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