We have modeled postseismic deformation from 1999 to 2003 in the region surrounding the 1999 Izmit and Düzce earthquake ruptures, using a three-dimensional viscoelastic finite element method. Our models confirm earlier findings that surface deformation within the first few months of the Izmit earthquake is principally due to stable frictional afterslip on and below the Izmit earthquake rupture. A second deformation process is required, however, to fit the surface deformation after several months. Viscoelastic relaxation of lower crust and/or upper mantle with a viscosity of the order of 2 to 5 × 1019 Pa s improves the models' fit to later GPS site velocities. However, for a linear viscous rheology, this range of values is inconsistent with highly localized interseismic deformation around the North Anatolian Fault Zone (NAFZ) that was well observed prior to the earthquake sequence. The simplest solution to this problem is to assume that the effective viscosity of the relaxing material increases with time after large earthquakes, that is, that it has a power law or Burger's body (transient) rheology. A Burger's body rheology with two characteristic viscosities (2 to 5 × 1019 Pa s and at least 2 × 1020 Pa s) in the mantle is consistent with deformation around the NAFZ throughout the earthquake cycle.
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 Deformation around the North Anatolian Fault Zone (NAFZ) has been monitored with high-precision space geodetic techniques since 1988 [e.g., Reilinger et al., 2006; Ergintav et al., 2002; Wright et al., 2001; McClusky et al., 2000; Kahle et al., 1998; Straub et al., 1997]. Data from these studies illustrate how the Earth's surface deforms around a plate boundary fault, throughout the earthquake cycle. Previously, we have modeled interseismic deformation around the 1940s rupture segment of the NAFZ, as well as coseismic and early postseismic deformation from the 1999 Izmit earthquake [Hearn et al., 2002a, 2002b; Reilinger et al., 2000; Hearn and Bürgmann, 2005]. We have concluded from these studies that the upper mantle viscosity must exceed 5 × 1020 Pa s (assuming linear viscoelasticity), that the NAFZ has an aseismic extension penetrating most or all of the crust, and that the aseismic creep along this shear zone (in the upper to middle crust) has a weakly velocity-strengthening rheology.
 Here, we model postseismic deformation following the Izmit-Düzce earthquake sequence over a longer time period (from 1999 to 2003). We develop plate boundary models incorporating viscous fault zone creep, stable frictional afterslip, viscoelastic relaxation of nonlinearly or linearly viscoelastic lower crust, and combinations of afterslip and viscoelastic relaxation. We confirm that the earliest Izmit postseismic deformation is consistent with velocity-strengthening frictional afterslip, though viscous shear zone creep is probably occurring at depth. After several months, none of the afterslip models produces high enough slip rates to explain the GPS postseismic velocities. To address this, we explore models with viscoelastic relaxation of the lower crust and upper mantle, in addition to afterslip, and we investigate circumstances under which such models can be consistent with the localized interseismic deformation around the NAFZ. The GPS data analysis and velocity fields from the first 6 years after Izmit postseismic deformation, and kinematic afterslip inversions based on these data, are described by the companion paper by Ergintav et al. .
 We use GAEA [Saucier and Humphreys, 1993], a 3-D viscoelastic finite element (FE) code, to model time-dependent postseismic deformation. The FE code solves for nodal displacements resulting from elastic deformation of the modeled volume in response to applied displacements or velocities. For the models presented here, kinematically imposed coseismic displacements across the Izmit and Düzce earthquake ruptures induce elastic stresses, which drive subsequent, distributed viscoelastic creep or fault zone slip. To calculate nodal displacements, GAEA uses a Galerkin weighted-residual method, incorporating 20-node block elements with quadratic shape and weighting functions. This allows smoothly varying fault geometries and slip distributions, and minimizes discontinuities in stresses between model elements. Distributed viscoelastic deformation is modeled by calculating the viscous strain that would occur over one time step (given the element stresses and viscosity), and using this information to update the element force vector. The same approach is employed in other FE codes, such as TECTON [Williams and Richardson, 1991] and PyLITH [Williams et al., 2005]. Postseismic fault zone creep is modeled by calculating the shear stress along a fault and using the constitutive relationship to calculate the slip per time step, and adding this to the slip at each fault node. Fault slip is imposed using the “split node” technique [Melosh and Raefsky, 1981]. Modeled surface displacements from GAEA are comparable to the elastic solutions of Okada  and Wang et al. , and the Savage  viscoelastic earthquake cycle solution [Hearn, 2003; Hearn et al., 2005; Hearn and Bürgmann, 2005].
 The finite element mesh covers a 1000 km by 900 km area, extends to a depth of 330 km, and is centered on the Izmit rupture (Figure 1). The side and bottom model boundaries are fixed and the surface is modeled as a stress-free boundary. The mesh is unstructured, with nodes at 1–2 km intervals along the Izmit rupture. The elasticity structure LAY2 from Hearn and Bürgmann  is assumed, as is the Izmit earthquake slip distribution obtained for this elastic model. In the LAY2 model, the Poisson's ratio is 0.25 and the Young's modulus is 30, 45, and 70 GPa at depth intervals of 0–16, 16–32, and 32+ km. In afterslip models, the crust viscosity is 1023 Pa s and the mantle viscosity is 1021 Pa s. The upper crust viscosity (0 to 16 km depth) is held at 1023 Pa s in all models. For the Izmit earthquake, the fault geometry is based on the geometry of the surface rupture, but represented as a single, continuous surface and extended downward. This surface is somewhat less smooth than that of Feigl et al. , which is also shown for reference on Figures 5, 7a, 7b, 9, and 11. For the Düzce earthquake, we use the slip distribution from Bürgmann et al. [2002a]. Because there are few GPS stations with postseismic velocities this area, we model this surface as vertical (which simplifies meshing). Coseismic slip is interpolated from the centers of kinematic slip model patches to the fault node coordinates in the FE mesh, and is imposed using the split node technique [Melosh and Raefsky, 1981]. The Izmit earthquake slip is imposed at t = 0 day and the Düzce earthquake slip at t = 87 days. Viscoelastic structure and fault rheology are varied in the simulations, as described in sections 4.1–4.4 and section 5.
