### Analytical equation for amplitude fluctuations

Following the mathematical framework of Ishimaru (1997, Chapters 16–18, pp. 321–379), we derived an analytical equation to evaluate the amplitude fluctuations of spherical waves radiated by a point source in 3D inhomogeneous acoustic media. For simplicity, we assumed an infinite inhomogeneous volume, in which velocity *c*(**r**) fluctuated as a function of position **r**. We assumed a weak velocity fluctuation, which was isotropic and uniformly distributed in space:

$$ c\left(\mathbf{r}\right)={c}_0\left(1+\xi \left(\mathbf{r}\right)\right) $$

(1)

where *c*
_{0} is the background velocity and *ξ*(**r**) is the fractional fluctuation of velocity. The fractional fluctuation should be small or |*ξ*(**r**)| ≪ 1 and 〈*ξ*(**r**)〉 = 0, where the symbol 〈〉 denotes the statistical ensemble average.

Under these conditions, wave propagation in 3D inhomogeneous acoustic media can be evaluated using a scalar wave equation:

$$ \varDelta u\left(\mathbf{r},t\right)-\frac{1}{c_0^2}\left(1-2\xi \left(\mathbf{r}\right)\right)\frac{\partial^2u\left(\mathbf{r},t\right)}{\partial {t}^2}=0 $$

(2)

where *u*(**r**,*t*) is the scalar wavefield, *t* is time, and ∆ is the Laplacian operator. For the solution of this equation, we assumed a time-harmonic spherical wave propagating outward from a point source. We employed a first-order Rytov approximation solution:

$$ u\left(\mathbf{r}\right)={u}_0\left(\mathbf{r}\right) \exp \varphi \left(\mathbf{r}\right) $$

(3)

$$ \varphi \left(\mathbf{r}\right)\equiv \chi \left(\mathbf{r}\right)+i\phi \left(\mathbf{r}\right)\equiv -2{k}^2{\displaystyle {\int}_{\mathrm{v}\hbox{'}}G\left(\mathbf{r}-\mathbf{r}\hbox{'}\right)\xi}\left(\mathbf{r}\mathbf{\hbox{'}}\right)\frac{u_0\left(\mathbf{r}\mathbf{\hbox{'}}\right)}{u_0\left(\mathbf{r}\right)}d\mathbf{V}\mathbf{\hbox{'}} $$

(4)

The exponential term in Eq. (3) represents the amplitude and phase modulation due to inhomogeneities superposed on the homogeneous background medium. For simplicity, the time-dependent term exp(−i*ωt*) is not explicitly written: the symbol *i* is the imaginary unit and *ω* is the angular frequency of a time-harmonic spherical wave. In Eq. (4), *k* is the wavenumber and symbols *u*
_{0}(**r**) and *G* are the wavefield and Green’s function, respectively, in the homogeneous background medium. The Green’s function is given by:

$$ G\left(\mathbf{r}-{\mathbf{r}}^{\mathbf{\prime}}\right)=\frac{ \exp \left(ik\left|\mathbf{r}-{\mathbf{r}}^{\mathbf{\prime}}\right|\right)}{4\pi \left|\mathbf{r}-{\mathbf{r}}^{\mathbf{\prime}}\right|} $$

(5)

The 3D integral of Eq. (4) becomes non-zero only for the volume element *d*
**V**′, for which a fractional fluctuation of velocity exists. Equation (4) is identical to equation (17–23) of Ishimaru (1997), except for the negative sign arising from the different definitions of the fractional fluctuation (Eq. (1)) and the fluctuation of the refractive index of Ishimaru (1997).

In Eq. (4), *χ*(**r**) and *ϕ*(**r**) are assumed to be real functions. The function *χ*(**r**) is related to the fluctuation in amplitude as follows:

$$ \chi \left(\mathbf{r}\right)=1\mathrm{n}\left|\frac{u\left(\mathbf{r}\right)}{u_0\left(\mathbf{r}\right)}\right| $$

(6)

It should be mentioned that, in case of |*χ*(**r**)| ≪ 1, |*χ*(**r**)| is approximately equal to |*u*(**r**) − *u*
_{0}(**r**)|/|*u*
_{0}(**r**)|. We here defined the variance of the amplitude level (hereafter, amplitude level variance) as:

$$ {\sigma}_{\chi}^2\equiv \left\langle {\left(\chi -\left\langle \chi \right\rangle \right)}^2\right\rangle $$

(7)

This quantity is one-fourth of the scintillation index (Ishimaru 1997), which represents the variance of wave intensity. Shapiro and Kneib (1993) derived a simple relationship between the amplitude spectra of *u*(**r**), *U*, and the amplitude level variance:

$$ {\sigma}_{\chi}^2=2\left(1\mathrm{n}\left\langle U\right\rangle -\left\langle 1\mathrm{n}U\right\rangle \right) $$

(8)

This relationship is useful for estimating \( {\sigma}_{\chi}^2 \) when we do not have rigorous information on *u*
_{0}(**r**).

