DEVOTI_corretto_Layout 6 781 Combination of coseismic displacement fields: a geodetic perspective Roberto Devoti Istituto Nazionale di Geofisica e Vulcanologia, Centro Nazionale Terremoti, Roma, Italy ANNALS OF GEOPHYSICS, 55, 4, 2012; doi: 10.4401/ag-6119 1. Introduction This study provides the mathematical framework for the rigorous combination of coseismic offsets observed by a global positioning system (GPS) network and investigates the results obtained on the occasion of the recent Emilia earth- quakes (Italy). This seismic sequence that affected northern Italy from May 20, 2012, allowed two offset fields to be com- puted, one with reference to the mainshock (M 5.9, followed by two other M 5.1 events on the same day), and a second with reference to the replicas that occurred on May 29, 2012 (M 5.8, M 5.3 and M 5.2; ISIDe data archive, http://iside.rm. ingv.it). The final displacement field is basically the result of a comparison and validation process with repeated feedback between the different analysis groups at the Istituto Nazionale di Geofisica e Vulcanologia (INGV; National In- stitute of Geophysics and Volcanology) that was established to obtain prompt coseismic displacement solutions, as pre- cise as possible, and in the first days after an event. This is im- portant for early seismic-source evaluation as it represents the most complete and validated dataset at the very early stage of a seismic crisis, and it is also extremely useful in re- ducing random and systematic errors in the estimated pa- rameters. This study is the result of a cooperative effort that involved different research groups at INGV, with the sharing of all of the collected GPS data. The intention was to com- pare these results and thus reducing sources of error associ- ated with individual processing strategies, to allow the final combination of the different displacement fields into a sin- gle consensus solution. The process assessed the robustness of each single GPS result, thus minimizing erroneous inter- pretations of individual solutions. For more information about the interpretation of the coseismic displacements, see Serpelloni et al. [2012, this volume]. The combination pro- cedure can be easily generalized to other geodetic parame- ters, and might be useful whenever a validated dataset is desirable or is intrinsically required (i.e., near real-time ob- servations of unknown surface processes, tectonic motion of large geographic areas, angular momentum observations of the whole Earth, coseismic surface deformation patterns). Other important advantages of the combination process in- clude efficient outlier detection via cross-checking of inde- pendent solutions, and realistic estimates of the displacement uncertainties. It is important to emphasize here that in this kind of activity, the cooperative behavior between individ- ual researchers and research groups should prevail over com- petitive and self-assertive behaviors, because the individuals (input solutions) become part of the same objective (valida- tion). On the other hand, the validation process should not overlap or replace the research activity; i.e., there should al- ways be space for individual initiatives. In this respect, the co- operative effort spent in constructing a prompt and validated product is a distinctive activity that involves both technolog- ical and scientific aspects, and it certainly stimulated the in- dividual research activities. Methods for comparison and combination of geodetic network solutions were introduced and developed in several studies in the early 1990's [Heflin et al. 1992, Boucher and Al- tamimi 1993, Davies and Blewitt 