3. GPS Velocities and Measures of Model Misfit
 We compare our model results with surface velocities at 55 GPS sites from the region surrounding the Izmit and Düzce ruptures (small circles on Figure 1). The GPS postseismic data analysis is described in detail by Ergintav et al. , who provide weekly velocities and errors at all continuous GPS sites in their analysis (which covers a broader region and a longer time span than we discuss here) in their auxiliary material. Ergintav et al. fit position measurements at survey GPS sites to a three-term logarithmic function, and they provide coefficients to this function which enable us to compute velocities and errors at any time [Ergintav et al., 2009, auxiliary material Data Set S1]. Secular GPS site velocities, computed with a block model [Reilinger et al., 2006], were subtracted from their GPS velocities to isolate the postseismic contribution.
 To measure the performance of our finite element models, we calculate the weighted-residual sum of squares (WRSS), which is a measure of model misfit to the GPS velocity data. A weighted residual is found by subtracting the finite element modeled velocity from the GPS site velocity, and dividing this residual by the one-sigma GPS velocity error. This quantity is squared and summed for both horizontal velocity components (east and north), for each GPS site, for 900 daily intervals (up to 900 days after the Izmit earthquake). Hence, this is a measure of how well the model fits the GPS velocity data throughout the entire 900-day time interval, not just a measure of misfit to a snapshot of the displacement over 900 days:
We distinguish WRSS from WRSSt, which is the measure of model misfit to GPS velocity at an individual time epoch:
Δt,s,d is the residual (GPS minus modeled) velocity, and σt,s,d is the GPS velocity error (that is, the 68% confidence interval). Subscripts t, s, and d refer to time, GPS site, and degree of freedom (east or north). In some cases, GPS sites were occupied for just part of the 900-day interval. The WRSS and WRSSt include weighted-residual contributions only from days within the time span covered by GPS observations. Vertical GPS site position data are provided by Ergintav et al.  in their auxiliary material Data Set S2, but these data are not converted to velocities. Using vertical velocities from a previous version of the Ergintav et al. analysis, we found that the vertical component contributed less than 10% to the WRSS and WRSSt values, and this contribution was insensitive to changes in model parameters. In the WRSS and WRSSt values presented here, only the horizontal velocity components are included.
Vo is the preearthquake slip rate, (a − b) is the empirical constant relating fault friction change to change in slip velocity, σ′n is the effective normal stress, and dτ is the time-dependent, earthquake-induced shear stress resolved onto the fault surface. dτ is τ − τo, where τ is the shear stress on the fault and τo is the shear stress on the fault prior to the earthquake, when the fault is assumed to have been slipping at a rate of Vo. The product of a − b and σ′n, called A-B, controls the increase in slip velocity due to a stress change. For positive A-B, aseismic afterslip may occur. For large, positive (A-B), a coseismic stress change has just a minimal effect on the rate of aseismic slip. In our models, velocity-strengthening afterslip is allowed at depths of 0–2 km and below 10 km, and values of parameter A-B for these intervals are sampled independently. We assume velocity-weakening friction (and no afterslip) at depths between 2 and 10 km the upper crust [e.g., Scholz, 1998]. In all of the models, A-B is doubled in value (relative to the lower crustal value) below the Moho. Viscous shear zone creep takes over at depths greater than 25 km if the rate of frictional afterslip falls below the viscous fault zone creep rate (assuming η/w = 1015 Pa s m−1). Figure 2a schematically illustrates the model.
4.2. Viscous Fault Zone Creep
 For the second class of afterslip models, we treat the fault zone at depths exceeding 10 km as a finite width zone with a Newtonian viscosity. The creep velocity V (i.e., the velocity of one side of the finite shear zone relative to the other) is
where η is viscosity and w is the horizontal width of the shear zone. Parameter η/w is varied in our models. V is relative to the rate at which the fault zone was creeping prior to the earthquake; η/w may increase or decrease with depth between 10 and 32 km (crust). The maximum crustal value is used at depths of 32+ km. The equation assumed for the variation of η/w with depth (z) in kilometers is
where (η/w)o is (η/w) at 10 km depth.
4.3. Viscoelastic Relaxation of the Lower Crust
 Models incorporating several configurations of Newtonian viscoelastic layers demonstrate that early postseismic deformation characteristic of large, strike-slip earthquakes is most compatible with relaxation of a thin, lower crustal layer [Hearn, 2003]. Hearn  also find that a nonlinear rheology fits the temporal evolution of postseismic deformation better than a Newtonian rheology but that unusual parameters (not consistent with available laboratory-derived flow laws) are required.
 Here, we model linear and nonlinear viscoelastic relaxation of a layer between 22 and 32 km depth (Figure 2c). In the nonlinear model, the effective viscosity is calculated for each model element as described by Hearn . The equation for effective viscosity of a model element is
where ηeff(t,elem) and σ(t,elem) are the effective viscosity and the differential stress (coseismic plus postseismic) in the model element at time t after the earthquake and σpre is the preearthquake differential stress. Since σ(t,elem) is calculated by the FE code prior to each time step and n = 3, the only free parameters are ηpre and σpre. We can compare ηpre with predictions of flow laws at lower crustal temperatures and σpre values to assess whether our required parameters are generally consistent with the rheology of crustal rocks.
 This approach does not account for any strain perturbation resulting from the background tectonic stress acting on coseismically weakened viscoelastic material. We are modeling deformation assuming
However, if we take the derivative of with respect to σ, that is, σpre + σ(t,elem), substituting equation (6) for ηeff, we get
If the coseismic stress change tensor and the preearthquake stress tensor have the same principal stress orientations, their differential stresses may be summed (so dσ(t,elem) equals the total dσ in equation (9)). In this case,
Equation (10) is just n times equation (7). If the principal stresses do not have the same orientation (as is usually the case), we cannot assume that dσ(t,elem) = dσ, and the difference between equations (7) and (10) is smaller. We provide ranges of inferred effective viscosities (assuming n = 3) in subsequent discussions of model results.
4.4. Dual-Process Models
 Using our finite element model, we may model afterslip and viscoelastic relaxation with various rheologies simultaneously. We limit our explorations to models with velocity-strengthening frictional afterslip in the crust (and viscous creep on a fairly stiff lower crustal shear zone with η/w = 1015 Pa s m−1). Figure 2d shows a schematic of a dual-process model in which both afterslip and viscoelastic relaxation are modeled. Afterslip is modeled as described in section 4.1, and the lower crust and upper mantle are modeled as Newtonian viscoelastic layers. In the lower crust and upper mantle, afterslip and viscoelastic relaxation may occur simultaneously. We use friction parameters from the best FAS model (FAS1), and we vary viscosities in the lower crust and mantle layers.
