On choosing spatial interpolation methods for data with noise and discontinuities

In this paper, data with significant noise and discontinuities is considered. Finding appropriate interpolation methods for these types of data poses several challenges. The main aims of this paper are to present spatial interpolation methods and to select an adequate interpolation method for the particular data. The results of different interpolation methods are implemented and tested in a case study of the Sai Gon river.

AGU International Journal of Sciences – 2019, Vol (4), 58 – 73 ON CHOOSING SPATIAL INTERPOLATION METHODS FOR DATA WITH NOISE AND DISCONTINUITIES Pham Thi Thu Hoa1, Pham My Hanh1 An Giang University, VNU - HCM Information: Received: 29/10/2018 Accepted: 03/1/2019 Published: 11/2019 Keywords: Spatial interpolation methods, linear interpolation, Inverse distance weighted interpolation, Spline interpolation, Kriging interpolation, data with noise and discontinuities ABSTRACT Spatial interpolation methods are used to predict values of spatial phenomena in unsampled locations These methods have been applied in many applications related to fluid dynamics, natural resources, environmental sciences and image processing In this paper, data with significant noise and discontinuities is considered Finding appropriate interpolation methods for these types of data poses several challenges The main aims of this paper are to present spatial interpolation methods and to select an adequate interpolation method for the particular data The results of different interpolation methods are implemented and tested in a case study of the Sai Gon river The main motivation is to apply the result of the paper to the sampled spatial data at Vam Nao region, which contains substantive noise and discontinuities INTRODUCTION methods, and implemented tools There are many previous studies applying different meshing methods to make inputs for hydraulic simulation For example, (Tran Ngoc Anh, 2011) created a flood inundation mapping for downstream region to study the flow systems of Thach Han and Ben Hai rivers in Quang Tri province using the MIKE FLOOD model In this study, the study area was discrete into a finite element mesh as input to the MIKE FLOOD model In another study, (Samaras, Vacchi, Archetti, & Lamberti, 2013) used Blue Kenue as a tool to build an input mesh for wave modeling and hydrodynamic modeling in coastal areas (Vu Duy Vinh, Kartijin Baetens, Patrick Luyten, Tran Anh Tu, & Nguyen Thi Kim Anh, 2013) built a rectangular mesh for the coastal area of the Red River Delta with a While modeling flows, we need to consider the evolution of hydrological phenomena at different points according to space (coordinates) On that platform, relevant parameters will be considered according to spatial variation From there, the equations express relationships as separate derivative equations to simulate hydrological phenomena containing space and time variables To be able to express these spatial properties, modeling needs to divide the space into cells in which each cell will be assigned its own characteristics of coordinates, hydrological parameters, and time The mesh generation depends on many factors such as topographic data, knowledge and experience of the implementers, meshing 58 AGU International Journal of Sciences – 2019, Vol (4), 58 – 73 resolution of 0.01 degrees In the study of twodimensional hydraulic modeling to simulate the flow at the estuary of Dinh An river, (Nguyen Phuong Tan, Van Pham Dang Tri, & Vo Quoc Thanh, 2014) created a value mesh on the coordinate system (Tung T Vu, Phuoc K T Nguyen, Lloyd H C Chua, & Law, 2015) used Blue Kenue to create a two-dimensional mesh to simulate two-dimensional hydrodynamics for flooding for a part of the Mekong river using the Telemac 2D model (Nguyen Van Hoang, Đoan Anh Tuan, & Nguyen Thanh Cong, 2015) interpolated the riverbed to simulate saline intrusion of the Hoa river in Thai Binh province using EFDC software Similarly, (Mabrouka, 2016) used Blue Kenue to create a computational mesh for the Medjerda river region in Tunisia, as an input for Telemac 2D system to model the flow in vegetation rivers With the approach of geographic information system, (Sai Hong Anh, Le Viet Son, Toshinori Tabata, & Kazuaki Hiramatsu, 2017) interpolated the elevation to create a mesh to serve flood simulation for the Red river area, Hanoi These studies show that elevation meshes are the inputs for flow modeling and therefore they affect the quality of simulation domain and the boundary nodes on the simulation results Some studies compared and evaluated interpolation methods For example, (Arun, 2013) made a comparison of interpolation techniques such as inverse distance weighting, Kriging, ANUDEM, nearest neighbors and Spline to evaluate the accuracy of the terrain models Similarly, (Panhalakr & Jarag, 2016) conducted an assessment of methods including inverse distance weighting, Kriging and Topo to raster in generating river bathymetry for the Panchganga river basin in the town of Kolhapur in India Most of the research has focused on the evaluation of interpolated results but there are few studies of the intervention in the