Mathematical Interpolation Method Research Articles

Accuracy Assessment of Spatial Interpolation Methods to Derives DEMs of Small Islands with Relative Topographic Variations

Responding to Climate Change in Small Island Developing States (SIDS), accurate Digital Elevation Model (DEM) can support the Sea level Rise (SLR) scenarios and sequenceit is impacts on coastal zone for proper adaptation. The DEM accuracy may vary to a certain degree following different interpolation algorithms and the data acquisition method. Indeed, numerous mathematical interpolation methods have been developed on spatial interpolation for topographic information densification and DEM restitution. The aim of this study focuses on the accuracy assessment of high spatial resolution DEM (at 2.5 m pixel size) regenerated from high topographic contour lines map at scale of 1:5,000 applying four different interpolation algorithms. Three deterministic methods were considered including the IDW with variable and fixed parameters, the Spline with regular and tension conditions, and the Natural Neighbor. While, for stochastic methods, the ordinary and simple Kriging were analyzed according to the semi-variogram adjustment considering five mathematical functions: Stable, Circular, Spherical, Exponential and Gaussian. For validation purposes, a datasets of 400 ground control points (GCPs) uniformly distributed over the study site, to cover all the existing altitude classes, were used. These were measured using Differential Global Position System (DGPS) with ± 1 cm and ± 2 cm for planimetric and altimetric accuracies, respectively. The results obtained show that ordinary and simple kriging methods, based on the exponential function, achieved a similar DEMs restitution with the best RMSE (± 0.65 m), which proved to be less than the tolerance or the total deviation (± 0.78 m). Consequently, these two Kriging methods are more accurate for DEM production for small island applications such as the evaluation of coastal zones vulnerability to SLR, flooding, detection of topographic features and hydrological modeling.

Shape Adaptation of the Inspection Guide Frame in Tunnels to Avoid Obstacles Detected by a Laser Range Finder

To advance the automated inspection and maintenance of the inner wall of tunnels, an advanced inspection system aimed at regulating traffic was developed. In this inspection system, a guide frame was installed along the tunnel ceiling wall that is above the protection frame built over the road and resembles a gantry crane. The inspection device was fitted with an inspection guide frame (IGF), which stabilized its movement and improved its accuracy. However, as this protection frame moves along the tunnel, the guide frame should have the capacity to avoid convex obstacles such as the duct fan, the lamp and road traffic signs within the tunnel. Therefore, the entire inspection guide frame is made of variable geometry truss (VGT), which makes it possible to suitably alter the shape of the guide frame whenever necessary and pass it through the tunnel. To enable the guide frame adapt to any shape, the inverse analysis method and mathematical interpolation method were applied. The orientation of each frame was reversely analyzed according to the shape of the obstacle measured using the laser range finder (LRF), and the frame’s actuator was controlled simultaneously. We investigated the construction of a system that can perform a series of tasks such as searching for obstacles and positioning, frame shape simulation, frame shape change, inspection of the device and movement. By applying spline interpolation, we managed to practically determine the shape of the guide frame that would avoid obstacles.

Improving Meteorological Input for Surface Energy Balance System Utilizing Mesoscale Weather Research and Forecasting Model for Estimating Daily Actual Evapotranspiration

Using Surface Energy Balance System (SEBS) to estimate actual evapotranspiration (ET) on a regional scale generally uses gridded meteorological data by interpolating data from meteorological stations with mathematical interpolation. The heterogeneity of underlying surfaces cannot be effectively considered when interpolating meteorological station measurements to gridded data only by mathematical interpolation. This study aims to highlight the improvement of modeled meteorological data from the Weather Research and Forecasting (WRF) mesoscale numerical model which fully considers the heterogeneity of underlying surfaces over the data from mathematical interpolation method when providing accurate meteorological input for SEBS model. Meteorological data at 1 km resolution in the Hotan Oasis were simulated and then were put into SEBS model to estimate the daily actual ET. The accuracy of WRF simulation was evaluated through comparison with data collected at the meteorological station. Results found that the WRF-simulated wind speed, air temperature, relative humidity and surface pressure correlate well with the meteorological stations measurements (R2 are 0.628, 0.8242, 0.8089 and 0.8915, respectively). Comparison between ET calculated using the meteorological data simulated from the WRF (ETa-WRF) and meteorological data interpolated from measurements at met stations (ETa-STA) showed that ETa-WRF could better reflect the ET difference between different land cover, and capture the vegetation growing trend, especially in areas with sparse vegetation, where ETa-STA intends to overestimate. In addition, ETa-WRF has less noise in barren areas compared to ETa-STA. Our findings suggest that WRF can provide more reliable meteorological input for SEBS model than mathematical interpolation method.

