Initial tests of point target geolocation for RADARSAT-2 ... · Additionally, higher resolution DEMs exists for most of North America. Future plans: The accuracy of the procedure

Initial tests of point target geolocation for RADARSAT-2 data using a DEM Pierre D. Beaulne and Ishuwa C. Sikaneta

Defence R&D Canada --- Ottawa Technical Memorandum

DRDC Ottawa TM 2008-344 April 2009

Initial tests of point target geolocation for RADARSAT-2 data using a DEM Pierre D. Beaulne and Ishuwa C. Sikaneta

Defence R&D Canada – Ottawa Technical Memorandum DRDC Ottawa TM 2008-344 April 2009

Principal Author

Original signed by P.D. Beaulne

Pierre D. Beaulne

Approved by

Original signed by D. Dyck

D. Dyck

Section Head

Approved for release by

Original signed by P. Lavoie

P. Lavoie

Chief Scientist

© Her Majesty the Queen in Right of Canada, as represented by the Minister of National Defence, 2009

© Sa Majesté la Reine (en droit du Canada), telle que représentée par le ministre de la Défense nationale, 2009

DRDC Ottawa TM 2008-344 i

Abstract ……..

This Technical Memo provides an algorithm for geolocating targets detecteded in RADARSAT-2 imagery and presents results from initial testing. The method iteratively solves a system of geometrically based equations that is adapted to include radar squint and a digital elevation model to improve accuracy. Analysis of data collected over Ottawa shows that the geolocated targets differ, on average, by 21 metres from the Google Earth® reference imagery. The improvement of geolocation accuracy with the use of a digital elevation model (DEM) is clearly illustrated using data collected in mountainous terrain near Vancouver. The results obtained are in accordance with other studies on the accuracy of RADARSAT-1 and RADARSAT-2 geolocation and are of the same order as the inherent error in Google Earth® imagery.

Résumé ….....

Ce Mémorandum propose un algorithme pour effectuer la géolocalisation de cibles identifiées dans des images de RADARSAT-2 et présente les résultats des premières études de son efficacité. Une méthode itérative est employée afin de résoudre un système d’équations à base géométrique adapté pour inclure un angle de strabisme et un modèle altimétrique numérique; ceci afin d’augmenter l’exactitude du processus. Une analyse de données radar collectionnées dans la région d’Ottawa démontre que la position des cibles géolocalisées de cette façon diffère, en moyenne, par 21 mètres par rapport aux images de référence Google Earth®. L’augmentation d’exactitude de géolocalisation possible avec l’emploi d’un modèle altimétrique numérique est clairement illustrée avec des données d’un endroit montagneux dans les environs de Vancouver. Les résultats obtenus sont en accord avec autres analyses de l’exactitude de géolocalisation, pour RADARSAT-1 et RADARSAT-2. L’estimation de l’erreur de résultats est du même ordre que l’erreur inhérente dans les images de Google Earth®.

ii DRDC Ottawa TM 2008-344

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DRDC Ottawa TM 2008-344 iii

Executive summary

Initial tests of point target geolocation for RADARSAT-2 data using a DEM

Beaulne, P.D. and Sikaneta, I.C. DRDC Ottawa TM 2008-344; Defence R&D Canada - Ottawa; April 2009

Background: This Memo presents a simple, general approach to the solution of the basic geolocation problem (transforming sensor coordinates to geographic ones) for Synthetic Aperture Radar (SAR) data. It uses the direct measurements made by the SAR, namely the range to target and the time of imaging, along with the RADARSAT-2 state vectors, a model of the imaging geometry and information on terrain height from a digital elevation model (DEM) to iteratively find a solution.

Results: Initial test results with RADARSAT-2 data of the Ottawa area show that the geographic coordinates of targets calculated with this algorithm differ, on average, by 21 metres from the Google Earth® reference imagery. This is of roughly the same order as the error inherent in Google Earth® referenced images, or typical 1:50000 scale digital or paper maps. The improved accuracy achievable by using a digital elevation model is demonstrated in mountainous terrain near Vancouver.

