Innovative data processing methods for gradient airborne

Innovative data processing methods
for gradient airborne geophysical data sets
DESMOND J. FITZGERALD, Intrepid Geophysics, Melbourne, Australia
HORST HOLSTEIN, University of Wales, Aberystwyth
H
ave you ever wondered whether the data you have collected, or have had collected for you, have been distorted
or contain misrepresentations due to poor software algorithms? Next-generation potential field data sets are arriving fast, yet few software providers have redesigned their
code to deal properly and formally with the vector and tensor nature of these data. We present several informative
examples to demonstrate how and why “noise” in the data
may not be all due to the hardware and why radical “rethinking” of the software can aid in exploration efforts.
We have adapted existing object-oriented software to
include a new series of classes that can be used when processing gradient data sets. These have the purpose of hiding the details (abstraction) of exactly what components of
a field have been observed in a survey data set. This avoids
the problem of doing a general rewrite of processing software 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 the
change is a dramatic shift to the world of vectors and tensors.
The new family of classes in this adapted software is
designed to honor all the commonly available airborne geophysical observation packages. Specifically, for magnetic
gradiometry systems, the magnetic intensity plus:
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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:
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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 the
task of enforcing any appropriate invariant relationships
(e.g., tensor symmetry, trace invariance, physical invariance) to changes in the coordinate system, and boost symmetry. This innate behavior can be relied upon to carry
through in any process involving a manipulation of a measurement with another reading. This greatly assists the
development of algorithms that work with all the various
systems in a physically consistent way.
This modified software uses vectors and tensors in an
object-oriented design where the details are mostly hidden
to the application software and exposed only where required.
The aim has been to see how traditional workflow patterns
need to be modified and to examine what has been done by
our peers working in the same area.
Test data sources. Survey areas used to test the new soft-
Figure 1. Object-oriented design
used by the adapted software to
hide details of the actual data
collected. This design example
shows gravity gradiometry as
the base class.
ware functions were the 2002 African airborne gravity data
set collected using the Bell system; the 2003 Timmins airborne gravity data set collected using the Sander system,
and the 2004 Baker airborne magnetic gradient data set collected by Firefly for Tanqueray Resources. Models were
both from the new tensor Holstein codes and from Potent.
The list of core functionality for a vector/tensor processing system includes:
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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-oriented design used by the adapted software to hide details
of the actual data collected is shown below in Figure 1.
Refinement of this design, once the base class (e.g., gravity gradiometry) is established, progresses with minimum
impact on applications and other libraries. Five variations
on the vector class are immediately required for service,
namely magnetic and gravity gradients, magnetic and gravity components, and a directional cosine vector.
Visualization. Patterns in measured field data and their
gradients are more easily grasped if graphical representaJANUARY 2006
THE LEADING EDGE
87
tions or mimics are used. Also, traditional
relationship between tensor components
image processing methods may be adapted
is accomplished using a Mohr (1900) circle
provided the resampling keeps the data
diagram, as shown in Figure 6. Engineers
coherent.
developed this graphical technique as a
Tensor. Several new techniques are
means of solving for the principal comneeded in geophysics to aid in assessing the
ponents before computers were invented.
quality and meaning of a tensor gradient
In the context of a processing system, a
signal. The simplest methods use the invarispreadsheet editor has been adapted to
ants. We recommend the adoption of a grid
show Mohr’s circles for an observed series
of tensors, with the new spherical interpoof tensor readings.
lation scheme (see below) working in real
In the case of gravity and magnetic gratime to honor the data during zoom and
diometry, the first invariant of the tensor is
pan. Alternatively, a grid of the cube root
supposed to be zero, and so a vertical axis
of the second invariant can be used as it has
is shown at this point. It would be expected
the units of Eötvös, shows edges where
that the raw gravity gradients major radius
they should be, and does not have “lobes.”
be approximately 3086 Eötvös, the normal
An RGB representation of the eigenvalues
free air vertical gravity gradient term. The
also shows promise (Figure 2).
tensors are symmetric, so only the top half
The profile view of each independent
needs to be shown. The horizontal axis repcomponent of a tensor can be useful for
resents the normal or principal components,
seeing inadmissible long-wave length Figure 2. A QC plot showing the cube and the vertical axis represents the rotatrends and also spikes. An alternative to root of the second invariant of a raw
tional gradients.
