The application of GPS precise point positioning technology in aerial triangulation

ISPRS Journal of Photogrammetry and Remote Sensing 64 (2009) 541–550

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The application of GPS precise point positioning technology in aerial triangulation Xiuxiao Yuan a,∗ , Jianhong Fu a , Hongxing Sun a , Charles Toth b a

Wuhan University, 129 Luoyu Road, Wuhan 430079, China

b

The Ohio State University, USA

article

info

Article history: Received 23 January 2008 Received in revised form 27 March 2009 Accepted 31 March 2009 Available online 28 April 2009 Keywords: GPS precise point positioning (PPP) GPS camera stations GPS-supported bundle block adjustment Error Accuracy

abstract In traditional GPS-supported aerotriangulation, differential GPS (DGPS) positioning technology is used to determine the 3-dimensional coordinates of the perspective centers at exposure time with an accuracy of centimeter to decimeter level. This method can significantly reduce the number of ground control points (GCPs). However, the establishment of GPS reference stations for DGPS positioning is not only labor-intensive and costly, but also increases the implementation difficulty of aerial photography. This paper proposes aerial triangulation supported with GPS precise point positioning (PPP) as a way to avoid the use of the GPS reference stations and simplify the work of aerial photography. Firstly, we present the algorithm for GPS PPP in aerial triangulation applications. Secondly, the error law of the coordinate of perspective centers determined using GPS PPP is analyzed. Thirdly, based on GPS PPP and aerial triangulation software self-developed by the authors, four sets of actual aerial images taken from surveying and mapping projects, different in both terrain and photographic scale, are given as experimental models. The four sets of actual data were taken over a flat region at a scale of 1:2500, a mountainous region at a scale of 1:3000, a high mountainous region at a scale of 1:32000 and an upland region at a scale of 1:60000 respectively. In these experiments, the GPS PPP results were compared with results obtained through DGPS positioning and traditional bundle block adjustment. In this way, the empirical positioning accuracy of GPS PPP in aerial triangulation can be estimated. Finally, the results of bundle block adjustment with airborne GPS controls from GPS PPP are analyzed in detail. The empirical results show that GPS PPP applied in aerial triangulation has a systematic error of halfmeter level and a stochastic error within a few decimeters. However, if a suitable adjustment solution is adopted, the systematic error can be eliminated in GPS-supported bundle block adjustment. When four full GCPs are emplaced in the corners of the adjustment block, then the systematic error is compensated using a set of independent unknown parameters for each strip, the final result of the bundle block adjustment with airborne GPS controls from PPP is the same as that of bundle block adjustment with airborne GPS controls from DGPS. Although the accuracy of the former is a little lower than that of traditional bundle block adjustment with dense GCPs, it can still satisfy the accuracy requirement of photogrammetric point determination for topographic mapping at many scales. © 2009 International Society for Photogrammetry and Remote Sensing, Inc. (ISPRS). Published by Elsevier B.V. All rights reserved.

1. Introduction Aerial triangulation (AT) is the basic method for analyzing aerial images in order to calculate the 3-dimensional coordinates of object points and the exterior orientation elements of images. Up until now, bundle block adjustment has been commonly employed for AT, and numerous ground control points (GCPs) are necessary for the adjustment computation (Wang, 1990). In the 1950s,

∗

Corresponding author. Tel.: +86 27 68778083; fax: +86 27 68778086. E-mail address: [email protected] (X. Yuan).

photogrammetrists began exploiting other auxiliary data to reduce the number of GCPs. However, investigation did not achieve an implementing result because of the many technological limitations at that time (Li and Shan, 1989). In the 1970s, with the application of Global Positioning System (GPS), the situation changed a lot. GPS can provide 3-dimensional coordinates of surveying points with centimeter accuracy in differential mode, it was therefore applied in AT to measure the spatial position coordinates of the projection centers (referred to as GPS camera stations or airborne GPS control points). In this way, the number of GCPs could be significantly reduced. Block adjustment of combined photogrammetric observations and GPS-determined positions of perspective centers is regarded as GPS-supported AT. Since the

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X. Yuan et al. / ISPRS Journal of Photogrammetry and Remote Sensing 64 (2009) 541–550

