Tsunami Detection by High-Frequency Radar Beyond the Continental Shelf

Abstract

Where coastal tsunami hazard is governed by near-field sources, such as submarine mass failures or meteo-tsunamis, tsunami propagation times may be too small for a detection based on deep or shallow water buoys. To offer sufficient warning time, it has been proposed to implement early warning systems relying on high-frequency (HF) radar remote sensing, that can provide a dense spatial coverage as far offshore as 200–300 km (e.g., for Diginext Ltd.’s Stradivarius radar). Shore-based HF radars have been used to measure nearshore currents (e.g., CODAR SeaSonde^® system; http://www.codar.com/), by inverting the Doppler spectral shifts, these cause on ocean waves at the Bragg frequency. Both modeling work and an analysis of radar data following the Tohoku 2011 tsunami, have shown that, given proper detection algorithms, such radars could be used to detect tsunami-induced currents and issue a warning. However, long wave physics is such that tsunami currents will only rise above noise and background currents (i.e., be at least 10–15 cm/s), and become detectable, in fairly shallow water which would limit the direct detection of tsunami currents by HF radar to nearshore areas, unless there is a very wide shallow shelf. Here, we use numerical simulations of both HF radar remote sensing and tsunami propagation to develop and validate a new type of tsunami detection algorithm that does not have these limitations. To simulate the radar backscattered signal, we develop a numerical model including second-order effects in both wind waves and radar signal, with the wave angular frequency being modulated by a time-varying surface current, combining tsunami and background currents. In each “radar cell”, the model represents wind waves with random phases and amplitudes extracted from a specified (wind speed dependent) energy density frequency spectrum, and includes effects of random environmental noise and background current; phases, noise, and background current are extracted from independent Gaussian distributions. The principle of the new algorithm is to compute correlations of HF radar signals measured/simulated in many pairs of distant “cells” located along the same tsunami wave ray, shifted in time by the tsunami propagation time between these cell locations; both rays and travel time are easily obtained as a function of long wave phase speed and local bathymetry. It is expected that, in the presence of a tsunami current, correlations computed as a function of range and an additional time lag will show a narrow elevated peak near the zero time lag, whereas no pattern in correlation will be observed in the absence of a tsunami current; this is because surface waves and background current are uncorrelated between pair of cells, particularly when time-shifted by the long-wave propagation time. This change in correlation pattern can be used as a threshold for tsunami detection. To validate the algorithm, we first identify key features of tsunami propagation in the Western Mediterranean Basin, where Stradivarius is deployed, by way of direct numerical simulations with a long wave model. Then, for the purpose of validating the algorithm we only model HF radar detection for idealized tsunami wave trains and bathymetry, but verify that such idealized case studies capture well the salient tsunami wave physics. Results show that, in the presence of strong background currents, the proposed method still allows detecting a tsunami with currents as low as 0.05 m/s, whereas a standard direct inversion based on radar signal Doppler spectra fails to reproduce tsunami currents weaker than 0.15–0.2 m/s. Hence, the new algorithm allows detecting tsunami arrival in deeper water, beyond the shelf and further away from the coast, and providing an early warning. Because the standard detection of tsunami currents works well at short range, we envision that, in a field situation, the new algorithm could complement the standard approach of direct near-field detection by providing a warning that a tsunami is approaching, at larger range and in greater depth. This warning would then be confirmed at shorter range by a direct inversion of tsunami currents, from which the magnitude of the tsunami would also estimated. Hence, both algorithms would be complementary. In future work, the algorithm will be applied to actual tsunami case studies performed using a state-of-the-art long wave model, such as briefly presented here in the Mediterranean Basin.

Appendices

Appendix 1: Directional Wave Number Spectrum

In the present simulations of the radar signal, we will use a standard analytical form of the directional wave energy density spectrum $\Psi ({\varvec{K}})$, as a function of the wavenumber vector. Assuming a fully developed sea, the spectrum will be constructed based on the one-parameter Pierson–Moskowitz frequency spectrum, which depends solely on wind speed, for instance $V_{19.5}$ measured at 19.5 m above the sea level, and an analytical angular spreading function.

A commonly used angular spreading function of direction $\theta$, given the dominant direction of wind waves $\theta _p$ is,

$$\begin{aligned} D(\theta ,\theta _p,s,\xi ) = \frac{\xi + (1-\xi )\, \cos ^{s}{\{\frac{\theta -\theta _p}{2}}\}}{N(s,\xi )}, \end{aligned}$$

(48)

where $\xi \in [0,1]$ represents the (asymmetric) fraction of the spectral wave energy associated with waves propagating in the opposite direction and s is an exponent controlling the peakedness of the spreading function. We use $\xi =0.1$ and $s=5$ in the present applications. The denominator of Eq. 48 is a normalization factor such that the integral of D over $[0,2\pi ]$ is equal to 1,

$$\begin{aligned} N(s,\xi ) = 2\pi \,\xi + 2\sqrt{\pi }(1-\xi )\, \frac{\Gamma (0.5(s+1))}{\Gamma (0.5s+1)} \end{aligned}$$

