JOINT AND COOPERATIVE INVERSION OF MAGNETIC AND TIME DOMAINELECTROMAGNETIC DATA FOR THE CHARACTERIZATION OF UXO

Leonard R. Pasion, Stephen D. Billings, and Douglas W. OldenburgUBC - Geophysical Inversion Facility

Department of Earth and Ocean Sciences, University of British ColumbiaVancouver, B.C., V6T 1Z4, CANADA

email: pasion@geop.ubc.ca

Abstract

Magnetics and electromagnetic surveys are the primary techniques used for UXO remediation projects.Magnetometry is a valuable geophysical tool for UXO detection due to ease of data acquisition and its abilityto detect relatively deep targets. However, magnetics data can have large false alarm rates due to geologicalnoise, and there is an inherent non-uniqueness when trying to determine the orientation, size and shape ofa target. Electromagnetic surveys, on the other hand, are relatively immune to geologic noise and are morediagnostic for target shape and size but have a reduced depth of investigation. In this paper we aim to im-prove discrimination ability by developing an interpretation method that takes advantage of the strengths ofboth techniques. We consider two different approaches to the problem: (1) Interpreting the data sets cooper-atively, and (2) Interpreting the data sets jointly. For cooperative inversion information from the inversion ofone data set is used as a constraint for inverting another data set. In joint inversion, target model parameterscommon to the forward solution of both types of data are identified and the model parameters from all thesurvey data are recovered simultaneously. We compare the confidence with which we can discriminate UXOfrom non-UXO targets when applying these different approaches to results from individual inversions. Inthis paper we focus on the details of the joint and cooperative inversion methodologies. Examples of the ap-plication of the methodology to TEM and magnetics data sets collected at the former Fort Ord in Californiaare presented. This work is funded in part by the U.S. Army Engineer Research and Development Centerand the Army Research Office.

Introduction

Electromagnetic and magnetic surveys are the standard geophysical techniques used for UXO reme-diation. Electromagnetic detection of a buried target is accomplished by illuminating the subsurface with atime varying primary field. If the buried target is conductive, eddy currents will be induced in the target, andsubsequently decay. These currents produce a secondary magnetic field which is then sensed by a receivercoil at the surface. Magnetometry is a passive detection system. The high magnetic susceptibility of a fer-rous target causes distortions to the Earth’s field which are measured by a magnetometer. Electromagneticsand magnetometry have proven to be successful in detecting UXO in recent UXO remediation projects andUXO technology demonstrations. However, the end goal of geophysical data interpretation is not simply thedetection of UXO, but rather the discrimination of anomalies originating from intact UXO from anomaliesdue to exploded ordnance and other metallic scrap. Current field production data interpretation techniquesare usually limited to gridding of data followed by defining a threshold level and visual inspection to selecttargets and therefore do not have any discrimination or identification capabilities.

Several data processing techniques for electromagnetics and magnetics survey data have been devel-oped that demonstrate an ability to discriminate between UXO and non-UXO items. These techniques relyon recovering the parameters in a realistic physics-based model that can reproduce the observed anomaly.The dipolar nature of the electromagnetic responses of compact metallic objects measured with sensor/target

1

geometries typical for UXO surveys has lead to a number of techniques for estimating the elements of themagnetic polarization tensor that define the induced dipole strength. The magnetic polarization tensor’scomponents are functions of the size, shape, location, orientation and material properties of the buried tar-get of interest and therefore provide a model vector from which the target characteristics can be inferred.Numerous examples of the application of this methodology to UXO data exist (for examples see blah, blah,blah). The accuracy with which the polarization tensor can be recovered depends on the noise levels ofthe induction sensor, the amount of geologic noise in the inverted data, and accurate accounting of surveyparameters such as sensor orientation and location.

As an illustration of the difficulty of UXO discrimination in conditions of lower signal-to-noise andpoor spatial coverage, consider the recovery of the polarization tensor components from a pair of syntheticdata sets generated from a Stokes mortar. Figure 1(a) is a photo of a Stokes mortar and the measured dipoledecay parameters of the mortar are listed in Figure 1(b). The TEM response is computed for a Stokes mortar

(a) Photo of a Stokes Mortar

k1 43.9α1 0.02β1 0.73γ1 9.1

k2 4.9α2 0.001β2 1.09γ2 10.8

(b) Polarization tensor pa-rameters used for forwardmodelling

60cm

100cm

1mV

(c) Dipole response at 1.105 ms of a Stokes Mortarlocated at depths of 60 and 100 cm.

