# Green LAI Mapping and Cloud Gap-Filling Using Gaussian Process Regression in Google Earth Engine

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. GPR Formulation for Vector Input

- Length-scale ${\sigma}_{b}$ describes the smoothness of $f\left(\mathbf{x}\right)$ dependence along the dimension b. Small ${\sigma}_{b}$ means $f\left(\mathbf{x}\right)$ changes quickly for variations of $\mathbf{x}$ along b; large values denote slow changes w.r.t. the b dimension. Alternatively, the inverse of ${\sigma}_{b}$ represents the relevance of band b in the prediction process. Intuitively, high values of ${\sigma}_{b}$ mean that relations largely extend along that band hence suggesting a lower informative content.
- Signal variance ${\sigma}_{s}^{2}$ is a scaling factor. It determines variation of $f\left(\mathbf{x}\right)$ from its mean. Small value of ${\sigma}_{s}^{2}$ characterize functions that stay close to their mean value, larger values allow more variation. If ${\sigma}_{s}^{2}$ is too large, the modeled function will be free to chase outliers.
- Noise variance ${\sigma}_{n}^{2}$ is formally not a part of the covariance function itself. It is used by the Gaussian process model to account for noise present in training data.

#### 2.2. GPR Formulation for Space-Spectrum (3D) Input

**X**; the third one is related to the information contained in the trained model, but not on the new input to be used for prediction. Accordingly, we can give one further step towards parallelization, and extend the estimation of ${\mathit{k}}_{*}$ to a $B\times M$ matrix ${\mathit{X}}_{*}=[{\mathit{x}}_{*1},..,{\mathit{x}}_{*M}]$ containing M new input vectors, being M the number of pixels of the multispectral image to be processed at once. Defining ${\mathit{K}}_{*}=[{\mathit{k}}_{*}^{1},{\mathit{k}}_{*}^{2},..{\mathit{k}}_{*}^{M}]$, it results:

#### 2.3. GPR Formulation for Space-Time (3D) Input

## 3. GPR Models Training

#### 3.1. Green LAI Model

#### 3.2. Gapfilling Model

## 4. GEE Implementation and Assessment

#### 4.1. LAI${}_{\mathit{G}}$ Mapping

#### 4.2. LAI${}_{\mathit{G}}$ Time Series Gapfilling

**t**common to all the pixels of S2 image collection over the same tile, we first convert each acquisition date in a numeric format. We choose to express it in terms of number of days with respect to the GPS time reference, i.e., 1 January 1970. As an example, according to this criterion the date 20 July 2019 becomes the number 18,097. Afterwards, we add (1) a new constant band to each S2 image of the collection corresponding to its numeric date (band ${\mathit{t}}_{*}$), and (2) a copy of this new band where we set to 0 (i.e., not masked anymore) all the pixels labeled as cloud. Cloudy pixel identify backs up on the S2 classification product [56] distributed along with L2A multispectral reflectance (band ${\mathit{t}}_{msk}$). Similarly, we set to zero also the reflectance values of these cloudy pixels in each element of the S2 collection. Summing up, in the time series of a generic pixels from the S2 collection, the time associated to cloudy samples is 0 as well as their multispectral radiometric values. Taking into account that the input time series for calculating the temporal Covariance matrix ${\mathit{K}}_{t}$ must not be time ordered, the contribution of all these cloudy captures to the estimation of LAI${}_{G}$ at any S2 acquisition date, i.e., the value of $k(0,{t}_{i})$ in Equation (10), becomes negligible. This is shown in Figure 5, where we analyze the LAIG time series of a pixel belonging to a vegetated area in Castille and Leon, Spain. As expected, the collection presents both meaningful (cloud-free) and non-valid (cloudy) samples. These two cases are described by the meaningful (MTS) and cloudy (CD) time series points. CD dates correspond to S2 acquisitions where the chosen pixel but not the whole tile is cloudy; completely cloudy captures over tile 30TUM have been discarded. The blue plot (GTS) describes the result obtained with the standard approach, i.e., all the cloudy samples are eliminated and only MTS is used for the GPR model prediction; in red the result provided by the proposed strategy (GTS2Z), i.e., by substituting (0,0) for each (date, LAIG) pair from a cloudy capture and performing the prediction using all the samples. The gapfilling has been performed over the whole observation period with a 5-day sampling step. It can be observed that GTS and GTS2Z plots perfectly match, and the overall difference between the two estimation is utterly negligible (RMSE = 2.9 × 10${}^{-14}$, R${}^{2}$ = 1). For the sake of completeness, we show in black (GTSZ) the result obtained when cloudy samples are set to 0 to demonstrate that this substitution leads to incorrect estimations of LAIG and should not be used.