Significance: A more time consuming but more effective method for referencing radar image targets to geographic coordinates uses contextual information in the radar image to georeference the entire radar image to a map. As this method computes an image warp function that maps known tie points in the radar data to easily identifiable points on a map, it can only be used when an accurate geographic map of the region of interest is available. The proposed geometry based approach does not need a reference map and so will work more effectively over regions where referenced maps do not exist, for example, over oceans, ice sheets or deserts. Greatly improved accuracy of target latitude and longitude is achievable using a digital elevation model. Since a DEM of the entire Earth's landmass has been created by the Shuttle Radar Topography Mission (SRTM), such data are now readily available. Additionally, higher resolution DEMs exists for most of North America.

Future plans: The accuracy of the procedure needs to be further tested, especially in mountainous terrain, with and without a DEM. The limits of the accuracy need to be examined by deploying radar corner reflectors with known Global Positioning System (GPS) positions. Finally, the geolocation accuracy of moving targets detected with the MODEX mode needs to be tested. This includes studies in maritime areas, where, highly accurate geolocation of ships detected in radar imagery is possible using the proposed method.

iv DRDC Ottawa TM 2008-344

Sommaire .....

Initial tests of point target geolocation for RADARSAT-2 data using a DEM

Beaulne, P.D. and Sikaneta, I.C. DRDC Ottawa TM 2008-344; Defence R&D Canada - Ottawa; avril 2009

Introduction ou contexte: Ce Mémorandum propose une approche simple et générale à la solution du problème de géolocalisation (la transformation de coordonnées de capteur en coordonnées géographiques) pour des données d’un Radar à Ouverture Synthétique (ROS). Cette méthode utilise les quantités directement mesurées par le ROS, soient la portée à la cible et son temps d’acquisition par le capteur. Celles-ci, avec les vecteurs d’état de RADARSAT-2, sont incorporés dans un modèle géométrique du processus d’acquisition de données ROS. La position géographique d’une cible d’intérêt est calculée itérativement avec l’aide de ce modèle et avec l’information topographique contenue dans un modèle altimétrique numérique de l’endroit en question.

Résultats: Les premiers tests de l’algorithme, employant des données RADARSAT-2 collectionnées dans les environs d’Ottawa, démontrent que les coordonnées géographiques calculées par cette méthode diffèrent, en moyenne, par 21 mètres par rapport aux images de référence Google Earth®. Cette différence est du même ordre que l’erreur de localisation présente dans les images Google Earth® elles-mêmes, ou présente dans n’importe quelle carte (en papier ou numérique) à une échelle de 1:50,000. L’augmentation d’exactitude achevable par l’emploi d’un modèle altimétrique numérique est illustrée avec un exemple de géolocalisation en terrain montagneux près de Vancouver.

Importance: Une approche plus précise pour référencer les cibles radar à des coordonnées géographiques utilise de l’information contextuelle pour géo-référencer l’image radar entière à une carte. Puisque cette méthode calcule une fonction de déformation qui établit une relation entre points de référence identifiables dans l’image radar et leurs points cartographiques correspondants, elle ne peut être utilisée que quand un carte géographique convenable est disponible. De plus, elle est plus longue et élaborée. L’approche géométrique proposée ne nécessite pas de référence externe et donc sera effective même dans les endroits pour lesquels une carte n’est pas disponible, tels que les océans, les déserts ou les endroits polaires. Une exactitude améliorée de coordonnées géographiques est possible en utilisant un modèle altimétrique numérique. Un tel modèle, du monde entier, a été créé avec la navette spatiale lors de la mission «Shuttle Radar Topography Mission (SRTM) ». En plus, des modèles à très haute résolution sont disponibles pour la plupart de l’Amérique du Nord.

Perspectives: L’exactitude de cette approche doit être vérifiée davantage, surtout dans les endroits montagneux, par des calculs effectués avec, et sans, emploi d’un plan altimétrique numérique. Les limites de cette exactitude doivent être examinées par le déploiement de réflecteurs radar ayant des coordonnées géographiques précisément mesurées par GPS. Enfin, l’exactitude du processus pour cibles mobiles identifiées par le mode MODEX doit aussi être mesurée. Ceci inclut des études d’endroits maritimes, ou, par cette méthode, la géolocalisation très exacte de navires détectés par radar est possible.

