______________________________________________________________________________ 

U.S. DEPARTMENT OF THE INTERIOR 

U.S. GEOLOGICAL SURVEY 

Concepts and Procedures Required for 

Successful Reduction of Tensor Magnetic 

Gradiometer Data Obtained from an 

Unexploded Ordnance Detection 

Demonstration at Yuma Proving Grounds, 

Arizona 

by 

Robert E. Bracken1 and Philip J. Brown1 

Open-File Report 2006-1027 

1

USGS, MS 964, Federal Center, Denver, Colorado 80225 

This report is preliminary and has not been reviewed for conformity with U.S. Geological Survey editorial standards. Although all 

software released in this report have been used by the USGS, no warranty, expressed or implied, is made by the USGS as to the 

functioning of the software. Any use of trade, product, or firm names is for descriptive purposes only and does not imply endorsement 

by the U.S. Government. 
Table of Contents 

Abstract.............................................................................................................................. 3 

Introduction....................................................................................................................... 4 

Description of the Survey ................................................................................................. 5 

Description of the TMGS ................................................................................................. 5

Brief History................................................................................. 5 

The Physical Unit........................................................................... 6

The Magnetic Field ........................................................................................................... 6

Magnetic Sources............................................................................7 

Superposition................................................................................. 7 

Scalar Field Measurements.................................................................8 

Scalar Gradient Measurements............................................................ 9

Vector Field Measurements................................................................ 9 

Magnetic Gradient Tensor Measurements................................................ 9

Important Concepts as Applied to the Prototype TMGS ........................................... 10

Extreme Attitude Sensitivity..............................................................11 

Common Mode and Angular Stability ......................................... 11 

Gradient Measurements................................................................... 12 

Simultaneous Measurement...................................................... 12 

Rejection of Motion-Related Common-Mode Signal........................ 12

Rejection of Diurnal and Other Geomagnetic Variations .................. 13

Slowly Varying with Attitude ................................................... 13 

Preferentially Sensitive to Nearby Sources ................................... 14

Leakage of the Earth's Common Mode Field into Gradient Measurements........14 

Thermal Drift............................................................................... 16 

Removing Thermal Drift ......................................................... 17

Magnetic-Field Curvature.................................................................17

A Linear Field Polynomial...................................................... 18 

Curvature of the Field Polynomial ............................................. 18 

Curvature of the Field from UXO Targets ....................................18 

Minimizing the Effects of Curvature by Collocation of Gradients ......... 19 

Data Collection................................................................................................................ 20

Spatial Data Collection in the High-Density Grid..................................... 20 

Spin Calibration............................................................................ 21 

The Coordinate Systems...................................................................21

Processing Procedures.................................................................................................... 22

Gradients vary too slowly as a function of attitude to warrant attitude 

Sensor axes are coincident ......................................................25

Data Presentation........................................................................... 22 

Determinant of the Tensor ...................................................... 22

Linear Volume-Model Engine ...................................................24

Assumptions Applied to the Spatial Data............................................... 24 

Positions match the ideal 0.25-m grid ......................................... 24

corrections ........................................................................ 25 

Sensor positions are coincident with the virtual tetrahedron vertices .... 25

Assumptions Applied to the Spin-Calibration Data....................................25 

1
The background field was stable and its tensor everywhere zero ......... 25

Deconvolving Bin Steps from Analog, and Matching Axes for Precision 

By Rotation ........................................................................37

Sensor 1 is perfectly aligned with the virtual tetrahedron ..................26

The sensor temperatures did not vary ......................................... 26

Raw-Data Conversion.....................................................................26 

Basic Data Forms ................................................................ 26

Recording Method ................................................................26

Conversion Program .............................................................27

Tracking..................................................................................... 27

Removal of Intervening Data Records and Noisy Samples...........................27 

Bin Conversions............................................................................27 

Application of Spin-Calibration Results for Converting Voltages to Fields....... 29 

Axis Polynomial.................................................................. 29 

Orthogonality Correction........................................................ 30

Attitude Correction ............................................................... 30

Temperature Corrections ........................................................31

Differencing Magnetic Fields to Obtain Initial Gradients............................ 31 

Ten-Second Averaging of Gradients to Obtain Station Gradient Data..............32 

Thermal-Drift Correction..................................................................33 

Unknown Constant............................................................... 34

In Lieu of Temperature-Dependent Spin-Cal Coefficients ................. 34

Gradient Collocation.......................................................................34 

Converting Sensor-Pair Gradients to a Tensor......................................... 37

By Linear Least-Squares Regression ..........................................38 

A Model of the Expected Tensor Determinant............................................................. 39 

Appendix A – Fortran routine called s_ab2h.f that applies the bin-size, axis 

polynomial, orthogonality, and attitude coefficients to basic TMGS analog and bin 

Conclusions...................................................................................................................... 45 

Acknowledgments ........................................................................................................... 45 

References Cited.............................................................................................................. 46 

data................................................................................................................................... 47 

2
Abstract 

On March 12, 2003, data were gathered at Yuma Proving Grounds, in Arizona, using a 

Tensor Magnetic Gradiometer System (TMGS). This report shows how these data were 

processed and explains concepts required for successful TMGS data reduction. Important 

concepts discussed include extreme attitudinal sensitivity of vector measurements, low 

attitudinal sensitivity of gradient measurements, leakage of the common-mode field into 

gradient measurements, consequences of thermal drift, and effects of field curvature. 

Spatial-data collection procedures and a spin-calibration method are addressed. 

Discussions of data-reduction procedures include tracking of axial data by 

mathematically matching transfer functions among the axes, derivation and application of 

calibration coefficients, calculation of sensor-pair gradients, thermal-drift corrections, 

and gradient collocation. For presentation, the magnetic tensor at each data station is 

converted to a scalar quantity, the I2 tensor invariant, which is easily found by calculating 

the determinant of the tensor. At important processing junctures, the determinants for all 

stations in the mapped area are shown in shaded relief map-view. Final processed results 

are compared to a mathematical model to show the validity of the assumptions made 

during processing and the reasonableness of the ultimate answer obtained. 

3
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 unexploded ordnance (UXO) applications, where 

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

commun., 2004). 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 nanoTeslas (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, which 

we briefly discuss herein (Pedersen and Rasmussen, 1990). 

We performed a survey to demonstrate the effectiveness of a prototype tensor magnetic 

gradiometer system (TMGS) for detection of buried 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, demonstrating that the prototype TMGS is a 

viable geophysical instrument and that direct measurement of the magnetic gradient 

tensor is a reasonable data-acquisition mode. 

This report shows how the demonstration survey data were processed and explains 

concepts required for successful TMGS data reduction. Although the procedures 

presented herein are applicable only to the prototype TMGS and to this particular data 

set, they serve as a vehicle for gaining a general understanding of the development and 

application of fluxgate tensor-magnetic gradiometer systems. They also form a solid 

4
foundation for future choices of techniques and design modifications for this specific 

system. 

Description of the Survey 

On March 12, 2003, the U.S. Geological Survey (USGS) used the TMGS to perform a 

demonstration survey over a known UXO target at the Standardized UXO Test Site in the 

Yuma Proving Ground (YPG), Arizona . The target was a 60-mm mortar shell buried 

0.25 m deep. We collected the data as 10-second stationary observations at five samples 

per second over a 3-m-square grid, called the “high-density 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 onsite: (a) primary measurements including 

tensor magnetics, gradiometer position, and gradiometer attitude; (b) spin-calibration 

measurements for the precise determination of a large group of calibration coefficients 

for the sensors and sensor array; and (c) thermal baseline measurements for correcting 

thermal drift in the sensors and sensor array. Some calibration data used in the reduction 

procedure were collected in the laboratory. The primary objectives of the survey were to 

show whether the TMGS could find the target UXO, to evaluate how well the target 

anomaly stands out against background noise, and to gain a feel for how accurately its 

location could be ascertained. 

Description of the TMGS 

Brief History: The TMGS was designed and built in the early 1990s for detection of 

volcano magnetic effects. These are time-varying magnetic fields produced in volcanoes 

by a variety of mechanisms (Zlotnicki and Le Mouel, 1988). The original instrument 

design required placement within about 3 km of expected volcanic events because 

gradients are preferential to nearby sources. In 1994 and 1995, the TMGS monitored 

near Kilauea Volcano, Hawaii and, after data-reduction, attained precision on the order of 

0.25 nanoTesla per meter (nT/m) over periods of weeks to months (Bracken and others, 

1998). However, over short periods of data collection, as would be applicable to UXO 

detection, the precision is expected to improve because long-period drift factors, which 

are difficult to remove, are not involved. 

The original design specifications required that the magnetic sources (targets) be at least 

25 m away from the sensor array, so that the gradients would remain nearly constant 

throughout the volume of the sensor suspension apparatus. However, for UXO detection, 

imaging, and discrimination, the sources will be much closer, resulting in significant 

gradient variation, or curvature, over the dimensions of the instrument. Therefore, in 

future redesigns, the sensor geometry and spacing will be appropriately modified to 

handle increased field curvatures. Additionally, the original design called for stationary 

measurements over long time periods. The UXO application requires a moving platform 

with measurements over short periods. Again, in future redesigns, the data-acquisition 

5
circuitry and data-reduction techniques will be modified. Notwithstanding, the original 

design was used here to collect the data for this report. 

The Physical Unit: The TMGS consists of two basic units: a tetrahedron that holds the 

magnetic sensors, called the tetrahedral em/mag sensor suspension apparatus (TESSA) 

shown in figure 1; and a cubical enclosure (called “Morris”) that houses the 

magnetometer circuitry and data-acquisition systems (not pictured). TESSA is designed 

to maximize thermal and mechanical rigidity for reasons that will become clear later. 

Four tri-axial magnetic sensors, spaced 0.97 m apart, are placed at the vertices of the 

tetrahedron and are oriented in various attitudes that facilitate calibrations. Morris 

provides a thermally controlled environment (35◦C) for the fluxgate magnetometer 

control circuitry. It also contains circuitry that measures the temperature of each sensor 

and samples the magnetometer output. The raw data are sent to a laptop computer for 

storage via an RS-232 link. 

 Figure 1. TMGS sensor array (TESSA) with sensors numbered. 

Currently, all data reduction occurs in postprocessing. However, the reduction procedures 

of future designs will most likely be carried out in real time on the laptop computer. This 

would provide an immediate readout of user-interpretable tensor derivatives, as well as 

downloadable files ready for analysis and archiving. 

The Magnetic Field 

The UXOs that are detectable by the current TMGS design produce a static magnetic 

field. That is, the field does not vary as a function of time; or at least it varies so slowly 

6
that Maxwell's magnetostatic equations are practically satisfied. The magnetic field is a 

vector field, meaning that it has both magnitude and direction. It is also a potential field, 

meaning that it can be described as the derivative of a potential function and thus, it 

obeys Laplace's equations. All of these qualities provide mathematical bases and 

constraints for analyzing magnetic sources, and hence, UXO targets. In other words, the 

magnetic anomaly produced by a particular UXO contains useful information about that 

UXO. 

Magnetic Sources: The source of the field is magnetization, within the target, of both 

induced and remanent (permanent) varieties. It should be noted that both forms of 

magnetization passively exist within the source material, and are not generated by active 

signals from the TMGS. The induced magnetization is caused by and proportional to the 

geomagnetic field. In earth materials, which have relatively low susceptibilities, the 

magnetization is also generally aligned with the field. However in high susceptibility 

materials, such as UXO, the direction of induced magnetization is difficult to predict due 

to complicated boundary conditions. Remanent magnetization, on the other hand, is 

independent of the Earth's field and can have essentially any orientation and magnitude. 

The smallest irreducible source of magnetic field is a magnetic dipole at the atomic level, 

and magnetization is a volumetric distribution of dipoles. We assume that dipoles are 

aligned throughout the target, which produces a net dipole moment together with a 

number of higher order moments. The suite of moments will be indicative of the 

distribution of magnetization, which in turn is controlled by the composition, shape, and 

distribution of materials within the UXO target. 

Therefore, by appropriately measuring the magnetic field at any point or set of points 

outside the source, characteristics of the magnetization can be ascertained, including its 

location (relative to the sample points) and possibly aspects of its distribution (within the 

UXO). Further analysis of the magnetization may yield specific information about mass, 

dimensions, orientation, and the like. However, it must be stressed that magnetization can 

vary widely among individuals of a given type of UXO, creating a likelihood of nonuniqueness and overlapping characteristics among differing types. Moreover, the 

magnetic field itself, no matter how precisely measured, does not uniquely determine the 

distribution of magnetization, much less the shape and size of the source material. 

