# Physically motivated moment-tensor decomposition for mining-induced
seismicity

Alex Rigby1
1 Institute of Mine Seismology  10 Church Street  Kingston  7050  Australia

###### Abstract

Compared to existing schemes, the decomposition of moment tensors for mining-
induced seismic events into closing-crack and double-couple components has the
advantage that each can be interpreted in terms of a physical source process
(excavation convergence and slip/shear, respectively). Obviously, not every
moment moment tensor permits such a decomposition, and we translate existing
bounds on those that do to the Hudson source-type plot. For moment tensors
falling within these bounds, it has previously been noted that there will be a
infinite set of possible decompositions in general. We derive an implicit
equation defining this set and suggest physically motivated constraints that
can be used to select a single decomposition from it. Furthermore, for moment
tensors falling outside the source-type bounds, we present a simple geometric
scheme for determining the closest moment tensor within them (which can then
be decomposed). To demonstrate the methods developed in this paper, we apply
them to two catalogues of mining-induced seismicity.

##  1 Introduction

The moment tensor gives a compact description of the low-frequency radiation
of a seismic source. Interpretation of a given moment tensor in terms of a
physical source process can be aided by decomposing it into various
components. The standard approach is to first decompose the moment tensor into
isotropic and deviatoric components. This deviatoric component is then
typically further decomposed into some combination of double-couple (DC)
and/or compensated linear-vector dipole (CLVD) components. This second step is
not unique, with a number of methodologies having been published based on
different assumptions and aims (Julian et al., 1998; Vavryčuk, 2015). A widely
adopted procedure, which was first proposed by Knopoff & Randall (1970), is to
decompose into DC and CLVD components with aligned P- or T-axes. However,
interpretation of such decompositions can be difficult: The coupling of DC and
CLVD orientations means that even if there is a significant DC component, its
nodal planes need not align with the actual plane of slip/shear in the
rockmass. Furthermore, there is no obvious source process that would yield
either isotropic or CLVD content in isolation (at least in the context of
mining-induced seismicity).

As proposed by Ryder (1988), seismic sources at mines can be broadly split
into two main categories. The first of these is slip/shear in the rockmass,
which can be described by a DC moment tensor. The second is “unstable
crushing” failure in the rockmass surrounding a mining void. The low-frequency
radiation of these so-called crush-type sources, which is dominated by the
convergence of the surrounding rockmass into the excavation, can be described
by the closing-crack moment tensor (Malovichko & Rigby, 2022). As we discuss
in Section 2, this categorisation motivates the development of a moment-tensor
decomposition into DC and closing-crack components without any restriction on
their relative orientation. Although the physical motivations are different,
this is essentially equivalent mathematically to the general crack plus
double-couple (CDC) decomposition considered by Tape & Tape (2013) into DC and
opening/tensile-crack components. While physically attractive, there are
complications that must be overcome for CDC decomposition to be applied
routinely in practice.

The first complication is that a CDC decomposition cannot be determined for
every possible moment tensor. Bounds on the moment-tensor eigenvalues that
permit a decomposition have been given by Tape & Tape (2013) in terms of the
lune. In Section 3, we translate these bounds to the more commonly used
source-type plot of Hudson et al. (1989). For moment tensors falling outside
these bounds, we provide a simple geometric procedure in Section 5 for
determining the closest CDC tensor based on results of Tape & Tape (2012a,
2013).

The second complication is that there are, in general, infinite possible CDC
decompositions for a moment tensor falling within these bounds (Tape & Tape,
2013). Intuitively, this is a result of the CDC decomposition being a function
of seven parameters, while the moment tensor only provides us with six.
Selection of a single decomposition requires the inclusion of an additional
constraint. In Section 4, we propose several physically motivated methods for
imparting this constraint based on knowledge of excavation geometry, stress,
or geology at the source. Furthermore, we apply these methods in Section 6 to
two catalogues of mining-induced events, where we also show how the resulting
decompositions can be interpreted in terms of physical source processes.

##  2 Preliminaries

Mining-induced seismic events are typically classified as being slip/shear-
type or crush-type (Ryder, 1988). We will give an overview these source types
in this section and define a moment-tensor decomposition based on them.

###  2.1 Slip/shear-type sources

The creation of mining voids results in the concentration and redistribution
of stresses, which can result in slip along previously clamped
structures/weaknesses or even in the shearing of intact rock (Ortlepp, 1997).
Assuming this episode of slip occurs relatively far from significant
excavations, the radiation can be described by the traditional DC moment
tensor

$\mathbf{M}^{D}=M_{D}\mathbf{R}_{D}\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&0\\\
0&0&-1\end{array}\right)\mathbf{R}_{D}^{T},$ (1)

where
$M_{D}=\left\|\mathbf{M}^{D}\right\|/\sqrt{2}=\sqrt{\sum_{ij}M_{ij}^{D}/2}$ is
the scalar moment of $\mathbf{M}^{D}$ and $\mathbf{R}_{D}$ is a rotation
matrix. The orientation of $\mathbf{M}^{D}$ can be described in terms of the
strike $\phi_{s}$ and dip $\delta$ of slip plane and the rake angle $\lambda$,
which gives the direction of slip in this plane (Aki & Richards, 2002).
Explicitly, in a south-west-down coordinate system, the components of
$\text{$\mathbf{M}^{D}$}(\alpha_{D},\phi_{s},\delta,\gamma)$ can be written as

$\displaystyle M_{\mathrm{SS}}^{D}$ $\displaystyle=M_{D}\left(-\sin\delta\sin
2\phi_{s}\cos\lambda-\sin 2\delta\sin\lambda\sin^{2}\phi_{s}\right),$
$\displaystyle M_{\mathrm{WW}}^{D}$ $\displaystyle=M_{D}\left(\sin\delta\sin
2\phi_{s}\cos\lambda-\sin 2\delta\sin\lambda\cos^{2}\phi_{s}\right),$
$\displaystyle M_{\mathrm{DD}}^{D}$ $\displaystyle=M_{D}\sin
2\delta\sin\lambda,$ $\displaystyle M_{\mathrm{SW}}^{D}$
$\displaystyle=M_{D}\left(\sin\delta\cos\lambda\cos 2\phi_{s}+\sin
2\delta\sin\lambda\sin\phi_{s}\cos\phi_{s}\right),$ $\displaystyle
M_{\mathrm{SD}}^{D}$ $\displaystyle=M_{D}\left(\sin\lambda\sin\phi_{s}\cos
2\delta+\cos\delta\cos\lambda\cos\phi_{s}\right),$ $\displaystyle
M_{\mathrm{WD}}^{D}$ $\displaystyle=M_{D}\left(-\sin\lambda\cos
2\delta\cos\phi_{s}+\sin\phi_{s}\cos\delta\cos\lambda\right).$ (2)