beginning of the 1980s, many papers have presented the significant research and experimental results of GPS-supported AT (Ackermann, 1984; Friess, 1986; Lucas, 1987). After about 20 years of these efforts, GPS-supported AT was extensively applied in aerial triangulation at many scales and in all types of terrain. It is particularly beneficial in areas where they are difficult to establish ground control (Ackermann, 1994). In the late 1990s, with the development of sensor technology, an integrated system of GPS / Inertial Navigation System (POS) was first used in AT to obtain the position and attitude information of aerial images directly. This technology, in theory, can eliminate the need for GCPs. However, research indicates that the digital orthophoto map can be made directly by image orientation parameters obtained via a POS (Cannon and Sun, 1996; Cramer et al., 2000; Heipke et al., 2001), but there will be larger vertical parallax when stereo models are reconstructed using these image orientation parameters and the height accuracy cannot satisfy the requirement of large scale topographic mapping. Therefore, a bundle block adjustment should be made, combined image orientation parameters obtained via the POS and photogrammetric observations (Greening et al., 2000). Whether exploiting GPS data or POS data in AT, DGPS positioning is necessary to provide the GPS camera stations at present. In the DGPS mode, one or more GPS reference stations should be emplaced on the ground and observed synchronously and continuously together with the airborne GPS receiver during the entire flight mission. Additionally, signals from GPS satellites should be received as few transmission interruptions as possible. Initialization surveying is also required before aircraft takes off and static surveying should be performed after landing. In the processing of GPS observations, carrier phase differential technique is used to eliminate or reduce GPS positioning errors, including satellite clock error, satellite orbit error, atmospheric delay error, and so on. Generally speaking, it is difficult to emplace proper GPS reference stations when the aerial photographic region is with large scope or difficult to access and communicate. In order to guarantee the quality of aerial images, a survey area must be photographed for a long period, which is result from the shortage of weather suitable for photography. GPS reference stations must therefore remain in place for a long time. Moreover, the accuracy of DGPS positioning is relevant to the length of baseline. The longer the baseline, the weaker the correlation between ionospheric refraction error and tropospheric delay error. Due to the need for spatial correlation of atmospheric delay errors, the lengths of GPS differential baselines are typically limited to within 20 km if centimeter level accuracy is required with high reliability (Sun, 2004). When it comes to aerial photogrammetry, this is difficult because the length of survey areas is typically more than 200 km and the distance between the survey area and the airport may be greater. For baselines with long length, the atmospheric delay mainly – composed of ionospheric delay and tropospheric delay – will degrade positioning accuracy significantly. In such cases, even the ionospheric delay can be almost removed by using dual frequency GPS receivers. However, there can still be a tropospheric delay within a few decimeters, meaning that for long baselines, the positioning accuracy is typically in the level of decimeters. At the same time, the establishment of GPS reference stations sometimes makes the implementation of a survey plan difficult due to traffic, communication and cost considerations. As a result, the method of replacing GPS reference stations by Continuous Operating Reference Stations (CORS) was proposed and obtained an accuracy in decimeter level compared with the results obtained by GPS reference stations (Bruton et al., 2001; Mostafa and Hutton, 2001). There are, however, no CORS in most of the survey areas, so this method cannot be applied extensively. With the development of GPS technology, the number of CORS is increasing all over the world and their distribution

is more and more reasonable. International GNSS Service (IGS) can provide precise satellite orbit and clock error products with accuracies of 5 cm and 0.1 ns (3 cm). Utilizing IGS products, if the atmospheric delay error can be removed, modeled or estimated at the centimeter level, it will be possible to obtain centimeter level positioning accuracy with only the observation of a single GPS receiver. Zumberge et al. presented a GPS precise point positioning (PPP) method based on an un-differenced mode and achieved centimeter level accuracy for static positioning (Zumberge et al., 1997, 1998). Later, Muellerchoen et al. presented a method for realizing GPS global precise real time kinematic positioning by using single epoch un-differenced dual frequency observations after initialization (Muellerchoen et al., 2000). In this way, centimeter to decimeter level accuracy can be achieved for aerial GPS kinematic positioning at present (Gao and Chen, 2004; Zhang et al., 2006). If GPS PPP technology is applied in GPS-supported AT, only one GPS receiver is mounted on the aircraft and GPS reference stations on the ground are no longer required. GPS-supported AT can therefore be implemented very easily and with great flexibility, which is obviously significant in large survey blocks or areas with difficult terrain. Therefore, GPS PPP technology is discussed in this paper based on the highly dynamic characteristic of aerial remote sensing. The error law of GPS camera stations obtained by this method is analyzed, and the positioning accuracy and the feasibility of GPS-supported AT using GPS PPP technology are discussed. The goal of this work is to eliminate the need for the GPS reference stations in GPS-supported aerial photography by the GPS PPP technology. This technology can not only reduce the cost of aerial photography but also increase the flexibility of aerial photographic operations, which is beneficial to the widespread use of GPS-supported AT. 2. GPS precise point positioning for aerial triangulation In contrast to DGPS positioning technology, GPS PPP is a type of absolute GPS positioning which uses IGS precise orbit parameters and clock error products. The main algorithms and correction models for the GPS PPP have been discussed in many papers (Han et al., 2001; Kouba and Heroux, 2001; Holfmann et al., 2003; Chen et al., 2004) and the most widely used data type is un-differenced ionosphere-free carrier phase measurements, or an ionospherefree combination with carrier phase and code pseudorange measurements. An alternative data type used by some studies is code-phase ionosphere-free combination that aims at accelerating the convergence speed for parameter solutions (Gao and Chen, 2004). In this paper, the single difference model is employed for reasons that will be discussed below. For simplification, error corrections including relativity, satellite phase center offset, satellite wind up, earth body tide, ocean load correction and so on, will not be discussed here. The original un-differenced data type is formed by an ionosphere-free combination of dual frequency GPS data (Kouba and Heroux, 2001):

ρ˜ j = ρ j + c · (dt − dt j ) + M j · T + ερ˜

(1)

Φ j = ρ j + c · (dt − dt j ) + N j + M j · T + εΦ .

