# Efficient Emulation of Radiative Transfer Codes Using Gaussian Processes and Application to Land Surface Parameter Inferences

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## Abstract

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## 1. Introduction

## 2. Gaussian Process Emulator

#### Extension to Full Spectrum Emulation

## 3. GP Emulation Examples

#### 3.1. Emulating Soil-Leaf-Canopy RT Models

**Table 1.**Parameters for PROSAIL and SEMIDISCRETE, their symbols, units, quasi-linearising transformations and boundaries (both in real and in transformed space). (${}^{*}$ Only PROSAIL, ${}^{\u2020}$ Only Semidiscrete.)

Parameter | Symbol | Units | Minimum | Maximum | Transformation | Transformed Min | Transformed Max |
---|---|---|---|---|---|---|---|

Leaf layers | xleafn | – | 0.8 | 2.5 | – | – | – |

Leaf chlorophyll
concentration | Cab | μ g·cm ${}^{-2}$ | 0.2 | 77 | $exp(-Cab/100)$ | 0.46 | 1 |

Leaf carotenoid concentration | Car | μ g·cm ${}^{-2}$ | 0 | 15 | $exp(-Ccar/100)$ | 0.95 | 1 |

Senescent fraction | Csen | – | 0 | 1 | – | – | – |

Equivalent water thickness | Cw | cm | 0.0043 | 0.0753 | $exp(-50\xb7Cw)$ | 0.028 | 0.81 |

Leaf dry matter | Cm | g·cm ${}^{-2}$ | 0.0017 | 0.0331 | $exp(-100\xb7Cm)$ | 0.037 | 0.84 |

Leaf area index | LAI | m ${}^{2}$·m${}^{-2}$ | 0 | 8 | $exp(-LAI/2)$ | 0.05 | 1 |

Average Leaf Angle | ALA | ${}^{\circ}$ | 0 | 90 | $ALA/90$ | 0.44 | 0.56 |

Soil brightness scalar ${}^{*}$ | Bs | – | 0 | 2 | – | – | – |

Soil moisture endmember ${}^{*}$ | Ps | – | 0 | 1 | – | – | – |

Weight of the first Price function ${}^{\u2020}$ | P1 | – | −0.5 | 1 | – | – | – |

Weight of the second Price function ${}^{\u2020}$ | P2 | – | −0.5 | 1 | – | – | – |

**Table 2.**Results of the validation for the SEMIDISCRETE emulator using an independent set of 1000 model realisations over the MODIS bands. The emulator was created by using 300 forward simulator model runs, selected with a Latin hypercube sampling design between the boundaries indicated in Table 1.

MODIS Band | Slope | Intercept | R | RMSE | MAE |
---|---|---|---|---|---|

1 | 1.002 | 1.222e-05 | 0.9995 | 9.774e-04 | 5.175e-03 |

2 | 0.999 | 1.562e-04 | 0.9998 | 5.602e-04 | 1.063e-02 |

3 | 1.004 | −3.362e-05 | 0.9996 | 9.181e-04 | 2.371e-03 |

4 | 1.000 | 2.784e-05 | 0.9999 | 5.255e-04 | 2.048e-03 |

5 | 0.993 | 1.591e-03 | 0.9990 | 1.437e-03 | 3.588e-02 |

6 | 0.998 | 4.182e-04 | 0.9995 | 9.490e-04 | 1.587e-02 |

7 | 1.000 | 1.183e-04 | 0.9997 | 8.194e-04 | 1.135e-02 |

**Figure 1.**Emulation performance for SEMIDISCRETE over the seven MODIS bands. The emulator was trained with 300 forward model runs, and validated with an independently produced set of 1000 parameters. Acquisition geometry was set to nadir illumination and ${30}^{\circ}$ viewing zenith.

**Figure 2.**Emulation performance for PROSAIL over the seven MODIS bands. The emulator was trained with 300 forward model runs, and validated with an independently produced set of 1000 parameters. Acquisition geometry was set to nadir illumination and ${30}^{\circ}$ viewing zenith.

**Table 3.**Results of the validation for the PROSAIL emulator using an independent set of 1000 model realisations over the MODIS bands. The emulator was created by using 300 forward simulator model runs, selected with a Latin hypercube sampling design between the boundaries indicated in Table 1.

