Abstract
Understanding the rheological properties of the upper mantle is essential to develop a consistent model of mantle dynamics and plate tectonics. However, the spatial distribution and temporal evolution of these properties remain unclear. Here, we infer the rheological properties of the asthenosphere across multiple great megathrust earthquakes between 2004 and 2014 along the Sumatran subduction zone, taking advantage of decadelong continuous GPS and tidegauge measurements. We observe transient mantle wedge flow following these earthquakes, and infer the temporal evolution of the effective viscosity. We show that the evolution of stress and strain rate following these earthquakes is better matched by a biviscous than by a powerlaw rheology model, and we estimate laterally heterogeneous transient and background viscosities on the order of ~10^{17} and ~10^{19} Pa s, respectively. Our results constitute a preliminary rheological model to explain stress evolution within earthquake cycles and the development of seismic hazard in the region.
Introduction
Great earthquakes generate large stress perturbations across a wide area in both the adjoining crust and upper mantle. The rheological structure of the lithosphere and asthenosphere plays an essential role in governing the relaxation of stress^{1}, and thus may modulate the intensity and frequency of earthquakes^{2,3}. The complex stress interactions between the lithosphere and asthenosphere, due to their rheological properties, control the postseismic strain and stress rates following great earthquakes^{2,4}. These enhanced strain and stress rates have been observed to modify global seismicity over long time spans^{3} and can also directly or indirectly trigger large earthquakes^{2,5}. However, the spatial distribution and temporal evolution of these rheological properties are poorly understood, since the pressure, temperature and strain rates at these depths preclude in situ measurements^{6,7,8}. The recording of many large earthquakes within the same geological setting, with multiple data sets encompassing high temporal resolution, affords us an opportunity to better constrain the rheology of the upper mantle.
Over the past decade, modern geodetic measurements have provided the opportunity to record how the surface of the crust moved, both during and after the earthquake. For example, the 2004 Mw 9.2 Sumatra–Andaman earthquake was the first giant earthquake that was observed with modern geodetic methods^{9,10,11}, providing detailed surface displacement measurements over the decade following the earthquake. Such a high resolution set of spatial and temporal geodetic measurements should have afforded tighter constraints on the rheological structure of the lower crust and upper mantle; however, numerous studies have proposed different rheological models to explain the data^{12,13,14,15,16,17,18,19}. Whether the early postseismic deformation can be explained by afterslip only or combination of afterslip and viscoelastic relaxation; or whether the longterm deformation can be explained by combination of afterslip and background viscoelastic relaxation or afterslip with a transient viscoelastic relaxation is still debated. Thus, the rheological structure of the lower crust and upper mantle is still poorly understood. This is similar to the 2012 Mw8.6 Wharton basin strike slip earthquake where the postseismic deformation has been explained by both a linear Burgers rheology^{20,21} and a nonlinear powerlaw rheology^{4}. The reasons behind these various interpretations are in part due to the natural tradeoffs between the multiple coexisting postseismic mechanisms, but largely because the significant tradeoffs arising from choices made when forward modelling viscoelastic deformation. It is crucial to address these model uncertainties clearly and precisely, through comparison with joint inversions of afterslip and viscoelastic relaxation^{22}.
Here we probe the rheological properties of the asthenosphere across multiple great megathrust earthquakes between 2004 and 2014 along the Sumatran subduction zone (Fig. 1). They are four Mw ≥ 7.8 events including the 2004 Mw 9.2 Sumatra–Andaman^{11}, the 2005 Mw 8.6 Nias–Simeulue^{23}, the 2007 Mw 8.4 Bengkulu^{24} and the 2010 Mw 7.8 Mentawai^{25} earthquakes. We develop a novel approach (Methods section) to image the time evolution of afterslip on the megathrust and viscous strain in the mantle wedge constrained by both nearfield continuous Sumatran Global Positioning System (GPS) Array (SuGAr), farfield continuous and campaign GPS measurement and farfield regional tide gauges (Figs. 1 and 2). We image the accelerated mantle wedge flow following these great earthquakes through direct inference of the temporal evolution of the effective viscosity, illuminating its transient behaviour. The rheology of the mantle arises from the complex interactions of pressure, temperature, stress, grain size, fluid and mineral content of the ambient rocks. We show that the transient flow following these earthquakes is well matched by a linear rheological model with biviscous behaviour at time scales from days to years, and we estimate the transient and background viscosities on the order of ~10^{17} and ~10^{19} Pa s, respectively.