(2)

Here, j denotes satellite; ρ˜ is the ionosphere-free combination of L1 and L2 code pseudorange; ρ j is the geocentric distance from the GPS receiver to the satellite j; dt is the GPS receiver clock error; dt j is the clock error of the satellite j, which can be obtained from IGS products; c is the vacuum speed of light; T is the zenith tropospheric delay; M j is the mapping function of tropospheric delay for satellite j, for which several models can be used; Φ j is the ionosphere-free combination of L1 and L2 carrier phase; j

X. Yuan et al. / ISPRS Journal of Photogrammetry and Remote Sensing 64 (2009) 541–550

N j is the non-integer ambiguity of ionosphere-free carrier phase combination; ερ˜ and εΦ are the noises. There are five unknown parameters in Eq. (1), including the 3-dimensional spatial coordinates of the receiver (X , Y , Z ) lying in ρ j , the zenith tropospheric delay T and the receiver clock error dt. Furthermore, in Eq. (2), besides the same parameters in Eq. (1), the ambiguity N j is unknown. For these unknown parameters, the ambiguity N j is constant if the cycle slip is repaired and the zenith tropospheric delay T changes very slowly or remains unchanged over a short time span, for example, over two hours. The receiver clock error dt changes very quickly and the coordinates of the receiver (X , Y , Z ) are dependent on the vehicle movement status. In aerial photogrammetric applications, the craft often carries out large maneuvers, the sequential filter based on the dynamic models of all parameters (Kouba and Heroux, 2001) can not be implemented for high accurate positioning because the process noises of the vehicle movement and the receiver clock error are very large. In this case, the recursive least squares algorithm can be used to separate the receiver coordinate and clock error from other parameters which remain constant or change very slowly (Chen et al., 2004). Assuming that X and Y are the two kinds of parameters to be estimated, the observation equation in matrix form can be written as: L = AX + BY + ε,

ε ∼ N (0, σ02 P −1 )

(3)

where L is observation vector; X is correction vector of the coordinates of GPS receiver antenna phase center and clock error; Y is vector of ambiguity parameters and the correction parameters to zenith tropospheric delay; A and B are design matrices; ε is the noise vector; σ0 is the standard deviation of the noise; P is the weight matrix of observations. The normal equation of the least squares solution for Eq. (3) can be expressed as: AT PA BT PA



AT PB BT PB

  X Y



N = 11 N21

N12 N22

  X Y

AT PL = T . B PL





(4)

Eq. (4) is equivalent to Eq. (5) that can first resolve Y :



N11 0

N12 N¯ 22

  X Y

AT PL = ¯T B PL



 (5)

where, 1 ¯ = B − AN − B 11 N12

(6)

¯. N¯ 22 = BT P B

(7)

If just the second function in Eq. (5) is considered including only the unknown parameter Y , a converted observation equation can be obtained:

¯ + ε, L = BY

ε ∼ N (0, σ02 P −1 ).

(8)

The receiver coordinates and clock error in Eq. (3) can be removed from Eq. (8), leaving only the ambiguity and zenith tropospheric delay. From Eq. (8), it is easy to implement the sequential filter as the system dynamic model can be established with very high accuracy. Even the method of the accumulation of normal equations can be exploited directly if the zenith tropospheric delay is considered as a constant. After the ambiguity and the zenith tropospheric delay are resolved, the coordinates and clock error of the GPS receiver can be calculated by observations of each epoch. For carrier phase observations in each epoch, the number of undetermined parameters is always more than that of observations. So N 22 is a rank deficiency matrix and the second equation in Eq. (5) cannot be resolved. Regarding the noninteger ambiguity parameter and the zenith tropospheric delay as

543

constants, when multiple epochs (GPS observed time is no more than 10 min) of no cycle-slip GPS observations are measured, the observation equations of all the epochs can be calculated simultaneously. The non-integer ambiguity parameter and the zenith tropospheric delay can be calculated firstly, and then they can be used in Eq. (3) to solve X . −1 T X = N11 (A PL − N12 Y ).

(9)

For un-differenced ionosphere-free carrier phase combination Eq. (2), the receiver clock error dt and the ambiguity N j are linearlycorrelated so that they cannot be parsed in algebra. This means the system of Eq. (8) based on only the carrier phase observation in Eq. (2) cannot converge even if the receiver clock error is removed by the recursive least squares technique. Of course, if Eqs. (1) and (2) are both used to establish Eq. (8), the system can converge because the receiver clock error dt can be determined with only the code pseudorange observations of each epoch. However, in this case, the final positioning accuracy is influenced by the noise of the code pseudorange observations, which is much larger than that of the carrier phase observations. In an optimal environment involving good satellite geometry and an unobstructed view of sky, less than ten minutes to half an hour observations will be enough for system convergence in decimeter level accuracy. However, in practical GPS kinematic applications, the satellite signal often loses lock and many cycle slips occur in the carrier phase observations due to environmental limitations, which results in a much longer system convergence time and a lower positioning accuracy than in an optimal environment. In this paper, the single difference-between-satellites ionosphere-free combination observation is used to remove the receiver clock error in Eqs. (1) and (2). Taking the difference between two satellite observations of Eqs. (1) and (2), the single difference ionosphere-free combination observation equations can be given as:

∇ ρ˜ ij = ∇ρ ij − c · ∇ dt ij + ∇ M ij · T + ∇ερ˜

(10)

