Assessing the effectiveness of data interpolation methods for 2D meshes and adjusting them for water flow modeling in Vam Nao area

AGU International Journal of Sciences – 2019, Vol. 7 (4), 49 – 57<br /> <br /> <br /> <br /> <br /> ASSESSING THE EFFECTIVENESS OF DATA INTERPOLATION METHODS<br /> FOR 2D MESHES AND ADJUSTING THEM FOR WATER FLOW MODELING<br /> IN VAM NAO AREA<br /> <br /> Chau Ngan Khanh1, Nguyen Tran Nhan Tanh1, Ngo Thuy An1<br /> 1<br /> An Giang University, VNU - HCM<br /> <br /> Information: ABSTRACT<br /> Received: 12/12/2018<br /> River water flow modeling often uses measured bottom elevation data for<br /> Accepted: 13/08/2019<br /> 2D-mesh interpolation. The interpolation quality affects water flow<br /> Published: 11/2019<br /> simulation outcome. Thus, methods of adjusting interpolated 2D-meshes<br /> Keywords: properly are needed. To support the mesh creation, our study analyzed<br /> River flow, interpolated mesh, effects of interpolation methods and adjusts improper mesh nodes. The<br /> mesh adjustments, mesh for research methods include: (1) linear interpolation method, (2) adjacent<br /> simulation, flow modeling, interpolation method, (3) distance inverse interpolation method, (4) creating<br /> QGIS, Blue Kenue,<br /> Telemac 2D contour plots to evaluate interpolation results, (5) selecting calculating<br /> domains and adjusting mesh nodes. Software used is QGIS 3.0 (in<br /> combination with Google Satellite), Blue Kenue 3.3.4, and Telemac 2D<br /> v7p3r1. The research results show that selecting appropriate interpolation<br /> methods help to create meshes that are in accordance with the actual<br /> situation of the flow topography (compared from Google Satellite), and that<br /> adjusting inappropriate mesh nodes contributes to improving the<br /> effectiveness of river flow modeling.<br /> <br /> <br /> 1. INTRODUCTION be the depth of water, the amount of rainfall, or<br /> Interpolation methods are numerical methods of the color of the points on an image. Spatial<br /> constructing new data points within a domain of interpolation methods are applied to determine<br /> a given data (Epperson, 2013 và Nguyen Duc missing values that were previously not able to<br /> Nhan, 2016). In this paper, we focus on be measured.<br /> presenting spatial interpolation methods (Pav, If the input data is detailed, results from<br /> 2005 và Ngo Van Thanh, 2009). The spatial different spatial interpolation methods are<br /> interpolation methods are currently applied in almost the same. However, actual measurement<br /> many fields such as hydro-meteorology, fluid to get detailed data is very expensive.<br /> dynamics, agricultural meteorological mapping Therefore, the input data are often sparse and<br /> and examination of soil, water and air discontinuous, especially when there are no<br /> distribution. data points at boundaries. For these types of<br /> The input data includes the spatial coordinates data, using better spatial interpolation methods<br /> and the values of the points. These values can with reduced levels of error is crucial for<br /> ensuring an optimized process.<br /> <br /> <br /> <br /> 49<br /> AGU International Journal of Sciences – 2019, Vol. 7 (4), 49 – 57<br /> <br /> In this paper, we present a summary of spatial The final purpose of the paper is to select the<br /> interpolation methods and the comparison of optimal interpolation method for the data at the<br /> these interpolation methods based on data with junction of Hau river and Vam Nao river (Chau<br /> noise and discontinuity. The research Ngan Khanh et al., 2018). The results of this<br /> discontinuous data are cut from the Sai Gon paper could be implemented for the on-going<br /> River data (see appendix). The paper focuses project "Application of the models in Telemac<br /> on the following three points. 2D and 3D to simulate the flow and transport of<br /> First, spatial interpolation methods in Telemac sediments at the junction of Hau and Vam Nao<br /> 2D and R software are introduced (Ata, 2017; rivers”. This project is conducted by the<br /> Rossiter, 2013; Akima et al., 2016 and Tran Thi department of science and technology of An<br /> Bang Tam, 2006). The materials for spatial Giang province and An Giang University.<br /> interpolation methods in Vietnamese language 2. SPATIAL INTERPOLATION<br /> liturature are few and inexplicit. METHODS<br /> Second, we will pursue methods to overcome In this section, we will present the spatial<br /> the drawbacks of spatial interpolation methods interpolation methods in Telemac 2D and R<br /> in Telemac 2D software when dealing with data software (Ata, 2017; Garnero và Godone, 2013;<br /> containing noise and discontinuities, based on Dorman, 2014; Pebesma và Graeler 2014 và<br /> using other interpolation methods in R Dumitru và cs., 2013)<br /> software. Good interpolation results are integral 2.1 Linear interpolation method<br /> to ensuring the accuracy of results in Telemac<br /> The linear interpolation method constructs new<br /> software. However, the input parameters of<br /> interpolated values by dividing the calculated<br /> spatial interpolation methods in Telemac 2D<br /> domain into triangles. The algorithm of this<br /> software are rarely changed. The interpolation<br /> delaunay triangulation method starts by<br /> functions in R software, meanwhile, are more<br /> selecting first point. We then look for the<br /> flexible with input parameters (Ata, 2017 và<br /> closest distance from the selected point to the<br /> Pebesma và Graeler, 2014). For data with noise<br /> sampled points and link adjacent points by<br /> and discontinuities, we will change the input<br /> straight lines.