geodetic data inversion based on bayesian formulation with direct

Geophys. J. Int. (2007) 171, 1342–1351 doi: 10.1111/j.1365-246X.2007.03578.xG

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Geodetic data inversion based on Bayesian formulation with directand indirect prior information

Mitsuhiro Matsu’ura, Akemi Noda and Yukitoshi FukahataDepartment of Earth and Planetary Science, University of Tokyo, 7–3-1 Hongo, Bunkyo-ku, Tokyo 113–0033, Japan. E-mail: [email protected]

Accepted 2007 August 3. Received 2007 July 30; in original form 2007 April 4

S U M M A R YMechanical interaction between adjacent plates, which causes crustal deformation in plateboundary zones, is rationally represented by tangential displacement discontinuity (fault slip)at plate interfaces. Given fault slip distribution, we can compute surface displacements on thebasis of elastic dislocation theory. Thus we can determine the functional form of a stochasticmodel to extract information about unknown fault slip distribution from observed surface dis-placement data. In addition to observed data we usually have prior information. For example,plate tectonics postulates that primary fault slip is parallel to relative plate motion. This isdirect prior information that bounds the values of model parameters within certain ranges.From physical consideration we may impose prior constraint on the roughness of fault slipdistribution. This is indirect prior information that regulates the structure of stochastic models.By combining the direct and indirect prior information with observed data in a proper way weconstructed a Bayesian model for geodetic data inversion, which has a hierarchic flexible struc-ture controlled by hyper-parameters. The optimum values of hyper-parameters are objectivelydetermined from observed data by using Akaike’s Bayesian Information Criterion (ABIC). Theinversion formula derived from the Bayesian model unifies the Jackson–Matsu’ura formulawith direct prior information and the Yabuki–Matsu’ura formula with indirect prior informationin a rational way. We demonstrated the effectiveness of the unified inversion formula throughthe analysis of the surface displacement data associated with the 1923 Kanto earthquake. In theanalysis with direct and indirect prior information we obtained the bimodal distribution of faultslip almost parallel to plate convergence on the North American–Philippine Sea Plate interface.If we ignore the direct prior information in the analysis, additional significant distribution offault slip perpendicular to plate convergence appears to the east, which is incomprehensiblefrom plate tectonics.

Key words: ABIC, Bayesian modelling, geodetic data inversion, plate motion, priorinformation.

1 I N T RO D U C T I O N

In plate boundary zones we can observe crustal movement on various

timescales from instantaneous coseismic change to long-term sec-

ular variation, caused by mechanical interaction at plate interfaces

(e.g. Sato & Matsu’ura 1992). On a long-term average, plates are in

steady relative motion with respect to each other. Therefore, both

coseismic fault slip and interseismic slip deficits at plate interfaces

may be regarded as the perturbation of steady relative plate motion.

Nowadays we can precisely determine 3-D plate interface geometry

from seismological observations (e.g. Hashimoto et al. 2004) and

relative plate motion from space-based geodetic measurements such

as GPS, SLR and VLBI (e.g. Sella et al. 2002). Thus, as demon-

strated by Matsu’ura & Sato (1989), we can rationally represent

plate-to-plate mechanical interaction by specifying spatiotemporal

changes in tangential displacement discontinuity (fault slip) at plate

interfaces. Tangential displacement discontinuity is mathematically

equivalent to the force system of two couples with no net force and

no net torque (Maruyama 1963; Burridge & Knopoff 1964). Such a

property must be satisfied for any force system acting on plate inter-

faces, because it originates from dynamic processes in the Earth’s

interior.

In general, given fault slip distribution on a plate interface, we can

compute surface displacements on the basis of elastic/viscoelastic

dislocation theory (e.g. Maruyama 1964; Yabuki & Matsu’ura 1992;

Fukahata & Matsu’ura 2005, 2006). Therefore, we can formulate

the inverse problem of estimating unknown fault slip distribution

from observed surface displacement data. When fault geometry is

unknown, the problem is essentially non-linear. Matsu’ura (1977)

has formulated the non-linear inverse problem of estimating fault

parameters from geodetic data with the sharp cut-off approach of

singular value decomposition for a coefficient matrix (Jackson 1972;

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Geodetic data inversion based on Bayesian formulation 1343

Wiggins 1972). When the fault geometry is known, which is the case

treated in the present paper, Matsu’ura et al. (1986) have developed

a method of geodetic data inversion based on the Bayesian for-

mulation with direct prior information about model parameters by

Jackson & Matsu’ura (1985). On the other hand, Yabuki &

Matsu’ura (1992) have developed another method of geodetic data

inversion based on Bayesian formulation with indirect prior con-

straint on the roughness of fault slip distribution. Their inversion

method has been widely used as a standard method to estimate co-

seismic fault slip distribution, and later applied to the problems

of estimating interseismic slip-deficit distribution at plate interfaces

(e.g. Yoshioka et al. 1993, 1994; Sagiya 1999, 2004). Fukahata et al.(2004) have extended the standard method to the case in which fault

slip distribution changes both in space and time, and applied it to

levelling data in Shikoku, southwest Japan, to reveal the interplate

slip history during one earthquake cycle including the 1946 Nankai

earthquake.

