In Bensen et al. [2007], some time is spent discussing the extraction of phase velocity, where they suggest that there was no known suitable method. For this reason, the au- thors suggest extraction of the group velocity using frequency-time analysis techniques [Dziewonski and Hales, 1972, Herrmann, 2013]. This in turn became the method of choice for many ambient noise studies because there are established codes and it is relatively straight forward albeit labour intensive.
The approach taken here estimates phase velocity dispersion and this is motivated by the fact that phase velocity is more useful than group velocity. Firstly, the group veloc- ity can be uniquely determined from phase velocity dispersion, since phase velocity is given by
cn(ω) = kn(ω)
ω , (2.1)
(a) −200 −100 0 100 200 Time (s) (b) 0.0 0.1 0.2 0.3 0.4 0.5 Frequency (Hz)
Figure 2.2: An example empirical Greens function is shown resulting from the minimalistic pre- processing approach before cross-correlating signals. In (a) is the empirical Green’s functions in the time domain highlighting the causal and acausal parts of the signal. In (b) is the complex spectrum with the real part shown as a dark line to highlight the spectral zero crossings, and the imaginary part shown as a feint line. It is clear from both the time domain and frequency domain that the noise is not spatial isotropic.
Un(ω) = ∂ ω
∂kn(ω), (2.2)
where cn is the phase velocity of moden as a function of frequencyω, kn is the wave
number, and Un is the group velocity. From these relationships, an expression can be
obtained for the group velocity in terms of phase velocity
Un(ω) = cn(ω) 1−c ω n(ω) ∂cn ∂ ω , (2.3)
hence given a differentiable phase velocity curve, the group velocity can be computed.
The reverse is not true because expression ofcnas a function ofUnbecomes
cn(ω) =
Z ω
ω0
Un(x)d x+cn(ω0), (2.4)
and the cn(ω0) is not known. Hence any extraction of phase velocity automatically
means group velocity is also available assuming differentiability, but the reverse is not true.
Secondly, surface wave ray paths are sensitive to phase velocity, not group velocity [Tanimoto, 1986]. In a non-linear inversion in which ray paths are recomputed, they should be recomputed based upon the phase velocity and not the group velocity. Many previous studies [Saygin et al., 2016, Galetti et al., 2016] have computed ray paths from group velocities which introduces a further approximation. For slowly varying, or nearly linear, dispersion of phase velocity, using the group velocity is a reasonable ap- proximation because the the group velocity dispersion will be a scaled version of the phase velocity curve to first order. Hence the relative change in group velocity and phase velocity coincide and the ray paths generated by each would be similar. Unfortu- nately, phase velocity dispersion for simple models and exemplar measured dispersion shows that there is a degree of gradient change in the phase velocity in frequency ranges of interest, hence this approximation may cause significant inaccuracies. In Figure 2.3, a reasonable but simple dispersion is shown illustrating the difference between phase
0.0 0.1 0.2 0.3 0.4 0.5 Frequency (Hz) 2.0 2.5 3.0 3.5 4.0 Ve lo cit y (k m /s )
Figure 2.3: An example dispersion curve showing difference between phase (solid black line) and group velocity (red dotted line). Updating rays using group velocity can bias the results because relative change in phase velocity and group velocity can change in the frequency range of interest for ambient noise tomography (grey shaded region).
and group velocity.
Others have similarly expressed the benefits of phase velocity over group, for example, Boschi et al. [2013] provide three motivations for phase velocity over group:
1. Group velocity is less precisely defined than phase velocity,
2. For the fundamental mode, phase velocity information is able to image deeper into the Earth than group velocity,
3. Group velocity measurements are more likely to be contaminated by interfering phases than phase velocity measurements.
more recently, different methods have been proposed for the extraction of phase veloc- ity dispersion from empirical Green’s functions. Firstly an image based time-domain technique [Yao et al., 2005, 2006], and a frequency domain technique [Ekström et al., 2009, Ekström, 2014] based on an analysis of the statistics of micro-tremor correlations between stations [Aki, 1957]. Both these techniques have been successfully applied to the recovery of phase velocity maps of regions from ambient noise correlations. To extract phase velocity, a frequency domain method is used, originally proposed by Aki for micro-tremor data [Aki, 1957] and reintroduced for ambient noise by Ekström [Ekström et al., 2009, Ekström, 2014]. Alternatives, such as image based methods exist for extracting phase velocity [Yao et al., 2005], however the frequency domain method does not use the far field approximation and therefore can extract longer period dispersion information from closer stations.
In the original Ekström [Ekström et al., 2009] approach, the zero crossings of the real component of the spectrum of the cross-correlograms were used to construct trial phase velocity curves directly. This uses the result from Aki [1957] that the real com- ponent of correlated noise between stations is of the form of a Bessel function of the first kind with order 0. Restating this result here
¯ ρ(ω0,r) =J0 ω0 c(ω0)r , (2.5)
where ρ¯ is the cross correlation spectrum, ω0 the angular frequency of the funda-
mental mode, c(ω0) is the frequency dependent phase velocity, and r is inter-station
distance. From the observed zero crossings in the empirical Greens function, i.e. a set
of zero crossings z1. . .zn at angular frequencies ω1. . .ωn, trial phase velocity curves
are constructed using
cm(ωn) = ωnr
zn+2m, (2.6)
dispersion curve mostly closely matching a reference dispersion curve for the region is chosen as the observed phase velocity curve. The problem with this approach is that noise inherent in the observations can cause spurious zero crossings as highlighted by Menke and Jin [2015] and this can result in phase velocity dispersion curves that drop precipitously to unfeasible values. These problems were recognised and a subsequent improvement to the method [Ekström, 2014] adds the extra step of fitting a piece wise spline to the real component of the spectrum in an effort to eliminate the spurious zero crossings. A further extension of this general approach was the inclusion of completely fitting the real part of the spectrum by Menke and Jin [2015], which improved the rejection of spurious zero crossings and added the ability to using residuals from the inversion of individual cross-correlated station pairs as quality factors.
Here the aim is to build on these advances by developing a Bayesian approach for ex- traction of phase velocity information. A key factor in Bayesian approaches is the in- clusion of prior information in both the formulation of the problem, the assumptions therein as probability distributions.