Fault slip distribution is usually retrieved from geodetic data assuming that the local crust is an elastic, homogeneous and isotropic half-space. In the last decades spatially dense geodetic data (e.g., DInSAR maps) have highlighted complex patterns of coseismic deformation that require new modeling tools, such as numerical methods, able to represent rheological and geometrical complexities of the Earth's crust. In this work, we develop a procedure to perform inversion of geodetic data based on the finite element method, accounting for a more realistic description of the local crust. The method is applied to the 2009 L'Aquila earthquake (Mw 6.3), using DInSAR images of the coseismic displacement. Results highlight the non-negligible influence of the medium structure: homogeneous and heterogeneous models show discrepancies up to 20% in the fault slip distribution values. Furthermore, in the heterogeneous models a new area of slip appears above the hypocenter. We also perform a resolution study, showing that the information about fault slip distributions retrieved from geodetic data should be considered as averaged on surrounding patches.
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 On April 6th 2009, at 3:32 local time (1:32 UTC), a Mw = 6.3 earthquake occurred in the Abruzzi region (central Italy), followed by five M > 5 aftershocks, the largest ones occurring on April 7th (Mw = 5.5) and 9th (Mw = 5.4) (Figure 1a). The mainshock, located at about 9 km depth, produced severe damage in the ancient city of L'Aquila and in many neighboring villages. Moment tensor solutions reveal normal faulting mechanism consistent with the NE-SW extensional trend of the central Apennines [e.g., Amato et al., 1998]. According to recent geodetic studies, surface geological observations and aftershocks relocations [Atzori et al., 2009; EMERGEO Working Group, 2010; Chiarabba et al., 2009], the main fault is identified as the NW striking and SW dipping Paganica Fault.
 Inversions of geodetic data for seismic source purposes are often performed implementing the analytical model of Okada  of a fault embedded in an elastic, homogeneous and isotropic half-space. It is reasonable to ask: how can material complexities affect slip distribution providing different surface displacement patterns? Analytical and semi-analytical codes such as Wang et al.  provide a partial answer to this question, since they allow describing the lithosphere with a 1D vertical layering. However, local asperities, anisotropy and the presence of topography are likely to be present in faulted zones. The above mentioned complexities can be managed only by means of numerical tools. Masterlark  tested the sensitivity of finite element (FE) computed slip distributions to common assumptions (e.g., homogeneous, isotropic, Poisson-solid and half-space medium), finding that differences in displacement predictions exceeded the uncertainty bounds. Hearn and Bürgmann  investigated the effect of depth-dependent elasticity on slip inversions using FE solutions of a shear dislocation in layered and uniform elastic Earth models. They found that incorporating realistic increases of shear modulus with depth in the inversions modifies the recovered centroid depth and seismic potency with respect to uniform elastic half-space models. Dubois et al.  analyzed the influence of structural and rheological heterogeneities on the earthquake deformation cycle, showing significant differences in slip distribution (beyond the associated uncertainty bounds) when using models with fault damage zones and elastic layering.
 In this paper, taking advantage of the FE method, we optimize the coseismic displacement data due to the L'Aquila earthquake to retrieve fault slip distributions. We also perform a resolution study to quantify the information we can extract from large coverage interferometric data.
 Differential SAR Interferometry (DInSAR), a technique commonly used to measure coseismic ground displacements, provides spatially dense deformation fields with great accuracy [e.g., Chini et al., 2010]. The data used in this work are from the COSMO-SkyMed and ENVISAT satellite missions and their characteristics are summarized in Table 1. Phase noise is mitigated by an adaptive filter [Goldstein and Werner, 1998], while unwrapped interferograms are obtained by a Region Growing algorithm [Reigber and Moreira, 1997]. The topographic contribution is removed by a 5 meter resolution Digital Elevation Model (DEM). The three resulting interferograms (Figure 1), generated with a pixel spacing of 80 m, measure a congruent deformation pattern [Stramondo et al., 2011]. The maximum displacement is located between the city of L'Aquila and the town of Fossa, reaching respectively −16 cm (COSMO-SkyMed), −24 cm (ENVISAT ascending) and −27 cm (ENVISAT descending) in the SAR Line of Sight. Variations are due to the different look angles and orbits. In addition to DInSAR data, we consider GPS horizontal and vertical measurements of 8 sites within about 30 km from the epicenter [Anzidei et al., 2009], from continuous stations (Figure 1a, AQUI, INFN, INGP, SMCO) and from GPS receivers installed a few days before the main event (CADO, ROIO, SELL, CPAG).
