Since the devastating earthquake and tsunami in 2004 offshore Sumatra, many source models have been put forward. Recent studies clearly show that modern GPS-processing could achieve high resolving power for slip in near real time, which is crucial for determining tsunami initial conditions, provided accurate GPS-processing and inversion. Here, we propose an inversion technique with improved representation of the subduction zone geometry and physically justified boundary conditions. We show that the discrepancy between the inversion of near- and far field GPS data for the 2004 event, which is often explained by postseismic slip, can be eliminated by using our inversion method and IASP91 earth model. Inverted source models, including versions with splay faulting, are shown to be consistent with satellite altimetry data of offshore tsunami wave height, suggesting that displacement at the splay fault might have been present but was likely a second order process.
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 The Sumatra-Andaman earthquake of December 2004 is probably the most extensively analyzed earthquake-tsunami event ever. Recently, a number of special issues of scientific journals (see introductory papers by Bilek et al. , Gu  and Tanioka et al. ) have been dedicated to the investigation of a very broad spectrum of aspects of this event. Tsunami source models have been proposed based on seismic, tide gauge, satellite altimetry, GPS-data and combinations thereof. Of first order importance as initial condition for the tsunami is the static deformation of the sea bed resulting from the coseismic relative motion between the subducted oceanic and the overriding continental plate It can be computed if the slip distribution at the fault zone is known. While teleseismic inversions yield a detailed picture of rupture timing and extent [Krüger and Ohrnberger, 2005], GPS-inversions provide a more direct measure for slip [Banerjee et al., 2007], are available shortly after an earthquake [Blewitt et al., 2006], and could even be used to follow rupture propagation in near-real time [Sobolev et al., 2007]. It is one of the goals of the German Indonesian Tsunami Early Warning System (GITEWS) to provide reliable information about expected arrival times, wave heights and inundation as quickly as possible to local warning centers in order to save lives and protect infrastructure. This is rendered difficult by the geomorphological settings in Indonesia: the trench is located closely to the coast, and the bathymetry is complex, including islands either protecting the main land or trapping tsunami energy in the forearc basin in case of a deep earthquake. Traditionally used tsunami source models based on epicenter and magnitude are not first choice: The epicenter does not necessarily coincide with the position of slip maximum, slip heterogeneities play an important role in the near-field [Geist and Dmowska, 1999], and the magnitude of large earthquakes is often underestimated during the first minutes. Thus, some events might not be recognized as being dangerous while at the same time the number of false alarms would be prohibitively high. These problems can be overcome using GPS [Sobolev et al., 2006, 2007], which is ideally suited for local Tsunami early warning.
 However, GPS based inversions for slip distribution still need to be improved. Recent inversions for the Sumatra 2004 event imply inconsistency between near- (<300 km) and far field (300–900 km) GPS data and tend to underestimate tsunami wave heights compared to satellite radar altimetry data [Chlieh et al., 2007; Pietrzak et al., 2007]. Chlieh et al.  attribute the apparent inconsistency in the GPS observations to large unconsidered postseismic slip, which, however, was not confirmed by the analyses of Banerjee et al. . Discrepancy between predicted and observed tsunami wave heights may arise due to activation of splay faults [Plafker et al., 2007], but may also result from an inappropriate earth model used in inversion (see below). To address these issues, we first present an enhanced inversion technique, and then apply this technique to the Sumatra 2004 event, focusing primarily on roles of postseismic slip and splay faulting by comparison of our modeling results with observations.
 We employ the following slip inversion procedure. First, the subduction interface is discretized into subfaults. A Green's functions approach is used to find the slip distribution which minimizes misfit between observed and modeled GPS displacements and which is in compliance with imposed physical boundary conditions. Finally, the slip distribution at the fault is used to compute the coseismic deformation of the sea floor, which provides the initial condition for the tsunami wave propagation code.