Figure 1: Photo and decay constants of a particular Stokes Mortar and the TEM responses at 1.105 ms and measuredparallel to the length of the Stokes Mortar.

buried at depths of 60 cm and 100 cm and oriented 30 degrees from horizontal. The TEM response at 1.105ms measured on a line parallel to the strike of the mortar are compared in Figure 1(c).

We assume that the TEM sensor has a noise floor of 0.5mv. When the mortar is buried at a depth of60 cm, the large signal-to-noise ratio results in an accurate recovery of the dipole parameters (Table 1), andan accurate prediction of the observed data (Figures 2(a) and 2(b)). The signal from a mortar buried at100 cm is much weaker and, as Figure 1(c) demonstrated, a significant portion of the data lies within thenoise level of the sensor. Inversion of these data result in recovered model parameters that, when modelled,accurately reproduce the data (Figure 3(a middle panels and Figure 3(b) green line) but not the parametersof the Stokes mortar (Table 1). Location and orientation were not correctly recovered. With the data locatedon the right side of the survey obscured by the sensor noise, the inversion attempted to place the location ofthe target at the center of the data peak.

2

Location Depth Azimuth DipX,Y (m) (m) (degrees) (degrees) k1 k2 k1/k2 Interpretation

A (0,0) 1.0 90.0 60.0 43.9 4.9 8.96 rod-likeB (-0.01,0.00) 0.60 90.0 59.9 43.28 4.91 8.81 rod-likeC (-0.38,0.05) 0.86 -16.7 62.7 7.86 11.57 0.68 plate-likeD ( 0.01,0.00) 0.99 89.3 62.3 39.77 6.08 6.54 rod-like

Table 1: Comparison of the recovered model parameters for: (B) a stokes mortar at a depth of 60 cm, (C) a Stokesmortar at a depth of 100 cm, unconstrained location, and (D) a Stokes mortar at a depth of 100 cm, withlocation constrained to ±5 cm of the real location. Row (A) lists the correct parameters.

(a) Synthetically generated ”observed” data (left) and pre-dicted (right) data from parameters recovered from an inver-sion.

(b) Observed data at t = 0.180 msand t = 1.105 ms, and along linesx = -0.5 m and x = 0.5 m.

Figure 2: Inversion of TEM data for a Stokes mortar at a depth of 60 cm. The signal-to-noise ratio is large, allowingfor accurate parameter recovery without constraints placed on the location.

3

(a) Comparison of observed data and data predicted from therecovered polarization tensor. Stokes location is indicatedby a star. When the location is unconstrained the inversionrecovers a location, indicated by the circle, coincident withthe peak of the signal (middle panels).

(b) Observed data at t = 0.180 ms and t = 1.105ms, and along lines x = -0.5 m and x = 0.5m. The green lines represent predicted data foran inversion with location unconstrained. Bluelines represent data predicted for an inversionwith location constrained.

Figure 3: Comparison of the inversion of a Stokes mortar at a depth of 100 cm. Without constraints on the targetlocation, it is possible to recover parameters unrepresentative of the target yet able to reproduce the observeddata.

The non-uniqueness demonstrated in this unsuccessful inversion can be reduced by accurate knowledgeof the target location. If we constrain the data to within ±5 cm of the real target location the recoveredparameters and location of the target are successfully obtained (Figure 2(c) right panel).

The magnetostatic secondary field response of typical UXO can also be well approximated with adipole. The magnetostatic polarization tensor for the dipole induced in a magnetic spheroid is well known(for example McFee (1989)) and enables one to forward model the magnetic dipole response of a spheroidof arbitrary size, shape, orientation, and location. However, inverting magnetics data directly for the size,shape, and orientation of the best fitting spheroid is not possible due to inherent non-uniqueness (see Billingset al. (2002) for further explanation). That is, for a spheroid at a particular orientation there exists an infinitenumber of spheroids that could produce the same dipole moment (Figure (3)). Ordnance discrimination us-ing magnetostatic data has been achieved by recognizing that intact ordnance tend to become demagnetizedafter impact while shrapnel tend to have a component of remnant magnetization. A level of discriminationis achieved by classifying targets as scrap when the direction of magnetization deviates from the direction ofEarth’s field by a large amount (Billings et al., 2002; Nelson et al., 1998; Lathrop et al., 1999). Billings et al.(2002) demonstrated identification ability when the different ordnance types expected in the survey area areknown. A ranking scheme was developed by assuming that a particular target type is more likely when lessremnant magnetization is required to fit the measured dipole moment.