<br /> parameters in R software so that the errors from<br /> different interpolation methods are reduced, and After the calculated domain is divided into<br /> the calculation time is shortened. triangles, the interpolation values are<br /> determined through the plane created by the<br /> The accuracy of these interpolation methods<br /> triangle. The equation of the plane passing<br /> will be tested by applying on the Saigon River<br /> through the three vertices of a triangle has the<br /> data. The spatial data of Saigon River are very<br /> following form<br /> detailed. From this detailed data, we proceed to<br /> cut off the data and banks, to obtain a<br /> discontinuous data set. Various interpolation where, z is the value to be interpolated at (x, y).<br /> methods are applied to reinterpret the cut The coefficients a, b and c of (1) are determined<br /> points. The results from these various by replacing the coordinates and the value at<br /> interpolation methods are compared to actual the vertices of the triangle which are<br /> sampled values. From those results,<br /> interpolation methods, which are suitable for , và .<br /> discontinuous data, will be defined. From the equation, we have the following<br /> equation system<br /> <br /> <br /> 50<br /> AGU International Journal of Sciences – 2019, Vol. 7 (4), 49 – 57<br /> <br /> <br /> is the value of the jth interpolated point,<br /> <br /> is the value of the neighboring point,<br /> <br /> The algorithm for delaunay triangulation and is the distance from the ith point to the jth<br /> determining the coefficients in linear equation point,<br /> (1) is simple. So, the calculations for the linear<br /> is the exponent number we choose to<br /> interpolation method are performed quite<br /> adjust the weight of the distance.<br /> rapidly. However, the linear interpolation<br /> method requires detailed input data for accurate IDW method is a simple method. Thus, it is<br /> interpolation results. easy to apply and has fast interpolation<br /> calculation time. However, this interpolation<br /> Notice: The linear interpolation method has no<br /> method is only accurate when we have detailed<br /> input parameters, except for input data and<br /> sampled data which has little change in its<br /> interpolated data. Therefore, we cannot adjust<br /> terrain. In the case of sparse data having varied<br /> parameters in linear interpolation method.<br /> terrain, the potential error of this method is<br /> 2.2 Inverse distance weighted (IDW) method large.<br /> The IDW method determines interpolation Notice: The input parameters of the IDW<br /> values by calculating the average values of method are the number of neighboring points I<br /> sampled points in the vicinity of an interpolated and the exponent n. Thus, the number of nearby<br /> point. The closer it is to the interpolated point, points I and the exponent index n in (3) could<br /> the more influential it is. To construct new data be adjusted.<br /> points, IDW method’s results are based on the<br /> 2.3 Nearest neighbour method<br /> measured values of the nearby points. The<br /> value of predicted points is close to the value of The nearest neighbour interpolation method is a<br /> neighboring points than those that are far away. specific case of linear interpolation method.<br /> The weight of nearby points is inversely This interpolation method determines new<br /> proportional to the distance of the predicted interpolated values by using the value of an<br /> point. The interpolation formula for the IDW adjacent data point which is nearest to the<br /> interpolated point. This interpolation method is<br /> method is as follows<br /> based on the comparison of the distance<br /> between an interpolated point and its adjacent<br /> data points.<br /> With a simple algorithm, the nearest point<br /> interpolation is implemented with very fast<br /> calculation speed. However, this interpolation<br /> ,<br /> method requires detailed input data for accurate<br /> where is the number of neighboring points of interpolation results, especially at the boundary<br /> the jth interpolated point, points.<br /> N is the number of interpolated points, Notice: The nearest point interpolation method<br /> has no parameters to adjust.