Seismic gradiometry uses a Taylor series expansion of the seismic wavefield in two-dimensional (2D) horizontal space. If we have observed waveform *u*
^{obs} at location (*x*
_{
S
}, *y*
_{
S
}) and time *t*, the wave *u* at a nearby location (*x*
_{
G
}, *y*
_{
G
}) can be approximated by a first-order approximation of the Taylor series:

$$ {u}^{\mathrm{obs}}\left({x}_S,{y}_S;t\right)\cong u\left({x}_G,{y}_G;t\right)+\frac{\partial u\left({x}_G,{y}_G;t\right)}{\partial x}\left({x}_S-{x}_G\right)+\frac{\partial u\left({x}_G,{y}_G;t\right)}{\partial y}\left({y}_S-{y}_G\right). $$

(1)

By using several observations from stations near the target grid at locations (*x*
_{
Si
}, *y*
_{
Si
}) (*i* = 1, ⋯, *N*), we can construct an observation equation of the wave amplitude for a given location as follows:

$$ \begin{array}{ll}{\boldsymbol{u}}^{\mathrm{obs}}\hfill & =\boldsymbol{Gm}\hfill \\ {}{\boldsymbol{u}}^{\mathrm{obs}}\hfill & \equiv \left(\begin{array}{c}\hfill {u}^{\mathrm{obs}}\left({x}_{S1},{y}_{S1};t\right)\hfill \\ {}\hfill {u}^{\mathrm{obs}}\left({x}_{S2},{y}_{S2};t\right)\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill {u}^{\mathrm{obs}}\left({x}_{SN},{y}_{SN};t\right)\hfill \end{array}\right)\hfill \\ {}\boldsymbol{G}\hfill & \equiv \left(\begin{array}{ccc}\hfill 1\hfill & \hfill {x}_{S1}-{x}_G\hfill & \hfill {y}_{S1}-{y}_G\hfill \\ {}\hfill 1\hfill & \hfill {x}_{S2}-{x}_G\hfill & \hfill {y}_{S2}-{y}_G\hfill \\ {}\hfill \hfill & \hfill \vdots \hfill & \hfill \hfill \\ {}\hfill 1\hfill & \hfill {x}_{SN}-{x}_G\hfill & \hfill \kern0.5em {y}_{SN}-{y}_G\hfill \end{array}\right)\hfill \\ {}\boldsymbol{m}\hfill & \equiv \left(\begin{array}{r}\hfill u\left({x}_G,{y}_G;t\right)\\ {}\hfill {\partial}_xu\left({x}_G,{y}_G;t\right)\\ {}\hfill {\partial}_yu\left({x}_G,{y}_G;t\right)\end{array}\right),\hfill \end{array} $$

(2)

where *N* is the number of stations used for the estimation. Liang and Langston (2009) used a very similar observation equation to estimate Rayleigh waves for the case that the grid locations are collocated with seismic stations. Though their approach simplifies gradiometry Eq. (2), the estimation of the seismic wavefield is limited to the station location. In our approach, we use seismic gradiometry to reconstruct the seismic wavefield as a continuous 2D field irrespective of the station location. We note that Spudich et al. (1995) performed a similar series expansion to estimate surface strain and stress tensors for the 1992 Landers earthquake (M7.4) recorded by a dense array.

The wave amplitude and its spatial gradients can be estimated by using the least squares method if we have input data from three or more stations. We note that we can significantly reduce the computational cost by using a previously computed kernel matrix for the least squares calculation because this inverse problem depends only on the station layout:

$$ \boldsymbol{m}={\left({\boldsymbol{G}}^T\boldsymbol{W}\boldsymbol{G}\right)}^{-1}{\boldsymbol{G}}^T\boldsymbol{W}{\boldsymbol{u}}^{\mathrm{obs}}\equiv \boldsymbol{K}{\boldsymbol{u}}^{obs}, $$

