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1 Geophysical Journal International Geophys. J. Int. (2010) 183, doi: /j X x Gravity changes due to the Sumatra-ndaman and earthquakes as detected by the GR satellites: a reexamination I. inarsson, 1. Hoechner, 1 R. Wang 1 and J. Kusche 2 1 GeoForschungsZentrum Potsdam, Telegraphenberg Potsdam, Germany. -mail: indridi@gfz-potsdam.de 2 Bonn University, Institute of Geodesy and Geoinformation, Nuβallee 17, Bonn, Germany ccepted 2010 July 30. Received 2010 July 28; in original form 2009 ugust 28 SUMMRY In this paper, we study the geoid changes caused by the Sumatra-ndaman earthquake (2004 December 26) and the earthquake (2005 March 28) using data from the gravity recovery and climate experiment (GR) satellites. New is that we explore the possibility to separate the effects of the Sumatra-ndaman and earthquakes. Based on correlation analysis we investigate the separability of the Sumatra-ndaman and signals, and the feasibility of separating the non-seismic linear trend from the time-dependent post-seismic relaxation effect. s input data, we use 76 months of global geopotential coefficients from the GFZ release 4 (RL04) of GR data. To suppress unphysical artefacts in GR data ( striping ), the GR coefficients are smoothed with an anisotropic decorrelation filter. Model parameters are estimated through a Bayesian approach. We compare our results with geophysical models of both earthquakes, calculated based on the viscoelastic-gravitational dislocation theory. The slip models used are inverted from GPS measurements. Our findings indicate that, the inversion of GR data for the Sumatra-ndaman effect will contain significant signal from the earthquake, and will benefit from removing the modelled effect from the data. omparing GR measurements to geophysical models reveals significant differences. We suggest, how the GPS-calibrated models may be improved to better fit the GR data. In the first months after the earthquake, we also find evidence for a fast process in the ndaman sea, possibly due to aseismic slip. Similar phenomena were observed in previous studies. Key words: Satellite geodesy; Transient deformation; Time variable gravity; Dynamics: gravity and tectonics GJI Gravity, geodesy and tides 1 INTRODUTION Over 5 years have passed since the 2004 earthquake and tsunami offshore Sumatra, which a few months later was followed by a large earthquake close to the island. arthquakes of this size cause mass-dislocation on a scale that is measurable by the gravity recovery and climate experiment (GR) gravity satellite mission. Displacements taking place iediately during the earthquake are called coseismic. The dislocated masses disturb the isostatic equilibrium of the crust and mantle, thus inducing a long-term post-seismic relaxation process. Pore fluid diffusion and viscous relaxation in the crust (see Jónsson et al. 2003) as well as deeper in the mantle (Ogawa & Heki 2007) have been suggested to cause post-seismic changes on relatively short timescales. Hence, an empirical model comprising of a single exponential function, as often chosen in GR inversions, may not be sufficient in describing the post-seismic effect. The GR mission provides measurements of the time-variable arth gravity field; for an overview see (Tapley et al. 2005). ombined with simultaneous and complementary observations and model results, it enables one to quantify interactions between atmosphere, hydrosphere and geosphere. GR is a twin-satellite formation, with two identical spacecrafts chasing each other in similar near-polar orbits. K-band ranging system provides the biased intersatellite range as well as its derivatives with respect to time. In addition, both satellites are equipped with geodetic GPS receivers as well as accelerometry for removing non-gravitational forces prior to data analysis. However, a problem that all users of monthly GR gravity field solutions face is the presence of correlated and wavelength-dependent noise in the provided spherical harmonic coefficients. This manifests itself in north south directed, elongated features in the derived gravity models, usually described as striping patterns. Simply truncating the spherical harmonic series at low degrees (long wavelengths), where the noise is not yet significant, causes the loss of an unacceptably large portion of the signal. This is therefore not an option when one is interested in signals of geographical extension of only a few 100 km, as with the signature of earthquakes. Several so-called de-striping postprocessing methods exist, but they need to be applied carefully when geophysical signals are of similar wavelength and spatial orientation as the GR stripes. Unfortunately, this is the case for 2010 GFZ Potsdam 733 Geophysical Journal International 2010 RS

2 734 I. inarsson et al. the Sumatra ndaman () earthquake. Fortunately, in our case we want to estimate co- and post-seismic parameters from several years of data, in which case much less smoothing is required than in the case of individual monthly solutions. Both the co- and post-seismic gravity effects of the earthquake have been identified in GR gravity measurements by various authors. Ogawa & Heki (2007) use time-series of pre-processed GR spherical harmonic coefficients from SR release 1, up to degree 80, filtered using an isotropic Gaussian filter with 350 km radius. They determine co- and post-seismic effects at single locations, considering various possible causes for the post-seismic recovery. They suggest, that a fluid diffusion from the region with increased pore pressure to the region of decreased pressure occurs not only in the uppermost crust, but also deep in the mantle. The high pressure and temperature in the mantle causes water to behave as supercritical fluid, accelerating the diffusion process. From the GR data, they conclude on a relaxation rate of about 0.6 yr = 219 days. The earthquake is not considered in Ogawa & Heki (2007). Han & Simons (2008) use Slepian s spatiospectral localizing basis functions, as an alternative to the usual spherical harmonic functions. These are estimated directly from in situ GR data, from the SR release 1. They are able to determine the coseismic jump at the time of the earthquake, but they do not try to solve for the earthquake. Han et al. (2008), also use Slepian s spatiospectral localizing basis functions estimated from in situ data. They determine a coseismic jump as well as an exponential-like post-seismic signal, using a constant relaxation rate of 150 days, but do not try to solve for the earthquake. Panet et al. (2007) analyse co- and post-seismic gravity footprints of both and earthquakes using wavelets at multiple spatial scales. They use the regularized NS GR solution up to degree 50 (Lemoine et al. 2007). They find both co- and post-seismic signals for the earthquake to be significant, with different postseismic relaxation rate (τ, see eq. 8) at different spatial scales, relating the different spatial scales to different depths in the crust. Moreover, they identify a detectable signal in the vicinity of the epicentre of the earthquake. De Viron et al. (2008) analyse the time-series of GR data using empirical orthogonal functions to identify footprints of several earthquakes, including the earthquake, using the regularized NS solution of Lemoine et al. (2007). Solving