2000, and references therein]. It is a particular least-squares task in which the input data types are the same as the output parameter types and the covariance matrices of the input datasets are fully popu- lated (because each dataset is the output parameter set of an earlier least-squares analysis). The parameters that can be combined can be three-dimensional (3-D) station positions at a stated epoch, constant 3-D station velocities, Earth-orienta- tion parameter time series, or coseismic station offsets. In principle, each of the aforementioned parameter types, or a subset of them, can be treated in the same mathematical framework and combined into a unique average solution with only slight modifications to the formulation given below. 2. Coseismic displacement estimation Each analysis group analyzed the GPS data using its own procedures and models. In our case, they provided three time series obtained with the BERNESE (http://www. bernese.unibe.ch/), GAMIT (http://www-gpsg.mit.edu/ ~simon/gtgk/), and GIPSY/OASIS (https://gipsy-oasis.jpl. nasa.gov/) software. Time series of daily station positions Article history Received July 19, 2012; accepted August 30, 2012. Subject classification: Crustal deformation, Data processing, GPS coseismic displacement, combination. 2012 EMILIA EARTHQUAKES DEVOTI 782 10 oE 11 oE 12 oE 44 oN 20' 40' 45 oN 20' 40' 46 oN BASS BLGN BOLC BOLG BOLO BOR2 BORR BRAS BREA BRIS BTAC CAFV CARP CODI COLL CONC DARF FAEZ FERR FERA FIREFOSC FOZA GRZM GUAS IMOL ITIM LEGN LERO MAGA MANT MEDI MO05 MODE MOPS MSEL MTRZ PARM PARR PERS PTO1 RAMS RAVE REBO REGG ROVRSALO SBPO SCHI SERM SGIP SIR2 TEOL TRE3 VERG VERO BOBB CAST CREM LDNS PADO PIA1 ROVI TARO VICE VR02 ASIA BOSC BRIX CGIA CITD CREA GARI ITRN LASP ODEZ PORT RMN2 RONC ROVE SDNA TGPO TVSO VALX VELO VEN1 10 mm 10 oE 11 oE 12 oE 44 oN 20' 40' 45 oN 20' 40' 46 oN BASS BLGN BOLC BOLGBOLO BOR2 BORR BRAS BREA BRIS BTAC CAFV CARP CODI COLL CONC DARF FAEZ FERR FERA FIREFOSC FOZA GRZM GUAS IMOL ITIM LEGN LERO MAGA MANT MEDI MO05 MODE MOPS MSEL MTRZ PARM PARR PERS PTO1 RAMS RAVE REBO REGG ROVR SALO SBPO SCHI SERM SGIP SIR2 TEOL TRE3 VERG VERO BOBB CAST CREM LDNS PADO PIA1 ROVI TARO VICE VR02 ASIA BOSC BRIX CGIA CITD CREA GARI ITRNLASP ODEZ PORT RMN2 RONC ROVE SDNA TGPO TVSO VALX VELO VEN1 SIR2 10 mm Figure 1. Combined (red arrows) and individual solution offsets of the horizontal (top) and vertical (bottom) components associated with the May 20, 2012, seismic events. The individual solutions are represented in green (BERNESE), cyan (GIPSY) and blue (GAMIT). Stars, epicenters of the M >5 events; circle, delimitation of the 50-km range from the mainshock (M 5.9); error ellipses, one-sigma confidence regions. 783 COMBINATION OF COSEISMIC DISPLACEMENTS 10 oE 11 oE 12 oE 44 oN 20' 40' 45 oN 20' 40' 46 oN 5603 5901 5902 6903 BASS BOLC BOLGBOLO BOR2 BORR BRAS BREA BTAC CAFV CARP CAST CODICOLL CONC DARF FAEZ FERR FERA FIRE FOZA GRZM ITIM LEGN LERO MAGA MANT MEDI MO05 MODE MSEL MTRZ PARM PARR PERS PTO1 RAMS REBO REGG ROVR SBPO SCHI SERM SGIP SIR2 TEOL VERG VERO 5904 BLGN CREM GUAS LDNS MOPS PADO RAVE ROVI VICE VR02 BOSC BRIX CGIA CITD GARI IMOL PALA PIA1 PIAC SALO TGPO 10 mm 10 oE 11 oE 12 oE 44 oN 20' 40' 45 oN 20' 40' 46 oN 5603 5901 5902 6903 BASS BOLC BOLG BOLO BOR2 BORR BRAS BREA BTAC CAFV CARP CAST CODI COLL CONC DARF FAEZ FERR FERA FIRE FOZA GRZM ITIM LEGN LERO MAGA MANT MEDI MO05 MODE MSEL MTRZ PARM PARR PERS PTO1 RAMS REBO REGG ROVR SBPO SCHI SERM SGIP SIR2 TEOL VERG VERO 5904 BLGN CREM GUAS LDNS MOPS PADO RAVE ROVI VICE VR02 BOSC BRIX CGIA CITD GARI IMOL PALA PIA1 PIAC SALO TGPO 10 mm Figure 2. Combined (red arrows) and individual solution