Topographic optimal network design of the unified control points in Korea
- Tae-Suk Bae^{1},
- Jay Hyoun Kwon^{2}Email author and
- Chang-Ki Hong^{3}
https://doi.org/10.5047/eps.2011.02.007
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2011
Received: 21 April 2009
Accepted: 11 February 2011
Published: 21 June 2011
Abstract
In general, the geometric approach based on the criterion matrix has focused on the optimal network design. At the scale of the local area, however, the topographic undulations should be considered for ground applications, such as geoid determination. Since the multi-purpose unified control points (UCPs) are planned for providing gravimetric information as well as the three-dimensional positions, the effect of high-frequency signals from topography need to be considered in the network optimization process. In this study, an optimization procedure incorporating the geometric and topographic configuration is presented. A Digital Elevation Model (DEM) that represents terrain information is combined with the second-order design algorithm with Taylor-Karman structure. The smoothed DEM data are removed from the original dataset, resulting in high-frequency data only; the root mean squares of the residuals are computed to create the weight matrix. As a result, the directional pattern is clearly seen in the weight matrix, and the final location of the network shows the north-south directional properties. Once the network selection process is complete, the Minimum Spanning Tree (MST) is created to examine the distribution of the baselines. The statistics on the MST were used for the criterion of optimal network validation.
Key words
Optimal network Taylor-Karman structure unified control point1. Introduction
Network optimization problem has been a popular issue in geodetic applications, especially in establishing new tracking or control stations. The well-known approach of optimal network design is to use the criterion matrix that is homogeneous and isotropic (Grafarend and Schaffrin, 1979; Schaffrin, 1985). This method is based on the geometric configuration of the candidate stations, i.e., either scattered or regularly gridded; thus, its final network is affected by only geometric relations between candidates. Without exception, however, the network stations are built on the physical surface of the Earth, which is closely connected with the topographic condition.
In 2007, the Korean government initiated a campaign to establish newly designed national control points. The control points, which are called unified control points (UCPs), are supposed to contain not only the traditional position information but also gravimetric information, such as gravity anomaly and geoidal height. The rationale for this campaign is to support scientific and engineering applications by providing all of the necessary information at common points. In fact, future plans are to expand the information at UCPs based on the various demands of users. For example, data on weather or environmental information can be added as contents provided at UCPs so that researchers can use and analyze the various data provided at common points. The current question revolves around the optimal distribution for the UCP networks. As the UCPs contain geometric and physical information, variations in topography and the geometric configuration have to be considered at the same time.
The primary aim of this study was to develop an optimal network design algorithm that considers both geometric and physical circumstances. Here, the geometric consideration is the traditional concept, namely, the weighting scheme based on the relative distances of points or stations. The physical consideration, on the other hand, involves the effect of topography, which could be crucial in collecting physical quantity such as gravity measurement. In other words, the network should be distributed in such a way that it can sense necessary frequency components of the intended signal. If the network in a mountainous area is not dense enough, the measured gravity will suffer from aliasing, which results in excessive amplitude at low frequency by smearing of the high-frequency signal. Therefore, topographic information, such as a Digital Elevation Model (DEM), should be used to design a network that is optimal in both geometrical and physical aspects.
The primary component of network optimization is still the geometric relations between stations, and information obtained from the DEM is used to assist the selection algorithm in our approach. Combining these components properly to achieve optimal solutions is considered to be a useful step, especially on the local area scale. The high-frequency part of the topography is essential for ground applications because the gravimetric quantity is likely to be more sensitive to the high-frequency signal; thus, high-frequency data were separated from the original DEM to determine the weights for each candidate station. Since there are no explicit criteria to stop the selection process, in this study, we controlled the process using the specific requirements of the baselines with the Minimum Spanning Tree (MST). As GPS measurements are widely used in such studies, the MST is considered to be a good way to quantify the chosen network.
The algorithms presented in this study are programmed in MATLAB script.
2. Theory of Network Optimization
From Eq. (1), it is clear that the criterion matrix is not dependent on the measurement types or linear models with specific rank-deficiency. The second order Design (SOD) was successfully implemented for the network design of ground tracking stations for the Low Earth Orbiter (LEO) orbit determination (Bae, 2005). This approach optimizes the cofactor matrix by determining the measurement weights between candidate and network stations. Other approaches, such as zero, first, and third order design (ZOD/FOD/TOD) can be referred to Schaffrin (1985) and/or Kuang (1996).
3. Digital Elevation Model
The DEM is a digital representation of topographic information on a computer in the form of orthometric height above geoid, which is an intuitive reference for height in the way water flows. The DEM is usually gridded on both north-south (N-S) and east-west (E-W) directions, not necessarily with the same interval. The DEM used in this study is the Shuttle Radar Topographic Map (SRTM) developed by NASA (National Aeronautics and Space Administration), which covers over 80% of the globe (http://srtm.csi.cgiar.org/). The original spatial resolution of SRTM is 3 arcseconds, from which the coarse DEM with 45 and 60 arcseconds for latitude and longitude, respectively, are extracted and used in this study. The DEM is restricted to within the range of 34-39° of latitude and 126-129.5° of longitude, which corresponds to the mainland of South Korea.
