The nature of data structures used for the representation of terrain has a great influence on the possible applications and reliability of consequent terrain analyses. This research demonstrates a concise review and treatment of the surface network data structure, a topological data structure for terrains. A surface network represents the terrain as a graph where the vertices are the fundamental topographic features (also called critical points), namely the local peaks, pits, passes (saddles) and the edges are the ridges and channels that link these vertices. Despite their obvious and widely believed potential for being a natural and intuitive representation of terrain datasets, surface networks have only attracted limited research, leaving several unsolved aspects, which have restricted their use as viable digital terrain data structures. The research presented here presents novel techniques for the automated generation, analysis and application of surface networks. The research reports a novel method for generating the surface networks by extending the ridges and channels, unlike the conventional critical points-based approach. This proposed algorithm allows incorporation of much wider variety of terrain features in the surface network data structure. Several ways of characterising terrain structure based on the graph-theoretic analysis of surface networks are presented. It is shown that terrain structures display certain empirical characteristics such as the stability of the structure under changes and relationship between hierarchies of topographic features. Previous proposals for the simplification of surface networks have been evaluated for potential limitations and solutions have been presented including a user-defined simplification. Also methods to refine (to add more details to) a surface network have been shown. Finally, it is shown how surface networks can be successfully used for spatial analyses e.g. optimisation of visibility index computation time, augmenting the visualisation of dynamic raster surface animation, and generating multi-scale morphologically consistent terrain generalisations.

1. Surface Networks: New techniques for their
automated extraction, generalisation and
application
Sanjay Singh Rana
Department of Geomatic Engineering
University College London
University of London
A revised and updated version of the thesis submitted for the degree of
Doctor of Philosophy in Geographical Information Science in 2004
2010

2. For fellow surface networks researchers

3. Contents
Abstract I
Acknowledgements II
List of Figures III
List of Tables VIII
Chapter 1 Introduction 1
1.1 Context 1
1.2 Unresolved issues and aims of the thesis 5
1.2.1 Model 9
1.2.2 Generation 14
1.2.3 Simplification 22
1.3 Structure of the thesis 26
Chapter 2 Automated Generation 27
2.1 Automated generation of surface networks 27
2.2 Novel solutions to unresolved issues 27
2.2.1 Stating semantic uncertainties in labelling terrain 27
2.2.2 Automated extraction of surface topology 32
2.2.3 Implementation 40
2.3 Discussion 40
Chapter 3 Structural Characterisation 54
3.1 Extending the description of surface networks 54
3.1.1 Standard graph measures 55
3.1.2 Two case studies 56
3.2 Simplification of surface networks 57
3.2.1 Surface Topology Toolkit 60
3.2.2 New weight measures 60
3.2.3 Non-sequential contractions 64
3.3 Refinement of surface networks 66
3.4 Discussion 69
Chapter 4 Applications 71
4.1 Proposal 71
4.2 Fast computation of visibility dominance in mountainous terrains 72
4.2.1 Proposal 73
4.2.2 Methodology 73
4.2.3 Results 76
4.2.4 Summary 79
4.3 Multi-scale and morphologically consistent terrain generalisation 79
4.3.1 Methodology and Results 81
4.3.2 Summary 81
4.4 Visualisation of the evolution of a terrain 81

4. Contents
4.4.1 Proposal 89
4.4.2 Methodology 89
4.4.3 Results and Summary 91
4.5 Scope for further applications 92
Chapter 5 Conclusions 95
5.1 Summary 95
5.1.1 Modelling terrain using surface networks: what’s possible and 95
what’s not
5.1.2 Revealing terrain structure using surface networks 97
5.1.3 Applications of surface networks 98
5.2 Future research 99
References 100
Appendix 110
A.1 SNG file format 110
A.2 SNM file format 111

