Estimating locations and total magnetization vectors of compact magnetic sources from scalar,
vector, or tensor magnetic measurements through combined Helbig and Euler analysis
Jeffrey D. Phillips*1, Misac N. Nabighian2, David V. Smith1, and Yaoguo Li2
1
U.S. Geological Survey, 2Colorado School of Mines
Summary
mx = −
The Helbig method for estimating total magnetization
directions of compact sources from magnetic vector
components is extended so that tensor magnetic gradient
components can be used instead. Depths of the compact
sources can be estimated using the Euler equation, and their
dipole moment magnitudes can be estimated using a least
squares fit to the vector component or tensor gradient
component data.
Introduction
The total magnetization of a magnetic source is the vector
sum of the remanent magnetization and the induced
magnetization. The Helbig method, as implemented by
Phillips (2005), is useful for estimating the horizontal
positions and the total magnetization directions of compact
magnetic sources, such as ore bodies, unexploded
ordnance, and abandoned wells, from measured or
calculated magnetic vector field components. Here we
extend the Helbig method in several ways. First, we show
how the method can be modified so that magnetic gradient
tensor measurements can be used instead of magnetic
vector component measurements. Next, we show how,
given the horizontal positions of the sources, a simplified
version of the Euler equation can be applied in small data
windows to estimate the depths to the compact sources by
assuming that they are magnetic dipoles. Finally, we show
how a linear least squares fit to the data can be used to
determine the dipole moments of the sources.
The Helbig Method
The Helbig method (Schmidt and Clark, 1997, 1998;
Phillips, 2005) is based on the observation (Helbig, 1963)
that the vector components (mx, my, mz) of the total
magnetization of a compact source can be estimated from
the first moments of the vector components (Bx, By, Bz) of
the anomalous magnetic field produced by the source.
Here we use the convention of geomagnetism that the xaxis points north, the y-axis points east, and the z-axis
points down. If necessary, the magnetic field components
can be derived from total-field magnetic data using Fourier
filters.
In theory, the first moments are evaluated as infinite
integrals over the horizontal plane:
my = −
mz = −
1
2π
1
2π
1
2π
∞ ∞
∫ ∫ x B ( x, y ) dx dy
z
−∞ −∞
∞ ∞
∫ ∫ y B ( x, y ) dx dy
(1)
z
−∞ −∞
∞ ∞
1
∞ ∞
∫ ∫ x B ( x, y ) dx dy = − 2π ∫ ∫ y B ( x, y ) dx dy.
x
−∞ −∞
y
−∞ −∞
The other two first moments are both zero, as are the
integrals (mean values) of Bx, By, and Bz over the plane.
We refer to these five integrals as Helbig’s vanishing
integrals.
Phillips (2005) has shown that it is possible to estimate
total magnetization directions using integrals (1) within a
finite data window, as long as a planar surface is first
removed from each magnetic vector component within the
window. Removing the surface assures that the mean
values of the components are zero within the window, and
that the two vanishing moments are approximately zero.
Using the Helbig integrals, total magnetization directions,
can be estimated in small data windows centered on each
grid node of the magnetic vector field component grids. If
the central grid node lies over an isolated compact source
that can be represented by a magnetic dipole, then the
calculated magnetization direction will accurately predict
the total magnetization direction of the dipole, and the
direction will remain relatively constant as the window size
is increased in small steps. On the other hand, if the central
grid node does not lie over an isolated source, the
calculated total magnetization direction will change with
increasing window size.
We can locate the grid nodes where the estimated
magnetization direction is relatively insensitive to window
size by setting a threshold for the maximum allowable
angular change in the direction between successive window
sizes, then counting the number of successive window sizes
that produce solutions with angular changes below this
threshold. Under ideal conditions the count will reach
maximum values at grid nodes over isolated sources. In
this case, the Helbig method can be used to estimate the
horizontal positions of the isolated compact sources as well
as their total magnetization directions.
By averaging the magnetization directions for the solutions
that fall below the threshold, we get an estimate of the total
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770
Combined Helbig and Euler analysis
magnetization direction at each grid node. This estimate
will be accurate only at grid nodes that lie over sources, i.e.
those with large counts. However, between sources the
estimated magnetization directions will vary smoothly as a
function of position. This means that the Helbig method
can also be used to produce a continuous estimate of the
magnetization direction over a magnetic survey area, which
can then be sampled at source locations determined using a
second method, such as Euler deconvolution (Reid et al.,
1990). A second source-location method might be required
if the sources are deep relative to the grid spacing
(therefore producing no isolated peaks in the count) or if
the sources are close together relative to their depth.
