2.1.

Estimating Optical Constants

It is well known that the underlying spectral features (absorption peaks in active spectroscopy) of both the chemical and surface derive from their complex optical constants.²⁰ While the determination of optical constants for liquids is relatively straightforward, their determination for solids is much more complicated. One of the more widely accepted approaches is to use single-angle reflectance spectroscopy followed by Kramers-Kronig transformation for estimating optical constants ($\tilde{n}$) for crystalline solids.^21–24 For solid minerals, DeVetter et al. have shown that using carefully designed mask apertures with low reflectance is more optimal for solid minerals.²⁵ The optical constants used in this research were measured and provided by Pacific Northwest National Laboratory (PNNL) through IARPA’s (Intelligence Advanced Research Projects Activity) SILMARILS (Standoff ILluminator for Measuring Absorbance and Reflectance Infrared Light Signatures) program.

2.2.

Trace Chemical Phenomenology

In measured data, estimating trace chemical reflectance is not as simple as calculating the reflectance of a solid chemical at the chemical/air boundary.²⁶ Instead, it varies greatly with a number of factors. Some of the more significant parameters are particle size²⁷ and shape for solids²⁸ or film thickness in the case of liquids,^29,30 sample morphology,³¹ surface roughness,³² and sampling angle (i.e. bidirectional reflectance function – BRDF).^9,33 In addition to these factors, reflectance spectra may also vary with chemical thermodynamic state,³⁴ molecular interactions,³⁵ and humidity.^36,37 Figure 3 demonstrates the expected variability of normalized reflectance for trace chemical films on surfaces. The curves show measurements for six identical samples (saccharin films on glass) collected by the same sensor at two slightly different measurement angles. There is high variability in the overall spectral shape and the depth of spectral features due to the differences in measurement geometry.

Fig. 3

Normalized reflectance measurements of six identical samples (saccharin film on glass at a concentration of $100\mu \mathrm{g}/{\mathrm{cm}}^2$) collected with the same sensor at two measurement angles, demonstrating the high variability of trace chemicals. Note the variability in not only overall spectral shape, but also the depth of the distinct spectral features.

2.3.

Existing Signature Models

The implementations of the Mie scattering particle model and TM uniform thin film model considered in this paper are defined by Myers et al.¹⁹ For the reader’s convenience, these models are summarized in the next two sections.

2.3.1.

Mie scattering models for particles

Mie scattering describes light scattering from an isolated spherical particle of known complex optical constant and diameter. In addition, it is often used to approximately describe the scattering of light from particles on a surface, which is schematically depicted in Fig. 4(a). For the case of particles on a surface, the effect of the substrate has a significant impact on the reflectance spectrum. The Mie scattering-based model accounts for multiple types of scattering – backscattering from the particle back to the sensor at varying angles depending on the particle shape and sensing geometry (e.g., angle) and forward scattering from the particle to the substrate (i.e., surface) first and back toward the sensor second [see Fig. 4(a)]. Finally, there is reflectance of the bare substrate itself in regions that are not covered in particles. The fraction of a pixel covered by particles is known as the fill factor. The fill factor (FF) depends on the chemical mass loading (i.e., concentration), $m$ ($\mu \mathrm{g}/{\mathrm{cm}}^2$), chemical density, $\rho$ ($\mu {\mathrm{g}/\mathrm{cm}}^3$), particle diameter mean, $\mu$ (cm), and standard deviation, $\sigma$ (cm), as

Eq. (1)

$$\mathrm{FF}=\frac{3m}{2\rho \sum _i{D}_{\text{particle},i}(m,\mu ,\sigma )},$$

where $D_{\text{particle},i}$ is a particle diameter with units (cm) sampled from a particle size distribution. We use a log-normal distribution, shown to be effective for modeling particle sizes^38–40 with mean $\mu$ and standard deviation $\sigma$. Let $R_{\text{particle}}$ be the particle reflectance based on Mie scattering