Results
Kinematic models of afterslip and viscoelastic flow
Building selfconsistent models of localised and distributed postseismic deformation is challenging partly due to the different physics involved in frictional sliding and viscoelastic flow. Traditionally, afterslip models are estimated following application of dynamic forwardmodelled viscoelastic flow corrections to the data, but this approach relies on an assumed rheological behaviour of the bulk rocks. To relax these assumptions, we invert for the kinematics of afterslip and viscous strain (Supplementary Fig. 1) simultaneously. Our inversion approach also intrinsically captures the mechanical coupling between afterslip and viscoelastic flow, including adjustments due to afterslip induced by viscoelastic flow and vice versa. The combined nearfield and farfield networks provide variable afterslip and viscous strain resolution along the megathrust (Supplementary Fig. 2), so we focus our discussion on the bestresolved features. Our estimates for the afterslip surrounding the coseismic rupture patches (Fig. 3a) are in general consistent with previous postseismic studies in this region^{13,26,27,28,29}. Incorporating the coupling with viscoelastic flow generally results in more afterslip at shallow depths and less afterslip at greater depths on the megathrust^{30} compared to aftersliponly inversions (Supplementary Fig. 3). Our model qualitatively explains both nearfield and farfield measurements (Supplementary Figs. 4 and 5). As expected, the majority of the nearfield postseismic displacements are driven by the frictional afterslip on the megathrust (Figs. 4 and 5), while the widespread viscoelastic flow in the mantle wedge contributes significantly to the vertical displacements not only for the nearfield, but also dominates the farfield vertical postseismic displacements (e.g. the tide gauges along the coast of Malaysia and Singapore (Fig. 4a), and the GPS vertical components at some example stations (Fig. 4b); we also show the horizontal components of the early time stage in Supplementary Fig. 6).
Our approach can, in principle, disentangle the contributions of afterslip and viscoelastic flow, provided enough data are available. To assess the potential and limitations of this experimental setting, we compare our kinematic results with dynamic simulations at four stations (PSKI, LNNG, MKMK and LAIS) in the relatively wellresolved Bengkulu area. We conducted three stressdriven simulations assuming a gravitational, vertically stratified viscoelastic Earth using VISCO1D^{31,32}. First we assumed the rheological structure adopted by ref. ^{16} for the 2004 Sumatra–Andaman earthquake (Burgers body, Simulation 1), second the same rheological parameters from ref. ^{16} but without the transient creep component (Maxwell body, Simulation 2), and finally a forward model with the inverted transient and steadystate viscosity from cuboid 2 (Burgers body, Simulation 3). The initial coseismic stress changes are calculated from ref. ^{24}. We show the predicted time evolution of the surface displacement, including contributions from both afterslip and viscoelastic flow following the 2007 Bengkulu earthquake (Fig. 5). We note that the vertical displacements provide a key discriminant between these two mechanisms e.g. station PSKI in Fig. 5c, often exhibiting the opposite sense of motion in the vertical. The surface displacements predicted by the dynamic model with our inverted rheological parameters (Simulation 3) and by the direct inversion are comparable with each other, with some minor differences, most likely due to the model employing a uniform rheology as opposed to the spatial variations discovered in the inversion. In addition, ref. ^{16} utilises a thinner elastic lid in the layered dynamic model than we have in our inversion; thus, it predicts larger surface displacement near stress concentration regions (Supplementary Fig. 7). Our modelled displacements sit half way between the Maxwell and the Burgers models, indicating some consistency with previous models of the region. The small differences in inferred viscosity may be due to our modelling assumptions. In particular, our inversion allows for the possibility of arbitrary three dimensional (3D) heterogeneities, while previous approaches were restricted to vertical stratification and did not give afterslip a consistent treatment.
Accelerated viscoelastic flow in the Sumatran mantle wedge
We capture the acceleration of viscoelastic flow in the mantle wedge following the large megathrust earthquakes of the last decade until 2014 (Fig. 3a; Supplementary Movie 1). As suggested from the resolution synthetic tests (Supplementary Fig. 2) and from the resolution matrix for the strain components ε_{13} and ε_{33} (Supplementary Figs 1 and 8), we are not able to estimate viscous strain and afterslip in the 2005 Nias–Simeulue rupture segment as accurately as the 2007 Bengkulu segment, due to the paucity of data directly down dip of the 2005 rupture above the viscous cuboids. Therefore, we focus on the bestresolved cuboids at the 2004 and 2007 rupture segments located in the shallowest portions of the megathrust (Fig. 3a,b). We track the temporal evolution of the stress and viscous strain rates (Supplementary Fig. 9a and b), and obtain the effective viscosity at each time step by dividing the estimated stress by the strain rate. The mechanical coupling between afterslip and viscoelastic flow is accounted for by keeping track of the timedependent stress change induced by the mainshocks and the postseismic deformation (see Methods section, Supplementary Fig. 9a–c).
