Corresponding author: M. G. Kogan, Lamont-Doherty Earth Observatory of Columbia University, Palisades, NY 10964, USA. (email@example.com)
 The 2006–2007 doublet of Mw > 8 earthquakes in the Kuril subduction zone caused postseismic transient motion in the asthenosphere, which we observed on the Kuril GPS Array in 2007–2011. Here we show that the Maxwell asthenospheric viscosity that best fits the geodetic data increased by nearly an order of magnitude over the interval of 4 years, from 2 × 1017 to 1 × 1018 Pa s. These effective values of viscosity can be explained by a power law rheology for which strain rate is proportional to stress raised to a power n > 1. The apparent change in viscosity can also be caused by other factors such as coupling between afterslip and viscoelastic flow. The open and intriguing question in connection with postseismic data after the Kuril earthquake doublet is the magnitude of the long-term asthenospheric viscosity, which shall be revealed by continued observations. An asthenosphere with viscosity of about 1 × 1019 Pa s is favored by the postseismic deformation still observed several decades after the 1960 Chile and 1964 Alaska Mw ~9 earthquakes. However, postseismic deformation associated with the 1952 southern Kamchatka Mw ~9 earthquake currently is not observed in the northern Kurils, an indication that the long-term asthenospheric viscosity in the Kurils is lower than that in Chile and Alaska.
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 Transient deformation following great subduction earthquakes is controlled by the rheology of the upper mantle and of the seismogenic fault. The physical mechanism of postseismic deformation can be understood from geodetic observations performed over years and decades following the earthquake. The Maxwell asthenospheric viscosity deduced from the data is an effective value to describe the dominant physical mechanism, dislocation creep for large earthquakes, or diffusion creep for smaller earthquakes [Bürgmann and Dresen, 2008]. For dislocation power law creep, the viscous strength depends significantly on the observation time interval with respect to the earthquake so that the effective Maxwell viscosity increases with time [Freed et al., 2006]. This interpretation implies that the changes in stress during the postseismic period are significant; otherwise, the effective viscosity would remain nearly constant.
 The long-term Maxwell viscosity of the upper mantle was evaluated from GPS observations performed several decades after the occurrence of two Mw ~9 megathrust earthquakes in the twentieth century: the 1960 Chile and the 1964 Alaska events [Hu et al., 2004; Suito and Freymueller, 2009]. The postseismic signal following these events was explained by a Maxwell mantle with viscosity on the order of 1019 Pa s in the wedge above the subducting plate. In contrast, long-lasting postseismic deformation was not observed in the region of the 1952 Mw ~9 Kamchatka megathrust earthquake (southern Kamchatka and northern Kuril Islands) from GPS observations performed 50 years later [Bürgmann et al., 2005; Steblov et al., 2008]. This suggests that the long-term asthenospheric viscosity in the Kuril-Kamchatka subduction zone is lower than that in the regions of Chile and Alaska.
 The initial, first-year postseismic deformation observed following the 2004 Mw 9.1 Sumatra earthquake was explained by viscoelastic relaxation in a biviscous Burgers upper mantle including a transient viscosity ~1017 Pa s and a long-term viscosity ~1019 Pa s [Pollitz et al., 2008]. Such a low transient viscosity does not affect deformation in later years [Kogan et al., 2011]. However, deformation in the near field, especially on Andaman Islands, required a dominant contribution of frictional afterslip downdip and updip of the coseismic rupture [Chlieh et al., 2007; Hu and Wang, 2012; Paul et al., 2012]. Duration of measurable afterslip varies widely; for example, the surface displacement attributed to afterslip is modeled by the logarithmic function with a time constant ranging from days [Hsu et al., 2006] to a year [Hu and Wang, 2012]. The 1964 Alaska earthquake provides evidence for longer-lived afterslip. Brown et al.  suggested a relaxation time of 2–3 years based on repeated leveling, and Suito and Freymueller  found that the uplift history of the Kodiak City tide gauge was best explained by afterslip with a ~10 year relaxation time. In the case of the Alaska event, the geometry of the slab meant that viscoelastic relaxation produced only a small vertical motion at these sites, so that afterslip dominated the uplift history.
 In 2006, the Russian Academy of Sciences installed an array of GPS stations on islands of the whole Kuril subduction zone, as a collaborative project with Lamont-Doherty Earth Observatory and University of Alaska Fairbanks. Subsequent to installation, on 15 November 2006, a thrust event with Mw = 8.3 occurred in the central Kurils previously known as a seismic gap in terms of great earthquakes (Figure 1). On 13 January 2007, an extensional event with Mw = 8.1 followed in the Pacific plate, about a hundred kilometers away from the thrust event. We refer to these events as the 2006 and 2007 earthquakes, respectively. In our earlier analysis of 2 years of GPS time series since 2007.5, we found a rapid postseismic response explained with the Maxwell viscosity of the asthenosphere ηa = 5 × 1017 Pa s [Kogan et al., 2011]. Here we analyze the postseismic response using a twice longer time series and invert the data for a sequence of yearly effective values of ηa.
