Prediction of Georeferencing Precision of Pushbroom Scanner Images

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 3, MARCH 2012 831

Prediction of Georeferencing Precision ofPushbroom Scanner Images

Önder Halis Bettemir

Abstract—The precision of georeferencing of pushbroom satel-lite images without using ground control points is estimated. Thestudy includes what-if scenarios for various precisions of startrackers, GPS receivers, image acquisition timing, interior cameraparameters, and digital elevation model (DEM). Moreover, theeffects of satellite altitude and sensor positions on the pushbroomarray are also taken into account. The probability distributionfunction of the input parameters is determined by consideringthe sensor characteristics, environmental effects, and assumptions.The precision of the georeferenced coordinates is estimated byMonte Carlo simulation and differential sensitivity analysis. Theanalyses provide the most probable georeferenced coordinates andtheir precision for a certain precision of sensors and imaginggeometry. The study is conducted based on the BilSAT orbit andSPOT 5 and Landsat optical system characteristics. An ShuttleRadar Topography Mission (SRTM) DEM and a DEM obtainedfrom stereo images are used in georeferencing.

Index Terms—Charge-coupled image sensors, error analysis,image registration, Monte Carlo methods, sensitivity.

I. INTRODUCTION

SATELLITE images are commonly used for geospatial ap-plications, 3-D city mapping, archaeology, city and re-

gional planning, agricultural and forest management, weatherforecast, etc. Each application requires different georeferencingaccuracies depending on the regulations or the desired quality.Direct georeferencing by colinearity equations is a fast andcheap way of georeferencing since site visits are avoided. Stud-ies conducted on direct georeferencing [1] show the importanceof uncertainty estimation. The effect of geometric conditions onthe georeferencing of space imagery including along track andacross track is examined by Jacobsen et al. [2]. Based on sensortypes, the most suitable affine transformation model is proposedfor pushbroom scanners. Zhu et al. [3], Mostafa and Hutton[4], Hongjian et al. [5], Reid and Lithopoulos [6], and Legat[7] provided contributions for the light detection and ranging or3-D city-mapping system by direct georeferencing algorithms.

Uncertainty estimation of georeferenced images without aground control point (GCP) is executed by Bouillon and Gigord[8], of SPOT images by Poli et al. [9] and Riazanoff [10], andof charge-coupled-device frame camera images by Bettemir

Manuscript received August 4, 2010; revised December 31, 2010, May 20,2011, and July 10, 2011; accepted July 17, 2011. Date of publicationAugust 30, 2011; date of current version February 24, 2012.

The author is with Yuzuncu Yil University, Van 65080, Turkey (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TGRS.2011.2163315

[11]–[13]. Research on frame camera images [11]–[13] con-sists of examination of different-precision sensors and effectof topography on the precision of georeferencing by errorpropagation law (EPL) [14]. Error analysis of a georeferencedimage by affine transformation based on EPL is executed byKutoglu [15], Sertel et al. [16], and Topan and Kutoglu [17].

In this study, precision of direct georeferencing is predictedby considering the uncertainties of camera attitude, cameraposition, interior camera parameters, image timing, and digitalelevation model (DEM). In addition to this, the effects of to-pography on the precision of both horizontal and vertical coor-dinates are also examined. The precisions of easting, northing,and elevation of the georeferenced coordinates are predicted.

The precision of direct georeferencing is estimated by bothdifferential sensitivity analysis (DSA) and Monte Carlo simu-lation (MCS). DSA provides uncertainties of output variables,sensitivity coefficients of input parameters, and ratio of thechange in output to the change in input, while all other param-eters remain constant [18], which is defined as the base case.A major drawback is that the localized behavior may not beapplicable for realms for the base case [19]. MCS is developedby Metropolis and Ulam [20] and until now adopted by manyresearchers for sensitivity analysis [21]–[26]. MCS predicts theuncertainty of output parameters by multiple model evaluationsin which the values of the input parameters are assigned ran-domly according to the probability distribution function (PDF)of their uncertainties. To conclude, DSA requires that basevalues of input parameters be kept fixed during the analysis, butin MCS, no input parameter is kept constant. However, DSAprovides more stable output for the sensitivity coefficients ofthe input parameters.