Evolution of effective viscosity following earthquakes
We directly evaluate the time series of effective viscosity from the kinematics of postseismic deformation. We see some consistency in the response of the mantle wedge following various megathrust earthquakes along the Sunda arc. We systematically infer a transient viscosity, beginning with smaller values (~10^{17} to ~10^{18} Pa s), and increasing by one order of magnitude within about 2 years (e.g. Fig. 6c,d). We note that temporal smoothing in the Kalman filter may bias the transient viscosity towards higher values, but smoothing is necessary to regularise the inversion. In subsequent years, the viscosities remain approximately constant (e.g. Fig. 6c,d). Such transient behaviour in the effective viscosity can be approximated to first order by a biviscous rheology. To extract the transient and steadystate viscosities, we fit our time series of effective viscosities with the exponential function (Methods section)
The choice of the function is for convenience only, to identify the steadystate and transient viscosities, and does not imply any particular underlying physical mechanism. Considering one cuboid region at a time, we conduct a grid search for the bestfit values for the background viscosity η_{M}, the transient viscosity η_{K}, and the time scale of the transient \(\tau _{\mathrm{K}} = \frac{{\eta _{\mathrm{K}}}}{{G_{\mathrm{K}}}}\), where G_{K} is the shear modular of the Kelvin body. Examples of the lowpass filtered time series of effective viscosities and bestfitting exponential model for cuboids 1 and 2 are shown in Fig. 6c,d (Supplementary Fig. 9c).
We note the reasonable agreement between the effective viscosity time series and the simplified model of Eq. (1) (e.g. Fig. 6c,d). Our estimated transient and background viscosities are on the order of ~10^{17} Pa s and ~10^{19} Pa s, respectively; with a characteristic time scale of 0.21 ± 0.05 yr for the transient. The time scale of the transient is commensurate with that found in other tectonic settings (e.g. ref.^{33}) and our effective viscosity estimates from our example cuboids are also in agreement with several published Burgers rheology models for the Sumatran subduction zone^{12,14,15,16,18,34,35} (Supplementary Table 1 and Supplementary Fig. 10) despite the different modelling assumptions and techniques. These results imply that the time evolution of the viscosity is reasonably well recovered using our approach. The remaining differences can be attributed to the longer time series used in this study, and the fact that we do not constrain a priori the lateral and temporal variations in viscosity.
Discussion
The full complexity of the rheology of the upper mantle originates from complicated interactions of temperature, pressure, stress levels, grain size, fluid and mineral content of the ambient rocks^{4,8,19,36,37}. However, the gross behaviour can be investigated through a simplified theoretical framework involving transient and steadystate creep. In the 2004 Sumatra–Andaman rupture region, we image spatially decreasing transient (Fig. 6a and Fig. 7) and background viscosities towards Thailand (Fig. 6b). This low viscosity could be associated with backarc spreading around the Andaman Sea, as modelled by ref. ^{19}, although we acknowledge the decreased spatial resolution in this region (Supplementary Fig. 2d). In the Mentawai segment, to the south, the steadystate viscosity is lower (Fig. 6b). We note the higher concentration of active volcanoes in the region, compared to farther north, which indicates a potential link between active volcanism and mantle mobility along the Sumatran arc^{37}.
To further address the constitutive behaviour of mantle rocks, we examine the in situ details of the stress–strain rate relationship at the location of our bestresolved cuboids (Fig. 8). The approach is similar to the analysis of laboratory data, where insight into the rheological behaviour is gained through controlled experiments, socalled creep tests, where the strain rate is enforced by the apparatus and stress is measured during the deformation of a small sample. Our approach is similar, except for the boundary conditions and scale of investigation: the stress varies because of a large coseismic perturbation and the subsequent relaxation, and we interrogate an experimental volume of 240,000 km^{3}.