Figure 3a shows afterslip along and below the sections of the NAFZ that ruptured in the Izmit and Düzce earthquakes, due to FAS driven by coseismic stress. The maximum slip rate two months after the Izmit earthquake isabout 1.5 m a−1. Afterslip on the rupture surface falls between areas of high coseismic slip, as expected from the initial slip distribution we use, which loads these sections of the fault. The total potency of afterslip as a function of time for the best FAS model (FAS1) at depths less than 2 km is similar to kinematic slip inversion results (Figure 3b). (The afterslip rate for 180 days shown on Figure 3b is from Ergintav et al. . The slip solutions for other time epochs were done by Hearn for Ergintav et al.  but were not ultimately included.) However, FAS1 yields more localized patches of shallow slip in different locations. This could be due to the coarse representation of the coseismic slip distribution in our FE model, and by smoothing of slip (and the use of 4-km fault patches) in our kinematic inversion. Two areas of deeper slip are indicated by the FAS and the kinematic model results. The high-slip areas are shifted to the east in the kinematic slip inversion relative to the FAS models, possibly due to the inversion routine's effort to fit velocities at GPS sites southeast of the Marmara Sea, which have anomalously large southward velocities (e.g., FIS1 and CINA).
 Model FAS1 cannot reproduce deep slip at the rates required by the kinematic inversions. At 180 days, the kinematic slip inversion requires a maximum afterslip rate of about 800 mm a−1, whereas FAS1 can only yield about 500 mm a−1 (Figures 3a and 3b). By 900 days after the Izmit earthquake, the FAS1 model yields perhaps one third the kinematically required deep afterslip rate. Deep afterslip triggered by the Düzce event is greater in FAS1 and the viscous fault zone creep (VSZ) models than indicated by the kinematic slip inversions. This may be because we used a coseismic slip distribution for a dipping surface [Bürgmann et al., 2002a] but modeled the fault as vertical.
5.1.2. Parameter Values and Fit to GPS Velocity Data
Figure 4 shows the sensitivity of WRSS to parameter variation for the FAS models. (See section 3 and equations (1) and (2) for definitions of WRSS and WRSSt.) In Figure 4, the WRSS reflects just an 80-day postseismic interval, and the minimum misfit is for a model (FAS1) in which velocity-strengthening parameter A-B (defined in section 4.1) for both depth intervals is about 0.5 to 0.7 MPa. This is comparable to, to somewhat larger than, our previous estimate (0.43 MPa, [Hearn et al., 2002a]). If we use data from the first 900 days, the best fit to the GPS velocities is obtained with (A-B) = 0.5 to 1 MPa in the shallow upper crust and significantly lower (up to 0.3 MPa) at 10 to 30 km depth (shaded region on Figure 4). For the 900-day time span, the sensitivity of WRSS to variations in A-B is low; WRSS values vary by less than 10% over the A-B range shown on Figure 4, and the minimum WRSS is about 7.2 × 106. All of the models reduce the WRSS by about 65% relative to a model with no site motion, probably because owing to the form of equation (3), all of the FAS models yield very slow afterslip and surface velocities after the first few months. The FAS models (and all of the models presented in this paper) explain east–west (fault-parallel) motion far better than north–south motion. The largest reduction to east-WRSS by any of the frictional afterslip models (for the first 80 days) is 72%, but none of the models fits the north–south velocities better than a model with no site motion (that is, the north velocity residuals are larger than the north velocities). For the 900-day cumulative WRSS, model FAS1 fits the north component slightly better, reducing the north component of the WRSS by up to 36%.
Figure 5 shows snapshots of modeled and GPS velocities, as well as their difference (residuals) 2.5 years after the Izmit earthquake. Similar plots for 1 year after the earthquake are shown in auxiliary material Figure S1. Table 1 summarizes how well the FE models fit GPS data at three postseismic time epochs. In all cases, most of the contribution to the WRSSt is from continuous GPS sites, which have very small velocity errors, especially after the first few weeks. Forty-five days after the Izmit earthquake, the best FAS model (FAS1) yields a 63% reduction in WRSSt relative to a model with fixed GPS sites. After 1 year and 2.5 years, the WRSSt value is reduced by 75% and 62% relative to a model with fixed sites. For FAS1, the mean velocity is 1/2 to 2/3 of the mean GPS velocity. Smaller velocities in the optimum FAS model are an indication that the model vectors are misoriented relative to the GPS vectors. The WRSS minimization selects models with smaller velocity amplitudes to minimize the WRSS. Because of this, WRSS during the first 80 days is somewhat larger than in our earlier model [Hearn et al., 2002a]. If we use the (A-B) values from Hearn et al. [2002a] to model afterslip, we do slightly better for the first 80 days but the WRSS is larger over the full 900-day interval.
The ratio /o is the ratio of mean modeled velocity amplitude to mean GPS velocity amplitude. WRSSo is the WRSS for a model in which all GPS sites are stationary.
For the elastic dislocation models [Ergintav et al., 2009], data from 68 GPS sites (rather than 55) were used. Hence, the WRSSo values are greater. Only strike-slip motion was allowed in these dislocation models (for more meaningful comparison with FE results). Including dip slip reduced the DM's WRSS by an additional 3% at 1 year and 0.5% at 2.5 years.
 Dynamically modeled afterslip from two viscous shear zone (VSZ) models is shown on Figures 3c and 3d. As is the case for model FAS1, these models produce two main patches of afterslip, and both of these patches are west of where the kinematic model places them. In the best fitting solution where η/w increases with depth (VSZ2, Figure 3d), the slip rate patterns are similar to those from the frictional aferslip model, except that no slip is allowed in the top two kilometers of the crust. The decay in slip velocities with time is slower, however, and there is a progressive deepening of the afterslip which is less apparent in model FAS1. This difference is not enough to be resolved by the GPS velocities. In the best fitting solution where η/w decreases with depth (VSZ1, Figure 3c), afterslip is evenly distributed over a broad depth interval and is faster than for the FAS and VSZ2 models (and comparable to the kinematic slip velocities). The slipping zone narrows late in the simulation, and becomes shallower (especially below the Düzce hypocenter). The western slip patch also disappears between years 1 and 2.5.
