Introduction

Bundle block adjustment is the most popular method in aerial triangulation compared with strip and independent model block adjustments. With the global positioning system (GPS) receiver installed on an airborne vehicle, the coordinates of the GPS antenna can be obtained and utilized in the bundle block adjustment (Ackermann 1984). GPS-supported aerial triangulation adopts GPS antenna coordinates to facilitate the estimation of exterior orientation parameters of aerial images and thus significantly reduce the number of ground control points as compared to conventional aerial triangulation (Friess 1986; Lucas 1987).

In 1990s, GPS-supported aerial triangulation mainly relied on the differential GPS (DGPS) mode, which required at least one GPS receiver with precise known coordinates established on the ground (Ackermann 1994). With synchronized observations of the ground GPS receiver, most common errors including the satellite clock error, troposphere and ionosphere errors for the airborne GPS receiver can be eliminated by observation differencing. Depending on the size of coverage, one or more ground GPS receivers are necessary, which consequently increases the logistic difficulties and the costs. The ground and airborne GPS receivers generally cannot be separated far, i.e., they should be within 20 km in order to retain the high accuracy of GPS kinematic positioning (Yuan et al. 2009). Zumberge et al. (1997) proposed a GPS precise point positioning (PPP) approach which can eliminate the need for ground receivers and provide millimeter-to-decimeter-level positioning accuracy using a standalone receiver. With IGS precise products to remove satellite orbit/clock errors and empirical models to calibrate systematic errors including antenna phase center offset and variation, phase windup, Sagnac and relativistic effects and tropospheric delay(Petit and Luzum 2010), PPP can offer millimeter-to-decimeter-level positioning accuracies depending on the receiver dynamics. More specifically, GPS PPP for airborne vehicles can provide centimeter-to-decimeter-level kinematic positioning accuracy (Gao and Chen 2004; Zhang and Forsberg 2007).

However, precise satellite orbit and clock products required by PPP always suffer a latency which prevents GPS PPP-supported aerial triangulation from being a time-critical application. Thanks to the availability of International GNSS Service (IGS) real-time service (RTS) since 2013, real-time PPP becomes feasible. Over the past years, most real-time PPP studies have been made for centimeter-level static positioning (Shi et al. 2014), rainfall monitoring (Shi et al. 2015a), numerical weather prediction (Dousa and Vaclavovic 2014), earthquake monitoring (Wright et al. 2012) and early warning systems (Geng et al. 2013). Yet no research has been reported to address real-time kinematic PPP to support aerial triangulation, which is the focus of this study.

The mathematics of real-time kinematic PPP-supported aerial triangulation is described first. The accuracy assessment of real-time orbit and clock products is carried out afterward, followed by the performance evaluation of real-time kinematic PPP to support aerial triangulation from both the GPS positioning and the aerial triangulation perspectives. Finally, conclusions and perspective for future research are provided.

Methodology

This section presents the mathematics of real-time kinematic PPP to determine the coordinates of the GPS antenna installed on the airplane, and the application of such coordinates in bundle block adjustment.

Real-time kinematic PPP

Traditional ionospheric-free (IF) PPP function model can be expressed as (Kouba 2009),

$$ P_{\text{IF}} = \rho + d_{\text{orb}} + c(dt^{\text{r}} - dt^{\text{s}} ) + d_{\text{trop}} {\kern 1pt} + \varepsilon_{{P_{\text{IF}} }} $$
(1)
$$ L_{\text{IF}} = \rho + d_{\text{orb}} + c(dt^{\text{r}} - dt^{\text{s}} ) + d_{\text{trop}} - \lambda_{\text{IF}} N_{\text{IF}} + \varepsilon_{{P_{\text{IF}} }} $$
(2)