To summarize, it is not possible to get shape information from magnetometer data alone, and TEMdata can also have difficulty in generalizing this information when data are incomplete and/or noisy. Thismotivates the research here to combine information from these two surveys.

It has been recognized that the performance of EM interpretation algorithms improve when locationinformation from magnetics is used as a constraint. Cooperative inversion has been applied to interpretmagnetometry data and single time channel TEM data (for example Nelson and McDonald (1999)) as well

4

(a) Spheroid dimensions that can produce the same in-duced dipole.

(b) Spheroid aspect ratio and angle from the Earth’s fieldthat can produce the same induced dipole. Constrainingthe angle at which the ordnance of the target lies willreduce the ambiguity of the spheroid solution.

Diameter Aspect Angle(mm) Ratio (degrees)

A 82 5 40.5B 138 5 84.0C 138 2.82 57.2D 327 0.06 53.4E 327 0.31 33.4

(c) Spheroid dimensions, and their angles relative to the Earth’s field, that produce the samedipole moment as a 105 mm projectile at 45o inclination.

Figure 4: Spheroid dimensions that can produce the same dipole as a 105 mm shell at 45 degrees inclination to theEarth’s field

as magnetometry data and multi-frequency EM data (for example Collins et al. (2001)). In these approachesthe ability of magnetometry to accurately determine the location of a target is exploited by fixing the locationof the target.

The objective of this research is to improve our ability to discriminate between UXO and non-UXOtargets by developing interpretation methods which can take advantage of the strengths as well as overcomesome of the shortcomings of both techniques. In this paper we consider two different approaches to in-terpreting multiple data sets: (1) Interpreting the data sets cooperatively, and (2) Interpreting the data setsjointly. In both cases the excellent locating ability of the magnetics method is used to stabilize the inver-sion of TEM data, and the ability of TEM to determine the orientation of a buried target is used to reducemagnetics’ implicit non-uniqueness such that the target shape and size can be inferred from the magnetics’data. Examples of methodology will be given for multiple time channel time domain electromagnetics andmagnetics data sets.

Dipole modelling of TEM and Magnetics Data

In order to invert measured TEM and magnetics data for the physical parameters of the target, it isnecessary to have a forward model to describe the TEM and magnetics response for a buried metallic object.

5

We restrict our forward model to axi-symmetric metallic targets, since this geometric subset adequatelydescribes all UXO and much of the buried metallic scrap encountered in a remediation survey. We alsoassume negligible contribution of the host medium to the measured signal.

Magnetics Forward Modelling

Spheroids have been used to approximate the magnetostatic response of ordnance by several au-thors (Butler et al., 1998; McFee, 1989; Altshuler, 1996). The magnetic field induced in a spheroid bythe Earth’s field can be decomposed into a multipole expansion. The dipole term of the field is

bS =µo

4πr3m ·

(3r̂r̂− ¯̄I

)(1)

where r̂ is the unit vector from the field measurement point and the spheroid center, ¯̄I, and m is the induceddipole moment. The quadrapole term of the multipole expansion is zero due to the symmetry of the spheroid.The next non-zero term is the octopole moment which, for distances from the target that exceed a few bodylengths, is negligible. Therefore, for many of the geometries encountered in UXO surveys, the response ofa spheroid is accurately modelled by the dipole moment. The induced dipole moment can be written as

mMag =V

µoATFMag A · bpMag (2)

where V is the spheroid volume, A is the Euler rotation tensor, bpMag is the Earth’s field, and FMag isthe magnetostatic polarization tensor. The spheroid shape information is contained in the magnetostaticpolarization tensor. We refer the reader to McFee (1989) for the functional relationship between the themagnetostatic polarization tensor and the aspect ratio e and spheroid diameter a .

Time Domain EM Forward Modelling

In the time domain electromagnetic induction method a time varying magnetic field is used to illumi-nate a conducting target. This primary field induces surface currents on the target which then generate asecondary magnetic field that can be sensed above ground. With time, the surface currents diffuse inward,and the observed secondary field consequently decays. The rate of decay, and the spatial behaviour of thesecondary field, are determined by the target’s conductivity, magnetic permeability, shape, and size.