(3)

where *W* is a diagonal weight matrix (e.g., Menke 2012). As a weighting factor, we empirically adopted the Gaussian function, which smoothly decreases weight amplitude with distance between the target grid point and the station with a variance of \( {\sigma}^2={\varDelta}_0^2/10 \), where *Δ*
_{0} = 50 km is a cutoff distance. Equation (3) can be formally decomposed as follows:

$$ \begin{array}{r}\hfill u\left({x}_G,{y}_G;t\right)={\boldsymbol{k}}_{\mathrm{u}}\cdot {\boldsymbol{u}}^{\mathrm{obs}}\\ {}\hfill {\partial}_xu\left({x}_G,\ {y}_G;t\right)={\boldsymbol{k}}_{\partial x}\cdot {\boldsymbol{u}}^{\mathrm{obs}}\\ {}\hfill {\partial}_yu\left({x}_G,\ {y}_G;t\right)={\boldsymbol{k}}_{\partial y}\cdot {\boldsymbol{u}}^{\mathrm{obs}}\end{array} $$

(4)

where *K*≡[*k*
_{
u
}, *k*
_{∂x
}, *k*
_{∂y
}]^{T} is a part of the solution of Eq. (3). Equation (4) suggests that the displacement and its spatial gradients at a grid point are obtained from weighted averages of the observed waveform at nearby stations, where the weighting factor *K* is obtained from the station layout. We note that this is an over-determined problem when three or more stations are used. Therefore, regularization by smoothing or damping is not necessary to solve this problem.

The quality of the estimation of the displacement and its spatial derivatives strongly depends on the locations of the stations relative to the grid point. The quality of the estimation improves as the number of stations used for Eq. (3) increases. On the other hand, the use of more distant stations may result in underestimation of the spatial derivatives of the waves, because we assumed a first-order Taylor series expansion of the wavefield in Eq. (1) (Langston 2007a). We therefore used only those stations within 50 km, with a smoothly decreasing weighting factor.

### Love and Rayleigh wave decomposition

By using a three-component seismogram and its derivatives, we can estimate the divergence and rotation of elastic vector motion. To eliminate the derivative with respect to depth, we used a traction-free boundary condition as proposed by Shapiro et al. (2000),

$$ \begin{array}{l}\mathrm{d}\mathrm{i}\mathrm{v}\boldsymbol{u}={\displaystyle \frac{\partial {u}_x}{\partial x}}+{\displaystyle \frac{\partial {u}_y}{\partial y}}+{\displaystyle \frac{\partial {u}_z}{\partial z}}={\displaystyle \frac{2\mu }{\lambda +2\mu }}\left({\displaystyle \frac{\partial {u}_x}{\partial x}}+{\displaystyle \frac{\partial {u}_y}{\partial y}}\right)\simeq {\displaystyle \frac{2}{3}}\left({\displaystyle \frac{\partial {u}_x}{\partial x}}+{\displaystyle \frac{\partial {u}_y}{\partial y}}\right)\kern1em \\ {}\mathrm{rot}\boldsymbol{u}=\left[{\displaystyle \frac{\partial {u}_z}{\partial y}}-{\displaystyle \frac{\partial {u}_y}{\partial z}},{\displaystyle \frac{\partial {u}_x}{\partial z}}-{\displaystyle \frac{\partial {u}_z}{\partial x}},{\displaystyle \frac{\partial {u}_y}{\partial x}}-{\displaystyle \frac{\partial {u}_x}{\partial y}}\right]=\left[2{\displaystyle \frac{\partial {u}_z}{\partial y}},-2{\displaystyle \frac{\partial {u}_z}{\partial x}},{\displaystyle \frac{\partial {u}_y}{\partial x}}-{\displaystyle \frac{\partial {u}_x}{\partial y}}\right],\kern1em \end{array} $$

(5)

where *λ* and *μ* are Lamé coefficients. We approximated the estimation of divergence by assuming a Poisson solid (*λ* = *μ*). The displacement vector can be represented by using potentials *ϕ*, *χ*, and *ψ* as

$$ \boldsymbol{u}=\mathrm{grad}\ \phi +\mathrm{rot}\left(\begin{array}{c}\hfill 0\hfill \\ {}\hfill 0\hfill \\ {}\hfill \chi \hfill \end{array}\right)+\mathrm{rot}\ \mathrm{rot}\left(\begin{array}{c}\hfill 0\hfill \\ {}\hfill 0\hfill \\ {}\hfill \psi \hfill \end{array}\right), $$