Uncertainties of input parameters are mainly caused byimperfections of the instruments and environmental effects.Precisions of the input data which are obtained via a sen-sor are assigned according to the declaration of precision oftheir vendors. Error sources of input parameters are brieflyexplained. Star tracker (STC) provides attitude of itself w.r.t.the inertial reference system. Error sources of STC are line-of-sight uncertainties, calibration errors, s-curve errors, andnoise equivalent angles. Due to the characteristics of the errorsources, an STC is four to six times less accurate in the opticalaxis [27]–[30].

GPS receivers are used for positioning. Based on the antenna,positioning algorithm, and signal quality, a single-frequencyabsolute positioning algorithm can provide 10-m root-mean-square error (rmse) [31]–[33]. Dual-frequency receivers canprovide rmse better than 1 m. Vertical precision of GPS

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832 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 3, MARCH 2012

positioning is significantly worse than horizontal precision[31]–[33]; thus, the defined precision ends up with a 3-D el-lipsoid in which its height is much larger than its width. Actualfocal length is the true distance from the optical center of thelens, measured along the optical axis, to the point of intersectionof the optical axis and plane of the pushbroom array. The focallength is estimated with a small amount of uncertainty in thecalibration process, which is called focal length error [29].

A linear pushbroom array may not be placed perfectly, andthe position of the optical center Δx Δy is estimated by acalibration procedure with a certain amount of error [34]. Apushbroom array may be fixed in such a way that it is notperfectly perpendicular to the optical axis. Slant θ in the push-broom array will lead to wrong measurement of the distancebetween the focus and the detector [29]. Due to imperfectionsof the lens, objects would be sensed by the pushbroom arrayin a different position than their actual position [34]. DEMproduced from a radar or an optical sensor would have certainamount of uncertainty due to terrain roughness, grid resolution,and interpolation methods [35]–[38].

A satellite clock is synchronized with the clock of the on-board GPS receiver. For this reason, clock bias of the satelliteclock is small enough to be neglected. However, the CPUinstalled on the satellite may perform multitask operationswhich are handled according to the preassigned priority rules.If the satellite CPU is busy during the image acquisition, theacquisition time may not be recorded correctly. Random delaysin the recording of the image acquisition time in the range of1–5 ms can be observed [13].

II. METHODOLOGY

Georeferencing requires a series of transformations from animage coordinate system SI to a ground coordinate systemSM . A multiband sensor would have more than one pushbroomarray. Therefore, it is not possible to place every array exactlyon the center in the y-direction. The array is assumed to beplaced at Δy in the y-direction. First transformation is fromSI to a photo coordinate system SP and uses the detector pitchc, the number of detectors in the pushbroom array w, and theprincipal coordinate Δx,Δy. Lens distortions are corrected by(2) in SP [40]

x′ =(x′′ − w

2

)∗ c−Δx y′ = Δy (1)

x =x′ −(x′ ∗ k1 ∗ r2 − x′ ∗ k2 ∗ r4 − x′ ∗ k3 ∗ r6

)×(−j1 ∗ sinφ0 ∗ (r2 + 2x′2) + 2j1 cosφ0xy

)p1r

2

y = y′ −(y′ ∗ k1 ∗ r2 − y′ ∗ k2 ∗ r4 − y′ ∗ k3 ∗ r6

)×(−2j1 sinφ0xy + j1 ∗ cosφ0 ∗ (r2 + 2y′2)

