Impact of UAV-Derived RTK/PPK Products on Geometric Correction of VHR Satellite Imagery

Abstract

Satellite imagery is a widely used source of spatial information in many applications, such as land use/land cover, object detection, agricultural monitoring, and urban area monitoring. Numerous factors, including projection, tilt angle, scanner, atmospheric conditions, terrain curvature, and fluctuations, can cause satellite images to become distorted. Eliminating systematic errors caused by the sensor and platform is a crucial step to obtaining reliable information from satellite images. To utilize satellite images directly in applications requiring high accuracy, the errors in the images should be removed by geometric correction. In this study, geometric correction was applied to the Pléiades 1A (PHR) image using non-parametric methods, and the effects of different transformation models and digital elevation models (DEMs) were investigated. Ground control points (GCPs) were obtained from orthophotos created by the photogrammetric method using precise positioning. The effect of photogrammetric DEMs with various spatial resolutions on geometric correction was investigated. Additionally, the effect of DEMs obtained using the photogrammetric method was compared with those from open-source DEMs, including SRTM, ASTER GDEM, COP30, AW3D30, and NASADEM. Two-dimensional polynomial transformation, the thin plate spline (TPS), and the rational function model (RFM) were applied as transformation methods. Our results showed that a higher-accuracy geometric correction process could be achieved with orthophotos and DEMs created using precise positioning techniques such as RTK and PPK. According to the results obtained, an RMSE of 0.633 m was achieved with RFM using RTK-DEM, while an RMSE of 0.615 m was achieved with RFM using PPK-DEM.

1. Introduction

Digital images captured by aerial platforms, such as satellites, aircraft, and unmanned aerial vehicles (UAVs), provide very high-resolution (VHR) datasets with spatial resolutions of only a few centimeters. The accuracy levels of these datasets surpass those of individual pixels when processed in a quality-controlled environment [1]. The acquisition of images through optical sensors mounted on UAVs yields significant data, which is utilized in the fields of photogrammetry and remote sensing. UAVs, capable of generating high-resolution orthophotos, digital elevation models (DEMs), and multispectral images, have been extensively employed in various fields, including wildfire monitoring [2], archeology [3], computer vision [4,5], geoscientific research [6,7] such as flood assessment [8], bathymetric studies [9,10], landslide and rockfall analysis [11], and earthquake damage assessment [12]. The traditional methods employed for generating DEMs often incur significant costs and require considerable time due to the land surveying processes involved. Recently, photogrammetry has emerged as a primary technique for efficiently producing DEMs [13].

In UAV-based photogrammetry, the choice of positioning techniques is crucial for the accuracy of DEMs. A significant contributor to this advancement is the incorporation of Global Navigation Satellite System (GNSS) technology, which has markedly improved the positional accuracy of UAV surveys. Improving positioning performance enhances the accuracy of spatial data. The integration of GNSS technology in UAV-based photogrammetric studies is rapidly growing, highlighting its critical importance in advancing this innovative field [14,15,16]. Post-Processed Kinematic (PPK) and Real-Time Kinematic (RTK) positioning techniques have been widely applied in UAV-based photogrammetry and provide accurate and reliable positioning solutions [17,18]. RTK and PPK solutions for UAVs have enhanced the accuracy of exterior orientation (EO) parameters, allowing for precise image georeferencing [19]. The increasing usage of these techniques demonstrates their benefits in acquiring precise geospatial data [20,21,22]. The aim of Ekaso et al.’s study was to evaluate the position accuracy provided by RTK and whether the theoretical RTK accuracy of 2–3 cm can be achieved without an additional ground control point (GCP) [23]. Cledat et al. focused on predicting how satellite visibility changes when a drone is in motion during mission planning, the drone’s ability to steer to maintain a minimum number of visible satellites, and the success of RTK or PPK ambiguity resolution [24]. Bertin et al. investigated the feasibility of using an RTK quadcopter (Phantom 4 RTK) alongside SfM (structure from motion) photogrammetry to monitor coastal topography at a fine scale, specifically at submeter accuracy [25]. Zhang et al. aimed to assess the repeatability, reliability, and effectiveness of the PPK–SfM framework in monitoring 4D Earth surface changes using time-lapse SfM photogrammetry with centimetric accuracy [26]. Martínez-Carricondo et al. took a step forward in the investigation of the accuracy of UAV photogrammetry projects based on onboard GNSS RTK using simultaneous differential corrections from multiple fixed GNSS base stations [15]. Forlani et al. conducted an experiment on the repeatability of DSM generation from UAVs [27]. They utilized RTK-measured camera positions to successfully orient photogrammetric blocks without GCPs, achieving a horizontal coordinate accuracy better than the GSD in most instances and within 1 cm of the accuracy obtained with GCPs. Taddia et al. evaluated a PPK approach for the reconstruction of photogrammetric models and DTMs of a coastal area in the North Adriatic Sea (Italy) using a DJI Phantom 4 RTK and GNSS data from a Continuously Operating Reference Station (CORS) station 15 km away [28]. In addition, in the field of coastal mapping, it was stated that even the PPK technique can provide accurate digital terrain models (DTMs) at the centimeter level. The aim of Dinkov et al.’s research is to assess the effectiveness of PPK-SfM-MVS in producing high-resolution digital topographic data of the Earth’s surface with accuracy at the centimeter level [29]. They stated that when georeferencing is performed using the GCP method, the horizontal accuracy of the points derived from the dense point clouds ranges from 2.6 to 3.8 cm for a GSD of approximately 1.6 cm, while the vertical accuracy falls between 5.2 and 6.1 cm. Atik et al. investigated the integration of positioning techniques with SfM under various GCP distribution scenarios and obtained an accuracy of 5 cm in the no-GCP case using the PPK method [30]. The integration of UAVs with GNSS RTK/PPK antennas enables the achievement of an accuracy of 3–5 cm through post-processing methods [31,32].

