Abstract
Satellite digital elevation models (DEMs) are used for decision-making in various fields. Therefore, evaluating and improving vertical accuracy of DEM can increase the quality of end products. This article aimed to increase the vertical accuracy of most popular satellite DEMs (i.e., the ASTER, Shuttle Radar Topography Mission [SRTM], Forest And Buildings removed Copernicus DEM [FABDEM], and Multi-Error-Removed Improved-Terrain [MERIT]) using the particle swarm optimization (PSO) algorithm. For this purpose, at first, the vertical error of DEMs was estimated via ground truth data. Next, a second-order polynomial was applied to model the vertical error in the study area. To select the polynomial with the highest accuracy, employed for vertical error modeling, the coefficients of the polynomial have been optimized using the PSO algorithm. Finally, the efficiency of the proposed algorithm has been evaluated by other ground truth data and in situ observations. The results show that the mean absolute error (MAE) of SRTM DEM is 4.83 m while this factor for ASTER DEM is 5.35 m, for FABDEM is 4.28, and for MERIT is 3.87. The obtained results indicated that the proposed model could improve the MAE of vertical accuracy of SRTM, ASTER, FABDEM, and MERIT DEMs to 0.83, 0.51, 0.37, and 0.29 m, respectively.
1 Introduction
The digital elevation model (DEM) is one of the important products in remote sensing. DEMs may be extracted from different methods with various levels of accuracy and cost. Traditionally, aerial stereo images have been the main source of DEM generation, but these models are now mainly extracted from satellite images [1]. The DEM (DTM according to some studies) is a three-dimensional view of the bare ground without any objects such as plants and buildings, while the digital surface model (DSM) shows the surface with all the objects on it; i.e., all satellite data are DSM model [2,3]. DSM generation methods based on the satellite imagery overweigh traditional methods because of their higher vastness and lower cost [1]. Since DEMs are obtained using different sensors and different geometries, choosing the most appropriate DEM that can accurately represent ground conditions is one of the main requirements in many studies [4]. Satellite DEMs like Shuttle Radar Topography Mission (SRTM) and ASTER have been widely used in environmental research, remote sensing, and GIS applications [5–11].
Despite many publications about the applications and accuracy assessment of DEMs, studies in the improvement of such products are limited. In the following, some works in this domain are presented. Jarvis et al. investigated the accuracy of SRTM and 1:50,000 scale topographic maps in Honduras and showed that SRTM and maps have average error levels of 8 and 20 m, respectively. They concluded that the vertical accuracy of SRTM is greater than 1:50,000 scale maps [12]. Rodriguez et al. studied the vertical accuracy of SRTM on a global scale using GPS data and other available altitudinal data. They indicated that the absolute error of this model is equal to 7.5 m [13]. Sefercik computed the vertical accuracy of ASTER DEM in Barcelona (Spain) and Zonguldak (Turkey). Based on his research, the vertical root mean square error (RMSE) of this DEM in comparison with reference elevation data is 4 and 8 m in Barcelona and Zonguldak, respectively. He showed that the accuracy of satellite DEMs has a strong correlation with the slope of the area [14]. Du et al. investigated the accuracy of ASTER and SRTM DEMs in two regions of China using statistical analysis. The results showed that the SRTM has 2.38 and 4.43 m RMSE error in these areas. Also, these errors for ASTER were equal to 6.98 and 4.83 m RMSE for these regions [15]. Elkhrachy evaluated the accuracy of ASTER and SRTM DEMs according to GPS data and topographic maps. Based on GPS data, the values of ±5.94 and ±5.07 m were achieved for SRTM and ASTER DEMs’ RMSE, respectively. Also, using the topographic map, the RMSE was ±6.87 and ±7.97 m for SRTM and ASTER DEMs, respectively [16]. Yue et al. have recruited ASTER GDEM v2 to generate a seamless DEM dataset blending SRTM1 [17]. Liu et al. studied the functioning of seven public freely available DEM datasets, including SRTM3 V4.1 DEM, ASTER GDEM V2, AW3D30 DEM, SRTM1 DEM, Multi-Error-Removed Improved-Terrain (MERIT) DEM, VFP-DEM, and Seamless SRTM1 DEM, over the area of High-Mountain Asia (HMA) through reference to elevation data with a great degree of accuracy obtained from ICESat altimetry. It was concluded that the AW3D30 DEM shows the greatest degree of accuracy in HMA land regions among these datasets [18]. González-Moradas and Viveen presented a study with the aim of accuracy assessment of ASTER GDEM2, SRTM v3.0, ALOS AW3D30 and TanDEM-X DEMs in Peru. To achieve this purpose, their elevation values were compared against values measured with a dual-frequency GNSS data. The overall results of the four areas showed that the RMSE was 6.907, 5.113, and 6.246 m RMSE for SRTM3, AW3D30, and ASTER GDEM2, respectively. The RMSE of 1.666 meters for the TanDEM-X was also estimated. The final results stated that the SRTM3 has appropriate performance, while ASTER GDEM2 had always low accuracy. Based on the obtained results, they concluded that SRTM3 delivers the most stable performance in almost all test areas [19]. Mingorance and Lopez reviewed more than 200 references to obtain knowledge about the DEM accuracy assessment methods applied in the last three decades. It was concluded that the relevant accuracy assessment techniques have been improved; however, plenty of factors should also be considered by investigators, proficient associations, and standard organizations, including a shared vocabulary, meta-quality assessment techniques, standardized assessment techniques, and practical quality indicators [20].