)p2r

2

(2)

where x′′ is the position of the image point in SI ; x′ andy′ represent the position of the image point in SP ; x and yare the distortion-corrected image coordinates in SP ; k1, k2,and k3 represent radial distortions; j1 and φ0 are decenteringdistortions; p1 and p2 are thin prism distortions; and r is theradial distance of the pixel from the image center. The next step

is transformation from SP to a camera coordinate system SC ,mapping from 2-D to 3-D, in which the third dimension is thenadir direction between the camera focus and the image center.For all pixel points, zi is equal to the focal length of the cameraf . In order to obtain a direction vector from the camera focusto image positive, (3) is implemented

rc =[xi, y, z]√x2i + y + z2i

, zi = −f (3)

where xi and zi represent the position vector of the ith detectorelement with respect to SC and f is the focal length of thecamera. The direction vector with respect to SC is transformedto a body-fixed reference system SB by using the rotationof the camera w.r.t. the direction of along track α. The nexttransformation is from SB to an orbital reference system SO.This transformation requires three attitude angles (roll, pitch,and yaw) supplied by the attitude determination system ofthe satellite. Finally, the direction line is transformed fromSO to an Earth-fixed reference system SE . The georeferencedcoordinates of the image point are obtained by finding theintersection point of the direction line in SE and the Earth-fixed reference ellipsoid. The intersection point is obtained bysubstituting the smaller root of s obtained in (4) to (5) [3][R2

X +R2Y

a2+

R2Z

b2

]∗ s2

+ 2 ∗[RX ∗Xcam +RY ∗ Ycam

a2+

RZ ∗ Zcam

b2

]∗ s

+X2

cam + Y 2cam

a2+

Z2cam

b2− 1 = 0 (4)

⎡⎣X0

Y0

Z0

⎤⎦ =

⎡⎣Xcam

Ycam

Zcam

⎤⎦+ s ∗

⎡⎣RX

RY

RZ

⎤⎦ (5)

where X0, Y0, Z0 represents the intersection point, Xcam,Ycam, Zcam represents the camera position in SE , a is thesemimajor axis of the reference ellipsoid, b is the semiminoraxis of the reference ellipsoid, s is the distance between thecamera focus and the surface of the reference ellipsoid, andR is the direction vector of the colinearity line. Initially, theintersection point has zero elevation which is not necessarilythe true elevation. Not only the elevation but also the reliefdisplacement horizontal position would be wrong. To computethe true elevation, the elevation of the intersection point isretrieved from DEM by using the initial horizontal position. Arelief displacement correction algorithm updates the horizontalposition. The elevation of the corrected horizontal position isobtained from DEM, and if the elevation difference is below thethreshold value, no correction is done; otherwise, the iterationcontinues. Orthoimage production is completed after projectingthe ellipsoidal coordinates onto the Universal Transverse Mer-cator projection system SM when monoimages are used (i.e.,the stereopairs are not available). Projection onto SM completesgeoreferencing. The next step is performing the uncertaintyestimation by means of MCS and DSA. MCS is introduced infive steps [25], [26].

BETTEMIR: PREDICTION OF GEOREFERENCING PRECISION OF PUSHBROOM SCANNER IMAGES 833

The precision and PDF of the input variables are determinedin the first step. Sensor precisions are obtained from calibrationresults and experiments [27]–[40]. PDFs are used to generatea sample set for the sensitivity analysis in the second step.The commonly used sample generation techniques are randomsampling, stratified sampling, and Latin hypercube sampling(LHS). Among the three methods, LHS is preferred since itdivides the uncertainty range of the input function into equalprobability intervals by ensuring that the sampling is uniformlydistributed among the uncertainty ranges. In this study, MCS isexecuted with 1000 evaluations. For this reason, the uncertaintyrange of each parameter is divided into 1000 equal probabilityranges, and the value of the input parameter is randomly as-signed within these ranges. This procedure is repeated for eachinput parameter. The randomly generated sets are randomlypaired, and a set of n-tuples is formed which can be repre-sented as xi = [xi1, xi2, . . . , xin], where i = 1, . . . ,m, with mbeing the sample size (1000), and n is the number of inputparameters (20).

