Block models
The problem (1/2) GPS velocity fields in plate boundary zones are very smooth Figure from Tom Herring, MIT What does this smoothness hide? Continuous deformation? Rigid block motions, with strain accumulation on faults?
The problem (1/2) The same holds at smaller scales Figure from Tom Herring, MIT How much can be explained by motion of (rigid) crustal blocks bounded by locked active fault accumulating elastic strain?
Solution: block models Idea: Velocity at any point on the surface is the sum of two contributions: A rigid body rotation. Elastic strain accumulation on faults Model depends on: Fault locking depth Fault coupling ratio Fault slip rates References: McCaffrey, R., Crustal Block Rotations and Plate Coupling, in Plate Boundary Zones, Geodynamics Series 30, S. Stein and J. Freymueller, editors, 101-122, AGU, 2002 (NEXT SLIDES) Meade, B.J., B.H. Hager, S.C. McClusky, R.E. Reilinger, S. Ergintav, O. Lenk, A. Barka, and H. Ozener, Estimates of Seismic Potential in the Marmara Sea Region from Block Models of Secular Deformation Constrained by Global Positioning System Measurements, Bull. Seis. Soc. Am. 92, 208-215, 2002.
Rigid body rotation Modeled using the linear, horizontal velocity of point X i within block b in the reference frame R: R r r V b (X i )= R " b # r X i Ω = rotation vector (or Euler pole) describing the rotation of the point X i in block b in the reference frame R (R can be another block or a geodetic reference frame, for example ITRF) X i = vector pointing from the center of the Earth to the point X i, at which the velocity is to be estimated.
Elastic strain accumulation Faults are discretized into patches Strain accumulation modeled as the sum of the contributions of each fault patch One coupling ratio can be associated to each patch (0 to 1) Contribution of patch p on fault f at the position of point X i is therefore modeled as: r 2 V fp (X i ) = #" fp $G $r j (Xi, X fp ) j =1 where: s fp = slip rate on the patch p, fault f; φ fp = coupling ratio on patch p, fault f; G = Green s function describing the effect at X i of a unit slip on patch p, fault f. s fpj
Fault slip rate Slip on a given patch is given by: r r r s = " # X fpj h f p where: Ω = rotation vector describing the rigid body rotation of footwall w.r.t. hanging wall: X p = position of patch p. h r r r " f = h " R # f " R
Final model Velocity at any given point is the difference between: Rigid body motion of the block carrying that point Displacement due to elastic strain accumulation on faults Velocity at point X i on the surface of block b (for P fault patches) is therefore given by: r r V X i = R V b " r V P # fp p=1
Final model For more than two blocks and more than one fault, the model becomes: r B r F P V X i = " RV b # " " V r fp b=1 Or, after substitution: r B r V X i = [ R " b # X r F P 2 $ i ]% H(X i & b)' $ $ $ {( fp % G %[ r j (X i, X fp ) h" f # X r fp ] j } where: B = number of blocks F = number of faults P = number of patches H=1 if X i belongs to block b H=0 if X i does not belong to block b. Linear model, unknowns = Ω and φ fp, solved by least-squares Fault slip rate are then derived from Ω f =1 p=1 b=1 f =1 p=1 j =1
Example 1: Oregon Data Residuals McCaffrey, 2002 Smooth velocity field Forearc behavior? Detached from overriding plate? Allow for along-strike coupling variation Solve for forearc rotation
Example 1: Oregon Coupling Slip deficit McCaffrey, 2002
Example 2: Marmara Sea Tectonic setting Anatolia / Eurasia North Anatolian fault and several branches Meade et al., 2001 GPS velocities w.r.t. Eurasia Question: can the data be fit by a block model accounting for all major known faults?
Example 2: Marmara Sea Estimate fault slip rates and velocity residuals Solid lines = preferred fault model. Dashed line in the Marmara Sea = alternate stepped geometry. Slip rate estimates and one-sigma uncertainties associated with our preferred shown as pairs. Upper values = strike-slip rates, with negative values indicate rightlateral motion; lower values = the fault-normal motion, negative values indicate opening. Solid black = straight fault geometry. Grey = stepped-fault geometry. White bell-curve = distribution of fault slip rates from Monte Carlo tests. Dashed line: slip rate as a function of locking depth Block model with straight fault geometry fits the data best Estimate slip rates consistent with geologic data Shallow best-fit locking depth (6-7 km) Meade et al., 2001
Example 3: Bay area GPS velocities (w.r.t. LUTZ) Major active faults (red lines) Some are locked, some are creeping (e.g., Hayward) Block model D Alessio et al., 2005
Example 3: Bay area Test 3 fault/block geometries
Conclusions Block models usually work very well But: Number of blocks? Size of blocks? Limit of continuum? Purely elastic, no postseismic.