Bayesian methods provide the means of integrating geologic,
paleoseismic, seismic, and geodetic data to improve estimates of
fault zone and lower-crustal properties (*e.g., Segall*, 2002)
that are important for predicting the long-term deformation of
plate boundary zones (*e.g., Shen et al.*, 2001),
understanding deformation and stress transfer following
earthquakes, and estimating future seismic hazard (*e.g.,
Hardebeck*, 2004). Herein, we develop a Bayesian methodology that
integrates geologic, geomorphic, and geodetic data to provide
probabilistic estimates of fault-zone and lower-crustal
properties.

We will apply this methodology to the Kunlun Fault in northern
Tibet (Figure 24.1), where estimates of lower-crustal
viscosity are generally lacking. Here, geologic and geomorphic
information constrain the range of permissible long-term slip
rates and coseismically generated offsets. We combined these a
priori estimates with GPS velocities using a Bayesian
implementation of an elastic-viscoelastic earthquake cycle model
(*Savage and Prescott*, 1978) to estimate fault-zone and lower
crustal properties in the area. The probabilistic nature of these
models allows straightforward assessments of the uncertainties
within, and covariance between model parameters. Our example
shows that these methods may aid in elucidating the active
tectonics of other areas, and improve seismic hazard assessments
by formally assimilating disparate data types (e.g., geologic,
geomorphic, seismic, or geodetic datasets) into crustal
deformation studies.

The Bayesian approach that we use to analyze data from northern
Tibet uses geologic and geomorphic estimates of fault slip rates
and observed offsets in conjunction with geodetic data to refine
estimates of fault slip rate, schizosphere thickness, recurrence
time of events, and viscous relaxation time of the plastosphere
(``chizosphere'' refers to the portion of the upper crust that
deforms elastically, while ``plastosphere'' refers to areas that
deform elastically over short time-scales but undergo viscous
stress relaxation over longer time-scales; *Scholz*, 1988).
Our method is based on Bayes' Rule (*Bayes*, 1763), which
allows quantitative refinement of initial estimates of model
parameters (in this case, fault zone and lower crustal properties)
given the information provided by the geodetic observations (*Segall*, 2002; *Johnson and Segall*, 2004):

In the context of this paper, denotes a vector of model
parameters, is a vector of the observed geodetic velocities,
denotes the probability that the set of model
parameters explains the geodetic velocities, is the
probability of observing the geodetic velocities given a
combination of model parameters , is the
probability that the chosen combination of model parameters
actually occur, and the denominator normalizes the probability to
all possible combinations of model parameters. In this context,
encapsulates the goodness of fit between observed
geodetic velocities and those that would be expected based on the
physical model that relates the surface velocity distribution to
fault-zone and lower crustal properties (described below).
Geodetic inversions that find the set of model parameters that
best explain the observed data maximize . In
contrast, the Bayesian approach not only considers the goodness of
fit of the velocity to model predictions, but also ,
which describes in a probabilistic sense, some prior knowledge
about what sets of model parameters are likely to occur. This
term may be used to incorporate information such as slip rates
along faults and recurrence times of earthquakes whose range may
be estimated based on geologic and geomorphic considerations (*e.g., Buck et al.*, 1996; *Biasi et al.*, 2002; *Segall*,
2002). Therefore, the Bayesian modeling strategy has a two-fold
advantage to conventional geodetic inversions: 1) Uncertainties in
model parameters and covariance between model parameters may be
straightforwardly obtained for complicated models (*e.g.,
Hargreves and Annan*, 2002); and 2) Other types of data (e.g.,
geologic, geomorphic, and paleoseismic) may be used in conjunction
with the geodetic data to improve estimates of model parameters.
Explicit evaluation of Equation 24.1 may be untenable
for potentially large, multidimensional parameter spaces that may
arise even in simple physical models such as the model described
below. Therefore, we solve Equation 1 using the
Metropolis-Hastings variant of the Markov-Chain Monte Carlo (MCMC)
simulation methods that allows approximation of the joint
distribution without an exhaustive search of the
parameter space (*Metropolis et al.*, 1953).

The mechanical model we employ (evaluated through )
idealizes earthquake-cycle deformation as resulting from elastic
strain release during rupturing events followed by viscoelastic
relaxation of the lower crust and upper mantle (*e.g., Savage
and Prescott*, 1978). The crust is idealized as two-dimensional
in cross-section, and so movement along the major strike-slip
faults occurs in the out-of-plane dimension. The rheology of the
crust is treated as a two-layered medium in which an elastically
deforming schizosphere overlies a linear, visco-elastically
deforming plastosphere (*e.g., Savage and Prescott*, 1978;
*Savage*, 2000; *Segall*, 2002; *Dixon et al.*, 2003).
The deformation during the earthquake cycle is defined by the
long-term, average strike-slip velocity of the modeled fault
(), the Maxwell relaxation time of the plastosphere (
, where is the viscosity and is the shear
modulus), the thickness of the schizosphere (), the time since
the last earthquake (), and the average recurrence interval
(). Following *Savage and Prescott* (1978) and *Segall*
(2002), we non-dimensionalize the model by introducing four
groups: ,
,
,
and
, where and are
arbitrary velocity and length scales set to 1 mm/yr and 1 km,
respectively. Although this is a simple representation of the
earthquake cycle, this model is appropriate for inferring slip
rates along the Kunlun Fault (and active strike-slip faults in
Tibet in general) because: 1) The model captures the basic
observation that the crustal deformation during the seismic cycle
consists of coseismic, elastic, deformation followed by
postseismic viscoelastic adjustment of the lower crust (*e.g.,
Savage and Prescott*, 1978; *Segall*, 2002); and 2) Several
previous studies have successfully employed these types of simple
models to provide first-order estimates of fault zone and
lower-crustal properties. Given the density of geodetic
measurements from this portion of Tibet, the earthquake cycle
modeling we employ allows us to consider velocity variations
during the earthquake cycle that cannot be assessed using the
types of dislocation models currently employed to interpret
interseismic geodetic velocities from this area (*i.e.,
Wallace et al.*, 2004; *Zhang et al.*, 2004).

Along the Kunlun Fault, numerous studies show consistency in
estimated slip rates over a variety of different time-scales
(10-100 kyr time-scales), and so the primary goal of the specific
example is to estimate lower-crustal properties beneath the Kunlun
fault based on geologic, geomorphic, and geodetic datasets. Along
the Kunlun fault, 1) Geologic and geomorphic studies indicate that
long-term slip rates are between 9-16 mm/yr (*Kidd and
Molnar*, 1988; *van der Woerd et al.*, 2001); 2) Geomorphic
studies and historic ruptures suggest that the slip during each
event is approximately 10 m (*van der Woerd et al.*, 2001);
and 3) The lack of historic seismicity places a minimum bound on
the recurrence time of earthquakes along the fault. Using this
information, we will construct prior joint probability densities
and apply our Bayesian methodology to refine estimates of each
of these parameters and the Maxwell relaxation time of the lower
crust.

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