rational function modeling for spaceborne sar datasets

ISPRS Journal of Photogrammetry and Remote Sensing 66 (2011) 133–145

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ISPRS Journal of Photogrammetry and Remote Sensing

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Rational function modeling for spaceborne SAR datasetsLu Zhang ∗, Xueyan He, Timo Balz, Xiaohong Wei, Mingsheng LiaoState Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan University, 129 Luoyu Road, Wuhan 430079, PR China

a r t i c l e i n f o

Article history:Received 1 April 2010Received in revised form27 July 2010Accepted 12 October 2010Available online 10 November 2010

Keywords:Rational function modelSARRange–Doppler model

a b s t r a c t

As a kind of generic sensor model, the rational function model (RFM) has been widely used in geometricprocessing of optical images, but has not yet been applied to SAR datasets. In this article the feasibilityand methodology of rational function (RF) modeling for SAR datasets are investigated. After a review ofthe mathematic formulation of the RF model and the Range–Doppler model for SAR systems, the feasibil-ity of applying RFM to SAR datasets is analyzed. Afterwards a two-stage approach is proposed as the keytechnique for SAR RFmodeling to solve unknownparameters of RFM in a fast and unbiasedway. The effec-tiveness and advantages of this approach are demonstrated by comparisonswith traditionalmethods. Ex-perimental results obtained for various spaceborne SAR datasets of different processing levels show thatRFM is a suitable replacement of the rigorous Range–Doppler model for spaceborne SAR images. Further-more, the impacts of several factors including the control point grid size, the number of elevation layers,and the orbit precision on SAR RFM solutions are evaluated quantitatively. The results show that the num-ber of elevation layers is a key factor in SAR RF modeling, and its value should be set carefully accordingto terrain conditions of study areas. Finally, potential applications of SAR RFM are discussed in brief.

© 2010 International Society for Photogrammetry and Remote Sensing, Inc. (ISPRS). Published byElsevier B.V. All rights reserved.

1. Introduction

Sensor models are essential components of digital photogram-metric and remote sensing systems as they present the functionalrelationships between the 2D image space and 3D object spacewhich is crucial for the geometric processing of remotely sensedimagery (Tao and Hu, 2001a). In general sensor models can be cat-egorized into two broad types: physical and generic sensor models(Toutin, 2004). Both types have their own advantages as well asdisadvantages.

Physical sensor models are rigorous in that they can exactly de-scribe the physical imaging procedures. Therefore high geometricmodeling accuracy can be achieved by physical sensor models. Pa-rameters in a physical model are normally uncorrelated becauseeach parameter has a unique physical meaning that is closely re-lated with the position and orientation of a sensor with respect toan object–space coordinate system (Tao and Hu, 2001a). However,different sensors usually have different forms of physical modelscorresponding to different imaging processes, which increases thedifficulty of developing geometric processing software that is ca-pable of handling multi-source remote sensing data. On the otherhand, physical model parameters are usually difficult to be de-termined with high accuracy due to intrinsic model complexity.

∗ Corresponding author.E-mail address: [email protected] (L. Zhang).

0924-2716/$ – see front matter© 2010 International Society for Photogrammetry anddoi:10.1016/j.isprsjprs.2010.10.007

Recently a simplified rigorous sensor model has been proposedfor satellite pushbroom sensors (Weser et al., 2008). However, thissimplified model is still complicated with a number of parametersandonly applicable to pushbroomsensors. Furthermore, somedatavendors are unwilling to disclose their physical sensor models tothe public for the sake of system security and commercial inter-est. Consequently, it is very appealing to develop a generic sen-sor model as a replacement of physical sensor models to describevarious remote sensing systems in a unified framework. There arefour different generic sensormodels defined in the OGC discussionpaper (OGC, 2004), namely, the polynomial model, the grid inter-polation model (GIM), the rational function model (RFM) and theuniversal image geometry model (UIGM). A generic sensor modelcan only be used if it can fit the physical model accurately enough.It has been revealed that the polynomial and GIM models usuallycould not produce adequate accuracy,which largely limits their ap-plication. The RFM uses the ratio of polynomials to delineate thetransformation between the image and the object spaces, and it hasbeen proved to be an ideal replacement of physical sensor models(Habib et al., 2007). The UIGM is in fact an extension to the RFM.Therefore, the RFM has become the most popular generic sensormodel and will be studied in this article.

The RF model has been intensively studied in the past decade,and has been successfully applied to various geometric processingof high-resolution optical imagery. For example, Tao and Hu(2001a) made a comprehensive study of the technical issuesfor RFM from a photogrammetric point of view. Habib et al.(2007) compared RFM with other generic sensor models as wellas physical models in aspect of geometric accuracy using real

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134 L. Zhang et al. / ISPRS Journal of Photogrammetry and Remote Sensing 66 (2011) 133–145

Abbreviations

BILS Batch Iterative Least-SquaresCKP Check PointCNP Control PointCUDA Compute Unified Device ArchitectureDEOS Department of Earth Observation and Space

SystemsDLR German Aerospace CenterDORIS Doppler Orbitography and Radio-positioning

Integrated by SatelliteECR Earth Centered RotationalESA European Space AgencyGCP Ground Control PointGCV Generalized Cross-ValidationGEC Geocoded Ellipsoid CorrectedGIM Grid Interpolation ModelGPS Global Positioning SystemICCV Iteration by Correcting Characteristic ValueIDKF Incremental Discrete Kalman FilteringMAE Maximum Absolute ErrorMGD Multilook Ground-range DetectedMSE Mean Square ErrorOGC Open Geospatial ConsortiumPRC Precise Orbit ProductPRF Pulse Repetition FrequencyRD Range–DopplerRF Rational FunctionRFM Rational Function ModelRMSE Root Mean Square ErrorRPC Rational Polynomial CoefficientRSR Range Sampling RateSAR Synthetic Aperture RadarSCS Single-look Complex SlantSLC Single Look ComplexSSC Single-look Slant-range ComplexSTDE Standard Deviation of ErrorUIGM Universal Image Geometry Model

satellite and aerial datasets. Another two studies were conductedto evaluate the geometric performance of RFM as a replacementof the rigorous sensor model when applied to the FORMOSAT-2and GeoEye-1 satellite datasets, respectively (Chen et al., 2006;Fraser and Ravanbakhsh, 2009). Different methods are proposedto solve the key problem of optimum determination of RFmodel parameters (Samadzadegan et al., 2005; Puatanachokchaiand Mikhail, 2008). Considering the limited accuracy of vendor-provided RFMparameters, a fewmethodswere developed to refinethese existing RF models through the usage of ground controlpoints (GCP) (Hu and Tao, 2002; Di et al., 2003; Fraser and Hanley,2003; Aguila et al., 2007; Xiong and Zhang, 2009; Tong et al.,2010). A variety of practical applications of RFM in geometricprocessing of high-resolution optical imagery were explored indepth, including sensor orientation (Fraser and Hanley, 2005;Fraser et al., 2006), block adjustment (Grodecki and Dial, 2003),geopositioning (Li et al., 2007), image rectification (Tao and Hu,2001b), and 3D reconstruction (Tao and Hu, 2002; Tao et al., 2004;Toutin, 2006).

However, it should be noted that up to now all these studiesare focused on handling optical imagery with RF models, andthere has been no systematic investigation on applying the RFmodel to synthetic aperture radar (SAR) datasets. Owing to itsunique capability of all-time and all-weather data acquisition,SAR remote sensing has been widely recognized as an important

Earth observation technology, and there have been vast successfulapplications of SAR remote sensing in areas such as Earth surfacemapping, environment and disaster monitoring, etc. Because ofdifferent imagingmechanisms and characteristics, SAR and opticaldatasets are generally considered to be complementary to eachother and can be synergistically utilized in many applications. Insuch cases, highly efficient geometric processing of remote sensingdatasets is urgently required. The RFM, as an established genericsensor model, may provide a perfect solution for this problem.