preinterpolation phase to get consistent results Therefore, this study was conducted to supplement to the defects of the interpolation We use Blue Kenue to create interpolated meshes for Vam Nao river in An Giang province To understand mesh creation more clearly, we analyze the effects of interpolation methods on results on the same interpolated dataset Besides, we also present the process of mesh adjustment through selecting calculated domain and adjusting boundary nodes The meshing appropriately will create a good mesh that can represent hydrological phenomena that are nearly identical to the reality after interpolation Mesh generation is a prerequisite step for simulation/modeling of hydrological problems based on spatial characteristics The generation of a reasonable mesh is vitally important which contributes to simulating and predicting hydrological phenomena more accurately The previous research mostly focused on applying interpolation methods to construct input meshes for computational models without paying much attention to the influence of interpolation methods on the mesh of model as well as the influence of the selection of the computational CHARACTERISTICS OF THE STUDY AREA Geographically, the Vam Nao river flows in the northeast - southwest direction; It flows through Phu My commune of Phu Tan district and Kien An commune of Cho Moi district, in An Giang province (Figure 1) Vam Nao river has long been an important waterway in the Mekong Delta because of its large basin It is about 7km long, 700m wide, 17m depth and is the only river that connects the Tien river with the Hau river Vam Nao is a river that creates many swirls which are enough to submerge large boats because it has many places where the bottom of the river is about 30m depth Known 59 AGU International Journal of Sciences – 2019, Vol (4), 58 – 73 catastrophes directly affecting the structure of the riverbanks is the status of the riverbed topography which have many fluctuations such as the erosion processes of deep canals distributed close to one side of the river and the strong accretion phenomenon that destroy the horizontal cross-sectional balance of each river section (Nguyen Nha Toan & Cao Van Be, 2010) for its strong currents and having very deep riverbed, Vam Nao river has brought a great amount of valuable seafood from the Mekong river such as giant barb, shark catfish, Giant pangasius, alligator, etc On the other hand, because the river section has strong currents and deep basins, it also creates whirlpool pits including some large and tens of meters deep vortices which cause landslides in the Vam Nao river area In addition to the socio-economic benefits that the Vam Nao river brings, the impacts of climate change on this area have significantly affected the lives of local people Therefore, this area has attracted the attention of authorities and researchers in recent years Vam Nao is located in An Giang province which has a diverse river bottom and double rivers due to the formation of islets and floating dunes in the middle of the river as well as single straight or meandering areas that create many bends One of the causes of geological Figure Vam Nao research area on QGIS 3.0 platform combined with Google Satellite 60 AGU International Journal of Sciences – 2019, Vol (4), 58 – 73 proximal interpolation and inverse distance weighted interpolation Next, we adjust the mesh’s nodes Then, to evaluate the interpolation techniques, we extract isolines of interpolated mesh and compare with Google Satellite maps using QGIS 3.0 software Finally, conclusions have been given This is a research process combining quantitative and qualitative analysis through the following steps (Figure 2): RESEARCH METHODS In this study, we conduct an evaluation of interpolation techniques Interpolation is a method of estimating the value of unknown points within the range of a discrete set of known points There are many different interpolation methods such as Kriging, polynomial regression, spline, etc Within the scope of Blue Kenue 3.3.4 software (Gardin, 2017), we interpolate the river bottom data using techniques including linear interpolation, Create 2D meshes Adjust raw data Interpolate the elevation data for the meshes Extract contour lines Identify differences between contour lines of the meshes Match with Google Satellite Conclude Figure Summary of the research process 3.1 Linear interpolation method the two known points are described by coordinates (x0, y0) and (x1, y1), for a value 𝑥 ∈ (𝑥0 , 𝑥1 ), the line equation can be expressed as follows: The linear interpolation method estimates the missing value between known values Linear interpolation assumes that the rate of change among known values is constant This is a straight line interpolation method in which each line segment connects two consecutive points (Lepot, Aubin, & Clemens, 2017) Therefore, each segment is interpolated independently If 𝑦−𝑦0 𝑥−𝑥0 𝑦 −𝑦 = 𝑥1−𝑥0 (1) From there, the value y along the straight line can be calculated as follows: 61 AGU International Journal of Sciences – 2019, Vol (4), 58 – 73 𝑦 = 𝑦0 + (𝑥 − 𝑥0 ) 𝑦1 −𝑦0 𝑥1 −𝑥0 In which, (x, y) are the coordinates of the interpolated point = 𝑦0 (𝑥1 −𝑥)+𝑦1 (𝑥−𝑥0 ) 𝑥1 −𝑥0 (2) other nearby points, this creates a constant interpolation (Franke, 1982) The proximal interpolation method works based on the principle of comparing the distance distribution between the point to be interpolated and the nearest neighbor of a randomly distributed dataset using the following formula: 3.2 Proximal interpolation method The proximal interpolation method, also called the nearest neighbor interpolation method, chooses the value of the closest known point and does not take into account the values of the 𝑑(𝑥, 𝑦) = ‖𝑥 − 𝑦‖ = √(𝑥 − 𝑦)(𝑥 − 𝑦) = (∑𝑖(𝑥𝑖 − 𝑦𝑖 )2 )1/2 (3) In which, (x, y) are the coordinates of the interpolated point; (xi, yi) are the coordinates of the ith point in the known value set For, (x, y) are the coordinates of the interpolated point; (xi, yi) are the coordinates of the ith point in the known value set 3.3 Inverse distance weighted interpolation method 3.4 Evaluating interpolated meshes using isolines The inverse distance weighted interpolation method is a multivariate technique, the value of an unknown point is determined by combining the linear weight of a set of values of known points, where the weight is a function of inverse distance This method results in the influence of nearby points and ignores unknown far points (Musashi, Pramoedyo, & Fitriani, 2018; D F J G Watson, 1985) Isolines, also called contour lines, are the lines shown on the topographic map of the locus of points on the natural ground, which depend on the ratio of the map to the actual topography Isolines connect points of the same height The sparse or near distance of the contour lines indicates the slope of the terrain being shown; the closer the contour lines is, the steeper the slope is, and vice versa (D Watson, 1992) The value of the unknown point is calculated by the following formula: To identify differences between interpolated meshes, this study compares isolines of 2D meshes applied different interpolation methods using BlueKenue 3.3.4 software (Gardin, 2017) Thereby, we identify unreasonable materials (we focus our attention on grooves, islets and accretion lines) Next, we match the irrational data with the GIS digital map (Dunham, 1962) to determine whether or not the data is suitable for the topography of the riverbed in the study area From there, we evaluate the influence of interpolation methods on generating 2D interpolated meshes 𝑍 ∗ = ∑𝑛𝑖=1 𝑤𝑖 𝑍𝑖 (4) ith Where Zi is the value of the point in the dataset including n points used in the interpolation process, and the wi weight is calculated by the following formula: ℎ𝑖 −𝑝 𝑤𝑖 = ∑𝑛 𝑗=1 ℎ𝑗 −𝑝 (5) Where p is the power parameter; hi is the distance from the ith point to the interpolated point, hi is calculated as follows: ℎ𝑖 = √(𝑥 − 𝑥𝑖 )2 + (𝑦 − 𝑦𝑖 )2 (6) 62 AGU International Journal of Sciences – 2019, Vol (4), 58 – 73 3.5 Selecting calculated adjusting nodes domain of the calculated domain is chosen so that the range must be within the river, and it is a smooth line We limit the selection of the edge of the bend area, where the incoming flow will be blocked (Figure 3) This will affect the outcome of the whole simulation system and Determining the scope of the computational domain is the first step in the process of building an interpolation mesh Based on the measured topograph, the boundary of the Vam Nao river basin was determined The boundary The bends cause obstruction of the flow Figure Boundary selection of a computational domain is not good Based on the collected topograph, the selected boundary of Vam Nao area originated from Chau Phu district (left branch, upper side) and Phu Tan district (right branch, upper side) to Long Xuyen city (left branch, bottom side) and Cho Moi district (right branch, bottom side) (Figure 4) 63 AGU International Journal of Sciences – 2019, Vol (4), 58 – 73 Figure The boundary of Vam Nao area From the identified domain, an overview 2D mesh has been created for the study area by using Blue Kenue software Thereafter, the bad boundary nodes on the inlet and outlet of the flow which may cause negative effects on the simulated results have been adjusted The boundary nodes whose two edges lie on two boundaries simultaneously are bad nodes (two edges are located simultaneously on both the open and close boundaries) (Figure 5) These boundary nodes need to be removed from the computational domain or to be adjusted so that no two simultaneous edges of each node are on two different boundaries, usually these nodes will be swapped with neighboring nodes (Figure 6, Figure 7) (Canadian Hydraulics Centre, 2011) 64 AGU International Journal of Sciences – 2019, Vol (4), 58 – 73 Figure Bad boundary nodes Figure Selecting bad boundary nodes to adjust Figure Permutation of boundary nodes 65 AGU International Journal of Sciences – 2019, Vol (4), 58 – 73 EXPERIMENTS AND RESULTS of Natural Resources and Environment of An Giang Province in 2017 as the interpolated data (Figure 8) In order to evaluate the interpolated meshes, we conducted to create 2D meshed for the study area and use the river topograph of Department Figure The topograph of Vam Nao river in 2017 on Blue Kenue combining with Google Maps The study area