2 GPS Observations and Processing
2.1 GPS Data
 The Kuril GPS Array has evolved from a single site established in 2005 to its current configuration of eight continuously operating stations (cGPS) and three survey-mode stations (sGPS), placed along the whole Kuril arc from Japan to Kamchatka (Figure 2). In 2005, the first ever GPS measurements were performed on the Kuril Islands at sGPS site VDLN (the southern tip of Urup I.). In 2006, five cGPS stations were installed along the Kuril arc several months before the great 2006–2007 Kuril earthquakes: KUNA (Kunashir I.), SHIK (Shikotan I.), ITUR (Iturup I.), URUP (Urup I.), and PARM (Paramushir I.). These sites were complemented, also before the earthquakes, with measurements at several sGPS sites: station VDLN on Urup I.; three stations in the central Kurils: KETC (Ketoy I.), MATC (Matua I.), and KHAC (Kharimkotan I.); and station PRM1 on the southern tip of Paramushir I. In 2007, following the 2006–2007 great Kuril earthquakes, stations KETC and MATC were upgraded to continuous operation, other sGPS sites were reobserved, and new sGPS site KOST was observed in the central Kurils (Simushir I.). In 2008, the sGPS station KHAC was upgraded to continuous operation, the last step in construction of the Kuril GPS Array. All cGPS stations were equipped with NetRS receivers and Zephyr Geodetic (TRM41249.00) antennas. This equipment on station KUNA was replaced with a NetR5 receiver and a Zephyr Geodetic II (TRM55971.00) antenna in 2010. In 2008–2011, measurements at sGPS sites VDLN, KOST, and PRM1 were repeated typically every year. Timespans of observations for the period 2006–2011 at each station of the array are shown in Figure 3.
2.2 Processing of GPS Observations
 The Kuril GPS Array is located along the margin of the Sea of Okhotsk on the hanging wall of the Kuril megathrust (Figure 1). DeMets  analyzed earthquake slip vectors in the Kurils and concluded that the Sea of Okhotsk either belongs to the North American plate (NAM) or moves insignificantly (slower than 5 mm/a) with respect to it. Bürgmann et al.  and Kogan and Steblov  analyzed GPS velocities in the Sea of Okhotsk region and concluded that the velocities of sites far from the Japan-Kuril-Kamchatka subduction zone are less than 2–3 mm/a relative to NAM. Here we assume that the Sea of Okhotsk belongs to NAM and that the movement of Kuril GPS sites relative to NAM represents the surface deformation of the overriding plate of the Kuril subduction zone.
 All GPS observations on the Kuril Array were processed over the time interval 2006.5–2011.5 using the GAMIT/GLOBK software (http://www-gpsg.mit.edu/~simon/gtgk/index.htm) and final satellite orbits available from Massachusetts Institute of Technology (MIT) (ftp://everest.mit.edu/pub/MIT_SP3). Details of the software package are given by Herring et al. . Next, we summarize the computation steps specific for the Kuril network.
 In step 1, we processed the 24 h spans of observations on the Kuril Array together with the 10–12 nearest sites of the International Global Navigation Satellite Systems Service (IGS) that have consistently uninterrupted time series, using GAMIT. This cluster of IGS stations was chosen so that they are widespread in azimuths relative to the Kuril Array to ensure good network geometry. We did not require the positions of IGS sites in the GAMIT solutions to be well determined in the International Terrestrial Reference Frame (ITRF): for example, these sites could be fairly recent and their velocities poorly known; also, their positions could be disturbed by earthquakes. We included these stations only to enable a strong connection of the Kuril Array to the global IGS network and, hence, connection with the stations well determined in ITRF2008 and unaffected by the seismic activity. The solution resulting from step 1 contains station positions, satellite orbits, and Earth orientation parameters (EOP). We included corrections for solid-Earth tides and for ocean loading. The regional GAMIT daily solutions are considered as loosely constrained in the sense that the covariance matrix allows the solution components to be constrained later from combination with a global solution and from imposing the reference frame condition.
 In step 2, the Kuril regional daily solutions were treated as quasi-observations and combined using a Kalman filter (realized in the GLOBK part of the software) with global daily solutions available from MIT (ftp://everest.mit.edu/pub/MIT_GLL). We call the result of step 2 the combined global solution. We retained in each combined solution only station positions and their covariance matrix of the Kuril network and of 23–25 globally distributed core IGS stations; orbits and EOP were dropped because they are reflected in station positions due to the global spread of the combined network [Dong et al., 1998]. The motion of selected core IGS stations is well determined in ITRF2008, and the stations are chosen such that their positions were not disturbed by any earthquake at a 1 mm level.
