# Estimating Coastal Winds by Assimilating High-Frequency Radar Spectrum Data in SWAN

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Models

#### 2.1. SWAN Wave Model

#### 2.2. HF Radar Scattering Model

**x**,

**s**) at a fixed time), $\mathbf{k}$ is the hydrodynamic wave vector, ${\mathbf{k}}_{b}=-2{k}_{0}(sin\varphi ,cos\varphi )$ is the wave vector associated with the Bragg waves propagating toward the radar (here, waves at $-{\mathbf{k}}_{b}$ are propagating away from the radar) and direction is idicated by $m=\pm 1$. The intrinsic radian wave frequency ${\omega}_{k}=\sqrt{gktanhkh}$, where $k=\left|\mathbf{k}\right|$ and $h=h\left(\mathbf{x}\right)$ is the water depth, and ${\omega}_{b}={\omega}_{k}$ evaluated for $k={k}_{b}$. The illumination patterns ${w}_{n}\left(\varphi \right)$ for the loop antennas are

## 3. The Assimilation Framework

#### 3.1. Cost Function

#### 3.2. Adjoint Models

#### 3.3. Implementation

- The adjoint solution is calculated using the error in the most recent prediction as input;
- Using the adjoint solution, the gradient is determined;
- The conjugate-gradient descent algorithm is used to calculate a new estimate of the wind field and ${\mathbf{U}}_{10}$;
- The SWAN model is run with corrected inputs and a new wave-spectrum prediction for the region is generated;
- The forward HF radar model is run with the new spectrum as input and a new prediction of the data is generated.

## 4. Results

#### 4.1. Problem Setup

#### 4.2. HF Radar Data Description

#### 4.3. Comparisons to Buoy Wave Data

#### 4.4. Comparisons to Buoy Wind Data

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

HFR | High-Frequency Radar |

SWAN | Simulating WAves Nearshore |

CODAR | Coastal Ocean Dynamics Applications Radar |

NDBC | National Data Buoy Center |

COAMPS | Coupled Ocean Atmosphere Modeling and Prediction System |

NCOM | Navy Coastal Ocean Model |

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**Figure 1.**Algorithm flow chart. The algorithm makes use of operation forecast data (as a first guess for winds), models and adjoint models, ancillary data (such as bathymetry), and HF radar Doppler data. The algorithm outputs are wind and wave products, improved wave spectra and improved estimates of the wind field.

**Figure 2.**Map of the Southern California Bight with 10 HF radar sites (red triangles) and 6 NDBC buoys (orange diamonds). Example input fields for the SWAN wave model from COAMPS and NCOM are on the bottom.

**Figure 3.**Range Doppler plots for the RFG1 HF radar site antennas 1 and 2 on 1 October 2017 at 0000z. The left plots are the predicted fields from the SWAN HF Doppler model. The center plots are the observed HF radar data. The right plots denote the difference between the first two fields. The first- and second-order Bragg peaks are visible in both the predicted and observed range Doppler plots.

**Figure 4.**Comparison of estimated to observed wave parameters for buoy 46217. Top left panel is a time series of significant wave height ${H}_{s}$; the middle left panel is a time series of mean wave period ${T}_{m}$; and the bottom left panel is a time series of mean wave direction ${\theta}_{m}$. In the time series, the open circles show the observed quantities, the red dots show the HFR assimilation estimate. The right panels are the associated scatter plots colored by experiment day.

**Figure 5.**Wind comparison for U and V 10-m winds for NDBC buoy 46053 (blue circles), COAMPS first guess (red) and SWAN HF assimilation model (green) over the first 23 days of October 2017. In order to construct a continuous wind record from the 24-hour assimilation cycles, only the middle 12 h from each cycle were used.

**Figure 6.**A 10-m wind field comparison for COAMPS first guess (left) and SWAN HF assimilation winds (right) for 3 October 2017 at 2300z. Notice the strong winds in the Santa Barbara Channel in the first guess that is not present in the assimilation estimate.

**Table 1.**Error statistics for significant wave height (${H}_{s}$), mean wave period (${T}_{m}$) and mean wave direction (${\theta}_{m}$) from comparison of the SWAN assimilation results to observation data used in the background. The mean error in quantity X is ${\overline{\u03f5}}_{X}$, the RMS error is ${\u03f5}_{X}^{rms}$. For comparison, the mean and standard deviation of the observations ($\overline{X}$ and ${\sigma}_{X}$) are included on the second line, in parentheses.

Station | ${\overline{\mathit{\u03f5}}}_{{\mathit{H}}_{\mathit{s}}}$ | ${\mathit{\u03f5}}_{{\mathit{H}}_{\mathit{s}}}^{\mathbf{rms}}$ | ${\overline{\mathit{\u03f5}}}_{{\mathit{T}}_{\mathit{m}}}$ | ${\mathit{\u03f5}}_{{\mathit{T}}_{\mathit{m}}}^{\mathbf{rms}}$ | ${\overline{\mathit{\u03f5}}}_{{\mathit{\theta}}_{\mathit{m}}}$ | ${\mathit{\u03f5}}_{{\mathit{\theta}}_{\mathit{m}}}^{\mathbf{rms}}$ |
---|---|---|---|---|---|---|

(${\overline{\mathit{H}}}_{\mathit{s}}$) | (${\mathit{\sigma}}_{{\mathit{H}}_{\mathit{s}}}$) | (${\overline{\mathit{T}}}_{\mathit{m}}$) | (${\mathit{\sigma}}_{{\mathit{T}}_{\mathit{m}}}$) | (${\overline{\mathit{\theta}}}_{\mathit{m}}$) | (${\mathit{\sigma}}_{{\mathit{\theta}}_{\mathit{m}}}$) | |

46217 | 0.67 m | 3.13 m | −0.16 s | 2056 s | −8.67° | 93.18° |

(1.05 m) | (0.40 m) | (6.77 s) | (1.36 s) | (180.80°) | (102.59°) |

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**MDPI and ACS Style**

Muscarella, P.; Brunner, K.; Walker, D.
Estimating Coastal Winds by Assimilating High-Frequency Radar Spectrum Data in SWAN. *Sensors* **2021**, *21*, 7811.
https://doi.org/10.3390/s21237811

**AMA Style**

Muscarella P, Brunner K, Walker D.
Estimating Coastal Winds by Assimilating High-Frequency Radar Spectrum Data in SWAN. *Sensors*. 2021; 21(23):7811.
https://doi.org/10.3390/s21237811

**Chicago/Turabian Style**

Muscarella, Philip, Kelsey Brunner, and David Walker.
2021. "Estimating Coastal Winds by Assimilating High-Frequency Radar Spectrum Data in SWAN" *Sensors* 21, no. 23: 7811.
https://doi.org/10.3390/s21237811