Advancement of Earth observation technologies and increasing availability of high-resolution satellite images have expanded the utilization of satellite images in various fields, including land observation, monitoring, and urban planning. A prerequisite for the wide utilization is the geometric accuracy of satellite images. Initial geometric models, such as the rational function model (RFM), contain position errors and need to be updated (Grodecki and Dial, 2003). Many studies confirmed that accurate position determination is possible by adjusting a RFM using ground control points (GCPs) whose ground coordinates are known precisely (Lee and So, 2009; Oh et al., 2011; Choi and Kang, 2012; Lee et al., 2013; Jeong et al., 2014).

The quality of GCPs is closely related to the accuracy of precision images. For example, in regions such as North Korea, the lower quality of GCPs results in less accurate precision images compared to South Korea (Lee et al., 2020). Maintaining and updating numerous of GCPs is challenging in terms of time and cost, but it is necessary to minimize errors caused by factors such as terrain changes (Ban and Kim, 2022).

Bundle block adjustment allows for estimation of precise sensor model parameters and ground coordinates of tiepoints observed in multiple images. In this study, we use this technology for updating inaccurate ground coordinates of GCPs observable in multiple overlapping satellite images to precise ground coordinates. We focus to apply bundle block adjustment for updating height values of GCPs, where errors occur more frequently. Additionally, we aim to test whether automated GCP updating is feasible. We match satellite images with GCP chip images to generate GCPs automatically. We then convert true height values of GCPs into initial height values with errors. Then, the GCP height values are iteratively estimated through bundle block adjustment. The estimated height values are compared with the true GCP height values.

In experiments, we used Korea Multi-Purpose Satellite (KOMPSAT)-3 and 3A images and Compact Advanced Satellite 500 (CAS-500) images. We tested GCP coordinate updates based on the number of satellite images used, imaging geometry, and initial weights. Results showed that the accuracy of GCP height values improved after adjustment compared to the initial heights across all datasets. These findings indicate that the proposed method can effectively improve the ground heights of GCPs in an automated manner.

Automated updates of GCPs’ ground coordinates are conducted through the following procedure. First, multiple satellite images and GCP chip images within the area of the satellite images are provided as input data. Then, the automatic matching process between the satellite images and GCP chip images is carried out. By combining matched image points over multiple images, tiepoints of GCPs are constructed. In order to convert true height values of GCPs into initial values with errors, height values are calculated from image points of the constructed tiepoints using initial RFMs of the multiple images and a digital elevation model (DEM). Subsequently, observation equations based on RFMs between tiepoints and their ground coordinates are formed and a bundle adjustment is performed. Finally, we analyze the adjusted GCP height values to determine the accuracy of our proposed method. Detailed explanations of each step are provided in the next sections. Fig. 1 shows a flowchart of the proposed method.

Figure 1. Flowchart of this study.

2.1. Automatic Generation of Tiepoints

Since block adjustment is performed based on tiepoints existing across multiple images, the accuracy of the constructed tiepoints can significantly impact overall adjustment results. Consequently, the step of constructing tiepoints is a crucial step in starting adjustment process. Therefore, in this study, automatic GCP chip matching developed in our previous studies (Yoon, 2019; Park et al., 2020) is utilized. This significantly reduces the time required to acquire GCP image coordinates and allows for automated construction of tiepoints.

The automated tiepoint construction process is as follows. Firstly, the initial image location of each GCP chip is back-projected to the satellite image using initial RFM, and interpolation is applied to geometrically align the GCP chip image with the satellite image. Secondly, pyramid images are created by gradually reducing the scale of the GCP chip image and the satellite image. Then, matching of the chip and satellite image is performed according to the pyramid level. The process is repeated for all GCP chips and all satellite images. Finally, the RANSAC algorithm is applied to the matching results to remove mismatched points, completing the matching process (Park et al., 2020). Fig. 2 shows the matching results between a portion of the satellite image and one GCP chip image. By combining matched image points over multiple images per each GCP chip, tiepoints of GCPs are constructed automatically.

Figure 2. Examples of satellite image and GCP chip image and their matching result. (a) KOMPSAT-3A satellite image. (b) GCP chip image. (c) Matching result.

