#### 2.1. Theory

[6] Owing to weak nonlinear effects, any pair of monochromatic ocean wave trains force water motions with frequencies *f*_{2} and wave number **k**_{2} that are the sum of the free ocean wave train frequencies and wave numbers. For equal frequency but opposing directions, the forcing is equivalent to horizontally uniform pressure oscillations applied at the sea surface [*Miche*, 1944]. Including water compressibility effects, *Longuet*-*Higgins* [1950] showed how seismic waves are excited with wave number **k**_{s} = **k**_{2} and frequency *f*_{s} = *f*_{2}. Here we use the extension to random waves given by *Hasselmann* [1963]. Proposed extensions to waves in finite depth [*Webb*, 2007] and low frequencies [*Tanimoto*, 2010] have no influence on the results presented here.

[7] The local sea state can be described by the directional wave spectrum *F*(*f*, θ). For practical purposes, this spectrum can be written as a product of the frequency spectrum *E*(*f*), which is the power spectrum of the sea surface vertical displacement usually obtained from the heave time series of a buoy, and a directional distribution *M*(*f*, θ) which gives, at each frequency the distribution of the surface elevation variance over all directions, from 0 to 2*π* and normalizes such that, for any *f*,

Hence, we have *F*(*f*, θ) = *E*(*f*)*M*(*f*, θ). From this spectrum, *Hasselmann* [1963] gives the wave-induced pressure at near-zero wave number (**k**_{2} ≃ 0) and twice the wave frequency (*f*_{2} = 2*f*). The quasi-equality **k**_{2} ≃ 0 means that **k**_{2} is much less than the surface gravity wave number, and thus practically zero for the evaluation of the spectral density of the wave-induced pressure. Obviously 2*π f*_{2}/*k*_{2} is still finite and equal to the seismic phase speed.

[8] *Hasselmann* [1963, equation (2.15)] (see Appendix A for details) can be transformed into

where *F*_{p3D} has S.I. units of Pa^{2} m^{2} s. In the following, we will also discuss the two-dimensional frequency-integrated spectrum *F*_{p2D} with S.I. units of Pa^{2} m^{2}.

[9] *I*(*f*) is a nondimensional function that depends only on the wave energy distribution *M* over the directions θ,

We note that *Farrell and Munk* [2008] used a definition of *I* with an integral over [0, 2*π*] which gives values of *I* that are twice as large. Since the integrand has a periodicity of *π*, we prefer to integrate only over [0, *π*], which simplifies the final expression for the noise source. This directional integral forbids generic and accurate relationships between the significant oceanic wave height

and the seismic sources.

[10] Using common oceanographic practice, as illustrated by Figure 1, we define the wind-sea as the part of the sea state with directions within 90° from the wind direction and associated with peaks for which the phase speeds are less than 1.2 times *U*_{10}, the wind speed at 10 m height. That second condition corresponds, in deep water, to *g*/(2*π f*) < 1.2*U*_{10}. This association of spectral components with a given peak is done using an watershed-type algorithm: all components that lie on a path of monotonically decreasing energy from the peak are associated with it. As a result, the wind-sea peak often involves some energy of wave components that would, by themselves, be rather identified as swell. Swells are all other wave components, which includes components generated by wind-sea reflection from shorelines. The swell part of the wave spectrum is usually further partitioned into different swell systems when the wave spectrum contains several narrow peaks. These peaks generally correspond to waves generated by different storms [e.g., *Hanson and Phillips*, 2001].

[11] In general, *I* is largest when swell from a distant source propagates against the locally generated wind-sea [e.g., *Kedar et al.*, 2008], in cases with rapidly turning winds [e.g., *Gerstoft et al.*, 2006], or when waves are reflected off the shoreline [e.g., *Bromirski et al.*, 1999]. Yet, *I* can still be significant for a wind that is constant and uniform, due to the generation of waves at very oblique angles (more than 90°) relative to the wind [*Donelan et al.*, 1985; *Long and Resio*, 2007]. Figure 2 summarizes these different conditions. The directional distribution of the wave energy is thus a critical characteristic of the wavefield, and the expected features of modeled and real sea states are discussed in section 3. The wave-induced pressure given by equation (2) may generate all sorts of seismic modes, including surface and body waves, because all seismic waves have very small wave numbers *k*_{s} ≪ *k*, for which the spectral density of the wave-induced pressure is uniform. Following Longuet-Higgins, we will now restrict our scope to the fundamental mode of seismic Rayleigh waves which is dominant in the band of periods from 5 to 12 s [*Haubrich and McCamy*, 1969; *Koper et al.*, 2010].

[12] With a uniform rock density *ρ*_{s} and shear wave velocity *β*, the wave-induced surface pressure yields an equivalent source for the power spectrum of the vertical ground displacement [*Hasselmann*, 1963],

where the nondimensional coefficient 0.05 < < 0.84 varies with the ratio of the acoustic wavelength to water depth *D*. We use = *c*_{1}^{2} + *c*_{2}^{2}, where *c*_{1} and *c*_{2} are the amplitude response functions for the first two normal modes, taken from *Longuet*-*Higgins* [1950]. For *f*_{s} = 0.15 Hz, peaks at *D* ≃ 2300 m, and this depth varies inversely proportionally to *f*_{s}. The units of *S*_{DF} (*f*_{s}) are meters times seconds. For display purposes we also define

with units of meters. *S*_{DF,I} is the local contribution to the vertical ground displacement variance, which has units of m^{2}, per unit distance of the seismic wave propagation.

