The present study aims at demystifying the dynamics of spacecraft imaging system including state vector, viewing orientation, attitude parameters (roll, pitch and yaw) and other related parameters, as a digital solution, with a full force rigorous orbital photogrammetric model.
In this approach satellite orientation parameters are modeled as keplerian orbital parameters in continuous time domain, as against conventional approaches that use position and velocity vector parameters of the imaging platform in discreet time domain,. The attitude parameters are, however, modeled as polynomials in discreet time domain. This hybrid time domain model offers an excellent insight and a better understanding of the dynamics of linear array sensor imagery that is illustrated with a Spot 2 data set.
The study brings out that the key components associated with dynamics of push broom imagery are two keplerian parameters true anomaly and ascending node, attitude parameters (roll, pitch and yaw) and distance between space craft and imaged ground point.
Keywords: Remote sensing; SPOT; IRS satellites; geometric rectification; DEM, ortho-photo; image orientation; stereo images; continuous time domain model; unified theory of least squares and rigorous orbital photogrammetric model.
1. Introduction
Earth observation satellites (EOS) such as Spot, IRS series of satellites, Quickbird, and IKONOS scan the earth surface with an array of CCD elements in push broom mode, while satellites such as Landsat scan the same in whisk broom mode. For push broom imaging, direction of scan is along the flight direction of satellite, whereas for whisk broom imaging, direction of scan is perpendicular to the flight direction.
Linear array sensors operating in push broom mode are widely used for acquiring images at high resolution for cartographic applications.
Up to the late 1970’s, control engineering concepts have been used to model the movement of photo plates of analytical instruments as per the dynamics of aerial scanner. During the late 1980’s, Kratky [16], Konecny [15] et al and others used analytical photogrammetric instruments for generating precision corrected products of Spot 2 with level 1B photo
products. These analytical instruments have been in vogue until the advent of digital photogrammetric workstations.
Digital emulation of the dynamics of spacecraft imagery has been influenced by aerial scanner imagery wherein position and velocity information is modeled in discreet time domain.
Other digital approaches have been based on rational polynomial functions, affine models and direct linear transformations. Several efforts have been made to understand the suitability of these models with respect to complexity, rigor and accuracy that are detailed in Dieter Fritsch et al., [7], and Dowman et al [6] and Daniela Poli [2].
Recent studies indicate that Kartky’s approach is far superior to other models and this is largely due to his strict adherence to emulate the movement of photo plates of analytical photogrammetric instruments using keplerian parameters of Spot2 orbit. Eminent photogrammetrists such as Edward Mikhail [19], Toutin [26] etc have noted that there are several inadequacies by modeling satellite imagery with aerial scanner and other digital approaches. GP Rao et al [12], GP Rao [13]
note the superiority of continuous time domain models from the simulation results of other disciplines such as control theory. In addition, these authors also indicate that Newton's, Faraday's, etc. are developed in continuous time domain and there is little possibility that these laws will be written in discreet time domain.
The focus of present study is on an analysis of dynamics of push broom spacecraft imagery based on Kartky’s approach and as a digital photogrammetric solution in hybrid time domain with SPOT 2 data. This analysis is relevant for extraction of metric information from other linear sensor arrays such as IRS series of satellites, IKONOS and Quickbird.
Accordingly, the paper is organized as six sections with this as an introduction. The need for rigorous orbital photogrammetric models is brought out as section 2. Sensor models based on discreet time domain and hybrid time domain are presented briefly as section 3. The parameters included for bundled adjustment are described in section 4. A critical analysis of pre-facto and post-facto parameters of the rigorous orbital photogrammetric model is presented as section 5 and is followed by a conclusion as section 6.
2.0 Need for Rigorous Photogrammetric Models It is well known that the very narrow field of view associated with push broom linear scanners on earth observation satellites
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(EOS) results in nearly parallel imaging rays along the direction of flight. This in turn causes a high correlation between projection centre coordinates and sensor view angle.