DRDC Ottawa TM 2008-344 v

Table of contents

Abstract …….. ................................................................................................................................. i

Résumé …..... ................................................................................................................................... i

Executive summary ....................................................................................................................... iii

Sommaire ..... .................................................................................................................................. iv

Table of contents ........................................................................................................................... iv

List of figures ................................................................................................................................ vi

1 .... Introduction ............................................................................................................................. 1

2 .... Geolocation of a Point Target in SAR data .............................................................................. 2 2.1 The system of equations ................................................................................................ 2 2.2 Solution via 3-D Newton’s Method ............................................................................... 3 2.3 Including terrain height from a DEM ............................................................................ 6

3 .... Testing the Algorithm with RADARSAT-2 Data .................................................................... 9 3.1 Test Scenes and Procedure ............................................................................................ 9 3.2 Test Results ................................................................................................................. 10

4 .... Conclusion .............................................................................................................................. 14

References ..... ............................................................................................................................... 15

Annex A .. C++ Code .................................................................................................................... 17

vi DRDC Ottawa TM 2008-344

List of figures

Figure 1: Geometry of the solution space. ...................................................................................... 5

Figure 2: Height above the ellipsoid. ............................................................................................. 6

Figure 3: EO image of the area for the RADARSAT-2 Data Acquisition over Ottawa. ................. 9

Figure 4: August 20, 2008 RADARSAT-2 SAR Image of the Ottawa Test Area. ...................... 10

Figure 5: Google Earth® EO image of a selected target. .............................................................. 11

Figure 6: Error Estimates for the calculated target position, relative to Google Earth® imagery, for the 20 targets identified in the Ottawa test area shown in Figures 3 and 4. .......................................................................................................................... 12

Figure 7: Google Earth® image of a mountainous area near Vancouver, demonstrating the accuracy improvement that is possible when using a DEM. ....................................... 13

DRDC Ottawa TM 2008-344 1

1 Introduction

Geolocation is the calculation of the geographic coordinates of a target (or any feature) from its image plane, or sensor, coordinates and from knowledge of the sensor position and beam pointing direction at the time of imaging. Use of an elevation model of the imaged area improves the accuracy of the process, especially in highly variable terrain.

For optical sensors, the sensor pointing direction is expressed in terms of angles. Radars, however, also measure the distance, or range from the sensor to a target or feature. A Synthetic Aperture Radar (SAR) additionally measures the Doppler history of all the returns from a target. This makes it possible to construct a simple model which relates these measured quantities to the (unknown) target position. Various approaches to this basic model are presented e.g. in [1] and [2].

This Technical Memo presents a simple, general approach to the solution of this basic model, which uses the direct measurements made by the SAR, namely the range to the target and the azimuth time of imaging, and information on terrain height from a digital elevation model (DEM). A necessary element of the required calculations is the determination of the range and the azimuth slow time from image sample numbers and SAR processing parameters, as well as the interpolation of satellite state vectors to the appropriate azimuth time. The former is discussed in [3] and the latter in [4].

The implemented solution is tested by selecting identifiable control points in SAR imagery, performing the geolocation, and plotting the results obtained using this method on a map to estimate the geolocation accuracy. A RADARSAT-2 data acquisition over Ottawa is analyzed in this manner and identified targets are compared to their corresponding images on Google Earth®. An example of the effect of terrain height on position estimation is also shown in a mountainous RADARSAT-2 scene near Vancouver. This serves only as an initial estimate of RADARSAT-2 geolocation accuracy using this algorithm, since radar corner reflectors with known Global Positioning System (GPS) coordinates were not deployed for the data acquisitions tested. The C++ code used in the analysis is presented in Annex A.

2 DRDC Ottawa TM 2008-344

2 Geolocation of a Point Target in SAR data

This section presents the equations for geolocation of a point target (or any sample) in SAR imagery and presents a simple implementation to solving them using 3-D Newton’s Method. The solution to the basic ellipsoidal geolocation is presented, along with the modifications necessary to incorporate terrain elevation information.

This method begins with the construction of a system of three equations that constrain the target

position, Tzyxx ,,

(where T denotes the transpose), and that allow the slant range plane coordinates (SAR samples corresponding to radar measurements) to be expressed in Earth Centered Cartesian coordinates. In the discussion below, the following assumptions and quantities are required:

All coordinates are expressed in the Cartesian Earth Centered Earth Fixed (ECEF) system. A discussion of this system and conversion from it to the familiar latitude, longitude and height can be found in [4] and [5].

The slant range from the sensor to the target (or sample), r , is known.

The azimuth (slow) time of imaging, 0t , is known or calculated from Doppler information.

The radar platform state vector position, Tpppp zyxtx ,,)( 0

, and velocity

Tzyxp vvvtv ,,)( 0

have been interpolated to the slow time 0t .