2.2. Calculation of Initial Heights of GCPs

The GCPs used for experiments have true height values. The primary goal of this study is to accurately update the ground height values of GCPs and analyze the updated performance. Therefore, we deliberately convert the true height values into initial height values with errors. Image coordinates, latitude, and longitude of a GCP and a DEM covering the image extent are used to calculate the initial height value of the GCP through ray tracing. This value will contain an error since the initial RFM possesses a significant amount of position error. Subsequently, the ray tracing is repeated for satellite images containing the same GCP. To reduce the uncertainty of a single observation, the average value of the estimated heights from multiple satellites is set as an initial height value.

2.3. RFM-based Bundle Block Adjustment

To perform RFM bundle block adjustment, mathematical observation equations must be constructed based on the observations of tiepoints. As shown in Fig. 3, the same ground point is observed within the overlapping areas of multiple satellite images. Points observed within these overlapping areas are configured as tiepoints. They play a crucial role in connecting the geometric relationships between images. The more overlap there is between images and the more uniformly distributed the tiepoints are without bias, the higher the uniformity of the model, thereby improving accuracy.

Figure 3. Tiepoint between multiple satellite images.

To update the ground coordinates of GCPs by adjusting multiple satellite images simultaneously, this study performs RFM-based bundle block adjustment. We update the initial height values of GCPs of the initial RFM models by adding polynomial correction terms (Yoon et al., 2018). The observation equation representing the transformation relationship between ground coordinates and image coordinates is given by Eq. (1) below.

where Line and Sample are the image coordinates calculated from the RPC and the ground coordinates of the GCP. ΔLine and ΔSample represent the offsets in the line and sample directions, respectively, and can be expressed in the form of an affine model as shown in Eq. (2) below. And ε_Line and ε_Sample represent the random errors occurring in the line and sample directions.

In the above equation, image adjustment parameter a₀ absorbs all in-track errors causing offsets in the line direction and satellite pitch attitude errors. b₀ absorbs cross-track errors causing offsets in the sample direction and satellite roll attitude errors in the sample direction. a_S, b_S absorb interior orientation errors such as focal length and lens distortion errors. a_L, b_L absorb small errors due to gyro drift during image scanning (Grodecki and Dial, 2003).

When a single GCP is observed in one satellite image, the equations for the line and sample image coordinates can be formulated using Eqs. (1) and (2), respectively, as shown in Eq. (3) below.

The RFM equations constructed as shown in Eq. (3) are nonlinear in the relationships between variables. Therefore, they are converted into a linearized model through a Taylor series expansion.

The model is iteratively calculated using the least squares method until the values of the image adjustment parameters and the GCP ground height adjustments converge. The increments of the image adjustment parameters and the GCP ground height values are calculated using the matrix Eq. (5) below.

where w, ẇ, and ẅ are the weights for the observations, image adjustment parameters, and initial ground heights of the tiepoints, respectively. Ḃ and B¨ are the partial derivatives of the observations F_Line, F_Sample with respect to the image adjustment parameters and ground heights of the GCPs. dx_RFM and dx_{TP Hgt} are the increments of the image adjustment parameters and heights of the GCPs to be ultimately calculated. M_model, M_RFM and M_{TP Hgt} are the misclosures with respect to the model equations, image adjustment parameters, and ground heights of the tiepoints (Ban and Kim, 2023).

In this study, the estimation is iteratively performed until the solution and covariance values converge. Since the bundle block adjustment results are sensitive to weights, an iterative process of re-estimating the weights is necessary. The covariance matrices of estimated parameters and residuals can be calculated using Eq. (6) proposed by Mikhail and Ackermann (1976). Then, by iteratively computing the corrections for the weights and stopping when the corrected weights converge, rigorous bundle block adjustment can be performed through this process.

It is notable that, in this study, the focus is on accurately estimating the height values of GCPs. Therefore, only the height is included as an adjustment parameter for the ground coordinates, while the horizontal position coordinates (latitude and longitude) are kept fixed.

