an analysis of mathematical and computer modelling of drainage basin characteristics

AN ANALYSIS OF MATHEMATICAL

AND COMPUTER MODELLING OF DRAINAGE

BASIN CHARACTERISTICS

Candidate number

41223

Word count

11,130

Final year

2007

1

TABLE OF CONTENTS Page

Abstract ........................................................................................................................... 3

Chapter 1: Introduction and objectives .................................................................. 4

1.1 Background .............................................................................................. 4

1.2 Mathematical modelling of drainage basins ............................................ 5

1.3 Computer modelling of drainage basins .................................................. 7

1.4 Aim and objectives .................................................................................. 9

Chapter 2: Study area ............................................................................................. 10

2.1 Choice of study area ............................................................................... 10

2.2 Study area background ........................................................................... 11

2.3 Naming convention ................................................................................ 12

Chapter 3: Mathematical modelling ...................................................................... 18

3.1 Methodology .......................................................................................... 18

3.2 Analysis of results .................................................................................. 31

3.3 Analysis of methodology ....................................................................... 32

Chapter 4: Computer modelling ............................................................................ 34

4.1 Methodology .......................................................................................... 34

4.2 Analysis of results .................................................................................. 38

4.3 Analysis of methodology ....................................................................... 39

Chapter 5: Comparison with objectives ................................................................ 42

5.1 Objective 1 ............................................................................................. 42

5.2 Objective 2 ............................................................................................. 45

5.3 Objective 3 ............................................................................................. 48

Chapter 6: Conclusions ........................................................................................... 71

Acknowledgements ...................................................................................................... 73

References .......................................................................................................................74

2

ABSTRACT

The study aimed to test basic mathematical and computer models of drainage basin

morphology, with the overall aim of allowing better-informed decisions to be made in

future analyses of a similar type. The key objectives were (1) to perform analyses of

stream long profiles using mathematical modelling and (2) to perform analyses of whole

drainage basin slope and elevation characteristics using computer modelling. The third

objective compared and contrasted these results to reach a conclusion.

Mathematical models were assessed by examining the extent to which stream long

profiles fit a concave-upward curve, which is diagnostic of equilibrium. Data was

collected through the use of fieldwork and analysed in Microsoft Excel.

Computer models were tested by analysing slope and elevation data output from a

Geographical Information System (GIS). The data for these analyses was downloaded

from Edina Digimap.

Several conclusions were found. (1) The mathematical model of stream profile is reliable.

However, research must ensure that error in the input data does not influence the results.

(2) The mathematical model allows analyses not possible through the use of GIS.

(3) Conversely, GIS allows analyses not possible through the use of the mathematical

model. (4) The best way to avoid error in both models is to use them in tandem and

compare the results.

Overall, the accuracy and reliability of the mathematical and computer models was found

to be a key consideration for future research, with both found to have strengths and

weaknesses.

Candidate number

41223

3

1. Introduction and Objectives

1.1 Background

The study of water courses and their drainage basins has been an important area of

research in Geography for over a century, having applications for a range of uses such as

water quality monitoring and projection (Ward and Robinson, 2000:p.296-342) and flood

and drought prediction (Jones, 1997:p.96-119). Modelling has been central to the

mapping, analysis and prediction of patterns within these areas, as their general trends

can be reapplied and tested many times in different areas. This is in comparison to studies

which examine new findings rather than test for results established by previous models,

which requires more time and has less scalability. Methods of collection, analysis and

modelling of data have evolved over time, encompassing mathematical and computer

modelling.

While the increasing ubiquity of computers blurs the line between mathematical and

computer models, they can be treated as loosely discrete entities. Mathematical models

are based in equations, calculations and other “traditional” mathematical logic. While

Microsoft Excel or Minitab may be used by researchers for ease, the underlying process

being carried out is based upon “first-principles” mathematics. However, questions

remain about the accuracy of many equations, including those of drainage basin form.

Computer models of drainage basins aim to interpret mathematical models by

representing them in specialist software. However, in addition to uncertainties about the

mathematical equations they have additional complexities. Computers are based upon the

fundamental mathematics of binary logic, but the intricate architectures built on top of

this in modern machines are far more difficult to define. The models which run on these

computers – Geographical Information Systems (GIS) – are mathematical models

adapted for calculation and display by computer. However, the best way to represent

these data in the virtual environment of GIS, rather than as written equations, poses

problems of accuracy and comparability with mathematical models.

4

There are definite issues with techniques used in modelling drainage basins. With

mathematical models, these centre on uncertainties over the accuracy of established

formulae. With computer models, these issues are with the representation of complex

observations in a virtual three-dimensional environment. While individual studies tend to

primarily use one method or another, there is a lack of holistic literature on the accuracy

and appropriateness of the methodologies when used together in the area of drainage

basin analysis. Literature is available, however, which studies the methods separately.

1.2 Mathematical modelling of drainage basins

The long profile of a water course (also referred to as the longitudinal profile, stream

profile, channel profile, river profile or long river profile) graphically describes the

relationship between height (H) and distance downstream (L) along the channel course

(e.g. Knighton, 1998:p.242). An important early text about the modelling of river long

profiles was Davis (1902) with a discussion of grading (pp.86-109) and base level (pp.77-

86). Both terms are key, with the former describing the standard equilibrium-graded

shape of a river profile and the latter marking the lowest extent in the area studied.

Since Davis (1902), various studies have demonstrated that a river profile is generally

graded into a concave-upward shape (Gregory, 1994:p.308-9), i.e. a power, logarithmic

or exponential function. Leopold and Langbein (1962:pp.2-7) study the entropy of river

profiles, demonstrating that the most efficient distribution of energy is the most probable

state (pp.2). There is ongoing discussion as to which function of the concave-upward

form best approximates stream profiles under varying conditions. Hack (1957: p.69-70),

for example, argues that H as a logarithmic function of L provides a good fit where

sediment particle size is relatively constant while a power function relationship between

slope (s) and L applies where particle size changes steadily downstream. Despite the

debate, it is widely accepted that concave-upward formulae provide an accurate fit for the

majority of rivers and streams (e.g. Snow and Slingerland, 1987:pp.15; Gregory,

1994:p.308-9; Knighton, 1998:p.242-245).

These concave-upward river profiles grade towards a base level (Gregory, 1994:p.308-9).

Davis’ (1902:pp.77-86) discussion, which aimed to clarify the definition of base level,

5

demonstrated that defining this term proves problematic due to its subjectivity. Davis’

(1902:pp.84) proposed that base level be defined as “simply … the level base with

respect to which normal sub-aerial erosion proceeds” and Kennedy (1994:p.47-8) notes

that this has become the generally accepted meaning, whether local or regional (Gregory,

1994:p.308-9). As an example, the base level when studying a drainage basin is the outlet

point, the lowest point in the long profile to which all water flows. For both base level

and grading, Davis’ (1902) ideas still prove valid and as such the concept of a concave-

upward river grading towards a base level forms the basis of models of river long profile.

While it is accepted, in line with Davis (1902), that water courses grade smoothly

towards base level, the conclusion that this is diagnostic of the equilibrium state – a

balance between the erosive force of water and resistance of the land – has been

challenged (Knighton, 1998:p.244-245). For instance, while Hack (1957: p.69-70)

demonstrates the applicability of concave-upward curves to river profiles, he concludes

that factors such as sediment size are important. Knighton (1998:p.245-260) discusses

factors which have previously or are currently thought to influence channel grading and

closeness to equilibrium (e.g. bed material size, geology, sediment load and channel

degradation), demonstrating that the variables are interlinked in their influence.

Equilibrium state will only occur when sediment discharge is roughly constant and in

balance with the river system, as Knighton (p.245) notes that channel slope depends on

adjustment caused by sediment. In turn, sediment load and sediment discharge are

affected by bed material characteristics such as size and geology (p.245).

Despite varying analyses of interlinkages between variables, established conclusions

regarding channel form nonetheless rely on the way in which these factors affect the form

of the river long profile, with constant sediment discharge giving a smooth shape. It is

clear that a simple analysis of long profile form does not accurately take into account

smaller-scale spatial factors (e.g. convexities such as pools and riffles – Gregory, 1994,

p.308-9) or temporal influence (i.e. change in profile form over time). However, the work

since Davis (1902) suggests that present long profile shape is a good indicator of current

state. As such, the principal that a concave-upward curve grading towards a base level is

6

indicative of the equilibrium or near-equilibrium remains unchanged and an assessment

of channel closeness equilibrium can be made from observed long profiles alone.

1.3 Computer modelling of drainage basins

While the concave-upward mathematical models discussed in section 1.2 can be easily

expressed and calculated, the adaptation of these to a virtual environment is more

difficult. Mathematical models form the basis of the calculations used in GIS, but often

represent only a select number of variables or pre-existing conditions. In addition to the

issues of expressing calculations within the computer, representation of these data for the

purposes of storage and display proves problematic. As Schuurman (2004:p.32) notes,

computerised models may present to the user an effortless representation of time and

space, but their implementation is restricted to binary digits. As such, literature on GIS

focuses on the underlying software architecture and resulting issues for the adaptation of

the mathematical model to the virtual binary one.

