Outline
Comptes Rendus

Internal geophysics (Physics of Earth’s interior)
Two-dimensional sensitivity kernels for cross-correlation functions of background surface waves
Comptes Rendus. Géoscience, Volume 343 (2011) no. 8-9, pp. 584-590.

Abstracts

Abstract

Ambient noise tomography has now been applied at scales ranging from local to global. To discuss the theoretical background of the technique, a simple form of a two-dimensional (2-D) Born sensitivity kernel was developed at a finite frequency for a cross-correlation function (CCF) of background surface waves. The use of far field representations of a Green’s function and a CCF in a spherically symmetric Earth model, assuming a homogeneous source distribution, is an efficient approach to the calculation of phase sensitivity kernels. The forms of a phase sensitivity kernel for major and minor arc propagations are the same as those for phase-velocity measurements of earthquake data. This result indicates the validity of ambient noise tomography under the given assumptions; however, the kernels are not equivalent in the case of an inhomogeneous source distribution.

Résumé

La tomographie à partir du bruit ambiant a, à présent, été appliquée à des échelles à la fois locales et globales . Pour discuter des fondements théoriques de la technique, une simple forme de noyau de sensibilité de Born à 2D et à fréquence finie a été développée pour une fonction de corrélation (CCF) d’un bruit consistant en ondes de surface. L’utilisation de représentations en champ lointain d’une fonction de Green et d’une CCF dans un modèle de Terre à symétrie sphérique, en supposant une distribution de source homogène, est une approche efficace pour le calcul de noyaux de sensibilité de la vitesse de phase. Les formes des noyaux de sensibilité pour les propagations le long d’arcs majeurs et mineurs sont les mêmes que celles qu’on observe dans les mesures de vitesse de phase avec les observations des séismes. Ce résultat indique la validité de la tomographie à partir du bruit ambiant dans les hypothèses données ici; cependant, les noyaux ne sont pas équivalents dans le cas d’une distribution inhomogène de sources.

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DOI: 10.1016/j.crte.2011.02.004
Keywords: Interférométrie, Noyau de sensibilité, Tomographie en bruit ambiant, Interferometry, Sensitivity kernel, Ambient noise tomography

Kiwamu Nishida 1

1 Earthquake Research Institute, University of Tokyo, Tokyo, Japan
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     title = {Two-dimensional sensitivity kernels for cross-correlation functions of background surface waves},
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Kiwamu Nishida. Two-dimensional sensitivity kernels for cross-correlation functions of background surface waves. Comptes Rendus. Géoscience, Volume 343 (2011) no. 8-9, pp. 584-590. doi : 10.1016/j.crte.2011.02.004. https://comptes-rendus.academie-sciences.fr/geoscience/articles/10.1016/j.crte.2011.02.004/

Version originale du texte intégral

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:

u=uL+uR.(1)
Love and Rayleigh wave displacement fields in laterally, slowing varying media can be written in terms of surface wave potentials (Tanimoto, 1990; Tromp and Dahlen, 1993). The Love wave part uL and the Rayleigh wave part uR are given by the surface wave potential χα, as follows:
uα=Dαχα,(2)
where the subscript α represents the Love wave (L) or Rayleigh wave (R). The spatial differential operators DR and DL are respectively defined as follows:
DR=U(r,ω)rˆ+kR1(rˆ,ω)V(r,ω)l(3)
DL=kL1(rˆ,ω)W(r,ω)(rˆ×l),(4)
where ∇l is the surface gradient operator, U(r, ω) is the local vertical eigenfunction, V(r, ω) is the local radial eigenfunction, and W(r, ω) is the local transverse eigenfunction as a function of surface location r, rˆ is a unit vector defined on a unit sphere, and kα(rˆ,ω) is the local wavenumber at the angular frequency ω. kα(rˆ,ω) can also be written in terms of the phase velocity cα(rˆ,ω), as follows: kα(rˆ,ω)=ω/cα(rˆ,ω). The convention for the Fourier transform is that exp (− iωt) appears in the Fourier integral when transforming from the time domain to the frequency domain. The eigenfunctions are normalized following Tromp and Dahlen (1993), as cα(rˆ,ω)Cα(rˆ,ω)I1α(rˆ,ω)=1, where Cα(rˆ,ω) is the group velocity and I1α is the energy integral.