this is to reconstitute the tensor into its tensor line data set. Flight-based
Each tensor has its principal compoleveling problems in the acquired data
geological signal strength and its rotational are clearly visible. A similar display
nents solved and used as a basis for drawparts (quaternion vector) and show pro- can also be generated from a tensor
ing each circle scaled to the maximum
files of the tensor expressed this way. grid with the display doing an on-the- difference in components for the current
Figure 3 shows profiles of tensor data com- fly calculation.
group. In addition, the rotational compoponents with the traditional and alternanent of the field tensor can be expressed
tive expression.
most economically in its quaternion representation. This is a
Vector mimic. For gravity components, as measured by a 4D space that allows the successive angular changes to be
system such as Sander, the predominant signal is the tradi- shown in the standard three views of plan, long, and cross.
tional vertical component. The maximum horizontal comFigure 7 shows a snapshot of the Intrepid Geophysics
ponent swings around all points of the compass, reflecting spreadsheet tool displaying each individual tensor before and
the lower signal-to-noise ratio as much as the density varia- after two filtering processes. This data is from a 2002 African
tion. The graphical mimic proposed for this case is shown in airborne gravity data set collected using the Bell system.
Figure 4.
The Grav_Lev channel is as delivered from Bell, the
Figure 5 shows a sample of the 2003 Timmins data set from Grav_2k2d channel is the tensor filtered by a low pass on each
Sander showing the gravity component displayed in a mimic individual component separately and finally, the Grav_RC is
of the above form.
an RC filter applied to the tensor as a whole. Subtle changes
Tensor mimic. A traditional means of understanding the are seen here. These are more obvious as one scrolls through
Figure 3. (a) Tensor represented by components XX, YY, ZZ, XY, YZ, ZX. (b) Tensor represented by eigenvalues and rotations.
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Figure 4. Gravity components mimic display. (a) A
“compass needle” in plan with the north component
vector having the arrow. (b) The residual vertical after
the mean for each line is subtracted.
Figure 5. 2003 Timmins gravity data set displayed using a gravity components mimic.
Table 1. 2002 African gravity gradiometry data set—tensor
statistics (eigenvectors).
Maximum
Mean
Minimum
1724
-148
-1537
Magnitude Eötvös
17
-9
53
Declination degrees
73
13
-13
Inclination degrees
Std. deviation degrees
3.5
12
3.5
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.
the records.
One possible critical approach is to examine the preserved
ratios of the invariants using the Mohr circles. If the tensor
signal has been compromised either during the acquisition or
in processing, it is immediately obvious when displaying the
data in a Mohr circle—the characteristic relationships between
the circles are not present and all you see are the axes.
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Statistical quantities.
Another immediate challenge is to create summary statistics for these
data types. There is a
very large and well-established set of methods
for directional field vector data that is used in
palaeomagnetism. Fisher
(1953) and McFadden
(1980) suggested that the
Figure 8. 2003 Timmins gravity data
distribution of vectors on
set statistics calculated using the
a unit circle is analogous
Fisher (1953) and McFadden (1980)
to a normal distribution.
techniques.
If one plots the
azimuth and dip of the potential field vectors onto a
Steroplot, the center of the cluster of points is the mean angular inclination and declination. The standard deviation or
angular dispersion is indicative of the radius of the cluster
of points. For tensor gradient data, the three principal components are assessed and reported as maximum, minimum,
and mean eigenvalues. The 2003 Timmins data set and 2002
African data set are used to illustrate this enhanced statistical reporting, as shown in Figure 8 and Table 1.
Signal quality measures. The quest for simple measures of
signal quality for each of the vector and tensor possibilities
continues. For airborne gravity, vertical acceleration is often
used as a trigger for reflights. This is not convincing, and
an alternative of looking at signal rotational variations is proposed.
Gravity gradient corrections. Taking a similar approach to
the scalar field, a theoretical gravity gradiometer component
should be subtracted from the observed gravity tensor to
calculate the free air tensor.
The theoretical first-order tensor correction at any elevation can be derived. For sea level it is approximately
assuming a downward vertical direction for the third
Cartesian measurement axis.
