geometric precision in space radar imaging - asprs

GEOMETRIC PRECISION IN SPACE RADAR IMAGING: RESULTS FROM TERRASAR-X Thomas P. Ager, System Engineer for Radar National Geospatial-Intelligence Agency 12310 Sunrise Valley Drive Reston, Virginia 20191 [email protected] Paul C. Bresnahan, Senior Photogrammetrist Observera Inc. 3856 Dulles South Court, Suite I Chantilly, Virginia 20151 [email protected]

ABSTRACT Imaging radars have the potential to provide unprecedented geometric accuracy. The accuracy of radar images is based on parameters that can be measured with precision. Radars make measurements of microwave echoes and from these the image formation chain calculates ranges and angles to the various ground resolution cells in the imaged area. These ranges and angles are referenced to positions within the velocity vector of the sensor. The errors associated with the state vectors, ranges, and angles can be engineered to be extremely small. Ground control point measurements made on images from the TerraSAR-X (TSX) commercial radar satellite are used to illustrate the operational implementation of radar geometric precision.

INTRODUCTION The geometric accuracy of imagery refers to the integrity of the conversion of measured image coordinates to their associated ground locations. For mono imagery this is done by intersecting image coordinates with a ground elevation surface. Ground coordinates can also be calculated using a stereo pair of images, without use of an elevation model. In this evaluation we focus on the intersection of range-Doppler arcs from mono radar images to surveyed ground elevations. This process isolates the error contributions of the image and its associated metadata.

THE RANGE-DOPPLER SAR GEOMETRY MODEL Passive remote sensing systems record reflected or emitted electromagnetic energy via a linear or frame array of detectors and it is a rather simple process to map these detector locations to image pixel coordinates. The relationship between image coordinates and their associated ground locations is based on light traveling from the ground to the sensor in a straight line, while accounting for phenomena such as the bending of light caused by atmospheric refraction. This relationship is expressed by the so-called collinearity equations that relate image coordinates, ground coordinates, and the sensor position and orientation at the moment of acquisition. (Mikhail, 2001) Radar image formation is fundamentally different than passive imaging systems in that a radar sensor does not directly measure pixel intensities and locations. A radar sensor transmits microwave pulses to the ground and measures the characteristics of the echoes. These data are processed to create the natural radar image coordinates: range and Doppler angle. The range is simply the distance between the antenna and the ground object, and the Doppler angle is bound by the sensor velocity vector and the vector connecting a reference sensor location to the ground resolution cell. For traditional spotlight radar acquisition, in which a fixed ground area is imaged via a long synthetic aperture, each resolution cell is imaged from multiple sensor locations. The reference sensor location is usually chosen to be the center location of the long aperture. For stripmap collections, in which the sensor collects at a fixed angle – usually broadside to the flight direction – the sensor location varies with each range line in the image. The spotlight collection mode of TerraSAR-X is actually a hybrid mode, called sliding spotlight, in which the beam angle varies slightly during the dwell, and the pulses ‘slide’ across the imaged ground area. For any ASPRS 2009 Annual Conference Baltimore, Maryland ♦ March 9-13, 2009

collection mode, the radar sensor geometry model is a range-Doppler model in which a range sphere is intersected with a Doppler cone, as shown in Figure 1. (Ager, 2001) This illustration shows a generalized case with a highly squinted collection. For broadside collections, the Doppler ‘cone’ is actually a circle and the cone angle is 90°.

Figure 1. SAR Geometry Model Sphere-Cone Intersection.

THE POTENTIAL OF RADAR IMAGERY ACCURACY The relevant geometric parameters for the Synthetic Aperture Radar (SAR) geometry model include the sensor velocity vector, the range, and the Doppler angle. It is possible to measure each with impressive precision.

