# A Semiempirical Transparency Model for Dual Energy Cargo Radiography
Applications

Peter Lalor  Areg Danagoulian  Department of Nuclear Science and Engineering,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA

###### Abstract

Cargo containers passing through ports are scanned by non-intrusive inspection
systems to search for concealed illicit materials. By using two photon beams
with different energy spectra, dual energy inspection systems are sensitive to
both the area density and the atomic number of cargo contents. Most literature
on the subject assumes a simple exponential attenuation model for photon
intensity in which only free streaming photons are detected. However, this
approximation neglects second order effects such as scattering, leading to a
biased model and thus incorrect material predictions. This work studies the
accuracy of the free streaming model by comparing it to simulation outputs,
finding that the model shows poor atomic number reconstruction accuracy at
high-$Z$ and suffers significantly if the source energy spectra and detector
response function are not known exactly. To address these challenges, this
work introduces a semiempirical transparency model which modifies the free
streaming model by rescaling different components of the mass attenuation
coefficient, allowing the model to capture secondary effects ignored by the
free streaming model. The semiempirical model displays improvement agreement
with simulated results at high-$Z$ and shows excellent extrapolation to
materials and thicknesses which were not included during the calibration step.
Furthermore, this work demonstrates that the semiempirical model yields
accurate atomic number predictions even when the source spectra and detector
response are not known exactly. Using the semiempirical model, manufacturers
can perform a simple calibration to enable more precise $Z$ reconstruction
capabilities, which has the potential to significantly improve the performance
of existing radiographic systems.

###### keywords:

Dual energy radiography , non-intrusive inspection, atomic number
discrimination , nuclear security

##  1 Introduction

In 2021, the Customs and Border Protection (CBP) processed more than 32.7
million imported cargo containers through U.S. ports of entry [1]. A principle
concern is that a terrorist might attempt to smuggle a nuclear or radiological
dispersal device through a port and subsequently detonate a weapon on U.S.
soil. Such an attack could have devastating economic consequences in excess of
$1 trillion due to infrastructure damage, trade disruption, and port shutdown
costs [2, 3]. The international atomic energy agency (IAEA) tracks incidents
of nuclear and other radioactive material out of regulatory control,
identifying 320 incidents of trafficking or malicious use since 1993 [4]. For
these reasons, U.S. congress passed the SAFE Port Act in 2006, mandating 100
percent screening of U.S. bound cargo and 100 percent scanning of high-risk
containers [5].

When a cargo container passes through a U.S. port, it is first passively
screened by a radiation portal monitor (RPM). This systems detect neutron and
gamma radiation which is being passively emitted from radiological sources
which may be hidden inside a cargo container [6, 7]. However, a smuggler could
evade passive detection through sufficient shielding of the radiological
device [8, 9]. As a result, radiography systems are also utilized to
complement passive detection [10]. These X-ray and gamma ray imaging systems
produce a sensitive density image of the scanned cargo, enabling the
identification of nuclear threats which have been shielded to avoid passive
detection. Furthermore, through the use of dual energy radiography, these
systems can produce a rough elemental analysis of cargo contents, since the
attenuation of photons is dependent on the atomic number of the material. This
enhances the capabilities of radiography systems to detect nuclear threats and
high-$Z$ shielding materials [9].

##  2 Background

###  2.1 Dual Energy Radiography Overview

When a radiography system scans a material of area density $\lambda$ and
atomic number $Z$, it measures a transparency $T(\lambda,Z)$, defined as the
detected charge in the presence of the material normalized by the open beam
measurement:

$T(\lambda,Z)=\frac{Q(\lambda,Z)}{Q_{\text{air}}}$ (1)

The exact functional form of Eq. 1 is arbitrarily complex, since it depends on
the precise geometric details of any particular scanning system. Importantly,
Eq. 1 depends on the atomic number of the imaged object through the mass
attenuation coefficient $\mu(E,Z)$, which describes the attenuation of photons
through matter by the Beer-Lambert law:

$\frac{I}{I_{0}}=e^{-\mu(E,Z)\lambda}$ (2)

where $E$ is the photon energy and $I/I_{0}$ is the ratio of transmitted to
initial photon intensity. The dual energy principle observes that the mass
attenuation coefficient depends on both energy and atomic number, and thus
multiple transparency measurements of the same material using different high
and a low energy beam spectra may allow for a determination of the atomic
number $Z$ of the imaged object. We use the subscript $H$ to refer to the high
energy beam, and the subscript $L$ to refer to the low energy beam. As such,
for a pair of measured transparencies $\\{T_{H},T_{L}\\}$, the corresponding
area density and atomic number estimates $\hat{\lambda}$ and $\hat{Z}$ are
determined by solving the following $2\times 2$ system:

$\hat{\lambda},\hat{Z}=\mathop{\text{Solve}}\limits\limits_{\lambda,Z}\begin{cases}T_{H}(\lambda,Z)=T_{H}\\\
T_{L}(\lambda,Z)=T_{L}\end{cases}$ (3)

Older works attempted to solve Eq. 3 directly [11, 12]. However, this approach
is generally too slow given the millions of pixels in a radiograph image and
often only yields approximate solutions. Instead, Eq. 3 is generally solved
through the use of reverse lookup tables [13]. In other words, for a
$(\lambda,Z)$ range of interest, $T_{H}(\lambda,Z)$ and $T_{L}(\lambda,Z)$ are
evaluated and tabulated. Then, for a pair of transparency measurements
$\\{T_{H},T_{L}\\}$ of an unknown material, $\\{\lambda,Z\\}$ are
reconstructed by reference to the lookup table. This approach is fast,
accurate, and conveniently simple.

The key remaining ingredient needed to solve Eq. 3 is an accurate choice of
transparency models $T_{H}(\lambda,Z)$ and $T_{L}(\lambda,Z)$. To do this,
existing dual energy literature can be broadly classified into two categories:
analytic methods and empirical methods. Analytic models are derived from the
exponential dependence of photon intensity (Eq. 2), which neglects scattered
photons and thus introduces a significant source of error. Empirical models
typically require a burdensome calibration step and show limited atomic number
selectivity for materials not included in the calibration. This work
introduces a hybrid method, which addresses both of these challenges by
defining a semiempirical transparency model.

###  2.2 Analytic Methods

Most authors define a transparency model by constructing a simple, closed form
expression for $T_{\\{H,L\\}}(\lambda,Z)$ [12, 13, 14, 15]. Eq. 4 shows the
typical choice, which we call the free streaming transparency model:

$\displaystyle\begin{split}&T_{H}(\lambda,Z)_{\text{free
streaming}}=\frac{\int_{0}^{\infty}D(E)\phi_{H}(E)e^{-\mu(E,Z)\lambda}dE}{\int_{0}^{\infty}D(E)\phi_{H}(E)dE}\\\
&T_{L}(\lambda,Z)_{\text{free
streaming}}=\frac{\int_{0}^{\infty}D(E)\phi_{L}(E)e^{-\mu(E,Z)\lambda}dE}{\int_{0}^{\infty}D(E)\phi_{L}(E)dE}\end{split}$
(4)

where $\phi_{\\{H,L\\}}(E)$ are the $\\{$high, low$\\}$ energy differential
photon beam spectra, and $D(E)$ is the detector response function, calculated
as

$D(E)=C\int_{0}^{E}R(E,E_{\text{dep}})E_{\text{dep}}dE_{\text{dep}}$ (5)

where $R(E,E_{\text{dep}})$ is the differential detector response matrix,
representing the probability that a photon with incident energy $E$ deposits
energy $E_{\text{dep}}$, and $C$ is a proportionality constant. Through direct
evaluations of Eq. 4, a large, high-resolution lookup table can be
constructed, enabling atomic number discrimination [13]. Other works perform a
variable transformation instead of using $T_{\\{H,L\\}}(\lambda,Z)$ directly,
and construct lookup tables on these new features [12, 14, 15]. These methods
are all expected to give similar results.