Aiming at this objective, the feasibility and methodology of RFmodeling for SAR datasets are comprehensively studied in thisarticle. First the mathematical formulation of RFM is given, anda rigorous physical model for SAR sensor is introduced and thefeasibility of RF modeling for SAR datasets is analyzed. Then themethodology of building RF models for SAR datasets is elaborated.Because the determination of RF model parameters is the keyproblem, it is described in detail and a new hybrid two-stageapproach is proposed to solve this problem efficiently. In the nextsection, various comparative analyses are carried out, and theexperimental results are presented to show the effectiveness of theproposed approach and the impacts of various factors in terms ofresulting accuracy and computation time cost. Finally, conclusionsare drawn and some promising applications of SAR RF models arediscussed.

2. RFM as a replacement sensor model for spaceborne SAR

2.1. Mathematical formulation of RF model

The commonly used RF model can be mathematically formu-lated as following equations:

r =Nr(P, L,H)

Dr(P, L,H)(1)

c =Nc(P, L,H)

Dc(P, L,H)(2)

where r and c are normalized image coordinates, i.e. row andcolumn indices of image pixels that have been offset and scaled tovary between −1.0 and +1.0. P, L and H represent the normalizedgeographic coordinates of the pixel, namely latitude, longitudeand elevation (height). The basic objective of the coordinates’normalization is to minimize the introduction of errors duringthe computations and to ensure the numerical stability of themodel solutions (Tao andHu, 2001a). The relationship between thenormalized and un-normalized coordinates is as below:

X =Xu − Xoffset

Xscale(3)

where X can be one of r, c, P, L and H . The subscripts u mean un-normalized coordinates. Xoffset and Xscale are the offset and scalefactors respectively.

Nr ,Nc,Dr andDc in Eqs. (1) and (2) are the polynomial functionsof geographic coordinates taken as numerators and denominatorsfor the ratio calculation. In their formulation, the maximum powerof each geographic coordinate is limited to 3, and the total powerof all coordinates is also limited to 3. Therefore, each polynomialtakes a cubic form with 20 terms

p =

m1−i=0

m2−j=0

m3−k=0

atP iLjHk= a0 + a1L + a2P + a3H

+ a4PL + a5LH + a6PH + a7L2 + a8P2+ a9H2

+ a10PLH + a11L3 + a12P2L + a13LH2+ a14PL2

+ a15P3+ a16PH2

+ a17L2H + a18P2H + a19H3, (4)

L. Zhang et al. / ISPRS Journal of Photogrammetry and Remote Sensing 66 (2011) 133–145 135

where p is one of Nr ,Nc,Dr and Dc . at (t = 0, 1, . . . , 19) arecalled the rational polynomial coefficients (RPC). For the twodenominator polynomials, the constant term a0 always equals 1 inorder to ensure the model’s consistency. Consequently, there are78 unknown RPCs to be solved in the general RF model.

The above RF model explicitly defines the ground-to-imagetransformation, i.e. from geographic coordinates to image coordi-nates. Hence it is called the forward RFM (OGC, 2004). In fact, theRF model can also be presented in the inverse form that gives theimage-to-ground transformation (Yang, 2000)

P =NP(r, c,H)

DP(r, c,H)(5)

L =NL(r, c,H)

DL(r, c,H)(6)

where the symbols have the same meaning as those in Eqs. (1)and (2). However, the inverse RFM is seldom used in practicalapplications, which may be due to the fact that the elevationinformation is usually not available in the image space.

Furthermore, the RF model can be considered as a generic formof polynomial functions. Eqs. (1) and (2) become regular third-order polynomialswhen the denominators constantly equal 1. Thishas two implications. One is that the RF model can be appliedto any sensor whose imaging geometry can be approximated bypolynomial functions, and the modeling accuracy of RF model isusually higher than that of the polynomial model. The other isthat the approaches for solving a polynomial model can also beemployed to solve the RF model, which will be described later.

2.2. Physical sensor model for spaceborne SAR

The Range–Doppler (RD) model is the most widely usedphysical sensor model for spaceborne SAR remote sensing systems(Curlander and McDonough, 1991). It provides the basic capabilityto transform between SAR coordinates (azimuth line and rangepixel) in image space and geographic coordinates (latitude,longitude and elevation) in object space, which can be usedto perform advanced geometric processing of SAR data such asgeoreferencing and geocoding. SAR geolocation can be carriedout in two ways, i.e. direct geolocation from SAR coordinates togeographic coordinates and indirect geolocation in the oppositeway. In general indirect geolocation is used more frequently inpractical applications.

The typical imaging geometry of spaceborne SAR systems isillustrated in Fig. 1. The ellipsoid represents the Earth model. Thecharacter O denotes the Earth center, and a Cartesian coordinatesystem XYZ with origin at O is set up. The Z axis points to theNorth Pole, while the X axis points to the intersection of the primemeridian and the equator, and the Y axis is orthogonal to X and Zaxes. The ground target T is observed by the SAR sensor on boardthe satellite S operated along the orbitwhich is usually representedby a series of orbit state vectors with time tags. The positions of Tand S in the Cartesian coordinate system are characterized by twovectors ROT and ROS respectively. The distance between T and S iscalled the slant range R.

In the RD model, the geometric characteristics of SAR imagingprocedure are rigorously described by two equations in azimuthand range (Curlander, 1982). The slant range equation, with ‖ asthe vector modulus operator, is:

R = |ROT − ROS |. (7)

For a given cross-track pixel j in the slant range image, the rangeto the jth pixel is calculated as

R = c · tr = c ·

tr0 +

j2RSR

, (8)

Fig. 1. Spaceborne SAR imaging geometry.

where c represents the light velocity, tr0 and tr are the one-waytimes from the SAR antenna to the first range pixel and the jth pixelrespectively, andRSR is the range sampling rate inHz. Similarly, theazimuth time location for the ith along-track line is computed as:

ta = ta0 +i

PRF(9)

where ta0 and ta are the azimuth time of the first line and the ithline respectively, and PRF is the pulse repetition frequency in Hz.Both the azimuth line number in (9) and the range pixel number in(8) are usually defined as zero-based.

The other equation for the RDmodel is the Doppler equation inazimuth dimension

fD =2λR

(VS − VT ) · (ROS − ROT ), (10)

where fD is the Doppler frequency, λ is the radar wavelength,VS and VT represent velocity vectors of satellite and targetrespectively, and the operator · is the inner product of twovectors. The Doppler frequency for a given pixel can be estimatedwith a reference function provided in the metadata of the SARdata product. Its value is zero for most spaceborne SAR systemsoperated in the so-called zero-Doppler imaging mode. When theEarth Centered Rotational (ECR) coordinate system is used, as isthe case for most satellite SAR data products, the target is alwaysconsidered to be stationary and its velocity is considered to be zero.The position and velocity of the SAR satellite at a given time can bedetermined by interpolation over its ephemeris data records.