Satellite imagery may be distorted by various causes, including projection, tilt angle, scanner, atmospheric conditions, ground curvature, and fluctuations [33]. Geometric correction is a widely used method for correcting related distortions. The most common non-parametric geometric correction model for VHR satellite imagery utilizes rational polynomial coefficients (RPCs) applied with a sufficient number of high-accuracy GCPs, a DEM, and a transformation model, which are necessary for the non-parametric geometric correction procedure that transforms raw data into orthoimages [34]. The accuracy of ground coordinates data affects the accuracy of geometric correction. In large areas, base maps such as topographic maps are used as GCP sources, eliminating the need for field surveys. Thanks to the use of precise positioning techniques such as RTK and PPK, together with UAV systems, UAV-derived orthophotos and DEMs with high accuracy are obtained. These products provide precise GCP coordinates for the geometric correction of satellite imagery. Various approaches to geometric correction have been explored in the literature. Zhang et al. proposed an RPC model that modifies a field-programmable gate array (FPGA) as an FP-RPC [35]. This model achieves promising results on two datasets in terms of process speed and accuracy. Marsetič et al. proposed a procedure that is automatically processed for the orthorectification of push broom sensors [36]. The model is primarily based on collinearity equations and was conducted using Level 1 satellite imagery. They examined the method on RapidEye images and achieved subpixel accuracy using the geometric model. Henrico investigated positional accuracy changes under different orthorectification scenarios to compare the distribution of GCPs and the quality of DEMs in 2016 [37]. The results showed that uniformly distributed GCP usage increases accuracy, and when increasing GCPs, the different DEM effects decrease. Cevik et al. searched for optimal GCP distributions for the geometric correction of Level 1 Pléiades images [38]. They applied the polynomial transformation model, the rational function model (RFM), and the thin plate spline (TPS) methods for transformation. This study demonstrated that proper distribution has a significant impact on positional accuracy in obtaining orthoimages. Satellite-based digital elevation models, including SRTM, ASTER, and ALOS, are readily accessible at no cost and are widely available. Ouyang et al. compared two traditional algorithms—linear regression and artificial neural network models—and found that the technique they presented reduced the root-mean-square errors of SRTM DEM by 28% and 12%, respectively, in their study [39]. They provide a method for reliably correcting SRTM or other DEM results across a large region. According to a study by Frey and Paul, both the SRTM DEM and the ASTER GDEM are appropriate for compiling topographic parameters in glacier inventories. The SRTM DEM produced partially better results than the ASTER GDEM; however, both DEMs are suitable for this purpose [40].

While existing studies [41,42,43] generally focus on publicly available global DEMs for geometric correction, a few studies [38,44] have used LiDAR data to produce high-spatial-resolution DEMs. However, obtaining LiDAR data may not always be possible due to limitations such as high cost. Global DEMs have low spatial resolution and low vertical accuracy. In particular, VHR images cannot provide sufficient accuracy for the geometric correction of satellite images. Today, UAV-based photogrammetry is a powerful technique for high-resolution DEM production [45]. UAVs offer the opportunity to model the topographic structure of large areas, providing advantages such as low sensor costs and high temporal resolution. Additionally, precise spatial data can be obtained without the need for fieldwork, thanks to precise positioning techniques such as RTK and PPK. However, to the best of our knowledge, we present a pioneering study that uses UAV-based DEM for satellite geometric correction. In particular, the applicability of the methodology presented in this study is increased by using direct georeferencing with RTK and PPK instead of indirect georeferencing. Thus, a practical UAV-based approach for the geometric correction of VHR satellite images has been presented.

In this study, several experiments were conducted to investigate the effect of different DEM sources and transformation models such as 2D polynomial model, TPS and RFM on the orthorectification of VHR satellite imagery. The objective was to investigate the effect of the accuracy of photogrammetric products obtained from UAVs with precise positioning techniques on the orthorectification of satellite imagery. The effect of precise positioning techniques, such as RTK and PPK, on the orthorectification of satellite imagery was examined. An RFM transformation was used to compare the effects of the DEM produced by the photogrammetric method and different open-source DEM data. The use of high-spatial-resolution and high-accuracy orthophotos produced by RTK as a GCP source in transformations was also investigated.

2. Methodology

2.1. Study Area

This study was conducted at the Ayazaga Campus of Istanbul Technical University in Türkiye (Figure 1). The campus encompasses a diverse landscape, including buildings, road networks, and dense vegetation. Within the study area, 25 points whose 3D positions were accurately determined were used: 10 points were used as check points (CP), and 15 points were used as control points. The acquisition time of the image that was used in this study was 24 October 2022, belonging to the Pléidas satellite as a Level 1 projected file. The image is very high-resolution, with a spatial resolution of 0.5 m and Red–Green–Blue (RGB) color composition.