5.2.2. Parameter Values and Fit to GPS Velocity Data
Figure 6 shows the sensitivity of WRSS to parameter variation for the VSZ models. The best fit to the GPS data is obtained in models where η/w either decreases or increases with depth (i.e., models VSZ1 and VSZ2). Most of the modest difference in fit between these models, and one in which η/w is constant with depth, arises from their superior performance in the later postseismic interval. A plot similar to Figure 6, covering just the first 180 days after the Izmit earthquake, shows a WRSS minimum in the same region, but with less of a preference for the models with depth-varying η/w. The east component of WRSS is reduced by up to 76%, and north component by up to 39%, relative to a model with no site motion. In model VSZ1, η/w increases (following equation (5)) from 1013 Pa s m−1 at 10 km depth to 1.5 × 1015 Pa s m−1 at 30 km depth. In model VSZ2, these values are 6 × 1014 Pa s m−1 and 4 × 1012 Pa s m−1, respectively. The best model with constant η/w requires a value of 5 × 1013 Pa s m−1.
 Modeled and GPS velocities, and residuals 2.5 years after the Izmit earthquake, are shown on Figures 7a and 7b. Similar plots for 1 year after the earthquake are shown in auxiliary material Figures S2 and S3. Like model FAS1, neither of the VSZ models fits the rapid, early postseismic velocities at near-field GPS sites. However, given the larger GPS velocity errors immediately after the Izmit earthquake, the WRSS penalty for this misfit is small. At 45 days, model VSZ1 reduces WRSSt by a mere 25% relative to a model with fixed GPS sites. After 1 and 2.5 years, the model reduces WRSSt by 73% and 67%, respectively, relative to a model with fixed GPS sites. VSZ2 reduces WRSSt by 61%, 85%, and 76% 45 days, 1 year, and 2.5 years after the Izmit earthquake. The better performance at 45 days is due to inhibited deep slip (and smaller modeled velocities). Both VSZ models yield velocity amplitudes which are closer to the GPS velocities than those produced by model FAS1.
5.3. Nonlinear Viscoelastic Relaxation Models
 We have also run models to assess whether relaxation of nonlinear viscoelastic lower crust (with stress exponent n = 3) could explain Izmit postseismic deformation. Figure 8 shows sensitivity of WRSS to variations in the preearthquake differential stress and viscosity. We find that for a nonlinear lower crust model to fit the GPS data as well as the best afterslip models, ηpre and σpre are about 2 to 6 × 1018 Pa s and 0.05 to 0.1 MPa. Previously [Hearn et al., 2002a], we made a rough estimate of σpre, based on equation (6), n = 3, a range of assumed, reasonable ηpre values, and the effective viscosity required by our best linearly viscoelastic lower crust model to fit the first 80 days of postseismic deformation (5 × 1016 Pa s). That estimate was 0.08 to 1.6 MPa, which is reasonable given that actual coseismic stresses in the lower crust are lower than the 5 MPa value we assumed in the 2002 calculation [Hearn et al., 2002a].
 We note that assuming n = 3 allows us to bracket admissible values of ηpre (section 4.3). To narrow this range further, and more precisely model the postseismic velocities, the postseismic deformation models should be embedded in a preexisting stress field [e.g., Freed et al., 2006b], ideally compatible with the rheology and the preearthquake deformation.
 The north component of the WRSS is reduced by up to 45%, and the east component by up to 80%, relative to a model with fixed GPS sites. As with the afterslip models, none of the models reduces the north component of the WRSS significantly during the first 6 months. Forty-five days after the Izmit earthquake, the best nonlinear lower crust model (NL1) yields a 58% reduction in WRSSt relative to a model with fixed GPS sites (Table 1). At 1 year, this reduction is 79% and at 2.5 years, the NL1 model yields just a 52% reduction in the WRSSt. Figure 9 shows modeled and GPS site velocities, and residuals 2.5 years after the Izmit earthquake. Similar plots for 1 year after the earthquake are shown in auxiliary material Figure S4. At 1 and 2.5 years, modeled velocities exceed GPS site velocities at 22 and 33 sites, respectively, and the mean velocities are comparable to the mean GPS velocities (/o = 0.86 and 1.16). Discrepancies between modeled and GPS vector orientations are similar to those produced by the afterslip models, but the larger velocities lead to larger residuals (and WRSSt values).
5.4. Dual-Process Models: Afterslip Plus Viscoelastic Relaxation
Figure 4 and Table 1 show that the FAS1 model does not yield rapid enough postseismic velocities in the vicinity of the Izmit rupture. The model's performance at later time epochs may be improved by supplementing FAS with relaxation of viscoelastic lower crust (or with viscous creep in a vertical, lower crustal shear zone with η/w lower than 1015 Pa s m−1) and viscoelastic relaxation of the upper mantle. Here, we present results for a model incorporating Newtonian viscoelastic relaxation of lower crust and upper mantle layers, with viscous shear zone η/w fixed at 1015 Pa s m−1. In this model, frictional afterslip and viscous shear zone creep are both modeled along the NAFZ, and below a depth of 25 km, the process yielding the greater slip rate over each time step is assumed to occur. Figure 10 shows the sensitivity of WRSS to mantle and lower crust viscosities for our dual-process (DP) models. The DP models outperform model FAS1 for a restricted range of mantle and crust viscosities (shaded region on Figure 10). The minimum permissible mantle or lower crust viscosity is about 2 to 5 × 1019 Pa s: for lower viscosity values, total WRSS increases dramatically. In models with the lowest mantle viscosities, a stiffer lower crust is required to fit the GPS velocity data, and for models with the lowest lower crust viscosities, a stiffer upper mantle is required. A model in which both the mantle and lower crust viscosities are both about 5 × 1019 Pa s also performs well. Hence, while our models provide firm lower bounds for the effective viscosities of the uppermost mantle and the lower crust, we cannot distinguish between the “jelly sandwich” and the “creme brulee” models of lithosphere viscoelastic structure [e.g., Burov and Watts, 2006].
