Geophys. J. Int. (2007) 171, 1342–1351 doi: 10.1111/j.1365-246X.2007.03578.x GJI Tectonics and geodynamics Geodetic data inversion based on Bayesian formulation with direct and indirect prior information Mitsuhiro Matsu’ura, Akemi Noda and Yukitoshi Fukahata Department 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 SUMMARY Mechanical interaction between adjacent plates, which causes crustal deformation in plate boundary zones, is rationally represented by tangential displacement discontinuity (fault slip) at plate interfaces. Given fault slip distribution, we can compute surface displacements on the basis of elastic dislocation theory. Thus we can determine the functional form of a stochastic model 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 is direct 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 slip distribution. 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 we constructed 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 objectively determined from observed data by using Akaike’s Bayesian Information Criterion (ABIC). The inversion formula derived from the Bayesian model unifies the Jackson–Matsu’ura formula with direct prior information and the Yabuki–Matsu’ura formula with indirect prior information in a rational way. We demonstrated the effectiveness of the unified inversion formula through the analysis of the surface displacement data associated with the 1923 Kanto earthquake. In the analysis with direct and indirect prior information we obtained the bimodal distribution of fault slip 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 of fault slip perpendicular to plate convergence appears to the east, which is incomprehensible from plate tectonics. Key words: ABIC, Bayesian modelling, geodetic data inversion, plate motion, prior information. 1 INTRODUCTION 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; 1342 C 2007 The Authors Journal compilation C 2007 RAS Downloaded from https://academic.oup.com/gji/article/171/3/1342/2034865 by guest on 25 November 2021
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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.
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).
ic.oup.com/gji/article/171/3/1342/2034865 by guest on 25 N
ovember 2021
1350 M. Matsu’ura, A. Noda and Y. Fukahata
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.
R E F E R E N C E S
Akaike, H., 1977. On entropy maximization principle, in Application ofStatistics pp. 27–41, ed. Krishnaiah, P.R., North-Holland, Amsterdam.
Akaike, H., 1980. Likelihood and the Bayes procedure, in Bayesian Statisticspp. 143–166, eds Bernardo, J.M., DeGroot, M.H., Lindley, D.V. & Smith,
A.F.M., University Press, Valencia.
Burridge, R. & Knopoff, L., 1964. Body force equivalents for seismic dis-