∇ Φ ij = ∇ρ ij − c · ∇ dt ij + ∇ N ij + ∇ M ij · T + ∇εΦ

(11)

where ∇ denotes single difference; i and j denote satellites, and other symbols are the same as those in Eqs. (1) and (2). Since two of the parameters in the carrier phase observation Eq. (2) – the receiver clock error and one un-differenced ambiguity – are removed from Eq. (11) by combining the observation equation of one satellite with that of other satellites, the single difference model has an advantage over the un-differenced model. The system based only on Eq. (11) can converge, the system based on Eqs. (10) and (11) can converge more quickly than that of Eqs. (1) and (2). 3. GPS-supported bundle block adjustment GPS-supported AT is a method for resolving the 3-dimensional coordinates of ground objects and the exterior orientation parameters of the images together, in which the coordinates of the GPS camera stations determined by GPS kinematic positioning technology are regarded as weighted observations for importing into the AT. The most popular GPS-supported AT is GPS-supported bundle block adjustment, whose detailed algorithm can be found in the bibliography (Yuan, 2000). In order to solve the datum problem and eliminate the systematic errors of GPS observations, a few GCPs should be distributed at both ends of the adjustment block. The typical distribution for the GCPs contains two schemes: (i) emplacing one cross strip on each end of the adjustment block and four full GCPs in the four corners around the adjustment block (Fig. 1(a));

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X. Yuan et al. / ISPRS Journal of Photogrammetry and Remote Sensing 64 (2009) 541–550

(a)

(b)

Fig. 1. Ground control schemes in GPS-supported bundle block adjustment: (a) 4 full GCPs in the corner + 2 cross strips; (b) 4 full GCPs in the corner + 2 rows height GCPs. Table 1 Technical parameters of aerial images in empirical blocks. Specifications

Test 1

Test 2

Test 3

Test 4

Date Aircraft Aerial camera Flight control system POS system Film Principal length Frame Average image scale Longitudinal overlap Lateral overlap Strips Cross strips Number of images GCPs Object points Area Maximum terrain undulation Airborne GPS receiver GPS receiver of reference station Sampling interval of airborne GPS receiver Sampling interval of ground GPS receiver Maximum range from aircraft to GPS reference station GPS initialization time Antenna–camera offset

Nov, 2004 Yun-12 Leica RC-30 Track Air POS AV 510 Kodak 2444 153.84 mm 23 cm × 23 cm 1:2500 61% 32% 9 2 255 72 3632 4 km × 5 km 38.6 m (flat) Trimble BD 950 Ashtech 0.1 s 2.0 s 28.99 km 5 min 0.303 m, −0.110 m, −2.029 m

Oct, 2005 Yun-12 Leica RC-30 Track Air POS AV 510 Kodak 2044 303.64 mm 23 cm × 23 cm 1:3000 63% 33% 10 2 405 155 6538 5 km × 8 km 181.6 m (mountainous) Trimble BD 950 Trimble 5700 0.1 s 0.5 s 28.46 km 5 min 0.303 m, −0.110 m, −2.002 m

Sep, 2005 Yun-8 Leica RC-30 CCNS 4 POS AV 510 Kodak 2402 154.06 mm 23 cm × 23 cm 1:32000 64% 33% 9 2 244 34 2957 47 km × 52 km 729.3 m (high mountainous) Trimble BD 950 Trimble 5700 0.1 s 1.0 s 187.59 km 5 min −2.015 m, −0.030 m, 3.102 m

Oct, 2005 Citation II Leica RC-30 CCNS 4 POS AV 510 Kodak 2402 153.53 mm 23 cm × 23 cm 1:60000 64% 30% 4 0 48 29 712 40 km × 57 km 109.3 m (upland) Trimble BD 950 Trimble 5700 0.1 s 1.0 s 145.40 km 5 min 2.034 m, −0.520 m, 1.320 m

(ii) emplacing four full GCPs in the four corners around the adjustment block and one row of height control points at each end of the adjustment block (Fig. 1(b)) (Ackermann, 1991). However, for the topographic mapping at medium and small scales or mountainous terrain, the accuracy of photogrammetric point determination obtained by GPS-supported bundle block adjustment without any GCPs can satisfy the requirements of topographic mapping specifications (Yuan et al., 2004). 4. Experimental results 4.1. Empirical test design In order to investigate the application feasibility of GPS PPP in GPS-supported AT and the accuracy of the AT with airborne GPS controls obtained by using the GPS PPP technique, four sets of aerial images taken from four actual photogrammetric mapping projects different in terrain and photographic scales were selected for experiments in this work. The main technical parameters of the empirical images are given in Table 1. After all negatives are scanned with a resolution of 21 µm to digitized images, the GPS-supported bundle block adjustment system WuCAPS (Yuan, 2008) is used for automatic point transfer. The corresponding image coordinates of all GCPs are measured manually in stereoscopic mode. The root mean square error (RMSE) of all image coordinates is statistically better than

±6.0 µm according to the results of the consecutive relative orientation with conditions for model connection and the function of gross errors eliminated by WuCAPS. All GCPs are determined by combined static GPS net surveying, and the planimetric coordinates are transformed to the Xian geodetic coordinate system 1980 under Gauss–Kruger projection, while the elevation coordinate system takes national height datum. The 3-dimensional coordinates of GCPs were supplied by four surveying and mapping corporations respectively. The planimetric accuracy is better than ±0.1 mm on the map in four test blocks. In Tests 1 and 2, elevation is measured by traditional leveling survey, and the accuracy is higher than ±0.1 m on the ground. In Tests 3 and 4, elevation is measured by a GPS geoid fitting method with accuracy of better than ±0.5 m on the ground. 4.2. The GPS PPP positioning The airborne GPS datasets of the four empirical blocks are processed by the GPS PPP kinematic positioning software Caravel PPP developed by authors according to the algorithm mentioned above. In order to reduce the magnitude of computation, airborne GPS data of 0.5 s sampling intervals are used. The final IGS satellite orbit and clock products are selected after 13 days of observation, with sampling intervals of 15 min and 5 min respectively. At the same time, the DGPS positioning of these projects is processed by POSPacTM (Applanix, 2007) to compare the results obtained by the GPS PPP.