The DJI Mavic 3M RTK UAV was used in this study. UAV-based photogrammetric flights were conducted at an altitude of 60 m. The images acquired during the flight were collected using 80% overlap and 70% sidelap ratios. The ground sampling distance (GSD) was approximately 1.6 cm/pixel, and the images were taken from nadir angles. In UAV flights performed in RTK mode, GNSS data were collected in RINEX format at 0.2 s intervals, and correction data were obtained from the ISKI-UKBS network, which is a regional network. With this technique, image coordinates were obtained in real time according to the relative positioning technique. In the other relative positioning technique, the PPK technique, the RTK mode of the UAV was switched off before the flight. Similarly, data in RINEX format were collected at 0.2 s intervals only by the GNSS integrated into the UAV. The data received from the ISKI-UKBS network synchronized with the flight time were used to analyze the UAV GNSS data. In this technique, UAV images were re-coordinatized by post-processing. The general flow of this study is shown in Figure 2.

2.2. UAV-Based Photogrammetric DEM

2.2.1. Real-Time Kinematic (RTK) Positioning

The Real-Time Kinematic (RTK) technique is a real-time differential positioning method that has been used in various applications since the mid-1990s. Multi-constellation and multi-frequency GNSS RTK positioning is a sophisticated real-time process aimed at achieving centimeter-level accuracy while minimizing data epochs, accommodating varying user movements even in challenging environments [46]. The principal limitation of this technique, similar to other differential methods, is the requirement for additional data, specifically corrections from at least one reference station. Additionally, it is important to note that the distance between the rover and its reference station should be restricted to approximately 50 to 100 km for optimal performance. As a result, greater distances between the receiver and the base can reduce positional accuracy during RTK positioning. This technique enhances accuracy and reliability over long distances, providing uniform accuracy in positioning. The most precise Real-Time Kinematic positioning method is Network RTK (NRTK), which utilizes corrections from a network of reference stations [47].

The fundamental formulas for network differential corrections (the pseudorange correction ($PRC$) and carrier phase correction ($CPC$)) were examined [48]:

$$\begin{array}{c}\begin{array}{c} PR{C}^j\left(t\right)={\rho }_A^{ j}\left(t\right)-R_A^{ j}\left(t\right)-c{\delta }^j\left(t\right)-c{\delta }_A\left(t\right)=I_A^{ j}\left(t\right)-T_A^{ j}\left(t\right)-E_A^{ j}\left(t\right)\\ \\ CPC\left(t\right)={\rho }_A^{ j}\left(t\right)-\lambda {\varphi }_A^{ j}\left(t\right)-\lambda N_A^{ j}-c{\delta }^j\left(t\right)-c{\delta }_A\left(t\right)=-I_A^{ j}\left(t\right)-T_A^{ j}\left(t\right)-E_A^{ j}\left(t\right)\end{array}\end{array}$$

(1)

where ${\rho }_A^{ j}\left(t\right)=\sqrt{{(X_A-X^J)}^2+{(Y_A-Y^J)}^2+{(Z_A-Z^J)}^2}$ is the geometrical distance determined using the established coordinates of both the base and the rover; $\lambda {\varphi }_A^{ j}\left(t\right)$ is the carrier phase data multiplied by wavelength; $\lambda N$ is the phase ambiguity multiplied by wavelength, fixed by the control center; $c{\delta }^j\left(t\right)$ is the satellite clock error; $c{\delta }_A\left(t\right)$ is the receiver clock error, estimated for all CORSs by the control center; $I_A^{ j}\left(t\right)$ is the ionospheric error; $T_A^{ j}\left(t\right)$ is the tropospheric error; and $E_A^{ j}\left(t\right)$ is the ephemerids error. It is essential for the network software to initially assess the ambiguities and errors in the clock of each station to standardize the differential corrections at the same level of ambiguity and timing.

RTK is recognized as a dependable and effective technique for the accurate positioning of outdoor vehicles in the open sky. Real-time positioning (RTK/NRTK) offers speed, accuracy, and affordability; however, it relies on GSM coverage, which might be inconsistent or non-existent, necessitating alternatives such as radio modems.

2.2.2. Post-Processed Kinematic (PPK) Positioning

In UAV-based photogrammetry, PPK positioning is a relative positioning technique that allows the usage of the baseline solution after field surveys. PPK-based direct georeferencing enhances accuracy and efficiency by eliminating the dependence on GCP distribution, overcoming the time and cost limitations of conventional georeferencing [26]. PPK solutions for UAVs, similar to RTK, provide accurate measurements of UAV–camera position and orientation. This reduces the degrees of freedom to three rotations for each image [that is, omega (ω), phi (φ), kappa (κ)], which can be derived from tie points and removes the need for establishing GCPs [49].