3.1. Used Dataset

In this study, satellite images from KOMPSAT-3, KOMPSAT-3A, and CAS-500 were used. To evaluate the improvement performance of bundle block adjustment in independent cases, satellite images taken over different regions were used. Images taken from Seoul, Incheon, and Sejong City in South Korea were used. Also, since block adjustment is performed based on tiepoints common to multiple images, images with the maximum possible overlapping areas among satellite images were selected. In the case of the Seoul images, 13 images were used to check the adjustment results according to the number of satellite images. They covered the widest range among the three experiment datasets and had many images with different imaging geometries. On the other hand, in the case of the Sejong images, there were two image pairs with similar imaging geometries. For the images used in the experiment, panchromatic band images processed at L1R and L2R level were used. Information on the attributes of the satellite images used for each region is provided in Tables 1, 2, and 3. Fig. 4 shows the imaging and overlapping areas of the satellite images.

Figure 4. Imaging and overlap areas of satellite images.

Table 1 . Specifications of satellite images used in experiments (Seoul).

Image	Satellite	Date of acquisition	Image size (pixels)	GSD (col/row)
1	KOMPSAT-3	2014. 03. 03	24,060 x 21,728	0.72 m / 0.74 m
2		2014. 12. 08	24,060 x 19,216	0.97 m / 0.83 m
3		2023. 01. 01	24,060 x 16,800	0.92 m / 0.95 m
4	KOMPSAT-3A	2015. 12. 11	24,060 x 17,520	0.76 m / 0.75 m
5		2015. 12. 11	24,060 x 17,560	0.76 m / 0.75 m
6		2016. 01. 08	24,060 x 16,960	0.68 m / 0.78 m
7		2017. 02. 06	24,060 x 18,840	0.74 m / 0.70 m
8		2017. 02. 06	24,060 x 18,880	0.74 m / 0.70 m
9		2017. 02. 14	24,060 x 18,440	0.62 m / 0.72 m
10		2017. 02. 23	24,060 x 21,280	0.58 m / 0.62 m
11		2017. 02. 23	24,060 x 21,280	0.58 m / 0.62 m
12		2017. 02. 24	24,060 x 21,680	0.65 m / 0.61 m
13		2019. 09. 18	24,060 x 15,880	0.84 m / 0.83 m

GSD: ground sample distance..

Table 2 . Specifications of satellite images used in experiments (Incheon).

Image	Satellite	Date of acquisition	Image size (pixels)	GSD (col/row)
1	KOMPSAT-3	2013. 05. 13	24,060 x 20,300	0.80 m / 0.79 m
2		2014. 03. 21	24,060 x 21,404	0.77 m / 0.75 m
3		2015. 03. 02	24,060 x 22,480	0.71 m / 0.71 m
4		2019. 01. 06	24,060 x 19,124	0.87 m / 0.84 m
5		2021. 03. 30	24,060 x 16,480	0.90 m / 0.97 m
6	KOMPSAT-3A	2018. 01. 27	24,060 x 18,800	0.67 m / 0.71 m

Table 3 . Specifications of satellite images used in experiments (Sejong).

Image	Satellite	Date of acquisition	Image size (pixels)	GSD (col/row)
1	CAS-500	2021. 10. 25	24,264 x 20,840	0.66 m / 0.58 m
2		2021. 12. 07	24,264 x 23,680	0.50 m / 0.51 m
3		2022. 02. 27	24,264 x 21,000	0.65 m / 0.57 m
4		2022. 03. 03	24,264 x 23,680	0.51 m / 0.51 m
5		2023. 03. 16	24,264 x 23,120	0.53 m / 0.52 m

Along with the satellite images, GCP chip images over the areas covered the images were used. The GCP chip images had a spatial resolution of 0.25 m from aerial orthoimages of Korean National Geographic Information Institute, and accurate ground coordinates (Park et al., 2020). Fig. 5(a) shows examples of GCP chip images used for South Korea, and Fig. 5(b) shows an example of the chip distribution in the Seoul images.

Figure 5. Examples of the South Korean GCP chip images and their distribution in Seoul. (a) Examples of the South Korean GCP chip images. (b) Distribution of the chips in Seoul.