offsets of the horizontal (top) and vertical (bottom) components associated with the May 29, 2012, seismic events. The individual solutions are represented in green (BERNESE), cyan (GIPSY) and blue (GAMIT). Stars, epicenters of the M >5 events; circle, delimitation of the 50-km range from the main event (M 5.8); error ellipses, one-sigma confidence regions. were computed in the International Terrestrial Reference Frame (ITRF)2008 reference framework [Altamimi et al. 2011], which provided three distinct time series of a regional GPS network located around the epicenter, in north-eastern Italy (see Figures 1, 2, for maps of the GPS network). To es- timate the coseismic offsets we considered only a single day of observations after the event, to minimize the effects of early post-seismic displacements, which can be eventually caused by afterslip or poroelastic rebound [e.g., Peltzer et al. 1996, Mahsas et al. 2008]. Fast exponential decays in the very early phase of the seismic cycle are commonly observed in a wide range of seismic events that cause deformations of up to several centimeters, depending on the magnitude and the driving mechanism [Segal et al. 2000, Yu et al. 2003, Devoti et al. 2012a]. Therefore, the coseismic offsets were estimated using 15 days before the epoch of the earthquake (whenever possible), and one day after the event, with the following days being used only to check the consistency of the coseismic displacement at the given station. We checked for outliers in the time series using the detrended 3-D position residuals, using different criteria for before and after the event. Before the earthquake, a point was rejected if its residual exceeded the weighted-root-mean-squared (WRMS) of all of the resid- uals by three times; after the event instead, if the difference between consecutive points exceeded 3 times the WRMS, we computed the weighted average between them. This latter condition was applied to only a few sites in the near field: CONC, MSEL and SBPO. All of the offsets were estimated directly in the same least-squares inversion, to obtain the es- timates of the vector and the associated network covariance matrix for each of the three processing schemes. Figures 3 and 4 show the residuals of the estimated station offsets (namely the WRMS of the residuals between pairs of solu- tions) in the near field (<50 km from the epicenters). This parameter measures the repeatability of the estimates across independent solutions, and it is extremely useful for the iso- lating of systematic errors of the individual solutions. For the May 20, 2012, event, the overall average WRMS was 2.1 mm in the horizontal plane and 4.2 mm in the vertical di- rection; for the May 29, 2012, event, the WRMS was 2.6 mm in both components. The worst case was observed for sta- tion MO05 (Finale Emilia), where the GIPSY solution dif- fered on average by 15 mm from the other two, in the vertical direction. Nevertheless, given the high uncertainties of the vertical GPS estimates, this difference can still be con- sidered acceptable. 3. Displacement field combinations Consider the linearized observation equation: (1) where y is the column vector of the observations, A is the design matrix, x is the column vector of the functional model DEVOTI 784 0 2 4 6 8 1 0 1 2 1 4 solution residuals (mm) M a y, 2 0 BLGN BOLG BOLO BTAC CONC FERR FERA GUAS LEGN MANT MO05 MODE MOPS PERS REGG SBPO SERM SGIP LDNS H o riz V e rt 0 1 2 3 4 5 6 M a y, 2 9 solution residuals (mm) BLGN BOLG BOLO BTAC CODI CONC FERR FERA GUAS LDNS LEGN MANT MO05 MODE MOPS PERS REGG SBPO SERM SGIP 5603 5901 5902 5904 6903 H o riz V e rt Figure 3. Cumulative