A review of the previous studies indicates that despite the high volume of studies performed on the accuracy assessment of DEMs that have provided a better understanding of the models, few studies have been conducted on improving the accuracy of these models. Thus, the present study is made up of two main parts. The first section examines the accuracy of some of the most common freely accessible satellite DEMs given the ground control points prepared by the Iranian Cartography Center. The second section aims to improve the accuracy of the DEMs by modeling error behavior in the study area utilizing an optimized polynomial. Recruiting the particle swarm optimization (PSO) optimization algorithm to fit an optimized polynomial to the error behavior across the region to estimate and reduce error in digital models can be considered the most significant contribution of the present research. Going through details, from 250 points, 190 points were used as control points, for the modeling step, and the rest of them were used as checkpoints, for the accuracy assessment step. It should be noted that although the altitudinal datum of the stated elevation models is different, the aim of this study is not to compare the accuracy of DEMs to recommend the best model for use in the study area, but rather to evaluate the height accuracy of common and existing models and improve their accuracy using the proposed algorithm. Therefore, the models with the accuracy of DEM with their initial height datum were evaluated and improved in this study. Therefore, changing the altitudinal datum has not been required to implement the proposed algorithm.
2 Study area
The study case, Khuzestan Province, has an area of 64,057 km2, located in the southwestern part of Iran between the coordinates of 47°40–50°33′E and 29°57–33°00′N. The elevation ranges from sea level to around 3,500 m above sea level. This province can be divided into two regions, namely the mountainous regions north of the Ahvaz Ridge and the plains to its south. Figure 1 shows the study area and distribution of the reference data of the elevation.
Location of the study area and ground truth data.
3 Materials and methods
As mentioned before, the purpose of this research is to increase the accuracy of some freely accessible satellite DEMs including SRTM, ASTER, Forest And Buildings removed Copernicus DEM (FABDEM), and MERIT. Therefore, the vertical accuracy of elevation models was estimated in comparison with elevation point data. The control point being utilized in this study was measured with an RMSE of about 5 mm. The RMSE, mean absolute error (MAE), and mean bias error (MBE) were employed to evaluate the accuracy of data. Then, a global polynomial has been utilized to improve the accuracy of these DEMs. To achieve better results, the coefficients of this polynomial were optimized by the PSO algorithm. Figure 2 shows the flowchart of the implemented algorithm.
Flowchart of the implemented algorithm.
3.1 SRTM DEM
In 2000, the SRTM was launched for collecting the data for 11 days. The emitted radar signals’ reflection was utilized by the shuttle within the X-band (wavelength of 3.1 cm) for the X-SAR sensor established by the German Aerospace Agency and the C-band (wavelength of 5.6 cm) for the SIR-C sensor established by the Jet Propulsion Laboratory. In the present study, the SRTM-ver3 DEM with a spatial resolution of 27.83 m has been recruited, which is also known as the 30 m DEM [21,22]. SRTM DEM was obtained from the data archive of USGS (Figure 3).
SRTM DEM in the study area.
3.2 ASTER DEM
The ASTER DEM is resultant from the TERRA satellite-carried ASTER sensor taking the Earth’s images within 14 various ultraviolet to infrared bands. Because of having such a wide range, the ground resolution is within the range of 15–90 m. Taking stereo images can help to create a DEM. The improved global DEM V3 adds 260,000 extra stereo pairs enhancing the coverage and decreasing the artifacts incidence. A superior spatial resolution is provided by the refined production algorithm increasing the vertical and horizontal accuracy and enhanced discovery and coverage of the water body. Through the ASTER GDEM V2, the Geo-TIFF format is maintained while providing the tile structure and gridding similar to V1, with 1° × 1° tile and 30-m postings [23]. ASTER DEM was obtained from the data archive of USGS.