4.1. Accuracy of GCP Height Estimation

We carried out experiments to confirm whether the height values of GCPs with errors could be updated to approximate the actual GCP height values. Table 4 shows the results of estimating the height values of GCPs using 13 satellite images from the Seoul area, 6 satellite images from the Incheon area, and 5 satellite images from the Sejong area. The root-mean-square-error (RMSE) values were calculated by comparing the initial and final adjusted height values of the GCPs with the actual height values.

Table 4 . Adjustment results by imaging area.

Imaging area	Used satellite image number	Tiepoint number	RMSE (m)
			Initial height	Updated height
Seoul	13	123	8.959	0.863
Incheon	6	43	4.176	1.052
Sejong	5	32	3.412	1.473

In the case of the Seoul images, the adjustment was performed using the most images among the three datasets. The initial estimated height RMSE was 8.959 m, and after adjustment, the RMSE was reduced to 0.863 m. This indicated a significant improvement in accuracy of approximately 90.7%. The adjusted height successfully approximated the actual GCP height values. For the Incheon images, using 6 images, the RMSE improved from 4.176 m to 1.052 m, showing an accuracy improvement of about 74.8%. Similarly, for the Sejong images, with 5 images used, the RMSE improved from 3.412 m to 1.473 m, indicating an accuracy improvement of approximately 56.8%.

The final adjustment results showed differences in performance improvement depending on factors such as the number of images used and the imaging geometry. Comparisons of adjustment results based on the number of images used and imaging geometry are detailed in the following sections.

Table 5 shows the adjustment results for 10 sample GCPs for the Seoul images. Each residual was calculated as the difference between the actual GCP height value and the initial estimated height value and the adjusted height value. The RMSE was calculated using all tiepoints, including the 10 sample GCPs.

Table 5 . Examples of GCP adjustment results (Unit: m).

GCP	True GCP height	Initial estimated GCP height	Residual	Final estimated GCP height	Residual
1	28.102	26.472	1.630	28.434	–0.332
2	32.678	31.789	0.889	32.976	–0.298
3	33.732	34.218	–0.486	33.771	–0.039
4	34.898	33.189	1.709	35.062	–0.164
5	39.08	40.174	–1.094	40.206	–1.126
6	110.45	104.590	5.860	109.654	0.796
7	115.992	108.422	7.570	115.700	0.292
8	127.52	117.986	9.534	126.650	0.870
9	128.04	111.272	16.768	128.885	–0.845
10	208.57	198.331	10.239	208.023	0.547
RMSE	-	8.959	-	0.863	-

Additionally, we examined the initial estimation errors and the adjustments according to the ground height values of the GCPs. Fig. 6 below illustrates the adjustment results for the (a) Seoul, (b) Incheon, and (c) Sejong regions based on GCP height values. The results indicate that higher elevation areas experienced larger initial estimation errors, which were significantly reduced after adjustment and showed a tendency to converge toward the true values. Conversely, in lower elevation areas, the errors sometimes propagated, leading to increased errors in some cases. Overall, while the adjustments improved the accuracy, not all GCPs reached values close to the true heights.

Figure 6. Variation of residual according to GCP height. (a) Seoul. (b) Incheon. (c) Sejong.

4.2. Effects of Number of Satellite Images

To estimate the GCP heights, multiple satellite images were used, and the adjustment results varied depending on the number of satellite images. Therefore, we compared the accuracy based on the number of satellite images used in the bundle block adjustment. The experiment used images from Seoul and Incheon regions. For Seoul, we varied the number of satellite images from 2 to 13, and for Incheon, from 2 to 6.

Table 6 shows the number of tiepoints constructed and the GCP height adjustment results as the number of satellite images increases. The RMSEs of the initial GCP height value and the height value after the final adjustment were indicated. According to the experiment results, when only 2 satellite images were used, the RMSE of the adjusted GCP heights for Seoul was 1.717 m. When 3 images were used, the RMSE decreased to 1.205 m. The error was further reduced to 0.873 m with 8 images. Although the RMSE slightly increased when using 9, 10, and 11 images, it decreased again with 12 and 13 images. The final RMSE of 0.863 m closely approximated the actual height values. Similarly, for Incheon, the RMSE slightly increased to 1.693 m when 3 images were used but gradually decreased as the number of images increased, reaching 1.052 m with 6 images.