The electromagnetic response of the target will be primarily dipolar (Casey and Baertlein, 1999;Grimm et al., 1997) for the target/sensor geometries of UXO surveys. The induced dipole has the sameform as the magnetostatic dipole of equation (2)

mEM (t) =V

µoATFEM (t) A · bPEM (3)

where A is the Euler rotation tensor, bpEM is the primary field generated by the sensor transmitter loop,and FEM is the magnetostatic polarization tensor. The target’s shape, size, and material properties (i.e.conductivity and magnetic susceptibility) are contained in FEM . The primary field in the TEM case (bPEM )will vary with transmitter/receiver location. In a typical survey TEM soundings will be acquired at a numberof different locations at the surface and the target will have been illluminated from several angles. As aresult the inherent ambiguity of the magnetic method is avoided. The polarization tensor FEM (t) for anaxi-symmetric target has the form

FEM (t) =

L2 (t) 0 0

0 L2 (t) 00 0 L1 (t)

(4)

6

The analytic expressions for the time domain response are restricted to a metallic sphere, and even an ex-pression for a permeable and conducting non-spherical axi-symmetric body is not available. Our approach,therefore, is to use an approximate forward model that can adequately reproduce the measured electromag-netic anomaly with minimal computational effort. In Pasion and Oldenburg (2001a) the following form forpolarization tensor elements was suggested:

Li = ki (t+ αi)−βi exp (−t/γi) (5)

The validity of this reduced modelling was verified through a series of empirical tests (Pasion and Oldenburg,2001b).

Inversion Methodology

For this presentation the Bayesian framework is used to formulate the inverse problem. The solution tothe inverse problem is the combination of the information known about the model parameters m prior to theexperiment and the ability of our physical model F to reproduce the experimental data. The prior informa-tion is represented as the probability distribution p (m). The ability to reproduce the experimental data dobsis described in the conditional probability density of the experimental data p (dobs|m). The prior and theconditional probability density (also called the likelihood function) of the experimental data are combinedvia Bayes theorem to form the a-posteriori conditional probability density p (m|dobs) of the model:

p (m|dobs) =p (m) p (dobs|m)

p (dobs)(6)

where p (dobs) is the marginal probability density of the experimental data. Equation (6) shows how theprior and the experiment data are combined, and therefore is a mathematical expression of the inversionphilosophy. That is, if we can regard the prior p (m) as the probability density assigned to m prior toexperiment, then the a-posteriori conditional probability density p (m|dobs) is the probability density weascribe to m after collecting the data. The a-posteriori conditional probability density encapsulates all theinformation we have on the model parameters and the model that maximizes it is usually regarded as thesolution to the inverse problem.

Characterizing Data Statistics

The likelihood function p (dobs|m) gives an indication of the misfit between and predicted and observeddata, and therefore depends on both the measurement errors and modelling errors. For this work we willassume that the errors follow a Gaussian distribution distribution:

p (dobs|m) =

√(2π)−N

detVdexp

[−1

2(dobs − F [m])T V −1

d (dobs − F [m])

](7)

where F [m] is the forward modelling operator and Vd is the covariance matrix of the data errors. Thisassumption is motivated by the ease in which the resulting inverse problem can be formulated and by thecentral limit theorem’s assertion that as the number of data approaches infinity, the distribution of errors ofthe data approaches the normal distribution. We do recognize that real field data have errors unaccounted forin the forward modelling operator (for example inaccurate sensor positioning, ”spikes” in the data, sensordrift) that can lead to non-Gaussian error distributions. Incorrect characterization of the data statistics canbias the values of the recovered parameters and also make the parameter variance analysis invalid (Billingset al., 2003).

Representing the Prior Information

We can incorporate information about the model parameters through the specification of the priorp (m).There are two variants which are important for our work:

7

Case I: Bounds on the parameters are known. Here we are provided with the maximum and minimumvalues that a parameter can achieve. Consider an individual parameter mj and bounds mL

j and mUj . In the

absence of knowing the true probability, we use a probability that is uniform on the interval[mLj ,m

Uj

]. The

prior is then

pb (mj) =

{const. if mL

j ≤ mj ≤ mUj ,

0 otherwise.(8)

The joint probability for all of the parameters is

p (m) =

np∏

j=1

pb (mj) (9)

The posterior pdf is thus equal to zero outside the supplied bounds.