###  2.2 Crush-type sources

Concentration of stress around mining voids can also lead to “unstable
crushing” failure in the surrounding rockmass (Ryder, 1988). In such cases,
failure is not concentrated along some predominant plane or structure, but is
rather composed of a number of small-scale episodes of shear/tensile rupture.
Importantly, the low-frequency radiation (that is, at wavelengths larger than
the diameter of the excavation) of a crush-type source is not dominated by
this small-scale behaviour of the rockmass (which can be expected to radiate
in the multi-$\mathrm{kHz}$ range). Instead, it is controlled by the overall
convergence of the surrounding rockmass into the excavation (Malovichko, 2020;
Malovichko & Rigby, 2022).

An example of such a source is shown in Fig. 1, where failure of a volume of
rock occurs at the highly stressed face of a tabular (planar) stope. In this
case, the low-frequency radiation is dominated by the convergence of the
footwall and hanging wall into the stope. Given the excavation’s geometry,
this radiation can be described in terms of the moment tensor for a closing
crack

$\mathbf{M}^{K}=\alpha_{\nu}M_{K}\mathbf{R}_{K}\left(\begin{array}[]{ccc}-\nu&0&0\\\
0&-\nu&0\\\ 0&0&\nu-1\end{array}\right)\mathbf{R}_{K}^{T},$ (3)

where $\alpha_{\nu}=2/\sqrt{4\nu^{2}+2(\nu-1)^{2}}$, $M_{K}$ is the scalar
moment, $\mathbf{R}_{K}$ is an rotation matrix, and $\nu$ is the Poisson’s
ratio of the rockmass (Wong & McGarr, 1991; Walter et al., 1997; Ford et al.,
2008). The orientation of $\mathbf{M}^{K}$ is defined entirely by that of its
P-axis, which will lie orthogonal to the plane of the excavation. In terms of
the azimuth $\phi$ and plunge $\theta$ of the P-axis, we can write the
components of $\mathbf{M}^{K}(M_{K},\phi,\theta)$ as

$\displaystyle M_{\mathrm{SS}}^{K}$
$\displaystyle=\alpha_{\nu}M_{K}\left[(2\nu-1)\cos^{2}\phi\cos^{2}\theta-\nu\right],$
$\displaystyle M_{\mathrm{WW}}^{K}$
$\displaystyle=\alpha_{\nu}M_{K}\left[(2\nu-1)\sin^{2}\phi\cos^{2}\theta-\nu\right],$
$\displaystyle M_{\mathrm{DD}}^{K}$
$\displaystyle=\alpha_{\nu}M_{K}\left[(2\nu-1)\sin^{2}\theta-\nu\right],$
$\displaystyle M_{\mathrm{SW}}^{K}$
$\displaystyle=\alpha_{\nu}M_{K}(2\nu-1)\sin\phi\cos\phi\cos^{2}\theta,$
$\displaystyle M_{\mathrm{SD}}^{K}$
$\displaystyle=\alpha_{\nu}M_{K}(1-2\nu)\cos\phi\sin\theta\cos\theta,$
$\displaystyle M_{\mathrm{WD}}^{K}$
$\displaystyle=\alpha_{\nu}M_{K}(1-2\nu)\sin\phi\sin\theta\cos\theta.$ (4)

(a)  (b)

Figure 1: (a) Plan view of a horizontal tabular stope. The hashed region
indicates a volume of dynamically fractured rock at the face. (b) Section view
showing the resulting convergence/closure of the footwall (FW) and hanging
wall (HW) to give the post-event stope profile (solid line). The pre-event
profile of the stope is given by the overlaid dotted line. Also shown is the
expected P-axis direction.

Usefully, the closing-crack model also gives a good approximation of the
radiation from crush-type sources associated with different excavation
geometries. This includes dynamic stress fracturing around tunnels as
considered by Malovichko & Rigby (2022). An example of such failure in the
back (roof) and floor of a horizontally loaded tunnel is shown in Fig. 2. More
generally, this failure will lie primarily in regions of high stress
concentration, which will be in the direction orthogonal to maximum in-plane
loading $\sigma_{\max}$. Radiation is largely controlled by resulting
convergence of the surrounding rockmass into the excavation, and the P-axis
will be approximately co-oriented with the direction of $\sigma_{\max}$.

Figure 2: Section view of a horizontally loaded tunnel that has undergone
dynamic stress fracturing in the back and floor as indicated by the hashed
regions. The pre- and post-event profiles of the tunnel are given by the solid
and dotted lines, respectively. Also shown is the expected P-axis direction,
which corresponds to the direction of maximum in-plane loading
$\sigma_{\max}$.

###  2.3 Source model

The slip/shear- and crush-type sources outlined above can be considered as two
extremes of a spectrum of possible source processes. In practice, there are
sources that combine aspects of both. For example, stress fracturing around an
excavation will cause a redistribution of stress that may result in near-
simultaneous slip along some nearby structure. Conversely, the dynamic stress
waves from slip along a structure may result in the triggering of stress
fracturing around a nearby excavation. More generally, the response
(convergence) of an excavation near an episode of slip/shear will contribute
to the resulting radiation even if there isn’t directly induced/triggered
failure around it (Sileny et al., 2001). An example of where this can be
expected is shown in Fig. 3, where lobes of high shear stress induce shear
failure ahead of the face of a tabular stope in either the footwall or hanging
wall.

Figure 3: Section view of shearing ahead of the face of a tabular stope, where
there are lobes of high shear stress (dashed lines). The pre- and post-event
stope profiles are given by the solid and dotted lines, respectively.

We can expect the moment tensor for a source composed of an episode of
slip/shear and/or excavation convergence to be a sum

$\mathbf{M}=\mathbf{M}^{K}+\mathbf{M}^{D}$ (5)

of closing-crack and DC moment tensors as defined in eqs. (3) and (1),
respectively. We note that mathematically, this is almost identical to the
general crack plus double-couple (CDC) model considered by (Tape & Tape,
2013), with the only difference being that they considered a tensile/opening
crack.

