In recent years the study of the ambient seismic noise band from 0.05 to 2 Hz has attracted sig- nificant interest. Ambient noise provides a continuous record which may be used in both seis- mic source and Earth structure investigations (Cessaro, 1994; Friedrich et al., 1998; Shapiro and Campillo, 2004; Gerstoft et al., 2006; Chevrot et al., 2007; Bensen et al., 2008; Gerstoft et al.,
2008; Harmon et al., 2010; Landès et al., 2010; Köhler et al., 2011; Tkalˇci´c et al., 2012). In ad- dition, ambient noise generated by ocean storms can be used to infer their location and possible climate-related trends in storm severity (Bromirski et al., 1999; Essen et al., 1999; Grevemeyer et al., 2000; Bromirski and Duennebier, 2002; Essen, 2003; Stutzmann et al., 2009; Aster et al., 2010; Hillers et al., 2012).
The ambient seismic vibrations in the given frequency band are also known as microseisms. Their generation mechanism was first proposed to be linked with onshore swell (Wiechert, 1904). We now divide microseisms into two groups: primary and secondary. Primary microseisms have an amplitude peak at around 0.07 Hz and are caused by the interaction between ocean waves and coastal geometry (Haubrich et al., 1963). Secondary microseisms have an amplitude peak at around 0.14 Hz and are generated by the interaction of two opposing wave trains with approxi- mately the same absolute value of the wave vectork(Longuet-Higgins, 1950). The microseism spectrum mainly consists of fundamental mode Rayleigh waves below a frequency of around 0.14 Hz, while body and surface waves are observed for higher frequencies (Lacoss et al., 1969).
Seismic array analysis has been driven throughout its history by both earthquake research and nuclear explosion detection (Schweitzer et al., 2002; Rost and Thomas, 2009). The lack of an impulsive onset for microseisms limits the capability of networks of single element stations in the determination of source backazimuth and slowness, hence, array analysis promises to contribute greatly to the study of the microseisms as it enables source backazimuth and slowness to be es- timated. Initial studies located microseism sources via triangulation between several arrays, each having moderate spatial resolution (Cessaro, 1994; Friedrich et al., 1998; Essen, 2003). More recent studies have been able to improve the source localisation (Schulte-Pelkum et al., 2004; Chevrot et al., 2007; Koper et al., 2009; Behr et al., 2013) and show that secondary microseisms generated near coastal locations remain stable over longer periods. The stationarity of the signal was exploited to perform cross correlations between interarray stations in order to generate sur- face wave travel times and extract structural information at the array location (Gerstoft et al., 2006; Bensen et al., 2008; Li et al., 2010; Köhler et al., 2011; Young et al., 2011). Studies of body waves generated by deep ocean storms confirm the location of wave generation to be near observed active storms (Gerstoft et al., 2008; Kedar et al., 2008; Landès et al., 2010). A numerical perspective on their generation is given by Ardhuin et al. (2011).
Methods used to estimate the slowness and backazimuth of wave sources arriving at seismic arrays are reviewed by Rost and Thomas (2002). A popular method for the analysis of seismic ambient noise is the modified frequency-wavenumber algorithm proposed by Capon (1969) which has high resolution capability but is limited to the estimation of narrowband signals. Kværna and Doornbos (1986) proposed that the slowness spectra be averaged over a specific frequency range. It was further shown by Kværna and Ringdal (1986) that this broadband extension improves the stability of the estimated spectrum. This approach is well suited for the study of ambient noise as a broadband analysis is desirable.
In underwater sound processing, great detail has been devoted to the study of broadband signals with matched-field processing (Bucker, 1976; Baggeroer et al., 1993), which is a general- ization of the conventional plane wave beamformer. The two main approaches for the extension of narrowband beamformers to broadband are the coherent and incoherent averaged signal methods.
While these extensions vary slightly between conventional beamforming, matched-field process- ing and subspace-based estimators, the central idea remains the same. The incoherently averaged signal (IAS) method in conventional beamforming (Kværna and Doornbos, 1986), matched-field processing (Baggeroer et al., 1993) and subspace-based estimators (Wax et al., 1984), averages over the narrowband direction of arrival spectra from each frequency bin to form a final result. For coherently averaged signal methods the spectrum is combined and focused on a single frequency prior to the slowness projection procedure. In the underwater sound literature, coherent averaging has been discussed in the context of matched-field processing (Westwood, 1992) and compared to the incoherent approach (Soares and Jesus, 2003). Coherent subspace-based methods (Wang and Kaveh, 1985; Chiou and Bolt, 1993) combine contributions from different frequencies. The application of standard narrowband techniques is then enabled by forming a modified cross-power spectral density matrix using a focusing matrix to project the narrowband estimates onto a single frequency (the focusing frequency). A disadvantage of this method emerges for multiple signals as an ideal focusing matrix does not always exist and approximations are needed. Other approaches for the estimation of broadband sources have been discussed by Krim and Viberg (1996).
The incoherent averaged signal method is well suited for the analysis of ambient noise as source locations are a function of frequency and a coherent averaging would be prone to smear- ing for closely spaced arrivals. In this work we re-examine the frequency averaged (approximate) approach for synthetic and observed ambient noise data and compare the performance to the in- coherently averaged signal or IAS method which we present in an implementation tailored for the analysis of ambient noise. The IAS method uses a summation of narrowband spectra, which pre- vents the introduction of errors into the slowness estimation. We also implement diagonal loading for the IAS Capon method as it was found to increase robustness for narrowband solutions (Capon, 1969; Featherstone et al., 1997) by loading the diagonal elements of a singular or near singular cross-power spectral density with a constant factor to reduce the bias on the direction of arrival estimation.
In Section 2, we outline a definition of the frequency-wavenumber spectrum. In Section 3 we show why frequency smoothing will result in a shifted slowness estimation in the power spectrum and outline the IAS method which results in a correct representation. Further, we briefly review the concept of diagonal loading which promises to be a very useful approach in the stabilisation of the IAS Capon method. Synthetic and real (observed) ambient seismic data examples are given in Section 4 followed by a discussion of considerations that must be made in implementing the IAS method.