A Study of Trajectory Models for Satellite Image Triangulation

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In-seong Jeong and James Bethel

Abstract metric camera. In common use, it generally encompasses Many Spaceborne imagery products are provided with both the internal camera geometry as well as any relevant metadata or support data having diverse types, representa- platform motions. For exploitation of a particular image, the tions, frequencies, and conventions. According to the vari- variables and parameters of that model must be assigned ability of metadata, a compatible physical sensor model numerical values, either from calibration, acquisition time approach must be constructed. Among the three components auxiliary sensors, triangulation, or some combination thereof. of the sensor model, i.e., trajectory model, projection equa- Generally, sensor models fall into two categories: models tions, and parameter subset selection, the construction of the based on the explicit physical characteristics of the system, position and attitude trajectory is closely linked with the and replacement models with generic, polynomial form availability and type of support data. In this paper, we show (RPCs), whose numerical values are obtained by means of a how trajectory models can be implemented based on support physical model. For the purposes of this paper, we will data from six satellite image types: QuickBird, Hyperion, exclude from consideration any polynomial based models SPOT-3, ASTER, PRISM, and EROS-A. Triangulation for each (rubber sheet warping) for which numerical parameters are image is implemented to investigate the feasibility and assigned without reference to a physical model. A physi- suitability of the different trajectory models. The results show cally-based, or rigorous, model has the advantage that all the effectiveness of some of the simple models while indicat- parameters have physical meaning. This is a more favorable ing that careful use of dense ephemeris information is environment for merging with other, related physical data necessary. These results are based on having a number of such as navigation sensors. high quality ground control points. For sensors that will be tested in this paper, many rigorous sensor models have been investigated: QuickBird (Weser et al., 2007; Robertson, 2003; Toutin, 2004), SPOT Introduction (Gugan and Dowman, 1988; Konecny et al., 1987; Makki, Experience has shown that there is a natural linkage 1991; Westin, 1990; Kim et al., 2007), ASTER (Dowman and between the type of metadata supplied with a raw, basic, Michalis, 2003), PRISM (Kocaman and Gruen, 2007) and or level-0 (no geometric correction) satellite image and the EROS-A (Chen and Teo, 2002; Tonolo and Poli, 2003; Westin models used for triangulation of that imagery. As technol- and Forsgren, 2001). Dowman and Michalis (2003) described ogy evolves, such metadata exhibits higher quality with a framework of the physical modeling process as consisting higher frequency sampling rates. Nevertheless, different of a satellite moving along a smooth elliptical orbit with vendors and suppliers make different choices about what image lines acquired sequentially, and with a provision to metadata to provide and about how to present it. Examples allow a dynamic scanning motion during the image capture. of such vendor choices include: reference coordinate Replacement models, on the other hand, may be easier system, Earth fixed or inertial, data frequency, ranging to implement in software, and have a faster execution time. from a few ephemeris points per image to hundreds, and For this approach, Rational Polynomial Coefficients (RPCs) parameterization of attitude data presented as Euler angles (Grodecki et al., 2003; Fraser et al., 2006) have been widely or as quaternions. Such diversity presents a challenge to used and studied. Some vendors, however, have used replace- sensor model developers in requiring a detailed study of ment models as means to avoid divulging proprietary informa- such metadata while constructing a compatible model. tion about physical characteristics of the system. This paper In an attempt to highlight these issues and to encourage a therefore will exclusively consider physical sensor models. movement toward standardization of metadata presentation, we have done a study involving imagery from six medium to Model Components in Satellite Photogrammetry high-resolution systems with the goal of evaluating different A rigorous, physical satellite sensor model can be thought of approaches for construction of the initial position and attitude as having three components: a time-dependent reference or trajectory and its refinement using the triangulation process. initial trajectory specification; a set of projection equations, usually over-parameterized; and an algorithm to select some subset of the trajectory and projection parameters for use in Satellite Sensor Model the actual estimation process. Note that when we speak of a The phrase sensor model refers to the mathematical relation- ships between image space and object space for a photogram- Photogrammetric Engineering & Remote Sensing Vol. 76, No. 3, March 2010, pp. 265–276. Geomatics Engineering, School of Civil Engineering, 0099-1112/10/7603–0265/$3.00/0 Purdue University, 550 Stadium Mall Drive, © 2010 American Society for Photogrammetry West Lafayette, IN, 47907 ([email protected]). and Remote Sensing