The STC provides attitude, the GPS receiver provides posi-tion, the DEM provides elevation, and the interior camera pa-rameters are assumed to be determined by a camera calibrationprocedure before the launch of the satellite. Therefore, inputparameters obtained from different sensors are not correlated,while the three attitude angles roll, pitch, and yaw, the threeCartesian coordinates X , Y , and Z, and the interior cameraparameters are self-correlated. Self-correlations occur becauseof the characteristics of the STC, GPS, and camera calibrationprocedure [27]–[34]. Control of correlation within a samplein Monte Carlo analysis is very important. Although sets arepaired randomly, there can be unenviable correlation in thesample set S. The correlation coefficient matrix of the inputparameters D in (6), shown at the bottom of the page, and thegenerated sample set E should be identical.

To make sure that S has the same correlation with D, thevalues of S are rearranged. This rearrangement is based onthe Cholesky factorization of D in which D = PPT, whereP is a lower triangular matrix. Cholesky factorization is pos-sible because D is a symmetric positive definite matrix. Thevariance–covariance matrix of S, i.e., E, is computed, and

Cholesky factorization is applied E = QQT. The matrix S∗ isdefined in [25]

S∗ = S(Q−1)TPT. (7)

In the third step, models are evaluated based on the generatedinput tuples in which the results of the analysis yi can be shownas yi = f(xi1, xi2, . . . , xin) = f(xi), where i = 1, 2, . . . ,m.The precisions of output parameters are predicted by using theobtained results as the fourth step of MCS. Since the sample setis generated by LHS, the most probable value and precision ofy can be estimated in

E(y) =m∑i=1

yim

(8)

V (y) =

m∑i=1

[yi − E(y)]2

m− 1. (9)

The fifth step is the computation of the sensitivity coefficientsSi by the cross-correlations of the zeroth lag of the normalizedcovariance function by (10). Sensitivity coefficients representcontribution of the input parameter on the uncertainty of theoutput parameter

SI =E [(Xij − E(Xj)) (Yi − E(Y ))]

σXjσY

(10)

where Xj is the jth input parameter and Y is the output. Thesecond method implemented for sensitivity analysis is DSA.The model is based on Taylor series expansion to approximatethe response of the model to changes in the model’s parameters.First-order Taylor series expansion approximating function y isshown in [25]

y = f(xo) = f(X0, Y0, Z0, ω, φ, κ,Δx,Δy, f, k1, k2,

k3, p1, p2, j1, φ1, α, θ, h, δt) (11)

y = y(xo) +

n∑i=1

[∂f(xo)

∂xi

](xi − xio) (12)

D =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

σ2X0

σX0σY0

σX0σZ0

σ2Y0

σY0σZ0

σ2Z0

σ2ω σωσφ σωσκ

σ2φ σφσκ

σ2κ

σ2Δx σΔxσf σΔxσk1

. . . σΔxσθ

σ2f σfσk1

· · · σfσθ

σ2k1

· · · σk1σθ

. . . σk2σθ

σ2θ

. . .σ2δt

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(6)


where xo is the vector of input parameters with base valuesand xi is the value of the ith input parameter within its range.The partial derivatives are numerically computed by a five-pointstencil method [41] shown in

f ′(x)≈−f(x+2h)+8f(x+h)−8f(x−h)+f(x−2h)

12h. (13)

A variance propagation technique is used to estimate theuncertainty in y. The computation of the variance of the out-put parameters by EPL with first-order Taylor series ends upwith [25]

V (y) =n∑

i=1

[∂f(xo)

∂xi

]2V (xj)

+ 2

n∑i=1

n∑j=i+k

[∂f(xo)

∂xi

] [∂f(xo)

∂xj

]D(xi, xj) (14)

where V is the variance of the output variables, D is thevariance–covariance matrix of the input parameters shown in(6), and ∂f(xo) is the partial derivative of the georeferencedcoordinates with respect to the input parameters, shown at thebottom of the page.

