Toward mapping surface deformation in three dimensions using InSAR

Transcription

1 GEOPHYSICAL RESEARCH LETTERS, VOL. 31, L01607, oi: /2003gl018827, 2004 Towar mapping surface eformation in three imensions using InSAR Tim J. Wright, 1 Barry E. Parsons, 1 an Zhong Lu 2 Receive 12 October 2003; revise 25 November 2003; accepte 28 November 2003; publishe 14 January [1] One of the limitations of eformation measurements mae with interferometric synthetic aperture raar (InSAR) is that an interferogram only measures one component of the surface eformation in the satellite s line of sight. We investigate strategies for mapping surface eformation in three imensions by using multiple interferograms, with ifferent imaging geometries. Geometries for both current an future missions are evaluate, an their abilities to resolve the isplacement vector are compare. The north component is always the most ifficult to etermine using ata from near-polar orbiting satellites. However, a satellite with an inclination of about 60 /120 woul enable all three components to be well resolve. We attempt to resolve the 3D isplacements for the 23 October 2002 Nenana Mountain (Alaska) Earthquake. The north component s error is much larger than the signal, but proxies for eastwar an vertical motion can be etermine if the north component is assume negligible. Inversions of hypothetical coseismic interferograms emonstrate that earthquake moel parameters can be well recovere from two interferograms, acquire on ascening an escening tracks. INDEX TERMS: 1243 Geoesy an Gravity: Space geoetic surveys; 1242 Geoesy an Gravity: Seismic eformations (7205); 1294 Geoesy an Gravity: Instruments an techniques; 6969 Raio Science: Remote sensing; KEYWORDS:. Citation: Wright, T. J., B. E. Parsons, an Z. Lu (2004), Towar mapping surface eformation in three imensions using InSAR, Geophys. Res. Lett., 31, L01607, oi: / 2003GL Introuction [2] In the past ecae, InSAR has prove a powerful technique for mapping surface eformation at an unpreceente spatial resolution [Massonnet an Feigl, 1998; Bürgmann et al., 2000]. A limiting factor in interpreting interferograms is that they are only sensitive to surface movements towars or away from the satellite. Fialko et al. [2001] recovere the 3D isplacements for the 1999 Hector Mine earthquake using ERS interferograms acquire on ascening an escening passes, an surface-parallel motion calculate by correlating the SAR amplitue images. This metho is only effective for events such as large earthquakes where the eformation signal is large they estimate an error of 10 cm on their north component of eformation. Here we investigate ways to resolve 3D surface isplacement fiels by using multiple interferograms. We use geometries that are possible with current, planne an propose SAR missions, an iscuss the implications for future image acquisition strategies. [3] In aition, we attempt to resolve the 3D surface isplacements for the 23 October 2002, Nenana Mountain (Alaska) earthquake, which to our knowlege is the first earthquake for which interferograms with 4 ifferent viewing geometries have been acquire in the epicentral area. One important question is whether 3D isplacements are necessary. We investigate whether the etermination of simple earthquake moels is improve by the use of interferograms with more than 2 viewing geometries. 2. Determining 3D Displacements [4] Before etermining 3D isplacements, it is necessary to etren all the interferograms, removing orbital errors, an to etermine a reference phase level. In many cases, this can be one using ata in the far fiel of the interferogram, away from the eforming area. In the subsequent iscussion, calculations are performe on a pixel by pixel basis. [5] Let ^p be the unit row vector ( p x, p y, p z ), pointing from the groun to the satellite in a local east; north; up reference frame. The observe range change r, with the positive being equivalent to motion away from the satellite in its line of sight, is then given by r = ^pu, where u is the column vector (u x, u y, u z ) T, containing the vector components of isplacement in the same reference frame. [6] Suppose now that a point on the groun is observe in interferograms with four ifferent look irections, for instance, with the antenna looking both right an left on both ascening an escening passes, or for two ifferent incience angles on ascening an escening passes. Defining R =(r 1, r 2, r 3, r 4 ) T, where r i are the line-of-sight isplacements for the ifferent look irections, then R = Pu where P is the 4 3 matrix given by 0 1 ^p 1 ^p P ¼ B 2 ^p A 3 ^p 4 If the covariance matrix for errors in the observe range changes is R, then the weighte least-squares (maximum likelihoo) solution for u is ^u ¼ P T 2 1 1P T 2 1 R P R R ð1þ ð2þ 1 COMET an Department of Earth Sciences, Oxfor University, UK. 2 USGS, EROS Data Center, SAIC, South Dakota, USA. Copyright 2004 by the American Geophysical Union /04/2003GL018827$05.00 an the covariance matrix for the estimate vector components is 2 u ¼ P T 2 1 R P 1 ð3þ L of5

