Inter-Source Interferometry of Seismic Body Waves: Required Conditions and Examples

Abstract

Seismic interferometry is widely applied to retrieve wavefields propagating between receivers. Another version of seismic interferometry, called inter-source interferometry, uses the principles of seismic reciprocity and expands interferometric applications to retrieve waves that propagate between two seismic sources. Previous studies of inter-source interferometry usually involve surface-wave and coda-wave estimations. We use inter-source interferometry to estimate the P-waves propagating between two sources rather than the estimation of surface waves and coda waves. We show that the recovered arrival times are dependent on the accuracy of the earthquake catalog of the two sources. Using inter-source interferometry, one can recover the waveform of the direct body waves and potentially reconstruct the waveform of coda waves, depending on the source-receiver geometry. The retrieval of these waveforms is accurate only when the wavefield is sampled with approximately 4 receivers per wavelength in the stationary phase zone. We show that using only receivers inside the stationary phase region for inter-source interferometry introduces the phase error of approximately 0.3 radians. In our study, we show an example of the P-wavefield reconstruction between two earthquakes using the seismic records from an array along San Andreas Fault. The retrieved P waves give a qualitative estimation of the thickness of the low-velocity zone of San Andreas Fault of approximately 4 km.

Appendix A: Phase Shift and Fresnel Integral

In Sect. 3, we show that one can reconstruct direct and scattered body waves using receivers in the stationary phase zones. Using an array with a limited aperture in the stationary phase region, the phase of the recovered waveforms differs slightly from the forward-modelled waveforms (Figs. 5, 6, and 7).

When we use the source-receiver geometry in Fig. 2, we can determine the time differences between the direct arrivals of the two sources at each receiver. We define the size of the stationary phase zone (SPZ) using a faction of the dominant period $T_{dominant}$ of our synthetic signals, where $T_{dominant} = 0.25$ s. In Sect. 3, our SPZ covers the receivers within $T_{dominant}/4$ from the maximum time difference (Fig. 19).

Fig. 19

Time differences between the direct arrivals of the two sources at each surface receiver in Fig. 2. The graph on the bottom left corner is the enlarged version of the area inside the black box

Figure 20 shows the direct wave that propagates between the two sources in Fig. 2, and the waves extracted from inter-source interferometry take only receivers into account with a delay time $T_{dominant}/4$ relative to the delay time at the stationary phase zone. One should note the slight phase change between the two waveforms. We show in this appendix that this phase error is associated with limiting the used receivers to the stationary phase region.

Fig. 20

The forward-modeled and the interferometric waveforms of the direct body waves

Fig. 21

Source and receiver positions in interferometry. x is the half-width of the stationary phase zone

Consider the source-receiver geometry in Fig. 21. Seismic interferometry for waves in a 2D homogeneous medium gives

$$\begin{aligned}&G(r_A,r_B,\omega ) - G^*(r_A,r_B,\omega ) \nonumber \\&\quad \approx \frac{i}{4\pi \rho } \oint _{S^\prime } \sqrt{\frac{1}{r_{SA}r_{SB}}}e^{ik(r_{SA}-r_{SB})} dS^\prime , \end{aligned}$$

(9)

where $G(r_A,r_B,\omega ) - G^*(r_A,r_B,\omega )$ is the difference between the causal and acausal parts of the Green’s function accounting for the wave propagation between the points A and B, $\omega$ is the angular frequency, $\rho$ is the mass density, k is the wave number, and S is the source position on a surface $S^\prime$ (Fan and Snieder 2009). We can approximate the term $r_{SA}-r_{SB}$ in Eq. (9) using a second-order Taylor series:

$$\begin{aligned} { r_{SA}-r_{SB} \approx (L_A - L_B) + \frac{1}{2}\bigg ( \frac{1}{2L_A} - \frac{1}{2L_B} \bigg )x^2. } \end{aligned}$$

(10)