The sensitivity coefficients of the input parameters can beestimated in [25]

Si =∂f(yi)

∂xi

[V (xi)

V (y)

]1/2. (15)

Fractional contributions to variance can be used to measureparameters’ contribution to the uncertainty in y. Contribution isbased on both the absolute effects of parameters, as measuredwith their partial derivatives, and the effect of distributionsassigned to the parameter, as measured by V (xi).

III. SENSITIVITY AND UNCERTAINTY ANALYSES

For the interior camera parameters, the values shown inTable I are assigned. The camera is assumed to be panchromaticand sensitive to wavelength between 450 and 745 μm. Atmo-spheric refraction corrections are performed for the specifiedwavelength [42]. The atmosphere is divided into 1000-m-thickportions. By assigning properties of a standard atmosphere,these portions are modeled. The temperature, air pressure, andwater vapor pressure are computed to determine the refractivityindex of the atmosphere and implement Snell’s law. STC isassumed to be assembled with an angle of 25◦ with respectto the Z-axis of SB (Fig. 1). At the instance of imaging, the

TABLE IASSUMED VALUES FOR THE INTERIOR CAMERA PARAMETERS

Fig. 1. STC position w.r.t. SB .

camera is assumed to be tilted 20◦ w.r.t. SB . For the satelliteposition and velocity, the orbit of BilSAT is used on August 3,2005, at 7:30:38 in UTC in which the satellite was close to theKiev capital of Ukraine.

Six STCs, two GPS receivers, two different-precision DEMs(SRTM DEM and precise stereo-image DEM), three differenttiming precisions, and three different satellite altitudes [27]–[39] are taken into account. Twelve analyses are conductedbased on the sensor combinations given in Table II. The preci-sion of only one input parameter is improved; then, the effect onthe precisions of easting, northing, and elevation is examined,and the sensitivities of the input parameters are monitored.Satellite altitudes are chosen within the low-Earth-orbit bound-ary, which are the altitudes of the NOAA-18, BilSAT, andWorldView-1 satellites, respectively. The corresponding orbitalvelocity is computed without changing the direction of the

∂E

∂xi=[

∂E∂X0

∂E∂Y0

∂E∂Z0

∂E∂ω

∂E∂φ

∂E∂κ

∂E∂Δx

∂E∂Δy

∂E∂f

∂E∂k1

∂E∂k2

∂E∂k3

∂E∂p1

∂E∂p2

∂E∂j1

∂E∂φ0

∂E∂α

∂E∂θ

∂E∂h

∂E∂δt

]

∂N

∂xi=[

∂N∂X0

∂N∂Y0

∂N∂Z0

∂N∂ω

∂N∂φ

∂N∂κ

∂N∂Δx

∂N∂Δy

∂N∂f

∂E∂k1

∂E∂k2

∂E∂k3

∂E∂p1

∂E∂p2

∂E∂j1

∂E∂φ0

∂E∂α

∂E∂θ

∂E∂h

∂N∂δt

]

∂H

∂xi=[

∂H∂X0

∂H∂Y0

∂H∂Z0

∂H∂ω

∂H∂φ

∂H∂κ

∂H∂Δx

∂H∂Δy

∂H∂f

∂E∂k1

∂E∂k2

∂E∂k3

∂E∂p1

∂E∂p2

∂E∂j1

∂E∂φ0

∂E∂α

∂E∂θ

∂E∂h

∂H∂δt

].


TABLE IISENSOR PRECISIONS AND INPUT PARAMETERS

Fig. 2. Effect of horizontal uncertainty on elevation.

satellite. The attitude angles of the satellite w.r.t. SO are 4.692◦,−1.254◦, and 0.8360◦ in roll, pitch, and yaw, respectively.