Table 1. Characteristics of the Interferograms Used
Perpendicular Baseline (m)
Incidence Angle (deg)
Fringe Rate (cm)
 The large amount of seismic data acquired during the L'Aquila sequence was subsequently used to investigate the characteristics of the local lithosphere. Di Stefano et al. (Fault zone properties affecting the rupture evolution of the 2009 (Mw 6.1) L'Aquila earthquake (Central Italy): insights from seismic tomography, submitted to Geophysical Research Letters, 2011) performed a 3D tomography study using 1276 aftershocks until the end of June 2009. The images reveal vertical and lateral vp heterogeneities. The main feature is a high vp volume placed below the city of L'Aquila between 4–10 km depth. Furthermore, a receiver function study by Bianchi et al.  shows lateral vs heterogeneities along the fault in the uppermost crust. The analysis of the AQU receiver function, the largest peak ground motion station [Cirella et al., 2009], highlights a high vs body (values up to 4.2 km/s) of 4–6 km thickness. This high velocity body, absent in the SE portion of the fault, may have influenced the kinematics of the L'Aquila seismic source.
3. FE Models of the Seismic Source
 We initially constrain the fault geometry using 5955 subsampled DInSAR data and 8 GPS data. We adopt Okada's  fault as forward model to solve the non-linear optimization problem. The fault inversion is performed by a two-step algorithm consisting in a global search (Neighbourhood Algorithm) followed by a Bayesian inference on the generated ensemble of models [Sambridge, 1999a, 1999b]. This procedure extracts the most probable set of solutions, instead of a single best-fit model. The method has been largely tested for inversions of geodetic data in volcanic areas [Trasatti et al., 2008]. We perform tests aimed to find a reasonable balance between DInSAR and GPS points in the computation of the total misfit function, which is composed by the weighted sum of the reduced chi-square of each dataset (the three interferometric images are considered as a unique dataset). The relative weight of each dataset is fixed at 95% DInSAR and 5% GPS (see Figure S1 of the auxiliary material). The choice of the weights is due to the low uncertainty of the GPS data (few mm) which ‘attracts’ the misfit function towards a lowering of the GPS chi-square. The best-fitting fault parameters resulting from the optimization are reported in Table 2. All the parameters are well defined since they show peaked bell-like Posterior Probability Density (PPD) curves (Figures S2 and S3 for 1D and 2D PPD distributions, respectively). The resulting fault plane dimension and position are compatible, within their uncertainties, with previous findings [e.g., Atzori et al., 2009]. Regarding the fault orientation of our best model, the strike value of 142° ± 2° follows the line of converging DInSAR fringes (strike between 140° and 145°), emphasizing the primary role of large coverage data for fault geometry determination. The fault trace is compatible with the Paganica Fault surface trace as well (see Figure 1a). The retrieved mean dip value is 42° ± 2, within the acceptable range of 40°–50°, typical for the NE trending structures of the Central Apennines [Amato et al., 1998].