Case II: Prior pdf’s are available For some parameters we may have more information. For instancemarginal pdf’s can be obtained from the magnetics inversion. For these parameters we characterize thefunctionals by an exponential:

pg (mj) = c exp (−f (mj − m̄j)) (10)

where the Gaussian prior of mj is centered on a prior model m̄j and its standard deviation σj:

f (mj −mj) =1

2σ2j

(mj − m̄j)2 .

The joint pdf is again obtained by multiplying the individual pdf’s as in equation (9).

Summarizing the Posterior Information

The a-posteriori conditional probability density defines the distribution of models posterior to thecollection of the data. A distribution is commonly characterized by its moments. The posterior mean modelfor the ith parameter mi is calculated by computing the first order moment of the posterior. If the posterioris Gaussian or “bell shaped”, then the mean would be equal to the maximum p (m|d) model. The covariancematrix is the second order moment of the estimate calculated about its mean. Monte Carlo techniques fornumerical evaluation of integrals for the moments of the a posteriori for the inversion of magnetics andTEM UXO data are demonstrated in Pasion et al. (2001).

Here we week a single model and we choose the model which is most likely to occur. We estimate avalue of m that maximizes the log of the a-posteriori conditional probability density

m∗ = maxm{log (p (m|dobs))} (11)

The solution to the inverse problem can be cast as an optimization problem

minimize φ (m) =∑

j

1

2σ2j

(mj − m̄j)2 +

1

2‖V −1/2

d

(dobs −F (m) ‖2

)

subject to mLi ≤ mi ≤ mU

i (12)

where j represents the index of parameters whose Gaussian pdf’s are known, and i represents model pa-rameters which have an upper and lower bounds. We note that if we have no prior information about theparameters, then the maximum likelihood solution is that which maximizes p (dobs|m), that is

minimize φ (m) =1

2‖V −1/2

d

(dobs −F (m)

)‖2 (13)

8

(a) Comparison of Observed Data and Predicted data for theMagnetics inversion

Parameter Recovered

X (m) 0.01± 0.004Y (m) -0.01± 0.004Z (m) 1.00± 0.005

Moment (Am2) 0.5879± 0.0058Azimuth (degrees) 86.3± 0.5Dip (degrees) -38.5± 0.4

Angle from Earth’sfield (degrees) 48.4± 0.3

(b) Recovered magnetic Dipole and Location

Figure 5: Results from the magnetics inversion step of the cooperative inversion.

Cooperative Inversion of TEM and Magnetics Data

We formulate the cooperative inversion of TEM and Magnetics as a three part procedure. Firstly themagnetic data are inverted to yield the best fit magnetostatic dipole mMag . The location of the dipole andthe variance of the location estimates are used as a priori information in the inversion of the TEM data. Thisresults in an improved recovery of parameter values from which to make TEM discrimination. In additionthe orientation of the item is obtained. This is the information required to obtain shape/size informationfrom the magnetic data.

We demonstrate the cooperative inversion procedure using the Stokes mortar example of Figure 3.The mortar is located at a depth of 100 cm (Z = 1.00 cm) and located at the center of the survey data((X,Y ) = (0, 0) cm). Synthetic Geonics EM63 TEM data was generated. The time channels range from0.180 ms to 25 ms. Two noise components were added to the forward modelled response to make the TEMsynthetic data set more realistic. Firstly a 5% random Gaussian noise was added. Secondly the sensornoise floor was emulated by adding an additional and a Gaussian noise with a standard deviation of 0.5mVa. Synthetic magnetics data was generated by representing the Stokes mortar with a spheroid with aneccentricity of e = 4.5 and radius a = 0.046 (volume = 0.0018m3). A normal error with standard deviationof 2cm was added to the station location of the magnetic measurements. For both the TEM and magneticsdata, stations were separated at 10 cm on lines separated by 50 cm.

1. Inversion of Magnetics Data

The first step of the cooperative inversion is to determine the dipole moment mmag = (mx,my,mz)that produces the best fit to the observed magnetic data, and the location R = (X,Y,Z) of the best fitdipole. We define a model vector m

m = [X,Y,Z,mx,my,mz, b] (14)

where the parameter b is a dc offset that is included to account for regional shifts in the data set. Theparameters are recovered by solving equation 13. Variance estimates of the parameters are obtained by localerror analysis. Figure 5(a) compare the observed synthetic data and response predicted by the recoveredparameters. The recovered parameters are listed in Figure 5(b).