Sensor State Vectors The motion of an orbiting satellite tends to be smooth and regular and it is possible to determine the orbital position and velocity vector to a high degree of accuracy using Global Positioning System (GPS) measurements. The accuracy depends upon the quality of the GPS receivers, the amount of input data used, and the rigor of the orbit determination process. Orbits with errors of 3 meters or less can be determined relatively quickly with modest quality GPS receivers and nominal data inputs. (Kahle, 2007) Extreme precision is possible with geodetic-quality receivers and rigorous processing. Such precision has been operationally demonstrated by the Jet Propulsion Laboratory (JPL) Gravity Recovery and Climate Experiment (GRACE) mission, in which orbit errors on the level of 10 centimeters were achieved. (Bertiger, 2002; Montenbruck, 2006) According to the JPL website, an operational positioning service, called the Global Differential GPS (GDGPS) System can provide real-time 10 centimeter level positioning accuracy and sub-nanosecond time transfer accuracy to space platforms that have geodetic quality GPS equipment. (JPL, 2008)

Range The range estimate is based on clock errors, delays in sensor hardware and atmospheric path delays. Range data can also be quite precise because the clock errors are minimal, the sensor hardware delays can be calibrated, and atmospheric refraction can be modeled to reduce errors. (DLR, 2008; Frey, 2004)

Doppler Angle The Doppler angle is defined relative to the platform trajectory tangent at the mapping reference time. The Doppler cone angle is not an orientation directly measured by star sensors or gyroscopes, as with the orientation angles of passive systems. The angle corresponding to cross range position in the SAR case is derived from extremely precise time measurements of returns from scatterers in the scene and knowledge of the platform position at the time of each pulse. The rate of change in these time measurements uniquely maps to the individual Doppler angles for stationary objects throughout the scene. The primary sources of error in the Doppler angle to an arbitrary scene point are uncertainty in instantaneous platform position and the radar center frequency. For TSX, these are negligible contributors, and the error in the derived Doppler cone angle is essentially zero. ASPRS 2009 Annual Conference Baltimore, Maryland ♦ March 9-13, 2009

EXPECTATIONS FOR TERRASAR-X GEOLOCATION ACCURACY Because of the negligible contribution of Doppler angle errors on geolocation accuracy, the expected TSX sensor accuracy depends primarily on the accuracy of the orbital state vectors and the range estimates. In the operational use of TSX, additional consideration must be given to Digital Elevation Model (DEM) error contributions.

TerraSAR-X Orbit Determination Several types of orbits are generated for TSX. Predicted orbits have a coarse required accuracy of 700 meters along track. Standard processing uses Rapid orbits with a required accuracy of 2 meters (3-D, 1σ). The most accurate orbits are Science orbits with a required accuracy of 20 centimeters (3-D, 1σ) and a goal of 10 centimeters. The German Aerospace Center (Deutsches Zentrum für Luft- und Raumfahrt, or DLR) reports that these specifications are being met. DLR notes that the orbit determination performance may suffer during times of solar maximum, but they should still be within the specifications. Rapid orbits are specified to be available within 15 hours after the GPS data is transmitted to the ground station, although DLR reports that most are available within 10 hours of image acquisition. Science orbits are specified to be available in 3 weeks, but they often are available within 5 days. Images ordered from archive will most often be processed using Science orbit data. (DLR, 2008) The precise orbit determination is possible because TSX carries a geodetic-quality, dual-frequency GPS receiver called the Integrated Geodetic and Occultation Receiver (IGOR) that provides low-noise carrier phase and pseudorange data. IGOR is an improved version of the JPL BlackJack GPS receiver used by the GRACE satellite. Rapid orbits are generated using the real-time GPS orbit and clock offset product from JPL. Science orbits are generated using GPS orbit and high-rate clock offset products from the Center for Orbit Determination in Europe (CODE) rapid service. DLR commissioning assessments indicate that the Rapid orbits are meeting the 20 centimeter level and Science orbits are at the 10 centimeter level. The TSX orbit is typically maintained within a 250 meter orbital tube to support interferometry, which requires frequent orbital maneuvering. The Rapid and Science orbit accuracy levels are still achieved because the orbit determination software can account for the maneuvers. (DLR, 2008; Kahle, 2007; Montenbruck, 2006; Yoon, 2008) In addition to providing high accuracy position and velocity, the IGOR GPS receiver is used as a radio occultation sensor, providing about 500 atmospheric profiles from the ionosphere down to the Earth’s surface each day. (Broad Reach Engineering, 2007) As a backup, TSX has a redundant pair of single-frequency MosaicGNSS receivers to generate Rapid orbits in the event of IGOR failure or interruption. Assessments indicate that the MosaicGNSS receiver-based orbit solutions are accurate to about 1 meter (3-D rms), which meets the 2 meter Rapid orbit requirement. (DLR, 2008; Montenbruck, 2006; Yoon, 2008) TSX also carries a Laser Retro Reflector (LRR) that is used to independently check the absolute accuracy of its orbit determination solution using Satellite Laser Tracking (SLR) sites from the International Laser Ranging Service (ILRS). (Yoon, 2008)