 We note here that if we assume a lower η/w (5 × 1014 Pa s m−1) for the viscous shear zone, and we allow viscous creep to take over when frictional afterslip slows down below a depth of 15 km (instead of at 25 km), higher mantle and lower crust viscosities are required. For a model with identical mantle and lower crust effective viscosities, an effective viscosity value of 1020 Pa s is required.
Figure 11 illustrates that, as is the case with all of our models, the east velocity component is fit much better than the north velocity component. (This is true at all time epochs.) For the entire 900-day period, the east WRSS is reduced by up to 83%, while the north WRSS is reduced by only up to 45%. For the best model, DP1, WRSS is 6 × 106, which corresponds with an RMS error of 60. This error is large because of the very small formal uncertainties for the GPS postseismic velocities [Ergintav et al., 2009]. For example, after 1 and 2.5 years, virtually all of the GPS one-sigma velocity errors (at both CGPS and SGPS sites) are less than 1.5 and 0.5 mm a−1, respectively. (If we included the vertical velocity component, with its larger uncertainties, the RMS error would be about 30% lower.)
 Adding viscoelastic mantle and lower crust relaxation increases the magnitude of modeled, late postseismic velocities to values approaching the GPS values. This viscoelastic relaxation only modestly reduces WRSSt and WRSS values, because as the modeled velocities approach parity with the GPS velocities, angular misfits cause larger residuals (Table 1). For the best DP model (DP1), WRSSt is 63% smaller than for a model with no site motion, 45 days after the earthquake. After 1 year, the WRSSt is 84% lower, which is comparable to the performance of the best dislocation model. However, /o is somewhat smaller than it is for the dislocation model (0.77 versus 0.86). After 2.5 years, the DP1 model's WRSSt is comparable to that of the FAS1 model for that time epoch. Unlike the FAS1 model, the DP model produces a mean, modeled GPS site velocity amplitude at 2.5 years which is close to the GPS value (/o = 1.05), and modeled velocity amplitudes exceed GPS velocities at 32 of the 55 sites.
6.1. Postseismic Deformation: Afterslip Plus Viscoelastic Relaxation
 The earliest Izmit postseismic deformation requires afterslip at seismogenic depths [Bürgmann et al., 2002b; Hearn et al., 2002a]; an extremely rapidly decaying postseismic transient component (with a characteristic decay time of 1 day), and a very fast initial afterslip rate [Ergintav et al., 2009; Cakir et al., 2003]. These constraints are consistent with velocity-strengthening frictional afterslip along the NAFZ. Shallow, stress-driven afterslip which is indicated by model FAS1 is also required by kinematic slip inversions [Ergintav et al., 2009]. Furthermore, postseismic slip at the Earth's surface was identified in Golcuk after the Izmit earthquake and along the Düzce rupture after that earthquake (O. Emre, personal communication, 2006), and the amplitude of this slip was roughly consistent with the kinematic inversions [Ergintav et al., 2009]. A combination of frictional afterslip and viscous fault zone creep at depth [e.g., Mehl and Hirth, 2008] is likely at work.
 Model VSZ1 shows that coseismic stress on the NAFZ was sufficient to drive afterslip at a fast enough rate to explain postseismic deformation for perhaps a year after the Izmit earthquake. However, none of the afterslip models can explain the deformation rates beyond about the first year. This suggests a second process, likely viscoelastic deformation of the upper mantle. Our dual-process models with mantle and lower crust viscosities of the order of 1019 to 1020 Pa s provide the best fit to the first 2.5 years of postseismic deformation. If a low-η/w viscous shear zone accommodates postseismic strain in the lower crust, rather than a relaxing viscoelastic layer, the mantle effective viscosity will be at the lower end of this range. Gravity surveys [Ergintav et al., 2007] also support a deformation contribution from viscoelastic relaxation in or below the lower crust. These data, spanning 2003 through 2005, suggest increasing gravity south of the NAFZ and decreasing gravity north of the NAFZ, along a profile crossing the eastern Marmara Sea. This is opposite the expected coseismic polarity and is consistent with relaxation of a lower crust or upper mantle layer [Hearn, 2003] but not afterslip or relaxation of a midcrustal detachment. (Afterslip on a south dipping, roughly NAFZ-parallel normal fault such as the Princes Island Fault might also cause this signature.)
 The only alternative to a multiprocess model is one in which most of the rapid postseismic deformation is due to temporarily low effective viscosity in the lower crust or upper mantle around the Izmit rupture, for example, due to a power law rheology. This has been proposed for Mojave Desert earthquakes [e.g., Freed and Bürgmann, 2004]. We were able to devise a nonlinear lower crust model with n = 3 that fit the earliest Izmit-Düzce postseismic deformation about as well as the afterslip models (after the first week). However, the low preearthquake stresses and strain rates required by this model are problematic.
Figures 12a and 12b show the temperatures at which flow laws representative of crustal rocks would yield the required ηpre (2 to 6 × 1018 Pa s) at the required σpre of 0.1 MPa. To generate Figure 12, we calculated effective viscosity for a randomly sampled set of synthetic flow laws with various values of A and Q, assuming that 2.5 < n < 3.5, σpre = 0.1 MPa, and T = 300 to 1200°C. Real flow laws for quartzite and crustal rocks of various types are plotted on Figures 12a and 12b for reference. Heat flux in the vicinity of the Izmit rupture is about 60 mW m−2 [Schindler, 1997], suggesting lower crust temperatures of 450 to 600°C. In this temperature range, some wet quartz flow laws yield the expected early postseismic effective viscosities. However, the required preearthquake stress is so low that linear diffusion creep, rather than nonlinear dislocation creep, should dominate. Another problem with the NL1 model is that much higher effective viscosities (at least 1020 Pa s) are required by surface deformation rates later in the earthquake cycle (as described in section 6.3). Finally, for the required interseismic differential stress of about 0.05 to 0.1 MPa, the strain rate in the lower crust must be about 10−16 s−1. This is too low to be consistent with deformation around a major plate boundary fault (or with the highly localized surface strain around the NAFZ throughout the earthquake cycle) unless a rheologically distinct, very low viscosity shear zone below the NAFZ controls levels of differential stress [e.g., Mehl and Hirth, 2008]. In this case, however, the shear zone would deform via diffusion creep, due to grain size reduction [e.g., Mehl and Hirth, 2008; Warren and Hirth, 2006]. Unless the grain size could adjust to the coseismic stress increment by further decreasing over postseismic timescales, the situation would reduce to a VSZ model.