##  3 Bounds

We can hope to gain insight into the nature of a source by decomposing an
inferred moment tensor into the crack and DC components of eq. (5). Obviously,
not every possible moment tensor can be decomposed in this manner. Bounds on
those that can have previously been derived by Tape & Tape (2013), which we
present here with slight modification to account for our use of a closing
rather than opening crack. We also translate these bounds to the source-type
plot of Hudson et al. (1989).

###  3.1 Lune

We write the eigenvalues corresponding to the T-, B-, and P-axes of a given
moment tensor $\mathbf{M}$ as the triple
$\boldsymbol{\Lambda}=(\lambda_{1},\lambda_{2},\lambda_{3})$ . The set of all
possible triples is the wedge

$\mathbb{W}=\\{\boldsymbol{\Lambda}\in\mathbb{R}^{3}:\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\\},$
(6)

and its projection onto the unit sphere $\mathbb{S}$ is the fundamental lune
$\mathbb{L}=p_{\mathbb{S}}(\mathbb{W})$, where

$p_{\mathbb{S}}(\boldsymbol{\Lambda})=\frac{\boldsymbol{\Lambda}}{\left\|\boldsymbol{\Lambda}\right\|}=\frac{\boldsymbol{\Lambda}}{\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}}}=\hat{\boldsymbol{\Lambda}}.$
(7)

$\mathbf{M}$ will have a CDC decomposition if its lune point
$\hat{\boldsymbol{\Lambda}}(\mathbf{M})$ falls within the region
$\mathbb{L}_{\mathrm{CDC}}\subset\mathbb{L}$ shown in Fig. 4a (Tape & Tape,
2013). The vertices of this region are the points

$\displaystyle\hat{\boldsymbol{\Lambda}}^{D}$
$\displaystyle=\hat{\boldsymbol{\Lambda}}(\mathbf{M}^{D})=p_{\mathbb{S}}(1,0,-1),$
$\displaystyle\hat{\boldsymbol{\Lambda}}^{K}$
$\displaystyle=\hat{\boldsymbol{\Lambda}}(\mathbf{M}^{K})=p_{\mathbb{S}}(-\nu,-\nu,\nu-1),$
$\displaystyle\hat{\boldsymbol{\Lambda}}^{1}$
$\displaystyle=p_{\mathbb{S}}(1-3\nu,\nu-1,\nu-1),$
$\displaystyle\hat{\boldsymbol{\Lambda}}^{2}$
$\displaystyle=p_{\mathbb{S}}(-2\nu,-1,-1).$ (8)

The edges of $\mathbb{L}_{\mathrm{CDC}}$ are segments of great circles joining
these vertices, which can be defined by planes through the origin with the
unit normal vectors

$\displaystyle\hat{\mathbf{n}}^{1D}$
$\displaystyle=p_{\mathbb{S}}(\hat{\boldsymbol{\Lambda}}^{1}\times\hat{\boldsymbol{\Lambda}}^{D})=p_{\mathbb{S}}(1-\nu,-2\nu,1-\nu),$
$\displaystyle\hat{\mathbf{n}}^{DK}$
$\displaystyle=p_{\mathbb{S}}(\hat{\boldsymbol{\Lambda}}^{D}\times\hat{\boldsymbol{\Lambda}}^{K})=p_{\mathbb{S}}(-\nu,1,-\nu),$
$\displaystyle\hat{\mathbf{n}}^{2K}$
$\displaystyle=p_{\mathbb{S}}(\hat{\boldsymbol{\Lambda}}^{2}\times\hat{\boldsymbol{\Lambda}}^{K})=p_{\mathbb{S}}(1,-\nu,-\nu).$
(9)

In terms of these normals, the CDC region is simply

$\mathbb{L}_{\mathrm{CDC}}=\\{\hat{\boldsymbol{\Lambda}}\in\mathbb{L}:\hat{\mathbf{n}}^{1D}\cdot\hat{\boldsymbol{\Lambda}}\leq
0,\hat{\mathbf{n}}^{DK}\cdot\hat{\boldsymbol{\Lambda}}\leq
0,\hat{\mathbf{n}}^{2K}\cdot\hat{\boldsymbol{\Lambda}}\geq 0\\}.$ (10)

(a)  (b)

Figure 4: (a) Region on the fundamental lune for which corresponding moment
tensors permit a CDC decomposition with $\nu=0.25$. (b) Corresponding region
on the Hudson plot.

###  3.2 Hudson

While there are advantages to plotting source types on the lune (Tape & Tape,
2012a, b, 2019), the plot of Hudson et al. (1989) shown in Fig. 4b sees more
frequent use for historical reasons. The horizontal and vertical coordinates
of a moment tensor on this plot are related to its eigenvalues by

$\displaystyle u$
$\displaystyle=-\frac{2}{3}\frac{1}{\lambda_{\max}}(\lambda_{1}+\lambda_{3}-2\lambda_{2}),$
$\displaystyle v$
$\displaystyle=\frac{1}{3}\frac{1}{\lambda_{\max}}(\lambda_{1}+\lambda_{2}+\lambda_{3}),$
(11)

where $\lambda_{\max}=\max(|\lambda_{1}|,|\lambda_{3}|)$ (Vavryčuk, 2015). For
values of $\nu\in[0,0.5]$, $\lambda_{\max}=-\lambda_{3}$ for the four lune
points given in eq. (8). It follows that the corresponding points on the
Hudson plot are

$\displaystyle(u_{D},v_{D})$ $\displaystyle=(0,0),$
$\displaystyle(u_{K},v_{K})$
$\displaystyle=\left(\frac{2}{3}\frac{1-2\nu}{1-\nu},-\frac{1}{3}\frac{1+\nu}{1-\nu}\right),$
$\displaystyle(u_{1},v_{1})$
$\displaystyle=\left(-\frac{4}{3}\frac{1-2\nu}{1-\nu},-\frac{1}{3}\frac{1+\nu}{1-\nu}\right),$
$\displaystyle(u_{2},v_{2})$
$\displaystyle=\left(-\frac{2}{3}\left[1-2\nu\right],-\frac{2}{3}\left[1+\nu\right]\right).$
(12)

Using eq. (11), it can be shown that a great circle arc
$\mathbf{n}\cdot\boldsymbol{\Lambda}=0$ on the lune for some
$\mathbf{n}=(a,b,c)$ becomes straight-line segments $h(u,v)=0$ on the Hudson
plot, where

