As the utilization of satellite images continues to expand across various fields, interest in remote sensing research and satellite data applications has significantly increased. Many countries and private companies are actively pursuing satellite research and development, resulting in numerous satellites being operated for diverse purposes. In accordance with this global trend, South Korea is advancing its space technology and expanding the use of satellite data through the development of the Compact Advanced Satellite series (Lee et al., 2016; Kim, 2023). Among these, the Compact Advanced Satellite 500-4 (CAS500-4), scheduled for launch in 2025, is expected to play a pivotal role in the monitoring of agricultural and forest resources. The CAS500- 4 will provide a daily revisit cycle over the Korean Peninsula, offer a spatial resolution of 5 m, and include five spectral bands, with a swath width of 120 km (Cha et al., 2023; Kim et al., 2024). This combination of wide coverage and frequent revisits will contribute significantly to resource management and enable swift decisionmaking in related industries.

The use of simulation images is widely acknowledged as a fundamental aspect of satellite mission planning (Eo, 2021). Similarly, the simulation of the CAS500-4 prior to its launch could facilitate the identification of potential issues and the development of effective response strategies. It is imperative that the entire process, from its inception, undergoes comprehensive validation and testing in order to guarantee the success of satellite operations. However, the lack of actual satellite data before the launch poses challenges for pre-launch research and verification, limiting the ability to predict and prepare for potential problems during early satellite operations. Preliminary studies for CAS500- 4 operations have utilized images from Sentinel-2 and RapidEye, which share similar spectral characteristics (Kim et al., 2023; Lee et al., 2024). However, these datasets do not fully reflect the unique characteristics of the satellite, leading to potential discrepancies between the study findings and actual operational results. Therefore, it is necessary to generate simulated images that accurately reflect the specifications of CAS500-4 to predict and address potential issues (Schott et al., 1999).

To generate CAS500-4 simulated images, an appropriate sensor model that defines the geometric relationship between satellite images and ground space is essential. Sensor modeling approaches are broadly divided into three types. Physical models require detailed information about the satellite and sensor. Abstract models simplify the relationships without needing detailed specifications. Generalized models mathematically describe the geometric relationship (McGlone, 1996; Kim et al., 2000). Park and Eo (2014) demonstrated the feasibility of restoring original images from orthoimages using orbital information and sensor parameters. Yun et al. (2002) validated the capability of generating simulated images with varying positions and attitudes. However, generating simulated images using physical sensor models requires precise details about the satellite and sensor, which may be challenging in the pre-launch stage.

As CAS500-4 is currently in its pre-launch phase, precise physical information is not yet available. To overcome this limitation, a generalized model that does not require detailed orbital or sensor specifications is more suitable. Among generalized models, the rational function model (RFM) is highly adaptable and does not require detailed sensor information, making it compatible with various sensors (Madani, 1999; Paderes et al., 1989). Additionally, RFM has demonstrated its utility in various applications by providing high accuracy and simplifying 3D position estimation (Tao and Hu, 2001; Fraser and Hanley, 2003).

This study proposes a method for generating CAS500-4 Level-1 Radiometric (L1R) simulated images through inverse orthorectification based on the rational function model (RFM). The proposed method utilizes orthoimages and rational polynomial coefficients (RPCs), which inherently encapsulate orbital and sensor information. The objective of this study is to meet the ground sample distance (GSD) requirement of CAS500-4 and simulate different viewing angles from the base orthoimage using adjusted RPCs. This research is expected to provide a foundation for validating operational processes prior to the launch of CAS500-4.

The simulated L1R images of CAS500-4 were generated by inverse orthorectification using the following data. Orthoimages containing ground coordinate information, RPCs representing the generalized model of the orbit, attitude, and sensor characteristics of the satellite, and a digital elevation model (DEM) quantifying three-dimensional terrain information. For this study, Sentinel-2 orthoimages, with spectral bands similar to those of the CAS500- 4, were used as the base orthoimage. The Level-1C (L1C) images with radiometric and geometric corrections were used (Drusch et al., 2012). Specifications for Sentinel-2 and the CAS500-4 are provided in Tables 1 and 2.

Table 1 . Specifications of the Sentinel-2 used in this study.