(49)

The directional PM spectrum used here is thus defined as,

$$\begin{aligned} \Psi _{PM}({\varvec{K}}) = \frac{\alpha }{2K^3}\,{\text {e}}^{-\left( \frac{K_p}{K}\right) ^2}\,D(\theta ,\theta _p,s,\xi ) \end{aligned}$$

(50)

with $\phi = {\text {atan}}{(K_y/K_x)}$, Phillips’ constant $\alpha = 0.0081$, and the spectral peak wavenumber is given by,

$$\begin{aligned} K_p = \sqrt{\frac{b\,g^2}{2\,V_{19.5}^4}} \end{aligned}$$

(51)

with $b=0.74$.

The $V_{19.5}$ value can be converted into the more standard parameter, $V_{10}$ (i.e., the wind speed at a 10 m elevation) by representing wind speed as a function of height, V(z), as a classical von Karman logarithmic atmospheric boundary layer profile, which yields the relationship, $V_{19.5} \simeq V_{10} (19.5/10)^{1/7}\simeq 1.10 V_{10}$. Figure 24 shows an example of a directional PM spectrum computed for $V_{10}=10$ m/s.

In deep water, the significant wave height corresponding to a given spectrum is obtained as a function of the zero-th moment of the spectral energy density, that is,

$$\begin{aligned} H_s = 4 \left\{ \int \int \Psi (K_x,K_y) {\text {d}}K_x\,{\text {d}}K_y \right\} ^{\frac{1}{2}} \end{aligned}$$

(52)

For the PM spectrum in Fig. 24, we find $H_s = 1.71$ m, with $K_p=0.0493$ m$^{-1}$ or $L_p = 127.4$ m, and in deep water (see Eq. 2), $T_p = 9.04$ s.

Fig. 24

Directional PM energy density spectrum (expressed in dB) for a wind speed $V_{10}=10$ m/s, with $s=5$, $\xi =0.1$ and $\theta _p = 0^\circ$, which corresponds to, $H_s=1.71$ m and $T_p=9.04$ s

Appendix 2: Second-Order Bragg Kernel and Radar Scattering

The second-order Bragg kernel is given by

$$\begin{aligned} {\mathbb {B}}_2^{\epsilon _1,\epsilon _2}({\varvec{K}}_1,{\varvec{K}}_2)={\mathbb {B}}_1\left( \Gamma _e({\varvec{K}}_1,{\varvec{K}}_2)+\Gamma _h^{\epsilon _1,\epsilon _2}({\varvec{K}}_1,{\varvec{K}}_2)\right) , \end{aligned}$$

(53)

in which $\Gamma _e$ is the electromagnetic, and $\Gamma _h^{\epsilon _1,\epsilon _2}$ the hydrodynamic, coupling coefficient. The latter is defined as

$$\begin{aligned} \Gamma _h^{\epsilon _1,\epsilon _2}({\varvec{K}}_1,{\varvec{K}}_2)&= \frac{1}{2}\left[ K_1 +K_2+ \frac{g}{\omega _1 \omega _2} (K_1 K_2-\varvec{{\varvec{K}}_1}\cdot \varvec{{\varvec{K}}_2})\right. \nonumber \\&\quad \times \left. \left( \frac{\sqrt{g|\varvec{K}_1+\varvec{K}_2|}+(\omega _1 + \omega _2)^2}{\sqrt{g|{\varvec{K}}_1+{\varvec{K}}_2|}-(\omega _1 + \omega _2)^2}\right) \right] , \end{aligned}$$

(54)

with $\omega _1=\epsilon _1\sqrt{gK_1}$ and $\omega _2=\epsilon _1\sqrt{gK_2}$ and the former is defined as

$$\begin{aligned} \Gamma _{e}({\varvec{K}}_1,{\varvec{K}}_2)=\frac{(\varvec{{\varvec{K}}_1}\cdot \varvec{{\varvec{u}}_i})(\varvec{{\varvec{K}}_2}\cdot \varvec{({\varvec{K}}_1-K_0{\varvec{u_i}})}}{2K_0{\text {cos}}^2(\phi _{bi})\left( \sqrt{\varvec{{\varvec{K}}_1}\cdot \varvec{({\varvec{K}}_1-2K_0{\varvec{u}}_i)}}+iK_0\Delta \right) }, \end{aligned}$$

(55)

where ${\varvec{u_i}}$ is an unit vector from pointing from the transmitter to the radar cell, $\Delta$ is the normalized surface impedance, and $\phi _{bi}$ is the bistatic angle (equal to zero in the monostatic case). See the appendix in Grosdidier et al. (2014) for more details.