3.3 FABDEM
FABDEM could be regarded as the first universal DEM capable of removing buildings and forests at a resolution of 30 m. A correction algorithm is applied by the relevant data to remove biases from the Copernicus GLO 30 DSM derived from the existence of objects on the surface of the earth. It leads to the improvement of datasets where natural hazardous events can be universally modeled by the users. These kinds of models are made in greater detail than ever seen before. The relevant data are generated in a tandem arrangement by the University of Bristol and are accessible at a grid spacing of 1 arc-second (about 30 m at the equator) throughout the universe. This kind of data looks quite perfect for those experts who require to comprehend flood risks on a universal scale or in worldwide areas with data scarcity [24].
3.4 MERIT DEM
The development of MERIT DEM was done by eliminating numerous error components, including stripe noise, absolute bias, tree height bias, and speckle noise, from the available spaceborne DEMs (i.e., AW3D-30m v1 and SRTM3 v2.1). It denotes the terrain elevations at a resolution of 3 s (∼90 m at the equator) and covers land regions between 90N–60S based on the EGM96 geoid reference. Stripe noise, absolute bias, tree height bias, and speckle noise were separated by applying numerous filtering methods and satellite datasets. After errors were eliminated, land regions mapped with 2 m or a finer vertical accuracy were improved from 39 to 58% [25].
Figure 3 demonstrates the SRTM DEM of the study area. This figure indicates that the this area can be categorized into three mountainous (region 1), foothill (region 2), and plain (region 3) regions. As the DEM demonstrates, elevation variations decline in these regions, respectively. The efficiency of the proposed model in each of the three regions will be examined separately to evaluate the model efficiency in regions with various elevation behaviors.
3.5 Accuracy assessment
To compute the vertical accuracy of DEMs, MAE, RMSE, and MBE indices were recruited to evaluate the uncertainty of the DEM data with reference data. The altitudinal data were evaluated based on the following equations [26,27]:
where O i is the observation altitude and P i is the DEM elevation.
3.6 Polynomial
A polynomial is mathematically a term including coefficients and containing some single terms. Polynomials are mostly applied in science and mathematics, for instance in forming polynomial equations to encode various elementary to complex scientific problems. Also, they are utilized in approximating simple to complicated functions. In this article, one global polynomial is used to approximate vertical error function in target satellite DEMs. To be more precise, the present study utilized a polynomial to model error behavior across the entire region after estimating the error in control points. In this section, the authors try to estimate the error distribution function in the study area to measure DEM elevation error in various points and correct them. Since the order of the polynomial is determined based on the complexity of the function and considering the relatively uniform error behavior across the region (Figure 4), a second-order polynomial was employed to model the error distribution function (equation (4)). Equation (4) expressed the polynomial employed in this article to model the error behavior of DEMs regarding their horizontal positions.
where Δz is the vertical error at points (x, y) in DEM. This parameter is computed by comparing the DEM with reference elevation data. Given that the coordinate values (x, y) were much higher than the altitudinal error (Δz), the normal value of these coordinates (normalized by linear method) was used in the equation. Here, a i (I = 1:5) are the coefficients of the polynomial. These coefficients determine how this polynomial approximates the error function.
Error distribution for (a) SRTM, (b) ASTER, (c) FABDEM, and (d) MERIt elevation data.
Determining the coefficients of the polynomial to ensure the best fitting to the desired function is among the most essential steps in using polynomials. To choose the polynomial order, a preliminary evaluation was done using first-, second-, and third-order polynomials, and the results showed that the second order has the most potential in error modeling in the study area. Therefore, this polynomial was used to optimize the coefficients. Provided that proper estimation of the error function across the region is made and deducted from the available DEM, the resulting DEM would be expected to have higher accuracy. Previous studies have used various techniques to determine the best polynomial coefficients, among which optimization algorithms have repeatedly been used over recent years [28]. Hence, the present study aims to use one of the most prevalent optimization techniques to calculate the optimized polynomial coefficient so that the polynomial error modeling function is the most similar to the error rate in the control points.
Therefore, a popular and dominant optimization method should be recruited to valuate these parameters. Given the aim of the present study and the requirement of optimization methods, the final RMSE of the DEM calculated at the control points after deducting the error (Δz) was considered the cost function. The optimization method, applied in this article, is presented in the following.