Nevertheless, statistical analysis of magnetization distributions may provide a basis for 

determining probabilities for classification of UXO (Billings, 2004). 

Superposition: In practice, the field being measured is a combination of magnetic fields 

produced by a number of sources, both nearby and distant and both small and large. The 

observable field at any point in space and time is actually the vector sum of all the 

contributing fields at that point. This principle is called superposition, and it is simply a 

statement that magnetic fields combine linearly, and that their combining can be 

described mathematically as a linear operation. This principle has bearing on instrument 

design and data reduction. 

7
An isolated measurement of magnetic field (at a particular location and a particular time) 

does not contain any information that identifies or separates its component fields. 

However, by taking multiple measurements, separated by space and (or) time, certain 

component fields can be separated and identified. This can be done if each field varies 

according to a unique and identifiable spatial and (or) temporal function. 

It can therefore be assumed that any magnetic field measurement will be a sum of several 

different fields, originating from differing sources and having differing characteristics. 

Some of these component fields will be of interest; the rest must be removed. The 

separation and removal of undesirable fields is done by specialized instrumentation, data 

acquisition and reduction, and interpretation methodologies that take advantage of unique 

and identifiable aspects of the superpositioned fields through multiple measurements 

taken over spans of length and time. (Some of these methods and the fields they handle 

are mentioned in the next few sections.) 

The TMGS is an example of this type of specialized instrumentation. Each sample is a set 

of simultaneous measurements that are spatially separated. Then, samples can be taken at 

a prescribed sample interval, during which time the instrument can be stationary or in 

motion. Thus, the TMGS is a tool for separating magnetic fields by looking at both 

spatial and temporal variations. 

The job of separating fields includes the requirement of being able to measure small 

fields that have been added to large fields. For example, the Earth's field is several orders 

of magnitude larger than the typical target anomaly; this predicates having 

instrumentation with a large dynamic range that can measure tiny variations riding on a 

huge signal. The magnetometers used in the TMGS have a dynamic range approaching 

140 decibels (dB), which, imagining magnetic fields to be analogous with sound, makes 

them about 1,000 times quieter than a home stereo CD player. More important, this range 

allows the TMGS to preserve the tiny signals from UXO targets while rejecting a huge 

common-mode signal from the Earth’s field, as will be discussed later. 

Scalar Field Measurements: The most common magnetometers in use today measure 

only the magnitude of the field and are insensitive to its direction. This measurement is 

useful primarily because the target's field is much smaller than the Earth's field, making 

the direction of the sum of the two fields nearly the same as the Earth's field. 

Consequently, most geophysical applications treat the magnetic field as a scalar quantity 

with its resultant simplifications. However, except under ideal circumstances where the 

scalar field is accurately known over an effectively infinite surface, scalar measurements 

loose the vector direction information of the target's field. 

Nevertheless, the target's anomaly often can be distinguished spatially from other 

anomalies and fields because measurements are taken at many locations. Also, temporal 

fields, such as the diurnal variation, can be removed by collecting base magnetometer 

data at a nearby stationary location. In spite of these techniques, distant sources can add 

8
noise and uncertainties to measurements of nearby targets, and multiple nearby targets 

can become indistinguishable from one another. 

Scalar Gradient Measurements: Because the magnetic field varies from place to place, 

it is appropriate to consider direct measurement of that variation. Operating two or more 

scalar magnetometers at a fixed distance apart produces a scalar gradient measurement, 

which is the spatial rate of change of the field magnitude in the direction of the baseline 

between the two measurement points. The gradient is preferentially sensitive to nearby 

sources because, while the magnitude of a dipolar field is proportional to the inverse cube 

of the distance to the source, the dipolar gradient goes by the inverse fourth power. 

Therefore, gradient measurements improve the spatial separation of anomalies, 

essentially eliminate interference from distant sources, and obviate the need for a base 

magnetometer. 

The scalar gradient has an associated direction, which depends on the orientation of the 

baseline between the two scalar magnetometers. By constructing a 3-axis scalar 

gradiometer (one with three independent gradient baselines), three of the five 

independent components of a gradient tensor can be found, assuming the direction of the 

total field is known by an independent means. This information is also helpful in 

separating anomalies and modeling nearby sources, but it does not form a complete 

characterization of the magnetic field. 

Vector Field Measurements: A vector magnetometer measures both the magnitude and 

direction of the magnetic field, as contrasted to the scalar magnetometer that is 

insensitive to direction. (Here, the direction is that of the magnetic field lines, not the 

baseline direction as in the scalar gradiometer.) Complete measurement of the vector 

field requires three simultaneous linearly independent measurements. This is the same as 

three different axis directions; hence the name, “tri-axial vector magnetometer.” If these 

axes are coincident and orthogonal (as in the rectilinear coordinate system with x, y, and 

z axes), then they represent the three components of the magnetic field vector, called bx, 

by, and bz. The field magnitude is then the square root of the sum of the squares of the 

three components, and the field direction is defined by the direction cosines. Of course, 

both the location and the precise attitude of the vector magnetometer must be known in 

order to define the field. The TMGS is based on vector field measurements. 

Magnetic Gradient Tensor Measurements: If vector field measurements are 

simultaneously made at two points in space, then three gradient quantities are obtained. 

They are the rates of change of each of the three magnetic field components ( bx, by, bz ) 

as functions of distance in the direction of the baseline between the two measurement 

points. For example, the gradient of the y component of the magnetic field in the x 

baseline direction is expressed as the partial derivative, ∂by/∂x = gyx. Of course there are 

three magnetic field directions; but there are also three possible baseline directions, 

resulting in nine gradients. These nine gradients are arranged into a 3-by-3 matrix, a 

9
mathematical object called the magnetic gradient tensor. It is symbolically written in row 

and column format as: 

 b   

x x by x bz x 

bx y by y bz y 

and interpretation. 

∂

∂

b z b z b ∂z x y z

Maxwell's magnetostatic equations require that both the curl and the divergence of the 

magnetic field be equal to zero in a sourceless region (Kaufman, 1992). This means that 

the gradient tensor will be symmetric and its trace components will sum to zero. In other 

words, only five of the nine tensor components are independent. This has implications 

that simplify and improve TMGS sensor design, noise characterization, data reduction, 

At a point in space and time, the magnetic gradient tensor together with the three vector 

imperfections and deviations from the ideal that necessitate corrections and 

compensations in data reduction processes. Of course, the development of any system 

attention must be paid to providing predictable behavior, stability, and repeatability 

within the narrowest limits of precision, while deferring corrections and compensations 

for accuracy to subsequent data-reduction processes. The output of the physical device 

can then be moved toward the theoretical and the ideal by applying knowable though not 

ideal parameters of the physical system in mathematical reductions. Consequently, proper 

identification of system parameters is critical to obtaining valid results and is possibly the 

most important aspect of developing a data-reduction procedure for the TMGS. 

2d and higher order spatial derivatives. 

∂

∂

∂

∂

∂

∂

∂

∂

∂

field components, consists of a complete description of the magnetic field to the 1st 

At the practical level, magnetometers and magnetic systems are found to have 

∂ ∂

∂ ∂

∂ ∂

order. That is, the TMGS directly measures 0th- and 1st-order spatial derivatives, but not 

Important Concepts as Applied to the Prototype TMGS 

strives toward the theoretical and the ideal. But in the physical device, the greatest 

For example, the Narod Ring Core magnetometers used in the TMGS are among the 

finest tri-axial fluxgates in the world. They are designed after the MAGSAT vector 

magnetometers (Acuna and others, 1978), and USGS magnetic observatories use them 

for reporting of the official HDZ vector data (Narod, 1987). Their output is precise and 

repeatable, yet no axes are perfectly orthogonal. Consequently, if the data reduction does 

not “know” and apply the relative axis-angle parameters—if it blindly assumes that the 

axes are orthogonal—then the resulting tensor will be erroneous. There are a number of 

subtleties like this one that the data-reduction procedures must take into account, and we 

must understand why these subtleties exist. Hence, this section on important concepts is 

provided. 

/ / / gxx g yx gzx 































G = / / / = gxy g yy gzy

/ / / gxz g yz gzz    

10
Extreme Attitude Sensitivity: Vector magnetometers in the Earth's field are extremely 

sensitive to attitude variations. For example, if an axis of a vector magnetometer is 

oriented approximately perpendicular to the Earth’s field and then its attitude is changed 

a slight amount, moving the axis, say, 20 micro-radians (μr) or 4 arc-seconds toward the 

Earth-field vector, then the axis will read an increase of 1 nT. Conversely, if the same 

axis will detect a 0.01-nT change in magnetic field, then it will also detect a 0.2-μr or 

0.04-arc-second change in attitude. Therefore, a vector magnetometer, such as is used in 

the TMGS, is like a high-precision tilt meter, returning attitude information relative to the 

direction of the magnetic field vector with precision down to 0.2 μr. 

If the intent is to measure magnetic-field components, the magnetometer's attitude must 

be known from an independent source with extremely high precision. Unfortunately, the 

best inertial guidance systems track attitude relative to a geographic reference frame only 

down to about 100 μr. If this level of precision was applied to vector measurements in the 

Earth's magnetic field, the attainable precision would be about 5 nT. 

Considering that the field from a 20-mm UXO may be on the order of 5 nT at a range of 

0.5 m, it can be concluded that the vector components of such a UXO cannot even be 

detected above the noise generated by attitude uncertainty, much less be modeled. This 

leads us to the idea of mechanically locking two (or more) sensors together. In this 

configuration, one sensor can be thought to act as a high-precision reference that tracks 

the attitude variations relative to the Earth’s field, while the other measures residual 

variations. However, we quickly see that, in a realizable system, no particular sensor will 

be removed far enough from the nearby sources to act as a pure Earth-field reference, nor 

do the residuals measure components of the magnetic field. Instead, by mechanically 

locking sensors together and differencing them, we obtain spatial gradients; two or more 

vector magnetometers affixed to a rigid platform become a magnetic gradiometer. 

Common Mode and Angular Stability: When the gradient is calculated, differencing 

removes any signal that is common to both magnetic sensors. (Normalizing this 

difference with respect to spatial separation yields a gradient.) In this case, the Earth’s 

field is by far the largest common field and therefore correlates between both vector 

magnetometers, producing huge common-mode signals among their various components. 

The common-mode signals from either magnetometer are then several orders of 

magnitude greater than any of the gradient-producing signals. This dynamic range puts 

rigorous demands on the gradiometer to maintain a precision balance between the 

magnetometers while the platform undergoes angular motion, mechanical shock, and 

thermal variation during data collection. The precision balance may also be called 

tracking and has to do with the ability of readings from one magnetometer to track 

precisely with readings from the other. 

One aspect of tracking involves angular stability, which equates to minimization of 

mechanical and thermal deformation. It was shown above that the attitude of a vector 

magnetometer needed to be known to about 0.2 μr, relative to a geographic reference 

11
frame. Now in the gradiometer configuration, the same figure applies, but it is relative to 

the other sensor instead of the geographic frame, assuming that a gradient is the desired 

quantity. While the geographic reference frame cannot be ascertained with sufficient 

precision, the attitudes of magnetometers in a gradiometer configuration can be 

maintained relative to one another to better than 0.2 μr. However, maintaining this level 

of stability challenges both the mechanical and thermal characteristics of materials and 

geometries used in the sensor suspension apparatus (SSA), or platform. 

The difficulty in achieving sufficient angular stability serves to illustrate the huge effect 

of the Earth’s field and is part of the pervasive issue of common-mode field rejection, 

which is of foremost importance and drives much of the gradiometer design and 

processing methodology. 

Gradient Measurements: The magnetic gradient tensor is derived by taking the 

difference between magnetic vector measurements and normalizing by the separation 

distance. (Devices that intrinsically measure gradients are not available.) Because of the 

Earth’s large field and the common-mode signals it produces in the gradiometer, it is of 

paramount importance that the magnetometer sensors be locked together by a suspension 

apparatus that minimizes relative temporal angular variations, whether due to thermal 

expansion or to mechanical strain. In the current design, the tetrahedral shape of TESSA 

balances these microscopic movements so that the net relative angular variations are 

minimized while providing the largest possible baseline for gradient sensitivity. The 

tetrahedron is a mechanically stable geometry because stresses are transmitted primarily 

along the edges (line segments separating the vertices), so that strains become mostly 

compressional or extensional, and not flexural. The symmetry helps minimize thermal 

angular changes because as the ambient temperature changes, all six edges will expand or 

contract by similar amounts. Meeting this angular-stability criterion together with the 

high quality of the magnetometers makes it possible to measure the magnetic gradient 

tensor. 