To monitor the effect of pixel position on the georeferencingaccuracy, five pixel positions are investigated: two edge pointsof the pushbroom array, one center point, and two intermediatepoints between the edges and the center. As a result of this,the 1st, 1536th, 3072nd, 4108th, and 6144th elements of thepushbroom array are analyzed.

Vertical uncertainty is computed indirectly because the truedirection of the colinearity line differs from the computeddirection obtained from telemetry. The solid line in Fig. 2represents the true direction of the light array, and the dashedline represents the computed direction. As this is the case,elevation retrieved from DEM will be h1, although h0 shouldbe retrieved. The difference between h1 and h0 is caused byhorizontal uncertainty. The effect of horizontal uncertainty isanalyzed on two different topographies; one is smooth, andthe other is steep. The maximum elevation difference in thesmooth topography is 13.8 m while it is 68.1 m for the roughtopography. Of the 50 randomly retrieved elevation values fromthe smooth and steep topographies, the standard deviations arecomputed as 4.6 and 22.7 m, respectively. MCS provides m el-evation values for a certain pixel. The standard deviation of them elevation values gives an idea about the elevation differencebetween h1 and h0, while in DSA, partial derivatives of theelevation with respect to the input parameters are examined toestimate vertical uncertainty.

Horizontal and vertical uncertainties form the 3-D uncer-tainty ellipsoid which is shown in Fig. 3, where N , E, and Hare northing, easting, and elevation, respectively. Probabilisti-cally, 68% of the randomly generated input data remain inside

Fig. 3. Three-dimensional uncertainty ellipsoid (Google Earth image).

the ellipsoid. The center of the ellipsoid is computed by the basevalues of the input parameters, and the volume constitutes theconfidence volume.

The uncertainty of the georeferenced coordinates and thesensitivity of the input parameters computed by MCS and DSAare shown in Fig. 4. In order to shorten the reported data, theaverage of the five inspected image points is plotted. Fig. 4(a)and (b) shows that easting has higher uncertainty than northingin the first three analyses. However, the situation is reversedin the following analyses. Fig. 4(c) and (d) shows that STC isthe most sensitive parameter and the precision characteristicsof the STC significantly affect the georeferencing precision.It is known that the precision of optical angle will be aroundfour to six times less precise than other axes [27]–[30]. Theassigned configuration of the satellite orbit and STC attitudemakes easting affected slightly more than northing. However,due to the effect of image timing uncertainty, the uncertainty ofnorthing becomes higher when more precise STC is installed.

There is notable difference on the vertical uncertainties ofthe smooth and steep topographies as shown in Fig. 4(a) and(b). Being the major uncertainty source of the horizontal uncer-tainty, STC has significant impact on the vertical uncertainty.If the horizontal uncertainty is large, DSA cannot provide goodestimations for the vertical uncertainty as it assumes that theslope of the region is constant and the uncertainty changesproportionally with the horizontal uncertainty. This drawbackis tried to be eliminated by computing the partial derivativesnumerically and taking the step size proportional with the hori-zontal uncertainty. However, MCS provides dependable resultsfor both sensitivities of significant parameters and uncertaintyof georeferencing. The analyses reveal that the effects of hor-izontal uncertainty and topography on the vertical uncertaintyare significant.

Both DSA and MCS provide sensitivity values between −1and 1. The magnitude of the sensitivity reflects the contributionof the input parameter on the uncertainty of georeferencing.Sensitivity differs from partial derivative in a way that partialderivative gives only the rate of change but sensitivity repre-sents the amount of contribution that the input parameter makeson the uncertainty of the output. The sign of the sensitivity givesthe relationship between the input parameter and the output.Positive sensitivity means that, if the input parameter increases,the output increases too and vice versa.

