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INTERIM REPORT 


Advanced Signal Processing for Detailed Site Characterization 

and Target Discrimination 


SERDP Project IVR-1669 


APRIL 2011 


Peter B. Weichman 
BAE Systems 


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Advanced Signal Processing for Detailed Site Characterization and 

Target Discrimination: EMI Model Validation Using NRL TEMTADS 
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Advanced Signal Processing for Detailed Site Characterization 
and Target Discrimination (Project 1669) 

Interim Report: EMI Model Validation Using NRL TEMTADS Data 

Peter B. Weichman 

BAE Systems, Advanced Information Technologies, 

6 New England Executive Park, Burlington, MA 01803 

This report details basic validation of our physics-based EMI models against data collected by the 
NRL TEMTADS system. The data was collected under laboratory-type conditions using artificial 
spheroidal targets. The models are essentially exact for these types of targets, and enable detailed 
comparison of theory and data in support of both model validation and measurement platform 
characterization. 


I. INTRODUCTION 

This document details successful validation of our 
physics-based “mean field” and “early time” approaches 
to modeling of time-domain electromagnetic (TDEM) re¬ 
sponses of compact, highly conducting targets. Specifi¬ 
cally, we apply our methods to the analysis of laboratory- 
style data collected by the NRL TEMTADS system using 
artificial spheroidal targets. The models use the detailed 
system parameters (transmitter and receiver coil posi¬ 
tion, orientation, and geometry; transmitted pulse wave¬ 
form; target position, orientation, geometry; target con¬ 
ductivity and permeability) to generate first principles 
predictions for the measured time-domain voltages. The 
models are designed to be essentially exact for spheroidal 
targets, and, as described in the remainder of this docu¬ 
ment, the remarkable agreement between measurements 
and predictions strongly supports this conclusion. 

The outline of the remainder of this document is as 
follows. Details of the EM theory underlying the models, 
and their numerical implementation, will be detailed else¬ 
where, but a basic overview is given in Sec. II. In Sec. Ill 
the basic parameters of the NRL TEMTADS system are 
detailed. In Sec. IV model predictions are compared with 
TEMTADS data for spherical targets, for which an ex¬ 
act analytic theory also exists (Sec. IV A); prolate (elon¬ 
gated) and oblate (discus-like) spheroidal targets (Sec. 
IV B). Finally, conclusions and directions for future work 
are presented in Sec. V. 


II. MODELING BACKGROUND 

The essence of the TDEM induction measurement is 
sketched in Fig. 1. The transmitter loop current pulse 
generates a magnetic field in the target region. This 
changing applied field, especially as the pulse terminates, 
induces currents in the target, generating a scattered 
magnetic field. The decaying scattered field, following 
pulse termination, induces the measured voltage in the 
receiver loop. 

There are three different regimes that one may identify 
in the voltage time traces: early, intermediate, and late 




FIG. 1: Sketch of target EMI response. Left: The transmit¬ 
ter current induces a magnetic field inside the target. Right: 
The transmitter pulse termination induces screening currents 
in the target that, via Lens’s law, oppose the change in the 
applied field. These currents are initially confined to the tar¬ 
get surface, but then diffuse inward, generating a decaying 
scattering magnetic field that is sensed though the induced 
voltage in the receiver loop. 


time. At very early time, immediately following pulse 
termination, the currents are confined to the immediate 
surface of the target. The initial diffusion of these cur¬ 
rents into the target interior leads to a power law decay 
(1/t 1 / 2 for nonferrous targets, 1/t 3 / 2 for ferrous targets 
[1, 2]). At intermediate time, as the currents penetrate 
the deeper target interior, the power law crosses over to 
a multi-exponential decay, representing the simultaneous 
presence of a finite set of exponentially decaying modes. 
Finally, at late time only the single, slowest decaying 
mode survives. 

At intermediate- to late-time our mean field algorithm 
models the dynamics by computing as large a number 
as possible of the modes, and determining the excitation 
level of each. At early time, the power law arises from 
a superposition of an essentially infinite number of ex¬ 
ponentials, and a complementary theory, based on the 
detailed dynamics of the initial very thin surface current 
sheet, has been developed instead. 












A. intermediate- to late-time modeling: mean field 
approach 

The solution to the Maxwell equations allows one to 
represent the electric field following pulse termination as 
a sum of exponentially decaying modes, 

oo 

E(x,t) = y4e W (x)e- A " f (2.1) 

n= 1 

where X n are decay rates, are mode shapes, and 
A n are excitation coefficients. The first two are intrinsic 
properties of the target, analogous to vibration modes 
of a drumhead. Only the excitation amplitudes actually 
depend on the measurement protocol. 

As time progresses, modes with larger values of A n de¬ 
cay more quickly, and so at any given time t the signal 
will be dominated by some finite set of modes, namely 
those modes with \ n <l/t. At very late time, t > 1/Ai, 
only the slowest decaying mode contributes, and the sig¬ 
nal becomes a pure exponential decay. Thus, the earlier 
in time one wishes to model quantitatively, the greater 
the number of modes that are required. The ultimate lim¬ 
itation turns out to be the rate at which the excitation in 
pulse is terminated. If the pulse is turned off on a time 
scale t r (see Sec. IIIB), then only modes with A n < l/t r 
have substantial amplitudes A nj and a finite set of modes 
suffices for a full description of the target electrodynam¬ 
ics. However, for large targets, this may require many 
thousands, or even tens of thousands, of modes, which is 
beyond current computational capability. However, the 
early time power law regime may extend out to 1 ms, 
or even 10’s of ms, and we will see that a few hundred 
modes is more than enough to overlap this regime. The 
early time (see Sec. IIB) and mean field approaches may 
then be combined to fully describe the target dynamics 
over the full measured time range. 

