Noise with periods 3 to 10 s, ubiquitous in seismic records, is expected to be mostly generated by pairs of ocean wave trains of opposing propagation directions with half the seismic frequency. Here we present the first comprehensive numerical model of microseismic generation by random ocean waves, including ocean wave reflections. Synthetic and observed seismic spectra are well correlated (r > 0.85). On the basis of the model results, noise generation events can be clustered in three broad classes: wind waves with a broad directional spectrum (class I), sea states with a significant contribution of coastal reflections (class II), and the interaction of two independent wave systems (class III). At seismic stations close to western coasts, noise generated by class II sources generally dominates, but it is intermittently outshined by the intense class III sources, limiting the reliability of seismic data as a proxy for storm climates. The modeled seismic noise critically depends on the damping of seismic waves. At some mid-ocean island stations, low seismic damping is necessary to reproduce the observed high level and smoothness of noise time series that result from a spatial integration of sources over thousands of kilometers. In contrast, some coastal stations are only sensitive to noise within a few hundreds of kilometers. This revelation of noise source patterns worldwide provides a wealth of information for seismic studies, wave climate applications, and new constraints on the possible directional distribution of wave energy.
 Ocean waves have long been suspected to cause microseisms [Wiechert, 1904] with a dominant seismic peak at twice the wave frequency [Bernard, 1941]. A realistic theory was only given by Longuet-Higgins  and extended to random waves by Hasselmann . Seismic noise sources could be located with a fair degree of accuracy using large aperture seismic arrays, as demonstrated by Haubrich and McCamy , and the contributions of the various seismic wave modes was also revealed, showing that fundamental Rayleigh and Love modes tend to dominate at frequencies less than 0.2 Hz. Still, without a full quantitative verification, the debate lingers on the distribution of noise sources, including the relative importance of shoreline reflections, and the general validity of the Longuet-Higgins–Hasselmann theory.
 A first detailed verification of this theory using the output of a numerical wave model was only performed recently by Kedar et al. . This work was limited to northwest Atlantic sources and coastal reflections were not included, although they are probably important for other regions [e.g., Zopf et al., 1976; Chevrot et al., 2007]. Also, recent progress in numerical wave modeling [WISE Group, 2007; Ardhuin et al., 2010] should also result in more accurate seismic noise applications.
 Here we extend the work by Kedar et al.  to the global ocean and also take into account coastal reflections. We show that a single global model of seismic noise energy radiation, with realistic ocean wave reflections and seismic attenuation factors, can explain most observed seismic noise in spectra of vertical ground motions. This validated model is then used to discuss the importance of various classes of sea state that contribute to seismic sources and to explain the different seismic noise climates observed at a few selected seismic stations.
 Our model is based on a state-of-the-art numerical wave model [Ardhuin et al., 2010] to which coastal reflection was added, but its seismic part is simplified in many aspects. Also, we only model the spectral distribution of the seismic wave energy, without any phase information. This paper is organized as follows. After a review of the noise generation theory, the practical modeling of microseisms is described in section 2. The validity of numerical wave models for the estimation of seismic noise is discussed in section 3. General model results are then presented in section 4 with validation at a few selected locations in section 5. Perspectives for improved models are given in section 6.
2. Seismic Noise Modeling
 Owing to weak nonlinear effects, any pair of monochromatic ocean wave trains force water motions with frequencies f2 and wave number k2 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  showed how seismic waves are excited with wave number ks = k2 and frequency fs = f2. Here we use the extension to random waves given by Hasselmann . Proposed extensions to waves in finite depth [Webb, 2007] and low frequencies [Tanimoto, 2010] have no influence on the results presented here.
 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 2π and normalizes such that, for any f,
Hence, we have F(f, θ) = E(f)M(f, θ). From this spectrum, Hasselmann  gives the wave-induced pressure at near-zero wave number (k2 ≃ 0) and twice the wave frequency (f2 = 2f). The quasi-equality k2 ≃ 0 means that k2 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 2π f2/k2 is still finite and equal to the seismic phase speed.
where Fp3D has S.I. units of Pa2 m2 s. In the following, we will also discuss the two-dimensional frequency-integrated spectrum Fp2D with S.I. units of Pa2 m2.
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  used a definition of I with an integral over [0, 2π] 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.
 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 U10, the wind speed at 10 m height. That second condition corresponds, in deep water, to g/(2π f) < 1.2U10. 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].
 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 ks ≪ 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].
 With a uniform rock density ρs 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 = c12 + c22, where c1 and c2 are the amplitude response functions for the first two normal modes, taken from Longuet-Higgins . For fs = 0.15 Hz, peaks at D ≃ 2300 m, and this depth varies inversely proportionally to fs. The units of SDF (fs) are meters times seconds. For display purposes we also define
with units of meters. SDF,I is the local contribution to the vertical ground displacement variance, which has units of m2, per unit distance of the seismic wave propagation.
 From this we obtain Fδ (fs), 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 RE being the Earth radius and U being the seismic group velocity. The bracketed term (RE2 sinϕ′dλ′dϕ′) is the Earth surface area element. RE 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 RE Δ used in flat Earth models [e.g., Hasselmann, 1963]. This expression assumes that UQ is the same for all the seismic modes.
 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 SDF,I = 10−16 m uniform over a square of 100 by 100 km located at a distance of 1000 km, gives a displacement variance δRMS2 = 1μ m2. With typical values of the parameters in equation (5), such a source can be given by a wave-induced pressure Fp2D ≃ 3 × 104 hPa2 m2, 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
 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.