 In step 3, we constrained each daily combined global solution with the reference frame ITRF2008 updated at MIT. The reference frame was transformed such that station positions and velocities are given relative to the North American plate, NAM. Transformation was based on the value of the rotation vector of NAM in ITRF2008 estimated by Altamimi et al. . The result of processing steps 1–3 are the time series of daily positions of stations of the Kuril network relative to NAM. These time series constitute our data set in the further analysis (Figures 4 and 5).
 We also tested the two-step processing: (1) performing daily GAMIT solutions from combination of the Kuril Array and globally distributed IGS stations well determined with respect to ITRF2008; (2) constraining the solution with ITRF2008. Daily station positions from a three-step and two-step solutions agree at a 1 mm level, but the three-step solution saves the total computation time by a factor of 2–3. This happens because processing using GAMIT, the most time-consuming procedure, is much faster with 10–12 regional IGS sites rather than with 23–25 global IGS sites.
3 Observed Surface Deformation After the 2006 and 2007 Earthquakes
3.1 Horizontal Deformation
 In the following text, the terms “near field” and “far field” are used to describe GPS station locations closer than and farther than ~200 km in the along-arc direction with respect to the epicenters of the 2006–2007 Kuril doublet. Hence, the near field denotes the region where a postseismic transient signal dominates, and the far field denotes the region where interseismic motion at a constant speed dominates.
 GPS observations after the 2006–2007 earthquake doublet exhibit three patterns of surface deformation (Figures 4 and 5):
Rapid transient motion during the first several months in the direction opposite to subduction. An early rapid postseismic transient of ~0.1 m was recorded at station URUP in the near field (Figure 5). Other stations in the near field were upgraded to continuous operation in mid-2007 (KETC and MATC) and in mid-2008 (KHAC) and therefore did not record this transient. For stations KETC and MATC, the total displacement between the latest preseismic epoch and first postseismic epoch is about 0.8 m; for station KHAC, such total displacement is 0.2 m. We cannot reliably separate these deformations into coseismic and postseismic contributions; the models indicate, however, that the effect of the 2006 thrust earthquake dominates, with roughly equal contributions from the coseismic static offset and the early postseismic transient displacement [Kogan et al., 2011]. A rapid postseismic transient was observed following only the 2006, but not the 2007, earthquake. In the far field, the initial transient deformation before mid-2007 was recorded at stations ITUR, SHIK, and KUNA, but the signal is very small at SHIK and KUNA (Figure 4). To summarize, due to a lack of continuous observations at many stations prior to mid-2007, we have only limited evidence of the details of the early postseismic motion usually associated with frictional afterslip or with transient viscosity in the Burgers body rheology.
Postseismic transient motion for 4 years since mid-2007 in the direction opposite to subduction. This signal was recorded at four cGPS (URUP, KETC, MATC, and KHAC) and one sGPS (KOST) stations in the near field, over a region extending for ~500 km along the Kuril arc (Figure 5). We denote the time intervals 2007.5–2008.5, 2008.5–2009.5, 2009.5–2010.5, and 2010.5–2011.5 as years 1, 2, 3, and 4, respectively. The highest speed of about 90 mm/a over year 1 was recorded in the center of the region at stations KETC, MATC, and KOST. At stations URUP and KHAC, which bound the near field, the speed is much smaller. Surface displacements in the near field decreased significantly in subsequent years with the highest speed of 20 mm/a in year 4. Such type of postseismic deformation is commonly associated with distributed viscoelastic flow in the weak asthenosphere [Pollitz et al., 2008]. These horizontal motions can also be explained by afterslip, but the vertical motions observed during 2007.5–2009.5 are incompatible with a model in which afterslip caused the observed horizontal signal [Kogan et al., 2011].
Motion at a constant speed of 20–30 mm/a in the direction of subduction both before and after the Kuril earthquakes caused by elastic strain accumulation. Elastic strain is driven by the mechanical locking between overriding and subducting plates at the subduction interface. This effect, commonly called the interseismic motion, is conspicuous in the far field of the 2006–2007 earthquakes (stations farther than ~200 km from the epicenters: KUNA, SHIK, ITUR, VDLN, PRM1, PARM, and PETS; see Figure 4). The effects of the locked interface should also exist in the near field as evidenced by occurrence of earthquakes; however, here the interseismic motion is overshadowed by much higher rate postseismic deformation.