The radar beam squint angle, sq (if applicable) is known.

A digital elevation model of the imaged area is available.

2.1 The system of equations

The unknown geographic position is described by three variables which are to be determined:

Tzyxx ,,

,

First the target is assumed to lie upon the surface of an ellipsoidal Earth. This is ensured by the constraint that its coordinates satisfy the equation of the ellipsoid:

01),,(2

2

2

2

2

2

1 bba r

z

r

y

r

xzyxf (1)

In this note, the WGS84 ellipsoid is used, with 0.137,378,6ar m and 3.752,356,6br m

being the equatorial and polar radii, respectively.

Next, given the measured slant range from the sensor to the target, r , it is required that


0)()()(),,( 2222222 zzyyxxrxxrzyxf pppp (2)

Equation (2) ensures that the calculated sensor to target distance (i.e. range) is equal to the measured slant range.

The radar beam is constrained to be side looking by requiring that the dot product of the slant range vector and the platform velocity vector correspond to the beam squint angle. If the data are not processed to zero Doppler, the residual Doppler centroid offset should also be subtracted. Thus,

0)sin()(),,(3 sqppp vrvxxzyxf (3)

The constraints of Equations (1) to (3) are neatly summarized in the following equation:

0

)sin(

)()()(

1

)(

)(

)(

)( 222

2

2

2

2

2

2

3

2

1

sqpzyxzpypxp

ppp

bba

vrzvyvxvvzvyvx

zzyyxxrr

z

r

y

r

x

xf

xf

xf

xf

(4)

2.2 Solution via 3-D Newton’s Method

To solve the vector Equation (4) one can use Newton’s iterative formula in multiple dimensions:

)()(11 nnnn xfxxx

J (5)

where )(x

J is the Jacobian matrix [6]:


zyx

ppp

baa

vvv

zzyyxxr

z

r

y

r

x

z

f

y

f

x

fz

f

y

f

x

fz

f

y

f

x

f

x

)(2)(2)(2

222

)(

222

333

222

111

J

(6)

To find the inverse of )(x

J , one can use the cofactor, Cij , formula for the inverse of a 3x3

matrix [6]:

211222113112321131223221

211323113113331131233321

221323123213331232233322

333231

232221

1312111

)(

1

)(

1)(

JJJJJJJJJJJJ

JJJJJJJJJJJJ

JJJJJJJJJJJJ

x

CCC

CCC

CCC

xx

T

J

JJ

(7)

The solution to Equation (5) requires an initial guess at the solution. As seen in Figure 1, there are two possible solutions to the system defined by Equation (4). If an initial guess is chosen close to the actual solution, the Newton Method will quickly converge to the desired solution.


Figure 1: Geometry of the solution space.

To set up the initial guess for x

, the following is proposed. First, the cross product pp vxu

is constructed. This vector is perpendicular to the orbit plane defined by px

and pv

. Then,

uuxw p /

is constructed. The positive sign corresponds to looking to port, and the

negative sign to looking to starboard. The point defined by w

now lies above the terrain so it is

moved back down, closer to the ellipsoid, by constructing 2/)(/0 ba rrwwx

. The

converged solution obtained from Newton’s Method can be tested to ensure it is on the correct side, by looking at the sign of the quantity ppp xvxx

)( . A negative sign indicates the

“starboard solution”, while a positive sign indicates the “port solution”.


2.3 Including terrain height from a DEM

A slight modification has to be made when the target of interest cannot be assumed to lie on the WGS84 ellipsoid. A correction must be made to account for the increment in the radius of the ellipse imparted by the additional height. This section proposes a method to modify Equation (1) to account for the extra height. As illustrated in Figure 2, let this height above the ellipsoid be denoted by h .

Figure 2: Height above the ellipsoid.