Using the mode orthogonality relation, 

J d 3 a:<7(x)e (m) * (x) • e (n) (x) = S mn , (2.2) 

where a(x) is the conductivity, the excitation amplitude 
can be shown to be given by 



in which the transmitter loop has been approximated by 
an ideal ID loop Ct with Nt windings, and 

pO 

ip =- dte Xnt d t I T (t ) (2.4) 

J — OO 

depends on the history transmitter loop current Irif) up 
until the beginning of the measurement window, taken 
here as t = 0. To gain some intuition, a single perfect 
square wave pulse of amplitude I® and duration t p , one 
obtains 

(2.5) 


t * 


M 


2 

-- 

>) 

M 

5 

K -^ 1 

6 

Y -V 

7 

8 

9 

( 10 

11 

12 

13 

14 

( 15 1 

( 16 

( 17 

( 18 

19 

(20 

( 21 ] 

M 

(23 

24) 


t 



Magnetometer 


EM Sensor 


GPS Antenna 


FIG. 2: Sketch of NRL TEMTADS array consisting of a 5 x 5 
array of 25 independent, concentric transmitter and receiver 
coils, numbered from 0 to 24 as shown. Due to rapid de¬ 
cay of signals with target depth, precise (cm level) geometry 
and placement of the coils (summarized in Table I) can have 
significant effect on the overall measured voltage amplitude. 


Sensor center horizontal separation 40 cm 

Transmitter coil center height 4.3 cm 

Transmitter diameter 35 cm 

Number of transmitter coil windings Nt 35 
Receiver coil center height 0.4 cm 

Receiver diameter 25 cm 

Number of receiver coil windings Nr 16 


TABLE I: NRL TEMTADS array geometry. The transmitter 
coil windings are 7.8 cm tall with 0.4 cm thick endcaps on top 
and bottom. Height is measured from the bottom side of the 
lower endcap, and the transmitters are then modeled as an 
idealized ID square loops at 0.4 +3.9 = 4.3 cm height. The 
receiver coils are vertically compact and lie at the bases of the 
transmitter coils, hence are modeled as idealized ID square 
loops at 0.4 cm height. 


For a mode that decays rapidly on the scale t p , one has 
A n t p 1, and I^ ~ /£. For a more slowly decaying 

modes, I^ will have a strong dependence on t p and n. 
In fact, for large targets one may actually encounter the 
regime A n t p <C 1 [e.g., t p = 25 ms and r n = 1/A n = 

0(100 ms)] where L^ will depend not only on but 
on previous pulses. 

Finally, the measured voltage takes the form 

oo 

V(t) = Vne~ Xnt (2.6) 

n= 1 

in which, approximating the receiver as well by an ideal 
ID loop Cr with Nr windings, the voltage amplitudes 
are given by the line integrals 

V n =A n N R [ e< n >(x)-dl. 

Jc R 


4 n) = 4(1 - e~ Xntp ). 


(2.7) 





































































































































































































Field profiles for k = 0 


Field profiles for k = 10 


Field at target surface (|R| = 0) 



Scaled distance |Ft| from target surface 


Scaled distance |Ft| from target surface 



FIG. 3: Illustration of the early time evolution of the surface density depth profile from the target surface for nonmagnetic 
(left) and magnetic (center) targets, beginning from a delta-function initial condition (perfect step function pulse termination). 
Distance R — r/L is scaled by the target size, time r — t/ru by the diffusion time, so that n here corresponds n n y/rc in (2.11), 
and is essentially the permeability contrast (g — fib)/gb- The profiles are plotted for a sequence of 26 equally spaced scaled 
times 10 -4 < r < 0.05 (earlier times corresponding to narrower profiles). The nonmagnetic profile exhibits a pure Gaussian 
spreading into the target interior, while the magnetic profile is much more complex due to the surface magnetic boundary 
condition. Its maximum is pushed inwards from the boundary, and decays more rapidly with time. The right plot shows the 
time trace for the current density at the surface, R = 0, and is essentially the profile H(Ky/r), equation (2.12), which appears 
in the measured voltage (2.11). For k = 0 (solid blue line) the r-dependence follows an exact 1/y/r power law. For n > 0 (solid 
red line) the r-dependence crosses over from the identical 1/y/r form at early-early time to the 1 /t 3 ^ 2 power law (dashed red 
line) at late-early time [the asymptotic forms displayed in (2.12)]. 


Equations (2.3)-(2.7) provide all the required ingredi¬ 
ents for generating predicted data based on a target and 
measurement platform model. Our “mean field” numer¬ 
ical code divides naturally into two parts. 

The internal code solves the Maxwell equations to pro¬ 
duce the intrinsic mode quantities A n and for a range 
of expected targets. With increasing A n , the modes have 
more complex spatial structure, and finite numerical pre¬ 
cision means that only a finite set (a few hundred) of 
slowest decaying modes are actually produced. 

The external code uses the mode data, along with the 
measurement platform data, to compute current inte¬ 
grals (2.4), the line integrals in (2.3) and (2.7), and then 
combines them to output the voltage amplitudes V n and 
hence the time series (2.6). Note that the line integral 
computation requires full knowledge of the relative posi¬ 
tion and orientation of the target and platform. 

For high precision, the internal code can take anywhere 
from minutes to hours to produce mode data for a single 
target. However, given this data, the external code takes 
at most a few seconds produce the full predictions. Pre- 
computation and storage of a rapidly accessible database 
of target data is therefore essential. 


B. Complementary early time modeling 

For a rapidly terminated transmitter pulse, the exter¬ 
nal electric field, and induced voltage, display an early 
time power law divergence [1, 2] (saturating at very early 
time only on the scale of the off-ramp time t r [4]). The 
boundary between the intermediate (multi-exponential) 
and late time (mono-exponential) regime occurs at the 


diffusion time scale 

t d = L 2 /D (2.8) 

where L is the characteristic target radius, and D = 
c 2 /Anger is the EM diffusion constant — this is the time 
scale required for the initial surface currents to diffuse 
into the center of the target. The early time regime cor¬ 
responds to times f < td (say, t < td/ 100), beginning 
deep into the multi-exponential regime where many (e.g., 
hundreds of) modes are excited. In this regime, for non- 
permeable, or weakly permeable targets (g ~ //&), one 
obtains the simple power law prediction prediction [1] 

V(t) = Ve/t 1 ' 2 , t « td, (2.9) 

with all of the target and measurement parameters en¬ 
compassed by the single amplitude V e , whose computa¬ 
tion requires the solution of a certain Neumann problem 
for the Laplace equation in the space external to the tar¬ 
get. 