X. Yuan et al. / ISPRS Journal of Photogrammetry and Remote Sensing 64 (2009) 541–550

(a)

545

0.40

δ E (m)

0.30

0.20

0.10

0

2000

4000

6000

8000

10000

12000

14000

16000

time (s)

(b)

0.40

δN (m)

0.30

0.20

0.10

0

2000

4000

6000

8000

10000

12000

14000

16000

10000

12000

14000

16000

time (s)

(c)

0.40

δH (m)

0.30

0.20

0.10

0

2000

4000

6000

8000 time (s)

Fig. 2. Internal accuracies of GPS precision point positioning in Test 1: (a) E direction; (b) N direction; (c) H direction.

4.2.1. Internal accuracy of GPS PPP positioning According to the description in Section 2, after solving the ambiguity and zenith tropospheric delay, the GPS PPP can be implemented by adopting least squares adjustment to the singledifference carrier phase observations of (m − 1) satellites observed in the same epoch. The theoretical accuracy δE ,N ,H can be set as an evaluation indicator of positioning accuracy, which is calculated by the standard deviation σˆ 0 of the residuals of GPS observations and the inverse coefficient matrix of normal equation when the adjustment has iterative convergence. It can also be defined to the internal accuracy of GPS PPP (Zhang et al., 2006).

q 1 δi = σˆ 0 (AT PA)− ii ,

(i = E , N , H )

(12)

where,

s σˆ 0 =

V T PV m−4

.

(13)

Internal accuracy can reflect the validity of adjustment model and the quality of GPS PPP positioning. The smaller the value, the higher the positioning accuracy of GPS PPP will be, and vice versa.

The change curve of internal accuracy in each epoch in Test 1 is shown in Fig. 2. It can be seen from Fig. 2 that the theoretical accuracies of 3-dimensional coordinates determined by GPS PPP technique in most epochs in Test 1 are better than ±0.15 m. Only several epochs are lower than ±0.20 m. That is because the change in the GPS satellites observed causes a jump between the few of epochs. The other tests have the same results as Test 1. This accuracy is similar to the best result reported by other papers (Gao and Chen, 2004; Zhang et al., 2006). If the correction to antenna phase center is more rigorously analyzed and ambulatory parameters are adopted to estimate the zenith tropospheric delay in mathematical model, the positioning accuracy may be further increased. 4.2.2. Result comparison between GPS PPP and DGPS In order to estimate the positioning accuracy of GPS PPP in more detail, positioning results of GPS PPP and DGPS are now compared. Table 1 shows the GPS reference station emplaced in the empirical block in Test 1 with the longest baseline 28.99 km. Therefore, the coordinate accuracies of DGPS positioning are very high. A whole flight mission including the empirical block of Test 1 is analyzed here. The actual flight track of the project is shown

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X. Yuan et al. / ISPRS Journal of Photogrammetry and Remote Sensing 64 (2009) 541–550

(a)

(b)

Fig. 5. The sky view of the GPS satellites (a) before and (b) after the time of coordinate difference jumping. Fig. 3. Flight tracks for a whole flight mission including Test 1.

in Fig. 3, in which the pane region expresses the nine strips of Test 1. The flight time of this project was about 5 h, 14 min and 23 s, and 188 630 epochs were observed. The change curves of the coordinate differences in three directions between GPS PPP and the DGPS are shown in Fig. 4. Fig. 4 demonstrates that there are differences in positioning results between GPS PPP and DGPS, which are less than ±0.8 m in East, ±0.5 m in North and ±1.2 m in Height. There is a large increase in the height due to the appearance of new satellite (PRN 6) which changes the geometrical distribution of GPS satellites (Fig. 5). The value of PDOP changes from 1.5 to 4.2, resulting in the instability of positioning results. In AT, GPS positioning results outside of the photographic strips are not adopted and the accuracies of their coordinates have no effect on GPS-supported bundle block adjustment. Therefore, only the positioning results in the photography strips are selected and used as the AT. Taking the photographic strip as the unit, the 3-dimensional coordinate differences between GPS PPP and DGPS of the nine strips (included in the dashed region in Fig. 4) are shown in Table 2. The empirical results show that the maximum absolute coordinate differences are less than 0.5 m in East, 0.3 m in North and 0.6 m in Height. The standard deviations are less than ±0.05 m in planimetry and ±0.09 m in elevation, the absolute mean value is close to their RMSE and is much greater than the standard deviations. This demonstrates the existence of evident systematic errors between the two kinds of coordinates, as well as showing that the positioning results of GPS PPP and DGPS have the same error law. 4.2.3. The absolute positioning errors of GPS PPP It is difficult to ascertain the absolute coordinate errors of GPS PPP because it is impossible to obtain the true coordinates of the