**1**denotes a matrix of ones; bold font defines variables which take into account all the pixels of the collection at once. The final prediction corresponds to the LAI${}_{G}$ cloud-free map at time ${\mathit{t}}_{*}$. Details on the GPR formulation here implemented can be found in [13]. The key steps are (1) the implementation of Equation (11) by defining the ${\mathbf{1}}_{msk}$ and ${\mathbf{1}}_{*}$ vectors, and (2) the parallel calculation of L matrix using the Cholesky Decomposition available in $ee$ library. In this case too, the matrix algebra operation can be performed by casting the multiband images from $image$ to $array$ data type. The final result is converted back to $image$ using the $arrayProject$ functions to add geocoordinates and turn it into a map. As dealing with $array$ type means asking for chunks of contiguous memory [57], care must be taken to the global volume of data involved in the process. Note that the theoretical computational complexity of LAI prediction as well as gapfilling steps is O(${n}^{3}$), being n the number of bands for the former and time series length for the latter. For this reason, the gapfilling task over multiple-tile areas must be carried out by looping over tiles and considering one date at a time. This way, memory exceeded errors are nicely avoided.

## 5. Cloud-Free Seamless Mapping of Wide Areas

## 6. Discussion

## 7. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Comparison of the RMSE evolution performance of all active learning (AL) methods (

**a**), and scatterplot with goodness-of-fit statistics of validation LAI${}_{G}$ data against estimations using the best Euclidean Diversity (EBD)-reduced LAI${}_{G}$ GPR model (

**b**).

**Figure 2.**Comparison between green LAI maps retrieved by Automated Radiative Transfer Models Operator (ARTMO) (

**a**) and GEE GPR coding (

**b**) from S2 acquisition over tile 30TUM on 2 May 2019.

**Figure 3.**Comparison of green LAI maps estimated by ARTMO and GEE GPR implementations: histograms (

**a**) and scatterplot density (

**b**). In (

**c**), the histogram of pixel-by-pixel difference between 10 m bands resampled at 20 m distributed by ESA and available in GEE.

**Figure 4.**Big Scale LAI${}_{G}$ Mosaic over Europe with 1 July to 15 July 2019 time span using maximum value strategy.

**Figure 5.**Result of GPR gapfilling of input LAI${}_{G}$ time series—meaning values (MTS) plus cloudy acquisitions (CD)—using different approaches: standard GPR prediction (GTS), CD substitutions with 0 (GTSZ) and the proposed parallel implementation (GTS2Z).

**Figure 6.**S2 acquisition over tile 30TUM on 27 May 2019: RGB (

**a**), $LA{I}_{G}$ map (

**b**) and gapfilled LAI${}_{G}$ map (

**c**) from GPR model applied in GEE to ± 35 dd span S2 collection. In (

**d**–

**f**), the same quantity zoomed onto the bottom-right corner.

**Figure 7.**Original and gapfilled LAI${}_{G}$ time series of a crop pixel with the area of study generated using GEE visualization APIs. The time span corresponds to the whole collection of S2 L2A imagery distributed by ESA available in GEE.

**Figure 8.**Mosaic of gapfilled green LAI maps over Iberian Peninsula from S2-tiling (

**a**) on 2 February (

**b**), 30 March (

**c**) and 30 June (

**d**), 2019. In (

**e**), zoom of 30 June mosaic over the Atlantic coast and Pyrenees Area.

**Table 1.**Overview of field campaigns for leaf area index (LAI${}_{G}$) collection used for training the Gaussian process regression (GPR) retrieval models.

Location | Period | #Points | Range | Instrument | Vegetation Type | Spectral Data |
---|---|---|---|---|---|---|

Barrax, Spain | 3 July | 102 | 0.4–6.2 | LAI-2000 | Alfalfa, corn, garlic, onion, potato, sugar beet, wheat | HyMap |

Valencia, Spain | May–17 November | 34 | 0.41–5.41 | LAI-2200 | Alfalfa, artichoke, lettuce, onion, potato | S2 |

Biely Kríž, Czech Republic | 16 August | 7 | 5.3–9.3 | LAI-2200 | Spruce forest | S2 |