only for the coseismic signal, they are not able to detect the earthquake, and suggest that the strong signal of the earthquake masks the nearby signal and makes separation of the two signals difficult. De Linage et al. (2008) estimate the earthquake signal of the earthquake. They use the regularized NS solution as well as the monthly SR release four solution. They examine the possible effect of hydrological signals on the estimated earthquake signals by using hydrological models. They are able to detect both co- and post-seismic signal of the earthquake, but disregard any possible effect of the earthquake. The estimated co- and post-seismic signals are compared to the results of modelling the earthquake using an elasto-gravitational non-rotating spherical earth model. In this study, we use GR data in the form of monthly spherical harmonic coefficients. To isolate the gravity effects of the two earthquakes, we have to separate them from other processes whose gravity footprint is also present in GR data. These include, for example hydrology, tidal model errors, residual oceanic signal. In Section 2, we introduce our mathematical model of the earthquakes as well as for other sources of temporal gravity signals. This consists of co- and post-seismic amplitudes, time of the earthquakes, relaxation rate, S2 tidal signal (see Melachroinos et al. 2009), semiannual and annual signals. We introduce a non-linear Bayesian method to constrain the estimation using aprioriknowledge. This is necessary in particular for the relaxation rate, on which the model depends in a non-linear way. In Section 3.1, we provide results of numerical experiments with real and modelled data. We analyse the estimated covariance matrix for different settings of the mathematical model, and discuss the risk of signal leaking between parameters due to high correlation. Based on this, we discuss whether the and earthquakes could be separated and modify our model accordingly. In Section 3.2, we fit the modified functional model to smoothed GR data on each point in a grid surrounding the area affected by the earthquakes. This is done for several scenarios, either using original GR data or reducing the GR data with a geophysical model, based on which of the two earthquakes we are trying to isolate. In Section 4, we then discuss and interpret the results. 2 MTHODS ND DT 2.1 Data The data used in this study are GR spherical harmonic coefficients from the GFZ-RL04 series (Schmidt et al. 2008), (Flechtner 2007) from the time span ugust 2002 to pril GR uses accurate distance measurements between two satellites, as well as accelerometers and GPS positioning, to recover the gravity field of the earth up to spherical harmonic degree 120 (see Schmidt et al. 2008). No data is provided for the months June 2003 and January 2004 due to GR data gaps. In the months 2004 July October, the ground track of GR was degraded because the satellites experienced a 4-day repeat cycle. Therefore, the solutions for these months are regularized. The higher degree coefficients of the regularized solutions tend to the mean field IGN-GL04 (see Flechtner 2009). 2.2 Signal recovery from GR spherical harmonic coefficient sets For recovering the geoid change signal, we use the GFZ-RL04 monthly GR Stokes coefficients, as described in Section Post-processing of GFZ-RL04 coefficients GR-derived Stokes coefficients, as provided by analysis centres, suffer from so-called striping. These are elongated patterns in north south direction (Schmidt et al. 2008), notable in spherical harmonic degrees above 30. There exist several methods to compensate for this. One is to convolve the signal with a smoothing kernel, in which case the filter is applied independently to each coefficient set. The kernel may be either isotropic, for example a Gaussian function (Swenson & Wahr 2002), or non-isotropic, as in (Han et al. 2005), where a different Gaussian radius is used for each degree, or in (Swenson & Wahr 2006) and (Kusche 2007), where information on the covariances of the estimated spherical harmonic coefficients is used to construct the filter. nother approach is to use statistical methods to separate geophysical signal and noise, e.g. using empirical orthogonal functions (Wouters & Schrama 2007). In that case, a time-series of coefficient sets is required GFZ Potsdam, GJI, 183, Geophysical Journal International 2010 RS

3 Sumatra-ndaman and earthquakes 735 We use the anisotropic decorrelation method described in (Kusche 2007). By going formally back to the least squares estimation of the coefficients, we have the vector of (unfiltered) SHcofficients x, the right-hand side vector b, the normal matrix N and an apriorisignal covariance matrix {xx T } =M 1. The filtered coefficients, x γ (a) are now obtained by x γ (a) = (N + am) 1 b = (N + am) 1 Nx = W a x, (1) using the weighting factor a to the signal-covariance matrix M (Kusche 2007) [eq. (20)]. The parameter a can be adjusted to tune the smoothness of the solution. The GR Stokes coefficients are filtered using the anisotropic filter described in Section using the filtering parameter a = We map the filtered coefficients to residual geoid heights N(θ, λ) (residuals taken w.r.t. a mean value over the time in question) on a grid around the area of interest (80 110, 10 S 20 N), using 0.25 grid intervals. For each grid node (θ i, λ j ), we set up a functional model, describing both the effects caused by the earthquake as well as dominating environmental effects (semiannual and annual variations, tidal effects and others), described as follows Functional model for the measurements The GR signal in this area contains a dominating annual component over land, most probably caused by hydrology variations. In addition, areas near the equator can have large semiannual signal. This enters our model as in N ann (θ i,λ j ; t) + N semiann (θ i,λ j ; t) (2) = ann (θ i,λ j ) sin(ω ann t + φ ann ) + semiann (θ i,λ j ) sin(ω semiann t + φ semiann ). (3) Recent studies Knudsen & ndersen (2002) and Knudsen (2002) have found a significant signal caused by aliasing of erroneous S2 tidal frequency with the orbit frequency to be present in this area. This aliasing period, ω S2 is approximately 162 days (Knudsen 2002), which we add to our model as N S2 (θ i,λ j ; t) = S2 (θ i,λ j ) sin(ω S2 t + φ S2 ). (4) The total geoid change of both the, as well as the earthquakes is modelled as the sum of the coseismic effect, and the post-seismic effect ( ) N ν (θ i,λ j ; t) = νco H tν (t) + νpost H tν (t) 1 e t tν τ (5) where the first term of the sum is the coseismic effect and the latter is the post-seismic effect. Thereby, ν = 1 or 1, 2 depending on whether we model only the earthquake (ν = 1) or also the earthquake (ν = 2). t ν is the time of the corresponding earthquake, and H tν (t) is the Heaviside step-function at time t ν. To account for the possibility of a long-term linear trend, as well as a constant bias, we may introduce N trend (θ i,λ j ; t) = trend (θ i,λ j )t, (6) N bias (θ i,λ j ; t) = bias (θ i,λ j ). (7) The bias ( bias ) is identified with the long-term mean geoid. In presence of the terms νco H tν (t) it will be the mean geoid height before the first event. It is difficult to judge, what the causes for a linear trend might be. It could be due to long-term hydrological phenomena (such as ground water storage changes) or