# Fast and Effective Techniques for LWIR Radiative Transfer Modeling: A Dimension-Reduction Approach

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## Abstract

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## 1. Introduction

- Employing machine learning techniques which: (1) are computationally faster than correlated-k calculation methods; (2) reduce the dimension of both the TUD and atmospheric state vectors; (3) produce the desirable latent-space-similarity property such that small deviations in the low-dimension latent space result in small deviations in the high-dimension TUD
- Developing a data augmentation method using PCA and Gaussian mixture models (GMMs) on real atmospheric measurements that lead to improved model training and generalizability
- Improving machine learning model training by introducing a physics-based loss function which encourages better fit models than traditional loss functions based on mean squared error
- Demonstrating an effective autoencoder (AE) pre-training strategy that leverages the local-similarity properties of the latent space to reproduce TUDs from atmospheric state vectors

#### Background

## 2. Methodology

#### 2.1. Data

#### 2.2. TUD Dimension-Reduction Techniques

#### 2.3. Metrics

#### 2.4. Radiative Transfer Modeling

#### 2.5. Atmospheric Measurement Augmentation

## 3. Results and Discussion

#### 3.1. Atmospheric Measurement Augmentation

#### 3.2. At-Sensor Loss Constraint

#### 3.3. Dimension-Reduction Performance

#### 3.4. Radiative Transfer Modeling

#### 3.5. Atmospheric Measurement Estimation

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The high-resolution LBLRTM transmittance, upwelling and downwelling vectors are shown with their downsampled counterparts for the Mako LWIR sensor. The downsampled vectors are the data used in the remainder of this study.

**Figure 2.**An example AE model where the TUD vectors are compressed through one or more encoder layers to a low-dimensional latent space. The decoder transforms the low-dimensional latent space back to the original TUD vector.

**Figure 3.**The first two principal components using a 6 component PCA model with the augmented TIGR data. Hot, humid atmospheric conditions vary with component 1 and 2 while cold, dry atmospheres are more dependent on component 2.

**Figure 4.**Autoencoder latent space when trained using 2 components. The points are scattered throughout the latent space with an overall clustering of similar atmospheric conditions. Both components appear dependent on surface temperature. Component 1 also appears more dependent on total water content versus component 2.

**Figure 5.**The RT model is created by first creating a low-dimensional representation of the TUD vectors with acceptable at-sensor radiance reconstruction errors. The latent space and decoder parameters are locked, and a sampling model is fit to correctly identify the low-dimensional components to map atmospheric measurements to their corresponding TUD vectors. This diagram is specific for the SAE approach, but the encoder and decoder can be replaced with equivalent PCA transformations.

**Figure 7.**Dimension-reduction techniques show improved results when using the augmented TIGR data. All models reduce the input data to 5 components in this plot; however, the number of components is an additional hyperparameter that will be considered later.

**Figure 8.**Comparison of SAE performance when using strictly MSE loss or the loss function described in Equation (11). Updating the model using information from the at-sensor radiance error improves reconstruction performance for reflective materials. The error bars represent the performance standard deviation when training multiple networks with identical architectures and random weight initialization.

**Figure 9.**Varying the number of latent components and calculating the area under the RMSE curve shown in Figure 8 shows how many components are necessary to reconstruct the TIGR data. Results are plotted for the validation set consisting of 158 samples using the augmented data for training and the loss function outlined in Equation (11) for SAE training. The PCA error bars correspond to performance standard deviation when using 5-fold cross-validation. The SAE errors bars show the performance standard deviation when random weight initialization is used.

**Figure 10.**The performance of the RT models is shown as a function of emissivity where it is clear the SAE derived RT models create a latent space that is easier to sample with a small neural network. In all cases performance improves as materials become more emissive since downwelling radiance plays a less significant role in these cases. The 15 component PCA model is also shown, where sampling the 15 components correctly becomes a complex problem resulting in lower overall performance.

**Figure 11.**The at-sensor radiance RMSE RT model errors for 176 test TIGR samples as a function of surface emissivity expressed in spectral brightness temperature. Model errors decrease with increasing emissivity, consistent with the findings in Figure 10.

**Figure 12.**The top 3 panels are the predicted TUD components plotted against the LBLRTM generated TUD components. The predicted TUD components were generated by optimizing the 4 latent components. The bottom 3 panels are the TUD component residual curves, showing low error across most spectral channels.

**Figure 13.**Predicted atmospheric measurements compared to the TIGR atmospheric measurements. The predicted atmospheric measurements will produce a close match to a given TUD vector but do show some deviations from the original TIGR measurement, specifically at high altitudes. Fortunately, high altitude error has less impact on the at-sensor radiance error because of lower air density.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Westing, N.; Borghetti, B.; Gross, K.C. Fast and Effective Techniques for LWIR Radiative Transfer Modeling: A Dimension-Reduction Approach. *Remote Sens.* **2019**, *11*, 1866.
https://doi.org/10.3390/rs11161866

**AMA Style**

Westing N, Borghetti B, Gross KC. Fast and Effective Techniques for LWIR Radiative Transfer Modeling: A Dimension-Reduction Approach. *Remote Sensing*. 2019; 11(16):1866.
https://doi.org/10.3390/rs11161866

**Chicago/Turabian Style**

Westing, Nicholas, Brett Borghetti, and Kevin C. Gross. 2019. "Fast and Effective Techniques for LWIR Radiative Transfer Modeling: A Dimension-Reduction Approach" *Remote Sensing* 11, no. 16: 1866.
https://doi.org/10.3390/rs11161866