A fundamental assumption made by the free streaming model is that only
noninteracting photons will be detected and thus ignores the effects of
scattered radiation. Past works have claimed that these approximations were
validated experimentally and through the use of Monte Carlo simulations [13,
14]. Detector systems are typically heavily collimated to reduce the
significance of scattered radiation, although it can sometimes be as much as
one percent of the primary detector beam [16]. Section 3 uses Monte Carlo
simulations to analyze the accuracy of the free streaming model, finding that
the model yields inaccurate atomic number reconstruction at high-$Z$.
Furthermore, the free streaming model shows significant bias if the beam
spectra and detector response are not known exactly. These results highlights
the primary disadvantage of analytic methods, as a biased transparency model
will yield inaccurate atomic number predictions.

###  2.3 Empirical Methods

To circumvent the biases introduced by the free streaming model, some authors
instead construct the transparency model empirically through the use of
calibration measurements [17]. To do this, transparencies of a known material
are measured for a range of different thicknesses, and the results are
tabulated. This is repeated for different materials to construct a lookup
table, and the $\\{\lambda,Z\\}$ of an unknown material are again determined
by reference to the lookup table. This is the most common approach used by
commercial dual energy cargo inspection systems.

The primary advantage of empirical approaches is the high degree of accuracy.
Since the transparency model is constructed directly from the data, there is
no inherent model bias. However, the data acquisition step can often be long
and laborious. Empirical methods require a significant quantity of high-
precision measurements to construct the lookup tables. Subsequently, if a
target material $\\{\lambda,Z\\}$ does not exist in the lookup table, an
interpolation scheme is necessary to predict the identity of the material,
introducing a source of error. Typically, only a few calibration materials are
used, and the task is simplified to discrimination between organics, organics-
inorganics, inorganics, and heavy substances [14]. However, such a coarse
label discretization reduces accuracy and can yield high variance results if a
target material is near the midpoint of two bins.

Lee et al. employs a different empirical approach by expressing a material-
selective projection image as a linear combination of polynomial basis
functions [18, 19]. Then, for a choice of basis materials, a set of material-
specific weighting factors are calculated through a least squares minimization
against a calibration phantom. After the system has been calibrated, material
classification is performed by evaluating the material-selective projections,
which indicate whether the corresponding basis material is present in the
image. The authors show that this approach can yield accurate material
discrimination capabilities without knowledge of the photon energy spectra or
the detector response function. However, this method is incapable of
identifying materials with atomic numbers that differ significantly from the
basis materials, resulting in limited atomic number selectivity.

###  2.4 A Semiempirical Transparency Model

To address the challenges described in sections 2.2 and 2.3, this work
introduces a semiempirical transparency model. First, we define a
semiempirical mass attenuation coefficient, $\tilde{\mu}(E,Z;a,b,c)$, as
follows:

$\tilde{\mu}(E,Z;a,b,c)=a\mu_{\text{PE}}(E,Z)+b\mu_{\text{CS}}(E,Z)+c\mu_{\text{PP}}(E,Z)$
(6)

where $\mu_{\text{PE}}(E,Z)$, $\mu_{\text{CS}}(E,Z)$, and
$\mu_{\text{PP}}(E,Z)$ are the mass attenuation coefficients from the
photoelectric effect (PE), Compton scattering (CS), and pair production (PP),
respectively. Values for the mass attenuation coefficients are calculated from
NIST cross section tables [20]. In Eq. 6, $a$, $b$, and $c$ are calibration
parameters which scale the relative importance of photoelectric absorption,
Compton scattering, and pair production on the net photon attenuation. In the
limit $a=b=c=1$, the nominal mass attenuation coefficient is recovered. The
motivation behind this approach is that by rescaling different components of
the mass attenuation coefficient, it may be possible to capture some of the
secondary effects ignored by the free streaming model. The semiempirical
transparency model is then adapted from the free streaming model (Eq. 4) as
follows:

$\displaystyle\begin{split}&\tilde{T}_{H}(\lambda,Z;a_{H},b_{H},c_{H})=\frac{\int_{0}^{\infty}D(E)\phi_{H}(E)e^{-\tilde{\mu}(E,Z;a_{H},b_{H},c_{H})\lambda}dE}{\int_{0}^{\infty}D(E)\phi_{H}(E)dE}\\\
&\tilde{T}_{L}(\lambda,Z;a_{L},b_{L},c_{L})=\frac{\int_{0}^{\infty}D(E)\phi_{L}(E)e^{-\tilde{\mu}(E,Z;a_{L},b_{L},c_{L})\lambda}dE}{\int_{0}^{\infty}D(E)\phi_{L}(E)dE}\end{split}$
(7)

As to be described in section 3, the semiempirical transparency model shows
improved agreement with simulated data compared to the free streaming model,
particularly when the source energy spectra and detector response function are
not known exactly. Additionally, the semiempirical transparency model requires
far less calibration data than fully empirical methods and shows excellent
extrapolation to elements which were not included in the calibration step.

##  3 Analysis

###  3.1 Accuracy of the Free Streaming Transparency Model

In order to quantify the accuracy of the free streaming model (Eq. 4),
transparency measurements were simulated in Geant4 [21, 22]. In each
simulation, dual energy bremsstrahlung beams with endpoint energies of
$10\text{MeV}$ and $6\text{MeV}$ were directed through targets composed of
different materials and thicknesses and measured by a stack of collimated
cadmium tungstate (CdWO4) scintillators. This outputs a simulated set of
transparency data points to compare to the model predictions. The simulation
geometry is described in more detail in section A.1.

Fig. 1(a) compares the simulated transparency measurements to the free
streaming model predictions, revealing an obvious model bias. To analyze the
atomic number discrimination capabilities of the model, Fig. 2(a) shows the
simulated measurements and the model predictions on an $\alpha$-curve. An
$\alpha$-curve is a plot of $\alpha_{H}-\alpha_{L}$ versus $\alpha_{H}$ for
different elements and area densities, where a log transform
$\alpha\rightarrow-\log T$ has been performed. Every element on an
$\alpha$-curve forms a characteristic $\alpha$-line, describing the relative
effect on $T_{H}$ and $T_{L}$ as area density is varied. An $\alpha$-curve is
a useful visualization tool because slight transparency differences between
the high and low energy measurements become more apparent, and the separation
between different $\alpha$-lines offers insights into atomic number
discrimination capabilities. Fig. 2(a) reveals that the free streaming model
accurately reconstructs the $\alpha$-lines of low- to med-$Z$ materials, but
becomes less accurate at high-$Z$.

###  3.2 Accuracy of the Semiempirical Transparency Model

First, $a$, $b$, and $c$ are calibrated by minimizing the squared logarithmic
error between the semiempirical model (Eq. 7) and simulated calibration
measurements:

$\displaystyle\begin{split}&a_{H},b_{H},c_{H}=\mathop{\mathrm{argmin}}\limits_{a,b,c}\sum_{i}\left(\log\tilde{T}_{H}(\lambda_{i},Z_{i};a,b,c)-\log
T_{H,i}\right)^{2}\\\
&a_{L},b_{L},c_{L}=\mathop{\mathrm{argmin}}\limits_{a,b,c}\sum_{i}\left(\log\tilde{T}_{L}(\lambda_{i},Z_{i};a,b,c)-\log
T_{L,i}\right)^{2}\end{split}$ (8)

In Eq. 8, the summation index $i$ labels the simulation of calibration
material $\\{\lambda_{i},Z_{i}\\}$ with corresponding output transparencies
$\\{T_{H,i},T_{L,i}\\}$. In this work, carbon $(Z=6)$, iron $(Z=26)$, and lead
$(Z=82)$ were chosen as calibration materials due to their availability in
real applications and because they span a wide range of $Z$. High resolution
simulations were performed at an area density of
$\lambda=150\text{g}/\text{cm}^{2}$ to generate the calibration dataset. At
least three calibration materials are required, and this work finds that
thicker materials yield a more accurate calibration. The calibration
parameters are determined separately for the high and low energy beams. The
best fit values of $a$, $b$, and $c$ are shown in the first two rows of table
1 for each incident beam spectrum.