For indirect geolocation of SAR data, the solution can beobtained by simultaneously solving Eqs. (7) and (10) in an iterativeway to determine the azimuth and range location of the target pixel(Hanssen and Kampes, 2008). However, for the direct geolocation,the third equation, called the Earth model equation, must beincluded together with the slant range and Doppler equationsto jointly determine the Cartesian coordinates (X, Y , Z) of thetarget point, fromwhich the geographic coordinates can be derived(Curlander, 1982). The Earth model equation assumes that targetsare always located on the Earth surface that can be modeled withan oblate ellipsoid as follows:

X2+ Y 2

(Re + h)2+

Z2

R2p

= 1 (11)


where Re is the equatorial radius of the Earth, h is the local targetelevation relative to the assumed model, and the polar radius Rp isgiven by

Rp = (1 − f )(Re + h), (12)

where f is the flattening factor of the ellipsoid.The equations above are primarily used to perform geolocation

for single look complex (SLC) products. The procedure of indirectgeolocation can be sketched as the following transformation chain:

(P, L,H) → (X, Y , Z) → (ta, tr) → (rSLC, cSLC). (13)

This means that for a given target with known geographiclocation, its geographic coordinates are first converted intoCartesian coordinates, then its corresponding azimuth and rangetime locations are determined by Eqs. (7) and (10), finally theline and pixel numbers for the target in the SLC SAR imagery arecalculated by (8) and (9).

The RDmodel can also be applied to SAR data products of otherprocessing levels, including multilook ground-range detected(MGD) and geocoded ellipsoid corrected (GEC) as defined by DLR(2006). However, for the MGD product, a projection from slantrange to ground range and amultilook processing in azimuth has tobe calculated. Consequently, the transformation chain of the MGDproduct is

(P, L,H) → (X, Y , Z) → (ta, RS) → (ta, RG) → (rMGD, cMGD), (14)

where RS and RG are the slant range and ground range respec-tively. Usually RG is computed as a simple polynomial of RS which isprovided within the metadata. The line number rMGD is propor-tional to the azimuth time interval ta − ta0, while the pixel numbercMGD is calculated by dividing RG by the column spacing of theMGDproduct.

The GEC product is actually a result of geocoding applied tothe SLC SAR data without terrain correction. During the geocodingprocedure, the terrain elevation is fixed to 0 instead of the realelevation height. Therefore, the transformation chain for the GECproduct is

(P, L,H) → (X, Y , Z) → (ta, tr) → (rSLC, cSLC)h=0−→

(X∗, Y ∗, Z∗) → (P∗, L∗)map projection

−→ (rGEC, cGEC),(15)

where the superscript * means the Cartesian or geographiccoordinates when the terrain elevation is assumed to be 0. Theline number rGEC and pixel number cGEC are determined by themap projection used for the GEC product which is specified in themetadata.

The accuracy of SAR geolocation depends strongly on theaccuracy of sensor position and velocity vectors, the measurementaccuracy of the pulse delay time, and the knowledge of thetarget height with reference to the assumed Earth model. Therelationship between geolocation accuracy and these factorshas been quantitatively analyzed by Curlander and McDonough(1991). It is noteworthy that the sensor’s attitude information isnot required during SAR geolocation. As a result, the location of theSAR pixel is inherently more accurate than that of optical sensorsfor which the sensor’s attitude information is a critical parameterduring geometric processing. This implies that the RFmodel shouldeven be more suitable for spaceborne SAR than for optical sensors.

2.3. Applicability of RFM to spaceborne SAR systems

Can spaceborne SAR data be geometrically processed withthe RF model like optical datasets? This question is difficult toanswer through theoretical deduction, but it can be explained by

the following comparison and analysis of imaging geometries foroptical and SAR sensors.

As mentioned above, the RF model has been applied success-fully to high-resolution optical satellite data.Most of these data arecollected by pushbroom sensors such as Ikonos, QuickBird, Geo-Eye,WorldView, EROS, etc. A pushbroom sensor consists of a lineararray of elementary detector cells arranged perpendicular to theflight direction of the spacecraft (Gupta and Hartley, 1997). A 2Dimagery is composed through a line-by-line scanning procedurewith a fixed time interval between adjacent lines along the satel-lite’s orbit. All pixels on the same scan line are simultaneously col-lected by the detector cell array subject to the perspective projec-tion. Each scan line corresponds to a projection center that variesfrom one scan line to another. Different areas of the Earth surfaceare imaged as the satellite flies forward.

The image acquisition principle of spaceborne SAR sensors issomewhat similar to that of pushbroom sensors. At each azimuthlocation an electromagnetic pulse is generatedwith a constant PRFand sent by the SAR antenna to the Earth surface, and then reflectedby ground resolution cells, and finally received as backscatteringsignals by the antenna. All the ground resolution cells illuminatedat the same azimuth time form a range line in the SAR image.During this procedure the slant range projection, instead of theperspective projection, is adopted, i.e. the pixel location of eachground resolution cell in a range line is determined by its distanceto the SAR antenna. This is a major difference between theimaging geometries of SAR and pushbroom optical sensors. Asthe SAR satellite moves in the azimuth direction, the ensembleof these range lines constitutes 2D imagery, which is comparableto pushbroom optical sensors. Therefore, spaceborne SAR can beconsidered as a generalized pushbroom sensor, and consequentlyits imaging geometry can be reasonably approximated by the RFmodel.

3. Methodology of RF modeling for SAR datasets

3.1. General workflow of developing RF model

The methodology of developing RF models for spaceborne SARdatasets is similar to that for optical remote sensing datasets.The basic principle is to determine RPCs, i.e. the unknown modelparameters, by fitting a 3D object grid to its corresponding imagegrid via techniques such as least-squares adjustment (Tao andHu, 2001a). The general workflow of developing the RF model isillustrated in Fig. 2. The procedure consists of three processingsteps, namely, setting up a control point (CNP) grid, RFM solutionand RFM refinement. The step of RFM refinement is optionaldepending on the availability of additional ground control points(GCPs).

3.1.1. Setting up the CNP gridThe first step is to set up the CNP grid in order to establish

the normal equations for the RF model. This can be accomplishedby employing the terrain-independent approach or the terrain-dependent approach, which depends upon the availability ofphysical sensor models.

When the physical sensor model is available, the terrain-independent approach can be applied, in which the control pointsare first distributed evenly on several elevation layers (horizontalplanes of different elevations) across the 3D object space coveredby the image. Afterwards their corresponding coordinates in the2D image space are determined by the physical sensor model. Inthis approach, the control points are collected in a virtual way andno terrain elevation information, except the range of elevation, isrequired. For civil spaceborne SAR systems, the physical sensormodels are always provided with their data products. Hence theterrain-independent approach can be used for spaceborne SARdatasets and will be employed in this study.


Fig. 2. General workflow of developing the RF model (Hu et al., 2004).

Contrarily, only the terrain-dependent approach canbe adoptedwhen the physical sensor model is not available. In such case,sufficient GCPs evenly distributed on the observed terrain surfacemust be utilized to act as the CNP grid. These GCPs are usuallycollected in conventional ways, i.e. measured on topographicmapsor by GPS, and unambiguously marked on the remotely sensedimagery. The minimum number of GCPs required depends onthe number of unknown RPCs which varies with different modelconfigurations (Tao and Hu, 2001b). For the general RF model ofthird order polynomials and unequal denominators, there are 78unknown RPCs, and accordingly at least 39 GCPs are needed.

3.1.2. RFM solutionThe second step is to solve the RF model by solving the normal

equation to determine the unknown RPCs. Assuming there are npoints in the CNP grid, in other words, n observations denoted as⟨ri, ci, Pi, Li,Hi⟩ (i = 1, 2, . . . , n) are to be fitted, then the normalequation can be formulated in matrix style:

∆ = WAX − Wl, (16)

where∆ is the vector of observation errors to beminimized towardzero. X is the column vector of the 78 RPCs to be determined

X =a0 a1 · · · a19 b1 b2 · · · b19 c0 c1 · · · c19 d1 d2 · · · d19

T(17)

where ai, bj, ci and dj (i = 0, 1, . . . , 19, j = 1, 2, . . . , 19) arethe coefficients of the numerator and denominator polynomials inEqs. (1) and (2) respectively, and the superscript T stands for thematrix transpose. Please note that the values of b0 and d0 are fixedas 1, as mentioned before.