2 Table 1. Relative Errors (Dilution of precision) in Vector Components of Groun Displacements Estimate From Observe Line-of-Sight Displacements With s = 1 Case a a q b L/R c e x 1 12, , 43 R a 12, L,R b 30, L,R , , 43 L,R a Satellite Azimuth. b Incience angle. c Look irection. L = Left-looking; R = Right-looking. Dilution of Precision in x, y, an z components. In the case where we assume that errors in range change are inepenent an have equal stanar eviations, s, we get 2 u ¼ s 2 P T 1 P ð4þ The square root of the iagonal terms of u give the stanar errors in the estimates of the components of the groun isplacement. If we set s = 1, then the iagonal terms provie a measure of the effect of geometry on these estimates in terms of the relative measurement error (equivalent to the ilution-of-precision use with GPS measurements [e.g., Strang an Borre, 1997]). [7] Below, we consier three cases of viewing geometry that are possible with current, planne or propose missions. In most geometries we consier a stanar, near-polar orbit, as use by current SAR satellites such as Envisat. Because the swath azimuth is approximately constant between 60 S an 60 N, where most volcanoes an fault zones occur, we calculate the ilution of precision assuming azimuths of 12 for ascening passes an 168 for escening passes, measure clockwise from local north Case 1: Right-Looking, 2 Incience Angles [8] We first consier the case where range changes are observe for two ifferent incience angles on both ascening an escening swaths. This is possible using satellites such as Envisat an Raarsat-1 that have multiple beam moes, recoring ata at ifferent incience angles, or with overlapping swaths from fixe beam satellites. If the two incience angles are assume to be 23 an 43, the relative errors of the vector components are given in Table 1. It can be seen that the error in the north component of the groun isplacement is much larger than in the other components. The ifficulty in resolving the north component results from the near-polar orbits an the small angular separation of the ifferent line-of-sight observations: 20 between ascening an escening passes an between the two incience angles. For a reasonable rms error of 10 mm in the range change observations, the north component of isplacement woul have an error of 12 cm comparable to the error in the north component of isplacement etermine using SAR azimuth offsets [Fialko et al., 2001] Case 2: Left & Right-looking [9] Some propose SAR missions, such as ECHO an EVINSAR [Wage et al., 2003], have the ability to rotate the spacecraft in orer to acquire ata with the raar looking left or right. The relative errors for a sun-synchronous orbit assuming an incience angle of 30 are given as case 2a in e y e z Table 1. The error in north component is improve by a factor of two with this geometry, but remains relatively poor compare to the east an vertical components because of the choice of a near-polar orbit. To emphasise this point, we consier case 2b with the same incience angle as before, but swath azimuths of 30 an 150 for ascening an escening passes respectively. This geometry, as propose for the EVINSAR mission, enables all three components to be well-resolve (Table 1). The isavantage of such a mission is that it acquires little ata outsie the latitue range 60 S to60 N Case 3: Left & Right-looking, 2 Incience Angles [10] In a few cases it might be possible to obtain two incience angles on ascening an escening passes with the raar both left-looking an right-looking. To investigate this extreme possibility, we extene the analysis for case 2a to eight range change measurements (Case 3; Table 1). The error in the north component has become relatively acceptable, although the result is best interprete as a 2 pffiffi improvement in errors through oubling the number of observations, rather than any stronger geometrical constraints. 3. The Nenana Mountain Earthquake [11] To our knowlege, the M6.7, 23 October 2002 Nenana Mountain (Alaska) Earthquake is the only earthquake for which interferograms have been acquire with 4 ifferent look irections. This is all the more remarkable in that the 4 Raarsat-1 post-event images were acquire within 6 ays of the event, before the much larger M7.9, 3 November 2002 Denali Earthquake, which occurre on the Denali Fault, immeiately east of the Nenana Mountain event (Figure 1f ). The 23 October 2002 event was rightlateral strike-slip on a vertical fault. Slip reache 90 cm at a epth of 12 km, but faile to break the surface [Wright et al., 2003]. We constructe 5 interferograms using ata from Raarsat-1 with 4 ifferent geometries split evenly between ascening an escening passes, an with incience angles between 24 an 45 (Auxiliary Table 1 1 ). Images were etrene, an a reference level was set using the far fiel of the interferograms. Unfortunately, the ascening interferograms i not acquire ata north of the fault, because of a change in the beam moe at that location. More etails of the InSAR ata an