1 Introduction
Shapiro et al. (2005) performed a cross-correlation analysis of long sequences of ambient seismic noise at around 0.1 Hz to obtain a group-velocity anomaly of Rayleigh waves due to the lateral heterogeneity of the crust in southern California. The authors inverted the measured anomalies to obtain a group-velocity map, employing a method that is now referred to as ‘ambient noise tomography’. The obtained group–velocity map at short periods (7.5–15 s) shows a striking correlation with the geologic structure.
Recently, phase velocity anomalies have also been measured using dense networks of seismic stations (Bensen et al., 2007). The anomalies are inverted to yield the three-dimensional S-wave velocity structure in the crust and in the uppermost mantle (Bensen et al., 2009; Nishida et al., 2008). The tomographic method was now been applied at scales ranging from local to global (Nishida et al., 2009).
The theoretical basis of cross-correlation analysis is the fact that a cross-correlation function (CCF) between a pair of stations provides the wave propagation between them (Snieder, 2004), as with the Green’s function. Assuming that a CCF has sensitivity along the ray path between a pair of stations (Lin et al., 2009), the measured phase or group velocity anomalies can be inverted to obtain maps of phase or group velocity. The ray approximation is justified by the high-frequency limit of the phase–velocity sensitivity kernel. The kernels for earthquake data have been evaluated by many researchers (Spetzler et al., 2002; Yoshizawa and Kennett, 2005; Zhou et al., 2004), but only one previous study has investigated ambient noise tomography (Tromp et al., 2010).
In the present study, a form of a two-dimensional (2-D) Born sensitivity kernel is obtained for a CCF, assuming the stochastic excitation of surface waves. For simplicity, potential representation is used for surface waves. The Born sensitivity kernel is then calculated in a spherically symmetric Earth model assuming a homogeneous source distribution. A simple expression of phase sensitivity kernels is derived from the Born sensitive kernel based on the Rytov approximation with the far-field approximation of a Green’s function and a CCF.
2 Theory of a synthetic cross spectrum of background surface waves between a pair of stations
For estimation of the sensitivity kernels, this section develops the theory of a synthetic CCF of background surface waves between a pair of stations.
It is assumed that a displacement field u can be represented by a fundamental Love wave and a fundamental Rayleigh wave, as follows:
(1) |
(2) |
(3) |
(4) |
The surface wave potentials satisfy the inhomogeneous spherical Helmholtz equation of a surface wave (Tromp and Dahlen, 1993), as follows:
(5) |
(6) |
(7) |
(8) |
A scalar Green’s function of Love and Rayleigh waves satisfies
(9) |
(10) |
The cross spectrum Φ of background surface waves between stations r1 and r2 can be given by
(11) |
The cross spectrum can be written as follows:
(12) |
(13) |
The excitation mechanism of ambient noise from 0.05 to 0.2 Hz, known as microseisms, is firmly established. Microseisms are identified at the primary and double frequencies: the primary microseisms at around 0.08 Hz have been ascribed to the direct loading of ocean swell onto a sloping beach (Haubrich et al., 1963). The typical frequency of secondary microseisms at around 0.15 Hz is approximately double the typical frequency of ocean swells, indicating the generation of the former via nonlinear wave–wave interactions among the latter (Longuet-Higgens, 1950). In both cases of the excitation mechanisms, the correlation length L can be characterized by the wavelength of ocean swell, on the order of 300 m, which is expected to be much shorter than the wavelength of seismic surface waves.
Supposing that the correlation length L(ω) is much shorter than the typical wavelength of background surface waves at ω, the cross spectrum Φαβ can be simplified as follows:
(14) |
(15) |
3 2-D Born sensitivity kernel for a CCF of background surface waves in the case of a heterogeneous source distribution
Employing a first-order Born approximation of a cross spectrum Φαβ (Eq. (14)), a 2-D Born sensitivity kernel is estimated for the cross spectrum, which is a representation of a CCF in the frequency domain.
The first-order perturbation of the cross spectrum δΦ can be written in terms of the perturbation of the Green’s function δG, as follows:
(16) |
(17) |
The above equation can then be simplified as follows:
(18) |
(19) |
This form of the above equation is similar to that of an adjoint kernel (e.g. Tarantola., 1984; Tanimoto, 1990; Tromp et al., 2010). For example, the first term of the kernel can be represented by convolution in the time domain between the time reversal of the CCF and the propagating Green’s function from to .