Terrain corrections. This is a very computation-intensive
operation. The availability of analytic gravity gradient formulae (Holstein 2004) as opposed to local differencing of
fields makes this operation more efficient. An existing terrain
correction implementation has been extended to compute the
potential, the field components, and the gradient tensor for
both magnetic and gravity terrain effects.
Figure 9. Sample magnetic data gridded with a missing observation
line—(a) before and (b) after gradient enhancement.
Gridding. Typical interpolation schemes in potential field
algorithms are:
1.
2.
3.
Akima spline (use observed gradient transformed to be
along direction of spline)
Minimum curvature (Briggs, 1974, and O’Connell et al.,
2005)
Nearest neighbors (blend gradient contribution with field
estimate)
The key question is how gradient information can be used
to create a superior representation of the field during interpolation. Each multiplication, addition, and division involved
is examined to see how this should be implemented when
vector or tensor components or gradients are involved. Having
looked at each case, it proved possible to define “appropriate
overloaded operator rules” for each case, and so in the preexisting application codes, there was very little change evident.
Figure 9a shows an image of standard gridding, while
Figure 9b shows gradient-enhanced gridding. The missing
observation line is used to stress-test the algorithm.
Figures 10a and 10b show a magnetic gradient data set
before and after gradient enhancement. The improvement is
most evident in the shape of the lineaments at acute angles
to the flight line direction. The maximum improvement occurs
on lines subparallel to the line direction.
Note that the higher gradient portion of the signal helps
define the dykes into tighter-thinner bodies. The above gradient enhancement is produced by an enhanced Akima spline
technique.
Tensor interpolation. The recommended method to interpolate between two observed tensors involves more than just
linear interpolation of each component. The current accepted
practice of linear manipulation of tensor components rapidly
compromises the observed signal. There are not only magnitude changes, but also angular variations. It is recommended
that the eigenvalues, together with the associated quaternion
of the tensor, be used for the filtering and interpolation
processes, as this will more correctly handle both the magnitudes and angular variations. After an interpolation, a “normalization” of the tensor reconstitutes the components. The
term spherical interpolation is used to describe the above
process. With the “observed data” object-oriented implementation, all of these details are hidden from the applications. This idea is pursued in the appendix.
Examples of these methods in action are very encouraging. Figure 11 is a schematic prismatic body model that has
been adapted to simulate a coal seam 100 m thick, buried 200
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Figure 10. Magnetic gradient grid (a) before and (b) after gradient
enhancement.
Figure 11. Schematic of model of simulated coal seam. Vertical and
horizontal scales are units of 10 m and 100 m, respectively.
m below the surface, with sharp angular fault-like edges. To
compare spherical interpolation against linear interpolation
the theoretical full tensor signal of the model is first sampled
along simulated flight lines 120 m above the surface at 400-m
spacings with 100-m sample intervals down each line. Using
only this simulated flight data, the full response at a regular
grid of points is then interpolated using both spherical and
linear interpolation. Figure 12 shows the comparison of the
two methods.
The two very encouraging improvements using the spherical interpolation are the sharpness of the edges and the tighter
dynamic range of the regional (less dispersion of the signal
strength). The green in Figure 12b, away from the edges of
the body, is an indicator of less dispersion during the interpolation process. The top left edge in 12a has a waviness, com-
Filtering. For profile data, the signal-to-noise ratio of measured gradients can be low, and there are
classical problems of enhancing the
signal while damping down the
noise as resampling from say 100
Hz to 2 Hz. Consequently, innovative noise reduction using IIR
Figure 12. (a) Linear interpolated cube root of second invariant of tensor. (b) Spherical interpolated
(recursive techniques) is suggested.
version. The spherical interpolation gives a result much closer to the theoretical response. A tensor
Tensor interpolation should be
grid in “semi” format of the model response is available upon request.
avoided at this stage so this probably precludes FFT methods.
Aspatial convolution, using an
odd operational length (e.g., 5), is
used in the filtering of vector and
tensor components. Filters currently tested include a moving
average, median, LaCoste RC and
Laplace curvature (damped).