GIS must allow representation of a variety of spatial and temporal information, ranging in

scope across a wide variety of geographical and non-geographical disciplines (Fisher and

Unwin, 2005:p.1-16). Drainage basin variables include elements such as vegetation,

settlement, topography, soil type, climate, biodiversity, water and geology, each with up

to 3 spatial dimensions and a temporal one. While temporal data can only vary in one

dimension, representation of spatial information is more complex. It must allow for a

wide range of uses (e.g. Wrigley et al., 1996, zonal and areal patterns in ecology;

Macmillan, 1996:p153-165, urban simulation) and be adaptable to varying completeness

and accuracy of information, not just within studies as a whole but in individual variables

(Walford, 2002:p.22-30).

The overlay method treats each element as a separate layer mapped to universal co-

ordinates (general overviews in Worboys, 1995:p.9-15; Lyon, 2003:p.1; Schuurman,

2004:p.3-4; software-specific examples in Maidment, 2002:p.17, ArcHydro for ArcGIS;

Allan and Peterson, 1994:pp.1-3, IDRISI). To build a model of the variables in a long

profile, a water course layer is overlaid on a topology layer. Both contain spatial

information in all three dimensions, and if layers were to contain data of change over

7

time (e.g. topographical alterations through erosion) temporal values would also be

included. The number of layers can become rather large, as when using ArcHydro for

delineating watersheds (Maidment, 2002:p.17).

Underlying this layered visual representation, the majority (Worboys, 1995:p.84) of GIS

software uses a relational database structure (Worboys, 1995:p.46-84; Schuurman,

2004:p.59-60). This allows cross-referencing of layers by storing data in tables and

creating linkages between columns. For example, within a drainage basin study

elevations may be stored in one table, while other tables of data (e.g. water courses,

slope) gain their information on elevation through this link. This relational model allows

variables being studied to interact, without duplicating information or treating a variable

as if it were unrelated to its surroundings (e.g. Worboys, 1995:p.46-84). This system

allows linkages between different data formats, as tables can store information

representative of a variety of file types.

The adaptation of the mathematical model to the GIS environment depends upon the way

in which mathematical formulae are translated into computer code. Not only must the

values of variables within mathematical formulae be calculated, but so too must

information that is taken for granted as easily-measurable in the world of mathematical

modelling. Key variables such as slope angles and stream courses, which can be

measured using instruments in the field (e.g. Tarboton, 1998:p1, slide 7) are complex to

establish and must be accurately calculated and modelled by GIS before mathematical

models can be used in the virtual environment.

The process of calculating these variables also increases the risk of error in computer

models. Elevations are often low in resolution or have dispersed sample points, and as

such may require interpolation to approximate values between known heights (ESRI,

2006a). Accuracy of these approximations is further compromised by the variety of

interpolation techniques and the complexities of selecting the most applicable one. This,

in turn, can cause problems calculating slope angles.

Flow directions of water are no easier to calculate. Problems with elevation and slope

angle data feed into flow direction calculations which can also be performed using

8

multiple formulae (Tarboton, 1997; Tarboton, 1998:p1, slide 7). The resultant stream

networks delineated by GIS thus vary depending on the methods employed.

As Fotheringham et al. (2000:p.31) note, changes to underlying software architecture of

GIS systems occurs at a variable pace as software developers change the code. Combined

with the multiple techniques of the computer models, it is clear that comparability and

accuracy may be a key problem when using this technique to study drainage basins.

1.4 Aim and objectives

When applied to the study of drainage basins, mathematical and computer models each

have clear strengths and weaknesses. Mathematical models suffer from doubts over the

accuracy of their central formulae (section 1.2) while computer models have problems

representing the complex natural phenomena being measured (section 1.3). Each provides

a unique approach to studying drainage basins, using formulae and methods which are

difficult to replicate in the other. At the same time, both have uncertainties regarding how

accurately they represent phenomena observed in the field. Because of the uniqueness of

each method, the majority of studies use only one technique. This study will examine

three key objectives, with the overall aim of testing which methodology is most reliable

and accurate for basic morphological analyses of drainage basins. This will allow better-

informed decisions to be made in future analyses. The objectives of this study are to:

1. Use mathematical modelling to perform morphological analyses of drainage

basins (i.e. examine long profile form), to test the ability and reliability of the

method through an examination of its strengths and weaknesses.

2. Use computer modelling to perform morphological analyses of drainage basins,

testing the ability and reliability of the method through an examination of its

strengths and weaknesses.

3. Compare and contrast the results of the mathematical and computer modelling to

allow conclusions to be drawn about the best use of both methodologies in future

morphological analyses of drainage basins.

9

2. STUDY AREA AND BACKGROUND

2.1 Choice of study area

To test the first objective, the study area needed to allow an assessment of the extent to

which streams matched a concave-upward curve. The area needed to be sufficiently small

to allow fieldwork to be carried out on all water courses, but of a sufficient size that

enough readings could be taken of each to reach a meaningful conclusion. Whether or not

they fit into a concave-upward pattern, rivers follow a graded curved from source to

mouth (Davis, 1902). The study of an entire river was unfeasible, and so the most

practical option was to study river reaches from source to points further downwater.

As well as being limited by size, the choice of drainage basin was limited by

accessibility. In order to carry out fieldwork on many consecutive days, it had to be

within walking distance of the nearest road. While this meant that the drainage basin

would be within a few hours of access, it did pose safety issues in terms of ensuring that

it was not inaccessible or too remote in an emergency. As such, possible study areas were

limited to those close to established hiking routes.

To test the second objective, the study area needed to be available as digital data. While

topographical data exists for much of the globe, the resolution was of key importance,

potentially affecting the accuracy and significance of results. It was noted that the 10m

resolution of the available British data is high in comparison with many areas, and as

such the study would be best carried out in Britain.

Other characteristics of the study area were also important considerations. The key area

of study was to be the curvature and elevation of the drainage basin. In order to test the

computer and mathematical models, the number of variables which could potentially

explain the results needed to be limited. Conclusions were to be drawn on the basis that

the models applied to the streams were valid for the study area and allowed comparison

with established literature. The mathematical and computer models base their results on

slope and elevation alone, and so keeping other variables constant was key. Factors such

10

as geology, vegetation, climate, tectonic influences and land use were taken into account

for a wide number of potential study areas.

2.2 Study area background

The study area chosen was the upper section of the Burn of Sorrow in Dollar Glen,

Clackmannanshire, Scotland. The stream runs from the Ochill Hills to the village of

Dollar, located at the base of Dollar Glen. The location of the study area is illustrated in

Figures 2.1 through 2.4. The British National Grid (BNG) was the co-ordinate system

used throughout this study (Figures 2.1 and 2.2). Splitting areas into 100km² blocks

referenced by letters, it subdivides these areas into 10km² blocks referenced by grid co-

ordinate. Dollar is in grid square NS 9699.

The drainage basin contains one main stream (the Burn of Sorrow) and four main

tributaries. A key factor in the decision to use this as the study area was strong similarity

between the tributaries. Of the four, three fork approximately 40% of the distance

between their source and the Burn of Sorrow and have the same aspect, facing in a south-

westerly direction. As with the rest of the drainage basin, geology, land use and climate

are the same. Such a significant number of constants made this a highly suitable site.

The study area is approximately one hour’s walk from the nearest road, with a path

running up the valley. However, the Burn of Sorrow becomes a deep gorge between

~150m and ~250m above sea level. The base level used for the study area began above

the gorge for two reasons. First, it would have been unfeasible and unsafe to measure in

the gorge itself, which has no access and lies around 50m below the surrounding land.

Second, the gorge is a convexity not found in most water courses, and so would negate

the applicability of the models. As the study aimed to keep variables fixed, the base point

of the study was around 20m upstream of the gorge. This point, the outlet of the drainage

area studied, was the lowest point by elevation at which measurements were taken.

11

2.3 Naming convention

In order to easily reference the various streams within the drainage basin, a labelling

system was used whereby each fork in the stream network was referenced (Figure 2.5).

The base point was labelled B; where the tributaries met the Burn of Sorrow points were

labelled T; where tributaries forked points were labelled F; sources were labelled S.

Streams were numbered from B upstream, with the tributary nearest to B named tributary

1, the second tributary 2, the third tributary 3 and the last tributary 4. These numbers were

appended to the letters to refer to individual points. For example, the confluence of the

stream closest to the base point and the Burn of Sorrow is labelled T1. Each tributary

(with the exception of tributary 2) has multiple sources. As such, an additional lowercase

letter identified each source, as shown in Figure 2.5. Water courses were referred to as

the reach from one point to the next. For instance, the reach from the source of tributary 2

to the base point was referred to as S2a-B.

12

Length of major BNG squares = 100km. North at top of image.

Primary data © Crown Copyright 2006. Manipulated in MapInfo.

Figure 2.1:

Location of Dollar in relation to major cities in Scotland,

showing major BNG squares

13

Length of minor BNG squares = 10km. North at top of image.

Primary data © Crown Copyright 2006. Manipulated in MapInfo.

Figure 2.2:

Location of Dollar in relation to the Firth of Forth,

showing both major (solid) and minor (dotted) BNG gridlines

14

ELEVATION

High: 720m

Low: 7m

Length of square = 20km. North at top of image.