<br /> <br /> <br /> <br /> <br /> 51<br /> AGU International Journal of Sciences – 2019, Vol. 7 (4), 49 – 57<br /> <br /> 2.4 Spline interpolation method linear interpolation method. After the calculated<br /> The spline interpolation method in R software domain is divided into triangles, the<br /> determines new interpolated values by dividing interpolation values are determined through the<br /> the calculated domain into triangles as in the plane having the following cubic equation<br /> <br /> <br /> (4)<br /> <br /> where, z (x, y) is an interpolation function at the The interpolation value z at a prediction<br /> point (x, y). The coefficient of the cubic<br /> location has the form<br /> function in (4) is determined by the values at<br /> the three vertices of the triangles and the values<br /> of the partial derivatives of the z (x, y)<br /> , (5)<br /> functions at the vertices of the triangle.<br /> In Arcgis software, the spline interpolation where and are the<br /> method is determined by the following th<br /> values and the weights at the i location from<br /> functions. neighborhood points of the given point. The<br /> <br /> ,<br /> sum of the weights equals 1, . The<br /> or<br /> estimate of z value is unbiased.<br /> In the kriging interpolation, the weights<br /> where R (r) is a function that depends on the<br /> are based not only on the<br /> distance from the point of interpolation to<br /> distance between measured points and<br /> points of data.<br /> prediction location but also on the overall<br /> The spline interpolation method has a complex spatial arrangement of the measured points.<br /> algorithm to determine the coefficients of the<br /> The Kriging interpolation process has two<br /> interpolation function. The calculation time<br /> steps. The first step is fitting a model which is<br /> depends on the selection of the number of<br /> the creation of semivariogram and covariogram<br /> adjacent points. This interpolation method has<br /> functions to estimate spatial autocorrelation<br /> relatively accurate results even when we have<br /> values. The second step is predicting the weight<br /> discontinuous measurement data and various<br /> terrain. parameters based on the spatial<br /> Notice: In the spline interpolation method, we autocorrelation values in the first step.<br /> could select the number of adjacent points and To estimate the weights in (5), many models are<br /> various interpolation functions from software used such as linear model, exponential model,<br /> such as R and Arcgis. Gaussian model, sphere model, nugget model<br /> 2.5 Kriging interpolation method and others. Kriging’s interpolation method has<br /> a complex algorithm, so that it takes time to<br /> The Kriging interpolation method is a method<br /> calculate the parameters in the model. The<br /> of estimating the value z (x, y) at the estimated<br /> calculating time depends on selecting the<br /> point (x, y) satisfying the following<br /> number of sampled points. This interpolation<br /> assumptions.<br /> method gives relatively accurate results in the<br /> <br /> <br /> <br /> 52<br /> AGU International Journal of Sciences – 2019, Vol. 7 (4), 49 – 57<br /> <br /> case that we do not have detailed measurement We compared the interpolation methods by<br /> data and terrain has been changed. using Telemac and R software for<br /> Notice: discontinuous data which is created from the<br /> detailed spatial data of Saigon river (Appendix).<br /> Parameters in kriging interpolation methods are<br /> Based on the results of these comparisons, we<br /> the variogram models and the number of<br /> will find out which interpolation methods<br /> sampled points.<br /> yielded large errors for discontinuous data. In<br /> Kriging interpolation methods have complex addition, we will point out that the interpolation<br /> algorithm, therefore calculation time might take methods can be implemented for data with<br /> longer than other interpolation methods. noise and discontinuities, especially river data<br /> 2.6 Method of analyzing the symmetry of lacking information at its bank lines.<br /> data The interpolation results were not compared<br /> The method of analyzing the symmetry of data through observing contour lines (Chau Ngan<br /> is presented and implemented in the report at Khanh, Nguyen Tran Nhan Tanh et al, 2018).<br /> the 21st national scientific conference on fluid Instead, the interpolation results are compared<br /> dynamics (Chau Ngaan Khanh et al.., 2018). to the original measured values. These<br /> This method requires and analyzes the comparisons are highly accurate and reliable.<br /> symmetry of the data. So that, sampled spatial 3.1 Remove information from detailed data<br /> data at the banks of Vam Nao river is cut to<br /> River bank data and river bed data of the Sai<br /> ensure the symmetry of the spatial data.<br /> Gon river are removed from the sampled spatial<br /> Adjusted data is symmetric through a line<br /> detailed data of the Sai Gon river (Figure 1) in<br /> which lies at the middle of the river and<br /> order to get discontinuous data. The extracted<br /> parallels to the banks of river. Linear<br /> data are presented in Figure 2. The distance of<br /> interpolation method is used for the adjusted<br /> each measuring line is 500 meters and the<br /> data. Interpolation results are compared through<br /> boundary data is asymmetric. The extracted<br /> observing contour lines.