Simultaneous Measurement: The current configuration of the TMGS has 4 tri-axial 

magnetic sensors, each with 3 axes, which makes a total of 12 vector field axes. After 

differencing of these 12 axes, there are 6 gradient baselines, only 3 of which are linearly 

independent; and this leads finally to 9 measured gradients. The key point here is that the 

12 vector field axes are sampled simultaneously during data collection, while the 

gradients are derived from the vector field data through subsequent data-reduction 

procedures. 

Rejection of Motion-Related Common-Mode Signal: Simultaneous sampling becomes 

critical when data are being collected on the fly. The platform yaws, pitches, and rolls, 

which leads to angular rates of change and, hence, higher frequency content in the 

common-mode signals. (The common-mode signals have their common modality among 

the 12 vector field signals by virtue of the fact that all 12 signals have most of their 

amplitude originating from the common magnetic field, which is the Earth’s field, and 

most of their variation due to angular rates of the platform in that field.) The higher 

12
frequency content implies rapid variation of signal amplitude. Consequently, if the 

signals were sampled sequentially instead of simultaneously, then there would be 

opportunity for a time-varying common-mode signal to have a different value on each 

axis. This would irreversibly mix temporal variations in with the spatial variations, 

defeating the spatial gradient measurement. Conversely, if the signals are sampled 

simultaneously, temporal variations are rejected, being separated and removed from the 

spatial measurements. 

However, simultaneous sampling is only part of the picture. Before sampling occurs, the 

magnetic field must first be converted to a measurable electrical signal by the 

magnetometer. The conversion effectively imposes a series of linear processes, or filters, 

that can be characterized by a transfer function that most likely will not have a flat 

spectrum nor zero-phase at all frequencies. Therefore, truly simultaneous measurements 

at all applicable frequencies can only be achieved by balancing the transfer functions 

among all of the axes on all of the magnetometers. Any application from a moving 

platform mandates this balancing procedure, which is required to ensure that the 

magnetometers will track precisely. 

Nevertheless, transfer-function balancing was not an essential data-reduction step for the 

data collected in the high-density grid because the platform was brought to rest at each 

station before sampling commenced. Consequently, further description is not included in 

the processing description to follow. 

Rejection of Diurnal and Other Geomagnetic Variations: Most forms of common-mode 

signal, other than those generated by attitude variations, are slowly varying, such as 

diurnal variations; for these, precise tracking is not as critical. Although they are 

normally a menace to magnetic field measurements, they have no effect on the gradients 

because of common-mode rejection. This, however, does not mean that a gradiometer can 

be used successfully in severe geomagnetic storms because high magnetic activity can 

cause temporal ground currents and secondary fields that may appear as nearby sources. 

Furthermore, a tri-axial base magnetometer should always be used in conjunction with 

TMGS surveys to remove temporal variations from the vector-field portion of the data. 

Slowly Varying with Attitude: While the vector field components are extremely sensitive 

to attitude variations, the gradients (components of the gradient tensor) are only mildly 

sensitive. Therefore, if the objective is to locate a magnetic source having a field that is 

small compared to the Earth’s field, then the attitude recovery system for a vector 

magnetometer must be far more precise than that of a tensor gradiometer doing an 

equivalent job. 

As an illustrative example, start with a purely dipolar source oriented perpendicular to the 

Earth’s 50,000-nT field and having a field strength of 8 nT along its 0.5-m equatorial 

radius. Establish an observation point at one of the two intersections of its 0.5-m 

equatorial radius and the Earth-field line that runs through the dipole. Now at the 

observation point, the dipole’s field is 8 nT and runs perpendicular to both the Earth’s 

field and the dipole’s equator. 

13
The total field strength is 50,000.00064 nT and the field direction is 160 μr (33 arcseconds) from the Earth-field direction. Therefore, if a vector magnetometer was 

associated with a state of the art attitude recovery system (with a precision of 100 μr) it 

would barely be able to sense the presence of the dipole from this observation point, 

showing its field strength to be somewhere in the range of 3 to 13 nT, or an error of 

62.5 percent of the full dipolar field expression. 

On the other hand, one of the gradients at the observation point ranges from –48 to 

+48 nT/m depending on its baseline direction. If the same 100-μr attitude precision is 

applied to a gradiometer, then the greatest expression of gradient error is in the range of – 

–0.0048 to +0.0048 nT/m, or an error of 0.01 percent of the full gradient expression. In 

this instance, the vector magnetometer is over 6,000 times more sensitive to attitude 

recovery error than the gradiometer. Therefore, with a nominal attitude recovery system, 

the TMGS is able to provide better data than a vector magnetometer using state-of-the-art 

attitude recovery. The gradiometer’s mild sensitivity to attitude is also a useful concept 

that has application to thermal corrections. 

Preferentially Sensitive to Nearby Sources: As previously described, gradients respond to 

nearby sources and tend to reject distant sources. Of course, this is why a tensor 

gradiometer is ideally suited for UXO work. Keeping this in mind, it becomes clear that 

in areas having magnetic materials near the surface, it is better to have an increased 

standoff (sensor height above the surface) to keep the surface magnetic signatures from 

swamping the UXO. 

Leakage of the Earth's Common Mode Field into Gradient Measurements: 

Assuming that, as alluded to above, (a) the magnetometers are quiet and stable, (b) the 

platform has minimized angular instabilities, and (c) the axis tracking errors can be 

removed, then the instrument is qualified to acquire gradient-tensor data. At this point, 

data-reduction procedures become important. This section describes the first of the 

subtleties, mentioned previously, that results from imperfections and deviations from the 

ideal and that requires application of system parameters in mathematical reductions. 

This subtlety is a leakage of the Earth's common-mode field into the gradients. It is 

caused by slight differences among the magnetometers, such as small disparities in 

calibrations, sensor axis inorthogonalities, and sensor attitude misalignments. All of these 

things can be corrected in data reduction, given accurate knowledge of the governing 

system parameters. For example, the relative attitudes of the sensors must be known 

precisely in order to align them mathematically. 

The leakage phenomenon is best described by way of an example that reflects typical 

TMGS behavior. Suppose that two truly parallel axes are being differenced for a gradient 

calculation as shown in figure 2. There is no gradient present, and the field difference 

between the axes correctly indicates a zero gradient both before and after the gradiometer 

is rotated one degree toward the Earth’s field vector. Suppose now that the two axes are 

14
one degree out of parallel as shown in figure 3. If gradient calculations are made, errantly 

assuming the axes to be parallel, then systematic errors, or false gradients, will appear. 

More insidiously, the false gradients will vary with sensor-platform attitude. Figure 3 

shows a platform rotation artifact that could falsely be interpreted as a 10-nT/m anomaly. 

This false anomaly is more than ample to swamp a real anomaly from a small UXO 

target. Furthermore, the false gradient is correlated to TESSA's attitude and is entirely a 

manifestation of common-mode leakage. In fact, any gradient reduced with incorrect 

parameters will likely have 2 or 3 times bigger errors because of similar effects 

compounded from calibration disparities, axis inorthogonalities, and attitude 

misalignments. 

Figure 2. An example of a gradiometer with no common-mode leakage. The 

sensor axes are parallel, and the sensor platform (the yellow box) is shown in two 

different attitudes relative to the earth’s main field vector. There are no false 

gradients and no platform rotation artifacts. 

15
Figure 3. An example of a gradiometer with common-mode leakage in a zerogradient space. The sensor axes are misaligned by one degree, and the sensor 

platform (the yellow box) is shown in two different attitudes relative to the 

Earth’s main field vector. There are false gradients of 622 nT/m and 612 nT/m; 

and a platform rotation artifact, or false anomaly, of 10 nT/m. 

Common-mode leakage is reduced to tolerable or insignificant levels by introducing 

coefficients derived from a special procedure, called spin calibration, which will be 

described later. If the calibration coefficients are not used, or incorrect calibration 

coefficients are used, the resultant tensor will be correlated to undulations in the ground 

surface and will bear no resemblance to any target anomaly or gradient. 

Thermal Drift: The magnetometer sensors used in the TMGS have a certain amount of 

temperature-dependent drift. Certain of the magnetometer calibration coefficients are 

actually functions of temperature, and should have been derived as such. However, time 

and facilities at YPG allowed only the constant portion of the coefficient functions to be 

measured. 

Another contributor to thermal drift is differential expansion of TESSA. If one side is 

exposed to solar heating, it will tend to expand more than the shaded side. This will tend 

to alter the measured gradients as the sensor attitudes change differentially in the Earth's 

field, as illustrated in figures 2 and 3. According to Bracken and others (1998), data taken 

in Hawaii showed that TESSA produces angular thermal drift of less than 0.05 nT/m per 

degree Celsius, when completely shaded. However, while collecting the high-density grid 

16
data, differential heating may have been a bigger factor because of direct sunlight on the 

face defined by sensors numbered 1, 3, and 4 (fig. 1), which was facing south. The future 

design of the sensor suspension apparatus, which will be specific for UXO work, is 

expected to improve substantially upon the tetrahedron in minimizing angular thermal 

drift. 

The technique used for reducing thermal drift in the high-density grid data removed both 

coefficient and angular drifts. The combined effect of these two causes was a more-orless linear drift by amounts on the order of 10 nT/m during the 4.5 hours from start to 

finish of the data collection. 

Without a thermal drift correction, the gradient tensor becomes increasingly inaccurate as 

a function of time with resultant striping parallel to the line direction and, in the highdensity grid data, an obvious discontinuity between the data on either side of a 100­

minute intermission in data acquisition. There is, however, no evidence that the thermal 

drift leads to significant common-mode leakage in the high-density data. Estimates show 

that the thermal drift would have to be an order of magnitude larger to cause noticeable 

common-mode leakage. 

Removing Thermal Drift: The method for removing thermal drift from the high-density 

data takes advantage of the fact that, unlike field vector components, gradients are not 

very sensitive to small attitude variations. Therefore, a particular location was selected to 

serve as a thermal base station. Data were collected here at the beginning, middle, and 

end of the high-density survey. Because TESSA's attitude varied by only a few tenths of 

a degree in the reoccupations of the thermal base station, the gradients measured there 

should have remained nearly constant. Therefore, any change in those gradients was a 

measure of the thermal drift. As a result, the thermal drift was removed as two sets of 

nine linear functions defined between the nine gradients measured at the beginning, 

middle, and end occupations of the thermal base station location. 

Magnetic-Field Curvature: As mentioned previously, TESSA was designed on the 

assumption that the gradient is effectively constant everywhere within the volume of a 

1-m tetrahedron. But, this assumption cannot be applied to UXO work because of the 

target proximity. The gradient tensor only describes an infinitesimal point in space. 

Surrounding points conceivably can have differing magnetic fields and tensors subject 

only to the constraints that there be no discontinuities and that the equations governing 

magnetostatic fields are not violated. For purposes of discussion, this nonlinear 

potentiality is called field curvature. 

The local magnetic field, within a volume of space (that is, TESSA) is fully described by 

an nth-order 3-dimensional vector polynomial that satisfies the magnetostatic laws, and 

the gradient tensor can be found easily by applying the gradient operator (V) to each 

component of the vector polynomial. This is essentially taking the 1st derivative in a 

volume, resulting in a reduction of order by 1. 

17
A Linear Field Polynomial: If the polynomial is order 1, the tensor will be order 0, which 

means that the field changes linearly as a function of location, and the tensor is 

everywhere the same; that is, each term of the tensor is a constant with respect to 

location. This system of equations can be solved completely with 8 vector field 

measurements (for example, three components at each of two sensors and two 

components at a 3d sensor). It is also the circumstance that TESSA was designed to 

handle because no field curvature is present. 

In this case the system is slightly overdetermined because the TMGS, with the tetrahedral 

configuration, provides 12 simultaneous vector field measurements and, thus, a direct 

measure of all 9 tensor components. However, only eight field measurements are 

required because only five of the nine tensor components are independent. Therefore, 

because the system is overdetermined, the degree to which the nine-measured-component 

tensor departs from being symmetric and traceless becomes an indicator of the severity of 

the curvature. 