2.3. Open-Source Satellite DEM

2.3.1. Shuttle Radar Topography Mission (SRTM)

The Shuttle Radar Topography Mission (SRTM) has an 80% global coverage of the Earth between 56° S and 60° N, which can be obtained from URL-1 [50]. It is a joint project of the National Aeronautics and Space Administration (NASA), the National Geospatial-Intelligence Agency (NGA), the German and Italian Space Agencies, and the Space Shuttle Endeavour from 11 to 22 February 2000, which collected synthetic aperture radar (SAR) images of land masses. In this mission, DEM data were generated using the single-pass InSAR technique with two sensors, and interferometric maps of the scanned areas were produced by combining the two radar datasets [51]. On 2 January 2015, the United States Geological Survey (USGS) processed the newly released tiles from the Land Processes Distributed Active Archive Center (LP DAAC) in NASA’s Shuttle Radar Topography Mission (SRTM) Version 3.0 Global 1 arc second dataset (SRTMGL1). In this study, SRTMGL1 DEM data were used, and Africa, Europe, North America, South America, Asia, and Australia are currently covered by SRTMGL1 in 1° × 1° tiles with a resolution of 1 arc second or roughly 30 m.

2.3.2. ASTER GDEM

Released in December 2008, the ASTER Global Digital Elevation Model (G-DEM) is a new global DEM set with a resolution of 1 arc second for the entire Earth. It can be downloaded from URL-2 [52]. NASA and Japan’s Ministry of Economy, Trade, and Industry (METI) collaborated to create this dataset, which is based on satellite imagery from the Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) sensor. The G-DEM’s official vertical accuracy is 7 m, which is less than half of the SRTM-3’s [53].

2.3.3. AW3D30 DEM

AW3D30 DEM was developed by the Japan Aerospace and Aeronautics Agency (JAXA) and is based on Advanced Land Observing Satellite (ALOS) (Panchromatic Remote-sensing Instrument for Stereo Mapping) PRISM stereo images, providing high accuracy and global coverage. This DEM product has a horizontal resolution of roughly 30 m (basically 1 arc-second). Its use is free for both business and non-commercial uses [54].

2.3.4. COP30 DEM

The European Space Agency (ESA) developed the Copernicus DEM (COP30), a Digital Surface Model (DSM) that depicts the Earth’s surface, including structures, infrastructure, and vegetation, as part of the Copernicus Program. The model offers global coverage with resolutions of 30 m (GLO-30) and 90 m (GLO-90) [55]. Radar interferometry was used to construct COP30, which is based on WorldDEM data [56]. Simard et al. investigated the accuracy of COP30 in a large-scale evaluation study of radar-derived global DEMs [57]. Accordingly, the average elevation bias and root-mean-square error (RMSE) of COP30 in bare ground areas were found to be below 0.05 m and 0.55 m, respectively.

2.3.5. NASA DEM

ASTER GDEM, ICESat GLAS, PRISM, and other ancillary data were combined with SRTM data to create NASADEM, a DEM reprocessing of NASA’s SRTM data [58]. The SRTM data were reprocessed using an optimized hybrid processing technique of interferometric SAR data [59].

2.4. Geometric Correction in Remote Sensing

From a mathematical perspective, the rectification of high-resolution satellite images is defined as a two-dimensional transformation problem. Geometric correction refers to the process of adjusting an image obtained through remote sensing so that its projection and scale align with those of a map using a mathematical transformation. This transformation employs a mathematical function to establish a relationship between two coordinate systems. The number of transformation parameters is a significant consideration while comparing and settling on a convenient approach [60]. The act of determining the corresponding position (coordinates) of an object in both the image space and the physical space of Earth is known as coordinate transformation. Coordinate transformation is represented by three shift parameters (X₀, Y₀, Z₀), rotation around three axes (ω, $\varphi$, κ), and scale factors $(S_X, S_Y,S_Z$) along three directional axes between two coordinate systems. As the scale between these parameters is generally considered to be unity, a total of seven transformation parameters must be estimated [61]. In cases where these seven parameters are unknown, common GCPs are used within both the image and ground coordinate systems. Within the scope of this study, the 2D polynomial model, TPS, and RFM were employed as coordinate transformation approaches.

2.4.1. 2D Polynomial Model

Polynomial transformation models are commonly used for coordinate conversion as they establish mathematical links between image and ground coordinates. These models are employed to facilitate the conversion between different coordinate systems. The core principle of the 2D polynomial model is to set a geometric relationship between the pixel coordinates of GCPs and their corresponding known ground coordinates. This transformation represents the mathematical function that links the coordinates of two distinct systems. The process of determining the reciprocal location (coordinate) of an object in both image space and object space is referred to as coordinate transformation.

The polynomial transformation defines the geometric relationship between the two coordinate systems through a linear model. Polynomial equations are applied to the x and y coordinates of the GCPs using the least-squares method, enabling the representation of image adjustments without explicitly identifying the source of distortions. The order of the polynomial function varies based on the structure of study area, the number of GCPs, and the required level of accuracy. Therefore, selecting the optimal polynomial order is crucial to achieving the desired accuracy in a given application. In the mathematical representation of the polynomial model, ${(u}_i,v_i)$ denotes the image coordinates, while $(X,Y,Z)$ represents the object coordinates, with a and b as the coefficients [38]. In the relation, the sum of j and k must be less than m. In theory, the value of m could potentially be infinite.

$$\begin{array}{c}{u}_i={\displaystyle \sum _{i=0}^m} {\displaystyle \sum _{j=0}^m} {\displaystyle \sum _{k=0}^m} a_{ijk}{X}^j{Y}^k{Z}^k\end{array}$$

(2)

$$\begin{array}{c}{v}_i={\displaystyle \sum _{i=0}^m} {\displaystyle \sum _{j=0}^m} {\displaystyle \sum _{k=0}^m} b_{ijk}{X}^j{Y}^k{Z}^k\end{array}$$

(3)

In this context, $m$ indicates the order of the polynomials. The determination of polynomial coefficients is generally performed using the conventional least-squares approach as outlined below:

$$\begin{array}{c}l=A\xi +\mathsf{\epsilon }\end{array}$$

(4)

where $l$ denotes the observation vector comprising the image coordinates, $A$ represents the design matrix, ε is the vector of observation errors, and $\xi$ is the unknown parameter vector.