For permeable targets, a new magnetic time scale 

Tmag = TD(Rb/R) 2 (2.10) 

emerges. For ferrous targets, g/gb = 0(100), and 
r mag /'Tc = 0( 10 -4 ) is tiny, and the early time voltage 
has a more complex magnetic surface mode structure, 

oo 

V(t) = J2 V nH(*nVi) ( 2 . 11 ) 

n= 1 

where the K n are surface mode eigenvalues, and the mode 























eigenvector 1 eigenvector 2 



eigenvectors 6,7 eigenvector 8 



eigenvectors 9,10 



eigenvectors 11,12 



FIG. 4: Contour plots for the first few magnetic surface modes 
eigenfunctions for an aspect ratio 4 prolate spheroid. Plotted 
is the stream function 'if n associated with each mode: red 
corresponds to positive ip n , blue to negative and green 
to near zero values (node lines). The level curves of ifn are 
the stream lines of the surface current. Higher order eigen¬ 
functions clearly have more complex structure with patterns 
of multiple, oppositely oriented current vortices (which cir¬ 
culate clockwise around blue patches, and counter clockwise 
around red patches). 


time trace profile 


H(s) 



erfc(s) 

s«l 

5 > 1 , 


( 2 . 12 ) 


where erfc(s) is the complementary error function, inter¬ 
polates between a 1/t 1 / 2 power law at early-early time, 
t < T mag , and a 1/t 3 / 2 power law at late-early time, 
t mag <C t <C td- For large ferrous targets, this latter in¬ 
terval is very large, and may, in fact, accurately represent 
the signal over nearly the entire measurement interval 
(see Sec. IV). 

Figure 3 illustrates the important features of the early 
time modeling, including the complex evolution of the 
surface current depth profile [which extends H(s) to a 
function of both time and space [2]] that ultimately gives 
rise to the externally measured voltage (2.11). 

The surface modes are special surface current profiles 
(see Fig. 4) that, instead of decaying exponentially, evolve 
according to the universal function H(s). They and the 
K n are solutions to an eigenvalue problem defined on the 
surface of the target [2]. They may be determined ana¬ 
lytically only for spherical targets, where one finds 


Kt = l/y/7 m^, Z = 1,2,3,..., (2.13) 


each (2 1 + l)-degenerate, with r mag = 47rcr/r 2 a 2 //m 2 , 
where a is the radius. The amplitudes V,f again require 
a solution to an external Laplace-Neumann problem. 

Unlike the bulk, exponential modes, under most con¬ 
ditions, only a very few surface modes are excited. The 
initial surface current pattern more-or-less follows the 
shape of the magnetic field generated by the transmit¬ 
ter coil. Unless the target is close to the coil, this field 
is fairly uniform, and the corresponding surface current 
density is fairly uniform as well, and can then be rep¬ 
resented by the first few (two or three) modes. There 
is a very heavy numerical overhead in computing these 
modes and their excitation amplitudes, all in pursuit of 
predicting the rather limited information content of just 
a few coefficients. Given the success of extending the 
mean field predictions into the intermediate-early time 
regime, we have therefore found that it is much more ef¬ 
ficient to extend the voltage curve by fitting the data at 
intermediate times to a one or two term series of the form 
(2.11), estimating K n « 1/yT^T for the first few modes. 
Although this precludes quantitative predictions at early- 
early time, it provides an enormously useful qualitative 
confirmation that the functional form H(s) accurately 
describes the data. 































Tx12: multiple bipolar pulse sequence 



Time (ms) 


FIG. 5: TEMTADS transmitter current bipolar pulse wave¬ 
form. Top: multiple periods. Bottom: single 100 ms period. 


III. TEMTADS PLATFORM 
A. Platform geometry 

The 5x5 NRL TEMTADS sensor array is sketched in 
Fig. 2, and its geometrical parameters are summarized in 
Table I. The loops Ct and Cr are all modeled as per¬ 
fect squares with 35 cm and 25 cm edges, respectively. 


The origin is taken to be at the base of the lower endcap 
for sensor 12, the positive x-axis towards sensor 13, the 
positive //-axis towards sensor 7, and the positive z-axis 
vertically upwards. The transmitter and receiver loop 
centers then all have x- and //-coordinates that are mul¬ 
tiples of 40 cm. The transmitters are all at z = 4.3 cm, 
and receivers are all at z = 0.4 cm. Target positions and 
orientations quoted in later sections are all in this frame 
of reference [3]. 

The precise overall voltage amplitudes, required at 
least for initial verification of the instrument calibration, 
turn out to be surprisingly sensitive to small changes in 
these numbers. The scattered fields may be thought of 
as approximately dipolar, and the voltage therefore de¬ 
creases roughly as 1/d 6 with depth d. For example, there¬ 
fore, a 1 cm error for a 30 cm deep target then leads to 
a 20% error in the voltage amplitude. A consistent sys¬ 
tematic error of this magnitude, in fact, is what led us to 
uncovering the existence of the endcaps, and the vertical 
offset between the transmitter and receiver loops! 

B. Transmitter waveform 

The TEMTADS pulse sequence is shown in Fig. 5, 
and its parameters are summarized in Table II. The se¬ 
quence is bipolar, meaning that the pulses alternate in 
sign. Each pulse is 25 ms long, followed by a 25 ms mea¬ 
surement window (“50% duty cycle”). Although square- 
wave-like, the pulses have a much more rapid termination 
time t r = 10 fi s than onset time (a few ms)—see Fig. 6 . It 
is important to understand which details of the waveform 
actually impact on the measurement prediction. 