Curvature of the Field Polynomial: If the field polynomial is order 2, then the tensor will 

no longer be constant. Each component now becomes a linear function of location 

because field curvature exists. In this case, 15 simultaneous field measurements are 

required and the tetrahedral configuration is slightly underdetermined. As the polynomial 

is increased to higher orders (n = 3, 4, ...), the curvature deepens and it describes an 

increasingly more complicated magnetic field. It would be desirable to use vector field 

data in formulating a polynomial of sufficiently high order to describe the field of a 

nearby UXO target. Unfortunately, the TMGS does not provide enough simultaneous 

independent field-vector measurements. 

Curvature of the Field from UXO Targets: Rudimentary modeling was performed to 

obtain a feel of scale for the curvature of nearby UXO targets. For a nearby (0.5 m) 

dipolar target it appears that, to obtain a sufficiently accurate tensor at the centroid of 

TESSA, the polynomial must be at least order 7 to 9. An order 7 polynomial has 80 

degrees of freedom, meaning that 80 field-vector measurements are required for a 

solution; the TMGS has only 12. Therefore, the system is severely underdetermined and 

cannot begin to characterize, by simultaneous measurements of the 12 axes, the 

curvatures of nearby targets. 

As a way out, one is tempted to think that 7 closely spaced TMGS readings (7 readings 

times 12 components per reading = 84 measurements) could be utilized in the polynomial 

solution. This cannot work, however, because the attitude of TESSA changes from 

reading to reading. Thus, common-mode leakage would dominate the solution, and the 

result would be correlated to ground-surface undulations instead of to the actual magnetic 

field. 

There may yet be mathematical procedures for resolving the curvature issue by 

combining gradients (instead of field components) from several consecutive TMGS 

readings, but this level of modeling has not been performed. In the meantime, the highdensity data were collected on a grid, allowing another option for resolving the curvature 

18
problem, which will be discussed next. Nevertheless, the best solution is a future redesign 

of the sensor suspension apparatus to have smaller dimensions and a geometry that 

handles curvature more effectively while retaining the same number of sensors. 

Minimizing the Effects of Curvature by Collocation of Gradients: With the tetrahedral 

configuration, the effective location of the measured gradient tensor is somewhere within 

the volume, nominally at the centroid, not in a direct line between any two sensors. 

However, the most accurate gradient measurements are those that lie on the baseline, and 

are assumed to be centered, between two sensors. The difficulty is that any single pair of 

sensors only provides three gradients when five are needed for the complete tensor. This 

means that the additional two gradients must be provided by another pair of sensors 

whose center is not in the same location as that of the first pair. 

If field curvature is significant, the gradient from the second sensor pair simply belongs 

to a different tensor. The solution to the problem hinges on the fact that, unlike field 

vectors, the gradients are not very sensitive to small attitude variations. This means that, 

for constructing a complete tensor, individual gradients that are not yet part of a tensor 

can be shared among stations (fig. 4). Combining gradients from three different stations 

in this manner will reduce curvature effects as long as the attitude of TESSA does not 

change much from station to station—exactly the situation with the high-density grid, 

which was a nearly flat, horizontal surface. 

19
Figure 4. Gradients from three pairs of sensors (red s3-s1, green s4-s1, blue s2­

s1) measured at three different stations (red, green, and blue tetrahedrons) are 

collocated at the point shown by a fuzzy black square to produce all nine 

components of a tensor. 

Data Collection 

Two different kinds of data were collected. The first kind was spatial data within the 

high-density grid. The second was angular data, derived from the spin calibration and 

used for deriving calibration coefficients applicable to all data collected at YPG during 

March 2003. 

Spatial Data Collection in the High-Density Grid: The high-density grid is a 3-by-3-m 

horizontal area aligned on magnetic north and approximately centered on a known 

60-mm UXO target, at 0.25-m depth. Also, eight additional known targets lie at regular 

intervals around the perimeter of the grid, four off the corners and four off the sides. 

20
According to information supplied by YPG, consisting of independent testing of random 

soil samples and a separate magnetics survey conducted with a scalar gradiometer, there 

also appears to be significant magnetism randomly distributed within the soils, including 

clutter (fragments of exploded ordnance) within and outside the east half of the south 

side. The square grid cells are 0.25 m on a side, resulting in 13 rows and 13 columns of 

data-acquisition locations (169 stations), some of which were occupied more than once. 

Stations were occupied beginning in the southwest corner, and proceeding north at 

0.25-m intervals. After completing the northern-most station on a given line, TESSA was 

returned to the south end of the next line, 0.25 m farther east. Station locations were 

established by using a tape measure and spray-painting marks on the ground for 

alignment with calibrated marks on the cart. The resulting accuracy of the station 

locations should be plus/minus a few centimeters in both northing and easting. At each 

station, data were sampled at 0.2-second intervals for a minimum of 10 seconds (50 

samples per station). 

The data were collected in two stages, resulting in the two data files, n0312c and n0312d. 

The first stage went from line 1, point 1 (L1P1) through line 8, point 10 (L8P10). After a 

100-minute hiatus, the second stage went from L8P10 through line 13, point 13 

(L13P13), which is the final station in the northeast corner. 

Spin Calibration: Data were collected for the express purpose of deriving precise 

coefficients used in reduction of TMGS data. The procedure is called spin calibration, in 

which TESSA is slowly rotated (within the Earth's large common-mode field) on a nonmagnetic turntable. A complete circle plus 10 degrees of overlap was completed in just 

under an hour for each of four designated tetrahedron attitudes. The location for this 

procedure, different from that of the high-density grid, was chosen to minimize magnetic 

gradients. Temporal variations in the geomagnetic field were recorded by nearby scalar 

and vector base magnetometers. Spatial variations of the local magnetic field were 

obtained by performing a densely spaced survey using a scalar gradiometer. 

The Coordinate Systems: The coordinate system for TESSA is Cartesian right-hand 

orthogonal with directions referenced to the positions of the magnetometer sensors. 

Refer to figure 1 and to one of the tetrahedrons shown in figure 4 to see how sensors s1, 

s2, s3, and s4 are arranged. The s1 sensor is at the top of the tetrahedron; s2, s3, and s4 

form a horizontal triangle at the bottom of the tetrahedron. The x, y plane is parallel to 

the s2-s3-s4 triangle and runs through the centroid, which is vertically below s1 by 

exactly 3⁄4 the tetrahedral height. The z-axis points down from the centroid toward the s2­

s3-s4 triangle. (In figure 4, the z-axis points into the page from s1.) The y-axis is parallel 

to the line defined by s3-s4, with the positive sense toward s3. (In figure 4, the y-axis 

points to the right.) The x-axis is arranged in a right-hand system. (In figure 4, the x-axis 

points toward the top of the page.) 

21
The origin of the high-density grid (geographic) coordinate system is at the grid point in 

the southwest corner at ground level. The positive z-axis points vertically down, the 

positive y-axis points magnetic east, and the positive x-axis points magnetic north. 

TESSA was oriented on the high-density grid with its x-axis pointing magnetic north, yaxis east, and z-axis down—approximately co-oriented with the high-density grid 

coordinate system. 

Processing Procedures 

In this section, the data-processing procedures are explained, with particular objectives to 

elucidate each processing step according to the important concepts given previously and 

to illustrate how each processing step affects the gradient tensor. A flow diagram is 

given in figure 5. 

Data Presentation: Considering that the gradient tensor itself is a complicated 

mathematical object that is not easily illustrated, other means of indicating data qualities 

must be sought. To this end, it should be noted that the data from the TMGS are 

comprehensive in the sense that they can be transformed into any of the four categories of 

magnetic field: scalar field, scalar gradient, vector field, or gradient tensor—the gradient 

tensor being the design objective and the most powerful of the categories. 

The scalar field and scalar gradient can be easily illustrated in color-shaded relief. But 

having no particular sensitivity to field direction, neither of these can be a comprehensive 

indicator of the tensor qualities. The vector field includes the field directional information 

but, lacking a state-of-the-art attitude recovery system, it will be dominated by attitudedependent noise. Also, having three components, it becomes difficult to illustrate. The 

remaining options are either (a) to prepare a large number of statistical devices that, while 

being quantitatively descriptive, rival the tensor's complexity, or (b) to find a 

qualitatively robust means of illustrating the tensor. 

Determinant of the Tensor: Examination of the tensor reveals an assortment of qualities 

that may be utilized. In particular, each magnetic gradient tensor has an orthogonal 

natural coordinate system into which it can be rotated, called the principle axis frame. 

Performing the rotation diagonalizes the matrix; that is, all six of the gradient components 

that are not in the trace become zero. Also, the threeremaining components 

(gxx,gyy,gzz) are a comprehensible set of gradients. (Only two of them are 

independent because gzz=-gxx-gyy, which is required by Laplace’s equation, 

V2Ф=0). 

If these three gradients are 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. Furthermore, this object is a scalar quantity that can be easily illustrated in 

color-shaded relief as well as representing an intrinsic quality of the magnetic gradient 

tensor. Finally, this quantity is a tensor invariant (called the I2 invariant) that can be 

22
Link

calculated easily from any reference frame by taking the determinant of the matrix— 

diagonalizing the matrix is not required (Pedersen and Rasmussen, 1990). 

Figure 6 

Figure 7 

Figure 8 

Figure 10 

Figure 11 

Figure 5. Flow diagram of data reduction showing the major processing stages 

for reduction to a tensor of the TMGS high-density data. 

23 
It can therefore be deduced that the magnitude of the determinant is in some way 

proportional to the proximity of a magnetic source. It actually goes by the 12th power of 

the inverse distance to the source. This results from the determinant being a product of 

three gradients, each of which follows an inverse 4th power law. The magnitude also has 

other characteristics dependent on the magnetic moments and the observation point. 

Evidently because of the influence of such an intense power law, the determinant has the 

greatest expression at the nearest approach to a source. In most cases, this means that a 

mapped determinant anomaly will reach a sharp (positive or negative) peak almost 

directly over its target. (Because of moment direction, the exact location of the peak may 

be offset slightly, or in some cases, the peak may be replaced with an abrupt positive-tonegative transition.) This is in contrast to a scalar-field anomaly that reaches an 

unfocused peak at a location somewhat offset from the target zenith. 

Linear Volume-Model Engine: The linear volume-model engine is a utility computer 

algorithm that displays TMGS data, at various stages of processing, in a common 

format—namely the determinant of a gradient tensor. It allows data in a particular 

premature stage of processing to be examined in a mapped determinant form by 

converting them directly, without the intervening reductions. For example, figure 7 uses 

the engine to show what the mapped determinant would look like if the thermal-drift 

correction and collocation stages of data reduction were not performed. 

The output is the determinant of the gradient tensor for each of the 169 stations in the 

high-density grid. The input, however, is allowed to be any of (a) raw TMGS data 

(analog voltages and bin numbers), (b) the 12 field-component values from the 4 TMGS 

sensors, or (c) the nine gradients obtained by subtracting the sensor readings (s2-s1, s3­

s1, s4-s1). Therefore, the linear volume-model engine guarantees a standard nonbiasing 

procedure for TMGS data examination. 

The linear volume model engine performs a two-stage process, the first of which converts 

the input data to 12-vector field components, and the second obtains a tensor (and its 

determinant) by applying a linear least-squares regression to the field components. 

Coefficients used in the first stage are nominal (for example, the magnetic sensor axes are 

assumed to be perfectly orthogonal). The regression in the second stage forces the trace 

to vanish by heavily weighting the trace summation equation, as suggested by Dr. 

Yaoguo Li, (Y. Li, oral commun., 2004). The regression, however, does not perform any 

other normalization. For example, weighting the least-squares system according to 

average field-component magnitudes introduces fictitious anisotropies; so, the engine 

does not perform this normalization. 

Assumptions Applied to the Spatial Data: To simplify the processing of the highdensity grid data, a number of assumptions were made about things that would not 

adversely affect the resulting tensor. These assumptions are the subheadings, as follows: 

Positions match the ideal 0.25-m grid: The position recovery for both TESSA and the 

high-density grid is simply to use the planned values from positioning TESSA and setting 

24
up the grid (for example, 1.25, 1.50, 1.75, ... see "Spatial Data Collection in the HighDensity Grid"). This means that there can be both systematic and random errors in the 

positions of TESSA amounting to plus/minus several centimeterss in x, y, and z. 