Photogrammetric models of push broom imaging systems are, therefore, rather different, compared to standard approaches usually applied for full frame imagery such as aerial photographs. The rigorous models are based on photogrammetry co-linearity condition and are iteratively modified to include any external and internal orientation. The output of these models is used to generate data products such as geo-coded and ortho-rectified (terrain relief corrected) products.
Well modeled push broom imaging systems from missions such as Spot2 and IRS -1C/IRS-1D have been used for across track stereo viewing applications. Subsequently, missions such as IRS P5 and Spot 5 have incorporated multi camera imaging systems for an along track stereo viewing capability. This stereo viewing capability with an appropriate photogrammetric model, facilitates realization of height information on a pixel by pixel basis, generation of anaglyph products, DEMs, ortho-rectified products and 3D visualization. A critical analysis on the performance of various models for push broom imaging systems, has, therefore, been a subject matter of serious research.
3.0 Sensor Models
Differences between discreet and hybrid time domain models are brought out in this section. It is noted that a hybrid time domain system is a superset encompassing orbital parameters in continuous time domain and attitude parameters in discreet time domain. These hybrid time domain systems are more flexible and are well suited even for “step and stare” imaging platforms used for acquiring very high resolution data.
3.1 Sensor Models in Discreet Time Domain In rigorous sensor model, physical properties of push broom sensor acquisition and photogrammetric co-linearity equations are used to describe the perspective geometry of imaging sensor on a line by line basis. Spacecraft sensor position, velocity and attitude information is available at fixed intervals of time as ephemeris and is used as an input with approximate values for modeling sensor exterior orientation (position, velocity and attitude).
Using the notation of Mikhale Edward et al [18], co-linearity equations for use with push broom imagery are as under:
U= m11*(XSat–XA ) + m12 * (YSat–YA ) + m13 * (ZSat– ZA) . .(1) V = m21*(XSat–XA ) + m22 * (YSat–YA ) + m23 * (ZSat– ZA) . .(2) W = m31*(XSat–XA ) + m32 * (YSat–YA ) + m33 * (ZSat– ZA) . .(3) x – xp = 0 = U / W and --- (4) y – yp = -f * V / W. --- (5)
where:
i xp, yp and f are respectively principal point co-ordinates and focal length of the imaging sensor,
ii x and y are the line and pixel number of the imaged ground point,
iii XSat, YSat and ZSat are satellite position coordinates associated with the ground control point A and
iv XA, YA and ZA are co-ordinates of ground control point A..
v The condition x - xp equal to zero describes a frame-let based perspective geometry.
The matrix M = [mij] consists of rotations associated with sensor, view angle and attitude parameters and is realized in the following order:
M = [M View] * [M Attitude] * [M Sensor] …(6)
Fig 1: Linear Array Imaging Process
The components of attitude are modeled as drift rate changes associated with roll, pitch and yaw from the central line of the imagery. Similarly, the position, velocity and cross product of velocity and position form the components of sensor orientation and are calculated using ephemeris values of that imaging instance. In other words, the above parameters are modeled as 2nd order polynomials sampled at discreet time intervals. This model for satellite imagery has been influenced by the models traditionally used for aerial imaging systems.
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3.2 Sensor Models in Hybrid Time Domain In Kratky’s [16] approach, satellite position is derived from known nominal orbit relations. Components of sensor orientation are right ascension of ascending node including earth rotation, inclination and traveled angle that is a sum of true anomaly and argument of perigee. In my model the composite rotation of these three matrices is as under:
MSenor = M3 * M2 * M1 …….(7)
Where:
i M3 is a rotation matrix about Y axis for orbital traveled angle at the instance of imaging.
ii M2 is a rotation matrix about X axis for inclination of orbit.
iii M1 is a rotation matrix about Z axis for ascending node.
The attitude variations are modeled by either a simple linear equation or quadratic polynomial and on the lines undertaken for discreet time domain models.
