# Removing Wave Bias from Velocity Measurements for Tracer Transport: The Harmonic Analysis Approach

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## Abstract

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## 1. Introduction

^{2}/s

^{3}or Watt/kg) [1,2,3]. Turbulent Kinetic Energy has a critical effect in transporting and changing the local concentration of ocean tracers such as dissolved oxygen, carbon dioxide, nutrients, plankton, and pollutants. Processes at the air–water interface can influence the transport and transformation of tracers in the water column. For instance, wind-driven waves can affect tracer concentration and distribution through advection and diffusion. In particular, diffusive processes should be dominated by turbulence, which may be biased by waves because of their overlap in frequency and scale. Thus, the study of diffusion and turbulence in the water column, and its identification from waves’ influence, should allow an understanding of vertical exchange processes that determine the fate of tracers in aquatic environments. It follows that estimating Turbulent Kinetic Energy, including its transport, production, and dissipation, is obscured by the presence of surface waves because, as mentioned above, waves and turbulence share spectral energy bands. A reliable approach for velocity measurements is needed to distinguish the signal related to waves and to turbulence.

## 2. Previous Methods for Removing Wave Bias

## 3. Harmonic Analysis (HA) Method

^{2}/Hz) (Figure 1A,D). Wave energy is mostly distributed at frequencies higher than 0.09 Hz (Figure 1D). The highest significant wave height (Figure 1B) is approximately 0.4 m with a dominant period of 6 s at 20:00, on 22 November 2009. The dominant periods of the waves vary from 2 to 10 s, and the average periods change from 3 to 6.4 s (Figure 1C).

_{0}, ${A}_{j}$, and ${\varphi}_{j}$ are obtained by fitting the harmonics ${\omega}_{j}$ to ${u}_{beam}$. For a more in-depth explanation of the least squares fit to Equation (17), see the Supplementary Material, Section 1. The harmonic orbital velocities are reconstructed with u

_{0}, ${A}_{j}$, and ${\varphi}_{j}$ from Equation (17) (red line in Figure 3A). This signal is taken as the wave orbital velocities $\tilde{u}\left(t\right)$, and then subtracted from the ${u}_{beam}$, i.e.,

## 4. Comparison between VAF and HA Method

## 5. Discussion

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Data from the northeastern Gulf of Mexico, in the Florida Big Bend region. (

**A**) Time series of water surface height (m), (

**B**) Significant wave heights (Hs), (

**C**) Dominant wave period (DPD) in red and Average wave period (APD) in green, and (

**D**) Power Spectral Density (m

^{2}/Hz) of the time series shown in (A).

**Figure 2.**(

**A**) Beam1 velocities at 9.04 m above the bottom (blue line) during a 37-min segment. Data windows to compute Power Spectral Density (PSDs) (m

^{2}/s, black lines), and ensembles used to calculate Reynolds stresses ($-\overline{{u}^{\prime}{w}^{\prime}}$, red line). (

**B**) PSDs calculated with the data windows denoted by black lines in (A); the shaded area indicates the frequency band in need of wave-bias removal. The red circles denote peaks at the frequencies identified to reconstrut wave signals.

**Figure 3.**(

**A**) Beam1 velocities (blue line) and wave orbital velocities (red line) computed by the Harmonic Analysis (HA) method; (

**B**) PSDs (m

^{2}/s) of the peaks sorted by descending order; and 0.08 PSD (m

^{2}/s) in the white line (

**C**) Frequencies corresponding with the sorted PSDs and (

**D**) Cumulative Sum of Normalized Power Spectral Density (CNPSD). In both C and D, the white line represents the number of harmonics with 95% energy.

