# Estimating Wave Direction Using Terrestrial GNSS Reflectometry

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## Abstract

**:**

## 1. Introduction

_{coh}at which the coherence is lost, while [13] have shown that the damping factor of the oscillating SNR data can be used to find the significant wave height.

_{coh}.

## 2. Theoretical Background

#### 2.1. Analysis of SNR Data

_{0}, c

_{1}and c

_{2}are the unknown parameters of a polynomial trend function of the time t, k is the wave number, λ is the wave length of the GNSS signal, d is the unknown damping coefficient, ε is the complementary angle of the incidence angle, Amp is the unknown amplitude of the SNR, and ${\mathsf{\varphi}}_{0}$ is the unknown phase offset of the oscillation. The oscillation is governed by the reflector height h

_{ref,t}, which is the height of the antenna above the reflecting surface at the position of the specular point, which might be variable in time. With h

_{APC}as the height of the antenna phase centre (APC) in a certain height datum and h

_{tide,t}as time-variable water surface height in the same height datum, h

_{ref,t}can be described as

_{sphere,t}corrects for the surface curvature in a spherical approximation. The complementary angle of the incidence angle ε defers from the elevation angle of the satellite due to the curvature of the reflecting surface and can be calculated according to [6,15]. The tropospheric refraction can be considered by a correction of ε derived from an astronomic refraction model [16,17,18,19].

_{APC}and h

_{tide}in Equation (2) are known, a non-linear least-squares adjustment for every satellite can be applied to estimate the individual unknown parameters. If h

_{ref}is likewise unknown, it can be assumed that it is constant for all satellites observed at the same time. Due to the multimodality of h

_{ref}, this parameter can be included in the non-linear adjustment only if good initial values are available. Otherwise, optimization techniques might be applied [20].

_{ε}becomes small in comparison to the noise of the SNR data and above a certain angle, it disappears in the noise. It is assumed that this is the cutoff angle ε

_{coh}, at which the coherence is lost. The threshold at which the loss of coherence is assumed is a matter of definition. We suppose to use a threshold that is related to the standard deviation of the SNR data σ

_{SNR}derived from the non-linear least-squares adjustment multiplied by a factor f. Under these assumptions the coherence is lost if

_{coh}is therefore deduced from Equations (3) and (4) as

_{coh}. The actual value of the factor f is of minor importance for the investigation of the anisotropic behavior of the cutoff angle ε

_{coh}as long as it is constant for all satellites involved in the investigation of a particular sea state.

#### 2.2. Scattered Reflection

_{h}is the standard deviation of the surface heights. For water surfaces, it is assumed to be approximately a quarter of the SWH [22]. Furthermore, λ is the wavelength of the GNSS signal, T is the correlation length of the reflecting surface, θ

_{i}is the incidence angle with θ

_{i}= 90°-ε, θ

_{r}is the reflecting angle, ϕ

_{r}is the azimuth of the reflection, X and Y are the dimensions of the reflecting area, A in the coordinate direction x and y, and k is again the wave number of the GNSS signal. We are interested in the scattered reflection in the direction of a specular reflection. Therefore, the incidence angle is equal to the reflecting angle and the azimuth of the reflection becomes zero. Since v

_{x}and v

_{y}become zero and D is 1 for this case, Equation (6) simplifies to

_{coh}. In accordance with [11] we use a value of 1 for the relation in this work.

_{coh}for correlation lengths between 2 to 60 m, and for different SWH at a reflector height of 12.3 m. It can be seen that for a particular SWH the cutoff angle ε

_{coh}reaches values of remarkable differences in dependence of the correlation length T. Hence, it should be possible to derive the wave direction if the cutoff angle can be derived from Equation (5) in several azimuthal directions and if the differences in the correlation length are large enough in these azimuthal directions. To clarify the range of the correlation length for sea surfaces, simulated wave fields will be used in Section 3.