Extending the Helbig Method to Magnetic Gradient
Measurements
To develop Helbig equations for derivatives of the
magnetic vector components, we can use integration-byparts on the integrals of (1). For example, the mx
component can be expressed as either
1
2π
∞ ∞
∫ ∫ x B ( x, y) dx dy
z
−∞ −∞
⎤
⎡∞
∫−∞ ⎢⎣−∫∞Bz ( x, y) x dx⎥⎦ dy
∞
∞
∞
1 ⎡
x 2 ∂Bz ⎤
x2
=−
−∫
dx⎥ dy
⎢
Bz ( x, y )
∫
2π − ∞ ⎣⎢
2 − ∞ − ∞ 2 ∂x ⎥⎦
1
2π
∞
=−
1
2π
∞ ∞
mx = −
∫ ∫ x B ( x, y ) dx dy
z
−∞ −∞
⎡∞
⎤
∫−∞ ⎢⎣−∫∞Bz ( x, y) x dy ⎥⎦ dx
∞
∞
∞
∂Bz ⎤
1 ⎡
dy ⎥ dx.
=−
⎢ Bz ( x, y ) xy − ∞ − ∫ xy
∫
∂y ⎦
2π − ∞ ⎣
−∞
∞
(3)
If the vertical component Bz of the magnetic field decays
more rapidly than the square of the distance, which is the
B
my =
1
4π
1
4π
1
mz =
4π
1
2π
∞ ∞
∫ ∫x
2
−∞−∞
∞ ∞
∫∫
y2
−∞−∞
∂Bz
1
dx dy =
∂x
2π
∂ Bz
1
dx dy =
∂y
2π
∞ ∞
∂B
1
∫−∞ −∫∞ x ∂xx dx dy = 4π
2
∞ ∞
∫ ∫ xy
−∞−∞
∂Bx
1
dx dy =
∂y
2π
∞ ∞
∫ ∫ xy
−∞−∞
∞ ∞
∫ ∫ xy
−∞−∞
∞ ∞
∫∫y
2
−∞−∞
∞ ∞
∫ ∫ xy
−∞−∞
∂Bz
dx dy
∂y
∂ Bz
dx dy (4)
∂x
∂B y
∂y
dx dy
∂B y
∂x
dx dy.
The partial derivatives represent components that can be
directly measured with a tensor magnetic gradient system.
In terms of standard symmetric tensor notation,
∂B
∂B
∂B
∂Bx
= Bxx , y = Byy , x = y = Bxy = Byx ,
(5)
∂y
∂y
∂x
∂x
∂Bz
∂B
= Bzx = Bxz , z = Bzy = Byz .
∂x
∂y
In addition to being useful for tensor magnetic gradient
measurements, equation (4) has the advantage over
equation (1) that, in order to evaluate the integrals within
finite data windows and still satisfy Helbig’s vanishing
integrals, it is only necessary to remove the mean value
from each tensor component within the window.
The Euler Equation
(2)
or
1
=−
2π
mx =
=
To extend the Helbig method, we can consider using firstorder derivatives of the magnetic vector components, rather
than the components themselves, for Helbig analysis. Firstorder derivatives of the components are measured directly
by tensor magnetic gradiometer systems, such as those
currently being evaluated for application to the detection of
unexploded ordnance (UXO) (Butler et al., 1998; Bracken
and Brown, 2005; Clem et al., 1996; Smith and Bracken,
2004) and for resource exploration (Schmidt et al., 2004).
mx = −
case for all compact magnetic sources, then the first terms
in the square brackets are zero. It follows that (1) can be
rewritten in terms of second moments of first derivatives of
the magnetic vector components as,
In the preceeding section, we showed how the Helbig
method can be extended to work with tensor magnetic
gradient components instead of vector magnetic field
components. In either case, the Helbig method can provide
estimates of the horizontal source locations and total
magnetization directions of the compact sources. To fully
characterize the sources, their depths and magnetic
moments must be estimated.
If the horizontal locations (x0, y0) of a dipole source are
known, the Euler equation (Reid et al., 1990) can be used
to estimate the depth z0 of the dipole and a constant β
representing interference from adjacent sources.
z 0 Bz + β = ( x − x0 ) Bx + ( y − y0 ) B y − nV .
(6)
Here Bx, By, and Bz are the vector components used for the
Helbig analysis, V is the magnetic potential computed as
B
B
B
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771
Combined Helbig and Euler analysis
the negative of the first vertical integral of Bz, and n=2 for a
dipole source. This equation can be solved in the leastsquares sense using data in small windows (3x3, 5x5,
7x7,…) centered on the known horizontal location of each
dipole source. Based on model studies, the solution that
produces the shallowest z0 is probably the best.