 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 R2 [e.g., Mei, 1989]. R2 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 R2 = 0.4 [O'Reilly et al., 1999] (see also http://www.coastalresearchcenter.ucsb.edu/cmi/files/2001-055.pdf).
 The model distinguishes three types of shorelines. We first define R2 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 . For icebergs (type C), R2 is again different.
 In a compromise between model simplicity and accuracy, we used values of R2 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 Fδ,R (fs) with R2 = 0.1 for type A, R2 = 0.2 for type B, which is fairly arbitrary but intended to give a higher reflection over usually steeper small-scale coral reefs, and R2 = 0.4 for type C because icebergs have nearly vertical sides. For R2 < 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 Fδ (fs) by a linear combination of noise spectra Fδ,R (fs) and Fδ,0 (fs) obtained with and without reflection, respectively. In order to capture some of the observed variability of R2 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. , with R2 = 0.06 at f = 0.04 Hz for type A shorelines. An independent validation of R2 is provided in the next section.
3. On the Accuracy of the Wave Model
 Since the work of Gelci et al. , numerical models have been used routinely to estimate and predict the directional wave spectrum F(f, θ) = E(f)M(f, θ), and the quality of the frequency spectrum E(f) is relatively well documented [e.g., Rogers et al., 2005; Ardhuin et al., 2007]. The model used here is known to have a slight (10 to 20%) positive bias for low-frequency energies f < 0.08 Hz along the U.S. west coast, which is noticeable in a high bias of modeled peak periods [Ardhuin et al., 2010]. The wave model was verified to produce accurate estimations of a wide variety of spectral moments (significant wave height and mean periods). Typical RMS errors for such moments are less than 10% for 3-hourly averages, with biases under 5%.
 On the contrary, the quality of modeled directional distributions M(f, θ) is seldom discussed. As a result, the accuracy of seismic noise sources is very uncertain because the I(f) integral (equation (3)) can have very different values with only small changes in the energy for some directions. Although no routine wave measurement can provide an estimation of I(f), we can use other quantities, measured by wave buoys, in order to get some confidence on the quality of modeled M(f, θ) distributions. We will here examine the first directional moment [e.g., Kuik et al., 1988],
where θ0 (f) is the mean wave direction at frequency f. m1 (f) is routinely obtained from heave-pitch-roll buoy data, such as measured by discus-shaped buoys used by the National Data Buoy Center (NBC), and three-axis acceleration data, such as provided by Datawell Waverider buoys, used by the Coastal Data Information Program (CDIP),
where Cxz is the cospectrum of x and z buoy displacements, with similar notations for the other cospectra.
 From m1 (f) it is customary to define the directional spread as
For a directional distribution with a single narrow Gaussian peak, σ1 is the standard deviation (half-width) of the distribution, and it asymptotes to 88° = rad as the spectrum broadens to become isotropic. We do not investigate other directional spreading parameters because their estimation from buoy data can be very noisy [O'Reilly et al., 1996].
3.1. Classification of Noise-Generating Sea States
 For any ocean wave frequency, we can expect three broad classes of sea states for which the directional term I is significant, as illustrated by Figures 2 and 3.
 In class I, the waves generated by the local wind are distributed with some energy at angles larger than 90° from the mean wave direction, which can then interact with waves in the opposite direction to generate seismic noise. The existence of this oblique energy is well established for wave frequencies typically larger than 0.2 Hz, based on high-frequency radar backscatter from the sea surface [e.g., Wait, 1966]. Hughes  used I = 1/16 for f > 0.5 Hz. Yet, the magnitude of this small amount of energy is very uncertain for wave frequencies below 0.2 Hz, at which most of the seismic noise energy is found.
 In class II, reflection from an obstacle (e.g., shoreline, iceberg) provides the source of waves in the opposing direction. Given the usually low reflection coefficient from shorelines, typically R2 < 0.05, the resulting value of I is expected to be less than 0.05.
 Class III corresponds to the much less common situation in which two distinct wave systems have overlapping frequency spectra and opposite directions. One of these systems is usually the wind-sea, forced by the local wind, and the other is a swell, usually propagating from a different generation area. In this case the values of I can be large, possibly more than 0.1. In the case of turning winds, there is a fuzzy boundary between class I (for rather slowly turning winds) and class III (for more rapidly turning winds) as the wind-sea in the old wind direction turns into swell while a new wind-sea grows in the new wind direction [e.g., van Vledder and Holthuijsen, 1993].
3.2. Wave Model Accuracy for Class I Situations
 For a single wind-sea system, Donelan et al.  proposed that M(f, θ) may be distributed like 1/cosh2 [b (θ − θ0)], but they acknowledged that there was virtually no information for ∣θ − θ0∣ > 90° and this was further restricted to frequencies less than 1.6 times the peak frequency fp.
Banner  used spatial data only, with a 180° ambiguity on the wave propagation direction, which gives no information at all on the relative values of M in opposing direction. Example of these questionable directional distributions, proposed by Donelan et al.  and Banner , are shown in Figure 4a.
 The only wave measurements of good enough quality to provide a reliable estimate of I may be those acquired by Long and Resio . Their spectra are derived from an array of wave gauges mounted on a surface-piercing support, using the iterative maximum likelihood (IML) method of Pawka , and were averaged for winds varying in a 20° sector. Some examples of average directional distributions are given in Figure 4b. The values of I indicated in Figure 4b were obtained after removing energy in a 40° angular sector opposite to the dominant wave direction, in order to remove a potential artifact of the IML estimator, which tends to fill in directional sectors with very little energy, and removes also possible reflections off the instrumentation (C. Long, personal communication, 2010). If the full directional distribution were kept, then the values of I would typically be ten times larger.
