# Gap-Filling Sentinel-1 Offshore Wind Speed Image Time Series Using Multiple-Point Geostatistical Simulation and Reanalysis Data

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## Abstract

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## 1. Introduction

- A novel TI selection method is presented to form the CTIS used to simulate the missing patterns at each time step. This method is based on the dependence between the coregistered coarse- and fine-scale information included in the training images. Once the CTIS is formed, fine-scale patterns are simulated and locally conditioned to the coarse-scale data.
- An MPS algorithm recently developed by [29], namely quick sampling (QS), is exploited for the first time in a spatiotemporal gap-filling application. The precise aim of this study is to take advantage of the robustness and computational efficiency of the algorithm to investigate its potential to provide realistic reconstructions of spatially complex patterns of continuous fields.
- A first real-world case study of image time-series expansion is provided in an offshore wind speed context by generating wind fields of realistic spatial and temporal variability while preserving the complex multivariate wind relationships. Considering the complex variability and dynamic nature of wind speed both in space and time, this endeavor is rather challenging and often fails to reproduce the inherent variability at the fine scale, especially when long-term training datasets are not available.

## 2. Area of Study and Training Datasets

#### 2.1. Sentinel-1A/B SAR Wind Retrievals

#### 2.2. UERRA-HARMONIE Regional Reanalysis

## 3. Methodology

#### 3.1. Multiple-Point Statistical (MPS) Simulation Framework

#### 3.2. Quick Sampling (QS) Algorithm

Algorithm 1: Quick Sampling | |

Inputs and parameters: | ti(s): training image(s), di: destination image (or simulation grid), sp: simulation path, dt: data type, k: number of best candidates, n: number of closest neighbors, ki: kernel |

Step 1: | For every uninformed node $x$ in the defined path $\left(sp\right)$ |

Step 2: | Retrieve data event ${d}_{n}\left(x\right)$ in the $di$ within the predefined radius |

Step 3: | Compute the mismatch map by calculating the distance/dissimilarity between data event ${d}_{n}\left(x\right)$ in di and ${d}_{n}\left(y\right)$ for every node in ti |

Step 4: | Rank distances in the mismatch map using quantile sorting to determine the $k$ best candidate(s) |

Step 5: | Sample among the $k$ best candidates and assign the selected value to node $x$ in the $di$ |

#### 3.3. Parametrization

#### 3.4. Conditional Training Image Set (CTIS)

#### 3.5. Offshore Wind Speed Image Time Series Simulation

#### 3.6. Evaluation Metrics

#### 3.6.1. Similarity and Divergence Measures

#### 3.6.2. Spatial Correlation

#### 3.6.3. Relative Bias (%)

## 4. Results and Evaluation

## 5. Discussion

#### 5.1. Challenges and Emerging Opportunities

#### 5.2. Alternative Auxiliary Data Sources

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$x$ | Pixel location in the simulation grid |

$y$ | Pixel location in the training image |

$F(\xb7)$ | Conditional cumulative distribution function (CCDF) |

$Z\left(x\right)$ | Random variable at location $x$ |

$z$ | Outcome of random variable $Z$, an attribute value |

$n$ | Subscript indicating the number of neighboring pixels |

i | Subscript indicating the index of a pixel over the entire image |

${\mathit{d}}_{n}$ | Data event ($z$ values) of $n$ neighboring pixels |

$\mathit{L}$ | Lag vectors of data events |

$m$ | Superscript indicating multiple variables |

$v$ | Variable index |

h | Kernel bandwidth (degrees) |

d | Euclidean distance from kernel center (degrees) |

$w$ | Relative variable weight |

${D}_{KL}\left(p||q\right)$ | Kullback–Leibler divergence (relative entropy) from $q$ to $p$ |

${p}_{b}\left(x\right)$ | Reference distribution for ${b}^{th}$ bin at location $x$ |

${q}_{b}\left(x\right)$ | Simulation distribution for ${b}^{th}$ bin at location $x$ |

$B$ | Number of bins |

b | Subscript indicating the index of a bin |

${\overline{z}}^{s}\left({x}_{i}\right)$ | Mean of realizations at ${i}^{th}$ grid cell |