The theory of plate tectonics postulates that oceanic plates de-

scend beneath continental plates at a constant rate on a long-term

average. On short and intermediate timescales, slip rates at plate in-

terfaces change both in space and time because of segmental fault-

ing and coupling there. In interseismic periods the increase of slip

deficits at a strongly coupled region of plate interfaces brings about

the accumulation of shear stress there (e.g. Hashimoto & Matsu’ura

2000, 2002). When the shear stress reaches a critical level, seismic or

aseismic fault slip occurs so as to cancel the slip deficits (Fukuyama

et al. 2002). Thus the primary components of coseismic fault slip

and interseismic slip deficits at plate interfaces should be almost

parallel to the direction of relative plate motion. In the inversion

analysis of crustal movement in plate boundary zones we need to

incorporate such postulate of plate tectonics with observed data as

direct prior information.

So far the direct and indirect prior information have been treated

individually in geodetic data inversion. In the present study, first, we

construct a Bayesian model for geodetic data inversion by incorpo-

rating both the direct and indirect prior information into observed

data in a proper way. Second, we derive an inversion formula from

the Bayesian model, which unifies the Jackson–Matsu’ura formula

with direct prior information and the Yabuki–Matsu’ura formula

with indirect prior information in a rational way. Finally, we exam-

ine the effectiveness of the unified inversion formula through the

analysis of the surface displacement data associated with the 1923

Kanto earthquake, central Japan.

2 M AT H E M AT I C A L F O R M U L AT I O N

On the basis of the entropy maximization principle (Akaike 1977),

Akaike (1980) has proposed a Bayesian information criterion for

objective model selection in statistical inference. The introduction

of Akaike’s Beysian Information Criterion (ABIC) enables us to

freely construct a stochastic model by combining various sorts of

prior information. The construction of Bayesian models for statis-

tical inference is usually performed in the following way. First, on

the basis of prior knowledge about a physical system (scientific the-

ory), we select the functional form of a stochastic model that relates

observed data with model parameters. Second, we represent prior

information about the model parameters in the form of a probability

density function (pdf), and combine it with the stochastic model by

Bayes’ rule. The prior information is generally classified into direct

and indirect ones. The direct prior information, which was originally

introduced by Jackson (1979) and later described in terms of prob-

ability theory by Jackson & Matsu’ura (1985), bounds the values

of model parameters within certain ranges on the basis of previous

studies or data analyses. On the other hand, the indirect prior infor-

mation, which was originally introduced by Akaike (1980) and later

extended by Yabuki & Matsu’ura (1992), regulates the structure of

the stochastic model in some way on the basis of physical consid-

eration to the problem concerned. Third, combining both the direct

and indirect prior information with observed data, we construct a

Bayesian model with a hierarchic flexible structure controlled by

hyper-parameters. For a family of parametric models ABIC gives

an objective measure of the goodness of the hypothetical predictive

distribution as an approximation to the true but unknown distribu-

tion. Therefore, we may use ABIC to select the optimum values of

hyper-parameters. Given the optimum values of hyper-parameters,

we can obtain the optimum values of model parameters by applying

a maximum likelihood algorithm.

2.1 Linear observation equations

We consider tangential displacement discontinuity (fault slip) w at

x = ξ on a plate interface Σ with a unit normal vector n as shown

in Fig. 1. In general, given the distribution of fault slip w (ξ) with a

magnitude w(ξ) and a unit direction vector ν(ξ) = [ν i(ξ)] on a plate

interface Σ(ξ) defined by ξ 3 = f (ξ 1, ξ 2), we can compute surface

displacements u(x) = [ui(x)] on the basis of elastic dislocation the-

ory as

ui (x) =3∑

j=1

3∑k=1

∫Σ

μGi j,k(x; ξ)w(ξ)[n j (ξ)νk(ξ)

+ nk(ξ)ν j (ξ)]dΣ(ξ) (i = 1, 2, 3) (1)

with⎧⎪⎪⎨⎪⎪⎩

n1 = −(∂ f /∂ξ1)/√

1 + (∂ f /∂ξ1)2 + (∂ f /∂ξ2)2

n2 = −(∂ f /∂ξ2)/√

1 + (∂ f /∂ξ1)2 + (∂ f /∂ξ2)2

n3 = 1/√

1 + (∂ f /∂ξ1)2 + (∂ f /∂ξ2)2,

(2)

where μ is the rigidity of the medium, G i j,k are the partial deriva-

tives of static Green’s tensor Gij(x; ξ) with respect to the source

Figure 1. Schematic representation of tangential displacement discontinuity

(fault slip) at a plate interface in a Cartesian coordinate system. The fault

slip vector w at a point x = ξ on a plate interface Σ with a unit normal vector

n is decomposed into the primary component wP parallel to the direction of

plate convergence and the secondary component wS perpendicular to it.

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1344 M. Matsu’ura, A. Noda and Y. Fukahata

coordinate ξ k , and ni (i=1, 2, 3) define the unit normal vector

n(ξ) of the plate interface Σ(ξ).