6.2. Comparison With Other Postseismic Deformation Studies
 We require frictional afterslip, viscous shear zone creep or viscoelastic relaxation of the lower crust, and viscoelastic relaxation of the upper mantle to explain the first 2.5 years of Izmit earthquake postseismic deformation. We have not examined all of the possibilities, for example, more combinations of FAS and VSZ creep (with lower η/w), and lateral heterogeneities, such as shear zones, in the upper mantle [e.g., Warren and Hirth, 2006]. Such lateral heterogeneities, or a transient rheology, or both, are required to make the observed postseismic deformation consistent with highly localized strain around NAFZ segments later in the earthquake cycle.
 Afterslip has been identified following earthquakes in several tectonic settings, including subduction zones [e.g., Heki and Tamura, 1997; Melbourne et al., 2002; Miyazaki and Larson, 2008]; and along thrust- and strike-slip faults within continental crust [e.g., Hsu et al., 2002; Podgorski et al., 2007]. In several cases, afterslip has been modeled as velocity-strengthening frictional slip [Perfettini and Avouac, 2004, 2007; Johnson et al., 2006, 2008, and references therein]. These studies give (A-B) values of 0.2 to 0.7 MPa, which are consistent with our results, and which likely indicate elevated pore pressures in the fault zone. Other studies, while not explicitly modeling frictional afterslip, show that afterslip occurs near the surface, or within the seismogenic zone between patches of high coseismic slip (where the fault was coseismically loaded, but is not hot enough for significant viscous creep [e.g., Podgorski et al., 2007; Miyazaki and Larson, 2008]). Given the near ubiquity of rapid afterslip following major earthquakes, it would be odd if this were not the cause of the earliest post-Izmit deformation.
 All of our models of stress-driven afterslip prior to the Düzce earthquake indicate a dominant patch of early afterslip near the Izmit hypocenter, and the maximum magnitude of afterslip prior to the Düzce event is about 2 m (depending on the model). The location and magnitude of this afterslip are consistent with kinematic afterslip results from Cakir et al. , obtained from interferometric synthetic aperture radar (InSAR) surface displacements from the first month after the Izmit earthquake. Our FE models suggest another patch of afterslip approximately centered between the Izmit and Düzce hypocenters, which correlates with a patch of afterslip centered about 50 km east of the Izmit hypocenter, in Figure 10 of Cakir et al. . Our stress-driven afterslip models and the Cakir et al.  inversions place the two afterslip patches to the west of their locations inferred from inversions of GPS data [Ergintav et al., 2009; Bürgmann et al., 2002a]. Our early afterslip results differ somewhat from what we presented in our previous study [Hearn et al., 2002a] because with the new GPS analysis and the higher-resolution mesh, we can now distinguish two patches of slip, rather than a continuous zone of afterslip extending the length of the coseismic rupture. Our VSZ models, and the kinematic slip inversions from Ergintav et al.  also suggest some pre-Düzce earthquake afterslip along or below the future Düzce rupture. This is not noted by Cakir et al. because the Düzce rupture is beyond the east limit of the region covered by their interferograms.
 Modeling studies of postseismic deformation following major earthquakes in other parts of the world also show that multiple processes contribute to postseismic deformation, and that afterslip (sometimes with other shallow processes) is likely the cause of the earliest deformation. For example, Johnson et al.  find that frictional afterslip and Newtonian viscoelastic relaxation of upper mantle below a depth of 45 to 85 km (with ηeff = 0.6–1.5 × 1019 Pa s) explain the first 4 years postseismic deformation following the 2002 Denali, Alaska earthquake. Freed et al. [2006a] explain Denali postseismic surface displacements in terms of afterslip, viscoelastic relaxation or fault zone creep in the lower crust, poroelastic rebound, and viscoelastic relaxation of the upper mantle. Freed et al. [2006a] find a mantle viscosity at the Moho (>1019 Pa s at 50 km, decreasing with depth) which is comparable to what we see, based on 2.5 years of postseismic deformation, in Anatolia. They also find, as we do, that the postseismic slip distribution is anticorrelated with coseismic slip, and that coseismic stresses along the fault cannot drive enough afterslip to explain all of the observed postseismic deformation. In a follow-up to their 2006 Denali paper, Freed et al. [2006b] conclude that the mantle beneath the Denali rupture cannot be Newtonian, and we summarize below why we arrive at the same conclusion for central Anatolia.
6.3. Interseismic Deformation Rules Out Maxwell Viscoelastic Mantle
 Moderate viscosities in the Anatolian upper mantle (i.e., less than 1020 Pa s) are consistent with marked attenuation of regional seismic phases (particularly Sn, which broadcasts upper mantle properties), and slow Vp in the mantle (7.7 km s−1) in central Turkey [Sandvol et al., 2001; Gok et al., 2000; Hearn and Ni, 1994]. (Viscosity of the lower crust is not as well constrained by geophysical methods.) Although some models of regional-scale secular deformation in the eastern Mediterranean region require a high, vertically averaged lithosphere viscosity consistent with block-like behavior over hundreds to thousands of years [Jiménez-Munt and Sabadini, 2002; Hubert-Ferrari et al., 2003; Fischer, 2006], others suggest that block-like deformation could be compatible with a moderate to low (1020 to 1019 Pa s) viscosity below the brittle-ductile transition (see Fischer , Anatolian block only, and Provost et al. ) if shear stresses are extremely low along the block-bounding faults. Upper mantle effective viscosity values required by models of postseismic deformation in other parts of the world cluster around 1018 to 1020 Pa s [Bürgmann and Dresen, 2008].
 If the mantle viscosity is moderate, as our postseismic models suggest, and linearly viscoelastic, strain around the NAFZ segments should depend strongly on the time since the last earthquake [Savage, 1990]. Earthquake cycle models incorporating Maxwell viscoelastic material below the brittle-ductile transition produce geodetically observable variations in the pattern of strain around the NAFZ at different times in the earthquake cycle, unless the viscosity is high (at least 5 × 1020 Pa s [Hearn et al., 2002b]). However, GPS observations show that strain concentration around NAFZ segments is insensitive to the elapsed time since the last great earthquake [e.g., Ayhan et al., 2002; Reilinger et al., 2006]. InSAR studies also show that strain is tightly localized around NAFZ rupture segments that have not experienced major earthquakes for over 60 years [e.g., Wright et al., 2001].
