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Supplementary material:
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I. Least-Square inversion for co- and post- seismic slip of 2004 Parkfield earthquake
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1. Method
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When the fault geometry is given, the inversion for fault slip is a linear problem:
y  Gb
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(s1)
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where G is the Green’s function, b is the unknown slip on the fault, and y is the measured
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surface displacements. When introducing the weighting matrix that accounts for the errors in
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the observations and Laplace regularization to slip distribution, the slip b can be obtained by
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minimizing
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f (b)   y1 (Gb  y)   2 Lb
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,
(s2)
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where  y1 is the covariance matrix about the observation errors, L is discrete Laplace’s
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operator,  2 represents the squared 2-norm. 2 is a regularization factor. Higher 2 value
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means stronger regularization to the slip distribution, and vice versa. When 2 is given, Eq. (s2)
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can be solved by Least-Square (LS) approach. For the Parkfield case, we apply the
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constrained Least-Square method, so that all of the slip is in right-lateral.
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Similar as the checkerboard test for the slip inversion in Bayesian approach, the test for LS
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inversion is shown in Fig. S1. In this test, 2 equals to 0.01. Figure S2 indicates the inversion
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results for the coseismic slip models of the Parkfield event using different 2 values. As
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shown, a smaller 2 value in the inversion leads to localized slip distribution (Fig. S2a), while
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a larger 2 value produces a widely distributed slip zone (Fig. S2c). In Fig. S2d, roughness
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(1/2) is plotted against model variance ( R T  1 R , with R  y  yˆ , and ŷ being the
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modeling results), and the results for 2=0.01 is marked by ‘*’.
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Figure S1. Checkerboard test for the LS slip inversion related to the Parkfield observation
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network. Plot (a) shows the input slip model (1-meter strike-slip on patches marked by green
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dots and 0 on the rest patches) to produce synthetic displacements (shown by black arrows in
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plot b), and the slip inverted (in black-white color), whose produced surface displacements
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are indicated by red arrows in plot (b). The red stars show the location of the Parkfield
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hypocenter.
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Figure S2. Coseismic slip models for the 2004 Parkfield earthquake using different 2 values
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in the LS inversion. The three slip distributions in plot (a), (b) and (c) correspond to the three
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smoothing factors 2=0.0005, 0.01, and 10, respectively. Plot (d) shows the model variance as
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a function of the smoothing parameter where the applied 2 values corresponding to (a), (b),
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and (c) are marked by a triangle, star, and circle, respectively.
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2. Co- and post- seismic slip of the Parkfield earthquake derived from LS inversion
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Based on the LS method described above, the slip models inverted from coseismic and
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postseismic displacements are shown in Fig. S3 and Fig. S4. Similar to the Bayesian approach,
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only the strike-slip component is considered during the inversion. Based on the coseismic slip
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model in Fig. S3, we calculate coseismic shear stress change. Three contour lines, -1.5, 0, and
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1.5 bar, are displayed in Fig. S4 by blue, black and red curves, respectively. It shows that
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inferred postseismic slip is basically observed in the coseismically stressed area.
A
B
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Figure S3. Coseismic slip model of the 2004 M6.0 Parkfield earthquake derived from LS
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inversion. In plot (a), the slip is color coded and the black dots indicate the aftershocks in the
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first 5 days after the Parkfield event. Plot (b) shows the coseismic displacements observed
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(black arrows) and estimated (red arrows). The dashed line marks the location of fault trace.
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The red stars display the location of the Parkfield mainshock.
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Figure S4. Postseismic slip following the Parkfield earthquake derived from LS inversion.
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The slip amount is color coded. The equivalent moment magnitude for each postseismic
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period is close to 5.6. The three dashed contour lines correspond to coseismic shear stress
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changes of -1.5 bar (blue), 0 (black) and 1.5 bar (red), respectively. Black circles indicate the
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aftershocks occurred in the six relevant time periods, while the stars refer to the Parkfield
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mainshock.
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Figure. S5 Plot of modeling errors (in the direction of postseismic displacement at each GPS
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site) against the perpendicular distance of each GPS site to the fault. The positive distances
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correspond to the sites on the northeast side of the fault. The different symbols display the
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modeling results for different postseismic time periods. The modeling errors indicate that the
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slip model derived in the homogenous half space uniformly overestimates the displacements
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on the southwest side of the fault, while it underestimates the displacements on the northeast
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side.
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II. Statistical tests to the physical constraints applied in the Bayesian inversion
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Here, we present the statistical tests of the physical constraints which are not shown in the
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main text. In particular, the following hypotheses are tested: (1) the early postseismic creep is
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spatially separated from large aftershocks (Fig. S6a); (2) the early postseismic creep is
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spatially separated from microseismicity (Fig. S6b); (3) the postseismic creep is located in the
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same area as the large aftershocks (Fig. S6c); (4) the postseismic creep is located in the same
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area as the microseismicity (Fig. S6d). The results indicate that hypothesis (1) and (2) can be
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accepted with high significance, while (3) and (4) are rejected.
(b)
(d)
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Figure S6. Histograms showing the distribution of log-likelihood (LL) values related to 106
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randomizations of the spatial constraints. The random spatial constraints are generated
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according to the four physical constraints: postseismic creep is spatially separated from large
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aftershocks (a), separated from microseismicity (b), locates in the same zone as large
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aftershocks (c), and locates in the same zone as microseismicity (d). The black arrow
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indicates the corresponding LL value of the physically constrained model.
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