Altitude	Revisit time	Spectral band	Center wavelength	Band width	Spatial resolution
786 km	5 day	Blue	490 nm	65 nm	10 m
		Green	560 nm	35 nm	10 m
		Red	665 nm	30 nm	10 m
		Red edge	705 nm	15 nm	20 m
		NIR	842 nm	115 nm	10 m

Table 2 . Specifications of the planned CAS500-4.

Altitude	Revisit time	Spectral band	Center wavelength	Band width	Spatial resolution
888 km	1–3 day	Blue	490 nm	65 nm	5 m
		Green	560 nm	35 nm
		Red	665 nm	30 nm
		Red edge	705 nm	15 nm
		NIR	842 nm	115 nm

The actual RPCs for CAS500-4 are not yet available, as the satellite has not yet been launched. Therefore, the RPCs from Korea Multi-Purpose Satellite-3A (KOMPSAT-3A) L1R were adjusted and used as a replacement for the RPCs of CAS500-4. Fig. 1 visually illustrates the Sentinel-2 base orthoimages used in the experiments, covering regions of North Korea, the Korean border, and South Korea. Table 3 provides detailed information about these Sentinel-2 base orthoimages. Additionally, as shown in Table 4, the dataset was constructed using Sentinel-2 base orthoimages along with KOMPSAT-3A original RPCs, which have different viewing angles. For each base orthoimage, the dataset was divided into two experiments. One uses RPCs with similar azimuth angles to the base orthoimage and the other uses RPCs with similar zenith angles.

Figure 1. Imaging and overlap areas of Sentinel-2.

Table 3 . Properties of Sentinel-2 base orthoimages used in this study.

Base image	Region	Date of acquisition	Image center latitude	Image center longitude	Azimuth angle	Zenith angle
A	North Korea	2023. 09. 22	39.21921072°	126.16062757°	281.6795°	5.7064°
B	Korea Border	2024. 06. 20	38.34078191°	127.33963247°	143.3725°	3.0451°
C	South Korea	2022. 04. 24	35.64733515°	128.50135770°	101.9912°	5.7146°

Table 4 . Properties of original KOMPSAT-3A RPCs used for base images in this study.

Base image	Original RPC	Date of acquisition	Column GSD	Row GSD	Azimuth angle	Zenith angle
A	1	2018. 10. 03	0.604 m	0.582 m	285.2397°	19.0457°
	2	2018. 05. 24	0.559 m	0.558 m	46.8354°	5.9221°
B	3	2017. 10. 30	0.570 m	0.581 m	145.7911°	13.8022°
	4	2019.	01.	29	0.543 m	0.541 m	260.7426°	3.2464°
C	5	2022. 11. 15	0.758 m	0.666 m	102.2763°	33.4465°
	6	2018. 09. 25	0.556 m	0.552 m	260.8570°	5.7353°

This study utilized the DEM provided by the National Geographic Information Institute (NGII) (Park et al., 2020). The 10 m resolution DEM was used for North Korea, and the 5 m resolution DEM was used for South Korea.

The proposed method adjusts the offset and scale parameters of the original KOMPSAT-3A RPCs to generate CAS500-4 L1R simulated images that meet the GSD requirements of CAS500- 4. Using the adjusted RPCs, an RFM is established, and inverse orthorectification is performed on Sentinel-2 base orthoimage to generate CAS500-4 L1R simulated images that reflect the viewing angle of the adjusted RPCs.

3.1. Adjustment of Offset and Scale Parameters in RPCs

RPCs are coefficients that model the nonlinear relationship between ground coordinates and image coordinates. They use offset and scale parameters to normalize these coordinates. RPCs consist of 20 numerator and denominator coefficients, forming rational polynomial equations with a total of 80 coefficients. In this study, RPCs collected from KOMPSAT-3A were used. These original RPCs represent the relationship between ground coordinates and image coordinates at the time of satellite imaging. Therefore, the original RPCs cannot be used directly to generate CAS500-4 simulated images. To address this, the RPCs were adjusted to satisfy the GSD requirements of CAS500-4 and to establish an RFM suitable for the Sentinel-2 base orthoimage.