3.2 Static Offsets in the Interval 2007.5–2011.5
 This study is focused on the long-term postseismic deformation caused by the 2006–2007 Kuril earthquakes over the interval 2007.5–2011.5. The 15 January 2009 Mw 7.4 earthquake (Figure 2) in the central Kurils produced static coseismic offsets of about 10 mm at stations KETC and MATC. We discarded the day of the earthquake from the GPS data and corrected the time series after that day for the coseismic offsets. The correction is a constant vector estimated by least squares and added to a station position to remove the offset.
 Surprisingly, the 11 March 2011 Mw 9.1 Tohoku earthquake did not result in observed coseismic offsets in the near field of the 2006–2007 Kuril earthquakes above the GPS noise level 1–2 mm (Figure 6), and we made no corrections for this effect. However, the Tohoku earthquake produced observed coseismic offsets as large as 10 mm in the southern Kuril Islands (the far field).
 Over the period of operation of the Kuril GPS Array, there has been a single volcanic eruption in the region: from Sarychev Peak Volcano on Matua Island. The associated motion of station MATC located 5.7 km away from the volcano started on 12 June 2009 and continued for 3 days at a nearly constant rate reaching a cumulative displacement of 31 ± 2 mm both horizontally and vertically. The motion stopped abruptly by 15 June. Horizontally, the station was displaced toward the crater, while vertically, the station subsided. Modeling the deformation using a Mogi source allows us to estimate the location of the deflating magma chamber at a depth of D ≈ 6 km from the ratio of vertical (δV) and horizontal (δH) displacements: D = H ⋅ (δV/δH), where H is a horizontal distance of the station from the volcano. We examined the time series rotated into components radial to the volcano and orthogonal to that direction to see if additional volcano-related signals were present. Aside from the eruption, the time dependence of the displacements in these two directions was similar, and we could find no evidence for significant volcanic deformation outside of the eruption itself. Therefore, we discarded the data from the 3 days of the eruption and corrected the posteruptive GPS time series to remove the step-like displacement due to the eruption.
4 Modeling Postseismic Deformation
 We analyze postseismic displacement of all five stations in the near field of the 2006–2007 great Kuril earthquakes relative to the North American plate using daily GPS time series. To isolate the postseismic signal, we seek to correct the observations for the background interseismic signal, that is, the constant station velocity caused by the strain due to subduction. This signal was not directly measured at the near-field stations because the preearthquake observations spanned only an interval of several months before the 2006–2007 earthquakes, or because the observations started after the earthquakes. The interseismic velocity was reliably estimated from several years of time series at stations in the far field; it varies by about 50% among stations at similar distances from the trench (that is, at all stations except SHIK). However, it is risky to interpolate the interseismic velocity to the near field since it can vary along the subduction zone due to variations in the strength of locking at the subduction interface [Fournier and Freymueller, 2007; Freymueller and Beavan, 1999].
 From analysis of trench-parallel gravity anomalies (TPGA) and of seismicity in the subduction zones, Song and Simons  suggested that the strength of locking is smaller at the subduction segments with positive TPGA. If this hypothesis holds, we could expect smaller interseismic velocities of stations in the near field of the 2006–2007 Kuril earthquakes compared with velocities in the far field because the near field coincides with the region of strongly positive TPGA. To address this source of uncertainty, we considered three approaches to correct the GPS time series for the interseismic background velocity Vint in the near field: (A) set Vint = 0; (B) set Vint equal to the mean interseismic velocity in the far field; (C) set Vint equal to 50% of the mean interseismic velocity in the far field.
 To suppress the unwanted short-period (periods of several days and weeks) and seasonal (periods of several months) variations in the signal, we fit the multiyear time series of GPS displacements with polynomials of degrees 1, 2, or 3 (Figures 4 and 5). The degree 1 parameterization (linear function) is used in the far field to match the interseismic motion, degrees 2 and 3 (quadratic and cubic functions, respectively) are used in the near field to match the postseismic transients. The root-mean-square (RMS) deviations of the data from best fit functions are about 2 mm. We compare our rheological models of postseismic displacement with these best fit analytical approximations of the data over the interval 2007.5–2011.5 at points 2007.5, 2008.5, 2009.5, 2010.5, and 2011.5.
 We assume in this study that viscoelastic relaxation with a Maxwell body rheology is appropriate to describe the postseismic motion observed on the Kuril GPS Array for the period starting about 6 months after the 2007 earthquake. The rapid initial relaxation, predicted by a Burgers rheology with constant and transient viscosities [Pollitz et al., 2006b], dies out in the first several months after the 2006 earthquake [Kogan et al., 2011]. We ignore contributions from afterslip because the model of afterslip that best fits the observed horizontal motion in the interval 2007.5–2009.5 predicts subsidence, opposite to the observed uplift [Kogan et al., 2011]. If afterslip was not the dominant component during 2007.5–2009.5, it is very unlikely that it would be significant after 2009.5. However, this assumption cannot be strictly proved because a decay time of afterslip is not well known [Pritchard and Simons, 2006].