A is the design matrix of size 2n by 78 (see Eq. (18) in Box I), l isthe observation column vector of size 2n

l =r1 r2 · · · rn c1 c2 · · · cn

T (19)

and W is a diagonal matrix of the weights for observations

W = diag[

1B21

1B22

· · ·1B2n

1D21

1D22

· · ·1D2n

](20)

where Bi and Di (i = 1, 2, . . . , n) are values of the denominatorpolynomials in Eqs. (1) and (2) for the ith observation respectively.

According to the classical ordinary least-squares estimationmethod, the solution to the normal equation is

X̂ =ATWA

−1ATWl (21)

where the superscript −1 means matrix inversion. It is worthnoting that in the above formula calculation of the weight matrixW is actually dependent on the solution vector X . Therefore, thesolution as (21) can only be obtained in an iterative way untilconvergence. The iteration is usually initiated by setting theweightmatrix W as an identity matrix, i.e. assigning equal weights to allobservations. The result of the first iteration in such case is namedas the direct solution as following

X̂ =ATA

−1AT l. (22)

In practical applications the direct solution can be used asthe RFM solution provided that it can achieve satisfactory fittingaccuracy which is often set as 0.05 pixels in the image space(Dolloff, 2004). There are two major benefits for doing so. First,the computational burden can be reduced significantly comparedwith that for the iterative weighted solution. Second, the problemof division by zero in the calculation of the weight matrix can beavoided.

However, it has been found that the normalmatrix ATWA or ATAis often ill-conditioned when the high order (i.e. more than secondorder) polynomials are used, which is known as the problem ofover-parameterization of the RFM (Dolloff, 2004). Consequentlythe solution of Eq. (21) or (22) turns out to be an ill-posed problem(Tao and Hu, 2001a). As a result, the stability of above direct oriterative solution is very low. Thus the RFM developed in this waycould hardly be used as a reliable replacement of the physicalsensormodel. A conventional approach to resolve such an ill-posedproblem is to utilize the regularization techniques, which will bedescribed later in detail.

To quantitatively evaluate the fitting accuracy of the developedRF model, a set of check points (CKP) must be exploited tocalculate the error statistics. These check points are collected inthe same way as those control points used to solve RPCs, and it isessential that they are independent from those control points. Thecommonly used error statistics include maximum absolute error(MAE), root mean square error (RMSE) and standard deviation oferror (STDE).

3.1.3. RFM refinementThe third processing step is to refine the original RFM by

incorporating additional GCPs into the model to improve thegeometric modeling accuracy. It is required that the GCPs used forRFM refinement must be collected independently from those GCPsused to solve the original RFM.

The RFM can be refined by employing direct or indirect meth-ods. The direct methods including batch iterative least-squares


8)
A =
1 L1 · · · H31 −r1L1 · · · −r1H3

1

1 L2 · · · H32 −r2L2 · · · −r2H3

2...

.... . .

......

. . ....

1 Ln · · · H3n −rnLn · · · −rnH3

n

0

0

1 L1 · · · H31 −c1L1 · · · −c1H3

1

1 L2 · · · H32 −c2L2 · · · −c2H3

2...

.... . .

......

. . ....

1 Ln · · · H3n −cnLn · · · −cnH3

n

(1

Box I.

(BILS) and incremental discrete Kalman filtering (IDKF) directlyupdate the original RPCs themselves (Hu and Tao, 2002). By con-trast the indirectmethods keep the original RPCs unchanged,whilecomplementary transformations such as affine transformation orsimple translation are introduced in image or object space to com-pensate the residual linear systematic errors existing in the origi-nal RFM (Hu et al., 2004; Fraser and Hanley, 2005). However, sucha step could not be performed in this study because no GCP isavailable.

3.2. Methods for solving ill-posed problems

In general RPCs in an RF model do not carry physical meaningas those in a physical sensor model, and they are often highlycorrelated with each other. As a result, the estimation of RPCsusually becomes a problem of solving ill-posed equations.

Solving ill-posed equations has long been a difficult problem inthe field of surveying data processing. Numerous methods havebeen developed to solve this problem. Most of these methodsutilize the regularization techniques. Among these methods theridge estimate has beenmost widely used in practical applications.Another promising method is the one named as iteration bycorrecting characteristic value (ICCV).

3.2.1. The ridge estimate methodThe ridge estimate method is developed based on the Tikhonov

regularization technique (Tikhonov and Arsenin, 1977). It can beformulated as

X̂ =ATWA + αI

−1ATWl, (23)

whereα is the ridge parameter that is usually set as a small positivequantity, I is an identitymatrix, and the other symbols are the sameas those in (21).

The ridge estimate method tries to slightly modify diagonal el-ements of the normal matrix to significantly reduce its conditionalnumber, so as to convert the ill-posed equation into amuch less ill-posed or even well-posed equation. As a result, the stability of thesolution can be improved remarkably, and with a properly chosenridge parameter, the result of ridge estimate usually has a smallermean square error (MSE) than that of ordinary least-squaresestimate.

It is self-evident that the result of ridge estimate dependsstrongly upon the value of the ridge parameter. Therefore, theoptimal determination of the ridge parameter becomes the keyproblem to successful applications of the ridge estimate method.A few methods have been proposed to resolve this problem,including ridge trace, GCV and L-curve.Ridge trace method

The ridge trace method first computes solutions for a groupof different α values. Then each solution component is plottedversus α value to form the so-called ridge trace. All the ridge traces

are jointly visualized and analyzed to determine the best α valueheuristically. The optimal ridge parameter should be located at thevalue of α from which all the solution components are roughlystabilized. This method is very simple and has been widely used.However, the determination of the optimal ridge parameter islargely subjective. Hence big errors may be introduced into theresult.GCV method

According to the GCV (generalized cross-validation) method,the optimal ridge parameter should be set as the value of α thatminimizes the GCV function as follows (Golub et al., 1979)

GCV(α) =

1n

I − A

ATWA + αI

−1 ATWl2

1n tr

I − A

ATWA + αI

−1 ATW2 , (24)

where n is the number of observations, tr represents the trace ofa square matrix, ‖ ‖ is the L2 norm in Euclidean space, and theother symbols are the same as those in (23). A major drawback ofthe GCV method is that sometimes the GCV function varies slowlywithα value or even does not converge. In such cases theminimumlocation of GCV function is very difficult or even not available,and consequently this method fails to determine the optimal ridgeparameter.L-curve method

The L-curve is a log–log plot of the norm of a regularizedsolution, i.e.

X̂, versus the norm of the corresponding residual

(fitting error), i.e.AX̂ − l

, as the ridge parameter α varies(Hansen, 1992). The L-curve is obtained by curve fitting for the setof 2D points

(ρ(α), η(α)) =

lg

AX̂ − l

α, lg

X̂α

, (25)

where lg represents logarithm, ρ and η are functions of α. Thecurve is L-shaped as it is approximately vertical for small α andhorizontal for large α. The best ridge parameter is determined bylocating the corner on the curve. Therefore, the key step to the L-curve method is to accurately find out the point with the biggestcurvature on the curve. The curvature κ is calculated as (Hansenand O’Leary, 1993)

κ = 2ρ ′η′′

− ρ ′′η′

3

(ρ ′)2 + (η′)22 , (26)

whereρ ′, ρ ′′, η′ andη′′ are the first and second derivatives ofρ andη on α, respectively. The value of α that maximizes the curvatureis chosen as the best ridge parameter.

The L-curve criterion combines information about the residualnorm with information about the solution norm, and achieves abalance between them through the use of the ridge parameter


(Wang and Ou, 2004). It has been found that the L-curve methodoutperforms the GCV and ridge trace in most cases, particularlywhen the noise level in observations is high (Choi et al., 2007;Yuan and Lin, 2008). However, it should be noted that the bestridge parameter determined by the L-curve method is not exactlythe real optimal ridge parameter, but a near optimal one (Hansen,1992).

