Multiquadric Interpolation Method Research Articles

A bounded randomly variable shape multi-quadric interpolation method in dual reciprocity boundary element method

The radial basis function interpolation is an effective method for approximation on scattered data. However, this interpolation method suffers from the contradiction between the accuracy and the numerical stability. With considering the distribution of the interpolation points, a bounded random shape variation scheme for the radial basis function is developed to circumvent the problem of numerical stability. In this scheme, the shape variation is bounded by the value that is determined by the maximum distance and the minimal distance which are applied to describe the average density of the interpolation centers. Within this bound, the shape of the MQ is modified through a random scheme. With applying this bounded randomly variable shape scheme, the accuracy and the stability of the MQ interpolation are balanced. Comparisons on the accuracy and the condition number of the interpolation matrix between the constant shaped MQ interpolation and this bounded randomly variable shaped MQ interpolation have been made to verify the conclusion. Furthermore, this scheme is integrated in the dual reciprocity boundary element method in the analysis of three dimensional elastic problems. Results of the numerical examples demonstrated that the developed scheme improved the accuracy of the dual reciprocity boundary element method stably.

강수량 관측망의 지역별 정량적 취약성 분석

본 연구에서는 기상청 관측망을 대상으로 지역별 정량적 관측망 취약성 분석을 실시하였다. 이를 위해 기상청 관측망 만을 이용하여 생산한 격자 자료 GRD(K)와 기상청과 국토교통부 관측망을 모두 사용하여 생산한 격자자료 GRD(A)의 강수량 차이를 비교하여 보았다. 격자자료 산정은 다중이차내삽법(Multi-quadric interpolation method; MQI)을 사용하였다. 도식적 분석으로 관측밀도와 정량적 분석으로 유역평균 강수량을 비교하였다. 비교 결과, 상대편차(GRD(A)의 중권역별 유역평균 강수량 대비 GRD(K)의 중권역별 유역평균 강수량 비)가 큰 지역들은 대략 동쪽의 주왕산에서 가야산, 덕유산, 내장산을 거쳐 서쪽의 월출산에 이르는 축을 기준으로 남쪽에 위치하고 있었고, 중심부의 덕유산에서 지리산에 이르는 남북축을 기준으로 평균적으로 GRD(K)가 GRD(A)보다 크게 산정되는 서쪽 지역과 GRD(K)가 GRD(A)보다 작게 산정되는 동쪽 지역으로 구분되었다. 즉, 소백산맥과 노령산맥 주변 및 이남지역이 GRD(K)와 GRD(A)의 차이가 큰 것으로 나타나 기상청 관측망 만을 이용할 경우 취약지역으로 평가되었다. This research was conducted to analyze the quantitative vulnerability of spatial estimation of precipitation for precipitation observation network of Korea Meteorological Administration (KMA). We compared the amount of precipitation of GRD(K) which were gridded data only using KMA's observation network and GRD(A) which were gridded data using both KMA and Ministry of Land, Infrastructure and Transport (MLIT)'s observation network. we used a multi-quadric interpolation method(MQI) to make a gridded data. We compared the observation densities as a schematic analysis and areal precipitation as a quantitative analysis. As a result, regions where the relative deviations (GRD(K) compared to GRD(A)) are large were located at south of the axis which lines across from Mt. Juwang, Gaya, Deokyu and Mt. Naejang at east to Mt. Wolchool at west. In addition, based on north and south axis which lines across Mt. Deokyu at mid region of South Korea to Mt. Jiri, areas were separated into west regions where GRD(K) were estimated to be bigger than GRD(A) and east regions where GRD(K) were estimated to be smaller than GRD(A). In other words, GDR(K) and GRD(A) showed the biggest gap in regions near or southern the Sobaek mountain range and the Noryung mountain range. Thus, these areas were evaluated as the most vulnerable in case of using KMA's observation network only.