Using this second order Tayor expansion for the phase and replacing the geometrical spreading in Eq. (9) by $1/\sqrt{L_A L_B}$, the integral in expression (9) is in the stationary phase approximation (Snieder and van Wijk 2015) given by

$$\begin{aligned}&G(r_A,r_B,\omega ) - G^*(r_A,r_B,\omega ) \nonumber \\&\quad \approx \frac{i}{4\pi \rho } \frac{e^{ik(L_A - L_B ) }}{\sqrt{L_A L_B}} \int _{-\infty }^{+\infty } \exp \left( \frac{ik}{2} \left( \frac{1}{L_A} - \frac{1}{L_B}\ \right) x^2 \right) dx \;, \end{aligned}$$

(11)

where we replaced the integration over a closed surface by an integration over the transverse x-coordinate. The integral in the right-hand side is of the same form as the Fresnel integral (Sandoval-Hernandez et al. 2018), which is defined as

$$\begin{aligned} { F(X) = \int _0^X \exp \left( {i \frac{\pi }{2}x^{\prime ^2}} \right) dx^\prime \;. } \end{aligned}$$

(12)

This integral is complex and is shown in the complex plane in Fig. 22 where the the origin corresponds to the integral for $X=0$, and the asymptotic point $0.5 + 0.5i$ is reached for $X \rightarrow \infty$. This graphical representation is called the Cornu spiral. The many windings of this spiral around the asymptotic point reflect that the Fresnel integral converges slowly.

Fig. 22

Fresnel integral from $t=0$ to $t \rightarrow \infty$ in the complex domain (blue). Black and red asterisks represent the integral over a large part of surface receivers and surface receivers located in the stationary phase zone, respectively

The real part of the integrand of the Fresnel integral (12) is shown in Fig. 23. The oscillatory nature of this integrand explains the shape of the Cornu spiral and the slow convergence of the Fresnel integral: the alternating positive and negative values of the integrand lead to the slow spiraling of the Fresnel integral in the complex plane, and one needs to integrate over a large interval for the integral to be close to its asymptotic value for $X \rightarrow \infty$.

In the waveforms of Fig. 20 the summation over receivers was limited to receivers that have a delay time of less than a 1/4 of the dominant period. That corresponds to a phase delay of less than $\pi /2$. In the Fresnel integral (12), this integration interval corresponds to the upper limit $X=1$. As shown by the black dot in Fig. 23, this upper limit is at the point where the real part of the Fresnel integral has its first zero crossing. (This is actually the rationale for restricting the integration to delay times less than 1/4 of the dominant period.) Since the real part of the integrand of the Fresnel integral changes sign, the real part of the Fresnel integral has a maximum at this point, and hence this upper limit corresponds to the rightmost point of the Cornu spiral that is indicated by the red star in Fig. 22.

The important point to note is that the phase of the Fresnel integral for $X=1$ differs from the phase of the integral computed for its asymptotic value for $X \rightarrow \infty$; the phase angle for the red star in Fig. 22 differs by about 0.273 rad from the phase angle for the black star. This phase difference leads to the phase difference in the waveforms shown in Fig. 20. The phase difference in the waveforms corresponds to about 0.201 rad. This is slightly less than the phase difference of 0.273 rad predicted above. This discrepancy is caused by the fact that the argument in this appendix ignores the decay of the interferometric integral that is due to variations of the geometrical spreading in the integration. Furthermore, the analysis in this appendix is applicable in the frequency domain, whereas the time-domain waveforms in Fig. 20 contain contributions from a range of frequencies. However, the main points from this appendix are that (1) the interferometric integral converges slowly and (2) truncating this integral can leads to phase errors of about 0.3 rad. Tapering of the interferometric integral will lead to a faster convergence of the integral. Since the phase error does not depend on the distance between the sources at locations A and B, the impact of the phase error on estimated velocities decreases with increasing distance between the sources.

Fig. 23

Real part of the integrand of the Fresnel integral (12) for $-5< x^\prime < 5$. Black dot denotes the first position where the real part changes from positive to negative values