<br /> data has similar measured distances to the<br /> 3. COMPARISON OF INTERPOLATION sampled spatial data of Vam Nao river, An<br /> METHODS FOR UNDETAILED DATA Giang province (Chau Ngan Khanh, et al.<br /> 2018).<br /> <br /> <br /> <br /> <br /> Figure 1. The sampled spatial data of the Sai Gon river (left hand side) and the research data which is cut<br /> from the sampled spatial data of the Sai Gon river data (right hand side)<br /> <br /> <br /> <br /> <br /> 53<br /> AGU International Journal of Sciences – 2019, Vol. 7 (4), 49 – 57<br /> <br /> The original detailed data of the Sai Gon river above extracted data to ensure the symmetry of<br /> is separated into two data sets. The first data is the sampled spatial data from Sai Gon river<br /> the extracted data obtained from the above through the axis which is at the middle of the<br /> method (left-hand side, Figure 2) and the river and parallel to the river banks. The data<br /> second data is the remaining data from the after cutting boundaries is shown in the right-<br /> research data after extracting the first data hand side, Figure 2. From this unboundary data,<br /> (right-hand side, Figure 2). we also created the two data sets including an<br /> For the method of analyzing the symmetry of extracted data and a remaining data with the<br /> data, we cut once again the boundary of the same method as above.<br /> <br /> <br /> <br /> <br /> Figure 2. The Saigon river data is removed with boundary (left hand side)<br /> and The Saigon river data do not have boundaries (right hand side)<br /> 3.2 Methodology 3.3 Results<br /> From the research data (Figure 1), we created To compare the results of the interpolation<br /> two sets of data including sparse Data 1 (left methods, we calculate the absolute value of the<br /> hand side, Figure 2) and the remaining data that differences between the sampled depth values<br /> need to be interpolated, as shown in Data 2. The from Data 2 and the interpolated depth values<br /> remaining points from Data 2 after extracting from Data 4 for each interpolation method.<br /> sparse data , we take the coordinates of the Then the differences among the interpolation<br /> points in Data 2 and name it as Data 3. methods are compared with each other. In each<br /> From sparse Data 1, we applied different interpolation method, we also compare two<br /> interpolation methods to construct new depth types of data, with the boundary and without<br /> values of Data 3. Then we get Data 4. the boundary. The results of the comparisons<br /> are presented in Table 1.<br /> The sampled depth values from Data 2 and the<br /> interpolated depth values from Data 4 are<br /> compared and tested in the next section.<br /> <br /> <br /> <br /> <br /> 54<br /> AGU International Journal of Sciences – 2019, Vol. 7 (4), 49 – 57<br /> Table 1. The results of interpolation methods applied for research data with boundary and without<br /> boundary, which is taken from the Saigon River data.<br /> <br /> Nearest neighbour method Average Sample variance<br /> Data without boundary 5.027 27.080<br /> Data with boundary 4.040 26.977<br /> Inverse distance weighted method Average Sample variance<br /> Data without boundary 4.098 16.073<br /> Data with boundary 2.504 9.704<br /> Kriging interpolation Method Average Sample variance<br /> Data without boundary 3.527 10.031<br /> Data with boundary 2.412 7.053<br /> <br /> <br /> In general, the method of analyzing the Notice: For sparse data, we advise against<br /> symmetry of the data without boundary data using these methods as they do not include<br /> gives poor results compared to the data with the extrapolation values.<br /> boundary throughout the interpolation methods. Inverse distance weighted method: This<br /> Nearest neighbour method: the method has method gives stable interpolated values,<br /> biggest error compared to other interpolation although the results are not as robust as the<br /> methods. Kriging method<br /> Linear interpolation method and Spline Kriging interpolation method with<br /> interpolation method: These interpolation exponential model: This method gives the best<br /> methods are based on the algorithm of triangle interpolation values of the above methods.<br /> division. If the points to be interpolated outside However, in the Kriging interpolation method,<br /> the divided triangles, they are considered as there are a multitude of existing models. Here,<br /> extrapolation points. we chose to utilize the simplest model: the<br /> In unboundary data, over 10% of interpolated exponential model.