Kratky’s model has been extensively investigated, extended by Baltsavias et al [9] for SPOT, Baltsavias et al [10] for MOMS-02/D2 and Baltsavias et al [11] for IKONOS, Daniela Poli et al [4] for MOMS-02/Priroda and Dieter Fritsch et al [7] for Landsat TM and JERS-1 data. These recent studies indicate that this model is very accurate and yet is flexible enough to integrate the parameters of more complex new push broom instruments. Other hybrid time domain implementations include Gugan et al [14], OíNeill et al [20], Dowman et al [6], Westin, [24] and Toutin [25].
Dieter Fritsch et al [7] note that Kratky’s rigorous photogrammetric bundle formulation includes imposing three additional constraints on geocentric distance “rs”, traveled angle “τ” and the geographic longitude λ realized as elliptical orbit parameters. In other words, if one were to rephrase Dieter Fritsch and Dirk Stallman observations from a control engineer’s perspective, the success of Kratky’s model is based on adoption of continuous time domain / hybrid time domain systems for Keplerian orbital dynamics.
4.0 Bundled Adjustment with Unified Least Squares Solution
Sensor parameters of Spot 2 push broom panchromatic imagery and bundled adjustment details are listed in this section. There are 6000 CCD elements in the push broom linear array that facilitates acquisition of image data at a 10 meter spatial resolution per pixel. This sensor is steer able at +/- 27 degrees from nadir at steps of 0.6 degrees to facilitate across track viewing, has a line integration time of 1.5 mille- seconds and admits of a swath width of 60kms. Orbits are determined regularly and the predicted ephemeris that is available along the video imagery.
The pre-facto and post-facto bundled adjustment values of 26 parameters comprising of full force model for Spot 2 scene acquired on 1st November 1988 with scene center at latitude 18.0092735 and long 73.5351357, are listed in table 1. Initial values of first eight parameters describing satellite orbit and imaging sensor are realized from satellite ephemeris files. The remaining eighteen parameters are realized as a part of the iterative bundled adjustment process with respect to translation (dx, dy and dz) and attitude (roll, pitch and yaw) in x, y and z dimensions. About seven ground control points were used as a part of bundled adjustment. It is noted that the translations are realized as refinements of keplerian parameters in continuous time domain while the attitude parameters are realized as 2nd order polynomials in discreet time domain whose sampling interval is a function of the detector integration time. All angular measurements are converted into radians and these parameters are listed as serials 1, 2, 3, 7, 18, 19, 20, 21, 22, 23, 24, 25 and 26 of table 1. The position and velocity parameters are converted to meters and are listed as serials 4, 9, 10, 11, 12, 13, 14, 15, 16 and 17 of table 1. The size of each CCD element is at 13 microns and the imaging sensor focal length of 1082mm is converted and listed as pixels in serial 8 of table 1.
Lee et al [13] have formulated the bundled adjustment equations corresponding to co-linearity equations by numerical differentiation. Our approach to formulate the bundled adjustment equations is, however, based on realizing the partial derivates as detailed by Slama et al in Manual of Photogrammet ry [18] and in Mikhail et al [14]. These partial derivates are linearized according to first-order Taylor decomposition with respect to unknown parameters as functions of time. The resulting system of bundled adjustment equations is solved iteratively with a unified least square adjustment. There is no known major difference either in performance or on the accuracy of the estimated values from either of these two approaches. As already noted in section 2, there is a high correlation between projection centre coordinates and sensor view angle. The bundled adjustment solution, therefore, has to address other issues such as inversion of singular matrices for realizing the refinements to adjustment parameters. The mathematical details are not detailed here as the objective of the present study is to understand the dynamics of imaging platform through an analysis of pre-facto and post-facto bundled adjustment parametric values.