where \( P_{\text{IF}} = \frac{{f_{1}^{2} }}{{f_{1}^{2} - f_{2}^{2} }}P_{1} + \frac{{ - f_{2}^{2} }}{{f_{1}^{2} - f_{2}^{2} }}P_{2} \) and \( L_{\text{IF}} = \frac{{f_{1}^{2} }}{{f_{1}^{2} - f_{2}^{2} }}L_{1} + \frac{{ - f_{2}^{2} }}{{f_{1}^{2} - f_{2}^{2} }}L_{2} \) are ionospheric-free code and phase observables with \( f_{1} = 154\;f_{0},\, f_{2} = 120\;f_{0},\, f_{0} = 10.23\;{\text{MHz}} \), respectively, \( \rho \) is the geometric distance as a function of the receiver and satellite coordinates, \( d_{\text{orb}} \) is the satellite orbit error, c is the speed of light in vacuum, \( dt^{\text{r}} \) is the receiver clock error, \( dt^{\text{s}} \) is the satellite clock error, \( d_{\text{trop}} \) is the troposphere delay, \( \lambda_{\text{IF}} \) is the carrier phase wavelength of the ionospheric-free ambiguity \( N_{\text{IF}} \) and \( \varepsilon \) contains residual errors including multipath and noises.

For real-time kinematic PPP, the IGS real-time service offers satellite coordinate and clock corrections of the broadcast ephemeris by

$$ {\mathbf{r}}_{{{\mathbf{SSR}}}} = {\mathbf{r}} - [\begin{array}{*{20}c} {{\mathbf{e}}_{{{\mathbf{radial}}}} } & {{\mathbf{e}}_{{{\mathbf{along}}}} } & {{\mathbf{e}}_{{{\mathbf{cross}}}} } \\ \end{array} ]\varvec{\delta}{\mathbf{O}} $$
(3)
$$ dt^{\text{s}} = dt_{\text{BRDC}}^{\text{s}} + \delta C/c $$
(4)

where \( {\mathbf{r}}_{{{\mathbf{SSR}}}} \) is the satellite coordinate vector after the real-time correction, \( {\mathbf{\delta O}} \) is the real-time satellite orbit correction vector in the form of the space state representative (SSR) in radial, along- and cross-track components, \( {\mathbf{e}}_{{{\mathbf{radial}}}} = {\dot{\mathbf{r}}}/\left| {{\dot{\mathbf{r}}}} \right| \), \({\mathbf{e}}_{\mathbf{along}} = ({\mathbf{r}} \times {\dot{\mathbf{r}}})/\left| {{\mathbf{r}} \times {\dot{\mathbf{r}}}} \right| \) and \( {\kern 1pt} {\mathbf{e}}_{{{\mathbf{cross}}}} = {\mathbf{e}}_{{{\mathbf{radial}}}} \times {\mathbf{e}}_{{{\mathbf{along}}}} \) are three direction unit vectors, r and \( {\dot{\mathbf{r}}} \) are satellite coordinate and velocity vectors calculated by the broadcast ephemeris, \( dt^{s} \) and \( dt_{\text{BRDC}}^{\text{s}} \) are corrected and original satellite broadcast clock error corrections, \( \delta C = C_{0} + C_{1} \Delta t + C_{2} \Delta t^{2} \) is the clock correction, \( \Delta t \) is the interval with respect to the reference time and C 0, C 1 and C 2 are three clock correction coefficients in the SSR message (RTCM 2011).

With the real-time satellite coordinate vector \( {\mathbf{r}}_{{{\mathbf{SSR}}}} \) and the real-time satellite clock error correction \( dt^{\text{s}} \) to remove the orbit and clock errors in (1) and (2), four types of unknown parameters are estimated in the PPP function model including the receiver coordinates \( \left[ {X_{\text{PPP}} ,{\kern 1pt} Y_{\text{PPP}} ,Z_{\text{PPP}} } \right]^{\text{T}} \), the receiver clock error \( dt^{\text{r}} \), the tropospheric parameter \( d_{\text{trop}} \) and the phase ambiguities \( N_{\text{IF}} \). Some general processing settings for real-time kinematic PPP for aerial triangulation are provided in Table 1.

