ANALYSIS OF THE GEOMETRIC ACCURACY PROVIDED BY THE FORWARD GEOCODING OF SAR IMAGES

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1 ANALYSIS OF THE GEOMETRIC ACCURACY PROVIDED BY THE FORWARD GEOCODING OF SAR IMAGES V. Karathanassi, Ch. Iossifidis, and D. Rokos Laboratory of Remote Sensing, Department of Rural and Surveying Engineering, National Technical University of Athens, Heroon Polytechniou 9, Zographos 15780, Athens, Greece ABSTRACT In recent years the SAR end users have increased significantly, however there is a big proportion of them being uninterested to getting involved with the raw SAR data and their parameters. Thus several remote sensing software packages have included SLC image processing in their modules, keeping users away from the raw SAR data processing and parameters. On the other hand, forward geocoding is rarely used and evaluated, even if the range value is precisely estimated by binary clock values included in raw data. In this paper, the potential of forward geocoding to providing true geolocated data is investigated. Geometric interpretation of the Newton-Rahson solutions and the produced errors is performed contributing to a)detect the ground points which are found on the Doppler centroid plane, b)certify the sensor calibration, c)estimate the ellipsoid used by ESA for SLCI ERS products, d)assess the influence of the input parameters, and e) investigate the accuracies achieved by the range values calculated by clock measurements included in the raw data. KEYWORDS Forward geocoding, geodetic coordinates, Newton-Raphson, precision, SAR 1. INTRODUCTION The geometric accuracy of geo-corrected SAR images is affected by the accuracy of the input parameters in the geocoding models (Curlander et al., 1991; Schreier, 1993). If Single Look Complex Images (SLCI) are used, these parameters are: orbit data, the range value with its associated accuracy provided by the header information of the SLC images, and approximate input values of the coordinates for every pixel in the image, for the forward (forward) geocoding model. azimuth time for first and last image pixel, and zero Doppler azimuth time per line with their associated accuracy provided by the header information of the SLC images, and ground coordinates for the backward geocoding model. In forward geocoding, the approximate input coordinate values required for every pixel of the image, contain errors due to topographic map, Digital Elevation Model, and gravimetric to orthometric model, with the most significant error being the pixel misidentification on the map.

2 If the Digital Elevation Model is estimated by interferometric phase, its accuracy depends on the scene, baseline, atmospheric conditions, etc. Although both geocoding methods provide solution for the SAR geocoding problem, the forward geocoding approach is rarely used. According to the literature, forward geocoding is considered a) much more difficult to quantify the final geolocation accuracy (Mohr and Madsen, 001) b) more complex and time consuming (Gelautz et al., 1998). Carrasco (1998) has studied the advantages and disadvantages of forward geocoding but his investigation on this item has focused only on the resampling techniques. This paper investigates the: Potential of Newton-Raphson method to providing solutions corresponding to the true geolocation, for each pixel of SLC ERS images; Spatial interpretation of the solutions and errors produced by forward geocoding; and Influence of the input parameters on the geometric accuracy provided by forward geocoding.. THE FORWARD GEOCODING METHOD Three equations relate the ground with the image coordinates: 1. The range (R) equation which defines a sphere centered in the satellite position. The radius R coincides with the distance between the satellite S and the position on the ground of a pixel P. P and S are expressed by the (x, y, z) Cartesian coordinates referred to the centre of the Earth. R = S P = ( S x Px ) + ( S y P y ) + ( S z Pz. The Doppler (f D ) equation which considers the relationship between the satellite velocities (Vs) and the ground points velocities (Vp) with the Doppler frequency of the target. Based on the header information of the image, i.e. the Doppler frequency f D equals zero, it is assumed that each pixel has been focused to its zero Doppler position, which is the minimum distance. This transforms the equation onto a plane which is perpendicular to the orbit. r r r r ( V s V p ) ( P S ) f D = () λ P S where λ is the wavelength. 3. The Earth surface equation which is the Earth ellipsoid equation with semi-axes a and b. The ellipsoidal height h of a point on the ground is introduced in this equation. Thus each point is considered to be found on an ellipsoid with a+h, and b+h semi-axes. P P x y Pz + + = 1 (3) ( a + h) ( a + h) ( b + h) In forward geocoding, the unknown Cartesian coordinates of a point on the ground are calculated by solving the system established by equations 1, and 3. For each pixel in the SLC ERS image, the known parameters are: a) image coordinates (azimuth, range); b) orbit information given by the state vectors of the satellite positions. In precise orbit data, vectors have a time gap equal to 30sec, while image capture duration is approximately 0sec. In restituted orbit data, the time gap equals to 4.167sec; c) the Earth reference ellipsoid parameters, semi major and semi minor axes a, b; and d) the range value. The knowledge of the ellipsoidal height (h) mainly depends on the DEM availability. If local topography is not known, height is assumed zero and the image is geocoded on the ellipsoid, like the GEC product of ESA. ) (1)