TerraSAR-X Range Calibration The vacuum of space, the ionosphere, and the troposphere have different refractive indices. The slant range error from the path delay is about 2-4 meters and depends upon the actual atmospheric conditions and the length of the path. The path lengths vary across the image as the incidence angles vary. The tropospheric path delay is nearly constant with time and is easily modeled. The ionospheric path delay is not as constant but its effect is relatively small for X-band imagery. Average path delays for the scene are modeled and applied in the product processing. For TSX the amount of atmospheric refraction is limited by the relatively steep collection angles. Some residual errors remain because of the varying incidence angles across the scene, but they are expected to be less than 0.5 meters. (DLR, 2008; Frey, 2004)

TerraSAR-X Expected Accuracy Due to the factors in the above discussion regarding orbit determination methods, precise sensor hardware calibration, and accurate delay and timing corrections applied during product processing, TSX geometric errors should be quite small. In fact, DLR commissioning results indicate a geolocation accuracy of 0.3 meters in range and 0.53 meters in azimuth. (Buckreuss, 2008) At this level of accuracy, repeated acquisitions are inherently coregistered to the pixel level. A preliminary geolocation accuracy evaluation indicated a 1 meter TSX accuracy level based on a slant-range complex image. (Nonaka, 2008)

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Impact of DEM Errors Because the superb accuracy performance described above, it is important to note operational performance will normally rely on intersection with an elevation model as opposed to a known elevation. Vertical errors in the elevation model will add a horizontal displacement in the image intersection process. Table 1 shows the horizontal ground error for combinations of incidence angle and elevation model error. The elevation model error contribution quickly dominates the orbit and timing error contributions. The ground errors are worse for lower incidence angles corresponding to steeper collection angles. At a 45° incidence angle, error in the horizontal position caused by the elevation model equals the error in elevation. Alternatively, stereo TSX images can be used to generate very precise coordinates for identifiable features without the need for an elevation model.

Elevation Error (m)

Table 1. Additional Horizontal Ground Displacement from DEM Error (DLR, 2008) Incidence Angle (°) 32 38 3.2 2.6

2

20 5.5

26 4.1

6

16.5

12.3

9.6

8

22

16

13

30

82

61

48

44 2.1

50 1.7

7.7

6.2

4.9

11

9

7

38

31

25

TERRASAR-X GEOLOCATION ACCURACY EVALUATION RESULTS The Civil and Commercial Applications Project (CCAP) group within the National Geospatial-intelligence Agency (NGA) performed a geolocation accuracy evaluation of TSX imagery. CCAP performs geolocation accuracy evaluations for civil and commercial imagery being purchased or considered for purchase by NGA. CCAP measured the locations of accurately-surveyed Ground Control Points (GCPs) on TSX imagery. Many of these points are located on or near airfields. Because the GCPs were originally developed for use with panchromatic imagery, not all points could be identified for measurement on the TSX images. CCAP used the difference between the imagery-derived ground coordinates and the coordinates of the GCPs to estimate accuracy statistics. As is common for imagery sources, TSX imagery can be processed into one of several geometric forms. To effectively quantify the inherent accuracy of TSX imagery, it is important to select the most suitable form for the geolocation accuracy evaluation. The most-pure form of a detected SAR image is one that is in a range-azimuth projection. Detected images can also be map projected – either onto an assumed constant ground elevation or onto a terrain model – the latter being an orthorectified image. While map projected products may seem desirable because the image rows and columns are aligned with the ground coordinate system, they have disadvantages for a geolocation accuracy evaluation. Firstly, they are resampled versions of the most-pure range-azimuth projection – causing some degradation in imagery interpretability. Secondly and more importantly, the map projection grid included with these images can be used for general positioning, but not for precise applications. Grids based on a constant ground elevation will have inherent horizontal positioning errors dependant upon the amount the actual terrain deviates from the assumed value. And orthorectified products include horizontal errors resulting from terrain model errors making it difficult to discern the error contribution from the source imagery itself. Precise geolocation is supported by the detected image form that is output in range-azimuth coordinates. Pixel locations on these images can be transformed to range-Doppler values for rigorous projection to an elevation surface. When the rigorous projection is to a surveyed-quality height, such as the height of a GCP, then the error contribution from the imagery itself can be quantified. For this reason, CCAP chose to evaluate the TSX range-azimuth projected form of detected image product, called the Multi-look Ground-range Detected (MGD) product. CCAP used BAE SYSTEMS SOCET Set© v5.4.1 photogrammetric software to evaluate the TSX imagery because it could import the MGD product – including sensor metadata – and perform the rigorous transformations to and from image coordinates, range-Doppler values, and horizontal ground coordinates at fixed heights of GCPs. CCAP evaluated 13 high resolution spotlight (HS) and 13 stripmap (SM) images. Table 2 shows the basic characteristics of the HS and SM images evaluated. All images were collected in single-polarization (VV) mode, included the Rapid orbit metadata, and were generated as MGD products with Spatially-Enhanced (SE) processing. The ground resolution of HS images is three times better in each dimension than SM images enabling CCAP to ASPRS 2009 Annual Conference Baltimore, Maryland ♦ March 9-13, 2009