B
For the case of magnetic gradient tensor measurements or
calculated derivatives of vector components, there are three
Euler equations (Zhang et al., 2000):
z0 Bzz + β z = ( x − x0 ) Bzx + ( y − y0 ) Bzy + nBz
z0 Bxz + β x = ( x − x0 ) Bxx + ( y − y0 ) Bxy + nBx
(7)
z0 Byz + β y = ( x − x0 ) Byx + ( y − y0 ) Byy + nBy .
In this case n=3 for a dipole source.
In a second approach, mentioned earlier, the Helbig method
can be used to provide a continuous estimate of the total
magnetization direction over the survey area, then this
estimate can be sampled at source locations determined
using a second magnetic source location method.
Candidates for this second method include full Euler
deconvolution as described by Reid et al. (1990) and
various modifications of Euler deconvolution using Hilbert
transforms (Nabighian and Hansen, 2001; Davis et al.,
2005) and multiple sources (Hansen and Suciu, 2002).
Dipole Moments
Once the total magnetization directions, the horizontal
locations, and the depths of the sources have been
established, the dipole moments of the sources can be
estimated by linear least squares inversion. For example, if
Ui(x,y) represents the calculated field component or tensor
component of the i-th source with unit dipole moment and
O(x,y) is the observed field component or tensor
component produced by all N sources, then the unknown
dipole moments mi will satisfy
N
∑ U ( x, y ) m
i =1
i
i
= O ( x, y ) + c
(8)
in the least squares sense, where c is an unknown constant
representing effects of the regional field.
Example
To test the method, we used the three dipole sources
described in Table 1. The calculated field component
grids, with 0.25 nT added instrument noise, are shown in
Figure 1a. The grids have a sample interval of 0.05 meter.
The Helbig method was applied to the field components
using the following approach. The Helbig integrals (1)
were evaluated in small windows centered on each grid
node and having odd integer dimensions increasing from
3x3 to 25x25. For field components, a planar surface was
removed from the component data within each window
before the integration. The algorithm looked for successive
solutions with increasing window size for which the total
magnetization direction varied by less than a specified
threshold of one degree. For the grid nodes corresponding
to the three dipole sources and for two other grid nodes
(labeled 1a and 1b in Table 2) adjacent to dipole source 1,
at least 6 successive solutions with variations of less than
one degree were found. For all other grid nodes, the
number of successive solutions with variations of less than
one degree was between zero and four. To estimate the
total magnetization direction, the successive solutions with
variations of less than the threshold were averaged. The
results, for averages with more than five solutions, are
shown in Table 2.
The simplified Euler equation (6) was used to estimate the
depths of the sources. The equation was evaluated in
windows of odd integer dimensions from 3x3 to 13x13
centered on the sources. The shallowest solution for each
source was selected. A magnetic potential calculated from
the Z-component was used. Results are shown in the
“depth” column of Table 2.
The dipole moments were estimated from the three field
components using a least squares inversion, and are shown
in the “moment” column of Table 2. The calculated
components from the estimated sources in Table 2 are
shown in Figure 1b.
Conclusions
Combined Helbig and Euler analysis can be used to locate
and characterize compact magnetic sources from either
magnetic vector field components or magnetic gradient
tensor components. Derived source characteristics include
the total magnetization directions and the magnetic dipole
moments. Accurately locating the sources, using either the
stability of the Helbig solutions or an independent sourcelocation method, is a critical requirement.
Acknowledgements
Research on magnetic gradient tensor algorithms for
unexploded ordnance applications has been funded by the
Strategic Environmental Research and Development
Program (SERDP). Additional funding has been provided
by the USGS Mineral Resources Program.
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Combined Helbig and Euler analysis
Table 1 – Dipole source parameters used to test the method.
Source
1
2
3
Easting (m)
1.0
2.0
1.5
Northing (m)
1.0
1.0
2.0
Depth (m)
.20
.25
.30
Inclination (deg)
0
60
90
Declination (deg)
30
-30
0
Moment (Am2)
.020
.025
.030
Table 2 – Results of inversion using the magnetic field components with added instrument noise. The eastings, northings,
inclinations, and declinations are from Helbig analysis. The depths are from Euler analysis. The dipole moments are from least
squares inversion.
Source
1
1a
1b
2
3
Easting (m)
1.00
1.05
0.95
2.00
1.50
Northing (m)
1.00
0.95
1.00
1.00
2.00
Depth (m)
.196
.199
.197
.249
.265
Inclination (deg)
0.521
4.193
5.196
60.258
89.385
Declination (deg)
30.284
30.892
29.633
-30.270
-32.443
Moment (Am2)
.0156
.0015
.0018
.0244
.0229
Figure 1. (a) X (north), Y (east), and Z (vertical) magnetic field components produced by the three test dipoles of Table 1 with
added instrument noise. Circles indicate the dipole locations. (b) Magnetic field components produced by the five solution
dipoles of Table 2.
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Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2007
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.
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