Gradients vary too slowly as a function of attitude to warrant attitude corrections: The 

heading, pitch, and roll of TESSA changed by a few degrees each, from station to station. 

If the TMGS is not leaking common-mode field, this amount of change in attitude has 

only minimal effect on the gradients. Therefore, all gradients are assumed to have been 

measured in a common reference frame. However, it is important to mention that this is 

not a generally applicable assumption. In anything but the flattest terrain, attitude 

recovery must be performed for the gradients to within a couple of degrees in heading, 

pitch, and roll. 

Sensor positions are coincident with the virtual tetrahedron vertices: As a foundation for 

tetrahedral calculations, an idealized tetrahedron with 97-cm edges is used and is called 

the virtual tetrahedron. Although they are fairly close, the true positions of the sensors do 

not exactly coincide with the vertices of the virtual tetrahedron. However, the errors are 

small, probably less than plus/minus 1 cm in each axis. This assumption would introduce 

less than a 1-percent error into gradient magnitude accuracies. 

Sensor axes are coincident: By design, the three axes in each of the four magnetic 

sensors used in TESSA are not coincident; the x-, y-, and z-axes occupy different 

physical locations on each sensor base. This design presumes that readings are being 

taken in low-gradient regions (recall the original purpose of TESSA), contrary to the 

UXO detection circumstance. Similarly, it violates one of the prerequisites mentioned 

earlier for obtaining the three components of the magnetic field vector; namely, that the 

axes must be coincident. 

However, the designing of a data-reduction procedure to correct this inadequacy was not 

performed. Indeed, such a procedure would have improved the results but was not critical 

in meeting the objectives of this study. Therefore, assuming the axes on each sensor to be 

coincident has introduced up to a plus-or-minus 4-percent error into the gradients. This 

has the potential to reduce the determinant accuracy by as much as plus-or-minus 12 

percent, a fairly large amount. However, a future design of the sensor suspension 

apparatus can minimize or obviate the need for a sensor-axis coincidence correction by 

taking advantage of the fact that the three axes on each sensor are collinear. 

Assumptions Applied to the Spin-Calibration Data: To simplify the processing of the 

spin-calibration data, a number of assumptions were made about things that should have 

minimal effect on the resulting tensor or could be reduced by other means. These 

assumptions are the italicized subheadings, as follows: 

The background field was stable and its tensor everywhere zero: For derivation of all the 

spin calibration coefficients, it was assumed that the geomagnetic field did not vary as a 

function of time. Also, it was assumed that the magnetic gradient tensor was zero 

25
everywhere within the volume of space required for the spin calibration. Though neither 

of these assumptions is strictly valid, measurements of both the temporal and spatial 

scalar field during the calibration showed that the variations would not significantly 

affect the coefficient values. Therefore, the spin-calibration data were reduced using the 

assumption that the scalar field was a temporal and spatial constant of 48,000 nT. 

However, the necessary temporal and spatial data were collected, so if it later becomes 

apparent that improved accuracy of the coefficients is necessary, these data can be 

applied. 

Sensor 1 is perfectly aligned with the virtual tetrahedron: Sensor 1 (s1) was used as a 

reference for converging the attitudes of the other sensors (s2, s3, s4) so that all four 

sensors can be mathematically rotated into a common coordinate system. However, no 

attempt was made to ascertain the attitude of s1 relative to the virtual tetrahedron. 

Therefore, it is quite likely that s1 is misaligned with the tetrahedron by a degree or so in 

heading, pitch, or roll. As a consequence in the final tensor, the field directions and the 

gradient baseline directions are slightly misaligned, which has only a very small effect on 

the tensor values. 

The sensor temperatures did not vary: Many of the coefficients derived from the spincalibration data are functions of temperature. However, the deeply involved dataacquisition procedures that were required to define these functions were not performed. 

Consequently, the coefficients were derived as if the temperatures were constant 

(although the temperatures were measured), and reliance was placed upon the thermaldrift correction to supplement the missing data. 

Raw-Data Conversion: The first processing stage is to convert from the data file 

recorded in the field to an in-house format for manipulation in postprocessing. 

Basic Data Forms: The raw data are recorded as an analog voltage that converts to 

magnetic field at the approximate rate of 100 nT/V. To maintain core linearity and signal 

precision, the magnetic flux through the core is minimized by applying an opposing 

magnetic field (a bucking field), which is incremented every few hundred nanoTeslas of 

field change. The amplitude of the bucking field is recorded as an integer value ranging 

from –256 to +255 and is called the bin, or bin number. The analog voltages and bin 

numbers from all 12 axes of the 4 magnetometers are recorded simultaneously every 

0.2 second with 16-bit precision to maintain tracking and 0.02-nT precision levels. In the 

high-density data, 1-Hz analog anti-alias (low-pass) filters were used before sampling. 

Auxiliary data streams also were recorded at the same rate for sensor temperatures, 

Morris temperature (temperature inside Morris), air temperature, and tilt (pitch and roll). 

Recording Method: All channels were recorded continuously for each data file with 

documentation fiducials (doc points) marking the beginning and end of each station 

collection period. The recording program, called KMGR7, is coded in C++ and produces 

several ASCII subfiles with hexadecimal values. Two subfiles will fit on a 1.38-Mb 

diskette. 

26
Conversion Program: The subfiles are interpreted, concatenated, and converted to an inhouse file format for further processing. In this file, analog voltages and bins are 

unmodified and are preserved as exactly the values that were recorded. Doc points and 

associated data-acquisition operator comments are separated into separate files for 

independent viewing and input to subsequent programs. 

Deconvolving Bin Steps from Analog, and Matching Axes for Precision Tracking: 

The TMGS can be understood as a linear system that responds as a function of frequency. 

Yet, the calibrations that are performed are effectively done at D.C. (wave number 0). 

This means that not all the calibrations hold for the low-frequency signals that would be 

expected during data acquisition from a moving platform. Consequently, it is necessary to 

establish a transfer function for each axis of each magnetometer that effectively ties the 

amplitudes and phases in the Nyquist interval back to the measured D.C. levels. This 

ensures that the 12 axes will track together up to a prescribed precision, and that step 

functions (injected into the analog voltage whenever a bin changes) can be handled 

without introducing significant discontinuities in the axis signals (see below, “Bin 

Conversions”). 

The transfer functions have been measured over the current Nyquist interval (0 to 2.5 Hz) 

and found to be quite stable. However, they have not been applied to the high-density 

data because the data were effectively collected at wave number 0. 

Removal of Intervening Data Records and Noisy Samples: Using the doc points, all 

records that were not acquired during the 10-second station-sampling period are removed. 

Any records contaminated with obvious spiking or other noise processes are also deleted. 

The x and y positions in the high-density grid coordinate system are added to the file at 

this time. Figure 6 illustrates the determinant that would result if the data were left in its 

raw form, without any further processing. 

Bin Conversions: For each of the 12 axes, the bin numbers are converted to equivalent 

voltages at a nominal rate of 5 V/bin for s1 and s2, and 3.3 V/bin for s3 and s4. (The 

differing conversion rates reflect two slightly different manufacturers’ model lines.) The 

equivalent voltages are then combined with the corresponding analog voltages to produce 

12 axis signals that can range from –800 to +800 V. (This voltage range, combined with 

the nominal 100 nT/V signal-to-field conversion, implies a –80,000-nT to +80,000-nT 

field range.) In general, the exact voltage associated with each bin number must be 

ascertained in the laboratory and is a complicated function of bin number, sensor 

temperature, and Morris temperature. However, rather than a direct measure of the bin 

number-to-voltage conversion, a more precise quantity is measured, called the bin-step 

size. It is simply the amount of voltage change associated with a step from a given bin to 

one of its neighboring bins. 

27
Although a comprehensive discussion of the dependencies between the bin conversion 

functions and the axis polynomial (described in the next section) is well beyond the scope 

of this writing, it suffices to say that accurate knowledge of each bin-step size reduces 

common-mode leakage, minimizes bin-step discontinuities, and improves tracking when 

taking measurements from a moving platform. An inaccurate knowledge of the bin-step 

size will introduce noise that manifests as a sawtooth of pseudo-random amplitude and 

wavelength riding on a changing field (typically, the Earth's common-mode field). 

While the laboratory bin-size calibrations have not yet been completed, it is expected that 

the bin-step sizes can be known to about 0.05 nT. Until then, preliminary bin voltage 

conversions have been used, with apparent bin-step size errors on the order of 1 nT. It 

should be noted that, in the high-density data, bins were fairly constant with occasional 

changes of plus/minus 1 bin step. Table 1 lists bin-step sizes used in reducing the highdensity grid data. 

Figure 6. Raw TMGS high-density data processed by the linear volume-model to 

a determinant of the magnetic gradient tensor. The map is correlated primarily to 

ground surface undulations due to common-mode leakage of the Earth's primary 

field, resulting from application of nominal calibration coefficient values. The 

color scale is nonlinear. 

28
 Table 1. Average bin sizes <volts/bin>. 

Axis Sensor 1 Sensor 2 Sensor 3 Sensor 4 

x 5.046971 5.050641 3.302770 3.308334 

y 5.048011 5.054041 3.311700 3.307056 

z 5.044710 5.056564 3.304325 3.312998 

Application of Spin-Calibration Results for Converting Voltages to Fields: The spin 

calibration procedure essentially nulls the Earth's common-mode field as a function of 

attitude by slowly spinning TESSA and then applying a nonlinear regression to the 

resulting data. The coefficients derived through the spin calibration are applied to the 

high-density grid axis signals (resulting from bin conversions) to calibrate and align the 

sensors. 

The proper application of the spin-calibration coefficients is critical to a successful data 

reduction. Because some of the associated derivations are also tedious, a Fortran 

subroutine called s_ab2h.f is provided in Appendix A to clarify how the spin calibration 

data are used. This subroutine applies the bin-size, axis polynomial, orthogonality, and 

sensor attitude coefficients to basic TMGS analog and bin data. A mathematical 

derivation of the spin-calibration procedure is provided in Bracken and others (2005). 

Axis Polynomial: The axis polynomial is a third-order fit of the axis signal to its 

corresponding magnetic field component. For each axis, it effectively calibrates the zerooffset, and the gain, which are respectively the 0th- and 1st- order terms of the 

polynomial. The axis polynomial and the orthogonality correction are solved 

simultaneously for all three axes on a given sensor by requiring that the scalar field be 

correct for all attitudes. Tables 2, 3, 4, and 5 list the coefficients of the axis polynomial. 

The zero-offset is the magnetic field present when the axis signal is 0. The gain is the rate 

of change in magnetic field with respect to the axis signal voltage. The quadratic and 

cubic terms are only a small contribution, indicating that the magnetometer response is 

extremely linear. (In each table, the least significant digit displayed is consistent with a 

0.01-nT precision level.) 

Table 2. Axis polynomial – constant term (zero-offset) <nanoTesla>. 

Axis Sensor 1 Sensor 2 Sensor 3 Sensor 4 

x –51.57 –34.78 5.84 20.28 

y –50.33 12.95 7.4 35.23 

z –56.54 –18.86 –25.86 5.18 

29
 Table 3. Axis polynomial – linear term (gain) <nanoTesla/volt>. 

Axis Sensor 1 Sensor 2 Sensor 3 Sensor 4 

X 99.95753 99.04721 99.36614 99.80745 

Y 101.12474 100.75199 100.41647 100.63311 

Z 99.06592 99.43129 99.76678 99.13475 

Table 4. Axis polynomial – quadratic term <nanoTesla/volt2>. 

Axis Sensor 1 Sensor 2 Sensor 3 Sensor 4 

X –0.00001834 –0.00000565 0.00001941 0.00000958 

Y –0.00002865 0.00000174 –0.00001841 –0.00003666 

Z 0.00018719 0.00007021 0.00004180 –0.00009373 

Table 5. Axis polynomial – cubic term <nanoTesla/volt3>. 

Axis Sensor 1 Sensor 2 Sensor 3 Sensor 4 

X 0.00000000514 –0.00000007088 0.00000000596 –0.00000013659 

Y 0.00000002123 0.00000000462 –0.00000000101 –0.00000016287 

Z 0.00000041569 0.00000007837 –0.00000007556 –0.00000029001 

Orthogonality Correction: The three axes of a given magnetometer are not quite 

perpendicular. Therefore, the orthogonality correction mathematically realigns them into 

a nearly perfectly orthogonal rectilinear coordinate system. Table 6 gives the actual 

angles between the three axes on each of the four sensors. 

