Henok

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UNESCO-IHE

INSTITUTE FOR WATER EDUCATION

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Reconstruction of 2D riverbed topography from

Scarce data.Case studies: Atrato River (Colombia) and Waal River

(The Netherlands).

Master of Science Thesisby

Henok Endale Abebe

Mentors

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Dedicated to my Family

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Reconstruction of riverbed topography from scarce data.

Abstract

Different numerical models have been developed to analyze morphodynamic evolution

of rivers. However, the functionality of these models is challenged due to lack of

accurate bathymetric data. The current practice to overcome this shortcoming is the use

of simple interpolation techniques to draw the bed topography where the data are

missing, however these methods are inaccurate (Schäppi, et al., 2010). This research

investigates whether the use of simple physics-based morphodynamic model

MIANDRAS and GIS interpolation techniques allow for reconstruction of riverbedtopography when data are scarce. The study is done to two meandering rivers. The first

one is the Waal River (the Netherlands), which has detailed measured bathymetry. The

other one is the Atrato River (Colombia), which is characterized by scarce measured

bathymetric.

Since the Waal River has detailed bathymetry, the performance of each technique is

evaluated on this river with the aim of making generalization on the results so that theycan be used later for the Atrato River. In case of GIS interpolation techniques (Kriging

and Spline), different scenarios are analyzed with different quantity of bed levels. The

data are synthesized by making several sailing tracks and centreline, each one

simulating a single-beam eco-sound bathymetric survey and the cross-sections are

designed as a function of river sinuosity and width. The results show that, when data are

scarce GIS interpolation (Kriging and Spline) tends to smooth out bars and pools,

whereas these are great relevance for identification of river navigability. Furthermore,

t th th d O di K i i i t l ti th d b tt lt

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Acknowledgements

Many people supported me to reach this stage. Therefore, one page could not accommodate

all my acknowledgements if I list the names of all people. For that reason I can only mention

some of them. I hope everybody understands, whereas at the same time I would like to

apologize for those who are not mentioned.

I would like to thank the Netherland University Foundation For International Cooperation(Nuffic) and my employer the Ministry of Water Resources, which is currently called

Ministry of Water and Energy of Ethiopia, for providing me the fellowship and leave of

absence, respectively.

I would like to thank Dr. Alessandra Crosato and Dr. Eric Mosselman, first for showing

interest to work with me and second for their persistent advices and guidance in the course of

the research. My gratitude also goes to Prof. Dano Roelvink for showing interest to be myprofessor, regardless of huge responsibility he had.

Technically how would this research be conducted without data? So that, I would like to

thank, the following people who tremendously cooperated in data acquisition process Dr.

Mohamed Yossef who provided me Waal River bathymetric data, Alejandro Montes and

Jaime Jimenez who provided the Atrato River data and Frank Melman and Sandra Post, who

shared with me some data of the Atrato River which they gathered during their field visit.

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Table of Contents

Contents

Abstract ................................................................................................................................... i

Acknowledgements ............................................................................................................... iii

List of Tables ........................................................................................................................ ix

List of Figures ........................................................................................................................ x

List of Symbols ................................................................................................................... xiii

CHAPTER 1 ................................................................................................................................. 1

1 INTRODUCTION .................................................................................................................. 1

1.1 Context ....................................................................................................................... 1

1.2 Problem statement ...................................................................................................... 1

1.3 Research questions ..................................................................................................... 2

1.4 Objectives ................................................................................................................... 2

1.5 Methodology .............................................................................................................. 3 1.5.1 General approach .................................................................................................... 3

1.5.2 Literature review .................................................................................................... 3

1.5.3 Data collection and analysis ................................................................................... 3

1.5.4 MIANDRAS and GIS applications ........................................................................ 3

1.5.5 Analysis of the results and integration with other studies ...................................... 3

1.5.6 Reporting ................................................................................................................ 3

CHAPTER 2 ................................................................................................................................. 5

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3.2.4 Reach- averaged water depth at bank-full condition ............................................ 27

3.2.5 Sediment Characteristics ...................................................................................... 28

3.2.6 Sinuosity ............................................................................................................... 28 3.2.7 Degree of non linearity of sediment transport formula ........................................ 29

3.2.8 Bar mode .............................................................................................................. 30

3.2.9 Theoretical bar wavelength .................................................................................. 31

3.2.10 Observed bar wavelength ................................................................................. 33

3.3 Atrato River .............................................................................................................. 36

3.3.1 Bank-full discharge .............................................................................................. 37

3.3.2 Chezy's coefficient and sediment grain size ......................................................... 37

3.3.3 Bank-full width .................................................................................................... 38

3.3.4 Reach-averaged depth at bank-full conditions ..................................................... 38

3.3.5 Sinuosity ............................................................................................................... 39

3.3.6 Degree of non linearity of sediment transport formula ........................................ 39

3.3.7 Bar mode .............................................................................................................. 39

3.3.8 Theoretical bar wavelength .................................................................................. 40

CHAPTER 4 ............................................................................................................................... 45

4 APPLICATION OF GIS INTERPOLATION METHODS TO THE WAAL RIVER...................... 45

4.1 Introduction .............................................................................................................. 45

4.2 Overview of interpolation methods .......................................................................... 45

4.3 Methodology ............................................................................................................ 46

4.4 Scenario one ............................................................................................................. 47

4.4.1 Statistical summary of synthesized input bed elevation....................................... 47

4 4 2

R lt d di i 48

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5 COMPUTATION OF WAAL RIVERBED TOPOGRAPHY USING MIANDRAS ..................... 73

5.1 Introduction .............................................................................................................. 73

5.2 Description of MINDRAS ....................................................................................... 73

5.3 Input and the calibration coefficients ....................................................................... 74

5.4 MIANDRAS application on the Waal River ............................................................ 74

5.4.1 Input data .............................................................................................................. 74

5.4.2 Results and discussions ........................................................................................ 76

5.4.3 Summary .............................................................................................................. 81

CHAPTER 6 ............................................................................................................................... 83

6 APPLICATION OF GIS AND MIANDRAS FOR ATRATO RIVER...................................... 83

6.1 GIS application for Atrato River .............................................................................. 83

6.1.1 Cons and pros of existing measured bathymetry of Atrato River ........................ 83

6.1.2 summary ............................................................................................................... 85

6.2 Application of MIANDRAS on the Atrato River .................................................... 86

6.2.1 Introduction .......................................................................................................... 86

6.2.2

Assumptions ......................................................................................................... 86

6.2.3 Input parameters used in MIANDRAS for Montano reach ................................. 86

6.2.4 Results and discussions ........................................................................................ 87

6.2.5 Summary .............................................................................................................. 88

CHAPTER 7 ............................................................................................................................... 89

7 CONCLUSIONS AND RECOMMENDATIONS........................................................................ 89

7.1 Conclusions .............................................................................................................. 89

7 1 1 C l i th li ti f GIS i t l ti t h i 89

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List of Tables

Table 2-1. The difference between measured and interpolated cross section coordinate

(Schäppi et al., 2010.). ..................................................................................................... 10

Table 2-2. The difference between measured and interpolated statistics (Schäppi et al., 2010).

.......................................................................................................................................... 13

Table 2-3. TSS results for all study areas (Merwade, 2009). ................................................... 14

Table 2-4. Calibration coefficients used for different rivers .................................................... 23

Table 3-1.Some characteristics of Waal River (Havinga et al, 2006). ..................................... 28

Table 3-2. Bar wavelength summary. ...................................................................................... 36

Table 3-3. Roughness coefficient and sediment size of Montano reach. ................................. 38

Table 3-4. Summary of theoretical bar wavelength. ................................................................ 43

Table 3-5. Comparison of Waal and Atrato Rivers in the study areas at bank-full conditions.

.......................................................................................................................................... 43

Table 4-1. Statistical summary of synthesized input bed elevation (scenario one). ................ 47

Table 4-2. Measured and interpolated centreline statistical summary (Scenario one). ............ 48

Table 4-3. Measured and interpolated cross-section one, statistical summary (Scenario one).49

Table 4-4. Measured and interpolated cross-section two, statistical summary (Scenario one).

.......................................................................................................................................... 50

Table 4-5. Measured and interpolated cross-section three, statistical summary (Scenario one).

.......................................................................................................................................... 51

Table 4-6. Comparison of cross-sectional area (Scenario one). ............................................... 52

Table 4-7. Statistical summary of synthesized input bed elevation (scenario two). ................ 57

T bl 4 8 M d d i l d li b d l i i i l ( i

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List of Figures

Figure 1-1. General methodology flow chart. ............................................................................ 4 Figure 2-1. Recent bed topography data surveyed in the Waal River (Crosato Lecture

material on Riverbed form, 2010). ..................................................................................... 5

Figure 2-2. Atrato River recently surveyed bed topography. ..................................................... 6

Figure 2-3. Atrato River location map. ...................................................................................... 7

Figure 2-4. Location of Waal River study reach (Hetzer, 2005). ............................................... 8

Figure 2-5. Problem of triangulation of cross section data if no break line is used (upstream

view), small river structures visible in cross section are extended along river, which

result in longitudinal stripe structure. (Schäppi et al., 2010). ............................................ 9

Figure 2-6. Comparison of different interpolation method (Schäppi et al., 2010). Z coordinate

corresponds to bed elevation............................................................................................ 10

Figure 2-7. Intersection of break line (Schäppi et al., 2010). .................................................. 12

Figure 2-8. Method of lateral and longitudinal interpolation (Schäppi et al., 2010). .............. 12

Figure 2-9. Trend surface for King Ranch reach (Merwade, 2009)......................................... 15

Figure 2-10. Cross section for King Ranch reach: a is at x location for Fig 2-9b, b: is y

location for Fig 2-9b and Z representing the base condition (Merwade, 2009). .............. 16 Figure 2-11. Semi-Variogram of Bed topography trend and residual surface for King Ranch

Reach a Bed topography and trend in all direction, b: bed topography and trend in s-

direction, c: residual in all direction, d: residual in s-direction and Z stand for base

case.(Merwade, 2009). ..................................................................................................... 16

Figure 2-12. Sinusoidal transverse variation of flow velocity and water depth in curved

channel (Crosato, 2008). .................................................................................................. 20

Figure 2-13 coordinate system of the model where s and n are orthogonal (Crosato, 2008) .. 20

Fi 2 14 Ri I ll h b d l l i h f h fi b d

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Figure 4-7. Measured and interpolated cross-section two (Scenario one). .............................. 51

Figure 4-8. Measured and interpolated cross-section three (Scenario one). ............................ 52

Figure 4-9. Cross-sectional areas of different cross-section. ................................................... 53 Figure 4-10. Reach-scale comparison of measured and interpolated data (Scenario one). ..... 53

Figure 4-11.Difference between measured and Ordinary Kriging in meandering reach

(scenario one). .................................................................................................................. 54

Figure 4-12. Difference between measured and Regularized Spline in meandering reach

(scenario one). .................................................................................................................. 54

Figure 4-13. Difference between measured and Tension Spline in meandering reach (scenario

one). .................................................................................................................................. 55

Figure 4-14. Difference between measured and Ordinary Kriging in straight reach (scenario

one). .................................................................................................................................. 55

Figure 4-15. Difference between measured and Tension Spline in straight reach (scenario one).

.......................................................................................................................................... 55

Figure 4-16.Difference between measured and Regularized Spline in straight reach (scenario

one). .................................................................................................................................. 56

Figure 4-17. Reach-scale interpolation error in terms of percentage of water depth (Scenario

one). .................................................................................................................................. 56 Figure 4-18. Conceptually surveyed bathymetry, for scenario two. ........................................ 57

Figure 4-19. Measured and interpolated centre line for scenario two. ..................................... 58

Figure 4-20. Measured and interpolated cross-section one for Scenario two. ......................... 59

Figure 4-21. Measured and interpolated cross-section two for scenario two. .......................... 60

Figure 4-22. Measured and interpolated cross-section three for scenario two. ........................ 61

Figure 4-23. Reach-scale difference between measured and interpolated bed elevations

(scenario two). .................................................................................................................. 61

Fi 4 24 Diff b d d O di K i i i d i h (

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Figure 4-39. Difference between measured and Ordinary Kriging, in straight reach (scenario

four). ................................................................................................................................. 71

Figure 4-40.Difference between measured and Regularized Spline, in straight reach ............ 71 Figure 4-41. Difference between measured and Tension Spline, in straight reach (Scenario

four). ................................................................................................................................. 72

Figure 4-42. Reach-scale interpolation error in terms of percentage of water depth (scenario

four). ................................................................................................................................. 72

Figure 5-1. General data preparation and processing methods . .............................................. 75

Figure 5-2. Measured and computed bed level in cross-section one. ...................................... 77

Figure 5-3. Measured and computed bed level in cross-section two. ...................................... 77

Figure 5-4. Measured and computed bed level in cross-section three. .................................... 78

Figure 5-5. Measured and computed bed level in cross-section four. ...................................... 78

Figure 5-6. Measured and computed bed level in cross-section five. ...................................... 79

Figure 5-7. Location of cross-sections on which to check the performance of MIANDRAS. 79

Figure 5-8. Wetted cross-sectional area for three cross-sections. ............................................ 80

Figure 5-9. Reach-scale comparison between measured and reconstructed bed topography. . 81

Figure 6-1. Typical bathymetry survey of Atrato River. .......................................................... 84

Figure 6-2 Recommended way of Bathymetry survey for Atrato River .................................. 84 Figure 6-3. Result of Kriging interpolation method: (a) using existing planimetry and b after

the river planimetry changed in to straight channel. ........................................................ 85

Figure 6-4 Comparison of measured and interpolated bed level at cross-section 47 ............... 87

Figure 6-5. Reach-scale comparison: (a) MIANDRAS bed level and (b) is Kriging

interpolation. .................................................................................................................... 88

Figure 7-1 Comparison of MIANDRAS and Ordinary Kriging method of interpolation ....... 90

Figure 7-2 MIANDRAS output for Montano reach ................................................................. 90

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List of SymbolsAbbreviation Description Unit

A Area m2

AK Ordinary Kriging with anisotropy -

B Bank-full width m

b Degree of none linearity in sediment

transport formula -

C Chezy roughness coefficient m0.5 /s

d50 Mean sediment size m

d90 Sediment size of 90 % passing m

DTM Digital terrain model -

g Acceleration due to gravity m/s2

GIS Geographical information system -

h0 Reach average normal depth (Bank-full depth) m

i Longitudinal bed slope m/m

IDW Inverse distance weighted -

m bar mode -

NAP North Amsterdam level -

NN Natural neighbour -

OK Ordinary Kriging -

QW B k f ll di h

3

/

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Chapter 1

1 Introduction

1.1 Context

Different numerical models have been developed to understand the morphodynamic evolution

of rivers. However, the functionality of these models is challenged due to different factors.