ocean mass change. The complete model N total (θ i, λ j ;t) now consists of N total = N ann (θ i,λ j ; t) + N semiann (θ i,λ j ; t) + N S2 (θ i,λ j ; t) + N ν Q (θ i,λ j ; t) + N trend (θ i,λ j ; t) + N bias (θ i,λ j ; t) (8) described by the coefficients ann,φ ann, semiann,φ semiann, S2,φ S2, trend, bias,τ, νco, νpost, t ν ν = 1or1, 2. The coefficients τ, νco, νpost, t ν, describe the earthquake(s), and the others account for other signals. The coefficients are estimated independently for each grid point oseismic and post-seismic effects s mentioned earlier, there is evidence of a post-seismic relaxation of the geoid lasting some years after the earthquake. This is believed to be caused by isostatic movements in the mantle (Panet et al. 2007). Pore fluid diffusion in the crust (Jónsson et al. 2003) as well as deeper in the mantle (Ogawa & Heki 2007) have also been suggested to cause post-seismic changes on relatively short timescales. In Vigny et al. (2005), it is suggested that in the first month after the earthquake, post-seismic movements are dominated by aseismic slip or coseismic slip caused by aftershocks. Because the characteristics of the post-seismic slip may be significantly different in the first month(s), the empirical model of a single exponential function describing the post-seismic relaxation may not be accurate. In the case of a relatively fast relaxation shortly after the earthquake, superimposed on a slower process stretching over several years, the measurement errors and low (monthly) sampling rate of the GR solutions make it difficult to separate the fast decay from a coseismic jump. We therefore choose to use only a single empirical post-seismic relaxation model as in eq. (5), but introduce the time t ν as a stochastic parameter, to be estimated using the measurement data. In this case, t ν is no longer interpretable as the epoch of the earthquake, because this is known with an accuracy of a few seconds due to seismic measurements. Instead, t ν marks a shift in the characteristics of the seismically caused geoid changes from a coseismic jump or relaxation with a fast decay rate of only several weeks or few months, to a postseismic relaxation according to the empirical model νpost (1 e t tν τ ) with τ of more than 1 year. The possible causes for a decay with a rate of around 3 months are discussed in some detail in Panet et al. (2007) Stochastic parameter estimation In the model in eq. (8), N depends linearly on all coefficients except for τ and t ν. For estimating the parameters, a variation of the Bayesian approach is used, while formulating the problem as a quasi-linear model (Gundlich & Kusche 2008). In keeping with the notation of (Gundlich & Kusche 2008), the vector of parameters is separated into β 1 = (τ, t ν ) T, on which there is non-linear dependence, and β 2 = ( ann,φ ann, semiann,φ semiann, S2,φ S2, trend, bias, νco, νpost ) T ν = 1or(1, 2) for the rest of the parameters on which the observation model depends linearly. For a given, constant value of β 1,wecansetupatraditional linear model describing the measurements, β1 β 2 = N GFZ Potsdam, GJI, 183, Geophysical Journal International 2010 RS

4 736 I. inarsson et al. For both τ and t ν, the a priori probability density functions p(τ)andp(t ν ) are introduced. These form the joint density function p(β 1 ) = p(τ)p(t ν ). The aprioridensity functions reflect our prior knowledge of the processes in question, thus restricting the results of an optimization to physically sensible values. In this case, the distribution p(t ν ) of the time is chosen as a normal distribution, but the probability density function p(τ)ofthe relaxation rate is a γ distribution, γ (k γ, θ γ )(seekotzet al. 1982). The γ distribution has the probability density function kγ 1 e p γ (τ) = τ τ θγ Ɣ(k γ )θ kγ γ which has a zero of multiplicity k γ 1atτ = 0, thus, by choosing k sufficiently large, the probability of drawing samples near τ = 0 can be made arbitrarily small. This is desirable, because the matrix β1 becomes near-singular for τ near zero. Given the aprioridensity functions, the posteriori density function for β 1 follows from the Bayes theorem p(β 1 N) p(β 1 )p( N β 1 ), where means proportionality. The expected value of β 1 given the measurements N ((β 1 N)) can now be calculated with the following integral ˆβ 1 = β 1 p(β 1 N)dβ 1 B 1 β 1 p(β 1 )p( N β 1 )dβ 1. (9) B 1 The expectation (β 2 N) for the parameters in β 2 obeys [ ] ˆβ 2 p(β 1 y) β 2 p(β 2 β 1, N)dβ 2 dβ 1 B 1 B 2 = p(β 1 y) ˆβ 2,β1 dβ 1, (10) B 1 where ˆβ 2,β1 is the expected value of β 2 given the measurements and a fixed value of β 1. Since in that case, the problem is linear, this is the traditional least-squares solution of β1 β 2 = N for a fixed value of β 1. In Gundlich & Kusche (2008), the problem is analysed in-depth, and a sampling based Monte arlo method for the calculation of the integrals in eqs (9) and (10) is proposed, as well as the respective proportionality constants. They also show how to obtain an estimation of the a posteriori covariance matrix (β 1, β 2 ) for the solution. Based on (β 1, β 2 ) we can draw conclusions on the quality of the estimated coefficients as well as their correlation to identify possible signal leakage. We have now finished describing the parametrization of the mathematical model describing the earthquake signatures as well as possible environmental effects present in the GR data. s will become clear, estimating both earthquake signatures from the GR data may be problematic due to the high formal correlations of the model parameters. We will therefore also investigate indirect methods of estimating the signature of each earthquake. This is done by subtracting a geophysical model describing one earthquake from the GR data prior to estimating the model coefficients. In Table 1, the four alternatives are listed w.r.t. which of the two events we try to extract from the data, and which (if any) gets subtracted beforehand. 2.3 Geophysical modelling Our geophysical model uses the coseismic slip model for the M w = 9.3 Sumatra 2004 earthquake by Hoechner et al. (2008a) which is a geodetic inversion of GPS data from Banerjee et al. (2007). The elastic parameters correspond to ISP91 (Kennett & ngdahl 1991). similar model is used for the M w = earthquake. Time dependence of the model is caused by viscous relaxation. The lithosphere is assumed to be purely elastic, the asthenosphere implements Burgers rheology, while the rest of the mantle employs standard Maxwell rheology. Burgers rheology introduces two additional parameters as compared to Maxwell rheology, which has the advantage that it can explain both short-term effects in the order of months or years as caused by earthquakes and observed by GPS, as well as effects lasting thousands of years like glacial rebound. The viscous parameters used in this study are derived from post-seismic GPS time series from the ndaman Islands by Paul et al. (2007), (Hoechner et al. 2008b). The arth layering parameters are listed in ppendix. ll computations are done with an updated version of the code PSGRN/PSMP by Wang et al. (2006). In Fig. 1 in ppendix, the modelled coseismic and post-seismic geoid change due to the and earthquakes are shown. In both cases, the modelled data has been filtered with the same filter as the GR data used in this study. The parametrization of both earthquakes is available online as Supporting Information. In Figs 1 and 8, we can compare the modelled geoid effect to the corresponding estimation from GR. The difference between the model and the GR data has several reasons. oncerning the data, there is significant noise in the GR signals, the artefacts in the corners of the upper panels in Fig. 1 are about 50 per cent of the earthquake signal. The geophysical model is based on GPS data and has been set up completely independently from the geoid data, a slight modification of the parameters α and η 1 (ppendix) could significantly improve accordance of the post-seismic behaviour between the GR and model data, but this is beyond the scope of this paper. Other potentially contributing processes, like mantle water diffusion Ogawa & Heki (2007) or post-seismic slip were not modelled in this study. 