WRMS of the station offset differences between the individual May 20, 2012, solutions (BERNESE, GAMIT and GIPSY) for the horizontal (gray) and vertical (open bar) components. Figure 4. Cumulative WRMS of the station offset differences between the individual May 29, 2012, solutions (BERNESE, GAMIT and GIPSY) for the horizontal (gray) and vertical (open bar) components. Ay x= + o 785 parameters, and o is a column vector of random errors. In our case, the observations and the model parameters are of the same nature, and therefore the linearized observation equations are simply: (2) where Xs and Xc are the offsets (observations) of the solu- tion s and the combined parameters c (estimates), respectively, while the design matrix is a simple reordering matrix, as observed by Davies and Blewitt [2000], which reparameterizes the parameter list from the combined offsets to the solution offsets (i.e., the component aij = 1 if element i of list c is the same as element j of list s). A notable property of the design matrix is that: (3) i.e., transposing the reordering matrix from list c to list s cor- responds to reversing the ordering from list s to list c. To sim- plify the mathematical formulation, we adopt the convention that ; i.e., the design matrix of the i-th solution (the reordering matrix from solution s = i to the combined solu- tion c) is denoted simply as Ai. The combination of the individual solutions is obtained by solving for the unknowns Xc in a least-squares sense, and a useful recursive solution can be easily derived: (4). From each input solution (usually in SINEX format), we can extract the estimate parameter vector Xi and its covariance matrix Ci, compute the reordering matrix Ai, and finally cu- mulate the normal matrix and vector, as the first and second factors, respectively, of the right-hand side of Equation (4). The relative weight of the input solutions is imposed by the condition that the contribution to the total |2 of each in- dividual solution is equally balanced [Devoti et al. 2012b]: (5). From the first preliminary combination, a variance fac- tor fi is estimated for each solution, and then applied to the covariance matrix of the corresponding solution in an itera- tive way, until the global |2 reaches unity. This procedure as- sures that each solution contributes equally to the global |2, which avoids an individual solution from dominating or any suffering from the combination of the individual solutions (i.e., over-weighted or down-weighted solutions). It might happen that the GPS stations are geographi- cally clustered, especially in urban areas where there are dif- ferent installations over inter-distances of a few kilometers. In this case, it is useful to constrain the clustered offsets, es- timating them only once. If the inter-distances of the GPS stations are small with respect to the coseismic deforma- tion pattern, i.e., of the order of 1 km, then clustered ob- servations will rather bias any inverse solution because repeated information at the same site might artificially over-weight that particular site, thus providing biased solu- tions. This can be handled during the combination estima- tion process by imposing the constraint: (6) whenever sites i and j have to be estimated together. The con- straints were imposed using the classical Lagrange multipli- ers technique [e.g., Tarantola 2005] that provides the constrained least-squares solution with a fully populated co- variance matrix of the combined parameters. 