 We model viscoelastic relaxation caused by coseismic slip using a combination of two open-source software packages:
 The VISCO1D code [Pollitz, 1997] which represents the deformation in terms of a spherical harmonic expansion of normal modes of a spherically stratified, self-gravitating, compressible, viscoelastic Earth.
 The RELAX code [Barbot and Fialko, 2010a, 2010b] which is based on a Fourier-domain Green's function in a half-space and on equivalent body-force representations of postseismic deformation. Computation by RELAX is much faster than by fully numerical approaches such as the finite element method (FEM) although it is difficult to apply RELAX to the very detailed rheological structures which can be incorporated in FEM.
 In contrast to spherically symmetric code VISCO1D, using RELAX allows us to consider the three-dimensional rheology of the subduction zone, including a dipping elastic slab and a low-viscosity mantle above the slab. On the other hand, RELAX cannot resolve long wavelength variations in deformation as accurately as VISCO1D. By superposing the results of these two programs, we can approximate a 3-D solution and examine a wide range of viscosity models without the large computational expense of a 3-D spherical finite element model. The accuracy of this approximation will be discussed later in the paper.
 Our approach is, first, to evaluate the deformation in simple spherically symmetric Earth models using VISCO1D for each year in the timespan 2007.5–2011.5. The best fit asthenospheric Maxwell viscosity is computed by grid search in the range 1 × 1016 to 1 × 1019 Pa s under the three different assumptions of the interseismic deformation outlined in the previous section. We prefer to start the analysis using such a simple rheological model because the slab structure (thickness, dip angle, curvature, and viscosity) is not well known. We then use RELAX to estimate corrections to postseismic surface deformation for the slab structure and to evaluate whether such corrections are large or significant.
 We measure the agreement between the GPS time series and the fitting model by a merit function defined as , the reduced chi-square with the zeroth order regularization, or the reduced chi-square for short [Press et al., 1994, p. 797]:
where dE and dN are the GPS displacements to the east and to the north, respectively; mE and mN are the model displacements to the east and to the north, respectively; M is the number of GPS stations; σE and σN are standard deviations of east and north components. Our definition of σE and σN accounts for the uncertainty in both the data and the model:
where σE _ obs is the standard deviation of the observed east component; r is the empirical factor set to 0.2 in this study. An analogous relation is used for the north component.
4.2 Viscoelastic Relaxation in a Spherically Symmetric Earth
 In modeling the viscoelastic relaxation using VISCO1D, the viscoelastic Earth model was adopted to be the same as in Kogan et al. , following Pollitz et al. [2006a]. It comprises 68 layers differing in elastic parameters grouped as four sets differing in Maxwell viscosity (Figure 7a). The Maxwell viscosity of the asthenosphere was the single varied parameter. The shear modulus of the asthenosphere was set to 67 GPa, as in Kogan et al. .
 Individual coseismic displacements from each of the 2006–2007 Kuril earthquakes are clearly visible in the observed time series [Steblov et al., 2008] because such signals occur within minutes and hours, while the events are separated by 2 months. In contrast, the postseismic displacements from both events in the 2006–2007 doublet overlap in the time series because their durations are at least years. It is possible, however, to model the viscoelastic relaxation from each event using its coseismic slip as an input and then to estimate the combined effect of both events. For the 2006 earthquake, we use the coseismic slip distribution of Kogan et al.  from inversion of GPS data on a 3 × 8 grid. This slip model compares well with the preferred model N6A of Steblov et al.  from inversion on a smaller grid. Influence of uncertainties in the slip model on a postseismic model for the 2006 earthquake was discussed by Vladimirova et al. . For the 2007 earthquake, we use the coseismic slip distribution J7B of Steblov et al.  which is based on a rupture width Wcoseis = 25 km.
 The postseismic displacements predicted for the 2006 and 2007 earthquakes differ significantly in azimuth. Accoring to the viscoelastic model, stations KOST, KETC, and MATC nearest to the hypocenters move to the southeast for the 2006 event (Figure 8a); in contrast, the stations move to the north for the 2007 event (Figure 8b). Observed motion is well matched by the postseismic model for the 2006 event, while the 2007 event produced only a small effect on all stations. The magnitude of postseismic motion predicted for the 2007 event is generally small compared to that of the 2006 event, but is quite sensitive to a value of Wcoseis, with a smaller speed of postseismic motion for a smaller Wcoseis. This is a poorly constrained parameter in the inversion of GPS coseismic offsets for the rupture model of the 2007 earthquake: values of Wcoseis = 25 km and Wcoseis = 50 km for a steeply dipping rupture plane both result in good fits to observed GPS coseismic offsets [Steblov et al., 2008]. Seismological coseismic inversion predicts most of the slip for the 2007 event at a shallow depth less than 20 km [Ammon et al., 2008] with the equivalent rupture width of 25 km. Our choice of the coseismic slip model J7B agrees with this seismological constraint.