More quantitatively, ignoring various small spike-like 
features, the pulse waveform is described by the following 
sequence of functional forms (dashed lines in Fig. 6 ): 


J 


m = < 


h( l-e-*/" 1 ), 

I{h) +1 2 [{1 - a){l - e^-^/ T2 

Tnax [1 ^2 )/^r]> 

0, 


with individual pulse length t p = 25 ms; very rapid ex¬ 
ponential time constant r\ = 2.5 /is over the interval 
0 < t < t\ fcs 10 /is; superposition of much slower expo¬ 
nential time constants r 2 = 0.33 ms, 73 = 4 ms over the 
interval t\ < t < t p ; and linear off-ramp time t r = 10 /is. 
The second half of the full bipolar pulse, beginning at 
t = 2is the same as the one above, but inverted. The 
current amplitudes are I\ ~ /(G) ~ 2.3 a, I 2 — 3.7 a, 
and / max = I(t p ) ~ Ii + I 2 ~ 6 a. The mixing co¬ 
efficient a = 0 . 01 - 0.02 is small, and varies substantially 
from pulse to pulse. However, it dominates the last 1-2% 
of the relaxation after the first couple of ms (see upper 


) +a(l 


0 < t < ti 
)], t\ < t <t p 
t 2 t tp I ty* 
t p + t r < t < 2t 


p-> 


(3.1) 


and lower left panels in Fig. 6 ). 

The functional forms in (3.1) are simple enough that 
analytic forms for the current coefficients (2.4) may 
be computed straightforwardly (though somewhat te¬ 
diously). If one were interested in quantitatively describ¬ 
ing the target dynamics through the entire pulse inter¬ 
val, as well as the measurement interval, all of this de¬ 
tail would indeed be important. However, as we will now 
show, the separation of time scales t\ <C r 2 <C 73 t p al¬ 
lows one (purely for convenience) to ignore most of these 
details without affecting the model fidelity in the mea¬ 
surement interval—it is really only the current amplitude 


















































Tx12: Pulse 1 turn-on 


Tx12: Pulse 1 turn-on detail 


Tx12: Pulse 1 turn-off 




25 25.02 25.04 

Time (ms) 


Tx12: Pulse 2 turn-on 


Tx12: Pulse 2 turn-on detail 




Tx12: Pulse 2 turn-off 



FIG. 6: Pulse waveform details, together with model fits, as described in the text, over various time intervals. The upper three 
plots show the positive pulse, and the lower three plots show the following negative pulse. As detailed in the text, different 
aspects of the shape of the pulse impact the measured (and predicted) response to differing degrees. 


Transmitter current amplitude 
Full, bipolar pulse period 
Individual pulse lengths 
Exponential onset time constants 

Pulse termination linear off-ramp width 


/max — 5.7 =b 0.3 a 
T p — 100 ms 

t p = 25 ms (50% “duty cycle”) 

n — 2.5 /is, 0 < t < t\ = 10 fis 

72 — 0.33 ms, 73 = 4 ms, t\ < t < t p 
tj' — 10 /xs tp ^ t ^ tp ~ |- tj' 


TABLE II: NRL TEMTADS pulse waveform parameters associated with the time traces in Fig. 6). The transmitter current 
amplitude varies by about 10% between pulse sequences. The pulse onset is quite complex, turning on rapidly from zero to 2.5 
a over a 10 /xs interval (with functional form given by an exponential with a 2.5 /xs time constant), followed by a superposition 
of exponentials with 0.33 and 4 ms time constants, saturating at about 6 a. The pulse terminates with a 10 /xs linear ramp. 
See equation (3.1) for precise analytic forms. Although not essential, for simplicity, the numerical model keeps only the 0.33 
ms exponential onset, and the linear off-ramp. 


/max, the pulse length and the off-ramp time t r that 
matter. 

To see this, note that for modes that decay rapidly 
enough that X n rs 1 , the e Xnt factor makes the inte¬ 
gral (2.4) insensitive to times prior to the pulse offset for 
which Ir(t) differs measurably from / max (i.e., times At 
such that At/rs <C 1 as well). In this case Z ^ // max can 
depend only on the details of the off-ramp. If, in addition, 
A n t r < 1 (which will be true for all modes computable 
using the mean field code unless the target is very small, 
perhaps a fraction of a cm or less in diameter), one will 

have /f n) /Xmax - 1 - 

On the other hand, for modes that decay slowly enough 
that A n T 2 < 1, the portion of the integral (2.4) arising 
from the pulse onset will be insensitive to the details of 


this onset. The factor e Xnt ~ e~ Xntp will be essentially 
constant over a time interval At which is up to several 
times t 2 in length. One may then approximate the inte¬ 
gral over this interval by e~ Xntp I(— t p + At). Using (3.1), 
I(—t p + At) may be expressed entirely in terms of the 
t 3 decay quantities, and may simply be approximated as 
/ max if one neglects a as well. Note that if, in addition, 
A t p < 1 , then Z ^ will be sensitive to multiple pulses. 
This condition is satisfied for the slowest decaying modes 
for large enough targets (e.g., 10 cm or more diameter 
steel targets). 

Since the two inequalities have an overlapping range, 
I/73 <C A n <C I/72, this confirms the claimed insensitiv¬ 
ity to the onset details for all modes. In light of this, our 
model neglects the small parameter O', and keeps only the 























































15 cm diameter Al sphere under Tx/Rx12: data and prediction 


15 cm diameter Al sphere under Tx/Rx12: data and prediction 




FIG. 7: Data and theory for a 15 cm diameter aluminum sphere with center lying 16.5 cm below the center of sensor 12 (see 
Fig. 2), which is also the only active sensor. The two plots differ only in the log vs. linear time scale. The solid red line is the 
data, the dashed red line the prediction from the exact analytic solution for the sphere, the dotted red line is the mean field 
prediction (based on 232 modes), and the dashed black line is the early time 1/y/t power law. The 1.03 overall multiplier listed 
in the legend has been applied to the data to optimize the fit, and is well within the expected 10% fluctuation in the current 
amplitude. The vertical dashed line marks the rough division between the early time and multi-exponential (< 100 modes) 
regimes, and it is seen that the mean field prediction pushes well into the early time regime. The slight deviation of the data 
from the analytic prediction at very early time, t < 0.1 ms, is likely an instrument saturation effect (seen much more clearly in 
Fig. 8, beginning roughly at the same voltage level). 