2. Inversion of TEM Data with Location Constraint

The second step is to use the location information from the magnetics inversion to help stabilize theTEM inversion. The inversion methodology outlined in the previous step is applied to TEM data collected

9

over the same target. The objective of this step is to obtain the following 13 model parameter vector

mEM = [X,Y,Z, φ, θ, k1, α1, β1, γ1, k2, α2, β2, γ2] . (15)

where the location (X,Y,Z) is constrained by the recovered magnetics location. We define a uniform priorcentered on the recovered magnetics location and with a width equal to the estimated variance. The TEMdipole parameters are constrained to be positive and have upper bounds that are large enough to allow for thelargest target expected in a typical survey. Figure 6 summarizes the cooperative inversion result. Figure 6(a)compares the data fit, and the table in Figure 6(b) compare the recovered and expected parameters. The

(a) Comparison of Observed Data and Predicted data for theTEM inversion

Parameter Expected RecoveredX (m) 0.00 0.01± 0.00Y (m) 0.00 -0.01± 0.00Z (m) 1.00 0.00± 0.00

Azimuth 90 89.7± 0.9Dip 60 63.2± 0.7

k1 43.9 41.76± 1.54α1 0.02 0.02± 0.04β1 0.73 0.78±0.08γ1 9.1 40.0 (constraint)

k2 4.9 6.40± 1.49α2 0.001 0.01± 0.1β2 1.09 0.96± 0.34γ2 10.8 8.89 ± 21.37

(b) Recovered magnetic Dipole and Location

Figure 6: Results from the TEM inversion step of the cooperative inversion.

location and orientation have been accurately recovered. The γ parameters is poorly recovered becauseit is constrained by the late time response which, for this example, is contaminated by the noise. The kparameters are accurately recovered and their values would be appropriate for characterizing the target.

3. Estimation of Shape and Size from magnetics Data

In The functional relationship of the ordnance orientation and the induced dipole moment is

m =(m⊥,m‖

)T=Boπea

3

6µo

[(F2 − F1) cos θ sin θF2 cos2 θ + F1 sin2 θ

](16)

where we choose a coordinate with axes paralell and perpendicular to the Earth’s field. Without any addi-tional information there are two known components of the dipole (magnitude and angle from Earth’s fieldθ) and three unknown parameters of the spheroid (a, e, θ). We need a constraint on the orientation in orderto uniquely determine the demagnetization factors F2 and F1, and therefore the aspect ratio and size of thebest fit spheroid.

When the orientation obtained from the TEM inversion (azimuth = 89.7o and dip = 63.2o) the recoveredspheroid is then 80 mm in diameter and has an aspect ratio of 5.5 (i.e. length = 44 cm, volume = 0.0015 m3).The spheroid used to forward model the data (i.e. diameter = 90 mm, aspect ratio = 4.5, volume = 0.0018m3) is very slightly longer and skinnier than the recovered spheroid, but approximately the same volume.

10

4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 655

60

65

70

75

80

Aspect ratio

Ang

le b

etw

een

the

sem

i−m

ajor

axi

s an

d E

arth

’s fi

eld

(a) Constraining spheroid size and shape withthe orientation from the TEM inversion.

(b) Comparing the possible dipole moments oriented between 0 and90 degrees from a collection of ordnance.

Figure 7: Using orientation information for constraining the recovered spheroid shape.

Figure 7(a) shows some of the possible spheroids that can generate the magnetic dipole, and how knowledgeof the orientation enables us to select a single spheroid. The solid line represents a suite of possible spheroidsand the dotted line represents the ordnance orientation (relative to the Earth’s field) recovered from the TEMinversion. The above method for determining the dimensions of the object will generally work well in theabsence of any remanent magnetization. When remanence is present a modified method of identificationusing magnetics is to (1) generate a list of possible ordnance, (2) determine the range of dipole momentsthat can be induced in each ordnance by varying the relative angle of the ordnance with the Earth’s field, and(3) find which target requires the least amount of additional magnetization to reproduce the magnetic dipolerecovered in the magnetics inversion. Figure 7(b) shows the possible induced dipole moments for a numberof ordnance. Each ordnance item sweeps out an arc as its orientation is varied. The recovered dipole momentis plotted as a black star. Without knowledge of the orientation of each ordnance, the ordnance items canbe ranked according to distance the recovered moment to the ordnances respective arcs. Knowledge of theordnance orientation reduces each arc to a single point (indicated by a symbol on each arc), thereby refiningour discrimination ability.