better identify GCPs on the imagery, but the larger scene size of SM enables more GCPs to fall within the image footprints for potential measurement. Table 2. TSX Test Image Product Characteristics Imaging Mode

Ground Resolution (m)

Scene Size (km)

Orbit Processing

High Resolution Spotlight Mode (HS) (300 MHz)

~1

~7 (Range) x ~5.5 (Azimuth)

Rapid

Multi-look GroundSpatially range Detected (MGD) Enhanced (SE)

VV

Stripmap Mode (SM)

~3

~32 (Range) x ~57 (Azimuth)

Rapid

Multi-look GroundSpatially range Detected (MGD) Enhanced (SE)

VV

Geometric Projection

Geometric Processing

Polarization

Table 3 contains a specific list of the test sites of the images and the collection dates, orbit directions, and incidence angles of the test images. The incidence angles range from 41° to 50° for the spotlight images and 35° to 45° for the stripmap images. Table 3. Test Sites and Images Test Site Argentina, Ministro Pistarini Iraq, Baghdad Japan, Kadena Kenya, Jomo Kenyatta Pakistan, Shabaz Panama, Howard Philippines, Clark United States, Edwards United States, Kaneohe Bay United States, Nellis United States, New River United States, Wheeler Sack United States, Whidbey Island

Spotlight (HS) Orbit Direction Descending Descending Ascending Descending Ascending Descending Ascending Ascending Ascending Descending Ascending Ascending Ascending

Collection Date 31 May 2008 13 May 2008 16 May 2008 13 May 2008 10 June 2008 13 May 2008 14 May 2008 10 May 2008 11 May 2008 31 May 2008 13 May 2008 30 May 2008 15 May 2008

Incidence Angle (°) 40.8 43.6 47.6 48.8 42.7 48.3 43.2 45.1 49.3 45.7 41.4 49.5 46.1

Stripmap (SM) Orbit Direction 25 May 2008 Ascending 24 May 2008 Descending 26 May 2008 Descending 7 June 2008 Ascending 30 May 2008 Ascending 17 May 2008 Ascending 14 May 2008 Descending 1 June 2008 Ascending 16 May 2008 Descending 22 May 2008 Ascending 23 May 2008 Descending 23 May 2008 Descending 23 May 2008 Descending

Collection Date

Incidence Angle (°) 37.3 42.8 37.3 39.2 42.8 35.3 44.5 44.5 37.3 37.2 37.3 39.2 42.8

Table 4 shows the horizontal errors for the test images. Several statistics are listed for each image: the number of check points that CCAP identified and measured, the mean delta Easting and Northing assessed using the check points, and the radial displacement of the mean Easting and Northing. Statistics are also listed for the 13 mean delta Easting and Northing values. These include mean, standard deviation, minimum, and maximum.


