Reducing Tensor Magnetic Gradiometer Data

for Unexploded Ordnance Detection

By Robert E. Bracken and Philip J. Brown

This research was supported by the U.S. Department of Defense through the 

Strategic Environmental Research and Development Program (SERDP)

Scientific Investigations Report 2005-5046

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

Gale A. Norton, Secretary

U.S. Geological Survey

Charles G. Groat, Director

U.S. Geological Survey, Reston, Virginia: 2005

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iii

Contents

Abstract.......................................................................................................................................................... 1

Introduction ................................................................................................................................................... 1

Description of the Instrumentation............................................................................................................ 1

Description of the Survey............................................................................................................................ 2

Method ........................................................................................................................................................... 2

Reduction Process .............................................................................................................................. 3

Results and Conclusions ............................................................................................................................. 5

Acknowledgments........................................................................................................................................ 6

References..................................................................................................................................................... 6

Figures

1. TMGS sensor array (TESSA).................................................................................................... 2

2. Flow chart of the TMGS data reduction procedure............................................................. 2

3. Determinant map of the raw data ........................................................................................... 3

4. Determinant map of the data reduced through the calibration stages ............................ 4

5. Determinant map of the final processed data....................................................................... 4

6. Modeled determinant map with locations of the dipolar sources 

used in the model....................................................................................................................... 5
Abstract

We performed a survey to demonstrate the effectiveness 

of a prototype tensor magnetic gradiometer system (TMGS) 

for detection of buried unexploded ordnance (UXO). In order 

to achieve a useful result, we designed a data-reduction procedure that resulted in a realistic magnetic gradient tensor and 

devised a simple way of viewing complicated tensor data, not 

only to assess the validity of the final resulting tensor, but also 

to preview the data at interim stages of processing.

The final processed map of the surveyed area clearly 

shows a sharp anomaly that peaks almost directly over the target UXO. This map agrees well with a modeled map derived 

from dipolar sources near the known target locations. From 

this agreement, it can be deduced that the reduction process is 

valid, making the prototype TMGS a foundation for development of future systems and processes.

Introduction

Many geophysical applications rely on magnetic field 

measurements to discern characteristics of geologic source 

materials and buried objects. Typically, scalar data are 

acquired but may be converted to vector data as required by 

certain modeling routines. Vector components are derivable if 

the scalar data were acquired over a large surface with sufficient resolution. However, in UXO applications, where sources 

are nearby, there is a practical limitation to this approach (D.V. 

Smith, oral commun.). Therefore, we consider vector-based 

acquisition methods.

Direct measurement of target-anomaly vector data is 

hampered by the huge Earth-field vector, which requires the 

attitude of the vector sensor to be recovered with practically 

unattainable levels of precision. For example, using the best 

available attitude-recovery systems along with a vector magnetometer, gives Earth-frame component accuracy to only about 

5 nT. This problem is circumvented in tensor gradiometry, 

which incorporates the vector information without requiring 

such stringent knowledge of the sensor-array attitude. 

Gradients are more useful for discerning nearby sources 

and source materials than are scalar or vector fields alone. 

Because gradient strength decreases with distance more 

rapidly than field strength, nearby sources are preferentially 

detected and distant sources are rejected. The tensor gradient can be manipulated to obtain source information through 

a number of methods that cannot be applied with fields only. 

One such method, called dipole mapping (Wynn and others, 

1975), uses a tensor to find the location, depth, and moment 

of a dipolar source. Other methods involve the use of tensor 

invariants (Pedersen and Rasmussen, 1990), which we briefly 

discuss here.

A gradient is measured by finding the difference between 

two magnetometer readings. When normalized with respect 

to sensor separation, the difference forms a gradient in the 

direction of the line between the two sensors. For a given line 

direction, there are three gradients corresponding to the three 

vector components, and there are three linearly independent 

line directions, making nine gradients. These gradients can 

be arranged into a matrix called a magnetic gradient tensor, 

mathematically expressed as:

Reducing Tensor Magnetic Gradiometer Data 

for Unexploded Ordnance Detection

By Robert E. Bracken and Philip J. Brown

�

�

� �

�

��b /�z �b /�z �b /�z x y z

x y z

�

� �

�

� � � � � �

� � � � � �

� b / y b / y b / y

b / x b / x b / x x y z

G

Because both the divergence and curl of a magnetostatic 

field must equal zero in a sourceless volume, this tensor 

is traceless and symmetric, having only five independent 

components. The magnetic gradient tensor and three field 

components comprise a complete description of the magnetic 

field to the first order, and this is what results from the TMGS 

measurements. 