Foggia, Italy | 17 March | 6 | 3.08–4.23 | LAI-2200 | Wheat | S2 |

Poznań, Poland | 17 July | 6 | 2.69–4.2 | LAI-2200 | Maize, triticale, wheat | S2 |

Kiev Oblast, Ukraine | 18 June | 3 | 0.27–0.56 | DHP | Maize, soybean | S2 |

Toulouse, France | 18 August | 1 | 1.77 | DHP | Maize | S2 |

**Table 2.**Overview of campaigns for LAI${}_{G}$ in-situ data collection used for LAI${}_{G}$ GPR model validation.

Location | Period | #Points | Range | Instrument | Vegetation Type | Spectral Data |
---|---|---|---|---|---|---|

Toulouse, France | Nov17/Mar-May-Jul-Aug 18 | 52 | 0.03–3.84 | DHP | Maize, soybean, sunflower | S2 |

Poznań, Poland | Apr-Jun-Aug 18 | 50 | 0.96–4.23 | LAI-2200 | Beetroot, maize, triticale, wheat | S2 |

Kiev Oblast, Ukraine | May-Jun-Aug 18 | 40 | 0.04–4.81 | DHP | Maize, soybean, sunflower, wheat | S2 |

**Table 3.**Averaged hyperparameters estimated using fixed crop-type and global approaches: ${\sigma}_{t}$ defines the gap-filled time series smoothness, ${\sigma}_{st}$ is the amplitude scaling factor and ${\sigma}_{n}$ accounts for the noise variance.

Wheat | Corn | Barley | Sunflower | Rape | Pea | Alfalfa | Beet | Potato | Global | |
---|---|---|---|---|---|---|---|---|---|---|

${\sigma}_{t}$ | 32.6018 | 41.0726 | 36.0351 | 23.0815 | 35.0548 | 23.9367 | 29.8602 | 47.3544 | 25.5081 | 32.7282 |

${\sigma}_{st}$ | 0.8776 | 1.0018 | 0.8395 | 0.5670 | 1.2058 | 0.8415 | 0.6465 | 1.1465 | 1.1870 | 0.9237 |

${\sigma}_{n}$ | 0.3377 | 0.4395 | 0.2833 | 0.2355 | 0.5085 | 0.2778 | 0.4028 | 0.3794 | 0.3620 | 0.3585 |

**Table 4.**Variation in percentage of LAI${}_{G}$ obtained with precalculated hyperparameters with respect to the original LAI time series (lowest values in bold). Last column exhibits the variance in the percentage.

Crop Type | Per-Pixel Hyperpar. | Averaged Hyperparameters | Variance | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Wheat | Corn | Barley | Sunflower | Rape | Pea | Alfalfa | Beet | Potato | Global | |||

Wheat | 9.064 | 10.078 | 10.845 | 10.240 | 9.072 | 10.372 | 8.851 | 10.519 | 10.996 | 8.995 | 10.097 | 0.787 |

Corn | 10.660 | 10.794 | 11.614 | 10.952 | 9.821 | 11.100 | 9.675 | 11.174 | 11.752 | 9.812 | 10.813 | 0.708 |

Barley | 8.206 | 8.593 | 9.282 | 8.707 | 7.932 | 8.834 | 7.739 | 9.058 | 9.435 | 7.826 | 8.608 | 0.580 |

Sunflower | 8.455 | 11.238 | 12.366 | 11.474 | 9.928 | 11.642 | 9.752 | 11.646 | 12.633 | 9.929 | 11.263 | 1.265 |

Rape | 10.222 | 10.798 | 11.483 | 10.950 | 9.576 | 11.070 | 9.351 | 11.207 | 11.598 | 9.573 | 10.816 | 0.799 |

Pea | 7.634 | 10.017 | 11.601 | 10.314 | 8.719 | 10.557 | 8.485 | 10.671 | 12.026 | 8.624 | 10.050 | 1.371 |

Alfalfa | 11.833 | 14.001 | 14.999 | 14.222 | 12.659 | 14.368 | 12.496 | 14.360 | 15.210 | 12.734 | 14.024 | 1.108 |

Beet | 8.975 | 8.975 | 9.629 | 9.083 | 8.207 | 9.223 | 8.054 | 9.389 | 9.714 | 8.149 | 8.991 | 0.577 |

Potato | 7.477 | 9.456 | 10.566 | 9.647 | 8.311 | 9.848 | 8.130 | 9.993 | 10.769 | 8.262 | 9.481 | 1.070 |

**Table 5.**Lines of pseudo-code of GPR core implementation in Google Earth Engine (GEE) with mathematical notation for variable easy identification.

