Combination of INSAR and GNSS Measurements for

Available online at www.sciencedirect.com

ScienceDirect Procedia Technology 16 (2014) 192 – 198

CENTERIS 2014 - Conference on ENTERprise Information Systems / ProjMAN 2014 International Conference on Project MANagement / HCIST 2014 - International Conference on Health and Social Care Information Systems and Technologies

Combination of INSAR and GNSS measurements for ground displacement monitoring Elisabeth Simonettoa*, Stéphane Duranda, Jordan Burdacka, Laurent Polidoria, Laurent Morela, Joëlle Nicolas-Duroya a

GeF/L2G, ESGT, 1 Bd Pythagore, 72000 Le Mans, France

Abstract Several attempts to combine GNSS and InSAR for deformation measurements have been published in the last years. In this work, we propose a global inverse method that combines phase measurements from GNSS and spaceborne radar interferometry to estimate the ground vertical displacement. It is based on the PS (permanent scatterer) and least squares approaches and considers non-ambiguous phases. The exploratory method is tested using simulated data.

© 2014 The The Authors. Authors.Published PublishedbybyElsevier ElsevierLtd. Ltd. This is an open access article under the CC BY-NC-ND license © 2014 (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of the Organizing Committees of CENTERIS/ProjMAN/HCIST 2014 Peer-review under responsibility of the Organizing Committee of CENTERIS 2014. Keywords: InSAR; PS; GNSS; least squares

1. Introduction Several attempts to combine GNSS and InSAR for deformation measurements have been published in the last years. Most of them consist in combining the displacements obtained separately with both techniques. Several authors show that the combination of GNSS and InSAR can densify the measurement network [1, 2, 3] or improve the InSAR measurement using both atmospheric corrections and GNSS measurements [4]. In all these experiments,

* Corresponding author. Tel.: +33 2 43 43 31 37; fax: +33 2 43 43 31 02. E-mail address: [email protected]

2212-0173 © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of the Organizing Committee of CENTERIS 2014. doi:10.1016/j.protcy.2014.10.083

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Elisabeth Simonetto et al. / Procedia Technology 16 (2014) 192 – 198

the anisotropy of the InSAR measurement is taken into account, for instance by projecting the GNSS data to the radar line of sight direction. Other experiments use GNSS data to improve the InSAR processing before the computation of a ground deformation. For instance, the interferogram can be registered on a low spatial frequency model built with GPS measurements in order to improve geophysical deformation monitoring [5], or the GNSS data can be injected into the InSAR processing to improve the precision of orbit determination [6]. Several authors have proposed to use GNSS data to perform the atmospheric correction in SAR interferograms, including directional considerations in the atmospheric delay [7, 8]. In this work, we propose a global inverse method that combines GNSS and radar interferometric measurements to estimate the DEM error, used in the InSAR processing, and the vertical ground displacement. It is based on the PS (permanent scatterer) approach and the STUN (Spatio-Temporal Unwrapping Network) algorithm [9]. STUN approach uses the least squares method. A common framework with GNSS phase processing seems possible. The idea is to combine respective normal equations. To test our approach, we use simulated InSAR and gnss data, and assume that ambiguities are known, that is radar differential interferometric phases are unwrapped and GNSS double-differenced ambiguities have already been determined. The assumption is strong and not realistic but allows us establishing a first solution. 2. Method 2.1. STUN algorithm In the STUN algorithm [9], N differential interferograms are computed using N+1 radar images. The master image is the same for all interferograms. PS are detected using the average signal to clutter ratio. Then, two kinds of PS are distinguished according to their amplitude dispersion index [10]. It derives from that a set of reference PS, named PSC, and a set of other PS. Using the PSC, the reference network is built, for instance by the Delaunay triangulation. A weighted integer least squares estimator is proposed to estimate both integer value of the phase ambiguity and float parameters (DEM error and displacement velocity) for each PSC. It consists in the processing the arcs of the reference network (the edges of the Delaunay triangulation) into three steps: a first problem inversion using the least squares adjustment leading to float parameter differences, the application of the LAMBDA (Least squares ambiguity decorrelation adjustment) method [11] to obtain integer values of the phase ambiguity coefficients and a second problem inversion to estimate final float parameters. DEM errors and displacement velocity for each PSC are then deduced using one PSC as reference. Neglecting noise and atmospheric signal and assuming a linear displacement model, the differential phase after unwrapping is modeled by:

Idiff ,P ,i 2k P ,iS Where :

4S

BA,i

O rP sin T P

eP

4S

O

t

i

t0 D P

(1)

P is a PS i is the interferogram index kP,i is the integer value of the phase ambiguity at P in i-interferogram BA,i is the perpendicular baseline for i-interferogram. It is assumed to be a constant in this work. rP is the radar distance for P eP is the DEM error at P t0 is the acquisition date of the master image ti – t0 is the temporal baseline for i-interferogram DP is the displacement velocity at P-pixel

