Have you ever wondered whether the data you have col-lected, or have had collected for you, have been distortedor contain misrepresentations due to poor software algo-rithms? Next-generation potential field data sets are arriv-ing fast, yet few software providers have redesigned theircode to deal properly and formally with the vector and ten-sor nature of these data. We present several informativeexamples to demonstrate how and why “noise” in the datamay not be all due to the hardware and why radical “rethink-ing” of the software can aid in exploration efforts.

We have adapted existing object-oriented software toinclude a new series of classes that can be used when pro-cessing gradient data sets. These have the purpose of hid-ing the details (abstraction) of exactly what components ofa field have been observed in a survey data set. This avoidsthe problem of doing a general rewrite of processing soft-ware from the ground up for each special case. Historically,codes have mostly been written to filter, level, and grid“scalar” line data (e.g., total magnetic intensity), so thechange is a dramatic shift to the world of vectors and ten-sors.

The new family of classes in this adapted software isdesigned to honor all the commonly available airborne geo-physical observation packages. Specifically, for magneticgradiometry systems, the magnetic intensity plus:

• vertical gradient only• transverse gradient (wingtip sensors)• transverse and longitudinal gradient (wingtip and tail

stinger)• all gradients (full triaxial system)• all components of a field• full second-order tensor gradients

For moving platform gravity, the vertical component (if available) plus:

• vertical component plus motion monitors (L&R/ZLS)• horizontal curvature tensor (Falcon system)• gravity components (Sander)• full second-tensor gradients (Bell)

With this approach, each derived class is delegated thetask of enforcing any appropriate invariant relationships(e.g., tensor symmetry, trace invariance, physical invari-ance) to changes in the coordinate system, and boost sym-metry. This innate behavior can be relied upon to carrythrough in any process involving a manipulation of a mea-surement with another reading. This greatly assists thedevelopment of algorithms that work with all the varioussystems in a physically consistent way.

This modified software uses vectors and tensors in anobject-oriented design where the details are mostly hiddento the application software and exposed only where required.The aim has been to see how traditional workflow patternsneed to be modified and to examine what has been done byour peers working in the same area.

Test data sources. Survey areas used to test the new soft-

ware functions were the 2002 African airborne gravity dataset collected using the Bell system; the 2003 Timmins air-borne gravity data set collected using the Sander system,and the 2004 Baker airborne magnetic gradient data set col-lected by Firefly for Tanqueray Resources. Models wereboth from the new tensor Holstein codes and from Potent.

The list of core functionality for a vector/tensor pro-cessing system includes:

• database support for new data types• residual anomaly calculation• mimic graphics• interpretation• group statistics• aircraft compensation• interpolation• leveling• filtering• terrain correction

Object-oriented design elements. The initial object-ori-ented design used by the adapted software to hide detailsof the actual data collected is shown below in Figure 1.

Refinement of this design, once the base class (e.g., grav-ity gradiometry) is established, progresses with minimumimpact on applications and other libraries. Five variationson the vector class are immediately required for service,namely magnetic and gravity gradients, magnetic and grav-ity components, and a directional cosine vector.

Visualization. Patterns in measured field data and theirgradients are more easily grasped if graphical representa-

Innovative data processing methods for gradient airborne geophysical data setsDESMOND J. FITZGERALD, Intrepid Geophysics, Melbourne, AustraliaHORST HOLSTEIN, University of Wales, Aberystwyth

JANUARY 2006 THE LEADING EDGE 87

Figure 1. Object-oriented designused by the adapted software tohide details of the actual datacollected. This design exampleshows gravity gradiometry as the base class.

tions or mimics are used. Also, traditionalimage processing methods may be adaptedprovided the resampling keeps the datacoherent.

Tensor. Several new techniques areneeded in geophysics to aid in assessing thequality and meaning of a tensor gradientsignal. The simplest methods use the invari-ants. We recommend the adoption of a gridof tensors, with the new spherical interpo-lation scheme (see below) working in realtime to honor the data during zoom andpan. Alternatively, a grid of the cube rootof the second invariant can be used as it hasthe units of Eötvös, shows edges wherethey should be, and does not have “lobes.”An RGB representation of the eigenvaluesalso shows promise (Figure 2).

The profile view of each independentcomponent of a tensor can be useful forseeing inadmissible long-wave lengthtrends and also spikes. An alternative tothis is to reconstitute the tensor into itsgeological signal strength and its rotationalparts (quaternion vector) and show pro-files of the tensor expressed this way.Figure 3 shows profiles of tensor data com-ponents with the traditional and alterna-tive expression.