3 RSULTS 3.1 Functional model We fit the functional model described in eq. (8) in Section for the cases i iii in Table 1. stimating τ independently, that is by Table 1. The different approaches to coefficient estimation. pproach Sumatra ndaman earthquake earthquake i stimated ii stimated Reduced (geophysical model) iii Reduced (geophysical model) stimated iv stimated stimated 2010 GFZ Potsdam, GJI, 183, Geophysical Journal International 2010 RS

5 Sumatra-ndaman and earthquakes 737 oseismic change Postseismic relaxation GR Model Figure 1. omparison of estimated co- and post-seismic effects using the GR monthly solutions for ugust 2002 pril 2009, and the geophysical model in monthly intervals for December 2004 December Both are filtered with the same anisotropic filter with a = Note that the parameters of the Monte arlo integration are estimated w.r.t. the estimated measurement noise, which is non-existent in the model data, leading to unrealistic results. Therefore, the Monte arlo integration was not applied to the model data. using a non-informative prior distribution p(τ) const proved to be unstable and often lead to physically meaningless results. We therefore introduce a γ distribution as aprioridistribution, as described in Section 2.2.4, with a mean of 540 days, or 18 months. This value provides the best fit to the data from the geophysical model in Section 2.3. With standard deviation of 300 days, this corresponds to shape parameter k γ = 3.24 and scale parameter θ γ = days. The prior distribution for the t ν parameter(s) is chosen as a normal distribution, centred on the day of the respective earthquake and with a standard deviation of 50 days. The Monte arlo analysis described in Section draws samples of random variables to estimate the coefficient vectors β 1 and β 2 independently in each grid point. lthough the aprioridistribution of the t ν parameter is relatively narrow compared to the monthly resolution of the data,the distribution for τ is much wider, reflecting the limited priori knowledge of the parameter. This causes higher uncertainty in the estimated values of the τ parameter, and because the values are calculated independently in each grid point using a new set of random numbers for each point, the uncertainty of the estimation is reflected in the difference of the estimated values in adjacent points. The resulting noisy estimation of τ contradicts the assumption that τ is assumed to depend on several physical processes which presumably would cause more smooth values of τ.we therefore smooth the estimated τ values using a Gaussian smoothing window with radius of 50 km to get more realistic values, and feed the result back into the procedure as a constant value, keeping all other parameters as before. The post-seismic decay rate of 18 months corresponds to a halflife of 374 days. The time elapsed between the and earthquakes on the other hand, is only 92 days. Because the elapsed time is short compared to the half-life, the correlation between the estimation of the two events will be considerable, per cent. This high correlation causes significant leakage effects between the estimation of the two events. s a result, the estimate of one particular coefficient (e.g. the coseismic effect) for one earthquake often has 2010 GFZ Potsdam, GJI, 183, Geophysical Journal International 2010 RS

6 738 I. inarsson et al. constant trend cos 365 sin 365 cos 182 sin 182 cos 162 sin 162 post orrelation matrix co -1 constant trend cos 365 sin 365 cos 182 sin 182 cos 162 sin 162 post co Figure 2. Model covariances, calculated for constant τ = 18 months and t ν=0, when estimating coefficients from monthly data on the interval ugust 2002 pril Linear trend /yr Postseismic geoid relaxation /yr Figure 3. stimated linear trend and post-seismic relaxation from a joint parameter estimation using monthly GR data from ugust 2002 to pril Parameters τ and t ν=0 as in the lower part of Fig. 1. picentres of the Sumatra ndaman and earthquakes are at points marked and, respectively. Note the main features of the left figure (bottom left and right, centre) are repeated in the right one with opposite sign. the same geometrical pattern as that for the other earthquake, with opposite sign in case of negative correlation. Because those results do not make sense in light of our knowledge of the earthquakes, we rule out alternative iv in Table 1 without further discussion. From now on, we will only try to extract one earthquake at a time. Therefore, in eq. (8) only the case ν = 1 will appear, but ν may refer to either earthquake, or the earthquake. In Fig. 2, we can see the correlation coefficients of the estimated parameters, using the full model (eq. 8) for constant values of τ = 540 days and t 0, ν = 1. s is evident from the correlation coefficients, there are two parameter pairs with noticeably high correlation, namely ρ( 1bias, 1trend ) = 0.75,ρ( 1post, 1trend ) = When considering noisy data, signal caused by one process of such a highly correlated pair is likely to be falsely attributed to the other one with an opposite sign in case the correlation is negative. In this study, the parameters νpost, νco are of main interest, the others serve merely to separate those from other signals also present in the data. Therefore, high correlation between νbias and νtrend is of no interest here. The same argument is valid for the pair (cos 182ω, cos 162ω) which has a negative correlation of 19 per cent, slightly higher than average. Because the correlation to the parameters describing the earthquake are low, this does not pose a problem. The strong correlation between the post-seismic signal, νpost, and the linear trend, trend, is the more problematic one. It reflects the high probability that in the presence of measurement errors, some part of the post-seismic relaxation will be falsely attributed to the long-term linear trend or vice versa, thereby affecting the estimated total effect of the earthquake. By comparing the figures of the estimated trend and the corresponding post-seismic relaxation, seen in Fig. 3, the same geometrical patterns are seen in various places but with opposite sign. The pattern estimated when including linear trend is weaker and does not follow the known fault line of the earthquake as well compared to the same calculations when omitting the trend. Therefore, we suspect that the linear trend also captures some of the earthquake signal GFZ Potsdam, GJI, 183, Geophysical Journal International 2010 RS

