Mathematical formulation
We consider vertical motion of the earth’s surface due to the periodic occurrence of two different types (Taisho and Genroku types) of earthquakes at a plate interface Σ, which cuts an elastic surface layer (lithosphere) overlying a Maxwell-type viscoelastic substratum (asthenosphere) under gravity (Fig. 2). For simplicity, we assume that these large interplate events occur periodically and Genroku-type events occur once every n large interplate events; that is, denoting the recurrence interval of the large interplate events in the southern Kanto region as T, the average recurrence intervals of Taisho- and Genroku-type events are given by T
T
= nT/(n − 1) and T
G
= nT, respectively. We suppose that the periodic occurrence of these earthquakes started at a time t = 0. From Matsu’ura and Sato (1989) and Matsu’ura et al. (2007), the vertical surface displacements z(x, t) due to fault slip is generally written as
$$ z\left(\mathbf{x},t\right)={\displaystyle \underset{0}{\overset{t}{\int }}}d\tau {\displaystyle \underset{\varSigma }{\int }}\left\{{q}_1\left(\mathbf{x},t-\tau; \boldsymbol{\upxi}, 0\right)\dot{w_1}\left(\boldsymbol{\upxi}, \tau \right)+{q}_2\left(\mathbf{x},t-\tau; \boldsymbol{\upxi}, 0\right)\dot{w_2}\left(\boldsymbol{\upxi}, \tau \right)\right\}d\boldsymbol{\upxi} \kern.5em t\ge 0 $$
(1)
Here, w
1(2)(ξ, τ) denotes fault slip components at a point ξ and a time τ parallel (1) and perpendicular (2) to relative plate motion and the dot indicates time derivative. The integral kernel q
1(2)(x, t − τ; ξ, 0) denotes the quasi-static vertical displacement response at a point x and a time t for the unit step increase of w
1(2)(ξ, τ).
We divide the plate interface Σ into a seismogenic region Σ
s
and the remaining steady-slip region Σ − Σ
s
. The fault slip vector w = (w
1, w
2) in the seismogenic region can be represented by the superposition of steady slip at the rate of relative plate motion V
pl
= (V
pl
, 0) and its perturbation w
s
;
$$ \mathbf{w}\left(\boldsymbol{\upxi}, t\right)=\begin{array}{cc}\hfill \left\{\begin{array}{ll}{\mathbf{V}}_{pl}\left(\boldsymbol{\upxi} \right)t+{\mathbf{w}}_s\left(\boldsymbol{\upxi}, t\right)\hfill & \mathrm{on}\ {\Sigma}_s\hfill \\ {}{\mathbf{V}}_{pl}\left(\boldsymbol{\upxi} \right)t\hfill & \mathrm{on}\ \Sigma -{\Sigma}_s\hfill \end{array}\right.\hfill & \hfill t\ge 0\hfill \end{array} $$
(2)
In the present case, we denote the seismogenic regions of the Genroku- and Taisho-type events by Σ
G
and Σ
T
, respectively, and the recurrence intervals of them by T
G
and T
T
, respectively. Then, the slip perturbation in each segment, w
G
(ξ, t) in Σ
G
or w
T
(ξ, t) in Σ
T
, can be written as
$$ \left\{\begin{array}{ccc}\hfill {\mathbf{w}}_G\left(\boldsymbol{\upxi}, t\right)=-{\mathbf{f}}_G\left(\boldsymbol{\upxi} \right)\left[\frac{t}{T_G}-{\displaystyle \sum_{k=1}^{n_G}}H\left(t-{t}_k^G\right)\right]\hfill & \hfill {t}_{k+1}^G > t\ \ge\ 0\hfill & \hfill\ \mathrm{on}\ {\Sigma}_G\hfill \\ {}\hfill {\mathbf{w}}_T\left(\boldsymbol{\upxi}, t\right)=-{\mathbf{f}}_T\left(\boldsymbol{\upxi} \right)\left[\frac{t}{T_T}-{\displaystyle \sum_{k=1}^{n_T}}H\left(t-{t}_k^T\right)\right]\ \hfill & \hfill\ {t}_{k+1}^T > t\ \ge\ 0\hfill & \hfill \mathrm{on}\ {\Sigma}_T\hfill \end{array}\right. $$
(3)
Here, f
G(T)(ξ) denotes the coseismic slip vector at a point ξ for the Genroku (Taisho)-type events, \( {t}_k^{G(T)} \) the time of the kth Genroku (Taisho)-type event, and n
G(T) the number of times of the latest Genroku (Taisho)-type event. H(t) indicates the Heaviside step function. Substituting Eqs. (2) and (3) into Eq. (1), we obtain