Computational routines for the automatic selection of the best parameters used by interpolation methods to create thematic maps

In precision agriculture, soil and plant variables are usually presented through thematic maps. The development of these maps is related to data collection, analysis, and interpolation. Although several mathematical interpolation methods are available (triangulation, natural-neighbor interpolation, inverse functions of distance, least-squares polynomials, and kriging), ordinary kriging (OK) and inverse distance weighting (IDW) are the most commonly used. However, farmers/agronomists are not highly trained in statistical methods to produce the best maps of soil and plant variables for precision agriculture. To ensure the best management approach is used, an automated method ready for application in an automated, easy-to-use mapping facility has the potential to be very useful. Thus, this study aimed to develop and apply computational routines capable of automatically identifying the best parameters for each interpolation method. Exhaustive tests were performed on these routines implemented in geoR software by using cross-validation methods and replacing the parameter values used by the interpolation techniques. The routines were applied to sample data encompassing corn and soybean yields as well as chemical and physical variables of soil collected in two agricultural areas located in the municipality of Serranópolis do Iguaçu, West Paraná, Brazil. For the semivariograms, 300 different adjustments were tested to identify the best parameters to interpolate the measured data using OK, and twelve different values were tested for the IDW exponent. As expected, the best results were obtained by OK when the variables exhibited spatial dependence. We concluded that the computational routines implemented are efficient and capable of identifying the interpolation method (OK with the spherical, exponential, Gaussian, and Matern family of semivariograms; IDW otherwise) with the best adjustment for each area as a function of the presence of spatial dependence.

Kriging: Methods and Applications

In this chapter we present Kriging also known as a Gaussian process (GP) model which is a mathematical interpolation method. To select the input combinations to be simulated, we use Latin hypercube sampling (LHS); we allow uniform and non-uniform distributions of the simulation inputs. Besides deterministic simulation we discuss random simulation, which requires adjusting the design and analysis. We discuss sensitivity analysis of simulation models, using functional analysis of variance (FANOVA) also known as Sobol sensitivity indexes. Finally, we discuss optimization of the simulated system, including robust optimization.

A mathematical spatial interpolation method for the estimation of convective rainfall distribution over small watersheds

Rainfall is one of crucial factors that impact on our environment. Rainfall data is important in water resources management, flood forecasting, and designing hydraulic structures. However, it is not available in some rural watersheds without rain gauges. Thus, effective ways of interpolating the available records are needed. Despite many widely used spatial interpolation methods, few studies have investigated rainfall center characteristics. Based on the theory that the spatial distribution of convective rainfall event has a definite center with maximum rainfall, we present a mathematical interpolation method to estimate convective rainfall distribution and indicate the rainfall center location and the center rainfall volume. We apply the method to estimate three convective rainfall events in Santa Catalina Island where reliable hydrological data is available. A cross-validation technique is used to evaluate the method. The result shows that the method will suffer from high relative error in two situations: 1) when estimating the minimum rainfall and 2) when estimating an external site. For all other situations, the method`s performance is reasonable and acceptable. Since the method is based on a continuous function, it can provide distributed rainfall data for distributed hydrological model sand indicate statistical characteristics of given areas via mathematical calculation.

Evaluation of Interpolants in Their Ability to Fit Seismometric Time Series

This article is devoted to the study of the ASARCO demolition seismic data. Two different classes of modeling techniques are explored: First, mathematical interpolation methods and second statistical smoothing approaches for curve fitting. We estimate the characteristic parameters of the propagation medium for seismic waves with multiple mathematical and statistical techniques, and provide the relative advantages of each approach to address fitting of such data. We conclude that mathematical interpolation techniques and statistical curve fitting techniques complement each other and can add value to the study of one dimensional time series seismographic data: they can be use to add more data to the system in case the data set is not large enough to perform standard statistical tests.