공간 강수량 정확도 향상을 위한 공간상세화 방법 평가

관측정밀도 보다 높은 강수량 공간상세화를 위해서는 공간 내삽법에 대한 체계적인 조사와 기본적인 특성 평가가 선행되어야, 향후 방법적인 면에서 논란의 여지를 줄이고 상세화된 자료에 대한 불확실성도 가늠할 수 있다. 따라서 본 연구에서는 공간내삽방법에 대한 평가를 수행하고자 한다. 본 연구에서는 특정 지역 혹은 특정 사상 선정으로 인한 결과의 임의성을 최대한 줄이기 위해 우리나라 전역의 10년(2001-2010년)치 기상청 AWS지점 일자료를 사용하였다. 그리고 역거리가중법(Inverse distance weighted method; IDW), 크리깅법(Ordinary kriging; KRG), 다중이차내삽법(Multi-quadric interpolation method; MQI)을 적용하여 <TEX>$5km{\times}5km$</TEX> 격자강수량을 재산정 하였다. 각 내삽 방법에 대한 정량적 평가를 위하여 기준 방법으로 본 연구에서는 티센면적가중법(Thiessen polygon method; TSN)을 사용하였다. 2001년부터 2010년까지 10년 동안의 일강수량 공간 상세화 결과를 중권역 단위로 비교하였을 때, 기상청 AWS 관측망을 사용하는 경우 전체 평균에 있어 방법 간 차이는 크지 않았다. 그러나 관측소가 희박한 지역과 개별호우사상의 경우는 지역특성별로 방법 간 유의한 차이가 나타났다. In order to estimate a reliable spatial precipitation which has high resolution sub-measuring scale, a systematical research for spatial interpolation methods and evaluation of characteristics of them are required. This will be able to reduce disputes on selection of methodologies and estimate uncertainties of downscaled data. Therefore, we assessed the spatial interpolation methods using automatic weather system (AWS)'s daily data of Korea Meteorological Administration (KMA) during past 10 years (2001-2010), so as to reduce arbitrary interpretations of the result by selecting particular regions or storm events. In addition, we adopted inverse distance weighted method (IDW), ordinary kriging method (KRG) and multi-quadric interpolation method (MQI) to produce <TEX>$5km{\times}5km$</TEX> grid precipitation. For quantitative evaluations of each interpolation method, thiessen polygon method (TSN) was used as a comparison method. When comparing the gridded daily precipitation for the past 10 years (2001-2010) by middle scale basins, differences of whole average for each methods were not significant when using KMA's AWS network. However, in case of regions where stations were sparse and each storm event, significant differences were founded caused by interpolation methods or characteristics of regions.

수자원분야 활용을 위한 최적 강수격자크기 및 관측소밀도 연구

보다 안전하고 경제적인 강수 관측망 운용과 공간 강수자료를 수자원분야에 효과적으로 활용하기 위한 목적에서, 현재 기상청 관할 관측소 밀도의 적정성 검토와 공간 강수량 생산 시 적정 격자크기에 대한 분석을 실시하였다. 이를 위해 우리나라 AWS 관측소가 가장 조밀한 서울지역에 대해 관측소 밀도별 격자 크기별 면적평균 강수량 및 추정된 격자강수량의 공간 정확도를 비교하여 보았다. 서울지역의 29개 관측소를 대상으로, 2001~2010년 기상청 AWS 및 ASOS 일강수량 자료와 다중이차내삽법(Multi-quadric interpolation method; MQI)을 사용하여 관측소 밀도별, 격자 크기별 격자강수량을 산정하였다. 적정 관측밀도 분석을 위해 29개 관측소 일 때(관측소간 평균거리 약 5 km), 20개 관측소 일 때(관측소간 평균거리 약 6 km) 그리고, 7개 관측소 일 때(관측소간 평균거리 약 10 km)로 구분하고, 적정 격자크기 분석을 위해 각각의 관측 밀도에 대해 10 km, 5 km, 1 km의 격자강수량을 산정하여 비교하였다. 관측소 밀도와 격자 크기에 따른 추정된 격자강수량의 정확도 분석 결과, 추정된 격자강수량의 공간 정확도를 높이기 위해서는 관측밀도가 가장 중요한 요소로 작용하였다. 그리고 우리나라의 경우 정확한 공간 강수량 추정을 위한 관측소 밀도는 대략 5 km 이하가 적절하고, 격자크기는 1 km 이하가 적절한 것으로 나타났다. In a purpose to manage precipitation monitoring network more safely and economically and to efficiently use spatial precipitation data in the water resources, we conducted a trade off study on density of Korea Meteorological Administration (KMA) stations and analysis on optimal grid size when estimating accurate spatial precipitation. For the analysis, we compared areal precipitation and spatial accuracy of estimated grid precipitation by station densities and grid sizes for Seoul region where Automatic Weather System (AWS) stations were most dense. Based on 29 stations near in Seoul, we estimated grid precipitation by station densities, grid sizes using daily precipitation data of KMA's AWS and Automatic Synoptic Observation System (ASOS) during 2001-2010. To find a optimal station density, we specified the case of 29 stations (average distance about 5km between stations), 20 stations (average distance about 6 km between stations) and 7 stations (average distance about 10km between stations). Then, we also calculated grid precipitation of 10 km, 5 km, 1 km size by each station density to find a optimal grid size. From the results of a accuracy tests on the estimated grid precipitation by station density and grid size, we found that station density acted as the most crucial factor to improve space accuracy. In addition, for an accurate spatial precipitation estimation, the appropriate station density and grid size should be sub-5 km and sub-1km respectively in South Korea.