 The wave model used here gives values of I that, for frequencies in the range 0.1 to 0.3 Hz, i.e., typically 0.5 < f/fp < 4, are of the order of 0.003 for a constant wind speed, where fp is the frequency of the ocean wave peak. This value of I is in fair agreement with the directional spectra recorded by Long and Resio  with the exception of the highest frequencies, for which our model is thus likely to underestimate the seismic noise. This is illustrated in Figures 4c and 4d with simple academic calculations with a uniform ocean and a constant 10 m/s wind. We find that the model results vary with the wave age, which is the ratio Cp/U10 of the phase speed at the peak frequency and wind speed U10. After 6 h of simulation Cp/U10 = 0.6, and grows to 1.0 after 24 h. The modeled directional distribution is broader for older seas, resulting in larger I values. Other wave model parameterizations typically give smaller values of I by at least 1 order of magnitude (see Appendix C). In real ocean conditions the nonhomogeneity in the wind will naturally enhance I compared to these academic tests, which probably reduces the differences between models. However, all wave models are most questionable for class I conditions.
3.3. Wave Model Accuracy for Class II Situations
 For class II, the model accuracy clearly depends on the parameterization of wave reflection. Besides some general knowledge on the decrease of R2 with increasing wave heights and mean wave frequencies [Elgar et al., 1994; O'Reilly et al., 1999], there is no published account of how reflection varies with frequency for a given sea state. There is also little knowledge on R2 variations with the depth profile of the shore face except for an increase of R2 over steeper topographies. There are accurate expressions for the reflections over a random bottom profile for intermediate depths [Ardhuin and Magne, 2007], but these are generally much weaker than the main reflection right at the shoreline. The directional distribution of the reflected wavefield is thus unknown. We have tested reflected distributions of the form cosp (θ − θr) where θr is the specular reflection direction and p is a power that we varied from 1 to 4, and these only changed the magnitude of seismic sources caused by very narrow incident spectrum (typical of swells), by about a factor of two (not shown). It is likely that the model error is mostly defined by the error on the magnitude of R2. This magnitude can be validated using directional spreads from buoy data.
 For statistical analysis, we use more robust frequency-integrated moments and spreadings:
with fmin = 0.04 and fmax = 0.16 Hz, consistent with our analysis of seismic noise data. At buoy 46214, Σ1 is generally well estimated with R2 = 0.1, together with the other wave parameters. We also verify that the model result with a resolution of 1/2° are very similar to the result with a resolution of 1/6° (Figure 5), even though the buoy is only 1/2° away from land.
 The wave model bias for Σ1 has a strong seasonal cycle, with no bias in summer (−10% bias without reflection). In winter the bias without reflection reaches −30%, and is reduced to −6% with a constant R2 = 0.1 applied at all frequencies. This seasonal cycle is correlated to a change in mean wave direction. In summer, dominant waves off this part of the coast are driven by alongshore winds, with a typical direction from 320° (Figure 6). This alongshore wind-sea is hardly reflected by the shoreline, whereas in winter the dominant wave directions ranges from 280 to 300.
 Still, the model with R2 = 0.1 occasionally overestimates the directional spread in winter. This is the case on 8 January, when 9.6 m high waves were recorded at buoy 46013 (buoy 46214 did not record this storm due to a power outage) and 24 February with 8 m waves recorded at 46214 (red star in Figure 6). This is consistent with the observed decrease of R2 with wave height [e.g., O'Reilly et al., 1999]. R2 = 0.1 is thus an overestimation for the highest sea states and that part of the coast.
 When looking at other locations along the California coastline, R2 = 0.1 typically leads to overestimations of the directional spread (once the buoy bias is corrected for). This is the case, for example, for the 3 m discus buoy 46042, offshore of Monterey Bay (see Table 1), where most of the shoreline consists of gently sloping beaches.
Table 1. Statistics of Model-Buoy Differences for Various Spectral and Directional Parameters at Buoys 51001 (1798 3 h Records in 2007), Northwest of Hawaii, and 46013 and 46214, on the Central California Shelf, Located 20 and 40 km From the Coast, Respectivelya
Values are given for the following statistical parameters: bias normalized by the root mean square observed value (NB), scatter index (SI), and Pearson's linear correlation (r). The units for the root mean square error (RMSE) are meters for Hs, hertz for frequencies, and degrees for Σ1, and we also give the value of the RMSE normalized by the root mean square observed value (NRMSE). Hs6 is the significant wave height defined from the wave spectrum but integrated only up to f = 0.15 Hz, corresponding to the frequency band used here for seismic noise analysis. fm0, −1 is the mean frequency obtained from the spectrum weighted by f−1.
With constant reflection, R2 = 0.1
With constant reflection, R2 = 0.1
With constant reflection, R2 = 0.1
With constant reflection, R2 = 0.1
3.4. Wave Model Accuracy for Class III Situations
 For class III, the value of I depends on the overlap Δθ of the directional sectors of opposing directions which is determined by the mean direction and directional spreading of the interacting wind-sea and swell systems. These are parameters that can be measured and for which the wave model can be validated. Mean directions exhibit random errors without bias, and these errors will only produce random errors in the noise.
 For opposed mean directions θ1 and θ2, positive biases on the directional spreading will generally reduce the noise, because then M(f, θ) M(f, θ + π) Δθ has lower values since M(f, θ) Δθ is conserved and M(f, θ + π) is diluted over a wider angular sector. Conversely, as θ2 moves away from θ1 + π, the noise is zero in the limit of very narrow spectra, and an increase in the spreading can augment considerably the noise level. The directional spreading is thus a key parameter for an accurate noise estimation.