## 3. Simulations

#### 3.1. Waves

_{i}= ω

_{i}

^{2}/g is the deep water wave number at angular frequency ω

_{i}, while g is the gravitational acceleration. ${\mathsf{\theta}}_{\mathrm{j}}$ is the direction of the elementary wave and ${\mathsf{\phi}}_{\mathrm{i}\mathrm{j}}$ is a random initial phase. The amplitude ${\mathrm{A}}_{\mathrm{i},\mathrm{j}}$ can be derived approximately from a directional wave spectrum as [25]

_{p}[22].

_{p}, as well as for ${\mathsf{\theta}}_{\mathrm{min}}$ and ${\mathsf{\theta}}_{\mathrm{max}}$, were derived from real data observed over a period of two months at the wave buoy ElbeWR in the North Sea at the Outer Elbe that was deployed by the German Federal Maritime and Hydrographic Agency (Bundesamt für Seeschifffahrt und Hydrographie, BSH). Besides other data, the buoy provides SWH, peak periods, wave principal directions, and the wave directional spreading with a temporal resolution of 30 min. Figure 3 depicts observed data for peak periods (a) and directional spreading (b). The peak period does not fall below a certain value for a specific SWH. Hence, a linear function was fitted to derive the wave peak period T

_{p,SWH}as a function of SWH, which will be used for the simulations. Likewise, the directional spreading does rarely exceed the plotted upper bound from which a piecewise linear function is derived. The directional spreading ${\mathsf{\theta}}_{\mathrm{S}\mathrm{W}\mathrm{H}}$ resulting from this function is used to define the minimal and a maximal wave direction as ${\mathsf{\theta}}_{\mathrm{min}}=-{\mathsf{\theta}}_{\mathrm{S}\mathrm{W}\mathrm{H}}/2$ and ${\mathsf{\theta}}_{\mathrm{max}}={\mathsf{\theta}}_{\mathrm{S}\mathrm{W}\mathrm{H}}/2$. It should be stated here, that the derived functions do not represent any physical relation between SWH and the peak period or the directional spreading, respectively. These functions can only be used to derive values for these parameters for the simulation of different wave fields.

#### 3.2. Calculation of Wave Direction

_{coh}from Equation (8) again by application of Newton’s method. Figure 6 depicts the cutoff angles for the different azimuthal directions for SWH values of 0.1 m, 1.3 m, and 2.5 m. The cutoff angles show a clear anisotropic behavior. We estimated the semi-minor and semi-major axes and the azimuth of the semi-major axis of a fitting ellipse from a least-squares adjustment. The difference of the axes’ length for the first ellipse at a SWH of 0.1 m is not significant. Hence, the direction of the semi-major axis defers from the downwind direction. For all $\mathrm{S}\mathrm{W}\mathrm{H}\ge 0.3\text{}\mathrm{m}$, the difference of the axes’ length is significant and the semi-major axes coincides with the downwind direction.

_{coh}can be calculated also for the minimum and maximum correlation length function presented in Figure 5b. As mentioned above, a random value with a standard deviation s of 5 cm was added to the heights of the simulated wave field. This will have a larger influence on the wave fields with smaller SWH values. We took this into account by calculating σ

_{h}in Equation (6) as of

_{coh}. At least for the simulated data, the semi-major axis coincides with downwind direction.

## 4. Validation with Experimental Data

_{coh}were than calculated according to Equation (5), whereby a factor f of 1.0 was used. Likewise, the standard deviation of the cutoff angles was derived from a covariance propagation of the results from the least-squares adjustment. All resulting cutoff angles together with their corresponding azimuth assigned to the same time slot were then used to fit an ellipse by means of a weighted least-squares adjustment, while the weights were derived from the standard deviation of the cutoff angles. For about 36% of all time slots the adjustment yielded significant differences of the semi-major and semi-minor axes of the ellipses. Figure 9 depicts the SWH at tide gauge together with significant time slots in red (at SHW = 1 m). About 73% of the time slots with significant differences show an average SWH of more than 0.3 m, while about 61% of the time slots with insignificant differences show an average SWH of less than 0.5 m. On day-of-year 191 and 192, the SWH reached values of more than 2 m but only some slots of these days show significant results. This seems to contradict the finding from the simulation in Section 3, that for higher SWH the results will be significant. However, in reality the oscillation of GNSS SNR observations at low elevation angles might become noisier due to shadowing effects or stronger tropospheric refraction. For the case of the tide gauge in the vicinity to the coast of the island, also breaking waves or converted shallow water waves might be included in the data set. An identification of such influence based on the existing data was not possible.