3.7 PSO
PSO algorithm was proposed by Eberhart and Kennedy [29] for optimization problems in continuous space. PSO is a computational technique optimizing a problem through iteratively attempting to enhance a candidate solution concerning a quality’s definite measure. This evolutionary algorithm starts with a random initial population. The primary values of the searched variables at the first generation, (a0, …, a5), are randomly formed within the range of upper and lower bounds:
In which R(0,1) indicates a uniformly distributed random number between 0 and 1, ai, 0 represents the searched variable at the first generation, and UB and LB are the two values for upper and lower bounds for parameters, respectively. Here, UB and LB were set as 5 and −5, respectively. these values were estimated utilizing a primary least squares error strategy. Each random possible answer is assumed as a particle that travels by a specific velocity that determined in any steps. Each particle’s coordination in the swarm is restricted in a definite searching space.
Furthermore, it is stated that the velocities and positions identify the particles within the swarm. All particles within each training iteration fly across the searching space to discover the superior location for each particle (the optimal fitness value). Equations (6) and (7) illustrate the alternations and updates in the position and velocity of the particles in each repetition [30].
where r
1 and r
2 denote random values between 0 and 1, c
1 and c
2 represent acceleration constants (equal to c
1 = c
2 = 1.496 in this study), w shows the inertia weight (equal to w = 1 in this study), x
id(t) signifies the ith particle position in a searching space with d-dimension indicating a candidate solution to the problem at repetition t, v
id(t) denotes the corresponding velocity, and
Then, the swarm’s best location is defined through the selection of the particle position with the lowest cost value. Over the optimizing procedure, numerous variables were explored by the PSO population as the polynomial parameters. Ultimately, the polynomial parameters’ best values are extracted as the coordination of the swarm’s best location. Ultimately, after identification of optimized values of polynomial parameters, the error value, calculated by polynomial, has been used to improve DEM accuracy in all DEMM pixels:
where Z corrected(x, y) is the corrected value of elevation at (x, y), Z DEM(x, y) is the elevation of (ASTER, SRTM, FABDEM, or MERIT) DEM, and Δz is the correction value calculated by polynomial at the (x, y) position (equation (8)).
4 Results and discussion
The first step is regarded as altitudinal accuracy assessment for the above-stated DEMs in the study area. For this purpose, 190 points in the Khuzestan Province were selected as ground truth data to evaluate the vertical accuracy in the above-mentioned numerical models. Table 1 demonstrates the results of the error estimation including altitudinal error for the SRTM, ASTER, FABDEM, and MERIT DEMs in comparison with ground truth checkpoints. The RMSE values for Khuzestan Province were 5.79, 6.23, 5.01, and 4.32 m, respectively.
Numerical results acquired from accuracy assessment of the ASTER, SRTM, FABDEM, and MERIT DEMs compared with the ground truth data
| Point no. | Latitude | Longitude | Z | Z | Z | Z | Z | Error | Error | Error | Error | Region number |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Geoid | ASTER | SRTM | FABDEM | MERIT | ASTER | SRTM | FABDEM | MERIT | ||||
| 1 | 32.01589 | 48.03482 | 137.938 | 143 | 142 | 140.631 | 140.843 | −4.938 | −1.938 | 2.693 | 2.905 | 2 |
| 2 | 31.99345 | 49.92993 | 876.254 | 885 | 883 | 882.125 | 881.914 | 8.746 | 6.746 | 5.871 | 5.66 | 1 |
| 3 | 32.71966 | 49.3284 | 2427.015 | 2,438 | 2,435 | 2433.581 | 2433.109 | 10.985 | 7.985 | 6.566 | 6.094 | 1 |
| 4 | 32.64732 | 48.59934 | 651.733 | 645 | 649 | 649.531 | 649.968 | −6.733 | −2.733 | −2.202 | −1.765 | 1 |