The difference of phase for an arc between two close pixels, P and Q, is written:

194


Idiff ,P ,i Idiff ,Q ,i 2'k P ,Q ,iS Where:

'k P ,Q ,i

k P ,i kQ ,i

4S

BA,i

O rP sin T P 'eP ,Q ,i

'eP ,Q

4S

O

eP ,i eQ ,i

t

i

t0 'D P ,Q

'D P ,Q ,i

(2)

D P ,i D Q ,i

DEM error and displacement velocity are then estimated for each secondary PS. For that purpose, an arc is defined between a secondary PS and the closest PSC. Using the same weighted method as before, the PS phases are unwrapped. This result is refined by a spatial unwrapping of residual phases that relies on a Minimal Cost Flow unwrapping algorithm. After this final unwrapping, the float parameters of PS are estimated by least squares adjustment according to equation 2. 2.2. GNSS phase model Let us consider a network of GNSS stations. A simplified observation equation for relating GNSS phase observation to Euclidian distance between receiver, i, and satellite, j, is as follows [12]:

OIi j

U i j cdti dt j Ti j I i j ON i j eij

(3)

Where: O is the carrier wavelength for the selected GNSS frequency c is the speed of light

Ii j

is the GNSS phase measurement expressed in cycles

Ui j

is the distance between i-station and j-satellite:

Uij

'X 2 'Y 2 'Z 2

Ti j is the tropospheric delay (in meters) I i j is the ionospheric delay (in meters) Ni j is the phase ambiguity for i-station and j-satellite eij includes other errors (noise, multipath, ...) In the context of GNSS positioning, a classical approach is to use the concept of the double differences, between two stations, 1 and 2, and two GNSS satellites, k and l. For each session and epoch, the GNSS phase model is:

OI12kl Where:

U 2l U1l U 2k U1k ON12kl e12kl

(4)

I12kl I2l I1l I2k I1k is the double difference of phase N12kl is the integer phase ambiguity for the double difference measurement between the two receivers, 1 and 2, and two satellites, k and l

e12kl is the remaining error in the double difference measurement Let us consider that each GNSS station, j, is located at the same place as a PSC, P. Considering differences of DEM errors and displacement velocities, the geometric distances are written in terms of the following variables:

195


With:

'X

§ DP ¨¨ N j hP eP t t0 cos T P ©

·¸¸ cos M

j

cos O j X s

'Y

§ DP ¨¨ N j hP eP t t0 cos T P ©

·¸¸ cosM

j

sin O j Ys

'Z

§ DP ¨¨ N j 1 exc2 hP eP t t0 cosT P ©

¹

¹

·¸¸ sin M ¹

j

(5)

Zs

Nj is the radius of curvature in the prime vertical exc the first eccentricity of ellipsoid Mj And Oj are the latitude and longitude of the receiver. They are replaced by the ones of P-pixel.

Indeed, the coordinate differences involve the ellipsoidal height of the receiver. This height is expressed using the DEM height and its error at P-pixel, together with the vertical displacement. It is written using the temporal baseline of the epoch and displacement velocity at P-pixel. 2.3. Global inverse method In this work, we want to assess the feasibility of combining GNSS and InSAR phase measurements into a unique inverse method. However, for simplification, we make the assumption that the radar differential phase are already unwrapped and that the GNSS double-differenced integer ambiguities have also been already computed. This means that, in a future work, the GNSS integer phase ambiguities, as well as the PSC radar phase ambiguities, should be resolved simultaneously with the float parameters. Here, GNSS measurements allow directly estimating the float parameters of each PS and not the difference of float parameters between pairs of PS. Besides, PSC and other PS parameters are adjusted in a single processing. Let us note:

U12kl

L With:

U 2l U1l U 2k U1k

1

H

1 1 1 , U

OHl

U

l 1

U1k

HU

U 2l

(6)

U 2k

T

and l

I

l 1

I1k I2l I2k

T

For an arc of the GNSS network (between two stations, named 1 and 2), the problem is nonlinear and resolved iteratively after linearization as follows:

With:

H Ol U X k | A X k . X k 1 X k

Bk

k the iteration index

e AX X

1

k

e2 D1

(7)

D2

T

H .J X k , J is the Jacobean matrix of the partial derivatives of U.