The stress–strain rate phase diagram in log–log space (Fig. 8) reveals two distinct domains that we interpret as the signatures of transient creep and steadystate creep. Following both the 2004 Mw 9.2 Sumatra–Andaman earthquake (Fig. 8b) and the 2007 Mw 8.4 Bengkulu earthquake (Fig. 8a), the transient domain reveals an apparent nonlinear relationship that dominates the deformation for about 2 years, followed by a contrasting steadystate creep that obeys a near linear stress–strain rate relationship. These results are compatible with a combination of steadystate creep and transient creep, which may be attributed to background diffusion creep plus temporary motion of the soft slip system (grain boundary sliding during the transient phase) of the olivine crystals^{4}. The stress/strain rate behaviour at cuboids 2 and 3 is well captured by the theoretical response of a Burgers material to a stress perturbation added to a background load (Fig. 8; Methods section), and we could not explain these data with a powerlaw rheology with a power exponent of 2 or greater. In addition, the estimated magnitude of the transient and steadystate viscosities of the resolved cuboids are not correlated with the magnitude of the spatial distribution of the coseismic stress changes (Supplementary Fig. 11a–c), further ruling out the possibility of the nonlinear powerlaw rheology. We conclude that a Burgers rheology involving linear workhardening transient creep and linear steadystate creep is a realistic description of mantle rock rheology within a time scale of days to years. We anticipate that future geodetic data sets, with greater spatial and temporal resolution both inland and offshore, will help us to gain more insight into the rheological structure of the region.
The transient rheology of the upper mantle has been widely proposed to explain postseismic deformation following large earthquakes at other subduction zones^{30,38}, and also in intracontinental tectonic settings^{39,40}. Our study highlights that at least two creep mechanisms accommodate the viscoelastic flow of the asthenospheric upper mantle, which can be captured by a linear biviscous Burgers rheology, at least to first order. Our findings provide new insight into the rheology of the mantle wedge, contributing to assessing seismic hazards and their associated tsunami hazards, over earthquake cycles in the Sumatran region.
Methods
GPS time series
The postseismic deformation following each of these events is well captured by the nearfield continuous Sumatran Global Positioning System (GPS) Array (SuGAr), farfield continuous and campaign GPS measurement, and farfield regional tide gauges (Figs. 1 and 2), resulting in a decadelong temporally dense time series. The combination of both nearfield and farfield data sets provides us favourable conditions to infer the probable rheology of the mantle wedge in this region.
We use continuous time series from 29 stations of the SuGAr network, 1 GPS station (SAMP) from the Indonesian National Coordination Agency for Surveys and Mapping, 1 International Global Navigation Satellite System (GNSS) Service (IGS) site (NTUS) in Singapore, 4 stations on the Andaman Islands^{28} and 12 sites in Thailand^{12}. We also derive the farfield vertical land elevation changes at 23 tidegauge locations around Singapore, Malaysia and Thailand, through a combination of tidegauge data and seasurface height altimetry Aviso time series. The resulting time series of displacements span the time period from early 2005 through to 2014, providing constraints for the temporal evolution of the rheological parameters of the various postseismic relaxation mechanisms.
We process the GPS data up to the end of 2013 from 29 SuGAr stations, 1 IGS station (NTUS), 1 station (SAMP) operated by the Indonesian National Coordination Agency for Surveys and Mapping and 4 stations on the Andaman Islands^{28} using the GPSInferred Positioning System and Orbit Analysis Simulation Software (GIPSYOASIS) version 6.2^{41}. In addition, we use the postseismic displacement at 12 GPS sites in Thailand from ref. ^{12}. Further details about our GPS processing methodology can be found in ref. ^{42}. We transform daily positions in the International Terrestrial Reference Frame 2008^{43} (ITRF2008) to the Sunda plate reference frame using the ITRF2008Sunda transformation^{44}. Locations of the GPS stations are shown in Figs. 1 and 2.
For our modelling, we use GPS time series that contain only the postseismic signals from the recent multiple Sumatran earthquakes. To obtain the postseismic time series, we remove other signals, i.e. longterm rates, annual and semiannual seasonal signals, coseismic offsets from all the megathrust events, and also the coseismic and postseismic displacement of the Mw 8.6 2012 Indian Ocean earthquake, which was not a megathrust event. All these signals are simultaneously estimated using a singleweighted nonlinear leastsquares optimisation procedure for each station. The resulting estimates are used to not only remove nonpostseismic signals but also to fill in data gaps. Further details about the signal estimation procedures are described in ref.^{42}.
Landheight displacement derivation
Tide gauges measure the relative sea level between the sea surface and the land. Thus, tide gauges capture changes of the land elevations. To extract the landheight changes, information on seasurface changes are required. Satellite altimetry measurements can provide this information. A combination of tidegauge and satellite altimetry data has been widely used to extract landheight changes^{45,46,47,48}. Landheight changes following the 2004 Sumatra–Andaman earthquake were successfully extracted from tide gauges by ref. ^{49}.