Eq. (2)

$$R_{\text{particle}}(\lambda )=S_B{Q}_B(\lambda )+S_{LT}{Q}_S(\lambda )R_{\text{sub}}(\lambda )+S_{QT}{Q}_S{(\lambda )}^2{R}_{\text{sub}}(\lambda ),$$

where $\lambda$ is the wavenumber (${\mathrm{cm}}^{-1}$) and $R_{\mathrm{sub}}$ is the substrate reflectance. $S_B$, $S_{LT}$, and $S_{QT}$ are the backscattering, linear transflection, and quadratic transflection strength parameters, respectively. $Q_B$ and $Q_S$ are the backward and forward scattering reflectance contributions calculated using the complex optical constants for the specific chemical. Then, the full model for a particle on a surface is defined as

Eq. (3)

$$R_P(\lambda )={\text{R}}_{\text{particle}}(\lambda )\mathrm{FF}+{\mathrm{SF}}_{\text{sub}}(1-\mathrm{FF})R_{\text{sub}}(\lambda ),$$

where ${\mathrm{SF}}_{\text{sub}}$ is the substrate scale factor, which may be used as a proxy for BRDF information. The user must define the particle diameter mean, $\mu$, standard deviation, $\sigma$, and substrate scale factor ${\mathrm{SF}}_{\text{sub}}$.

Fig. 4

(a) A diagram of the types of scattering captured by the Mie scattering particle model. Backscatter interacts with the particle and reflects back toward the sensor while forward scatter reflects off the particle, onto the substrate, and back toward the sensor. Areas without particles will only show substrate reflectance. (b) The TM method models the light refraction as it travels through and back out of the liquid film on the substrate, as well as scattering within the film as the light interacts with the substrate itself. (c) The STM model includes films of nonuniform thickness sparsely covering the pixels. The film contributions are calculated using the TM method. The reflectance is a linear combination of film and substrate reflectance.

Figure 5 shows comparisons of the Mie scattering model predictions to actual measurements of cyclotrimethylenetrinitramine (RDX) and pentaerythritol tetranitrate (PETN) particles on glass (samples prepared by the Naval Research Laboratory). These results were generated using $\mu =12\mu \mathrm{m}$, $\sigma =10\mu \mathrm{m}$, and ${\mathrm{SF}}_{\text{sub}}=1.0$.

Fig. 5

The Mie scattering model (blue curve) provides strong fits to measurements (black curve) of (a) RDX and (b) PETN particles on glass.

2.3.2.

Transfer matrix model for liquids

The TM method is a standard approach for calculating the reflection and transmission properties through a stack of uniform thin films with each layer having a known complex refractive index thickness [see Fig. 4(b)]. Recall the complex optical constant is denoted by $\tilde{n}$. We define the optical constants at each uniform thin film layer interface as: ${\tilde{n}}_{\text{air}}$ for air, ${\tilde{n}}_{\mathrm{chem}}$ for the chemical, and ${\tilde{n}}_{\text{sub}}$ for the substrate. The complex reflection coefficients at each layer interface are defined by $r_1$ and $r_2$ (for the single chemical case)^16–18,26

Eq. (4)

$$r_1(\lambda )=\frac{{\tilde{n}}_{\text{air}}^{*}-{\tilde{n}}_{\mathrm{chem}}^{*}}{{\tilde{n}}_{\text{air}}^{*}+{\tilde{n}}_{\mathrm{chem}}^{*}},$$

and

Eq. (5)

$$r_2(\lambda )=\frac{{\tilde{n}}_{\mathrm{chem}}^{*}-{\tilde{n}}_{\text{sub}}^{*}}{{\tilde{n}}_{\mathrm{chem}}^{*}+{\tilde{n}}_{\text{sub}}^{*}},$$

where * indicates the complex conjugate. Let $r_3$ be defined as

Eq. (6)

$$r_3(\lambda )=\frac{r_1(\lambda )+r_2(\lambda )\exp (-2i\delta )}{1+r_1(\lambda )r_2(\lambda )\exp (-2i\delta )},$$

where $\delta$ is the optical depth through the chemical film.^17,18,26 Finally, the reflectance from a uniform thin film is given as^14,15

Eq. (7)

$$R_F(\lambda )=r_3(\lambda )r_3{(\lambda )}^{*}.$$

Note that $r_1$ and $r_2$ are calculated at normal incidence. This is an approximation, as knowledge of the incidence angle is not guaranteed in standoff active spectroscopy applications.