We decompose the fault slip vector w into the primary compo-

nent wP parallel to the direction of plate convergence νP and the

secondary component wS perpendicular to it:

w(ξ) = wP (ξ) + wS(ξ) = wP (ξ)ν P (ξ) + wS(ξ)ν S(ξ) (3)

with

ν S(ξ) = n(ξ)×ν P (ξ). (4)

Then, we can rewrite eq. (1) as

ui (x) =∫

Σ

H Pi (x; ξ)wP (ξ)dΣ(ξ) +

∫Σ

H Si (x; ξ)wS(ξ)dΣ(ξ) (5)

with

H Pi (x; ξ) =

3∑j=1

3∑k=1

μGi j,k(x; ξ)[n j (ξ)νP

k (ξ) + nk(ξ)νPj (ξ)

], (6)

H Si (x; ξ) =

3∑j=1

3∑k=1

μGi j,k(x; ξ)[n j (ξ)νSk (ξ) + nk(ξ)νS

j (ξ)], (7)

where HPi (x; ξ) and HS

i (x; ξ) are the slip-response functions, which

represent the surface displacements caused by unit fault slip in the

directions of νP and νS , respectively. Our problem is to estimate

the fault slip distribution wP(S)(ξ) in eq. (5) from observed surface

displacement data.

In order to discretize the problem we represent the fault slip dis-

tribution wP(S)(ξ) on Σ(ξ) by the superposition of a finite number of

known basis functions Φ j (ξ 1, ξ 2) defined on a ξ 1-ξ 2 plane parallel

to the Earth’s surface (x 3 = 0); that is,

wP(S)(ξ)dΣ(ξ) = 1

n3(ξ)

m∑j=1

a P(S)j Φ j (ξ1,ξ2)dξ1dξ2. (8)

Then, substituting eq. (8) into eq. (5), we obtain a set of linear

observation equations in vector form to be solved for the expansion

coefficients a P(S)j :

d = Ha + e, (9)

where d is a n×1 dimensional data vector composed of horizontal

and/or vertical displacements at observation points (x = xo) dis-

tributed in some area, e is the n×1 dimensional corresponding error

vectors, a is a 2m×1 dimensional model parameter vector composed

of aP with the elements a Pj ( j = 1, . . . , m) and aS with the elements

aSj ( j = 1, . . . , m), and H = [HP , HS] is a n×2m coefficient matrix

whose elements are numerically calculated from

H P(S)i j (xo) =

∫∫1

n3(ξ)H P(S)

i (xo; ξ)Φj (ξ1, ξ2)dξ1dξ2

(i = 1, 2, 3; j = 1, . . . , m) (10)

with ξ 3 = f (ξ 1, ξ 2). As to the slip response functions HP(S)i (x; ξ), if

we are interested in short-term crustal movement due to coseismic

slip or episodic transient slip, we can use the analytical expressions

for an elastic half-space (Yabuki & Matsu’ura 1992). If we are inter-

ested in long-term crustal movement due to interseismic slip deficits

or steady plate subduction, we must take into account the effects of

viscoelastic stress relaxation in the asthenosphere (Matsu’ura & Sato

1989). Recently, Fukahata & Matsu’ura (2005, 2006) have obtained

general expressions for static/quasi-static internal displacements

due to a dislocation source in an elastic/viscoelastic multilayered

half-space by extending the expressions for surface displacements

by Matsu’ura et al. (1981).

From eq. (9), assuming the data errors e to be Gaussian with zero

mean and a covariance matrix σ 2E, we obtain a stochastic model

that relates the observed data d with the model parameters a as

p(d | a; σ 2) = (2πσ 2)−n/2 ‖E‖−1/2

× exp

[− 1

2σ 2(d − Ha)TE−1(d − Ha)

], (11)

where σ 2 is an unknown scale factor of the covariance matrix, and

‖E‖ denotes the absolute value of the determinant of the n×n matrix

E. With this stochastic model we can extract quantitative information

to estimate the model parameters a from the observed data d.

2.2 Direct and indirect prior information

In addition to the observed data d we usually have some prior in-

formation about the model parameters a. The prior information is

generally classified into the direct prior information that bounds the

values of model parameters within certain ranges and the indirect

prior information that regulates the model structure in some way.

The direct prior information is given, in some cases, as the prior

data obtained from previous data analyses and in other cases, more

commonly, as the prior knowledge based on previous studies. In ei-

ther case we can write the direct prior information in the following

form:

a = a + f. (12)

Here a = [a j ] represent the most likely values of a before getting

observed data, and f = [ fj ] denote their expectation errors. Assum-

ing the expectation errors f to be Gaussian with zero mean and a

covariance matrix ε2F, we obtain the pdf form of direct prior infor-

mation as

r1(a; ε2) = (2πε2)−l/2‖ΛF‖−1/2 exp

[− 1

2ε2(a − a)TF−1(a − a)

],

(13)

where l is the rank of a 2 m×2 m symmetric matrix F, and ‖ΛF‖denotes the absolute value of the product of the non-zero eigenvalues

of F. If the prior information comes from the prior knowledge, ε2 is

regarded as an unknown scale factor (hyper-parameter) that should

be determined from observed data through inversion analysis. If the

prior information is based on the prior data, on the other hand, we

can take ε2 to be 1 without loss of generality. In this case the pdf

form of direct prior information in eq. (13) is reduced to

r2(a) = (2π )−l/2‖ΛF‖−1/2 exp

[−1

2(a − a)TF−1(a − a)

]. (14)

To measure variations of physical quantity, in general, we need

some reference. Actually, we can consider the linear observation

equations (eq. 9) also to be written for the variations of model pa-

rameters measured from some hidden references, including the case

of a = 0. Therefore, unless we have sufficient data, the solution of

eq. (9) inevitably depends on the hidden references (Jackson 1979;

Matsu’ura & Hirata 1982). The explicit use of direct prior informa-

tion is necessary to resolve this problem. If some model parameter

has infinitely large uncertainty, direct prior information about the

model parameter becomes non-informative. Including such a case,

we can always write direct prior information in the form of eq. (13)

or (14).