Table 1 General settings of real-time kinematic PPP for aerial triangulation
Full size table

Real-time kinematic PPP to support aerial triangulation

In the real-time kinematic PPP-supported aerial triangulation, the GPS antenna coordinates \( \left[ {X_{\text{PPP}} ,{\kern 1pt} Y_{\text{PPP}} ,Z_{\text{PPP}} } \right]^{\text{T}} \) should first be interpolated to the coordinates at the camera exposure epoch \( \left[ {\begin{array}{*{20}c} {X_{\text{PPP}} } & {Y_{\text{PPP}} } & {Z_{\text{PPP}} } \\ \end{array} } \right]_{\text{exposure}}^{\text{T}} \) and then be considered as quasi-observations in the bundle block adjustment by

$$ \left[ \begin{aligned} X_{\text{PPP}} \hfill \\ Y_{\text{PPP}} \hfill \\ Z_{\text{PPP}} \hfill \\ \end{aligned} \right]_{\text{exposure}} = \left[ \begin{aligned} X_{S} \hfill \\ Y_{S} \hfill \\ Z_{S} \hfill \\ \end{aligned} \right] + {\mathbf{R}}\left[ \begin{aligned} u \hfill \\ v \hfill \\ w \hfill \\ \end{aligned} \right] + \left( {\left[ \begin{aligned} a_{X} \hfill \\ a_{Y} \hfill \\ a_{Z} \hfill \\ \end{aligned} \right] + (t - t_{0} ) \cdot \left[ \begin{aligned} b_{X} \hfill \\ b_{Y} \hfill \\ b_{Z} \hfill \\ \end{aligned} \right]} \right) $$
(5)

where \( \left[ {\begin{array}{*{20}c} {X_{S} } & {Y_{S} } & {Z_{S} } \\ \end{array} } \right]^{\text{T}} \) are the coordinates of the perspective center of the aerial camera, or exterior orientation parameters, \( \left[ {\begin{array}{*{20}c} u & v & w \\ \end{array} } \right]^{\text{T}} \) are GPS antenna offsets with respect to the camera perspective center, R is an orthogonal rotation matrix as a function of three angle elements of exterior orientation parameters \( \omega ,\phi ,\kappa \), \( \left[ {\begin{array}{*{20}c} {a_{X} } & {a_{Y} } & {a_{Z} } \\ \end{array} } \right]^{\text{T}} \) and \( \left[ {\begin{array}{*{20}c} {b_{X} } & {b_{Y} } & {b_{Z} } \\ \end{array} } \right]^{\text{T}} \) are compensation coefficients of GPS positioning translation and drift errors in XYZ directions, t 0 is normally taken as the epoch of the first exposure within one strip and t is the epoch of the current exposure (Yuan 2008).

The collinearity equation of the aerial triangulation is

$$ \begin{aligned} x = x_{0} - f\frac{{a{}_{1}(X - X_{S} ) + b{}_{1}(Y - Y_{S} ) + c{}_{1}(Z - Z_{S} )}}{{a{}_{3}(X - X_{S} ) + b{}_{3}(Y - Y_{S} ) + c{}_{3}(Z - Z_{S} )}} \hfill \\ y = y_{0} - f\frac{{a{}_{2}(X - X_{S} ) + b{}_{2}(Y - Y_{S} ) + c{}_{2}(Z - Z_{S} )}}{{a{}_{3}(X - X_{S} ) + b{}_{3}(Y - Y_{S} ) + c{}_{3}(Z - Z_{S} )}} \hfill \\ \end{aligned} $$
(6)

where (x, y) are measured image point coordinates, \( \left[ {X,Y,Z} \right]^{\text{T}} \) are coordinates of ground object points, \( \left( {x_{0} ,y_{0} ,f} \right) \) are three interior orientation parameters and \( \left( {a_{i} ,b_{i} ,c_{i} } \right),i = 1,2,3 \) are functions of the three angle elements of exterior orientation parameters \( (\omega ,\phi ,\kappa ) \).