5.0 Analysis Of Spacecraft Parameters Post-facto bundled adjustment analysis reveals that there is hardly any difference in the pre-facto and post-facto values of first eight parameters as listed in table 1. Likewise, the contributions of all translation parameters with an exception of dz0 are within a sub-pixel level. The contribution of dz0 corresponds to satellite to ground surface look-vector distance.
The post facto values of attitude parameters describe orientation of imaging sensor and are, as expected, significantly different from pre-facto values
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Table 1
# Parameter
Description Pre-facto Values *
1.0e+005 Post-facto Values * 1.0e+005
1 Right Ascension of Ascending Node
0.0000430934813 0.00004309348133
2 Inclination of
orbit 0.0000021386431 0.00000213864318 3 Argument of
Perigee
0.0000280836226 0.00002808362265 4 Semi major axis
of orbit 0.0739927669678 0.07399276696788 5 Eccentricity of
orbit 0.00000026549
857 0.00000026549858
6 Time At Frame Center
0.21428115234 400
0.21428115234400 7 Across Track
View Angle 0.00000354301
838 0.00000354301838
8 Focal length in
pixels 0.83230769230
769 0.83230769230769
9 dx0 0 0.00000596263014
10 dx1 0 0.00000000001572
11 dx2 0 0.00000000101476
12 dy0 0 0.00003954055208
13 dy1 0 0.00000000016379
14 dy2 0 0.00000000321556
15 dz0 0 8.24991705845123
16 dz1 0 -0.00000000019978
17 dz2 0 -0.00000004068430
18 do0 0 0.00003732575930
19 do1 0 0
20 do2 0 0
21 dp0 0 -0.00001177291907
22 dp1 0 0
23 dp2 0 0
24 dk0 0 0.00002426113540
25 dk1 0 0
26 dk2 0 0
Residual parameters of 7 GCPs in pixles Point ID Residual in X Residual in Y
1 0.0006984 -0.00000024985699
2 -0.031852 -0.00000674963979
3 -0.055803 0.00014686304797
4 -0.089006 -0.00002321667466
5 0.1661612 -0.00012481995714
6 0.000555 0.00000052777432
7 0.009247 0.00000893070342
RMS Error
Along Lines 0 .0753 Not applicable
RMS Error
Along Pixels Not applicable 7.3497e-05
It is a common practice to list root mean square residual error (RMS error) in pixels as an indicator to convergence of bundled adjustment equations. Most of the models terminate their iteration process if RMS threshold fixed at a pixel level or at an acceptable integral multiple thereof is attained. CS Fraser et al [1], note that a 3D point positioning accuracy corresponding to 0.3 of the pixel footprint is possible but under practical test conditions accuracies of between 0.5and 2 pixels are more commonly encountered.
6.0 Conclusions
The output of post-pass Spot instrumentation is known for its excellent quality and its imaging process is a far simpler compared to other missions. Dynamics of this push broom scanner imagery has been specifically studied as a digital photogrammetric solution in hybrid time domain. The output listed as table 1 is that of a data set picked up at random.
For my model, RMS error has been separately computed as RMS error along lines and along pixels. It is noted that the RMS error along pixels is very small and can safely be equated as zero. The RMS error along lines is at second decimal of a pixel, which indicates that the model has attained a very high degree of convergence. In particular, it is interesting to note that even pixel no 5 that has got maximum root mean square residual error along lines, is also at a sub-pixel level.
The complete photogrammetric solution requires the full force 26 parameters listed in table 1. A careful analysis is indicative that the main contribution to the dynamics of imaging system is based on satellite movement along its orbital path and is described by true anomaly and right ascension of ascending node. The angular changes of these two parameters are included in co-linearity equations as linear changes with time.
Further, as expected, attitude parameters also contribute significantly to orientation imaging sensor as with aerial photographs and aerial scanning systems. The look vector slant range to the pixel on earth surface is function of the altitude of spacecraft above ground and is an important parameter contributing to faster convergence of the photogrammetric model. Similar results are available for many other missions that acquire data at very high resolutions.
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