One of these factors is the availability of bathymetric data. Detailed bathymetric data are

rarely available. The reasons for this are many, to mention some of them, lack of funding to

do extensive survey, difficult accessibility of rivers etc.

The current practice to overcome this shortcoming is using of simple interpolation techniques,

but these are inaccurate (Schäppi, et al., 2010). This is because meandering rivers are

characterized by the presence of alternate and large point bars. To overcome this, it isimportant to study the effectiveness of combining field data and modelling tools. One

possibility is the application of GIS interpolation techniques and another possibility is the

application of the MIANDRAS model.

This research is aimed to investigating the best way to reconstruct meandering riverbed

topography. Two tools are used: GIS interpolation methods and the model MIANDRAS. The

functionality of these tools is first checked on a data-rich study area which is the Waal River

( h N h l d ) d h h i i d f hi d i li d d

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case, the 2D bed topography is constructed using different interpolation techniques. However,

bed topography developed by simple interpolation technique is inaccurate (Merwade, 2009).

This is because rivers bed topography have special trend which is governed by themorphodynamic of the locality. For instance in the meandering rivers cross-section, the bed

elevation is lower in the outer bend (pool) than in the inner bend (bar) (Crosato, 2008).

Furthermore most of the time the error incurred due to the method of interpolation used is not

quantified. As a result it would be nice to investigate best method to reconstruct meandering

river 2D bed topography. Furthermore, depending on the quantity of measured data it would

be nice to have a guideline which tells which method to use for reconstruction of meandering

riverbed topography.

1.3 Research questions

The issue addressed in the previous section can be synthesized into the following research

questions.

1. What is the optimum sailing track for meandering rivers bed topographic surveys in

case of using single a beam echo sounder (Atrato and Waal River)?

2. To what extent can MIANDRAS be used for reconstruction of meandering rivers bed

topography?

3. To what extent can GIS interpolation techniques be used for reconstruction of

meandering riverbed topography?

4. What is the minimum amount of data required, for a sufficiently accurate

reconstruction of meandering rivers bed topography, using GIS interpolation

h i ?

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1.5 Methodology

1.5.1 General approach

To address all the research questions and objectives, the work is organized step wise. As a

result, the following methodology is used.

1) Literature review.

2) Data collection and analysis.

3) MIANDRAS and GIS application.

4) Analysis of the results and integration with other studies.

5) Reporting.

1.5.2 Literature review

In order to understand the theoretical background of the topic, different literature is reviewed.

To take advantage of previous findings, the results of different researchers are reviewed. For

instance, the calibration coefficients of MIANDRAS model are not easy to fix. As a result, tostart with calibration, the values that have been used in other similar rivers are used. This

helps in optimization of the calibration coefficients.

1.5.3 Data collection and analysis

Data analysis is integrated in different chapters. For instance, the bed topography data

analysis is reported in Chapter 4 and the sediment, bank-full discharge analyses is reported in

h

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Waal

River

MIANDRAS application

Summary

of the

results

Run one

Run two

Run three

Literature review

Data analysis

GIS (Kriging and

Spline

interpolation)

Scenario one

Scenario two

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Chapter 2

2 Literature Review

2.1 Introduction

To meet the objective of this research, it is imperative to review the works that have been

done before. Literature review is one of the ways, to organize the experience from other

studies and incorporate them in new studies. Furthermore, it gives an idea how to improve the

effectiveness of each of the processes that may play a role in the course of the study.

This chapter starts with description of study areas and goes to summarization of experience

gained in previous applications of GIS interpolation techniques. Furthermore, the basic

assumption and mathematical equation of MIANDRAS are described. The last but not the

least, some of the experience in using MIANDRAS model is discussed.

2.2 Description of the study areas

The study is carried out on two rivers: the Atrato River (Colombia) and the Waal River (the

Netherlands). Both rivers are meandering so the model MIANDRAS can be used. These

rivers differ in availability of data, in particular bed topography data. As shown in Figure 2-1,

the Waal River has detailed bed topography data, whereas the Atrato River has only limited

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Figure 2-2. Atrato River recently surveyed bed topography.

2.2.1 Atrato River

The Atrato River is located in the north-western part of Colombia. It rises on the western

slopes of the Cordillera Occidental of the Andes and flows northward to finally discharge its

water in the Gulf of Urabá of the Caribbean Sea. It is 670 km long and its maximum

Flow

http://www.britannica.com/EBchecked/topic/95846/Caribbean-Sea

http://www.britannica.com/EBchecked/topic/95846/Caribbean-Sea

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reconstruction of 2D riverbed topography. The experience that will be gained in this river will

give idea how to reconstruct the 2D bed topography of river which has scarce data.

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sectional data and the DTM points located outside channel (Schäppi, et al., 2010). The

accuracy of this method is low and highly depending on the spacing of the measured cross-

sections in lateral and longitudinal direction. As a result, this method is not reliable in riversthat have complex bed topography. It is recommended to introduce break lines by hand in

order to generate a realistic image of the river banks and bars. The result of interpolation with

and without break lines is shown in Figure 2-5.

Without break

line

Flow

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of abrupt changes in the slope in case of randomly spaced data. Figure 2-6 shows the result of

different interpolation techniques. The z-coordinate is the value of bed elevation.

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2.3.3 Description of algorithm

The algorithm replaces the grid point of water surface DTM with the corresponding riverbedelevation, which is interpolated from cross-sectional data. As input, the program demands the

GIS raster file of water surface DTM and the coordinates of the cross section point. To

delineate the boundary of the point to be replaced two break lines at the bank of the river need

to be defined. The program checks for each DTM grid point the location between these two

break lines. If it is located at the boundary, it will be replaced with new interpolated

elevation.The accuracy of the interpolation is improved by introducing additional break lines.

The interpolation of cross-sectional points into grid points is done in two steps, first in lateral

and second in longitudinal direction. For each grid point, the cross-sectional data points on theneighbouring profiles are used and the longitudinal direction is computed from the upper and

lower break lines. In order to perform a linear interpolation in longitudinal direction, the

corresponding locations on the neighbouring cross sections need to be identified. Figure 2-7

illustrates how break lines and cross sections are created.

2.3.4 Lateral interpolation

The lateral point interpolation for the coordinate of point P i*,j and Pi*,J+1 of two neighbouring

cross-sections is computed by linear interpolation of the Z coordinate (elevation) of cross

sectional data point to X,Y-coordinate of Pi*,j and Pi*j+1 (for more information see Figure 2-8):

Equation 2-1

Where shows the interpolated Z coordinate in cross section J, t and (t-1) are horizontal

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Figure 2-7. Intersection of break line (Schäppi et al., 2010).

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2.3.5 Longitudinal interpolation

For longitudinal direction stream wise a linear interpolation of bed elevation of correspondingpoints P j and P j+1on the cross section of grid point G is calculated according to the following

equation:.

Equation 2-3

Where shows the interpolated Z coordinate (elevation) in grid point G. S and (1-S) the

fraction of horizontal distance between grid point G and intersection point and 1 J iP onneighbouring cross section, Zi*,j and Zi*,j+1

is elevation of intersection point on cross section j

and j+1 respectively. For more information see Figure 2-8.The parameter S calculated as

follow.

Equation 2-4

2.3.6 Findings

Schäppi et al method has been applied on the Thur River (Canton Thurgau, Switzerland) in

the frame work of a restoration project. However differences were observed between

measured and interpolated bed elevations for exposed, submerged and entire riverbed. The

statistical value is presented in Table 2-2

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In order to mimic the meandering nature of riverbed topography, data stored in Cartesian

coordinate system is transformed to coordinate system defined by s-axis along the flow

direction and n-axis across the flow direction which is perpendicular to s-axis. Spatialinterpolation is under taken for each data set in (s, n) coordinates for two variables of zi and ei,

where zi is bed topography at any point (ith

) measured plus ei is the corresponding residual

after fitting the trend function. In the study seven interpolation techniques is used (IDW, RS,

TS, TG, NN, OK and AK).

After spatial interpolation of points Zi in testing data set the resulting surface area validated

with comparing the observed bed topography. In the validation data set as well as quantifying

the accuracy of interpolation by calculating RMSE (root mean square error).

Equation 2-5

Where Zi is measured at ith

location, Z^is estimate of Zi and n is the total points in validation

data set. To make clear the content of the following section the terminology is abbreviated as

follow:BT: Beta trend. GT: Global polynomial trend

LT: Local polynomial trend.

The traditional technique such as student t-test and TSS (total sum square) performed for

comparing the trend surface. The result of TSS shown BT and LT capture more than 80% of

the variance in measured bed topography for all data set on other hand GT captured only 50%

of the variance in measured bed topography for most data sets. The lower TSS for global

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interpolation of residuals. Further more Experimental Semi-variogram plot of BT, LT, GT

and the corresponding residual showed how the spatial correlation in the bed topography is

affected see Table 2-3. The result of Semi-variogram experiment for King Ranch reach isconsistent with the TSS and t-test statistical analysis. Both BT and LT produce similar Semi-

variograms, which suggest that these two surfaces have similar spatial distribution of semi-

variance; on other hand GT is unable to capture most of the variance in the bed topography.

Semi -variogram plot of bed topography and trend surface in s-direction have higher range

compared to other direction. It is due to smaller Variations or higher correlation in bed

topography along the flow direction. Similarly Semi-Variograms of residual showed that BT

and LT capture about 75% of Semi-Variance in the bed topography where as residual from

GT produce a Semi-Variogram that looks similar to observed bed topography in terms of range and sill.

The outcome of this method showed that exclusive treatment of spatial trend while

interpolating riverbed topography provides better result moreover it is applicable to many

rivers. On the other hand it is motioned that more research needed to be done in order to see

some other factor that could affect the results.

Bathymetry of the entire reach.

Close up view of the surface in a.

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Figure 2-10. Cross section for King Ranch reach: a is at x location for Fig 2-9b, b: is y location for Fig

2-9b and Z representing the base condition (Merwade, 2009).

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the bed shear stress can be related to the depth-averaged flow velocity using Chezy's equation.

The final equations are obtained by linearization and simplification.

Starting equation for water flows are:

Equation 2-6

Equation 2-7

Equation 2-8

Where

v depth-averaged transverse velocity (m/s)

h water depth (m)

Zw water level (m)

Rc radius of curvature of the channel centreline (m), positive if the centre of

curvature lies at smaller n

g acceleration due to gravity (m/s2)

Cf = g/C2

friction factor (-), in which C is the Chezy coefficient (m1/2/s)

The water level Zw is eliminated from Equations 2-6 and 2-7 by cross differentiation, applying

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The basic equations of the model, Equation 2-8 and 2-11 are non-linear. Non-linear equations

are complex and difficult to solve. The way to simplify them is by linearization, which can be

done by applying a perturbation method. Linear behaviour is simulated at a point whichmeans representing the reference condition or along a small interval. This means that the

linearized equation strictly apply to the situations that are close to the reference conditions

(Crosato, 2008).

The simplification of the model by linearization require the definition of the reference

conditions, which are described by zero order equations, perturbation parameter ( ) as

indicator of the order of magnitude and the effect of the perturbation, which will be described

by first-order equations. In case of mildly curved meandering rivers, it can be assumed thatthe unperturbed system describes the flow an infinitely long straight channel with constant

discharge, for which the following conditions are valid.

Equation 2-12

As a result, the zero-order set of equations describe the condition for uniform flow in which

the values of the variables correspond to their reach-average values and the first-order set of equations describes the small deviation from the normal flow condition which is caused by

small disturbance to the flow. All variables are given by the sum of two terms: reach average

value (zero-order) plus perturbation term (first-order).For instance, some of the variable are

described as follow.

Equation 2-13

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channel centreline ( , which means close to the reference situation, the angle between

the s-directions and the sediment transport direction is small ( Crosato,2008)

Equations 2-8 and 2-11 are linearized by substituting each variable with the sum of its zero-

order and first-order. All non-linear terms and all terms of O () are disregarded. Linearizedequations are the following

Equation 2-15

Equation 2-16

Where which is called curvature ratio

The above linearized equations work when bar mode is equal to one, which means this

equation is meant for analyse of meandering river.

2.4.2 Assumptions

MIANDRAS computes the longitudinal profile of the near bank excess flow velocity (U) and

water depth (H) with respect to the reach averaged values; u0 and h0, respectively see Figure

2-12. Excess H and U are caused by the local channel curvature and by upstream disturbances,

such as change of channel curvature, which under certain conditions may lead to the

formation of alternate bars in the river channel. The local bank erosion rate is assumed to be a

function of both U and H. The transverse profile of water depth and velocity are assumed to

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Figure 2-12. Sinusoidal transverse variation of flow velocity and water depth in curved channel

(Crosato, 2008).