An attempt is made to honor
the invariants of a tensor while a
noise or spike-filtering operation
Figure 13. (a) Model grid of free air. (b) Computed free air by integration of TXZ-TYZ-TZZ.
is performed. The LaCoste RC filter code is used as a test on observed Bell tensor data to
Table 2. 2002 African gravity gradiometry data set misclosure
explore the possibilities. This filter is recursive and dampens
tensor statistics based on 1200 crossovers.
the noise. The Laplace filter is adapted from the LaCoste filMaximum
Mean
Minimum
ter for curvature calculation and filtering. It uses a stack of
original and transformed or filtered observed values and uses
Magnitude Eötvös
489
3.41
-490
the time in seconds (sampling time say 1 s or 10 s). It is sim31
Declination degrees
68
27
ilar in operation to the RC and a Kalman filter but works on
76
Inclination degrees
55
13
the whole tensor/vector.
Std. deviation
56
45
60
pared to the same point in 12b, that is characteristic of linear
versus spherical interpolation. Nearest-neighbor techniques
are used to produce the above. The data range and color clipping are near enough to be considered the same for both
images. Extensions to minimum curvature methods are now
in place. It appears that for tensor data, the use of minimum
curvature is not as necessary, as you know a lot more about
the gradients.
Tensor integration. Vassiliou (1986) showed how to create a
transfer function in Fourier space to calculate the TZ (or say
an equivalent gravity free air grid) from up to three tensor
components (TXZ - TYZ - TZZ). Getting a satisfactory result from
the process has had to wait until the process of properly resampling onto a grid was sorted out, otherwise the “signal noise”
in the final product is unacceptable. Proof of the method is
illustrated for the model above. Figure 13 shows on the left,
the modelled free air grid and on the right, the computed free
air from the TXZ, TYZ, and TZZ component grids. The dynamic
range is almost identical, but the DC component for the computed product is unknown.
Leveling. This process along with aircraft compensation, presents some of the biggest challenges.
Magnetic gradient field data. Misclosure at a crossover point
for a field becomes the vector difference. Some questions to
ask would be, does the observation instrumentation’s calibration drift in time? If so, how?
Tensor gradients. The misclosure tensor has a much lower
signal/noise ratio than the signal down each profile. For example, compare the direction statistics shown in Table 1 with the
direction statistics for their misclosures, as shown in Table 2.
For the eigenvector of the maximum eigenvalue, the angular
standard deviation goes from 3 to 56º.
Workflows. Due to the newness of tensor data sets we observe
that groups may have a workflow for gravity gradient tensor
processing that appears to be distorted by the limitations of
the tools available to them. As a result of the work above, the
authors urge that more conventional workflows for the newer
data types be adopted using appropriate tools.
The classical example of this is the disregard for a Nettletonstyle gravity reduction workflow when dealing with gravity
tensor data. It is recommended that the workflow follow an
observed, theoretical earth model correction, free air correction, Bouguer, and terrain correction path. This will also aid
the users of this data.
To illustrate this point, consider the following list of tools
the authors have updated to support the tensor and vector
data being acquired:
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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 T XZ-T YZ-T ZZ to
estimate TZ
terrain corrections
forward and inverse geology constrained 3D modeling
Conclusions. Smarter computational support for new generation geophysical data sets is here to stay. A greater use of
object-oriented methods can contribute to controlling the software complexity of each process. Processing the “observed”
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package as an object can be achieved. This helps to hide
details from processes that do not need to know them and
thereby presents field physics issues in a more natural manner.
Instrumentation engineers should be encouraged to
gather still more real-time characteristics and to report them,
to help in extracting the most value from the data post-mission. Aircraft compensation can now be rethought in order
to take the field nature of the signal and its rotational parts
into account.
The current practice of manipulating gravity tensor components independently during filtering, leveling, gridding,
etc., is flawed. A blurring and dispersion of the geological
signal content results. Having collected either a full tensor
or more than one component, it makes sense to use all the
data collected in the interpretation phase. The distortion
being introduced by poor interpolation is generally close to
the wavelengths of most interest to the diamond explorers.
With full-tensor data, the opportunity to adjust/level and
grid a more reliable representation of your field is enhanced
as you have all the gradients.
If you do not have full-tensor data, you are better off
using profile interpretation methods as there is less distortion in the signal due to interpolation errors. This supports
collecting full tensor even at a lesser resolution over collecting just one or two components.
Suggested reading. “Machine Contouring Using Minimum
Curvature” by Briggs (GEOPHYSICS, 1974). “Dispersion on a
sphere” by Fisher (Proceedings of the Royal Society, 1953).