5. Abstract
The nature of data structures used for the representation of terrain has a great influence
on the possible applications and reliability of consequent terrain analyses.
This research demonstrates a concise review and treatment of the surface network
data structure, a topological data structure for terrains. A surface network represents the
terrain as a graph where the vertices are the fundamental topographic features (also
called critical points), namely the local peaks, pits, passes (saddles) and the edges are the
ridges and channels that link these vertices. Despite their obvious and widely believed
potential for being a natural and intuitive representation of terrain datasets, surface
networks have only attracted limited research, leaving several unsolved aspects, which
have restricted their use as viable digital terrain data structures. The research presented
here presents novel techniques for the automated generation, analysis and application of
surface networks.
The research reports a novel method for generating the surface networks by
extending the ridges and channels, unlike the conventional critical points-based approach.
This proposed algorithm allows incorporation of much wider variety of terrain features in
the surface network data structure.
Several ways of characterising terrain structure based on the graph-theoretic analysis
of surface networks are presented. It is shown that terrain structures display certain
empirical characteristics such as the stability of the structure under changes and
relationship between hierarchies of topographic features. Previous proposals for the
simplification of surface networks have been evaluated for potential limitations and
solutions have been presented including a user-defined simplification. Also methods to
refine (to add more details to) a surface network have been shown.
Finally, it is shown how surface networks can be successfully used for spatial analyses
e.g. optimisation of visibility index computation time, augmenting the visualisation of
dynamic raster surface animation, and generating multi-scale morphologically consistent
terrain generalisations.

6. Acknowledgements
The research that I have managed to do in the last five years could not have been possible
without the help of many people. My heartfelt thanks go to:
• I think I can now *safely* write that for good reasons, initially neither of my
supervisors was probably too keen on the topic as the topic of the research was only
partially related to their research interests, and there were several rather tricky issues
in this topic. So, I am grateful of my PhD Supervisors, namely Jeremy Morley and Mike
Batty for their patience and support throughout the research.
• My family for being there for me in times of thrills and sorrows at the end of telephone
line and Christmas dinners.
• Brilliant research colleagues, namely Gert Wolf, John Pfaltz, Jo Wood, Shigeo
Takahashi, Jason Dykes, Lewis Griffin, David Unwin, David Mark, and many others who
provided intellectual and materials support.
• All the colleagues at CASA, every one of whom helped to make the long journey seem
so short and enjoyable.
• Close friends, namely Anupam, Bittu, TK, and Jim for their company and faith.
• The Association of Commonwealth Universities, UCL, and Ordnance Survey for
providing the funding at various stages of the research. I also thank the colleagues at
University of Leicester for their support and encouragement.
• Isle of Man dataset was obtained from Jo Wood. Cairngorm data were obtained from
the Digimap Archive under the CHEST agreement and are a crown copyright. Ordnance
Survey provided the Salisbury terrain data. Gert Wolf provided the Latschur Mountains
surface network.
• PhD Thesis examiners, namely David Unwin and David Kidner for their questions and
suggestions during the PhD examination.

7. List of Figures
1.1 Representation of a terrain into two different models depending upon the 3
application.
1.2 (a) A contour representation of a terrain and (b) the corresponding surface 7
network. Numbers in the parentheses of features in (a) are their respective
elevations and the number on the links of the graph in (b) are weights associated
with edges.
1.3 Inconsistent topology (from Wolf 1990). 10
1.4 (a) Two channel junctions (in circles) in a surface network and (b) their 12
decomposition into an infinitesimally closely located pair of pits and passes
(modified after Wolf 1990).
1.5 (a) A ridge bifurcation (in circle) in a surface network and (b) their decomposition 13
into an infinitesimally closely located pair of pass and peak (modified after Wolf
1990).
1.6 A mountain with gullies on slope faces (Source: Anonymous). 13
1.7 Point p(i,j) in a grid and its 8 adjacent neighbours (after Takahashi et al. 1995). 15
1.8 Point p in a grid (analytical view) and its 7 adjacent neighbours (hollow circles) 16
(after Takahashi et al. 1995).
1.9 Decomposition of a degenerate pass (modified from Takahashi et al. 1995). Figure 17
shows the neighbours and their heights. Higher neighbours are placed inside a
grey region. (a) The original neighbour list, (b) the reduced neighbour list, (c) the
list in the first turn of the loop in the algorithm, and (d) the final set of
neighbours which will define the pass.
1.10 Critical points of the surface and the configuration of their eigenvalues and 20
eigenvectors. R1 and R2 are the real parts of the eigenvalues.
1.11 Scale dependency of the Wood (1998) method. (a) The top-left corner of a raster 21
and the first cells that could be classified based on different filter window sizes
and (b) two peaks with similar local shape but with different extents.
1.12 (a) (yo–zo)-contraction and (b) (xo–yo)-contraction of a surface network. 24
2.1 Quantitative vs. Qualitative model uncertainty. The loss of height and shape 28
information during a conversion of a raster surface to a TIN as shown in (a) can
be evaluated to some precision however the representation of a raster (assumed
to be a continuous surface) as a set of feature lines and points as shown in (b)
involves unknown and subjective choice during conversion hence uncertainty will
remain qualitative or at best a probability.