processing are presente in Wright et al. [2003], along with a source moel for the event. [12] We etermine the vector components of isplacement using equation (2), an their errors using equation (3), because the noise varie between interferograms (Auxiliary Table 1 1 ). The stanar errors in east, north an up components are 6, 286, an 41 mm respectively. The large errors in north an vertical components for this geometry mean that the noise swamps the signal, although the east component is well-etermine an the simplicity of the inversion results suggest the 6 mm error is realistic (Figures 1a 1c). However, because the earthquake is approximately east-west in orientation, the expecte north 1 Auxiliary material is available at ftp://ftp.agu.org/apen/gl/ 2003GL of5

3 Figure 1. Determining the 3D isplacement fiel for the Nenana Mountain Earthquake. (a c) Estimates of u x, u y, an u z respectively. Note that the stanar errors in u x, u y an u z are 6, 286 an 41 mm respectively, hence the noisy appearance of u y an u z ;(, e) Solutions for u x 0 an u z 0, etermine assuming u y =0.Ina e, the contour interval is 10 mm, except in b where it is 100 mm. The extents of the figure are shown as a ashe box in f; (f ) Location map for the Nenana Mountain earthquake, whose aftershocks are shown in white. Focal Mechanisms for the 23 October an 3 November 2002 earthquakes are from Harvar CMT, an their epicentres are inicate by the stars. Black lines elimit the extents of the InSAR ata available for this stuy, an the region of overlap where 3D isplacements were etermine is highlighte in yellow. White lines are mappe faults, an the re line is the surface rupture of the 3 November 2002 earthquake. component of eformation is small. We can therefore etermine proxies for eastwar an vertical eformation, u x 0, u z 0, by setting u y = 0 (Figures 1 an 1e). The errors for these proxies are 6 an 4 mm respectively, an a ramatic improvement is noticeable in the systematic pattern of u z 0, with 4 cm of uplift an subsience evient in the convergent an extensional quarants respectively. The east component of surface eformation reaches a maximum of 10 cm, some 10 km south of the fault. 4. Do We Nee 3D Displacements? [13] Although it may not be possible to acquire 4 line-ofsight components routinely, it is possible to o so for 2 lineof-sight components right-looking on ascening an escening passes. We illustrate the esirability of oing so by looking at the trae-offs in earthquake moel parameters that exist when only a single line-of-sight measurement is available, as is the case in the majority of earthquakes stuie using near-polar satellites such as ERS-1/2. [14] Figure 2 shows synthetic interferograms for the 4 possible line-of-sight irections for a near-polar orbiting mission, calculate using an elastic islocation moel [Okaa, 1985] with source parameters similar to those of the 1999 Düzce (Turkey) Earthquake, i.e., a right-lateral strike slip earthquake with 5 m of slip on an E-W fault ipping north [Bürgmann et al., 2002]. As in case 2a above an incience angle of 30 was assume, with azimuths of Figure 2. Synthetic interferograms for a Düzce-like earthquake [Strike/Dip/Rake/Depth Range/Length/Slip = 262 /52 / 175 /0 11 km/20 km/5 m], calculate with ifferent viewing geometries (Case 2a; Table 1). Each fringe is equivalent to a range change of 100 mm, half of the wavelength of an L-ban mission. 3of5

4 Figure 3. Trae-offs among slip, rake an moment for Monte-Carlo inversions of noisy, synthetic ata for the Düzce earthquake (Figure 2). Black points were obtaine using only escening, right-looking interferograms, re points using interferograms from both ascening an escening tracks with a right-looking raar, an the cyan points using interferograms from both ascening an escening track with both right-looking an left-looking raar. 12 an 168. With these azimuths, the ascening, rightlooking an escening, left-looking interferograms are very similar, as are the escening, right-looking an ascening, left-looking images. [15] The synthetic interferograms were subsample using the quatree algorithm [e.g., Jónsson et al., 2002], an a series of Monte-Carlo inversions [Wright et al., 1999] were then carrie out in which the synthetic ata were perturbe ranomly for each inversion, the noise being base on a 1-imensional covariance function erive from real interferograms in the Düzce area [Wright et al., manuscript in preparation, 2003; Hanssen, 2001]. Figure 3 shows the trae-offs that exist between the slip, rake an moment for the moel when only the escening, right-looking ata are use (as woul be the case for most applications of ERS), when both ascening an escening right-looking ata are use, an with 4 components (ascening & escening, left & right-looking). Trae-offs for other fault parameters are shown in Auxiliary Figure 1 1. [16] There is a marke trae-off among parameters when only the escening track is use, which is substantially reuce when inverting ata from both ascening an escening tracks. Use of left-looking ata in aition only prouces ap ffiffi small extra benefit an again is best interprete as a 2 reuction in errors through a oubling of the observations. We repeate the calculation using a fault with an N-S strike, with all other parameters ientical (Auxiliary Figure 2) 1. Despite the largest component of eformation being sub-parallel to the satellite azimuths, an hence harer to etect, earthquake parameters can again be reliably recovere using only ascening an escening, right-looking interferograms. It seems likely that this combination of ascening an escening, right-looking interferograms is sufficient to resolve the parameters of islocation source moels for events large enough to eform the surface by more than a few centimeters. 