We previously used this model as part of a hyperspectral imaging simulator. As shown in Fig. 6, the simulator was able to duplicate the main characteristics of the hyperspectral image of a sample that depicted the logo of IARPA using two different chemicals.¹⁹ In particular, the TM model effectively predicted the spectra of two chemicals, silicone oil and triethyl phosphate (TEP), on a plastic surface is shown in Fig. 6.

Fig. 6

The TM model accurately captures the spectral shape and features of thin films of silicone oil and TEP on HDPE. (a) A cartoon representation of the sample, (b) the measured spectra, and (c) the simulated representation.

4.1.

Description of the Measured Data

Various substrate samples with chemical contaminations at a range of concentrations were prepared and provided by Johns Hopkins University Applied Physical Laboratory (JHU/APL). The solid chemicals were first dissolved in a solvent and then evenly airbrushed over the substrates using a mechanical arm. The active MIR hyperspectral reflectance measurements were collected by the system developed by Block MEMS for the IARPA SILMARILS program.^5,6,41 In total, JHU/APL prepared six different chemicals on eight different substrates, though not all of the chemicals were used on all of the substrates. To avoid biasing the classification results for a particular chemical–substrate combination, we limit the data used for these experiments to those chemicals and substrates for which we have at least one measurement for each unique pair (three chemicals and four substrates in this case). The breakdown of measured samples per chemical, substrate, and concentration are shown in Table 1.

Table 1

The number of measured samples and their concentrations for each unique chemical/substrate combination under consideration in this study.

As shown in Table 1, there are an unequal number of measurements for each chemical–substrate class. This can lead to biased parameter tuning and results if not addressed properly. For this research, we focus on overall performance metrics. That is, fit and classification performance results are averaged within each class prior to averaging performance results across the different classes in Table 1.

Recall from Secs. 2.3.1 and 3 that the user must define several parameters before applying the Mie scattering or STM models: the particle diameter mean and standard deviation, $\mu$ and $\sigma$, respectively, and the substrate scale factor, ${\mathrm{SF}}_{\text{sub}}$. The selected parameters should be relevant and physically realistic for the trace chemical detection application. Ideally, a range of values for each parameter should be used such that the simulations capture the full variability. Solid particles with a mean diameter of $10\mu \mathrm{m}$ were dissolved to produce the samples discussed. Though dissolved particles may be $<0.1\mu \mathrm{m}$ in diameter, the scattering from such particles is negligible in the MIR where the illumination waves are on the order of $1\mu \mathrm{m}$. Similarly, we only consider particle diameter standard deviations of 0.1 to $1.25\mu \mathrm{m}$. The substrate scale factor in the Mie scattering and STM models provides a proxy for the substrate BRDF as this information is not necessarily readily available. We consider a scale factor ranging from 0.1 to 10.0, which is the range of BRDF values measured from a clean sample of high-density polyethylene (HDPE) (measurements provided by PNNL under the IARPA SILMARILS program). These parameter ranges are summarized in Table 2. Both the simulated and real data used for this analysis consist of 200 wavenumbers from 980 to $1290{\mathrm{cm}}^{-1}$ with an approximate $1.56{\mathrm{cm}}^{-1}$ spacing. Reflectance signatures are normalized to avoid any calibration inconsistencies as well as to show the differences in the overall shape and location of spectral features, which are the discriminating features in active spectroscopy detection and classification applications.

Table 2

Tunable model parameters and their range of values used for our experiments.

4.2.

Comparisons of Simulated and Measured Data

The plots in Fig. 7 provide qualitative comparisons of the synthetic spectra generated by the STM simulation tool (gray curves) with their corresponding measurements (black curves). The variability in the simulated spectra can be attributed to the wide range of parameter values summarized in Table 2. Of perhaps more interest to this research is the quantitative comparison of the abilities of the various signature models discussed to accurately model real measured reflectance signatures. For this comparison, we calculate the overall root mean square error (RMSE) of the outputs of the Mie scattering, TM, and STM models with their corresponding measurements.

Fig. 7

Comparisons of STM-simulated spectra (gray curves) with their corresponding measurements (black curves). All data are shown in normalized reflectance units.

