlated tothe volume, so the overall quantitative improvement in pre-diction quality is relatively weak. In the regions the model ismost incorrect, it is also assigning a high variance.

By comparing the side-profiles of the reconstructed bodywith the gravity only case and the synthetic truth, there is avaluable qualitative difference — the incline of the body isresolved. These side-profiles are compared in Figure 6. Table1 shows a quantitative evaluation of our GP inversion method(a) having additional drill-hole data and (b) compared to theUBC solution.

5.2 Fusing Magnetic ObservationsMagnetic field measurements on the Earth’s surface vary sig-nificantly with respect to location, and are weakly influencedby induced magnetism from dense, iron-rich minerals belowthe surface. While drill-holes and gravity provide indirectmeasurements of a common attribute (density), additional ob-servations of Total Magnetic Anomaly (TMA) (which is theanomaly of the Earth’s magnetic field magnitude) can only befused with gravity and drill-holes if a relationship can be cap-tured between the physical attributes of magnetic susceptibil-ity and density. The current GP model was extended to fuseTMA by learning a simple covariance relationship betweenthese attributes, producing a multi-sensor multi-task covari-

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Figure 6: Side-profiles of the dipping body in the mean pre-dictive density before (left) and after (middle) fusing infor-mation from the drill-holes. (right) Ground truth.

(a) (b)

Figure 7: Simulated magnetic data along with gravity anddrill-hole data. (a) Forward magnetism simulation for thedipping body (no drill-holes were simulated; glyphs indicatemagnetic field direction). (b) Incomplete gravity coveragewas simulated over the dipping body, with two full drillholes.

ance model consistent with the physical sensors.The benefit of fusion is most clearly seen when the cov-

erage pattern differs between sensors. The dipping body hasbeen assigned a positive susceptibility, and gravity observa-tions were simulated over only half of the test region, accom-panied by drill-holes (Fig. 7b). TMA was available over thefull region, but without any direct measurements of suscep-tibility. The magnetic anomaly observations were forwardsimulated above ground level for the Earth field vector at theCooper Basin’s location, yielding the observations in Fig. 7a.

The inversions were initially run independently, leadingto poor predictions of both density and susceptibility fromlack of information. By then allowing the GP to model astochastic coupling between the density and susceptibility,the joint inversion was able to use complementary informa-tion from each dataset. This has led to inversion results thatclosely resemble the full coverage inversions. In addition,the predictive variance assigns maximum confidence to loca-tions where all sensor modalities were available together. Thefused density (Fig. 8a and Fig. 8b) and susceptibility (Fig. 9aand Fig. 9b) are provided for this case.

5.3 Cooper Basin InversionThe inversion algorithm was applied to the real Cooper Basindata to obtain the two predictive means shown in Figure 11.The location of this data is shown in Figure 10 (left) markedas a black rectangle, along with the processed observations(center) that show the incomplete and irregular gravity cover-age available. The figure on the right shows the gravity mea-surements forward simulated from the GP predictive mean,which are both consistent with the observations and provide

(a) Fused density mean

(b) Fused density variance

Figure 8: Joint inversion results for density with fusion modelthat includes gravity and magnetic data of the dipping body.

Inversion method RMSE (g/cm3)Gaussian process inversion 0.029UBC inversion 0.073

Table 2: Cross-validation Errors for our GP inversion methodand UBC codes on the Cooper Basin dataset.

a plausible spatial interpolation between the observations.The results of the inversion are presented in Figure 11 with

gravity only and with drill-hole data in the same region. Itappears that the features in the predictive mean are geolog-ically reasonable and can be compared against existing geo-logical knowledge of the area. For example, the low densityzone in blue in Figure 11 is likely to be controlled by a com-bination of the base of the trough depo-centers described inSection 4.2 and the low density granitic material that intrudesthe basement and deeper basin sediments. The NE strike ofthe northern margin of this anomaly is consistent with dom-inant NE structural trend in the region and the strike of thefaults that are likely to bound the basement highs. The higherdensity anomalies that have been resolved at deeper levelsvisible in Fig. 11 (right) are likely to relate to basement rocksbetween intrusive bodies although this needs to be tested fur-ther against other available datasets and interpretations.

Furthermore, while the ground truth density in the CooperBasin is not known, it is possible to obtain limited quantita-tive results by cross-validating on the drill-holes (Table 2).Crustal rock densities are close to 2.67g/cm3, so the er-ror standard deviation is approximately 1% and 3% for ourmethod and the UBC result respectively. Cross-validationwas conducted using the GP inference algorithm, and as apoint of comparison the UBC software was run on the samedata. The results in this case suggest that our approach out-performs our UBC setup. See section 6 below for a general

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(a) Fused susceptibility mean

(b) Fused susceptibility variance

Figure 9: Joint inversion results for susceptibility with fusionmodel that includes gravity and magnetic data of the dippingbody.

comparison between these techniques.

6 Relation between UBC and GP inversionsBoth UBC inversion and GP inversion can be framed intoa single Gaussian prior - Gaussian likelihood model. TheUBC solution is the mode of the posterior (MAP), whichcan be obtained by minimizing the unnormalized negative logposterior. However, UBC considers the geophysical param-eters, e.g. densities, uncorrelated (a priori) and in the bestcase having different but fixed variances (which depend ondepth). Such variances are set beforehand in order to achievea weighting effect that compensates the decay of the sensorsensitivity with respect to the depth. GP inversion, on theother hand, allows these parameters to be correlated. Thesecorrelations are learned from the data through maximizationof the marginal likelihood (i.e. hyper-parameter learning).

7 DiscussionThis project has applied Bayesian machine learning methodsto the problem of joint geophysical inversions. Our approachallows for an assessment of uncertainty associated with inver-sions of geophysical datasets, enabling the geophysicists andgeologists who will use the results in decision-making work-flows to robustly assess the likelihood of predictions being ac-curate throughout the model volume. In the longer term thiswill allow interpretation workflows to include a quantifiableassessment of uncertainty, something geophysical surveyorsas a community are only just beginning to tackle.

In future work we will investigate low-rank approxima-tions to Kyy to address the cost of GP inference once the co-variance matrix has been computed (see e.g. Quinonero Can-dela and Rasmussen [2005] for an overview). We will also

Figure 10: Satellite view of Australia with the Cooper Basinregion of interest highlighted by the blue bounding box (be-low). The measured gravity anomalies (middle) and the pre-dicted gravity anomalies (right) correspond to this blue box.

Figure 11: The predictive mean outputs of the Gaussian pro-cess inversion algorithm on the Cooper Basin data. Gravityonly (left) and gravity with drill-holes (right).

investigate likelihood approximations that exploit the struc-ture of spatial problems, such as those presented in Stein et al.[2004]. Given that the inversion problem is ill posed, it is crit-ical to form priors that capture the knowledge of geologistsabout the plausible structure of rocks. Their prior knowledgeis also critical for characterizing non-linear dependencies be-tween different rock properties, which are poorly constrainedby geophysical data alone.

AcknowledgementsThe authors gratefully acknowledge funding for the project“Data Fusion and Machine Learning for Geothermal TargetExploration and Characterization” by the Australian Renew-able Energy Agency (ARENA) and National ICT Australia(NICTA). NICTA is funded by the Australian Government asrepresented by the Department of Broadband, Communica-tions and the Digital Economy and the Australian ResearchCouncil through the ICT Centre of Excellence program.

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