First, the latitude and longitude offset and scale of the KOMPSAT-3A RPCs were adjusted to fit the ground coverage of the Sentinel-2 base orthoimage. Then, the line and sample offset and scale were adjusted to meet the GSD requirement of 5 m for the CAS500-4. Using the adjusted RPCs, an RFM was established, which was used to estimate ground coordinates corresponding to image coordinates, and the GSD was calculated. The adjustment process involved iteratively modifying the line and sample offset and scale until the GSD converged to 5 m. The transformation relationship between ground coordinates and image coordinates represented by the RFM with adjusted offset and scale parameters is given in Eq. (1). The normalization equations for image and ground coordinates, which improve the mathematical stability of the RFM, are described in Eqs. (2) and (3).

where (Line, Sample) represent the line and sample of the image coordinates, respectively, while (λ, φ, h) represent longitude, latitude, and height of the ground coordinates, respectively. (Linen, Samplen) are the normalized image coordinates, and (X_n, Y_n, Z_n) are the normalized ground coordinates.

Line_OFFSET and Sample_OFFSET are offset parameters used to normalize the image coordinates by adjusting the reference point or aligning it to the origin. Line_SCALE and Sample_SCALE are scale parameters used to adjust the size and proportion of the image coordinates, ensuring compatibility with the normalized coordinate system.

For ground coordinates, λ_OFFSET, φ_OFFSET and h_OFFSET are offset parameters used to set the reference points for longitude, latitude, and height, respectively. Additionally, λ_SCALE, φ_SCALE and h_SCALE are scale parameters that convert ground coordinates into normalized ranges, contributing to the stability of the model. These offset and scale parameters were adjusted to fit the coverage of the base orthoimage and meet the GSD requirements of CAS500-4. Using the normalized ground coordinates and adjusted RPCs, a third-order polynomial RFM was constructed to calculate image coordinates. The polynomials P_i (i = 1~4) used in the RFM take the form shown in Eq. (4).

where the coefficients c1~20 represent the rational polynomial coefficients, and they are expressed as aijk, bijk, cijk and dijk.

3.2. RFM Establishment Based on Adjusted RPCs

In Section 3.1, we establish an RFM using the equations from Eq. (1) to Eq. (4) to generate a CAS500-4 L1R simulated image based on the adjusted RPCs. By adjusting the offset and scale parameters of the original KOMPSAT-3A RPCs, the alignment with the coverage area of the Sentinel-2 orthoimage was achieved, meeting the 5 m GSD requirement and enabling the modeling of CAS500-4 simulated images. Furthermore, as the RPCs inherently contain satellite orbital information, the RFM is established with geometric characteristics corresponding to the viewing angle of the adjusted RPCs, rather than those of the base orthoimage. The changes in the RFM based on the adjusted RPCs are shown in Fig. 2.

Figure 2. Change in the RFM based on adjusted RPCs.

3.3. CAS500-4 L1R Simulated Image Generation by Inverse Orthorectification

Inverse orthorectification is performed using the RFM established based on the adjusted RPCs to generate CAS500-4 L1R simulated images. Fig. 3 shows the process of generating L1R-simulated images through inverse orthorectification. This process involves estimating the ground coordinates corresponding to the image coordinates of the L1R simulated image. Based on these ground coordinates, the image coordinates of the base orthoimage are calculated. Subsequently, the corresponding pixel values are interpolated and assigned back to the image coordinates of the L1R simulated image.

Figure 3. Process of generating CAS500-4 L1R simulated images by inverse orthorectification.

To generate simulated images, the RFM was used to estimate the ground coordinates corresponding to the image coordinates of the simulated images. In this process, accurate height values were of critical importance. To achieve this, the ray tracing method proposed by O’Neill and Dowman (1988) was applied. This method involves back-projecting a ray emitted from the sensor to its point of intersection with the ground surface. For the image coordinates of the simulated images, an initial height value is assumed, and the RFM is used to estimate the corresponding ground coordinates. The actual height value for the estimated ground coordinates is then extracted from DEM, and the ground coordinates are re-estimated using the RFM with the updated height value. This iterative process continues until the final ground coordinates are determined. Fig. 4 shows the ray tracing method used to iteratively determine the ground coordinates corresponding to the image coordinates of the simulated images.

Figure 4. Ray tracing method.