Table 4. Horizontal Errors for Spotlight and Stripmap Images Test Site Argentina, Ministro Pistarini Iraq, Baghdad Japan, Kadena Kenya, Jomo Kenyatta Pakistan, Shabaz Panama, Howard Philippines, Clark United States, Edwards United States, Kaneohe Bay United States, Nellis United States, New River United States, Wheeler Sack United States, Whidbey Island Mean Standard Deviation Minimum Maximum

# of check points 2 2 14 7 10 19 7 10 6 5 3 8 14

Spotlight (HS) Mean Δ Mean Δ Δr Easting (m) Northing (m) (m) 0.8 -0.8 1.2 -0.4 -0.9 1.0 0.7 -0.3 0.7 0.0 -0.2 0.3 0.9 -0.2 0.9 -0.3 0.3 0.4 0.4 -0.1 0.4 -0.1 -0.6 0.6 0.3 -0.3 0.4 -0.4 -0.1 0.4 0.4 0.0 0.4 0.1 0.2 0.2 0.8 -0.5 1.0 0.3 -0.3 0.5 0.4 -0.4 -0.9 0.9 0.3

# of check points 5 2 6 2 5 12 5 11 4 4 6 9 10

Stripmap (SM) Mean Δ Mean Δ Northing Easting (m) (m) 0.0 -2.1 0.5 -2.3 -1.2 -1.3 -0.4 -2.3 1.2 -2.3 1.5 -0.7 0.7 -0.2 0.6 -0.9 -1.1 -1.4 2.0 -2.6 0.7 0.9 0.5 -0.8 0.1 -1.6 0.4 -1.4 0.9 1.0 -1.2 -2.6 2.0 0.9

Δr (m) 2.1 2.4 1.8 2.4 2.6 1.7 0.7 1.1 1.8 3.3 1.1 1.0 1.6

Horizontal error can be reported as a circular error at a 90 percent probability level (CE90). We computed this value by sorting the radial displacements from Table 4 and determining the 90th percentile statistic, as shown in Table 5. CCAP defined the 90th percentile position as n * p + 0.5 = 13 * 0.9 + 0.5 = 12.2 where n is the number of images and p is the probability level. Therefore the CE90 is linearly interpolated between the 12th and 13th ordered radial displacement values. (Ager, 2004; Bresnahan, 2007) Table 5. Sorted Radial Errors for Spotlight and Stripmap Images Spotlight (HS) Test Site United States, Wheeler Sack Kenya, Jomo Kenyatta Panama, Howard United States, Nellis United States, Kaneohe Bay United States, New River Philippines, Clark United States, Edwards Japan, Kadena Pakistan, Shabaz Iraq, Baghdad United States, Whidbey Island Argentina, Ministro Pistarini Estimated HS CE90 (m)

Sorted Δ r (m) 0.2 0.3 0.4 0.4 0.4 0.4 0.4 0.6 0.7 0.9 1.0 1.0 1.2 1.0

Stripmap (SM) Test Site Philippines, Clark United States, Wheeler Sack United States, Edwards United States, New River United States, Whidbey Island Panama, Howard United States, Kaneohe Bay Japan, Kadena Argentina, Ministro Pistarini Kenya, Jomo Kenyatta Iraq, Baghdad Pakistan, Shabaz United States, Nellis Estimated SM CE90 (m)

Sorted Δ r (m) 0.7 1.0 1.1 1.1 1.6 1.7 1.8 1.8 2.1 2.4 2.4 2.6 3.3 2.8

The scatter plots in Figure 2 show the mean displacements for each image in blue, and the associated centroid of all of the errors in red. The overall means are close to zero. The CE90 circles are shown in magenta.


3

3

2

2

1

1

Image Error Centroids

0 -3

-2

TerraSAR-X-1 Stripmap Monoscopic Absolute Geolocation Accuracy (Rigorous Sensor Model Data)

-1

0

1

2

3

Mean

Δ N (m)

Δ N (m)

TerraSAR-X-1 Spotlight Monoscopic Absolute Geolocation Accuracy (Rigorous Sensor Model Data)

Image Error Centroids

0 -3

-2

-1

0

-1

-1

-2

-2

-3

1

2

3

Mean

-3

Δ E (m)

Δ E (m)

Figure 2. Horizontal Errors for Spotlight and Stripmap Images. The results of this evaluation are consistent with the expected TSX accuracy. The CCAP estimate for the spotlight CE90 is 1 meter, and it is plausible that the TSX accuracy is even better. This is because the estimated spotlight CE90 is on the same order as the geolocation accuracy of the surveyed GCPs and the pixel resolution for spotlight images. CCAP did not attempt to quantitatively remove these influences. The influence of the pixel resolution can be seen in the ratio of the stripmap CE90 to the spotlight CE90, which is 2.8. This is close to the ratio of 3 of the respective ground resolutions of the products. The coarser resolution of the stripmap imagery likely caused increased image measurement error that raised the estimated CE90 value for stripmap images.