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Figure 2-14. River Irwell the bed level with respect to water surface the figure above and below the

satellite Image (Crosato, 2008).

Other study was carried out for Tigris River in Iraq (Bakker & Crosato, 1989). The objective

of the study was to study the planimetric changes and the possibility of cut-off occurrence

near the intake of Al-Shemal hydropower station. The results showed that the 2012 predicted

planimetric changes gives satisfactory result when compared with 2008 satellite image. As

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inaccurate, because the number of computational grid cells used in a cross section was not

enough. Increasing number of grid cells per cross-section demand large computational times.

For this reason, Delft3D is not appropriate to study long river reaches. For this caseMIANDRAS provides better results and is much simpler to apply as well as faster.

Figure 2-16. Blue Nile River near the city of Sinja (Singa Sudan): A bove bed topography computed

using 2D model (Delft3D). Below bed topography computed using MIANDRAS for a discharge of

2000m3 /s (Ali, 2008).

Calibration coefficients that have been used for different rivers in the world are summarized

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Table 2-4. Calibration coefficients used for different rivers

(These values are gathered from the work of various students)

ParameterIrwell

River

Geul

River

Allier

River

Blue NileRiver

( b/n

Rosereire

s and

Sinnar)

Dhaleswari

River

Tigris

River

Country UKNetherla

ndsFrance Sudan Bangladesh Iraq

Bank-full width (m) 11.8 8 65 400 250 300

Bank-full discharge(m3 /s) 45 22 325 8000 1300 -

Bank-full depth (m) 1.45 2.26 2.61 - - -

Chezy coefficient (m1/2

/s) 22.53 20 47 50 65 50

Valley slope (m/m) 0.0053 0.40E-2 8.33E-4 1.2E-04 -0.64E-

3

Sediment

size (mm)

D50 11.2 0.0250 0.005 0.15 0.160 0.03

D90 30.5 0.0280 0.03 0.25 0.270 0.06

t i o n c o e f f i c i e n t s

Eu

Natural

bank 1.0E-5 0.4E-05 - 1.0E-7 0.9575E-07 0.80E-04

Vegetated

bank 5.0E-6 - - - - -

Eh

(s-1

)

Natural

bank 2.0E-5 0.6E-05 - 9E-9 1.858E-06

0.22E-

04

Vegetated

bank 1.0E-5 - - - -

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Chapter 3

3 Basic characteristics of the study areas

3.1 Introduction

In chapter two, previous work in reconstruction of 2D riverbed topography reviewed. To start

with this new method of reconstruction of river bathymetry, first it is important to analyses

basics characteristics of the study reaches.

In this chapter, the characteristics of the study areas are described and also most of the input

parameters of MIANDRAS are computed. Computing of these parameters will help in

implementing the experience gained from this research to other similar rivers, further more it

also assist in adopting MIANDRAS calibration coefficients from previous work.

In Section 3.2 the basic characteristics of Waal River are analysed, such as bank-fulldischarge, bank-full width, theoretical and measured bar wavelength etc. In Section 3.3

similar analyses is done for Atrato River. At the end of every section the results of analyses

are summarized with tables, this highly assists while referring the experience of this research

to implement for other similar rivers.

3.2 Waal River

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3.2.1 Bank-full discharge

The bank-full discharge is tried to estimate from the measured cross-section, however thecross-section was measured during low flow so that could not represent the bank-full

condition. The initial value is referred from the work of Zeekant (1983).According to Zeekant

the bank-full discharge is Q10 which is round up value of 2150 m3 /s. However, after running

the MIANDRAS it is realize that the bank-full discharge used was a bit lower, as a result it is

adjusted by trial and error to the reasonable value of 2500 m3 /s.

3.2.2 Chezy's coefficient

The Chezy's coefficient is taken from the work of Hetzer (2005), as shown in Figure 3-2,

Chezy coefficient varies in the study reach. The variation is highly pronounced at Nijmegen

and Erlecom which are the area embedded with bank and bed protection structures,

furthermore Figure 3-3 show these places are characterized by a sharp bend; which attribute

for lower Chezy coefficient (rough). Taking in to account the variability in the reach,

representative value of 47 m1/2

/s is taken (for bank-full discharge).

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Figure 3-3. Locations with lower Chezy roughness coefficient.

3.2.3 Bank-full width

Figure 3-4 shows the bank-full widths measured from Google Earth. The bank-full width is

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Equation 3-1

Where

C Chezy's coefficient (m1/2

/s)

A cross-sectional area (m2)

Qw bank-full discharge (m3 /s)

h0 reach average normal depth (m)

i longitudinal bed slope (-)

B bank-full width (m)

Rh hydraulics radius (-)

By rearranging Equation 3.1, the following equation obtained.

Equation 3-2

By substituting the value of each parameter in Equation 3.2, reach-averaged normal depth

computed as follow.

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Meandering reach Straight reach

Figure 3-5.General diagram of sinuosity.

3.2.7 Degree of non linearity of sediment transport formula

The sediment size indicated in Table 3-1 tells the study reach is characterized by very coarse

sand. As a result, the most suitable sediment transport formula is Meyer-Peter and Muller.

According to Meyer-Peter and Muller, the degree of non linearity of sediment transport with

respect to velocity (for bank-full discharge), can be calculated using the Equation.3.3

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By substitute the result of Equation 3-5 in to Equation 3-4 the following result obtained:

By substituting the value of all unknown parameter in Equation 3-3, the degree of non-

linearity computed as follow.

N.B the value of θ 0 (Shield parameter) is computed from Equation 3.12

For the bank-full discharge the degree of non linearity, b is 5, which means that sediment

transport is far from initiation of motion (a lot of sediment is transported).

3.2.8 Bar mode

The bar mode tells what type of bars exist in the river cross-section. The bar mode can be

computed using simple physics-based predictor equation (Crosato and Mosselman, 2009).

This equation works well for rivers that have width-to-depth ration up to 100. At bank-full

discharge the Waal River has width-to-depth ratio of 33, which means that it is possible to use

Crosato and Mosselman's, equation (Equation 3-6).

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3.2.9 Theoretical bar wavelength

There are different methods to calculate the bar wavelength. One of the methods is physics-based, which consists of equations that relate different properties of the river. The equation is

developed by relating the physical process which involve in the river. The other method is

purely empirical which is developed based on field experience or laboratory experiment.

Physics-based calculation

According to Struiksma et al, (1985) Equation 3-7 can be used to calculate steady bars

wavelength. This formula is a function of bar mode so that it works for all types of bar

Equation 3-7

Where

b the degree of non-linearity (-) which is equal to 5 for Waal River

λw the stream wise adaptation length for perturbation in transverse direction (m).

λs the steam wise adaptation length for perturbation in the cross-sectional riverbed (m)

Profile.

Lp bar wavelength (m)

To solve Equation3-7 first λs and λw need to be computed. These two parameters can be

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Equation 3-11

Where E is a calibration coefficient and 0 is the reach-averaged Shield parameter. For Waal

River E can be taken as 0.5 (Crosato experience).The Shield parameter can be calculated by

the following formula: Equation 3-12

Where u is flow velocity and calculated as follow

Equation 3-13

From Equation 3-1 and Equation 3-2 Qw, B, and h0 are 2500 m3 /s, 250 m and 7.68 m

respectively. By substituting the value of each unknown in Equation 3-13 the flow velocity

computed as follow.

Similarly by substituting each of the parameter in Equation 3-12, Shield parameter computed

as follow

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Empirical formula (alternate bars only)

Ikeda (1984) checked different formula for the estimation of migrating alternate barwavelength. The formulas are classified based on Froude number (F).

Equation 3-14

The Froude number for Waal River is less than 0.8. According to Ikeda, for Froude number

less than 0.8 following formula can be used to estimate the alternate bar wavelength:

Equation 3-15

Where Cf is the resistance coefficient and calculated in Equation 3-9, B is bank-full width and

h0 is mean depth, all are computed in Equation 2. By substituting the value of each unknown

in Equation 3-15, the wavelength of migrating alternate bar computed as follow.

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the longitudinal profile is approximately set 50 m off-set from left bank (facing downstream).

Due to continuous existence of groynes and other bank protective structures (Figure 3-9), it

was difficult to distinguish the bars (Figure 3-6 and Figure 3-8). The existences of thesestructures cause erosion and deposition in their vicinity which ultimately makes difficult to

distinguish the bar. Furthermore this process is dynamic which varies with discharge.

However the bar wavelength is approximately measured from Figure 3-7 and Figure 3-8 . The

number of bars observed from Figure 3-6 is approximately 8 to 9 within 18.3 km which

means the average wavelength of these bars is 2 to 2.3 km. Since this area is located in

meandering reach the type of bar that is observed is alternate bar which is superimposed on

point bars, the wavelength of point bars is approximately 6.1 km. The approximation is done

by taking the three top bars in Figure 3-6. In straight reach the wavelength of free alternatebars is computed similar way and it becomes 3.6 km (Figure 3-7). However, these values are

approximated value and may not help to make comparison with the theoretical one, because

the theoretical method may not work well for trained river.

1.00

2.00

3.00

4.00

5.00

e v e l e l e v a t i o n ( N A P ) ( m )

Approximated

barswavelength

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Figure 3-7. Longitudinal profile from chainage 2.1 to 40 km, straight reach.

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

20.1 22.1 24.1 26.1 28.1 30.1 32.1 34.1 36.1 38.1

B e d L e v e l e l e v a t i o n ( N A P ) ( m )

Longitudinal length from upstream to downstream(km)

Approximated

bar wavelength

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Figure 3-9. Location of training work in Waal River (Havinga et al, 2006).

Measured and theoretical bar wavelengths are summarized in Table 3-2. Furthermore, because

of aforementioned reason the magnitude two methods may not match.

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`

Montaño

reach

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two samples do not differ much as a result for this research the values of samples taken in

1989 are used. The Chezy coefficient used is 47 m1/2/s. This is because it is used as a

calibration coefficient to optimize the water depth.

Table 3-3. Roughness coefficient and sediment size of Montano reach.

Sample Date D50 (mm) D90 (mm)

Normal

depth (m)

Manning

Coefficient

Chezy

coefficient

(m1/2/s)

High flow 13-12-88 0.332 0.500 10.46 0.0328 45

Low flow 10-03-89 0.336 0.404 7.522 0.0289 50

NB the Chezy coefficient is derived from manning coefficient and the normal depth by

assuming the channel is wide.

3.3.3 Bank-full width

The length of the river is long and as a result it was not easy to get one representative bank-

full width. It was necessary to measure some representative widths and then make average to

get one representative width. The following widths are measured from Google Earth 216 m,191.92 m, 249.27 m, 145.47 m, 272.08 m, 246.99 m, 186.74 m and 204.45 m. The bank-full

width is taken as the average of the aforementioned widths and becomes 214 m. However,

there is uncertainty with this value because the approximation is highly subjective and the

bank-full width observed in Google Earth might not match with real one.

3.3.4 Reach-averaged depth at bank-full conditions

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Figure 3-11. Atrato River Montano reach bed slope map.

3.3.5 Sinuosity

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

0 10000 20000 30000 40000 50000 60000 70000 80000 90000

S l o p e ( m / m )

longitudinal distance (m)

Slope Average slope

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Equation 3-19

Where

m the bar mode (-)

g acceleration due to gravity (m/s2)

b degree of non-linearity of sediment transport formula (-).

B bank-full width (m)

Qw bank-full discharge (m3 /s)

C Chezy's coefficient (m1/2

/s)

relative density of sediment under water (-) and it is 1.65

The bar mode computed as follow:

A bar mode is equal to one, which indicate the formation of alternate bars in addition to point

bars. This is an important information for the analysis of the straight reach, where

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profile.

Lp bar wavelength (m)

In order to solve Equation3-20 first λs and λw need to be computed. These two parameters

can be derived as follow (Crosato and Mosselman, 2009):

Equation 3-21

Equation 3-22

The following result is obtained by substituting the value of h0, and Cf in to Equation 3-21.

Equation 3-23

Where f ( ) account the effect of gravity on the direction of sediment transport over

transverse bed slope.

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By substituting the values of each parameter in Equation 3-23, λs Computed as follow.

By substituting the value of each parameter in Equation 3-20, the bar wavelength computed as

follow.

Empirical formula (alternate bar only)

Ikeda (1984) had checked different formula for estimation of migrating alternate bar

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Ikeda states that, the value obtained by this formula has possible error of plus 80% and minus

40%. It is suggested that migrating bars wavelengths are 2-3 times smaller than steady bars

(Crosato experience).

Another empirical formula is the one suggested by Leopold and Wolman (1960). Actually this

formula is used for the estimation of meandering wavelength. To apply this formula for

estimation of bar wavelength, sinuosity value is taken unity. By doing so, the alternate bar

wavelength can be calculated. This wavelength is equivalent to alternate bar wavelength when

the river was straight or at inception of meandering. So Leopold and Wolman's equation gives

wavelength of steady alternate bars.

Equation 3-29

Where S is sinuosity which is unity and B is the bank-full width.

Table 3-4. Summary of theoretical bar wavelength.

Ikeda,1984

Migrating alternate

bars

Leopold and

Wolman,1960

Steady alternate bars

Struiksma et al, 1985 steady

alternate bars

Unit Km km km

Bar 2 731 2 333 5 122

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Chapter 4

4 Application of GIS interpolation methods to the WaalRiver

4.1 Introduction

In previous chapter, the basic characteristics of the study areas were analyzed. This gives a

chance to see how the bed form looks like and tell what type of bed form predominately exists.