The simulated transparency measurements are compared to the semiempirical
model predictions in Fig. 1(b), finding that the semiempirical formulation
reduces the model bias by a factor of ${\approx}10$ over the free streaming
model. Fig. 2(b) plots the model predictions on an $\alpha$-curve, revealing
that the semiempircial model shows improved agreement with the simulations at
high-$Z$. In order to calculate the atomic number reconstruction accuracy of
the free streaming and the semiempirical transparency models, the $Z$ of best
fit is calculated for each of the simulated materials by finding the atomic
number which minimizes the squared logarithmic error between the simulated
measurements and the model predictions. The results of this analysis are shown
in the first three rows of table 2, along with the true material $Z$. Both
models yield accurate atomic number predictions for low- to med-$Z$ materials.
At high-$Z$, the free streaming model becomes unstable and is unable to
identify any material which matches the simulation outputs. On the contrary,
the semiempirical model maintains a relatively high degree of accuracy at
high-$Z$, although the solution is not always unique for reasons described in
our previous works [23]. This high-$Z$ degeneracy is a fundamental property of
dual energy X-ray scanners whereby different high-$Z$ elements can yield
identical transparency measurements.

###  3.3 Uncertainty in the Source Energy Spectra and Detector Response
Function

The source energy spectra $\phi_{\\{H,L\\}}$ and detector response function
$D(E)$ serve as implicit parameters in both the free streaming and the
semiempirical transparency models (Eqs. 4 and 7). These parameters are unique
to each experimental system and it has thus far been assumed that they are
known exactly. However, due to the angular dependence of linac based X-ray
sources, the beam spectra change at different beam angles. Furthermore,
variability within the linac may result in photon beam spectra which differ
from an idealized model. Similarly, the true detector response of a real
system may differ from simulation output. As a result, it is not reliable to
assume the source energy spectra and detector response function are all known
exactly.

To probe this effect, we intentionally introduce a discrepancy between the
implicit parameters used during transparency calculations and those used
during transparency simulations. This procedure will enable us to benchmark
the effectiveness of the semiempirical transparency model in a realistic
setting in which the model parameters slightly differ from the experiment.
During transparency simulations, we will continue to use $\\{10,6\\}$ MeV
bremsstrahlung beam spectra and record the true detector response. However,
during transparency calculations, we use bremmstrahlung beam spectra with
endpoint energies of $9.7\text{MeV}$ and $6.3\text{MeV}$ and the following
approximate formulation of the detector response function:

$D(E)\approx
E\left(1-e^{-\mu_{\text{det}}(E)\lambda_{\text{det}}}\right)\frac{\mu_{\text{det}}^{\text{en}}(E)}{\mu_{\text{det}}(E)}$
(9)

where $\lambda_{\text{det}}$ is the area density of the detector,
$\mu_{\text{det}}$ is the mass attenuation coefficient of the detector, and
$\mu_{\text{det}}^{\text{en}}$ is the mass energy absorption coefficient of
the detector [20]. We use the label “mismatched” to distinguish instances with
an intentional shift between the model parameters and the simulation
parameters. It is expected that a mismatched beam spectra and detector
response will worsen the atomic number reconstruction capabilities of the free
streaming model. It would be useful if the semiempirical model were robust to
this discrepancy by capturing these effects in the calibration procedure. In
other words, perhaps through a particular selection of $a$, $b$, and $c$, it
may be possible to rescale different components of the mass attenuation
coefficient to counteract the mismatched model parameters. For instance, by
choosing $c_{L}<1$, we can artificially decrease the number of pair production
interactions in our calculations, which may allow for the $6.3$ MeV
semiempirical transparency model to match the results of the $6$ MeV
simulations.