$$ \begin{array}{c}\hfill z\left(\mathbf{x},t\right)={\displaystyle \underset{0}{\overset{t}{\int }}}{U}_s\left(\mathbf{x},t-\tau \right)d\tau -\frac{1}{T_G}{\displaystyle \underset{0}{\overset{t}{\int }}}{U}_G\left(\mathbf{x},t-\tau \right)d\tau -\frac{1}{T_T}{\displaystyle \underset{0}{\overset{t}{\int }}}{U}_T\left(\mathbf{x},t-\tau \right)d\tau \hfill \\ {}\hfill +{\displaystyle \sum_{k=1}^{n_G}}{U}_G\left(\mathbf{x},t-{t}_k^G\right)+{\displaystyle \sum_{k=1}^{n_T}}{U}_T\left(\mathbf{x},t-{t}_k^T\right)\hfill \\ {}\hfill \max \left({t}_{n_G}^G,\ {t}_{n_T}^T\right)\le t< \min \left({t}_{n_G+1}^G,\ {t}_{n_T+1}^T\right)\hfill \end{array} $$
(4)
with
$$ {U}_s\left(\mathbf{x},t\right)\equiv {\displaystyle \underset{\Sigma}{\int }}{q}_1\left(\mathbf{x},t;\boldsymbol{\upxi}, 0\right){V}_{pl}\left(\boldsymbol{\upxi} \right)d\boldsymbol{\upxi} $$
(5)
$$ {U}_{G(T)}\left(\mathbf{x},t\right)\equiv {\displaystyle \underset{\Sigma_{G(T)}}{\int }}\left\{{q}_1\left(\mathbf{x},t;\boldsymbol{\upxi}, 0\right){f}_{G(T)1}\left(\boldsymbol{\upxi} \right)+{q}_2\left(\mathbf{x},t;\boldsymbol{\upxi}, 0\right){f}_{G(T)2}\left(\boldsymbol{\upxi} \right)\right\}d\boldsymbol{\upxi} $$
(6)
The first term on the right side of Eq. (4) represents permanent displacements due to steady slip at V
pl
(ξ) over the whole plate interface. The second and third terms are displacements due to steady slip deficit in source regions. The fourth and fifth terms are the effects of past events, which include coseismic displacements and postseismic transient displacements due to viscoelastic stress relaxation in the asthenosphere.
The height of the 1703 Genroku paleo-shoreline is given by the difference between the vertical displacements in Eq. (4) at the present and just before the 1703 Genroku event (\( t={t}_{n_G}^{G-} \));
$$ \varDelta z\left(\mathbf{x},t\right)=z\left(\mathbf{x},t\right)-z\left(\mathbf{x},{t}_{n_G}^{G-}\right) $$
(7)
The intrinsic stress relaxation time is calculated as about 5 years for a typical value of viscosity (5 × 1018 Pa s) of the earth’s asthenosphere (e.g., Matsu’ura and Iwasaki 1983), but it takes more than a hundred years for the complete decay of the transient vertical motion of the lithosphere-asthenosphere system (Sato and Matsu’ura 1988). The recurrence interval of large interplate earthquakes at the Sagami trough is several hundred years, and the pre-1703 large interplate earthquake occurred in 1293 (e.g., Shimazaki et al. 2011). So, replacing the viscoelastic response functions in Eqs. (5) and (6) for t > 300 year with their values at t = ∞, we can finally obtain a simple equation to analyze paleo-shoreline data as
$$ \begin{array}{c}\hfill \varDelta z\left(\mathbf{x},t\right)=\left(t-{t}_{n_G}^G\right){U}_s\left(\mathbf{x},\infty \right)-\left(t-{t}_{n_G}^G\right){U}_G\left(\mathbf{x},\infty \right)/{T}_G-\left(t-{t}_{n_G}^G\right){U}_T\left(\mathbf{x},\infty \right)/{T}_T\hfill \\ {}\hfill +{U}_G\left(\mathbf{x},t-{t}_{n_G}^G\right) + {U}_T\left(\mathbf{x},t-{t}_{n_T}^T\right)\hfill \\ {}\hfill \max \left({t}_{n_G}^G,\ {t}_{n_T}^T\right)\le t< \min \left({t}_{n_G+1}^G,\ {t}_{n_T+1}^T\right)\hfill \end{array} $$
(8)
In the above equation, the first term on the right side, the permanent displacements due to steady plate subduction, should be determined from the heights of older paleo-shorelines. We can calculate the third and fifth terms, because coseismic slip distribution f
T
(ξ) of the 1923 Taisho event has been already obtained from geodetic data. Then, given the present heights of the 1703 Genroku paleo-shoreline Δz(x, t), we can estimate coseismic slip distribution f
G
(ξ) of the 1703 Genroku event from Eq. (8).