Joint Inversion of TEM and Magnetics Data

In both the TEM and magnetostatic forward modelling the response is approximated by the dipoleproduced by a spheroid. Ideally the joint inversion procedure would be to recover the location, orientation,and spheroid properties that can best reproduce the TEM and magnetostatic dipoles. The model vector inthis ideal case would be

m = [X,Y,Z, φ, θ, a, e, µ, σ,mREM ] (17)

where (X,Y,Z) is the location, φ and θ represent the orientation. The spheroid is characterized by a semi-major axis a, an eccentricity e, the magnetic permeability µ, and the conductivity σ. However the TEMforward model do not allow the model parameter to written as an explicit function of the spheroid dimensionsand material properties. Therefore the only parameters common to both the TEM and magnetostatic forwardmodelling will be the location and orientation. Consequently the model parameter vector which we seek torecover in the joint inversion procedure is then

m = [X, Y, Z, φ, θ, F1, F2, k1, α1, β1, γ1, k2, α2, β2, γ2] (18)

11

We invert for the magnetostatic polarization tensor components F1 and F2 (or equivalently the demagnetiza-tion factors) instead of the spheroid eccentricity and size, because the objective function is a much simplerfunction of F1 and F2 than the spheroid dimensions a and e.

The magnetics and electromagnetic surveys are independent geophysical experiments. The applicationof Bayes theorem for independent probability density functions is:

p(

m|dMagobs , dEMobs

)=p (m) p

(dMagobs |m

)p(dEMobs |m

)

p (dobs)(19)

where we have defined a new observed data vector as dobs =(

dMagobs , dEMobs

)T. We again choose to maximize

the log of the a-posteriori

m∗ = maxm{log

(p(

m|dMagobs , dEMobs

))} (20)

= maxm{log (p (m)) + log

(p(

dMagobs |m

))+ log

(p(dEMobs |m

))} (21)

When assuming normally distributed errors in the data, the maximization of equation (21) is equivalent tominimizing the following objective function:

Φ (m) =α‖V magd

−1/2 (FMag [m]− dmagobs

)‖2 + (1− α) ‖V tem

d−1/2 (FTem [m]− dTemobs

)‖2(22)

=α ΦMag (m) + (1− α) ΦEM (m) (23)

subject to constraints. (24)

The parameter α is introduced since we often only have an idea of the relative difference in errors and notthe absolute errors of each data set. The parameter α controls the relative degree to which we fit the misfitof the magnetics data and the TEM. If we know the value of the data errors for both data sets and can use anaccurate value for the data covariance matrices, the expected value for the least squares data misfit is equalto the number of data. The value of α that would make the magnetics and TEM objective functions equal atsolution would be

α =NEM

NEM +NMag. (25)

In general it is not possible to accurately specify the variance of the data and modelling errors of the sensordata. For this paper we estimate the value of the magnetics and TEM data misfit at the solution inverting thedata sets individually, then using the final misfits as estimates. The value of α is then estimated as:

α =ΦEM (mem

∗ )

ΦEM (mem∗ ) + ΦEM (mmag∗ )

. (26)

where mem∗ and mmag

∗ are the models recovered from inversion of TEM and magnetics data sets individually.We recognize that there will be situations where we may have poor estimates for the target misfit of theobjective functions, where a more rigorous exploration of the weighting parameter α is required. Techniquesfor automatically choosing the weighting parameter have been developed in the multi-objective optimizationfield (for example Mathworks (2002)).

The recovered parameters when applying this joint inversion methodology to the synthetic data setare listed in figure 8. As was the case in the cooperative inversion, the low signal-to-noise in the late timechannels do not allow for reliable recovery of the γ parameters.

12

mi Expected RecoveredX (m) 0.00 0.0Y (m) 0.00 0.0Z (m) 1.00 1.0

Azimuth 90 90.0Dip 60 61.6

(a) Location and Orientation

mi Expected Recovered

F1 21667 22435F2 3065 3356

(b) Magnetostatic polarization tensorcomponents multiplied by volume.

mi Expected Recovered

k1 43.9 40.60α1 0.02 0.02β1 0.73 0.78γ1 9.1 40.0 (constraint)

k2 4.9 6.44α2 0.001 0.01β2 1.09 0.97γ2 10.8 8.33

(c) TEM Dipole parameters

Figure 8: Parameters recovered by joint inversion.

