Inversion of regional gravity gradient data over the Vredefort Impact Structure, South Africa∗Cericia Martinez† and Yaoguo Li††Center for Gravity, Electrical, and Magnetic Studies, Department of Geophysics, Colorado School of Mines

ABSTRACT

We present the results for 3D inversion of gravity gradiome-try data over the Vredefort Impact Structure in South Africa.With the rapidly growing field of gravity gradiometry, an in-vestigation into the extraction of information from multiplecomponents is warranted. Though the gradient tensor has fiveindependent components, any combination of the componentscan be used to invert for a structural representation of the sub-surface. In an effort to understand how differing componentcombinations contribute to the resolution of the model, we usetwo measured and one calculated gravity gradient componentfrom the Falcon system for inversion. This inquiry focuseson regional geologic features in the context of inversion withselect gravity gradient components over the Vredefort impactstructure in South Africa.

INTRODUCTION

Gravity has long been used to study and characterize regionalstructural features. Changes in rock type and lithology giverise to density contrasts and ultimately anomalous gravity mea-surements relating to the regional subsurface features. His-torically, gravimeters have been the tool of choice for explo-rationists. Though the torsion balance was one of the firstinstruments able to measure the changes in the gravity field,technological developments in gravimeters made absolute andrelative gravity measurements more accessible while gradiome-ters slowly became relics of history. In the past few decades,technological innovations have been commercialized that havebrought gravity gradiometry back to the forefront as an explo-ration tool (Jekeli, 1993; Bell et al., 1997).

Gravity gradients are a measure of the spatial rate of change inthe gravity field. The unit of measure for the gravity gradientis the Eotvos (Eo), named for Lorand Eotvos, where 1 Eo isequal to 0.1 mGal/km. The gravity gradient obeys Laplace’sequation making the tensor symmetric and traceless, leavingonly five independent components. Gravity gradiometry is avaluable alternative to typical vertical gravity measurements insome applications. The generally sparse collection of verticalgravity measurements limits the lateral resolution and extentto which a target can be characterized. In measuring the gra-dient of the gravity field at denser spacing, lateral resolution isincreased and the target body can potentially be better charac-terized. The resolution is increased because the gradient of thegravity field is being measured and decays as inverse distancecubed, compared to the decay of the vertical gravity field asinverse distance squared.

Unlike vertical gravity measurements, gravity gradients do notrequire multiple corrections to the measured data in order toobtain the anomaly values. Since the gradient of the gravity

Figure 1: Geologic map of the Vredefort impact structureshowing main geologic units; from Henkel and Reimold(2002)

field is being measured, the topography is still one of the ma-jor contributors to the measured values particularly in areas ofhigh relief. For this reason, removing the terrain effect is animportant step in reducing the data to an interpretable product.Furthermore, the density value used to remove the terrain ef-fect from the data is crucial. With a density that is too high,signal from the target may be removed while a density that istoo low fails to fully remove the terrain effect. For this rea-son, care must be taken in selecting the proper representativedensity value of the topographic region.

The horizontal gradient components contain information thatis useful in detecting target edges and delineating structure.The vertical gravity gradient contains information about thesize and depth to the target, and is typically the most read-ily interpreted component due to its relation to vertical grav-ity. Prevalent methods for interpreting various components ofgravity gradient data are focused on enhancing edges, linea-ments, and using invariants (Murphy and Brewster, 2007; Ped-ersen and Rasmussen, 1990).

Just like vertical gravity and magnetic data, gravity gradientdata can be exploited to obtain a 3D model of density contrasts.In addition to characterizing the geometry and density of thetarget, a 3D density model can potentially be used to estimatevolume of the target; infer about lithologic changes; or providefurther knowledge for drilling purposes.

Here we explore the utility of inverting single and multiple

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Vredefort impact structure

gravity gradient components for regional geologic features ac-cording to the methodology set forth by Li (2001). The gravitygradient data is over the Vredefort impact structure in SouthAfrica.

GEOLOGY

The Vredefort impact structure is in the Witwatersrand Basinof South Africa, as seen in the geologic map of Figure 1. Theimpact structure is the largest and one of the oldest known im-pact structures on earth. The structure is thought to be around2023 Ma years old (Reimold and Gibson, 1996). The originalsize of the impact structure is thought to be close to 300 kilo-meters in diameter. The Vredefort Impact Structure is in theheart of the Witwatersrand Basin, known for its economic goldreserves. It is believed that downfaulting of the Witwatersrandstrata due to the impact helped preserve the vast economic re-sources within the basin today.