$z\left({x}_{i}\right)$ | Reference attribute value at ${i}^{th}$ grid cell |

$N$ | Number of pixels over the entire image |

## Abbreviations

SAR | Synthetic aperture radar |

MPS | Multiple-point statistics |

NWP | Numerical weather prediction |

SCAT | Scatterometers |

GMF | Geophysical model function |

NRCS | Normalized radar cross section |

WRF | Weather research and forecasting |

TI | Training image |

CTIS | Conditional training image set |

UERRA | Uncertainties in ensembles of regional reanalyses |

QS | Quick sampling |

EU | European Union |

ECMWF | European Centre for Medium Forecast |

IW | Interferometric wide |

VV | Vertical–vertical |

VH | Vertical–horizontal |

UTC | Coordinated universal time |

ASF | Alaska Satellite Facility |

OWI | Ocean wind fields |

Probability distribution function | |

CCDF | Conditional cumulative distribution function |

SG | Simulation grid |

DS | Direct sampling |

FFT | Fast Fourier transform |

RBF | Radial basis function |

RMSE | Root mean square error |

MAE | Mean absolute error |

LOOCV | Leave-one-out cross validation |

PSS | Perkins skill score |

KL | Kullback–Leibler |

MRB | Median relative bias |

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**Figure 1.**Outline of the study area (red polygon) and typical Sentinel tiles within a one-week period (white polygons). (Satellite base map is hosted by Esri).

**Figure 2.**Sentinel-1 and UERRA image time series over the 2-year period of interest. Images with a red outline (left) represent Sentinel-1 gap datetimes.

**Figure 5.**Visual comparison between wind speed (m/s) images from UERRA, Sentinel-1, 3 realizations and the mean of 50 realizations on 10 June 2017.

**Figure 6.**Visual comparison between wind speed (m/s) images from UERRA, Sentinel-1, 3 realizations and the mean of 50 realizations on 15 June 2017.

**Figure 7.**Scatterplots and marginal distributions between wind speed values (m/s) from UERRA, Sentinel-1 (reference) and one realization each on 10 June 2017 (first row) and 15 June 2017 (second row).

**Figure 8.**Experimental omnidirectional and directional (along 90° and 180° directions) variograms for the reference and 5%–95% envelope of 50 realizations on 10 June 2017 (first row) and 15 June 2017 (second row).

**Figure 9.**Kullback–Leibler divergence (

**left**) and Perkins skill score (

**right**) used to quantify the divergence and similarity of the reference and simulation distributions per grid cell, respectively.

**Figure 10.**Median relative bias plot (%) resulting from the relative bias between the reference and the mean of 50 realizations at each simulation time step.

Variables | |||||
---|---|---|---|---|---|

Parameter | UERRA | Sentinel-1 | Longitude | Latitude | Distance to the Coast |

n | 25 | 75 | 1 | 1 | 1 |

ki | 103 × 103 RBF (h = 0.001, w = 0.01) | 103 × 103 RBF (h = 0.001, w = 1) | 103 × 103 RBF (h = 0.001, w = 0.1) | 103 × 103 RBF (h = 0.001, w = 0.1) | 103 × 103 RBF (h = 0.001, w = 0.1) |

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## Share and Cite

**MDPI and ACS Style**

Hadjipetrou, S.; Mariethoz, G.; Kyriakidis, P. Gap-Filling Sentinel-1 Offshore Wind Speed Image Time Series Using Multiple-Point Geostatistical Simulation and Reanalysis Data. *Remote Sens.* **2023**, *15*, 409.
https://doi.org/10.3390/rs15020409

**AMA Style**

Hadjipetrou S, Mariethoz G, Kyriakidis P. Gap-Filling Sentinel-1 Offshore Wind Speed Image Time Series Using Multiple-Point Geostatistical Simulation and Reanalysis Data. *Remote Sensing*. 2023; 15(2):409.
https://doi.org/10.3390/rs15020409

**Chicago/Turabian Style**

Hadjipetrou, Stylianos, Gregoire Mariethoz, and Phaedon Kyriakidis. 2023. "Gap-Filling Sentinel-1 Offshore Wind Speed Image Time Series Using Multiple-Point Geostatistical Simulation and Reanalysis Data" *Remote Sensing* 15, no. 2: 409.
https://doi.org/10.3390/rs15020409