$h(u,v)=\begin{cases}v-\frac{1}{2(b+2c)}\left[(c-b)u+2(c-a)\right]&\mathrm{if}\,v>\frac{1}{4}u,\\\
v-\frac{1}{2(b+2a)}\left[(a-b)u+2(c-a)\right]&\mathrm{otherwise}.\end{cases}$
(13)

The line $v=u/4$ is is dotted in Fig. 4 and corresponds to
$\lambda_{1}=-\lambda_{3}$. For the segments of interest,
$\lambda_{1}\leq-\lambda_{3}$ (that is, $v\leq u/4$), meaning that the arcs of
defined by eq. (9) become

$\displaystyle h_{1D}(u,v)$ $\displaystyle=v-\frac{\nu+1}{4(1-2\nu)}u=0,$
$\displaystyle h_{DK}(u,v)$ $\displaystyle=v+\frac{\nu+1}{2(1-2\nu)}u=0,$
$\displaystyle h_{2K}(u,v)$
$\displaystyle=v-\frac{\nu+1}{2(2-\nu)}u+\frac{\nu+1}{2-\nu}=0.$ (14)

It follows that a moment tensor $\mathbf{M}$ will have a CDC decomposition if
and only if its corresponding point $(u,v)$ on the Hudson plot satisfies
$h_{1D}(u,v)\leq 0$, $h_{DK}(u,v)\leq 0$, and $h_{2K}(u,v)\geq 0$.

##  4 Decomposition approach

As previously noted by Tape & Tape (2013), a moment tensor falling within the
region described in Section 3 will have infinite CDC decompositions in
general. That this is the case is not surprising given that (assuming a fixed
$\nu$) describing the crack and DC tensors individually requires a total of
seven parameters ($M_{K}$, $\phi$, $\theta$ for $\mathbf{M}^{K}$ and $M_{D}$,
$\phi_{s}$, $\delta$, $\lambda$ for $\mathbf{M}^{D}$), while their sum
$\mathbf{M}$ requires only six. In this section, we provide a method for
generating a subset of the infinite possible decompositions and propose
physically motivated approaches for selecting a decomposition from this set.

###  4.1 Generation

Supposing that we can decompose a given moment tensor $\mathbf{M}$ according
to eq. (5), we can constrain the parameters of the crack component
$\mathbf{M}^{K}(M_{K},\phi,\theta)$ in terms of invariants of the remaining DC
component $\mathbf{M}^{D}=\mathbf{M}-\mathbf{M}^{K}$. The first of these
constraints is that $\mathbf{M}^{D}$ must have zero trace, yielding

$\displaystyle\mathrm{tr}(\mathbf{M}^{K})$
$\displaystyle=\mathrm{tr(}\mathbf{M})$ $\displaystyle\Rightarrow M_{K}$
$\displaystyle=-\frac{\mathrm{tr(}\mathbf{M})}{(\nu+1)\alpha_{v}}.$ (15)

The second constraint is that the determinant of $\mathbf{M}^{D}$ must also be
zero, which gives the implicit equation

$d(\phi,\theta)=\det\left[\mathbf{M}-\mathbf{M}^{K}\left(-\frac{\mathrm{tr(}\mathbf{M})}{(\nu+1)\alpha_{v}},\phi,\theta\right)\right]=0$
(16)

defining possible P-axis orientations of $\mathbf{M}^{K}$.

A simple way of determining a set of crack P-axis orientations that
approximately satisfy eq. (16) is to first calculate $d(\phi,\theta)$ for
gridded values of $\phi$ and $\theta$ and then approximate the
$d(\phi,\theta)=0$ contour using the marching-squares algorithm (Lorensen &
Cline, 1987). The results of this procedure for a randomly generated CDC
moment tensor

$\text{$\mathbf{M}$}=\left(\begin{array}[]{ccc}-0.56&0.21&-0.71\\\
0.21&-0.52&0.28\\\ -0.71&0.28&-0.39\end{array}\right)$ (17)

are shown in Fig. 5a, with the $d(\phi,\theta)=0$ contour shown in red. This
contour is also plotted in a lower-hemisphere stereographic projection
(stereonet) in Fig. 5b. For each valid crack P-axis orientation, we can
determine the nodal planes of the remaining DC component [a procedure for
doing this is given by Gasperini & Vannucci (2003)], which we present in terms
of their poles in Fig. 5c. Fig. 5d shows the beachball diagrams of a selected
decomposition that corresponds to the marked orientations of the crack P-axis
and nodal-plane poles in Figs. 5b and 5c, respectively. Normalised scalar
moments $M_{K}/M$ and $M_{D}/M$ are listed above the crack and DC components,
where $M$ is the scalar moment of $\mathbf{M}$. Note that while $M_{K}$ is
fixed according to eq. (15), $M_{D}$ will vary between different
decompositions.

(a)  (b)  (c)  (d)

Figure 5: (a) Values of $d(\phi,\theta)$ for $\mathbf{M}$ given in eq. (17),
with $d(\phi,\theta)=0$ in red. (b) Stereonet of valid crack P-axes. (c)
Setereonet of corresponding DC nodal-plane poles. (d) Beachball diagrams for a
selected decomposition $\mathbf{M}=\mathbf{M}^{K}+\mathbf{M}^{D}$ as marked in
(b) and (c) with black crosses. The radii of these beachballs is proportional
to the listed scalar moment values, which have been normalised to that of
$\mathbf{M}$.

###  4.2 Selection

Depending on the information available about a given event, there are a number
of ways in which a single decomposition could be selected:

  * •

If there is knowledge about the excavation geometry and stress state at/near
the source, we can determine the expected P-axis orientation of the crack
component (see Section 2.2). The decomposition with a crack P-axis closest to
this can then be selected (that is, the one minimising the angle to the
expected axis).

  * •

Alternatively, there may be information from geological mapping, seismic data,
or some other means available about geological structures at/near the source.
This can then be used to determine an expected nodal-plane orientation of the
DC component. The decomposition with a DC component that has a nodal plane
most similar to this expectation can then be selected. There are a number of
ways in which this similarity could be quantified; a simple metric is the
angle between the poles.