## 5. Conclusions and Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Data Availability Statement

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**Figure 1.**Detrended signal-to-ratio (SNR) data from a global navigation satellite systems (GNSS) receiver at a tide gauge station (grey dots) forday-of-year 190 in 2018, GPS PRN 8. The red line shows the adjusted oscillation, the horizontal continuous blue lines show the threshold according to Equation (4) for the factor f, and the vertical dashed blue lines present the corresponding cutoff angle ε

_{coh}with 7.52° for f = 0.5 and 6.06° for f = 1.0.

**Figure 2.**Cutoff angle ε

_{coh}as a function of different correlation lengths T and significant wave heights (SWH). The cutoff angle was derived for incoh = 1.

**Figure 3.**Observed peak period (

**a**) and directional spreading (

**b**) plotted over SWH for the wave buoy ElbeWR. Red lines show the functions used to derive T

_{p}and ${\mathsf{\theta}}_{\mathrm{min}}$ and ${\mathsf{\theta}}_{\mathrm{max}}$ for simulations.

**Figure 5.**Correlation lengths for 100 simulated wave fields for a SWH of 2.5 (

**a**). The red line shows the average correlation length for the corresponding azimuth. Minimum and maximum of the mean correlation length for the specific SWH and their quadratic fit (

**b**).

**Figure 6.**Cutoff angles for three SWH values and different azimuth (blue dots). Ellipses in red present the least-squares fit with their semi-major (red line) and semi-minor (green line) axes. The axes at SHW of 0.1 m do not show a significant difference. All other semi-major axes coincide with the west-to-east downwind direction.

**Figure 7.**Differences of the maximum and minimum cutoff angles in relation to the minimum cutoff angle plotted over SWH. The difference is becoming larger and more pronounced with increasing SWH values.

**Figure 8.**Position of GNSS station north of the island of Wangerooge (

**a**) and GNSS antenna installed atop of the tide gauge station TGW2 in the North (

**b**) (photo BfG).

**Figure 10.**Cutoff angles (grey dots, shading with respect to their weights), adjusted ellipses, and average wave directions derived from buoy data, (

**a**) SWH = 1.2 m, (

**b**) SWH = 0.35 m.

**Figure 11.**Wave direction from the buoy (grey dots, shading with respect to SWH), azimuth of semi-major axis derived from the analysis of the cutoff angles (red) and wind direction (blue).

**Figure 12.**Correlation between the wave direction from the buoy and the wind direction (

**a**) and the wave direction from the buoy and the wave direction from the cutoff angles (

**b**). The shading of the dots is with respect to SWH.

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**MDPI and ACS Style**

Reinking, J.; Roggenbuck, O.; Even-Tzur, G.
Estimating Wave Direction Using Terrestrial GNSS Reflectometry. *Remote Sens.* **2019**, *11*, 1027.
https://doi.org/10.3390/rs11091027

**AMA Style**

Reinking J, Roggenbuck O, Even-Tzur G.
Estimating Wave Direction Using Terrestrial GNSS Reflectometry. *Remote Sensing*. 2019; 11(9):1027.
https://doi.org/10.3390/rs11091027

**Chicago/Turabian Style**

Reinking, Jörg, Ole Roggenbuck, and Gilad Even-Tzur.
2019. "Estimating Wave Direction Using Terrestrial GNSS Reflectometry" *Remote Sensing* 11, no. 9: 1027.
https://doi.org/10.3390/rs11091027