The acquisition of a 3D displacement field can help to understand the crustal deformation pattern of seismogenic faults and deepen the understanding of the earthquake nucleation. The data for 3D displacement field extraction are usually from GPS/interferometric synthetic aperture radar (InSAR) observations, and the direct solution method is usually adopted. We proposed an iterative least squares for virtual observation (VOILS) based on the maximum a posteriori estimation criterion of Bayesian theorem to correct the errors caused by the GPS displacement interpolation process. Firstly, in the simulation examples, both uniform and non-uniform sampling schemes for GPS observation were used to extract 3D displacement. On the basis of the experimental results of the reverse fault, the normal fault with a strike-slip component, and the strike-slip fault with a reverse component, we found that the VOILS method is better than the direct solution method in both horizontal and vertical directions. When a uniform sampling scheme was adopted, the percentages of improvement for the reverse fault ranged from 3% to 9% and up to 70%, for the normal fault with a strike-slip component ranging from 4% to 8% and up to 68%, and for the strike-slip fault with a reverse component ranging from 1% to 8% and up to 22%. After this, the VOILS method was applied to extract the 3D displacement field of the 2008 Mw 7.9 Wenchuan earthquake. In the East–West (E) direction, the maximum displacement of the hanging wall was 1.69 m and 2.15 m in the footwall. As for the North–South (N) direction, the maximum displacement of the hanging wall was 0.82 m for the southwestern, 0.95 m for the northeastern, while that of the footwall was 0.77 m. In the vertical (U) direction, the maximum uplift was 1.19 m and 0.95 m for the subsidence, which was significantly different from the direct solution method. Finally, the derived vertical displacements were also compared with the ruptures from field investigations, indicating that the VOILS method can reduce the impact of the interpolated errors on parameter estimations to some extent. The simulation experiments and the case study of the 3D displacement field for the 2008 Wenchuan earthquake suggest that the VOILS method proposed in this study is feasible and effective, and the degree of improvement in the vertical direction is particularly significant.

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The 2010 Mw 8.8 Maule earthquake occurred offshore central Chile and ruptured ∼500 km along the megathrust fault resulting from the oceanic Nazca plate subducting beneath the continental South American plate. The Maule earthquake produced remnant crustal displacements captured by a vast set of geodetic observations. However, given the nature of the observational techniques, it is challenging to extract its accurate three-dimensional coseismic deformation field with high spatial resolution. In this study, we modified the extended simultaneous and integrated strain tensor estimation from geodetic and satellite deformation measurements (ESISTEM) method with variance component estimation algorithm (ESISTEM-VCE) to retrieve the three-dimensional surface displacement field of this event by integrating the interferometric synthetic aperture radar (InSAR) and global positioning system (GPS) measurements. The ESISTEM-VCE method accounts for the spatial correlation of surface displacement among the adjacent points and determine the accurate weight ratios for different data sets, but also uses the uncertainties of GPS data and considers the different spatial scales from the different datasets. In the simulation experiments, the RMSEs of the ESISTEM-VCE method are smaller than those of the ESISTEM and ESISTEM-VCE (same d0) methods, and the improvements of 97.1%, 3.9%, and 84% are achieved in the east-west, north-south, and vertical components, respectively. Then, we apply the proposed methodology to the 2010 Mw 8.8 Maule earthquake, to obtain a three-dimensional displacement field that could provide fine deformation information. In the east-west component, the significant deformation is in the north of the epicenter, closed to the Constitución, with a maximum westward displacement of 495.5 cm. The displacement in the north-south component is relatively small compared to that in the east-west component. The maximum uplift reaches 211.8 cm, located at the southwest of the Concepción. Finally, the derived vertical displacements are also compared with field investigations, indicating that the ESISTEM-VCE method can obtain more accurate weight of different datasets and perform better than the ESISTEM method. The results highlight that the earthquake ruptured along the NE-SW direction, with a dominant thrust and a relatively small component of right-lateral strike-slip, coinciding with the characteristics of subduction and right-lateral shear. The experiments with the simulated and real data suggest that the improved ESISTEM-VCE method in this study is feasible and effective.

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