We carefully select tidegauge data that have as few gaps as possible because multiple gaps or shorter tide recordings can bias the longterm trend estimation. Our final selected tide gauges are located in Singapore, Malaysia and Thailand, covering a wide area in the farfield of the Sunda megathrust (Figs. 1 and 2). We downloaded the tidegauge data from the Permanent Service for Mean Sea Level database^{50} (PSMSL, 2016, from website http://www.psmsl.org/data/).
For the satellite altimetry data, we use the Aviso data product from ftp.aviso.altimetry.fr. The altimeter products were produced by Ssalto/DUACS and distributed by Aviso, with support from Cnes (http://www.aviso.altimetry.fr/duacs/). The Aviso data are from a combination of all altimeter missions, and have been corrected for instrument, orbit errors and errors resulting from signal propagating through the atmosphere, as well as perturbation errors from surface reflection. Tides, such as pole tides, solid, ocean and loading tides are corrected as well. Crosscalibration is also applied to yield a consistent and homogenous highquality data set^{51}. This data set has been successfully and widely implemented for sea level studies^{45,52,53,54,55,56,57}. We use the delaytime mode of the global mean sea level anomalies product of the Aviso data set because of its accuracy as compared with realtime mode. These global, gridded, multimission data span from 1993 until May 2014 with daily solutions. In order to extract the landheight information by subtracting altimetry data with the monthly tidegauge data, we use the daily solutions to estimate monthly average sea surface height anomalies.
A previous study in this region from ref. ^{49} reports that if the distance between the tide gauge location and the satellite observation grid data is less than 100 km, a good estimate of landheight changes can be achieved. Using this criterion, we select 24 tide gauges with nearby Aviso grid points to extract the landheight changes following the megathrust earthquakes at the Sumatran subduction zone between 2004 and 2014.
Before subtracting the Aviso data from the tidegauge data to obtain the landheight changes, we remove seasonal cycles from both data sets. To do this, we use a simple but robust technique as described by ref. ^{58}. We average the wholetime period data for each month, and then subtract the monthly averages from each month of the time series. This technique is better than a leastsquares approach, because the leastsquares method estimates constant seasonal parameters for the wholetime period of each tide gauge. Using this technique we find that the seasonal signals for each month can vary considerably. We show one example of seasonal signals from both Aviso and the tide gauges at Johor Baru (location shown in Fig. 2) in Supplementary Fig. 12.
We minimised the uncertainty by carefully removing the variable seasonal signals, and by finding the collocated tide gauge and Aviso grids pairs. However, there are a number of uncertainties that are still contained within the resultant landheight time series due to the lowspatial resolution of the Aviso grids. For example, the local complexities of shallow bathymetric effects and manmade landheight signals that would be captured by the tide gauges are absent in the coarse altimetry Aviso grids. These localised signals would then be mapped into the landheight time series, potentially producing abnormalities in the landheight variation when compared to neighbouring tide gauges that were less affected by the local signals. For example, the high rate of subsidence at tide gauge BPAKBang Pakong (Fig. 2) is comparable or even larger than that of tide gauge TAPN Ko Taphao Noi (Fig. 2), which was closer to the 2004 Sumatra–Andaman earthquake, and strongly affected by the postseismic deformation process (Supplementary Fig. 5). This abnormally high rate of subsidence likely contains contamination from humaninduced subsidence at Bangkok, Thailand (see refs. ^{59,60}), ~50 km away. Tidegauge RAYGRayong (see Fig. 2) shows strong variations preceding the 2004 Mw 9.2 Sumatra–Andaman earthquake (Supplementary Fig. 5). Similar abnormal signals were also found at tide gauges east of Malaysia, which were less affected by the 2004 rupture. We also noted a systematic scatter in the vertical displacement time series for tide gauges around Singapore over time period ~2010 to 2011 (Supplementary Fig. 5). We are not clear what may have caused this offset, but we saw significant amplitude decline of the Aviso time series over this time period, which results this offset across all the tide gauges in Singapore region. To completely and accurately deal with these abnormal signals and uncertainties presents a substantial challenge. However, the tide gauge data do clearly show changes in their linear trend after the great megathrust events (vertical dashed lines in Supplementary Fig. 5), which provides a strong constraint on the direction of land movement, if not its magnitude, affording a crucial constraint for the mechanisms of the postseismic relaxation processes (see red profiles in Supplementary Fig. 5).