Description of the Instrumentation

The TMGS consists of two basic units. Figure 1 shows 

the sensor array, called TESSA (tetrahedral em/mag sensor 
2 Reducing Tensor Magnetic Gradiometer Data for Unexploded Ordnance Detection

suspension apparatus), which is a 1-m tetrahedron with four 

triaxial magnetic sensors at its vertices. The second unit is an 

enclosure that provides a thermally controlled environment for 

the magnetometer circuitry. It also contains circuitry that measures the temperature of each sensor and samples the magnetometer output. The raw data are sent, via an RS-232 link, to a 

laptop computer for storage. The magnetometers are ring-core 

triaxial fluxgates manufactured by Narod Geophysics (Narod, 

1987) for use in magnetic observatories by the U.S. Geological Survey (USGS).

m deep. We collected the data as 10-s stationary observations 

at five samples per second over a 3-m-square grid centered on 

the target and having 0.25-m grid cells. Smith and Bracken 

(2004) describe the survey in detail. Three categories of data 

were collected on-site: (a) primary measurements including 

magnetics, position, and attitude; (b) spin calibration measurements; and (c) thermal baseline measurements. Some calibration data used in the reduction procedure were collected in the 

laboratory.

Method

In the reduction process, we converted the survey data to a 

13-by-13 grid of tensors. However, to help describe the reduction and to present the results, the tensors were further reduced 

to a set of scalar values—the determinants of the tensors, which 

provide a more intuitive display for figures 3–6. Any gradient 

tensor can be rotated into a natural coordinate system, called the 

principle axis frame, in which all but the diagonal components 

are zero. If the three diagonal gradients are then multiplied 

together, the result can be imagined to represent the volume of 

a box in gradient space—the bigger the box, the more gradient 

strength it contains. This product is a scalar quantity, called 

a tensor invariant, that can be calculated from any reference 

Figure 1. TMGS sensor array (TESSA).

The TMGS was originally designed for detection of 

volcano magnetic effects from a stationary observation point 

(Bracken and others, 1998). Therefore, no provision was 

made for the sensors to be on a moving platform, as would be 

required for production UXO work. Consequently, this demonstration was designed as a grid of stationary observations. 

Nevertheless, provision has been made in the reduction procedure for data collected from a moving platform. In addition, 

the sensor geometry is inappropriate, and the sensor separation 

is too great, for the intense field curvatures of proximate UXO 

sources. To compensate for the curvature, a “gradient collocation” stage has been incorporated into the data-reduction 

procedure shown in figure 2.

Description of the Survey

On March 12, 2003, the USGS used the TMGS to perform an experimental survey over a known UXO target at the 

Standardized UXO Test Site in Yuma Proving Ground, Arizona (YPG). The target was a 60-mm mortar shell buried 0.25 

Figure 2. Flow chart of the TMGS data-reduction procedure.
frame by taking the determinant of the matrix—diagonalizing 

is not actually required (Pedersen and Rasmussen, 1990). The 

determinant will have the greatest expression at the nearest 

approach to a source. A mapped determinant anomaly will normally reach a sharp (positive or negative) peak almost directly 

over its target, but sometimes it may have an abrupt positive-tonegative transition instead. This is in contrast to a scalar field 

anomaly that reaches an unfocused peak at a location significantly offset from the target.

Reduction Process

A detailed description of the reduction process is given 

in Bracken and Brown (in press). The reduction addresses 

three significant issues: drift, leakage, and tracking. Drift is a 

tendency of the instrumentation to change its calibration as a 

function of temperature and time. Leakage is an incomplete 

cancellation of a field common to all sensors (i.e., the Earth’s 

field), leaving a small percentage interfering with the gradients. Tracking is the ability of the system to produce accurate 

gradients and not to leak while the sensor array is undergoing 

attitude changes. 

Tracking is handled in the “tracking xfer function” reduction stage shown in figure 2. However, no tracking corrections 

were required for this survey because TESSA was stationary 

during data collection. Figure 3 shows a determinant map of 

how the survey would look if the data were processed only 

through the tracking stage—the raw data in our case. Because 

the magnetometers were not calibrated, the tensor anomalies 

are dominated by leakage that is correlated to sensor-attitude 

variations. 

Leakage is handled in the “bin conversion” and “apply 

cal. coefficients” reduction stages shown in figure 2. If two 

sensors give different readings in the same field, that error will 

be included in the measured gradient and will be indistinguishable from a true gradient. Typically, these types of errors will 

swamp the rather small gradients being sought. Consequently, 

it is extremely important to derive a comprehensive and precise set of calibration coefficients for the magnetometers being 

used. The “bin conversion” stage is specific to our magnetometers and uses coefficients that we derived in the laboratory. 

The “apply cal. coefficients” stage is applicable to any vector 

magnetometer and requires coefficients derived from a “spin 

calibration,” which we did by slowly rotating TESSA in the 

Earth’s field. We derived the calibration coefficients by applying a nonlinear regression that required the magnitude resulting from each sensor to equal the true magnitude of the Earth’s 

field as monitored by a base magnetometer. More information 

on spin calibration is available in Bracken and Smith (2005).

Figure 4 shows a determinant map of how the survey 

would look if the data were processed only through the leakFigure 3. Determinant map of the raw data.

Description of the Survey 3
4 Reducing Tensor Magnetic Gradiometer Data for Unexploded Ordnance Detection

Figure 4. Determinant map of the data reduced through the calibration stages.

Figure 5. Determinant map of the final processed data.
age-handling calibrations. The leakage has gone away, but 

thermal-drift effects dominate so heavily that no hint of the 

anticipated target anomaly is yet visible.