As a result, it helps in planning the type and intensity of bathymetry survey required in the

rivers. Because, the bed topography of a river dictated by the nature of bed form. The intent of

this chapter is to optimize reconstruction of 2D bed topography of meandering river from

scare data. For analysis, GIS interpolation methods are used specifically; Kriging and Spline

interpolation methods are used.

Since Waal River has detail bathymetry, artificially scarce bed levels data are synthesized.The scarce data are synthesized using four different scenarios. These scenarios are developed

based on current bathymetry survey practice. Later the scarce data are used to reconstruct the

2D bed topography. The accuracy of each of interpolation methods are checked both in reach

and cross-section scale. The comparison is done between measured and interpolated

bathymetry.

In Section 4.2 selected interpolation methods are discussed and in Section 4.3 methodology

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(Environmental Systems Research and Canada, 1998). Consequently, this method can easily

predict ridges and valleys in the data; furthermore, it is the best method for representing

smoothly varying surfaces. There are two types of Spline interpolation methods: Regularized

and Tension. A Regularized Spline interpolation method incorporate first derivative (slope),

second derivative (rate of change of slope) and third derivative (rate of change of second

derivative) in order to minimize the calculation time steps. Tension Spline interpolation

method use only first and second derivatives. It includes more points in the calculation, which

means it takes more computational time. For detail information, see Appendix A.3.

4.3 MethodologyGeneral methodology used in data preparation and analysis for this chapter is schematized in

Figure 4-1.The methodology valid for all scenarios.

Measured

bathymetry

Export to

AutoCAD

For each scenario

extract x and y

coordinates

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4.4 Scenario one

To mimic the practical way of data collection, the input bed levels of this scenario are

prepared by creating conceptual sailing track and cross-sections. The sailing track goes from

one bank to the other with a wavelength four times the river width (1000 m). The cross-

sections are set every four times the river width (1000 m). The distance between consecutive

points on the sailing track is 100 m and 20 m for cross-sections. Figure 4-2 shows how bed

elevation is conceptually surveyed.

Sailing track points cross-section points

Figure 4-2 conceptually surveyed bathymetry, for scenario one.

4.4.1 Statistical summary of synthesized input bed elevation

To check how much the variability in the riverbed topography captured in the synthesized

data, the statistical summary of synthesized bed elevation is shown in Table 4-1 and the raster

file of measured bed topography in Figure 4-3. As it is clearly observed synthesized bed

elevation did not capture the extremes value of measured bed elevation. This might hinder the

output of interpolation in representing the extremes. The elevation is expressed reference to

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4.4.2 Results and discussion

The input parameter used for each of the interpolation method are indicated in Appendix B.

Interpolated bed elevations are compared at three cross-sections, centreline (Figure 4-4), and

on the reach-scale with the measured bed elevation. Reach-scale comparison is made for each

of interpolation method by plotting Graph of residual (measured elevation minus interpolated

elevation) versus frequency of occurrence of the residual. Lastly, the result obtained discussed.

Figure 4-4. Location of cross-sections (scenario one).

4.4.3 Comparison of centrelines

As it is observed in Table 4-2 the difference between measured and interpolated mean,

maximum, minimum and standard deviation for all methods of interpolation is in the order of

magnitude less than 0.01 times the water depth, which is less than 0.07 m. On the other hand

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-3

-2

-1

0

1

2

3

4

5

0 10000 20000 30000 40000 50000

B e d e l e v a t i o n

r e f e r e n c e t o N A P

( m )

Chainage (m)

Measured Spline (regularized) Kriging (ordinary) Spline (tension)

Figure 4-5 Measured and interpolated centreline (Scenario one)

4.4.4 Comparison at cross-section one

At this cross-section, Tension and Regularized Spline interpolation methods gave more or less

the same result. Higher difference is observed between measured and interpolated maximum

and minimum bed elevations, this can be easily observed in Figure 4-6. Except at one location,

which is the last point near the right bank, Ordinary Kriging interpolation methods gave errors

of less than 0.07 times the water depth. This means that, except at one location which is 1.5 m

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Figure 4-6 Measured and interpolated cross-section one for scenario one (Scenario one).

4.4.5 Comparison at cross-section two

As it is seen from Table 4-4 and Figure 4-7 none of the interpolation methods could capture

the measured maximum and minimum bed elevations. Ordinary Kriging gave higher values

all along the section with order of magnitude of 0.14 times the water depth in the deepest part

and 0.11 times the water depth in the remaining parts. Similarly, Spline interpolation (both

Regularized and Tension) estimate higher bed levels in the deepest part with the order of

-1.0

-0.5

0.0

0.5

1.0

1.5

2.02.5

0 50 100 150 200 250 B e d e l e v

a t i o n r e f e r e n c e t o

N A

P

( m )

Chainage in transeverse direction (m)

Spline(tension) MeasuredKriging (ordinary) Spline (regularized)

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Figure 4-7. Measured and interpolated cross-section two (Scenario one).

4.4.6 Comparison at cross-section three

At this cross-section, similar to the other cross-sections discussed above the measured

maximum and minimum bed elevations are not captured in the interpolation results. Table 4-5

shows that the measured values have higher standard deviation compared to the other ones.

As a result, the interpolations gave a flatter cross-section. Tension and Regularized Spline

gave the same result.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 50 100 150 200 250 B e d e l e v a t

i o n r e f e r e n c e t o N A P

( m )

Chainage in transverse direction (m)

Measured Spline (tension)Spline (regularized) Kriging (ordinary)

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Figure 4-8. Measured and interpolated cross-section three (Scenario one).

The accuracy of 1D models depend on the cross-sectional wetted area. As a result, the

accuracy of the interpolation methods in representing the cross-sectional area at bank-full

conditions is also checked. The wetted areas are computed by connecting the right and left

bank levels to the bank-full level (Figure 4-9). In case when bank levels are lower than bank-

full level, they are connected by extending a vertical line to the bank-full level. In general the

results show except Ordinary Kriging method at cross-section two, all methods of

interpolation come up with error less than 10 % of measured area (Table 4-6). As it is

0.0

1.0

2.0

3.0

4.0

5.0

0 50 100 150 200 250 B e d e l e v a t i o

n r e f e r e n c e t o N A P

( m )


Measured Spline (tension)

Spline (regularized) Kriging (ordinary)

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NB The vertical axis is ten times exaggerated

Figure 4-9. Cross-sectional areas of different cross-section.

4.4.7 Reach-scale comparison

At the Reach-scale, the difference between measured and interpolated bed elevations are

indicated in Figure 4-10. The results show the frequency and range of elevation difference

between observed and interpolated bed elevation is almost the same for all method of

interpolation Figure 4-11 to Figure 4-16 show the areal distribution of elevation difference

Bank-full level

0

20

40

60

80

-10 B e d

e l e v a t i o n r e f e r e n c e t o N A P

( m )

0 50 100 150 200

Cross-sectional distance (m), Cross-section three

100 Legend

Measured

Ordinary Kriging

Spline ( Tension and Regularized)

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Figure 4-11.Difference between measured and Ordinary Kriging in meandering reach (scenario one).

i f i b d h f d

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Figure 4-13. Difference between measured and Tension Spline in meandering reach (scenario one).

R i f i b d h f d

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Figure 4-16.Difference between measured and Regularized Spline in straight reach (scenario one).

4.4.8 Summary

Regularized and Tension Spline interpolation methods gave almost the same results at the

cross-sectional scale. Furthermore, they gave the same result at reach-scale. For instance 70 %

of the area reconstructed with error less than 10% of the water depth (Figure 4-17 ). On the

other hand Kriging interpolation method gave better result both at reach and cross-sectional

scale. The data used for scenario one were not enough to describe the riverbed topography in

the meandering reach, because, the largest errors are observed in this reach (Figure 4-11, 12

R t ti f i b d t h f d t

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4.5 Scenario two

This scenario is similar to scenario one except that the cross-sections are taken more often:

every two times the river width. The intent of reducing the spacing between the cross-sections

is to improve the result of interpolations and see how the output of interpolations is influenced

by the spacing between measured cross-sections. The distance between consecutive points in

sailing track is 100 m and 20 m for cross-sections (as in the previous scenario). Figure 4-18

shows how the bathymetry is conceptually surveyed.

Figure 4-18. Conceptually surveyed bathymetry, for scenario two.

4.5.1 Statistical summary of synthesized input bed elevations.

Detailed bed topography shows that bed elevations range between 7.07 and -6.14 m.

Table 4-7 that indicates the maximum and minimum values of the synthesized input bed

Cross-section

Conceptual Sailingtrack


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deviation closer to measured one. Figure 4-19 shows more clearly the result of Regularized

Spline interpolation is out of the range of measured bed elevation.

Table 4-8. Measured and interpolated centre line bed elevation statistical summary (scenario two).

-4

-2

0

2

4

6

d e l e v a t i o n r e f e r n c e t o N A P

( m )

Measured Kriging (Ordinary) Spline (Tension) Spline (Regularized)

Description

Centreline bed elevation reference to NAP

Mean (m) Standard

deviation (m)

Maximum (m) Minimum (m)

Measured 0.824 1.357 3.902 -1.855

Spline

Tension 0.733 1.381 3.803 -2.647

Regularized 0.559 1.550 5.366 5.095

Kriging (ordinary) 0.833 1.357 3.820 -2.419


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Figure 4-20. Measured and interpolated cross-section one for Scenario two.


As it is shown in Table 4-10 significant differences are observed between measured and

interpolated minimum and maximum bed elevations. Interpolated cross-sections could not

capture the deepest part (Figure 4-21). In most part of the cross-section Ordinary Kriging

interpolation method over estimate the bed elevation. Except a few points, Regularized and

Tension Spline interpolation methods under estimate the bed elevation. This may be resulted

due to missing of extremes in synthesized bed elevations. Because, it is clearly illustrated in

Section 4.5.1 the extremes are not included in synthesized data.

-1

0

1

2

3

0 50 100 150 200 250 B e d e

l e v a t i o n r e f e r e n c e t o

N A P ( m )

Chainage in transvers direction (m)

Measured Kriging ( Ordinary)

Spline ( Tension) Spline (Regularized)

Reconstruction of riverbed topography from scarce data

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Figure 4-21. Measured and interpolated cross-section two for scenario two.


This cross-section is located in meandering reach near the city of Erlecom. Table 4-11 shows

in all interpolation methods the maximum bed elevation is under estimated. The reason is

similar to the previous section; the bed elevations used for analysis of interpolation did not

capture all the variability (bars and pools). However, Figure 4-22 shows except a few

locations all interpolated cross-sections gave results close to measured one.

0.00

1.00

2.00

3.00

4.00

5.00

0.00 50.00 100.00 150.00 200.00 250.00 300.00 B e d e l e v a t i o n r e f e r e n c e t o

N A P ( m )


Measured Kriging (Ordinary)

Spline (Tension) Spline (Regularized)


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Figure 4-22. Measured and interpolated cross-section three for scenario two.

4.5.7 Reach-scale comparison

As clearly seen from Figure 4-23, most of the results of Tension Spline and Ordinary Kriging

interpolation methods are concentrated with less deviance from measured bed elevations. On

other hand most of the results of Regularized Spline interpolation method located are within

higher deviance. To illustrate the areal distribution of the error, Figure 4-24 to 4-29 show the

mapping of the difference s between measured and interpolated bed topography.

0.0

1.0

2.0

3.0

4.0

5.0

0 50 100 150 200 250 300 B e d e l e v

a t i o n r e f e r e n c e t o N A

P

( m )





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Figure 4-24. Difference between measured and Ordinary Kriging in meandering reach (run two).

Figure 4-25 shows most of the area has higher difference, which means the performance of Regularized Spline interpolation method is not good in meandering reach. Even in some area

(chainage 5 km to 6 km and 2 km and 3 km) it gave very strange value.


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Figure 4-26. Difference between measured and Tension Spline in meandering reach (scenario two).

In the straight reach Ordinary Kriging interpolation method give similar result with themeandering reach.


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Figure 4-28. Difference between measured and Regularized Spline in straight reach (scenario two).

In the straight reach Tension Spline interpolation method comes with more area with higherelevation difference.


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Figure 4-30. Reach-scale interpolation error in terms of percentage of water depth (scenario two).

4.6 Scenario three

Similar to the other scenarios the input bed elevation of this scenario is synthesized in the

same way. The input points are created with the combination of centreline points every ten

metres and cross-section every four times the river width. The distance between consecutive

points in the cross-section is 20 m. The output of this scenario showed poor quality. It gave

raster cell size of 787*787 m which is too coarse. When the output cell size is reduced it could

0

10

20

30

40

50

60

< 5% 5 % to

10%

10 % to

20%

20% to

30%

30% to

50%

50% to

80%

50% to

80%

>100

% o

f a r e a

Error in % of water depth

Tension Spline interpolation Regularized Spline Ordinary Kriging interpolation


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4.7.1 Statistical summary of input bed elevations

When compare detailed measured bed elevation (Figure 4-3), which range between 7.07 and -

6.14 m. and the statistical summary of synthesized input bed elevation, which is shown inTable 4-12 synthesized input bed elevations could not capture the maximum and minimum

bed elevations that exist measured data. Absence of the variability in synthesized input bed

elevations means poor quality of interpolated surface in representing the pools and bars.

Table 4-12. Statistical summary of extracted input bed elevation (scenario four).

4.7.2 Results and discussion

For all interpolation methods the parameters used, is similar to the other scenarios and listed

in Appendix B. Since the centreline is taken as input it is not used for comparison. Instead, in

cross-section scale three cross-sections and interpolated cross-sectional area compared with

measured one. Furthermore in reach-scale the difference between measured and interpolated

surface compared. The locations of the three cross-sections are indicated in Figure 4-4. Lastly,

in the summary part, Reach-scale error in reconstruction of the bed bathymetry discussed.