On the other hand, from physical consideration about the finite-

ness of shear strength of faults (or shear stress acting on faults),

we may impose prior constraint on the roughness of fault slip dis-

tribution. Such constraint can be regarded as the indirect prior

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information that regulates the structure of the stochastic model

p(d | a; σ 2) in eq. (11). Following Yabuki & Matsu’ura (1992), we

define the roughness R of fault slip distribution by the sum of the

squares of the second-order spatial derivatives of each fault slip

component:

R = R P + RS (15)

with

R P(S) =2∑

k=1

2∑l=1

∫Σ

(1

sk(ξ)sl (ξ)

∂2ΔwP(S)(ξ)

∂ξk∂ξl

)2

dΣ(ξ), (16)

where si (i = 1, 2) are the scale factors defined by

si (ξ) =√

1 + n2i (ξ)/n2

3(ξ), (17)

and ΔwP(S) represent the variations of the primary and sec-

ondary fault slip components wP(S)(ξ) measured from the most

likely fault slip distribution wP(S)(ξ) before getting observed

data:

ΔwP(S)(ξ) = wP(S)(ξ) − wP(S)(ξ). (18)

Representing wP(S)(ξ) by the superposition of the basis functions

Φj(ξ) in the same form as wP(S) (ξ) in eq. (8),

wP(S)(ξ)dΣ(ξ) = 1

n3(ξ)

m∑j=1

a P(S)j Φ j (ξ1,ξ2)dξ1dξ2, (19)

we can rewrite eq. (18) as

ΔwP(S)(ξ)dΣ(ξ) = 1

n3(ξ)

m∑j=1

[a P(S)

j − a P(S)j

]Φ j (ξ1,ξ2)dξ1dξ2.

(20)

Then, substituting the above expressions into eq. (16), we obtain

the roughness R written in the positive-definite quadratic form of

the model parameter variations, a P(S)j − a P(S)

j , as

R =m∑

i=1

m∑j=1

(a P

i − a Pi

)G P

i j

(a P

j − a Pj

)

+m∑

i=1

m∑j=1

(aS

i − aSi

)GS

i j

(aS

j − aSj

)(21)

or, in vector form,

R = (aP − aP )TGP (aP − aP ) + (aS − aS)TGS(aS − aS)

= (a − a)TG(a − a) (22)

with the ij elements of the m×m symmetric matrices GP(S) defined

by

G Pi j = GS

i j

=2∑

k=1

2∑l=1

∫ ∫1

n3(ξ)

1

s2k (ξ)s2

l (ξ)

∂2Φi (ξ1, ξ2)

∂ξk∂ξl

∂2Φ j (ξ1, ξ2)

∂ξk∂ξldξ1dξ2.

(23)

Here it should be noted again that ξ= (ξ 1, ξ 2, ξ 3) represents the

coordinates of a point on the plate interface Σ:

ξ3 = f (ξ1, ξ2). (24)

Thus, we can represent the prior constraint on the roughness of fault

slip variations ΔwP(S)(ξ) in pdf form with an unknown scale factor

(hyper-parameter) ρ2 as

q(a; ρ2) = (2πρ2)−k/2‖ΛG‖1/2 exp

[− 1

2ρ2(a − a)TG(a − a)

],

(25)

where k is the rank of G, and ‖ΛG‖ denotes the absolute value of

the product of the non-zero eigenvalues of G.

If the direct prior information in eq. (13) or (14) and the indirect

prior information in eq. (25) are independent of each other, we can

obtain the proper pdf form of total prior information by the simple

product of them as

p1(a; ρ2, ε2) = q(a; ρ2)r1(a; ε2) or p2(a; ρ2) = q(a; ρ2)r2(a). (26)

Actually, they are not independent of each other, and so the above

expression is improper; it is impossible to correctly normalize the

total prior pdf defined by eq. (26). In such a case, as demonstrated by

Fukahata et al. (2004), a proper pdf form of total prior information

is given by

p1(a; ρ2, ε2) = (2π )−m‖ρ−2G + ε−2F−1‖1/2

× exp

[−1

2(a − a)T(ρ−2G + ε−2F−1)(a − a)

](27)

or

p2(a; ρ2) = (2π )−m‖ρ−2G + F−1‖1/2

× exp

[−1

2(a − a)T(ρ−2G + F−1)(a − a)

]. (28)

2.3 Bayesian modelling and ABIC

Now we combine the total prior information with observed data

by Bayes’ rule, and construct a Bayesian model with a hierarchic

flexible structure controlled by hyper-parameters. As pointed out

in Section 2.2, we have two different cases. In the first case, the

total prior information is obtained by unifying the prior constraint

q(a; ρ2) and the prior knowledge r 1(a; ε2), and in the second case,

by unifying the prior constraint q(a; ρ2) and the prior data r 2(a).