3 The range value has to be estimated for each pixel. For the results presented in this paper the range value was generated either by: a) interpolating the ranges of the first, center and last pixel of the image, ( header based range), or b) using the sensor and target (ground area corresponding to the pixel) Cartesian coordinates approach ( target based range). The ranges of the first, center and last pixel required by the first approach ( header based range), are resulted from the multiplication of the speed of light by the half of the zero Doppler range time of first, center, last range pixel, which are reported in the header information. The target Cartesian coordinates, required by the second approach ( target based range), are the approximate start values, which initially activate the Newton Raphson method applied for the system solution. For the forward geocoding solution, the Newton Raphson (Charpa and Canale, 1988) iterative method is usually chosen because the numerical system established by equations 1, and 3 is non linear. For its application, the Jacobian matrix is firstly required.for each pixel, approximate start values of its Cartesian coordinates (Px, Py, Pz) are introduced. The algorithm delivers new start values for the next iteration step. For the results presented in this paper, the iteration stops when the start values converge to the output values with an accuracy at the order of 10-9 m. and met convergence after 8-1 repetitions. This means that almost always in the geocoding problem, the Newton-Raphson method finds the roots of the system. For each pixel, the start values which initially activate the Newton-Raphson method result usually from a map and DEM, after taking into consideration discrepancies between the gravimetric and orthometric (ellipsoidal) heights. Consequently, they yield errors due to topographic map and DEM accuracy, as well as, to the accuracy of the gravimetric to orthometric model used. But the major issue that affects the accuracy of the start values is the precise identification of the pixel on the map in order to generate them. Although the accuracies of the map, DEM, and gravimetric to orthometric models affect the precision of the start values, their contribution is much smaller in relation to the error produced by the misidentification of the pixel on the map. In fact, this error may reduce the precision of the start values up to some hundred meters. 3. SPATIAL INTERPRETATION OF THE FORWARD GEOCODING SOLUTION The three equations reported in section present three surfaces: a sphere, a plane and an ellipsoid. The roots of the system established by them are the coordinates of a point, which is the intersection of the three surfaces. From the three surfaces, the Doppler centroid plane, which in our case coincides with the zero Doppler plane, is the most accurately defined. Once the position of the sensor - from where a point is imaged with the minimum distance to the orbit - is estimated, the plane which is perpendicular to the orbit, is also well defined. But this does not occur for the other two surfaces. The sphere is depended on the range value, which is the radius of the sphere. Different range values define different size spheres, centered in the same position. The intersection of the zero Doppler plane with the sphere is an arc. Different range values produce different arcs, all of them found on the zero Doppler plane (figure 1).

4 zero Doppler plane orbit R1 R arc1 arc R3 arc3 Figure 1. Intersections of different range spheres with the zero Doppler plane The ellipsoid is depended on the value of the height (h) of the point imaged. Different height values produce different ellipsoids, each one intersecting the arc of each range, in a different point (figure ). Consequently, for each pixel of the image, the system established by the three equations depends on the range value introduced in the range equation and the height value introduced in the ellipsoid. Although for each system, the roots provided by the Newton Raphson method are unique in the local area, these cannot be considered as the solution of the geocoding problem, since the appropriate system for the geolocation of each pixel is not known. arc1 3 F D 1 C B 4 R A E h 4 =650m h 3 =67.44m arc arc3 h =600m h 1 =550m Figure. Intersections of the arcs defined by figure 1 with the various ellipsoids If the range R is accurately defined in the header information of the image, then the number of the potential systems for each point is limited, and depends only on the various height values introduced in the ellipsoid equation. This number can be further limited if a precise DEM is used and height values ranging up to ±50m from the true value are introduced. Their roots are points produced from the intersection of a single arc (e.g. figure : arc) with the various ellipsoids. Even in this case, the most appropriate system for each pixel is not known. 4. IMPLEMENTATION OF THE FORWARD GEOCODING For the tests described in this section we have used: a Single Look Complex image captured by ERS on 19 September 1999 with orbit 3086 and frame 0756 and its header information; high precision orbit data; a reflector whose coordinates were estimated with an accuracy at the order of 1cm by using differential GPS measurements (GRS-80 reference ellipsoid: φ = 38 ' , λ = 1

