# Wavelet Analysis for Wind Fields Estimation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{3}spline scaling function, in addition to other wavelet bases as Gabor and Mexican-hat. The purpose is to extract more reliable directional information, when wind speed values range from 5 to 10 ms

^{−1}. Using C-band empirical models, associated with the estimated directional information, we calculate local wind speed values and compare our results with QuikSCAT scatterometer data. The proposed approach has potential application in the evaluation of oil spills and wind farms.

## 1. Introduction

^{−1}when deriving wind fields from ERS-2 SAR images. Cameron et al. [3] combined SAR and scatterometer data to characterize wind farms and their potential energy output around coastal areas. Their investigation included the method in [2] as an alternative inversion scheme for wind vectors retrieval from SAR backscatter, using a Bayesian approach to combine trial wind vectors and weather predicted data. The method has proven to be adequate for both moderate and high winds. The range of strong (high) wind speeds according to [4], is higher than 11 ms

^{−1}.

^{−1}. However, Solberg et al. [6] noticed a high probability of false slicks for wind speeds less than 5 ms

^{−1}; this analysis also reported fewer dark spots from local low-wind areas when in the range between 5 and 10 ms

^{−1}. Pavlakis et al. reported in [7] that under low wind speed conditions, such as 3 to 7 ms

^{−1}, oil spills could yield detectable radar backscattering contrast signals. These authors assumed that medium winds are within the interval of 7 ms

^{−1}to 13 ms

^{−1}and high winds are above 13 ms

^{−1}.

^{−1}. Our algorithm takes a SAR image as input, decomposes it by using wavelet functions, transforms the wavelet coefficients into their spectral version and finally detects peaks in the spectrum domain to recover the orientation of the streaks. The motivation for choosing undecimated wavelets is: Mexican-hat presents suitable selectivity in position and the Gabor wavelet can be tuned to detect directional features. Our algorithm estimates the wind direction using the Fourier spectrum, although the wavelet transform provides good localization in both spatial and spectral domains. Our method takes the wavelet coefficients of the decomposed SAR image as input to peak detection using spectral energy, while it attenuates the undesirable high frequencies and maintains the main spectral energy, located perpendicular to the orientation of streaks [16]. The image decomposition by wavelets enables detection of wind streaks at a certain spatial scale and later identification of wind orientation and wind speed estimation.

## 2. SAR Images and QuikSCAT Data

^{−1}in wind speed and 20° in wind direction. This accuracy depends on the distance from the shore, wind speed range and atmospheric conditions [19].

^{−1}), wind direction (degrees) and rain flag, respectively, and fourth index is ascending or descending orbit. Figure 2a shows QuikSCAT wind vectors over Atlantic, Tropical, South extracted on September 29, 2006. The first cell of the matrix is at longitude 0.125° E and in latitude −89.875°, in the 0.25° × 0.25° space resolution. The QuikSCAT wind speed is relative to a height of 10 m above sea level and to a neutral atmospheric stability [20]. The wind direction data follows the oceanographic convention, indicating the direction the wind blows, and are used as input variables in C-band models for calculating the wind speed.

^{−1}(see Table 1). From the available data set (about 14 SAR images), 5 images were selected with time difference between 7 and 12 hours, 4 with approximately 4 hours, and 3 with less than 1 hour. The interpretation of QuikSCAT region of interest relies on geographic coordinate transformation, according to Equations 2 and 3.

## 3. Methods

#### 3.1. The WDWaT Method for Wind Direction Estimation

#### 3.2. The LG-method for Wind Direction Estimation

#### 3.3. Wind Speed Retrieval Models for Assessment and Comparison Purposes

_{o}to the local near-surface wind speed v, wind direction versus antenna look direction Φ, and incidence angle θ. The general form of the function is given by

_{o}decrease as the incidence angle increases and the increasing wind speed sensitivity to the error from the wind direction.

#### 3.4. Undecimated Wavelets

#### (A) The à Trous Wavelet Transform

_{3}-spline basis.

_{3}-spline.