8. List of Figures
2.2 Three types of approaches for modelling the scale-space. 29
2.3 (a) Topographic map of a part of Salisbury area in England showing a pass feature 31
in the box (courtesy: Streetmap.co.uk) and (b) the aerial photograph of area
(courtesy: GetMapping.com).
2.4 A topological network of ridges (white lines) and channels (black lines) bounded 33
by pits, peaks and passes (from Wood 1998). Note, many ridges and channels do
not follow true ridge and channel locations.
2.5 Flowchart for stages II- IV to construct surface networks from raster terrain. 34
2.6 (a) Positive plan curvature with divergent flow lines and (b) negative plan 35
curvature with convergent flow lines (modified after Peschier 1996).
2.7 3x3 feature extraction filter window and the criteria for feature classification. 36
2.8 Comparison of feature extraction methods in a terrain around Salisbury, UK. (a) 43
Elevation based flow assessment, (b) descent of plan curvature, and (c) conic
section analysis. All the methods used the same input DEM and parameters. Red
cells are ridges and blue cells are channels. The contour overlay can be used to
observe the robustness of the feature extraction algorithm. The cell size of DEM
is 10 m and has 502x501 cells.
2.9 Fill-in-the-gap filter used to connect broken features. (a) Simple case, where the 43
filter produces desired output and (b) where it produces problematic outputs.
2.10 (a) Broken surface network topology. Snapping the unconnected downstream 44
ends of channels and unconnected loose end of ridges by (b) linear extension, (c)
steepest downslope path, and (d) aspect path. Solid lines are original links and
dashed lines are extensions. Blue lines are channels and red lines are ridges.
Compare the methods for accuracy for links in the red box in NE part of the
area. The small boxes are an artefact of raster to vector conversion.
2.11 Conversion of a ridge and channel segments to elementary surface network. 45
Dashed edges are artificial.
2.12 The configuration of leader nodes at the various types of ridge and channel 46
intersections and their respective decompositions. See text for the commentary.
Dashed edges are artificial.
2.13 Representation of a loop of ridge segments around a crater by decomposition 48
into elementary surface networks. Dashed edges are artificial.
2.14 (a) An arbitrary configuration of leader nodes at the various types of ridge and 49
channel intersections and (b) their respective decompositions. See text for the
commentary.
2.15 (a) Hill-shaded view of a part of Isle of Man raster terrain (cell size 100 m, 50
IV

9. List of Figures
135x121 cells) and the corresponding (b) surface network model (peaks, pits and
passes are not displayed to reduce clutter) with contours (20 m interval).
2.16 (a) Hill-shaded view of a part of Salisbury raster terrain (cell size 10m, 329x379 51
cells) and the corresponding (b) surface network model (peaks, pits and passes are
not displayed to reduce clutter) with contours (20 m interval). See text for
commentary on the features in the circle.
2.17 The effect of smoothing on the feature classification of the Salisbury area terrain. 52
(a) The four linear trends (red lines) have been derived visually and the
corresponding numbers indicate the order in which they appear during smoothing
and (b) feature maps. Blue areas are channels and red areas are ridges.
2.18 The effect of smoothing on the feature classification of the Isle of Man area 53
terrain. (a) The four linear trends (red lines) have been derived visually and the
corresponding numbers indicate the order in which they appear during smoothing
and (b) feature maps. Blue areas are channels and red areas are ridges.
3.1 (a) Surface Network of central Isle of Man, (b) distribution of mean depth values, 58
(c) frequency plot of degree, and (d) variations in graph diameter with
contractions.
3.2 (a) Surface Network of a part of Latschur mountains in Austria, (b) distribution of 59
mean depth values, (c) frequency plot of degree and, (d) variations in graph
diameter with contractions.
3.3 Graphical User Interface of Surface Topology Toolkit. 61
3.4 Comparison of the effectiveness for selection of points in the surface network 63
between (b) maximum of elevation difference criterion and (c) and maximum of
edge length criterion. Note that criterion (b) selects a long ridge due to its low
drop in elevation (350).
3.5 Comparison of the effectiveness for selection of points in a surface network, 63
between (b) sum of elevation difference criterion and (c) valency criterion,
showing how criterion (b) can mislead about the ridge/channel crossings.
Numbers at peaks in (a) are sum of elevation differences and their valences (in
parentheses).
3.6 Cascading contraction. In the original form in nature as shown in (a) the two 64
channels at the bottom with weights equal to 8 exist because of the flow from the
8 channels upstream. But when two upstream channels are contracted, the
number of contributing channels to the minor channels is reduced to 6 as shown
in (b). This could mean that the two minor channels dry up and are then removed
from the network as shown in (c).
3.7 Generating artificial changes in terrain, in this case a large valley using User 66
Defined Contraction on a part of Latschur surface network.
3.8 Generating artificial changes in terrain, in this case erosion of minor features to 66
V