5 41' , h = , ITRF96 -epoch datum: X (m) = , Y(m) = , Z(m) = ) a DEM of 5m resolution, generated by 1: topographic maps with 100m contour lines for steep slopes; the EGM96 model, proved as the most appropriate for the study area (Pagounis, 000), for the orthometric height estimation. Forward geocoding was implemented for the pixel of the ERS image, which represents the reflector. The geocoding was performed many times for the pixel of the reflector, each time providing the Newton-Raphson method with different approximate start coordinates. These were produced by the reflector coordinates as following (figure 3): x x Figure 3. The sampling used for the estimation of approximate input coordinates for the reflector of the SAR image Considering the reflector being the central point of a grid (point labeled 1, figure 3), 3 new points were defined with an increasing horizontal, or vertical, or horizontal and vertical step. The step value each time is twice its previous value. The initial horizontal and vertical step for the calculating procedure was 0.15 (4.60m) in φ, and 0. (4.85m) in λ, respectively. Figure 3 shows 4 of the 3 defined points. The experiments implemented in this study are classified into three categories and described in the following 3 subsections. 4.1 Experiments with Target Based Range Value In the first category, the defined as above φ, λ coordinates and the height 67.44m of the reflector were introduced as approximate start coordinates in the Newton- Raphson method. The height of the reflector was also introduced in the ellipsoid equation and the range was calculated based on the approximate start coordinates. Thirty-two different nonlinear systems, each one having a different range value, were consequently established. Each system has each own set of roots that is the intersection of each range arc with the ellipsoid defined by the height 67.44m (figure ). Thus the solutions move on this ellipsoid (figure : section AB) and due to the sampling way, yield points that lay symmetrically to the reflector (R). Their distance from the

6 reflector is m (table ) and is equal to the distance between the projections of the approximate start point and the reflector on the zero Doppler plane, respectively. The estimated by Newton-Raphson method errors range from 14cm to 48.78m. For each system, the respective error shows the distance of the start coordinates introduced in the Newton- Raphson method to the roots of the system. Due to the fact that all the roots of the established systems are found on the zero Doppler plane the estimated error pronounces the norm of the projection of the point that is used as start point in the Newton-Raphson method, on this plane. In figure 3, the points labeled 1, 13, 5, 4, 3, 10 and 5 are found very close to the zero Doppler plane. Table 1 shows their errors. However this error cannot pronounce the system being the most appropriate for the true geolocation of the pixel. The same experiment was repeated four times by changing the value of the height to 550, 600, 650, and 700m, and the semi axes of the ellipsoids, respectively. The number of the systems established as previously and their respective set of roots were increased by four. In figure, these set of roots define points which are found on the respective ellipsoids and lay symmetrically to the point defined with the same φ, λ coordinates as the reflector (sections CD, EF etc.). The displacements of the Newton-Raphson solutions in relation to the reflector are shown in table. For each height h, the solution with the minimum displacement corresponds to the point which has the same φ, λ coordinates as the reflector (figure : points 1, etc.). In figure, such points lay on the diagonals, e.g. 1R, of the quadrilaterals defined by two range arcs e.g. arc1, arc, and two ellipsoids, e.g. h 4, h 3. Based on the errors of the Newton-Raphson method, only the points of figure 3 which are close to the zero Doppler plane can be indicated. These are the same (points 1, 13, 5, 4, 3, 10 and 5) as those indicated by the ellipsoid 67,44. Table 1 shows their errors. Table 1: Forward geocoding errors of points found on the zero Doppler plane Height Error (m) (m) Point 1 Point 13 Point 5 Point 4 Point 3 Point 10 Point Experiments with Header Based Range Value In the second category of experiments, the sets of φ, λ coordinates produced by the sampling described above and the height of the reflector (67.44m) were introduced as approximate start coordinates in the Newton- Raphson method. The height of the reflector was also introduced in the ellipsoid equation and the value of the range was based on the header information of the SLC images. Consequently, this value was constant for all the subsequent experiments. Since the three surfaces, plane-sphere-ellipsoid, were kept the same during all the experiments, the system to be solved by the Newton-Raphson method is unique. But its roots yield the appropriate geolocation of the pixel only if the range value is accurately defined. For our experiment its accuracy was estimated by introducing as start coordinates in the Newton- Raphson method, the coordinates measured with GPS. The error was 98.15m, which pronounces the accuracy of the range value of the header information of the SLC images. By using the other 3 points of the sampling, and interpreting their errors it was found that the header based range value yields a point which is found on the top right of the samples of figure 3. Based on the header based range value, no indication on the accuracy of the geocoding can be given.