 All buoy data were averaged over 3 h before estimating these parameters. We first consider the 3 m discus-shaped buoy 51001, offshore of Hawaii, representative of open ocean conditions, and a different kind of buoy, a Datawell Waverider, number 46214, located 40 km from the central California coastline, in 550 m depth. Basic validation statistics at the two buoys are listed in Table 1, using data for 2008 at 46214 and 2007 at 51001, because directional data is not available for 2008 at that buoy.
 Of particular interest are the variations of Σ1 with and without reflection at the coast. The fact that red and blue symbols in Figure 5a are almost superimposed clearly shows that reflection has little effect on waves recorded at the Hawaii buoy, because that buoy is located 300 km from small islands.
 For 3 m discus buoy such as 46013, 46042 and 51001, the estimation of σ1 is generally noisy and biased high by at least 5° for which is typically 14% of the RMS measured value, the same should apply to Σ1. No bias was found for Datawell waverider buoys such as 46214 [O'Reilly et al., 1996]. Hence our 10 to 15% bias relative to the 51001 buoy data probably corresponds to a near-zero bias compared to the real waves.
4. Spatial Distributions of Seismic Noise Sources
 The above analysis of the wave model performance gives us confidence in the order of magnitude of the estimation of the seismic noise generation given by equation (2). A quantitative validation will be performed below using observed seismic noise. Figure 7 shows the average spatial distribution of modeled seismic sources, in the seismic frequency band 0.08–0.32 Hz. In this average, wave-induced pressure fluctuations are largest in the midlatitudes from 40° to 60°.
 More powerful and less frequent storms are needed to make longer-period waves, hence the encounter of two wave trains of frequency 0.05 Hz from two distinct storms (class III) is a rare source of seismic noise at fs = 0.1 Hz, even though low-frequency swells can propagate to longer distances with minimal energy loss [Ardhuin et al., 2009]. Noise at the lowest frequencies is thus mostly associated with reflections (class II).
 As previously noted by Kedar et al. , the water depths on the North Atlantic mid-oceanic ridge correspond to the sharp maximum of in equation (5) for average wave periods. Combined with high waves from extratropical cyclones, this feature makes the North Atlantic ridge the most powerful source of seismic noise (Figure 7b). Because the relative importance of coastal reflection is debated, the contribution of reflections, probably overestimated given our choice of reflection coefficients, is isolated in Figure 7c. Reflections significantly increase seismic sources along western coasts, within 1000 km from shore, in particular in Europe, from northern California to Alaska, and off the south Chile and south Australia coasts.
 In the Southern Ocean, reflections off the Antarctic peninsula, islands, and icebergs are predicted to generate large sources (Figure 7a), in a background of already high seismic generation due to highly variable winds. Although icebergs are particularly numerous in the South Pacific in 2008, compared to other years [Ardhuin et al., 2011], we have not yet been able to isolate their contribution in seismic records due to the large distance between Global Seismic Network stations and the main iceberg concentrations. Iceberg-related sources may be more easily seen in body waves for which the attenuation is weaker. We now verify and discuss the other model predictions.
 In the following, we chose a constant but realistic U = 1.8 km s−1. For Q = 605 (Q = 151), seismic energy at fs = 0.15 Hz is halved during propagation every 800 km (200 km).
 We adjust Q separately for each seismic station to set the value of UQ. This approach allows us to account for some of the spatial variability of Q, but it should only make sense for stations to which noise has traveled across relatively uniform crust. Faced with the limited knowledge of UQ in the main seismic noise frequency band, values of Q for the model were selected with the objective to reproduce the temporal variability of the seismic noise, maximizing the linear correlation r (Figure 8), while keeping realistic noise levels (Figure 9). Here r is Pearson's linear correlation coefficient defined as
where the overbar denotes the arithmetic average, i.e., = (Xi)/N, and X and Y are series of the two quantities considered. Here X and Y are the observed and modeled spectral densities at a given frequency.
 For high Qs, seismic sources from a vast area of the ocean contribute to the seismic record, while a low Q means that sources are concentrated in the vicinity of the seismic station. Since dominant ocean waves generally have a strong variability at scales larger than 300 km [e.g., Tournadre, 1993] we expect a similar variability in noise sources given by the model. When sources are widely distributed over distances from a seismic station, there should only be a limited range of Q values that will put the proper weight on nearby and remote stations and yield realistic noise time series.
 After calibrating the wave model, Q factors, and frequency dependence of R2 for the year 2007, we illustrate the model validation with data from the year 2008, with four seismic stations: KIP (Hawaii), BORG (Iceland), BKS (central California) and SSB (France). This choice was motivated by the desire to span a wide range of seismic noise climates, from noisy island stations (KIP and BORG) to calmer mainland stations further (SSB) or closer (BKS) to shore. We have also tested the model at many other locations with a similar degree of performance except for Dumont d'Urville (DRV), where the wave model performance near the sea ice edge may be an issue, and Mexico City (UNM) where high-frequency noise (fs > 0.2 Hz) is not well reproduced, possibly due to seismic amplification associated with the local sedimentary basin. (These locations include the seismic stations CMLA (Azores), ESK (Scotland), EFI (East Falkland Islands), SCZ (Salinas, California), JTS (Costa Rica), RPN (Easter Island), NNA (Nana, Peru), COR (Corvallis, Oregon), RER (Reunion Island), PPT (Papeete, French Polynesia), INU (Inuye, Japan), FDF (Fort de France, Martinique), KDAK (Kodiak Island, Alaska).)