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trajectory, it means a path through a six dimensional space model become parameter elements in the projection equations. including both position and attitude. Also, the trajectory model influences the parameter subset selection according to the quality of support data used to (Initial) Trajectory Model construct the trajectory. And clearly, the parameterization of As Dowman and Michalis (2003) summarized, unlike frame the projection equations limits the possible range of variables camera geometry, due to the dynamic nature of pushbroom present in the subset model. imaging geometry, each line has its own exterior orientation (EO) parameters. However, those parameters cannot be individually considered in the model because information to recover Two Approaches for the Trajectory Model explicitly the parameters of all scan lines is insufficient. As suggested by Ebner (1999), there are generally two Therefore, we assume that the EO of adjacent lines is highly approaches for the trajectory model. These are (a) the orienta- correlated and may be modeled by a low order function. tion point approach, and (b) the orbital constraint approach. This calls for a model to specify an initial trajectory of both With the orientation point approach, at certain regular or position and attitude. This initial trajectory is important since irregular time intervals, position and attitude are determined subsequent estimation often entails making corrections or (usually by auxiliary sensors on the satellite) and provided as refinements to this initial path. The position and attitude orientation points, or ephemeris points. For any scan lines in along this trajectory should be a function of time, so that time- between the observed points, a low order piecewise interpola- tagged line numbers can be unambiguously referenced to it. A tion may be used to interpolate a position and attitude. Ebner trajectory model can be very complete and accurate with only (1999) points out that whereas this reduces the number of small corrections required, or it can be very rough or simplis- unknowns to a manageable number, it leaves much of the tic, with significant departures built up during the estimation trajectory un-tethered to any physical model for the motion. process. Such an initial trajectory can consist of, for example, The orbital constraint approach (Ohlhof et al., 1994) Kepler elements or a sequence of discrete positions and assumes that the imaging satellite moves along a smooth attitudes. One may have to distinguish between the camera mathematical curve. All scan line exposure stations would and the platform if they are not the same entity. Such a therefore be constrained on this orbit path. For a short arc, the trajectory model describes the initial approximation of the assumption of a “two-body” orbit may be used. This may be time varying exterior orientation of the camera system. parameterized with six elements of a state vector or, equiva- lently, six Kepler elements. For more extended arcs, additional Projection Equations force model parameters may be used. The basic idea of the As described, a rigorous sensor model tries to reflect the orbital constraint was originally introduced in the early days geometry and physics of how the image is formed based of satellite photogrammetry (Case, 1961). This concept has on the well-known collinearity condition. Unlike the been exploited in many published sensor models. frame camera model, the satellite sensor model should Regarding the attitude trajectory, older, strictly nadir contain sufficient parameters to accommodate any permis- looking cameras could derive attitude information from the sible scanning motions. These scanning motions may position trajectory and its relation to the Earth. For modern, be present in the initial trajectory or they may have to agile, body-scanning instruments, such assumptions are clearly be built up during the estimation process. Also, it is a not valid. These instruments completely decouple the scan- common strategy to model the small variations from the ning motion from travel along the orbit path. In these cases, nominal trajectory as a low order polynomial with regard an explicit sampling of the attitude trajectory, analogous to an to scan line time. All of those factors are considered as orientation point, often coincident with it, is essential. known, unknown, or observed parameters in the projection equations. These equations treat the parameters of both interior orientation (IO) and EO. Note that this strategy of Types of Support Data refining a given trajectory requires external information When deriving the trajectory for a particular satellite, it such as high-quality ground control points. is important to check the available support data because its characteristics, e.g., quality and sampling rate, may Parameter Subset Selection influence the triangulation method (McGlone, 2004). Since The photogrammetric projection equations are often over- there are multiple ways to specify a position and attitude parameterized and may therefore include highly correlated, trajectory, the trajectory model chosen for the triangulation or even dependent parameters which may lead to singularity is often closely related to the specification in the support or solution instability. Including these dependent parameters data. For example, if ephemeris data is provided at a very should not be viewed as flawed model construction; rather low frequency, it is usually not a good choice to select the it provides flexibility for the user to select an independent orientation point approach. Conversely, if the data rate subset. This step is such an essential part of the modeling and quality are high, it may be convenient to adopt the process that we formalize it as a separate process. The orientation point approach rather than the orbit constraint selection involves designating some variables as known and approach, which requires an adjustment procedure to find fixed, others as completely unknown, and still others as the best fitting orbit to the discrete points of the short arc. observed with a quantifiable uncertainty. Guidance for the According to the decisions of the vendor, each sensor selection process may come from the metadata characteris- provides different types of support data having different tics from the experience of the analyst, from the analysis of formats, reference coordinate systems, date rates, representa- the dependency pattern of the refinement parameters, or tion, units, quality, statistical completeness, conventions, and from analysis of columns in the condition equation matrix, other characteristics. This also means that the three model so that what remains after elimination is full rank. components, described earlier for satellite photogrammetry, are sometimes slightly or greatly influenced by these factors, Relation between the Three Model Components and have to be modified or adapted according to the support The three model elements just described are closely related data type. Cooperation between the vendor and the pho- when implementing photogrammetric triangulation. A good togrammetric engineer using the data is essential to ensure a quality trajectory model ensures better performance of the complete and rigorous implementation of the sensor model projection equations. Often factors considered in the trajectory for a specific sensor (de Venecia et al., 2006).