SUMMARY In summary, DLR claims that TSX orbit errors are as small as 20 centimeters for routine images available within 1 day of collection, and that range errors are well under 1 meter. We believe these claims because our test results show that TSX spotlight imagery has an overall ground accuracy of 1 meter and possibly even less, when projected to known elevations. Science orbits with 10 centimeter accuracy are computed within 1 week after collection, and these are applied to all archived products. Such images should slightly exceed the outstanding performance we have quantified here.

ACKNOWLEDGEMENTS The authors would like to thank Patrick Cyphert of Observera Inc. for his support in measuring TSX images for CCAP and Dr. Fidel Paderes of BAE SYSTEMS for providing a software patch and troubleshooting support for SOCET Set that enabled CCAP to complete the TSX evaluation. The authors would also like to thank Dr. Steve Gayer and Bob Johnston of Lockheed Martin Corporation for their helpful discussions regarding Doppler angle errors.

REFERENCES Ager, T., 2001. Active sensing systems, Chapter 11 in Mikhail, E.M., J. S. Bethel, and J. C. McGlone, Introduction to Modern Photogrammetry, John Wiley & Sons, New York, pp. 301-349. Ager, T., 2004. An analysis of metric accuracy definitions and methods of computation, NIMA InnoVision whitepaper, 29 March 2004. Bertiger, W., Y. Bar-Sever, S. Desai, C. Dunn, B. Haines, G. Kruizinga, D. Kuang, S. Nandi, L. Romans, M. Watkins, S. Wu, and S. Bettadpur, 2002. GRACE: millimeters and microns in orbit, Proceedings of ION GPS 2002, Portland, Oregon, September 2002. ASPRS 2009 Annual Conference Baltimore, Maryland ♦ March 9-13, 2009

Bresnahan, P. C., and T. Jamison, 2007. A Monte Carlo simulation of the impact of sample size and percentile method implementation on imagery geolocation accuracy assessments, Proceedings of ASPRS 2007 Conference, Tampa, Florida, May 7-11, 2007. Broad Reach Engineering, 2007. Broad Reach Engineering GPS receiver launched On TerraSAR-X mission, Press Release, dated 1 November 2007, http://www.broadreachengineering.com/pressreleases.html, accessed 4 December 2008. Buckreuss, S., and R. Werninghaus, W. Pitz, 2008. The German satellite mission TerraSAR-X, IEEE Radar Conference (RadarCon), Rome, Italy, May 26-30, 2008. DLR, 2008. TerraSAR-X ground segment basic product specification document, TX-GS-DD-3302, v1.5, February 24, 2008. Frey, O., E. Meier, D. Nüesch, and A. Roth, 2004. Geometric error budget analysis for TerraSAR-X, Proceedings of the 5th European Conference on Synthetic Aperture Radar EUSAR, May 2004, pp. 513-516. JPL, 2008. The NASA global differential GPS system, http://www.gdgps.net/index.html, accessed 2 December 2008. Kahle R., B. Kazeminejad, M. Kirschner, Y. Yoon, R. Kiehling, and S. D’Amico, 2007. First in-orbit experience of TerraSAR-X flight dynamics operations, 20th International Symposium on Space Flight Dynamics, Annapolis, Maryland, 24-28 September 2007. Mikhail, E.M., J. S. Bethel, and J. C. McGlone, 2001. Introduction to Modern Photogrammetry, John Wiley & Sons, New York, 479p. Montenbruck, O., Y. Yoon, E. Gill, and M. Garcia-Fernandez, 2006. Precise orbit determination for the TerraSARX mission, 19th International Symposium on Space Flight Dynamics, Kanazawa, Japan, 4-11 June 2006. Nonaka, T., Y. Ishizuka, N. Yamane, T. Shibayama, S. Takagishi, and T. Sasagawa, 2008. Evaluation of the geometric accuracy of TerraSAR-X, International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol. XXXVII, Part B7, Beijing, China, pp. 135-140. Yoon, Y., M. Eineder, N. Yague-Martinez, and O. Montenbruck (2008). TerraSAR-X precise trajectory estimation and quality assessment, DLR Electronic Library, http://www.weblab.dlr.de/rbrt/pdf/TGRS_08_TSX.pdf.