 Seismic spectra are obtained from the LH channel (1 sample per second) of broadband STS1 or STS2 seismometers. In the frequency of interest there are negligible differences between the two instruments. Seismic displacement spectra were computed with overlapping windows, instrumental responses were removed, and spectra were averaged over 4 h segments [e.g., Stutzmann et al., 2000], giving 28 degrees of freedom for each spectral estimate at a spectral resolution of 0.001 Hz.
 Because ocean wave properties change only on a time scale of a few hours, segments with earthquakes or instrument glitches were filtered out as follows. Using the low-frequency ground displacement variance Elf (in the band 0.05–0.1 Hz where wave effects should be minimal), records were discarded when Elf jumped by a factor 8 over 4 h, provided that Elf was larger than a station-dependent threshold. This threshold was set to 3 × 10−12 m2 for BORG, 1 × 10−13 m2 for KIP, 1 × 10−15 m2 for SSB, and 1 × 10−14 m2 for BKS. After this step, seismic spectra were interpolated in time on the 3 h time step of the wave model output.
 Spectral levels from all four stations (Figure 9) suggest that Q should decrease from about 200–1000 at fs = 0.12 Hz, to about 50–300 at fs = 0.3 Hz, consistent with previous estimations [Mitchell, 1995]. A reasonable correlation and spectral level is obtained at seismic stations KIP, BORG, BKS, and SSB using 580, 180, 260, and 88 for Q, respectively. Since we have neglected the three-dimensional seismic propagation effects that can reduce appreciably the energy level at the seismic station, these effects are partially compensated by artificially lower values for Q. Any bias in the numerical wave model may similarly be compensated. Still this variation of Q is generally consistent with results at longer periods by Mitchell , who showed that Q is generally higher for older crust, as found around Hawaii, and lower for younger crust, as found offshore of California. A systematic investigation of the variability of Q with frequency and location will be performed in another paper.
 Because Q varies spatially, sources contribute differently to different seismic stations. Figure 10 shows the average expected source distributions for the four chosen stations. We shall now discuss each station specifically.
5.1. Where Reflection Is Usually Negligible
 Our tour begins with BORG and KIP, where shoreline reflection makes little contribution, as shown by the small difference between the solid red and dashed black curves in Figure 9. Increasing further the shoreline reflection coefficient to unrealistically high values will not change this result. In the case of KIP, reflection is negligible because the Hawaiian islands are very small compared to the area of the noise sources. On some occasions the effect of reflections can be seen, in particular for the lowest frequencies, but the resulting noise is often too weak to dominate other sources. Figure 11 shows one example of observed and modeled wave spectra at the wave buoy number 51001 which is the closest to the KIP station, and the recorded and modeled seismic noise at KIP. Contrary to coastal data from central California coast [e.g., Bromirski et al., 1999], there is little resemblance between the wave and seismic spectra which are both fairly well represented by the model. Although they are blurred in the seismic record, one can still track the same swell events in both seismic and wave records.
 Focusing on 13 March 2008 at 0300 UTC, the modeled noise map (Figure 12a) can be explained by a careful examination of the directional wave spectra around Hawaii (Figures 12b and 12c). The broad region of relatively strong seismic sources to the northeast of Hawaii has a seismic peak period of 7 s, corresponding to 14 s waves. The wind-sea at that time is unable to contribute to these frequencies but, in the modeled spectrum at the location of the southeast buoy 51004, there are four swell systems that could contribute to this noise. Only swells number 3 and 4 have opposite directions. At the location of buoy 51001 there is one more swell system, number 7, generated by the reflection of swell 1 off the islands (it is absent in the model run without reflection).
 Swells 1 and 7 do interact to produce some noise but this source has a very limited spatial extension, about 300 by 800 km in the vicinity of the westernmost island. Swell 1 was generated by a northwest Pacific storm on 9 March 3500 km away from Hawaii, consistent with the slope of the long-dashed lines in Figure 11. This same storm was not a very strong source of noise by itself, with modeled values of Fp2D (ks ≃ 0) under 200 hPa2 m2.
 According to the model, the strongest source of noise on 9 and 10 March was a Gulf of Alaska storm which peaked at 10000 hPa2 m2 on 9 March, and sent waves to the northeast, crashing on the Alaskan shoreline, with very little swell radiated in the Pacific. This other storm was joined by a deep extratropical depression that moved very fast to the east, from 40 N 150 W on 9 March at 1200 UTC to 51 N 135 W by 11 March at 0000 UTC. This storm is the source of swell number 4. As for swell number 3, it was generated on 3 March in a Southern Ocean storm located near 59 S 158 W, 10000 km away from Hawaii, which is consistent with the dotted line in Figure 11. This swell 3 is much weakened when arriving around Hawaii, yet it meets swell 4 in the exact opposite direction. These two swells give the largest noise source recorded at KIP station in Hawaii on 16–20 March, although they only give a very small contribution to the wave energy recorded at the 51001 wave buoy.
 With R2 irrelevant for KIP, Q remains the only tunable parameter in the model, and model-data correlations are very sensitive to it (Figure 9b). This sensitivity is consistent with the fact that modeled sources span a wide range of distances from KIP (Figure 10). The high values of Q needed to fit the observed noise reveals that KIP effectively integrates sources from a very large area in the North Pacific, which explains the smoother appearance of the KIP time series (Figure 13b) compared to other stations. We finally note that the highest correlations are given by Q ≃ 1000, which would produce too high spectral levels for fs > 0.15 Hz. Adding a frequency-dependent refraction term N in equation (6) decreasing with N ≃ 1 at fs = 0.15 Hz and N < 0.4 at fs = 0.15 Hz, would produce a more accurate empirical model.