The horizontal uncertainty is very high in the first threeanalyses; for this reason, the uncertainty graph is scaled


Fig. 4. Uncertainty of georeferencing and sensitivity of the input parameters.

semilogarithmically. The sensor configuration of the first caseis appropriate for an inexpensive research satellite. Fig. 4(c)–(f)shows that the STC output is the most sensitive parameter,which means that much of the uncertainty of georeferencingis caused by STC. The second analysis is conducted with abetter STC, and the third analysis is conducted with a lowerorbit. The horizontal uncertainty is significantly improved;however, the STC is still the most sensitive input parameter,and more precise STC is adopted for the fourth analysis. Withinthis configuration, the uncertainty of image acquisition timingbecomes the most sensitive input parameter in northing whileSTC is still the most sensitive parameter in easting. In thefifth analysis, fewer workload is assumed for the main controlunit of the satellite, and the sensitivity of Δt is decreased.In the sixth analysis, more precise STC is assigned, and atypical configuration for small satellites is obtained. In theseventh analysis, Δt is improved since it was the most sensitiveparameter in northing. The satellite altitude was decreased inorder to diminish the effect of STC in the eighth analysis.This modification decreased the effect of STC but increasedthe effect of Δt. In the ninth and tenth analyses, the precisionof STC is improved, but significant improvements cannot beobtained. In the eleventh analysis, the precision of positioningis improved, which also does not significantly affect the georef-

erencing precision. The reason of this can be explained by theuncertainty characteristics of the GPS in which the elevationis significantly less precise. Because of high satellite altitude,the effect of imprecise vertical positioning within a range ofa few meters is not significant. Improvement of the horizontaluncertainty also improves the vertical uncertainty. However, thepresent horizontal uncertainty is too little that improvementof it will hardly have an effect on the vertical uncertainty. Inthe last analysis, georeferencing is performed by using moreprecise DEM.

There are some notable inferences with the analyses whichcan be briefly explained. After a certain precision of STC,it is difficult to increase the precision of georeferencing byimproving the STC precision. The remaining imprecise sensorswill become more sensitive and prevent a notable improvementin georeferencing. The tenth analysis shows this situation inwhich there is little improvement in georeferencing precisionwhen the STC precision is improved from 3 to 1 arcsec.

If the horizontal uncertainty is large, its effect on the verticalposition is notable. However, since DSA is not a proper methodfor estimating the uncertainty of elevation, particularly forsteep-sloped surfaces, uncertainty predictions of DSA are notwithin reasonable ranges. Sensitivity coefficients of relativelyinsensitive parameters fluctuate in MCS since the analyses are


Fig. 5. Projection of uncertainty ranges onto surfaces.

conducted by generating random numbers. For this reason,by chance, small amount of sensitivity can be encountered,although the input data are decorrelated. Sensitivity valuessmaller than 0.1 are considered as insensitive and ignored inMCS. However, sensitivity values are stable enough for theinput parameters which have apparent effect on the outputvariables.

The most sensitive parameter for northing is STC. How-ever, in ensuing analyses, the uncertainty of the imaging timebecomes more sensitive. Orbital characteristics and imaginggeometry make imaging time uncertainty more sensitive thanattitude angles. The most sensitive parameter for easting isattitude in the initial analyses and principal coordinate in thefinal analyses. A low altitude orbit provides more precise geo-referencing since the effect of uncertainty of STC fades away.On the other hand, the velocity of the satellite increases, whichincreases the effect of image timing uncertainty.

Uncertainties of interior orientation parameters become sig-nificant if very precise attitude and position determinationsystems are assembled. For further improvements of the pre-cision of georeferencing, precise knowledge of interior cameraparameters is required. On the other hand, it is difficult to obtaingeoreferencing accuracy better than 8 m for small satellitesbecause of the limitations on power consumption, weight, andcomputational demand. The power consumption of a 1-arcsecSTC is around 20 W, and it weighs 20 kg, which are challengingrequirements for a small satellite [28].