7 Sumatra-ndaman and earthquakes Long term trend /yr northernmost part of the post-seismic effect in Fig. 1 (around point ) is not affected by any leakage from hydrological signal. 3.2 stimated earthquake gravity footprint s mentioned earlier, we have not been able to separate the gravity footprints of the and earthquakes from the GR data, because of the high correlation of the corresponding parameters. Note that this is due to the earthquakes being close both in time and space it does not necessarily mean that the earthquake is too small to be detectable in the data. To determine whether that is the case, we have to revert to indirect methods, based on combining modelled data and the functional model for the Sumatra earthquake described earlier. We use a geophysical model describing the and earthquakes separately. We start by analysing the alternative i from Table 1, where the signal is estimated using original GR data. In Fig. 1, the co- and post-seismic signals are shown, estimated by only solving for the earthquake. comparison of the GR time-series and the signal predicted by the geophysical model, as well as the functional model described by the estimated coefficients is shown in Figs 5 7. In Fig. 8, the estimated total effect of the earthquake, that is coseismic plus post-seismic, is shown for pril Figure 4. stimated long-term geoid trend from the WGHM hydrological model, using monthly intervals for ugust 2002 February 2008, filtered using a = 11. xperiments were done to subtract from the data an estimated long-term trend, derived from a hydrological model WGHM, (Döll et al. 2003), and subsequently omitting trend from eq. (8). This procedure shows no significant improvement compared to ignoring any possible trend altogether, neither for co- nor post-seismic parameter estimate. Some patterns, that were suspected to be of hydrological origin did indeed disappear, but in other areas, patterns from WGHM were clearly visible in the results, suggesting that the long-term trend from the model is less accurate than the GR measurements, at least in those areas. We therefore decide not to subtract any hydrological model from the GR measurements, and not to include the term trend in our stochastic modelling in eq. (8). co and post, estimated ignoring linear trends and modelling only the earthquake, are shown in the lower part of Fig. 1, compared to the same results using the geophysical model. We should note, that it may contain some leakage effects from hydrological trends. s an example, we could take the geoid low over south-central Sumatra, south-east of the southern tip of the positive ridge in the left Fig. 1. In Fig. 4, we recognize a long-term trend in geoid height derived from the WGHM hydrological model (Döll et al. 2003). lthough we find the trend too unreliable to correct the GR gravity fields, Fig. 4 gives an indication of how large a possible hydrological longterm trend might be. Using the maximum value of 0.3 yr 1 and assuming a one-to-one leakage into the relaxation coefficient post over the entire GR timespan after the earthquakes ( 5 yr), we obtain a maximum effect of 1.5. However, the maximum effect is only to be expected over land, while over ocean the maximum effect is reduced by half, giving 0.8, which is almost a third of the modelled effect of the earthquake. Hydrological signal may therefore affect the southernmost part of the post-seismic signal. On the other hand, the kernel corresponding to the anisotropic filtering has a half-radius of approximately 200 and 250 km in the north south and east west directions, respectively. Therefore, the geoid change [] Geoid change, at point: =(7.04,96.33) Fitted model GR data Geophysical model - Sumatra + Geophysical model - Sumatra only time [year] Figure 5. Geoid change by the Sumatra ndaman/ earthquakes as described by model, compared to the fitted parameters and GR data in point. geoid change [] Geoid change, at point: =(7.75,91.25) Fitted model GR data Geophysical model - Sumatra + Geophysical model - Sumatra only time [year] Figure 6. Geoid change by the Sumatra ndaman/ earthquakes as described by model, compared to the fitted parameters and GR data in point GFZ Potsdam, GJI, 183, Geophysical Journal International 2010 RS

8 740 I. inarsson et al. 8 6 Geoid change, at point: =(3.0,92.6) Fitted model GR data Geophysical model - Sumatra + Geophysical model - Sumatra only geoid change [] time [year] Figure 7. Geoid change by the Sumatra ndaman/ earthquakes as described by model, compared to the fitted parameters and GR data in point. In Fig. 9, the estimated relaxation rate, τ, for the unreduced GR data is shown. The apriorivalue of 540 days (18 months) was chosen in compliance with the geophysical model, which is calibrated using measurements independent from GR. s we see, the estimated value deviates from the aprioriwest of point, near to where the post-seismic relaxation is the strongest, where it reaches a minimum of around 507 days. One feature in Fig. 9 attracts our attention namely the local minimum north of, exactly at the epicentre of the earthquake. The progression of the estimated τ along the fault line contains a large minimum near point, and a small one near. This does not resemble the progression of the estimated post along the fault (Fig. 1, top right). Due to the nature of the Monte arlo method, we would expect resemblance if τ had a constant value, lower than the apriorivalue. This may suggest, that τ is indeed not constant, but rather a variable dependent on, for example material properties (e.g. thickness) or on stress or strain. GR τ (relaxation rate) days Figure 9. stimated relaxation rate, τ, using monthly GR data from ugust 2002 to pril It is consistent with this suggestion, that the northern minimum is located where the earthquake caused the most dislocation, and the southern minimum where all the dislocation due to took place. further argument is that the minimum disappears when removing the modelled effect (Fig. 12). lthough this is not enough information to separate the signal of the earthquake from that of the earthquake, it is an indication that there is a significant signal from the earthquake present in the data. Subtracting the modelled gravity footprint of the earthquake (case iii in Table 1) to subsequently estimate the earthquake is Model Total geoid change Total geoid change Figure 8. Left panel: Total geoid change, including co- and post-seismic effects, caused by earthquakes after 4 years, calculated by evaluating the co- and post-seismic parts of the functional model at t = 4 years. Right panel: Total geoid change according to the geophysical model 4 years after the Sumatra ndaman earthquake. This includes both co- and post-seismic effects GFZ Potsdam, GJI, 183, Geophysical Journal International 2010 RS