Geometry of the fault and mantle wedge
Our fault geometry is derived from a model used by ref. ^{15}, who modified and extended the Slab 1.0 model to 16° North along the Sunda megathrust. We place the strain volumes (cuboids with ~100 km length, 60 km width and 40 km height) in the locality of the mantle wedge with a depth range from ~50 to ~230 km, consistent with previous postseismic studies of this region^{12,13,14,15,16,18,34,35,61}, with strike aligned to the strike of the megathrust. We ignore the viscoelastic relaxation of the oceanic asthenosphere, which will have limited influence on our observations over the time frame considered in this work^{4}.
Joint inversion with the Extended Network Inversion Filter
With both the nearfield and farfield postseismic time series (observed motions corrected for interseismic, seasonal and other earthquakerelated deformations), we invert for localised slip on the megathrust and distributed strain in finite volumes^{22,27,62} (Supplementary Fig. 1), corresponding to afterslip on the megathrust and ductile flow in the mantle wedge, respectively. We do not consider localised and shortterm deformation generated by pore pressure changes (poroelastic rebound^{63}) due to coseismic stress changes, since we are focusing on longterm and widespread postseismic relaxation. We apply a modified version of the Extended Network Inversion Filter^{64,65} (ENIF) to the time series data to infer the temporal evolution of afterslip on the megathrust and viscous strain in the mantle wedge.
The ENIF simultaneously inverts for both fault slip and viscous strain within a finite cuboid volume at asthenosphere depths. The original version only inverts for fault slip, whereas the modified version can invert for both fault slip and strain within a finite volume. Our modified ENIF models the cumulative surface displacement (u_{ i }(r_{ j }, t)) of GPS sites at location \({\mathbf{r}}_{\mathbf{j}}\), time t, as
where Einstein summation notation is implied and (u_{ i }(r_{ j }, t)) represents the ith component of displacement at time t. The first term on the right hand side is the surface displacement at time t due to slip s_{ k }(ξ, t) on a fault patch A(ξ), where G_{ ik }(r_{j}, ξ) represents the Green’s functions that map unit slip on a fault patch to surface displacement at a location r_{j}. The second term is the surface displacement at time t due to strain component ε_{ ml }(ϑ, t) of a cuboid V(ϑ) located at R_{ j } in the subsurface. Each cuboid has six independent components of strain (i.e. ml = 11, 12, 13, 22, 23, 33) (Supplementary Fig. 1). \(G_{iml}(R_j,\vartheta )\) are the 3D Green’s functions that map the unit strain components to the surface displacement. L(r_{ j }, t) are the local nontectonic random benchmark motions e.g. the local motion of GPS monument. f(t) is the common mode error of the wholeGPS network such as seasonal loading, ocean loading, hydraulic loading, etc. E contains the observation errors that are not taken into account during the GPS processing (e.g. multiple path errors and azimuthally varying path delays). This error is assumed to follow a Gaussian distribution with zero mean and covariance \(\sigma ^2{\mathrm{\Sigma }}_n\), where ∑_{ n } is the GPS data covariance matrix derived from GPS data processing.
We discretise the fault plane into rectangular subpatches. We calculate the Green’s functions (surface displacement at GPS stations) due to unit slip on the subpatches, assuming that they are buried in a homogeneous, isotropic, elastic halfspace, using the analytic solution of Okada^{66}. We calculate the 3D Green’s functions that map the strain components of the deformable cuboids to surface displacement at GPS stations through a novel analytic solution^{62}. For each cuboid, there are six independent components of strain. We show an example of surface displacement associated with a single cuboid in Supplementary Fig. 1. To reduce computational burden and remain within the limit of the computational capacity, we downsample the daily solution time series of both GPS and tide gauge data to every ten days. The instantaneous rates (e.g. slip and strain) might be affected slightly; however, the final accumulated slip on the megathrust and strain in the mantle wedge should remain unchanged.
Afterslip penalisation and spatial and temporal smoothing
Imaging the spatial distribution of afterslip with two directions of slip on fault patches, and ductile flow with six components of strain within cuboid volumes (Supplementary Fig. 1) results in a glut of free parameters in the inversion. To avoid overinterpreting the available data, we apply extra constraints for both the afterslip and viscous strain to regularise them moving in a sensible way.