3.2.2. The ICCV methodThe ridge estimate method employs the regularization tech-

nique to reduce the ill-conditioning degree of the normal matrixby imposing constraints to its diagonal elements, so as to signif-icantly improve the stability of solutions. Nevertheless, solutionsobtained by the regularization technique are intrinsically biasedbecause in the ridge estimate the normal equation is changed sub-stantially. Therefore, a new method named ICCV (iteration by cor-recting characteristic value) is developed upon the original normalequation to produce unbiased results (Wang et al., 2001;Wang andLiu, 2002).

The ICCV method can be deduced by adding the unknownsolution item X̂ to both sides of the normal equation, i.e.ATWA + I

X̂ = ATWl + X̂ . (27)

This equation can only be solved in an iterative way as below

X̂ (k)=

ATWA + I

−1ATWl + X̂ (k−1)

(28)

where the superscript (k) represents the solution obtained by thekth iteration.

Let q = (ATWA+ I)−1, then the above solution can be rewrittenas

X̂ (k)=

q + q2 + · · · + qk

ATWl + qkX̂ (0) (29)

where the superscript (0) means the initial value of the solution.Eqs. (28) and (29) are expressions of the iteration by correcting

the characteristic value. The convergent and unbiased propertiesof the ICCV method have been theoretically proved by Wang et al.(2001). And the effectiveness of this method in RF modeling forSPOT-5 and QuickBird imagery has been demonstrated by Lin andYuan (2008).

3.2.3. A hybrid two-stage approachUnbiased solutions can be obtained by the ICCV method in

an iterative way. Nevertheless, the speed of convergence for thismethod depends primarily on the initial value of solutions. Whenthe initial value is very close to the true solution, the solution mayconverge quickly after a few iterations. Otherwise, the iterationswill cost a large amount of computation time or in an even worsecase the iterations could not converge at all.

An intuitive strategy is to use the direct solution of ordinaryleast-square estimate shown in (17) as the initial value. But whenthe normal matrix is ill-conditioned, the least-squares solution isso bad that the iterations may be divergent. Considering the factthat RPCs for third-order RFM are usually close to zero, a practicalway to initialize the solution is to set the initial RPC values to zero(Lin and Yuan, 2008). However, the computation may take a longtime as the number of iterations required to converge is usuallyquite large.

In order to overcome the aforementioned shortcomings of boththe ridge and the ICCV estimate, we propose to adopt a hybrid two-stage approach to solve the ill-posed equations. According to thisapproach, the ridge estimate with the ridge parameter determinedby the L-curve method is employed in the first stage to produce anear optimal solution, and then in the second stage this solutionis used as the initial value for the ICCV method to obtain the finalsolution. In this way, the merits of both methods can be combined

to ensure an unbiased final solution with high accuracy and lowcost in computation time. The effectiveness and high performanceof this approach will be demonstrated by experiments later.

4. Experimental results and evaluation

4.1. Experiment design

In order to perform a comprehensive evaluation of RFmodelingfor spaceborne SAR datasets, a few experiments are designed forfollowing purposes:• to evaluate the overall fitting accuracy of the RFM for SAR

datasets at both CNPs and CKPs;• to compare RFM solutions derived by using different methods

in terms of fitting accuracy and computation time;• to assess the impact of the CNPgrid size onRFM fitting accuracy;• to assess the impact of the number of elevation layers on RFM

fitting accuracy;• to evaluate the applicability of RFM for datasets acquired by

various spaceborne SAR sensors in different imagingmodes andprocessing levels;

• to evaluate the sensitivity of RFM to the orbit precision.

It has been revealed that the highest fitting accuracy can beachieved with the third-order RF model with unequal denomina-tors when compared with other configurations of RF model (Taoand Hu, 2001a,b). Therefore in this study only the third-order RFmodel is tested.

Two error statistics, namely MAE and RSME, are used in thisstudy to characterize the fitting accuracy of RFM solutions. Theywill be calculated in azimuth line and range pixel dimensions ofSAR imagery. Planimetric error statistics will also be computed toevaluate the overall fitting accuracy.

4.2. Test datasets

In this study 15 SAR datasets are tested with RF modeling.These datasets are acquired by different spaceborne SAR sensorsincluding ERS-1/2, ENVISAT ASAR, ALOS PALSAR, TerraSAR-X,COSMO-SkyMed and RADARSAT-2. Test areas covered by thesedatasets range from flat plains to mountainous areas, located inChina and Canada (see Fig. 3). The rectangles in Fig. 3 outline thecoverage of these datasets.

Basic information of these test datasets, such as acquisitionmode and terrain condition, are summarized in Table 1. It is worthnoting that the processing levels including L1.1 for ALOS PALSAR,SSC for TerraSAR-X and SCS for COSMO-SkyMed are in fact thesame as SLC. For convenience, we indexed these datasets withletters from A to O, also shown in Fig. 3. The ranges of elevation aredetermined from the GTOPO30 DEM of 30 arc second resolution(Gesch and Larson, 1996).

4.3. Experiments and results

As stated before, in all the experiments for this study theterrain-independent approach is employed to establish the CNPgrid for solving RFM and the CKP grid for fitting accuracyassessment. Both CNP and CKP grids are regularly distributed inthe 3D object space occupied by the SAR imagery. The CKP grid isorganized in followingway tomaximize its independence from theCNP grid: in horizontal dimension each CKP grid node is located atthe center of the rectangle formed by four neighboring CNP gridnodes, and in vertical dimension the elevation of each CKP gridnode is set as the mean value of the two adjacent elevation layers.To ensure adequate number of observations, the minimum sizeof CNP grid is defined as 10 × 10 (10 rows by 10 columns in ahorizontal plane) with at least 4 elevation layers.


a b

c d

Fig. 3. Geographic location and coverage of test datasets. (a) Three Gorges, China, (b) Shanghai, China, (c) Snow Mount Meili, China, (d) Toronto and Waterloo, Canada.

Table 1Basic information of test datasets.

SAR sensor Data index Observed area Acquisition date Imaging mode Processing level Image size(line by pixel)

Elevation range (m)

ERS-2 A Three Gorges, China 1996.01.21 IM SLC 26 588 × 4900 39–2962ERS-1 B Shanghai, China 1996.06.03 IM SLC 25 651 × 4900 0–28

ENVISAT ASAR C Three Gorges, China 2003.08.17 IM SLC 26 893 × 5187 39–2962D Shanghai, China 2008.03.17 IM SLC 26 905 × 5166 0–28

ALOS PALSAR E Three Gorges, China 2007.11.14 FBS L1.1 18 432 × 9344 43–2962

TerraSAR-X

F Three Gorges, China 2008.08.12 SM SSC 28 887 × 17 136 56–2565G Shanghai, China 2008.04.21 SM SSC 28 857 × 15 584 0–15

HToronto, Canada 2007.12.15 SL

SSC 8363 × 950475–190I MGD 14 016 × 15 571

J GEC 15 000 × 15 500

COSMO-SkyMed K Shanghai, China 2008.03.15 HIMAGE SCS 23 136 × 18 427 0–17L Snow Mount Meili, China 2008.04.19 HIMAGE SCS 21 984 × 19 070 1900–6740

RADARSAT-2M

Waterloo, Canada2009.05.28 Standard SLC 19 796 × 6778 67–530

N 2009.04.22 Fine SLC 9818 × 8721 199–473O 2009.04.26 Ultra-fine SLC 10 449 × 8857 262–420

4.3.1. Comparison of different methods for solving RPCsIn the first experiment, five methods for solving RPCs, as

described in Section 3.2, are compared in terms of fitting accuracyand computation time cost. Test dataset F , i.e. TerraSAR-X imageof Three Gorges area, is used as the experimental data. The CNPgrid size is fixed as 50 × 50 with 11 elevation layers. For the ICCVmethod, the initial values of RPCs are set as zero. The results arelisted in Table 2.