At the center of the impact structure is the Vredefort Dome,which is the central uplift due to the impact. The VredefortDome has two major structural pieces: the core and the col-lar. The core of the dome is composed of Archean granitoidswhile the collar is composed of overturned supracrustal strata(Reimold and Gibson, 2006).

Many geophysical surveys have been conducted within the Vre-defort impact structure to characterize the geologic features.Density of the core area was constrained by a seismic refrac-tion survey across the Dome (Green and Chetty, 1990). Henkeland Reimold (2002) investigated the magnetic anomaly of theVredefort Dome thought to be the result of a thermal overprintfrom the impact event. More recently, Muundjua et al. (2007)implemented a ground magnetic survey in an attempt to char-acterize the geologic structure.

GRAVITY GRADIENT DATA

The gravity gradient data were acquired using the Falcon R©systemwith north - south trending flight lines spaced 1 kilometer apart.Only a subset of the collected data is presented here and cov-ers approximately 1200 square kilometers. The survey is semi-draped with flight heights ranging from 50 to 280 meters abovethe topographic surface. The acquired data underwent rou-tine proprietary processing and corrections for residual aircraftmotion and self-gradient. Prior to data delivery, the data aredemodulated, filtered, and leveled. The terrain effect was re-moved using a density of 2.67 g/cc.

For this investigation, the two observed components are thecurvature gravity gradients Txy and Tuv = (Txx−Tyy)/2. Thesetwo observed components are displayed in Figure 2. To ex-plore what information can be extracted from the data, we cal-culate a third, more common component, Tzz. The Tzz compo-nent was calculated using an equivalent source method. Theequivalent source layer consisted of 1000 meter cubic cells lo-cated 1000 meters below the lowest elevation point. Both ofthe observed data components (Txy and Tuv) were used to ob-tain the equivalent source layer.

An L-curve critera (Hansen, 1992) was used to select the op-timal equivalent source layer. The model objective function isplotted against the data misfit on a log-log scale and illustratesthat models with increasingly complex model structure tendto fit the data to a higher degree than those of simple struc-tures. The point of maximum curvature on the plot is consid-ered to yield the optimal tradeoff between fitting the data andobtaining a reasonably realistic model.The forward modeledTzz component is shown in Figure 3. The equivalent sourcemethod doubles its utility by removing noise from the data dueto the inability of the equavilent source layer to reproduce therandom fluctations that are characteristic of noise. For this rea-son, the Tzz component is expected to have less noise than thatof the observed components.

(a)

(b)

Figure 2: Gravity gradient components: a)Txy, b)Tuv = (Txx −Tyy)/2

RESULTS

Three models were obtained by inverting Tzz, Txy and Tuv, andTxy, Tuv, and Tzz together. The mesh is composed of rectan-gular prisms with constant density contrast within each cell.The mesh discretization used has cell sizes of 500 meters in

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Vredefort impact structure

Figure 3: Tzz calculated using an equivalent source generatedfrom the Txy and Tuv = (Txx −Tyy)/2 components.

the easting and northing directions. The depth cells start with25 meter thickness near surface to accomadate the topographyand gradually increase to 500 meter thinckness. Padding cellsextend beyond the data area. The rectangular mesh has cell di-mensions of 86 cells in the easting by 80 cells in the northingby 87 cells in depth giving a total of 598560 cells.

The three models were obtained via blind inversion of the com-ponents. A zero reference model is used with an initial modelof 0.0 g/cc so that we are not assuming anything about the ge-ologic features. The lower and upper bounds placed on thedensity contrast are -5.0 g/cc and 5.0 g/cc. These bounds wereselected to allow for a somewhat unrestricted recovery of den-sity contrasts while maintaining a plausible range of densities.The length scales in each direction are two times the cell sizesuch that LE =LN=1000 meters and LZ=500 meters. Topog-raphy is used in the inversion to remove model cells that layabove the ground surface.

Tzz inversion

We first invert the Tzz component to obtain a density contrastdistribution. Though the Tzz component was calculated, it servesas a base model to compare whether the addition of more com-ponents (observed or calculated) increases the quaility of themodel. A series of inversions using a range of values were car-ried out in order to select the optimal regularization parameter.Again, we use an L-curve critera and plot the data misfit versusthe model objective function (Hansen, 1992).

For brevity, we omit the predicted data and difference datafrom the model shown. The range of values seen in the dif-ference map are an indication of the noise in the data whichthe recovered model does not fit. The standard deviation cal-culated from the entire difference map is 0.96 Eo. We notethat a low noise level is expected from inversion of the Tzz datasince it is calculated from the other two observed components.