MODIS Band | Slope | Intercept | R | RMSE | MAE |
---|---|---|---|---|---|

1 | 0.984 | 0.000 | 0.999 | 1.669e-03 | 3.679e-02 |

2 | 1.000 | −0.000 | 1.000 | 2.681e-04 | 1.307e-02 |

3 | 0.947 | 0.001 | 0.990 | 4.349e-03 | 4.678e-02 |

4 | 0.977 | 0.001 | 0.998 | 1.987e-03 | 5.585e-02 |

5 | 1.001 | −0.000 | 1.000 | 3.523e-04 | 1.722e-02 |

6 | 1.000 | 0.000 | 1.000 | 3.210e-04 | 7.791e-03 |

7 | 1.000 | 0.000 | 1.000 | 2.416e-04 | 2.392e-03 |

#### 3.2. Emulation of a Coupled Soil-Leaf-Canopy-Atmosphere RT Model Combination: PROSAIL and 6S

**Figure 3.**Validation result of the single band coupled PROSAIL and 6S models of top-of-atmosphere reflectance over the MODIS bands. The coupling is done under the assumption that the land surface is Lambertian, and using a continental aerosol model. The emulator uses the PROSAIL parameters as inputs, extended by aerosol optical thickness (AOT), atmospheric water vapour content (WVC) and ozone concentration (${O}_{3}$). The validation has been carried out on a set of 1000 parameter values drawn from uniform independent distribution over the parameter ranges.

**Table 4.**Results of the validation for the coupled PROSAIL + 6S emulator using an independent set of 1000 model realisations over the MODIS bands. The emulators were trained running the PROSAIL + 6S model combination 400 times over a sampling pattern determined by Latin hypercube design. The coupling between the surface and atmosphere is assumed Lambertian.

MODIS Band | Slope | Intercept | R | RMSE | MAE |
---|---|---|---|---|---|

1 | 0.985 | 0.000 | 0.998 | 2.087e-03 | 3.889e-02 |

2 | 1.001 | −0.000 | 1.000 | 1.888e-04 | 5.694e-03 |

3 | 1.061 | −0.001 | 0.989 | 4.996e-03 | 2.635e-02 |

4 | 0.981 | 0.001 | 0.998 | 1.730e-03 | 5.136e-02 |

5 | 0.999 | 0.000 | 1.000 | 2.282e-04 | 5.256e-03 |

6 | 1.000 | 0.000 | 1.000 | 2.342e-04 | 4.954e-03 |

7 | 0.999 | 0.000 | 1.000 | 2.885e-04 | 1.857e-03 |

#### 3.3. Spectral Emulation of a Coupled Soil-Leaf-Canopy RT Model Combination

**Figure 4.**(Top panel) A set of ten simulated PROSAIL spectra (full lines) and their emulated spectra using 250 forward model runs as a training set (dashed lines). The topmost emulated spectrum has a grey area around it showing the $\pm \sigma $ emulation error. (Bottom panel) Distribution of the residuals (full model minus emulator) of the 1000 model realisations validation dataset. We show the mean (orange line), median (green line), and the 5%–95% (light grey) and 25%–75% (dark grey) percentile ranges.

**Figure 5.**The emulated gradient of the PROSAIL RT model (estimated using a finite difference approximation with a step size of which was set to ${10}^{-5}$ of the transformed parameter range). We show the gradient with respect to leaf chlorophyll concentration at 650 nm (orange triangles) and with respect to $LAI$ at 850 nm (green triangles). The gradient is calculated around the 1000 independent samples used to validate the model in Figure 4.

#### 3.4. Spectral Emulation of Atmospheric Effects

**Figure 6.**Sample spectra for $1/{x}_{a}$, ${x}_{b}$ and ${x}_{c}$ simulated with 6S, assuming “Continental” aerosol type, nadir looking illumination geometry and view zenith of ${30}^{\circ}$. The simulation was done for day of year 138 (mid-may), and shows two spectra for two different atmospheric compositions, having different water vapour, ozone and aerosol optical depths.

#### 3.5. Full BRDF Coupling of a Soil-Leaf-Canopy Model and Atmospheric Spectral RT Model

**Figure 7.**Three examples of emulation of the 6S RT model. The three columns show three different input parameter sets (indicated at the top). The set-up was for day 138 (mid-may), with nadir illumination geometry and view zenith or ${30}^{\circ}$, aerosol properties are selected as “Continental”, and we show the output spectra from the model and the emulation (note that these particular model configurations were not part of the data used to train the emulators).

**Figure 8.**Validation of the 6S spectral emulation. The black line shows the mean difference over the 500 sample validation set, whereas the grey areas show the RMSE.

**Figure 9.**Three examples of emulation of the coupled PROSAIL and 6S RT models. The three traces show the full model (full line) for a particular set of input parameters, and the emulator prediction for the same input parameters (dashed lines). The set-up was for day 138 (mid-may), with nadir illumination geometry and view zenith or ${30}^{\circ}$, aerosol properties are selected as “Continental”.

**Figure 10.**Validation of the coupled PROSAIL+6S spectral emulation. The mean RMSE (blue line) shows the mean RMSE, whereas the grey shaded areas show different quantiles (95, 90 and 50%).