15 cm diameter St sphere under Tx/Rx12: data and predictions 



15 cm diameter St sphere under Tx/Rx12: data and predictions 



FIG. 8: Data and theory for a 15 cm diameter steel sphere with center lying 16.5 cm below the center of sensor 12 (see Fig. 
2), which is also the only active sensor. The two plots differ only in the log vs. linear time scale. The solid red line is the data, 
the dashed red line the prediction from the exact analytic theory solution for the sphere, the dotted red line is the mean field 
prediction (based on 232 modes). The dashed black line is a two term fit to the early time form (2.11) using the known values 
(2.13), and the dotted black line is a single term fit using ki as a fit parameter. The vertical dashed line marks the rough 
division between the early time and multi-exponential (< 100 modes) regimes, and is much later here than in Fig. 7 because 
the EM time scale is proportional to the product cr/i, which is an order of magnitude larger here. For reasons described in 
the text, the mean field prediction has a more complex structure for ferrous targets, and penetrates only to the edge of the 
early time regime (it is the fact that it is accurate beyond about 20 ms that is the real figure of merit here, as would be more 
evident if the data extended to later time). The 1.3 multiplier listed in the legend is that applied to the data to optimize the fit, 
and lies outside the expected 10% fluctuation in the current amplitude. The difference is likely the result of small positioning 
errors. Sensor saturation is apparent below about 0.5 ms. The late-early time 1/t 3 ^ 2 power law is evident in the data, but full 
convergence to the 1/y/t early-early time power law is incomplete, and not expected until about 10 fi s. 


















































t 2 decay, leading to the much simpler pulse waveform 

I(t) = Jmax(l - e-*/ T2 ), 0 < t < t p , (3.2) 

plus the identical linear off-ramp for t p < t < t p + t r . As 
shown, the separation of time scales built into the pulse 
waveform ensures that this simplification produces only 
negligible errors in the data predictions. 


IV. DATA COMPARISONS 
A. Spherical targets 

Having described the electromagnetic model, and the 
platform model required to implement it, we now turn to 
its validation with real data. We will begin with spher¬ 
ical targets, for which exact analytic solutions exist in 
both the early time [1, 2] and multi-exponential regimes 
[5]. This allows one to validate the sensor model under 
conditions where the target model is fully specified. 

Figure 7 shows results for a 15 cm diameter alu¬ 
minum sphere, plotted on both linear and log time 
scales—the latter much more clearly verifies the asymp¬ 
totic 1/Vi early time power law. The agreement is 
quite remarkable—note that the vertical scale is in mil¬ 
livolts, not some arbitrary scaled unit. The only real fit¬ 
ting parameter is the conductivity, and the chosen value 
a = 3 x 10 7 S/m is well within the range expected for 
aluminum. As discussed in Sec. Ill, the overall pulse-to- 
pulse transmitter current amplitude is stable only at the 
10% level. This leads to an identical uncertainty in the 
overall voltage amplitude. In the figure, an overall factor 
of 1.03 has been applied to the data to obtain an optimal 
fit, well within this uncertainty. The slowest decaying 
mode for this target is t\ = 21.5 ms, so the measurement 
window here barely enters the late time regime t > t\. 
The mean field prediction, based on an approximate cal¬ 
culation of the first 232 modes, is seen to accurately de¬ 
scribe the data well into the early time regime. 

Figure 8 shows results for a 15 cm diameter steel 
sphere, again plotted on both linear and log time scales. 
The only real fitting parameters are the conductivity and 
relative permeability, and the chosen values a = 5 x 10 6 
S/m and /x = 100 are well within the ranges expected for 
steel. The overall 1.3 multiplier applied to the data lies 
well outside that expected based on current amplitude 
fluctuations alone. Fine tuning of a and /x might account 
for some of this error, but, as alluded to in Sec. Ill A, the 
likely culprit is small (0.5 cm level) target positioning 
errors. 

The mean field prediction has much more interesting 
structure for ferrous targets. Due to the nature of the 
EM boundary conditions in the large permeability con¬ 
trast limit, rather than computing only the slowest de¬ 
caying modes, two distinct sets of slow (169 modes in this 
case, with time constants larger than 3.01 ms) and fast 
(63 modes in this case, with time constants smaller than 


0.74 ms) decaying modes are produced, with large gap be¬ 
tween that would only be filled if one pushed the compu¬ 
tation to higher order. This is the source of the S-curve- 
like structure seen in the right panel of Fig. 8. The reduc¬ 
tion in the number of slowly decaying modes reduces the 
accuracy of the theory near the early-intermediate time 
boundary (as compared to the nonmagnetic case shown 
in Fig. 7), but the presence of the more rapidly decaying 
modes at least provides an improved trend at very early 
time. The slowest decaying mode for this target has a 
time constant t\ = 180 ms, indicating a late time regime 
an order of magnitude beyond the measurement. 

The early time prediction, which follows both the ex¬ 
act solution and the data over a significant fraction of the 
time interval, deserves some comment. As described in 
Sec. IIB, to obtain the solid black lines in Fig. 7) we use 
the known eigenvalues (2.13), but determine the ampli¬ 
tudes Vn in (2.11) by fitting to the data. We keep only 
two terms 


V(t) 




(1 - a)H 



“b OlH 



(4.1) 

with the known value £ mag = 0.35 ms, and fit the ampli¬ 
tude Vo = 83 V, and mixing parameter a = 0.4. The one 
term series Vo = 60 V, a = 0 provides an adequate, but 
worse fit. 

However, a better fit than both of these is provided by 
a single term series in which one allows the eigenvalue k\ 
to be adjusted. The dotted black lines in Fig. 7) shows 
the result obtained using = Z e ff/\Amag with Z e ff = 
1.2, along with amplitude Vf = 80 V. This will be our 
fitting method of choice for non-spherical targets, where 
the eigenvalues K n have not yet been computed. 