It is assumed that there is a vector y

that satisfies the equation defining the ellipsoid (Equation (1)). Rewritten in vector notation:


1yyT A (8)

where

2

2

2

00

00

00

b

a

a

r

r

r

A (9)

From Figure 2, it is seen that the point of interest (at height h ) satisfies the relation:

y

yhyx

)( (10)

From Equations (8) and (10), it can be deduced that:

2

2

y

hy

yyy

hyxx TT

AA

(11)

Taking the magnitude of Equation (10), combining it with Equations (11) and (1), and explicitly stating the coordinates:

0

),,(

2222

222

2

2

2

2

2

2

2

1

hzyx

zyx

r

z

r

y

r

x

hx

xxxzyxf

baa

T

A

(12)

Substituting Equation (12) into the vector function (Equation (4)) the following is obtained:


0

)sin(

)()()(

)(

)(

)(

)( 2222

2222

222

2

2

2

2

2

2

3

2

1

sqpzyxzpypxp

ppp

baa

vrzvyvxvvzvyvx

zzyyxxrhzyx

zyx

r

z

r

y

r

x

xf

xf

xf

xf

(13)

Analogously to Equation (6), the Jacobian matrix for Equation (13) is derived as:

zyx

ppp

baa

vvv

zzyyxxu

zh

r

z

u

yh

r

y

u

xh

r

x

)(2)(2)(2

222222323232

J (14)

where

][ 222 hzyxu (15)

The solution to Equation (13), just as that for Equation (4) is found using the 3-D Newton’s Method (Equation (5)). Note that for every iteration, the terrain height is updated based upon the current estimate of the target position nx

.


3 Testing the Algorithm with RADARSAT-2 Data

3.1 Test Scenes and Procedure

Two test scenes are considered. The first, collected on August 20, 2008, over Ottawa, is used to demonstrate the geolocation accuracy achievable with the method described in Section 2. The second, collected on November 4, 2008, near Vancouver, is used to demonstrate the effect of using a DEM where terrain elevation becomes a potentially significant source of error.

Figure 3: EO image of the area for the RADARSAT-2 Data Acquisition over Ottawa. The section used in the target analysis is bounded in red. Imagery is ©2008 Google-Imagery Image©2008 Digital Globe,©2008 Europa Technologies, ©2008 Tele Atlas.


Figure 3 shows an electro-optical (EO) image from Google Earth®. The area bounded in red was imaged by RADARSAT-2 on August 20, 2008, and is the test area from which reference targets were selected. The SAR image of this area is shown below in Figure 4. Twenty easily identifiable targets were selected (e.g. intersections and building corners), and their radar image space coordinates were converted to geographic coordinates as described in Section 2. An estimate of the geolocation error was then calculated as the difference between the estimated position and the reference position, as obtained from the Google Earth® image.

Figure 4: August 20, 2008 RADARSAT-2 SAR Image of the Ottawa Test Area.

3.2 Test Results

Figure 5 shows a closeup image of one of the target locations selected in the radar image. The corner of the building shown is the reference point that was identified in the SAR image. The target position estimated by the algorithm is denoted by the red triangle. In this case, the two positions differ by 23.5 metres. This procedure was carried out for all 20 selected reference targets.


Figure 5: Google Earth® EO image of a selected target. Imagery is ©2008 Google- Imagery is ©2008 Google-Imagery Image©2008 Digital Globe,©2008 Europa Technologies, ©2008 Tele Atlas.

The results from the analysis of these targets are summarized in Figure 6. The mean difference between the geolocation as determined by the method described here and that of the Google Earth® imagery is 21 metres. This is of the same order as the error inherent in any paper or digital 1:50000 scale map [7].


Figure 6: Error Estimates for the calculated target position, relative to Google Earth® imagery, for the 20 targets identified in the Ottawa test area shown in Figures 3 and 4.

Standard georeferenced RADARSAT-2 products are generated using a mean terrain height over the scene [8]. However this could result in significant error in high relief terrain. To examine this, RADARSAT-2 data of a mountainous area near Vancouver are examined (Figure 7). In this figure, the intersection at the center (near the red pin) was chosen as a reference point and its geographic position was calculated with (green pin) and without (red pin) the use of a DEM. The position calculated without the DEM differed by 160 metres from that of the reference point, while the one calculated with a DEM differed by only 8 metres. This clearly illustrates the effect terrain height can have on geolocation, and the improvement possible with the use of a DEM.


Figure 7: Google Earth® image of a mountainous area near Vancouver, demonstrating the accuracy improvement that is possible when using a DEM. The point identified in the radar imagery was the intersection near the red pin. Imagery is ©2008 Google-Imagery Image©2008 Digital Globe,©2008 Europa Technologies, ©2008 Tele Atlas.