## 4. Examples of the Use of GPs in Inverse Problems in Remote Sensing

#### 4.1. The Synthetic “Satellite” Observations

Parameter | Value | ${p}_{0}$ | ${p}_{1}$ | ${p}_{2}$ | ${p}_{3}$ | ${p}_{4}$ | ${p}_{5}$ |
---|---|---|---|---|---|---|---|

N | 2.1 | – | – | – | – | – | – |

${C}_{ab}$ | 60 | 5 | 55 | -0.3 | 122 | 0.1 | 240 |

${C}_{car}$ | 7 | – | – | – | – | – | – |

${C}_{sen}$ | 0.5 | – | – | – | – | – | – |

${C}_{w}$ | 0.0176 | 0.004 | 0.043 | −0.31 | 121 | 0.07 | 235 |

${C}_{m}$ | 0.002 | – | – | – | – | – | – |

$LAI$ | 2 | 0.1 | 5 | −0.29 | 120 | 0.1 | 240 |

$ALA$ | 70 | – | – | – | – | – | – |

${B}_{s}$ | 1.0 | – | – | – | – | – | – |

${P}_{s}$ | 0.3 | – | – | – | – | – | – |

**Figure 11.**Temporal evolution of the forward modelled reflectances for the three sensors considered in this work over a year.

**Table 6.**Description of the different parameters used for simulating the different sensors in the synthetic experiments. Noise model parameters are in units of reflectance.

Parameter | Sentinel2/MSI | Sentinel3/SLSTR | Proba-V |
---|---|---|---|

Number of bands | 12 | 5 | 4 |

Revisit frequency | 5 days | 2 days | 2 days |

Overpass time | 10:30 AM | 10:00 AM | 11:00 AM |

Variation in viewing zenith angle (VZA) | ${7}^{\circ}$ | ${0}^{\circ}$ | ${50}^{\circ}$ |

Percentage valid observations | 30% | 30% | 30% |

Average temporal window | 12 days | 10 days | 11 days |

Noise model slope m | 0.0042 | 0.0075 | 0.108 |

Noise model intercept c | 0.0028 | 0.005 | 0.007 |

#### 4.2. Generic Strategy for the Inverse Problem

**Figure 12.**(Top) Shows the difference between the cost function for three different acquisition periods using the Sentinel3/SLSTR synthetic data. The full lines show the cost function value as a function of $LAI$ using the full PROSAIL model, whereas the dashed lines show the value of the cost function using the emulators (and using the emulator with acquisition geometry closest to the observations). The circle represents the minimum of the cost function, and the grey call out is the mean value of $LAI$ for each period. (Bottom) The difference between the gradient of the cost function calculated with the full PROSAIL model (approximated by finite differences, full line), and the gradient approximated by using the emulators (dashed line) for the same cost functions shown in the top panel.

#### Solving for the Posterior

#### 4.3. Inversion Using Markov Chain Monte Carlo (MCMC)

#### 4.4. Variational DA Solution: EO-LDAS Framework

**Figure 13.**Results of inferred posterior parameter distributions for the inversion with simple gradient descent algorithm and NUTS MCMC sampler, in both cases using GP emulators in lieu of the full RT model. The observations of every eight day period have been inverted assuming that the parameters are constant within the inversion window. A Gaussian uninformative prior is used to constrain the observations. The dashed line shows the ground truth, the dots show the value of the minimisation, and the grey areas represent the posterior parameter distribution. The dash indicates the median of the posterior parameter distribution.

**Figure 14.**Results of inferred posterior parameter distributions using the

`eoldas_ng`package, supplementing the fit to the observations by a prior distribution and a first order regularisation model. The orange line shows the value of the maximum a posteriori (MAP) estimate, with the grey area representing the 5%–95% posterior interval. The green dashed line shows the truth.

#### 4.5. Simple Particle Filter

**Figure 15.**Evolution of the land surface parameters (from top to bottom: leaf chlorophyll concentration, leaf equivalent water thickness and leaf area index) derived from the synthetic observations introduced in Section 4.1. The parameters were inferred using a particle filter. The dynamic model used is a first order regularisation model, and we have used 10,000 particles. The plots show the mean particle, the 5%–95% and 25%–75 % posterior intervals, as well as the ground truth.

## 5. Discussion and Outlook

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Gómez-Dans, J.L.; Lewis, P.E.; Disney, M. Efficient Emulation of Radiative Transfer Codes Using Gaussian Processes and Application to Land Surface Parameter Inferences. *Remote Sens.* **2016**, *8*, 119.
https://doi.org/10.3390/rs8020119

**AMA Style**

Gómez-Dans JL, Lewis PE, Disney M. Efficient Emulation of Radiative Transfer Codes Using Gaussian Processes and Application to Land Surface Parameter Inferences. *Remote Sensing*. 2016; 8(2):119.
https://doi.org/10.3390/rs8020119

**Chicago/Turabian Style**

Gómez-Dans, José Luis, Philip Edward Lewis, and Mathias Disney. 2016. "Efficient Emulation of Radiative Transfer Codes Using Gaussian Processes and Application to Land Surface Parameter Inferences" *Remote Sensing* 8, no. 2: 119.
https://doi.org/10.3390/rs8020119