| 5 | 32.75017 | 48.2441 | 487.644 | 497 | 495 | 494.417 | 493.319 | 9.356 | 7.356 | 6.773 | 5.675 | 1 |
| 6 | 31.56039 | 50.11553 | 2880.705 | 2,887 | 2,888 | 2885.825 | 2885.182 | 6.295 | 7.295 | 5.12 | 4.477 | 1 |
| 7 | 30.48187 | 49.62003 | 18.123 | 16 | 17 | 16.316 | 16.519 | −1.123 | −0.123 | −1.807 | −1.604 | 3 |
| 8 | 30.80467 | 49.06381 | 9.299 | 7 | 8 | 7.614 | 8.0591 | −1.299 | −0.299 | −1.685 | −1.2399 | 3 |
| 9 | 31.30483 | 48.64483 | 24.005 | 21 | 23 | 22.829 | 22.549 | 0.995 | 1.995 | −1.176 | −1.456 | 3 |
| 10 | 30.75944 | 48.2789 | 8.715 | 11 | 10 | 12.531 | 12.337 | 1.285 | 0.285 | 2.816 | 2.622 | 3 |
| 11 | 31.27695 | 47.98609 | 15.262 | 12 | 14 | 13.673 | 14.149 | −0.262 | −1.262 | −1.589 | −1.113 | 3 |
| 12 | 30.22265 | 49.76137 | 17.67 | 15 | 17 | 16.206 | 16.591 | −1.67 | −0.67 | −1.464 | −1.079 | 3 |
| 13 | 30.85111 | 49.82291 | 211.468 | 207 | 210 | 210.481 | 210.572 | −4.468 | −1.468 | 1.013 | 1.104 | 2 |
| 14 | 31.56393 | 49.36068 | 215.481 | 220 | 219 | 218.182 | 217.893 | 3.519 | 2.519 | 1.701 | 1.412 | 2 |
| 15 | 32.2184 | 49.43259 | 693.153 | 687 | 691 | 691.018 | 691.32 | −6.153 | −2.153 | −2.135 | −1.833 | 1 |
| 16 | 32.27739 | 48.51004 | 106.569 | 103 | 105 | 105.749 | 105.807 | −3.569 | −1.569 | −0.82 | −0.762 | 2 |
| 17 | 30.49058 | 48.77732 | 12.13 | 15 | 14 | 16.081 | 16.241 | 1.87 | 0.87 | 2.951 | 3.111 | 3 |
| 18 | 31.87507 | 48.55341 | 41.533 | 45 | 46 | 43.769 | 43.934 | 2.467 | 3.467 | 2.236 | 2.401 | 2 |
| 19 | 31.24402 | 49.18028 | 34.39 | 31 | 33 | 33.619 | 33.495 | −3.39 | −1.39 | −0.771 | −0.895 | 2 |
| 20 | 30.2996 | 50.38949 | 238.128 | 234 | 236 | 237.437 | 237.518 | −3.128 | −1.128 | −0.691 | −0.61 | 2 |
| Min | — | — | 8.715 | 7 | 8 | 7.614 | 8.0591 | −6.733 | −2.733 | −2.202 | −1.833 | — |
| Max | — | — | 2880.705 | 2,887 | 2,888 | 2885.825 | 2885.182 | 10.985 | 7.985 | 6.773 | 6.094 | — |
| Mean | — | — | 455.3608 | 455.95 | 456.75 | 456.5808 | 456.566 | 0.43925 | 1.18925 | 1.17 | 1.155205 | — |
| Standard deviation | — | — | 799.1949 | 801.9891 | 801.4647 | 800.8867 | 800.6845 | 5.193281 | 3.534206 | 3.064291 | 2.767891 | — |
To prepare the error map of DEMs in the study area, the inverse distance weighting interpolation method was employed (Figure 4). The values of this map represent the distribution of error for study area estimating the difference between the satellite elevation data and the reference altitudinal data in all control points. Based on the results, the highest elevation error for all DEMs was observed in the eastern and northern parts of Khuzestan Province. According to the findings of this study, error distribution manifested the same behavior in SRTM and ASTER data. The largest positive error was thus observed in northern and northeastern regions, while the largest negative error was observed in the center of the province. Meanwhile, southern low-altitude regions had the smallest error. Although the overall error of SRTM data in most parts of the study area is less than ASTER data, SRTM data in some areas such as the western and southeast parts of the province demonstrate a higher error rate than ASTER. Moreover, in the two MERIT and FABDEM models, the largest positive error is observed in regions with high elevation. The error magnitude declines as the elevation drops and reaches its minimum in the center of the province. As the elevation keeps declining in southern areas, the error tends to have negative values in this region (Figure 4).
In contrast with the vast range of elevation in the study area, the range of altitudinal error behaves more uniformly. Therefore, a second-order polynomial could reveal proper efficiency for modeling error behavior in such areas. As coefficients’ estimation could be regarded as an important step in polynomial employment, a popular optimization method has been recruited to approximate these parameters properly.