^{j−1}zeros at each decomposition scale j. The reconstruction of the original signal s(k) is obtained by adding the last smoothed signal s

_{N}(k) with the set of wavelet coefficients [34],

#### (B) The Gabor Wavelet Transform

_{0}, y

_{0}) is the center of the spatial domain and (ξ

_{0}, ν

_{0}) is the optimal spatial frequency of the filter in the frequency domain. Here, σ

_{x}and σ

_{y}are the standard deviations of the modulated Gaussian along x and y axes.

#### (C) The Mexican-hat Wavelet Transform

#### 3.5. Proposed Spectral Algorithm for Wind Direction Estimation

_{3}-spline), Gabor and Mexican-hat, as illustrated in Figure 3. We extend the algorithm in [8] by using other wavelet transforms, which have the potential to improve the streak detection results.

## 4. Results

_{3}-spline, (3) UWT with Gabor, (4) UWT with Mexican-hat, (5) WDWaT, (6) LG and (7) QuikSCAT. This figure illustrates the wind direction using each of the 7 methods for 3 SAR images; each of these images contains a different number of valid imagettes. The color code for the direction vectors can be blue (B), green (G), yellow (Y), magenta (M) and white (W), and they correspond to one method, in conjunction to a numerical identifier as pointed out above. As an example, the code G:1, indicates green arrows, which represent the wind directions calculated by method (1). Notice that each row of Figure 5 shows the same SAR image, but with arrows representing the wind direction result of different methods over each imagette. Table 2 presents the mean and standard deviation of wind direction for the different methods, with bold numbers indicating high similarity to the QuikSCAT values.

_{x}= σ

_{y}= 6.95, ξ

_{0}= 3.14 and ν

_{0}= 0, tuned according to the dimension of the streaks (200 to 1,600m) in our dataset. The Mexican-hat wavelet uses parameter a set to 0.3π, which resulted in noise suppression and streak recovery.

^{−1}(32 imagettes).

^{−1}while between August and December, the winds are expected to be stronger (around 9.0 ms

^{−1}).

^{−1}.

_{3}-spline base function achieves comparable results to the 2-D Mexican-hat results. Table 3 shows that it outperforms the other methods regarding the RMSE and correlation measures, particularly for data set with wind speeds up to 10 ms

^{−1}. Wind directions estimated by this method are highly correlated (0.61) with QuikSCAT data and thus present the lowest RMSE (31.15°) and standard deviation (23.31°).

^{−1}) and it is less accurate when performed in areas of low to moderate wind speeds (4–9 ms

^{−1}).

_{3}-spline and Mexican-hat. We use wind direction results of these two methods as the inputs to CMOD models for wind speed estimation. Thus, for each imagette we compare the CMOD results with the corresponding QuikSCAT speed data.

^{−1}. At moderate wind speeds they agree fairly well. The CMOD-IFR2 and CMOD5 models are very similar to each other. The main difference occurs at very high wind speeds >20 ms

^{−1}, where CMOD5 tends to output higher winds [13].

^{−1}and 0.99 ms

^{−1}). Table 4 displays the estimated speeds with CMOD4 for the RADARSAT-1 SAR standard images. They are highly correlated (0.79 and 0.9) with QuikSCAT data.

_{3}-spline) and FFT algorithm with Mexican-hat wavelet. In this paper, we apply the PR model called Elfouhaily scattering to estimate the NRCS for SAR images with the HH polarization, as suggested in [27]. Such a model allows estimation of wind speed in fairly agreement with wind speed values at several meteorological observation stations. Table 4 displays that CMOD4 outperformed the other C-band based models concerning RMSE and correlation values. It implies that the estimated speed values are close to the QuikSCAT values. At low to moderate wind speed values, CMOD4 is the best choice to retrieve SAR wind speed in high resolution SAR images acquired at C-band [26]. However, especially at high wind speed, CMOD4 underestimates the wind speeds significantly. Also CMOD-IFR2 and CMOD5 output better estimations at high wind speed values, but still underestimate the wind speed [13].

## 5. Conclusions

_{3}-spline, obtaining better results, given the wind speed range under inspection.