Table 2. Fault Parameters Retrieved by Non-linear Inversion
 In the following, we build FE models with the fault geometry determined above. Even if the fault plane is not fully optimized in the FE models, it is reasonable to suppose that potential variations of dip and strike values are within their previously determined uncertainty. Once the fault plane is constrained, our inverse problem is linear, provided we use linear elasticity. We set up a method consisting of three steps: i) building FE models of a fault embedded in a 3D domain, including topography and other characteristics; the fault is mapped by small patches; ii) computing a matrix of elementary ‘Green's functions’ for every slip patch; iii) performing linear inversions of the FE-computed matrix to retrieve the slip distribution. The FE models are composed of about 100'000 3D-brick elements contained in a cylindrical domain having radius of 100 km and height of 80 km. The fault plane dimensions in Table 2 are expanded up to 20 km in length and 15 km in width in order to map the slip distribution without edge artifacts, and subdivided into 300 1 × 1 km patches. The FE models also include the elastic structure, computed from the vp and vs anomalies resulting from the tomography by Di Stefano et al. (submitted manuscript, 2011). We use a density profile of 2600 kg/m3 above 4 km depth and 2800 kg/m3 below it [Di Luzio et al., 2009]. The resulting Lame constants assume values such as μ = 18 GPa and λ = 30 GPa at shallow depths (1–3 km) and μ = 30 GPa and λ = 50 GPa below 10–14 km depth. The asperity found below the L'Aquila city at 4–10 km depth [Bianchi et al., 2010; Di Stefano et al., submitted manuscript, 2011] is reproduced by our FE model and is characterized by very high elastic parameters: μ = 40 GPa and λ = 54 GPa. The models include the topography of the area, generated using a DEM with pixels of 90 m. Four different models are considered: HOF, HOmogeneous with Flat topography medium, HEF, HEterogeneous with Flat topography, HOT, HOmogeneous with Topography and HET, HEterogeneous with Topography. For each model we perform the steps outlined above. We compute four ‘Green's function’ matrices, by collecting all the surface displacements for each model considered. The linear inversion is performed adopting the Occam's smoothing scheme [deGroot-Hedlin and Constable, 1990], minimizing the chi-square and the second order derivative (Laplacian) to avoid large oscillations in slip values. Positivity constraints are placed on slip to elude thrust components during the inversion, in contrast with the normal mechanism of the fault.
 Slip distribution results are shown in Figure 2. The misfit lowers from 6.2, in case of the non-linear inversion, to 4.2 (HEF), 4.3 (HET) and 4.7 (HOF, HOT). Smoothing curves are shown in Figure S4. Results for all the models show that there is a large slip area in the central-SE part of the fault. The effect of topography can be appreciated by comparing results from HOF and HOT models, but differences are of second order. The small effect on the slip distribution could be related to the small topographic variations. Indeed, in areas with a strong topography, the effects on the slip distribution are larger [e.g., Kyriakopoulos et al., 2010]. More interesting features appear in the presence of an elastic structure. In the heterogeneous model HEF, the slip is distributed along the strike direction and it is concentrated between 3 and 7 km depth. A new local maximum of slip appears above the hypocenter, according to seismological inversions [Cirella et al., 2009]. The presence of the slip concentration above the hypocenter may be attributed to the presence of the high rigidity body, absent in the homogeneous models. The rake varies in the range −90° ± 4° in the high slipping areas. In the HET model the topography partially masks out the effect of the elastic structure, so that the resulting new high slip area due to the internal heterogeneities is more visible in the HEF model that in the HET model. The maximum slip is about 120 cm in the homogeneous models and 110 cm in the heterogeneous models. To better appreciate the changes introduced by the 3D elastic structure, Figures 2c and 2f show the differences between HEF and HOF, and between HET and HOT, respectively, reaching up to 20% of the maximum slip in both cases. The fault's hanging wall area shows low residual values (from 0 to 2.5 cm) for all three DInSAR images (Figure S5). In the foot wall area there are positive residuals for ENVISAT descending data and negative residuals for ENVISAT ascending data, reaching an absolute value of 5 cm in both cases.