The searching space used for implementing PSO is regarded as all candidate polynomial coefficients, and the primary population is based on a random assignment. Every particle position is indicated by a vector with a length equivalent to the polynomial coefficient number. To delve into the issue, in the model-training stage, using PSO optimization, the pre-mentioned cost function (equation (2)) is minimized. In this study, to improve the exploration feature and achieve a novel solution in the searching space, the inertia weight damping ratio = 0.9997 is regarded for the calculation of w in each repetition (w = w × damping ratio), where an increase in the repetition number leads to a reduction in the w value [30]. Moreover, in this study, the quality of measure, or instead the cost function, was chosen in conformity with the final accuracy of DEM and evaluation criteria (equation (2)). Therefore, the cost function was computed by the implementation of the polynomial in every repetition and the use of the validation technique for validation and training subsets. Discovering the global best values (optimal features) is a kind of repetitive process until the termination criterion is satisfied. Consequently, these optimal features are subsets with the greatest accuracy. Going through details, at the optimization stage, an inferior set of parameters is discarded in the PSO algorithm method and it progressively directs the population toward areas with a convincing set of parameters. According to this mechanism, the training stage of the polynomial can be performed using better solutions to the problem.
To estimate the proposed method perfectly, this method was evaluated using ground truth data, which are not used in the training step. The results are summarized in Table 2. After calculating polynomial coefficients using control points, as well as evaluating it using checkpoints, this polynomial has been used to calculate the corrected height in all raster points.
Numerical results obtained from accuracy assessment of the corrected ASTER and SRTM DEM utilizing proposed method
| Point no. | Latitude | Longitude | Z | Z | Z | Z | Z | Error | Error | Error | Error | Region number |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Geoid | ASTER | SRTM | FABDEM | MERIT | ASTER | SRTM | FABDEM | MERIT | ||||
| 1 | 32.01589 | 48.03482 | 137.938 | 143 | 142 | 140.631 | 140.843 | 0.025 | 0.057 | −0.058 | −0.024 | 2 |
| 2 | 31.99345 | 49.92993 | 876.254 | 885 | 883 | 882.125 | 881.914 | −0.044 | −0.058 | −0.132 | −0.049 | 1 |
| 3 | 32.71966 | 49.3284 | 2427.015 | 2,438 | 2,435 | 2433.581 | 2433.109 | 0.081 | 0.068 | −0.087 | −0.070 | 1 |
| 4 | 32.64732 | 48.59934 | 651.733 | 645 | 649 | 649.531 | 649.968 | 0.218 | 0.118 | 0.206 | 0.044 | 1 |
| 5 | 32.75017 | 48.2441 | 487.644 | 497 | 495 | 494.417 | 493.319 | −0.043 | −0.146 | −0.400 | −0.148 | 1 |
| 6 | 31.56039 | 50.11553 | 2880.705 | 2,887 | 2,888 | 2885.825 | 2885.182 | 0.008 | 0.140 | 0.038 | 0.006 | 1 |
| 7 | 30.48187 | 49.62003 | 18.123 | 16 | 17 | 16.316 | 16.519 | 0.169 | 0.201 | 0.260 | 0.104 | 3 |
| 8 | 30.80467 | 49.06381 | 9.299 | 7 | 8 | 7.614 | 8.0591 | −0.145 | −0.083 | −0.112 | 0.082 | 3 |
| 9 | 31.30483 | 48.64483 | 24.005 | 21 | 23 | 22.829 | 22.549 | 0.067 | 0.103 | −0.070 | −0.111 | 3 |
| 10 | 30.75944 | 48.2789 | 8.715 | 11 | 10 | 12.531 | 12.337 | −0.353 | −0.266 | 0.011 | 0.005 | 3 |
| 11 | 31.27695 | 47.98609 | 15.262 | 12 | 14 | 13.673 | 14.149 | −0.024 | 0.001 | 0.117 | 0.035 | 3 |
| 12 | 30.22265 | 49.76137 | 17.67 | 15 | 17 | 16.206 | 16.591 | 0.013 | 0.024 | 0.008 | −0.009 | 3 |
| 13 | 30.85111 | 49.82291 | 211.468 | 207 | 210 | 210.481 | 210.572 | −0.141 | −0.145 | 0.086 | −0.002 | 2 |
| 14 | 31.56393 | 49.36068 | 215.481 | 220 | 219 | 218.182 | 217.893 | 0.046 | −0.022 | 0.073 | −0.007 | 2 |
| 15 | 32.2184 | 49.43259 | 693.153 | 687 | 691 | 691.018 | 691.32 | 0.085 | −0.012 | 0.005 | −0.013 | 1 |
| 16 | 32.27739 | 48.51004 | 106.569 | 103 | 105 | 105.749 | 105.807 | 0.042 | −0.042 | −0.087 | −0.027 | 2 |
| 17 | 30.49058 | 48.77732 | 12.13 | 15 | 14 | 16.081 | 16.241 | 0.030 | 0.061 | −0.163 | −0.010 | 3 |
| 18 | 31.87507 | 48.55341 | 41.533 | 45 | 46 | 43.769 | 43.934 | −0.085 | −0.091 | 0.188 | −0.049 | 2 |