Primary data downloaded from Edina Digimap and manipulated

using ESRI ArcGIS. © Crown Copyright/database right 2006. An

Ordnance Survey/EDINA supplied service.

Figure 2.3:

Location of the Burn of Sorrow in relation to Dollar

15

Dollar

Burn of Sorrow

ELEVATION

High: 720m

Low: 7m

Length of each side of base = 20km. North illustrated on image.




Figure 2.4:

Three dimensional view of the area surrounding Dollar Glen,

with the study area shaded in light blue

16

N

Not to scale or orientated to North.

See Figure 2.3 for actual representation.

Original image

Figure 2.5:

Schematic diagram of the streams within the study area

17

T1 F1

T2 F2

T3 F3

T4

S1a

S1b

S2a

S3a

S3b

S4a S4b S4c

B

3. MATHEMATICAL MODELLING

3.1 Methodology

Mathematical modelling of the long profile was used to assess the extent to which

streams within the drainage basin were in equilibrium. In order to plot long profiles of all

watercourses in the study area, measurements of distance from source and height above

base level were required for the Burn of Sorrow and its tributaries.

A Garmin eTrex Global Positioning System (GPS) device was to be used to read the

BNG reference of each point, as GPS provides the most accurate possible location in the

field. GPS could provide grid references for any point, whether taken at regular intervals

along the channel paths or at breaks in slope. With GPS able to measure location

regardless, the ability to measure elevations at each point was the critical factor. The GPS

device included an altimeter, but non-differentiated GPS (non-DGPS) devices such as the

model used are unreliable for accurate height measurements. Initially, it was thought that

data could be gathered by taking elevation and location readings at regular intervals along

each channel. However, without the ability to measure elevation for any arbitrary point,

this was impossible.

Instead, a clinometer was used to record angles between consecutive breaks of slope.

Following the path of a stream, each angle was taken by measuring from the current

position to the next break of slope, measuring from that break of slope to the next etc. At

each point, GPS was used to record the grid reference. Height change and thus

cumulative height (i.e. elevation of each point) could later be derived from this

information. Using variable distances between points was a more feasible option than

marking points at regular intervals and reading between them regardless of whether there

was a change of slope. In addition, the variable distance between breaks of slope allowed

accurate recording of areas where change was very rapid, and eliminated superfluous

readings where slope stayed constant.

18

The GPS device logged locations as 10-figure BNG references, allowing co-ordinates of

1m accuracy. However, despite high accuracy due to variable distances between points,

error was introduced by the GPS. This is dependent upon factors such as weather

conditions and clarity of line-of-sight to GPS satellites. When measuring co-ordinates, the

GPS device gives this error as a possible range within which the true point lies. For

example, an error of 6m would place the real location of the point somewhere within a

6m radius. The recording of this data in addition to location and angle meant that at each

break of slope, five values were recorded – BNG grid square (NN or NS), BNG x co-

ordinate, BNG y co-ordinate, GPS range and angle to next break of slope.

Once readings had been taken along all streams in the drainage basin, these data were

converted into a format suitable for plotting long profiles, i.e. cumulative height for each

point, with each point references as a distance from source. First, level horizontal

distances (the distance between points in the x and y planes) between consecutive GPS

19

ΔH

d

α

Figure 3.1:

Diagram showing measured angle (α), horizontal

distance (d) and change in height (ΔH), which were

used as values in the Sine Rule

Original diagram

Line of sight

points were calculated. To do this, x and y co-ordinates were altered to 6-figures to

represent BNG tiles, as readings crossing the between NN and NS caused the y co-

ordinate to change from 99999 to 00000. In order to correct this numerically, the 1 was

prefixed to NS y values and 0 to NN y values so that, for example, moving from NS

94955 99989 to NN 94960 00026 became 094955 099989 to 094960 100026. This

additional number has no meaning for the BNG co-ordinate, but allowed subtraction of

NS y co-ordinates from NN. Using this method, level horizontal distances were

calculated by subtracting the lower x and y co-ordinates from higher ones. These

distances, viewed from above, represent two sides of a right-angled triangle. The

hypotenuse – the direct distance between the measured points – was computed using

Pythagoras’ Theorem.

Second, height change (ΔH) between GPS points was calculated. Again, this can be

visualised as a right-angled triangle, but positioned in the z plane. The known values were

the measured angle from one break in slope to the next (α) and horizontal distance (d) as

calculated in the previous step. The unknown change in height (ΔH) is marked along with

the known variables in Figure 3.1. As the sizes of the right angle (90º) and a second angle

(α) were known, the third angle could be calculated. As the angles of a triangle total 180º,

the third angle equals (90 – α)º.

Substituting these values into the Sine Rule,

(Equation 3.1)

gives,

(Equation 3.2)

(Equation 3.3)

Microsoft Excel was used to repeat this process for the angle between each set of two

consecutive measured points on each stream, providing cumulative height measurements

20

with locations corresponding to each distance measurement. The distance was cumulated

to give distance from source rather than distance between points, and long profiles plotted

for each water course (Figures 3.2, 3.4, 3.6, 3.8, 3.10, 3.12, 3.14 and 3.16).

However, these long profile graphs neglect GPS error, an important consideration for the

reliability of the data. To illustrate the error ranges, Pythagoras’ Theorem and the Sine

Rule (Equation 3.1) were repeated twice again. However, in each case d was adjusted to

represent either the minimum distance between two measured points or the maximum.

The minimum distance was calculated by subtracting the error values of each set of two

consecutive points from d, the distance between them. Maximum distance was calculated

by adding the two error ranges to d. When the Sine Rule was repeated using the altered

values, the result was the highest and lowest possible height change between each set of

points. These values were represented on long profiles as y-error bars, with a line

extending vertically through each point to illustrate the possible range of its real position

(Figures 3.3, 3.5, 3.7, 3.9, 3.11, 3.13, 3.15, 3.17).

The long profiles shown in Figures 3.2 through 3.18 (and Figures 3.19 and 3.20,

discussed momentarily) were plotted in Microsoft Excel. The software only allowed the

addition of trendlines where statistically relevant. Interestingly, all trendlines were

concave-upward, in line with the literature (e.g. Gregory, 1994:p.308-9; Davis,

1902:pp.77-86; Hack, 1957: p.69-70).

Similar analysis and long profiles could potentially have been produced using computer

models. Reasons for using fieldwork are discussed in Chapter 4.

21

B

T1

F1

S1a

0

50

100

150

200

250

300

350

400

0 500 1000 1500 2000 2500 3000 3500

Distance from Source (m)

Hei

gh

t ab

ove

Bas

e (m

)

Figure 3.2: Long profile from S1a to B

0

50

100

150

200

250

300

350

400

0 500 1000 1500 2000 2500 3000 3500


Hei

gh

t ab

ove

Bas

e (m

)

Figure 3.3: Data points from S1a to B with trendline and error bars

H = 856.71 – 105.64 logeL r2 = 0.9914

(Both diagrams plotted from measured data)

22

B

T1

F1

S1b

0

50

100

150

200

250

300

350

0 500 1000 1500 2000 2500 3000 3500

Distance from source (m)

He

igh

t a

bo

ve

ba

se

(m

)

Figure 3.4: Long profile from S1b to B

0

50

100

150

200

250

300

350

0 500 1000 1500 2000 2500 3000 3500


Hei

gh

t ab

ove

bas

e (m

)

Figure 3.5: Data points from S1b to B with trendline and error bars

H = 679.06 – 82.678 logeL r2 = 0.961


23

B

T1

T2

F2

S2a

0

50

100

150

200

250

300

350

0 500 1000 1500 2000 2500 3000 3500


Hei

gh

t ab

ove

Bas

e (m

)


0

50

100

150

200

250

300

350

400

0 500 1000 1500 2000 2500 3000 3500


Hei

gh

t ab

ove

Bas

e (m

)


H = 596.79 – 73.571 logeL r2 = 0.9862


24

B

T1

T2

T3

F3

S3a

0

50

100

150

200

250

300

0 500 1000 1500 2000 2500 3000 3500


Hei

gh

t ab

ove

Bas

e (m

)


0

50

100

150

200

250

300

350

0 500 1000 1500 2000 2500 3000 3500


Hei

gh

t ab

ove

Bas

e (m

)


H = 465.61 – 56.186 logeL r2 = 0.9851


25

B

T1

T2

T3

F3

S3b

0

50

100

150

200

250

300

350

0 500 1000 1500 2000 2500 3000 3500


He

igh

t a

bo

ve

Ba

se

(m

)


0

50

100

150

200

250

300

350

400

450

0 500 1000 1500 2000 2500 3000 3500


Hei

gh

t ab

ove

Bas

e (m

)

Figure 3.11: Data points from S3b to B with trendline and error bars

H = 648.95 – 79.704 logeL r2 = 0.9852


26

B

T1

T2

T3

T4

S4a

0

50

100

150

200

250

300

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000


Hei

gh

t ab

ove

Bas

e (m

)


0

50

100

150

200

250

300

350

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000


Hei

gh

t ab

ove

Bas

e (m

)