Waal River extracted bed elevation reference to NAP

Mean Standard deviation Maximum Minimum

m m m m

0.819 1.360 5.870 -2.735


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Figure 4-32. Measured and interpolated cross-section one for scenario four.


At this cross-section, all interpolation methods gave flatter cross-sections. In contrary, the

cross-section is located in the meandering reach, which is characterized by a deep and a

shallow part. Figure 4-33 shows that none of the interpolation methods could capture the

deepest part. To avoid this special attention needs to be given while bathymetry is surveyed.

Table 4-14. Measured and interpolated cross-section two, statistical summary (scenario four).

-1.0

-0.5

0.0

0.5

1.0

1.5

2.02.5

0 50 100 150 200 250 B e d e l e v a t i o n r e f e r e n c e t o N A P

( m )





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Likewise, with the previous section this cross-section is also located in the meandering reach

(near the city of Erlecom). The result of interpolations (Figure 4-34) shows that theinterpolated cross-section is almost horizontal whereas the measured cross-section shows

defined feature with deep and shallow part at this cross-section none of the interpolation

methods could show that. The reason is similar to what explained in previous section.

Table 4-15. Measured and interpolated cross-section three, statistical summary (Scenario four).

4.0

5.0

e t o N A P



DescriptionCross-section three bed elevation reference to NAP

Mean (m) Standard

deviation (m)

Maximum (m) Minimum (m)

Measured 2.290 0.787 4.170 1.546

SplineTension 2.507 0.252 3.051 2.317

Regularized 2.474 0.292 2.942 1.959

Kriging (ordinary) 2.435 0.160 2.820 2.278


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Figure 4-35. Reach-scale Comparison of the three methods of interpolation.

Figure 4-36 shows except, chainage between 6 km and 7 km all the areas result uniform

difference (error). Huge difference observed near the in inner and outer bend might not be due

to interpolation method, because sometimes this kind of huge difference may happen due to a

050

100150200250

300

- 6 . 2

- 5 . 3

- 4 . 4

- 3 . 5

- 2 . 6

- 1 . 8

- 0 . 9 0 . 0 0 . 9 1 . 7 2 . 6 3 . 5 4 . 4

F r e q u e n c y

Elevation difference (m)

Measured minus ordinary Kriging

a

0

2000

4000

6000

8000

- 5 8 . 0

- 4 8 . 8

- 3 9 . 7

- 3 0 . 5

- 2 1 . 4

- 1 2 . 3

- 3 . 1 6 . 0

1 5 . 1

2 4 . 3

3 3 . 4

4 2 . 5

F r e q u e n

c y


Measured minus Regularized spline

b

0500

100015002000

- 5 . 9

- 5 . 2

- 4 . 4

- 3 . 6

- 2 . 8

- 2 . 1

- 1 . 3

- 0 . 5 0 . 2 1 . 0 1 . 8 2 . 6 3 . 3 4 . 1 4 . 9 5 . 6

F r e q u e n c y


Measured minus Tension spline

C


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Figure 4-37 shows that the Regularized Spline interpolation method results in a large area

with high bed elevation difference.

Figure 4-37. Difference between measured and Regularized Spline in meandering reach (scenario

four)


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p g p y

In straight reach Ordinary Kriging interpolation method (Figure 4-39) has similar feature as

for the meandering reach. The reason of having higher difference near the bank might not be

due to interpolation method, because sometimes this kind of huge difference may happen due

to a mismatch of the measured and interpolated boundary.

Figure 4-39. Difference between measured and Ordinary Kriging, in straight reach (scenario four).

Figure 4-40 shows similar features as Figure 4-37, which means that Regularized Spline

performed similarly on both straight and meandering reach.


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p g p y

Figure 4-41. Difference between measured and Tension Spline, in straight reach (Scenario four).

4.7.7 Summary

Increasing the amount of input bed elevation highly influence the output of interpolation.

However, this is not always true. For instance, the accuracy of Regularized Spline

interpolation decreased as the amount of data increase. Furthermore the output of the other

methods of interpolations did not show significant difference when the amount of input bed

elevations increased. It is rather the way the data collected what highly matters. Scenario one

and two captured the extremes (maximum and minimum) in better way than scenario three

and four Ultimately the performance of scenario one and two found better This lead to the


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p g p y

Chapter 5

5 Computation of Waal Riverbed topography usingMIANDRAS

5.1 Introduction

In the previous chapter, GIS interpolation techniques were analysed for the reconstruction of

2D riverbed topography, specifically for the Waal River. It is learned that this method work well when data are rich. What can still be done if data are inadequate for the application of

GIS interpolation techniques? One of the possibilities could be checking the functionality of a

simple morphodynamic model for reconstruction of riverbed topography. One of simple

model is MIANDRAS, which can compute the equilibrium 2D bed topography. The amount

of data MIANDRAS model requires are relatively small. In this section it is investigated

whether MIANDRAS offers an acceptable alternative for the reconstruction of 2D riverbed

topography.

MIANDRAS demands reach-scale river characteristics and knowledge of at least 3-5 cross-

sections; furthermore the model is easy to use. Actually, this model gives the bank-full water

depth at nine points per cross-section. These water depths are computed based on the

curvature of the river centreline and on morphodynamic characteristics at bank-full discharge.

The model computes higher water depth at the outer bends and lower water depth at the inner

bends. Moreover, the model is able to predict the presence of steady alternate bars. Since only


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p g p y

with small to moderate Froude number (Crosato, 2008). Furthermore, the channel is assumed

mildly curved. For detail, see Section 2.4.1. The numerical solution of equation does not give

problems. However, the accuracy of computation influenced by various factors such as,

truncation errors, distance between sections, time-steps length and calibration coefficient etc.

5.3 Input and the calibration coefficients

MIAMDRAS demands the following input parameters: the bank-full discharge, bank-full

width, water depth at bank-full conditions, sediment grain size (d50 and d90), valley slope and

the coordinates of the channel centreline. These values are supposed to be representative of

the reach. Especially the bank-full discharge should be accurate enough. Furthermore, the

bank level should be measured at least at three locations (actually it depends on the length of

the reach). Otherwise, the accuracy of the projected bed level might be questionable.

Assumed bank level is one of the parameters which determine the accuracy of this method.

To calibrate the computation five calibration coefficients are used. They are E, α1, δ, Eu and

Eh. These parameters are weigh the influence of sloping bed in sediment transport direction,

channel curvature on bed shear stress direction, secondary flow convection, flow inducedbank erosion and bank retreat due to cohesive bank failure, respectively. There are formulas

for computation initial values. However, while running the model the right value is obtained

iteratively. The formulas are indicated in Appendix C. Bank erosion is not analyzed for this

research so that the bank erosion coefficients (Eu and Eh) are kept zero. All the process

involved in this chapter is summarized in Figure 5-1.


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g y

run two the experience from Blue Nile River worked pretty well. Which means in this way the

calibration coefficients can be optimized. The calibration coefficients used for four runs are

indicated in Table 5-2.

Bank-full width and

discharge

Centreline Coordinate

Data analysis

Computation of

Calibration

coefficients

Negative

water depth

and velocity

Yes

Change the calibration

coefficients

No


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Table 5-2. Calibration coefficients used for four runs.

Run Eu Eh α1 σ E Remarks

1 0 0 0.1 2 1 -

2 0 0 0.5 4 3 Blue Nileexperience

3 0 0 0.1 4 1.2 -

4 0 0 0.5 2 3 -

5.4.2 Results and discussions

As discussed in Section 5.3 the outcome of MIANDRAS can be calibrated by changing the

calibration coefficients. However, it is found difficult to get the combination of thesecoefficients which give good result. As a result, these coefficients optimize by comparing the

result of each run with measured data and then each new run is set by increasing or decreasing

the calibration coefficients of the previous run. The performance of each run is compared with

measured cross-section at five locations (Figure 5-7). To identify a run which gives good

result (result close to measured cross-section) a weight is given to each run. The best run is

selected based on score of highest cumulative weight. In this analysis the results are also

compared at the Reach-scale.

All runs gave more or less the same result at cross-section one (Figure 5-2) and centreline

(Figure 5). Nevertheless, the result in the remaining cross-sections is quite different (Figure 5-

3, 5-5 and 5-6). When the results are compared at the straight reach, little differences are

observed. For instance, the results observed at cross-section one and cross-section four show

little differences (Figure 5-2 and Figure 5-5). Since one run over estimate the bed level at one

side and underestimate at the other side, differences observed counter balance each other. On


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Figure 5-2. Measured and computed bed level in cross-section one.

-1.5

-1.0-0.5

0.0

0.5

1.0

1.5

2.0

2.5

0 50 100 150 200 250

B e d e l e v

a t i o n r e f e r e n c e t o N A P ( m

)

Cross-sectional distance (m)

MIANDRAS (run-1) MIANDRAS (run-2) MIANDRAS (run-3)

Measured Bed level MIANDRAS (run-4)

3.0

4.0

5.0

c e t o N A P ( m )

MIANDRAS (run-1) Measured Bed level MIANDRAS (run-2)

MIANDRAS (run-3) MIANDRAS (run-4)


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Figure 5-4. Measured and computed bed level in cross-section three.

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

0 50 100 150 200 250 B e d e l e v a t

i o n r e f e r e n c e t o N A P ( m )




0.5

1.0

1.5

2.0

e r e n c e t o N A P ( m )




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Figure 5-6. Measured and computed bed level in cross-section five.

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0 50 100 150 200 250 300 B e d e l e v a t i o n r e f e r e n c e t o N A P ( m )




2.0

3.0

4.0

5.0

e t o N A P ( m )




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The accuracy of 1D models depend on the cross-sectional wetted area. As a result, the

accuracy of the MIANDRAS in representing the cross-sectional area at bank-full conditions is

also checked for selected three cross-sections. The wetted areas are computed by connecting

the right and left bank levels to the bank-full level (Figure 5-8). In case when bank levels arelower than bank-full level, they are connected by extending a vertical line to the bank-full

level. The result shows that in all three cross-sections the maximum error occurred is 11 % of

measured area which is at cross-section two run four whereas in the remaining cross-sections

observed error is less than 5 %. This means that application of MIANDRAS for 1D model

application is seems good. However, this kind of computation needs continuous bank-full

level. Furthermore measured cross-section which is used for comparison is not measured

during bank-full conditions so that the result obtained might be altered if the one measured atbank-full condition is used. To make binding conclusion further research is needed.

Table 5-4. Wetted cross-sectional area of MIANDRAS.

Run

Cross-sectional area (m2)

Cross-section 1 Cross-section 2 Cross-section 3

Measured 1547.8 1568.9 1553.2

Run one 1605.4 1733.2 1588.3

Run two 1601.9 1747.6 1679.4Run three 1601.3 1723.7 1586.2

Run four 1606.0 1739.5 1571.6

100

Bank-full level


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the reach-scale, the accuracy of each run is expressed in terms of error (measured minus

computed bed level). This error is expressed in terms of percentage of water depth. A run that

gives less percentage of error with large percentage of area nominated as the best run. Similar

to the cross-sectional scale, run four performs well at the reach-scale too. As a result, 67.5 %of the total area is reconstructed with an error less than 10% of the water depth (0.77m).

Furthermore, the area of the river channel reconstructed with an error more than 30 % of the

water depth is less than 2.8 % of the total area. Figure 5-9 clearly shows that, the difference

between each run is little. For instance, in run two, which is the least less accurate run, 64.3 %

of the total area is reconstructed with an error less than 10% of the water depth (0.77m).

Furthermore, the area reconstructed with an error more than 30 % of the water depth is less

than 4.7 % of the total area.

Compare to cross-sectional scale, magnitude of difference between each run is less at the

reach-scale. As a result, if the conditions do not allow for calibration, it might be possible to

use MIANDRAS without calibration using the standard values of calibration coefficients

derived from Table 3-5. Because it is observed that, the accuracy obtained without calibration

is not far from calibrated one. Here it is very important to be aware that calibration in this

context means the run which come with better result. This is because it is observed that

through calibration the result could not enhanced beyond certain accuracy. This lead to theconclusion that, MIANDRAS calibration coefficient has little significant on reach-scale

reconstruction of riverbed topography.

40 0

45.0

Run-1 Run-2 Run-3 Run-4


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deciding the location of cross-sections and designing the layout of the sailing track is one of

the challenges. Since MIANDRAS identifies the locations of pools and bars, it could assist in

designing the best sailing track and deciding the location of the cross-sections to be surveyed.

In this research it is observed that, the calibration coefficients have little influence on the

reach-scale reconstruction of 2D riverbed topography. This is proofed by comparing the result

of different calibration coefficients. As the results show, the difference observed is little.

Furthermore, there is little chance to know which combination of these coefficients give the

best result unless there are some cross-sections for calibration. Even when there is cross-

section for calibration, there is uncertainty related to the location of cross-sections. This isbecause it observed that the performance of the calibration coefficients is different from cross-

sections to cross-sections. Furthermore, it is not always true best result at the cross-sectional

scale be the same at the reach-scale. Therefore, calibration at the cross-sectional scale is

uncertain. This leads to the conclusion that MIANDRAS could be used without calibration for

reconstruction of meandering riverbed topography.

To maintain the accuracy of MIANDRAS for reconstruction of 2D rivers bed topography, the

input data should be measured precisely. Furthermore, the floodplain level (bank-full level)needs to be measured precisely. It might be very good to measured it at least once per 5-10

kilometres. Actually if there is, some other way to obtain the water surface level, throughout

the reach, no need to measure this level. However, these water surface levels should be the

one at bank-full condition. The performance of MIANDRAS for reconstruction of river

bathymetry might be higher than the one observed in this analysis. Because the Waal River is

a trained river, so that the characteristics that expected in natural rivers might not exist.