4 2-D Born sensitivity kernel in a spherically symmetric Earth model for a homogeneous source distribution
For simplicity, the focus is on 2-D Born sensitivity kernels in a spherically symmetric Earth model for a homogeneous source distribution. A scalar Green’s function in a homogeneous model can be simplified in the following form:
(20) |
In this section, it is assumed that homogeneous and isotropic sources excite background surface waves. This approximation enables us to simplify the α component of a cross spectrum in the following form:
(21) |
Figure 1 shows a typical example of the Born sensitivity kernel of a Rayleigh wave at 5.61 mHz with the source spectrum of an empirical model (Fukao et al., 2002). The sensitivity is concentrated within the first Fresnel zone. The figure also shows the side lobes of the kernel, which are suppressed when considering band-limited kernels (Yoshizawa and Kennett, 2005), as shown in the following section.
![](/geoscience/article/CRGEOS_2011__343_8-9_584_0/jpg/src/tex/figures/gr1.jpg)
Born sensitivity kernel of a Rayleigh wave at 5.61 mHz, with amplitude normalized by the cross spectrum Φαα(Θ12, ω). Each station is located on the equator. The longitude of station 1 is 40°; that of station 2 is 140°. The kernel was calculated using PREM (Dziewonski and Anderson, 1981). S1, S2, A1, and A2 indicate the locations of station 1, station 2, the antipode of station 1, and the antipode of station 2, respectively. This kernel is not singular, even near stations and near the antipodes of the stations.
Fig. 1. Noyau de sensibilité de Born d’une onde de Rayleigh à 5,61 mHz, d’ amplitude normalisée par le spectre croisé Φαα(Θ12, ω). Chaque station est localisée sur l’Equateur. La longitude de la station 1 est 40°; celle de la station 2, 140°. Le noyau a été calculé en utilisant PREM (Dziewonski and Anderson, 1981). S1, S2, A1 et A2 indiquent la localisation de la station 1, de la station 2, l’antipode de la station 1 et l’antipode de la station 2, respectivement. Le noyau n’est pas singulier, même près des stations et des antipodes des stations.
To obtain a more comprehensive form of the kernel, a far-field approximation of the Green’s function and the CCF is considered. A far-field representation of Green’s function is given as follows (Tromp and Dahlen, 1993):
(22) |
(23) |
(24) |
Similarly to the approximation of the Green’s function (Dahlen and Tromp, 1998, chapter 11.1), a far-field representation of the CCF is evaluated. Using the Poisson sum formula (Dahlen and Tromp, 1998, eq. 11.4, p. 408) to convert the summation over the angular degree l to an integral over the wavenumber k, the following representation is obtained:
(25) |
The above equation is transformed into a traveling representation of the CCF (Dahlen and Tromp, 1998, chapter 11.2), as follows:
(26) |
The analysis employs a relation of the transformation into a traveling wave representation (Dahlen and Tromp, 1998, appendix B.11), as follows:
(27) |
(28) |
Following (Dahlen and Tromp, 1998, chapter 11.3), the far-field approximation of the CCF is obtained as follows:
(29) |
(30) |
The symmetry between the causal and acausal parts is broken in the case of heterogeneous distribution of sources (Cupillard and Capdeville, 2010; Kimman and Trampert, 2010; Nishida and Fukao, 2007). Source heterogeneity also causes a bias of the phase from 0 to π/4 (Kimman and Trampert, 2010).
These far field representations were used to calculate an asymptotic 2-D Born sensitivity kernel for a minor arc propagation (R1 or G1), and for a major arc propagation (R2 or G2), as follows:
(31) |
(32) |
![](/geoscience/article/CRGEOS_2011__343_8-9_584_0/jpg/src/tex/figures/gr2.jpg)
Schematic map of the geometry of stations and their antipodes. Star symbols show the locations of stations at and , and those of their antipodes at and . The open circle shows the location of a phase velocity anomaly at .