4 Conclusion

This Technical Memo has presented an algorithm to translate slant range target coordinates in a satellite radar image into geographic coordinates on the ground. The method presented is geometrically based and iterative, converges quickly, and incorporates a digital elevation model to improve accuracy. The inclusion of a DEM into the model, and the resulting improvement in geographic location estimation are improvements over currently used algorithms, such as that used for standard RADARSAT-2 products which, at best, use an average terrain height over a scene, as determined from a DEM. To demonstrate the algorithm and to test the geolocation accuracy of RADARSAT-2, two data scenes are analysed. The first scene, taken over Ottawa, is used to test RADARSAT-2 geolocation accuracy using twenty reference targets. These targets include street intersections, building corners and other recognizable features. The mean estimated error of the calculated positions is 21 metres. The second scene over high relief terrain near Vancouver clearly shows that local terrain height estimates are very important for geolocation estimation in this type of terrain. It is stressed that this Technical Memo only provides preliminary estimates of RADARSAT-2 geolocation estimation using this methodology. The “ground truth” for the targets in this analysis is provided by Google Earth® and is subject to its own inherent error (as with all paper and digital ‘maps’). Therefore, a proper accuracy assessment would require known point targets such as corner reflectors, with accurately measured geographic coordinates.

The appendix of this Technical Memo provides a C++ implementation of the algorithm.


References .....

[1] Schreier, G., Editor (1993), SAR Geocoding: Data and Systems, Karlsruhe, Germany: Wichmann.

[2] Olmsted, C. (1993), Alaska SAR Facility Scientific SAR User’s Guide, (ASF-SD-003), Fairbanks, USA.

[3] Curlander, J.C. and McDonough, R.N. (1991), Synthetic Aperture Radar: Systems and Signal Processing; Chapters 3-4, New York: John Wiley and Sons.

[4] Beaulne, P.D. and Sikaneta, I.C (2005)., A simple and precise approach to position and velocity estimation of low earth orbit satellites, (DRDC Ottawa TM 2005-250) Defence R&D Canada - Ottawa.

[5] Vallado, D.A. (2001), Fundamentals of Astrodynamics and Applications, 2nd ed.; Chapters 3-4, El Segundo, USA: Microcosm Press and Kluwer Academic Publishers.

[6] O’Nana, M and Enderton, H.(1990), Linear Algebra, 3rd ed. San Diego, USA: Harcourt Brace Jovanovich.

[7] USGS National Map Accuracy Standards, www.rockyweb.cr.usgs.gov/nmas.html (Access date: 1 Dec, 2008).

[8] (2008) RADARSAT-2 Product Format Definition, (RN-RP-51-2713, Issue 1/7), Richmond, Canada: MacDonald Dettwiler and Associates.


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Annex A C++ Code

// Necessary Headers #include <iostream> #include <stdlib.h> #include <string.h> #include <iomanip> #include <math.h> #include "myPosition.hpp" #include "radar.h" const double myPosition::majaxis = 6378137.0; const double myPosition::minaxis = 6356752.3142;; const double myPosition::pi = 3.1415926535; //Function to transform Latitude, Longitude and Ellipsoidal height // to x,y,z cartesian coordinates void myPosition::LL2xyz() { double N;// Prime Vertical radius of curvature double f;// Flattening double e;// Eccentricity f = (majaxis-minaxis)/majaxis; e = 2.0*f-f*f; N = majaxis/sqrt(1-e*pow(sin(latitude),2)); x = (N+height)*cos(latitude)*cos(longitude); y = (N+height)*cos(latitude)*sin(longitude); z = (N*(1-e)+height)*sin(latitude); } //Function to transfer x,y,z cartesian coordinates //to Latitude, Longitude and Ellipsoidal height void myPosition::xyz2LL() { double P;//Distance between points??? double eprime; double latitudeP; // utility variable double f;// Flattening double e;// Eccentricity double N;// Prime Vertical radius of curvature P = sqrt((xt*xt)+(yt*yt)); latitude = atan(zt*majaxis/(P*minaxis)); eprime = (pow(majaxis,2)-pow(minaxis,2))/pow(minaxis,2); f = (majaxis-minaxis)/majaxis; e = 2*f-f*f; longitude = atan2(yt,xt); do{ latitudeP = latitude; latitude = atan((zt+eprime*minaxis*pow(sin(latitudeP),3))/(P-e*majaxis*pow(cos(latitudeP),3)));