A few findings can be obtained by this experiment. First, allthe five methods can produce satisfactory results with overallfitting errors (RMSE) less than 10−3 pixels, which clearly illustratesthe feasibility and effectiveness of using RFM as a replacementsensor model for spaceborne SAR systems. Furthermore, for eachmethod the fitting errors at CKPs are larger than those at CNPs.However, they are still at the same level, which shows the goodgeneralization capability of the established RF models.

Second, among the three ridge estimate methods, the GCVmethod is better than the ridge trace and the L-curve in lineand planimetric fitting accuracy, but worse in pixel dimension.Meanwhile, the GCV method costs more computation time thanthe other two ridge estimation methods. The L-curve method isevidently better than the ridge trace in both fitting accuracy andcomputation time cost.

Thirdly, the unbiased ICCVmethod achieves significantly higherfitting accuracy than the biased ridge estimate methods in almostall dimensions at both CNPs and CKPs. However, its cost incomputation time is much higher than those of the ridge estimatemethods, which constitutes a big disadvantage for practicalapplications of this method.

Finally, the hybrid two-stage approach (L-curve + ICCV)produces much better fitting accuracy than the ridge estimatemethods in all dimensions. Compared with the ICCV method, the


Table 2Results of different methods for solving RPCs.

Methods Computation time (s) RSME (MAE) at CNPs (10−3 pixels) RSME (MAE) at CKPs (10−3 pixels)Line Pixel Planimetric Line Pixel Planimetric

Ridge trace 5.50 0.78 0.07 0.79 0.98 0.09 0.99(2.16) (0.27) (2.16) (2.59) (0.35) (2.59)

GCV 10.00 0.14 0.08 0.16 0.25 0.16 0.30(0.62) (0.46) (0.62) (1.69) (1.11) (2.02)

L-curve 2.37 0.25 0.07 0.26 0.43 0.08 0.44(1.25) (0.28) (1.26) (2.29) (0.35) (2.30)

ICCV 216.04 0.10 0.03 0.11 0.18 0.09 0.20(0.38) (0.26) (0.42) (0.98) (0.86) (1.11)

L-curve +ICCV 2.47 0.10 0.01 0.10 0.18 0.03 0.18(0.38) (0.06) (0.38) (0.98) (0.18) (0.99)

Table 3Results with different sizes of CNP grid for test dataset F .

CNP grid size Computation time (s) RSME (MAE) at CNPs (10−3 pixels) RSME (MAE) at CKPs (10−3 pixels)Line Pixel Planimetric Line Pixel Planimetric

10 × 10 0.10 0.11 0.04 0.12 0.14 0.11 0.18(0.26) (0.15) (0.26) (0.75) (0.77) (1.02)

20 × 20 0.35 0.11 0.02 0.11 0.16 0.07 0.17(0.29) (0.10) (0.29) (0.88) (0.43) (0.94)

30 × 30 0.75 0.10 0.02 0.11 0.17 0.05 0.18(0.33) (0.08) (0.34) (0.94) (0.30) (0.96)

40 × 40 1.36 0.10 0.01 0.10 0.17 0.04 0.18(0.36) (0.07) (0.36) (0.97) (0.23) (0.98)

50 × 50 2.14 0.10 0.01 0.10 0.18 0.03 0.18(0.38) (0.06) (0.38) (0.98) (0.18) (0.99)

60 × 60 3.17 0.10 0.01 0.10 0.18 0.03 0.18(0.39) (0.05) (0.39) (0.99) (0.15) (1.00)

70× 70 4.38 0.10 0.01 0.10 0.18 0.02 0.18(0.40) (0.05) (0.40) (1.00) (0.14) (1.01)

80 × 80 6.03 0.10 0.01 0.10 0.18 0.02 0.18(0.40) (0.04) (0.40) (1.01) (0.13) (1.01)

90 × 90 7.80 0.10 0.01 0.10 0.18 0.02 0.19(0.41) (0.04) (0.41) (1.01) (0.12) (1.02)

100 × 100 9.88 0.10 0.01 0.10 0.19 0.02 0.19(0.41) (0.04) (0.41) (1.02) (0.11) (1.02)

hybrid approach can attain noticeably better accuracy in pixeldimension and identical or even slightly better accuracy in lineand planimetric dimensions. Meanwhile, the computation timecost of this approach is much less than that of the ICCV methodand just slightly larger than that of the L-curve method. Herebywe are convinced that this hybrid approach is the best methodto solve unknown RPCs for RF modeling. Consequently, in thefollowing experiments only this hybrid approach is employed forcomputation.

4.3.2. Evaluation of the impact of CNP grid size on RFM solutionIn this experiment, a group of scenarios are tested with dataset

F to evaluate the impact of CNP grid size (in one elevation layer)on RFM solution. These scenarios share the same settings exceptusing different sizes of CNP grid as from 10×10 to 100×100 withthe number of elevation layers being fixed as 11.

The experimental results are listed in Table 3. It can be seenthat for both CNPs and CKPs the fitting errors in pixel dimensionare gradually reduced along with the increasing size of CNP grid.By comparison, the variations of fitting errors in line dimensionare different. The RSME in line at CNPs is nearly constant, whileall other error statistics in line are increasing. Furthermore, itis obvious that fitting errors in line are much bigger than theircounterparts in pixel. As a result, the planimetric fitting errors aredetermined primarily by disagreements in line.

In Table 3 we can see that the RSME at CNPs can almost beconsidered as constant since it is decreasing very slowly, whilethe MAE at CNPs is increasing towards convergence. At CKPs thevariation of RSME is so small that it can be considered as being not

affected by CNP grid size, whilst the MAE decreases significantlyfrom 10× 10 to 20× 20, then increases gradually from 20× 20 to100 × 100. However, the variation of MAE at CKPs is smaller thanthat at CNPs.

To conclude, the CNP grid size has a small impact on RFMfitting accuracy for spaceborne SAR datasets. Furthermore, a rapidincrease in the computation time cost along with increasing CNPgrid size can be clearly observed in Table 3. Therefore, the CNP gridsize can be usually set as 10×10 or 20×20 for the sake of keepinghigh fitting accuracy and saving computation time cost.

4.3.3. Evaluation of the impact of number of elevation layers on RFMsolution

The objective of this experiment is to assess the impact of thenumber of elevation layers in the 3D CNP grid on RFM solutions.Datasets F and L are tested because of the undulating terrainconditions in these test areas. The CNP grid size in one elevationlayer is fixed as 20 × 20 according to the previous evaluationresults.

For dataset F , the number of elevation layers is varied from 4to 20. The experimental results are given in Table 4. When thenumber of elevation layers increases, the variations of fitting errorsat CNPs are small, while at CKPs the fitting errors change a littlein line but change much in pixel. Fig. 4 shows the variations ofplanimetric RSME and MAE at CKPs along with increasing numberof elevation layers. A rapid decrease in both error statistics canbe observed until 11 layers being applied. Afterwards, the fittingerrors gradually approach convergence. Therefore, for this datasetthe number of elevation layers must be set to at least 11 to ensure


Table 4Results with different numbers of elevation layers for test dataset F .

