Definition of the Subject
Volcano seismicity produces a wide variety of seismic signals that provide glimpses of the internal dynamics of volcanic systems. Quantitativeapproaches to analyze and interpret volcano‐seismic signals have been developed since the late 1970s. The availability of seismic equipments withwide frequency and dynamic ranges since the early 1990s revealed a further variety of volcano‐seismic signals in oscillation periods longerthan a few seconds. Quantification of the sources of volcano‐seismic signals is crucial to achieving a better understanding of thephysical states and dynamics of magmatic and hydrothermal systems.
Introduction
Volcanoes occur in tectonically active regions of the earth where magmatic and hydrothermalfluids display complex interactions with volcanicrocks and the atmosphere. Volcano seismicity is the manifestation of such complex interactions occurring inside ofvolcanic edifices. Volcano seismology aims at achieving a better...
Abbreviations
- Moment tensor:
-
A point seismic source representation defined by the first-order moment of the equivalent body force or the stress glut. Slip on a fault as well as volumetric changes such as an isotropic expansion and tensile crack can be represented by the moment tensor.
- Waveform inversion:
-
An approach to estimate source mechanisms and locations of seismic events by finding the best fits between observed and synthesized seismograms.
- Autoregressive equation:
-
A difference form of the equation of motion of a linear dynamic system, which is a basic equation to determine the complex frequencies (frequencies and Q factors) of decaying harmonic oscillations in observed signals.
- Crack wave:
-
A dispersive wave generated by fluid-solid interactions in a crack. The phase velocity of the crack wave is smaller than the acoustic velocity of the fluid in the crack.
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Acknowledgment
I am deeply grateful to Masaru Nakano for numerous discussions on all the subjects presentedin this manuscript. I thank Yasuko Takei for constructive comments on the phenomenological source representation. Comments from Pablo Palacios, LucaD'Auria, Takeshi Nishimura, and an anonymous reviewer helped improve the manuscript. I used the Generic Mapping Tools (GMT) [130] in the preparation of figures.
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Appendices
Appendix A: Green's Functions
Here, I briefly explain the relationship between the displacement and Green's functions. To simplify the explanation, I use two‐dimensional equations in an infinite medium. The extension of the equations into three‐dimension is straightforward. Green's functions defined by Eq. (5) are explicitly written as
Note that \( { (G_{11},G_{21}) } \) represents the x 1 and x 2 components of the wavefield at \( { (\boldsymbol{x},t) } \), which is excited by the impulse in the x 1 direction applied at \( { \boldsymbol{x}=\boldsymbol{\eta} } \) and \( { t=\tau } \). Similarly, \( { (G_{12},G_{22}) } \) represents the x 1 and x 2 components of the wavefield excited by the impulse in the x 2 direction at \( { \boldsymbol{x}=\boldsymbol{\eta} } \) and at \( { t=\tau } \). Therefore, i and j in \( { G_{ij} } \) represent the component and direction of the impulse, respectively. To specify the relationship between the receiver and source, we use the notation \( { G_{ij}(\boldsymbol{x},t;\boldsymbol{\eta},\tau) } \). We obtain \( { G_{ij}(\boldsymbol{x},t;\boldsymbol{\eta},\tau) = G_{ij}(\boldsymbol{x},t-\tau;\boldsymbol{\eta},0) } \), since Green's functions are independent of the time of origin.
The displacement \( { u_i(\boldsymbol{x},t) } \) satisfies Eq. (3), which is explicitly written as
Equation (6) indicates that \( { u_i } \) is described by the following relationships:
We can verify that the displacement given by Eqs. (A7) and (A8) satisfies Eqs. (A5) and (A6) in the following way. We can rewrite Eqs. (A5) and (A6) as
We denote the right-hand side of Eq. (A9) as L 1, which is derived from Eqs. (A7) and (A8) as
This equation can be modified as follows:
From Eq. (A3), we find that the second integral of Eq. (A12) is zero. Therefore, Eq. (A12) from Eq. (A1) becomes
Similarly, we can show that the right-hand side of Eq. (A10) is equivalent to \( { f_2^S(\boldsymbol{x},t) } \), and thus the displacement in the forms of Eqs. (A7) and (A8) satisfies the equation of motion (A5) and (A6).