Inversion algorithm
In order to discretize the problem, following Yabuki and Matsu’ura (1992), we represent the spatial distribution of each slip-vector component f
G1(2)(ξ) by the superposition of a finite number of known basis functions Φ
j
(ξ) as
$$ {f}_{G1(2)}\left(\boldsymbol{\upxi} \right)d\boldsymbol{\upxi} =\frac{1}{n_3\left(\boldsymbol{\upxi} \right)}{\displaystyle \sum_{j=1}^m}{a}_{1(2)j}{\Phi}_j\left(\boldsymbol{\upxi} \right)d\boldsymbol{\upxi} $$
(9)
where n
3 denotes the vertical component of normal vector of the plate interface Σ and m is the number of basis functions. Substituting Eq. (9) into Eq. (8), we obtain a linear observation equation in vector form to be solved for the coefficients a
1(2)j
;
$$ \mathbf{d}=\mathbf{H}\mathbf{a}+\mathbf{e} $$
(10)
Here, d is a N × 1 data vector defined as
$$ {d}_i=\varDelta z\left({\mathbf{x}}_i,t\right)-\left(t-{t}_{n_G}^G\right){U}_s\left({\mathbf{x}}_i,\infty \right)+\frac{\left(t-{t}_{n_G}^G\right)}{T_T}{U}_T\left({\mathbf{x}}_i,\infty \right)-{U}_T\left({\mathbf{x}}_i,t-{t}_{n_T}^T\right) $$
(11)
where e is a N × 1 error vector, a is a 2m × 1 model parameter vector composed of a
1 with the elements a
1j
(j = 1,…, m) and a
2 with the elements a
2j
(j = 1,…, m), and H = [H
1, H
2 ] is a N × 2m matrix whose elements are numerically calculated from
$$ {H}_{1(2)ij}={\displaystyle \underset{\Sigma}{\int }}\frac{1}{n_3\left(\boldsymbol{\upxi} \right)}{\Phi}_j\left(\boldsymbol{\upxi} \right)\left\{{q}_{1(2)}\left({\mathbf{x}}_i,t-{t}_{n_G}^G;\boldsymbol{\upxi}, 0\right)-\frac{\left(t-{t}_{n_G}^G\right)}{T_G}{q}_{1(2)}\left({\mathbf{x}}_i,\infty; \boldsymbol{\upxi}, 0\right)\right\}d\boldsymbol{\upxi} $$
(12)
From Eq. (10), assuming the errors e to be Gaussian with zero mean and a covariance matrix σ2
I, we obtain a stochastic model that relates the data d with the model parameters a as
$$ p\left(\mathbf{d}\Big|\mathbf{a};{\sigma}^2\right)={\left(2\pi {\sigma}^2\right)}^{-N/2} \exp \left[-\frac{1}{2{\sigma}^2}{\left(\mathbf{d}-\mathbf{H}\mathbf{a}\right)}^T\left(\mathbf{d}-\mathbf{H}\mathbf{a}\right)\right] $$
(13)
Here, σ2 is an unknown scale factor of the covariance matrix.