 At BORG in Iceland, a lower Q, compared to KIP, with stronger North Atlantic sources, compared to the North Pacific, yields the observed similar noise levels (Figures 9a and 9b). Varying Q at BORG makes little change to model-data correlations for each frequency (Figure 8) and globally for δRMS (0.87 < r < 0.90), but it dramatically impacts the noise level (Figure 9a). This insensitivity of r in the model results from the fact that contributing sources are at the same distance from BORG, in a small section of the mid-Atlantic ridge (Figure 10a), to the southwest of Iceland. The model appears to reproduce as well the strong noise events, with δRMS up to 14 μm for durations less than a day, and the low noise background with δRMS under 2 μm. In this respect the present model appears more accurate than the previous noise model by Kedar et al. [2008, Figure 6].
5.2. Where Reflection Is Important or Dominant
 The French SSB station is representative of western Europe, with an intermediate noise climate. There, model results are highly sensitive to both Q and R. A uniform Q = 260 gives acceptable correlations (Figure 13c), with powerful mid-Atlantic sources and closer coastal sources contributing equally. As in Hawaii, the correlation can be improved by increasing Q, here up to 320, but this produces noise levels that are a factor of two too high for fs > 0.15 Hz. Here again, seismic wave refraction is a likely omitted factor. The model adjustment is also complicated by the important role of coastal reflection, which is only poorly approximated by a spatially uniform reflection coefficient R2.
 Our journey ends in central California where, like at other stations [e.g., Stutzmann et al., 2000, 2009], lower frequencies dominate in winter, with here a peak at fs = 0.13 Hz, while summer energetic signals are above fs = 0.16 Hz. After comparing the synthetic noise spectra to seismic data from BKS, and based on directional measurements at nearby buoys (section 3), we increased R2 by 50%, relative to all other model calculations. Values of the seismic ground displacement δRMS estimated for BKS are as high as 4.2 and 5.4 μm in January and February 2008, compared to maximum recorded values of 2.2 and 1.6 μm only (red stars in Figure 13d). These overestimations occur for the two strongest storms of 2008, with ocean wave heights of 9.5 and 7 m measured at the 46013 buoy, while wave heights are less that 6 m for the rest of the year. Clearly our shoreline reflection coefficient R2 (f) is overestimated for these two events. The buoy data (Figure 14a) and the seismic noise model (Figure 13d) suggests that R2 should decrease with wave height, as expected [Elgar et al., 1994; O'Reilly et al., 1999].
 We also note that any constant Q leads to a poor shape of the synthetic noise spectrum (Figure 9d). In fact, for fs < 0.15 Hz (dominant in winter) the time series of synthetic noise levels are well correlated with the BKS records for a very wide range of Qs, from 70 to 520, whereas the higher frequencies require a lower Q, in the range 60–160, to produce any decent correlation (Figure 8d). These two frequency domains may correspond to different seismic modes [Haubrich and McCamy, 1969], but they also correspond to different classes of seismic generation events. Indeed, according to the model, reflection only plays a minor role in summer (the dashed and red lines are almost identical for June to September in Figure 13d), and class III is relatively rare, thus the summer noise is mostly produced by class I events with fs > 0.15 Hz. This is easily understood from the wind directions which blows alongshore in summer and thus generates waves that mostly propagate alongshore (Figure 6a) with very little coastal reflection. Swells from the southwest, that are also present in summer, do not apparently contribute much to the noise recorded at BKS, contrary to southern California stations such as Pasadena (not shown).
 In winter the wind and wave directions have typical directions that differ from summer wave directions by 20°–30° (Figure 6b), hence resulting in stronger reflected wave components. Because the modeled noise without reflection is often near zero, we can conclude that class II dominates in winter.
 For the lower frequencies, the large winter storms hit the U.S. west coast rather uniformly, creating strong noise sources offshore of northern California and Oregon. It is possible that seismic attenuation is weak, with Q ≃ 500, and that the seismic energy that reaches BKS is strongly reduced by refraction (by about a factor 10). In this case, BKS records could be influenced by sources as far as the Oregon shelf edge. It is also possible that the noise is mostly coming from central California waters, i.e., Q ≃ 100 and that refraction does not play an important role. For the higher frequencies (fs > 0.15 Hz) the sources must be fairly local, within 300 km, because increasing Q to 320 already degrades the correlations between modeled and recorded noise level time series.
 Even in winter, some strong noise events are unrelated to reflections and correspond to class III events (blue stars in Figure 13d). For such an event, on 26 January 2008, δRMS reached 1.8 μm (Figure 13d), the second largest recorded value for 2008. The modeled wave-induced pressure for that event is shown in Figure 15.