Equations (5) and (6) compose the function model of the real-time PPP-supported aerial triangulation. Suppose the number of flight strips, camera exposures, image points and ground object points are n, m, k and l, respectively. In general, the triple overlap is required in order to construct stereo images and connect stereo images between conjunct strips. The function model of real-time PPP-supported aerial triangulation consists of 3 m quasi-observations (5) and 2 k collinear Eqs. (6). Unknown parameters include three flight-dependent GPS antenna offset parameters \( \left[ {\begin{array}{*{20}c} u & v & w \\ \end{array} } \right]^{\text{T}} \), 6n strip-dependent linear drift compensation coefficients \( \left[ {\begin{array}{*{20}c} {a_{X} } & {a_{Y} } & {a_{Z} } \\ \end{array} } \right]^{\text{T}} \) and \( \left[ {\begin{array}{*{20}c} {b_{X} } & {b_{Y} } & {b_{Z} } \\ \end{array} } \right]^{\text{T}} \), 6m image-dependent exterior orientation parameters \( \left[ {\begin{array}{*{20}c} {X_{S} } & {Y_{S} } & {Z_{S} } \\ \end{array} } \right]^{\text{T}} \) and \( (\omega ,\phi ,\kappa ) \) and 3l point-dependent coordinates of ground object points \( \left[ {X,Y,Z} \right]^{\text{T}} \).

Experiments and analyses

In order to verify the feasibility of real-time kinematic PPP for aerial triangulation, three tests are assessed with detailed specifications given in Table 2. Tests 1, 2 and 3 cover about 110, 140 and 30 km2 with maximum terrain undulations of approximately 100 (hilly region), 250 (mountainous region) and 40 m (flat region), respectively. Although the coverage and terrain undulation do not affect GPS kinematic PPP solution, these ground conditions influence the performance of aerial triangulation. Specifically, the flat region has the most rigorous accuracy requirement for ground check points, whereas the mountainous region has the least. If these three tests featuring different ground conditions can be proven to satisfy the corresponding accuracy requirement, the real-time kinematic PPP can be considered practical for aerial photogrammetric mapping applications.

Table 2 Experiment description for real-time kinematic PPP-supported aerial triangulation used in this study
Full size table

Airborne GPS observations were collected by the NovAtel OEM4 at the sampling rate of 0.5, 0.5 and 0.2 s for three tests. In Tests 1 and 2, ultralight aircrafts were utilized to load the GPS receiver and the Leica RCD30 aerial camera at an average altitude of about 1100 m, while in Test 3, an unmanned aerial vehicle and Nikon D800 were adopted at an altitude of about 1050 m. A total number of 15, 17 and 16 flight strips were carried out for each test, among which two cross strips were designed to determine strip-dependent linear drift compensation coefficients. All forward and side overlaps were greater than 75 and 34 %. For the three tests, we set up 5, 5 and 4 ground orientation points to provide the coordinate datum for the test region and used 11, 24 and 7 ground check points to evaluate the accuracy of real-time kinematic PPP-supported aerial triangulation.

Assessment of real-time orbit and clock products

The IGS real-time service coordinates the distribution of real-time GPS satellite orbit and clock products from several participating analysis centers along with IGS-combined products. Details of contributors and the daily monitoring can be found at http://igs.org/rts/contributors and http://igs.org/rts/monitor, respectively. In this study, two types of real-time satellite products, namely the IGS-combined product (IGC) and the CNES product (CNT), are selected. The reason why we select CNT products is that the CNT products feature the smallest clock standard deviation, which have been proven to produce the best PPP-based troposphere estimation (Shi et al. 2015b), and would probably lead to the best performance for real-time kinematic PPP-supported aerial triangulation.

Unlike real-time PPP for static positioning, meteorological and coseismic applications, the observation period of real-time kinematic PPP for aerial triangulation does not take too long for a single flight. For example, no more than 4 h are needed for each test in this study. As a result, only visible satellites in each test are assessed. The IGS final products are selected as the reference to evaluate the accuracy and the precision of real-time orbit and clock products. The satellite orbit accuracy evaluation is performed in XYZ directions. The satellite clock precision evaluation is carried out by a single-difference process in order to remove the clock datum inconsistency among various clock products (Shi et al. 2014).