  * •

In the absence of information from which the expected orientation of either
the crack or DC can be inferred, we can constrain their relative orientations.
In particular, we can select the decomposition that for which the two
components have the nearest P-axis orientations. This is the method used for
selecting the decomposition shown in Fig. 5d. Alternatively, if a source is
expected to be essentially pure crush- or slip-type, then we could select
decomposition whose DC component has the largest or smallest scalar moment
$M_{D}$, respectively.

##  5 Approximate decomposition

A moment tensor $\mathbf{M}$ falling outside the CDC region defined in Section
3 cannot be decomposed directly following the procedure of Section 4. However,
we can determine and decompose the closest CDC moment tensor
$\mathbf{M}^{\mathrm{CDC}}$ to $\mathbf{M}$, which we take to be the one
minimising $\left\|\mathbf{M}-\mathbf{M}^{\mathrm{CDC}}\right\|$. In other
words, $\mathbf{M}^{\mathrm{CDC}}$ minimises the scalar moment of the
remaining non-CDC component
$\mathbf{M}^{\mathrm{NCDC}}=\mathbf{M}-\mathbf{M}^{\mathrm{CDC}}$. We outline
a relatively simple procedure for determining $\mathbf{M}^{\mathrm{CDC}}$ in
this section.

###  5.1 On the lune

As shown by Tape & Tape (2012a),
$\left\|\mathbf{M}-\mathbf{M}^{\mathrm{CDC}}\right\|$ is minimised when
$\mathbf{M}^{\mathrm{CDC}}$ has the same eigenframe as $\mathbf{M}$, in which
case it equal to the distance
$\left\|\boldsymbol{\Lambda}-\boldsymbol{\Lambda}^{\mathrm{CDC}}\right\|$
between the corresponding eigenvalue triples. This distance satisfies

$\displaystyle\left\|\boldsymbol{\Lambda}-\boldsymbol{\Lambda}^{\mathrm{CDC}}\right\|^{2}$
$\displaystyle=\left\|\boldsymbol{\Lambda}\right\|^{2}+\left\|\boldsymbol{\Lambda}^{\mathrm{CDC}}\right\|^{2}$
$\displaystyle-2\left\|\boldsymbol{\Lambda}\right\|\left\|\boldsymbol{\Lambda}^{\mathrm{CDC}}\right\|\cos\angle(\hat{\boldsymbol{\Lambda}},\hat{\boldsymbol{\Lambda}}^{\mathrm{CDC}}),$
(18)

where
$\angle(\hat{\boldsymbol{\Lambda}},\hat{\boldsymbol{\Lambda}}^{\mathrm{CDC}})=\angle(\boldsymbol{\Lambda},\boldsymbol{\Lambda}^{\mathrm{CDC}})$
is the angle between the triples. For a given angle, this is minimised for

$\left\|\boldsymbol{\Lambda}^{\mathrm{CDC}}\right\|=\left\|\boldsymbol{\Lambda}\right\|\cos\angle(\hat{\boldsymbol{\Lambda}},\hat{\boldsymbol{\Lambda}}^{\mathrm{CDC}}),$
(19)

which gives

$\left\|\boldsymbol{\Lambda}-\boldsymbol{\Lambda}^{\mathrm{CDC}}\right\|=\left\|\boldsymbol{\Lambda}\right\|\sin\angle(\hat{\boldsymbol{\Lambda}},\hat{\boldsymbol{\Lambda}}^{\mathrm{CDC}}).$
(20)

Noting that
$\angle(\hat{\boldsymbol{\Lambda}},\hat{\boldsymbol{\Lambda}}^{\mathrm{CDC}})\in[0,\pi/2]$,
this means that minimising
$\left\|\boldsymbol{\Lambda}-\boldsymbol{\Lambda}^{\mathrm{CDC}}\right\|$
reduces finding the closest point
$\hat{\boldsymbol{\Lambda}}^{\mathrm{CDC}}\in\mathbb{L}_{\mathrm{CDC}}$ to
$\hat{\boldsymbol{\Lambda}}$.

###  5.2 Cases

There are five cases to consider in determining
$\hat{\boldsymbol{\Lambda}}^{\mathrm{CDC}}$, which correspond to the partition
of $\mathbb{L}$ into the regions shown in Fig. 6a. In addition to
$\mathbb{L}_{\mathrm{CDC}}$ as defined in eq. (10), the remaining four regions
are

$\displaystyle\mathbb{L}_{D}$
$\displaystyle=\\{\hat{\boldsymbol{\Lambda}}\in\mathbb{L}:\hat{\mathbf{n}}^{1D\perp}\cdot\hat{\boldsymbol{\Lambda}}\geq
0,\hat{\mathbf{n}}^{DK\perp}\cdot\hat{\boldsymbol{\Lambda}}\geq 0\\},$
$\displaystyle\mathbb{L}_{DK}$
$\displaystyle=\\{\hat{\boldsymbol{\Lambda}}\in\mathbb{L}:\hat{\mathbf{n}}^{DK\perp}\cdot\hat{\boldsymbol{\Lambda}}<0,\hat{\mathbf{n}}^{DK}\cdot\hat{\boldsymbol{\Lambda}}>0\\},$
$\displaystyle\mathbb{L}_{2K}$
$\displaystyle=\\{\hat{\boldsymbol{\Lambda}}\in\mathbb{L}:\hat{\mathbf{n}}^{2K}\cdot\hat{\boldsymbol{\Lambda}}>0\\},$
$\displaystyle\mathbb{L}_{1D}$
$\displaystyle=\\{\hat{\boldsymbol{\Lambda}}\in\mathbb{L}:\hat{\mathbf{n}}^{1D\perp}\cdot\hat{\boldsymbol{\Lambda}}<0,\hat{\mathbf{n}}^{1D}\cdot\hat{\boldsymbol{\Lambda}}>0\\}.$
(21)

where the normals

$\displaystyle\hat{\mathbf{n}}^{1D\perp}$
$\displaystyle=p_{\mathbb{S}}(\hat{\mathbf{n}}^{1D}\times\hat{\boldsymbol{\Lambda}}^{D})=p_{\mathbb{S}}(\nu,1-\nu,\nu),$
$\displaystyle\hat{\mathbf{n}}^{DK\perp}$
$\displaystyle=p_{\mathbb{S}}(\hat{\boldsymbol{\Lambda}}^{D}\times\hat{\mathbf{n}}^{DK})=p_{\mathbb{S}}(1,2\nu,1).$
(22)