F th t l ti th b d d d A lt it i ht t b d t k


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Chapter 6

6 Application of GIS and MIANDRAS for Atrato River

6.1 GIS application for Atrato River

In Chapter 4, the best method of GIS interpolation technique for reconstruction of riverbed

topography was Ordinary Kriging method of interpolation. In this section, application of this

method for Atrato River studied. Although this method is valid for the whole reach of theAtrato River, it is only analyzed for 10-kilometer of Montano reach (Figure 6- ). This is done

because MIANDRAS is also analyzed for the same reach so that the results can be compared.


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Cons of the current measured bathymetry of the Atrato River

Most of measured points appeared situated near one side of the river (Figure 6-1),

which means that this kind of bathymetry survey tends to miss either pool or bars.

Accordingly, the interpolated survey comes with the same shortcoming. Instead it

would be nice the bathymetry survey to be done like Figure 6-2, which is the sailing

track touching the both side of the banks alternatively.

The distance between successive measured cross-sections is too large, since it is

approximately measured one cross-section per 20 km. Since cross-sections are the

only way to bring in the bed topography variability all together insufficiencies of thecross-sections highly affect the outcome of the interpolations.

Some of measured cross-sections are incomplete.

Some of the cross-sections include the floodplain. The way the cross-sections

measured should be consistent all along the reach. This is because Ordinary Kriging

method of interpolations is based on area correlation of the measured points

(Variogram). Therefore inconsistency in measuring the cross-sections may bias the

result of interpolation. Consequently, the cross-section should be measured uniformlythroughout the reach (with or without the floodplain).

At some location there are redundant measured points. This kind of points may affect

the results of interpolation.

On average

B/2


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Since there are measured cross-sections and sailing track points, the bed topography

can be reasonably constructed with a few additional cross-sections.

6.1.2 summary

From Waal River, it is learned that there are two possible ways to reconstruct the Atrato

Riverbed topography. The first is using existing measured data and the other one is adding

extra cross-sections. However, the selection between the two alternatives depends on the level

of accuracy required. The result of the second alternative would be more accurate. In case of

the first alternative the cross-sections, which include the floodplain should be screened. This

is because all measured cross-sections should contain the same feature. In the analysis, onlymeasured bed level in the main river channel should be used. Furthermore, to improve the

outcome of interpolations it would be nice if the curved river planimetry is converted in to a

straight one (Figure 6-3 b). This is because ordinary Kriging interpolation method considers

the extent of interpolation for the area between upstream and downstream longitude (Figure

6-3 a). Therefore, the areas out of the river channel will have interpolated values. This may

affect the interpolated values of the river channel. Consequently, to confine the interpolation

in the river channel and enhance the result of interpolation it is advised to convert the

meandering planimetry in to straight before interpolation analysis started.(a)


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increased. As a result, the distance between cross-sections governs the level of accuracy to be

achieved.

6.2 Application of MIANDRAS on the Atrato River

6.2.1 Introduction

In this section, application of MIANDRAS on the Atrato River is analyzed. Due to data

limitation, the analysis is done for a selected reach (see Section 6.1). However, if all the

necessary information (see Section 5.4) were available, it would be possible to apply for the

other reach too. The result of analysis is not calibrated. This is because there are no enoughcross-sections for calibration. However, from the analysis on Waal River it is observed that

the calibration process does not make significant changes on the result.

In this analysis, the result of MIANDRAS bed topography is compared with the measured

bathymetry. However, the amounts of measured data are very scarce. Therefore, to cover the

whole area measured data are interpolated. This means the comparison is made between the

interpolated surface and MIANDRAS output. For comparison one cross-section and reachscale raster data is used.

6.2.2 Assumptions

It was difficult to get most recent information on the hydrodynamic characteristics of Atrato

River in the study reach. Thus, it was necessary to assume or use the result of past studies.

The parameters either assumed or referred from past studies are listed in the following table


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Table 6-2. MIANDRAS input parameters used for Atrato River.

Bank-full slope Sediment size

Chezy

Coefficient

Calibration coefficients

Qw B h0

Valley Chann

el D50 D90

C α1 σ E Eh Eu

m3/s m m m/m m/m mm mm m1/2

/s - - - s-1

-

2000 214 6.2 2.07E-

4

1.67E-

4

0.332 0.404 47 0.4 3.2 3.2 0 0

6.2.4 Results and discussions

The intent of this analysis is not to compute bed topography of the study reach; it is rather to

show all the possibility to do it. This is because the choice between different methods depends

on the level of accuracy desired to achieve. As a result, if the alternatives are set the choice

would be up to the user.

The result shown in cross-section 47 (Figure 6-4) do not show significant difference between

each method. However, this might not be true for the whole reach. For instance, reach-scale

bed topography reconstructed by MIANDRAS (Figure 6-5 a) is shown distinct bars and pools

whereas the output of Ordinary Kriging (Figure 6-5 b) shown diffused pools and bars. In this

case, the result of Ordinary Kriging is supposed to be identical to the measured cross-section

because the same cross-section is used for interpolation Furthermore at the beginning of the


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(a) (b)

Figure 6-5. Reach-scale comparison: (a) MIANDRAS bed level and (b) is Kriging interpolation.

6.2.5 Summary

The aim of reconstruction of Atrato Riverbed topography is to identify the navigation


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Chapter 7

7 Conclusions and recommendations

In this study, different methods to enhance the method of reconstruction of 2D riverbed

topography from scarce data were studied. Special attention is given to the reproduction of

bars because these mega forms hinder navigation. Cross-sectional areas are important for 1D

modelling of long river reaches, for this the study also analyzed cross-sectional area accuracy

for interpolated cross-sections. This chapter describes the results of the experience gained inthe study. Furthermore, recommendations to improve the result of this research and to draw

attention on the limitations of this research are also elaborated.

7.1 Conclusions

7.1.1 Conclusions on the application of GIS interpolation techniques

In this research, Ordinary Kriging interpolation method appeared the best method of

interpolation for reconstruction of riverbed topography. However, the accuracy of this

method depends on the quality and quantity of the input bed elevations.

The performance of Ordinary Kriging interpolation method declines as the amount of

input bed elevation declines. Consequently, it is not wise to use this method when data

are scarce In this regard the shortcoming observed was the smoothing out of bars and


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MIANDRAS can be used for reconstruction of meandering riverbed topography

without calibration with little difference from calibrated one.

Bank-full water surface level is one of the parameters, which affect the performance

of MIANDRAS in reconstruction of riverbed topography. This is because after the

water depth is computed by MIANDRAS these water depths are projected to bed

levels by deducting it from water surface level. Therefore, if these levels are

assumed and or measured incorrectly, all the result would be wrong. So that the

bank-full water surface level need to be measured precisely.

0

10

20

30

40

50

60

70

< 5% 5 % to10% 10 % to

20%

20% to

30%

30% to

50%

50%

to80%

80% to

100%

>100%

% o

f a r e a

MIANDRAS Run four Kriging Scenario four


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7.2 Recommendations

7.2.1 Recommendations on the application of GIS interpolation techniques

Ordinary Kriging method of interpolations is based on the areal correlation of

measured data. The floodplain and the main river channel have different trend in bed

elevation, which result in different areal correlation (Variogram). As a result, it would

be nice to treat the floodplain and main channel separately. Even in the river channel,

the characteristics of bed elevation are different in straight and meandering reaches.

Accordingly, it would be nice to treat the meandering and straight reaches separately.

Since the distance between measured points highly influences the outcome of

Ordinary Kriging interpolation method therefore, it might be helpful to convert curved

river planimetry to straight before interpolation (Figure 6-3 a) and (Figure 6-3 b).

While bathymetry survey, one of the challenges is to set a reasonable distance between

cross-sections and identifying the location of cross-sections, which is governed by the

nature and type of bed form that exist in the river. As a result, if the wavelength of the

bars reasonably computed, it would assist in selecting suitable spacing and distance of cross-sections.

Since the output of Ordinary Kriging method depends on the quality and quantity of

input data, it would be nice first check adequacy of measured data. As a starting point,

the output of this research can be used to check the adequacy of measured points. In

general, to increase the accuracy of interpolation in cross-section scale more data are

required than the reach-scale


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depth in cross-sections and the smoothing also governs the distribution of the

computational grid points in the model.

In this research it is assumed that the computational time step is insignificant for this

kind of computation in MIANDRAS, so it would be nice to check this in the future

research.

The computation done for the Atrato River (Montano reach) is based on a lot of

assumptions. As a result, the outcome might be improved if measured data are used. It

also hold true for Waal River, which only one bed topography was used. This did not

allow distinguishing migrating bars from steady bars.

Wetted cross-sectional area which is computed by MIANDRAS shown promisingresults when compared to measure cross-sectional area. Consequently, it would be

nice to do further research in rivers that have bank-full cross-section (Table 5-4).

MIANDRAS is not accurate at sharp bend, island and in presence of backwater. Since

some locations of Atrato River are characterized with such thing it would be nice to

exclude these reaches in computation

MIANDRAS is best for reach-scale reconstruction of riverbed topography. If cross-

sectional scale accurate bathymetry is required it would be nice to use Ordinary

Kriging interpolation method. Though it need extensive measured bathymetry data.


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8 References

Ali YSA (2008). Morphological processes in the Blue Nile River between Roseires and Sinnar MScthesis Unesco-IHE, Delft.

BAKKER B. & CROSATO A.,1989. Meander studies Tigris River. Report Q985 November 1989,

WL | DelftHydraulics, The Netherlands.

Biju-Duval B, Le Quellec P, Mascle A, Renard V, Valery P (1982). Multibeam bathymetric survey

and high resolution seismic investigations on the Barbados ridge complex (eastern Caribbean):

a key to the knowledge and interpretation of an accretionary wedge. Tectonophysics 86: 275-

282.

Charlton ME, Large ARG, Fuller IC (2003). Application of airborne LiDAR in river environments:

the River Coquet, Northumberland, UK. Earth Surface Processes and Landforms 28: 299-306

DOI 10.1002/esp.482.

Colombia geographic map[ Image] (2007). Cited September 2010:

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Crosato A. & Desta F.B. 2009. Intrinsic steady alternate bars in alluvial channels; Part 1:

experimental observations and numerical tests. In: Proc. of the 6th

Symp. on River Coastal and

Estuarine Morphodynamics (RCEM 2009), 21-25 Sept. 2009, Santa Fe, Argentina, RCEM

2009, Taylor & Francis Group, Vol. 2, pp. 759-765.

Crosato A. & Mosselman E., 2009. Simple physics-based predictor for the number of river bars and

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century Waal River

by means of a 2D physics-based numerical model. Journal of Hydrology, published online by

Wiley InterScience, DOI: 10.1002/hyp.7804.

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Digital Terrain Models. Journal of Water Resources Planning and Management 127: 394-402.

R.J.Hey JC (1996). Water Surface topography in River Channel and Implication for Meandering

Development.University of Califorina , Califorina , USA.

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Knovel ,Norwich, NY.

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distribution in unsurveyed rivers. Int.J. Sediment Res. 24: 127-144.

Schäppi B, Perona P, Schneider P, Burlando P (2010). Integrating river cross section measurements

with digital terrain models for improved flow modelling applications. Computers &

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channels. Journal of Hydraulic Research, IAHR, Vol. 23, No. 1, pp. 57-79.

Tassi P (2007). Numerical modelling of river processes: flow and riverbed deformation.ThesisTwente,Enschede.

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Appendixes

Appendix A: Description of Kriging and Spline interpolationmethods

A.1 Introduction

Interpolation is one of GIS application that predicts values for cells in a raster from a limited

number of sample data points. It can be used to predict unknown values for any geographicpoint data, such as elevation, rainfall, chemical concentrations, and etc. The assumption that

makes interpolation practical tools is that, spatially distributed objects are spatially correlated

or things that are close together tend to have similar characteristics. This is the basis of

interpolation. A typical use for point interpolation is to produce an elevation surface from a

set of sample measurements.

The interpolation tools are generally divided into deterministic and geostatistical methods.

The deterministic interpolation methods give values to locations based on the surroundingmeasured values and use specified mathematical formulas that determine the smoothness of

the resulting surface. The deterministic methods include IDW (inverse distance weighting),

Natural Neighbour, Trend, and Spline. The geostatistical methods are based on statistical

models that include autocorrelation (the statistical relationship among the measured points).

Because of this, geostatistical techniques not only have the capability of producing a

prediction surface, but also provide some measurement of the certainty or accuracy of the

di i i i i i i l h d f i l i


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A.2 Kriging

Kriging is one of geostatistical interpolation technique which is based on statistical models

that include autocorrelation, which calculate the statistical relationships among the measuredpoints. Because of this, Kriging techniques not only have the capability of producing a

prediction surface but also provide some measure of the certainty or accuracy of the

predictions. Kriging assumes, the distance or direction between sample points reflects a

spatial correlation that can be used to explain variation in the surface. The Kriging tool fits a

mathematical function to a specified number of points, or all points within a specified radius,

to determine the output value for each location. Kriging is a multistep process; it includes

exploratory statistical analysis of the data, variogram modelling, creating the surface, and

(optionally) exploring a variance surface.