_{3}-spline relies on the fact that the former enhances the streak patterns, as well as the latter, but it also enhances undesirable noise and small-scale fluctuations when deriving wind fields from SAR images. It is a particular characteristic of the Mexican-hat wavelet. Both methods performed similarly when discarding imagettes containing wind speed values >10 ms

^{−1}. In this case, the algorithms achieved the lowest RMSE and the highest correlation values. Our investigations suggested that it was accomplished by the multiscale blurring effect, provided by the B

_{3}-spline and Mexican-hat wavelet bases, which reduced undesirable noise, and small-scale surface roughness, in the range of low to moderate wind speeds. In addition, this blurring effect preserved relevant information (e.g. streaks) for direction estimation for several scales. Our results also suggested that the wavelet coefficients, obtained with the B

_{3}-spline base function, were more suitable to characterize wind-induced streaks oriented in the wind direction in scales higher than 200 m. It means that the à trous decomposition with triangular function in low to moderate wind speed areas is more sensitive to small-scale roughness than B

_{3}-spline base function, as we expected.

_{3}-spline and Mexican-hat functions to extract wind-induced streaks and ignore surface small-scale intensity variations. It is noteworthy that the proposed method also smoothed speckle when applied to our dataset of multi-look SAR images. The combination of smoothing effect and multi-look processing, with streak pattern enhancement for wind fields estimation, improved the algorithm accuracy. Due to the ability of these masks to smooth variations of intensity at small-scales, the performance of the algorithm was superior in areas of low to moderate wind speeds in comparison with areas of high wind speeds. On the other hand, we observed that the energy of the Gabor wavelet function could have been tuned differently, probably improving wind direction estimates if considering a more extensive exploration of the parameters for better alignment with the streak patterns.

## Acknowledgments

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**Figure 1.**SAR images over the coast of Rio Grande do Norte, Northeast Brazil. (a) RADARSAT-1 SAR, acquired on September 29, 2006 with HH polarization. (b) Extract of the SAR image (4096 × 4096 pixels) referenced in latitude and longitude (decimal degrees) representing 51.2 × 51.2 km. (c) ENVISAT ASAR, acquired on February 01, 2005 with HH polarization. (d) ALOS PALSAR, acquired on July 20, 2007 with HH polarization.

**Figure 2.**(a) QuikSCAT wind direction and wind speed estimation on September 29, 2006. (b) QuikSCAT over ROI.

**Figure 3.**Algorithms under investigation for wind direction detection: proposed algorithms (top and center) and the Fichaux and Ranchin’s algorithm [8] (bottom).

**Figure 4.**(a) Original SAR image I. (b) Image of the dominant directions of the induced streaks in the ocean detected by the Gabor wavelet.

**Figure 5.**Wind direction vectors from 3 different SAR images: (a–c) RADARSAT-1 SAR image, on September 29, 2006, (d–f) ALOS PALSAR image, on July 20, 2007 and (g–i) ENVISAT ASAR image, acquired on February 01, 2005. White arrows indicate the ground-truth value, from QuikSCAT in all images; color-method associations appear on the label of each image.

**Figure 6.**Comparison between QuikSCAT (abscissa) and SAR-based methods (ordinate) for two data sets: (a, c, e, g) after removing the low-confidence (rain cells) from QuikSCAT data and (b, d, f, h) regions with wind speeds less than 10ms

^{−1}; the FFT methods differ from their wavelet decompositions: à trous, triangular base (a, b), à trous B

_{3}-spline (c, d), Mexican-hat (e, f) and Gabor (g, h).

**Figure 7.**Comparison of wind speed retrieval results and QuikSCAT scatterometer winds. (a, c, e) Wind direction estimated by the FFT method using B

_{3}-spline function. (b, d, f) Wind direction estimated by the FFT method using Mexican-hat function.