The analysis performed earlier in this section was then repeated under these
new considerations. The new values of the calibration parameters $a$, $b$, and
$c$ are shown in the last two rows of table 1. The subsequent transparency
calculations using both the free streaming model and the semiempirical model
were compared to the simulated dataset. Fig. 2(c) reveals that using
mismatched model parameters significantly worsens the accuracy of the free
streaming model. On the contrary, Fig. 2(d) demonstrates that the
semiempirical transparency model continues to show excellent agreement with
the simulated transparency measurements. These results are quantified in the
last two rows of table 2, where the semiempirical model shows significantly
improved atomic number reconstruction capabilities compared to the free
streaming model. As before, the solution becomes non-unique at high-$Z$ due to
a fundamental and unavoidable ambiguity between heavy metals [23]. This result
shows promise for the semiempirical transparency model to enable accurate
material discrimination capabilities in realistic cargo scanning applications.

(a) Comparing the free streaming model (dashed lines, Eq. 4) to transparency
simulations (points), indicating noticeable discrepancies.

(b) Comparing the semiempirical model (dashed lines, Eq. 7) to transparency
simulations (points), indicating strong agreement.

Figure 1: Photon beam transparency as a function of area density for different
materials using the high energy $10$ MeV bremsstrahlung beam. The
semiempirical model shows significantly reduced bias compared to the free
streaming model.

(a) Comparing the free streaming model (dashed lines, Eq. 4) to transparency
simulations (errorbars), indicating a discrepancy at high-$Z$.

(b) Comparing the semiempirical model (dashed lines, Eq. 7) to transparency
simulations (errorbars), indicating strong agreement.

(c) Comparing the free streaming model (dashed lines, Eq. 4) to transparency
simulations (errorbars) using mismatched model parameters, indicating large
discrepancies.

(d) Comparing the semiempirical model (dashed lines, Eq. 7) to transparency
simulations (errorbars) using mismatched model parameters, indicating strong
agreement.

Figure 2: $\alpha$-curves for the analysis cases presented in section 3. Different materials correspond to different $\alpha$-lines, and the separation between these $\alpha$-lines enables atomic number discrimination. Compared to the free streaming model, the semiempirical model shows significantly improved agreement with the simulated data, especially when using mismatched model parameters. Materials shown are hydrogen $(Z=1)$, aluminum $(Z=13)$, iron $(Z=26)$, silver $(Z=47)$, gadolinium $(Z=64)$, lead $(Z=82)$, and uranium $(Z=92)$. Bremsstrahlung endpoint energy | a | b | c
---|---|---|---
10 MeV | 1.1158 | 0.9616 | 0.9973
6 MeV | 0.9427 | 0.9687 | 0.9984
10 MeV (simulation), 9.7 MeV (model) | 1.0659 | 0.9788 | 1.0031
6 MeV (simulation), 6.3 MeV (model) | 1.1224 | 1.0343 | 0.9089
Table 1: Calibration parameters $a$, $b$, and $c$ for each of the analysis cases considered in this study. For the first two rows, the beam spectra and detector response used in the model exactly match the simulations. For the last two rows, there is an intentional mismatch between the beam spectra and detector response used in the model compared to the simulations. | H | C | Al | Ca | Fe | Ge | Zr | Ag | Cs | Gd | W | Pb | U
---|---|---|---|---|---|---|---|---|---|---|---|---|---
True $Z$ | 1 | 6 | 13 | 20 | 26 | 32 | 40 | 47 | 55 | 64 | 74 | 82 | 92
Free streaming $Z$ | 1 | 6 | 13 | 20 | 26 | 32 | 40 | 48 | 57 | 69, 91 | N/A | N/A | N/A
Semiempirical $Z$ | 1 | 6 | 13 | 20 | 26 | 32 | 41 | 48 | 56 | 64 | 76, 99 | 81, 95 | 90
Free Steaming $Z$ (mismatched) | N/A | 1 | 8 | 15 | 20 | 26 | 34 | 40 | 47 | 55 | 64, 98 | 67, 96 | 69, 94
Semiempirical $Z$ (mismatched) | 1 | 6 | 13 | 20 | 26 | 32 | 41 | 48 | 56 | 64 | 76, 100 | 80, 97 | 85, 94
Table 2: Comparing the true material atomic number to the reconstructed atomic
number using the free streaming transparency model (Eq. 4) and the
semiempirical transparency model (Eq. 7). In the second and third rows, the
models and the simulations use identical photon beam spectra and detector
response functions. In the last two rows, the models use mismatched beam
spectra and detector responses. Cells containing two entries indicate both
materials match the simulations, whereas cells containing “N/A” indicate no
material was found to reproduce the simulated measurements.