Number of elevation layers Computation time (s) RSME (MAE) at CNPs (10−3 pixels) RSME (MAE) at CKPs (10−3 pixels)Line Pixel Planimetric Line Pixel Planimetric

4 0.15 0.11 0.04 0.12 0.23 8.20 8.20(0.30) (0.19) (0.32) (1.50) (38.60) (38.60)

5 0.16 0.11 0.04 0.11 0.16 3.10 3.10(0.30) (0.17) (0.32) (0.89) (15.50) (15.50)

6 0.21 0.11 0.03 0.11 0.16 1.40 1.40(0.30) (0.15) (0.32) (0.90) (7.10) (7.10)

7 0.21 0.11 0.03 0.11 0.16 0.63 0.65(0.29) (0.13) (0.31) (0.91) (3.40) (3.40)

8 0.24 0.11 0.03 0.11 0.16 0.30 0.34(0.30) (0.12) (0.31) (0.90) (1.70) (1.80)

9 0.31 0.11 0.03 0.11 0.16 0.14 0.21(0.30) (0.12) (0.31) (0.89) (0.90) (1.20)

10 0.32 0.11 0.03 0.11 0.16 0.07 0.17(0.30) (0.11) (0.31) (0.89) (0.48) (0.95)

11 0.35 0.11 0.02 0.11 0.16 0.04 0.17(0.30) (0.10) (0.31) (0.89) (0.31) (0.90)

12 0.38 0.11 0.02 0.11 0.16 0.04 0.16(0.30) (0.10) (0.31) (0.89) (0.29) (0.89)

13 0.44 0.11 0.02 0.11 0.16 0.03 0.16(0.30) (0.09) (0.30) (0.89) (0.29) (0.89)

14 0.48 0.11 0.02 0.11 0.16 0.03 0.16(0.30) (0.10) (0.30) (0.89) (0.29) (0.89)

15 0.48 0.11 0.02 0.11 0.16 0.03 0.16(0.30) (0.10) (0.30) (0.88) (0.29) (0.89)

16 0.54 0.11 0.02 0.11 0.16 0.03 0.16(0.30) (0.10) (0.30) (0.88) (0.29) (0.89)

17 0.56 0.11 0.02 0.11 0.16 0.03 0.16(0.30) (0.10) (0.30) (0.88) (0.29) (0.89)

18 0.59 0.11 0.02 0.11 0.16 0.03 0.16(0.30) (0.10) (0.30) (0.88) (0.29) (0.89)

19 0.63 0.11 0.02 0.11 0.16 0.03 0.16(0.30) (0.11) (0.30) (0.88) (0.30) (0.89)

20 0.65 0.11 0.02 0.11 0.16 0.03 0.16(0.30) (0.11) (0.30) (0.88) (0.30) (0.89)

Fig. 4. Plot of RFM fitting errors versus number of elevation layers for test dataset F .

a high fitting accuracy. The computation time cost with differentnumbers of elevation layers is also given in Table 4. Although itincreases from 4 to 20 layers, the variation is much smaller thanthat for different CNP grid sizes. In other words, the impact of thenumber of elevation layers on computation time cost is limited.

The dataset L is tested with the number of elevation layersranging from 4 to 30. The planimetric fitting errors at both CNPsand CKPs as well as computation time costs are given in Table 5.Fig. 5 shows the plot of planimetric RSME and MAE at CKPsversus the number of elevation layers from 10 to 30. A similarvariation pattern as that for dataset F can be observed. However,theminimumnumber of elevation layers required for convergenceof fitting errors is 25, which is much more than that for dataset F .This should be due to the fact that test area of Snow Mount Meilihas a much larger elevation range than the Three Gorges area.

Fig. 5. Plot of RFM fitting errors versus number of elevation layers for test dataset L.

Above experimental results show that the number of elevationlayers is a key parameter for the RFM solution for spaceborneSAR datasets since it affects fitting errors at CKPs significantly.Furthermore, the minimum number of elevation layers requiredprimarily depends on the terrain conditions of the test areas. Forflat areas, 5–7 layers is enough to attain a high fitting accuracy,while for mountainous areas with large terrain relief, more than10 layers are required to produce reliable solutions.

4.3.4. Applicability evaluation of RFmodeling for different spaceborneSAR sensors

In this experiment, all the datasets listed in Table 1 are tested toevaluate the applicability of RF modeling for different spaceborneSAR sensors. The 3DCNPgrid size is set as 10×10×7 or 10×10×11according to terrain conditions of the test area. For the ERS-1/2


Table 5Results with different numbers of elevation layers for test dataset L.

Number of elevation layers Computation time (s) Planimetric fitting errors atCNPs (10−3 pixels)

Planimetric fitting errors atCKPs (10−3 pixels)

RSME MAE RSME MAE

4 0.16 0.08 0.36 686.70 1848.205 0.16 0.08 0.35 308.80 838.906 0.18 0.08 0.36 165.50 453.707 0.21 0.08 0.35 96.00 265.608 0.25 0.08 0.35 58.90 164.109 0.28 0.08 0.35 37.30 104.80

10 0.31 0.08 0.35 24.20 68.4011 0.34 0.08 0.35 15.90 45.2012 0.38 0.08 0.35 10.50 30.1013 0.42 0.08 0.35 7.00 20.2014 0.45 0.08 0.35 4.70 13.6015 0.49 0.08 0.35 3.10 9.2016 0.54 0.08 0.35 2.10 6.3017 0.55 0.08 0.35 1.40 4.3018 0.59 0.08 0.35 0.91 3.0019 0.63 0.08 0.35 0.61 2.2020 0.66 0.08 0.35 0.42 1.6021 0.72 0.08 0.35 0.29 1.3022 0.74 0.08 0.35 0.22 1.1023 0.78 0.08 0.35 0.18 1.0024 0.81 0.08 0.35 0.16 0.9925 0.84 0.08 0.35 0.15 0.9726 0.88 0.08 0.35 0.14 0.9627 0.90 0.08 0.35 0.14 0.9628 0.9470 0.08 0.35 0.14 0.9629 0.9763 0.08 0.35 0.14 0.9630 1.0425 0.08 0.35 0.14 0.96

Table 6Fitting accuracy of results for different test datasets.

Test dataset 3D CNP grid size(rows × columns × elevation layers)

Planimetric fitting errors atCNPs (10−3 pixels)

Planimetric fitting errorsat CKPs (10−3 pixels)

RSME MAE RSME MAE

A 10 × 10 × 11 1.40 2.60 1.70 10.00B 10 × 10 × 7 1.00 1.80 1.20 6.40C 10 × 10 × 11 6.70 11.70 8.50 42.80D 10 × 10 × 7 5.10 8.80 6.20 32.30E 10 × 10 × 11 0.85 1.90 1.00 6.40G 10 × 10 × 7 0.05 0.18 0.07 0.33H 10 × 10 × 7 0.01 0.02 0.01 0.02I 10 × 10 × 7 0.02 0.04 0.03 0.16J 10 × 10 × 7 0.0015 0.0061 0.0016 0.0047K 10 × 10 × 7 0.05 0.17 0.06 0.36M 10 × 10 × 7 0.30 0.57 0.38 1.93N 10 × 10 × 7 0.06 0.13 0.06 0.15O 10 × 10 × 7 0.09 0.24 0.11 0.67

and ENVISAT ASAR datasets, only the original orbit records areused. The experimental results are shown in Table 6. Results forthe datasets F and L are excluded here since they have been givenin Tables 4 and 5 respectively.

First of all, the planimetric fitting errors for all the datasetsare within 0.05 pixels. This effectively proves the applicabilityof RF modeling for various spaceborne SAR sensors. Second, itis noteworthy that the fitting accuracies for the datasets from Gto O are much higher than those for datasets A, B, C and D. Thismay be due to the fact that the new generation of SAR sensor(including TerraSAR-X, COSMO-SkyMed and RADARSAT-2) usuallyhas a higher geometric accuracy than the old sensors such asERS-1/2 and ENVISAT ASAR. Third, it can be seen that for thesame scene of TerraSAR-X data acquisition, the three products ofdifferent processing levels (namely SSC, MGD and GEC) producedissimilar fitting accuracies. The highest accuracy is achieved by RFmodeling for the GEC product, followed by the SSC product, whilefor the MGD product the fitting accuracy is lowest. However, more

experiments and further investigations are needed to validate ifthis is a regular pattern for all cases with different SAR sensors andvarious terrain conditions.