Satellite | Mode Beam | Orbit | Image Time UTC | Wind Conditions ^{1} |
---|---|---|---|---|

RADARSAT-1 | Standard 7 | 39713 | 2003/06/14 07:56 | M/9.1 |

RADARSAT-1 | Standard 2 | 39756 | 2003/06/17 08:09 | M/6.3 |

RADARSAT-1 | Standard 7 | 56863 | 2006/09/26 07:55 | H/11.2 |

RADARSAT-1 | Standard 2 | 56906 | 2006/09/29 08:07 | M/9.8 |

RADARSAT-1 | Standard 3 | 56906 | 2001/02/03 20:42 | M/6.1 |

RADARSAT-1 | Standard 6 | 56906 | 2001/02/07 07:53 | L/4.0 |

ENVISAT | IMG | 11779 | 2004/06/01 00:39 | M/9.4 |

ENVISAT | IMG | 15286 | 2005/02/01 00:38 | M/9.8 |

ENVISAT | IMP | 19566 | 2005/11/29 00:41 | H/11.0 |

ENVISAT | IMP | 25342 | 2007/01/04 12:13 | M/6.9 |

ALOS | FBS8 | 7905 | 2007/07/20 01:16 | M/10 |

ALOS | FBS8 | 12602 | 2008/06/06 01:13 | M/8.2 |

ALOS | FBS8 | 18641 | 2009/07/25 01:18 | H/10.5 |

ALOS | FBS8 | 19064 | 2009/08/23 01:16 | M/9.7 |

^{1}L, low wind (< 5 ms

^{−1}); M, moderate wind (5 ms

^{−1}< v < 10 ms

^{−1}); H, high wind (> 10 ms

^{−1}). Mean value of speed wind provided by QuikSCAT.

FFT | WDWaT | LG | ||||||
---|---|---|---|---|---|---|---|---|

SAR images | Measures | à trous | Gabor | Hat | Haar | Gradient | QuikSCAT | |

Triangular | B_{3}-spline | |||||||

2003/06/14 | Mean (°) | 352.6 | 306.2 | 270 | 287.66 | 355.0 | 294 | 314.2 |

Std. dev. (°) | 0.3 | 2.1 | 0 | 20.4 | 5.8 | 4.8 | 2.6 | |

2006/09/26 | Mean (°) | 334.3 | 328.5 | 270.0 | 180.84 | 335.0 | 264.3 | 277.5 |

Std. dev. (°) | 22.2 | 1.9 | 0 | 169.1 | 5.8 | 10.4 | 10.4 | |

2006/09/29 | Mean (°) | 322.6 | 316.3 | 279.0 | 311.61 | 295.0 | 279.4 | 316.5 |

Std. dev. (°) | 11.1 | 0.07 | 10.6 | 2.3 | 5.7 | 4.7 | 0.0 | |

2001/02/03 | Mean (°) | 275.76 | 246.33 | 272.33 | 225.12 | 303.33 | 270.46 | 259.5 |

Std. dev. (°) | 58.26 | 15.84 | 76.96 | 7.42 | 56.86 | 0.09 | 2.59 | |

2001/02/07 | Mean (°) | 328.57 | 327.99 | 327.53 | 321.67 | 190 | 270.35 | 292.5 |

Std. dev. (°) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

2005/11/29 | Mean (°) | 270 | 275.53 | 360 | 270 | 360 | 241.89 | 283.5 |

Std. dev. (°) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

2007/01/04 | Mean (°) | 252.66 | 290.39 | 317.03 | 217.28 | 280 | 254.84 | 235.5 |

Std. dev. (°) | 152.9 | 6.13 | 46.03 | 13.94 | 58.06 | 11.33 | 5.01 | |

2005/02/01 | Mean (°) | 284.51 | 284.83 | 237.83 | 325.74 | 287.5 | 309.49 | 282 |