Since the estimated ground coordinates do not exactly match the image pixels, interpolation is required to correct the pixel values. To achieve this, the center point of the image coordinates in the orthoimage corresponding to the estimated ground coordinates is calculated. Bicubic interpolation is applied to preserve the highfrequency components of the orthoimage while ensuring smooth results in the simulated image with modified viewing angles (Titus and George, 2013). The interpolated pixel values are then assigned to the corresponding image coordinates in the simulated image. This process is repeated for all image coordinates of the simulated image to generate the CAS500-4 L1R simulated image.

In this study, we applied the proposed inverse orthorectification method using RPCs with different geometric characteristics corresponding to the varying viewing angles of each base orthoimage. To evaluate the generated L1R simulated images, we verified both the GSD and the changes in viewing angles. The results indicate that the CAS500-4 L1R simulated images reflect various viewing angles, confirming the effectiveness of the proposed method.

4.1. Comparison of CAS500-4 L1R Simulated Images Generated by RPCs

To calculate the GSD and viewing angles, the RFM was used to estimate the ground coordinates corresponding to the image coordinates. By assuming two distinct height values for the same image coordinates, two distinct ground points were obtained. These points were interpreted as lying on a direction vector relative to the projection center (Seong et al., 2021). Using this direction vector, the azimuth and zenith angles, which represent the viewing angle, were calculated. The calculated GSD and viewing angles of the CAS500-4 L1R simulated images were compared with those of the base orthoimage and the original RPCs used for the simulation. The results are presented in Tables 6 to 8.

Table 6 . Results of CAS500-4 L1R simulated image (Image A).

		Base	Original		Simulated
		Image A	RPC 1	RPC 2	Image A - RPC 1	Image A - RPC 2
GSD	X	10 m, 20 m	0.604 m	0.559 m	5.001 m	5.000 m
	Y	0.582 m	0.558 m	5.001 m	5.000 m
Viewing angle	Azimuth	281.6795°	285.2397°	46.8354°	289.3677°	47.1175°
	Zenith	5.7064°	19.0457°	5.9221°	26.9055°	9.4714°
		Processing time			1,655 sec	1,380 sec

The results demonstrated that the generated L1R simulated images meet the 5 m GSD requirement of CAS500-4. Additionally, it was observed that the viewing angles of the original RPCs were better replicated compared to those of the base orthoimage. A comparison was conducted between L1R simulated images generated with odd-numbered RPCs, which had azimuth angles similar to the base orthoimage, and those generated with evennumbered RPCs, which had zenith angles similar to the base orthoimage. The results showed that simulated images generated with even-numbered RPCs more accurately replicated the viewing angles of the original RPCs compared to those generated with odd-numbered RPCs. Furthermore, the ground coordinate estimation for even-numbered RPCs converged more quickly, resulting in shorter processing times.

Figs. 5 to 7 show the results of composing the R, G, and B bands of the generated CAS500-4 L1R simulated images, along with magnified comparisons with the base orthoimage. The simulated images generated using odd-numbered RPCs exhibited relatively larger differences compared to the base orthoimage due to the greater zenith angle disparity, as opposed to those generated using even-numbered RPCs.

Figure 5. Generated CAS500-4 L1R simulated images (Image A).

4.2. Accuracy Analysis of CAS500-4 L1R Simulated Images Using GCPs

CAS500-4 L1R simulated images were generated using RPCs with different viewing angles, and the 3D ground coordinates estimated from the simulated images were compared with ground control points (GCPs) to evaluate accuracy. The analysis was conducted using five GCPs provided by NGII for each region. In the North Korean region, the lack of available GCPs precluded an assessment of accuracy. The locations of the GCPs are shown in Fig. 8, and the image coordinates of the simulated images containing the GCPs, as well as the ground coordinates of the GCPs, are presented in Tables 9 and 10. The discrepancy in the image coordinates of the simulated images containing the GCPs indicates that each image was generated with different viewing angles.

Figure 8. Position of GCPs.

Table 9 . Image coordinates of CAS500-4 L1R simulated image and ground coordinates of GCPs (Image B).