Fig. 2. Carte schématique de la géométrie des stations et de leurs antipodes. Les symboles en étoile montrent la localisation des stations à et et celle de leurs antipodes et. Le cercle vide montre la localisation d’une anomalie de vitesse de phase à .
5 2-D phase sensitivity kernel in a spherically symmetric Earth model for a homogeneous source distribution
To obtain a phase sensitivity kernel for phase–velocity perturbations, the causal part of an R1 or G1 wave packet is isolated as follows:
(33) |
(34) |
The Rytov approximation is employed to obtain a phase sensitivity kernel for phase–velocity perturbations (e.g. Yoshizawa and Kennett, 2005; Zhou et al., 2004). In the Rytov method, the logarithm of the cross spectrum is considered instead of the wavefield itself. By taking the logarithm, can be divided into real (amplitude) and imaginary (phase) parts, as follows:
(35) |
(36) |
Using the asymptotic kernel , a phase sensitivity kernel for R1 or G1 can be written as
(37) |
Actual phase-velocity anomalies are measured with a finite frequency band (e.g. Bensen et al., 2007). To consider a sensitivity kernel for phase measurements, an averaged phase-velocity kernel in a certain frequency band is better than that at a single frequency. An averaged kernel is defined as follows:
(38) |
In the same manner, a phase sensitivity kernel for R2 or G2 is given by
(39) |
(40) |
Figure 3 shows a typical example of the phase sensitivity kernels and at a central frequency of 7.5 mHz for a frequency band width of 5 mHz. The sensitivity is concentrated within the first Fresnel zones. The side lobes of the kernels are suppressed due to averaging in the frequency domain. The expression of the R2 kernel is equivalent to that for earthquake data (Spetzler et al., 2002), which validates its application of ambient noise tomography using the observed phase-velocity anomalies of R2 data (Nishida et al., 2009).
![](/geoscience/article/CRGEOS_2011__343_8-9_584_0/jpg/src/tex/figures/gr3.jpg)
Imaginary part of the Rytov sensitivity kernels ( and ) of the fundamental Rayleigh wave at a central frequency of 7.5 mHz (from 5 to 10 mHz). Each station is located on the equator. The lower panel and the left-hand panel show slices of the kernels along the paths indicated by solid lines (an equatorial path, and the lines of longitude at 90° and 140°). The lower panel shows that the kernels have negative values along the minor and major arcs. Fig. 3. Partie imaginaire des noyaux de sensibilité de Rytov ( et ) du mode fondamental de l’onde de Rayleigh, à une fréquence centrale de 7,5 mHz (de 5 à 10 mHz). Chaque station est localisée sur l’Equateur. Le panneau inférieur et le panneau de gauche montrent des coupes des noyaux le long de trajets indiqués par des lignes continues (trajet équatorial et sur les lignes de longitude à 90° et 140°). Le panneau inférieur montre que les noyaux ont des valeurs négatives le long des arcs mineurs et majeurs.
6 Effects of heterogeneous distribution of sources on a Born sensitivity kernel
This section considers the effects of heterogeneous distribution of sources on the Born sensitivity kernel. For simplicity, the kernel is considered in a spherically symmetric case for the minor arc. It is simply assumed, phenomenologically, that the cross spectrum has only azimuthal dependency, as follows:
(41) |
(42) |
Of course, the coefficient a(φ) may have a imaginary part because an incomplete source distribution also causes a bias in the phase from 0 to π/4 (Kimman and Trampert, 2010). The imaginary part also causes a severe bias of the Born sensitivity kernel.
7 Conclusion
A theory of Born and phase sensitivity kernels was developed for a CCF using potential representation for surface waves. Simple forms were shown of Born and phase sensitivity kernels of a CCF in a spherically symmetric Earth model, assuming a homogeneous source distribution. The expression of the resultant phase sensitivity kernel is equivalent to that for phase measurements of earthquake data. This equivalence indicates the validity of ambient noise tomography under the given assumptions. The incomplete source distribution defines a hyperbolic pattern with foci at the pair of stations in a spherically symmetric case, which would generate a bias in the measured phase-velocity anomaly.
Acknowledgment
The author thanks Dr. Anne Seiminski, an anonymous reviewer, and the associate editor Dr. Michel Campillo for their constructive comments.