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McGlone (2004) designates three categories of support data for satellite imagery: (a) conventional orbit and attitude data, (b) highly accurate, time-series ephemeris and attitude data, and (c) moderately accurate time-series ephemeris and attitude data. All of these approaches sample the position and attitude at a slower rate than the line rate. Therefore, some kind of interpolation is always required to infer position and attitude for a particular line. Likewise, the low-order, time-dependent polynomials which are often used, allow correction/refinement of sampled data, based on information from ground control points as conveyed by the extended collinearity equations.

Study Motivation and Significance In order to investigate these issues in a practical way that is Figure 1. Initial and refined trajectory illustration. tied to real data, we have done the following experiment. We have selected six different pushbroom type satellite sensors, QuickBird, Hyperion, SPOT-3, ASTER, PRISM, and EROS-A, to give a representative sampling of the types of where the dotted lines show the initial exposure stations and support data that one encounters. We have set up eight directions of the optical axis as the sensor moves during different initial trajectory models (two for position times scene acquisition. The solid lines show the refined location four for attitude equals eight) to accommodate the diversity of exposure stations and refined attitudes. One immediate of support data that was given. We then implemented a hypothesis can be suggested: a smooth but inaccurate initial collinearity-based, rigorous sensor model for each sensor trajectory may after iterative refinement produce a better with each relevant initial trajectory model (not all trajectory result than a more accurate but noisier initial trajectory. This models are possible with all sensors). For each combina- concept will be revisited later. tion, a single image resection was performed, estimating We have examined eight possibilities for the initial sensor/trajectory parameters from measured control points, trajectory model, formed by all (eight) combinations of two with the results tabulated. Our purpose was to explore the position trajectory models and four attitude trajectory models. impact of the initial trajectory model selection on the final We next describe each of these, summarized in Table 2. triangulation outcome. All of the images were taken over nearby terrain in the vicinity of West Lafayette, Indiana. Position 1: Spline Interpolation from Ephemeris (X,Y,Z) Data This model is an application of the orientation point approach. Satellite position (X,Y,Z) is given at regular or Test Data Description irregular intervals. The epochs of the given (X,Y,Z) do not Table 1 gives a summary of the sensors and the correspon- necessarily coincide the epochs of any scan line. Mostly, ding images that were used in the study. For all except they are not coincident. Therefore, piecewise cubic spline Hyperion, the panchromatic image layer was used. For image interpolation is performed to assign an initial position to point observation of Hyperion, a hyperspectral sensor, a all scan lines. three-band RGB composite image was used assuming good inter-band co-registration. For ASTER, the visible and near- Position 2: Interpolation by Kepler Elements (after Fitting infrared (VNIR) 3B band was used for test. And, for PRISM, Orbit to X,Y,Z Points) CCD-4 (nadir looking image) was used for the test. This model is an application of the orbital constraint approach. Using state vector observations which consist of Overview of Trajectory Models position and velocity (X,Y,Z,Vx,Vy,Vz), one Kepler orbital During the course of the triangulation or resection, an initial parameter set which best fits all the observations is obtained trajectory model will be refined (for short arcs) by low order by the least squares method. Additional parameters may be polynomial corrections. This process is illustrated in Figure 1 added to account for non-compliance with the two-body