The root mean square (RMS) and the standard deviation (STD) statistics are calculated for real-time orbit and clock products, respectively. In Test 1, we observed 17 GPS satellites during the flight. Figure 1 depicts the CNT and IGC orbit RMS and the clock STD for Test 1. It is clear that satellite orbit accuracies of real-time CNT and IGC products are better than 5 cm in XYZ directions, i.e., 3.23/3.39/2.93 and 2.86/2.14/2.77 cm, respectively. Furthermore, overall satellite clock precisions of 0.15 and 0.19 ns for CNT and IGC products indicate the real-time satellite clock products are precise to 0.1 to 0.2 ns, which might have a great potential to support real-time kinematic PPP for aerial triangulation.

Fig. 1
figure 1

Real-time CNT (top) and IGC (bottom) orbit and clock accuracies with respect to the IGS final product in Test 1

Full size image

Accuracy assessment from the perspective of GPS positioning

The real-time kinematic PPP estimation using real-time CNT and IGC products is conducted by the P3 software package of the University of Calgary (http://people.ucalgary.ca/~point/p3.html). The PPP solution using IGS final products is adopted as reference. In order to reduce the computational burden and retain the high accuracy of kinematic PPP, the interval of airborne GPS observations is re-sampled to one second to match the five-second interval of aerial camera exposure.

The flight trajectory of 15 strips in Test 1 is illustrated in Fig. 2. The number of available GPS satellites and the corresponding point dilution of precision (PDOP) throughout the flight are shown in Fig. 3 with average values of 7.5 and 2.2, respectively.

Fig. 2
figure 2

Flight trajectory in Test 1

Full size image
Fig. 3
figure 3

Number of satellites and PDOP in Test 1

Full size image

The flight in Test 1 lasts 4 h and 23 min which result in a series of PPP coordinate solutions in 15,780 epochs with the interval of one second. The Lagrange interpolation method is adopted to obtain GPS antenna coordinates at 483 aerial camera exposure epochs in nine strips. The real-time PPP coordinate differences at these 483 camera exposure epochs with respect to those using IGS final products are depicted in the upper row of Fig. 4 on a strip-by-strip basis.

Fig. 4
figure 4

PPP coordinate differences using real-time CNT (left column) and IGC (right column) products with respect to those using IGS final products at 483 camera exposure epochs before (upper row) and after (lower row) the bundle block adjustment in aerial triangulation (AT) in Test 1. The black vertical dashed line separates the nine strips

Full size image

With the strip separation by the black dash line in Fig. 4, the strip-dependent pattern is quite visible for both CNT and IGC coordinate difference series, which well matches the finding that the GPS coordinate error compensation should be performed on a strip-by-strip basis as described in the function model (5). Moreover, it can also be identified that the CNT coordinate difference series are much more stable than the IGC counterpart. Almost all CNT coordinate differences in XYZ directions are smaller than 0.5 m. However, the maximum coordinate difference is greater than 1.0 m, reaching up to 1.1 m in the Y direction of the IGC series. The relatively better stability of the CNT series within each strip (less than 0.1 m) indicates a high probability of better performance by the GPS coordinate error compensation in the following bundle block adjustment.

Accuracy assessment from the perspective of aerial triangulation

The bundle block adjustment is carried out using the WuCAPS software package of Wuhan University (Yuan 2008). The GPS antenna coordinates at 483 camera exposure epochs are treated as quasi-observations in (5), whereas plane coordinates of image points are measured by homologous points matching with an accuracy of 1.0 μm (an equivalent of a 1.0/60 pixel) and are used as observation in (6).

After the GPS coordinate error compensation in the bundle block adjustment, the real-time CNT and IGC PPP coordinate difference series with respect to the reference are depicted in the lower row of Fig. 4. First, compared with those series in the upper row, the coordinate series after the error compensation have been significantly improved. More specifically, the strip-dependent pattern has been removed which results in the smooth CNT and IGC coordinate series in the lower row. Similar to the contrast between CNT and IGC series prior to the error compensation, the CNT coordinate difference series are also more consistent than the IGC counterpart after the error compensation.