Figure 3.13: Data points from S4a to B with error bars

Trendline not statistically relevant


27

B

S4b

F4T4

T3

T2

T1

0

50

100

150

200

250

300

350

0 500 1000 1500 2000 2500 3000 3500 4000 4500


He

igh

t a

bo

ve

Ba

se

(m

)


0

50

100

150

200

250

300

350

0 500 1000 1500 2000 2500 3000 3500 4000 4500


Hei

gh

t ab

ove

Bas

e (m

)

Figure 3.15: Data points from S4b to B with error bars



28

B

T1

T2

T3

T4F4

S4c

0

50

100

150

200

250

300

0 500 1000 1500 2000 2500 3000 3500 4000


Hei

gh

t ab

ove

bas

e (m

)

Figure 3.16: Long profile from S4c to B

0

50

100

150

200

250

300

0 500 1000 1500 2000 2500 3000 3500 4000


Hei

gh

t ab

ove

bas

e (m

)

Figure 3.17: Data points from S4c to B with error bars



29

B

T1

T2

T3

T4

0

50

100

150

200

250

300

0 500 1000 1500 2000 2500 3000 3500 4000


He

igh

t a

bo

ve

ba

se

(m

)

Figure 3.18: Long profile from T4 to B

0

50

100

150

200

250

300

0 500 1000 1500 2000 2500 3000 3500 4000


Hei

gh

t ab

ove

bas

e (m

)

Figure 3.19: Data points from T4 to T3 with trendline and error bars

H = 245.65 – 0.1429L r2 = 0.9898


30

3.2 Analysis of results

As Figures 3.2 through 3.11 show, the long profiles of the first three tributaries (S1a-B,

S1b-B, S2a-B, S3a-B and S3b-B) all produced a graded curve from source to base.

Figures 3.3, 3.5, 3.7, 3.9 and 3.11 illustrate that when a concave-upward trendline is

added, all three tributaries provide statistically significant fits (S1a-B, r2 = 0.9914; S1b-B,

r2 = 0.961; S2a-B, r2 = 0.9862; S3a-B, r2 = 0.9851 and S3b-B, r2 = 0.9852). Trendlines

pass through the majority of y-error bars.

Figures 3.12 through 3.17 show that this was not the case for the fourth tributary (S4a-B,

S4b-B and S4c-B). None of these profiles demonstrate a graded curve or statistically

significant trendline. All have convexities at T4-T3, with S4b-B and S4c-B having

convexities in their upper reaches. Without a statistically significant fit, there is no way to

assess whether a trendline would pass through the GPS y-error bars.

These results produce interesting evidence surrounding the development of the study

area. The three lower tributaries (Figures 3.2 to 3.11) all exhibit textbook concave-

upward curves (e.g. Davis, 1902:pp.77-86; Hack, 1957: p.69-70; Gregory, 1994:p.308-9)

at significance level p=5. None of these profiles contain convexities. This analysis,

despite the debate discussed in section 1.2, suggests that this mathematical model is

accurate. If a concave-upward fit is taken to show equilibrium, S1a-B, S1b-B, S2a-B,

S3a-B and S3b-B are in steady-state (Knighton, 1998, p.244).

Analysis of points upstream of T3 produces a less certain result. Figures 3.12 through

3.17 do not have statistically significant trends, concave-upward or otherwise. However,

the failure of these stream courses (S4a-b, S4b-B and S4c-B) to produce a significant

result can, in part, be attributed to a central cause. It is clear from Figures 3.12 through

3.17 that a convexity exists at T4-T3, influencing all three streams. When T4-B is plotted

(Figure 3.18) T4-T3 clearly produces a break in the graded profile. A linear trend (Figure

3.19) is most significant for T4-T3 (r2=0.9898). It appears that T4-T3 has a large

influence on the concavity of tributary 4.

31

In addition, convexities exist in S4a-T4, S4b-T4 and S4c-T4, and are likely to also have a

significant impact on curvature. The short length of these reaches means it is impossible

to assess the most accurate trendline. There are several explanations: T4-T3 may be the

only section in disequilibrium; T4-T3 may be a zone of transition between stable and

unstable, causing convexities in S4a-T4, S4b-T4 and S4c-T4; convexities in S4a-T4, S4b-

T4 and S4c-T4 may cause the T4-T3 convexity; or convexities in T4-T3, S4a-T4, S4b-T4

and S4c-T4 may all be influential. Whichever is correct, the fourth tributary is not in

equilibrium (Davis, 1902:pp.77-86; Hack, 1957: p.69-70; Gregory, 1994:p.308-9;

Knighton, 1998, p.244).

3.3 Analysis of methodology

Fieldwork to collect data for mathematical modelling is advantageous where doubts exist

about the accuracy of computer models (Chapter 4). However, fieldwork has key

disadvantages due to the possibility of error. While location and slope are theoretically

easily measurable, practicalities of fieldwork mean that there is a degree of error in each,

either with instrumentation (GPS error range; clinometer accuracy) or human error.

BNG co-ordinates allowed theoretical 1m accuracy. However, error introduced by GIS

meant that the accuracy of recorded co-ordinates was mixed due to variable error ranges.

The actual accuracy of the GPS co-ordinates varied between more and less accurate than

the digital data (Chapter 4). The error bars in Figures 3.3, 3.5, 3.7, 3.9, 3.11, 3.13, 3.15,

3.17 and 3.19 illustrate this. The accuracy gained through GIS will vary for other studies

with topology, location of study area and position of GPS satellites, but it has the

potential to be more accurate than the 10m-resolution digital data. GPS measurement can

therefore be either more accurate or less accurate in comparison to computer modelling.

Accurate measurement of elevation can be problematic. While DGPS devices provide

precise readings, they require two people to operate and can be time-consuming to use.

Standard GPS height readings, however, are unreliable. The methodology employed was

thus to use a clinometer to measure angles between breaks of slope, converting readings

into cumulative height. As with the use of GPS to measure location, there is potential for

this to be advantageous or otherwise. Clinometers have the advantage that the sight can

32

be aimed at any break in slope. However, various factors reduced the accuracy of this

method. The design of the clinometer meant that readings could only be taken to the

nearest 0.5º. Furthermore, accuracy of readings depends upon the accuracy of the user,

and careful attention to technique was required.

When problems with accuracy of angles are combined with issues of GPS error, there is

the potential for a large disparity between actual values and measured values. As noted in

section 3.1, this was reduced by re-running the calculations to give minimum and

maximum distances, and thus a range of possible heights. For instance, when measuring

the angle from A to B, if the GPS accuracy of A was 6m and the GPS accuracy of B was

17m, the total length of the measured gradient line has a range of ((B ± 6) – (A ± 17))m,

which equates to a potential error of ±23m.

In the case of this study, the closeness of S1a-B, S1b-B, S2a-B, S3a-B and S3b-B to a

concave-upward curve, and the confidence afforded to them due to high r2 values,

suggests that the readings were accurate. The logical explanations for S4a-B, S4b-B and

S4c-B being in disequilibrium mean it is unlikely that error had a significant effect.

In any case, while GPS and clinometer readings do have the potential advantage of high-

resolution measurements (in some cases, more so than computer models) issues of

accuracy must be borne in mind with similar studies.

33

4. COMPUTER MODELLING

4.1 Methodology

GIS was used to analyse slope and elevation within the drainage basin. Various software

packages are available, including MapWindow, MapInfo, ESRI ArcGIS and IDRISI.

Some (e.g. MapWindow, MapInfo) are designed for the display of data. They are capable

of data analysis, but to a limited degree. Others (e.g. ArcGIS, IDRISI) provide detailed

analytical abilities.

In addition to these software packages, pieces of software which add extra functionality

(“plugins”) are available. Two with good functionality for drainage basin analysis are

ESRI ArcHydro and TauDEM. ArcHydro is compatible with ArcGIS, while TauDEM is

compatible with both ArcGIS and MapWindow. For this reason, and because the

Geography Department holds a software license for both ArcGIS and ArcHydro, ArcGIS

9.1 was used in conjunction with both the ArcHydro and TauDEM plugins.

Raw data for the analysis were downloaded from Edina Digimap [accessed 10/07/06] as a

Digital Terrain Model (DTM). DTMs consist of a point shapefile, which contains a list of

co-ordinates and elevation data for numerous individual spatial points. Ordnance Survey

(OS) points are discrete, spaced evenly vertically and horizontally and have no missing

points. OS Land-Form Profile 1:10000-scale data were used, with heights represented

digitally as points at 10m intervals, with an elevation accuracy of ±1.8-5m (Cutler, 2005;

OS Land Form Profile Technical Information [accessed 12/12/06]), the most accurate

digital data available. It should be noted that despite this high resolution, Cutler

(2005:p.2) highlights that point-data format is not as precise as contour data, which is

stored as smooth, joined contour lines with an accuracy of ±1-1.8m (OS Land Form

Profile Technical Information [accessed 12/12/06]). However, this data is incompatible

with ArcGIS.