There are two kinds of Kriging: ordinary and universal. The ordinary Kriging is the most

common and widely used method. It assumes there is no constant mean of the sample points

over the reach. The Universal Kriging assumes there is overriding trend in the measured data

points. To make clear let us see the following example, prevailing wind and this can be

modelled by deterministic function (polynomial). This polynomial is subtracted from the

original data points and the autocorrelation is modelled from random errors. Once the modelis fit to the random errors, the polynomial adds back to the original data to make meaning full

result. Universal Kriging is recommended to use when there is a trend in the measured data

points, which can be scientifically explained.

computation steps

K i i i i il t IDW (I di t i ht d) it i i ht f th di


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then calculate prediction surface. Because of these two distinct steps Kriging being called use

the data twice.

Variography

To start with the computation the first step is fitting the model or spatial modelling is also

known as structural analysis or variography. In order to analysis the spatial distribution or

correlation of the measured points, the empirical semi-variogram is computed for all pair of

points separated with distance (h). The distance can be calculated using the following formula

where J and i are two points location

Semi-variogram (distance (h)) = 0.5*(average ((valuei-value j)2) Equation A 2

As you see from Equation A.2 the semi-variogram is calculating the square of difference of

pair of measured point's values. In Figure A 2 shown how the semi-variogram is calculated,

the red point is paired with other blue points and this process is done for other points also.

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Figure A 5. Exponential model example (http:\\ www.arcgis help, December, 2010).

Exponential model is applied when spatial autocorrelation decreases exponentially with

increasing distance. In this case the autocorrelation disappears completely only at an infinitedistance. The exponential model is also a commonly used model. The choice of which model

to use is based on the spatial autocorrelation of the data and on prior knowledge of the

phenomenon.

Understanding a semi-variogram: Range, sill, and nugget

As previously discussed, the semi-variogram show the spatial autocorrelation of the measured

sample points. As basic principle of geography things that are closer are more alike and also,measured points that are close will generally have a smaller difference squared than those

farther apart. Once each pair of locations is plotted after being classified (binned), a model is

fit through them. Range, sill, and nugget are commonly used to describe these models.

Range and sill

When you see the semi-variogram model, the curve attains a constant horizontal value, as aresult the distance in x-axis in which the model attains constant y value is called range. The

sample points separated by the distance less than the range are auto correlated otherwise not.

The value in y axis the semi-variogram model attain the range is called sill. A partial sill is the

sill minus the nugget. Nugget is described in the following section.


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Mathematical Model

In the following section some of the general shape and equation of mathematical model which

are used for analysis of semi-variance is shown.

Figure A 7. Spherical semi-variance model.0(0)y

h c(h)y

h0

2

1

2

3c(h)

0

3

0

c

hhc y

0(0)

0

exp1c(h) 0

y

h

r

hc y


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Figure A 10. Linear semi-variance model.

A.3 Spline methods

Spline is one of interpolation technique which uses mathematical function to predict the value

of unmeasured points. The mathematical function should reduce surface curvature, pass

through the measured point and result in smooth surface. Usually spline applied for prediction

of smoothly varying surface, like elevation and water table height. To minimize the surface

curvature the following condition should be fulfilled in the interpolated surface.

1. The surface should pass exactly through the measured points.

2. The surface must have minimum curvature or the cumulative sum of the squares of the

second derivative terms of the surface taken over each point on the surface must be a

minimum.

There are two type of spline interpolation, Regularized and Tension. The Regularized methodcreate smooth and gradually varying surface in which the resulted data point may lie outside

the measured data. The Tension method relatively less smooth surface, but the resulting

surface will be bounded in the range of measured data points

The other parameters that control the smoothness of output surface are the number of points

and weighting factor. For the Regularized Spline method, the weight parameter defines the

i ht f th thi d d i ti f th f i th t i i i ti i Th


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2ln1

2ln

42

1 (r) 0

22 r

cr

K cr r

R

For the case of Tension

T (x, y) = a1

)(Kc

2ln

2

1-(r) 02

r

r R

Where

and22

Parameter entered in the command line

r the distance between the required point and sample point

K0 modified Bessel function

C constant equal to 0.577215

ai coefficients found by the solution of a system of linear equations.


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Appendix B: input parameters for GIS Runs

General summary of the input dialogue box for each run is indicated below, since the method

of interpolation does not, only the general dialogue box is shown below For the all methodraster cell size is taken as the minimum of the of the input cell.

Figure B 1. Ordinary Kriging input dialogue box.


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Figure B 3. Spline Regularized input dialogue box.


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Appendix C: Description of MIANDRAS calibration coefficients

C.1 Introduction

There are five calibration coefficients to calibrate MIANDRAS. The value of these

coefficients is not easy to obtain, however there are formulas to compute the initial value. The

final value might be 10 times higher than the initial one.

C.2 Erosion coefficient

There are two erosion coefficients: flow induced bank erosion coefficient (Eu) and bank

retreat due to cohesive bank failure erosion coefficient (Eh). These coefficients vary

depending on the characteristics of the reach. If the morphology of the river varies from reach

to reach, it is advisable to compute these coefficients separately for each reach. The cohesive

bank failure erosion coefficient (Eh), highly depend on the slope of the bank. In the area, with

mild slope, It can be assumed the erosion is mainly contributed by near bank flow and flow

induced erosion coefficient computed by Equation C.1.Where as in the area the bank is steep,

the flow induced erosion coefficient can be computed by dividing Equation C.1 by two andsimilarly bank instability induced bank erosion coefficient is computed using Equation

C.2.The assumption is, half of the erosion is due to flow and other half, due to bank instability.

y

y

ut u

n E

0Equation C 1


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greater than two means negative friction which means the transverse shift of the maximum

velocity toward the outer bed. The value equal to two means, there is no transverse friction.

The initial value can be computed using Equation C.3

2

g

C901

B

ho Equation C 3

Where

σ Coefficient weighing the secondary flow momentum convection (-)

C Chezy coefficient (m1/2 /s)g Acceleration due to gravity (m/s

2)

B Channel width (m)

ho Reach average water depth (m)

C.4 Coefficient weighing the effect of transverse bed slope in sedimenttransport direction

E is introduced in the equation for the angle between sediment transport direction and the

main flow direction, in which it weigh the influence of transverse bed slope. The value of, E,

has been derived from flume experiment, however to apply for real river, need to multiply by

factor of two. The possible reason for this can be counterbalance the scale effects induced by

presence of bed form The first estimate can be computed using Equation C 4


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kc

g A 1k 2

2-

1 Equation C 5

Where

A Calibration coefficient (-)

k Von Karman constant (-)

g Acceleration due to gravity (m/s2)

C Chezy coefficient (m1/2

/s)

An increase of 1 leads to an increase of the effect of curvature on the bed deformation,

which means it facilitate the occurrence of spiral flow. The problem like negative water depth

and flow velocity can be alleviated by reducing the value of 1 . Usually 1 range between 0.4

to 1.2.The initial value can be computed using Equation C.6.

3.0

50

0.10

D

h E o Equation C

6

Where

D50 The mean grain size (m)

ho Reach average water depth (m)


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Appendix D: Results

D.1 GIS interpolation methods

D.1.1 Waal River Scenario one


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Figure D 1. Results of GIS interpolation (Scenario one) for Waal River (The bed elevation is reference

to NAP).

D.1.2 Waal River Scenario two


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D.1.3 Waal River Scenario four


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D.2 MIANDRAS

D.2.1 Results of MIANDRAS Runs for Waal River


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Figure D 4. Results of measured and MIANDRAS RUNS for Waal River.


A di E Mi ll

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Henok Endale Abebe 113

Appendix E: Miscellaneous

E.1. Waal River centreline coordinate.

Original Centreline coordinateNr Longitude Latitude Nr Longitude Latitude Nr Longitude Latitude Nr Longitude Latitude

1.0 196879.2 432196.2 25.0 194708.1 429712.1 49.0 192026.1 431864.1 73.0 189089.1 430263.9

2.0 196782.6 432081.5 26.0 194561.7 429744.7 50.0 191876.3 431871.9 74.0 189032.1 430125.2

3.0 196689.3 431964.0 27.0 194419.4 429791.8 51.0 191726.4 431876.1 75.0 188992.4 429980.9

4.0 196599.4 431844.0 28.0 194281.8 429851.6 52.0 191576.4 431875.2 76.0 188932.0 429844.1

5.0 196513.0 431721.4 29.0 194150.3 429923.6 53.0 191426.6 431867.5 77.0 188844.6 429722.2

6.0 196432.2 431595.0 30.0 194034.5 430018.3 54.0 191277.2 431854.5 78.0 188745.9 429609.4

7.0 196358.0 431464.7 31.0 193939.8 430134.6 55.0 191128.3 431836.3 79.0 188635.8 429507.6

8.0 196290.6 431330.7 32.0 193854.0 430257.6 56.0 190980.2 431812.4 80.0 188515.6 429418.09.0 196229.9 431193.5 33.0 193775.5 430385.3 57.0 190833.9 431779.6 81.0 188386.5 429342.4

10.0 196175.8 431053.6 34.0 193696.3 430512.8 58.0 190689.9 431737.6 82.0 188242.5 429301.5

11.0 196128.1 430911.4 35.0 193613.7 430638.0 59.0 190547.8 431689.5 83.0 188093.8 429282.4

12.0 196087.1 430767.2 36.0 193527.7 430760.9 60.0 190408.5 431633.9 84.0 187943.9 429281.8

13.0 196052.8 430621.1 37.0 193439.1 430881.9 61.0 190274.3 431567.1 85.0 187794.2 429291.4

14.0 196019.9 430474.9 38.0 193349.7 431002.3 62.0 190143.6 431493.5 86.0 187645.4 429310.2

15.0 195965.7 430335.2 39.0 193259.3 431122.1 63.0 190015.7 431415.2 87.0 187500.0 429346.4

16.0 195889.1 430206.4 40.0 193168.2 431241.2 64.0 189892.0 431330.3 88.0 187357.9 429394.4

17.0 195794.3 430090.3 41.0 193074.4 431358.3 65.0 189774.5 431237.2 89.0 187217.9 429448.418.0 195687.0 429985.6 42.0 192975.5 431471.0 66.0 189663.8 431136.0 90.0 187081.7 429511.1

19.0 195567.3 429895.4 43.0 192864.3 431571.5 67.0 189561.2 431026.6 91.0 186951.2 429585.0

20.0 195436.8 429821.6 44.0 192740.4 431655.9 68.0 189467.8 430909.3 92.0 186823.1 429663.0


Nr Longitude Latitude Nr Longitude Latitude Nr Longitude Latitude Nr Longitude Latitude

21 0 195298 8 429763 0 45 0 192607 0 431724 2 69 0 189386 4 430783 4 93 0 186696 7 429743 8

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21.0 195298.8 429763.0 45.0 192607.0 431724.2 69.0 189386.4 430783.4 93.0 186696.7 429743.8

22.0 195155.0 429720.7 46.0 192466.3 431776.1 70.0 189318.3 430649.8 94.0 186573.3 429829.0

23.0 195006.9 429697.9 47.0 192321.9 431816.5 71.0 189235.7 430525.3 95.0 186453.5 429919.2

24.0 194857.0 429695.2 48.0 192175.0 431846.7 72.0 189157.1 430397.6 96.0 186334.7 430010.9

97.0 186216.5 430103.2 121.0 183215.5 432025.7 145.0 179747.2 432925.6 169.0 176374.2 434099.4

98.0 186099.2 430196.7 122.0 183069.4 432059.7 146.0 179600.7 432958.2 170.0 176226.8 434127.0

99.0 185983.9 430292.6 123.0 182921.7 432085.5 147.0 179453.5 432986.7 171.0 176078.1 434146.7

100.0 185869.7 430389.8 124.0 182773.4 432108.5 148.0 179305.3 433010.2 172.0 175928.6 434158.5

101.0 185756.0 430487.7 125.0 182625.0 432129.9 149.0 179156.9 433032.1 173.0 175778.8 434166.1

102.0 185642.3 430585.5 126.0 182476.3 432149.9 150.0 179009.0 433056.8 174.0 175628.8 434169.8

103.0 185528.1 430682.7 127.0 182327.4 432167.7 151.0 178861.5 433084.3 175.0 175478.8 434169.0

104.0 185413.3 430779.3 128.0 182178.2 432183.6 152.0 178714.6 433114.5 176.0 175328.9 434163.3

105.0 185298.0 430875.2 129.0 182029.3 432201.8 153.0 178568.7 433149.2 177.0 175179.3 434153.3

106.0 185182.1 430970.5 130.0 181881.1 432224.5 154.0 178425.4 433193.3 178.0 175029.4 434148.7

107.0 185065.7 431065.1 131.0 181733.9 432253.5 155.0 178286.3 433249.3 179.0 174879.4 434150.7

108.0 184948.2 431158.4 132.0 181588.0 432288.3 156.0 178150.0 433311.9 180.0 174729.5 434155.2

109.0 184827.3 431247.1 133.0 181443.5 432328.6 157.0 178015.7 433378.7 181.0 174579.6 434161.4

110.0 184703.5 431331.8 134.0 181300.3 432373.2 158.0 177883.0 433448.6 182.0 174429.8 434169.2

111.0 184578.2 431414.3 135.0 181158.4 432421.6 159.0 177751.2 433520.4 183.0 174280.1 434178.4

112.0 184451.3 431494.2 136.0 181017.6 432473.5 160.0 177619.4 433591.9 184.0 174130.5 434189.4

113.0 184321.9 431570.2 137.0 180878.1 432528.5 161.0 177486.5 433661.6 185.0 173981.1 434203.1

114.0 184190.2 431642.0 138.0 180739.6 432586.3 162.0 177352.9 433729.7 186.0 173832.1 434220.0

115.0 184056.7 431710.2 139.0 180600.9 432643.3 163.0 177218.6 433796.5 187.0 173683.5 434240.3

116.0 183921.3 431774.7 140.0 180461.0 432697.5 164.0 177083.5 433861.6 188.0 173535.8 434266.6