In the first case, combining p1(a;ρ2,ε2) in eq. (27) with

p(d | a; σ 2) in eq. (11) by Bayes’ rule, we can construct a Bayesian

model as

p(a; σ 2, ρ2, ε2|d) = c p(d|a; σ 2)p1(a; ρ2, ε2). (29)

Introducing new hyper-parameters, α2 = σ 2/ρ2 and β2 = σ 2/ε2, in-

stead of ρ2 and ε2, we obtain the explicit expression for the Bayesian

model as

p(a; σ 2, α2, β2|d) = c (2πσ 2)−(m+n/2)‖E‖−1/2

× ‖α2G + β2F−1‖1/2 exp

[− 1

2σ 2s(a)

](30)

with

s(a) = (d − Ha)TE−1(d − Ha)

+ (a − a)T(α2G + β2F−1)(a − a), (31)

where it should be noted that the hyper-parameters α2 and β2 control

the relative weights between observed data, indirect prior constrain,

and direct prior knowledge.

For certain fixed values of α2 and β2 we can obtain the solution

that maximizes the posterior pdf in eq. (30), or minimizes s(a) in

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eq. (31), with the following maximum likelihood algorithm (Jackson

& Matsu’ura 1985; Yabuki & Matsu’ura 1992). For any solution that

minimizes s(a) the variation of s(a) with respect to a must vanish:

HTE−1(d − Ha) − (α2G + β2F−1)(a − a) = 0. (32)

The solution a∗ that satisfies the above equation is given by

a∗ = a + (HTE−1H + α2G + β2F−1)−1HTE−1(d − Ha). (33)

Then, we can rewrite s(a) in eq. (31) as

s(a) = s(a∗) + (a − a∗)T(HTE−1H + α2G + β2F−1)(a − a∗). (34)

This means that the posterior probability density distribution in

eq. (30) is Gaussian with the mean a∗ and the covariance matrix

C(a∗) = σ 2(HTE−1H + α2G + β2F−1)−1. (35)

Actually, α2 and β2 are adjustable hyper-parameters. In order to

determine the optimum values of these hyper-parameters we can use

ABIC defined by

ABIC(σ 2, α2, β2|d) = −2 log L(σ 2, α2, β2|d) + C, (36)

where L(σ 2, α2, β2|d) denotes the marginal likelihood, whose ex-

plicit expression is give by

L(σ 2, α2, β2|d) ≡∫ +∞

−∞p(a; σ 2, α2, β2|d)da

= c (2πσ 2)−n/2‖E‖−1/2‖α2G + β2F−1‖1/2

× ‖HTE−1H + α2G + β2F−1‖−1/2 exp

[− 1

2σ 2s(a∗)

]. (37)

In this case we can obtain an analytical relation σ 2 = s(a∗)/n from

∂L/∂σ 2= 0. Then, substituting eq. (37) together with the above an-

alytical relation into eq. (36), we obtain the explicit expression of

ABIC as

ABIC(α2, β2|d) = n log s(a∗) − log ‖α2G + β2 F−1‖+ log ‖HTE−1H + α2G + β2F−1‖ + C ′, (38)

where C ′ is a term independent of α2 and β2. The values of α2 and β2

that minimize ABIC in eq. (38) can be found by numerical iterative

search in the 2-D hyper-parameter space. Once the optimum values

of α2 and β2 were found, denoting them by α2 and β2, we can obtain

the optimum solution a and the covariance matrix C(a) of estimation

errors from eqs (33) and (35) as

a = a + (HTE−1H + α2G + β2F−1)−1HTE−1(d − Ha), (39)

C(a) = σ 2(HTE−1H + α2G + β2F−1)−1 (40)

with σ 2 = s(a)/n.

In the second case, where the total prior information is given by

unifying the prior constraint q(a; ρ2) and the prior data r 2(a), we

construct a Bayesian model by combining p2(a; ρ2) in eq. (28) with

p(d|a; σ 2) in eq. (11):

p(a; σ 2, ρ2|d) = c p(d|a; σ 2)p2(a; ρ2). (41)

Using the hyper-parameter α2 = σ 2/ρ2 instead of ρ2, we obtain the

explicit expression for the Bayesian model as

p(a; σ 2, α2|d) = c (2πσ 2)−(m+n/2)‖E‖−1/2‖α2G + σ 2F−1‖1/2

× exp

[− 1

2σ 2s(a)

](42)

with

s(a) = (d − Ha)TE−1(d − Ha)

+ (a − a)T(α2G + σ 2F−1)(a − a). (43)

For certain fixed values of σ 2 and α2, the solution that maximizes

the posterior pdf in eq. (42), or minimizes s(a) in eq. (43), is given

by

a∗ = a + (HTE−1H + α2G + σ 2F−1)−1HTE−1(d − Ha). (44)

Then, rewriting s(a) in eq. (43) as

s(a) = s(a∗) + (a − a∗)T(HTE−1H + α2G + σ 2F−1)(a − a∗), (45)

we obtain the explicit expression for the marginal likelihood:

L(σ 2, α2|d) ≡∫ +∞

−∞p(a; σ 2, α2|d)da

= c (2πσ 2)−n/2‖E‖−1/2‖α2G + σ 2F−1‖1/2

× ‖HTE−1H + α2G + σ 2F−1‖−1/2 exp

[− 1

2σ 2s(a∗)

]. (46)

Thus, the explicit expression of ABIC is given by

ABIC(σ 2, α2|d) = n log σ 2 − log ‖α2G + σ 2F−1‖+ log ‖HTE−1H + α2G + σ 2F−1‖ + s(a∗)/σ 2 + C ′, (47)

where C ′ is a term independent of σ 2 and α2. The values of σ 2 and α2

that minimize ABIC in eq. (47) can be found by numerical iterative

search in the 2-D hyper-parameter space. For the given optimum

values of σ 2 and α2, denoting them by σ 2 and α2, we obtain the

optimum solution a and the covariance matrix C(a) of estimation

errors as

a = a + (HTE−1H + α2G + σ 2F−1)−1HTE−1(d − Ha), (48)

C(a) = σ 2(HTE−1H + α2G + σ 2F−1)−1. (49)

3 A P P L I C AT I O N T O A N A C T UA L C A S E

In order to examine the applicability of the inversion method

to actual cases, we analyze the surface displacement data as-

sociated with the 1923 Kanto earthquake (M7.9), which oc-

curred at the interface between the North American (NAM) and

Philippine Sea (PHS) Plates (e.g. Matsu’ura et al. 1980). The

coseimic surface displacements at the 1923 Kanto earthquake

have been revealed from the comparison of the pre- and post-

seismic levelling (1884–1898 to 1923–1927) and triangulation

(1884–1899 to 1924–1925) reported by Military Land Survey

(1930). In Fig. 2 we show the vertical and horizontal displace-

ments associated with the 1923 Kanto earthquake. In Fig. 2(a),

the white and grey bars indicate uplift and subsidence at bench-

marks along levelling roots, respectively. The vertical displacement

data have already been corrected by subtracting the coseismic height

change (–8.6 cm) of the Tokyo standard datum. For the horizon-

tal displacements we have two different data sets. Military Land

Survey (1930) has converted angle-change data of triangulation sta-

tions into horizontal displacement vectors by fixing the reference

point Teruishi and the direction from it to Tsukuba. On the other

hand, Sato & Ichihara (1971) have converted the angle-change data

into horizontal displacement vectors by fixing five reference points,

Tsukuba, Teruishi, Dodaira, Kokusi II and Kenashi. The converted

horizontal displacement vectors include systematic errors due to the

coseismic rotation and extension of reference base lines. According

to Matsu’ura et al. (1980), the systematic errors are very large in the

former data set, but not in the latter data set. In Fig. 2(b) we show

the horizontal displacement vectors converted by Sato & Ichihara

(1971), which are assumed to be free from systematic errors in the

following inversion analysis for simplicity.

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Figure 2. The surface displacements associated with the 1923 Kanto earth-

quake. (a) Vertical displacements at benchmarks obtained from the com-

parison of the pre- and post-seismic levelling (1884–1898 to 1923–1927).

The white and grey bars indicate uplift and subsidence, respectively. (b)

Horizontal displacement vectors at triangulation stations obtained from the

comparison of the pre- and post-seismic triangulation (1884–1899 to 1924–

1925) by fixing five reference points, TSU (Tsukuba), TER (Teruishi), DOD

(Dodaira), KOK (Kokushi II) and KEN (Kenashi).

3.1 Tectonic setting and plate interface geometry

in the Kanto region

The Kanto region, central Japan, is in a complex tectonic setting,

where the Pacific (PAC) Plate is descending beneath the NAM and

PHS Plates, and the PHS Plate is descending beneath the NAM

Plate and running on the PAC Plate at its eastern margin. In order

to correctly estimate fault slip distribution on the plate interfaces,

we need a realistic model of plate interface geometry. Recently,

Hashimoto et al. (2004) have constructed a 3-D digitized plate in-

terface model in and around Japan from ISC (International Seis-

mological Center) hypocentre data and JMA (Japan Meteorological

139˚ 140˚ 141˚ 142˚ 143˚

34˚

35˚

36˚

37˚

20

40

20

40

60

80

60

PACPHS

NAM

Boso

Izu

Figure 3. 3-D geometry of plate interfaces in the Kanto region. The thick

and thin iso-depth contours (in km) represent the upper boundaries of the

Philippine Sea (PHS) Plate, descending beneath the North American (NAM)

Plate, and the Pacific (PAC) Plate, descending beneath the NAM and PHS

Plates, respectively. We took the light-grey region on the NAM–PHS Plate

interface as the potential source region. The thick solid arrow indicates the

motion of the NAM Plate relative to the PHS Plate.

Agency) unified hypocentre data. Fig. 3 shows the 3-D geometry of

plate interfaces beneath the Kanto region. Here, the thick iso-depth

contours represent the upper surface of the PHS Plate descending be-

neath the NAM Plate, and the thin iso-depth contours represent that

of the PAC Plate descending beneath the NAM and PHS Plates. In

the analysis of the surface displacement data associated with the

1923 Kanto earthquake, we took the light-grey region on the

NAM–PHS Plate interface as the potential source region. The di-

rections of relative plate motion νP(ξ) at the plate interface were

calculated from the global plate motion model NUVEL-1A (DeMets

et al. 1994). The spatial distribution of the primary and secondary

fault slip components, wP(ξ) and wS(ξ), were each represented by

the superposition of 366 normalized bi-cubic B-splines, distributed

over the potential source region at 8-km intervals. We used the an-

alytical expressions of slip-response functions for an elastic half-

space (Yabuki & Matsu’ura 1992) to calculate the elements of the

coefficient matrix H in eq. (9).