Vector mimic. For gravity components, as measured by asystem such as Sander, the predominant signal is the tradi-tional vertical component. The maximum horizontal com-ponent swings around all points of the compass, reflectingthe lower signal-to-noise ratio as much as the density varia-tion. The graphical mimic proposed for this case is shown inFigure 4.

Figure 5 shows a sample of the 2003 Timmins data set fromSander showing the gravity component displayed in a mimicof the above form.

Tensor mimic. A traditional means of understanding the

relationship between tensor componentsis accomplished using a Mohr (1900) circlediagram, as shown in Figure 6. Engineersdeveloped this graphical technique as ameans of solving for the principal com-ponents before computers were invented.In the context of a processing system, aspreadsheet editor has been adapted toshow Mohr’s circles for an observed seriesof tensor readings.

In the case of gravity and magnetic gra-diometry, the first invariant of the tensor issupposed to be zero, and so a vertical axisis shown at this point. It would be expectedthat the raw gravity gradients major radiusbe approximately 3086 Eötvös, the normalfree air vertical gravity gradient term. Thetensors are symmetric, so only the top halfneeds to be shown. The horizontal axis rep-resents the normal or principal components,and the vertical axis represents the rota-tional gradients.

Each tensor has its principal compo-nents solved and used as a basis for draw-ing each circle scaled to the maximumdifference in components for the currentgroup. In addition, the rotational compo-nent of the field tensor can be expressed

most economically in its quaternion representation. This is a4D space that allows the successive angular changes to beshown in the standard three views of plan, long, and cross.

Figure 7 shows a snapshot of the Intrepid Geophysicsspreadsheet tool displaying each individual tensor before andafter two filtering processes. This data is from a 2002 Africanairborne gravity data set collected using the Bell system.

The Grav_Lev channel is as delivered from Bell, theGrav_2k2d channel is the tensor filtered by a low pass on eachindividual component separately and finally, the Grav_RC isan RC filter applied to the tensor as a whole. Subtle changesare seen here. These are more obvious as one scrolls through

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Figure 3. (a) Tensor represented by components XX, YY, ZZ, XY, YZ, ZX. (b) Tensor represented by eigenvalues and rotations.

Figure 2. A QC plot showing the cuberoot of the second invariant of a rawtensor line data set. Flight-basedleveling problems in the acquired dataare clearly visible. A similar displaycan also be generated from a tensorgrid with the display doing an on-the-fly calculation.

the records.One possible critical approach is to examine the preserved

ratios of the invariants using the Mohr circles. If the tensorsignal has been compromised either during the acquisition orin processing, it is immediately obvious when displaying thedata in a Mohr circle—the characteristic relationships betweenthe circles are not present and all you see are the axes.

Statistical quantities.Another immediate chal-lenge is to create sum-mary statistics for thesedata types. There is avery large and well-es-tablished set of methodsfor directional field vec-tor data that is used inpalaeomagnetism. Fisher(1953) and McFadden(1980) suggested that thedistribution of vectors ona unit circle is analogousto a normal distribution.

If one plots theazimuth and dip of the potential field vectors onto aSteroplot, the center of the cluster of points is the mean angu-lar inclination and declination. The standard deviation orangular dispersion is indicative of the radius of the clusterof points. For tensor gradient data, the three principal com-ponents are assessed and reported as maximum, minimum,and mean eigenvalues. The 2003 Timmins data set and 2002African data set are used to illustrate this enhanced statis-tical reporting, as shown in Figure 8 and Table 1.

Signal quality measures. The quest for simple measures ofsignal quality for each of the vector and tensor possibilitiescontinues. For airborne gravity, vertical acceleration is oftenused as a trigger for reflights. This is not convincing, andan alternative of looking at signal rotational variations is pro-posed.

Gravity gradient corrections. Taking a similar approach tothe scalar field, a theoretical gravity gradiometer componentshould be subtracted from the observed gravity tensor tocalculate the free air tensor.

90 THE LEADING EDGE JANUARY 2006

Figure 8. 2003 Timmins gravity dataset statistics calculated using theFisher (1953) and McFadden (1980)techniques.

Figure 6. Tensor mimic Mohr (1900) circle diagram showing the relationship between tensor components.

Figure 7. 2002 African gravity tensor gradients mimic display, showing original and filtered gravity data.

Table 1. 2002 African gravity gradiometry data set—tensorstatistics (eigenvectors).

Magnitude EötvösDeclination degreesInclination degreesStd. deviation degrees

Maximum172417733.5

Mean-148-91312

Minimum-1537

53-133.5

Figure 4. Gravity components mimic display. (a) A“compass needle” in plan with the north componentvector having the arrow. (b) The residual vertical afterthe mean for each line is subtracted. Figure 5. 2003 Timmins gravity data set displayed using a gravity components mimic.