5. Conclusions [17] We have shown that it is possible to resolve 3D isplacements to a high egree of accuracy with an optimally configure InSAR satellite. A satellite that only covers the earth between latitues of 60 S an 60 N woul cover most continental volcanoes an fault zones, an woul enable north-south eformation to be etermine with an error only twice as large as the error in range changes (i.e., 20 mm for typical atmospheric conitions), if it acquire left an right looking images on ascening an escening passes. Stacking multiple interferograms coul reuce this error further such that slow north-south eformation coul be measure. For near-polar orbiting satellites resolving the north component of eformation is more ifficult ue to lack of iversity in viewing geometry. Again, a satellite that looks both left an right woul be the best option, an the north error coul be reuce to 30 mm if images with multiple incience angles coul be acquire; this might not be accomplishe without a constellation of raar satellites. [18] With a single, polar-orbiting satellite, it is likely that such a strategy may only be possible in exceptional circumstances, for very specific targets. In the general case, acquiring both ascening an escening imagery over volcanoes an fault zones shoul be straightforwar, without causing any programming conflicts in a eicate mission. We show that this is sufficient to etermine earthquake moel parameters, an strongly recommen that such acquisition strategies be implemente for current an future SAR missions. [19] Acknowlegments. COMET is a NERC-supporte Earth Observation Centre. TJW is supporte by a NERC postoctoral research fellowship. RADARSAT-1 images are #2002 Canaian Space Agency an were provie by the Alaska SAR Facility (ASF). Part of this research was performe at the SAIC, EROS Data Center uner USGS contract O3CRCN0001 an funing from NASA (NRA-99-OES-10 RADARSAT ). Some figures were prepare using the public omain Generic Mapping Tools GMT. References Bürgmann, R., P. Rosen, an E. Fieling (2000), Synthetic Aperture Raar interferometry to measure Earth s surface topography an its eformation, Ann. Rev. Earth. Planet. Sci., 28, Bürgmann, R., M. E. Ayhan, E. J. Fieling, T. J. Wright, S. McClusky, B. Aktug, C. Demir, O. Lenk, an A. Türkezer (2002), Deformation uring the 12 November 1999, Düzce, Turkey earthquake, from GPS an InSAR ata, Bull. Seismol. Soc. Am., 92, Fialko, Y., M. Simons, an D. Agnew (2001), The complete (3-D) surface isplacement fiel in the epicentral area of the 1999 M w 7.1 Hector Mine earthquake, California, from space geoetic observations, Geophys. Res. Lett., 28(16), Hanssen, R. (2001), Raar interferometry: Data interpretation an error analysis, Kluwer Acaemic Publishers, Netherlans. 4of5

5 Jónsson, S., H. Zebker, P. Segall, an F. Amelung (2002), Fault slip istribution of the 1999 Mw 7.1 Hector Mine earthquake, California, estimate from satellite raar an GPS measurements, Bull. Seismol. Soc. Am., 92, Massonnet, D., an K. L. Feigl (1998), Raar interferometry an its application to changes in the earth s surface, Rev. Geophys., 36, Okaa, Y. (1985), Surface eformation ue to shear an tensile faults in a half-space, Bull. Seismol. Soc. Am., 75, Strang, G., an K. Borre (1997), Linear algebra, geoesy, an GPS, Wellesley-Cambrige Press, Wellesley MA, USA. Wage, G., B. Parsons, an the EVINSAR science team (2003), Achieving the EVINSAR objectives with TerraSAR-L, in Proc. Thir International Workshop on ERS SAR Interferometry, Frascati, Italy, 2 5 December. Wright, T. J., B. Parsons, J. Jackson, M. Haynes, E. Fieling, P. Englan, an P. Clarke (1999), Source parameters of the 1 October 1995 Dinar (Turkey) earthquake from SAR interferometry an seismic boywave moelling, Earth Planet. Sci. Lett., 172, Wright, T. J., Z. Lu, an C. Wicks (2003), Source moel for the M w 6.7, 23 October 2002, Nenana Mountain Earthquake (Alaska) from InSAR, Geophys. Res. Lett., 30(18), 1974, oi: /2003gl T. J. Wright an B. E. Parsons, Centre for the Observation an Moelling of Earthquakes an Tectonics, Department of Earth Sciences, Parks Roa, Oxfor, OX1 3PR, UK. ([email protected]; barry.parsons@ earth.ox.ac.uk) Z. Lu, U.S. Geological Survey, EROS Data Center, SAIC, 47914, 252n St. Sioux Falls, SD 57196, USA. ([email protected]) 5of5