(var) calculate_LAI_GREEN = function(image){ |

(var) ${\mathit{X}}_{*}^{T}\mathit{D}$ = image.multiply(D).toArray().toArray(1); |

(var) ${\mathit{X}}^{*}$ = image.toArray().toArray(1); |

(var) Term1 = ${\mathit{X}}_{*}^{T}\mathit{D}$.matrixTranspose().matrixMultiply(${\mathit{X}}_{*}$).arrayProject([0]).multiply(−0.5).exp().multiply(${\sigma}_{f}^{2}$) |

(var) PtTDX = ee.Image(X).matrixMultiply(${\mathit{X}}_{*}^{T}\mathit{D}$).arrayProject([0]).arrayFlatten([TS_ID]); |

(var) ${\mathit{K}}_{*}$ = PtTDX.subtract($\mathit{X}\mathit{D}\mathit{X}$.multiply(0.5)).exp().toArray() |

(var) f(${\mathit{X}}_{*}$) = ${\mathit{K}}_{*}$.arrayDotProduct($\mathit{\alpha}$.toArray()).multiply(Term1).toArray(1).arrayProject([0]).arrayFlatten([[‘LAIG’]]); |

return image.select(‘LAIG’)} |

**Table 6.**Lines of pseudo-code of GPR gap-filling core implementation in GEE with mathematical notation for variable easy identification.

(var) N${}_{t}=$LAIG${}_{c}$.size(); |

(var) ${\mathbf{1}}_{msk}={t}_{msk}$.multiply(0).add(1.0); |

(var) ${\mathbf{1}}_{*}={t}_{*}$.multiply(0).add(1.0); |

(var) I = ee.Image(ee.Array.identity(N${}_{t}$)); |

(var) prod = ${t}_{msk}$.matrixMultiply(1${}_{msk}$.matrixTranspose()); |

(var) K${}_{t}=prod$.subtract(prod.matrixTranspose()).pow(2).multiply(${D}_{t}$).multiply($-0.5$).exp().multiply(${\sigma}_{ft}$); |

(var) L = I.multiply(${\sigma}_{nt}^{2}$).add(${\mathit{K}}_{t}$).matrixCholeskyDecomposition(); |

(var) ${\mathit{\alpha}}_{tmp}$ = L.matrixInverse().matrixMultiply($LAI{G}_{c}$.toBands().unmask().toArray().toArray(1)); |

(var) ${\mathit{\alpha}}_{t}$ = L.matrixTranspose().matrixInverse().matrixMultiply(${\mathit{\alpha}}_{tmp}$); |

(var) ${\mathit{T}}_{*}=t*$.matrixMultiply(${\mathbf{1}}_{msk}$.matrixTranspose()); |

(var) ${\mathit{T}}_{msk}$ = ${t}_{msk}$.matrixMultiply(${\mathbf{1}}_{.}$matrixTranspose()).matrixTranspose(); |

(var) $\mathit{K}*$ = ${\mathit{T}}_{*}$.subtract(${\mathit{T}}_{msk}$).pow(2).multiply(${D}_{t}$).multiply($-0.5$).exp().multiply(${\sigma}_{ft}$); |

(var) $LAIG({t}_{*}$) = $\mathit{K}*$.matrixMultiply(${\mathit{\alpha}}_{t}$).arrayProject([0]).arrayFlatten([[‘LAIG’]]); |

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## Share and Cite

**MDPI and ACS Style**

Pipia, L.; Amin, E.; Belda, S.; Salinero-Delgado, M.; Verrelst, J.
Green LAI Mapping and Cloud Gap-Filling Using Gaussian Process Regression in Google Earth Engine. *Remote Sens.* **2021**, *13*, 403.
https://doi.org/10.3390/rs13030403

**AMA Style**

Pipia L, Amin E, Belda S, Salinero-Delgado M, Verrelst J.
Green LAI Mapping and Cloud Gap-Filling Using Gaussian Process Regression in Google Earth Engine. *Remote Sensing*. 2021; 13(3):403.
https://doi.org/10.3390/rs13030403

**Chicago/Turabian Style**

Pipia, Luca, Eatidal Amin, Santiago Belda, Matías Salinero-Delgado, and Jochem Verrelst.
2021. "Green LAI Mapping and Cloud Gap-Filling Using Gaussian Process Regression in Google Earth Engine" *Remote Sensing* 13, no. 3: 403.
https://doi.org/10.3390/rs13030403