It is worth emphasizing the importance of the fact that 
analytic functional forms of the type (4.1) fit the data so 
well. The log-time plot demonstrates that the data span 
the full range over which the argument s in (2.12) inter¬ 
polates between the two power laws [6]. The data there¬ 
fore has significant structure through this time range, 
but this does not reflect any deep structure of the target 
(beyond the fact that it is ferrous). Quite the contrary: 
as illustrated in Fig. 3 it represents the dynamics of a 
laterally very smooth surface current sheet as it begins 
to penetrate the first centimeter or so into target. The 
complexity arises strictly from the interplay between the 
electric and magnetic field boundary conditions at the 
surface. This serves to confirm that the early time regime 
provides limited target discrimination ability (again, be¬ 
yond the fact that it is ferrous). 

Figures 9 and 10 show consolidated plots of data and 
theory for 10 cm and 15 cm diameter spheres at various 
depths centered below sensor 12. Agreement continues 
to be excellent. The instrument noise floor is evident for 
deeper targets. Further discussion may be found in the 
captions. All of the data curves display a tendency to 
flatten out at very early time, £ < 0.1 ms, even for curves 


well below the obvious saturation regime. There is likely 
some more subtle instrument effect at work here. 



Txl 2, Rxl 2: 15, 10 cm diameter Al spheres under Tx/Rxl 2 at various depths-to-center: data and predictions 


10 


10 ' 


> 

E 


o 10" 


10 " 


10 " 



" t 1/2 power law 
-16.5 cm (xl.03) 

■ 16.5 cm prediction 
-22.5 cm (xl.28) 

■ 22.5 cm prediction 
30 cm, 0° (xl.06) 

30 cm, 30° (xl.06) 

30 cm, 60° (xl.06) 

30 cm, 90° (xl.06) 

30 cm prediction 

-31.5 cm (x0.87) 

■31.5 cm prediction 
45 cm (xl.1) 

45 cm prediction 

- 49.5 cm (xl) 

■ 49.5 cm prediction 

- 74.5 cm (x0.9) 

■ 74.5 cm prediction 

10 cm sphere, 22 cm (xl .3) 

10 cm sphere, 22 cm prediction 


\ 


W A I 
v >)'% 


10" 


10 

Time (ms) 


10 


FIG. 9: Consolidated plots showing data (thin solid and dashed lines) and theory (thick dashed lines) for 10 cm and 15 cm 
diameter aluminum spheres at various depths centered below sensor 12. The upper curves (16.5 cm depth) are repeated from 
Fig. 7. The legend shows the multipliers used to scale the data curves for optimal fit. In most cases these lie within the 10% 
error expected from the variability of the transmitter current. Larger deviations are again likely due to small positioning errors. 
The four 30 cm depth curves demonstrate the expected invariance of the signal with sphere orientation (“north pole” of the 
sphere tilted by the indicated angles towards the center of sensor 11). The single pair of 10 cm sphere traces demonstrate the 
impact of target size on the late-time regime, displaying a distinctly earlier downturn. 


B. Prolate and oblate spheroidal targets 

Having verified instrument calibration and several 
other quantitative details under conditions where an ex¬ 
act solution exists, we now move on to spheroidal targets. 

Figure 11 shows a consolidated plot of data and theory 
for various prolate (elongated) spheroidal aluminum tar¬ 
gets at various depths and orientations. Spheroid aspect 
ratios a z /a xy vary between 2 and 5. 

The theoretical plots (thick dashed lines) are the mean 
field predictions based on the first 232 modes. It is evi¬ 
dent from the plots that this large a number of modes en¬ 


ables one to push the mean field predictions well into the 
early time regime. For smaller targets (e.g., the 4 x 4 x 20 
cm and 5 x 5 x 20 cm spheroids) this can cover nearly the 
entire measurement window. The multi-exponential time 
series eventually saturates and falls below the data, but 
not before the 1 /y/t power law begins to be established. 
Interpolating between the mean field prediction and this 
power law clearly enables one to accurately match the 
data over the full range. 

Most of the target discrimination information occurs at 
intermediate to late time. The traces are all more-or-less 
parallel at early time, and variations in the overall ampli- 











Txl 2, Rxl 2: 15, 10 cm diameter steel spheres under Tx/Rxl 2 at various depths-to-center: data and predictions 


10 ' 


10 - 


10 ' 


> 

CD 

O) 

03 


10 "* 


10 "' 


10" 


10”' 



■ t 1/2 power law 
1 1 _3/2 power law 

- Early time prediction (1=1,2) 
-16.5 cm (xl .3) 

■ 16.5 cm prediction 
-22.5 cm (xl .4) 

■ 22.5 cm prediction 
30 cm, 0° (xl .02) 

30 cm, 30° (xl .02) 

30 cm, 60° (xl .02) 

30 cm, 90° (xl .02) 

30 cm prediction 
45 cm (xl) 

45 cm prediction 
-51.5 cm (xl) 

-51.5 cm prediction 
-74.5 cm (xl) 

1 74.5 cm prediction 
-10 cm sphere, 29.5 cm (xl) 
1 10 cm, 29.5 cm prediction 

1 1 10- 1 


J_ I I I _ 

10 ° 

Time (ms) 


FIG. 10: Consolidated plots showing continuing quantitative agreement between data and theory for the 10 cm and 15 cm 
diameter steel spheres at various depths centered below sensor 12. The upper curves (16.5 cm depth) are repeated from Fig. 
8. The legend shows the multipliers used to scale the data curves for optimal fit. In most cases these lie within the 10% error 
expected from the variability of the transmitter current. Larger deviations are again likely due to small positioning errors. The 
four 30 cm depth curves demonstrate the expected invariance of the signal with sphere orientation (“north pole” of the sphere 
tilted by the indicated angles towards the center of sensor 11). This invariance is not completely obvious for ferrous spheres, 
since a small remnant magnetization could break the symmetry. The single pair of 10 cm sphere traces demonstrate a much 
more subtle impact of target size since the data never enter the late-time regime. 


tilde could equally well come from variation in depth or 
size of the target. On the other hand, at later time, the 
traces for smaller targets (e.g., again, the 4 x 4 x 20 cm 
and 5 x 5 x 20 cm spheroids) drop off much more quickly 
than those of larger targets. 