TABLE 1. SUMMARY OF TESTED TENSORS AND THE CORRESPONDING IMAGES USED

Tested sensors

QuickBird Hyperion SPOT-3 ASTER PRISM EROS-A

GSD 0.6 m 30 m 10 m 15 m 2.5 m 1.8 m Nominal orbit altitude 450 km 705 km 822 km 705 km 691.65 km 480 km Digitization 11 bits 16 bits 8 bits 8 bits 8 bits 11 bits Platform - EO-1 - TERRA ALOS - CNES JAXA(Japan Operating Digital NASA, (French AEROSpace ImageSat organization Globe USGS space NASA Exploration International agency) Agency) Line scan rate 6,900 Hz 224 Hz 665 Hz 455 Hz 2,703 Hz 254 Hz Scene acquisition date 07 October 12 April 21 April, 02 July 02 December 11 April 2006 2003 1996 2001 2006 2004

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TABLE 2. TRAJECTORY MODEL TYPES: EIGHT CASES

Trajectory Position Attitude model Code Examples of Use

Dense Quaternion MOMS-02 (Ebner et al., interpolation (SLERP) SD 1992) Spline Kepler elements (nadir interpolation from looking assumed) SK Ephemeris (X,Y,Z) data Centerline attitude SC Interpolation between start and end attitude SSE Dense Quaternion interpolation (SLERP) KD SPOT (Gugan and Dowman, 1988; Westin, Kepler elements (nadir KK 1990; Makki, 1991); Interpolation by looking assumed) Landsat (Salamonowicz, 1986) Kepler elements ASTER (Dowman and (after fitting orbit Michalis, 2003): assumes to X,Y,Z points) Center line attitude KC constant angle, not necessarily centerline attitude Interpolation between start and end attitude KSE

assumption, in cases of a long arc. Then, at each epoch Attitude 3: Centerline Attitude when a line or framelet image is captured, the instantaneous Because the attitude trajectory during a typical image position is determined from the time or epoch of the line. acquisition is a relatively short arc compared to the entire This model is available for cases where either sparse or orbit, Konecny (1987) assumed a straight-line path and dense state vectors are provided. constant attitude for all scan lines. Such a constant initial attitude, now taken from the center of the image, can be Attitude 1: Dense Quaternion Interpolation (SLERP) later augmented by low order, time-dependent polynomials. This model is an application of the orientation point In this model, therefore, every scan line will have the same approach. For the given time series of quaternions, an initial attitude. Note that the straight-line path assumption is interpolation method is applied. However, simply applying an unnecessary simplification. piecewise cubic spline interpolation individually to quaternion components is not appropriate. Because attitude Attitude 4: Interpolation between Start (First Scan Line) is expressed as a rotation matrix and it is a function of all and End (Last Scan Line) Attitude four quaternion elements, it may not be a good approach This model looks similar to the centerline attitude model in to interpolate individual quaternion elements and then terms of the small number of attitude points used. This is compute rotation matrices. Therefore, to ensure a smooth appropriate for agile sensors rather than classical constant rotation between the sampled quaternions, spherical linear pointing sensors, since it is better to use a start and end interpolation, abbreviated as SLERP, is adopted. SLERP attitude rather than one centerline attitude to accommodate produces an arc of the geodesic between two quaternions large attitude sweeps between the first and last scan line. on the four-dimensional unit hypersphere, and the result- Even though the start and end attitudes may be given by ing interpolated quaternion moves at a constant angular Euler angles (sequential rotations), it is usually best to make velocity along the path. Details are outlined in Watt and the interpolation using spherical linear interpolation (SLERP). Watt (1992). So, for each scan line, the interpolated attitude from given start and end attitude will be assigned. Attitude 2: Kepler Elements (Nadir Looking Assumed) There are some other attitude trajectory approaches that Although Kepler elements implicitly express the position are not included in Table 2. One of those approaches tries to trajectory, they also implicitly provide attitude information combine Euler angles with respect to an orbital reference with the assumption of a nadir looking sensor. The satellite frame with Kepler elements to provide more detailed attitude body is often controlled to be aligned with the orbital trajectory (Radhadevi, 1998; Westin and Forsgren, 2001). reference frame, a moving reference frame defined by the nadir vector, the velocity vector, and the vector perpendi- cular to orbit plane. Since, for many imaging satellites, Description of Experiment the attitude difference between this nadir looking model In order to evaluate the applicability or suitability of the and the actual satellite orientation is quite small, Kepler different specified initial trajectory models to a variety of elements can be used to give initial orientation parameters data sets, we selected the six designated satellite data with the nadir looking assumption. This attitude model is sources for study. This set of data represents a subset of often coupled with a prior knowledge about the off-nadir the imagery that we have archived over recent years cover- look angle to give a better attitude trajectory. In cases of ing our own geographic area in the vicinity of West recent, agile, body scanning satellites employing large off- Lafayette, Indiana. The objectives of the experiment were to nadir look excursions, this model cannot be adopted. (a) investigate how triangulation results are influenced by