	Simulated
	Image B - RPC 3		Image B - RPC 4
	X (pixel)	Y (pixel)	X (pixel)	Y (pixel)
GCP 1	5278.88	11479.34	5215.21	11414.25
GCP 2	9596.56	17860.44	9486.78	17825.48
GCP 3	10163.85	12594.27	10090.49	12568.82
GCP 4	14750.92	14903.38	14663.18	14910.85
GCP 5	17155.73	14089.18	17073.43	14119.74

	GCP
	X (m)	Y (m)	Z (m)
GCP 1	212119.2061	623614.2426	144.6135
GCP 2	239970.6579	596863.2089	309.3466
GCP 3	237230.0526	623189.7085	271.1904
GCP 4	262161.3669	616719.272	212.7188
GCP 5	273084.6579	623224.6644	383.6943

Table 10 . Image coordinates of CAS500-4 L1R simulated image and
ground coordinates of GCPs (Image C).

	Simulated
	Image C - RPC 5		Image C - RPC 6
	X (pixel)	Y (pixel)	X (pixel)	Y (pixel)
GCP 1	2696.18	10657.88	2511.48	9997.12
GCP 2	6197.84	6314.93	6188.82	5828.46
GCP 3	8050.95	14490.14	7718.94	14062.15
GCP 4	12322.42	5989.77	12314.88	5784.24
GCP 5	12433.48	12778.86	12162.76	12556.14

	GCP
	X (m)	Y (m)	Z (m)
GCP 1	121962.306	328447.5435	91.9672
GCP 2	135813.782	352639.5231	66.9611
GCP 3	151530.8966	313841.4464	44.0908
GCP 4	165728.4525	359157.1005	106.2105
GCP 5	171810.778	325813.9	102.5681

The results of estimating the 3D ground coordinates corresponding to the image coordinates using the RFM and comparing them with the GCPs are shown in Tables 11 and 12. Additionally, the root-mean-square-error (RMSE) results for the 3D coordinates of the L1R simulated images generated using each base orthoimage and RPCs are presented in Fig. 9. A comparison of the accuracy of the L1R simulated images generated using RPCs with different viewing angles revealed that the simulated images generated with even-numbered RPCs, where the zenith angle between the base orthoimage and the original RPCs was similar, demonstrated higher accuracy across the X, Y, and Z axes. This finding confirms that using data with similar zenith angles enables the generation of more accurate CAS500-4 L1R simulated images.

Figure 9. Accuracy comparison of CAS500-4 L1R simulated images.

Table 11 . Accuracy results of CAS500-4 L1R simulated images (Image B).

	Simulated
	Image B - RPC 3			Image B - RPC 4
	Δ X (m)	Δ Y (m)	Δ Z (m)	Δ X (m)	Δ Y (m)	Δ Z (m)
GCP 1	–8.1552	–78.2844	–3.997	–9.3616	–5.6032	–1.6618
GCP 2	–57.7254	–24.8229	–5.6832	14.5077	–10.9847	–1.7189
GCP 3	–46.585	–57.7748	–5.2409	11.6579	–12.0522	–5.2229
GCP 4	–10.2035	–43.4555	–9.005	–2.5107	–9.577	–5.2077
GCP 5	–8.2248	–10.8887	–3.738	8.085	–2.4643	–1.3045
RMSE	33.8841	49.1721	5.8448	10.0567	8.8894	3.5162

Table 12 . Accuracy results of CAS500-4 L1R simulated images (Image C).

	Simulated
	Image C - RPC 5			Image C - RPC 6
	Δ X (m)	Δ Y (m)	Δ Z (m)	Δ X (m)	Δ Y (m)	Δ Z (m)
GCP 1	46.3938	-96.077	10.4536	-52.9364	-9.5401	3.6782
GCP 2	3.7571	-141.9745	28.4048	-53.5696	-4.1089	1.2027
GCP 3	-141.7501	-95.908	34.2295	-44.784	-3.5614	5.7423
GCP 4	-54.6253	-143.0382	30.6228	-38.8406	-1.2892	2.0581
GCP 5	-145.0776	-95.3015	34.7159	-46.5422	-3.2206	-2.7649
RMSE	96.2196	116.7289	7.4164	47.6494	5.15	3.4592

