Enhancement of magnetic data by stable downward continuation for UXO applications
Yaoguo Li and Sarah Devriese*
Center for Gravity Electrical, and Magnetic Studies, Department of Geophysics, Colorado School of Mines
Summary
We present an algorithm for enhancing magnetic data in
UXO applications using a stable downward continuation
method. The algorithm formulates the downward
continuation as an inverse problem using Tikhonov
regularization and has the flexibility of incorporating the
expected power spectrum of UXO anomalies. The degree
of regularization is estimated automatically using the wellestablished methods in linear inverse problems. Numerical
tests show that the algorithm can reliably estimate the noise
in the data and reconstruct the magnetic anomaly at ground
surface within the limitation imposed by the noise. The
reconstructed field at the ground surface exhibits
significant enhancement compared to the original data. This
presentation will discuss the stable downward continuation
algorithm, estimation of expected power spectrum, the
choice of regularization level, and demonstrate the method
with both synthetic and field data sets.
height in a typical man-portable system or a vehicle-towed
system may be between 0.15 to 0.5 m above the ground,
and a helicopter-based survey would be flown at a height
about 1.5 to 2.0 m above the ground (Doll, et al., 2006).
Thus, there is a need for enhancing magnetic data to
alleviate the effects of observation height and to estimate
the noise characteristics. We propose to address this
specific aspect of magnetic data by developing and
applying a stable downward continuation method.
In this paper, we first describe a stable downward
continuation algorithm formulated as a regularized
inversion using Tikhonov regularization in the Fourier
domain. We will discuss the details of the algorithm such
as the choice of spectral weighting and determination of
regularization parameter. We will use a synthetic data set to
illustrate the algorithm and its performance in the
enhancement of UXO magnetic data. We will also present
the noise estimation and data enhancement achieved in
several field data sets from airborne systems acquired in
UXO demonstration surveys.
Introduction
The magnetic method is one of the two most effective
geophysical techniques currently in use for UXO detection
and discrimination. The use of magnetic data in UXO
applications has been well studied and there are welldocumented successful examples of locating and
discriminating UXO. Detection of UXO through magnetic
data depends on identifying dipole-like anomalies, while
discrimination depends upon inversions to recover reliably
the magnetic source parameters. These magnetic data are
acquired with sensors at some height above ground surface.
It is well known that height significantly affects magnetic
anomaly shape, amplitude, and spatial extent.
Consequently, anomalies due to multiple metallic targets
may overlap at a given height above the ground surface,
and the acquisition noise may significantly decrease the
signal-to-noise ratio (SNR) of data. These adverse effects
will decrease the effectiveness of current detection and
discrimination methods and may hamper the use of
magnetics in UXO clearance.
The data resolution, quantified by the distance between
separable anomalies, decreases linearly with increasing
observation height. The ideal data are those acquired with
a sensor at zero height above the ground provided that the
ground surface is not significantly magnetic. However,
because of the intrinsic limitations imposed on field data
acquisition, all platforms collect data at some height above
the ground surface. For example, the magnetic sensor
Stable downward continuation
The magnetic data at two observation heights are related by
the upward continuation operation (Blakely, 1995),
Th ( x, y, ∆h) =
1
2π
∞ ∞
∫ ∫ [( x'− x)
− ∞− ∞
2
T0 ( x' , y ' )∆h
,
dx' dy '
+ ( y '− y ) 2 + ∆h 2 ]3 / 2
(1)
where T0 ( x, y ) and Th ( x, y, h) are respectively the magnetic
data at two observation heights separated by a vertical
distance ∆h. Applying a two-dimensional Fourier transform
to eq.(1) yields a simpler form in which the Fourier
transforms of the two quantities are related to each other by
a simple upward continuation operator
~
~
Th (ω x , ω y , ∆h) = e − ∆hωr T0 (ω x , ω y ) ,
(2)
where T~0 (ω x , ω y ) denotes the Fourier transform of T0 ( x, y ) ,
(ωx , ω y ) are wavenumbers in x- and y-direction, and
ω r = ω x2 + ω y2
is the radial wavenumber. The upward
continuation operator attenuates
frequency content of a magnetic
can be partially reversed by
continuation, which is the inverse
with height the highanomaly. These effects
performing downward
of eq.(2), to reconstruct
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1464
Stable downward continuation of UXO magnetic data
the field at the ground surface. The ground surface is the
lowest possible observation height.