The surface wave potentials satisfy the inhomogeneous spherical Helmholtz equation of a surface wave (Tromp and Dahlen, 1993), as follows:

l2χα(rˆ,ω)+ξα2(rˆ,ω)χα(rˆ,ω)=Fα(rˆ,ω),(5)
where ξα is the following complex wavenumber,
ξα(rˆ,ω)=kα(rˆ,ω)ω2Qα(rˆ,ω)Cα(rˆ,ω)i,(6)
where Qα is a quality factor. The equivalent surface traction Fα is defined as follows:
FL(rˆ,ω)=kL(rˆ,ω)W(r,ω)f3(rˆ,ω)R2,(7)
FR(rˆ,ω)=(U(r,ω)f1(rˆ,ω)+kR(rˆ,ω)V(r,ω)f2(rˆ,ω))R2,(8)
where R is the radius of the Earth. Here, f1 and f2 are spheroidal components of the equivalent surface traction f, and f3 is a toroidal component of f in the form of f=rˆf1+lf2rˆ×lf3.

A scalar Green’s function of Love and Rayleigh waves Gα(rˆ,rˆs,ω) satisfies

l2Gα(rˆ,rˆs,ω)+ξα2Gα(rˆ,rˆs,ω)=δ(rˆ,rˆs).(9)
A scalar potential function χα can be written as
χα(rˆ,ω)=ΣGα(rˆ,rˆs,ω)Fα(rˆs,ω)dΣ,(10)
where Σ is the unit sphere.

The cross spectrum Φ of background surface waves between stations r1 and r2 can be given by

Φ(r1,r2,ω)=u(r1,ω)u(r2,ω)=DL1DL2ΦLL(rˆ1,rˆ2,ω)+DR1DR2ΦRR(rˆ1,rˆ2,ω)+DL1DR2ΦLR(rˆ1,rˆ2,ω)+DR1DL2ΦRL(rˆ1,rˆ2,ω)(11)
where ∗ represents a complex conjugate, 〈〉 denotes an ensemble (statistical) average, an αβ component of the cross spectrum Φαβ(rˆ1,rˆ2,ω) is χα(rˆ1,ω)χβ(rˆ2,ω), and Dαi is the spatial derivative at point rˆi. For simplicity, the αβ component of the cross spectrum Φαβ(rˆ1,rˆ2,ω) is evaluated below.

The cross spectrum Φαβ(rˆ1,rˆ2,ω) can be written as follows:

Φαβ(rˆ1,rˆ2,ω)=ΣΣGα(rˆ1,rˆ,ω)Gβ(rˆ2,rˆ,ω)Ψαβ(rˆ,rˆ,ω)dΣdΣ,(12)
where Ψαβ is the cross spectrum of surface traction Fα(rˆ,ω)Fβ(rˆ,ω) between points rˆ and rˆ. Assuming that the excitation sources of the background surface waves are spatially isotropic but heterogeneous, the cross spectrum Ψαβ(rˆ,rˆ,ω) is expressed in the following form:
Ψαβ(rˆ,rˆ,ω)=Ψˆαβ(rˆ,ω)Ψˆαβ(rˆ,ω)h1|rˆrˆ|L(rˆ,ω),(13)
where Ψˆαβ(r,ω) is a power spectrum of surface traction at rˆ (Fukao et al., 2002; Nishida and Fukao, 2007), L(rˆ,ω) is the frequency-dependent coherent length, and the function h(x) is the Heviside step function.

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:

Φαβ(rˆ1,rˆ2,ω)=4π2ΣΨˆαβe(rˆ,ω)Gα(rˆ1,rˆ)Gβ(rˆ2,rˆ)dΣ.(14)
Here, the power spectrum of effective surface traction per unit wavenumber Ψˆαβe(ω) is defined as follows:
Ψˆαβe(rˆ,ω)L2(rˆ,ω)4πR2Ψˆαβ(rˆ,ω).(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:

δΦαβ(rˆ1,rˆ2,ω)=4π2ΣΨˆαβe(rˆ,ω)δGα(rˆ1,rˆ,ω)Gβ(rˆ2,rˆ,ω)+Gα(rˆ1,rˆ,ω)δGβ(rˆ2,rˆ,ω)dΣ.(16)
δGα can be written as follows (e.g. Yoshizawa and Kennett, 2005):
δGα(rˆ1,rˆ,ω)=Σ2kα2(rˆ3,ω)δc(rˆ3,ω)c(rˆ3,ω)Gα(rˆ1,rˆ3)Gα(rˆ3,rˆ)dΣ3.(17)

The above equation can then be simplified as follows:

δΦαβ(rˆ1,rˆ2,ω)=ΣKαβ(rˆ1,rˆ2,rˆ3,ω)δc(rˆ3,ω)c(rˆ3,ω)dΣ3.(18)
A 2-D Born sensitivity kernel for phase–velocity anomalies is defined as follows:
Kαβ(rˆ1,rˆ2,rˆ3,ω)=2kα2Φβα(rˆ2,rˆ3,ω)Gα(rˆ1,rˆ3)+kβ2Φαβ(rˆ1,rˆ3,ω)Gβ(rˆ2,rˆ3).(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 rˆ2 to rˆ3.