Assuming a line-of-sight propagation of a spherical wave (Ishimaru 1997), we specifically considered the amplitude level variance at distance *L* from the point source:

$$ {\sigma}_{\chi}^2=\left\langle {\left(\chi -\left\langle \chi \right\rangle \right)}^2\right\rangle =4{\pi}^2{k}^2{\displaystyle \underset{0}{\overset{L}{\int }}d\eta {\displaystyle \underset{0}{\overset{\infty }{\int }}\upsilon { \sin}^2}\left[\frac{\gamma \left(L-\eta \right)}{2k}{\upsilon}^2\right]}{\varPhi}_{\xi}\left(\upsilon \right)d\upsilon $$

(9)

where *γ*≡*η*/*L*, and *Φ*
_{
ξ
} is the power spectral density function of *ξ*. Equation (9) is same as equation (18–5) of Ishimaru (1997). Assuming a von Kármán-type random media (e.g., Sato et al. 2012) for characterizing the stochastic properties of *ξ*, *Φ*
_{
ξ
} can be expressed by a function of wavenumber *υ* as follows:

$$ {\varPhi}_{\xi}\left(\upsilon \right)=\alpha {\varepsilon}^2{\left({\upsilon}^2+{a}^{-2}\right)}^{-\beta } $$

(10)

where *ε*
^{2}≡〈*ξ*(**r**)^{2}〉, *a* is the correlation distance, and parameters *α* and *β* (3/2 ~ 5/2) are positive constants. The value of *α* may be set in relation to *β* as follows:

$$ \alpha ={\pi}^{-\frac{3}{2}}{a}^{-2\left(\beta -\frac{3}{2}\right)}\varGamma \left(\beta \right)\varGamma {\left(\beta -\frac{3}{2}\right)}^{-1} $$

(11)

where *Γ*() is the gamma function. It should be noted that this definition of the power spectral density function, used by Ishimaru (1997), differs in (2*π*)^{− 3} with that of Sato et al. (2012). Substitution of equation (10) into (9) and integration over *υ* gives the amplitude level variance at distance *L* as a simple integral form:

$$ {\sigma}_{\chi}^2=\alpha {\pi}^2{k}^2{\varepsilon}^2{a}^{2\left(\beta -1\right)}{\displaystyle \underset{0}{\overset{L}{\int }}d\eta \left[\frac{1}{\beta -1}-\mathrm{R}\mathrm{e}\uppsi \left(1,2-\beta, i\frac{\gamma \left(L-\eta \right)}{k{a}^2}\right)\right]} $$

(12)

where Reψ() is the real part of the confluent hypergeometric function of the second kind (Abramowitz and Stegun 1964, equation 13.2.5, p. 505). If we assume exponential-type random media (*α* = *π*
^{−2}
*a*
^{−1} and *β* = 2; e.g., Sato et al. 2012), Eq. (12) can be simplified to:

$$ {}^{Exp}\sigma_{\chi}^2={k}^2{\varepsilon}^2a\left[L-{\displaystyle \underset{0}{\overset{L}{\int }}d\eta \mathrm{R}\mathrm{e}\uppsi \left(1,0,i\frac{\gamma \left(L-\eta \right)}{k{a}^2}\right)}\right] $$

(13)

From the same geometric consideration of wave propagation as in Ishimaru (1997), Eqs. (12) and (13) can be used for a plane wave case when *γ* = 1.

### Evaluation of amplitude fluctuations

Power spectral density of von Kármán-type random media varies with parameter *β* (Fig. 3a). Reflecting this property, the amplitude level variance increases with increasing *β* value (Fig. 3b). For the comparative analysis using numerical and observational data in the following sections, we assumed stochastic parameters of *a =* 1 km and *ε* = 0.03 (Kobayashi et al. 2015) for crustal inhomogeneities. The hypocentral distance dependence of amplitude level variance arises from the increase in the power spectral density components of inhomogeneity at small wavenumbers, which causes the phase modulation of a propagating wave (Uscinski 1977). The dotted line indicates the saturation level of \( {\sigma}_{\chi}^2, \) which is equal to 0.17, in the strong wavefield fluctuation regime, where the log-amplitude variance tends to the constant, or scintillation index saturates (e.g., Müller and Shapiro 2003). Below this saturation level, estimations from the analytical equation derived in this study may be useful for data analysis.

To better understand the *k* dependence of \( {\sigma}_{\chi}^2 \) for exponential-type random media, in relation with the contribution of the integral part of Eq. (13), different *k* values of *π*/4, *π*/2, *π*, 2*π*, and 4*π* km^{−1} (corresponding to frequencies of 3/4, 3/2, 3, 6, and 12 Hz, respectively, for the inhomogeneous medium with a background velocity of 6.0 km/s) are considered (Fig. 4). The results show that the amplitude level variance increases for waves with large wavenumbers or short wavelengths. For example, in the case of *k* = *π*, a spherical wave propagating outward from a point source reaches the strong wavefield fluctuation regime at a hypocentral distance of ~30 km, indicating that the deterministic analysis of wave amplitude is no longer applicable.