# Spatial Surface Reflectance Retrievals for Visible/Shortwave Infrared Remote Sensing via Gaussian Process Priors

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## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. Optimal Estimation of Surface Reflectance

#### 2.2. Forward Model and Uncertainty

#### 2.3. Baseline Optimal Retrievals

#### 2.4. Prior

Algorithm 1: Simplified Optimal Spatial Inversion. |

#### 2.5. Naive Spatial Retrieval Structure

#### 2.6. Efficient Implementation

**x**denote the concatenated version of the latent state vector. For the update step shown in Equation (2) with $\alpha \approx {\left[{\nabla}_{x}^{2}Q\right]}^{-1}$ representing the constant matrix that results from the Levenberg-Marquardt approximation in (A3), we have

#### 2.7. Complexity

#### 2.8. Other Practical Considerations

## 3. Results

#### 3.1. Simulation Study

- Sample multiple surface reflectance states of vegetation, the most common of the priors described in Section 2.4. The atmospheric states are correlated according to their predetermined orientation following the technique outlined in Section 2.5.
- Simulate noisy AVIRIS-NG instrument radiance measurements corresponding to the sampled joint state using the built-in methods and configuration of the ISOFIT code [23]; the noise model is described in Section 2.2.
- Invert the simulated radiance measurements according to the implementation outlined in Section 2.6. Setting prior cross-pixel covariances to 0 results in individual retrievals as a special case.

Algorithm 2: Simulation Procedure: generate n pixels, compute correlated radiances, and invert. Repeat ${m}_{iter}$ times. |

#### 3.2. Application to Real Data

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

VSWIR | Visible/Shortwave InfraRed |

AVIRIS-NG | Airborne Visible-Infrared Imaging Spectrometer-Next Generation |

ATREM | Atmosphere Removal |

LUT | Look-up Table |

UQ | Uncertainty Quantification |

OE | Optimal Estimation |

AOD | Aerosol Optical Depth |

RTM | Radiative Transfer Model |

## Appendix A. Appendix/Proofs

#### Appendix A.1. Iterative Optimization

#### Appendix A.2. Parameter Estimation

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**Figure 1.**Representative radiance and reflectance spectra. Red, green, and blue lines indicate visible color channels.

**Figure 2.**Inversions of simulated data showing the water vapor (units of g/cm${}^{2}$) and aerosol optical depth estimates across 10 pixels in 1D. The retrieved fields are more realistic for spatial (Spatial_Post) than for individual retrievals (Posterior). The post-hoc smoothing (Smooth_Post) can improve the individual retrievals but cannot overcome the bias.

**Figure 3.**Inversions of simulated data showing the aerosol optical depth estimates across 9 pixels on a $3\times 3$ grid. The spatial prior is more accurate than the independent inversions and comparable to post-hoc smoothing.

**Figure 4.**Inversions of simulated data showing the water vapor estimates (units of g/cm${}^{2}$) across 9 pixels on a $3\times 3$ grid. The spatial field better represents the truth; smoothing cannot overcome the bias in the independent estimates.

**Figure 5.**Prior score plots for 25 simulated realizations. The posterior estimates for the spatial model are usually closer to their priors than the independent models. The effect is weaker for the 2D case, suggesting that the improvement tends to be most pronounced with highly correlated data. (

**a**) Prior score results for a 1D array of 10 pixels. (

**b**) Prior score results for a 2D grid of 9 pixels.

**Figure 6.**The top box plots show the difference in log score between the spatial and independent models across 25 simulations. The $(25\%,50\%,75\%)$ quantile values are $(21.8,52.8,69.1)$ for the 1D case and $(-19.2,10.4,39.6)$ for the 2D case. The two lower box plots show the difference in log score between the spatial and smoothed independent model. Quantiles are $(-37.9,91.8,128.0)$ and $(-3326.4,16.9,67.3)$ for 1D and 2D respectively. Note the discontinuous x-axis.

**Figure 7.**The aerosol optical depth prediction for validation data at Ivanpah. The predictions are effectively identical, but the spatial retrievals are closer to the in situ measurement of 0.043.

**Figure 8.**The surface reflectance profiles are nearly identical for the Cuprite data, with scaling changes due to the estimation of atmospheric parameters. This suggests that independent inversions may be overestimating reflectance. Pixels 105, 254, and 255 are adjacent and the reflectance can be interpreted as a percent, so at a particular wavelength a reflectance of 0.4 means 40% of the incoming radiant energy is reflected.

**Figure 9.**The water vapor (units of g/cm${}^{2}$) estimates are noticeably smoother under the spatial models. The predicted fields are qualitatively more realistic and are a principled alternative to post-hoc smoothing.

**Figure 10.**For the Cuprite dataset, the aerosol optical depth prediction is susceptible to the surface state prediction (

**bottom right**), but smoothing with a spatial prior decreases the noise. The

**top left**figure shows inversions done 1 pixel at a time, compared to stripes of 1 × 5 pixels in the

**top right**figure and squares of 2 × 2 pixels in the

**bottom left**figure. The

**bottom right**figure shows topography visible at a single wavelength, 1600 nm.

**Figure 11.**A retrieved aerosol field under a spatial model is smoother than the independent retrievals and spreads out large estimates.

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## Share and Cite

**MDPI and ACS Style**

Zilber, D.; Thompson, D.R.; Katzfuss, M.; Natraj, V.; Hobbs, J.; Braverman, A.
Spatial Surface Reflectance Retrievals for Visible/Shortwave Infrared Remote Sensing via Gaussian Process Priors. *Remote Sens.* **2022**, *14*, 2183.
https://doi.org/10.3390/rs14092183

**AMA Style**

Zilber D, Thompson DR, Katzfuss M, Natraj V, Hobbs J, Braverman A.
Spatial Surface Reflectance Retrievals for Visible/Shortwave Infrared Remote Sensing via Gaussian Process Priors. *Remote Sensing*. 2022; 14(9):2183.
https://doi.org/10.3390/rs14092183

**Chicago/Turabian Style**

Zilber, Daniel, David R. Thompson, Matthias Katzfuss, Vijay Natraj, Jonathan Hobbs, and Amy Braverman.
2022. "Spatial Surface Reflectance Retrievals for Visible/Shortwave Infrared Remote Sensing via Gaussian Process Priors" *Remote Sensing* 14, no. 9: 2183.
https://doi.org/10.3390/rs14092183