 Most GPS based approaches, this study included, rely on a-priori knowledge of the potential fault zone geometry because, on the one hand, inversion for fault geometry itself is highly nonlinear, computationally intensive [Maerten et al., 2005] and requires a number of measurements which is rarely achieved with nowadays distribution of GPS-receivers. On the other hand, especially for large subduction earthquakes being of importance in the tsunami early warning context, slip can be assumed to be localized fairly well near the top of the Benioff zone, with exception maybe of particularities like splay faults. Discretization of the fault should provide resolution high enough to allow for realistic slip heterogeneities [Geist and Dmowska, 1999], overlap of adjacent subfaults should be minimized and variation of the dip angle along and perpendicular to the trench has to be taken into account. We propose an algorithm to perform automatic discretization of the subduction interface which will be made available, and works as described in the auxiliary material. A splay faulting option is included in the form of assigning a fix dip angle to selected subfaults (or using these faults additionally).
 Our reference fault geometry model for the 2004 event consists of 12 × 36 = 432 subfaults ranging from 5 to 53 km depth (models ranging deeper revealed no slip below 50 km). Additionally, we constructed two models featuring splay faulting: model Sp260 and model Sp845a are described in the ‘Results’ section, inversions using other geometries can be found in the auxiliary material.
 For the forward model we use FORTRAN code EDGRN/EDCMP by Wang et al.  which is based on a semi-analytical approach for a layered half-space and applies an orthogonalization scheme to accurately compute the layering effects. We restrict ourselves to GPS-stations which are located closer than 900 km from their nearest subfault, hence sphericity does not play a significant role [Banerjee et al., 2005; Chlieh et al., 2007], especially since some distance correction is taken into account by using a stereographic projection from spherical to cartesian coordinates. Earth layering models are based on IASP91, PREM (continental) and CRUST2 (oceanic) seismic velocity models.
 The inversion concept is similar to the one used by other authors, but provides some changes concerning regularization to enhance physical justification of the boundary conditions. Green's functions, being the response of the GPS-stations (three or two components) to unit dip- and strike-slip at the subfaults, are computed. The number of model parameters is larger than the number of observations, which renders the inversion an optimization problem. The cost-function to be minimized is set up in the following way:
The weighting factors λ1 and λ2 for the smoothing terms are calibrated using synthetic scenarios with checkerboard, homogenous (end members) and realistic slip distributions. Summation for χ2 is over the available displacement components at all GPS-stations, dobs are the observed, dmod are the modeled (predicted) displacements and σ are the 1-sigma standard deviations. Summation in the smoothing terms is over all subfaults (SF) and their respective nearest neighbors (NN). Additionally, for the slip-smoothing term a ‘virtual’ subfault layer is placed at the lower and lateral boundaries (BND) which is set to zero to avoid physically impossible infinite stress concentration at the fault boundaries. The slip smoothing regularization with above boundary conditions enables us to skip global moment minimization or driving moment towards an a-priori target as done e.g. by Subarya et al.  or Chlieh et al. . The first would lead to patchy patterns with high slip at measurement points and low slip in-between, while the second is hard to justify, because seismic and geodetic moment are not easily comparable as being derived from very different frequencies and measuring different processes (energy release vs. acoustic luminosity [Menke et al., 2006]). Instead of applying smoothing to rake angles, we apply it to azimuth of slip vectors. This makes some difference for large events with strongly varying strike- and dip angles, and is in accordance to the relative motion of the tectonic plates. No minimum/maximum constraints on slip and rake are required. Minimization of the cost function is performed with a quasi-Newton line-search algorithm from the Matlab Optimization Toolbox (The MathWorks, Inc.).
 Usually, timing of the rupture can be neglected for tsunami initial conditions. For this exceptionally large event though, which lasted around 10 minutes, some influence of the timing is reflected in the sea surface height as recorded by satellite JASON1 (see Figure S4). Here, we base the timing of the source on a GPS-inversion for rupture propagation by Vigny et al. . It is generally consistent with teleseismic analysis by Krüger and Ohrnberger  and shows slip having started at 94.7°E/3.1°N and rupture velocity being around 3.7 km/s for the first 200 km and then slowing down to about 2 km/s. Rise time is assumed to be 60 s. Effect of timing on JASON1-track is shown in Figure S4.