Afterslip is the continuation of slip as accelerated creep following the coseismic motion, upon which we enforce a nobackslip or positivity constraint (see ref. ^{65}). In addition, we apply spatial smoothing to both afterslip and strain components with a Laplacian matrix to minimise large jumps between neighbouring patches. The spatial and temporal smoothing hyperparameters are estimated together with the model parameters for each model run, which is also one of the advantages of the ENIF^{64,65} technique. The Kalman filter is a Bayesian filter, where the parameters are predicted through a transition matrix and updated with the measurement from the current step. Then we step through to the next time epoch. As more and more information is incorporated into the system through time, the estimates of the parameters consistently improve. The filter initialises with a prescribed starting value, then adjusts it within the specified uncertainty to improve the fit to the measurement. There is a tradeoff between the spatial and temporal smoothing hyperparameters. To address this potential issue, we conducted a few preliminary runs to determine a reasonable spatial smoothing hyperparameter that reliably produces both a smooth slip distribution and good fit to the data. We then employ this value with a small covariance, thereby only allowing its value to be minimally changed by the Kalman filter, while using a large covariance for the temporal smoothing hyperparameter which allows for a broad range of adjustments when necessary during the updating step. We finally obtain the initial guess for the temporal smoothing hyperparameter, through a recursive algorithm where each previous estimate is updated to the current input, until model results from the current estimates and previous ones converge to within a specified tolerance (see ref. ^{65}).
Isotropic strain and strain directions penalisation
Viscoelastic flow in the asthenosphere is assumed to be deviatoric. To model the viscous strain, we penalise isotropic strain as well as strain directions that are perpendicular to the induced coseismic stress changes at each strain volume. Accordingly, we penalise the trace of the viscous strain tensor of each cuboid, encouraging, but not strictly enforcing it to be zero:
Here \({\cal E}\)_{11}, \({\cal E}\)_{22}, \({\cal E}\)_{33} are the strain components along three mutually orthogonal directions^{27}. This results in a majority deviatoric solution, but does not prevent some degree of dilatation in the cuboid volumes^{67}. Viscoelastic flow following earthquakes is driven by coseismic stress perturbations, as is the case for any postseismic dynamic models. In our inversion, we also implement a direction penalisation for the strain tensor of each cuboid to encourage the viscous strain to point in approximately the same direction as the coseismic stress changes^{8}.
Checkerboard tests
To investigate how well our combined nearfield and farfield networks can resolve slip on the megathrust and strain within each cuboid, we perform a suite of checkerboard tests, illustrated in Supplementary Fig. 2. The input model is given in Supplementary Fig. 2a, where we use a checkerboard with ~220 km×~180 km rectangles of constant slip, synthetic input strain and displacements on the nearfield and farfield networks. For the synthetic strains within the cuboids, we use the spatial distribution of the coseismic stress changes following the 2004 Sumatra–Andaman, the 2005 Nias–Simeulue and the 2007 Bengkulu earthquakes with these synthetic viscous strains scaled by the norm of the deviatoric coseismic stress changes. We first test both the nearfield GPS network (SuGAr, Andaman GPS) and farfield (Thailand) GPS stations only, and the inverted results are shown in Supplementary Fig. 2b. In general, the slip pattern is relatively wellresolved in places near the GPS stations. The inverted strain distribution does not match the input model, with some additional strain occurring down dip of the 2005 rupture area and also in the northern tip of the mantle wedge. Due to there being no farfield sites for both the 2005 and 2007 rupture segments, the missed strain signals are mapped into the slip, causing overestimates on slip magnitude and a slight distortion of the spatial distribution of slip. We then test the nearfield GPS network without the farfield GPS stations and tide gauges (Supplementary Fig. 2c). We note that the general pattern of the slip is relatively wellresolved excepting the middle section of the 2004 rupture area. However, the strain within the 2004 rupture region is missing, and there is additional artificial strain in the northern part of the 2004 down dip rupture area and the down dip region of the 2005 rupture region. Finally, we explored the combination of both the nearfield and farfield networks (all GPS stations and tide gauges) in Supplementary Fig. 2d, which affords the best resolution among all these tests for both slip on the megathrust and strain within the cuboids. In general, the slip patterns are wellresolved, with the exception of the middle section of the 2004 rupture due to a paucity of data within this region. In particular, the afterslip is best resolved at the shallow and deep part of the 2007 Bengkulu rupture area, the shallow and part of the deep part of the 2005 Nias–Simeulue, and the northern part of the 2004 Sumatra–Andaman rupture regions. For the strain, the bestresolved sections are the 2004 Sumatra–Andaman and the 2007 Bengkulu rupture segments. In conclusion, incorporation of the farfield tide gauges and the consequent increase in the total amount of available data can help to stabilise the inversion and improve the model resolution for each individual mechanism.