antenna phase center of the airborne GPS receiver. In fact, only GPS camera stations at exposure time are taken into account in AT. Therefore, traditional bundle block adjustment with dense GCPs is performed in the empirical blocks including the nine strips in Test 1. The 3-dimensional coordinates of the perspective center of each image can be calculated with WuCAPS by emplacing 23 full GCPs and 16 height GCPs as orientation points. According to the antenna–camera offset, the perspective center coordinates can be transformed to the coordinates of GPS antenna phase center. Treating these coordinates as ‘‘true value’’, the results of GPS PPP are compared with them one-by-one. The 3-dimensional coordinate differences can be set as the ‘‘true errors’’ of GPS PPP positioning. The change curves of ‘‘true errors’’ of GPS camera stations of 9 trips in Test 1 are shown in Fig. 6. Fig. 6 shows that the ‘‘true errors’’ of the GPS camera stations obtained by GPS PPP are systematic and change in the range of ±0.2 m in each strip, while there are clear jumps between the adjacent strips. This is due to atmospheric error. In addition, zenith tropospheric delay, which mostly affects elevation coordinates, cannot be estimated accurately in GPS PPP. The zenith tropospheric delay, in particular, is not a constant, and it changes with time. Therefore, a set of parameters to estimate this error for all strips in GPS PPP cannot reflect practice absolutely. Moreover, there are errors within precise satellite orbit parameters and clock errors too. When the coordinates of GPS camera stations are interpolated by adjacent ephemeris, there will be some errors. All these errors result in systematic error in the positioning results of GPS PPP. Fig. 6 also demonstrates that coordinate errors in East, in particular, change systematically along the strips. The errors in strips of the same flight direction are also very similar. The main reason is that the camera has an exposure time delay. In addition, the flight velocities are not the same in different directions. Moreover, in order to ensure the camera obliquities remain within limits, gimbal PV 30 is used in the flight mission, which will result in a change of the vector of antenna–camera offset in the image

Fig. 4. The coordinate difference between DGPS and GPS PPP.

X. Yuan et al. / ISPRS Journal of Photogrammetry and Remote Sensing 64 (2009) 541–550

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Table 2 Coordinate differences between GPS PPP and DGPS in the strip of Test 1 (unit: m). Strip no.

1

2

3

4

5

6

7

8

9

E N H

−0.238

−0.203

−0.500

−0.442

−0.416

0.088 0.368

−0.290 −0.246

−0.500

0.170

−0.155 −0.113

0.111

0.104

0.189

0.156

−0.094

−0.241 −0.142 −0.295

0.070

0.146

−0.552

−0.439

−0.523

−0.277

E N H

−0.133

−0.077

−0.178 −0.038 −0.001

−0.385

−0.284

−0.354

−0.343

0.000 0.234

−0.147 −0.021 −0.024

−0.079

0.000 −0.001

0.029 −0.425

0.013 −0.202

0.088 −0.284

−0.145

E N H

−0.176

−0.141

−0.113 −0.051

−0.405

−0.388

0.003

−0.237 −0.200 −0.011

−0.341

0.011 0.308

−0.195 −0.110 −0.187

−0.451

0.081 −0.011

0.074 −0.487

0.053 −0.318

0.154 −0.393

−0.225

RMSE

E N H

0.178 0.094 0.046

0.146 0.058 0.312

0.197 0.112 0.204

0.114 0.064 0.042

0.238 0.201 0.044

0.452 0.076 0.488

0.346 0.059 0.325

0.405 0.155 0.396

0.388 0.126 0.226

Std. deviation

E N H

0.024 0.048 0.044

0.039 0.057 0.045

0.027 0.018 0.081

0.017 0.038 0.042

0.025 0.023 0.043

0.034 0.019 0.031

0.054 0.028 0.070

0.016 0.020 0.050

0.017 0.018 0.022

4

Maximum

Minimum

Mean value

0.000 0.000

0.094

0.125

Table 3 GPS PPP positioning errors in each strip in Test 1 (unit: m). Strip no.

1

2

3

E N H

−1.290 −0.386

−0.329

−1.382

0.486 0.833

0.354 0.852

E N H

−1.042 −0.125

−0.003

−0.944

0.101 0.425

0.179 0.621

E N H

−1.154 −0.278

−0.101

−1.251

0.586

0.263 0.534

RMSE

E N H

1.156 0.287 0.588

Std. deviation

E N H

0.069 0.072 0.044

Maximum

Minimum

Mean value

0.656

0.450

5

6

0.209 0.404 0.625

−1.283

0.003 0.079 0.500

−1.007

0.005 0.243 0.576

−1.147

0.252 0.744

0.142 0.282 0.543

1.255 0.256 0.747

0.102 0.106 0.097

0.099 0.046 0.060

7

8

0.348 0.166 0.655

−1.397

−0.051 −0.005

−1.157 −0.002

0.501

0.276

0.214 0.067 0.585

−1.260

0.234 0.555

0.077 0.250 0.577

1.150 0.239 0.557

0.079 0.063 0.037

0.084 0.054 0.037

0.322 0.618 0.080 0.465

9 0.287 0.126 0.583

−1.188 −0.369

0.042 0.001 0.268

−0.759

0.047 0.382

0.159 0.026 0.421

−1.006 −0.114

0.233 0.085 0.586

1.262 0.067 0.385

0.173 0.055 0.428

1.012 0.167 0.415

0.095 0.054 0.041

0.070 0.049 0.051

0.070 0.050 0.079

0.120 0.125 0.149

0.141 0.464

0.677 0.018

−0.067

0.388

Fig. 6. GPS camera station coordinate errors in Test 1.

