It is practically impossible and unnecessary to obtain spatial-temporal information of any given continuous phenomenon at every point within a given geographic area. The most practical approach has always been to obta...It is practically impossible and unnecessary to obtain spatial-temporal information of any given continuous phenomenon at every point within a given geographic area. The most practical approach has always been to obtain information about the phenomenon as in many sample points as possible within the given geographic area and estimate the values of the unobserved points from the values of the observed points through spatial interpolation. However, it is important that users understand that different interpolation methods have their strength and weaknesses on different datasets. It is not correct to generalize that a given interpolation method (e.g. Kriging, Inverse Distance Weighting (IDW), Spline etc.) does better than the other without taking into cognizance, the type and nature of the dataset and phenomenon involved. In this paper, we theoretically, mathematically and experimentally evaluate the performance of Kriging, IDW and Spline interpolation methods respectively in estimating unobserved elevation values and modeling landform. This paper undertakes a comparative analysis based on the prediction mean error, prediction root mean square error and cross validation outputs of these interpolation methods. Experimental results for each of the method on both biased and normalized data show that Spline provided a better and more accurate interpolation within the sample space than the IDW and Kriging methods. The choice of an interpolation method should be phenomenon and data set structure dependent.展开更多
In this paper, we study weighted mean integral convergence of Hakopian interpolation on the unit disk D. We show that the inner product between Hakopian interpolation polynomial Hn（f;x,y） and a smooth function g（x,...In this paper, we study weighted mean integral convergence of Hakopian interpolation on the unit disk D. We show that the inner product between Hakopian interpolation polynomial Hn（f;x,y） and a smooth function g（x,y） on D converges to that of f（x,y） and g（x,y） on D when n →∞ , provided f（x,y） belongs to C（D） and all first partial derivatives of g（x,y） belong to the space LipM^α（0 〈 α ≤1）. We further show that provided all second partial derivatives of g（x,y） also belong to the space LipM^α and f（x,y） belongs to C^1 （D）, the inner product between the partial derivative of Hakopian interpolation polynomial δ/δx Hn（f;z,y） and g（x,y） on D converges to that between δ/δxf（x,y） and g（x,y） on D when n →∞. oo.展开更多
In this paper, we study the weighted (0,2;0)-interpolation on infinite interval, which means to determine a polynomial of degree ≤3n-1 when the function values are prescribed at two sets of points namely the zeros of...In this paper, we study the weighted (0,2;0)-interpolation on infinite interval, which means to determine a polynomial of degree ≤3n-1 when the function values are prescribed at two sets of points namely the zeros of H n(x) and H ′ n(x) and the wieghted second derivative at the zeros of H n(x).展开更多
In the paper, a result of Walsh and Sharma on least square convergence of Lagrange interpolation polynomials based on the n-th roots of unity is extended to Lagrange interpolation on the sets obtained by projecting ve...In the paper, a result of Walsh and Sharma on least square convergence of Lagrange interpolation polynomials based on the n-th roots of unity is extended to Lagrange interpolation on the sets obtained by projecting vertically the zeros of (1-x)2=P (a,β) n(x),a>0,β>0,(1-x)P(a,β) n(x),a>0,β>-1,(1+x)P P(a,β) n(x),a>-1,β0 and P(a,β) n(x),a>-1,β>-1, respectively, onto the unit circle, where P(a,β) n(x),a>-1,β>-1, stands for the n-th Jacobi polynomial. Moreover, a result of Saff and Walsh is also extended.展开更多
为了解决工业发展导致的灌区土壤投入品残留污染问题,给出一种基于地理信息系统(geographic information system,GIS)的土壤污染监测预警系统。该系统结合VOC-PF1型传感器、STM32主控芯片和GSM通信模块,实现了高效的数据采集和通信功能...为了解决工业发展导致的灌区土壤投入品残留污染问题,给出一种基于地理信息系统(geographic information system,GIS)的土壤污染监测预警系统。该系统结合VOC-PF1型传感器、STM32主控芯片和GSM通信模块,实现了高效的数据采集和通信功能。通过反距离加权(inverse distance weighted,IDW)插值法进行空间分析,并设立预警阈值,实现对灌区土壤投入品残留污染的实时监测和预警。实验结果表明:该系统的监测精度高达98%,监测时长最高为49 s,具有很高的实用性和效率。研究结果不仅为灌区土壤投入品残留污染监测提供了有效手段,也为环境保护和农业可持续发展提供有力支持。展开更多
文摘It is practically impossible and unnecessary to obtain spatial-temporal information of any given continuous phenomenon at every point within a given geographic area. The most practical approach has always been to obtain information about the phenomenon as in many sample points as possible within the given geographic area and estimate the values of the unobserved points from the values of the observed points through spatial interpolation. However, it is important that users understand that different interpolation methods have their strength and weaknesses on different datasets. It is not correct to generalize that a given interpolation method (e.g. Kriging, Inverse Distance Weighting (IDW), Spline etc.) does better than the other without taking into cognizance, the type and nature of the dataset and phenomenon involved. In this paper, we theoretically, mathematically and experimentally evaluate the performance of Kriging, IDW and Spline interpolation methods respectively in estimating unobserved elevation values and modeling landform. This paper undertakes a comparative analysis based on the prediction mean error, prediction root mean square error and cross validation outputs of these interpolation methods. Experimental results for each of the method on both biased and normalized data show that Spline provided a better and more accurate interpolation within the sample space than the IDW and Kriging methods. The choice of an interpolation method should be phenomenon and data set structure dependent.