Drift is handled in the “thermal-drift correction” reduction stage shown in figure 2. We collected data for this correction by occupying a base location three times during the 

survey—at the beginning, middle, and end. The correction was 

simply to require the gradients at the base location to be constant and to adjust the intervening data with linear functions.

Results and Conclusions

The final tensors resulting from the reduction process 

have been converted to determinants and are shown in figure 5. 

Of particular importance are the obvious circular anomaly in 

the center of the map and the near coincidence of its peak with 

the plotted location of the target UXO. The amplitude and 

sharpness of the anomaly is significant. It is not obscured in a 

sea of noise but, rather unmistakably, detects both the presence 

and the location of the source.

In order to confirm that the final processed map is reasonable, we produced a model of magnetic dipolar sources near 

the known locations of the UXO targets in the survey area, calculated ideal tensors at the survey station locations, and made 

a map of the determinants, as shown in figure 6. The magnetic 

fields of the center target and eight edge targets were modeled using depth and location specifications provided by YPG. 

Positions of dipolar sources were adjusted horizontally, up to 

a quarter of a meter from the specified locations, in order to 

align anomalies. We obtained a reasonable moment magnitude 

for the center target by adjusting the peak value of its anomaly 

to match the reduced anomaly.

In conclusion, the similarity of the reduced and modeled 

results indicates that the reduction process is producing real 

tensor data. This preliminary analysis is convincing for two 

compelling reasons. First, the determinant uses all nine tensor 

components, making its value representative of the whole tensor. Second, the reduced and modeled invariant maps are quite 

similar, which means that the system consistently produced 

realistic values under a large variety of target-field conditions. 

Several stages of the data reduction are critical. If any 

stage is missing or its coefficients are inaccurate, the resultant 

tensor map will be incomprehensible. This is in contrast to a 

typical scalar magnetics survey in which the basic anomalies 

are usually visible from the raw data and simply improve as 

reductions are performed. The TMGS processing stages are 

Figure 6. Modeled determinant map with locations of the dipolar sources used in the model.

Results and Conclusions 5
6 Reducing Tensor Magnetic Gradiometer Data for Unexploded Ordnance Detection

analogous to a keyed lock—if any tumbler is out of place, 

the lock will not open. This observation is well illustrated in 

figure 4, which shows how the final processed map would 

have appeared if all but the last two reduction stages were 

performed. The map is completely devoid of any hint that the 

reduction is moving toward a reasonable result.

Future work on our system includes changing the geometry and reducing the size of the sensor array to handle field 

curvature and modifying the data-acquisition systems to facilitate data collection on the fly. We expect further research on 

the reduction process to validate proposed tracking algorithms 

and to allow an abbreviated spin calibration. We are also 

studying tensors and tensor invariants for detecting, locating, 

and modeling sources.

Acknowledgments

This research was supported by the U.S. Department of 

Defense through the Strategic Environmental Research and 

Development Program (SERDP). We acknowledge YPG personnel for facilitating our work at the Standardized UXO Test 

Site and Professor Yaoguo Li of Colorado School of Mines for 

providing consultation.

References

Bracken, R.E., and Brown, P.J., in press, Concepts and procedures required for successful reduction of tensor magnetic 

gradiometer data obtained from a UXO detection demonstration at Yuma Proving Ground, Arizona: U.S. Geological 

Survey Open-File Report.

Bracken, R.E., Grover, T.P., and Puniwai, G., 1998, Development and testing of a tensor magnetic gradiometer system 

with trial monitoring near the Kilauea Volcano, Hawaii: 

U.S. Geological Survey Open-File Report 98-773, 163 p.

Bracken, R.E., and Smith, D.V., 2005, Calibrating a tensor 

magnetic gradiometer using spin data: U.S. Geological Survey Scientific Investigations Report 2005-5045, 5 p., available online at URL <http://pubs.usgs.gov/sir/2005/5045/>.

Narod, B.B., 1987, Ring core fluxgate magnetometers for use 

as observatory variometers [abs.]: International Union of 

Geodesy and Geophysics 19th General Assembly, v. 2, no. 

GA5.3-3, 673 p.

Pedersen, L.B., and Rasmussen, T.M., 1990, The gradient tensor of potential field anomalies: Some implications on data 

collection and data processing of maps: Geophysics, v. 55, 

no. 12, p. 1558–1566.

Smith, D.V., and Bracken, R.E., 2004, Field experiments with 

the magnetic gradiometer system at Yuma Proving Ground, 

Arizona, in Proceedings of the Symposium on the Application of Geophysics to Engineering and Environmental 

Problems (SAGEEP), February 2004. 

Wynn, W.M., Frahm, C.P., Carroll, P.J., Clark, R.H., Wellhoner, J., and Wynn, M.J., 1975, Advanced superconducting gradiometer/magnetometer arrays and a novel signal 

processing technique: IEEE Transactions on Magnetics, v. 

MAG-11, no. 2, p. 701–707.

Manuscript approved for publication February 24, 2005

Published in the Central Region, Denver, Colorado

Editing, layout, photocomposition—Richard W. Scott, Jr.

Graphics by the authors