space coordinate system. This means that the errors of GPS PPP are systematic in each strip and not the same between strips. The ‘‘true errors’’ of nine strips in Test 1 are given in Table 3 respectively. The theoretical coordinate accuracy of perspective centers obtained by traditional bundle block adjustment reached mXs = ±3.4 cm, mYs = ±3.9 cm, mZs = ±2.4 cm. The accuracy is much higher than that of GPS positioning. Therefore, the ‘‘true errors’’ in Table 3 can be regarded as the absolute errors of GPS PPP positioning. It can be seen from Table 3 that the maximum of the absolute coordinate difference is less than 1.4 m in East, 0.5 m in North and 0.9 m in Height. Moreover, the standard deviations of the coordinate differences are less than ±0.10 m in all three directions, the absolute mean values of these errors are close to the RMSE and are much higher than the standard deviations (excepted for the 9th strip). This empirical result demonstrates that there are evident

systematic errors in the positioning results of GPS PPP in each strip. Fig. 7 is the enlarged figure of the first strip in Test 1. It can be seen from Fig. 7 that the ‘‘true errors’’ of GPS camera stations obtained by GPS PPP are systematic and changes linearly with flight time. The 3-dimensional coordinates of GPS camera stations are used as a kind of weighted observations in bundle block adjustment. Their effects on adjustment can then be eliminated by adopting proper systematic error compensation model strip-by-strip in the overall adjustment. 4.3. Accuracy of GPS-supported bundle block adjustment In order to confirm the feasibility of GPS PPP applied in AT, the 3-dimensional coordinates of GPS camera stations obtained by GPS

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X. Yuan et al. / ISPRS Journal of Photogrammetry and Remote Sensing 64 (2009) 541–550 1.0

error (m)

0.5 0.0 -0.5 -1.0 -1.5

1

3

5

7

9

11

13

15

17

19

21

camera station number

Fig. 7. GPS PPP positioning errors of the first strip in Test 1.

Table 4 The accuracy statistic of bundle block adjustment in AT. Test

Control modela

σ0 (µm)

1

Ground DGPS PPP

5.7 6.9 6.9

Ground DGPS PPP

2

3

4

a b c

Number of orientation points

Number of check points

Maximum residuals of check point coordinates (m)b

RMSE (m)c

Plan.

Height

Plan.

East

Plan.

Height

East

North

Plan.

Height

23 4 4

39 4 4

49 68 68

33 68 68

0.14 0.23 0.23

0.18 0.23 0.19

0.198 0.232 0.231

−0.241

0.06 0.09 0.09

0.08 0.09 0.09

0.103 0.125 0.128

0.079 0.111 0.103

4.8 6.6 6.6

39 4 4

69 4 4

116 151 151

86 151 151

0.17 0.19 0.18

−0.20 −0.29 −0.25

0.205 0.328 0.305

−0.382

0.06 0.07 0.06

0.06 0.11 0.11

0.087 0.131 0.129

0.126 0.157 0.169

Ground DGPS PPP

6.4 7.3 7.3

20 4 4

6 4 4

14 30 30

9 30 30

−0.98 −1.59 −1.60

−1.42 1.39 1.48

1.037 1.804 1.741

−1.002 −1.113

0.48 0.76 0.77

0.67 0.82 0.82

0.824 1.116 1.120

0.559 0.548 0.479

Ground DGPS PPP

6.9 6.4 5.8

15 4 4

19 4 4

14 25 25

10 25 25

−2.78

3.12 4.02 4.03

3.384 4.893 4.696

−2.644 −2.599

1.29 1.88 1.73

1.28 1.46 1.45

1.818 2.375 2.257

1.416 1.350 1.544

Height

4.68 4.13

North

0.281 0.267 0.351 −0.390 0.843

2.027

‘Ground’, ‘DGPS’ and ‘PPP’ denote the bundle block adjustment with GCPs, DGPS airborne controls and PPP airborne controls respectively. The residual ∆ denotes the difference between the results of AT and the known coordinatesq of the check points. q

The RMSE is calculated as µi =

P

∆2i /n, where n is number of the check points; µPlan. =

PPP are used in GPS-supported bundle block adjustment (referred to as GPS PPP-supported AT) processing with WuCAPS. Moreover, GPS-supported bundle block adjustment with DGPS airborne controls (referred to as DGPS-supported AT) and conventional bundle block adjustment with dense GCPs are processed together with the same data to evaluate the accuracy of photogrammetric point determination. The ground control scheme shown in Fig. 1(a) is exploited for Tests 1, 2 and 3, and the scheme shown in Fig. 1(b) is used for Test 4 because of the absence of control strips. The systematic error parameters are considered for each strip in the GPS-supported bundle block adjustment to compensate for the systematic errors of GPS PPP and DGPS. Some GCPs were used as check points in the AT processing. Table 4 lists the detailed statistic results of three models of bundle block adjustment. Table 1 shows that images in Tests 1 and 2 can be used as topographic mapping at scales of 1:500 to 1:2000, while images in Test 3 can be used as topographic mapping at scales of 1:5000 to 1:10000 and images in Test 4 can be used as topographic mapping at a scale of 1:50000 respectively. According to topographic mapping specifications for the aerophotogrammetric office operation of China, Test 1, Test 2, Test 3 and Test 4 can be classified as flat region, mountainous region, high mountainous region and upland region. It can be seen from Tables 1 and 4 that: (1) For Test 1, the coordinate RMSE of all check points is better than ±0.15 m in planimetry and elevation respectively, and the maximum coordinate unconformities are 0.232 m in planimetry and 0.281 m in elevation. This result meets the accuracy requirement of the AT for the flat topography mapping at the scale of 1:500, which requires that the tolerance of check point coordinate unconformities should be