Table 6 . Adjustment results by the number of used satellite images.

Used satellite image number	Seoul		Incheon
	Tiepoint number	RMSE (m)	Tiepoint number	RMSE (m)
2	32	1.717	17	1.488
3	52	1.205	30	1.693
4	62	0.959	32	1.501
5	66	0.934	38	1.241
6	70	0.926	43	1.052
7	75	0.899	-
8	81	0.873
9	88	0.881
10	91	0.897
11	104	0.933
12	119	0.895
13	123	0.863

As a result, we observed a tendency where the error progressively decreases as the number of images increases. Therefore, using a larger number of images is advantageous for more accurate estimation compared to using fewer images. However, the results also indicate that simply having more satellite images and tiepoints does not linearly correlate with higher accuracy. The accuracy may be influenced by other factors such as the precision of the matched tiepoints.

4.3. Effects of Imaging Geometry

Since the degree of adjustment varied by region, we compared the results based on the imaging geometry of the satellite images used to analyze the factors contributing to performance differences. The imaging geometry of a satellite image represents the spatial relationship between the satellite and the ground when the image is acquired, and it is expressed in terms of the satellite’s attitude and angles. Depending on imaging geometry, the direction of ground observation changes, leading to variations in the degree and direction of relief displacement in the images. The satellite images used for the Seoul area in this study had the most images and thus exhibited various imaging geometries. In contrast, the satellite images for the Incheon and Sejong areas used only 6 and 5 images, respectively, resulting in some pairs having similar imaging geometries. In particular, among the 5 images of Sejong, two pairs had almost identical imaging geometries. Table 7 shows the imaging angles of the satellite images used, and Figs. 7, 8, and 9 illustrate examples of the imaging geometries for the satellite images of Seoul, Incheon, and Sejong, respectively. Additionally, to focus on the differences in imaging geometry, the number of images and the number of tiepoints used were kept constant across the experiments.

Figure 7. Imaging geometry of Seoul satellite images.

Figure 8. Imaging geometry of Incheon satellite images.

Figure 9. Imaging geometry of Sejong satellite images.

Table 7 . Imaging angles of satellite images.

Imaging area	Date of acquisition	Roll (°)	Pitch (°)	Yaw (°)	Azimuth angle (°)	Incidence angle (°)	Off-nadir angle (°)
Seoul	2014. 03. 03	4.373	9.839	-2.993	146.180	12.099	10.898
	2014. 12. 08	-25.064	14.764	-17.057	229.887	32.438	28.954
	2015. 12. 11	25.289	-22.842	14.514	207.888	36.973	33.722
	2016. 01. 08	-14.591	-29.548	-5.051	146.954	35.910	32.813
	2017. 02. 06	-26.064	-18.719	-6.984	119.330	34.783	31.792
Incheon	2013. 05. 13	-15.069	14.845	-3.513	215.197	23.579	21.168
	2014. 03. 21	-13.584	9.840	-3.310	224.139	18.712	16.838
	2015. 03. 02	6.442	4.839	-2.963	117.331	9.007	8.129
	2019. 01. 06	-19.322	17.849	-3.781	216.976	29.282	26.205
	2021. 03. 30	-13.716	27.841	-4.386	194.956	34.746	30.956
Sejong	2021. 10. 25	-26.688	-1.078	2.354	79.179	28.995	26.672
	2021. 12. 07	-0.565	0.125	2.967	82.282	0.564	0.523
	2022. 02. 27	-26.850	-1.092	2.323	79.225	29.155	26.834
	2022. 03. 03	5.703	0.429	2.934	260.835	6.209	5.752
	2023. 03. 16	-11.837	-0.431	2.856	80.056	12.766	11.808

When using the 5 satellite images of Seoul, the RMSE decreased after adjustment from 5.451 m to 0.931 m. Using the 5 satellite images of Incheon, the RMSE decreased from 4.240 m to 1.228 m. When using the 5 satellite images of Sejong with the most similar imaging geometries, the RMSE decreased from 3.412 m to 1.473 m. The adjustment results for each region showed accuracy improvements of 82.9% for Seoul, 71% for Incheon, and 56.8% for Sejong. The initial height error was the largest for the Seoul images and the smallest for the Sejong images, but the adjusted height error was closest to the true value for the Seoul images. The adjustment results based on imaging geometry are presented in Table 8.