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TABLE 3. SUMMARY OF SUPPORT DATA FOR EACH IMAGE IN THE STUDY

Support data rate (period)

Number of used Time No. of Sensors Support data used support data elements (sec) scan lines

Dense state vectors QuickBird (ECEF) 211 state vectors 0.02 138 Dense quaternion 211 quaternions 0.02 138 Dense state vectors Hyperion (ECEF) 34 state vectors 1 (average) 224 Dense quaternion 34 quaternions 1 (average) 224 Sparse state vectors SPOT-3 (ECEF) 1 state vector 60 - Intermediate frequency 15 state vectors ASTER state vectors (ECEF)(VNIR 3B band) 0.8 360 Sparse state vectors (ECEF) PRISM (Payload Correction Data Auxiliary) 9 state vectors 1 2700 Dense quaternion 61 quaternions 0.1 270 Sparse state vectors (ECIN) 6 state vectors 3.62 920 EROS-A Euler angles 2 Euler angles (orbital reference frame) (start-stop) - -

the choice of trajectory model, (b) evaluate the applicability specifications of support data are identifiable using product of different trajectory models to the specific support data manuals. For example, the quaternions of Hyperion and provided with each image, and (c) based on the studies just PRISM do not have clear specification of the related, refer- described, determine a “best” trajectory model considering ence coordinate system and rotation direction. Also, the the image and support data characteristics. same kind of data is provided in different support data A summary of the support for each image in the study sections. For example, PRISM also provides 60 second rate is given in Table 3. The second column makes a distinc- state vector data in the Precision Orbit Determination File, tion between reference coordinate systems: Earth Centered which is far sparser than the 1 second state vectors from the Earth Fixed (ECEF) versus Earth Centered Inertial (ECIN). Payload Correction Data Auxiliary section. Also, in the second column a distinction is made between Table 4 shows the applicable trajectory model cases dense versus sparse sampling of the trajectory data. according to the given support data type. QuickBird and The criterion of sparse versus dense data is based on the Hyperion provide dense support data, and thus six trajectory number of image scan lines between successive support data models can be applied. Excluded is the attitude model using points (column 5). For sensors having less than 300 lines start and end Euler angles. For the other sensors, at least between ephemeris point pairs, we declare them to be dense two models that utilize Kepler orbital elements are available support data. The support data of QuickBird, Hyperion and to use. PRISM can use the dense quaternion interpolation the quaternions of PRISM belong to this group. A line rate model with 0.1 second rate quaternion data. EROS-A provides of more than 900 lines between ephemeris point pairs falls Euler angle information such using start and end attitude, into the sparse data category. ASTER falls in between those and piecewise polynomial coefficients (although undocu- two cases. mented) to account for its non-synchronous scanning For some sensors, there are other support data fields motion. For each model, a single image resection was that we did not use. Not all of the format information or performed using ground control points.