We usually have some prior information about the model parameters a. In the present case, we use the following two different types of prior information. One is that a roughness of fault slip distribution should be small, and the other is that the most likely direction of slip vectors is parallel to relative plate motion. Then, from Eq. (51) in Matsu’ura et al. (2007), the total prior information can be written as
$$ {p}_p\left(\mathbf{a};\ {\rho}^2,{\varepsilon}^2\right)={\left(2\pi \right)}^{-m}{\left|{\rho}^{-2}\mathbf{G}+{\varepsilon}^{-2}{\mathbf{F}}^{-1}\right|}^{1/2}\times \exp \left[-\frac{1}{2}{\mathbf{a}}^T\left({\rho}^{-2}\mathbf{G}+{\varepsilon}^{-2}{\mathbf{F}}^{-1}\right)\mathbf{a}\right] $$
(14)
with
$$ {\rho}^{-2}\mathbf{G}+{\varepsilon}^{-2}{\mathbf{F}}^{-1}=\left[\begin{array}{cc}\hfill {\rho}^{-2}{\mathbf{G}}^P\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {\rho}^{-2}{\mathbf{G}}^S+{\varepsilon}^{-2}\mathbf{I}\hfill \end{array}\right] $$
(15)
where ρ
2 and ε
2 are unknown scale factors, G
P and G
S are the m × m symmetric matrices defined by Eq. (23) in Matsu’ura et al. (2007). We obtain a Bayesian model by combining the total prior information, Eq. (14), with the stochastic model, Eq. (13);
$$ p\left(\mathbf{a};\ {\sigma}^2,{\alpha}^2,{\beta}^2\Big|\ \mathbf{d}\right)=c{\left(2\pi {\sigma}^2\right)}^{-\left(m+N/2\right)}{\left|{\alpha}^2\mathbf{G}+{\beta}^2{\mathbf{F}}^{-1}\right|}^{1/2} \exp \left[-\frac{1}{2{\sigma}^2}s\left(\mathbf{a}\right)\right] $$
(16)
with
$$ s\left(\mathbf{a}\right)={\left(\mathbf{d}-\mathbf{H}\mathbf{a}\right)}^T\left(\mathbf{d}-\mathbf{H}\mathbf{a}\right)+{\mathbf{a}}^T\left({\alpha}^2\mathbf{G}+{\beta}^2{\mathbf{F}}^{-1}\right)\mathbf{a} $$
(17)
Here, α
2≡σ
2/ρ
2 and β
2≡σ
2/ε
2 are hyper-parameters that control the relative weights of the first- and second-type prior information to the observed data.
To obtain the optimum solution, which maximizes p(a; σ
2, α
2, β
2| d) in Eq. (16), we need to determine the optimum values of α
2 and β
2. For this purpose, we can use Akaike’s Bayesian Information Criterion (ABIC) (Akaike 1980). Matsu’ura et al. (2007) give the explicit expression of ABIC as
$$ \mathrm{ABIC}\left({\alpha}^2,{\beta}^2\Big|\ \mathbf{d}\right)=N\ \log\ s\left({\mathbf{a}}^{*}\right)- \log \left|{\alpha}^2\mathbf{G}+{\beta}^2{\mathbf{F}}^{-1}\right| $$
$$ + \log \left|{\mathbf{H}}^T\mathbf{H}+{\alpha}^2\mathbf{G}+{\beta}^2{\mathbf{F}}^{-1}\right|+C^{\prime } $$
(18)
with
$$ {\mathbf{a}}^{*}={\left({\mathbf{H}}^T\mathbf{H}+{\alpha}^2\mathbf{G}+{\beta}^2{\mathbf{F}}^{-1}\right)}^{-1}{\mathbf{H}}^T\mathbf{d} $$
(19)
where C ′ is a term independent of α
2 and β
2. The values of α
2 and β
2 that minimize ABIC can be found by numerical search in the 2D hyper-parameter space. Denoting the optimum values of α
2 and β
2 by \( {\widehat{\alpha}}^2 \) and \( {\widehat{\beta}}^2 \), respectively, we can obtain the optimum solution â and the covariance matrix C (â) of estimation errors as
$$ \widehat{\mathbf{a}}={\left({\mathbf{H}}^T\mathbf{H}+{\widehat{\alpha}}^2\mathbf{G}+{\widehat{\beta}}^2{\mathbf{F}}^{-1}\right)}^{-1}{\mathbf{H}}^T\mathbf{d} $$
(20)
$$ \mathbf{C}\ \left(\widehat{\mathbf{a}}\right)=\kern0.5em {\widehat{\sigma}}^2{\left({\mathbf{H}}^T\mathbf{H}+{\widehat{\alpha}}^2\mathbf{G}+{\widehat{\beta}}^2{\mathbf{F}}^{-1}\right)}^{-1} $$
(21)
with
$$ {\widehat{\sigma}}^2=s\left(\widehat{\mathbf{a}}\right)/N. $$
(22)