Fig. 4(a) and (b) shows that the steep-sloped topographyhas slightly less uncertainty in the direction of northing andmore uncertainty in the direction of easting. It is clear that thesteep topography decreases the horizontal uncertainty in thedirection of northing, which is an unexpected result. If Fig. 5is analyzed, it can be seen that steep topography may reducethe horizontal uncertainty. In the figure, horizontal uncertaintyis represented by the range within the two parallel lines. Thelines are drawn parallel since the directions of the true and thepredicted colinearity lines are very close. The colinearity linesare projected onto three different surfaces: inclining, horizontal,and declining. It is seen that projection onto the inclining sur-face ends up with a closer horizontal distance, while projectiononto the horizontal surface ends up with the same horizontaldistance, and finally, projection onto the declining surface endsup with a further horizontal distance between the parallel lines.In the direction of northing, the steep topography is inclining,and in the direction of easting, it is declining.

Apart from STC and image timing uncertainties, the sensitiv-ity coefficients of the input parameters are not very significant.However, the principal point coordinate Δx has significanteffect on easting after the ninth analysis. The imaging geometrymakes this parameter sensitive on the easting, which becomesmore apparent when the precision of other input parametersis improved. A similar phenomenon is observed for Δy onnorthing. Therefore, precise camera calibration is mandatory ifhighly precise sensors are installed on-board.

IV. CONCLUSION

This study provides uncertainty ranges for georeferencedimages without GCPs. In the context with this aim, two sen-sitivity analysis methods are implemented on two differenttopographies for 12 sensor combinations. For each analysis,the input parameter having the highest sensitivity is improved.Sensor combinations are determined by only considering thesensitivities of input parameters. In the successive analysis,a better sensor available in the market is used for the inputparameter which has the highest sensitivity. Budget of thesatellite, available power, and space are not considered. Forthis reason, this study is not a satellite design and can beuseful for end users who work on georeferenced images withoutGCPs. By considering the analysis results, the precision ofa georeferenced image can be predicted by an interpolationbetween two-closest-sensor configuration and geometry.

Sensor combinations belong to a fictitious satellite which isformed by currently available sensors. Sensor precisions aredeclared by their vendors or obtained from tests performed byresearchers. For this reason, results of analyses can be discussedin terms of precision. If the users are sure that sensors have nobias, then the results can be considered as accurate.

The study reveals the effects of horizontal uncertainty andtopography on the vertical uncertainty. Moreover, the effect ofeach input parameter on the precision is measured. The studypresents valuable information for the prediction of precision ofgeoreferencing.

DSA is a stable uncertainty and sensitivity analysis method.The sensitivity coefficients obtained from DSA were moredependable when compared with the output of MCS. However,it was not possible to properly differentiate the topography withrespect to input parameters. Therefore, DSA did not provideintense information. On the other hand, with the help of ran-domized evaluations, the topographies of both smooth and steepregions are very well scanned by MCS, and valuable results re-lated to the uncertainty of elevation are obtained. MCS providesmore influential sensitivity coefficients of input parameters forelevation. Furthermore, the characteristics of the derivatives ofinput parameters are very different. The uncertainty of imagetiming is a line of projection of the satellite orbit while thepartial derivative of a lens distortion parameter is a planedepending on the position of the pixel. Unfortunately, DSAcannot take the geometry of partial derivatives into account; itconsiders only the magnitude.

Only an instance of image acquisition of a pushbroom arrayis considered. It is known that precision of telemetry would notbe the same during the acquisition of the image. In addition


to this, changes of satellite attitude and imaging geometryshould also be considered. For this reason, a detailed study ofprediction of the whole image is better to be conducted as afurther study.

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He is currently an Assistant Professor with theDepartment of Civil Engineering, Yuzuncu YilUniversity, Van, Turkey. His study subjects arephotogrammetry, geographic information systems,remote sensing, inertial navigation systems, and

machine control and optimization.Dr. Bettemir is a member of the Turkish Chamber of Civil Engineers.

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Prediction of Georeferencing Precision of Pushbroom Scanner Images