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

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.

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

1

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

1 INTRODUCTION

Interpolation methods are numerical methods of

constructing new data points within a domain of

a given data (Epperson, 2013 và Nguyen Duc

Nhan, 2016) In this paper, we focus on

presenting spatial interpolation methods (Pav,

2005 và 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

The input data includes the spatial coordinates

and the values of the points These values can

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

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

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

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

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 và

Pebesma và 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

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

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

2 SPATIAL INTERPOLATION METHODS

In this section, we will present the spatial interpolation methods in Telemac 2D and R software (Ata, 2017; Garnero và Godone, 2013;

Dorman, 2014; Pebesma và Graeler 2014 và Dumitru và 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

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

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

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

method is as follows

, where is the number of neighboring points of

the jth interpolated point,

N is the number of interpolated points,

is the value of the jth interpolated point,

is the value of the neighboring point,

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 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

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

Notice: The nearest point interpolation method

has no parameters to adjust

2.4 Spline interpolation method

The spline interpolation method in R software

determines new interpolated values by dividing

the calculated domain into triangles as in the

linear interpolation method After the calculated domain is divided into triangles, the interpolation values are determined through the plane having the following cubic equation

(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

In Arcgis software, the spline interpolation

method is determined by the following

functions

,

or

where R (r) is a function that depends on the

distance from the point of interpolation to

points of data

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

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

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

The interpolation value z at a prediction location has the form

, (5)

values and the weights at the ith location from neighborhood points of the given point The

sum of the weights equals 1, The estimate of z value is unbiased

In the kriging interpolation, the weights

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 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

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

case that we do not have detailed measurement

data and terrain has been changed

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 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 COMPARISON OF INTERPOLATION

METHODS FOR UNDETAILED DATA

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

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

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 2 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)

Figure 1 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)

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)

For the method of analyzing the symmetry of

data, we cut once again the boundary of the

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 right-hand side, Figure 2 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

Figure 2 The Saigon river data is removed with boundary (left hand side) and The Saigon river data do not have boundaries (right hand side)

3.2 Methodology

From the research data (Figure 1), we created

two sets of data including sparse Data 1 (left

hand side, Figure 2) and the remaining data that

need to be interpolated, as shown in Data 2 The

remaining points from Data 2 after extracting

sparse data , we take the coordinates of the

points in Data 2 and name it as Data 3

From sparse Data 1, we applied different

interpolation methods to construct new depth

values of Data 3 Then we get Data 4

The sampled depth values from Data 2 and the

interpolated depth values from Data 4 are

compared and tested in the next section

3.3 Results

To compare the results of the interpolation methods, we calculate the absolute value of the differences between the sampled depth values from Data 2 and the interpolated depth values from Data 4 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 1

Table 1 The results of interpolation methods applied for research data with boundary and without

boundary, which is taken from the Saigon River data

Inverse distance weighted method Average Sample variance

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

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

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

0

In boundary data, NA values fall below 5% of

the data

Notice: For sparse data, we advise against

using these methods as they do 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

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 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 2

Table 2 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

Nearest neighbour method and

Kriging interpolation method

Inverse distance weighted method

and Kriging interpolation method

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

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

4 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

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

Acknowledgments: We would like to express

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

APPENDIX

Research data taken from Saigon River is

attached in the xyz file

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