2.4.2. Thin Plate Spline (TPS)

TPS is a widely recognized interpolation technique that has been developed to address a significant issue in geosciences: the modeling of scattered data [62]. As a method for smoothly fitting noisy data, TPS has gained widespread acceptance. TPS is typically a type of radial basis function (RBF) that performs interpolation based on distances. TPS is primarily used for multivariate data approximation and is effective in calculating interpolation parameters from GCPs for coordinate transformation while remaining robust against local distortions [63]. TPS enables image registration by deforming an image to align with another while generating a smooth interpolation surface between control points. The mathematical expressions for TPS are presented as follows [64]. Equation (5) provides the explicit form of TPS given n points ($u_i,v_i$) in a plane.

$$\begin{array}{c}\begin{array}{c}f\left(u,v\right)=a_0+a_{1}u+a_{2}v+{\displaystyle \sum _{i=1}^N} f_i{r}_i^2\mathrm{l}\mathrm{n}{r}_i^2\end{array}\end{array}$$

(5)

where r_i is the Euclidean distance:

$$\begin{array}{c}\begin{array}{c}{r}_i=\sqrt{(}{\left(u-u_i\right)}^2+{\left(v-v_i\right)}^2)\end{array}\end{array}$$

(6)

$N$ represents the number of GCPs, and $f_i$ denotes the coefficient (row vectors), which must be determined by minimizing the following expression:

$$\begin{array}{c}E\left(f\right)+\lambda R\left(f\right)\end{array}$$

(7)

$$\begin{array}{c}E\left(f\right)={\displaystyle \sum _{i=1}^N} {‖z_i-f\left(u_i,v_i\right)‖}^2\end{array}$$

(8)

$$\begin{array}{c}R\left(f\right)=\int \int \left({\left({\displaystyle \frac{{\partial }^{2}f}{{\partial }^{2}u}}\right)}^2+2{\left({\displaystyle \frac{{\partial }^{2}f}{\partial u\partial v}}\right)}^2+{\left({\displaystyle \frac{{\partial }^{2}f}{\partial v^2}}\right)}^2\right)\mathrm{d}u\mathrm{d}v\end{array}$$

(9)

where $z_i$ represents the observation at ($u_i,v_i$), and λ denotes the smoothing parameter that regulates the smoothness of the allowable deformation. When λ is set to 0, there are no smoothness constraints, and the TPS function interpolates the data points. If λ differs from zero, the TPS function incorporates smoothness constraints while interpolating the data points. This condition results in a smoother spline function, which aids in controlling overfitting.

2.4.3. The Rational Function Model (RFM)

Mathematically, the RFM serves as a generalized sensor model and has been widely employed in the geometric processing of optical images. It utilizes the ratio of polynomials to establish the transformation between image and object spaces, providing an effective alternative to physical sensor models [65,66]. The RFM extends the polynomial model, encompassing polynomial models as a specific case. It offers flexibility in coordinate systems, allowing the use of object coordinates in various frameworks, including geocentric, geographic, or any map projection system [67]. Additionally, it is derived from the ratio of two polynomials, as expressed in Equations (10) and (11).

$$\begin{array}{c}l={\displaystyle \frac{\sum _{g=0}^m \sum _{j=0}^m \sum _{k=0}^m a_{jk}{X}^g{Y}^j{Z}^k}{\sum _{g=0}^m \sum _{j=0}^m \sum _{k=0}^m b_{jk}{X}^g{Y}^j{Z}^k}}\end{array}$$

(10)

$$\begin{array}{c}s={\displaystyle \frac{\sum _{g=0}^m \sum _{j=0}^m \sum _{k=0}^m c_{jk}{X}^g{Y}^j{Z}^k}{\sum _{g=0}^m \sum _{j=0}^m \sum _{k=0}^m d_{jk}{X}^g{Y}^j{Z}^k}}\end{array}$$

(11)

where $(l,s)$ denote the image coordinates, and $(X,Y)$ represent the object coordinates, with $a, b,$ and $c$ as the corresponding coefficients. The subscripts n, 0, and s indicate normalization, translation, and scale, respectively. Additionally, the polynomial order must not exceed three, satisfying the condition where (g + j  + k) < 3 [68].

2.5. Experimental Details

This study was carried out with UAV images, satellite images, and DEM data obtained for the Istanbul Technical University Ayazaga Campus. Two different flights were carried out on the campus to obtain aerial images: the RTK and non-RTK modes. The RINEX data collected in non-RTK mode were then analyzed using PPK. The photogrammetric evaluation was performed using Pix4Dmapper software (version 4.3.33). As a result, orthophotos and DEM images coordinated with RTK and PPK were produced.