TABLE 4. APPLICABLE TRAJECTORY MODEL CASES ACCORDING TO THE GIVEN SUPPORT DATA (A = APPLICABLE, N/A = NOT APPLICABLE)

Applicable models of sensors

Trajectory model Quickbird Hyperion SPOT-3 ASTER PRISM EROS-A

SD A A n/a n/a n/a n/a SK A A n/a n/a n/a n/a SC A A n/a n/a n/a n/a SSE n/a n/a n/a n/a n/a n/a KD A A n/a n/a A n/a KK AAAAAA KC AAAAAA KSE n/a n/a n/a n/a n/a A

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Table 5 shows the number and source of Ground TABLE 5. NUMBER AND SOURCE OF GROUND CONTROL POINT (GCP) Control Points (GCP) and Checkpoints (CKP) used in the AND CHECKPOINT (CKP) experiment. Their accuracies can be summarized as Tested sensors No. of GCP No. of CKP Acquisition method follows. The control points determined by GPS have a standard deviation of 3 to 5 cm. Regarding the control QuickBird 13 7 GPS surveying points obtained from USGS 7.5-minute topographic maps, the horizontal accuracy standard requires that the posi- Hyperion 23 22 Extracted from 1:24 000 scale map tions of 90 percent of all points tested must be accurate within 0.5 mm on the map. At 1:24 000 scale, this corre- SPOT-3 28 25 Extracted from sponds to 12.2 meters. The vertical accuracy standard 1:24 000 scale map requires that the elevation of 90 percent of all points ASTER 13 8 Extracted from tested must be correct within half of the contour interval. 1:24 000 scale map On the maps used with a contour interval of 10 feet, the PRISM 77GPS surveying map must correctly show 90 percent of all points tested EROS-A 15 7 GPS surveying within 5 feet (1.5 meters) of the actual elevation. Such map control was appropriate for the coarse resolution sensors. The extended, collinearity-based photogrammetric model has the form: applied to the interpolated result. The attitude 3 model falls into both approaches, and selection of the approach differs according to the availability of dense orientation Ϫ x x0 X XL support data. Ϫ ϭ Ϫ y y0 lMA MP MN Y YL Equation 1 has been modified from the usual collinear- Ϫ f Z Z ity equation form by pre-multiplying the position correc- L T tions by (MAMPMN) . Thus, all of the position and angular J K J P Q P Q corrections are applied to an initial trajectory referenced to ¢ XL the sensor system. This change provides a better basis to Ϫ (M M M )T ¢Y (1) perform the Parameter Subset Selection procedure, and this A P N L topic will be explored in a companion paper. Figures 2 ¢Z L through 7 show the distribution of GCP and CKP within the P Q K test images. In these Figures, the white triangles mark the GCPs and the black squares mark CKPs. where the variables are defined as follows: (x,y) are the image coordinates within line or framelet, (X,Y,Z) are the ground coordinates with respect to ECEF, (XL,YL,ZL) is the nominal position of sensor perspective center with respect to ECEF, MN Experiment Results and Evaluation is the nominal rotation from ground (ECEF) to sensor, MA is the additional corrections to nominal rotation, function of ⌬v, Each of the six images was triangulated (resected) using all ⌬ ⌬ of the possible variations of trajectory models (the support f and k, MP is the rotation considering camera pointing angle, ⌬v,⌬f,⌬k are the attitude corrections, ⌬X , ⌬Y , ⌬Z data of each image permitted only a subset of the models to L L L be tested). The results are summarized in Tables 6 through are the position corrections, and x0, y0, f are the interior N orientation parameters. 11. s0 in those Tables means the a posteriori reference ⌬ ⌬ ⌬ ⌬ ⌬ standard deviation, a post-adjustment statistic obtained by Exterior orientation corrections ( v, f, k, XL, YL, ⌬ Equation 3: ZL) are modeled with up to a third order polynomial function of time as follows: T N ϭ v Wv s0 (3) ¢v ϭ dw0 ϩ dw1 # t ϩ dw2 # t2 ϩ dw3 # t3 A r ¢f ϭ df0 ϩ df1 # t ϩ df2 # t2 ϩ df3 # t3 ¢k ϭ dk0 ϩ dk1 # t ϩ dk2 # t2 ϩ dk3 # t3 where v is residuals from condition equations, W is the ¢ ϭ ϩ # ϩ # 2 ϩ # 3 weight for observations, and r is the adjustment redundancy XL dX0 dX1 t dX2 t dX3 t . (2) ¢ ϭ ϩ # ϩ # 2 ϩ # 3 As to the stochastic model for the adjustment, an a YL dY0 dY1 t dY2 t dY3 t ¢Z ϭ dZ0 ϩ dZ1 # t ϩ dZ2 # t2 ϩ dZ3 # t3 priori standard deviation of one pixel is assumed for all L image observations, and therefore the weight of all the Depending upon the specific sensor, the coordinate observations is assumed as 1. system on which the given support data are based may vary. Appropriate coordinate transformations must therefore be applied to set up the given collinearity equations correctly. Here, we adopted the ECEF frame as a reference QuickBird system. Regarding the above-mentioned trajectory models: In Table 6, the Quickbird case, the best result is the SC position 2 and attitude 2, which use orbital constraint, model in terms of total RMSE of checkpoints. Although when ECIN-based state vectors are given, Kepler elements the dense quaternion (SLERP) model makes the smallest are first computed. Then, the position and attitude trajec- initial mis-closure, it has a larger RMSE than the centerline tory are derived from the Kepler elements. Then the attitude model. This is probably because the dense quater- trajectory based on the ECIN frame is converted to the ECEF nion data rate, 50 Hz, provides a more realistic initial frame. When ECEF-based state vectors are given, they are attitude trajectory, but it also has some noise that limits its first converted to ECIN state vectors. Then the same proce- accuracy versus a less constrained refinement. Between the dure is performed to obtain the ECEF based trajectory. two position trajectory models, the spline interpolation Regarding the models, position 1, attitude 1, and attitude 4 model provides better checkpoint RMSE than the Kepler which use orientation points, when ECIN orientation points element interpolation model, but their differences are not are given, a coordinate transformation to ECEF is directly significant.