117.0 183783.8 431834.7 141.0 180320.3 432749.4 165.0 176947.2 433924.2 189.0 173389.0 434297.5

118.0 183644.4 431890.0 142.0 180178.8 432799.3 166.0 176807.3 433978.4 190.0 173242.3 434328.6



119.0 183502.9 431939.9 143.0 180036.3 432845.9 167.0 176664.6 434024.6 191.0 173095.5 434359.6

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119.0 183502.9 431939.9 143.0 180036.3 432845.9 167.0 176664.6 434024.6 191.0 173095.5 434359.6

120.0 183359.8 431985.0 144.0 179892.5 432888.5 168.0 176520.2 434065.1 192.0 172948.7 434390.6

193.0 172802.0 434421.7 217.0 169222.2 434574.0 241.0 165694.8 434455.3 265.0 162310.4 433410.5

194.0 172655.2 434452.7 218.0 169074.7 434601.1 242.0 165558.4 434393.0 266.0 162160.8 433421.9

195.0 172508.5 434483.7 219.0 168926.7 434625.5 243.0 165425.5 434323.5 267.0 162011.3 433434.3

196.0 172361.0 434511.1 220.0 168778.6 434649.1 244.0 165295.1 434249.5 268.0 161862.0 433448.3

197.0 172212.4 434531.2 221.0 168630.4 434672.4 245.0 165165.4 434174.1 269.0 161712.8 433464.4

198.0 172063.0 434544.9 222.0 168482.4 434696.7 246.0 165034.1 434101.6 270.0 161563.6 433479.2

199.0 171913.2 434553.1 223.0 168335.1 434725.3 247.0 164901.5 434031.4 271.0 161413.9 433489.2

200.0 171763.3 434557.9 224.0 168188.7 434757.9 248.0 164768.4 433962.2 272.0 161263.9 433491.0

201.0 171613.3 434560.2 225.0 168041.5 434786.9 249.0 164634.7 433894.2 273.0 161114.0 433488.2

202.0 171463.3 434560.2 226.0 167893.4 434810.0 250.0 164500.2 433827.8 274.0 160964.1 433481.8

203.0 171313.4 434558.5 227.0 167744.1 434824.2 251.0 164363.9 433765.3 275.0 160814.5 433470.8

204.0 171163.4 434555.9 228.0 167594.3 434832.4 252.0 164225.0 433708.6 276.0 160665.3 433455.2

205.0 171013.4 434554.6 229.0 167444.3 434835.4 253.0 164083.9 433657.8 277.0 160516.6 433436.1

206.0 170863.4 434556.3 230.0 167294.4 434832.2 254.0 163940.7 433613.2 278.0 160369.0 433409.1

207.0 170713.5 434562.4 231.0 167144.7 434822.1 255.0 163795.7 433574.7 279.0 160222.2 433378.5

208.0 170563.6 434565.9 232.0 166995.8 434804.1 256.0 163649.5 433541.3 280.0 160075.5 433347.1

209.0 170413.7 434561.8 233.0 166847.8 434780.2 257.0 163502.4 433512.2 281.0 159929.5 433312.9

210.0 170265.1 434541.8 234.0 166700.4 434751.9 258.0 163354.6 433486.3 282.0 159787.2 433265.7

211.0 170116.3 434522.9 235.0 166553.8 434720.2 259.0 163206.4 433463.1 283.0 159647.4 433211.3

212.0 169966.7 434512.9 236.0 166408.0 434685.2 260.0 163057.8 433442.5 - - -

213.0 169816.7 434508.4 237.0 166262.9 434647.3 261.0 162908.9 433424.6 - -

214.0 169666.7 434508.2 238.0 166118.6 434606.1 262.0 162759.6 433409.9 - - -

215.0 169517.3 434520.2 239.0 165975.6 434560.7 263.0 162610.0 433399.4 - - -

216.0 169369.4 434545.1 240.0 165834.3 434510.6 264.0 162460.1 433401.3 - - -


Transformed Centreline Coordinate for Waal River

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1.0 278.9 327.6 25.0 1997.1 3138.6 49.0 5008.0 1478.9 73.0 7620.3 3567.0

2.0 354.2 457.3 26.0 2146.8 3130.5 50.0 5156.9 1497.2 74.0 7651.9 3713.6

3.0 425.7 589.1 27.0 2295.0 3107.6 51.0 5305.3 1518.9 75.0 7665.7 3862.7

4.0 493.5 722.9 28.0 2440.5 3071.5 52.0 5452.8 1546.0 76.0 7704.3 4007.25.0 557.3 858.7 29.0 2582.0 3022.0 53.0 5599.0 1579.6 77.0 7770.4 4141.7

6.0 614.9 997.2 30.0 2711.3 2946.5 54.0 5743.9 1618.3 78.0 7849.3 4269.3

7.0 665.2 1138.5 31.0 2823.7 2847.3 55.0 5887.4 1662.1 79.0 7941.1 4387.8

8.0 708.1 1282.2 32.0 2928.8 2740.4 56.0 6029.0 1711.5 80.0 8044.8 4496.0

9.0 743.9 1427.8 33.0 3027.9 2627.8 57.0 6167.3 1769.4 81.0 8160.3 4591.0

10.0 772.9 1575.0 34.0 3127.9 2516.0 58.0 6301.8 1836.0 82.0 8296.1 4654.3

11.0 795.0 1723.3 35.0 3230.9 2406.9 59.0 6433.3 1908.1 83.0 8439.6 4697.4

12.0 810.3 1872.6 36.0 3336.9 2300.8 60.0 6560.6 1987.3 84.0 8587.4 4722.8

13.0 818.6 2022.3 37.0 3445.0 2196.8 61.0 6681.1 2076.6 85.0 8736.6 4738.414.0 826.8 2172.0 38.0 3553.8 2093.6 62.0 6797.0 2171.8 86.0 8886.4 4744.5

15.0 857.8 2318.6 39.0 3663.3 1991.1 63.0 6909.3 2271.2 87.0 9035.8 4732.5

16.0 912.7 2458.0 40.0 3773.6 1889.4 64.0 7016.1 2376.5 88.0 9184.0 4709.2

17.0 987.2 2588.1 41.0 3886.3 1790.4 65.0 7115.5 2488.8 89.0 9331.0 4679.6

18.0 1075.9 2708.9 42.0 4003.5 1696.9 66.0 7206.6 2607.9 90.0 9475.8 4640.3

19.0 1179.3 2817.4 43.0 4131.0 1618.1 67.0 7288.3 2733.7 91.0 9616.8 4589.4

20.0 1296.0 2911.5 44.0 4268.0 1557.3 68.0 7359.5 2865.6 92.0 9756.3 4534.2

21.0 1422.5 2992.0 45.0 4411.5 1514.0 69.0 7417.2 3004.0 93.0 9894.6 4476.1

22.0 1557.5 3057.3 46.0 4559.2 1487.9 70.0 7460.5 3147.6 94.0 10030.6 4412.823.0 1699.9 3103.8 47.0 4708.4 1473.5 71.0 7522.1 3283.9 95.0 10164.0 4344.3

24.0 1847.4 3130.9 48.0 4858.4 1469.7 72.0 7576.9 3423.5 96.0 10296.6 4274.3


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97.0 10428.9 4203.5 121.0 13716.1 2827.3 145.0 17287.3 2535.2 169.0 20811.5 1957.0

98.0 10560.4 4131.4 122.0 13865.8 2819.5 146.0 17437.2 2528.5 170.0 20961.5 1955.5

99.0 10690.4 4056.4 123.0 14015.8 2819.8 147.0 17587.1 2525.9 171.0 21111.3 1962.1

100.0 10819.5 3980.2 124.0 14165.8 2822.6 148.0 17737.1 2528.5 172.0 21260.6 1976.5101.0 10948.3 3903.2 125.0 14315.7 2827.0 149.0 17887.1 2532.2 173.0 21409.5 1994.9

102.0 11077.1 3826.3 126.0 14465.6 2832.9 150.0 18037.0 2533.0 174.0 21557.8 2017.2

103.0 11206.3 3750.2 127.0 14615.4 2841.0 151.0 18187.0 2531.0 175.0 21705.4 2044.0

104.0 11336.0 3674.8 128.0 14765.1 2851.0 152.0 18337.0 2526.3 176.0 21852.0 2075.6

105.0 11466.0 3600.1 129.0 14914.9 2858.3 153.0 18486.6 2516.6 177.0 21997.8 2111.1

106.0 11596.5 3526.1 130.0 15064.8 2860.9 154.0 18635.3 2496.9 178.0 22144.8 2140.7

107.0 11727.5 3452.9 131.0 15214.8 2857.2 155.0 18781.8 2464.9 179.0 22292.9 2164.3

108.0 11859.3 3381.3 132.0 15364.5 2847.5 156.0 18926.7 2426.2 180.0 22441.4 2185.4

109.0 11993.8 3315.0 133.0 15513.7 2832.3 157.0 19070.4 2383.1 181.0 22590.1 2204.9110.0 12130.4 3252.9 134.0 15662.4 2812.6 158.0 19213.1 2336.9 182.0 22739.1 2222.8

111.0 12268.0 3193.4 135.0 15810.5 2788.9 159.0 19355.1 2288.7 183.0 22888.2 2239.2

112.0 12406.9 3136.6 136.0 15958.0 2761.7 160.0 19497.3 2240.9 184.0 23037.4 2253.9

113.0 12547.4 3084.2 137.0 16104.9 2731.3 161.0 19640.2 2195.1 185.0 23187.0 2265.8

114.0 12689.5 3036.3 138.0 16251.1 2697.9 162.0 19783.5 2151.0 186.0 23336.7 2274.5

115.0 12832.9 2992.2 139.0 16397.6 2665.7 163.0 19927.3 2108.3 187.0 23486.6 2279.6

116.0 12977.4 2952.1 140.0 16544.7 2636.4 164.0 20071.6 2067.4 188.0 23636.6 2278.6

117.0 13123.2 2916.8 141.0 16692.3 2609.5 165.0 20216.8 2029.5 189.0 23786.5 2273.2

118.0 13270.2 2886.7 142.0 16840.3 2584.8 166.0 20363.9 2000.7 190.0 23936.4 2267.8119.0 13418.1 2862.1 143.0 16988.7 2563.5 167.0 20512.5 1980.0 191.0 24086.3 2262.3

120.0 13566.8 2842.5 144.0 17137.8 2546.5 168.0 20661.8 1965.3 192.0 24236.2 2256.9



193.0 24386.1 2251.5 217.0 27939.0 2714.9 241.0 31392.4 3441.5 265.0 34549.3 5046.5

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194.0 24536.0 2246.0 218.0 28089.0 2713.7 242.0 31515.8 3526.8 266.0 34698.6 5060.8

195.0 24685.9 2240.6 219.0 28239.0 2715.0 243.0 31634.6 3618.4 267.0 34848.0 5074.1

196.0 24835.9 2239.4 220.0 28389.0 2717.2 244.0 31750.2 3713.9 268.0 34997.6 5085.8

197.0 24985.8 2245.5 221.0 28539.0 2719.6 245.0 31865.2 3810.2 269.0 35147.3 5095.4

198.0 25135.2 2258.0 222.0 28689.0 2720.9 246.0 31982.3 3903.9 270.0 35296.9 5106.6

199.0 25284.2 2275.8 223.0 28838.9 2717.6 247.0 32100.9 3995.8 271.0 35446.0 5122.9

200.0 25432.7 2297.0 224.0 28988.8 2710.6 248.0 32220.2 4086.7 272.0 35593.9 5147.2

201.0 25580.8 2320.6 225.0 29138.7 2707.5 249.0 32340.3 4176.5 273.0 35741.2 5175.9

202.0 25728.6 2346.4 226.0 29288.7 2710.7 250.0 32461.6 4264.8 274.0 35887.7 5208.1

203.0 25876.0 2373.8 227.0 29438.2 2722.8 251.0 32585.4 4349.4 275.0 36033.1 5245.0

204.0 26023.3 2402.0 228.0 29587.1 2740.7 252.0 32712.7 4428.8 276.0 36177.3 5286.1

205.0 26170.9 2428.9 229.0 29735.3 2763.8 253.0 32843.3 4502.6 277.0 36320.6 5330.6

206.0 26319.0 2452.7 230.0 29882.4 2793.1 254.0 32976.9 4570.7 278.0 36461.1 5383.0

207.0 26467.7 2472.3 231.0 30028.0 2829.1 255.0 33113.3 4633.2 279.0 36600.5 5438.5

208.0 26616.0 2494.8 232.0 30171.5 2872.8 256.0 33251.8 4690.8 280.0 36739.6 5494.5

209.0 26762.8 2525.3 233.0 30313.2 2922.0 257.0 33391.8 4744.5 281.0 36877.3 5553.8

210.0 26905.6 2571.2 234.0 30453.4 2975.3 258.0 33533.0 4795.1 282.0 37009.1 5625.3

211.0 27049.1 2614.7 235.0 30592.3 3032.0 259.0 33675.1 4843.2 283.0 37137.4 5703.1

212.0 27194.9 2649.8 236.0 30729.9 3091.6 260.0 33818.0 4888.8 - - -

213.0 27341.9 2679.6 237.0 30866.3 3154.0 261.0 33961.7 4931.8 - - -

214.0 27489.7 2705.2 238.0 31001.2 3219.5 262.0 34106.3 4971.6 - - -

215.0 27639.1 2717.7 239.0 31134.2 3288.9 263.0 34252.1 5007.0 - - -

216.0 27789.1 2718.1 240.0 31264.7 3362.8 264.0 34400.2 5030.2 - - -

Henok

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