As point shapefiles can contain data such as population distribution, which are rarely

evenly spaced, they do not necessarily comprise a regular grid of points as the OS data

34

do. As such, the shapefile stores the location of every point, which is inefficient for a

regularly-spaced grid. The OS data were converted into a raster, a grid of cells each of a

fixed size and a single co-ordinate from which the location of all cells can be calculated

(ESRI, 2006b). This is a more efficient storage method as the evenly-spaced grid requires

only elevation data to be saved. In the case of an unevenly-spaced grid, interpolation is

required to predict heights of areas in between cells of known elevations (section 1.3;

ESRI, 2006b). This technique is required when converting contour data to raster form,

but the regular structure and point shapefile data format of the OS tiles rendered this step

unnecessary. A raster was created directly from the point data, converting each point into

a raster cell. An example is shown in Figure 4.1 (original point data) and Figure 4.2

(raster).

As the study area crosses between two BNG squares, the OS data was downloaded as two

tiles, an NN and NS point shapefile, before being converted into two separate rasters. The

rasters were merged to create a single file containing the entire study area. Merging the

rasters was necessary to eliminate the artificial seam between sets of data, which

calculations of slope and water flow would treat as a solid edge. This merged raster was

loaded into ArcGIS and a colour scale applied (Figure 4.3).

The next step was pre-processing, a set of techniques which prepare the data for analysis

(Tarboton, 1998; Maidment, 2002:p.68-80; Maidment and Robayo, 2002:pp.13-31;

McDonnell, 2006; Tarboton, 2005). Pre-processing was carried out using both ArcHydro

and TauDEM, allowing the best match to the stream and drainage basin structure

observed in the field to be manually chosen. The merged raster formed the initial input

data.

35

Figure 4.1:

Example OS Land-Form Profile point shapefile,

with elevations of each point displayed

17 14 12 13 14

16 16 13 13 12

16 17 14 14 13

17 17 16 15 14

14 16 16 16 15

Figure 4.2:

Raster version of Figure 4.1’s shapefile,

with fixed cell size and elevations in each cell

Figures 1.1 and 1.2 adapted from

“Creating raster surfaces from points”

(ESRI, 2006b)

36

17 14 12 13 14

16 16 13 13 12

16 17 14 14 13

17 17 16 15 14

14 16 16 16 15

ELEVATION

High: 720m

Low: 7m





Figure 4.3:

Raster of study area

37

First, sinks were filled to remove errors from the DTM such as cells surrounded by other

cells of slightly higher elevations or seams produced by the merge. These would

otherwise trap the simulated flow of water. This step minutely altered elevations as

appropriate (e.g. Maidment and Robayo, 2002:pp.14). to the naked eye, the resulting

coloured raster looked identical to Figure 4.3.

Second, probable flow direction of water was calculated using the pit-filled raster as

input. As with many GIS methods (e.g. Tarboton, 1998:p1, slide 7) there is debate as to

the best way to achieve this (Tarboton, 1997, Tarboton, 1998:p1, slide 7). The most

prevalent methods for this process are D∞ and D8, with D8 being the current standard.

While D8 splits flow directions into 45º segments (i.e. 8 discrete directions), D∞

represents flow as a continuous angle between 0 and 2Π radians. While D∞ has the

advantage of increased accuracy of angle, and thus increased accuracy of flow direction,

there are complex mathematical problems with it (Tarboton, 1997). As such, both

methods were used where supported (TauDEM). The ArcHydro D8 (Figure 4.4),

TauDEM D8 (Figure 4.5) and TauDEM D∞ (Figure 4.6) rasters are displayed.

Third, flow accumulation (ArcHydro) or contributing area (TauDEM) was calculated

using the flow direction rasters. TauDEM produced both D8 and D∞ contributing area

rasters. The flow accumulation and contributing area functions essentially produce the

same output, calculating the upslope area and accumulated flow from upslope cells. The

area and flow is represented as a number of upslope grid cells per raster cell. ArcHydro

(Figure 4.7), TauDEM D8 (Figure 4.8) and TauDEM D∞ (Figure 4.9) outputs are

displayed.

38





Figure 4.4:

ArcHydro D8 flow direction raster

39





Figure 4.5:

TauDEM D8 flow direction raster

40

DIRECTION ANGLE

High: 6.5°

Low: 0°





Figure 4.6:

TauDEM D∞ flow direction raster

41

UPSTREAM CELLS

High: 3017

Low: 0





Figure 4.7:

ArcHydro flow accumulation raster

42

UPSTREAM CELLS

High: 5047

Low: 1





Figure 4.8:

TauDEM D8 upslope contributing area raster

43

UPSTREAM CELLS

High: 198661

Low: 1





Figure 4.9:

TauDEM D∞ upslope contributing area raster

44

The final pre-processing step was the creation of a stream raster. ArcHydro and TauDEM

each output a single grid, using all previous rasters as input. TauDEM combines the D8

and D∞ functions in the calculation. This is a key stage, defining the probable paths of

channels. Through comparison to a determined threshold of flow accumulation, the raster

assesses and delineates stream channels within the area. ArcHydro does not allow the

setting of this threshold, so only one raster was produced (Figure 4.10). However,

TauDEM allows this parameter to be altered. Various thresholds were entered and the

model re-run. Key results are shown in Figure 4.11 (threshold=50), 4.12 (threshold=10)

and 4.13 (threshold=5). This demonstrates that, as would be expected, a reduction in the

threshold required to assess that a stream exists results in more streams on the raster.

None of the outputs in Figures 4.10, 4.11, 4.12 and 4.13 are satisfactory. While the

majority correctly calculate the paths of key tributaries, none correctly calculates the

forks. Analysis of these rasters prompted the decision to use fieldwork for measurement

of long profiles (Chapter 3) rather than computer modelling. An accurate output of stream

long profile would be impossible based on these rasters, and so despite the variable

changes in GPS accuracy in comparison to the fixed 10m resolution of the DTM,

fieldwork the most valid option.

The failure of pre-processing to correctly delineate streams did not, however, affect the

accuracy of the computer modelling relation to whole drainage basin characteristics.

While a stream raster relies on calculations at the cell scale, with elevation errors of

single units affecting the overall output, an assessment of the whole drainage basin is not

so affected by individual cells. A second group of techniques called watershed processing

was used to complete the computer modelling (Maidment, 2002:p.68-80; Maidment and

Robayo, 2002:pp.13-31; Tarboton, 2005).

45

DELINEATED STREAMS

Watercourse





Figure 4.10:

ArcHydro stream raster

46

DELINEATED STREAMS

Watercourse





Figure 4.11:

TauDEM stream raster (threshold = 50)

47

DELINEATED STREAMS

Watercourse





Figure 4.12:


48

DELINEATED STREAMS

Watercourse





Figure 4.13:


49

First, watersheds were delineated using both plugins. The pre-processing rasters were

used as inputs to allow the plugins to calculate divisions between neighbouring drainage

basins. TauDEM used the stream raster produced by an accumulation threshold of 10 (the

most accurate) for this step. Outputs are displayed in Figures 4.14 (ArcHydro) and 4.15

(TauDEM). TauDEM produced significantly more drainage basins than ArcHydro due to

the larger number of streams on the stream raster. When ArcHydro watersheds were

overlaid on the TauDEM raster, however, their general borders were relatively similar.

At this stage, a decision was made over which dataset to use for the final watershed

processing steps. ArcHydro provides more complete analysis tools than TauDEM, and as

such was to be used for the analysis (Maidment, 2002:p.68-80). However, either the

ArcHydro or TauDEM rasters could be used as input (Maidment, 2002:p.68-80;

Maidment and Robayo, 2002:pp.13-31; Tarboton, 2005). If one of plugins had produced

a significantly more accurate stream or watershed raster, it would have been prudent to

choose to use that dataset. As this was not the case, the ArcHydro dataset was chosen for

two reasons. First, the final stages of analysis were designed to work most closely with an

ArcHydro dataset. Secondly, the additional complexity of the TauDEM rasters provided

no more valuable data than the ArcHydro ones. The inclusion of extra watersheds

provided only a visual advantage.

The second step was slope analysis (Maidment, 2002:p.68-80; Maidment and Robayo,

2002:pp.13-31). Slope angle was calculated for the entire raster, giving a coloured output

scale corresponding to angle in degrees (Figure 4.16). Each watershed was then coloured

to give a clear display of slope (Figures 4.17, 4.18 and 4.19). The choice of the ArcHydro

dataset meant that the third and fourth tributaries are grouped together, but this does not

affect accuracy of slope values and is simply a visual difference.

Finally, in addition to the slope rasters, lists of elevations for each drainage basin were

exported from the pit-filled DTM. These elevations were plotted as hypsometric curves

(Figures 4.20, 4.21 and 4.22), graphs of elevation against frequency of occurrence

(Goudie et al., 1994). While Microsoft Excel was used to plot the data, the data itself was

obtained via computer modelling.