4. Smoothing Technique
Many smoothing techniques exist, but as discussed earlier, the DEM is smoothed along the same latitude using one-dimensional Fourier transform. The smoothing process is regarded as a convolution of the data and a window function (see Eq. (9)). Also, the convolution of two signals in the time domain can be performed in the frequency domain by the convolution theorem, which requires three applications of the Fourier transform. The Fourier transform of the convolution equals the product of the spectra (Fourier transforms) of the convolved functions.
Figure 2 shows the residuals of DEM after the smoothed data have been removed, revealing a clear illustration of topographic information on Korea. For the weights of each candidate station based on the DEM residuals (high-frequency part), the root mean squares (RMS) of a total of 5 pixels (the pixel for the computation and two pixels on each side of the E-W direction) are calculated. Since the RMS values can range widely, these are normalized rowwise to be on a uniform scale (maximum of 1 for the highest values at each row). This RMS can then be used for the topographic weight information in determining the network stations. A strong tendency of variation in DEM weighting along the E-W direction can be seen (see Fig. 8) as well as the dominant pattern in the N-S direction, which is closely related to the topographic distribution of Korea.
5. Selection of Optimal Network
In addition to the above algorithm, it is necessary to incorporate the topographic information into the network optimization algorithm to customize for the specific conditions described in this study. Thus, the topographic weight information (see Fig. 8) is used for the selection process. That is, under the geometric condition, the top ten stations with most uniform weight variation are selected, and the station with the highest topographic weight is chosen for the next network station. Details on the algorithm follow in next section.
6. Test Results
The optimal network selection process begins with two stations of the International Ground Station (IGS) network, SUWN and DAEJ, although these two stations are rather close to each other. The number of stations in the final network is dependent on the budget of the project, but this process can be continued until the baseline length reaches the specified length. The average trace of the cofactor matrix, however, implies that the network may not be improved significantly over 40 stations.
7. Minimum Spanning Tree
Statistics of the minimum spanning tree of the chosen network stations (N = Number of candidate stations).
N = 5 | N=10 | N = 20 | |
---|---|---|---|
Mean [km] | 60.846 | 59.135 | 59.152 |
Std. dev. [km] | 6.464 | 6.460 | 8.004 |
Min. [km] | 49.064 | 44.497 | 42.734 |
Max. [km] | 76.327 | 71.725 | 74.127 |
The maximum baseline of the network comes from the pair of nodes connecting the mainland and the island. On the other hand, the baseline on the east coast is the shortest, which is due to the high weight around the mountain chain. Excluding these exceptions, the baseline length is fairly well distributed and shows a small variation around the mean value of 59 km. Therefore, the MST and mean and/or maximum baseline length can also be a criterion for completion of the network selection process.
8. Discussion
The homogeneous and isotropic network design reported here was performed as a geometric approach to optimization. The topography, however, is also an important factor when the aim is to provide a better explanation of the localized network in Korea, combined with the geometric solution. The Korean peninsula is topographically characterized by the presence of high mountain chains on the east coast and south-west area, and this is clearly indicated in the weight matrix (see Fig. 8). Thus, the topographic information was incorporated into the optimal network design for an optimization that is more appropriate to Korea.
Once the smoothed data are removed, the DEM residuals are used to determine the weights for each candidate station. The weight map based on DEM shows the variation along the E-W direction of Korea, and the chosen network stations follow the pattern of the mountain chain along the N-S direction that better conforms to Korea. The stations are more densely located on the east coast due to the rough terrain in that area.
The number of stations in the network is dependent on the budget, but the MST can be one of the alternative criteria to control the number of stations. Since the MST connects the entire network with a minimum cost (baseline length), it is possible to have the uniform baseline lengths throughout the network, resulting in high possibilities of stable solution. Other approaches, such as finding a location using the least-squares solution instead of selecting a station from the candidates, need to be investigated further.
Declarations
Acknowledgments
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0069542).
Authors’ Affiliations
References
- Bae, T.-S., Optimized network of ground stations for LEO orbit determination, ION2005 NTM, 515–522, 2005.Google Scholar
- Cormen, T., C. Leiserson, R. Rivest, and C. Stein, Introduction to Algorithms (2nd), 1216 pp., McGraw-Hill, 2001.Google Scholar
- Jekeli, C., Fourier Geodesy, The Ohio State University, Columbus, Ohio, 2007.Google Scholar
- Grafarend, E., Genauigkeitsmaße geodatischer Netze, Publ. DGK A-73, München, 1972.Google Scholar
- Grafarend, E. and B. Schaffrin, Kriterion-Matrizen I—zweidimensionale homogene und isotrope geodätische Netze, Z. Vermessungswesen, 104, 133–149, 1979.Google Scholar
- Kuang, S., Geodetic Network Analysis and Optimal Design: Concepts and Applications, 368 pp., Sams Publications, 1996.Google Scholar
- Schaffrin, B., Network design, in Optimization and Design ofGeodetic Networks, edited by Grafarend, E. and F. Sanso, 548–597, Springer-Verlag, 1985.Google Scholar
- Schmitt, G., Second order design of free distance networks considering different types of criterion matrices, Bull. Geodetica, 54, 531–543, 1980. 10.1007/BF02530711View ArticleGoogle Scholar
- Wimmer, H., Ein Beitrag zur Gewichtsoptimierung geodätischer Netze, 254 pp., Publ. DGK C-269, München, 1982.Google Scholar