| 19 | 31.24402 | 49.18028 | 34.39 | 31 | 33 | 33.619 | 33.495 | 0.068 | 0.055 | 0.015 | −0.001 | 2 |
| 20 | 30.2996 | 50.38949 | 238.128 | 234 | 236 | 237.437 | 237.518 | 0.104 | 0.157 | 0.067 | 0.017 | 2 |
| Min | — | — | 8.715 | 7 | 8 | 7.614 | 8.0591 | −0.353 | −0.266 | −0.4 | −0.148 | |
| Max | — | — | 2880.705 | 2,887 | 2,888 | 2885.825 | 2885.182 | 0.218 | 0.201 | 0.26 | 0.104 | |
| Mean | — | — | 455.36075 | 455.95 | 456.75 | 456.58075 | 456.565955 | 0.00605 | 0.006 | −0.00175 | −0.01135 | |
| Standard deviation | — | — | 799.1949 | 801.9891 | 801.4647 | 800.8867 | 800.6845 | 0.123928 | 0.116457 | 0.148331 | 0.058088 |
In the next step, the error distribution maps of corrected DEMs were produced. Figure 5 shows the error map of satellite DEMs. Based on this figure, the vertical error of SRTM, ASTER, FABDEM, and MERIT based on RMSE index has been improved to 0.95, 0.63, 0.44, and 0.36 m, respectively. The results showed that, using the proposed algorithm, the accuracy of DEMs in the study area increased significantly. Due to the fact that error behavior in these areas is a bit different in small parts such that a second-order polynomial cannot model this error properly, in some parts, the accuracy has not increased much and even may be decreased.
Error distribution for corrected (a) SRTM, (b) ASTER, (c) FABDEM, and (d) MERIt elevation data.
The resulting maps from the interpolation of error distribution in the four studied DEMs indicated that the error range in MERIT DEM was smaller so that in most areas the residual error after correction with the proposed model ranged between −0.014 and 0.07 m. This indicates that the proposed model was the most efficient in this DEM. On the other hand, the accuracy improvement was more tangible in ASTER compared to the other models. In other words, although the initial ASTER DEM had the lowest accuracy among the selected ones, its final DEM presented a higher accuracy than SRTM. The error range in the corrected FABDEM model in most parts of the study area was also between −0.06 and −0.03 m; however, positive errors are also observed across the entire region. These errors indicate the non-compliance of the error behavior with the second-order polynomial; however, the small error values indicate that there is no need to increase the order of the polynomial. The corrected STRM output also indicates positive errors in the eastern and northwestern areas of the region, whereas the largest negative errors were observed in the central and southern areas. According to Yamazaki et al., The MERIT DEM was developed by removing multiple error components (absolute bias, stripe noise, speckle noise, and tree height bias) from the existing DEMs (SRTM3 v2.1 and AW3D-30m v1). Although, in Merit's model, many errors have been removed from the elevation model, this model still has an average error of 1.83 m in the studied area. On the contrary, the error has been significantly reduced by the proposed method, in comparison with the mentioned study.
To evaluate the relationship between elevation and vertical error in the digital elevation/surface models, linear regression has been used (Figure 6a and b). According to Figure 6a, after applying the linear regression between elevation and vertical error, a strong positive relationship between these two parameters can be clearly witnessed. This means that with the increase in elevation in all models, the error will also increase. To be more precise, this relationship is stronger in the SRTM and ASTER models than in the other two models. On the contrary and according to Figure 6b, after applying the proposed algorithm to reduce the vertical error, in both SRTM and ASTER models, a positive relationship between the stated parameters can still be observed, but this relationship is very weak and can be ignored. In the case of FABDEM and MERIT models, a very weak negative relationship can be seen between such parameters. So these two parameters can be assumed to be independent in the corrected models.
Linear regression relationship between elevation and error (a) before algorithm implementation (b) after algorithm implementation.