4. Results for the Emila earthquake Figures 1 and 2 show the combined offsets (red arrows) of the two seismic events (the May 20 and 29, 2012, events, respectively) projected in the local horizontal and vertical directions. The maximum horizontal offset was observed at MO05, the closest site to the mainshock epicenter (hori- zontal, 30 mm; vertical, 73 mm). The offsets caused by the May 29, 2012, events were generally smaller than the off- sets caused by the mainshock, although it has also to be noted that the available GPS network did not correctly sam- ple the expected surface deformation, and therefore the measured offsets might be aliased. The individual solutions are also shown with different colors in Figures 1 and 2, to have a first glance of their repeatability. Apart from any ap- parently random differences, we will show that systematic bias might also characterize the individual solutions. A first validation of the individual solutions was carried out by plotting the offsets measured outside an area of 50-km ra- dius from the epicenter. Ideally, the far-field offsets should be randomly distributed (if not affected by the earthquake), and they represent a good test case to determine any sys- tematic errors. Figures 5 and 6 show the histograms of the far-field offsets for the two events, where the most notable features are non-zero medians for the individual solutions; e.g. for May 20, 2012, a median vertical component of -2.1 mm (GAMIT), 3.0 mm (GIPSY) and north component of 2.1 mm (GIPSY), and less pronounced bias for the May 29, 2012, event. The combined offsets show instead a general bias reduction of the far-field offsets, indicating that the combined solution is in fact the best unbiased representa- tion of the displacement field. The combined solution also shows a significant reduction in the formal errors (Table 1): both events show (on average) 1.2 mm one-sigma error in the horizontal plane, and about 3.1 mm in the vertical di- rection, whereas the individual solutions show (on average) a 50% higher sigma. COMBINATION OF COSEISMIC DISPLACEMENTS X A Xs cc s= + o" A c s" A A c s T cs = " "^ h A Ai i c= " ... ... X A C A A C A A C X A C X 1 1 1 1 2 2 1 2 1 1 1 1 2 2 1 2 c T T T T 1= + + + + + -- - - -^ ^h h X X C X X X X C X X f f 1 1 c i T i i c i c j T j j c j = = - - - - - - ^ ^ ^ ^ ^ ^ h h h h h h X Xi j= DEVOTI 786 0 2 4 6 8 10 0 2 4 6 8 10 -10 -5 0 5 10 0 2 4 6 8 10 COMBINED 0 2 4 6 8 10 U p 0 5 10 15 E a st -10 -5 0 5 10 0 2 4 6 8 N o rt h GAMIT 0 1 2 3 4 5 0 1 2 3 4 -10 -5 0 5 10 0 1 2 3 4 5 Residuals (mm) GIPSY 0 1 2 3 0 1 2 3 4 5 -10 -5 0 5 10 0 1 2 3 BERNESE 0 2 4 6 8 10 U p 0 2 4 6 8 10 E a st -10 -5 0 5 10 0 2 4 6 8 10 12 N o rt h GAMIT 0 2 4 6 8 0 2 4 6 8 -10 -5 0 5 10 0 2 4 6 8 GIPSY Residuals (mm) 0 2 4 6 0 1 2 3 4 -10 -5 0 5 10 0 1 2 3 4 BERNESE 0 5 10 15 0 5 10 -10 -5 0 5 10 0 5 10 15 COMBINED Figure 5. Distribution of the far-field offsets for the individual (BERNESE, GAMIT and GIPSY) and COMBINED May 20, 2012, solutions. Red line, best- fit Gaussian curve. The median (m) and dispersion ( ) are computed for the three spatial directions (up, east, north). Figure 6. Distribution of the far-field offsets for the individual (BERNESE, GAMIT and GIPSY) and COMBINED May 29, 2012, solutions. Red line, best- fit Gaussian curve. The median (m) and dispersion (v) are computed for the three spatial directions (up, east, north). Event BERNESE GAMIT GIPSY COMBINED May 20, 2012 1.8 / 5.9 1.6 / 4.0 2.1 / 5.1 1.2 / 3.1 May 29, 2012 1.8 / 6.1 1.7 / 4.5 1.8 / 4.4 1.2 / 3.2 Table 1. Average formal errors (1-sigma) of the estimated coseismic offsets. The two values represent the horizontal and vertical components, respectively, of the uncertainties. 