 We model postseismic displacements in four annual segments over the entire interval 2007.5–2011.5 (Figures 9 and 10). In each time segment, misfits between the data and the model are calculated for a range of asthenospheric viscosities from 1 × 1016 to 1 × 1019 Pa s (Figure 11). Contributions to postseismic displacement from each earthquake in the 2006–2007 doublet are tested in the following manner: the effect of only the 2006 earthquake (Figure 9) and the summed effect of both earthquakes (Figure 10) are compared with the data. To correct the GPS time series for the interseismic background in the near field, we considered three alternatives: 0% correction, 50% correction, and 100% correction (see section 'Approach').
 According to the curves of misfit, the data favor the viscosity increasing with time from 2 × 1017 Pa s (year 1) to 1 × 1018 Pa s (year 4) (Figure 11). This result holds regardless of whether or not the effect of the 2007 earthquake is included. It also holds regardless of the interseismic correction.
4.3 Effect of the Slab and of Lateral Variations in Viscosity
 The effects of the slab and of reduced viscosity in the mantle wedge above the slab on the surface deformation were studied by Pollitz et al.  using coupled normal modes and by Suito and Freymueller  and Hu and Wang  using the finite element method. Here we consider both effects by combining: (a) the spherical harmonic expansion of normal modes (VISCO1D) and (b) the equivalent body-force representation in a half-space (RELAX).
 Solutions by use of RELAX and VISCO1D should agree for laterally uniform Earth models with similar viscoelastic layering. Disagreement arises for the following reasons: (1) a spherical Earth in VISCO1D versus a half-space in RELAX; (2) differences in the range of resolved wavelengths, with VISCO1D better resolving long wavelengths and RELAX better resolving short wavelengths; (3) effect of elastic layering accounted for in VISCO1D according to Dziewonski and Anderson  while omitted in RELAX (Poisson medium is adopted). We compared postseismic displacements calculated using VISCO1D and RELAX in two cases: (1) a simple synthetic thrust fault considered by Rundle  and used for benchmarking by F. F. Pollitz (http://earthquakes.usgs.gov/ research/software/#VISCO1D) (Figures S1 and S2 in the supporting information); (2) the rupture model of the 2006 Kuril earthquake (Figure S3). A laterally uniform rheology was chosen. In both cases, departures between RELAX and VISCO1D are 10–20% at sites with the largest displacement. The azimuths of displacements agree well. Probably, most of the discrepancy between the two curves in Figure S2 is due to neglect of the vertical layering in elastic parameters in RELAX.
 To suppress systematic differences between the predictions of VISCO1D and RELAX, we correct the solution from VISCO1D for the effect of the slab in two steps. In step 1, we take a difference of two solutions by RELAX, with and without the slab, to estimate the effect of the slab. In step 2, we add the result of step 1 to the spherically symmetric solution from VISCO1D.
 Details of the solutions from RELAX are as follows. The structure without the slab has three horizontal layers: the elastic lithospheric lid with thickness Tlid = 62 km, a Maxwell asthenosphere to a depth of 220 km, and a Maxwell mantle below the asthenosphere to a depth of 512 km, the depth of the computational domain (Figure 7b). The value of Tlid was chosen to agree with flexural studies of the Pacific plate near the Kurils [Levitt and Sandwell, 1995]. The viscosity of the asthenosphere was varied. The viscosity of the underlying mantle was set to 1 × 1020 Pa s. Deformation was computed on a rectangular three-dimensional domain with grid 512 × 512 × 256 and with grid step 4 km in the horizontal dimensions and 2 km in the vertical dimension (Figure 12). This structure mimics the rheology of the spherical Earth model adopted in computations by the use of VISCO1D to a depth of 670 km. The time step of numerical integration is 1/10 of the Maxwell relaxation time of the asthenosphere. The computational domain is centered in map view on the center of the coseismic rupture of the 2006 Kuril earthquake. The rupture model of 8 × 3 patches is the same as in Kogan et al. . The elastic moduli are set λ = μ = 67 GPa.
 We add an elastic slab dipping 22° downdip the coseismic rupture to a depth of 220 km, the bottom of the asthenosphere. The assumed dip approximates the geometry of the subduction interface beneath the Kurils to the chosen depth [Hayes et al., 2012]. The slab length was set to 920 km, and the slab thickness Tslab was set to 62 km. This is the largest reasonable Tslab because the slab cannot be thicker than the lithospheric lid. The actual value of Tslab is not well known, and we therefore also tested a 42 km thick slab with the other geometrical parameters unchanged.