Dynamic clearance measure to evaluate locomotor and perceptuo-motor strategies used for obstacle circumvention in a virtual environment

Circumvention around an obstacle entails a dynamic interaction with the obstacle to maintain a safe clearance. We used a novel mathematical interpolation method based on the modified Shepard’s method of Inverse Distance Weighting to compute dynamic clearance that reflected this interaction as well as minimal clearance. This proof-of-principle study included seven young healthy, four post-stroke and four healthy age-matched individuals. A virtual environment designed to assess obstacle circumvention was used to administer a locomotor (walking) and a perceptuo-motor (navigation with a joystick) task. In both tasks, participants were asked to navigate towards a target while avoiding collision with a moving obstacle that approached from either head-on, or 30° left or right. Among young individuals, dynamic clearance did not differ significantly between obstacle approach directions in both tasks. Post-stroke individuals maintained larger and smaller dynamic clearance during the locomotor and the perceptuo-motor task respectively as compared to age-matched controls. Dynamic clearance was larger than minimal distance from the obstacle irrespective of the group, task and obstacle approach direction. Also, in contrast to minimal distance, dynamic clearance can respond differently to different avoidance behaviors. Such a measure can be beneficial in contrasting obstacle avoidance behaviors in different populations with mobility problems.

제주도 북동부 한동지역의 MCP 회귀모델식을 적용한 AEP계산에 대한 연구

풍력발전 단지의 설계시 풍력 자원 평가 과정은 필수적인 과정이다. 풍력 자원 평가를 위해 장기풍황(20년)자료를 이용하여야 하지만 장기간 관측하는 것은 어렵기 때문에 예정지의 1년 이상의 관측데이터로 평가를 실시하였다. 예정지의 단기 풍황탑(Met-Mast; Meteorology Mast) 자료를 주변의 장기관측 자료인 자동기상관측(AWS; Automatic Weather Station)데이터를 이용하여 수학적 보간법으로 예정지의 데이터를 장기 데이터로 변환한 것을 MCP(Measure-Correlative-Predict)기법이라 한다. 본 연구에서는 MCP기법 중 선형 회계방법을 적용하였다. 선택된 MCP 회귀 모델식에 따라 제주 북동부 구좌지역의 AWS데이터를 제주 북동부 한동 지역의 Met-mast 데이터에 적용하여 연간 에너지 생산량을 예측 하였다. 예정지의 단기 풍황을 이용하였을 때와 보정된 장기 풍황을 이용하여 때 연간 에너지 생산량을 비교하였다. 그 결과 연간 약 3.6 %의 예측오차를 보였고, 이는 연간 약 271 MW의 에너지 생산량의 차이를 의미한다. 풍력발전기의 생애주기인 20년을 비교 하였을 때 약 5,420 MW의 차이를 나타내었으며, 이는 약 9개월 정도의 에너지 생산량과 비슷한 수준이다. 결과적으로, 제안 된 선형 회귀 MCP 방법을 이용하는 것이 단기관측 자료를 통한 불확식성을 제거하는 합리적인 방법으로 판단된다. Wind resource assessment is necessary when designing wind farm. To get the assessment, we must use a long term(20 years) observed wind data but it is so hard. so that we usually measured more than a year on the planned site. From the wind data, we can calculate wind energy related with the wind farm site. However, it calculate wind energy to collect the long term data from Met-mast(Meteorology Mast) station on the site since the Met-mast is unstable from strong wind such as Typhoon or storm surge which is Non-periodic. To solve the lack of the long term data of the site, we usually derive new data from the long term observed data of AWS(Automatic Weather Station) around the wind farm area using mathematical interpolation method. The interpolation method is called MCP(Measure-Correlative-Predict). In this study, based on the MCP Regression Model proposed by us, we estimated the wind energy at Handong site using AEP(Annual Energy Production) from Gujwa AWS data in Jeju. The calculated wind energy at Handong was shown a good agreement between the predicted and the measured results based on the linear regression MCP. Short term AEP was about 7,475MW/year. Long term AEP was about 7,205MW/year. it showed an 3.6% of annual prediction different. It represents difference of 271MW in annual energy production. In comparison with 20years, it shows difference of 5,420MW, and this is about 9 months of energy production. From the results, we found that the proposed linear regression MCP method was very reasonable to estimate the wind resource of wind farm.

Use of turbulent flow statistical properties for correcting erroneous velocity vectors in PIV

For particular turbulent flows, PIV measurements technique provides more than 10% of spurious velocity vectors at each time step. To correct these vectors, we propose to use the Linear Stochastic Estimation (LSE) dealing with the spatial correlation tensor of the velocity. If this tensor cannot be determined in some measurement zones, Proper Orthogonal Decomposition is used to model this tensor. Both reconstruction methodologies are tested from PIV measurements performed in a Spark Ignition engine flow. We show that for coherent structures analysis, the LSE reconstruction method provides better results than classical mathematical interpolation methods. To cite this article: Ph. Druault, Ph. Guibert, C. R. Mecanique 332 (2004).