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:

Gα(rˆ1,rˆ2,ω)=l2l+14π(ξα2l(l+1))Pl(cosΘ12),(20)
where Θ12 is the angular distance between rˆ1 and rˆ2, and Pl is the Legendre function of the l’th order.

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:

Φαα(rˆ1,rˆ2,ω)=Ψˆααe(ω)πl2l+1(ξα2l(l+1))(ξα2l(l+1))Pl(cosΘ12).(21)
The cross terms Φαβ are omitted for α ≠ β because they take values of zero in the case of a homogeneous source distribution.

Figure 1 shows a typical example of the Born sensitivity kernel of a Rayleigh wave at 5.61 mHz with the source spectrum Ψααe 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.

Fig. 1

Born sensitivity kernel of a Rayleigh wave at 5.61 mHz, with amplitude normalized by the cross spectrum Φαα12, 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 Φαα12, 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):

Gα(rˆ1,rˆ2,ω)=s=1gαs(Δ12s,ω).(22)
The Green’s function of the s’th orbit gαs is defined as follows:
gαs(Δ12s,ω)=eikαΔ12s+(s1)π2π4eωΔ12s2CαQα8πkαsin|Δ12s|,(23)
where the integer s( = 1, 2,  … ) represents the surface wave orbits. The quantity Δ12s, which is the total angular distance traversed by a given arrival, is given by explicitly by,
Δ12s=Θ12+(s1)π,soddsπΘ12,seven.(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:

Φαα(rˆ1,rˆ2,ω)=Ψˆααe(ω)πs=(1)s02(ξα2k2)(ξα2k2)Pk12e2iskπkdk.(25)

The above equation is transformed into a traveling representation of the CCF (Dahlen and Tromp, 1998, chapter 11.2), as follows:

Φαα(rˆ1,rˆ2,ω)=2Ψˆααe(ω)πs=1,3,(1)(s1)/2Qk1/2(1)ei(s1)kπ(ξα2k2)(ξα2k2)kdk+s=2,4,(1)s/2Qk1/2(2)eiskπ(ξα2k2)(ξα2k2)kdk.(26)

The analysis employs a relation of the transformation into a traveling wave representation (Dahlen and Tromp, 1998, appendix B.11), as follows:

Pk12(cosΘ)=Qk12(1)(cosΘ)+Qk12(2)(cosΘ),(27)
where Qk1/2(1,2) corresponds to waves propagating in the direction of increasing and decreasing Θ, respectively. The following equation is also employed (Dahlen and Tromp, 1998, eq. 11.13):
Qk12(1,2)(cosΘ)=e±2ikπQk12(1,2)(cosΘ)+e±ikπtankπPk12(cosΘ).(28)

Following (Dahlen and Tromp, 1998, chapter 11.3), the far-field approximation of the CCF is obtained as follows:

Φαα(rˆ1,rˆ2,ω)=s=1ϕαs,(Δ12s,ω),(29)
where ϕαs(Δ12s,ω) is the cross spectrum of the s’th orbit, as follows:
ϕαs(Δ12s,ω)=Ψˆααe(ω)2π2QαCαωgαs(Δ12s,ω)eπ2i+gαs(Δ12s,ω)eπ2i.(30)
Because the cross spectrum Φαα12, ω) is a real function, the corresponding CCF is an even function, which has a causal part and an acausal part. The first term represents the causal part of the cross spectrum; the second term represents the acausal part. This equation shows the phase retreat (π/2) of the causal part of the corresponding CCF from the Green’s function (Nakahara, 2006; Sanchez-Sesma and Campillo, 2006).

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 Kα1(rˆ1,rˆ2,rˆ3,ω) for a minor arc propagation (R1 or G1), and Kα2(rˆ1,rˆ2,rˆ3,ω) for a major arc propagation (R2 or G2), as follows:

Kα1=2πΨˆααeQαCα/cαsinΘ13sinΘ23sin(kα(Θ13+Θ23))eω(Θ13+Θ23)2QαCα(31)
Kα2=2πΨˆααeQαCα/cαsinΘ13sinΘ23cos(kα(π+Θ13+Θ23))eω(π+Θ13+Θ23)2QαCα+cos(kα(π+Θ13+Θ23))eω(π+Θ13+Θ23)2QαCαsin(kα(2π+Θ13+Θ23))eω(2π+Θ13+Θ23)2QαCα,(32)
where Θij is the angular distance between the i’th and j’th points, as shown in Fig. 2.

Fig. 2

Schematic map of the geometry of stations and their antipodes. Star symbols show the locations of stations at rˆ1 and rˆ2, and those of their antipodes at rˆ1 and rˆ2. The open circle shows the location of a phase velocity anomaly at rˆ3.