 An uncertainty analysis is performed to assess the reliability of our results. We compute the data covariance matrix covd and calculate the model covariance matrix covm [Menke, 1989]. The diagonal terms of covm are the variances of the slip values. Figure 3a shows the standard deviation associated with the slip distribution. Single slip values are acceptable since their uncertainty amounts to 2–4 cm for most part of the fault, degrading at fault bottom and borders. Following the approach outlined by Funning et al. , we define a linear dimension of model resolution in order to assess the degree to which single slip values are averaged with the surroundings. In Figure 3b we observe that the resolving power rapidly decreases with depth. At hypocentral depths, the model resolves slip features larger than about 6 km. Thus, we must be aware that only general features of slip predictions based on geodetic data (even large coverage DInSAR data) should be considered, according to the intrinsic data resolving power and smoothing artifacts.
 As a final remark, we perform inversions of new HOT and HET models with a steeper fault characterized by dip equal to 50°, based on recent seismological studies (e.g., Di Stefano et al., submitted manuscript, 2011). Results are in general agreement with those previously obtained (Figure S6).
4. Discussion and Conclusions
 All the slip distributions show a wide slip area located in the central-SE part of the fault, in agreement with previous results [Atzori et al., 2009; Cirella et al., 2009]. However, Cirella et al.  retrieved a slip distribution deeper than the hypocenter, making use of seismological data. This result cannot be obtained from geodetic data. Indeed, in Figure 3 we show that even dense DInSAR data are unsuitable to resolve deep slip features. We also show that using heterogeneous models such as HEF and HET, the maximum slip area is flattered and shifted NW, and a new patch of slip appears above the hypocenter. This feature doesn't appear in geodetic inversions with Okada's  models [e.g., Atzori et al., 2009] but it is common only to inversions in non-homogeneous models [e.g., Cirella et al., 2009]. We can compare the results for the slip distributions with the elastic characteristics close to the fault plane (Figure 4). In the HEF model, the slip is concentrated between 5–10 km depth, and distributed along the strike direction. This could be due to the low shear modulus of the shallow layers, which amplifies the deformation in extension and absolute value. However, the inversion algorithm, trying to fit the observed data, produces an opposite behavior. Indeed, in the heterogeneous models the slip distribution is more concentrated, we get a lower maximum slip and a new slip area appears in correspondence of the high rigidity body below L'Aquila city. The most important result of our inversions is that the 3D elastic structure introduces differences in the slip distribution that are not negligible. We obtain differences up to 20%. Topography does not have a critical influence on slip distribution since the main differences are due to the presence of elastic heterogeneities.
 The resulting scalar moments are M0 = 2.5 × 1018 Nm for HOF and HOT, using an average shear modulus value of 30 GPa, corresponding to Mw = 6.2, and M0 = 1.2 × 1018 Nm for HEF and HET, corresponding to Mw = 6.0. The value in the heterogeneous models is lower than the observed Mw = 6.3 because of the low shear modulus values near the fault at shallow depths. While the ‘Green's function’ feels the effect of the whole elastic structure, as described above, the moment density is proportional to the local shear modulus and slip. Therefore, the scalar moment computed in the heterogeneous models is lower than the homogeneous ones because of the combined effect of the slip concentration and of the low shear modulus values at shallow depths.
 Dense and large coverage DInSAR data is very useful to determine the fault geometry, but we must keep in mind that geodetic data, even DInSAR data, is characterized by poor resolution at large depths. Furthermore, the information about fault slip distribution should be considered as averaged on surrounding patches, even in complex models such as those shown.
 Many thanks to the LabTel group from INGV-Rome for the scientific and technical support. We thank M. Bonafede and R. Di Stefano for helpful discussions. C. Chiarabba and P. De Gori are acknowledged for providing tomographic data. Two anonymous reviewers improved the quality of the manuscript. J. P. Merryman Boncori improved the English of the manuscript. This work is partially funded by the Italian Space Agency (ASI) through the SiGRiS project.
 The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.