}while(fabs(latitude-latitudeP)>1e-9); N = majaxis/sqrt(1-e*pow(sin(latitude),2)); height = (P/cos(latitude))-N; } // Newton function to compute coordinates double myPosition::advance(double *u, double *v, double *w) { // Get the terrain height at the current position. Maybe from DEM double h = getTerrainHeight(*u, *v, *w); // Compute function values at current point double f1 = range*range - (*w-z)*(*w-z) - (*u-x)*(*u-x) - (*v-y)*(*v-y); double f2 = vx*(x-*u)+vy*(y-*v)+vz*(z-*w) + sqrt(vx*vx+vy*vy+vz*vz)*sin(squint)*range; double f3 = pow(*u,2.0)/pow(majaxis,2.0)+pow(*v,2.0)/pow(majaxis,2.0)+pow(*w,2.0)/pow(minaxis,2.0) -(pow(*u,2.0)+pow(*v,2.0)+pow(*w,2.0))/pow(sqrt(pow(*u,2.0)+pow(*v,2.0)+pow(*w,2.0))-h,2.0); // Compute Jacobian matrix at current point double j11 = -2.0*(*u-x); double j12 = -2.0*(*v-y); double j13 = -2.0*(*w-z); double j21 = -vx; double j22 = -vy; double j23 = -vz; double j31 = 2.0*(*u)/pow(majaxis,2.0)+2.0*(*u)*h/pow(sqrt(pow(*u,2.0)+pow(*v,2.0)+pow(*w,2.0))-h,3.0); double j32 = 2.0*(*v)/pow(majaxis,2.0)+2.0*(*v)*h/pow(sqrt(pow(*u,2.0)+pow(*v,2.0)+pow(*w,2.0))-h,3.0); double j33 = 2.0*(*w)/pow(minaxis,2.0)+2.0*(*w)*h/pow(sqrt(pow(*u,2.0)+pow(*v,2.0)+pow(*w,2.0))-h,3.0); // Compute the determinant of the Jacobian double jdet = j11*(j22*j33-j32*j23) - j12*(j21*j33-j23*j31) + j13*(j21*j32-j22*j31); // Compute the inverse Jacobian matrix double inv11 = +1.0*(j22*j33-j32*j23)/jdet; double inv21 = -1.0*(j21*j33-j23*j31)/jdet; double inv31 = +1.0*(j21*j32-j22*j31)/jdet; double inv12 = -1.0*(j12*j33-j13*j32)/jdet; double inv22 = +1.0*(j11*j33-j13*j31)/jdet; double inv32 = -1.0*(j11*j32-j12*j31)/jdet; double inv13 = +1.0*(j12*j23-j13*j22)/jdet; double inv23 = -1.0*(j11*j23-j13*j21)/jdet;


double inv33 = +1.0*(j11*j22-j12*j21)/jdet; // Compute the difference from current point double delu = inv11*f1+inv12*f2+inv13*f3; double delv = inv21*f1+inv22*f2+inv23*f3; double delw = inv31*f1+inv32*f2+inv33*f3; *u -= delu; *v -= delv; *w -= delw; return delv*delv+delu*delu+delw*delw; } // Compute the coordinates of the target in ECEF int myPosition::findcoord(int side) { // The member variables (x,y,z) for the radar position and // (vx,vy,vz) for the radar velocity vector are already set // First determine a vector that points perpendicularly to the // track vector and the sub platform point to radar point vector // Use the cross product to determine this vector double ux=y*vz-z*vy; double uy=z*vx-x*vz; double uz=x*vy-y*vx; double un=sqrt(ux*ux+uy*uy+uz*uz); ux/=un; uy/=un; uz/=un; // Compute an initial guess by using (ux,uy,uz) // extended to range on desired side. // If side = 1, then point to starboard, if -1 then point to port double srange = sqrt(range*range-pow(sqrt(x*x+y*y+z*z)-majaxis/2.0-minaxis/2.0,2.0)); xt=x-side*srange*ux;// This will be the initial guess of the x coordinate yt=y-side*srange*uy;// This will be the initial guess of the y coordinate zt=z-side*srange*uz;// This will be the initial guess of the z coordinate // Improve the initial guess double shrink = (majaxis+minaxis)/2.0/sqrt(xt*xt+yt*yt+zt*zt); xt*=shrink; yt*=shrink; zt*=shrink; int icount = 0;// counter variable double tol = 0.1; // Tolerance of convergence // Compute the position vector using Newton Method while((advance(&xt,&yt,&zt)>tol) & (icount++<100)) {/*Do nothing*/};