787 Acknowledgements. I would like to offer my special thanks to all of those who helped in carrying out the GPS analysis at the INGV, to all who archived and provided GPS observations: RING, EUREF, ASI, IREALP, Re- gione Veneto, IGM, Italpos Leica Geosystems, Fondazione Geometri e Geometri Laureati dell'Emilia Romagna, Department of Earth Sciences of the University of Siena, and Department of Physics of the University of Bologna, for providing data from the MO05 station, and GeoTop s.r.l. for pro- viding data from the CONC station. Finally, my thanks also go to all of those who participated in the post-seismic GPS campaign and the field assistance. References Altamimi, Z., X. Collilieux and L. Métivier (2011). ITRF2008: an improved solution of the international terrestrial ref- erence frame, J. Geodesy, 85, 457-473; doi:10.1007/s00190- 011-0444-4. Boucher, C., and Z. Altamimi (1993). Development of a con- ventional terrestrial reference frame, In: D.E. Smith and D.L. Turcottep (eds.), Contributions of Space Geodesy to Geodynamics: Earth Dynamics, Geodynamics Series, 24, 89-97. Davies, P., and G. Blewitt (2000). Methodology for global ge- odetic time-series estimation: a new tool for geodynam- ics, J. Geophys. Res., 105, 11,083-11,100. Devoti, R., L. Anderlini, M. Anzidei, A. Esposito, A. Galvani, G. Pietrantonio, A.R. Pisani, F. Riguzzi, V. Sepe and E. Serpelloni (2012a). The coseismic and post-seismic de- formation of the L'Aquila 2009 earthquake from repeated GPS measurements, Ital. J. Geosci. (B. Soc. Geol. Ital.), 131 (3), in press. Devoti, R., E. Flammini, G. Pietrantonio, F. Riguzzi and E. Serpelloni (2012b). Toward a Dense Italian GPS Velocity Field: Data Analysis Strategies and Quality Assessment, In: N. Sneeuw et al. (eds.), VII Hotine-Marussi Sympo- sium on Mathematical Geodesy, International Associa- tion of Geodesy Symposia, 137, doi:10.1007/978-3-642- 22078-4_51. Heflin, M., et al. (1992). Global geodesy using GPS without fiducial sites, Geophys. Res. Lett., 19 (2), 131-134; doi: 10.1029/91GL02933. Mahsas, A., K. Lammali, K. Yelles, E. Calais, A.M. Freed and P. Briole (2008). Shallow afterslip following the 2003 May 21, Mw = 6.9 Boumerdes earthquake, Algeria, Geophys. J. Int., 172, 155-166; doi: 10.1111/j.1365-246X.2007.03594.x. Peltzer, G., P. Rosen, F. Rogez and K. Hudnut (1996). Post- seismic rebound in fault step-overs caused by pore fluid flow, Science, 273, 1202-1204. Segall, P., R. Bürgmann and M. Matthews (2000). Time-de- pendent triggered afterslip following the 1989 Loma Pri- eta earthquake, J. Geoph. Res., 105, 5615-5634. Serpelloni, E., L. Anderlini, A. Avallone, V. Cannelli, A. Cava- liere, D. Cheloni, C. D'Ambrosio, E. D'Anastasio, A. Espo- sito, G. Pietrantonio, A.R. Pisani, M. Anzidei, G. Cecere, N. D'Agostino, S. Del Mese, R. Devoti, A. Galvani, A. Massucci, D. Melini, F. Riguzzi, G. Selvaggi, and V. Sepe (2012). GPS observations of coseismic deformation fol- lowing the May 20 and 29, 2012, Emilia seismic events (northern Italy): data, analysis and preliminary models, Annals of Geophysics, 55 (4); doi:10.4401/ag-6168. Tarantola, A. (2005). Inverse Problem Theory and Model Pa- rameter Estimation, SIAM (Society of Industrial and Ap- plied Mathematics), ISBN 0-89871-572-5. Yu, S-B., Y-J. Hsu, L-C. Kuo, H-Y. Chen and C-C. Liu (2003). GPS measurement of postseismic deformation following the 1999 Chi-Chi, Taiwan, earthquake, J. Geophys. Res., 108, 2520; doi:10.1029/2003JB002396. Corresponding author: Roberto Devoti, Istituto Nazionale di Geofisica e Vulcanologia, Centro Nazionale Terremoti, Roma, Italy; email: roberto.devoti@ingv.it. © 2012 by the Istituto Nazionale di Geofisica e Vulcanologia. All rights reserved. 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