Although the operation of downward continuation is
mathematically valid in the source free region above the
ground surface (Blakely, 1995), it is numerically unstable
because of the presence of high-frequency noise. However,
we can stabilize the process by formulating it as an inverse
problem and derive a regularized operator (Huestis and
Parker, 1979). We will carry out this operation in the
wavenumber domain because of the numerical efficiency.
Let us treat the Fourier transform T~ (ω , ω , ∆h) of observed
h
x
y
magnetic anomaly at height ∆h as the data, and the sought
Fourier transform of anomaly at the ground surface as the
model. The upward continuation operator e − ∆hωr then
defines a forward mapping. The inverse problem is then
stated as finding a well-behaved T~0 (ω x , ω y ) that has certain
spectral characteristics and reasonably reproduces the data
~
Th (ω x , ω y , ∆h) by eq.(2). Assuming the magnetic data that
would be observed at the ground surface has an expected
power spectrum of P0 (ω x , ω y ) , we can define a model
objective function φm to quantify the conformity of the
model with the expected spectral property,
~
φ m = ∫∫ P0 (ω x , ω y ) −1 T0 (ω x , ω y ) dω x dω y ,
2
(3)
where the reciprocal of the expected power spectrum
provides the weighting function. An assumed P0 (ω x , ω y )
that decays with increasing wavenumber corresponds to an
increased weight at higher wavenumber for the final model.
Consequently, the final model from the inversion will have
the similar spectral decay. Such a spectral decay
corresponds to a smooth magnetic field in the space
domain. Thus, the use of inverse spectral decay as a
weighting function provides a versatile approach to
incorporate prior information about the magnetic data. We
will discuss this aspect in more detail in the next section.
We define a data misfit function as,
~
~
φ d = ∫∫ T0 (ω x , ω y )e −∆hω − Th (ω x , ω y , h) dω x dω y ,
r
2
(4)
which is in effect the scaled version of l2 data misfit in the
space domain by Parseval’s theorem. The lack of a
normalization factor in this definition also implies the
assumption of independent noise in the space domain,
which translates to a constant noise level across the
wavenumber band of the data.
Utilizing the formalism of Tikhonov regularization, the
inverse problem of reconstructing the magnetic anomaly at
the ground surface then becomes one of optimization:
minimize: φ = φ d + µφm
(5)
where µ is the regularization parameter and it determines
the trade-off between fitting the data and the conformity of
model with the expected spectral properties. Solving the
minimization with the optimal value of regularization
parameter µ produces the desired Fourier transform of
downward continued data.
The solution of eq.(5) involves both complex data and a
complex model. Therefore, we resort to the calculus of
variation to carry out the formal minimization. The result is
a simple decoupled system and we have a closed-form
inverse solution:
~
T0 (ω x , ω y ) =
e ∆h ω r
~
.
Th (ω x , ω y , ∆h )
1 + µP0 (ω x , ω y )e 2 ∆hωr
(6)
Performing a 2D inverse Fourier transform of T~0 (ω x , ω y )
then yields the sought magnetic data at the ground surface.
The choice of regularization parameter µ determines the
level of data fit and therefore the smoothness of the
downward continued field. If we have precise knowledge
of the statistics of errors in the data, we can define the
expected value of the data misfit in eq.(4) and the optimal
regularization parameter must be the one that yields the
expected misfit. In general, we have little precise
information about the noise characteristics, and this
approach is not applicable. In such cases, we must resort to
other methods to determine µ. The process then is
equivalent to estimating the noise level in the data.
There are several approaches for performing the estimation.