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Figure 2. GCP and CKP distribution of the QuickBird image.

Figure 3. GCP and CKP distribution of Hyperion image.

Hyperion than does the Kepler element interpolation model, but their In Table 7, the Hyperion case, the best result is the SC difference is very small. model in terms of the total RMSE of checkpoints. Although the Kepler attitude model makes the smallest initial mis- closure, it has the largest final RMSE among the three attitude models. This represents another instance in which ranking SPOT-3 by initial mis-closure does not predict ranking of the final In Table 8, the SPOT-3 case, the best result comes from the mis-closure. Between the two position trajectory models, the KK model in terms of total RMSE of check points. But, spline interpolation model provides better checkpoint RMSE differences between KK and KC model are not significant.

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Figure 4. GCP and CKP distribution of SPOT-3 image.

ASTER angle excursion in this case was larger than the others. That In Table 9, the ASTER case, the best result comes from the KC is likely related to the improved performance when the start model in terms of total RMSE of checkpoints. But, differences and end attitudes are specified. In the case of the KK and KC between KC and the KK model are not significant. models, though it may not fit to EROS-A, a body scanning agile sensor, the RMSE result looks reasonable. This is probably because the Kepler attitude model still provides PRISM initial angles within the convergence range. This was an In Table 10, the PRISM case, the best result comes from the KC unexpected but interesting outcome. model in terms of total RMSE of checkpoints. This is reminis- cent of the QuickBird case, in that although the dense quaternion (SLERP) model makes the smallest initial mis-closure, it Conclusions has a larger final RMSE than the centerline attitude model. The The overall accuracy differences between the two position reason is likely similar to that stated for QuickBird. trajectory models are quite small. That conclusion, of course, assumes that one has high-quality controlpoints, as in this study. If that were not the case, then the conclusions would EROS-A be quite different. In Table 11, the EROS-A case, the best result comes from the Despite the simplicity of the centerline attitude model, KSE model in terms of total RMSE of checkpoints. The scan the RMSE from that model is the best among others in the

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Figure 5. GCP and CKP distribution of ASTER image.

Figure 6. GCP and CKP distribution of PRISM image.

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Figure 7. GCP and CKP distribution of EROS-A image.

TABLE 6. TRIANGULATION RESULT OF QUICKBIRD

Initial Final