Effective viscosity estimation
We estimate the effective viscosity within each cuboid through
where Δσ is the norm of deviatoric stress evolution for each block, calculated from the accumulated viscous strain at each epoch estimated by the Kalman filter combined with the stress kernel associated with each cuboid (see refs. ^{22,27}), \({\mathrm{\Delta }}\dot \varepsilon\) is the norm of deviatoric strain rate (as estimated by the Kalman filter), σ_{0} and \(\dot \varepsilon _0\) are the background stress and strain rate of the mantle wedge flow under the steadystate stage condition, respectively. For example, ref. ^{22} has demonstrated a static inversion for viscosity immediately following an earthquake, and in this study we expand it to capture the timedependent features. It is impossible to directly evaluate the magnitude of stress and background strain rate in the mantle; thus, we explore a range of strain rates from 10^{−16} to 10^{−14} s^{−1}, a range of steadystate viscosity from 10^{18} to 10^{22} Pa s, and the background stress corresponds to the background strain rate multiplied by the steadystate viscosity. We loop through these parameters within the range, and calculate the misfit between Burgers rheology model and the effective viscosity. The minimum misfit provides the most likely combination of both background viscosity and strain rate that matches the trend of stress relaxation within the cuboid, and thus the corresponding time series of effective viscosity.
Phase diagram
We model the evolution of the stress and stress rate in a cuboid by using a simple springdashpot model. We consider the circumstance where the mechanical system is first at equilibrium under a constant stress τ_{0} but then subjected to a new stress τ_{0} + Δτ at time t = 0. The set of governing equation for the system is given by
where G, \(\eta\) and \({\mathbf{\varepsilon }}\) are the rigidity, viscosity and anelastic strain. The subscripts M and K are for the Maxwell and Kelvin materials, respectively. The evolution of stress is provided in closed form below for t ≥ 0,
with
The evolution of total anelastic strain rate \({\dot{\mathbf \varepsilon }}^i = {\dot{\mathbf \varepsilon }}_{\mathrm{M}} + {\dot{\mathbf \varepsilon }}_{\mathrm{K}}\) is given by
We compared the analytic solution to a numerical integration using a fourthorder Runge–Kutta method, which showed agreement to numerical precision. We model the phase diagram of Fig. 8 using Eqs. (5–8) by adjusting the physical parameters manually. The bestfit parameters for cuboid 2 are Δτ = 24.5 kPa, τ_{0} = 23 kPa, η_{M} = 2.45 × 10^{18} Pa s, η_{K} = 2.9 × 10^{17} Pa s, \(G_{\mathrm{M}}\) = 30 GPa and G_{K} = 80 GPa, respectively. For cuboid 3, they are Δτ = 30 kPa, τ_{0} = 31 kPa, η_{M} = 4.6 × 10^{18} Pa s, η_{K} = 3 × 10^{17} Pa s, G_{M} = 30 GPa and G_{K} = 100 GPa.
Data availability
All geodetic data used in this study are available on request due to privacy or other restrictions.
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Acknowledgements
This research was supported by the National Research Foundation Singapore under its Singapore NRF Fellowship scheme (National Research Fellow Award Nos. NRFNRFF2010064 and NRFNRFF201304), the Earth Observatory of Singapore via its funding from the National Research Foundation Singapore, and the Singapore Ministry of Education under the Research Centres of Excellence initiative. We thank Kelin Wang, Roland Bürgmann and an anonymous reviewer for their helpful comments. We are grateful to Prof. Ray J. Weldon for commenting on tide gauge data processing. We also would like to thank Jessica Murray and YaJu Hsu for useful discussions about the ENIF. The modified ENIF code is available on request. Figures are produced by using Generic Mapping Tools (GMT)^{68}. This is EOS paper number 142.
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Q.Q., E.M.H. and S.B. designed the study. Q.Q. conducted the study and wrote the manuscript with contributions from all other coauthors. L.F. prepared the GPS time series. J.D.P.M. derived the analytic solutions.
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Qiu, Q., Moore, J.D.P., Barbot, S. et al. Transient rheology of the Sumatran mantle wedge revealed by a decade of great earthquakes. Nat Commun 9, 995 (2018). https://doi.org/10.1038/s41467018032986
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DOI: https://doi.org/10.1038/s41467018032986
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