“Gravimagnetic field tensor gradiometry formulas for uniform
polyhedra” by Holstein et al. (SEG 2004 Expanded Abstracts).
“The best estimate of Fisher's precision parameter k” by
McFadden (Geophysical Journal International, 1980). “Welche
Umstände bedingen die Elastizitätsgrenze und den Bruch eines
Materials?” by Mohr (Z. Ver. Dt. Ing., 1900). “Gridding aeromagnetic data using longitudinal and transverse horizontal
gradients with minimum curvature operation” by O’Connell
et al. (TLE, 2005). “Comparison of methods for the processing
of gravity gradiometer data” by Vassiliou (Proceedings of
Gravity Gradiometer Conference, 1986). TLE
Acknowledgment: This project is proudly supported by the International
Science Linkage programme established under the Australian
Government’s innovation statement, Backing Australia’s Ability.
Corresponding author: [email protected] or [email protected]
Appendix—Eigen-representation of gravimagnetic field
gradient tensors for interpolation and filtering purposes.
Gravity and magnetic field gradient tensors are known to
be symmetric and of zero trace (sum of diagonal components). This admits five degrees of freedom among the nine
components present in the full 3D tensor of rank 2.
Interpolation and filtering processes must honor these
dependencies.
Analogously to vectors, tensor components depend on
the choice of measurement coordinate system. It is, however, possible to describe the tensor in terms of disjoint
structural and orientational properties. The structural properties are independent of the choice of coordinate system,
while the orientational properties depend only on the choice
of coordinate system. This gives rise to the possibility of
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interpolating these two properties separately, in a manner
that allows reconstruction of an interpolated tensor at an
“in between” field measurement point. These remarks similarly extend to filtering.
Let T be the 3ǂ3 matrix associated with a field gradient
tensor in a given Cartesian coordinate system. The matrix
T will be symmetric and satisfy trace (T)=0. For such a
matrix, there exists a 3ǂ3 rotation matrix R satisfying
RTT R = Ǽ
(A1)
where Ǽ is a 3ǂ3 diagonal matrix containing the three eigenvalues of T, all real. This is a result from standard eigensystem construction. Moreover, the three eigenvalues sum
to zero on account of the preserved trace,
trace(T) = trace(Ǽ) = 0
(A2)
and the columns of R form three orthonormal vectors that
define the unit axes of an orthogonal coordinate system. In
this coordinate system, the tensor has the diagonal matrix
representation Ǽ. Standard mathematical software can determine matrices R and Ǽ. The structural information of the
target that gives rise to the field gradient anomaly is contained in the matrix Ǽ. This information can be conveniently
interpolated or filtered in terms of operations on each of the
three diagonal components, subject to the trace condition.
The orientation information is contained in the rotation
matrix R. It is known from the theory of rotation operators
that matrix R has a unit 4-vector representation, (strictly
speaking, a unit quaternion). Thus, the sequence of observed
field gradient tensors can be associated with a path traced
out on the surface of a unit 4-sphere. Interpolation is to be
carried out in this manifold, to yield a new unit 4-vector
interpolant. This interpolant allows the corresponding rotation matrix to be reconstructed. Effectively, we have an interpolation process for rotation matrices that yields only
rotation matrices. Similar remarks hold for the filtering operation.
The decomposition into structural and rotational parts
yields 2 plus 3 independent quantities respectively, conforming to the five degrees of freedom of the original formulation. The decomposed form, however, allows
interpolation or filtering processes to admit only consistent
matrix representations of the underlying tensors with regard
to their structural and rotational field gradient information
content. Thus, an interpolated rotation matrix Rint and an
interpolated diagonal matrix Ǽ int together determine the
reconstructed matrix Tint of the tensor via
Tint = Rint Ǽ int RintT
(A3)
By way of simple illustration, consider the effect of orientation error in the measurement system for a constant geology. Tensor eigen-interpolation as proposed above will
preserve the geology, while tensor component linear interpolation will not. To demonstrate this another way, imagine two tensor observations made close to each other where
the eigenvalues are identical, but there is a slight offset in
the rotations by 5°.
The spherical interpolation method allows you to continuously estimate intermediate tensors that yield correct
eigenvalues, while linear interpolation does not.