<br /> data are non-numeric values (NA: not a In conclusion, the method of analyzing the<br /> number). Those values are located near the symmetry of data is not appropriate. Therefore,<br /> boundary and are called extrapolation points we only use interpolated results with boundary<br /> because in this interpolation method, the data. The interpolation results from different<br /> extrapolated points (points outside triangles) interpolation methods are tested. The null<br /> will not be calculated and is set to NA values. statistical hypothesis is that the averages of the<br /> When conducting interpolation by linear difference between interpolated values and<br /> method on Telemac software, this software will sampled values among different interpolation<br /> set the values for these extrapolation points to methods are equal. The results of the hypothesis<br /> 0. testing are presented in Table 2.<br /> In boundary data, NA values fall below 5% of<br /> the data.<br /> <br /> <br /> <br /> <br /> 55<br /> AGU International Journal of Sciences – 2019, Vol. 7 (4), 49 – 57<br /> Table 2. Hypothesis testing for the difference between interpolated values and sampled values among different<br /> interpolation methods.<br /> <br /> P-value<br /> Nearest neighbour method and One side Two side<br /> Inverse distance weighted method 7.37233E-16 1.47447E-15<br /> <br /> <br /> Nearest neighbour method and One side Two side<br /> Kriging interpolation method<br /> 6.91801E-15 1.3836E-14<br /> <br /> <br /> Inverse distance weighted method One side Two side<br /> and<br /> Kriging interpolation method<br /> 0.200230608 0.400461216<br /> <br /> <br /> The P value is very small in the hypothesis interpolation results from data without<br /> testing between the nearest neighbour method boundary have significant errors. Second, for<br /> with the inverse distance weighted method and discontinuous data, the linear interpolation<br /> the nearest neighbour method with the Kriging method and the nearest neighbour method<br /> interpolation method. The averages of the should not be used because the interpolation<br /> difference between interpolated values and results will be biased. If we want to use the<br /> sampled values from the nearest neighbour linear interpolation method or the Spline<br /> method is larger than from the inverse distance interpolation method, we should have boundary<br /> weighted method and the Krige interpolation data to eliminate extrapolation points. Third,<br /> method. Therefore, the nearest neighbour the inverse distance weighted method and<br /> method is not as good as the inverse distance Kriging methods should be used for<br /> weighted method and the Krige interpolation discontinuous data.<br /> method in discontinuous data, such as the Interpolation methods play an important role in<br /> sparse data taken from the Saigon River data. constructing new data points in Telemac<br /> The P value is greater than 0.05 in the software. It is time-consuming to run models in<br /> hypothesis testing between the Kriging the software. It can take several months or<br /> interpolation method and the nearest neighbour maybe years. Therefore, the accuracy of<br /> method. There is reasonable evidence to interpolated data is very important to get<br /> support that the averages of the difference accurate results with output data. Our future<br /> between interpolated values and sampled values work will focus more on the Spline method to<br /> of the two methods are equal. find ways to overcome extrapolation points.<br /> 4. CONCLUSION Different models in the Kriging interpolation<br /> method need to be investigated further in order<br /> There are three major conclusions drawn from<br /> to apply to different types of spatial data.<br /> the study. First, we should not cut boundary<br /> from the spatial data as in the method of Acknowledgments: We would like to express<br /> analyzing the symmetry of data. The our sincere thanks to Mr. Mai Anh Vu, TSC<br /> <br /> <br /> <br /> 56<br /> AGU International Journal of Sciences – 2019, Vol. 7 (4), 49 – 57<br /> <br /> consulting and constructing services company, Epperson, J. (2013). An introduction to<br /> for providing data of Saigon River. We would numerical methods and analysis. Canada:<br /> like to thank Ms. Chau Ngan Khanh, faculty of Wiley.<br /> information technology of An Giang University Garnero, G. & Godone, D. (2013).<br /> for supporting us about Telemac software. Comparisons between different<br /> APPENDIX interpolation techniques. Proceedings of the<br /> Research data taken from Saigon River is international archives of the<br /> attached in the .xyz file. photogrammetry, remote sensing and spatial<br /> REFERENCE information sciences XL-5 W, 3:27_28.<br /> <br /> Akima, H., Gebhardt, A., Petzold, T. & Knott, G. (2012). Interpolating cubic splines.<br /> Maechler, M. (2016). Akima: Interpolation Springer Science & Business Media, 18.<br /> of irregularly and regularly spaced data. R Mattle, O. (2017). TELEMAC 3D user manual<br /> package version 0.5-4. version 7.2. EDF-DRD.<br /> Ata, R. (2017). TELEMAC 2D user manual Nguyen Duc Nhan. (2016). Numerical methods.<br /> version 7.2. EDF-DRD. Post and telecommunication institution. 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