Conclusion

In this paper we considered two approaches to interpreting magnetics and TEM datasets: (1) coopera-tive inversion, and (2) joint inversion. Both approaches utilize the ability of magnetics to accurately locateburied targets and the ability of TEM to recover the orientation of the target. Knowledge of the orienta-tion of the target is required in order to uniquely determine the size and shape of the best fit spheroid. Wedemonstrated these techniques on synthetic magnetics and TEM data sets collected over a Stokes mortar.After demonstrating that individual inversions of the simulated data sets provided limited information on theburied target, we showed that both joint and cooperative inversions were able to estimate the size and shapeof the buried target. Although tests of the respective algorithms need to be conducted to assess performancein a field setting, the joint and cooperative inversion techniques have the potential to improve the currentcharacterization and identification ability.

References

T.W. Altshuler. Shape and orientation effects on magnetic signature prediction for unexploded ordnance. InProceedings of the 1996 UXO Forum, 1996.

S.D. Billings, L. R. Pasion, and D. W. Oldenburg. Discrimination and identification of uxo by geophysicalinversion of total-field magnetic data. Technical report, U.S. Army Research and Development Center,Vicksburg, MS, 2002. ERDC/GSL TR-02-16.

S.D. Billings, L.R. Pasion, and D.W. Oldenburg. Discrimination and classification of uxo using magnetom-etry: inversion and error analysis using robust statistics. In Proceedings from SAGEEP 03, 2003.

D.K. Butler, E.R. Cespedes, C.B. Cox, and P.J. Wolfe. Multisensor methods for buried unexploded ordnancedetection, discrimination, and identification. Technical Report T.R. SERDP-98-10, U.S. Army Corps ofEngineers Waterways Experiment Station, 1998.

K. F. Casey and B. A. Baertlein. An Overview of Electromagnetic Methods In Subsurface Detection, chapterThe Magnetic Polarizability Dyadic and Point Symmetry, pages 9–46. Taylor and Francis, 1999.

13

L.M. Collins, Y. Zhang, J. Li, H. Wang, L. Carin, S.J. Hart, S.L. Rose-Pehrsson, H.H. Nelson, and J.R.McDonald. A comparison of the performance of statistical and fuzzy algorithms for unexploded ordnancedetection. IEEE Transactions on Fuzzy Systems, 9(1):17–30, February 2001.

R. E. Grimm, M. W. Blohm, and E. M. Lavely. Uxo characterization using multicomponent, multichanneltime-domain electromagnetic induction. In Proceedings of UXO Forum ’97, 1997.

J.D. Lathrop, H. Shih, and W.M. Wynn. Enhanced clutter rejection with two-parameter magnetic classifi-cation for uxo. In SPIE Conference on Detection and Remediation Technologies for Mines and MinelikeTargets IV, volume 3710, pages 37–51. The International Society for Optical Engineering, 1999.

The Mathworks. Optimization Toolbox For Use with MATLAB, 2nd edition, 2002.

J.E. McFee. Electromagnetic remote sensing: Low frequency electromagnetics. Technical Report 124,Defence Research Establishment Suffield, Ralston, Alberta, January 1989.

H.H. Nelson, T.W. Altshuler, B. Barrow E.M. Rose, J.R. McDonald, and N. Khadr. Magnetic model-ing of uxo and uxo-like targets and comparison to targets measured by mtads. In Proceedings of theUXO/Countermine Forum, 1998.

H.H. Nelson and J.R. McDonald. Target shape classification using the mtads. In Proceedings of UXO Forum’99, 1999.

L. R. Pasion and D. W. Oldenburg. A discrimination algorithm for uxo using time domain electromagnetics.Journal of Engineering and Environmental Geophysics, 6(2):91–102, 2001a.

L. R. Pasion and D. W. Oldenburg. Locating and characterizing unexploded ordnance using time domainelectromagnetic induction. Technical report, U.S. Army Research and Development Center, Vicksburg,MS, 2001b. ERDC/GSL TR-01-10.

L.R. Pasion, S.D. Billings, and D.W. Oldenburg. Inversion of magnetic and electromagnetic data for the dis-crimination of uxo. In Partners in Environmental Technology Technical Symposium and Workshop. Strate-gic Environmental Research and Development Program (SERDP) and Environmental Security Technol-ogy Certification Program (ESTCP), 2001.

14