define great circles perpendicular at $\hat{\boldsymbol{\Lambda}}^{D}$ to
those corresponding to $\hat{\mathbf{n}}^{1D}$ and $\hat{\mathbf{n}}^{DK}$ ,
respectively. If $\hat{\boldsymbol{\Lambda}}\in\mathbb{L}_{\mathrm{CDC}}$,
then we trivially have
$\hat{\boldsymbol{\Lambda}}^{\mathrm{CDC}}=\hat{\boldsymbol{\Lambda}}$. Also
simple is the case of $\hat{\boldsymbol{\Lambda}}\in\mathbb{L}_{D}$, where
$\hat{\boldsymbol{\Lambda}}^{\mathrm{CDC}}=\hat{\boldsymbol{\Lambda}}^{D}$. If
$\hat{\boldsymbol{\Lambda}}\in\mathbb{L}_{DK}$, then
$\hat{\boldsymbol{\Lambda}}^{\mathrm{CDC}}$ will lie on the great circle
defined by the plane with normal $\hat{\mathbf{n}}^{DK}$. Given that
$\angle(\hat{\boldsymbol{\Lambda}},\hat{\boldsymbol{\Lambda}}^{\mathrm{CDC}})<\pi/2$
in this region, we can find this $\hat{\boldsymbol{\Lambda}}^{\mathrm{CDC}}$
by projecting onto this plane and then onto the unit sphere, giving
$\hat{\boldsymbol{\Lambda}}^{\mathrm{CDC}}=p_{\mathbb{S}}(\hat{\boldsymbol{\Lambda}}-[\hat{\boldsymbol{\Lambda}}\cdot\hat{\mathbf{n}}^{DK}]\hat{\mathbf{n}}^{DK})$.
The remaining two cases can be treated in the same way. In summary,

$\hat{\boldsymbol{\Lambda}}^{\mathrm{CDC}}=\begin{cases}\hat{\boldsymbol{\Lambda}}&\mathrm{if\,}\hat{\boldsymbol{\Lambda}}\in\mathbb{L}_{\mathrm{CDC}},\\\
\hat{\boldsymbol{\Lambda}}^{D}&\mathrm{if\,}\hat{\boldsymbol{\Lambda}}\in\mathbb{L}_{D},\\\
p_{\mathbb{S}}(\hat{\boldsymbol{\Lambda}}-[\hat{\boldsymbol{\Lambda}}\cdot\hat{\mathbf{n}}^{DK}]\hat{\mathbf{n}}^{DK})&\mathrm{if\,}\hat{\boldsymbol{\Lambda}}\in\mathbb{L}_{DK},\\\
p_{\mathbb{S}}(\hat{\boldsymbol{\Lambda}}-[\hat{\boldsymbol{\Lambda}}\cdot\hat{\mathbf{n}}^{2K}]\hat{\mathbf{n}}^{2K})&\mathrm{if\,}\hat{\boldsymbol{\Lambda}}\in\mathbb{L}_{2K},\\\
p_{\mathbb{S}}(\hat{\boldsymbol{\Lambda}}-[\hat{\boldsymbol{\Lambda}}\cdot\hat{\mathbf{n}}^{1D}]\hat{\mathbf{n}}^{1D})&\mathrm{if\,}\hat{\boldsymbol{\Lambda}}\in\mathbb{L}_{1D}.\end{cases}$
(23)

(a)  (b)

Figure 6: (a) Partition of the lune into the regions defined in eqs. (10) and
(21) for $\nu=0.25$. (b) Corresponding regions on the Hudson plot.

For completeness, we also show the five regions considered on the Hudson plot
in Fig. 6b. The straight-line segments corresponding to the normals of eq.
(22) are

$\displaystyle h_{1D\perp}(u,v)$ $\displaystyle=v+\frac{1-2\nu}{2(\nu+1)}u=0,$
$\displaystyle h_{DK\perp}(u,v)$ $\displaystyle=v-\frac{1-2\nu}{4(\nu+1)}u=0.$
(24)

###  5.3 Contours

Rearranging eq. (20) gives the normalised scalar moment

$\gamma_{\mathrm{NCDC}}=\frac{\left\|\mathbf{M}^{\mathrm{NCDC}}\right\|}{\left\|\mathbf{M}\right\|}=\frac{\left\|\boldsymbol{\Lambda}^{\mathrm{NCDC}}\right\|}{\left\|\boldsymbol{\Lambda}\right\|}=\sin\angle(\hat{\boldsymbol{\Lambda}},\hat{\boldsymbol{\Lambda}}^{\mathrm{NCDC}}),$
(25)

which quantifies the amount of non-CDC content in a given moment tensor. We
plot isolines of constant $\gamma_{\mathrm{NCDC}}$ on the lune in Fig. 7a.
Geometrically, a given isoline is composed of small circle sections in the
regions defined by eq. (21). In $\mathbb{L}_{D}$, this circle is the points at
a constant angle $\sin^{-1}(\gamma_{\mathrm{CDC}})$ from
$\hat{\boldsymbol{\Lambda}}^{D}$. In $\mathbb{L}_{DK}$, $\mathbb{L}_{2K}$, and
$\mathbb{L}_{1D}$ they are at a constant angle
$\pi/2-\sin^{-1}(\gamma_{\mathrm{CDC}})$ from the normals
$\hat{\mathbf{n}}^{DK}$, $-\hat{\mathbf{n}}^{2K}$, and
$\hat{\mathbf{n}}^{1D}$, respectively. Alternatively, we can define an isoline
in terms of plane intersections as

$\begin{cases}\hat{\boldsymbol{\Lambda}}^{D}\cdot\hat{\boldsymbol{\Lambda}}-\sqrt{1-\gamma_{\mathrm{NCDC}}^{2}}=0&\mathrm{if}\,\hat{\boldsymbol{\Lambda}}\in\mathbb{L}_{D},\\\
\hat{\mathbf{n}}^{DK}\cdot\hat{\boldsymbol{\Lambda}}-\gamma_{\mathrm{NCDC}}=0&\mathrm{if}\,\hat{\boldsymbol{\Lambda}}\in\mathbb{L}_{DK},\\\
\hat{\mathbf{n}}^{2K}\cdot\hat{\boldsymbol{\Lambda}}+\gamma_{\mathrm{NCDC}}=0&\mathrm{if}\,\hat{\boldsymbol{\Lambda}}\in\mathbb{L}_{2K},\\\
\hat{\mathbf{n}}^{1D}\cdot\hat{\boldsymbol{\Lambda}}-\gamma_{\mathrm{NCDC}}=0&\mathrm{if}\,\hat{\boldsymbol{\Lambda}}\in\mathbb{L}_{1D}.\end{cases}$
(26)

Fig. 7b shows these isolines translated to the Hudson plot using eq. (11).
They are significantly more complicated than those on the lune as small
circles do not map to straight-line segments like great circles do.