3.2 Inversion analysis with direct and indirect

prior information

We invert the vertical and horizontal displacement data for the pri-

mary and secondary components of fault slip on the NAM–PHS

Plate interface simultaneously. In general, triangulation is less reli-

able measurement than levelling, and so we assumed that the errors

of horizontal displacement data are twice as large as those of vertical

displacement data. From plate tectonics we postulate that the most

likely values of the secondary fault slip components wS (ξ) are zero

(aS = 0); that is,

r1(aS ; ε2) = (2πε2)−m/2 exp

[− 1

2ε2(aS − 0)T(aS − 0)

]. (50)

In addition, from physical consideration we impose the indirect

prior constraint on the roughness of fault slip distribution given in

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Figure 4. The contour map of ABIC in the 2-D hyper-parameter space. The

contour intervals are taken to be 10. The cross indicates the minimum point

that gives the optimum values of the hyper-parameters.

eq. (22). Then, the total prior information can be written as

p1(a; ρ2, ε2) = (2π )−m‖ρ−2G + ε−2F−1‖1/2

× exp

[−1

2aT(ρ−2G + ε−2F−1)a

](51)

with

ρ−2G + ε−2F−1 =[

ρ−2GP O

O ρ−2GS + ε−2I

]. (52)

Combining the total prior information in eq. (51) with the stochastic

model in eq. (11), and using α2 = σ 2/ρ2 and β2 = σ 2/ε2 instead of

ρ2 and ε2, we obtain a Bayesian model:

p(a; σ 2, α2, β2|d) = c (2πσ 2)−(m+n/2)‖E‖−1/2

×‖α2G + β2F−1‖1/2exp

[− 1

2σ 2s(a)

](53)

with

s(a) = (d − Ha)TE−1(d − Ha) + aT(α2G + β2F−1)a, (54)

α2G + β2F−1 =[

α2GP O

O α2GS + β2I

]. (55)

In this case, the formal expressions of ABIC, the optimum solu-

tion, and the covariance matrix of estimation errors are given in

eqs (38), (39), and (40), respectively.

We show the contour map of ABIC(α2, β2) in Fig. 4, where the

cross indicates the minimum point that gives the optimum values

of hyper-parameters α2 and β2. For these values we computed the

optimum model a and its covariance matrix C(a) from eqs (39) and

(40), respectively, and then the optimum fault slip distribution from

eq. (8). In Fig. 5(a) we show the inverted coseismic slip of the 1923

Kanto earthquake, which extends to 30 km in depth and has a bi-

modal distribution with the 5 km-deep western and the 15 km-deep

eastern peaks of about 8 m. The slip vectors are almost parallel to the

direction of plate convergence except for their clockwise rotation

near the Sagami Trough. Fig. 5(b) shows the estimation errors of the

inverted fault slip distribution. In the main slip area, the estimation

errors are about 1–2 m, and so the bimodal coseismic slip distribu-

tion with 8 m peaks is reliable. From the comparison of the surface

Figure 5. The coseismic slip distribution of the 1923 Kanto earthquake and its uncertainty estimated from the inversion analysis with direct and indirect prior

information. (a) Inverted fault slip distribution. The thick arrows indicate fault slip vectors on the NAM–PHS Plate interface, represented by the iso-depth

contours. The magnitude of fault slip vectors is shown by the grey-scale contours. The white star indicates the epicentre of the 1923 Kanto earthquake. (b) The

contour map of estimation errors for the inverted fault slip distribution in (a).

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139˚ 140˚ 141˚

35˚

36˚

1.0 m up

1.0 m down

Observed Computed

(a)

139˚ 140˚ 141˚

35˚

36˚

2.0mObserved

Computed

(b)

Figure 6. Comparison of the surface displacements computed from the in-

verted slip distribution with the observed data. (a) Vertical displacements.

The thin grey and black bars indicate the computed uplift and subsidence,

respectively. The thick white and grey bars indicate the observed uplift and

subsidence, respectively. (b) Horizontal displacements. The black and white

arrows represent the observed and computed horizontal displacement vec-

tors, respectively.

displacements computed from the inverted slip distribution with the

observed data in Fig. 6, we can see that the optimum model well

explains both the vertical and horizontal displacement data, except

for the horizontal displacement at the Oshima Island (east off the

Izu Peninsula) on the PHS Plate.