There are also interesting dependencies on target ori¬ 
entation in this regime (green, red, magenta, and cyan 
curves for the 10 x 10 x 20 cm spheroid [7]). For a verti¬ 
cal target, the excited modes are dominated by currents 
the circulate around the symmetry axis, while for a hori¬ 
zontal target the currents tend to circulate along it. The 
horizontal target mode has a slower decay rate (time con¬ 


stant Th = 13.7 ms vs. r v = 12.0 ms), and couples differ¬ 
ently to the transmitted field, and this is visible in the 
later-time traces. 

Identical conclusions are evident from the data on 
oblate (discus-like) spheroidal aluminum targets (aspect 
ratios a = 0.2,0.4) shown in Fig. 12. Here we have 
overlayed segments of 1 j\ft power law on each curve, 
explicitly demonstrating successful interpolation (with, 
perhaps, 5-10% errors in the overlap regime). 

The dependence on orientation is much stronger for 
oblate spheroids (green, red, magenta, and cyan curves 
for the 20 x 20 x 8 cm spheroid [7]). Because it is be- 









Txl 2, Rxl 2: Al spheroids under Tx/Rxl 2 at various depths-to-center and orientations: data and predictions 


1 t 1/2 power law 



10x10x20, 27 cm depth, 90 u (xl.1) 

27 cm prediction 

6x6x30, 29 cm depth, 90° (xl .1) 

29 cm prediction 

6.67x6.67x20, 28.67 cm depth, 90° (xl .05) 

28.67 cm prediction 

5x5x20, 24.5 cm depth, 90° (xl .25) 

24.5 cm prediction 

4x4x20, 25 cm depth, 90° (xl .2) 

25 cm prediction 

10x10x20, 21 cm depth, 0° (xl.15) 

21 cm prediction 

10x10x20, 22 cm depth, 30° (xl .1) 

22 cm prediction 

10x10x20, 24.4 cm depth, 60° (xl.1) 

24.4 cm prediction 

10x10x20, 26 cm depth, 90° (xl.1) 

26 cm prediction 


10 

Time (ms) 


FIG. 11: Consolidated plot of data and theory for a range of artificial aluminum prolate spheroidal targets. The dimensions 
listed in the legend are diameters. Orientation angles indicate symmetry axis declination (toward the center of sensor 11), 
so that 0° corresponds to vertical and 90° to horizontal. The multipliers are again the overall factors applied to the data to 
obtain optimal agreement with the prediction. The thick dashed lines are the mean field predictions, which show remarkable 
agreement well into the early time regime, where the onset of the 1 /y/t power law is evident. 


ing “squeezed” vertically, the horizontal target (discus on 
edge) mode now has significantly faster decay rate than 
vertical target (discus lying flat) mode (time constant 
Th = 13.0 ms vs. r v = 23.9 ms). Because the the latter 
mode is not excited at all when the target is horizontal, 
the 90° (cyan) curve in Fig. 12 dies much more quickly 
at late time than the other curves. 

In both Figs. 11 and 12 the multipliers used to scale 
the data for optimal fit appear to have a small (^ 10%) 
systematic bias that cannot be explained by random vari¬ 
ation in the transmitter loop current. A combination of 
small conductivity and positioning errors is the likely ex¬ 
planation. 

Figures 13 and 14 show data and theory for steel pro¬ 
late and oblate spheroidal targets. As for spherical tar¬ 
gets (Figs. 8) and 10), the early time regime dominates, 
and the mean field results (dotted curves; with S-curve 


behavior excised in this case so as not to busy up the 
plots too much) are valid only over a small part of the 
time interval where the data is already becoming quite 
noisy. In most cases, however, the fact that the data is 
dropping below the early time curve is evident, pointing 
to the necessity of a multi-exponential description. As 
before, these predictions actually push quite deeply into 
the early time regime, but the measurement window, and 
instrument dynamic range, are such as to strongly limit 
the information content of the multi-exponential part of 
the signal. 


V. SUMMARY AND CONCLUSIONS 

The results presented in this document demonstrate 
the unprecedented accuracy available from our first prin- 















Tx12, Rx12: Al spheroids underTx/Rx12 at various depths-to-center and orientations: data and predictions 



Time (ms) 


FIG. 12: Consolidated plot of data and theory for a range of artificial aluminum oblate spheroidal targets. The dimensions 
listed in the legend are diameters. Orientation angles indicate symmetry axis inclination (toward the center of sensor 11), 
so that 0° corresponds to horizontal and 90° to vertical. The multipliers are again the overall factors applied to the data to 
obtain optimal agreement with the prediction. The thick dashed lines are the mean field predictions, which show remarkable 
agreement well into the early time regime, where the 1/y/t power law takes over. 


ciples, physics based models covering the entire mea¬ 
surement window, from the early time multi-power law 
regime, all the way through the multi-exponential regime 
to the late time mono-exponential regime. Prior to the 
mean field code’s current upgrade [8], the number of 
accurately computed modes used to describe the multi¬ 
exponential regime was limited to perhaps a few dozen 
[9]. As seen in the validation results presented, this up¬ 
grade is absolutely critical to the success of the predic¬ 
tions, by generating the required overlap of the early time 
and multi-exponential regimes. 

It should be emphasized that the increase in predic¬ 
tive power continues to operate with extremely high nu¬ 
merical efficiency. The creation of the mode data for a 


given target cannot be performed in real time, but once 
this data is made available in a database that spans the 
expected target geometries, its acquisition and use for 
measurement predictions can be performed in real time— 
operating at essentially the same speed as predictions us¬ 
ing the exact solution for the sphere. 