 A low pressure offshore of California produced winds over 25 m/s which, to the west, amplified swells from the northwest (generated by another storm off Washington State) into high seas (Hs up to 8 m at point C in Figure 15). To the east of the low pressure, rough seas are generated (Hs ≃ 4 m), that propagate as swell to the north of the cold atmospheric front (line with triangles). This marine weather situation yields two maxima in the wave-induced pressure. At point A (Figure 15), Fp2D reaches 2600 hPa2 m2 as the wind-sea interacts with swell from the northwest. The corresponding directional wave spectrum is shown in Figure 1. The weaker maximum to the north at B, is due to the interaction of two swells. We note that the highest waves at point C yield weak pressure fluctuations (200 hPa2 m2) because there is no opposing swell to give a strong contribution to I(f) (equation (3)). If I(f) at C had been the same as at A, the wave-induced pressure at C would have been 200 times larger. Minor sources due to shoreline reflections are also predicted, mostly along the Oregon coast. The model was verified at the buoy 46013 (green star) the data of which was processed into directional spectra using the Maximum Entropy Method [Lygre and Krogstad, 1986]. The wave-induced pressure estimated from the buoy data is 110 hPa2 m2, and the local model estimation is 160 hPa2 m2. For 26 January, the estimation of wave heights from the seismic records, using empirical relationships based on average reflection-dominated conditions [Bromirski et al., 1999] yields Hs values in the range 8 to 11 m for this event, depending on the method details, whereas observations did not exceed 5.5 m. This kind of overestimation was already noted by Zopf et al. . This particular seismic record is thus a poor proxy for the coastal ocean wave climate, unless outside information is added. Noise models, such as described here, can help decipher the complex ocean wave patterns from seismic records.
6. Summary and Perspectives
 Building upon recent improvements in numerical wave modeling and adding a parameterization for wave reflections at shorelines both critical for a proper restitution of the directional wave properties, we have presented the first numerical model of seismic noise that is able to reproduce all three classes of sea states that lead to significant seismic noise levels. These are the generation by a single wind-sea (class I), the generation due to reflections (class II) and the generation due to two opposing wave systems, a wind-sea opposing a swell or two opposing swells (class III) than can be generated by the same moving storm (class IIIa) or by two well separated storms (class IIIb). The only similar previous model, by Kedar et al. , did not include coastal reflections and thus could not reproduce some of the recorded noise events. Although the details of the modeled directional wave spectra cannot be validated to the required level of accuracy for class I generation, and in spite of a simplified seismic propagation model, the consistency of modeled noise with observations demonstrate that our model captures the dominant features of seismic noise generation.
 Namely, the strongest noise sources occur in the middle of ocean basins, where crossing wavefields are more common, and where deep enough waters can amplify the wave forcing to a detectable level. This result is consistent with the previous North Atlantic analysis by Kedar et al. . We further quantified secondary sources, located at the edge of the continental shelf, that are often associated with shoreline reflections, and dominate the records at some seismic stations close to the coast. This is only a summary of a very dynamic and variable field of noise sources which changes on the time scale of a few hours. The high correlation between measured and modeled time series of noise levels demonstrates that the location of sources are well captured by the model. This statement is essentially supported by class III events which are the most intense and require specific oceanic conditions fulfilled only in time-dependent and limited areas.
 More questions arise on the magnitude of these sources. The model clearly requires a seismic wave damping that, when represented by a quality factor Q, varies by at least a factor 2 between oceanic regions around Hawaii and coastal regions off California, with intermediate values in the North Atlantic and southwest Pacific (based on Azores and Easter Island data, not shown). This spatial variation of Q for seismic periods in the range 3 to 10 s is comparable with estimates from earthquake analysis for periods longer than 30 s [e.g., Mitchell, 1995]. In general Q increases with the age of the crust. Our model also supports a general decrease of Q by a factor 2 to 4 from low (0.15) to high (0.3 Hz) frequencies.
 The shoreline reflections used in the wave model are only expected to provide a correct order of magnitude that will be improved by allowing the reflection to vary spatially and as a function of wave amplitude. Yet the most important limitation of the present model is probably the use of a horizontally uniform seismic model, which means that seismic wave focusing or defocussing and scattering due to crustal heterogeneities are not represented. In reality, these effects likely reduce the noise level at stations in Hawaii and California, explaining why the highest model-observation correlations are found for values of Q that tend to overestimate the observed noise level. Further progress will therefore require the use of more realistic seismic propagation models to quantify this effect. When this is done, the seismic noise might be used to estimate wave model errors on the energy level and directional wave distribution. Figures 5 and 14 already illustrate how the reflection coefficient can be adjusted from buoy data, which was validated here with seismic noise data. The noise data could be used already to perform this adjustment where directional buoy data is not available.
 The classification of the noise-generating sea states proposed here should provide a useful guide for a detailed calibration of direct seismic noise models, because the wave model performs differently for different classes. We have also found that this classification helps in the retrieval of wave parameters from seismic noise because the empirical relationships between ocean waves and seismic noise differ for different classes. This will be the topic of another paper.
 With our notations and σ = ω/2, equation (2.15) of Hasselmann  gives the spectral density of the equivalent wave-induced pressure at the surface:
This is also exactly equation (1) in the work of Webb  in the limit of deep water waves. This can be rewritten in terms of the usual spectral density F(f, θ) = fζ (σ, θ)/(2π), provided by numerical wave models. Also, the directional integral from 0 to 2π is equal to twice the integral between 0 and π, giving
Because it is more common to work with spectral densities in frequency, we define f2 = ω/(2π), and Fp3D (K ≃ 0, f2) = 2π Fp (K ≃ 0, ω), and obtain our equation (2).
 If there was any doubt on the accuracy or interpretation of equation (2.15) of Hasselmann , it can be lifted by recovering the result of Longuet-Higgins  for monochromatic waves. This requires to specify the seismic wave number K in order to properly treat the singularity at K = 0 in this case. One thus needs to step back to equation (2.13) in the work of Hasselmann . We assume from the start that the wave numbers of the two interacting wave trains are nearly opposite k ≃ −k′ to simplify the integrand, we neglect difference interactions (middle line of his equation (2.13)), and correcting for a few typographic errors, we get
where δ is Dirac's function, and σ and σ′ are the radian frequencies of the two interacting wave trains.