We apply the L-curve criterion (Hansen, 1992). The Lcurve criterion is a heuristic method based on the behavior
of Tikhonov curve. It was observed that, when plotted on a
log-log scale, the data misfit as a function of model
objective function exhibits a characteristic corner. As the
degree of regularization decreases towards this corner
point, the model objective function changes very little
while the misfit is being reduced greatly. Further decrease
in the degree of regularization beyond this point would
result in rapid increase in the model complexity with little
reduction in the data misfit. The L-curve criterion considers
the corner point to be the best compromise between
extracting maximum amount of signal and being least
affected by noise. It therefore states that the optimal
solution for a given regularized inverse problem is the one
that corresponds to the corner point of the Tikhonov curve.
The corner is defined numerically as the point with the
maximum curvature.
Power spectrum of magnetic data in UXO
The formulation presented in the preceding section is
general, and one can potentially use any assumed form of
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1465
Stable downward continuation of UXO magnetic data
power spectrum to define the model objective function in
order to stabilize the inversion. The weighting function
based the reciprocal of assumed power spectrum has the
ultimate effect of re-shaping the spectral content of the
downward continued magnetic data. The faster decaying
power spectrum corresponds to a smoother final result. For
example, assuming P0 = 1 /(ω x2 + ω y2 ) is equivalent to
over a 30 m by 30 m area. The red line is the radially
averaged power spectrum consisting of two distinct
ensembles and a constant noise power. The first ensemble
average depth is consistent with the buried UXO in the area
and the second corresponding to the observation height
above the ground. The agreement between the actual and
theoretical power spectra is remarkable.
requiring the downward continued data to be smooth by the
first-order derivative in the space domain. In the absence of
specific information regarding the nature of the source,
such a generic assumption is useful for obtaining first order
results. To obtain the best possible result, one must
incorporate the correct spectral property.
For our current application, we choose to use the theoretical
radially averaged power spectrum. Consider a set of UXO
items producing the magnetic data in question. Each UXO
is represented by a magnetic dipole to a high degree of
accuracy. Thus, we have an ensemble of random dipoles
with varying dipole moments and burial depths. Spector
and Grant (1970) showed elegantly that, within a scaling
factor, the radially averaged power spectrum of the
magnetic data produced by an ensemble is the same as that
of the data produced by an “average” member of the
ensemble. For a dipole source, this power spectrum has the
following form,
P(ωr ) ∝ Aωr2 e −2 hωr ,
(7)
where h denotes the vertical distance between the ensemble
average and the observation plane and A is the scaling
factor depending on the source strength. It follows naturally
that we use eq.(7) to define the spectral weighting for
enhancing magnetic data in UXO applications.
For practical applications, one must estimate the depth of
the ensemble average below the observation plane. This
can be easily accomplished by performing a least-squares
fit between the eq.(7) and the radially averaged power
spectrum of the data. To account for the presence of noise,
we may also include a constant noise power to eq.(7). In
more complicated cases, we may also need to include more
than one ensemble average. Once the average depth h of
the UXO ensemble is obtained through this process, the
power spectrum P0 (ω x , ω y ) used in defining the model
objective function is given by,
P0 (ω x , ω y ) = ω r2 e −2 ( h−∆h )ωr ,
(8)
where the vertical distance h − ∆h defines the ensemble
depth below the ground surface to which we downward
continue the data.
As an example, Figure 1 illustrates the power spectral
estimation using a field data set from the former Camp
Sibert in Alabama, US. The blue line is the actual power
spectrum calculated from the FFT of the magnetic data
Figure 1: Radially averaged power spectrum and its
parametric representation. The blue curve is the
numerically calculated power spectrum of a subset of
magnetic data at the former Camp Sibert. The red line
represents the radially averaged power spectrum consists of
two distinct ensemble averages plus a constant noise
power.
Synthetic example
We now proceed to illustrate the algorithm using a
synthetic example. We first illustrate the details of the
algorithm and show the effectiveness of the stable
downward continuation. We will then use this example to
evaluate the range of observation height from which the
enhancement is viable.