 In general, our inversion is in agreement with previous studies, e.g. Chlieh et al. . Inverted slip, together with observed and modeled GPS displacements for different crustal models is shown in Figure 1. Clearly localized patches with maximum slip of approximately 24 m are located around 4°N and between 6 and 9°N. Smaller, but still significant amount of slip is found in the Andaman segment of the rupture. The upper model in Figure 1 has been computed using earth layering corresponding to IASP91 crustal structure (continental), the lower corresponding to CRUST2 (oceanic). The IASP91 model smoothly fits the whole dataset, whereas the CRUST2 model over predicts near- and under predicts far field data. Chlieh et al.  attributed this deficiency to post-seismic slip that occurred before the near-field measurements were collected and which has not been accounted for correctly. However, our inversion shows that this discrepancy is drastically reduced if a continental crust earth model (IASP91) is used, leaving only minor need for postseismic slip. GPS-residuals and an additional model using PREM layering are shown in the auxiliary material, together with energy release projected on latitude.
Plafker et al.  suggested that a splay fault offshore northern Sumatra could be the secondary source responsible for the severe inundation which struck the Aceh province. In order to check this hypothesis, we performed an inversion based on following geometry. An additional splay fault connects to the 9th subfault layer of the reference geometry at 37 km depth and has a dip angle of 45°, which places the upper edge at around 105 km from the trench (model Sp845a). This corresponds approximately to splay faulting as suggested by Sibuet et al.  for the Aceh basin. Figure 2 illustrates inversion results for slip and rake obtained from this model. Indeed we find surface slip on the splay fault around 4.5°N. Other significant patches of slip are located around 9°N and 13.5°N. However, we note that due to the relatively low number of measurements in the near field, these findings should be considered with care, hence it would be very desirable to have independent confirmation on splay faulting in the northern part of the rupture, e.g. from seismic, bathymetric or ROV surveys. Note also that slip on the main fault is only slightly changed by the presence of the splay fault (compare Figures 1 and 2). Detailed inundation modeling for Banda Aceh, assessing the effect of splay faulting will be published elsewhere.
 Next we use the inverted slip distribution to compute the corresponding sea floor deformation. We include the effect of horizontal displacements [Tanioka and Satake, 1996] into initial conditions for the tsunami-propagation model, but find it to be small in this case (see auxiliary material). JASON1 satellite radar altimetry data (obtained from T. Schoene) provide an independent check for our inversion. Since run-up and inundation is not required here, relatively low resolution is sufficient to correctly reproduce the satellite data. Fast, robust and well-tested code TUNAMI-N2 by Imamura et al.  with spatial and temporal resolution of 5 arcmin (ETOPO2) and 3 s respectively is used and takes less than 10 min of computation time on a single processor. Figure 3a shows the measured and modeled sea surface elevation along the JASON1 track approximately two hours after the earthquake. The CRUST2 model (magenta) clearly yields too small amplitudes of the leading waves in the south, whereas the IASP model (blue) performs quite well.
Figure 3b shows the models with splay faulting (using IASP crust). Model Sp260, in which the upper two subfault layers are assigned a fix dip angle of 60°, fits the Jason data slightly better than the model without splay fault, as it is able to better reproduce the splitting of the first wave, which is due to the two separated slip maxima, one offshore Aceh and one around the Nicobar islands (see discussion by Pietrzak et al. ). The second model including splay faulting (Sp845a) results in a too pronounced first trough. All in all, our modeling implies that JASON1 data does not appear to be sufficient to deduce or to reject splay faulting.
 Application of an accurate discretization of the subduction zone geometry and introduction of physical boundary conditions in the inversion process significantly enhances the quality of GPS based inversion, as is shown by checking against independent radar altimetry data. The discrepancy resulting from near- and far-field data, which is often attributed to postseismic effects, can largely be resolved by using a continental crustal structure. GPS and satellite altimetry data of offshore tsunami wave height are compatible with splay faulting in seaward direction with several meters of slip, but data are not sufficient to make stringent conclusions. Splay faulting, if present, is likely to be a second order effect in respect to magnitude of sea floor deformation and contribution to the tsunami, at least as recorded by JASON1 satellite.
 This is publication 18 of the GITEWS project (German Indonesian Tsunami Early Warning System). The project is carried out through a large group of scientists and engineers from GeoForschungsZentrum Potsdam (GFZ) and its partners from DLR, AWI GKSS, IFM-GEOMAR, UNU, BGR, GTZ, as well as from Indonesian and other international partners. Funding is provided by the German Federal Ministry for Education and Research (BMBF), grant 03TSU01. We thank R. Burgmann for careful and constructive reviews.