# Transformation of Water Wave Spectra into Time Series of Surface Elevation

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## Abstract

**:**

## 1. Introduction

## 2. Fourier Analysis Background

- Axiom 1:
- x is a real-valued function if and only if the Fourier transform of x is Hermitian;
- Axiom 2:
- x is a Hermitian function if and only if the Fourier transform of x is real-valued.

## 3. Transformation of the Spectrum

Oleinik, P.H.; Tavares, G.P.; Machado, B.N.; Isoldi, L.A. Transformation of Water Wave Spectra into Time Series of Surface Elevation: base implementation. Earth2021, 1, 1–9. Available online: https://gist.github.com/PhelypeOleinik/39803f2385a18d0e86dc3ff8fe02af7b (accessed on 18 September 2021).

## 4. Verification of the Method

`random_number`intrinsic function, which uses the xorshift1024∗ algorithm [21]. These were then transformed into normally-distributed random number streams using the polar form [22] of the Box–Muller transform [23].

#### 4.1. Verification of Average Values

#### 4.2. Reversing the Process to Obtain the Spectrum from $\eta $

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Histogram and box plot of the ratio between ${H}_{\sigma}$, ${H}_{{}^{1}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{/}_{\phantom{\rule{-0.166667em}{0ex}}3}\uparrow}$, and ${H}_{{}^{1}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{/}_{\phantom{\rule{-0.166667em}{0ex}}3}\downarrow}$ and the reference ${H}_{{m}_{0}}$ for the 5000 sampled spectra.

**Figure 4.**Wave spectra at the point selected for analysis. “Original” is the original spectrum used for the analysis, “Equation (9)” is the result of Equation (9) applied on the time series of $\eta $ obtained from the original spectrum, and “Result” is the result of Equation (9) after noise removal treatment.

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**MDPI and ACS Style**

Oleinik, P.H.; Tavares, G.P.; Machado, B.N.; Isoldi, L.A.
Transformation of Water Wave Spectra into Time Series of Surface Elevation. *Earth* **2021**, *2*, 997-1005.
https://doi.org/10.3390/earth2040059

**AMA Style**

Oleinik PH, Tavares GP, Machado BN, Isoldi LA.
Transformation of Water Wave Spectra into Time Series of Surface Elevation. *Earth*. 2021; 2(4):997-1005.
https://doi.org/10.3390/earth2040059

**Chicago/Turabian Style**

Oleinik, Phelype Haron, Gabriel Pereira Tavares, Bianca Neves Machado, and Liércio André Isoldi.
2021. "Transformation of Water Wave Spectra into Time Series of Surface Elevation" *Earth* 2, no. 4: 997-1005.
https://doi.org/10.3390/earth2040059