7 These experiments were repeated by changing the height of the points and using four more ellipsoids defined by the heights 550, 600, 650 and 700. The respective errors for points having the same φ,λ coordinates with the reflector are shown in table. By the estimation of the Newton-Raphson method errors for different ellipsoids and the use of a reflector, the parameters (axes, height) of the ellipsoid used by ESA for the region can be extracted. 4.3 Experiments with Range Value derived by GPS Measurements Finally, experiments using the range value produced by the reflector s true coordinates were performed. For the φ, λ coordinates of the reflector, the 3 new points of the sampling were generated having the height of the reflector (67.44m). When these were introduced as approximate start coordinates in the Newton-Raphson method they produced the same set of roots, which yield a point with coordinates very close to the true coordinates of the reflector. The distance of this point from the reflector, which is only 4.67m, demonstrates the accurate calibration of the ERS system. For each point used as approximate start point, the error produced by the Newton Raphson method shows the accuracy of its coordinates. This is the only case that the Newton-Raphson method estimates errors that are indicative for the accuracy of the approximate start coordinates. The experiments were repeated four more times, changing the height of the points to 550, 600 and 650 and 700, and their respective ellipsoids. According to figure the Newton-Raphson method yields four new solutions, each one having a different distance from the solution of the ellipsoid produced by height 67.44m. These are shown in table. Since each solution yields a displacement from the intersection of the ellipsoid with the true range arc, all the solutions are found on the true range arc (figure : points 3,4 etc.). These displacements are linearly linked to the value of the height introduced in the ellipsoid equation but are also depended on the true value of the height. For each solution, the errors estimated by the Newton-Raphson method show the distance of the approximate start coordinates from the solution. Therefore, even if the accurate range is known for a pixel, the user cannot distinguish the true height for this pixel. Table. Displacements of the solutions of the Newton-Raphson method from the reflector Height Displacement of the solutions of the Newton-Raphson method from the reflector (m) (m) Target based range Header based range reflector based range Table shows that in all cases that height is not known accurately, forward geocoding which uses target based range values produces higher accuracies compared to the errors produced by range values derived by GPS. High accuracy ranges for a pixel can be produced either by the use of a reflector or by clock values provided by raw data (Mohr and Madsen, 001). Based on the geometry of figure the same conclusions can be derived. 5. DISCUSSION AND CONCLUSIONS In this paper, an analytical geometric interpretation of the solution and the errors produced by forward geocoding was implemented. The Newton Raphson method gives solutions which rarely coincide with the true geolocation of a point. The errors estimated by the Newton-Raphson

8 method, indicate the distance of the introduced approximate start coordinates from the solution. Precise orbit data and the method of estimation of the range value are crucial for the accuracy of the solution: The value of the range estimated by the approximate start coordinates which are introduced in the Newton-Raphson method ( target based range) produces errors, which indicate the distance of the approximate start coordinates from the zero Doppler plane. Consequently, points very close to this plane can be defined by forward geocoding, but conclusions on the true geolocation of the pixels cannot be derived. The range value provided by the header information of the image produces unsatisfactory results. If a reflector is available, the parameters of the ellipsoid used by ESA can be determined. The range provided by a reflector or clock values which are included in the raw SAR data, cannot produce accurate results since the knowledge of height for every pixel is also significant for the establishment of the non linear system which leads to the true geolocation of the pixel. Among the three range estimations the target based range produces the most accurate results. Only in the case that the accurate height of the point is known, the range provided by a reflector or clock values produces the highest accuracy, and test of the ERS calibration can be performed. ACKNOWLEDGEMENTS The authors are grateful to the Laboratory of Higher Geodesy of the National Technical University of Athens for kindly providing the reflector s coordinates. REFERENCES Carrasco, D., SAR Interferometry for Digital Elevation Model Generation and Differential Applications, Ph.D.Dissertatation, Polytechnic University of Catalonia, Dept. of Signal Theory and Communications, Barcelona,1998. Charpa, S.C. and Canale R.P., Numerical Methods for Engineers, nd edn, McGraw-Hill Inc., USA, Curlander, J.C., and McDonough, R.N., Synthetic Aperture Radar Systems and Signal processing, 1 st edn, John Wiley & Sons, New York, Gelautz, M., Frick H., Raggam, J., Burgstaller, J., Leberl, F., SAR image simulation and analysis of alpine terrain. ISPRS Journal of Photogrammetry & Remote Sensing, 53(1998), pp.17-38, Meler, E., Frei, U., Nϋesch, D., Precise Terrain Corrected Geocoded Images. In: SAR Geocoding: Data and Systems, (Scheier G. ed.), Herbert Wichmann Verlag GmbH, Karlstuhe, pp , Mohr, J.J. and Madsen, S.N., Geometric Calibration of ERS Satellite SAR Images, IEEE Transactions on Geoscience and Remote Sensing, 39 (4), pp , 001. Pagounis, B.N., The contribution of the processing of the geodetic heights in the local geodynamic. The Greek case study. Ph.D.Dissertatation, National Technical University of Athens, Dept. of Rural and Surveying Engineering, Athens, 000. Schreier, G., SAR Geocoding: Data and Systems, 1 st edn, Herbert Wichmann Verlag GmbH, Karlstuhe, 1993.