The theoretical first-order tensor correction at any ele-vation can be derived. For sea level it is approximately

assuming a downward vertical direction for the thirdCartesian measurement axis.

Terrain corrections. This is a very computation-intensiveoperation. The availability of analytic gravity gradient for-mulae (Holstein 2004) as opposed to local differencing offields makes this operation more efficient. An existing terraincorrection implementation has been extended to compute thepotential, the field components, and the gradient tensor forboth magnetic and gravity terrain effects.

Gridding. Typical interpolation schemes in potential fieldalgorithms are:

1. Akima spline (use observed gradient transformed to bealong direction of spline)

2. Minimum curvature (Briggs, 1974, and O’Connell et al.,2005)

3. Nearest neighbors (blend gradient contribution with fieldestimate)

The key question is how gradient information can be usedto create a superior representation of the field during inter-polation. Each multiplication, addition, and division involvedis examined to see how this should be implemented whenvector or tensor components or gradients are involved. Havinglooked at each case, it proved possible to define “appropriateoverloaded operator rules” for each case, and so in the pre-existing application codes, there was very little change evi-dent.

Figure 9a shows an image of standard gridding, whileFigure 9b shows gradient-enhanced gridding. The missingobservation line is used to stress-test the algorithm.

Figures 10a and 10b show a magnetic gradient data setbefore and after gradient enhancement. The improvement ismost evident in the shape of the lineaments at acute anglesto the flight line direction. The maximum improvement occurson lines subparallel to the line direction.

Note that the higher gradient portion of the signal helpsdefine the dykes into tighter-thinner bodies. The above gra-dient enhancement is produced by an enhanced Akima splinetechnique.

Tensor interpolation. The recommended method to interpo-late between two observed tensors involves more than justlinear interpolation of each component. The current acceptedpractice of linear manipulation of tensor components rapidlycompromises the observed signal. There are not only magni-tude changes, but also angular variations. It is recommendedthat the eigenvalues, together with the associated quaternionof the tensor, be used for the filtering and interpolationprocesses, as this will more correctly handle both the magni-tudes and angular variations. After an interpolation, a “nor-malization” of the tensor reconstitutes the components. Theterm spherical interpolation is used to describe the aboveprocess. With the “observed data” object-oriented imple-mentation, all of these details are hidden from the applica-tions. This idea is pursued in the appendix.

Examples of these methods in action are very encourag-ing. Figure 11 is a schematic prismatic body model that hasbeen adapted to simulate a coal seam 100 m thick, buried 200

m below the surface, with sharp angular fault-like edges. Tocompare spherical interpolation against linear interpolationthe theoretical full tensor signal of the model is first sampledalong simulated flight lines 120 m above the surface at 400-mspacings with 100-m sample intervals down each line. Usingonly this simulated flight data, the full response at a regulargrid of points is then interpolated using both spherical andlinear interpolation. Figure 12 shows the comparison of thetwo methods.

The two very encouraging improvements using the spher-ical interpolation are the sharpness of the edges and the tighterdynamic range of the regional (less dispersion of the signalstrength). The green in Figure 12b, away from the edges ofthe body, is an indicator of less dispersion during the inter-polation process. The top left edge in 12a has a waviness, com-

92 THE LEADING EDGE JANUARY 2006

Figure 9. Sample magnetic data gridded with a missing observationline—(a) before and (b) after gradient enhancement.

Figure 10. Magnetic gradient grid (a) before and (b) after gradientenhancement.

Figure 11. Schematic of model of simulated coal seam. Vertical andhorizontal scales are units of 10 m and 100 m, respectively.

pared to the same point in 12b, that is characteristic of linearversus spherical interpolation. Nearest-neighbor techniquesare used to produce the above. The data range and color clip-ping are near enough to be considered the same for bothimages. Extensions to minimum curvature methods are nowin place. It appears that for tensor data, the use of minimumcurvature is not as necessary, as you know a lot more aboutthe gradients.

Tensor integration. Vassiliou (1986) showed how to create atransfer function in Fourier space to calculate the TZ (or sayan equivalent gravity free air grid) from up to three tensorcomponents (TXZ - TYZ - TZZ). Getting a satisfactory result fromthe process has had to wait until the process of properly resam-pling onto a grid was sorted out, otherwise the “signal noise”in the final product is unacceptable. Proof of the method isillustrated for the model above. Figure 13 shows on the left,the modelled free air grid and on the right, the computed freeair from the TXZ, TYZ, and TZZ component grids. The dynamicrange is almost identical, but the DC component for the com-puted product is unknown.