Geometric correction was applied using The OrthoEngine module of Catalyst Professional software (version 3.0.3). A total of 15 GCPs and 10 CPs with homogeneous distribution in the study area were selected for orthorectification. These points were marked both on the orthophoto and on the Pleaideas satellite image by selecting prominent and fixed objects on the ground. The RMSE value was calculated using 10 CPs for each method.

The RMSE values of the CPs were calculated as the difference between the point coordinates in the rectified image obtained after the transformations and the orthophoto generated with the photogrammetric technique. In Equation (12), $\epsilon \epsilon$ refers to the sum of the squares of the calculated coordinate differences, and the n variable represents the number of CPs. The RMSE is an absolute value. The reference coordinates of the GCPs were taken from the UAV-based products.

$$RMSE=\sqrt{{\displaystyle \frac{\epsilon \epsilon }{n}}\mathrm{}}$$

(12)

3. Results

DEMs generated from UAV images using RTK and PPK positioning techniques were compared with open-source DEM images in the literature. Table 1 presents the altitude RMSE values calculated using 21 GCPs. When comparing the RTK and PPK methods, a DEM that is 0.8 cm more accurate was produced with the PPK method. The DEM produced with RTK has an accuracy of 4.7 cm. The spatial resolution of open-source DEM data is 30 m, and the elevation accuracy is at the meter level. Among these data, ASTER GDEM has the lowest vertical accuracy with 3.806 m, while COP30 has the highest accuracy with 2.684 m. DEM images produced by each method are shown in Figure 3.

Table 1. Altitude differences (unit is meter).

Value	NASA DEM	Aster GDEM	AWD30	COP30	SRTM	RTK-DEM	PPK-DEM
Altitude	3.181	3.806	3.007	2.684	3.388	0.047	0.039

Table 1. Altitude differences (unit is meter).

Value	NASA DEM	Aster GDEM	AWD30	COP30	SRTM	RTK-DEM	PPK-DEM
Altitude	3.181	3.806	3.007	2.684	3.388	0.047	0.039

3.1. Results of 2D Polynomial Model

The performance of the methods used for the orthorectification of the projected Level 1 Pléiades image is compared with the RMSE in Table 2, Table 3, Table 4 and Table 5. For the 2D polynomial model, the results of the 1st-, 2nd-, and 3rd-order models are compared in Table 2.

When the values obtained as a result of orthorectification using a 2D polynomial model were evaluated, it was observed that PPK had lower error values than RTK for the first order. When the 2nd-order quadratic model was used, a significant increase in error values was observed compared to the 1st-order linear model. It was observed that the error values increased even more in the third-order cubic model. In this case, it was observed that the accuracy in orthorectification decreased as the complexity of the model increased when the 2D polynomial model was used. According to the experiments, the PPK results were generally slightly better than those of RTK. The best results in the 2D polynomial model were usually obtained in the first order.

Table 2. Results of 2D polynomial model (unit is meter).

Method	RMSE	1st Order	2nd Order	3rd Order
RTK	X	0.359	0.643	0.706
	Y	0.815	1.345	1.420
	Total	0.890	1.490	1.586
PPK	X	0.347	0.664	0.692
	Y	0.811	1.332	1.386
	Total	0.882	1.488	1.549

Table 2. Results of 2D polynomial model (unit is meter).

Method	RMSE	1st Order	2nd Order	3rd Order
RTK	X	0.359	0.643	0.706
	Y	0.815	1.345	1.420
	Total	0.890	1.490	1.586
PPK	X	0.347	0.664	0.692
	Y	0.811	1.332	1.386
	Total	0.882	1.488	1.549

3.2. Results of Thin Plate Spline

TPS is generally a flexible interpolation method for irregular data. As seen in Table 3, the PPK method has slightly lower error values compared to RTK. TPS generally has higher error values than the 2D polynomial model. Therefore, the TPS method may not be suitable for this study’s dataset.

Table 3. Results of thin plate splines (unit is meter).

Method	Coordinate	RMSE
RTK	X	0.523
	Y	1.199
	Total	1.308
PPK	X	0.515
	Y	1.188
	Total	1.295

Table 3. Results of thin plate splines (unit is meter).

Method	Coordinate	RMSE
RTK	X	0.523
	Y	1.199
	Total	1.308
PPK	X	0.515
	Y	1.188
	Total	1.295

3.3. Results of Rational Function Model

The RFM takes into account the DEM source for geometric correction. The effect of the spatial resolution of the photogrammetrically generated DEM on the geometric correction is analyzed. DEMs with 2 cm, 50 cm, 1 m, and 2 m spatial resolution were produced using RTK and PPK positioning techniques. The 5th-order RFM provides the lowest RMSE values for all DEMs. The 4th- and 5th-order RFMs have similar results. In the 6th-order RFM, overfitting occurs, resulting in higher RMSE values. In photogrammetric DEMs, the effect of spatial resolution on accuracy is limited. When DEMs with 2 cm and 2 m spatial resolution are used, there is no significant difference in RMSE values of CPs. For the DEM produced with PPK, there is a slight improvement in the RMSE value as the spatial resolution decreases. With the 50 cm spatial resolution PPK-DEM, the lowest RMSE of 0.615 m was achieved. When the 6th-order RFM was applied with RTK-DEM with a 1 m spatial resolution, the highest RMSE of 8.845 m was obtained. The results of the RFMs using photogrammetric DEMs with different spatial resolutions are presented in Table 4.