µ2E + µ2N .

less than 0.25 m for planimetry and less than 0.30 m for elevation determination (GB 7930-87, 1998). (2) For Test 2, the coordinate RMSE of all check points is better than ±0.15 m in planimetry and is better than ±0.20 m in elevation, and the maximum coordinate unconformities are 0.328 m in planimetry and −0.390 m in elevation. This result also satisfies the accuracy requirement of the AT for the mountain topography mapping at the scale of 1:500, which requires that the tolerance of check point coordinate unconformities should be less than 0.35 m for planimetry and less than 0.40 m for elevation determination (GB 7930-87, 1998). (3) For Test 3, the coordinate RMSE of all check points is better than ±1.2 m in planimetry and is better than ±1.0 m in elevation, and the maximum coordinate unconformities are 1.804 m in planimetry and −1.113 m in elevation. This result meets the accuracy requirement of the AT for the mountain topography mapping at the scale of 1:5000, which requires that the tolerance of check point coordinate unconformities should be less than 2.5 m for planimetry and less than 2.0 m for elevation determination (GB/T 13990-92, 1993). (4) For Test 4, the coordinate RMSE of all check points is better than ±3.0 m in planimetry and is better than ±1.5 m in elevation, and the maximum coordinate unconformities are 4.893 m in planimetry and −2.644 m in elevation. This result meets the accuracy requirement of the AT for the hill topography mapping at the scale of 1:50000, which requires that the tolerance of check point coordinate unconformities should be less than 17.5 m for planimetry and less than 3.0 m for elevation determination (GB 13340-90, 1991).

X. Yuan et al. / ISPRS Journal of Photogrammetry and Remote Sensing 64 (2009) 541–550

In short, the coordinate accuracies of the object points in GPS PPP-supported AT are slightly lower than those of conventional bundle block adjustment with dense GCPs. They do, however, satisfy the specifications of topographic mapping in aerial photogrammetry. Moreover, if four full GCPs are emplaced in the four corners of the adjustment region, there is no obvious difference between the accuracies in ATs with GPS PPP and that with DGPS, which embodies that even there are more errors in positioning results with GPS PPP than DGPS. In spite of the presence of systematic errors in each strip, the final results of photogrammetric point determination will not be influenced if the systematic error in each strip is rectified in the process of overall adjustment. It accordingly demonstrates the validity of the error analysis in Section 4.2. However, these empirical images are actual aerial images for topographic mapping engineering, and all GCPs are outstanding object points, but not artificial target points. The corresponding image coordinates of all GCPs are measured manually in the stereoscopic mode, containing certain interpretation errors. Especially for these images at a small scale of 1:60000 in Test 4, the image measurement errors of GCPs are bigger, which affect the final adjustment results. 5. Conclusions The theory of GPS PPP and its application in AT are discussed in this paper. The empirical results of actual projects and the comprehensive analysis validate the feasibility of adopting GPS PPP in AT. Our conclusions are: (1) Owing to the errors aroused by tropospheric delay, interpolating orbit parameters and clock errors, there are some differences in positioning results between GPS PPP and DGPS. When the continuous photography time in one strip is less than 15 min, coordinate differences almost present a systematic characteristic changing linearly with flight time. (2) The 3-dimensional coordinates of GPS camera stations determined by GPS PPP are regarded as a kind of weighted observations in bundle block adjustment. When a systematic error compensation model changing linearly with time are adopted strip-by-strip in the process of overall adjustment and four full GCPs are emplaced in the four corners of the adjustment block, the effects of positioning errors of GPS camera stations on photogrammetric point determination can be almost eliminated. The coordinate accuracies of object points are the same as those obtained by DGPS-supported bundle block adjustment. Therefore, GPS PPP can replace DGPS to be applied in topographic mapping according to the existing specifications of topographic mapping in aerial photogrammetry. (3) GPS PPP eliminates the need to establish GPS reference stations, so only one airborne GPS receiver is required. GPS PPP can not only reduce the difficulty and cost of aerial photography, but also increase the flexibility of aerial photography. The performance model of GPS-supported AT can also be replaced by GPS PPP, which makes more extensive applications. GPS-supported AT will play a very important role in national fundamental surveying and mapping, especially in non-mapped areas, inaccessible regions and on the frontiers. As is well known, the accuracy of DGPS-supported AT has been already proved and DGPS-supported AT has been employed for topographic mapping in regions that are difficult to map in China for about 10 years. The aim of the work is to validate the feasibility of GPS PPP-supported AT and reach the same accuracy with DGPSsupported AT, avoiding the use of GPS reference stations in GPSsupported aerial photography and simplifying the workflow. Of course, work still needs to be done on improving the accuracy of the GPS-supported bundle block method.

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