Table 6. Orthogonality correction angles <degrees> (x is fixed in space; y 

"moves" only in the x-y plane). 

Axis Sensor 1 Sensor 2 Sensor 3 Sensor 4 

y-x 89.89108 90.11640 89.82039 90.24590 

z-y 90.75768 90.01508 91.26369 89.72196 

z-x 89.72650 89.56956 90.21198 89.10476 

30
Attitude Correction: The attitude correction is performed as a separate regression from 

the axis polynomial and orthogonality, but uses the same spin-calibration data sets. It 

aligns sensors s2, s3, and s4 with sensor s1 (see previous section "Assumptions Applied 

to the Spin-Calibration Data"). In other words, it mathematically rotates all four sensors 

into a common reference frame, the coordinate system for TESSA. Table 7 shows the 

ideal, and table 8 shows the measured attitude correction angles derived from the spincalibration data. 

Table 7. Ideal attitude angles <degrees>. (Positive is CW rotation about axis as 

viewed from sensor origin). 

Axis Sensor 1 Sensor 2 Sensor 3 Sensor 4 

z''' 0.00000 0.00000 120.00000 –120.00000 

y'' 0.00000 –109.47122 –109.47122 –109.47122 

z' 0.00000 180.00000 60.00000 –60.00000 

Table 8. Actual measured attitude angles <degrees>. (Positive is CW rotation 

about axis as viewed from sensor origin). 

Axis Sensor 1 Sensor 2 Sensor 3 Sensor 4 

z''' 0 assumed –0.43250 121.59830 –119.26004 

y'' 0 assumed –109.35126 –110.02656 –109.88480 

z' 0 assumed 180.12615 60.42545 –59.63062 

Temperature Corrections: The coefficients in the axis polynomial are functions of sensor 

temperature rather than simple constants. Similarly, the angles used in the attitude 

correction are functions of outside temperature, actually the temperature of TESSA. 

However, these coefficients were measured as constants rather than functions of 

temperature. As a result, the gradients from the high-density grid had to be corrected for 

thermal drift as is described in the section “Thermal-Drift Correction.” 

Differencing Magnetic Fields to Obtain Initial Gradients: From this point, subsequent 

reductions must be performed on gradients. Therefore, three gradient baseline directions 

are found for each 0.2-second sample. (There are six baselines, but only three form a 

basis for a coordinate system because the remaining three are linearly dependent upon the 

first three.) Because of reductions already performed (simultaneous sampling, transferfunction matching, proper bin conversions, and spin calibration), these gradients are 

sealed against a variety of common-mode leakage problems, including the Earth's 

common-mode field. 

31
The gradients are then found by subtracting the three field components of sensor s1 from 

each of the three remaining sensors, s2, s3, and s4, and then dividing by the sensor 

separation distance of 0.97 m. The gradient baseline directions in the TESSA coordinate 

system are listed as follows in table 9. 

Table 9. Azimuths and dip in degrees of the three sensor-pair gradient baselines. The 

coordinate system is geographic with TESSA level and pointing magnetic north (azimuth 

0 deg). For clarity, the specific gradients associated with each of the three sensor pairs are 

shown as well. 

Sensor pair Azimuth Dip Gradients 

S2-s1 0o 54.7o ∂bx/∂21, ∂by/∂21, ∂bz/∂21 

S3-s1 120o 54.7o ∂bx/∂31, ∂by/∂31, ∂bz/∂31 

S4-s1 240o 54.7o ∂bx/∂41, ∂by/∂41, ∂bz/∂41 

Obviously, these gradient baselines have not been rotated into an orthogonal coordinate 

system, as have been the field components. Because subsequent processing steps need 

only work on individual gradients, however, this rotation need not be done. Also, where 

the field components were extremely sensitive to the attitude of TESSA, these gradients 

change very slowly with attitude and are therefore much more easily handled. 

Ten-Second Averaging of Gradients to Obtain Station Gradient Data: For each 

station in the high-density grid, each of the nine gradients calculated in the previous step 

was averaged over a 10-second period (50 samples at 0.2 second per sample). Assuming 

that the noise is Gaussian, averaging 50 samples increases the precision and confidence 

of the station by a factor of 7 over that of any single sample, using the √n rule. In theory, 

this averaging step can only be performed on the gradients, not on the field components, 

because the combining of field components in any manner before spatial differentiation 

breaks the seal and allows common-mode leakage to reenter the system. In practice, with 

the high-density data, had the averaging been done before the gradients were calculated 

(which it was not), the error would have been almost insignificant. Figure 7 illustrates 

the determinate that would result if the data were processed only up through 10-second 

averaging. 

32
Figure 7. Calibrated TMGS high-density data processed by the linear volumemodel to a determinant of the magnetic gradient tensor. Application of the 

calibration coefficients (from the spin calibration) to the field components, and 

subsequently finding the gradients by differencing the field components in each 

sample, removed the common-mode leakage, leaving thermal drift as the 

dominant of the remaining nongradient effects. Therefore, this map is correlated 

primarily to thermal drift while the ground-surface undulation effects of the 

previous figure have been removed. It is instructive to observe that the major 

feature on this map coincides with the 100-minute hiatus between data sets, 

during which time the temperatures of the sensors continued to climb. 

Thermal-Drift Correction: Thermal drift was removed from the gradients, as described 

previously. The thermal base station, in the southwest corner of the high-density grid, 

was used as the reference location and was occupied three times during data collection: 

at the beginning, in the middle, and at the end. Each of the nine gradients described 

previously (∂bx/∂21, ∂by/∂21, ∂bz/∂21, ∂bx/∂31, and so forth) was treated independently 

of the others in setting up a thermal baseline for each of the two data files (n0312c and 

n0312d). 

33
The baselines required that each gradient at the thermal base station remain constant over 

the duration of the data collection, and the resulting correction was applied as a linear 

function of sensor temperature to the intervening stations. The thermal drift rates ranged 

from 0 to 2.6 nT/m per degree Celsius (sensor temperature) depending on which gradient 

was being considered. The sensor head temperatures increased at about 2.3 degrees 

Celsius per hour over the 4.5-hour duration of the combined data sets. 

Unknown Constant: Because there was an unknown amount of thermal drift between the 

high-density and the spin-calibration data collections, the true value of each gradient at 

the thermal base station had to be considered an unknown constant value. The constant 

was estimated for each of the nine gradients by subtracting the gradient's average (over 

the entire high-density grid) from its thermal base station value after removal of the 

thermal drift. In other words, each of the nine gradients was adjusted by removing its 

average value. 

In Lieu of Temperature-Dependent Spin-Calibration Coefficients: Had the coefficients 

derived from the spin-calibration data contained the appropriate thermal dependencies, 

this entire thermal drift correction would have been obviated. Although it is anticipated 

that future TMGS data reductions will use thermally dependent coefficients, good field 

practice predicates that provisions for thermal corrections always be made. That is, a 

base location should be reoccupied every few hours of data collection. Figure 8 illustrates 

the determinant that would result if the data were processed up through the thermal drift 

correction. 

Gradient Collocation: Because of field curvature, each of the nine sensor-pair gradients 

(∂bx/∂21, ∂by/∂21, ∂bz/∂21, ∂bx/∂31, and so forth) is potentially in a different location 

than any of the others. Due to the mean value theorem (a basic theorem of calculus), a 

gradient calculated by differencing does exist on the line segment between the two 

bounding sensors, but its exact location is not known (fig. 9). (The collocation procedure 

described here cannot mitigate this effect of curvature between sensor pairs, but it can 

move the sensor pairs into the same volume of space.) If the gradient-producing system is 

random, however, the most probable gradient location is halfway between the sensors. 

This is the assumption that must be made unless future modeling shows that the probable 

gradient location is skewed due to sources always being below TESSA. Therefore, the 

most probable gradient locations (halfway between sensors) are given as follows, in 

table 10. 

34
Table 10. Expected x, y, and z coordinates in meters of the locations that are 

halfway between sensors. The origin is at the centroid in the TESSA coordinate 

system. The specific gradients associated with each of the three sensor pairs are 

included. 

Sensor pair X (m) Y (m) Z (m) Gradients 

s2-s1 0.280 0.000 -0.198 ∂bx/∂21, ∂by/∂21, ∂bz/∂21 

s3-s1 -0.140 0.243 -0.198 ∂bx/∂31, ∂by/∂31, ∂bz/∂31 

s4-s1 -0.140 -0.243 -0.198 ∂bx/∂41, ∂by/∂41, ∂bz/∂41 

Figure 8. Thermal-drift corrected TMGS high-density data processed by the 

linear volume-model to a determinant of the magnetic gradient tensor. With the 

dominating effect of thermal drift removed, the true anomalies have become 

visible. Yet, they are distorted and unfocused due to the remaining nongradient 

effects of field curvature. 

35 
Figure 9. Illustrates the implications of mean value theorem, that (a) if a gradient 

is calculated by taking the difference of field components from two sensors, then 

the gradient exists and it lies somewhere between the two sensors; and (b) the 

location of the calculated gradient is not constrained to be at any particular place 

between the two sensors. 

Apparently, the set of three gradients calculated from any pair of sensors is nearly 0.5 m 

from the gradients of either of the other two sensor pairs. Considering that the entire 

anomaly is scarcely more than 1 m in diameter, it is not difficult to see that these gradient 

pairs must be mathematically collocated before they can be combined into a coherent 

tensor. 

To implement the methodology described previously (“Minimizing the Effects of 

Curvature by Collocation of Gradients,” and figure 4), each of the nine gradients was 

treated individually as a scalar function of x-y position. Initially, the positions used were 

the centroid's x and y, which are the same as the station locations. Each of the 13 northsouth data-collection lines was smoothly (1-d) interpolated to a 1-cm interval, preserving 

gradient values, and all derivatives. Then at a given station location, each gradient was 

replaced with an appropriately offset value from the same line or from an adjacent line. 

36
For example, to accumulate the collocated gradients at station L7P7 (line 7 point 7, in the 

exact center of the high-density grid), the s2-s1 gradients were replaced with values from 

the same line interpolated at 28 cm south. This location is near L7P6, which is 25 cm to 

the south. In fact, the actual location of the s2-s1 gradients from station L7P6 is about 3 

cm north of station L7P7. 

The s3-s1 gradients were replaced with values from line 6 interpolated at 14 cm north. 

This location is just over halfway between L6P7 and L6P8. The actual location of the s3­

s1 gradients from station L6P8 is about 11 cm north and 0.75 cm west of L7P7. 

The s4-s1 gradients were replaced with values from line 8 interpolated at 14 cm north. 

This location is just over halfway between L8P7 and L8P8. The actual location of the s4­

s1 gradients from station L8P8 is about 11 cm north and 0.75 cm east of L7P7. 

Therefore, if it can be assumed that the gradient values are located halfway between 

sensors, the collocation algorithm actually implemented is fairly precise in the northsouth direction, but it smears the gradients over a 1.5-cm-wide east-west zone. 

Nevertheless, this is an acceptably small error. 

Finally, extrapolation was performed around the edges of the high-density grid to supply 

missing information for bordering rows and columns. This may have reduced the tensor 

accuracies on lines 1 and 13, and on points 1, 2, and 13 of all lines, covering nearly 

40 percent of the high-density grid area around the edges. This means that any statistical 

chart of the collocated high-density grid data will be skewed unless these rows and 

columns are first eliminated from the statistical analysis. 

Converting Sensor-Pair Gradients to a Tensor: At this point in the processing of the 

high-density grid data, the gradients (s2-s1, s3-s1, s4-s1) are ready for conversion to a 

tensor. Because all nine gradients have been measured and only five are required, the 

tensor is slightly overdetermined. At least two ways exist for computing the tensor. 

By Rotation: Although this method is not described in detail here, the most obvious way 

of computing the tensor is to rotate the gradients first into an orthogonal coordinate 

system and then rotate that into the coordinate system of TESSA, in order to coincide 

with the coordinate system of the field components. Both of these rotations can be done 

simultaneously by standard matrix multiplication between the matrix of column vectors 

(∂bi/∂21 ∂bi/∂31 ∂bi/∂41) and an appropriately derived rotation matrix. It should 

be noted that rotating a finalized tensor requires a similarity transform to rotate both the 

field components and the gradient baseline directions; but here, only the baseline 

directions need to be rotated. 