[13] From this we obtain *F*_{δ} (*f*_{s}), the power spectrum of the vertical ground displacement, at any seismic station of latitude *λ* and longitude ϕ, as the instantaneous combination of sources at (*λ*′, ϕ′). We propagate the seismic (Rayleigh) waves in a vertically symmetric earth model, neglecting all three-dimensional propagation effects, and parameterizing seismic wave scattering and dissipation with a uniform quality factor *Q*. The seismic power attenuation is thus a function of the spherical distance Δ between source and station, giving,

with *R*_{E} being the Earth radius and *U* being the seismic group velocity. The bracketed term (*R*_{E}^{2} sinϕ′d*λ*′dϕ′) is the Earth surface area element. *R*_{E} sinΔ in the denominator is the geometrical spreading factor for wave energy that follows geodesics on the sphere [e.g., *Kanamori and Given*, 1981], replacing the distance *R*_{E} Δ used in flat Earth models [e.g., *Hasselmann*, 1963]. This expression assumes that *UQ* is the same for all the seismic modes.

[14] Our model yields the power spectrum of the ground displacement, and for quantitative validation we will use the root mean square (RMS) vertical ground displacement

For infinitely high *Q*, i.e., without loss of seismic energy, a source of power *S*_{DF,I} = 10^{−16} m uniform over a square of 100 by 100 km located at a distance of 1000 km, gives a displacement variance *δ*_{RMS}^{2} = 1*μ* m^{2}. With typical values of the parameters in equation (5), such a source can be given by a wave-induced pressure *F*_{p2D} ≃ 3 × 10^{4} hPa^{2} m^{2}, which is of the order of the highest sources given by our model. The variance actually recorded at a station is the sum of the variances contributed by all such elementary squares. Hence the noise level increases significantly with the spatial extension of the source area.

#### 2.2. Numerical Wave Modeling for Seismic Noise

[15] Wave spectra are provided by a numerical wave model based on the version 3.14 of the WAVEWATCH III^{(R)} code [*Tolman*, 1991, 2009], using improved parameterizations for wind-wave generation and dissipation. These parameterizations have been carefully adjusted to reproduce as well as possible a wide variety of observations, including directional properties [*Ardhuin et al.*, 2010; *Delpey et al.*, 2010]. The better quality of these parameterization has been validated independently by 2 years of routine analysis and forecasting [e.g., *Bidlot*, 2009] (see http://tinyurl.com/3vpr7jd). Our global model, with a resolution of 0.5° in latitude and longitude, is forced by 6-hourly wind analysis from the European Centre for Medium range Weather Forecasting (ECMWF), daily ECMWF sea ice concentration analysis, and monthly Southern Ocean distribution statistics for small icebergs [*Tournadre et al.*, 2008]. Adding icebergs was critical for improving the model quality for latitudes south of 45°S [*Ardhuin et al.*, 2011]. Using higher spatial resolution for coastal areas did not significantly change the model results. We use a spectral resolution with 24 directions by 31 frequency, and the noise sources are only stored for 15 frequencies (0.04 to 0.17 Hz) in order to limit the amount of stored information. All the computed seismic noise sources are available in NetCDF format from the IOWAGA project (http://tinyurl.com/yetsofy), and cover the years 1994 to 2010, with an ongoing extension into the future.

[16] A novel aspect of our wave model is the introduction of reflections both at the ocean-land boundary, the usual shoreline, and also around icebergs, which provide additional “shorelines,” which are the sides of the icebergs. Technical details are described in Appendix B, and we focus here on the magnitude of the reflection. Shoreline reflections are observed to decrease with wave height and wave frequency, and strongly increase with bottom slope. Using a nondimensional reflection coefficient *R* for the wave amplitudes, the energy reflection coefficient is *R*^{2} [e.g., *Mei*, 1989]. *R*^{2} inferred from measurements is less than 0.05 for most wave conditions over a moderately steep beach [*Elgar et al.*, 1994]. Only for nearly vertical cliffs we may expect higher values, up to *R*^{2} = 0.4 [*O*'*Reilly et al.*, 1999] (see also http://www.coastalresearchcenter.ucsb.edu/cmi/files/2001-055.pdf).

[17] The model distinguishes three types of shorelines. We first define *R*^{2} for pieces of the shoreline that are longer than the wave model grid spacing (type A: continents and large islands). The shorelines of all smaller islands (type B) are taken from the shoreline database of *Wessel and Smith* [1996]. For icebergs (type C), *R*^{2} is again different.

[18] In a compromise between model simplicity and accuracy, we used values of *R*^{2} that are spatially uniform for each shoreline type, and independent of wave amplitude. Our goal here is not to provide the most accurate modeling of coastal reflection but rather to test a range of plausible values and their impact on seismic noise. We thus obtain *F*_{δ,R} (*f*_{s}) with *R*^{2} = 0.1 for type A, *R*^{2} = 0.2 for type B, which is fairly arbitrary but intended to give a higher reflection over usually steeper small-scale coral reefs, and *R*^{2} = 0.4 for type C because icebergs have nearly vertical sides. For *R*^{2} < 0.4, a double reflection would only give less than 16% of the incoming energy, which may be neglected. We can thus treat reflection as a linear process and obtain the modeled noise spectra *F*_{δ} (*f*_{s}) by a linear combination of noise spectra *F*_{δ,R} (*f*_{s}) and *F*_{δ,0} (*f*_{s}) obtained with and without reflection, respectively. In order to capture some of the observed variability of *R*^{2} we have chosen a linearly varying frequency dependence given by the empirical function *P*,

so that our estimation of the seismic noise spectrum is

This reflection model gives reflection coefficients in the upper range of those inferred from the beach data of *Elgar et al.* [1994], with *R*^{2} = 0.06 at *f* = 0.04 Hz for type A shorelines. An independent validation of *R*^{2} is provided in the next section.