50

ELEVATION

High: 720m

Low: 7m





Figure 4.14:

Delineated ArcHydro watershed on base raster

51

ELEVATION

High: 720m

Low: 7m





Figure 4.15:

Delineated TauDEM watershed on base raster

52

SLOPE ANGLE

High: 85°

Low: 0°





Figure 4.16:

Entire slope angle raster

53

SLOPE ANGLE

High: 85°

Low: 0°





Figure 4.17:

Slope angle raster with first tributary’s drainage basin highlighted

54

SLOPE ANGLE

High: 85°

Low: 0°





Figure 4.18:

Slope angle raster with second tributary’s drainage basin highlighted

55

SLOPE ANGLE

High: 85°

Low: 0°





Figure 4.19:

Slope angle raster with third and fourth tributaries’ drainage basins highlighted

56

0

50

100

150

200

250

300

350

400

0 50 100 150 200 250 300 350 400 450 500

Height above base (m)

Fre

qu

ency

Figure 4.20: Hypsometric curve of first tributary’s drainage basin

0

50

100

150

200

250

300

350

400

0 50 100 150 200 250 300 350 400 450 500


Fre

qu

ency

Figure 4.21: Hypsometric curve of second tributary’s drainage basin


57

0

50

100

150

200

250

300

350

400

0 50 100 150 200 250 300 350 400 450 500


Fre

qu

ency

Figure 4.22: Hypsometric curve of third and fourth tributaries’ drainage basins

(Diagram plotted from measured data)

4.2 Analysis of results

The results of pre-processing and watershed processing were mixed. TauDEM provided

the most accurate output stream raster (Figure 4.11). ArcHydro, however, produced a

poor result, completely missing the third tributary (Figure 4.10). As such, fieldwork was

used to analyse the long profile (Chapter 3).

The results of modelling drainage basin slope were more successful. Figures 4.17 and

4.18 demonstrate that tributaries 1 and 2 have the most consistent slope. The third

tributary (Figure 4.19) also shows consistently high slope, although an area towards its

base suggests a flattening as it T3. This variation may suggest that it has not yet reached

equilibrium.

Tributary 4 (Figure 4.19) is highly variable. Slope is unpredictable, not even varying in a

constant fashion from steep to slight, but including convexities along all three minor

streams. When compared to the other tributaries and the first two in particular, it would

58

seem that the fourth tributary shows little pattern in change of slope, and lacks the

gradual change that would be expected at equilibrium.

The quantitative data from hypsometric curves confirmed the qualitative analysis of

slope. Slope and elevation are clearly related, and an area with greater variation in slope

would be expected to have greater variation in elevation. This can be seen to be true, with

the first (Figure 4.20) and second (Figure 4.21) tributaries showing a closely-packed

range of frequencies. Tributary 1 has a range of 0-60 occurrences, while tributary 2 has a

range of 0-51 (excluding the five individual erroneous points). This suggests a smooth

and steady surface in both cases.

Figure 4.22, on the other hand, shows that the third and fourth tributaries have a greater

range of frequencies, from 0-253 (excluding the highest 8 points). This dramatic range

suggests that the surface includes more convexities, resulting in non-smoothed contours.

4.3 Analysis of methodology

The initial decision over the software package clearly has a potential impact on the

results. Opting to use ArcGIS with ArcHydro and TauDEM was driven by licensing and

availability practicalities. Likewise, the use of OS data over other data was based upon

access to the Land-Form Profile DTMs. Results which would have been achieved with

other software packages and data are unknown, but the comparison of TauDEM and

ArcHydro stream rasters in section 4.2 illustrates that it can potentially make a large

difference. This could only be assessed by repeating this study with various software

configurations – a prohibitively time-consuming task.

The conversion of point shapefiles to raster and the pre-processing stages were carried

out in a standard order (Maidment, 2002:p.68-80). Whether or not there is a requirement

to merge rasters is dependent upon study area location and size of digital data tiles, but

caused no known problems with the results. Any imperfections due to the merge were

annulled by the sink-filling stage.

59

Although there is debate surrounding many of the techniques (summary in Tarboton,

1998:p1, slide 7) the models used were dictated by limitations of the software. The

exception to this was the use of both D8 and D∞ in TauDEM calculations, which

nonetheless seemed to make little difference in terms of visible dissimilarities at later

stages (Figures 4.8 and 4.9). It should be noted however, that dissimilarities which are not

visible may still be important; the sink-filled raster looks identical to Figure 4.3 but is a

key stage.

The main area of doubt in the methodology was the discrepancy between the stream

rasters. ArcHydro (Figure 4.10) and TauDEM (Figures 4.11, 4.12 and 4.13) results were

widely different, and illustrated problems with analysis of streams rather than drainage

basins. The study of the water courses would not have been possible without fieldwork,

with the GIS stream rasters showing unsatisfactory results. This illustrates a key reason

for ongoing debate in all areas of GIS – as discussed in section 1.3, calculation and

representation of that which may seem relatively trivial in mathematical modelling is

complex in GIS. The failure of the stream rasters to represent the observed streams

illustrates severe problems with a system designed to heavily supplement, if not replace,

fieldwork.

The problems with the precisely-calculated stream rasters are unlikely to have had a

significant effect on the holistically-calculated watershed boundaries (section 4.2) and so

results obtained can be judged significant, especially since hypsometric analysis

quantitatively confirmed the qualitative assessment of slope. However, the added

complexity required to create a more aesthetically-intuitive set of boundaries for the

drainage basin using TauDEM rather than ArcHydro was restrictive. While the TauDEM

boundaries would more clearly show tributary 3, they would make no difference to the

results of the slope analysis. If more detailed analysis was required for a future study, the

time-intensive process of analysing outlines for many drainage basins could be very

restrictive.

Aside from the major problem of stream raster accuracy and the minor problem of

processing drainage basin outlines, the use of GIS was definitely advantageous. It gave

60

slope angle results which could not have been obtained through fieldwork. Although the

results of the slope analysis are accurate, with slope variation increasing towards the third

and fourth tributaries and being especially erratic in the fourth, the problems with outline

calculation, the vast inaccuracies in the stream rasters and the potential differences

between software must be taken into account in future studies.

61

5. COMPARISON WITH OBJECTIVES

5.1 Objective one

To test the first objective (analysis of ability and reliability of mathematical models of

long profile) the use of either GIS or fieldwork was considered. If computer modelling

(GIS) could produce accurate readings of elevation and distance for each stream in the

study area, the data could be used as input to test the mathematical model and examine

whether or not streams within the drainage basin fit a concave-upward curve (e.g. Snow

and Slingerland, 1987:pp.15; Gregory, 1994:p.308-9; Knighton, 1998:p.242-245). Stream

rasters were produced using established techniques (Maidment, 2002:p.68-80; Maidment

and Robayo, 2002:pp.13-31; Tarboton, 2005) in ArcGIS with the ArcHydro and

TauDEM plugins. However, neither ArcHydro (Figure 4.10) nor TauDEM (Figures 4.11,

4.12, 4.13) correctly identified all streams. While the GIS data had a horizontal accuracy

of 10m and a vertical accuracy of ±1.8-5m (Cutler, 2005; OS Land Form Profile

Technical Information [accessed 12/12/06]) the lack of an adequate stream raster meant

that establishing a series of points along the course of each stream was impossible

(section 3.1). Instead, data were collected in the field.

As described in section 3.1, it was initially hoped that both pieces of data could be

collected using GPS at regular intervals along the streams. This would provide location

and elevation data as input to the mathematical model. However, due to the unreliability

of height data from the GPS device’s altimeter, measuring of arbitrary points in this

fashion was not possible. Instead, angles were taken between consecutive breaks of slope

along each stream, with a GPS co-ordinate taken at each point. The location data were

converted into distances downstream and cumulative elevations calculated from the

angles through use of Pythagoras’ Theorem and the Sine Rule (Equation 3.1). These data

were plotted as long profiles using Microsoft Excel (Figures 3.2 through 3.17).

In addition to the unreliability of the GPS elevation readings, two more issues arose with

the method used for collecting data. First, the GPS was potentially erroneous in its

readings of location. The potential location error range which it gave was converted into

62

potential elevation error by altering values of distance between consecutive points in line

with the measured error ranges, and re-running the Sine Rule. This was represented on

the long profiles in Figures 3.3, 3.5, 3.7, 3.9, 3.11 and 3.17 as y-error bars. The second

key methodological issue was the potential for error through use of the clinometer, which

lacked enough precision to give angles to more than the nearest 0.5º. Unlike GPS, no

error range can be calculated for the clinometer, so the possibility of error introduced had

to be borne in mind when analysing the results.

With the initial problems establishing a technique and then potential error introduced by

the GPS and the clinometer, collecting data for input into the mathematical model proved

to be the most problematic part of the process. However, as discussed below it seems that

the data collected was relatively accurate. GIS, on the other hand, failed to produce the

same reliable data.

The findings surrounding data collection have significance for studies of long profile

form. As well as informing future research of the limitations of GIS and fieldwork, they

raise key questions about studies which use digitally derived data as input to their

mathematical models. While authors such as Knighton (1998:p.242-245) and Snow and

Slingerland (1987:pp.15) note that studies have found that the concave-upward trendline

is generally accepted as diagnostic of equilibrium, they give no explicit consideration to

the accuracy of input data for mathematical modelling of stream profiles. The conclusion

that a study area is in equilibrium simply because it fits a concave-upward shape should

not be drawn unless the source data is examined for correctness. While it is accepted that

a concave-upward long profile denotes equilibrium (Hack, 1957: p.69-70; Gregory,

1994:p.308-9; Knighton, 1998:p.242-245) this is logically dependent upon reliability of

the source data. Studies of long profiles must be sure to question the possibility of

cumulative error from their data collection, especially as it seems that no obvious method

is free from potential error. This is not to say that the concave-upward fit is not diagnostic

of equilibrium. Rather, the finding is that it is important that research should only

diagnose equilibrium if the curve is highly significant, to avoid the possibility of

cumulative error giving a false positive.