Table 8 . Adjustment results by imaging geometry.

Imaging area	Used satellite image number	Tiepoint number	RMSE (m)
			Initial height	Updated height
Seoul	5	32	5.451	0.931
Incheon			4.240	1.228
Sejong			3.412	1.473

4.4. Effects of Weights

Due to the risk of overestimation or stagnation in estimation based on the initial weights assigned, assigning appropriate weights to the image coordinate observations, image adjustment parameters, and the initial heights of the GCP tiepoints may be crucial. Although the proposed method included readjusting the weights iteratively, the initial weights significantly influenced the adjustment results. Therefore, a sensitivity analysis of the weights was conducted to determine the optimal weights based on the characteristics of the variables for accurate estimation of the actual GCP heights.

The experiment was conducted using 13 satellite images of the Seoul area and 123 extracted tiepoints. We confirmed that the adjustment results varied greatly depending on the weights assigned to the image adjustment parameters and the heights of the tiepoints. When high weights were assigned to both the image adjustment parameters and the initial heights of the tiepoints, neither parameter was adjusted, resulting in the same RMSE of 8.959 m as the initial value. When low weights were assigned to the image adjustment parameters and high weights to the initial heights of the tiepoints, the GCP heights were not accurately adjusted due to the high weights, resulting in an RMSE of 3.240 m. When low weights were assigned to both parameters, the RMSE was reduced to 3.417 m compared to the initial RMSE, indicating an improvement towards the actual height values, but the adjustment for the GCP heights was still minimal.

Therefore, to achieve the objective of this study, it was recommended to have high weights to the image adjustment parameters and low weights to the GCP heights. Under these conditions, the RMSE was reduced to 0.863 m, providing the closest approximation to the true values. However, if the model configuration changes or additional factors are considered, adjusting the initial weight settings may be necessary. Accordingly, further consideration is required to establish criteria for determining the optimal initial weights. Table 9 presents the comparison results of the adjustments based on the initial weight settings for each parameter, with the values indicating the RMSE of the final adjusted heights.

Table 9 . Adjustment results by initial weights.

Initial weight		Image adjustment parameter
		High	Low
Tiepoint height	High	8.959 m	3.240 m
	Low	0.863 m	3.417 m

In this study, we proposed a method to update the height values of GCPs using an RFM-based bundle block adjustment technique with multiple satellite images. Experiment results showed that the RMSE of GCP heights after adjustment was within 1m of the true values. This demonstrated the capability to estimate the actual GCP height values.

We analyzed the effects of the number of images used, imaging geometries, and initial weights on the adjustment results. The RMSE after adjustment showed a decreasing tendency as the number of images used in the adjustment increased. This was because additional images provided more observations, thereby enhancing the accuracy of the adjustment. Additionally, the dataset with its diverse imaging geometries enabled more accurate estimations compared to the datasets with similar imaging geometries. Diverse imaging geometries provided a broader range of observations, contributing to a more accurate estimation of the GCP ground heights. Weights played a crucial role in performing rigorous block adjustments, and the adjustment results were highly sensitive to the assigned initial weights. The most accurate results were obtained when low weights were assigned to the initial GCP heights and high weights to the image adjustment parameters. This approach resulted in more significant adjustments to the GCP initial heights than to the image adjustments, thereby leading to more precise estimations.

We highlight that tiepoints over multiple images were constructed through automated matching between satellite images and GCP chips. This strongly supports the automated update of GCPs’ ground coordinates precisely. Experiment results in this study indicated that the proposed method could be effectively utilized for practical GCP management. In particular, this method is expected to contribute to the improvement of GCP quality in areas where accurate field surveys are challenging or regions with changing terrain. In future research, the effects of matching errors need to be addressed in detail. The effects of terrain types also need further investigation.

This work is supported by the Korea Agency for Infrastructure Technology Advancement (KAIA) grant funded by the Ministry of Land, Infrastructure and Transport (Grant RS-2022-00155763).

No potential conflict of interest relevant to this article was reported.