(6)

where the terms on the right-hand side correspond to *P*-, *SH*-, and *SV*-motion, respectively (Aki and Richards 2002). The divergence and rotation components are therefore written as

$$ \begin{array}{l}\mathrm{d}\mathrm{i}\mathrm{v}\ \boldsymbol{u}={\mathit{\nabla}}^2\phi \hfill \\ {}\mathrm{rot}\ \boldsymbol{u}=\left(\begin{array}{c}\hfill {\partial}_x{\partial}_z\chi -{\partial}_y\left({\mathit{\nabla}}^2\psi \right)\hfill \\ {}\hfill {\partial}_y{\partial}_z\ \chi +{\partial}_x\ \left({\mathit{\nabla}}^2\ \psi \right)\hfill \\ {}\hfill -\left({\partial}_x^2+{\partial}_y^2\ \right)\chi \hfill \end{array}\right).\hfill \end{array} $$

(7)

The divergence and the vertical component of the rotation vector therefore consist of Rayleigh (*P*-*SV*) and Love (*SH*) waves, respectively, whereas the horizontal components of the rotation are a mixture of Love and Rayleigh waves. By estimating these terms by seismic gradiometry, we can separate the Love and Rayleigh wave components from the observed ground velocity or displacement signal.

### Slowness estimation

By modeling the observed displacement as the amplitude term multiplied by the propagation phase shift term, we can separate a term related to the geometrical spreading and radiation pattern from the slowness vector. To estimate the arrival direction, we need to know ground velocity, displacement, and its spatial derivatives. Following Langston (2007b), we model the observed displacement as follows:

$$ u\left(x,y;t\right)=G\left(x,y\right)f\left[t-{p}_x\left(x-{x}_0\right)-{p}_y\left(y-{y}_0\right)\right], $$

(8)

where *G* denotes the amplitude term and *x*
_{0} and *y*
_{0} are a reference location. By taking derivatives with respect to spatial variables, we obtain

$$ \begin{array}{l}\frac{\partial u\left(x,y;t\right)}{\partial x}={A}_x(x)u\left(x,y;t\right)+{B}_x(x)v\left(x,y;t\right)\hfill \\ {}\frac{\partial u\left(x,y;t\right)}{\partial y}={A}_y(x)u\left(x,y;t\right)+{B}_y(x)v\left(x,y;t\right),\hfill \end{array} $$

(9)

where *v*(*x*, *y*; *t*) = ∂_{
t
}
*u*(*x*, *y*; *t*) is ground velocity, obtained from the derivative of the ground displacement wavefield with respect to time, and coefficients *A* and *B* (hereinafter referred to as the gradiometry parameters) are defined as follows:

$$ \begin{array}{l}{A}_x(x)\equiv \frac{1}{G\left(x,y\right)}\frac{\partial G\left(x,y\right)}{\partial x},\ {A}_y(y)\equiv \frac{1}{G\left(x,y\right)}\frac{\partial G\left(x,y\right)}{\partial y}\hfill \\ {}{B}_x(x)\equiv -\left[{p}_x+\frac{\partial {p}_x}{\partial x}\left(x-{x}_0\right)\right]\simeq -{p}_x,\ {B}_y(y)\equiv -\left[{p}_y+\frac{\partial {p}_y}{\partial y}\left(y-{y}_0\right)\right]\simeq -{p}_{\mathrm{y}}.\hfill \end{array} $$

(10)

Notice that the spatial derivatives of the slowness vector are omitted in Eq. (10). In other words, we postulated that the slowness field changes slowly and that it can be assumed that the slowness is nearly constant at a grid point and the surrounding stations used for gradiometry estimation.