##  4 Conclusion

This work presents a semiempirical transparency model for cargo radiography
measurements. The model introduces three calibration parameters which scale
different components of the mass attenuation coefficient. The semiempirical
model is highly flexible in its ability to be applied to different system
geometries by recalculating the calibration parameters. Compared to analytic
models, the semiempirical model shows significantly less bias and improved
atomic number reconstruction accuracy for high-$Z$ elements. Furthermore, the
semiempirical model maintains strong agreement with simulated data even when
the source energy spectra and detector response function are not known
exactly. Compared to empirical methods, the semiempirical model requires
significantly less calibration data and shows accurate extrapolation to
elements and thicknesses which were not included in the calibration step. For
these reasons, the semiempirical transparency could enable improved atomic
number reconstruction capabilities for dual energy cargo radiography
applications. Future research should apply the semiempirical transparency
model to real transparency data taken by dual energy systems to verify the
accuracy in a practical setting.

##  5 Acknowledgements

This work was supported by the Department of Energy Computational Science
Graduate Fellowship (DOE CSGF) under grant DE-SC0020347. The authors
acknowledge advice from Brian Henderson for his expertise on the technical
nuances of dual energy radiography systems. The authors declare no conflict of
interest.

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## Appendix A

###  A.1 Simulation Details

Simulations of the incident beam spectra $\phi_{\\{H,L\\}}$ and detector
response matrix $R(E,E_{\text{dep}})$ were based on the work of Henderson [24]
and designed to be simple, generalizable, and representative of typical cargo
scanners. In order to model the bremsstrahlung beam spectra, electrons were
directed at a 0.1cm tungsten radiator backed by 1cm of copper. The
bremsstrahlung photons were then recorded by a tally surface
$(r=0.2\text{cm})$ placed 10cm behind the tungsten target, subtending a half-
angle of 20 milliradians $(\approx 1.15^{\circ})$. The resulting dual energy
beam spectra are shown in Fig. 3. The detector response matrix was calculated
by directing photons along the long axis of a $15.0\times 4.6\times 30.0$ mm
cadmium tungstate (CdWO4) crystal. The incident energy and total deposited
energy of each photon was binned to produce the detector response matrix.

Figure 3: Simulated $10$ and $6$ MeV endpoint energy bremsstrahlung beam
spectra ($\phi_{H}$ and $\phi_{L}$, respectively).

To simulate transparency measurements of a target material, photons with
energy sampled from $\phi_{H}$ or $\phi_{L}$ were directed in a fan beam
through a large target box. The photons were then detected by a two meter
stack of CdWO4 detectors with a $10$cm lead collimator to filter scattered
radiation. The simulation geometry is shown in figure 4. This simulation was
repeated for a large number of target area densities and atomic numbers to
obtain a simulation dataset.

Figure 4: A bremsstrahlung fan beam (left) is directed towards a target
(center). A collimated stack of CdWO4 detectors (right) measures the
transparency of the photon beam.