**Figure 4.**PSD (m

^{2}/s) of original Beam1 velocities (m/s) at (

**A**) 9.04 m, (

**B**) 7.04 m, (

**C**) 4.04 m, and (

**D**) 1.04 m above the bottom; PSD of Beam1 velocities of the Vertical Adaptive Filtering (VAF) method at (

**E**) 9.04 m, (

**F**) 7.04 m, (

**G**) 4.04 m, and (

**H**) 1.04 m and PSD of Beam1 velocities of the HA method (

**I**) 9.04 m, (

**J**) 7.04 m, (

**K**) 4.04 m, and (

**L**) 1.04 m. Line plots: (

**M**–

**P**), show the PSD corresponding to the black vertical line in the contour plots.

**Figure 5.**(

**A**) Time-averaged PSD (m

^{2}/s) as a function of height above bottom with original Beam1 velocities. (

**B**) Difference between the VAF method and the original time-averaged PSD. (

**C**) Difference between the HA method and original time-averaged PSD.

**Figure 6.**Time-averaged cospectra (${\mathrm{Co}}_{{u}^{\prime}{w}^{\prime}},$m

^{2}/s) of ${u}^{\prime}$ and ${w}^{\prime}$ (

**A**) original; (

**B**) corrected by VAF method; and (

**C**) corrected by HA method.

**Figure 7.**Turbulent Knetic Energy (TKE) estimated by (

**A**) Original data, (

**B**) VAF, (

**C**) HA method with 95% wave energy removed, and (

**D**) HA method with 100% wave energy removed.

**Figure 8.**Reynolds stresses ($-\overline{{u}^{\prime}{w}^{\prime}}$) computed by original data (blue dotted line), VAF method (red dotted line), and HA method (black solid line) at (

**A**) 9.04 m; at (

**B**) 5.04 m; and at (

**C**) 1.04 m.

**Figure 9.**Standard deviation (

**A**) of VAF method Reynolds stress estimates ($-{u}^{\prime}{w}^{\prime}$) and (

**B**) of HA method Reynolds stress estimates, normalized by that of original Reynold stress estimates. The white line represents the value 1.

**Figure 10.**Data from a shoal off Cape Canaveral, Florida. (

**A**) Significant wave heights (H

_{s}) with a blue solid line marked on the left of the y-axis and dominant wave period (T

_{P}) marked on the right of the y-axis. (

**B**) The PSD corresponding to the black vertical line in (

**C**) and (

**F**). The green and black line denotes the PSD of original data and 95% wave-removed data, respectively. Red line is a reference line for –5/3. The PSD of original data are shown (

**C**) at 7.35 m, (

**D**) at 4.35 m, and (

**E**) at 0.75 m height. The PSD of 95% wave-removed data are given (

**F**) at 7.35 m, (

**G**) at 4.35 m, and (

**H**) at 0.75 m height.

**Figure 11.**Comparison of TKE with original data to TKE with the HA method (

**A**) at 0.75 m, (

**B**) at 2.85 m, (

**C**) at 4.95 m, and (

**D**) at 7.35 m. Red and blue circles denote the estimation of 95% and 100% wave-removed data, respectively. Red and blue solid lines denote the corresponding linear-fitting line of the estimations. Black dotted line indicates the reference line of equilibrium between TKE with original data and TKE with the HA method.

**Table 1.**Performance indicators of wave-removal methods in the estimates of turbulence parameters at different distances from the bottom. The HA method was more effective in reducing wave-related values of Turbulent Kinetic Energy (TKE) and Reynolds stress than the VAF method.

Method | TKE (% Reduction) | Reynolds Stress (Standard Dev × 10^{−3} m^{2}/s^{2}) | |||
---|---|---|---|---|---|

1.04 m | 5.04 m | 9.04 m | 1.04 m | 9.04 m | |

VAF | 24 | 34 | 60 | 0.55 | 0.87 |

HA | 55 | 63 | 67 | 0.46 | 0.58 |

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**MDPI and ACS Style**

So, S.; Valle-Levinson, A.; Laurel-Castillo, J.A.; Ahn, J.; Al-Khaldi, M. Removing Wave Bias from Velocity Measurements for Tracer Transport: The Harmonic Analysis Approach. *Water* **2020**, *12*, 1138.
https://doi.org/10.3390/w12041138

**AMA Style**

So S, Valle-Levinson A, Laurel-Castillo JA, Ahn J, Al-Khaldi M. Removing Wave Bias from Velocity Measurements for Tracer Transport: The Harmonic Analysis Approach. *Water*. 2020; 12(4):1138.
https://doi.org/10.3390/w12041138

**Chicago/Turabian Style**

So, Sangdon, Arnoldo Valle-Levinson, Jorge Armando Laurel-Castillo, Junyong Ahn, and Mohammad Al-Khaldi. 2020. "Removing Wave Bias from Velocity Measurements for Tracer Transport: The Harmonic Analysis Approach" *Water* 12, no. 4: 1138.
https://doi.org/10.3390/w12041138