Leveling. This process along with aircraft compensation, pre-sents some of the biggest challenges.

Magnetic gradient field data. Misclosure at a crossover pointfor a field becomes the vector difference. Some questions toask would be, does the observation instrumentation’s cali-bration drift in time? If so, how?

Tensor gradients. The misclosure tensor has a much lowersignal/noise ratio than the signal down each profile. For exam-ple, compare the direction statistics shown in Table 1 with thedirection statistics for their misclosures, as shown in Table 2.For the eigenvector of the maximum eigenvalue, the angularstandard deviation goes from 3 to 56º.

Filtering. For profile data, the sig-nal-to-noise ratio of measured gra-dients can be low, and there areclassical problems of enhancing thesignal while damping down thenoise as resampling from say 100Hz to 2 Hz. Consequently, innova-tive noise reduction using IIR(recursive techniques) is suggested.Tensor interpolation should beavoided at this stage so this proba-bly precludes FFT methods.

Aspatial convolution, using anodd operational length (e.g., 5), isused in the filtering of vector andtensor components. Filters cur-rently tested include a movingaverage, median, LaCoste RC andLaplace curvature (damped).

An attempt is made to honorthe invariants of a tensor while anoise or spike-filtering operationis performed. The LaCoste RC fil-

ter code is used as a test on observed Bell tensor data toexplore the possibilities. This filter is recursive and dampensthe noise. The Laplace filter is adapted from the LaCoste fil-ter for curvature calculation and filtering. It uses a stack oforiginal and transformed or filtered observed values and usesthe time in seconds (sampling time say 1 s or 10 s). It is sim-ilar in operation to the RC and a Kalman filter but works onthe whole tensor/vector.

Workflows. Due to the newness of tensor data sets we observethat groups may have a workflow for gravity gradient tensorprocessing that appears to be distorted by the limitations ofthe tools available to them. As a result of the work above, theauthors urge that more conventional workflows for the newerdata types be adopted using appropriate tools.

The classical example of this is the disregard for a Nettleton-style gravity reduction workflow when dealing with gravitytensor data. It is recommended that the workflow follow anobserved, theoretical earth model correction, free air correc-tion, Bouguer, and terrain correction path. This will also aidthe users of this data.

To illustrate this point, consider the following list of toolsthe authors have updated to support the tensor and vectordata being acquired:

• gridding• profile editing of the complete tensor for damping noise• loop leveling• spreadsheet editor—mimic displays• project manager support• visualization—support for tensor grids and dynamic

resampling• import into new persistent database types (including

gdbs)• statistical improvements• free air corrections—integration of TXZ-TYZ-TZZ to

estimate TZ• terrain corrections• forward and inverse geology constrained 3D modeling

Conclusions. Smarter computational support for new gener-ation geophysical data sets is here to stay. A greater use ofobject-oriented methods can contribute to controlling the soft-ware complexity of each process. Processing the “observed”

JANUARY 2006 THE LEADING EDGE 93

Figure 13. (a) Model grid of free air. (b) Computed free air by integration of TXZ-TYZ-TZZ.

Figure 12. (a) Linear interpolated cube root of second invariant of tensor. (b) Spherical interpolatedversion. The spherical interpolation gives a result much closer to the theoretical response. A tensorgrid in “semi” format of the model response is available upon request.

Table 2. 2002 African gravity gradiometry data set misclosuretensor statistics based on 1200 crossovers.

Magnitude EötvösDeclination degreesInclination degrees

Std. deviation

Maximum489685556

Mean3.41271345

Minimum-490317660

package as an object can be achieved. This helps to hidedetails from processes that do not need to know them andthereby presents field physics issues in a more natural man-ner.

Instrumentation engineers should be encouraged togather still more real-time characteristics and to report them,to help in extracting the most value from the data post-mis-sion. Aircraft compensation can now be rethought in orderto take the field nature of the signal and its rotational partsinto account.

The current practice of manipulating gravity tensor com-ponents independently during filtering, leveling, gridding,etc., is flawed. A blurring and dispersion of the geologicalsignal content results. Having collected either a full tensoror more than one component, it makes sense to use all thedata collected in the interpretation phase. The distortionbeing introduced by poor interpolation is generally close tothe wavelengths of most interest to the diamond explorers.With full-tensor data, the opportunity to adjust/level andgrid a more reliable representation of your field is enhancedas you have all the gradients.