The RMS of real-time CNT and IGC coordinate differences for nine strips before and after the bundle block adjustment is summarized in Fig. 5. Apparently, the GPS PPP coordinate error compensation in the bundle block adjustment has significantly removed a majority of GPS PPP coordinate errors. Almost all PPP coordinate RMS accuracies of both CNT and IGC solutions have improved except for five out of 27 coordinate components in the IGC solution. Although overall accuracies of these five strips have decreased in the X direction, the corresponding accuracies in Y and Z directions have been improved and so have the three-dimensional positioning accuracies. In other words, the GPS PPP coordinate error compensation in the bundle block adjustment can greatly reduce GPS PPP positioning errors and thus provide accurate exterior orientation parameters for aerial triangulation. In addition, the overall RMS accuracies of CNT and IGC coordinate series for all nine strips are calculated as 0.0534/0.0806/0.0492 and 0.1779/0.1327/0.0889 m, respectively, which also reflects the superiority of CNT solutions over IGC solutions after the bundle block adjustment.

Fig. 5
figure 5

PPP coordinate differences using real-time CNT (top) and IGC (bottom) products with respect to those using IGS final products for nine strips in Test 1 before and after the bundle block adjustment in aerial triangulation (AT)

Full size image

The performance evaluation is conducted with two references, i.e., coordinates of 11 ground check points calculated by the bundle block adjustment using IGS final products and those by GPS real-time kinematic (RTK). Coordinates of ground check points calculated using the IGS final products serve as first reference. Figure 6 illustrates the coordinate differences using real-time CNT and IGC products in the horizontal and vertical directions. All coordinate differences in the E direction are below 10 cm for both CNT and IGC solutions. A similar phenomenon can also be detected in the N direction except for the ground check points 3 and 9 using the IGC product. As to the vertical component, the magnitude of coordinate differences is much larger than that of the horizontal components. It can also be identified that the overall coordinate differences using the real-time CNT products are significantly smaller than those using the real-time IGC products. This phenomenon is the same as the fact detected in Figs. 4 and 5 for the analysis of the GPS PPP coordinate series.

Fig. 6
figure 6

Coordinate differences of PPP coordinate differences in ENU directions using real-time CNT and IGC products with respect to those using IGS final products in Test 1

Full size image

The coordinates of ground control points obtained by GPS RTK are then used to evaluate the accuracies of real-time kinematic PPP-supported aerial triangulation. The overall accuracies in ENU directions are summarized in Table 4 and Fig. 7. First, the accuracies of east, north and horizontal components using real-time CNT products are very similar to those using IGS products, i.e., 0.075/0.122/0.143 versus 0.076/0.119/0.141 m. The discrepancies (0.096/0.142/0.171 m) of the coordinate solution using real-time IGC products with relatively worse clock stability are much larger than those of the CNT solution. Second, discrepancies among real-time CNT, IGT and the post-mission IGS products are more significant in the vertical component than in the east and north, i.e., 0.134/0.211/0.095 m in vertical versus 0.075/0/096/0.076 m in east and 0.122/0.142/0.119 m in north. Also, similar to the phenomenon identified in the PPP coordinate solution, the coordinates of ground check points determined by the real-time CNT product with better clock stability are more consistent with those of the post-mission IGS product than the real-time IGC counterpart.

Fig. 7
figure 7

Horizontal and vertical accuracies of PPP-supported aerial triangulation using real-time (CNT and IGC) and final (IGS) products for three tests

Full size image

Discussion

Analyses of Tests 2 and 3 with different aerial and ground conditions summarized in Table 2 are carried out in terms of the real-time orbit accuracy, clock precision and the coordinate accuracy of ground check points by kinematic PPP-supported aerial triangulation.

First, the real-time orbit accuracies and clock precisions with respect to the post-mission IGS products are listed in Table 3. The real-time IGC product was unavailable during the observation period of Test 3, which to some extent agrees with Hadas and Bosy (2015) that the availability of current IGS real-time products is 92 %, i.e., not always available all the time. Both real-time CNT and IGC orbits are accurate to 5 cm, whereas the real-time clocks are within 0.1 to 0.3 ns. The <5-cm orbit accuracy and 0.1- to 0.3-ns clock precision on a flight basis can be considered comparable to the IGS-reported orbit accuracy and clock precision on the daily basis (http://igs.org/rts/monitor).