A media-based assessment of damage and ground motions from the january 26th, 2001M 7.6 Bhuj, India earthquake

We compiled available news and internet accounts of damage and other effects from the 26th January, 2001, Bhuj earthquake, and interpreted them to obtain modified Mercalli intensities at over 200 locations throughout the Indian subcontinent. These values are used to map the intensity distribution using a simple mathematical interpolation method. The maps reveal several interesting features. Within the Kachchh region, the most heavily damaged villages are concentrated towards the western edge of the inferred fault, consistent with western directivity. Significant sedimentinduced amplification is also suggested at a number of locations around the Gulf of Kachchh to the south of the epicenter. Away from the Kachchh region intensities were clearly amplified significantly in areas that are along rivers, within deltas, or on coastal alluvium such as mud flats and salt pans. In addition we use fault rupture parameters inferred from teleseismic data to predict shaking intensity at distances of 0–1000 km. We then convert the predicted hard rock ground motion parameters to MMI using a relationship (derived from internet-based intensity surveys) that assigns MMI based on the average effects in a region. The predicted MMIs are typically lower by 1–2 units than those estimated from news accounts. This discrepancy is generally consistent with the expected effect of sediment response, but it could also reflect other factors such as a tendency for media accounts to focus on the most dramatic damage, rather than the average effects. Our modeling results also suggest, however, that the Bhuj earthquake generated more high-frequency shaking than is expected for earthquakes of similar magnitude in California, and may therefore have been especially damaging.

The 26 January 2001 M 7.6 Bhuj, India, Earthquake: Observed and Predicted Ground Motions

Although local and regional instrumental recordings of the devastating 26, January 2001, Bhuj earthquake are sparse, the distribution of macroseismic effects can provide important constraints on the mainshock ground motions. We compiled available news accounts describing damage and other effects and interpreted them to obtain modified Mercalli intensities (MMIs) at >200 locations throughout the Indian subcontinent. These values are then used to map the intensity distribution throughout the subcontinent using a simple mathematical interpolation method. Although preliminary, the maps reveal several interesting features. Within the Kachchh region, the most heavily damaged villages are concentrated toward the western edge of the inferred fault, consistent with western directivity. Significant sediment-induced amplification is also suggested at a number of locations around the Gulf of Kachchh to the south of the epicenter. Away from the Kachchh region, intensities were clearly amplified significantly in areas that are along rivers, within deltas, or on coastal alluvium, such as mudflats and salt pans. In addition, we use fault-rupture parameters inferred from teleseismic data to predict shaking intensity at distances of 0–1000 km. We then convert the predicted hard-rock ground-motion parameters to MMI by using a relationship (derived from Internet-based intensity surveys) that assigns MMI based on the average effects in a region. The predicted MMIs are typically lower by 1–3 units than those estimated from news accounts, although they do predict near-field ground motions of approximately 80%g and potentially damaging ground motions on hard-rock sites to distances of approximately 300 km. For the most part, this discrepancy is consistent with the expected effect of sediment response, but it could also reflect other factors, such as unusually high building vulnerability in the Bhuj region and a tendency for media accounts to focus on the most dramatic damage, rather than the average effects. The discrepancy may also be partly attributable to the inadequacy of the empirical relationship between MMI and peak ground acceleration (PGA), when applied to India. The MMI–PGA relationship was developed using data from California earthquakes, which might have a systematically different stress drop and therefore, a different frequency content than intraplate events. When a relationship between response spectra and MMI is used, we obtain larger predicted MMI values, in better agreement with the observations.

110 二値化画像相関法PTVに関する二・三の数学的考案

Cross-correlation function used in the binary image cross-correlation method developed by Uemura, T., Yamamoto, F., and Koukawa, M. is used without mathematical strict proof, although their method is thought to be convenient and efficient to analyze the particle images due to the high speed algorithm for particle identification. In the present study, the binary image crosscorrelation function is derived from mathematical foundations, and the generality is discussed. Since the velocity vectors are computed discretely at the particle positions, some interpolation methods are needed to rearrange the velocity vectors at the grid points in order to extract the other physical informations. The three following new mathematical interpolation methods (differential form of equation of continuity, integral form of equation of continuity, and boundary element method) are proposed in the present paper.