63

With this in mind, the mathematical model was tested on each stream by applying

trendlines to the long profiles. The most significant fit for the first three tributaries was a

concave-upward line in each case, relevant to a high confidence level (S1a-B, r2 =

0.9914; S1b-B, r2 = 0.961; S2a-B, r2 = 0.9862; S3a-B, r2 = 0.9851 and S3b-B, r2 =

0.9852). In line Davis (1902) and Snow and Slingerland (1987:pp.15), and with the

knowledge that a high significance is needed to eliminate the possibility of error affecting

the long profile, the close fit to a concave-upward trendline suggests that these three

streams are in equilibrium (Gregory, 1994:p.308-9; Knighton, 1998:p.242-245).

In contrast, S4a-B, S4b-B and S4c-B did not fit the mathematical model. This was

attributed to convexities both at T4-T3 and in the upper reaches of S4a-B, S4b-B and

S4c-B. While trend lines for these stream reaches were not significant, trendlines were

fitted to sections T4-T3, S4a-T4, S4b-T4 and S4c-T4. However, the short length of these

reaches meant that more than one type (concave-upward and linear) was significant in

each case. This may be because GPS error was relatively large towards the sources, and

confirms that error in the data may have large effects on validity of results. It serves to

emphasise that errors in source data, whether obtained through GIS or fieldwork, are of

critical importance and can influence the result of the mathematical model.

The reliability of the model of equilibrium stream profiles (the concave-upward curve)

does not need to be questioned. Sufficient literature agrees that a concave-upward curve

is representative of equilibrium (e.g. Hack, 1957: p.69-70; Snow and Slingerland,

1987:pp.15; Gregory, 1994:p.308-9; Knighton, 1998:p.242-245). However, the ability of

the method to produce accurate results was shown to be a key consideration for future

studies of this type. Error is likely to be introduced whichever method is used to source

the data, and as such both GIS and fieldwork have advantages and disadvantages for this

purpose. While GIS data were unreliable for this study, very high resolution data and/or

different software may produce acceptable input. Fieldwork, though with the potential for

error, allowed highly accurate mapping and generally showed breaks in slope more

precisely than the 10m resolution of GIS data. While the mathematical model is reliable

for morphological analysis of stream network form, possible issues with accuracy of

source data must be examined in similar research.

64

5.2 Objective two

The second objective (analysis of ability and reliability of computer models of drainage

basins) required a choice over which GIS software to use for the study. Dictated largely

by availability of data and licensing agreements, ArcGIS 9.1 with the ArcHydro and

TauDEM plugins was used. The digital data was downloaded from Edina Digimap

[accessed 10/07/06] as OS Land-Form Profile 1:10000-scale point shapefiles. This data

had a horizontal accuracy of 10m and a vertical accuracy of ±1.8-5m (Cutler, 2005; OS,

Land Form Profile Technical Information [accessed 12/12/06]).

Longley and Batty (1996) note that point shapefile data may be used for a variety of

spatial information (e.g. Wrigley et al., 1996, zonal and areal patterns in ecology;

Macmillan, 1996:p153-165, urban simulation) and as such point shapefiles are not

optimised to store a the regularly-spaced OS elevation data. For this reason, the raw data

were converted into raster form through use of established techniques (e.g. Maidment,

2002:p.68-80; ESRI, 2006a). This converted data could then be used in the GIS using the

overlay method (Worboys, 1995:p.9-15; Maidment, 2002:p.17; Lyon, 2003:p.1;

Schuurman, 2004:p.3-4). The capability to convert data formats for use in a wide variety

of applications demonstrates the ability and adaptability of the GIS. This is an important

function, allowing other studies of drainage basins to use various formats of input data.

The next stage in the preparation of the data for use with the computer model was

watershed pre-processing (Tarboton, 1998; Maidment, 2002:p.68-80; Maidment and

Robayo, 2002:pp.13-31; Tarboton, 2005; McDonnell, 2006). This established set of

techniques filled sinks in the input raster, calculated flow directions (Figures 4.4 through

4.6), calculated flow accumulation (upstream contributing area – Figures 4.7 through 4.9)

and produced stream rasters (Figures 4.10 through 4.13).

Pre-processing illustrated two key findings with respect to the ability of the method. The

first was that imperfections (such as sinks in the original raster) can have a large effect on

the final result. The sink-filling stage was of critical importance to later calculations, but

the difference between the original (Figure 4.3) and sink-filled raster was not visible to

the naked eye. In a similar fashion to the potential cumulative error found with the use of

65

GPS and clinometer readings for mathematical modelling, attention must be paid to what

could be perceived as relatively unimportant processes. In order to produce reliable

results, similar research which employs the methodology used in this study must ensure

that all stages of watershed pre-processing are carried out in line with the literature

(Tarboton, 1998; Maidment, 2002:p.68-80; Maidment and Robayo, 2002:pp.13-31;

Tarboton, 2005; McDonnell, 2006).

The second finding of pre-processing related to uncertainties over the wide range of

methods which GIS applications employ in order to perform each stage (e.g. Tarboton,

1997; Tarboton, 1998:p.1, slide 7; Fotheringham et al., 2000:p.31). This was illustrated

by the variation between stream rasters created using the ArcHydro (Figure 4.10) and

TauDEM (Figures 4.11 through 4.14), none of which were a satisfactory representation of

water courses observed in the field. Confirming the above finding that accumulation of

error or variation can result in widely differing results, variations in the methodologies

employed by each plugin at the flow direction stage were carried through to the flow

accumulation (Figures 4.4 through 4.6) and stream raster stages. The four stream rasters

which these methods produced varied widely. This has important implications for future

studies. The choice of GIS software and plugins clearly has the ability to result in

different findings, and is an important consideration. Analysis of results achieved through

the use of a GIS program (including the results found by this study) should be viewed

critically in light of the potential difference which could have been found had other

software been used. Furthermore, the lack of consistent results from either plugin

introduces a high degree of doubt about reliability of the use of the GIS method for

stream delineation. This is concerning, as GIS is often perceived as being able to replace

the need for fieldwork.

Despite the inaccuracies of the stream rasters, which prompted the use of fieldwork to

collect input data for the mathematical model, the failure of pre-processing to correctly

identify streams within the drainage basin did not influence the ability of the software to

correctly analyse whole drainage basin characteristics. While stream rasters rely on

individual cell calculations, drainage basin delineation is concerned with larger area

characteristics.

66

Both ArcHydro and TauDEM were used to delineate watersheds (Figures 4.14 and 4.15).

As logically expected, TauDEM (with its greater number of streams) delineated a greater

number of watersheds. Either of these outputs could have been used for the final stages

(Maidment, 2002:p.68-80; Maidment and Robayo, 2002:pp.13-31; Tarboton, 2005).

However, due to ArcHydro being primarily designed to analyse ArcHydro-delineated

watersheds, and because the additional TauDEM watersheds added nothing more to the

study than a clearer visual representation of the boundary between the watersheds of the

third and fourth tributaries, the ArcHydro results were used.

The fact that inaccuracy of stream rasters has little effect on drainage basin delineation is

important for other studies of this type. While, as has already been noted, cumulative

error is indeed an important consideration, it is equally important to realise that error in

earlier stages may in some cases have little effect on later stages. In the case of watershed

delineation, it was valid to select the results which reduced complexity in terms of the

number of drainage basins and to ensure compatibility with the final ArcHydro analysis.

Watershed processing was completed by performing slope analysis (Figure 4.16) of each

drainage basin (Figures 4.17 through 4.19) in line with Maidment (2002:p.68-80) and

Maidment and Robayo (2002:pp.13-31), and creating hypsometric curves for each

(Figures 4.20 through 4.23).

The analysis of the results in section 4.2 provided evidence of patterns of drainage basin

slope characteristics. The first (S1a-T1 and S1b-T1) and second (S2a-T2) tributaries

showed consistently similar patterns, with a high frequency of steep slope values. This

qualitative analysis was confirmed by the quantitative analysis of the hypsometric curves,

which showed that elevation (logically related to slope) was closely-packed in these

basins. The third tributary (S3a-T3 and S3b-T3), while similar to the first two, showed

slightly more variation, especially towards its confluence with the Burn of Sorrow. The

fourth tributary was much more variable. Both the qualitative (slope) and quantitative

(elevation) analyses demonstrated high variability throughout S4a-T3, S4b-T3 and S4c-

T3. While tributaries 1 and 2 showed elevation frequency ranges of 0-60 and 0-51

respectively, the fourth tributary showed a much greater range of 0-253. The large

67

difference in frequencies suggests that while the first two tributaries are definitely in, the

fourth tributary has many convexities and is in disequilibrium.