(a)  (b)

Figure 7: (a) Isolines of constant non-CDC amount as defined by eq. (25) on
the lune for $\nu=0.25$. (b) Same on the Hudson plot.

##  6 Case studies

The methods outlined in Sections 4 and 5 provide us with a relatively simple
means of determining a CDC decomposition for a given moment tensor. In this
section, we demonstrate these methods on moment tensors inferred from real
seismic events recorded at two mines.

###  6.1 Case A

The first catalogue we consider comes from a South African mine and is
summarised in Fig. 8. It is composed of 43 events with moment magnitude
$M_{W}\geq 1.0$ that were recorded in the first half of 2022 near the two
advancing fronts of a tabular stope. This stope, the footprint of which is
shown in Figs. 8a-8c, has a height of approximately $1\,\mathrm{m}$, strike of
$60^{\circ}$, and dip of $22^{\circ}$. Also shown is the spatial distribution
of the events, which are rendered as three-dimensional beachballs. We have
labeled the clusters corresponding to the south-western and north-eastern
fronts as A and B, respectively. Note that the spread of event locations in
the direction normal to the stope is mostly an artefact of the configuration
of the mine’s seismic array, which lies largely in the plane of the stope. As
shown in Fig. 8d, the moment tensors for events in both clusters lie either in
or near the CDC region (for $\nu=0.25$) described in Section 3. Their
orientations are reasonably consistent within each cluster as shown in Figs.
8e-8g.

(a)  (b)

(c)  (d)

(e)  (f)

(g)

Figure 8: Case study A data summary. (a) View looking at an azimuth of
$330^{\circ}$ for case. The footprint of the stope is shown in gray, with
darker areas being mined in the period of interest. Seismic events are
rendered as beachballs centered at their inferred hypocenter, with red dipoles
indicating the P-axis direction. Events are grouped spatially into clusters A
and B. (b) Plan view. (c) View at an azimuth of $240^{\circ}$, with the
direction of _in situ_ maximum a principal compressive stress $\sigma_{1}$
shown. (d) Hudson source-type plot with colouring according to cluster. The
contours indicate amount of non-CDC content as defined in Sec 5.3. (e)-(g)
Stereonet plots showing the P-, B-, and T-axis orientation, respectively, with
colouring according to cluster.

To aid in their interpretation, we have determined a CDC decomposition for
each moment tensor $\mathbf{M}$ in three steps: First, the closest CDC moment
tensor $\mathbf{M}^{\mathrm{CDC}}$ to $\mathbf{M}$ is found following the
procedure of Section 5.2. A set of possible decompositions of
$\mathbf{M}^{\mathrm{CDC}}$ is then determined as outlined in Section 4.1.
Lastly, we select a single decomposition from this set using the first
criteria given in Section 4.2; in particular, we choose the decomposition for
which the closing-crack component’s P-axis is as close as possible to being
orthogonal to the plane of the stope, which has a normal vector with an
azimuth of $330^{\circ}$ and plunge of $68^{\circ}$.

The closing-crack P-axis orientations for cluster A are shown relative to the
stope’s normal vector in Fig. 9a. The greatest deviation from this normal is
by approximately $30^{\circ}$ degrees. However, this is for a moment tensor
whose decomposition has a small closing-crack component; in particular, it has
$M_{K}/M=0.24$. As such, it is reasonable to interpret the source of this
event as being a near-pure episode of slip/shear, with the closing-crack
component being poorly constrained and possibly just an artefact of the
inversion/decomposition procedure. We have rendered the P-axes for all
decompositions with $M_{K}/M<0.3$ translucently to indicate that they should
be interpreted with caution. It can be seen in Fig. 9e that there are two such
decompositions.

The nodal planes of the DC components for cluster A are shown in Fig. 9b.
Here, the sole decomposition with $M_{D}/M<0.3$ is rendered transparently and
is reasonably interpreted as near-pure crush-type source. For the majority of
the remaining decompositions, it can be seen that they have a DC nodal plane
that dips towards the face of the stope (to the north east), which is shown in
red. These can be interpreted as shearing ahead of the face in the hanging
wall as shown in Fig. 3. Alternatively, these sources could be interpreted as
episodes of slip/shear in the plane of the stope, which is shown in blue in
Fig. 9b; however, this is a less likely mode of failure given that the
direction of maximum loading is approximately orthogonal to the plane of the
stope as shown in Fig. 8c.

The closing-crack P-axis orientations for cluster B are shown in Fig. 9c,
which are all within $15^{\circ}$ of the stope’s normal. Again, as shown in
Fig. 9d, the majority of the DC components have a nodal plane dipping towards
the face of the stope (to the south west), which leads to an interpretation of
shearing in the hanging wall ahead of the face. However, there are also two
that have nodal planes dipping away from the face of the stope, which instead
points towards shearing in the footwall.

(a)  (b)  (c)  (d)  (e)

Figure 9: Case study A decomposition summary. (a) Stereonet of closing-crack
P-axis orientations for cluster A (red squares). Those that are translucent
correspond to decompositions with a small closing-crack component
($M_{K}/M<0.3$). The black cross gives the direction orthogonal to the plane
of the stope. The contours give angle to this direction. (b) Stereonet of the
DC nodal planes (black lines) and poles (magenta squares) for cluster A. Those
that are translucent correspond to decompositions with a small DC component
($M_{D}/M<0.3$). The blue and red lines give the orientation of the stope and
its face, respectively. (c) Same as (a) for cluster B. (d) Same as (b) for
cluster B. (e) Normalised scalar moments for the closing-crack and DC
components coloured by cluster.