 Now taking a monochromatic wavefield with variance at two opposite wave numbers k0 and −k0,
we can remove the integral over kx and ky in (19), because the integrand is now only nonzero for k = k0 and k = −k0, where Fζ (k) gives the surface elevation variance E1 and E2, respectively. Since we are only treating deep-water waves, σ2 becomes σ02 = g∣k0∣ = gk0, and we get
We now do the same for k′x and k′y:
Because there are no seismic waves with both K = ±2k0 and ω = 2σ0, the last two lines are irrelevant, and we get
 The pressure variance is now given by integrating over the spectral components of the pressure, which naturally removes the Dirac functions, and gk0 can be replaced by σ02 for deep water waves,
We can now identify this expression with the variance from Longuet-Higgins's second-order pressure (his equation (31)):
where E1 = a12/2 and E2 = a22/2 are the surface elevation variances of the two interacting wave trains (expressed in m2 when using S.I. units). This variance of the second-order pressure was verified in the laboratory by Cooper and Longuet-Higgins .
Appendix B:: Parameterization of Reflections in the Wave Model
 The numerical implementation of reflections was straightforward. In the case of icebergs and subgrid islands, the reflected energy is redistributed evenly in all directions within 90° of the direction opposite to the incoming waves. This corresponds to a Lambertian surface approximation, which is used for wave scattering from rough surfaces. For resolved lands, a mean direction perpendicular to shore θn was defined from the land or sea status of the 8 grid points surrounding the local point (Figure B1).
 For each model grid point adjacent to land, the analysis of the land-sea geometry gives one value of θn among 16 possible directions. Together with any incoming wave direction θi this defines a specular reflection direction θr = 2θn − θi + π. For each spectral component of direction θi going toward the coast (i.e., such that cos(θi − θn) > 0), the total reflection is R2 times the incoming energy. This reflected energy R2E(f) M(f, θi) is redistributed over directions around the specular reflection direction θr, with a broad distribution taken proportional to cosn (θ − θr), where the power n is a function of the local shoreline geometry.
 For this purpose we distinguish three different shoreline geometries relative to the local point as illustrated by Figure B1: we set n = 2 for a straight coast (three connected land points among the neighbors), n = 1 for a mild corner (two land points among the neighbors), and n = 0 at a sharp corner (only one land point, among the 4 closest neighbors) which corresponds to the same treatment done for subgrid islands and icebergs. Changing these values of n in the range 0 to 2 has little effects on our results. A more rigorous treatment should use the distribution of the shoreline orientation at the scale of the ocean wavelength, namely of the order of 100 m.
Appendix C:: Sensitivity of Noise Results on Wave Model Parameterization
 The previous work by Kedar et al.  used the same noise generation theory but a different wave model, with wave generation and dissipation parameterizations given by Tolman and Chalikov . Because so little is known about the directional distribution of wave energy at large oblique angles relative to the mean wave direction, it is important to discuss the impact of different possible choices for wave model parameterizations and numerical schemes. We have thus tested the parameterizations in use today in the two largest operational centers, namely the European Center for Medium range Weather Forecasting, based in Reading, UK [Bidlot et al., 2005], and the U.S. Marine Meteorology Branch (MMAB) of the National Center for Environmental Prediction (NOAA/NCEP) based in Camp Springs, MD [Tolman and Chalikov, 1996]. Results for these two parameterizations are shown in Figure C1 for the same simple academic case of a uniform wind and ocean.
 The striking result is that the values of I obtained with these two parameterizations are at least 1 order of magnitude less than with the present model. In realistic cases the variability of the wind field may compensate for the otherwise narrow spectrum, but we generally expect other parameterizations to give less realistic results for class I generation. This may be the reason why the dynamic range of the seismic noise predicted by Kedar et al.  is much larger than observed, as they probably underestimate the background noise level due to class I generation (see their Figure 6).
 The reason for this order of magnitude difference between various parameterizations is that they all use an isotropic dissipation term, except for Ardhuin et al.  who introduced a breaking wave dissipation time scale that varies with the wave direction, on the grounds that waves in the dominant directions are steeper and should break more frequently [Filipot et al., 2010]. This feature was introduced in order to be able to reproduce directional spreadings observed during the SHOWEX experiment [Ardhuin et al., 2007]. For realistic conditions, other model differences play an important role, such as the large bias for swell energies when using the parameterizations by Bidlot et al. .
 Preliminary comparisons of modeled noise using the wave model parameterizations by Tolman and Chalikov  or Bidlot et al.  suggest that the average noise level may differ by a factor of two compared to our reference simulation, but the correlation with the observed noise time series remain high. There is thus a need for a thorough investigation of these differences, but this goes beyond the scope of the present paper.
 We thank the operators of GEOSCOPE, Berkeley Digital Seismic Network, and IRIS/IDA for providing excellent broadband seismic data. Data were taken from both GEOSCOPE and IRIS data centers. Wind and wave data were provided by ECMWF, and wave buoy data were taken from the U.S. National Data Buoy Center and the Coastal Data Information Program, while experimental directional spectra were kindly provided by C. Long and D. Resio. A. Balanche performed the spectral analysis of all the seismic stations not shown here but used in our analysis (KDAK, COR, FDF, INU, CMLA, ESK, etc.). F.A. is funded by ERC grant 240009 “IOWAGA” and by the U.S. National Ocean Partnership Program under grant N00014-10-1-0383. M.S. acknowledges the Consolider-Ingeno 2010 (CSD2006-00041, Topo-Iberia). The advice of two anonymous reviewers helped clarify the manuscript.