Both the ability and reliability of computer modelling were shown to be variable, in line

with concerns raised in the literature (e.g. Fotheringham et al., 2000:p.31; Schuurman,

2004:p.32). There were clear issues surrounding inaccurate delineation of stream

locations, which raise doubts about the use of GIS in this area of study. Research reliant

on GIS for this purpose must ensure that stream rasters used accurately represent streams

observed in the field. This inaccuracy was clearly related to the finding that current

methods of slope and flow direction calculation are variable and dependant upon

technique (e.g. Tarboton, 1997; Tarboton, 1998:p.1, slide 7; Fotheringham et al.,

2000:p.31), illustrating potential issues with cumulative error being carried through the

various stages of the computer model. This finding raises questions about differences

between GIS software, plugins and source data (Fotheringham et al., 2000:p.31) and this

choice must be taken into account when performing similar research.

The computer model did, however, clearly have wide-ranging ability. The capacity to use

different data formats and techniques allows an informed choice to be made about the

research methods to use. While doubts remain about the accuracy of stream delineation,

the ability and reliability of watershed delineation was high. The computer model allowed

an in-depth analysis of both slope and elevation for each drainage basin, something which

would not have been possible through fieldwork. As a whole, the study found wide-

ranging ability within the computer model to carry out a variety of analyses using various

software and data While ability was high, reliability was more variable, and as such the

theoretical ability of the software to carry out a technique such as stream delineation

should not be taken to mean that the results produced are reliable.

5.3 Objective three

The third objective was to compare and contrast the findings of the first two objectives.

It has been established that the reliability of the mathematical model of stream profile

form is no in doubt (Hack, 1957: p.69-70; Gregory, 1994:p.308-9; Knighton, 1998:p.242-

68

245) but that possible error in the source data must be taken into account when assessing

the extent to which a long profile shows equilibrium (section 5.1). This error may take the

form of unreliable stream delineation or unreliable input data with GIS, or errors in

fieldwork due to the use of GPS or clinometer readings. Similarly, the resolution of the

data has the potential to induce error as it could produce a false result if it is significantly

inaccurate (section 5.2). Cumulative error was shown to have a large effect in both cases,

illustrating that only a highly significant concave-upward curve should be taken to

indicate equilibrium; curves of borderline significance may only fall within the

confidence level due to error.

Similarly, cumulative error was an issue when performing pre-processing of data for use

with the computer model. Differences between software (ArcHydro, TauDEM) resulted

in differing flow direction, flow accumulation and stream rasters. This is in contrast to the

cumulative error of fieldwork, where technique was established but input data were

unreliable.

Watershed delineation using the computer model was successful. The results showed

consistent slope and elevation values in the first two tributaries and to a lesser extent in

the third. This correlates with the findings of the mathematical model, which found

equilibrium long profiles in the lower tributaries but disequilibrium characterised by

convexities in the fourth tributary. The computer and mathematical models each confirm

the results of the other, reducing the possibility of error and indicating a high probability

of equilibrium in the first three tributaries. This comparison of techniques is a useful

method for future studies of drainage basins, as agreement of two separately calculated

sets of results allows error to be discounted in both models.

While this comparison method is highly desirable, significantly reducing the possibility

of error, both modelling techniques provided functions which the other could not

replicate. While GIS proved unreliable in producing accurate stream rasters, it is feasible

that different GIS software would produce correct data. However, mathematical models

would still be required in order to analyse the extent to which the long profile data fits an

equilibrium concave-upward curve. The GIS (computer modelling) software was not

69

capable of performing this analysis, requiring the use of long profile graphs

(mathematical modelling). Similarly, the morphological data of slope and elevation

calculated by the GIS was not something that could be assessed using fieldwork or

mathematical models. It is clear that both mathematical and computer modelling provide

opportunities for analysis which the other cannot perform. In both cases, the reliability of

conclusions based upon the models was high where input data were reliable.

Comparison of the results of the first two objectives illustrates advantages and

disadvantages in both methods. Mathematical modelling clearly has the ability to produce

meaningful results which are unobtainable through computer modelling. While it is clear

that a concave-upward upward curve is indicative or equilibrium (Hack, 1957: p.69-70;

Gregory, 1994:p.308-9; Knighton, 1998:p.242-245), questions were raised over the

reliability of the curves produced by mathematical modelling due to potential error in the

source data. Computer modelling had more variable ability and reliability, producing

slope and elevation readings which were unobtainable through fieldwork but incorrectly

delineating stream rasters. However, the poor ability of the GIS in this area was found to

depend upon the software used for the analysis, and as such a separate study using

different software may have achieved more reliable results in this area.

Accuracy and reliability clearly need to be taken into account when using either model.

Comparison of the results of mathematical and computer modelling proved to be a useful

technique in order to improve reliability and accuracy. The long profile form of each of

the tributaries corresponded to the variation in slope and elevation in their respective

drainage basins. As such, the possibility of error was largely discounted from these

results. While each technique (computer and mathematical modelling) has abilities that

the other does not, it seems that, where possible, the most desirable approach is to use

both models in tandem and compare their results to each other to confirm the findings.

70

6. CONCLUSION

The overall aim of this study was to test which methodology (computer or mathematical

modelling) is most reliable and accurate for basic morphological analyses of drainage

basins, in order to allow better-informed analyses to be made in future studies. The study

has several key findings.

First, the reliability of the mathematical model of stream profiles at equilibrium is

supported by a large volume of literature (Hack, 1957: p.69-70; Gregory, 1994:p.308-9;

Knighton, 1998:p.242-245) and as such is taken to be reliable. However, consideration

must be given to the ability of the model to produce accurate results from data obtained

through GIS or fieldwork (section 5.1). In order to ensure that any concave-upward trend

found is significant, and therefore truly represents equilibrium in practice, future studies

must ensure that error is rigorously eliminated from the source data. It was found that

when source data is likely to be accurate, the mathematical model is highly reliable.

Second, it was found the mathematical model allowed analyses which were not possible

through the use of GIS. The stream rasters created using GIS were largely incorrect

(figures 4.10 through 4.13) and did not represent stream patterns observed in the field.

Even if this can be corrected through the use of different GIS software, it was found that

the 10m resolution of the digital data is likely inferior to results that can be obtained

through use of GPS. While it is unsurprising that the GIS results were of stream raster

delineation were not completely accurate due to continuing debate over methodologies to

employ (e.g. see Tarboton, 1997; Tarboton, 1998; Maidment, 2002:p.68-80; Schuurman,

2004:p.32) it is concerning that they were so largely wrong. If, however, future studies do

opt to use GIS to delineate streams, they must ensure that the stream raster delineated

accurately represents the study area. As noted in section 5.2, the theoretical ability of the

software to carry out a technique should not be taken to mean that the results are reliable.

Third, it was found that despite the severe problems with stream raster delineation,

computer modelling of watersheds as a whole was accurate (Section 5.2; Figures 4.16

through 4.19). Furthermore, GIS provided analytical techniques of slope and elevation

71

which it is not possible to replicate using fieldwork. Future studies of similar

morphological characteristics should note that they will require the use of GIS to analyse

whole drainage basin slope and elevation characteristics.

Fourth, a key finding was that while the mathematical and computer models can provide

unique analyses when used on their own (e.g. assessment of long profile form;

assessment of whole drainage basin slope and elevation), a particularly useful technique

for future studies of this type is the use of them in tandem. As noted in section 5.3, when

the long profile form (as created using mathematical modelling) was compared to

elevation and slope characteristics (output by the computer model), the extent to which

streams were in equilibrium corresponded to slope and elevation values. Where slope was

high and elevation frequency closely-packed, river long profiles were found to be in

equilibrium. Conversely, where slope and elevation frequency were both variable, river

long profiles were found to be in disequilibrium. This comparison has high potential in

the area of drainage basin analysis. As previously noted, the possibility of error in the

results of both the mathematical and computer models introduces doubt into the findings

when assessed on their own. However, when the results of both models are found to

demonstrate the same conclusion, the chance of error causing a false result is

significantly reduced.

Overall, the accuracy and reliability of both models was found to be a key consideration

for future research. Both computer and mathematical models were found to have

strengths and weaknesses. With the computer model, these centred on its ability to

delineate streams. With the mathematical model, they centred on its ability to overcome

errors in source data. It was found that the way to improve confidence in the results of

either model is not to use them individually, but rather compare and contrast them.

Comparison of the results of mathematical and computer modelling proved to be a useful

technique in order to improve reliability and accuracy, with the possibility of error largely

discounted from the results. Whether future research will choose to use both methods in

this way is unknown. However it is clear that, in any case, the most important finding is

that studies must pay close attention to the possibility of error.

72

ACKNOWLEDGEMENTS

The author would like to thank the following people

for their help getting started with the GIS software:

Dr Josh Fisher

Oxford University Centre for the Environment,

University of Oxford

Dr Rachael McDonnell



Dr Ruth Robinson

Department of Geography and Geosciences,

University of St Andrews

[Added this final acknowledgement for Scribd

upload, November 2012 – was unable to include in

original document due to candidate anonymity]

In addition, sincere thanks go to my supervisor for

all his help and advice:

Dr Giles Wiggs



73

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an analysis of mathematical and computer modelling of drainage basin characteristics

Documents

creating raster

t3 t2 t1

basic morphological

primary data

river long

drainage basin

river long

long profile