 Figures 13a–13c show postseismic displacement fields for the interval 2007.5–2008.5 (year 1) calculated with the following rheologies: laterally uniform (Figure 13a), laterally nonuniform with the 42 km thick slab (Figure 13b), and laterally nonuniform with the 62 km thick slab (Figure 13c). In Figures 13a–13c, the Maxwell viscosity of the asthenosphere was set to 2 × 1017 Pa s, the best fitting value from the solution for the laterally uniform Earth by use of VISCO1D (Figure 11). Overall, the patterns of postseismic displacement are similar for all three rheological structures. We are primarily interested in displacements in the Kuril island arc, the only region where they can be observed geodetically. We focus on the islands in the center of the map where the signal is the largest. For the model with a 42 km thick slab, the horizontal displacement is reduced by less than 10% relative to the laterally uniform rheology. For the 62 km thick slab, the horizontal displacement is reduced by about 20%.
 In Figure 13d, we consider a combined effect of the 62 km thick slab and of a lateral variation in the asthenospheric viscosity from 2 × 1017 Pa s in the wedge to 1 × 1019 Pa s beneath the Pacific Ocean (see Figure 7b for the rheological structure). Comparison of Figures 13d and 13c shows that the damping effect of assigning a higher viscosity to the Pacific asthenosphere is about 40% in the horizontal postseismic displacement of the Kuril Islands. If the viscosity of 1 × 1019 Pa s was set uniformly in the whole region on both sides of the slab, the postseismic displacement would be near zero [Kogan et al., 2011]. Therefore, the postseismic motion observed on the Kuril arc is mostly controlled by the asthenospheric viscosity within the wedge.
 Figure 14 shows observed and modeled postseismic displacements of GPS stations for the rheological structure with the 62 km thick slab during years 1 and 4. Misfit is computed with respect to observations corrected and uncorrected for the interseismic motion. The values of reduced chi-square are all small (2–5) for models with the slab (Figure 14) and without it (Figure 9). Also, the values of are similar for models with the slab and without it as long as the time interval, asthenospheric viscosity, and interseismic correction are kept the same (for example, compare Figures 14a and 9b or Figures 14c and 9c). Therefore, we regard postseismic modeling with the laterally uniform rheology as a reasonable proxy for more realistic nonuniform rheology of the subduction zone.
4.4 The Postseismic Vertical Signal Caused by the Slab
 There is a general problem with the vertical GPS component: it is three to four times noisier than the horizontal components at short periods (several days) and at long periods (pronounced seasonal fluctuations). It is worth noting that vertical GPS observations were not used in the published analysis of postseismic deformation after several great earthquakes: the 1960 Chile [Hu et al., 2004], the 1964 Alaska [Suito and Freymueller, 2009], and the 2004 Sumatra [Hu and Wang, 2012].
 Modeled vertical displacements are significantly damped if the slab is included in the rheological structure (Figure 13). There is a specific problem with understanding the vertical signal over the Kuril GPS Array: its stations are located on the steep slope from the region of postseismic uplift over the Sea of Okhotsk to the region of postseismic subsidence southwest of the island arc (Figure 13). Therefore, the amplitude and even the direction of modeled vertical motion (uplift versus subsidence) are sensitive to details of the rheological structure, such as parameters of the slab and location of the coseismic rupture. This sensitivity to details limits capability of the vertical signal to serve as a diagnostic test to discriminate among different rheologies.
Kogan et al.  analyzed cumulative vertical displacements observed on the Kuril GPS Array for the time period 2007.5–2009.5 and showed that these data are incompatible with afterslip because the model of afterslip predicts subsidence, opposite to the observed uplift. Kogan et al.  also showed that viscoelastic relaxation for a Maxwell asthenosphere with viscosity 1 × 1018 Pa s predicts general uplift of 0–40 mm in reasonable agreement with the data.
 Here we show important aspects of the vertical component observed for the time period twice longer than that in Kogan et al. . Figure 15 shows plots of vertical time series on the Kuril GPS Array. Only the mean vertical velocity over the total 4 year period is resolved. Figure 16 compares observed vertical velocities with predictions from two viscoelastic models: a laterally uniform model and a model with the slab (see deformation fields in Figures 13a and 13c). Formal agreement estimated by the reduced chi-square clearly favors the model with the slab, although deviations of the model from the data are still significant. The misfit between the models and the data also occurs from the unknown interseismic vertical signal which is uplift of 5–10 mm/a [Aoki and Scholz, 2003].
5 Discussion and Conclusions
 The GPS observations in the Kuril subduction zone showed rapidly decreasing deformation for the period 2007–2011 after the 2006–2007 doublet of great earthquakes. The highest observed postseismic speed of about 90 mm/a in 2007.5–2008.5 decreased significantly in subsequent years, with the highest speed of 20 mm/a in 2010.5–2011.5. Such type of postseismic deformation is commonly associated with distributed viscoelastic flow in the asthenosphere.