A volume rendered image of the density contrast model isshown in Figure 4 with all cells below 0.15 g/cc removed forclarity. The remaining high density contrast elucidates two

dense features. The rounded feature to the west is near thecore of the Vredefort Dome while the linear feature to the eastlies near the collar of the Vrederfort Dome.

Figure 4: Volume rendered image of 3D density constrastmodel constructed from Tzz with values below 0.15 g/cc re-moved. Values are in g/cc.

Txy and Tuv inversion

We next invert the two observed components Txy and Tuv =(Txx −Tyy)/2 together using the same mesh as that of the sin-gle Tzz component. The errors in the data are estimated in thesame manner as above. The estimated noise level for the Txycomponent is 5.42 Eo with a similar value of 5.53 for that ofthe Tuv component.

The density contrast volume is displayed in Figure 5 with allcells below 0.15 g/cc removed for comparison. The remaininghigh density contrast again identifies two main features. Thetwo features maintain a similar density contrast to that of theTzz model. The two features are more compact with the highdensity contrast boundaries being significantly tightened to thesource locations of the structures.

Figure 5: Volume rendered image of 3D density constrastmodel constructed from the two observed components withvalues below 0.15 g/cc removed. Values are in g/cc.

Txy, Tuv, and Tzz inversion

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Vredefort impact structure

We next invert the two observed components Txy and Tuv =(Txx−Tyy)/2 and the calculated Tzz component using the samemesh. Again, the errors in the data are estimated in the samemanner. The estimated noise level for the Txy component is5.63 Eo with a similar value of 5.82 for Tuv component. Thenoise level estimated for the Tzz component is 1 Eo, which isconsistent with the noise level seen from the Tzz inversion of0.96 Eo. This lower standard deviation of the data noise isexpected since the equivalent source method is not expected toreproduce the data errors.

The density contrast volume obtained from inverting three com-ponents is displayed in Figure 6 with cells below 0.15 g/cc re-moved. We again see the rounded feature and linear feature.We observe density contrast values consistent with the previ-ous two models. The recovered structures are comparable tothose of the two component model with the exception of thesmaller spurious features that have been eliminated with theincorporation of a third component.

Figure 6: Volume rendered image of 3D density constrastmodel constructed from inverting three components with val-ues below 0.15 g/cc removed. Values are in g/cc.

The gravity and magnetic modeling performed by Henkel andReimold (2002) provide a framework for the geologic featuresseen within the Vredefort Dome. Their fitted geologic model isshown in Figure 7. The location of this cross section is shownon the geologic map in Figure 1. Along the 300 kilometer crosssection, roughly 22 kilometers fall within the data area that wasinverted (marked in Figure 3 as a black line). The approximatelocation of this 22 kilometer stretch is denoted on the sectionof Figure 7 as a red line. From the reconstructed section, it canbe seen that there are likely massive fragments of crust in thefirst 15 kilometers of the subsurface. The two main featuresseen in the density contrast model may correspond to theseshallow crustal remanents.

CONCLUSIONS

With the three obtained models, we gain an understanding ofthe effect single and multiple components have on the result-ing inverted model structure and recovered density contrast.

Figure 7: Cross section through Vredefort impact structureconstructed from gravity and magnetic data, modified fromHenkel and Reimold (2002).

We see a marked improvement in the model structure whenusing multiple components rather than just Tzz. The improve-ments gained in the model when using three instead of twocomponents are noticable and notworthy considering the thirdcomponent was derived from two observed components. Theimprovements seen in the three component model from the twocomponent model may be related to the components them-selves. The anomalies of the two observed components aremore spread out whereas the Tzz component anomalies are co-nentrated with distinct boundaries. Given that the third compo-nent is calculated, the slight changes in the model from two tothree components may be a reflection of the distinct anomalyboundaries characteristic of the Tzz component. In this exam-ple, one component gives a satisfactory model where two mainfeatures are distinguishable. Two components delineates thestructure with more compact high density contrast while threecomponents slightly changes the model by removing remna-nent features, leaving two clearly defined structural features.

Beyond improvements in the model structure, we have demon-strated the utility of inversion for regional scale investigationsusing gravity gradient data. Given the lateral resolution and de-cay typified by gravity gradient data, large scale features suchas those seen here within the Vredefort dome can be resolved.

ACKNOWLEDGMENTS

Thank you to Mark Dransfield and Fugro for the Vredefort dataand for helpful information in completing this study. We thankthe sponsors of the Gravity and Magnetics Research Consor-tium (GMRC) for supporting this work: Anadarko, BP, BGP,Chevron, ConocoPhillips, Fugro, Marathon Oil, Petrobras, andVale.

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