Table 4. Results of RFMs according to photogrammetric DEMs with various spatial resolutions (unit is meter).

DEM (Spatial Resolution)	RMSE	4th Order	5th Order	6th Order
RTK-DEM (2 cm)	X	0.530	0.515	3.070
	Y	0.397	0.372	0.345
	Total	0.662	0.635	3.089
RTK-DEM (50 cm)	X	0.589	0.516	2.211
	Y	0.340	0.367	0.437
	Total	0.680	0.633	2.254
RTK-DEM (1 m)	X	0.591	0.517	8.276
	Y	0.334	0.383	3.120
	Total	0.679	0.643	8.845
RTK-DEM (2 m)	X	0.588	0.520	1.645
	Y	0.340	0.392	0.395
	Total	0.679	0.651	1.692
PPK-DEM (2 cm)	X	0.588	0.514	2.202
	Y	0.340	0.379	0.321
	Total	0.679	0.639	2.225
PPK-DEM (50 cm)	X	0.587	0.514	0.676
	Y	0.358	0.337	0.301
	Total	0.688	0.615	0.740
PPK-DEM (1 m)	X	0.587	0.513	1.923
	Y	0.351	0.345	0.407
	Total	0.684	0.618	1.966
PPK-DEM (2 m)	X	0.590	0.513	2.086
	Y	0.338	0.345	0.304
	Total	0.680	0.618	2.108

Table 4. Results of RFMs according to photogrammetric DEMs with various spatial resolutions (unit is meter).

DEM (Spatial Resolution)	RMSE	4th Order	5th Order	6th Order
RTK-DEM (2 cm)	X	0.530	0.515	3.070
	Y	0.397	0.372	0.345
	Total	0.662	0.635	3.089
RTK-DEM (50 cm)	X	0.589	0.516	2.211
	Y	0.340	0.367	0.437
	Total	0.680	0.633	2.254
RTK-DEM (1 m)	X	0.591	0.517	8.276
	Y	0.334	0.383	3.120
	Total	0.679	0.643	8.845
RTK-DEM (2 m)	X	0.588	0.520	1.645
	Y	0.340	0.392	0.395
	Total	0.679	0.651	1.692
PPK-DEM (2 cm)	X	0.588	0.514	2.202
	Y	0.340	0.379	0.321
	Total	0.679	0.639	2.225
PPK-DEM (50 cm)	X	0.587	0.514	0.676
	Y	0.358	0.337	0.301
	Total	0.688	0.615	0.740
PPK-DEM (1 m)	X	0.587	0.513	1.923
	Y	0.351	0.345	0.407
	Total	0.684	0.618	1.966
PPK-DEM (2 m)	X	0.590	0.513	2.086
	Y	0.338	0.345	0.304
	Total	0.680	0.618	2.108

Table 5 shows the results of the DEM-based orthorectification process using RFMs. Error analysis of the 4th-, 5th-, and 6th-order RFMs was performed using different DEM sources. In the 4th- and 5th-order RFMs, the X and total error values of PPK-DEM were higher than those of RTK-DEM, but the Y error values were slightly lower. When the 6th-order model was used, a significant increase in the number of errors was observed. Here, the model may have become overly complex and overlearned. In general, RTK-DEM showed a more stable performance compared to PPK-DEM.

Table 5. Results of the rational function model (unit is meter).

DEM	RMSE	4th Order	5th Order	6th Order
RTK-DEM *	X	0.530	0.515	3.070
	Y	0.397	0.372	0.345
	Total	0.662	0.635	3.089
PPK-DEM *	X	0.588	0.514	2.202
	Y	0.340	0.379	0.321
	Total	0.679	0.639	2.225
ASTER	X	0.970	0.552	-
	Y	1.015	1.692	-
	Total	1.404	1.780	-
SRTM	X	0.640	-	-
	Y	0.614	-	-
	Total	0.887	-	-
AW3D30	X	0.672	0.592	2.473
	Y	0.549	0.588	0.898
	Total	0.868	0.834	2.631
COP30	X	0.597	0.608	0.927
	Y	0.739	0.715	0.865
	Total	0.950	0.939	1.268
NASADEM	X	0.619	0.627	-
	Y	0.709	0.718	-
	Total	0.941	0.953	-

* The spatial resolution of the DEMs is 2 cm.

Table 5. Results of the rational function model (unit is meter).

DEM	RMSE	4th Order	5th Order	6th Order
RTK-DEM *	X	0.530	0.515	3.070
	Y	0.397	0.372	0.345
	Total	0.662	0.635	3.089
PPK-DEM *	X	0.588	0.514	2.202
	Y	0.340	0.379	0.321
	Total	0.679	0.639	2.225
ASTER	X	0.970	0.552	-
	Y	1.015	1.692	-
	Total	1.404	1.780	-
SRTM	X	0.640	-	-
	Y	0.614	-	-
	Total	0.887	-	-
AW3D30	X	0.672	0.592	2.473
	Y	0.549	0.588	0.898
	Total	0.868	0.834	2.631
COP30	X	0.597	0.608	0.927
	Y	0.739	0.715	0.865
	Total	0.950	0.939	1.268
NASADEM	X	0.619	0.627	-
	Y	0.709	0.718	-
	Total	0.941	0.953	-

* The spatial resolution of the DEMs is 2 cm.