9 Sumatra-ndaman and earthquakes oseismic geoid change Postseismic geoid relaxation /yr Figure 10. stimated coseismic (left panel) and post-seismic (right panel) effect of the earthquake,using monthly GR data from ugust 2002 to December 2008, modelled effect of the Sumatra ndaman earthquake removed. a theoretically feasible approach. However, the model may not be accurate. The corresponding estimated co- and post-seismic effects (of the earthquake) are shown in Fig. 10. We see a noticeable positive area at the northern tip of Sumatra, very likely caused by a discrepancy between the model and the true gravity footprint of the earthquake. Note that the gravity footprint of the earthquake is about a fourth that of. Furthermore, the signal is spatially more concentrated than that of the earthquake, therefore more of its signal is contained in the higher degree spherical harmonics which derived from GR are known to be prone to errors. It is therefore likely, that a per cent modelling error of the earthquake would have nearly the same effect as the entire earthquake, rendering any conclusions from this kind of analysis impossible. We have now eliminated the possibility to directly estimate the earthquake gravity footprint using the GR data. s already mentioned, this does not mean that it is too small to be visible, only that it cannot be separated from the earthquake. It remains to be investigated, what effect the presence of the signal has on the estimated signal. Until now, this has been neglected by authors extracting the signal from GR data. We subtract the model from the GR measurements and use the reduced data to estimate the gravity footprint (case ii in Table 1). The result can be seen in Fig. 11 and we clearly see a difference when compared to the results of the same analysis carried out using the original GR data (Fig. 1). To argue that the reduced data are indeed better described by the functional model, the estimated posterior signal variances may be used. In Fig. 12, we show σ orig σ res, where σ orig is the posterior signal variance of the original GR data, and σ res that of the reduced data. It is clear, that by subtracting the model, we get an improvement of about 5 per cent in the posterior signal variance. The improvement in signal variance oseismic geoid change /yr Postseismic geoid relaxation Figure 11. stimated coseismic (left panel) and post-seismic (right panel) effect of the Sumatra ndaman earthquake,using monthly GR data from ugust 2002 to December 2008, modelled effect of the earthquake removed GFZ Potsdam, GJI, 183, Geophysical Journal International 2010 RS

10 742 I. inarsson et al Δσ τ days Figure 12. Left panel: Difference of the estimated a posteriori error variance for using original data and monthly GR data from ugust 2002 to December 2008, reduced by the model. ˆσ orig ˆσ. Right panel: stimated relaxation rate τ using reduced GR data from ugust 2002 to December /yr Figure 13. Difference of the amplitude of the coseismic (left panel) and post-seismic (right panel) geoid signal when calculated with unreduced data and monthly GR data from ugust 2002 to December 2008, reduced with modelled earthquake, co,orig co, or post,orig post,, respectively. suggests that the modelled geoid effect describes the reality with an accuracy well within a factor of two. Hence, we can assume that the co- and post-seismic effects from the reduced data (Fig. 11) contain significantly less contribution from the earthquake and are therefore a better approximation to the effect of the earthquake only than the results in the upper row of Fig. 1. In the right part of Fig. 12, we see the estimated relaxation time τ using the reduced data. bove, we noticed that when using unreduced data, the relaxation rate had two minima, one near the epicentre of the earthquake (Fig. 9). When using the reduced GR data, this minima mostly disappears. This is a further argument, that the effect of the earthquake is indeed significant in GR data. ssuming the geophysical model correctly represents the effect of the earthquake, the difference w.r.t. the estimation using unreduced data (Fig. 12) can be interpreted as that part of the signal that leaks into the estimation of the signal. 3.3 Time delay s described earlier, the time t 0 is estimated alongside other parameters. s a stochastic parameter, this marks the transition from the coseismic or fast post- or aseismic relaxation, to the long-term post-seismic characteristics. lthough the time of the earthquake is known with an accuracy of seconds, we introduce the time t ν as a stochastic parameter is to look for evidence for faster processes than the post-seismic relaxation rate of 540 days. Introducing a second exponential function is not feasible because the data is too noisy and the temporal resolution too low. Therefore, we use the parameter t ν to absorb fast processes near the apriorivalue of t ν (the exact time of the earthquake). Because a second coseismic jump due to the earthquake would attract the t ν estimate, we choose the aprioridistribution with standard deviation of 50 days, which is wide enough for the 2010 GFZ Potsdam, GJI, 183, Geophysical Journal International 2010 RS

11 Sumatra-ndaman and earthquakes days t days t 0 Figure 14. stimated t 0 deviation from the apriorivalue using unreduced (left panel) and -reduced (right panel) GR monthly data from ugust 2002 to December The epoch of the earthquake is 92. coseismic signal to affect the estimation of t ν, if it is present in the data. The estimated t 0, using both original and reduced data (case ii), is shown in Fig. 14. Two facts attract our attention. In the first place, while the estimated t 0 deviates only slightly from the apriori value on large areas, a significant shift away from the apriori value is noticeable in the ndaman sea. The shift is geologically consistent and always in the same direction, namely around 30 days after the earthquake. The area with the largest deviation is in the ndaman sea, and is of similar size and shape as the area which Panet et al. (2007) (Figs 7 and 9) associate with a transient signal. Panet et al. (2007) find this transient signal to level out after about three months. Because our estimation of t 0 is of course also affected by the coseismic change, which has its largest values in this area, the estimation of the parameter t ν=0 will be a compromise between the fast relaxation and the coseismic effect. Secondly, the absence of any deviation from the apriorivalue near is interesting. Because the apriorivalue is the epoch of the earthquake, the existence of a significant (coseismic) jump three months later would have an attracting effect on the estimation of t ν. The fact that there is no anomaly evident in the area affected by the earthquake, suggests that the coseismic effect of the earthquake is too small to stick out against the effects and the measurement noise also evident in the data. 4 DISUSSION We have established a functional model to describe the measured total gravity variation in the area of the earthquake using GRderived Stokes coefficients. We adjust the model parametrization to minimize the effect of correlations, yet take into account those dominant environmental effects that are present in the area over the timespan in question. s has already been noted by various authors (De Linage et al. 2008), the co- and post-seismic effects of the earthquake are clearly visible in the GR data. Regarding the earthquake, we conclude, that the and earthquake signals cannot be reliably separated using only GR measurements. This is mainly due to the fact that they are only a few months apart, and that at the spatial resolution of GR the gravity footprints overlap. The first causes high formal correlation between the model parameters that have to be estimated. The latter makes the correlation problematic. When both signals are present at the same point, the highly correlated model parameters have to be estimated at the same grid