The rotation approach yields a nine-component tensor that is asymmetric and has a nonvanishing trace to the extent that there was residual curvature or other noise in the 

measurement and reduction processes. It has the potential to be a good diagnostic tool 

except that the rotation tends to mix up noise from the various sensor-pair gradients. 

37
Nevertheless, the determinant of the tensor derived in this manner is still fairly 

representative of the system capabilities. 

By Linear Least Squares Regression: Another method of converting the (s2-s1, s3-s1, s4­

s1) gradients into a tensor is by backing out relative field components at the sensors and 

then applying to them a linear least-squares regression, as is done by the Linear VolumeModel Engine (previously described). Therefore, the Linear Volume-Model Engine is 

used here to calculate the final tensor and determinant from the collocated (s2-s1, s3-s1, 

s4-s1) gradients. This is illustrated in figure 10 with corroborative modeling shown in 

figure 11. 

Figure 10. Collocated TMGS high-density data processed by the linear volumemodel to a determinant of the magnetic gradient tensor. Collocation of the 

gradients measured by TESSA has largely mitigated the distortions caused by 

field curvature. This result approaches the map produced by modeling fields from 

the targets. 

38
Figure 11. Model of the high-density grid targets, processed to a determinant of 

the magnetic gradient tensor. The magnetic fields of the center target, a 60 mm 

UXO at 0.25-m depth, and the eight edge-targets were modeled by assuming the 

depth and approximate location specifications provided with the Calibration Grid. 

Horizontal positions were adjusted slightly to account for positioning inaccuracies 

of both the target and TESSA. Magnetic dipole moment directions were assumed 

to be in alignment with the Earth's field (preliminary modeling shows this 

assumption to be somewhat in error, but use of the modeled directions would 

become circular reasoning in this application); but moment magnitudes were 

adjusted to match anomaly peaks. This figure is preliminary, intended only for 

qualitative comparisons with figures 6, 7, 8, and 10. A more precise model is 

described in the next section and shown in figure 13, which is for comparison 

with figure 12. 

A Model of the Expected Tensor Determinant 

To illustrate the capabilities of the TMGS, a model was produced of the expected fields 

from the nine UXOs that were buried in and around the high-density grid. Because this 

39
model was not rigorously derived, using it for in-depth quantitative analyses is not 

recommended. However, it does demonstrate conclusively that the TMGS, in its current 

form, produces a realistic tensor with a better than expected detection capability. 

A forward modeling routine was used that calculates the magnetic field, the magnetic 

gradient tensor, and the determinant of the tensor generated by any given spatial 

arrangement of point-dipole sources. The location and dipole moment of each source 

must be specified. An object of any shape can be modeled by specifying a sufficient 

quantity of point sources within its boundaries. Here, only one source was used for each 

of the nine UXO, assuming the targets to be sufficiently represented by a single point 

dipole. No attempt was made to model anomalies caused by clutter and geologic sources. 

Parameters of the forward model are: 

 The height of the measurement surface in negative meters above the ground 

surface; 

 the x, y location of the observation point; 

 the magnitude, declination, and inclination of the Earth’s magnetic field; 

 the depth in positive meters below the ground surface of the dipole being 

modeled; 

 the dipole’s x, y location; and 

 the dipole’s moment magnitude, declination, and inclination. 

The height of the measurement surface was fixed at –0.88 m, corresponding to the height 

of the most probable gradient measurement locations (see the z coordinate in table 10). 

The depth of each dipole was fixed at the known depth of its corresponding UXO target 

obtained from the “ground truth” information supplied for the YPG calibration grid (table 

11). 

40
Table 11. Values used for the dipole modeling of the UXO targets. The UXO ID 

corresponds to the numbers listed on the target locations in figures 12 and 13. 

The X-Location and Y-Location model parameters were adjusted within a 0.25-m 

boundary of expected positional error. The Dipole Moment magnitude was 

adjusted to match modeled anomaly amplitudes to measured amplitudes. 

UXO ID 

X 

Location 

(meters) 

Y 

Location 

(meters) UXO Type 

UXO Depth 

(meters) 

Dipole 

moment 

(AmpereMeter×Meter) 

1 –0.30 –0.30 M75 0.15 0.0500 

2 1.60 –0.40 

60 mm w/ 

clutter 0.25 0.1380 

3 3.40 –0.40 81 mm 0.50 0.1380 

4 3.40 1.77 81 mm 0.50 0.1380 

5 3.30 3.20 8# shot 0.91 0.0780 

6 1.50 3.30 8# shot 0.20 0.0780 

7 –0.80 3.50 8# shot 0.20 0.0780 

8 –0.70 1.50 M75 0.15 0.1380 

9 1.31 1.71 60 mm 0.25 0.0866 

It was estimated that the recovered x and y locations of both TESSA and the buried UXO 

had small systematic and random errors accumulatively totaling 0.25 m. Therefore, the x 

and y positions of the dipoles were adjusted within this bound to match the positions of 

the anomalies. It is encouraging that positional matches were established without 

exceeding the error bound. 

Because the UXO targets were degaussed before being buried, we assumed that no 

remanent magnetism existed, only induced magnetism; therefore, the dipole moment 

direction was held constant, in alignment with the Earth's field at declination 0 and 

inclination 58.7 degrees. Although this assumption could be in error by 20 to 30 degrees 

in both declination and inclination, it suits the qualitative nature of this application, 

especially considering that the position of the determinant anomaly is not affected much 

by the moment direction. 

There were no independent data available to evaluate the moment magnitudes of the 

buried dipoles. Consequently, we chose to adjust the moment magnitude of the central 

dipole so that the peak value of the modeled determinant (fig. 11) matched the peak value 

of the measured determinant (fig. 10). The moment magnitudes of the flanking dipoles 

were similarly adjusted to match the flanks of their corresponding anomalies (their peaks 

were beyond the high-density grid boundaries). It is again encouraging that the resulting 

moment magnitudes were of reasonable values for all of the targets. 

41
Map views of the collocated TMGS data (physical) and modeled results for the highdensity grid can be seen in figures 12 and 13. The similarities between the physical and 

modeled results are striking, particularly of the central 60-mm UXO (ID 9). In addition, 

the dipole moments are reasonable. Values around the perimeter of the physical data, 

while reasonable, could be subject to significant edge effects. Therefore, the fact that any 

of them have similarities to the model gives further confidence in the TMGS. The origin 

of a huge negative anomaly in the southeast corner of the physical grid is not altogether 

clear. However, it is likely due to a real source, possibly clutter, because edge effects 

around other parts of the grid do not lead to such intense anomalies. 

Figure 12. Determinant plot of physical data using the same color scale as the 

modeled data seen in figure 13. The UXO IDs correspond to the data presented in 

table 11. 

42
Figure 13. Determinant plot of modeled data using the same color scale as the 

physical data seen in figure 11. The UXO IDs correspond to the data presented in 

table 11. 

Modeled values are plotted against physical values in figure 14, and table 12 shows 

statistics for the residual of the modeled and physical data. Essentially, these statistics 

represent the deviation of the model values from the ideal value represented by the 

fuchsia line drawn in the figure 14 cross plot. These statistics are skewed both by edge 

effects (as described in "Gradient Collocation") and by the fact that no attempt was made 

to model the clutter in the southeast corner. 

43
Figure 14. Cross plot of modeled versus physical magnetic tensor determinant 

data. The statistics for the deviation of the model from the ideal data line drawn 

in fuchsia can be seen in table 12. This plot is skewed by edge effects and clutter 

in the southeast corner. 

Table 12. Statistics for the model and physical 

data residual. These statistics are skewed by edge 

effects and clutter in the southeast corner. 

Modeling residual 

statistics 

Geometric mean 95.15 

Standard error 6.75 

Standard deviation 572.39 

Mean –154.93 

Median –64.87 

Mode 15.5 

44
Conclusions 

The similarity of the physical (measured) 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 physical 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 analogous to a keyed lock—if any tumbler is out of place, the lock will not open. This 

observation is well illustrated in figure 7, 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 the Colorado School of Mines for providing consultation. 

45
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46
Appendix A – Fortran routine called s_ab2h.f that applies the bin-size, 

axis polynomial, orthogonality, and attitude coefficients to basic TMGS 

analog and bin data. 

C 

C________________________________________________________________ 

C 

C SUBROUTINE S _ A B 2 H 

C________________________________________________________________ 

C 

C SUBROUTINE S_AB2H RETURNS THE MAGNETIC FIELD COMPONENTS AND A 

C GRADIENT TENSOR IN A FIXED REFERENCE FRAME USING DATA AND 

C COEFFICIENTS FROM THE USGS TMGS PROTOTYPE SYSTEM. 

C 

C THIS SUBROUTINE IS DESIGNED PRIMARILY TO SHOW HOW THE 

C COEFFICIENTS DERIVED FROM THE YUMA, 13MAR03 SPIN CALIBRATION

C ARE TO BE APPLIED TO THE RAW TMGS DATA. THE PRELIMINARY 

C COEFFICIENTS MAY BE FOUND IN AN ASCII TEXT FILE CALLED: 

C 

C "tmgs_coefs_n0430_yuma_prelim.txt".

C 

C 

C INPUT ARGUMENTS (SIGNALS)

C VANA - REAL*8. ARRAY OF DIMENSIONS (3,4) CONTAINING THE

C ANALOG FIELD MEASUREMENTS <VOLTS> FROM THE 3 AXES OF 

C THE 4 MAGNETOMETERS. ROWS 1,2,3 CORRESPOND TO THE

C X,Y,Z AXES AND COLUMNS 1,2,3,4 CORRESPOND TO THE

C MAGNETOMETER NUMBERS. IT IS EXPECTED THAT THESE 

C NUMBERS WILL HAVE BEEN CORRECTED FOR SPIKES AND 

C NOISE, AND THAT ANALOG FILTER RESPONSES WILL HAVE

C BEEN DECONVOLVED (PARTICULARLY IMPORTANT IMMEDIATELY

C AFTER A BIN STEP.) PRIOR TO CALLING THIS SUBROUTINE.

C TYPICALLY, THESE SIGNALS WILL RANGE FROM +-5 BUT MAY

C RAIL AT +-6.55 VOLTS. THESE SIGNALS TRANSLATE TO 

C MAGNETIC FIELD NOMINALLY AT ABOUT 100 nT/VOLT (SEE

C ARGUMENT CANA BELOW).

C VBIN - REAL*8. ARRAY OF DIMENSIONS (3,4) CONTAINING THE

C BIN NUMBERS <BIN#> FROM THE 3 AXES OF THE 4 

C MAGNETOMETERS. ROWS 1,2,3 CORRESPOND TO X,Y,Z AXES

C AND COLUMNS 1,2,3,4 CORRESPOND TO THE MAGNETOMETER

C NUMBERS. IT IS POSSIBLE TO HAVE CORRECTED THESE 

C NUMBERS FOR BIN-CURRENT VARIANCES AND THEREFORE THEY 

C ARE NOT NECESSARILY INTEGERS. TYPICALLY, BIN

C NUMBERS WILL RANGE FROM +-150 BUT MAY RAIL AT +-255. 

C BIN NUMBERS INDICATE THE AMOUNT OF OFFSET IN THE 

C ANALOG FIELD MEASUREMENT VOLTAGE AND TRANSLATE TO 

C VOLTS NOMINALLY AT ABOUT 5.0 VOLTS/BIN# FOR MAGS 1

C AND 2 AND 3.3 VOLTS/BIN FOR MAGS 3 AND 4. (SEE

C ARGUMENT CBIN BELOW).

C 

C INPUT ARGUMENTS (COEFFICIENTS)

C CBIN - REAL*8. ARRAY OF DIMENSIONS (3,4) CONTAINING THE

C AVERAGE BIN SIZES (S) (VOLTS/BIN#). ROWS 1,2,3

C CORRESPOND TO X,Y,Z AXES AND COLUMNS 1,2,3,4

C CORRESPOND TO THE MAGNETOMETER NUMBERS. TYPICALLY, 

47
C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

CANA 

FOR MAGS 1 AND 2, CBIN IS ABOUT 5.0 AND

FOR MAGS 3 AND 4 IT'S ABOUT 3.3 VOLTS/BIN#.