Now, in order to examine the effectiveness of the direct prior

information, we invert the same data set without the direct prior

information. In this case the Bayesian model is given by

p(a; σ 2, α2‖d) = c (2πσ 2)−(n+k)/2(α2)k/2‖E‖−1/2‖ΛG‖1/2

× exp

[− 1

2σ 2s(a)

](56)

Figure 7. The values of ABIC plotted as a function of the hyper-parameter

α2.

with

s(a) = (d − Ha)TE−1(d − Ha) + α2aTGa. (57)

The expressions of ABIC, the optimum solution, and the covariance

matrix of estimation errors are given in Yabuki & Matsu’ura (1992)

as

ABIC(α2∣∣ d) = (n + k − 2m) log s(a∗) − k log α2

+ log ‖HTE−1H + α2G‖ + C ′, (58)

a = (HTE−1H + α2G)−1HTE−1d, (59)

C(a) = σ 2(HTE−1H + α2G)−1 (60)

with σ 2 = s(a)/n.

We show the ABIC plotted as a function of the hyper-parameter

α2 in Fig. 7, where the minimum point gives the optimum value α2.

For this value we computed the optimum model a and its covari-

ance matrix C(a) from eqs (59) and (60), respectively, and then the

optimum fault slip distribution from eq. (8). In Fig. 8 we show the

inverted coseismic slip distribution together with the estimation er-

ror distribution. From the comparison of Fig. 8(a) with Fig. 5(a) we

can see that if the direct prior information is ignored, additional sig-

nificant distribution of fault slip perpendicular to plate convergence

appears east off the Boso Peninsula, which is incomprehensible from

plate tectonics.

4 D I S C U S S I O N A N D C O N C L U S I O N S

In Section 2 we properly constructed a Bayesian model by combin-

ing both direct and indirect prior information with observed data.

From the Bayesian model we derived two inversion formulae, corre-

sponding to the case where the total prior information consists of the

indirect prior constraint and the direct prior knowledge or the direct

prior data. The expressions of ABIC in these two cases, eqs (38) and

(47), are different from each other, but the optimum solutions, eqs

(39) and (40) and eqs (48) and (49), are formally the same. Actually,

the latter can be regarded as a special case of the former, and so we

focus the following discussion on the former case.

When the total prior information consists of the indirect prior

constraint and the direct prior knowledge, the optimum solution is

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Figure 8. The coseismic slip distribution of the 1923 Kanto earthquake and its uncertainty estimated from the inversion analysis without direct prior information.

(a) Inverted fault slip distribution. The thick arrows indicate fault slip vectors on the NAM–PHS Plate interface, represented by the iso-depth contours. The

magnitude of fault slip vectors is shown by the grey-scale contours. The white star indicates the epicentre of the 1923 Kanto earthquake. (b) The contour map

of estimation errors for the inverted fault slip distribution in (a).

given by eqs (39) and (40). If we have no direct prior information,

taking the limit of β2F−1→O in eqs (39) and (40), we obtain

a = a + (HTE−1H + α2G)−1HTE−1(d − Ha), (61)

C(a) = σ 2(HTE−1H + α2G)−1. (62)

It should be noted that the inversion formula by Yabuki & Matsu’ura

(1992) in eq. (59) is obtained by taking a = 0 in eq. (61). In other

words, the Yabuki–Matsu’ura inversion formula should be modified

as eq. (61) correctly. The difference between the inversion formulae

(59) and (61) becomes essential in the estimation of interseismic

slip-deficit distribution at plate interfaces. On the other hand, if we

have no indirect prior constraint, taking the limit of α2G→O in eqs

(39) and (40), and regarding σ 2 and β2 as constants, we obtain the

inversion formula by Jackson & Matsu’ura (1985):

a = a + (HTE−1H + β2F−1)−1HTE−1(d − Ha), (63)

C(a) = σ 2(HTE−1H + β2F−1)−1. (64)

Furthermore, taking the limit of α2G→O in eqs (61) and (62) or the

limit of β2F−1→O in eqs (63) and (64), we obtain the well-known

least-squares solution,

a = a + (HTE−1H)−1HTE−1(d − Ha), (65)

C(a) = σ 2(HTE−1H)−1 (66)

with σ 2 = (d − Ha)TE−1(d − Ha)/(n − 2m), if it exists. Then, we

can conclude that the inversion formula derived in Section 2.3 unifies

the Jackson–Matsu’ura formula with direct prior information and

the Yabuki–Matsu’ura formula with indirect prior information in a

rational way.

In Section 3 we demonstrated the effectiveness of the unified in-

version formula through a comparison between two different analy-

ses of the same surface displacement data associated with the 1923

Kanto earthquake. First, we incorporated both the direct prior knowl-

edge about model parameters, based on the postulate of plate tec-

tonics, and the indirect prior constraint on the roughness of slip

distribution, based on physical consideration, into the analysis. From

the inversion analysis we obtained the bimodal fault slip distribution

with the 5 km-deep western and 15 km-deep eastern peaks of about

8 m on the NAM–PHS Plate interface. The slip vectors are almost

parallel to the direction of plate convergence. These features of co-

seismic slip distribution are consistent with our expectations from

plate tectonics. In the second analysis we inverted the same data set

without the direct prior information. Then, we obtained additional

significant distribution of fault slip perpendicular to the direction of

plate convergence, which is incomprehensible from plate tectonics.

These inversion results demonstrate that the unified inversion for-

mula enables us to incorporate the postulate of plate tectonics into

geodetic data inversion in a quantitative way.

A C K N O W L E D G M E N T

We thank Chihiro Hashimoto for providing us the digital data of a

3-D model of plate interface geometry in the Kanto region.

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