We use a model consisting of two buried dipoles separated
horizontally by 2 m and buried at the same depth of 0.5 m
below the surface. Assuming an observation height of 2 m
above the ground surface and adding 0.5 nT of Gaussian
noise, we simulated the observed data with 0.1-m spacing
shown in Figure 2. At this height, the magnetic data shows
one broad anomaly. In UXO applications, this might
indicate the presence of clutters, but it would be difficult to
discern how many buried targets are present.
Estimating a radial power spectrum and then applying the
stable downward continuation yields the result shown in the
same figure. The continued data are smooth and well
behaved. Two expected anomalies are clearly visible. For
comparison, we also show the predicted data, which is a de-
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Stable downward continuation of UXO magnetic data
noised version of the observed data. The estimated noise
given by the difference between the observed and predicted
data is representative of the statistics of the actual noise.
objective function decreases monotonically. The Tikhonov
curve has a well defined corner, as indicated by a single
peak in curvature. This optimal regularization level yielded
the result in Figure 2.
As a final assessment in this example, we compare the true
anomaly from the two dipoles at the ground surface with
the downward continued data in Figure 4. We note that the
downward continued data are smoother and the amplitude
is greatly reduced. This is expected since the continuation
process must attenuate a great deal of high-frequency
content to obtain a stable solution. Despite the smooth
appearance and low amplitude, the enhancement is
significant since the two anomalies are clearly visible.
Figure 2: Synthetic example illustrating the effectiveness of
stable downward continuation applied to UXO magnetic
data. The observed data are produced at a 2-m observation
height by two dipoles buried at 0.5 m below the surface.
The top-right panel displays the result of downward
continuing to the ground surface.
Figure 4: Comparison between the true anomaly at the
ground surface and the stably downward continued
anomaly.
Lastly, we have carried out the simulation for a arne of
different observation heights in order to evaluate the
limitations of the downward continuation. The criterion is
the ability to identify the presence of two separate
anomalies. The results suggest that the maximum height
from which we can enhance the data to reveal two
anomalies is about 2.2 m for the current data set and dipole
separation. In general, this height depends on the dipole
separation, noise level, and acquisition geometry.
However, the observation height comparable to the dipole
separation is a reasonable limit given the current quality of
magnetic data in UXO application.
Conclusion
Figure 3: The top panels show respectively the data misfit
( φd ) and model objective function ( φm ) for the downward
continuation shown in Figure 2. Tikhonov curve in the
lower-left panel has a well-defined corner that is located by
the peak curvature point.
Figure 3 shows more details regarding the performance of
the algorithm. To search for an optimal regularization
parameter, we have tested a range of values. The data misfit
exhibits the typical monotonic increase with the
regularization parameter, while the corresponding model
We have presented a stable downward continuation method
for enhancing magnetic data acquired for UXO detection
and discrimination. The key components are the expected
power spectrum and automatic estimation of noise level in
the data. Numerical tests using synthetic and field data sets
have demonstrated the effectiveness of the approach.
Acknowledgements
This work is supported in part by the Strategic
Environmental Research and Development Program
(SERDP) through the project MM-1642.
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EDITED REFERENCES
Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2009
SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for
each paper will achieve a high degree of linking to cited sources that appear on the Web.
REFERENCES
Blakely, R. J., 1995, Potential theory in gravity and magnetic applications: Cambridge University Press.
Doll, W. E., T. J. Gamey, L. P. Beard, and D. T. Bell, 2006, Airborne vertical magnetic gradient for near-surface applications:
The Leading Edge, 25, 50–53.
Hansen, P. C., 1992, Analysis of discrete ill-posed problems by means of the L-curve: Society of Industrial and Applied
Mathematics Review, 34, 561–580.
Huestis, S. P., and R. L. Parker, 1979, Upward and downward continuation as inverse problems: Geophysical Journal of the
Royal Astronomical Society, 57, 171–188.
Spector, A., and F. S. Grant, 1970, Statistical models for interpreting aeromagnetic data: Geophysics, 35, 293–302.
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1468