Fig. 2. Carte schématique de la géométrie des stations et de leurs antipodes. Les symboles en étoile montrent la localisation des stations à rˆ1 et rˆ2 et celle de leurs antipodes rˆ1 etrˆ2. Le cercle vide montre la localisation d’une anomalie de vitesse de phase à rˆ3.

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 ϕαc,1(rˆ1,rˆ2,,ω) is isolated as follows:

ϕαc,1(rˆ1,rˆ2,ω)=Ψˆααe(ω)2π2Qα(ω)Cα(ω)ωgα1(Θ12,ω)eπ2i.(33)
The causal part of the Born sensitivity kernel is defined as follows: Kαc,1=12(Kα1H(Kα1)i), where H represents the Hilbert transform in the frequency domain. For simplicity, the source term Ψˆααe is assumed to be a smooth function in the frequency domain. Then, perturbation of the causal part δϕαc,1 can be written as
δϕαc,1=Σδcα(rˆ3,ω)cαKαc,1(rˆ1,rˆ2,rˆ3,ω)dΣ3.(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 ϕαc,1 is considered instead of the wavefield itself. By taking the logarithm, ϕαc,1 can be divided into real (amplitude) and imaginary (phase) parts, as follows:

lnϕαc,1=ln(Aα1exp(ψα1i))=lnAα1ψα1i,(35)
where Aα1 is the amplitude of the causal wave packet, and ψα1 is its phase. The phase perturbation δψα1 for propagation along the minor arc (R1 or G1) is given by
δψα1=Kp,α1(rˆ1,rˆ2,rˆ3,ω)δccdΣ3,(36)
where the phase sensitivity kernel Kp,α1 is the imaginary part of Kαc,1/ϕαc,1.

Using the asymptotic kernel Kαc,1, a phase sensitivity kernel for R1 or G1 can be written as

Kp,α1=kα322πsinΘ12sinΘ13sinΘ2312coskα(Θ13+Θ23Θ12)π4eω(Θ13+Θ23Θ12)2QαCα.(37)
This expression of the phase sensitivity kernel is the same as that for phase measurements of earthquake data (Yoshizawa and Kennett, 2005). The equivalence of the expressions serves as validation of ambient noise tomography. Of course, this discussion is valid only under the assumptions of one-dimensional background structure and a homogeneous source distribution.

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 K¯p,α1 in a certain frequency band is better than that at a single frequency. An averaged kernel K¯p,α1 is defined as follows:

K¯p,α1=1Δff0Δf/2f0+Δf/2Kp,α1df,(38)
where f0 is the central frequency and Δf is the frequency band width.

In the same manner, a phase sensitivity kernel Kp,α2 for R2 or G2 is given by

Kp,α2=kα322πsinΘ12sinΘ13sinΘ2312sinkα(Θ13+Θ23Θ12)π4eω(Θ13+Θ23Θ12)2QαCα(39)
+sinkα(Θ13+Θ23Θ12)π4eω(Θ13+Θ23Θ12)2QαCαcoskα(Θ13+Θ23Θ12)π4eω(Θ13+Θ23Θ12)2QαCα.(40)
It is also possible to define an averaged phase-velocity kernel K¯p,α2.

Figure 3 shows a typical example of the phase sensitivity kernels K¯p,α1 and K¯p,α2 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).

Fig. 3

Imaginary part of the Rytov sensitivity kernels (K¯p,α1 and K¯p,α2) 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 (K¯p,α1 et K¯p,α2) 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 Kα1 is considered in a spherically symmetric case for the minor arc. It is simply assumed, phenomenologically, that the cross spectrum ϕα1 has only azimuthal dependency, as follows:

ϕα1(Δ121,ω)=Ψˆααe2π2QαCαωa(ϕ12)gα1(Δ121,ω)e12πi+a(ϕ21)gα1(Δ121,ω)e12πi,(41)
where a(φ) is a real coefficient that varies as a function of azimuth, and φ12 is the back-azimuth to rˆ2 at rˆ1. Because the focus is on the perturbation of a time-symmetric part of a CCF, the real part of the Born sensitivity kernel Kα1 is evaluated, as follows:
[Kα1]{a(ϕ23)+a(ϕ13)}sin(kα(Θ23+Θ13)){a(ϕ23)a(ϕ13)}cos(kα(Θ23Θ13)),(42)
where ℜ indicates the real part. The first term shows an elliptic pattern with foci at the stations, whereas the second term shows a hyperbolic pattern with foci at the stations. The second term vanishes in the case of a homogeneous source distribution, as in Eq. (31). If the time symmetry of the CCF is broken because of a heterogeneous source distribution, the second term would take on a hyperbolic pattern. Thus, the antisymmetry causes a bias in phase velocity maps, even without phase changes in the causal and acausal parts. Note that the second term vanishes in the case of |rˆ3||rˆ2rˆ1|.

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.


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