 The simplest way around this problem is to model an increase in effective viscosity with time between major earthquakes, due to either a power law or a transient rheology. Nonlinear flow laws for crust and mantle rocks at the temperatures we expect in the lower crust and upper mantle can theoretically produce an effective viscosity of the order of 2 to 5 × 1019 Pa s within a range of stress levels exceeding the coseismic stress change (Figures 13a and 13b). If the power law rheology stress exponent n is about 3, a factor of 10 effective viscosity increase over the earthquake cycle would require a background differential stress of a few MPa, comparable to the coseismic stress change. Figure 13a shows that under these conditions, some wet peridotite flow laws can provide the required viscosity. Deeper in the mantle (at 800°C), several flow laws relevant to mantle rock can give this effective viscosity. Preliminary NAFZ earthquake cycle models show that if a wet peridotite flow law is assumed, upper mantle differential stresses of the order of megapascals are reasonable [Hearn et al., 2005]. However, these models suggest surface strain rates that still vary more with time between large earthquakes than we observe (unless an unreasonably large stress exponent n is assumed in the flow law). Furthermore, the modeled, interseismic differential stresses (about 2 MPa) are too low to allow significant dislocation creep unless very large mineral grains are present. This is unlikely in a plate boundary region experiencing a high strain rate.
 The simplest alternative possibility is that the mantle or lower crust has a transient Burger's body rheology, which responds to a stress step with two relaxation times. In laboratory experiments on the deformation of mantle rocks, a low, initial effective viscosity gives way to a higher effective viscosity as strain proceeds [e.g., Post, 1977], and the two characteristic viscosities differ by about a factor of 10 or less [Chopra, 1997]. The greatest differences between the two viscosity values, and the most rapid transition from the lower to the higher effective viscosity, are seen in experiments conducted at lower temperatures [Chopra, 1997; Peltier et al., 1980].
 Earthquake cycle models incorporating a Burger's body rheology in the mantle can fit the NAFZ fault-parallel velocities over the early postseismic and late interseismic intervals [Hetland, 2006; Hearn et al., 2005]. Hetland  finds that the effective Maxwell times for the two viscosities in his Burger's Body model are 2 to 5 years (corresponding to an ηeff of 0.4 to 1 × 1019 Pa s, assuming G = 60 MPa) and at least 400 years (ηeff of 8 × 1020 Pa s). Our model DP1 indicates a greater initial ηeff of 2 to 5 × 1019 Pa s, because much of the most rapid postseismic deformation is due to afterslip (which the Hetland model does not include). In preliminary earthquake cycle models, Hearn et al.  used this initial ηeff and represented transient rheology by varying ηeff with time following equation (1) of Peltier et al. . This is
where η∞ is the long-term effective viscosity, ηo is the effective viscosity immediately after the earthquake, and to is the characteristic time over which the effective viscosity evolves toward η∞. Assuming η∞/ηo = 10, we found that velocity profiles were fairly time-invariant for most of the earthquake cycle. However, a very rapid characteristic time (to = one decade) for the viscosity evolution function is required. This is consistent with other postseismic and interseismic deformation models incorporating transient rheology [e.g., Pollitz, 2003, 2005; Hetland, 2006], but it is far smaller than timescales typical of postglacial rebound studies [e.g., Peltier et al., 1980]. More constraints on transient mantle rheologies, and specifically, the relevance of such rheologies to postseismic deformation timescales, are sorely needed. Though this model provides an elegant way to explain the late postseismic and interseismic deformation, the rheology we have assumed requires more justification.
6.4. NAFZ-Normal Extension
 Postseismic deformation following a strike-slip earthquake on a straight, vertical fault within a layered Maxwell viscoelastic medium cannot result in fault-normal extension along the fault or beyond its ends. However, Figure 14 shows that after we subtract deformation predicted by our best model, DP1, from the GPS velocity field 2.5 years after the Izmit earthquake, a residual component of fault-normal (north–south) extension is present. Horizontal strain rates shown on Figure 14 are computed by first interpolating velocities (or residuals) to a grid of equally spaced points, and then using a central difference approximation to compute the velocity gradient (and then the strain rate) at each point.
 Around the Düzce rupture, we may explain this fault-normal extension in terms of postseismic dip-slip downdip of the Düzce rupture surface. For example, slip inversions in our companion paper show that 1 year after the Izmit earthquake, dip slip continues in this location at a rate of up to 200 mm a−1 at depths of 10 to 20 km [Ergintav et al., 2009]. Residual velocity fields (after subtraction of velocities from dislocation models including dip slip) show no residual extension across the Düzce rupture. A component of postseismic dip slip is reasonable because coseismic dip-slip occurred along the north dipping Düzce rupture [Bürgmann et al., 2002a]. Our finite element models do not produce this postseismic dip slip because the Düzce rupture is modeled to be as vertical.
 Neither our models of stress-driven afterslip nor the kinematic dislocation models of Ergintav et al.  can explain a component of postseismic, north-south extension centered on the NAFZ in two other areas: the Lake Sapanca region and the Marmara Sea. This is why our finite element models and the kinematic slip inversions of Ergintav et al.  do not fit the north postseismic velocity component as well as they fit the east component. Ergintav et al.  show that this extension, which integrates to about 3 mm a−1 of north–south lengthening across the Marmara Sea, persists throughout a 6-year postseismic interval.
Cakir et al.  suggest that the NAFZ is not vertical in the Lake Sapanca region, and that the Izmit earthquake rupture had a component of dip slip in this area. This might be driving postseismic dip slip, which would cause extension normal to the rupture trace. In the Marmara Sea, there is no evidence for coseismic dip slip which could drive significant postseismic dip slip and NAFZ-normal extension. We speculate in our companion paper [Ergintav et al., 2009] that though the coseismic stress change would not encourage significant postseismic slip on these structures, coseismic softening of these faults might allow dip slip driven by the background tectonic stress. An alternative hypothesis involves a coseismic reduction in effective viscosity of lower crust or upper mantle (or broad shear zones at these depths) with a power law or transient viscous theology. This would result in the transfer of background tectonic stress to the upper crust in the Marmara region, adding a component of north–south extension to the upper crustal stress field, and driving the observed extension. The extension might occur via dip slip on east–west oriented dipping structures or by distributed extension of the elastic upper crust. We note that Ergintav et al.  tried (without success) to explain NAFZ-normal extension kinematically, with dip slip on known normal faults. Postseismic stress transfer depends on exactly how this NAFZ-normal extension is occurring, so characterizing and understanding it is important for assessing seismic hazard in the Marmara Sea.