To perform a more accurate investigation of the algorithm in regions with various elevation features, the following table demonstrates the error of satellite models before and after the implementation of the proposed algorithm in three mountainous, foothill, and plain regions.
Table 3 shows that the proposed model makes a great contribution to reducing bias in elevation models. Going through details, before applying corrections, high rates of positive and negative bias could be witnessed in all models. In contrast, after applying them, the bias values will decrease significantly.
Comparison of error propagation in three regions with various altitudinal behaviors
| Region 1 | Region 2 | Region 3 | |
|---|---|---|---|
| Max-Z_Geoid | 2944.2848 | 468.1456 | 41.5192 |
| Min-Z_Geoid | 174.3882 | 33.3606 | 17.0099 |
| Mean-Z_Geoid | 887.94435 | 141.740397 | 25.432011 |
| Standard deviation-Z_Geoid | 653.95891 | 111.415 | 6.109 |
| MAE-SRTM-before | 9.427 | 4.585 | 1.460 |
| MAE-SRTM-after | 1.031 | 0.737 | 0.312 |
| MAE-ASTER-before | 10.15 | 5.388 | 1.960 |
| MAE-ASTER-after | 0.811 | 0.693 | 0.313 |
| MAE-FABDEM-before | 8.283 | 1.189 | 4.470 |
| MAE-FABDEM-after | 0.552 | 0.391 | 0.271 |
| MAE-MERIT-before | 7.314 | 0.940 | 3.877 |
| MAE-MERIT-after | 0.484 | 0.346 | 0.190 |
| MBE-SRTM-before | 6.099 | −3.985 | 2.460 |
| MBE-SRTM-after | 0.140 | −0.136 | 0.096 |
| MBE-ASTER-before | 6.966 | −3.888 | 2.060 |
| MBE-ASTER-after | 0.145 | −0.144 | 0.106 |
| MBE-FABDEM-before | 5.198 | −1.889 | −2.170 |
| MBE-FABDEM-after | 0.043 | −0.046 | 0.038 |
| MBE-MERIT-before | 4.819 | −0.917 | −1.377 |
| MBE-MERIT-after | 0.0136 | −0.0334 | 0.024 |
It should be noted that the proposed model could be implemented in other areas with different roughness and also different error rates of DEMs. The only possible required changes will be to determine the polynomial order. As stated earlier, in this study, due to the relatively uniform error behavior, the second-order polynomial has been utilized. If the error behavior is more uniform in a region, the first-order polynomial can achieve proper accuracy. On the other hand, if the error rate in a region is more complex, there will probably be a need to use higher-order polynomials.
5 Conclusion
In this study, the accuracy of SRTM, ASTER FABDEM, and MERIT DEMs was assessed by comparing them with high-accuracy altitudinal points prepared by the Iranian Cartography Center in the Khuzestan Province. Although all elevation data models have almost same horizontal resolution, they accounted for various rates of altitudinal accuracies in the study area. As a result, based on the MAE index, the error values of 4.83, 5.35, 4.28, and 3.87 m were obtained for SRTM, ASTER, FABDEM, and MERIT data, respectively. The evaluation results obtained by the RMSE index for SRTM, ASTER, FABDEM, and MERIT data were 5.79, 6.23, 5.01, and 4.32 m, respectively. Based on the findings of the present study, it can be concluded that in most parts of the study area, the DEM obtained by MERIT explained higher performance in comparison with other digital models. In the next section, the accuracy of these models was improved by the optimized second-order polynomial. PSO algorithm was employed to optimize the coefficients of the polynomial. The corrected DEM was computed by subtracting the estimated error by polynomial from DEM elevation. The results illustrated that the altitudinal accuracy (MAE) was increased to 0.83, 0.51, 0.37, and 0.29 m for SRTM, ASTER FABDEM, and MERIT DEMs, respectively. Therefore, according to the results obtained in the study area, which consists of three separate parts, mountainous, foothills, and plains, the proposed algorithm has obtained acceptable results in different areas.
Acknowledgements
We are grateful to the Research Council of Shahid Chamran University of Ahvaz for financial support (SCU.EG1400.26151).
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Author contributions: MK, KR and MA designed the algorithms and AS carried them out. MK and AS developed the model code and performed the simulations. AS prepared the manuscript with contributions from all co-authors. The authors applied the SDC approach for the sequence of authors.
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Conflict of interest: The authors certify that they have NO affiliations with or involvement in any organization or entity with any financial interest, or non-financial interest in the subject matter or materials discussed in this manuscript.
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Data availability statement: Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.
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