We calculate overall RMSE while varying each of the model parameters to capture the sensitivity of the models considered. Using Eq. (1), we find that the fill factor is only $<1$ for mean particle diameters $>2\mu \mathrm{m}$. Therefore, the substrate scale factor has no effect for small particle sizes [see Eqs. (3) and (8)]. We begin the sensitivity analysis by varying the mean particle diameter, $\mu$, within the range in Table 2. The standard deviation, $\sigma$, and substrate scale factor, ${\mathrm{SF}}_{\text{sub}}$, are set to $0.5\mu \mathrm{m}$ and 1.0, respectively. The average RMSE for each of the reflectance models is shown as a function of $\mu$ in the left-hand plot in Fig. 8. Because the TM model only varies with the input concentration, its RMSE does not vary with $\mu$. STM outperforms the other two models in terms of overall RMSE. Both STM and Mie scattering demonstrate minimum RMSE for mean particle diameters $>2\mu \mathrm{m}$ (i.e., fill factor $<1$) indicating that the uniform thin film assumption of the TM model is less valid for residue samples.

Fig. 8

Overall RMSE as a function of each of the model parameters for each of the three signature models considered. The STM model (red dotted curves) consistently outperforms the other models in terms of overall fit to the measured data.

We continue the sensitivity analysis while allowing $\sigma$ to vary. For this result, $\mu$ is set to $5.46\mu \mathrm{m}$ to jointly minimize the RMSE for both the Mie scattering and STM models. Figure 8(b) shows that the particle size standard deviation has less effect on the RMSE than the mean particle size. Finally, we set $\sigma$ to $1.14\mu \mathrm{m}$ to minimize the RMSE of both Mie scattering and STM and allow ${\mathrm{SF}}_{\text{sub}}$ to vary. The result is shown in Fig. 8(c). Overall, the STM model achieves an average 25% reduction in overall RMSE.

The RMSE provides a measurement of the overall fit of the simulated data to the measured data, but does not tell us how well the models capture the phenomenology of the samples. Figure 9 compares example spectra simulated by each of the models with real measurements. As with the previous results, we select model parameters that jointly minimized the RMSE for both the Mie scattering and STM models for this result: $\mu =5.46\mu \mathrm{m}$, $\sigma =1.14\mu \mathrm{m}$, and ${\mathrm{SF}}_{\text{sub}}=1.0$. The examples include all three chemicals on three different substrates. The Mie scattering and TM models capture some of the spectral features in each sample, but the STM models provides an overall better match to the phenomenology. Some of the differences between the STM and TM models in particular appear minor (e.g., the feature at $1025{\mathrm{cm}}^{-1}$ in the pentaerythritol plot). Recall that an active spectrometer has many applications, and the spectral library often contains hundreds or more chemicals. Minor differences such as this are critical for accurate classification when considering many chemicals that share the same or very similar features.

Fig. 9

Comparisons of measured data (black solid curves) with example spectra generated by the three signature models considered. The model parameters were selected to jointly minimize the overall RMSE of all models. The STM model (red dotted curves) captures more of the features and provides an overall better fit to the phenomenology of the measured data than the Mie scattering (blue dashed curves) and TM (green dashed curves).

4.3.

Classification Results on Real Data

Next, we test the ability of the reflectance models to improve classification results on real measurements. For these results, we use RMSE as a classification metric. We compare each measurement to the full spectral library generated by each model and select the chemical that minimizes the RMSE. The model parameters are varied in the same way as in Sec. 4.2. The overall classification accuracy as a function of each model parameter is shown for each model in Fig. 10. Again, we see that the Mie scattering and STM models perform better for larger particle sizes, suggesting that substrate effects must be considered for capturing the phenomenology of residue samples. Again, performance is averaged over each class prior to averaging across the different chemical–substrate classes. On average, STM achieves an overall classification accuracy 12% to 15% greater than TM and Mie scattering, respectively. The peak classification accuracy of the STM model is 74% as compared to the TM and Mie scattering models at 46% and 64%, respectively.

Fig. 10

Overall classification accuracy as a function of each of the model parameters for each of the three signature models considered. The STM model (red dotted curves) consistently outperforms the other models in terms of overall fit to the measured data. In particular, both the STM and Mie scattering models achieve higher accuracy for larger particle diameters (fill factor $<1$), indicating that the nonuniform film assumption is valid.