point. The correlation may cause one signal to be mistaken for the other, especially in the presence of the measurement noise of the GR mission, which is large relative to the size of the signals we are trying to extract. We are therefore left with the option of estimating only the effect of the earthquake. Using GR data, reduced for the modelled effect of the earthquake, to subsequently estimate the signal proved unsuccessful. This suggests, that either the signal is not strong enough to stick out against the measurement noise, or that the model error of the earthquake dominates the signal of the earthquake. However, estimating the signal from reduced GR data yields significantly different results compared to using unreduced GR data. Using GR data reduced by a geophysical model of the earthquake, the a posteriori variance is reduced, and the variance of the co- and post-seismic effect is reduced by up to a maximum of 45 and 30 per cent respectively, where the effect of the earthquake is largest. This suggests, that the modelled effect of the earthquake provides a better explanation for the source of the GR data than the null-hypothesis, that is that there is no trace of the earthquake contained in the data. However, the data is too noisy to draw any further conclusions on the quality of the model, especially because the effect of the earthquake is present in the same area. Furthermore, the estimations of the relaxation rate τ has a minimum at the epicentre of the earthquake, of a similar nature as what is present where the strongest post-seismic relaxation of the earthquake west of the ndaman islands. Together with the fact, that the minimum at disappears when using the -reduced GR data, indicates that the post-seismic effect of the effect is indeed significant in the GR data. The estimation of the epoch of a possible jump seems not to be affected by the earthquake. This suggests that the 2010 GFZ Potsdam, GJI, 183, Geophysical Journal International 2010 RS

12 744 I. inarsson et al. coseismic signal is too small to stick out against the measurement noise on its own. On the other hand, the post-seismic signal has the same characteristics in time (hence the correlation) for both earthquakes. Therefore, the post-seismic signal complements the one from and so makes a significant difference when added to it. In this work, we present a geophysical model of the earthquake, based on seismic and GPS measurements, which can be used as a first estimate of possible leakage effect. The size of the post-seismic relaxation is based on GPS measurements, which are scarce, both in space and time. In addition, they are dependent on simplifying assumptions on the viscoelastic properties of the crust and mantle. Therefore, the calculated geoid relaxation may still contain significant errors. On the other hand, the geographical extent of the model is based on seismic measurements, with considerably higher accuracy than the wavelength of the GR-derived gravity field. Therefore, the extent of the relaxation due to the earthquake may be assumed to be very similar to what is shown in the bottom right part of Fig. 1. We therefore have quite strong information on where leakage may (and does) occur, but only weak information on the strength. Because of the uncertainty of the geophysical model, Fig. 11, compared to Fig. 1, merely gives an idea of the extent of possible leakage effect. In any case it is clear, that when estimating the geoid effect of the earthquake using GR data without regarding the effect of the earthquake, the estimation will contain a significant contribution from the earthquake. In Fig. 8, we see the total effect of the earthquake after 4 years of relaxation, for GR and the geophysical model. We see, that the minimum in the ndaman sea, mostly caused by the coseismic jump, has similar size for both model and GR data. However, the centre of the modelled effect is shifted to the east w.r.t. the GR data. The other distinguishing feature is the maximum to the west of the trench. lthough there are some deviations between the model and GR results, the magnitude of the effect is the same, as well as the main geometric feature, the uplift effect above the fault. This indicates, that the viscoelastic model used in this work is an adequate approximation of the processes at work in the crust and mantle in the aftermath of the earthquakes. ccording to the model, the viscoelastic relaxation of the coseismically induced stress change causes a viscous flow in the asthenosphere and upper mantle, resulting in a continuation of the coseismic crust movements in depth like a diffusion process. Both the coseismic crust motion and the induced viscous flow in the asthenosphere are dominated in the horizontal direction towards the subduction zone. s a consequence we can observe a continuous uplift of the focal area during the post-seismic period. The phenomenon has been demonstrated by a simple viscoelastic dislocation model [see figs 6 and 7 in Rundle (1982) or fig. 3 in Wang et al. (2006)]. In general, the Kelvin and Maxwell elements are used to describe the short-term transient and long-term steady-state relaxation processes, respectively. In this study, the best-fit viscosity we found for the Kelvin element is about Pa s that corresponds to a characteristic relaxation time of about 1.5 yr. The Maxwell element has a relaxation time one order larger than the Kelvin element. ompared with the 5 years period of the GR data, we can conclude that the observed post-seismic geoid uplift is dominantly caused by the short-term transient relaxation process in the asthenosphere. fterslip events arise generally within a few months after the earthquake. The timescale is similar or smaller as for the transient relaxation mentioned above, however the geoid change caused by afterslip amounting to 10 per cent of the coseismic slip is much smaller than that caused by relaxation processes, as we show elsewhere (Hoechner et al., in preparation). We notice, that the relaxation above the fault is considerably larger in the viscoelastic model than in GR. From Fig. 1, it is clear that the area of positive post-seismic relaxation amplitude reaches into the ndaman sea for both the geophysical model and the GR estimation. Overestimating the post-seismic relaxation can therefore cause the minimum in Fig. 8 to be shifted to the east, explaining the difference of the minima that we noticed earlier. In chapter 2.3, we suggested, that by modifying parameters in the rheology model the accordance of the post-seismic behaviour between the GR data and the model could be significantly improved. Finally, when estimating the epoch of a possible jump, we came across an area in the ndaman sea, where the estimated epoch deviates from that of the earthquake. This has a similar shape and position as the transient signal found by (Panet et al. 2007), contributed to fast post-seismic relaxation or aseismic slip. KNOWLDGMNTS The authors thank Roland Schmidt for many discussions. RFRNS Banerjee, P., Pollitz, F., Nagarajan, B. & Burgmann, R., oseismic slip distributions of the 26 December 2004 Sumatra-ndaman and 28 March 2005 earthquakes from GPS static offsets, Bull. seism. Soc. m., 97(1), S86 S102. De Linage,., Rivera, L., Hinderer, J., Boy, J.