6.5. Implications for the Mechanics of the Anatolia-Eurasia Plate Boundary
 We can begin to assemble a model for part of the Anatolia-Eurasia plate boundary, which is consistent with observations of central NAFZ coseismic, postseismic, and interseismic deformation, as well as longer-term geological slip rates.
 Stresses in the middle crust around the central NAFZ must be close to the Coulomb failure threshold to a depth of about 24 km, to permit the deep coseismic rupture noted for the 1999 Izmit earthquake [Hearn et al., 2002a; Hearn and Bürgmann, 2005]. Even if the brittle-ductile transition deepens coseismically [Rolandone et al., 2004; Ellis and Stöckhert, 2004], downward propagation of the coseismic rupture to 22 km depth is consistent with a near-Coulomb stress state in the upper to middle crust as a starting point. The earliest post-Izmit deformation is most likely due to velocity-strengthening frictional afterslip, which also requires stresses along the fault zone in the middle crust to be at the Coulomb threshold. Small values of (A-B) (= (a − b)σ′n) are required to model the high rate of early afterslip. This is consistent with deep coseismic rupture: a strongly velocity-strengthening fault zone would stop downward propagation of the coseismic rupture. For (A-B) to be low, it is likely that σ′n is low, probably because of high pore pressures along the fault zone. This would be consistent with low effective friction along at least the postseismically slipping parts of the fault.
 In the lower crust, creep along a viscous shear zone is likely, based on laboratory studies and geological evidence [e.g., Mehl and Hirth, 2008]. Distinguishing between such creep and distributed relaxation of the lower crust, based on surface deformation data, is often not practical with available GPS data [Freed et al., 2006a], and we do not address this issue in models also incorporating FAS in the upper crust and viscoelastic mantle. All of our VSZ models yield η/w of about 5 × 1013 to 1014 Pa s m−1 in the lower crust. Regardless of the width of the shear zone, if the relative rate across it is 20 mm a−1, the shear stress is about 6–7 MPa in the lower crust. A shear zone with a strongly nonlinear rheology could allow rapid postseismic strain at higher, background differential stress. However, studies of fault zone rocks indicate that shear zones are substantially softer than laboratory flow laws for lower crustal rocks would suggest [Mehl and Hirth, 2008], rendering high differential stresses in the lower crust unlikely.
 Below the Moho, we may choose between distributed flow of mantle material with a power law or transient rheology, and creep within a localized shear zone (which should broaden with depth, becoming rheologically indistinguishable from mantle rock at T > 950°C [Warren and Hirth, 2006]). If a viscous shear zone accommodates relative plate motion in the lower crust, later postseismic deformation requires some contribution from the upper mantle. A transient Burgers body rheology allows the required viscoelastic relaxation to be consistent with relatively high and invariant strain rates around the NAFZ throughout the earthquake cycle.
 Taken together, these arguments suggest that the central NAFZ is a “weak” plate boundary fault, embedded in a strong lithosphere. This idealization is consistent with some regional dynamic models of secular deformation [e.g., Jiménez-Munt and Sabadini, 2002; Fischer, 2006] and NAFZ propagation [Hubert-Ferrari et al., 2003]. It is also consistent with recent models of postseismic deformation following large strike-slip earthquakes in other regions [e.g., Johnson et al., 2008]. Laboratory studies of lower crust and upper mantle rocks also suggest weak shear zones embedded in stronger lithosphere at plate boundaries [e.g., Mehl and Hirth, 2008; Warren and Hirth, 2006], though these studies cannot address the strength of the fault zone in the brittle upper crust.
 To complete our understanding of this plate boundary, we need more geological justification for the Burger's body rheology we have assumed, specifically, for the short characteristic time over which the viscosity evolves toward its higher, “steady state” value. Also, earthquake cycle models incorporating rheologically distinct lower crust and upper mantle shear zones embedded in nonlinearly viscoelastic diabase and peridotite [e.g., Mehl and Hirth, 2008; Warren and Hirth, 2006] should be investigated. In the case of wide shear zones, such models should produce lateral variations in the effective plate thickness, which localize interseismic strain [Chery, 2008]. If these models also incorporate afterslip in the crust, they should produce high, early postseismic strain rates and localized, invariant strain rates through most of the earthquake cycle, while being consistent with recent studies of exhumed shear zone rocks and laboratory rheology experiments.
 Another factor that should be explored more in future models of postseismic deformation in this region is lateral, rheological contrasts; for example, the reduced effective elastic plate thickness in the Marmara Sea region and to the south, where the lithosphere is relatively hot [e.g., Schindler, 1997]. Such a contrast might lead to faster postseismic, viscoelastic relaxation of lower crust and/or upper mantle in the Marmara region and to the south, setting up relatively rapid southward motion of this region (and NAFZ-normal extension).
 These models will not immediately explain the disagreement between geologic (Holocene) and geodetic estimates of the central NAFZ slip rate. The Holocene slip rates are about 30% slower than the geodetic rates [e.g., Kozaci et al., 2009]. Earthquake cycle models will provide a means to test possible explanations for this discrepancy, such as long-term variations in effective fault friction or driving stresses [Chery and Vernant, 2006], or variations in lower crustal rheology related to the elevated rate of large NAFZ earthquakes during the twentieth century [Kozaci et al., 2009].
 This research was supported by NSERC Discovery Grant RGPIN 261458-07 to Elizabeth Hearn at UBC and by NSF grants EAR-0337497, EAR-0305480, and INT-0001583 to Robert Reilinger and Simon McClusky at MIT. We gratefully acknowledge all of the research groups who have participated in measuring surface deformation around the NAF since 1988. We would also like to thank James Dolan and an anonymous reviewer for their helpful suggestions, which which improved the clarity of this manuscript.