###  6.2 Case B

The second catalogue we consider, which is shown in Fig. 10, comes from a
Western Australian mine. It consists of 204 events from 16 May 2019 to 21 June
2019, which range from $M_{W}\,\mathrm{-1.4}$ to $M_{W}\,\mathrm{0.9}$. As can
be seen in Figs. 10a-10d, these events locate to the backs (roofs) of two
tunnels that are relatively isolated from stopes or other large excavations
(note that these locations are the point of failure initiation). Spatially,
they form three clusters: one along the upper tunnel (labeled cluster A) and
two along the lower tunnel separated by an intersection (labeled cluster B to
the south and cluster C to the north). Figs. 10e-10h show that the moment
tensors for all events lie in or near the CDC region (for $\nu=0.23$) and are
oriented relatively consistently within each cluster (particularly the
P-axes).

(a)  (b)

(c)

(d)  (e)

(f)  (g)

(h)

Figure 10: Case study B data summary. (a) View looking west, where the grey
wireframes indicate mined-out tunnels and stopes. Seismic events are rendered
as beachballs centered at their inferred hypocenter, with red dipoles
indicating the P-axis direction. Events locating near two tunnels are grouped
spatially into clusters A, B, and C. (b)-(c) Plan views of the top and bottom
tunnels, respectively, with the direction of _in situ_ maximum and
intermediate principal compressive stress $\sigma_{1}$ and $\sigma_{2}$,
respectively, indicated. (d) View looking south. (e) Hudson source-type plot
with colouring according to cluster. The contours indicate amount of non-CDC
content as defined in Sec 5.3. (f)-(h) Stereonet plots showing the P-, B-, and
T-axis orientation, respectively, with colouring according to cluster.

We have determined a CDC decomposition of each moment tensor following the
same three-step procedure outlined in Section 6.1, with the only modification
being to the direction of the closing-crack P-axis orientation. As noted in
Section 2.2, convergence of the surrounding rockmass into a tunnel is expected
to produce a moment tensor with a P-axis in the direction of maximum loading
orthogonal to the tunnel’s axis. Given that they are oriented horizontally at
an azimuth of approximately $10^{\circ}$ and both $\sigma_{1}$ and
$\sigma_{2}$ in the area are horizontal (orientations given in Figs. 10b and
10c), we select decompositions with closing-crack P-axis orientations as close
as possible to an azimuth of $100^{\circ}$ and plunge of $0^{\circ}$.

For cluster A, it can be seen in Fig. 11a that for the majority of cases, it
was possible to construct a decomposition with a closing-crack P-axis close to
the desired orientation. Furthermore, as shown in Fig. 11g a large fraction of
the decompositions have a relatively small DC component, with approximately
half having $M_{D}/M<0.3$. It is reasonable to interpret these sources as
episodes of stress fracturing in the back of the tunnel accompanied by
convergence of the surrounding rockmass into the excavation. Fig. 11b shows
that DC components of those decompositions with $M_{D}/M>0.3$ do not share a
common orientation. Instead, as indicated by the fill colour of the poles,
there are at least two orientation clusters (unfilled poles do not belong to a
cluster). These have been identified by clustering according to Kagan angle
(Kagan, 1991) using DBSCAN ($\varepsilon=18^{\circ}$, $N_{\min}=12$) as
proposed by Cesca et al. (2014).

For spatial clusters B and C, it was also possible to determine decompositions
with closing-crack P-axis close to the desired orientation as shown in Figs.
11c and 11e, respectively. On average, it can be seen from Fig. 11g, that
these decompositions have more DC content than those found for cluster A
(particularly in the case of cluster B). Furthermore, as shown in Figs. 11d
and 11f, these DC components largely share a common orientation. It is
therefore reasonable to interpret the sources in these clusters as being
episodes of subhorizontal or subvertical slipping/shearing in the back of the
lower tunnel combined with convergence of the surrounding rockmass.

(a)  (b)  (c)  (d)  (e)  (f)  (g)

Figure 11: Case study B decomposition summary. (a) Stereonet of closing-crack
P-axis orientations for cluster A, which are translucent for $M_{K}/M<0.3$.
The contours show the angle to the desired orientation, which is given by the
black cross. (b) Stereonet of the DC nodal planes and poles for cluster A,
which are translucent for $M_{D}/M<0.3$. The poles are coloured by Kagan-angle
cluster as described in the text. (c) Same as (a) for cluster B. (d) Same as
(b) for cluster B. (e) Same as (a) for cluster C. (f) Same as (b) for cluster
C. (g) Normalised scalar moments coloured by cluster.

##  7 Conclusion

Motivated by the existing classification of mining-induced events into crush-
and slip/shear-type sources (Ryder, 1988), we have investigated the CDC
decomposition of moment tensors into closing-crack and DC components. In doing
this, we have built on the results of Tape & Tape (2013), who considered the
essentially equivalent (mathematically speaking) case of decomposition into
opening-crack and DC components. They noted that not every moment tensor
permits such a decomposition and expressed the set that do in terms of bounds
on the lune. We have translated these bounds to the source-type plot of Hudson
et al. (1989), which sees more widespread use for historical reasons. We note
that this result, and all others presented in the paper, can be readily re-
expressed in the context of opening cracks for application to different
physical source processes (such as tensile earthquakes, hydraulic fracturing,
etc.).

Tape & Tape (2013) also noted that the moment tensors falling within the CDC
region permit an infinite number of decompositions in general. We have derived
an implicit equation that defines this set in terms of the azimuth and plunge
of the closing-crack P-axis orientation. To determine a finite subset of
approximate solutions to this equation, we have made use of the marching-
squares algorithm. While this is a relatively robust and performant approach,
there is potential for optimisation here (or perhaps even the derivation of
parametric solutions). We have also outlined a number of physically motivated
criteria that can be used for the selection of a single decomposition from
this set.

For moment tensors falling outside the CDC region, we have presented a simple
geometric method of determining the closest CDC moment tensor, which can then
be decomposed. This is a useful tool and allows for the analysis of moment
tensors obtained from unconstrained inversion. However, it is not necessarily
the case that the CDC moment tensor found by this procedure is the one that
gives the best fit to waveforms. Where possible, it may therefore be better to
utilise an inversion scheme that is constrained to the CDC region.

As a demonstration of the methods developed, we have analysed two catalogues
of mining-induced seismic events. In both cases, we showed that constraining
the expected orientation of the closing-crack’s P-axis can provide insight
into the physical process responsible for the remaining DC component. The
information about excavation geometry and stress state near the source that is
required to do this is typically readily available, meaning that there is
potential for such procedures to be implemented for the routine interpretation
of seismic sources at mines.

## Acknowledgments

We are thankful to the mines that provided permission to use their data as
presented in Section 6.

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