-P., Rogister, Y., Lambotte, S. & Biancale, R., Separation of coseismic and postseismic gravity changes for the 2004 Sumatra-ndaman earthquake from 4.6 yr of GR observations and modelling of the coseismic change by normalmodes suation, Geophys. J. Int., 176, , doi: /j x x. De Viron, O., Panet, I., Mikhailov, V., Van amp, M. & Diament, M., Retrieving earthquake signature in GR gravity solutions, Geophys. J. Int., 174, doi: /j X x. Döll, P., Kaspar, F. & Lehner, B., global hydrological model for deriving water availability indicators: model tuning and validation, J. Hydrol., 270, Flechtner, F., GFZ Level-2 Processing Standards Document For Level- 2 Product Release 0004 (Rev. 1.0, 2007 February 19), GeoForschungsZentrum Potsdam, Germany. Flechtner, F., Release notes for GFZ RL04 GR L2 products ( ), GeoForschungsZentrum Potsdam, Germany. Gundlich, B. & Kusche, J., 2008, Monte arlo integration for quasi-linear models, in VI Hotine-Marussi Symposium on Theoretical and omputational Geodesy, Wuhan, hina, ed. Xu, P., Liu, J. & Dermanis,. Han, S.-. & Simons, F., Spatiospectral localization of global geopotential fields from the Gravity Recovery and limate xperiment (GR) reveals the coseismic gravity change owing to the 2004 Sumatra-ndaman earthquake, J. geophys. Res., 113, B01405, doi: /2007jb Han, S.-., Shum,.K., Jekeli,., Kuo,.-Y. Wilson,.R. & Seo, K.-W., Non-isotropic filtering of GR temporal gravity for geophysical signal enhancement, Geophys. J. Int., 163, 18 25, doi: /j X x. Han, S.-., Sauber, J, Luthcke, S.B, Ji,. & Pollitz, F.F., Implications of postseismic gravity change following the great 2004 Sumatra-ndaman earthquake from the regional harmonic analysis of GR intersatellite tracking data, J. geophys. Res., 113, B11413,doi: /2008JB Hoechner,., Babeyko,.Y. & Sobolev, S.V., 2008a. nhanced GPS inversion technique applied to the 2004 Sumatra earthquake and tsunami, Geophys. Res. Lett., 35, L08310, doi: /2007gl GFZ Potsdam, GJI, 183, Geophysical Journal International 2010 RS

13 Sumatra-ndaman and earthquakes 745 Hoechner,., Wang, R., inarsson, I. & Sobolev, S.V., 2008b. nalysis of postseismic effects of the Sumatra 2004 earthquake: contribution to GPS and geoid height change from viscoelasticity, OS, Trans. m. geophys. Un., 89(53), Fall Meet. Suppl., bstract G Jónsson, S., Segall, P., Pedersen, R. & Björnsson, G., Post-earthquake ground movements correlated to pore-pressure transients, Nature, 424, Kennett, B.L.N. & ngdahl,.r., Traveltimes for global earthquake location and phase identification, Geophys. J. Int., 105(2), Knudsen, P., Ocean tides in GR monthly averaged gravity fields, Space Sci. Rev., 108(1 2), doi: /: Knudsen, P. & ndersen, O., orrecting GR gravity fields for ocean tide effects, Geophys. Res. Lett., 29 8 (19)1 4. Kotz, S., Johnson, N.L. & Read,.B., ncyclopedia of Statistical Sciences, Wiley-Interscience, Hoboken, NJ. Kusche, J., pproximate decorrelation and non-isotropic smoothing of time-variable GR-type gravity field models, J. Geod., 81(11), doi: /s Lemoine, J.M., Bruinsma, S., Loyer, S., Biancale, R., Marty, J.., Perosanz, F. & Balmino, G., Temporal gravity field models inferred from GR data, dv. Space Res., 39(10), Melachroinos, S.., Lemoine, J.M., Tregoning, P. & Biancale, R., Quantifying FS2004 S 2 tidal model from multiple space-geodesy techniques, GPS and GR, over North West ustralia, J. Geodesy, 83, doi: /s Ogawa, R. & Heki, K., Slow postseismic recovery of geoid depression formed by the 2004 Sumatra-ndaman earthquake by mantle water diffusion, Geophys. Res. Lett., 34, L06313,doi: /2007GL Panet, I. et al., oseismic and post-seismic signatures of the Sumatra 2004 December and 2005 March earthquakes in GR satellite gravity, Geophys. J. Int., 171, Paul, J., Lowry, R., Bilham, R., Sen, S. & Smalley, R., Jr, Postseismic deformation of the ndaman Islands following the 26 December, 2004 Great Sumatra ndaman earthquake, Geophys. Res. Lett., 34, L19309, doi: /2007gl Rundle, J.B., Viscoelastic-gravitational deformation by a rectangular thrust fault in a layered arth, J. geophys. Res., 87(B9), Schmidt, R., Flechtner, F., Meyer, U., Neumayer, K.-H., Dahle,., König, R. & Kusche, J., Hydrological signals observed by the GR satellites, Surv. Geophys., 29(4 5), doi: /s Swenson, S. & Wahr, J., Methods for inferring regional surfacemass anomalies from gravity recovery and climate experiment (GR) measurements of time-variable gravity, J. geophys. Res., 107, doi: /2001JB Swenson, S. & Wahr, J., Post-processing removal of correlated errors in GR data, Geophys. Res. Lett., 33, L08402, doi: /2005gl Tapley, B.D., Reigber,., Bettadpur, S., Flechtner, F., Ries, J. & Watkins, M. 2005, The GR Mission Status, merican Geophysical Union, Spring Meeting 2005, bstract #G Vigny,. et al., Insight into the 2004 Sumatra ndaman earthquake from GPS measurements in southeast sia, Nature, 436, 7048, Wang, R., Lorenzo-Martín, F. & Roth, F., 2006, PSGRN/PSMP a new code for calculating co-and post-seismic deformation, geoid and gravity changes based on the viscoelastic-gravitational dislocation theory, omput. Geosci., 32-4, Wouters, B. & Schrama,.J.O., Improved accuracy of GR gravity solutions through empirical orthogonal function filtering of spherical harmonics, Geophys. Res. Lett., 34, L23711, doi: /2007gl PPNDIX : RHOLOGIL RTH MODL The geoid changes induced by the earthquake models are computed using code PSGRN/PSMP (Wang et al. 2006) implementing Burgers rheology (a Kelvin Voigt body and a Maxwell body in series connection) for relaxation of shear modulus. lthough the Maxwell element explains long-term relaxation processes (for instance, glacial rebound), the Kelvin element enables modelling of short-term effects. No relaxation of compressional modulus is considered. n additional tool POTON by R. Wang is used to extrapolate the theoretical co- and post-seismic geopotential changes from the solid earth surface (i.e. the ocean bottom) to the geoid (including the effect of the ocean). The Table 1 contains the parametrization of the viscoelastic, gravitational half-space. lastic parameters Table 1. Parametrization of the viscoelastic, gravitational half-space. No. Depth v p v s ρ η 1 η 2 α GFZ Potsdam, GJI, 183, Geophysical Journal International 2010 RS

14 746 I. inarsson et al. oseismic Postseismic 1 1 Sumatra ndaman Figure 1. Modelled co- and post-seismic effects for the Sumatra ndaman and earthquakes separately. In contrast to Fig. 1, these results are not estimated by fitting a functional model to the models, but are direct results of the geophysical models (December 2004 December 2008). Note the different colour scales for the earthquake. correspond to ISP (Kennett & ngdahl 1991), viscous parameters are from Hoechner et al. (2008b) based on analysis of post-seismic GPS time-series. The elastic layer has a thickness of 40 km, the transient (Burgers) zone goes to 210 km depth, below is Maxwell rheology. The following parameters are listed in Table 1. See Fig. 1 for an illustration of a Burgers body. In Fig. 1, we see the co- and post-seismic effects for the Sumatra ndaman and earthquakes separately. η 1 Transient viscosity (dashpot of the Kelvin Voigt body.η 1 0means infinity value) (Pa s) η 2 Steady state viscosity (dashpot of the Maxwell. body.η 2 0means infinity value) (Pa s). α = μ 1 μ 1 +μ 2 (> 0and 1).Ratio between the effective and the unrelaxed shear modulus. α 1 α μ 2 μ 1 = μ 2 = vs 2ρ Figure 2. Burgers body GFZ Potsdam, GJI, 183, Geophysical Journal International 2010 RS