Modified quadratic shepard and radial basis functions interpolations for determination of orthometric heights from global positioning system (GPS) heights

We can obtain the position of any point on earth easily, accurately and quickly as geocentric coordinates (X, Y, Z) or geodetic coordinates (O, λ, h) in the World Geodetic System (WGS84) coordinate system with global positioning system (GPS) technique widely used in engineering studies today. However, we need to convert the ellipsoid height (h) to orthometric height to be able to use it in the applications. In this study, radial basis functions (RBF) and modified quadratic shepard (MQS) interpolation methods were used to determine orthometric heights from GPS measurements in two different study fields. It has been observed that multiquadric interpolation method gave better results when statistical values of test points in the study fields were considered.

A COMPUTATIONAL APPROACH TO THE ROBINSON PROJECTION

The Robinson Projection is the most preferred World map projection in the atlas cartography. There are no analytical formulas except Robinson's look-up table for this projection. This deficiency has led a number of requests for the plotting formulas and cartographers have studied to derive analytical equations using different algorithms. In these works, different interpolation algorithms are applied to Robinson's table values and solutions are presented including some critics about the deformations on this projection. In this study, a summary of these computation algorithms is collected. The multiquadric interpolation method is suggested and applied to the Robinson's tabular coordinates. A series of numerical evaluations are presented then for the controversies and for comparison between these computation algorithms.

A COMPUTATIONAL APPROACH TO THE ROBINSON PROJECTION

The Robinson Projection is the most preferred World map projection in the atlas cartography. There are no analytical formulas except Robinson's look-up table for this projection. This deficiency has led a number of requests for the plotting formulas and cartographers have studied to derive analytical equations using different algorithms. In these works, different interpolation algorithms are applied to Robinson's table values and solutions are presented including some critics about the deformations on this projection. In this study, a summary of these computation algorithms is collected. The multiquadric interpolation method is suggested and applied to the Robinson's tabular coordinates. A series of numerical evaluations are presented then for the controversies and for comparison between these computation algorithms.

Spatial characteristics of thunderstorm rainfall fields and their relation to runoff

The main aim of this study was to assess the ability of simple geometric measures of thunderstorm rainfall in explaining the runoff response from the watershed. For calculation of storm geometric properties (e.g. areal coverage of storm, areal coverage of the high-intensity portion of the storm, position of storm centroid and the movement of storm centroid in time), spatial information of rainfall is needed. However, generally the rainfall data consists of rainfall depth values over an unevenly spaced network of raingauges. For this study, rainfall depth values were available for 91 raingauges in a watershed of about 148 km 2. There was a question about which interpolation method should be used for obtaining uniformly gridded data. Therefore, a small study was undertaken to compare cross-validation statistics and computed geometric parameters using two interpolation methods (kriging and multiquadric). These interpolation methods were used to estimate precipitation over a uniform 100 m×100 m grid. The cross-validation results from the two methods were generally similar and neither method consistently performed better than the other did. In view of these results we decided to use multiquadric interpolation method for the rest of the study. Several geometric measures were then computed from interpolated surfaces for about 300 storm events occurring in a 17-year period. The correlation of these computed measures with basin runoff were then observed in an attempt to assess their relative importance in basin runoff response. It was observed that the majority of the storms (observed in the study) covered the entire watershed. Therefore, it was concluded that the areal coverage of storm was not a good indicator of the amount of runoff produced. The areal coverage of the storm core (10-min intensity greater than 25 mm/h), however, was found to be a much better predictor of runoff volume and peak rate. The most important variable in runoff production was found to be the volume of the storm core. It was also observed that the position of the storm core relative to the watershed outlet becomes more important as the catchment size increases, with storms positioned in the central portion of the watershed producing more runoff than those positioned near the outlet or near the head of the watershed. This observation indicates the importance of interaction of catchment size and shape with the spatial storm structure in runoff generation. Antecedent channel wetness was found to be of some importance in explaining runoff for the largest of the three watersheds studied but antecedent watershed wetness did not appreciably contributed to runoff explanation.