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 7 (4), 58 – 73

ON CHOOSING SPATIAL INTERPOLATION METHODS FOR DATA WITH NOISE

AND DISCONTINUITIES

Pham Thi Thu Hoa1, Pham My Hanh1

1

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

1 INTRODUCTION

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

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

resolution of 0.01 degrees In the study of

two-dimensional 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

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

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 pre-interpolation 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

2 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

AGU International Journal of Sciences – 2019, Vol 7 (4), 58 – 73

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

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

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)

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

Figure 1 Vam Nao research area on QGIS 3.0 platform combined with Google Satellite

3 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,

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):

Figure 2 Summary of the research process

3.1 Linear interpolation method

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

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:

𝑦−𝑦0 𝑥−𝑥0=𝑦1 −𝑦0

𝑥1−𝑥0 (1) From there, the value y along the straight line can be calculated as follows:

Create 2D meshes

Interpolate the elevation data

for the meshes

Extract contour lines

Identify differences between contour lines of the meshes

Match with Google Satellite

Conclude

Adjust raw data

AGU International Journal of Sciences – 2019, Vol 7 (4), 58 – 73

𝑦 = 𝑦0+ (𝑥 − 𝑥0)𝑦1 −𝑦0

𝑥1−𝑥0=𝑦0 (𝑥1−𝑥)+𝑦1(𝑥−𝑥0)

In which, (x, y) are the coordinates of the

interpolated point

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

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:

𝑑(𝑥, 𝑦) = ‖𝑥 − 𝑦‖ = √(𝑥 − 𝑦)(𝑥 − 𝑦) = (∑ (𝑥𝑖− 𝑦𝑖)2

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

3.3 Inverse distance weighted interpolation

method

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)

The value of the unknown point is calculated by

the following formula:

𝑍∗= ∑𝑛 𝑤𝑖𝑍𝑖

𝑖=1 (4) Where Zi is the value of the ith point in the

dataset including n points used in the

interpolation process, and the wi weight is

calculated by the following formula:

𝑤𝑖= ℎ𝑖−𝑝

∑𝑛𝑗=1ℎ𝑗−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)

For, (x, y) are the coordinates of the interpolated point; (xi, yi) are the coordinates of the ith point in the known value set

3.4 Evaluating interpolated meshes using isolines

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)

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

3.5 Selecting calculated domain and

adjusting nodes

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

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

Figure 3 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)

The bends cause obstruction of the flow

AGU International Journal of Sciences – 2019, Vol 7 (4), 58 – 73

Figure 4 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)

Figure 5 Bad boundary nodes

Figure 6 Selecting bad boundary nodes to adjust

Figure 7 Permutation of boundary nodes

AGU International Journal of Sciences – 2019, Vol 7 (4), 58 – 73

4 EXPERIMENTS AND RESULTS

In order to evaluate the interpolated meshes, we

conducted to create 2D meshed for the study

area and use the river topograph of Department

of Natural Resources and Environment of An Giang Province in 2017 as the interpolated data (Figure 8)

Figure 8 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

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

Figure 9 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

Erosion pit