If you do not have full-tensor data, you are better offusing profile interpretation methods as there is less distor-tion in the signal due to interpolation errors. This supportscollecting full tensor even at a lesser resolution over col-lecting just one or two components.

Suggested reading. “Machine Contouring Using MinimumCurvature” by Briggs (GEOPHYSICS, 1974). “Dispersion on asphere” by Fisher (Proceedings of the Royal Society, 1953).“Gravimagnetic field tensor gradiometry formulas for uniformpolyhedra” by Holstein et al. (SEG 2004 Expanded Abstracts).“The best estimate of Fisher's precision parameter k” byMcFadden (Geophysical Journal International, 1980). “WelcheUmstände bedingen die Elastizitätsgrenze und den Bruch einesMaterials?” by Mohr (Z. Ver. Dt. Ing., 1900). “Gridding aero-magnetic data using longitudinal and transverse horizontalgradients with minimum curvature operation” by O’Connellet al. (TLE, 2005). “Comparison of methods for the processingof gravity gradiometer data” by Vassiliou (Proceedings ofGravity Gradiometer Conference, 1986). TLE

Acknowledgment: This project is proudly supported by the InternationalScience Linkage programme established under the AustralianGovernment’s innovation statement, Backing Australia’s Ability.

Corresponding author: des@intrepid-geophysics.com or hoh@aber.ac.uk

Appendix—Eigen-representation of gravimagnetic fieldgradient tensors for interpolation and filtering purposes.Gravity and magnetic field gradient tensors are known tobe symmetric and of zero trace (sum of diagonal compo-nents). This admits five degrees of freedom among the ninecomponents present in the full 3D tensor of rank 2.Interpolation and filtering processes must honor thesedependencies.

Analogously to vectors, tensor components depend onthe choice of measurement coordinate system. It is, how-ever, possible to describe the tensor in terms of disjointstructural and orientational properties. The structural prop-erties are independent of the choice of coordinate system,while the orientational properties depend only on the choiceof coordinate system. This gives rise to the possibility of

interpolating these two properties separately, in a mannerthat allows reconstruction of an interpolated tensor at an“in between” field measurement point. These remarks sim-ilarly extend to filtering.

Let T be the 3�3 matrix associated with a field gradienttensor in a given Cartesian coordinate system. The matrixT will be symmetric and satisfy trace (T)=0. For such amatrix, there exists a 3�3 rotation matrix R satisfying

RTT R = � (A1)

where � is a 3�3 diagonal matrix containing the three eigen-values of T, all real. This is a result from standard eigen-system construction. Moreover, the three eigenvalues sumto zero on account of the preserved trace,

trace(T) = trace(�) = 0 (A2)

and the columns of R form three orthonormal vectors thatdefine the unit axes of an orthogonal coordinate system. Inthis coordinate system, the tensor has the diagonal matrixrepresentation �. Standard mathematical software can deter-mine matrices R and �. The structural information of thetarget that gives rise to the field gradient anomaly is con-tained in the matrix �. This information can be convenientlyinterpolated or filtered in terms of operations on each of thethree diagonal components, subject to the trace condition.

The orientation information is contained in the rotationmatrix R. It is known from the theory of rotation operatorsthat matrix R has a unit 4-vector representation, (strictlyspeaking, a unit quaternion). Thus, the sequence of observedfield gradient tensors can be associated with a path tracedout on the surface of a unit 4-sphere. Interpolation is to becarried out in this manifold, to yield a new unit 4-vectorinterpolant. This interpolant allows the corresponding rota-tion matrix to be reconstructed. Effectively, we have an inter-polation process for rotation matrices that yields onlyrotation matrices. Similar remarks hold for the filtering oper-ation.

The decomposition into structural and rotational partsyields 2 plus 3 independent quantities respectively, con-forming to the five degrees of freedom of the original for-mulation. The decomposed form, however, allowsinterpolation or filtering processes to admit only consistentmatrix representations of the underlying tensors with regardto their structural and rotational field gradient informationcontent. Thus, an interpolated rotation matrix Rint and aninterpolated diagonal matrix � int together determine thereconstructed matrix Tint of the tensor via

Tint = Rint � int RintT (A3)

By way of simple illustration, consider the effect of orien-tation error in the measurement system for a constant geol-ogy. Tensor eigen-interpolation as proposed above willpreserve the geology, while tensor component linear inter-polation will not. To demonstrate this another way, imag-ine two tensor observations made close to each other wherethe eigenvalues are identical, but there is a slight offset inthe rotations by 5°.

The spherical interpolation method allows you to con-tinuously estimate intermediate tensors that yield correcteigenvalues, while linear interpolation does not.

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