Several methods, for example Fourier domain (Langston 2007a) or analytic signal (Langston 2007c; Liang and Langston 2009) methods, have been proposed for estimating these coefficients. Here, we propose a simple alternative method for estimation of the gradiometry parameters. We use a time window consisting of *M* discretized time samples, and assume that the parameters *A*
_{
i
} and *B*
_{
i
} (*i* = *x*, *y*) do not change within the time window:

$$ \begin{array}{c}\hfill {\partial}_iu\left(x,y;{t}_1\right)={A}_i(x)u\left(x,y;{t}_1\right)+{B}_i(x)v\left(x,y;{t}_1\right)\hfill \\ {}\hfill {\partial}_iu\left(x,y;{t}_2\right)={A}_i(x)u\left(x,y;{t}_2\right)+{B}_i(x)v\left(x,y;{t}_2\right)\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill {\partial}_iu\left(x,y;{t}_M\right)={A}_i(x)u\left(x,y;{t}_M\right)+{B}_i(x)v\left(x,y;{t}_M\right).\hfill \end{array} $$

(11)

From *M* discrete time samples of data for ground displacement *u*(*x*, *y*; *t*), ground velocity *v*(*x*, *y*; *t*),_{,} and the estimated spatial gradient of ground displacement ∂_{
i
}
*u*(*x*, *y*; *t*), the coefficients *A*
_{
i
}(*x*) and *B*
_{
i
}(*x*) are estimated by the least squares method. Because it is a simple two-parameter estimation, it has the following analytic solution:

$$ \begin{array}{l}{A}_i(x)={\displaystyle \frac{\left({\boldsymbol{v}}_t\cdot {\boldsymbol{v}}_t\right)\left({\partial}_i{\boldsymbol{u}}_t\cdot {\boldsymbol{u}}_t\right)-\left({\boldsymbol{u}}_t\cdot {\boldsymbol{v}}_t\right)\left({\partial}_i{\boldsymbol{u}}_{\boldsymbol{t}}\cdot {\boldsymbol{v}}_t\right)}{\left({\boldsymbol{u}}_t\cdot {\boldsymbol{u}}_t\right)\left({\boldsymbol{v}}_t\cdot {\boldsymbol{v}}_t\right)-{\left({\boldsymbol{u}}_t\cdot {\boldsymbol{v}}_t\right)}^2}}\\ {}{B}_i(x)={\displaystyle \frac{\left({\boldsymbol{u}}_t\cdot {\boldsymbol{u}}_t\right)\left({\partial}_i{\boldsymbol{u}}_t\cdot {\boldsymbol{v}}_t\right)-\left({\boldsymbol{u}}_t\cdot {\boldsymbol{v}}_t\right)\left({\partial}_i{\boldsymbol{u}}_{\boldsymbol{t}}\cdot {\boldsymbol{u}}_t\right)}{\left({\boldsymbol{u}}_t\cdot {\boldsymbol{u}}_t\right)\left({\boldsymbol{v}}_t\cdot {\boldsymbol{v}}_t\right)-{\left({\boldsymbol{u}}_t\cdot {\boldsymbol{v}}_t\right)}^2}},\end{array} $$

(12)

where *u*
_{
t
}, *v*
_{
t
}, and ∂_{
i
}
*u*
_{
t
} are vectors constituted by discrete time samples of ground displacement, ground velocity, and spatial gradients of ground displacement, respectively. We note that Liu and Holt (2015) used a very similar least squares method for estimating gradiometry parameters but with a damping constraint.

Because the estimation (12) is a pure time-domain method that uses a finite time window width, it does not require a long-duration time window as do methods using a Fourier transform (Langston 2007b) or an analytic signal with a Hilbert transform (Langston 2007c). The estimation should be stable except for the case in which the denominator of Eq. (12), (*u*
_{
t
} ⋅ *u*
_{
t
})(*v*
_{
t
} ⋅ *v*
_{
t
}) − (*u*
_{
t
} ⋅ *v*
_{
t
})^{2}, is equal to or very close to zero. Such instability can occur only if the displacement time series *u*
_{
t
} is parallel to the ground velocity *v*
_{
t
}. However, such a situation does not arise if a single wave packet arrives at the array as assumed in Eq. (8), because the ground velocity phase of a packet always differs from that of the displacement by *π*/2. Thus, the gradiometry parameter estimation can be unstable only if multiple wave packets arrive at the array within the same time window. This drawback is intrinsic to the seismic gradiometry method itself, owing to the single wave packet assumption, not to the particular estimation method proposed here.