// Test for convergence if(icount=100) { cout "Error computing coordinate" << endl; return 0; } // Return 1 for starboard coordinate, -1 for port coordinate return((x*(vz*(yt-y)-vy*(zt-z)) +y*(vx*(zt-t)-vz*(xt-x)) +z*(vy*(xt-x)-vx*(yt-y)))>0.0)? 1 : -1; }

DOCUMENT CONTROL DATA (Security classification of title, body of abstract and indexing annotation must be entered when the overall document is classified)

1. ORIGINATOR (The name and address of the organization preparing the document. Organizations for whom the document was prepared, e.g. Centre sponsoring a contractor's report, or tasking agency, are entered in section 8.)

Defence R&D Canada – Ottawa Ottawa, ON K0A 1Z4

2. SECURITY CLASSIFICATION (Overall security classification of the document including special warning terms if applicable.)

UNCLASSIFIED (U)

3. TITLE (The complete document title as indicated on the title page. Its classification should be indicated by the appropriate abbreviation (S, C or U) in parentheses after the title.)

Initial Tests of Point Target Geolocation for RADARSAT-2 Data using a DEM

4. AUTHORS (last name, followed by initials – ranks, titles, etc. not to be used)

Pierre D Beaulne and Ishuwa C Sikaneta

5. DATE OF PUBLICATION (Month and year of publication of document.)

April 2009

6a. NO. OF PAGES (Total containing information, including Annexes, Appendices, etc.)

20

6b. NO. OF REFS (Total cited in document.)

8

7. DESCRIPTIVE NOTES (The category of the document, e.g. technical report, technical note or memorandum. If appropriate, enter the type of report, e.g. interim, progress, summary, annual or final. Give the inclusive dates when a specific reporting period is covered.)

Technical Memorandum

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9a. PROJECT OR GRANT NO. (If appropriate, the applicable research and development project or grant number under which the document was written. Please specify whether project or grant.)

15eg

9b. CONTRACT NO. (If appropriate, the applicable number under which the document was written.)

N/A

10a. ORIGINATOR'S DOCUMENT NUMBER (The official document number by which the document is identified by the originating activity. This number must be unique to this document.)

DRDC Ottawa TM 2008-344

10b. OTHER DOCUMENT NO(s). (Any other numbers which may be assigned this document either by the originator or by the sponsor.)

N/A

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Unlimited Distribution

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Full Unlimited Announcement

13. ABSTRACT (A brief and factual summary of the document. It may also appear elsewhere in the body of the document itself. It is highly desirable that the abstract of classified documents be unclassified. Each paragraph of the abstract shall begin with an indication of the security classification of the information in the paragraph (unless the document itself is unclassified) represented as (S), (C), (R), or (U). It is not necessary to include here abstracts in both official languages unless the text is bilingual.)

This Technical Memo provides an algorithm for geolocating targets detected in RADARSAT-2 imagery and presents results from initial testing. The method iteratively solves a system ofgeometrically based equations that is adapted to include radar squint and a digital elevation model to improve accuracy. Analysis of data collected over Ottawa shows that the geolocatedtargets differ, on average, by 21 metres from the Google Earth® reference imagery. Theimprovement of geolocation accuracy with the use of a digital elevation model (DEM) is clearly illustrated using data collected in mountainous terrain near Vancouver. The results obtained arein accordance with other studies on the accuracy of RADATSAT-1 and RADARSAT-2 geolocation and are of the same order as the inherent error in Google Earth® imagery.

14. KEYWORDS, DESCRIPTORS or IDENTIFIERS (Technically meaningful terms or short phrases that characterize a document and could be helpful in cataloguing the document. They should be selected so that no security classification is required. Identifiers, such as equipment model designation, trade name, military project code name, geographic location may also be included. If possible keywords should be selected from a published thesaurus, e.g. Thesaurus of Engineering and Scientific Terms (TEST) and that thesaurus identified. If it is not possible to select indexing terms which are Unclassified, the classification of each should be indicated as with the title.)

Geo-location, Geo-referencing, Geolocation, RADARSAT-2, SAR, DEM

Documents

Initial tests of point target geolocation for RADARSAT-2 ... · Additionally, higher resolution DEMs exists for most of North America. Future plans: The accuracy of the procedure