文摘In this paper, we study weighted mean integral convergence of Hakopian interpolation on the unit disk D. We show that the inner product between Hakopian interpolation polynomial Hn（f;x,y） and a smooth function g（x,y） on D converges to that of f（x,y） and g（x,y） on D when n →∞ , provided f（x,y） belongs to C（D） and all first partial derivatives of g（x,y） belong to the space LipM^α（0 〈 α ≤1）. We further show that provided all second partial derivatives of g（x,y） also belong to the space LipM^α and f（x,y） belongs to C^1 （D）, the inner product between the partial derivative of Hakopian interpolation polynomial δ/δx Hn（f;z,y） and g（x,y） on D converges to that between δ/δxf（x,y） and g（x,y） on D when n →∞. oo.
文摘反距离加权插值方法(Inverse Distance Weighted,IDW)是生成数字高程模型(Digital Elevation Model,DEM)的常用内插手段之一,不同的地形应使用合适的IDW距离指数进行插值。本文选取了平原、丘陵、小起伏山地、中起伏山地和大起伏山地5种地形,设计了2组试验,从地形宏观形态和地形微观形态2个方面研究了地形对IDW插值中最优距离指数(Optimal order of distances,OOD)的影响。首先使用狼群算法(Wolf pack algorithm,WPA)计算不同地形区下IDW插值的OOD,分析不同地形之间OOD的分布差异;其次选取坡度、坡向、曲率3个地形因子,计算各采样点的OOD,分析不同地形因子对采样点OOD的影响。结果表明,从平原地区到大起伏山地地区,随着区域内地形起伏度的增加,OOD减小。采样点的OOD在高值区的占比随坡度增大而减小;OOD随坡向变化差异不大;随着地形曲率的增大,OOD在高值区的占比增加,在低值区的占比减小。在较为平坦的地区,例如平原地区,丘陵地区建议使用OOD在3≤a≤4范围内取值进行IDW插值,而在小起伏山地、中起伏山地和大起伏山地等山地区建议采用OOD在1≤a≤2范围内取值进行IDW插值。
文摘In this paper, we study the weighted (0,2;0)-interpolation on infinite interval, which means to determine a polynomial of degree ≤3n-1 when the function values are prescribed at two sets of points namely the zeros of H n(x) and H ′ n(x) and the wieghted second derivative at the zeros of H n(x).
基金NSFC under grant1 0 0 71 0 3 9and by Education Committee of Jiangsu Province under grant0 0 KJB1 1 0 0 0 5 .
文摘In the paper, a result of Walsh and Sharma on least square convergence of Lagrange interpolation polynomials based on the n-th roots of unity is extended to Lagrange interpolation on the sets obtained by projecting vertically the zeros of (1-x)2=P (a,β) n(x),a>0,β>0,(1-x)P(a,β) n(x),a>0,β>-1,(1+x)P P(a,β) n(x),a>-1,β0 and P(a,β) n(x),a>-1,β>-1, respectively, onto the unit circle, where P(a,β) n(x),a>-1,β>-1, stands for the n-th Jacobi polynomial. Moreover, a result of Saff and Walsh is also extended.