4.3. Applying Stereoscopic Viewing to CAS500-4 L1R Simulated Images

The results demonstrate that the proposed method enables the generation of simulated images with viewing angles different from those of the base orthoimage. To analyze the results of the simulation for the North Korean region, the potential for stereoscopic viewing using the generated CAS500-4 L1R simulated images was examined with two images simulated at distinct viewing angles. The results of applying stereoscopy to the simulated images with different viewing angles are shown in Fig. 10, which demonstrates the variations in parallax caused by the distinct viewing geometries. The analysis confirmed the feasibility of stereoscopic viewing by demonstrating the elimination of Y parallax in the magnified comparison images, ensuring geometric alignment between the images. This result demonstrates that the proposed method successfully generates simulated images with viewing angles different from those of the base orthoimage, providing valuable input for stereoscopic analyses in satellite imaging applications.

Figure 10. Results of stereoscopic viewing using CAS500-4 L1R simulated images (Image A).

This study proposed a method for generating CAS500-4 L1R simulated images using inverse orthorectification based on the RFM. The application of this method resulted in the successful generation of L1R simulated images that met the 5 m GSD requirement for CAS500-4. Additionally, the study demonstrated that higher accuracy in simulated images could be achieved by using RPCs with zenith angles that are similar to those of the base orthoimage. The feasibility of stereoscopic viewing was also confirmed by generating L1R-simulated images with different viewing angles. These results validated the capability of the proposed method to generate CAS500-4 L1R simulated images that not only meet the GSD requirement but also simulate various viewing angles effectively.

The findings suggest that the generated L1R simulated images can serve as valuable material during the pre-launch development and research stages of CAS500-4, supporting the validation of satellite operations and the assessment of various operational scenarios. However, simply adjusting the offset and scale parameters of the original KOMPSAT-3A RPCs has limitations in establishing a precise sensor model for the base orthoimage. To address these limitations, further research is needed to refine RPCs to match the specific ground coverage of the base orthoimage and to develop RPCs that can accurately simulate CAS500-4 with higher precision. Additionally, further studies on precise geometric correction methods will be essential to enhance the overall accuracy and reliability of the simulated images.

Figure 6. Generated CAS500-4 L1R simulated images (Image B).

Figure 7. Generated CAS500-4 L1R simulated images (Image C).

Table 5 . Specifications of DEM used in this study.

Region of use	North Korea	South Korea
Ellipsoid	GRS80	WGS_1984
Coordinate system	Transverse Mercator	Transverse Mercator
Image size X	68,697 pixels	134,974 pixels
Image size Y	111,519 pixels	132,972 pixels
GSD X	10 m	5 m
GSD Y	10 m	5 m
Upper left X	–54,468.3 m	–3,021.48 m
Upper left Y	1,173,149 m	659,460 m
Lower right X	632,501.7 m	671,848.5 m
Lower right Y	57,959.13 m	–5,399.96 m

Table 7 . Results of CAS500-4 L1R simulated image (Image B).

		Base	Original		Simulated
		Image B	RPC 3	RPC 4	Image B - RPC 3	Image B - RPC 4
GSD	X	10 m, 20 m	0.570 m	0.543 m	5.000 m	5.000 m
	Y	0.581 m	0.541 m	5.000 m	5.000 m
Viewing angle	Azimuth	143.3725°	145.7911°	260.7426°	147.6901°	263.0117°
	Zenith	3.0451°	13.8022°	3.2464°	18.5727°	7.0896°
		Processing time			954 sec	781 sec

Table 8 . Results of CAS500-4 L1R simulated image (Image C).

		Base	Original		Simulated
		Image C	RPC 5	RPC 6	Image C - RPC 5	Image C - RPC 6
GSD	X	10 m, 20 m	0.758 m	0.666 m	5.000 m	5.000 m
	Y	0.556 m	0.552 m	5.000 m	5.000 m
Viewing angle	Azimuth	101.9912°	102.2763°	260.8570°	107.3229°	262.9362°
	Zenith	5.7146°	33.4465°	5.7353°	41.9673°	7.4069°
		Processing time			2,869 sec	702 sec

This work was carried out with the support of “Cooperative Research Program for Agriculture Science and Technology Development (Project No. PJ 016233)” Rural Development Administration, Republic of Korea.

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