 Analysis of viscoelastic relaxation after the Kuril doublet of earthquakes could be affected by uncertainties in corrections for the interseismic motion and for the 3-D rheological structure. We found that the best fit Maxwell asthenospheric viscosity ηa only weakly depends on these uncertainties. The best fit value of ηa grows by a factor of 5 over the interval of 4 years, from 2 × 1017 to 1 × 1018 Pa s. This suggests that the computed viscosities are effective values. They can be explained by a power law rheology, for which strain rate is proportional to stress σ raised to a power n > 1. Effective viscosity for such rheology is ηeff = Cσ1 − n where C is a power law parameter, assumed constant during a single postseismic cycle [Freed et al., 2006]. The power law exponent n is typically believed to be in the range 2.5 to 4 [Kirby and Kronenberg, 1987]. As coseismic stresses σ relax with time after the earthquake, effective asthenospheric viscosity grows to eventually reach the preseismic value. Our estimated changes in effective viscosity imply substantial changes in the stress within the mantle wedge over this period. For a power law exponent n = 2.5, the stress would need to decrease by a factor of 2.9 to increase the effective viscosity by a factor of 5. For n = 4, the increase in effective viscosity implies a decrease in stress by a factor of 1.7. Although the computational models we have used do not allow us to compute stress at depth reliably, we can infer from these results that the coseismic stresses must be substantially larger than the ambient deviatoric stresses within the mantle wedge. Apart from the power law rheology, two alternative mechanisms can be considered to explain the evolution of the Maxwell viscosity from the Kuril postseismic GPS data over the 4 year interval: (1) a more complicated linear rheology such as the biviscous Burgers body, and (2) afterslip on the coseismic rupture and downdip. Pollitz et al.  explained the first year of postseismic deformation following the 2004 Sumatra earthquake using a Burgers rheology of the asthenosphere with the constant viscosity η1 = 1 × 1019 Pa s and a transient (Kelvin) viscosity η2 = 5 × 1017 Pa s. However, the process controlled by the transient viscosity mostly relaxes during the initial several months. Kogan et al.  showed that deformations predicted for the period 2007.5–2008.5 are identical for the Maxwell rheology and for the Burgers rheology with the best fitting value of η1 and with the value of η2 as in Pollitz et al. . Probably, the Kuril postseismic observations over the 4 year interval can be explained with a significantly larger value of η2. However, both rheological bodies, Maxwell and Burgers, are purely phenomenological and are not directly diagnostic of a specific physical mechanism [Bürgmann and Dresen, 2008].
 Most likely, afterslip was active in the first several months following the 2006 Kuril earthquake as evidenced by station URUP, the only site in the near field with observations spanning this time interval. However, for the later period, viscoelastic relaxation most likely dominates because afterslip predicts vertical postseismic deformation that is opposite in sign to the data in 2007.5–2009.5 [Kogan et al., 2011]. This assumption cannot be strictly proved because a decay time of afterslip is not well known [Pritchard and Simons, 2006].
 The open and intriguing question in connection with postseismic data following the 2006–2007 Kuril earthquakes is whether the long-term asthenospheric viscosity is on the order of ~1 × 1019 Pa s for the Kuril subduction zone, the value constrained by the postseismic deformation still observed several decades after the 1960 Chile and 1964 Alaska Mw ~9 earthquakes. Probably, this question can be clarified from several more years of postseismic observations of the Kuril GPS Array. However, postseismic deformation associated with the 1952 southern Kamchatka Mw ~9 earthquake is currently not observed on the northern Kurils. This suggests that the long-term asthenospheric viscosity in the Kurils is lower than in Chile and Alaska.
 We greatly appreciate generous help of UNAVCO, especially of J. Normandeau who provided instruments and expertise to perform space geodesy on the Kurils. We thank R. W. King and T. A. Herring for suggesting how to use effectively GAMIT/GLOBK in order to compute the plate boundary deformation. We thank F. F. Pollitz for advice on VISCO1D and S. Barbot for advice on RELAX. Comments of S. Barbot and of the anonymous reviewer allowed us to improve the manuscript. This work was supported by NSF grants EAR-1212723/1215933 and EAR-1141792/1141873 to Lamont-Doherty Earth Observatory of Columbia University and to University of Alaska Fairbanks, and by the JPL grant JPLCIT 1452786 and the IRIS Consortium grant IRIS 39-GSN both to Lamont-Doherty Earth Observatory. Figures were drawn with the GMT software [Wessel and Smith, 1998]. This is the Lamont-Doherty Earth Observatory contribution 7693.