Appendix B: Moment Tensor for a Spherical Source
Let us consider the moment density tensors for three tensile cracks in the planes \( { \xi_1=0, \xi_2=0 } \), and \( { \xi_3=0 } \). If we sum and average these three tensors and assume that \( { [u_1]=[u_2]=[u_3]=D_s } \), we obtain the following moment density tensor:
This represents the moment density tensor for the isotropic expansion of a cubic element. Following Müller [89], we consider a spherical crack surface of radius R where a constant radial expansion \( { D_s = (d_s + \Delta_s) } \) occurs (Fig. A1). Here, the inner wall of the crack moves inward by \( { d_s } \) and the outer wall moves outward by \( { \Delta_s } \). The moment tensor of the spherical expansion may be obtained by integration of the moment density tensor (B1) over the surface R:
where \( { \Delta V } \) is the volume given as
The volume \( { 4\pi R^2 d_s } \) is caused by the inward motion, which compresses the sphere. The volume \( \Delta V_s = 4\pi R^2 \Delta_s \), on the other hand, is caused by the outward motion, which excites seismic waves in the region outside the sphere. \( { \Delta V_s } \) can be determined by solving an elastostatic boundary‐value problem in the following way. We assume an isotropic medium, and denote the regions inside and outside the sphere as regions 1 and 2, respectively. Since the motion is radial only, the equation of motion is given as e. g., [121]
and
where u is the radial displacement and \( { \sigma_{rr} } \), \( { \sigma_{\theta\theta} } \), and \( { \sigma_{\phi\phi} } \) are the stress components in the spherical coordinate. Equations (B4) and (B5) apply to both regions 1 and 2. Substituting Eq. (B5) into Eq. (B4) and setting \( { \rho (\partial^2 u_r/\partial t^2)=0 } \), we obtain the following static equilibrium equation:
This equation has two solutions: \( { u=ar } \) and \( { u=b/r^2 } \), where a and b are constants. The former is the interior solution for region 1 (\( { r \leq R } \)), and the latter is the exterior solution for region 2 (\( { r \geq R } \)). I denote the interior and exterior solutions as \( { u_i } \) and \( { u_e } \), respectively. The constants a and b are determined by the boundary conditions for the radial displacement and the continuity of the radial stress at \( { r = R } \):
where \( { \sigma_{rr}^i } \) and \( { \sigma_{rr}^e } \) are the radial stresses in regions 1 and 2, respectively. Accordingly, we obtain
Then, we obtain
Equation (B3) can be modified as
Finally, we obtain the moment tensor for the spherical expansion as
Appendix C: Moment Tensor for a Cylindrical Source
We consider the moment density tensors for two tensile cracks in the planes \( { \xi_1=0 } \) and \( { \xi_2=0 } \), where ξ1 and ξ2 are two horizontal axes. If we sum and average these two tensors and assume that \( { [u_1]=[u_2]=D_c } \), we obtain
This represents the moment density tensor for the expansion of a cubic element in the two horizontal directions. Let us consider a vertical cylinder of length L and radius R. The cylinder surface at radius R can be regarded as a cylindrical crack, where the radial expansion \( { D_c=(d_c+\Delta_c) } \) occurs (Fig. A2). The moment tensor for the radial expansion of the cylinder may be obtained by integration of the moment density tensor (C1) over the surface R:
where \( { \Delta V } \) is the volume given as
We obtain the static equilibrium equation for the radial expansion of the cylinder u as
This equation has internal and external solutions, which are given as \( { u=ar } \) and \( { u=b/r } \), respectively. The constants a and b are determined by the boundary conditions
where superscripts i and e denote the interior and exterior solutions. We then obtain
and
The moment tensor for the vertical cylinder is therefore given as
Let us rotate the vertical cylinder with axis orientation angles ϕ and θ (Fig. 8b). This can be done in the following steps: (1) fix the cylinder and (2) rotate the ξ1 and ξ2 axes around the ξ3 axis through an angle \( { -\phi } \), and (3) further rotate the ξ1 and ξ3 axes around the ξ2 axis through an angle \( { -\theta } \). The rotation matrix \( { \boldsymbol{R} } \) is given as
Using the matrix \( { \boldsymbol{R} } \), we obtain the moment tensor for a cylinder with the axis orientation angles \( { (\phi,\theta) } \) as
which leads to Eq. (34).
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Kumagai, H. (2009). Volcano Seismic Signals, Source Quantification of. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_583
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