is meshed with a distance of 30m using three different interpolation techniques including the inverse distance weighted interpolation, the linear interpolation, and the proximal interpolation The computational domain of the study area is discrete into 31926 triangle elements and 16911 nodes The minimum depth of a 2D mesh describes elements which are near the riverside The maximum depth of a 2D mesh describes elements which are around the erosion pit The interpolated results are tested using contour lines in conjunction with Google Satellite 66 AGU International Journal of Sciences – 2019, Vol (4), 58 – 73 Erosion pit Figure 2D mesh of the study area using inverse distance weighted interpolation method The minimum depth of an element is -1.947m and the maximum depth of an element is -42.028m (Figure 10) when using the linear interpolation method 67 AGU International Journal of Sciences – 2019, Vol (4), 58 – 73 Virtual accretion lines Virtual islet Virtual river trenches Figure 10 Contour lines of 2D mesh using linear interpolation method 68 AGU International Journal of Sciences – 2019, Vol (4), 58 – 73 The minimum depth of an element is -1.980m and the maximum depth of an element is -42.300m (Figure 11) when using the proximal interpolation method The islets are not correct The shorelines are not correct Figure 11 Contour lines of 2D mesh using proximal interpolation method 69 AGU International Journal of Sciences – 2019, Vol (4), 58 – 73 The minimum depth of an element is -1.963m and the maximum depth of an element is -42.132m (Figure 12) when using the inverse distance weighted interpolation method The islets describe actual terrain Actual accretion lines Figure 12 Contour lines of 2D mesh using inverse distance weighted interpolation method The contour lines of the meshes are interpolated with linear method (Figure 10), proximal method (Figure 11), and inverse distance weighted method (Figure 12) have been compared with the river flow topography and the riverbed of Vam Nao area using Google Satellite show that (1) the mesh is interpolated using the linear method causing the appearance of grooves and islets or accretion lines that are not accurate with the actual topographic flow, (2) the proximal interpolation method has improved the interpolated mesh by removing virtual grooves and virtual accretion lines that appear in mesh using the linear interpolation method, (3) however, this method still has islets and accretion lines along the river banks that 70 AGU International Journal of Sciences – 2019, Vol (4), 58 – 73 not match the actual topography of the study area, (4) the inverse distance weighted interpolation method has shown the advantage of interpolation in case of the topograph data is discrete and quite sparse, this method has eliminated virtual grooves and virtual accretion lines as well as described the river topography more accurately than the linear interpolation and the proximal interpolation Thereby, the interpolation methods significantly affect the interpolated results used in the flow simulation In addition, to demonstrate the efficiency of selecting the calculated domain and adjusting mesh’s nodes, we conducted a flow simulation of the study area using the Telemac 2D system version v7p3r1 (Ata, 2018; Mattic, 2018) The experimental results show that the flow through these bad nodes will cause congestion after certain time steps (Figure 13) Therefore, it is proved that the selection of the calculated domain and the adjustment of the boundary nodes play an important role in the construction of the simulation system The flow is blocked Figure 13 Illustration of the blocked flow at the crooked nodes CONCLUSION islets as well as accurately describe the islets and the accretion lines with the actual topography of the riverbed This research used the same type of triangular mesh during interpolation and evaluation The selection of the calculated domain and adjustment of mesh’ nodes play an important role in the construction of the simulation system Generating mesh plays a core role in the model analysis, a good mesh will accurately represent the model being simulated and improve the quality of the simulation system This study has shown the effects of interpolation methods on 2D interpolated results The study also presented the process of selecting and adjusting boundary nodes to minimize errors during the simulation progress Based on the obtained results, the inverse distance weighted method helps improve the efficiency of the 2D interpolated mesh on discrete and fairly sparse topograph through the removal of virtual grooves, accretion lines, and 71 AGU International Journal of Sciences – 2019, Vol (4), 58 – 73 REFERENCES Data in Malang Region CAUCHY, 5(2), 4854 Arun, P V (2013) A comparative analysis of different DEM 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interpolation Thereby, the interpolation methods significantly affect the interpolated results used in the flow simulation In addition, to demonstrate... using three different interpolation techniques including the inverse distance weighted interpolation, the linear interpolation, and the proximal interpolation The computational domain of the study