- REAL*8. ARRAY OF DIMENSIONS (4,3,4) CONTAINING THE

POLYNOMIAL COEFFICIENTS (A,B,C,D) THAT TRANSLATE

ANALOG VOLTAGE TO MAGNETIC FIELD <nT/VOLT^3,

nT/VOLT^2, nT/VOLT, nT>. ROWS 1,2,3,4 CORRESPOND TO

THE 3RD ORDER, 2ND ORDER, LINEAR, AND CONSTANT TERMS

RESPECTIVELY OF THE POLYNOMIAL. COLUMNS 1,2,3

CORRESPOND TO X,Y,Z AXES AND LEVELS 1,2,3,4

CORRESPOND TO THE MAGNETOMETER NUMBERS. TYPICALLY,

THE 3RD ORDER TERM IS EXTREMELY SMALL, THE 2ND

ORDER TERM IS VERY SMALL, THE LINEAR TERM IS NEAR

100, AND THE CONSTANT TERM IS SMALL. THESE 

COEFFICIENTS ARE AUGMENTED BY THE SCALE DIVISOR THAT 

C 

C 

C 

C 

SCALE 

WAS APPLIED IN THEIR DERIVATION (SEE ARGUMENT

SCALE).

- REAL*8. SCALAR NUMBER (SCALE) THAT WAS USED DURING

DERIVATION OF THE 3RD-ORDER-POLYNOMIAL COEFFICIENTS 

C DESCRIBED IN ARGUMENT CANA. THIS SUBROUTINE WILL 

C CORRECTLY APPLY SCALE TO THE COEFFICIENTS. BUT IF 

C IT IS DESIRED TO APPLY IT BEFORE THE SUBROUTINE 

C 

C 

C 

C 

C 

C 

C 

C 

C 

CORT 

CALL, SIMPLY DIVIDE THE 3RD-ORDER TERM BY SCALE^2,

DIVIDE THE SECOND ORDER TERM BY SCALE, AND MULTIPLY

THE CONSTANT TERM BY SCALE; THEN SET SCALE, ITSELF,

EQUAL TO 1.

- REAL*8. ARRAY OF DIMENSIONS (3,4) CONTAINING THE

ORTHOGONALITY CORRECTION ANGLES (DEGREES). ROWS 

1,2,3 CORRESPOND TO ANGLES (ALP), (BET), (GAM), AND

COLUMNS 1,2,3,4 CORRESPOND TO THE MAGNETOMETER

NUMBERS. ALPHA IS THE ANGLE BETWEEN THE X AND Y 

C AXIS. BETA IS THE ANGLE BETWEEN THE Y AND Z AXIS. 

C 

C 

C 

AND, GAMMA IS THE ANGLE BETWEEN THE X AND Z AXIS.

TYPICALLY, EACH OF THESE ANGLES VARIES FROM 88 TO 92

DEGREES. 

C 

C 

C 

C 

C 

CATT - REAL*8. ARRAY OF DIMENSIONS (3,4) CONTAINING THE

ATTITUDES (DEGREES) OF THE MAGS RELATIVE TO MAG1.

ROWS 1,2,3 CORRESPOND TO ANGLES (DEL), (EPS), (ZET),

AND COLUMNS 1,2,3,4 CORRESPOND TO THE MAGNETOMETER

NUMBERS. IN TRANSFORMING FROM EACH MAGNETOMETER'S 

C 

C 

PHYSICAL ATTITUDE TO THE MAG1 ATTITUDE, THERE ARE 3

ROTATIONS. THE POSITIVE SENSE IS ALWAYS CLOCKWISE 

C ABOUT THE ROTATION AXIS AS VIEWED FROM THE ORIGIN. 

C DELTA IS THE FIRST ROTATION ANGLE AND PROCEEDS ABOUT 

C THE Z''' AXIS. EPSILON IS THE SECOND ROTATION ANGLE 

C AND PROCEEDS ABOUT THE Y'' AXIS. ZETA IS THE THIRD 

C ROTATION ANGLE AND PROCEEDS ABOUT THE Z' AXIS. 

C 

C 

C 

THEREFORE, THE UNPRIMED SYSTEM IS THE MAG1 ATTITUDE.

TYPICALLY, EACH OF THESE ANGLES VARIES BY +-2

DEGREES ABOUT THE IDEAL. THE IDEAL ANGLES ARE AS 

C FOLLOWS: 

C 

C ROTATION MAG1 MAG2 MAG3 MAG4 

C DEL 0 0 +120 -120 

C EPS 0 -109.471 -109.471 -109.471 

C ZET 0 180 +60 -60 

C 

C 

48
C OUTPUT ARGUMENTS: 

C 

C 

C 

C 

HFLD - REAL*8. ARRAY OF DIMENSIONS (3,4) CONTAINING THE

MAGNETIC FIELD COMPONENTS (nT). ROWS 1,2,3

CORRESPOND TO X,Y,Z AXES AND COLUMNS 1,2,3,4

CORRESPOND TO THE MAGNETOMETER NUMBERS. THE FIELDS 

C 

C 

ARE CALIBRATED, THE AXES ARE ORTHOGONAL, AND THE

MAGNETOMETERS ARE IN THE MAG1 COMMON REFERENCE 

C FRAME. THE ARRAY IS LAYED OUT PICTORIALLY AS 

C FOLLOWS: 

C 

C X1 X2 X3 X4 

C Y1 Y2 Y3 Y4 

C Z1 Z2 Z3 Z4 

C 

C 

C 

C 

GTSR - REAL*8. ARRAY OF DIMENSIONS (3,3) CONTAINING THE

MAGNETIC GRADIENT TENSOR (nT/m) DERIVED FROM HFLD,

ASSUMING A TETRAHEDRAL CONFIGURATION OF THE 4 

C MAGNETOMETERS WITH 0.97 METERS DISTANCE BETWEEN 

C 

C 

CENTERS. WITH 4 SENSORS, A DIRECT DERIVATION OF

EACH OF THE 9 COMPONENTS OF THE TENSOR IS AVAILABLE. 

C 

C 

C 

C 

C 

C 

C 

HOWEVER, IF A) THE FIELD IS STATIC, B) ALL OF THE

GRADIENTS ARE CONSTANT WITHIN THE VOLUME OF TESSA,

AND C) THERE IS NO MEASUREMENT NOISE, THEN THE

TENSOR WILL BE TRACELESS AND SYMMETRIC, HAVING ONLY

5 INDEPENDENT COMPONENTS. THEREFORE, TO THE DEGREE

THAT IT IS NOT TRACELESS AND SYMMETRIC, ONE OR MORE

OF THESE CONDITIONS HAS NOT BEEN REACHED. THE ARRAY 

C IS LAYED OUT PICTORIALLY AS FOLLOWS: 

C 

C 

C 

C 

C 

Gxx Gxy Gxz

Gyx Gyy Gyz

Gzx Gzy Gzz 

C WHERE: 

C 

C 

C 

Gij = dHi/dj = gtsr(i,j) 

C 

C SUBROUTINE S_AB2H WRITTEN BY ROB BRACKEN, USGS.

C FORTRAN 77, HP FORTRAN/9000, HP-UX RELEASE 11.0

C VERSION 1.0, 20030430 ( ORIGINAL CODE ).

C 

C 

subroutine s_ab2h(vana,vbin, cbin,cana,scale,cort,catt

& ,hfld,gtsr )

C 

C 

C DECLARATIONS 

C 

C INPUT ARGUMENTS (SIGNALS)

real*8 vana(3,4),vbin(3,4)

C 

C INPUT ARGUMENTS (COEFFICIENTS)

real*8 cbin(3,4),cana(4,3,4),scale,cort(3,4),catt(3,4)

C 

C OUTPUT ARGUMENTS (FIELDS AND TENSOR)

real*8 hfld(3,4),gtsr(3,3) 

49
C 

C 

C INTERNAL VARIABLES 

C 

C AXIS VOLTAGE RECONSTRUCTION 

C 

real*8 va(3,4) 

C FIELD CORRECTION 

C 

real*8 hf(3,4) 

C ORTHOGONALITY CORRECTION 

real*8 hfl(3,4)

real*8 alp,bet,gam

real*8 calp,salp, cbet, cgam

real*8 coseta 

C 

C ATTITUDE ROTATION TO MAG1 REFERENCE FRAME 

C 

real*8 del,eps,zet

real*8 cdel,sdel, ceps,seps, czet,szet 

C GRADIENT TENSOR CALCULATIONS 

C 

real*8 edge, face, cent

real*8 edge2,face3,cent4 

C 

C RECONSTRUCT AXIS VOLTAGE FROM ANALOG VOLTAGE AND BIN NUMBER 

C 

do k=1,4

do j=1,3

va(j,k)=

& vana(j,k)

& +vbin(j,k) * cbin(j,k)

enddo 

enddo 

C 

C 

C CONVERT AXIS VOLTAGE TO CORRECTED MAGNETIC FIELD (nT)

C 

do k=1,4

do j=1,3

hf(j,k)=

& 1 * cana(4,j,k) * scale

& +va(j,k) * cana(3,j,k) * 1 

& +va(j,k)**2 * cana(2,j,k) / scale

& +va(j,k)**3 * cana(1,j,k) / scale**2

enddo 

enddo 

C 

C 

C APPLY ORTHOGONALITY CORRECTION 

C 

do k=1,4

C 

C X TO Y ANGLE 

alp=cort(1,k)

calp=dcosd(alp)

salp=dsind(alp) 

50
C 

C Y TO Z ANGLE 

bet=cort(2,k)

cbet=dcosd(bet)

C 

C X TO Z ANGLE 

gam=cort(3,k)

cgam=dcosd(gam)

C 

C COSINE OF THE ANGLE BETWEEN Z AND Z-PRIME 

coseta=dsqrt(1.d0-cbet**2-cgam**2)

C 

C X-AXIS CANNOT "MOVE" 

hfl(1,k)=

& hf(1,k)

C 

C Y-AXIS CAN "MOVE" ONLY IN THE X-Y PLANE 

hfl(2,k)=

& ( -hf(1,k)*( calp )

& +hf(2,k) ) / salp

C 

C Z-AXIS CAN "MOVE" ANYWHERE 

hfl(3,k)=

& ( hf(1,k)*(cbet*calp/salp-cgam)

& -hf(2,k)*(cbet /salp )

& +hf(3,k) ) / coseta

C 

enddo 

C 

C 

C APPLY ATTITUDE XFRM (ROTATION) TO COMMON (MAG1) REFERENCE FRAME

C 

do k=1,4

C 

C ROTATION 1 IS ABOUT Z''' 

del=catt(1,k)

cdel=dcosd(del)

sdel=dsind(del)

C 

C ROTATION 2 IS ABOUT Y'' 

eps=catt(2,k)

ceps=dcosd(eps)

seps=dsind(eps)

C 

C ROTATION 3 IS ABOUT Z' 

zet=catt(3,k)

czet=dcosd(zet)

szet=dsind(zet)

C 

C FIND THE ROTATED COMPONENTS 

hfld(1,k)=

& hfl(1,k)*( czet*ceps*cdel-szet*sdel )

& +hfl(2,k)*( czet*ceps*sdel+szet*cdel )

& +hfl(3,k)*(-czet*seps )

hfld(2,k)=

& hfl(1,k)*(-szet*ceps*cdel-czet*sdel )

& +hfl(2,k)*(-szet*ceps*sdel+czet*cdel ) 

51
C 

 & +hfl(3,k)*( szet*seps )

hfld(3,k)=

& 

& 

& 

enddo 

hfl(1,k)*(

+hfl(2,k)*(

+hfl(3,k)*( 

seps*cdel

seps*sdel

ceps 

)

)

) 

C 

C 

C FIND GRAD TENSOR WHERE: gtsr(i,j) = Gij = dHi/dj 

C Note: With 4 sensors, a direct measurement of all 9

C components of the tensor is available. However, if

C a) the field is static, b) all of the gradients are

C constant within the volume of TESSA, and c) there is

C no measurement noise, then the tensor will be

C traceless and symmetric, having only 5 independent

C components. Therefore, to the degree that it is not

C traceless and symmetric, one or more of these

C conditions has not been reached. 

c 

call s_h2g(hfld, gtsr)

C 

C 

C EXIT PROCEDURE 

C 

990 return 

end 

52