4. Discussion

The accuracy of positional data is generally classified into horizontal and vertical components, which, in GNSS, are represented by distinct values. The vertical positioning accuracy is typically lower than the horizontal accuracy due to unfavorable satellite geometry, atmospheric delays (particularly tropospheric effects), inaccuracies in the geoid model, and multipath interference. Additionally, signal delays caused by tropospheric refraction and water vapor further degrade vertical accuracy, leading to increased positioning errors. A widely acknowledged principle in GNSS states that the vertical positioning error is generally twice the size of the horizontal error [69]. In this study, the RTK and PPK positioning techniques demonstrated lower accuracy in the vertical component compared to the horizontal component. Additionally, it has been shown that PPK offers greater flexibility than RTK. Utilizing PPK allows for the same dataset to be processed in various ways multiple times, thereby facilitating the minimization of errors. A comparison of the obtained results reveals that PPK provides superior performance compared to RTK. The improved accuracy of PPK is attributed to the use of precise satellite orbit and clock products in the PPK process. According to Table 4, the effect of photogrammetric DEMs with various spatial resolutions on geometric correction with RFM was investigated. Since the horizontal coordinates are taken from the high-spatial-resolution orthophoto, the effect of reducing the spatial resolution of the DEM on the geometric correction is limited. There is no significant difference when DEMs with spatial resolutions of 2 cm and 2 m are used.

The total error increases considerably when ASTER GDEM is used. The DEM could not provide sufficient accuracy at this stage. When the SRTM DEM source was used, only the 4th-order result was produced in the RFM. The reason why calculations could not be made at other order is that the error tolerance was exceeded. When AW3D30 DEM was used, lower error values were obtained in the 4th and 5th order compared to the 6th order. This can be interpreted as utilizing the AW3D30 DEM data source, which may be suitable for low-grade models. When COP 30 DEM was used, the errors obtained by the RFM in the 4th and 5th order were stable. This DEM generally provided an average accuracy. Similar errors were received in the 4th and 5th order when the NASA DEM source was used. These results indicate that when the RFM was used for orthorectification, the best results were obtained with RTK-DEM and PPK-DEM as DEM sources. Lower-degree models produced better results compared to those of the 6th-order models. Since complexity generally increased in the 6th order, error values changed negatively. When the 6th-order RFM is applied, a dramatic increase in RMSE values occurs. Considering the data used, if the 6th-order RFM is applied, overfitting happens. The most significant disadvantage of RFMs is overfitting, which negatively impacts the accuracy of RFMs [70]. Considering the results in Table 4 and Table 5, it is concluded that it is appropriate to use the 4th- and 5th-order RFM. However, the RMSE increases because overfitting occurs in high-order RFMs. Data from other DEM sources, such as ASTER, should be used carefully and controlled in these models. While AW3D30 DEM provides a reliable alternative for low-order models, it should be used with caution in applications where the high-order RFM algorithm is used.

Although open-source DEM data have a high vertical error, a lower RMSE is achieved when used for geometric correction. The horizontal coordinates of the control points are determined from the orthophoto produced by RTK. The horizontal coordinates remain the same regardless of the DEM used. This guarantees a certain level of accuracy. However, the accuracy is significantly affected by the different precision of open-source DEMs and DEMs produced by the UAV-based method. As a result, the higher the vertical accuracy of the DEM, the more effective the geometric correction that can be applied.

In [38], a meter-level error value was obtained in the orthorectification process performed in the same study area. An indirect georeferencing method was used to produce the orthophoto, which served as a GCP source. This reduced the orthophoto accuracy compared to RTK. In addition, the vertical accuracy of the DEM was 8 cm. Compared with the DEM accuracies produced in our study, this value reduced the accuracy. In another study [43] examining the effects of open-source DEMs, an RMSE of more than 2 m was obtained with ALOS PALSAR, SRTM, and ASTER DEM data. Google Earth and the General Directorate of Mapping—Globe (HGM), which have lower accuracies than that of orthophotos, were used as GCP sources and had a higher RMSE. Thus, the importance of the base map selected as the GCP source for effective orthorectification was demonstrated. UAV imagery, when positioned using RTK and PPK techniques, can achieve submeter-level orthorectification accuracy, even when combined with open-source DEMs.

5. Conclusions

Precise position information is necessary to obtain accurate and reliable results for non-parametric geometric correction methods. In this study, the use of DEM data generated from UAV imagery, utilizing precise positioning techniques for the geometric correction of satellite imagery, was investigated. Generally, it is concluded that high-accuracy DEM data produced by RTK and PPK provide the most successful results for geometric correction. However, when using horizontal position information measured from the orthophoto, the effect of low-accuracy height information is limited. For effective geometric correction, using DEM data with high spatial resolution and vertical accuracy, together with precise horizontal position information, is the optimum solution. RTK and PPK methods enable the application of high-accuracy geometric correction without requiring any GCP information. In future studies, the effects of geometric correction on applications involving satellite image processing are planned to be investigated.