Std. dev. (°) | 1.18 | 1.22 | 64.8 | 1.77 | 39.55 | 40.95 | 7.94 | |

2007/07/20 | Mean (°) | 316.58 | 286.04 | 270 | 326.54 | 240 | 241.81 | 268.5 |

Std. dev. (°) | 47.25 | 9.30 | 0 | 1.30 | 42.43 | 21.73 | 6.36 | |

2008/06/06 | Mean (°) | 294.44 | 335.56 | 270 | 264.92 | 230 | 209.79 | 328.5 |

Std. dev. (°) | 16.57 | 16.57 | 0 | 73.59 | 0 | 8.59 | 0 | |

2009/07/25 | Mean (°) | 342.17 | 340.34 | 315 | 321.89 | 320 | 224.99 | 330 |

Std. dev. (°) | 4.37 | 4.19 | 63.64 | 1.4 | 28.28 | 15.34 | 0 | |

2009/08/23 | Mean (°) | 344.64 | 285.99 | 270 | 243.2 | 340 | 263.75 | 282.75 |

Std. dev. (°) | 0.4 | 1.31 | 0 | 37.89 | 28.28 | 6.05 | 1.06 |

**Table 3.**Statistical parameters of the comparison of the scatter plot shown in Figure 6.

Measures | Total data set (41 imagettes) | Only imagettes with wind speed values < 10 ms^{−1} | ||||||
---|---|---|---|---|---|---|---|---|

Triangular | B_{3}-spline | Gabor | Hat | Triangular | B_{3}-spline | Gabor | Hat | |

bias (°) | 19.75 | 17.01 | −0.13 | −12.24 | 19.90 | 16.39 | −1.82 | −10.25 |

RMSE (°) | 72.13 | 31.15 | 60.68 | 63.66 | 82.60 | 31.24 | 69.00 | 38.82 |

correlation | 0.35 | 0.57 | −0.11 | 0.47 | 0.35 | 0.61 | −0.22 | 0.62 |

std. dev. (°) | 73.92 | 24.57 | 49.24 | 70.61 | 85.50 | 23.31 | 53.22 | 47.44 |

mean (°) | 301.39 | 298.65 | 281.50 | 269.39 | 298.23 | 294.71 | 276.51 | 268.07 |

maximum (°) | 353.54 | 347.28 | 360 | 328.28 | 353.39 | 347.28 | 360 | 327.46 |

QuikSCAT parameters | ||||||||

mean (°) | 281.63 | 278.33 | ||||||

std. dev. (°) | 30.54 | 33.50 | ||||||

maximum (°) | 330 | 328.5 |

**Table 4.**Statistical parameters of the comparison of the scatter plot shown in Figure 7.

Measures | B_{3}-spline | Mexican-hat | ||||
---|---|---|---|---|---|---|

CMOD-IFR2 | CMOD4 | CMOD5 | CMOD-IFR2 | CMOD4 | CMOD5 | |

bias (ms^{−1}) | 0.79 | 0.12 | 0.64 | 0.63 | 0.06 | 0.68 |

RMSE (ms^{−1}) | 1.75 | 1.34 | 1.71 | 1.34 | 0.99 | 1.26 |

correlation | 0.72 | 0.79 | 0.69 | 0.85 | 0.90 | 0.87 |

std. dev. (ms^{−1}) | 2.06 | 2.05 | 1.91 | 2.17 | 2.29 | 2.09 |

mean (ms^{−1}) | 9.71 | 9.05 | 9.57 | 9.56 | 8.98 | 9.60 |

maximum (ms^{−1}) | 11.33 | 10.87 | 10.97 | 11.61 | 11.07 | 11.48 |

QuikSCAT parameters | ||||||

mean (ms^{−1}) | 8.92 | |||||

std. dev. (ms^{−1}) | 2.11 | |||||

maximum (ms^{−1}) | 11.2 |

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**MDPI and ACS Style**

Leite, G.C.; Ushizima, D.M.; Medeiros, F.N.S.; De Lima, G.G.
Wavelet Analysis for Wind Fields Estimation. *Sensors* **2010**, *10*, 5994-6016.
https://doi.org/10.3390/s100605994

**AMA Style**

Leite GC, Ushizima DM, Medeiros FNS, De Lima GG.
Wavelet Analysis for Wind Fields Estimation. *Sensors*. 2010; 10(6):5994-6016.
https://doi.org/10.3390/s100605994

**Chicago/Turabian Style**

Leite, Gladeston C., Daniela M. Ushizima, Fátima N. S. Medeiros, and Gilson G. De Lima.
2010. "Wavelet Analysis for Wind Fields Estimation" *Sensors* 10, no. 6: 5994-6016.
https://doi.org/10.3390/s100605994