4.3.5. Evaluation of the impact of orbit precision on RFM solutionAs shown in Table 6, the fitting accuracies for ERS-1/2 and

ENVISAT ASAR datasets are relatively lower compared with othertest datasets. One possible way to improve the accuracy is to usethe external precise orbit data in the RDmodel during RFmodeling.

There are two kinds of precise orbit data available for ERS-1/2and ENVISAT ASAR respectively, i.e. DELFT and PRC orbits for ERS-1/2, DELFT and DORIS orbits for ENVISAT ASAR. The DELFT orbitdata is freely provided by the Department of Earth Observationand Space Systems (DEOS) at the Delft University of Technology(Scharroo et al., 1998; Doornbos et al., 2002). The PRC and DORISorbit data are released by the European Space Agency (ESA)(Zandbergen et al., 1995, 2003). In this experiment the DELFT orbitdata is used in comparison with original orbital records delivered


Table 7Fitting accuracy of results for test datasets with/without precise orbit.

Test dataset 3D CNP grid size(rows×columns×elevation layers)

Planimetric fitting errors without precise orbit(10−3 pixels)

Planimetric fitting errors with precise orbit(10−3 pixels)

RSME (MAE) at CNPs RSME (MAE) at CKPs RSME (MAE) at CNPs RSME (MAE) at CKPs

A 10 × 10 × 11 1.40 1.70 2.20 2.40(2.60) (10.00) (9.00) (10.00)

B 10 × 10 × 7 1.00 1.20 0.52 0.52(1.80) (6.40) (1.90) (2.20)

C 10 × 10 × 11 6.70 8.50 0.01 0.02(11.70) (42.80) (0.05) (0.16)

D 10 × 10 × 7 5.10 6.20 0.12 0.16(8.80) (32.30) (0.37) (0.84)

with the SAR datasets to evaluate the impact of orbit precision onRFM solution. The results are given in Table 7.

As expected, dramatic improvements in fitting accuracy for thetwo ENVISAT ASAR datasets are obtained by using the preciseorbit data. Therefore, the orbit precision is a key factor in solvingRFM for ENVISAT ASAR datasets. Meanwhile, with the DELFT orbitsapplied there is only a limited improvement for the ERS-1 datasetand even an unexpected degradation in fitting accuracy for theERS-2 dataset.Why this occursmay be attributed to the differencesin orbit determination procedures for ERS-1/2 and ENVISATsatellites. Generally the DELFT orbit data for ENVISAT ASAR ismoreaccurate than that for ERS-1/2. Hence, using precise orbits maynot help RF modeling for ERS-1/2 datasets as much as for ENVISATASAR datasets.

5. Potential applications of SAR RFM

Two major advantages of RFM over rigorous physical sensormodels are its uniformity across multiple sensors and high com-putation efficiency. At the same time, the RF models establishedfor spaceborne SAR datasets can achieve high fitting accuracieswith reference to the rigorous RDmodel according to above exper-imental results we obtained. Consequently, the RFM can be usedinstead of the RD model in various applications where geometricprocessing of SAR data is required, including SAR geopositioning,SAR geocoding, SAR image simulation, SAR interferometry, radar-grammetry, etc.

The benefits of using RFM instead of the RD model inthese applications are twofold. First, it can remarkably speed uptransformations between 2D image coordinates and 3D geographiccoordinates without substantial loss of computation accuracy.Calculation using RFM only requires a series of multiplications,additions and divisions. The process is massively parallel, becauseeach pixel position can be calculated independently based on theRPC coefficients. Therefore, the RFM is an ideal candidate for animplementation on modern graphics hardware using, e.g., NVIDIACUDA (Compute Unified Device Architecture), allowing for real-time or near real-time SAR geocoding performance.

Second, due to its uniformity, RFM allows sensor-independentgeopositioning of SAR and optical data using the same procedurewith different RPC values. This allows the implementation of aunified, fast, and accurate geopositioning technique independentof sensor type. Using this approach, we can jointly geocode datafrom various remote sensing sensors and easily integrate them forinterpretation and data fusion applications. For example, the RFMcan be used as a basic tool in block adjustment between datasetsacquired by multiple spaceborne SAR sensors just like that foroptical sensors (Grodecki and Dial, 2003).

6. Concluding remarks

The RFM is a generic sensor model that has been intensivelystudied and used in geometric processing of high-resolution

optical images. However, it has never been widely used forSAR datasets. In this article, the feasibility and methodology ofRF modeling for spaceborne SAR datasets are comprehensivelystudied, and possible applications of SAR RFM are discussedbriefly. According to the experimental results and analyses, a fewconclusions as valuable guidance for SAR RFmodeling can bemadeas following:

• First, the rational functional model (RFM) can be successfullyapplied to spaceborne SAR datasets of different processinglevels (SLC, MGD and GEC) acquired by various SAR satellites.The planimetric fitting errors (RSME and MAE) of RFM withreference to the rigorous Range–Doppler model are usuallysmaller than 10−3 pixels in image space.

• Second, a hybrid two-stage approach is proposed to combinethe L-curve ridge estimate method and the unbiased ICCVmethod to efficiently solve the ill-posed problem of RPCestimation as the key step of RF modeling. Compared withtraditional ridge estimation methods and ICCV method, thisapproach can obtain unbiased solutions with higher fittingaccuracy at low computation time cost.

• Third, it is revealed that the number of elevation layers in the3D CNP grid plays a key role in RF modeling for spaceborne SARdatasets. The minimum number of elevation layers requiredprimarily depends on the terrain condition of the study area. Bycontrast, the planar size of the CNP grid is found to have littleimpact on the fitting accuracy of RFM, while the computationtime costwill be increased significantlywith large CNP grid size.Therefore, a grid size of 10 × 10 or 20 × 20 is a good choice forRF modeling.

• Finally, for ENVISAT ASAR datasets using precise orbit datais verified to be able to improve the fitting accuracy ofRFM dramatically compared with the results of using originalorbit records. However, it is demonstrated that the impact oforbit precision on RF modeling is not significant for ERS-1/2datasets.

In this study all the RF models for spaceborne SAR datasets areestablished in the terrain-independent approach. Consequently,these models fit well to the rigorous RD model. However, theircapabilities in absolute geolocation have not been evaluatedquantitatively using GCPs. This problem together with themethodology of RFM refinement will be investigated in our futureresearch works. Furthermore, we will explore the applications ofSAR RFM as discussed previously, including near real time SARgeocoding, SAR interferometry and radargrammetry.

Acknowledgements

This work is financially supported by the National NaturalScience Foundation of China (Grant No. 40701122 and 60910180),the National High-tech Research and Development Program ofChina (Grant No. 2007AA120203) and the Key Laboratory of Geo-informatics of State Bureau of Surveying and Mapping (Grant


No. 2009-05). The ERS-1/2 and ENVISAT ASAR datasets areprovided by ESA through the EU-China cooperative Dragon2program (id. 5297). The TerraSAR-X datasets are provided by DLRthrough a scientific research project (id. GEO0606) and by InfoterraGmbH. The authors would also like to give thanks to Prof. YongWang of East Carolina University, Prof. Jonathan Li of WaterlooUniversity, and the Beijing Earth Observation Inc. for providing thetest datasets of ALOS PALSAR, RADARSAT-2 and COSMO-SkyMedrespectively.

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