As seen in the figures, the dominant regimes visible 
in the data depend very strongly on the target size and 
physical properties. Increasing target size and magnetic 
permeability expands the early time regime to later phys¬ 
ical time. Smaller aluminum targets (e.g., blue lines in 
Fig. 11) are completely described by the mean field ap¬ 
proach over the full time range, while even the smaller 
steel targets barely enter multi-exponential regime (see 








Tx12, Rx12: St prolate spheroids under Tx/Rx12 at various depths-to-center: data and predictions 


10 " 


> 

E 



1 1 0 power law 

— 10x10x20, 49 cm depth, 90° (x0.6) 

1 49 cm MF prediction 

. - 49 cm early t prediction (l eff = 0.75) 

— 6x6x30, 51 cm depth, 90° (x0.75) 

■ 51 cm MF prediction 

. _ 51 cm early t prediction (l gff = 0.7) 

6x6x30, 46 cm depth, 90° (x0.8) 

46 cm MF prediction 

46 cm early t prediction (l gff = 0.7) 

6.67x6.67x20, 33.67 cm depth, 90° (x0.85) 

33.67 cm MF prediction 

33.67 cm early t prediction (l eff = 0.8) 

— 5x5x20, 34.5 cm depth, 90° (x0.9) 

111 34.5 cm MF prediction 


— 4x4x20, 35 cm depth, 90 (x0.95) 

1 35 cm MF prediction 
. _ 35 cm early t prediction (I = 0.75) 


10 

Time (ms) 


FIG. 13: Consolidated plot of data and theory for a range of steel prolate spheroidal targets. Orientation angles indicate 
symmetry axis inclination (toward the center of sensor 11), so that 0° corresponds to horizontal and 90° to vertical. The 
multipliers are again the overall factors applied to the data to obtain optimal agreement with the prediction. The thick dashed 
lines are the early time predictions, which show remarkable agreement over nearly the entire measurement window. The latter 
take the form (2.11) with a single term, in which the amplitude Vf and eigenvalue ki = l e j?/y/T mag are adjusted to optimize the 
fit. Here r mag is defined by (2.10) and (2.8), with the choice L = min {a xy ,a z } = a xy . The mean field predictions are shown by 
the dotted lines. If extended over the full time interval, they also would display the S-curve behavior seen in Fig. 8. For these 
smaller targets, their region of validity begins only as the signal levels are falling into the noise floor. The early time regime 
therefore pretty much encompasses the full range of useful data. 


Figs. 13 and 14) before the signal fades into the noise 
floor [10]. 

All of these features, whose quantitative interpretation 
is enabled by the present models, will be used in pursuit 
of robust target discrimination and identification in later 


stages of this project. The code efficiency becomes es¬ 
pecially critical for this purpose, as searches through the 
database for the target whose response best matches the 
data may require hundreds, or even thousands, of itera¬ 
tions. 


[1] P. B. Weichman, Universal early-time response in high- 
contrast electromagnetic scattering, Phys. Rev. Lett. 91 , 
143908 (2003). 

[2] P. B. Weichman, Surface modes and multipower-law 
structure in the early-time electromagnetic response of 
magnetic targets, Phys. Rev. Lett. 93 , 023902 (2004). 

[3] The external code actually takes the frame of reference 
issue several steps further. It allows one to specify a hor¬ 
izontal survey grid, with azimuth specified relative to a 
chosen “north”. The height of the platform center (above 


each grid point) may then be chosen arbitrarily, and the 
3D platform orientation about this raised center may also 
be specified arbitrarily (through specification of three Eu¬ 
ler angles). The target 3D position and orientation may 
then be specified as well. 

[4] The divergence eventually saturates even for a perfect, 
instantaneously terminated, square wave pulse, but only 
due to speed of light retardation effects in the background 
medium. 

[5] See, e.g., J. D. Jackson Classical Electrodynamics (John 















Txl 2, Rxl 2: St oblate spheroids under Tx/Rxl 2 at various depths-to-center: data and predictions 



FIG. 14: Consolidated plot of data and theory for a range of steel oblate spheroidal targets. Orientation angles indicate 
symmetry axis inclination (toward the center of sensor 11), so that 0° corresponds to horizontal and 90° to vertical. The 
multipliers are again the overall factors applied to the data to obtain optimal agreement with the prediction. The thick dashed 
lines are the early time predictions, which show remarkable agreement over nearly the entire measurement window. The latter 
take the form (2.11) with a single term, in which the amplitude Vf and eigenvalue k>i = l e ff/ ^/T mag are adjusted to optimize 
the fit. Here r mag is defined by (2.10) and (2.8), with the choice L — min {a xy ,a z } = a z . The mean field predictions are shown 
by the dotted lines. If extended over the full time interval, they also would again display the S-curve behavior seen in Fig. 8. 
For these smaller targets, their region of validity again begins only as the signal levels are falling into the noise floor, and the 
early time regime pretty much encompasses the full range of useful data. 


Wiley and sons, New York, 1975). 

[6] This is far less obvious on the linear-time plot, in which 
only the 1/t 3 ^ 2 late-early time power law is clearly visible. 
The log-time scale is key to elucidating the multi-scale 
nature of the target electrodynamics. 

[7] For the 10 x 10 x 20 cm and 20 x 20 x 8 cm spheroid mea¬ 
surements, the bottom of the target rested on a platform 
at 31 cm depth, and so the depth-to-center varies with 
orientation: between 26 cm (horizontal target) and 21 cm 
(vertical target) for the former; between 27 cm (vertical 
target) and 21 cm (horizontal target) for the latter. 

[8] A full theoretical description of the algorithm is con¬ 
tained in P. B. Weichman, Chandrasekhar Theory of El¬ 
lipsoidal Scatterers, BAE Technical Report (2010). 

[9] P. B. Weichman and E. M. Lavely, Study of inverse prob¬ 


lems for buried UXO discrimination based on EMI sen¬ 
sor data, Proc. SPIE Vol. 5089 Detection Technologies 
for Mines and Minelike Targets VIII (SPIE, Bellingham, 
WA, 2003), p. 1139. 

[10] This signal fading is exacerbated by the more rapid 1/t 3 ^ 2 
decay of the late-early time signal for steel targets, as 
compared to the much slower 1/t 1 ^ 2 aluminum targets. 
The latter then have greater tendency to maintain strong 
signals into the multi-exponential regime (see Figs. 11 
and 12). As decribed in Sec. IIB (see especially Fig. 3), 
this more rapid decay has its orgin in the surface cur¬ 
rent dynamics, which, in magnetic targets, tends to more 
quickly push the currents away from the target surface.