The A-vectors and B-scalars of all arcs, considering all pairs of visible satellites, and all epoch of one session are concatenated into a matrix and a column vector. Considering all sessions and after GNSS system convergence, the matrices and vectors are also concatenated leading to a matrix, named AGNSS, and a column vector, named BGNSS. Interferometric observations of PSC are also put in the form of a matrix, APSC, and a column vector, BPSC, considering each arc in the reference network (equation 2). Then, one can write:

196


BPSC APSC X k

APSC . X k 1 X k

(8)

In the same way, the equation system for interferometric observations of other PS is built. Here, a PS is linked to the closest PSC. This leads to the matrix, APS, and the vector, BPS. The three matrices, AGNSS, APSC and APS are concatenated, as well as the three column vectors, BGNSS, APSC and APS. This leads to a global system described by the matrix, AGLOBAL, and the column vector, BGLOBAL. Let us remark that in this work, weighting has not been used. 3. Experiments InSAR phase data (Table 1) are simulated with respect to real ERS data (satellite ephemerides, radar system parameters), a real SRTM DEM, a simulated DEM error map, and a simulated deformation map (figure 1). This tool is implemented in Matlab [13], [14]. Although the simulation tool is quite simple and suffers from a lack of realism (no SAR processing, no errors on orbit data, etc.), it permits testing our approach with unwrapped phases and known parameters. Here, DEM errors are uniform random values between -10 m and 10 m. Vertical displacement velocities are uniform random values between 0 and 25 mm per year. We simulate 8 unwrapped differential interferograms whose characteristics are presented in the table 1. The ground pixel size is around 40 m. The area covers a surface of around 1250 km². Five hundred PS have been randomly selected, corresponding to a density of around 0,4 PS / km². PSC are selected according to a regular grid with a mesh size of 50 pixels.

Fig. 1. Simulated DEM error and displacement velocity for each PS. Table 1. Simulated interferogram characteristics. Interferogram index

1

2

3

4

5

6

7

8

Temporal baseline (days)

140

175

210

350

385

525

700

735

Perpendicular baseline (m)

-643

-590

-433

-543

-853

-1558

-29

-907

For GNSS data, we consider a configuration where a PSC is linked to at least two PSC equipped with a GNSS receiver (figure 2). Then, to test the influence of the number of stations, we can keep some of these GNSS locations.


197

Fig. 2. Configuration of PSC and GNSS receivers. On the left: PSC reference network. On the right: GNSS network.

A Matlab code [15] is used to simulate GNSS measurements in a similar form of a classical RINEX file per session. Input data are the receiver geographic coordinates and satellite broadcasted ephemerids. Nine sessions are simulated for each PSC equipped with a GNSS receiver. Each session has a given duration and is time centered on the radar image acquisition time, with a given sampling interval. Both L1 and L2 phase measurements for the GPS only constellation are simulated. In this work, the atmosphere is not taken into account. We can compare estimated float parameters for each PS with the simulated ones, with a simplified STUN algorithm (without unwrapping and without weighting) and with the proposed combination algorithm combining. We obtain 179 PSC and 48 GNSS stations. The Delaunay triangulation of the reference network consists in 522 arcs. Without GNSS measurements, the mean of the DEM error estimation is close 0,03 m and standard deviation is 0,09 m. The average estimation error for displacement velocities is 0,86 mm/yr and the standard deviation is -0,05 mm/yr. With GNSS measurements, we obtain a null average and standard deviation of DEM error estimation. The mean estimation error and standard deviation for the displacement velocities are near zero. Using only one third of GNSS stations, that is 16, errors are still negligible. 4. Conclusions and perspectives This work is an exploratory attempt of combining two kinds of measurements, GNSS and InSAR. However, in this approach, the inverse system is ill-conditioned and adding noise, as it is the case for real data, will lead to bad estimated values. In future work, conditioning will be considered to propose an adapted approach and to study the interest or not of such combination for the processing of real data. Besides, the ambiguities are not estimated here and may be integrated in future works. Then, parameter estimation may be improved using the variance-covariance matrix for component weighting as proposed in the STUN algorithm. Then, many questions remain about the influence of the number of radar images, PS density or number and spatial distribution of the GNSS receivers. Acknowledgements This work is based on Jordan Burdack's master dissertation at ESGT in 2013. This presentation is partially funded by the French Space Agency CNES in the frame of the TOSCA 2014 program. References [1] Wright T, Lu Z, and Wicks C. Constraining the slip distribution and fault geometry of the Mw 7.9, 3 November 2002, Denali Fault Earthquake with InSAR and GPS. Bulletin of Seismological Society of America 2004;94(6B):S175-S189. [2] Droz P, Fumagalli A, Novali F, Young B. GPS and INSAR technologies: a joint approach for the safety of Lake Sarez. In: 4th Canadian Conference on Geohazards 2008, Quebec, OC, Conference CD;8p. [3] Hammond WC, Li Z, Plag HP, Kreemer C, Blewitt G. Integrated INSAR and GPS studies of crustal deformation in the Western Great Basin, Western United States. Int. A. of the Ph., Rem. Sens. and Spatial Information Sc., Kyoto Japan, 2010;XXXVIII(Part 8):39-43. [4] Catalão J, Nico G, Hanssen R, Catita C. Integration of INSAR and GPS for vertical deformation monitoring: a case study in Faial and Pico islands. In: Proc. Fringe 2009 Workshop, Frascati, Italy, 30 November – 4 December 2009;ESA SP-677. [5] Wei M, Sandwell D, Smith-Konter B. Optimal combination of InSAR and GPS for measuring. J. Adv. Space Res. 2010; doi:10.1016/j.asr.2010.03.013.

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