Table 3 Average number of satellites, PDOP, satellite orbit accuracy and clock precision of real-time CNT and IGC products with respect to the post-mission IGS products for each test
Full size table

Second, coordinate accuracies of ground check points by kinematic PPP-supported aerial triangulation using real-time (CNT and IGC) and post-mission (IGS) products are provided in Table 4. All three tests meet the accuracy requirement of aerial photogrammetric mapping at the scale of 1:1000 in China (GB/T 7930-2008 2008), i.e., 0.50/0.40 m for horizontal/vertical components in Test 1, 0.70/0.60 m for Test 2 and 0.50/0.28 m for Test 3. Comparison among these three products represents comparable positioning accuracy for aerial photogrammetric mapping. The consistency between the real-time (CNT and IGC) and post-mission (IGS) products also tells that the real-time satellite product with better clock stability can lead to better precision with respect to the post-mission IGS product. In Test 1, the real-time CNT product with better clock stability (0.15 ns) provides more consistent solution with the post-mission IGS product. Similarly, the real-time IGC product with relatively smaller clock STD of 0.22 ns in Test 2 also leads to more consistent accuracies than the real-time CNT product with clock STD of 0.25 ns. This phenomenon agrees with Shi et al. (2015b) that the real-time product with better clock stability is able to provide better performance for real-time PPP-based troposphere parameter estimation.

Table 4 Accuracies of ground check points determined by kinematic PPP-supported aerial triangulation using real-time (CNT and IGC) and post-mission (IGS) products for each test
Full size table

Conclusions and future work

Since 2013, IGS has published real-time GPS satellite orbit and clock products to enable real-time PPP at a global scale, which have already been proven to satisfy many static applications. But no work had been reported on the feasibility of real-time kinematic PPP for aerial triangulation.

The mathematics of real-time kinematic PPP-supported aerial triangulation is described. Three tests featuring various aerial and ground conditions are evaluated in this study. Some conclusions are summarized as below.

First, two types of real-time orbit and clock products (CNT and IGC) are assessed with respect to post-mission IGS final products. Less than 5-cm orbit accuracy and 0.1- to 0.3-ns clock precision are obtained during the observation period in three tests.

Second, the real-time kinematic PPP can result in comparable accuracy for ground check points as does post-mission kinematic PPP using IGS final products. Depending on the observation condition and the utilized real-time product, millimeter-to-centimeter-level differences and centimeter-to-2-decimeter differences are identified for horizontal and vertical coordinates of ground check points, respectively. The accuracy requirement of three tests featuring different ground conditions has been satisfied, which indicates the real-time kinematic PPP can be widely adopted for 1:1000 aerial photogrammetric mapping within flat, hilly and mountainous regions.

Third, the comparison between the real-time CNT and IGC products unveils that the real-time product with better clock precision can lead to more consistent coordinates of ground check points.

In summary, the evaluation of real-time orbit/clock products and the application of real-time kinematic PPP to aerial triangulation in this study have confirmed the feasibility of real-time kinematic PPP-supported aerial triangulation based on IGS real-time service, which can satisfy aerial photogrammetric mapping at the scale of 1:1000 and greatly shorten the responding time for time-critical aerial photogrammetric applications such as responding to geo-hazards.

The performance of real-time kinematic PPP-supported aerial triangulation can still be improved in the future. The relatively worse accuracy of the vertical coordinates in real-time kinematic PPP-supported aerial triangulation could be improved by better consideration of the tropospheric parameter in the PPP model. Shi et al. (2014) have already demonstrated the improvement of zenith troposphere augmentation for static PPP applications. A three-dimensional troposphere tomography augmentation (Chen and Liu 2014) could be anticipated in the future to reduce or eliminate the troposphere effect on real-time kinematic PPP coordinate solutions and subsequently improve the vertical accuracy in real-time kinematic PPP-supported aerial triangulation.