10. List of Figures
yield a large ridge using User Defined Contraction on a part of Isle of Man
surface network.
3.9 Refining a long ridge with a (y0-z0)-splitting. Note how the choice of configuration 68
ensures a topological consistency after the addition of the new edges.
3.10 A sequence of 2 refinements on longest ridges of a hypothetical surface network 68
with repeated (y0-z0)-splitting. Blue lines are channels and orange lines are ridges.
Red dots are peaks, green dots are passes and black dots are pits.
3.11 (a) Uncertainty regarding an elevation value in the case of scattered points and (b) 70
in the case of surface network.
4.1 (a) Hill-Shaded terrain of SE Cairngorm Mountains, Scotland. Minimum Elevation 74
= 395 m and Maximum Elevation = 1054 m and (b) 910 topographic feature
targets with an overlay of contours (20 m interval).
4.2 Comparison between the (a) Golden Case based visibility dominance and (b) 77
topographic features based visibility dominance. Darker coloured areas have more
visual dominance than lighter coloured areas.
4.3 Uncertainty assessment based on the entire DEM. (a) Absolute vs. estimated 77
visibility dominance of all locations and (b) residuals based on the linear
regression between absolute and estimated visibility dominance values of all
locations.
4.4 Uncertainty assessment based on selective sampling. (a) Absolute vs. estimated 78
visibility dominance of the topographic features and (b) correlation coefficient vs.
errors at 19 sets of 418 random locations, with the average value shown with the
dotted line.
4.5 Comparison between the AVI and EVI based on the reduced observers strategy. 78
4.6 (a) Original 50m spatial resolution DEM and (b) its resampling to 200 m spatial 80
resolution with simple averaging filter. Note the smoothness in (b) at the expense
of structural losses for example at the point indicated by the arrow.
4.7 (a) Landform PROFILE contours of a part of Salisbury (top) and hill-shaded 10 82
m Salisbury raster terrain (bottom) produced using the TOPOGRID function.
Note the pronounced terracing effect on slopes located in the northeast and
northwest of the area.
4.8 Comparison of the preservation of elevation, morphology and surface continuity 83-87
after different types of generalisation of the 10 m cell size Salisbury raster terrain
shown in Figure 4.7b to 100 m cell size. The area in circles shows that MMTG
method preserves both elevation and morphology. Also note that ordinary
interpolation using TOPOGRID is not sufficient to ensure feature preservation.
MMTG generalisation method also produces the best continuity in generalised
DEM.
VI

11. List of Figures
4.9 Digital elevation models of a sand spit at Scolt Head Island, North Norfolk, UK. 90
Two situations are shown representing the results of surveying the feature in
1997.
4.10 Increase in the structural information delivery with the addition of contours and 91
surface network overlays.
4.11 22 Intermediate surfaces (microsteps) generated by blending the February, 1997 93
surface (Situation 1) into the September, 1997 (Situation 2) surface.
4.12 Use of the surface network representation to visualise the changes in the 94
morphology of the sand spit. The box indicates an area of interest. Note that the
surface network variations highlight changes that are not evident from the
representation that uses color to show variation in elevation.
VII

12. List of Tables
1.1 Criteria for classification of critical points in the eight-neighbour method (after 15
Takahashi et al. 1995).
1.2 Criteria for the classification of non-degenerate critical points based on Delaunay 16
triangulation (after Takahashi et al. 1995).
1.3 Morphometric Features described by second derivatives (after Wood 1996). 20
VIII

13. "Never again," cried the man, "never again will we wake up in the morning and think Who
am I? What is my purpose in life? Does it really, cosmically speaking, matter if I don't get
up and go to work? For today we will finally learn once and for all the plain and simple
answer to all these nagging little problems of Life, the Universe and Everything!"
…..
"An answer for you?" interrupted Deep Thought majestically. "Yes. I have."
…..
"Forty-two," said Deep Thought, with infinite majesty and calm.
…..
"Forty-two!" yelled Loonquawl. "Is that all you've got to show for seven and a half million
years' work?"
"I checked it very thoroughly," said the computer, "and that quite definitely is the answer. I
think the problem, to be quite honest with you, is that you've never actually known what the
question is."
"But it was the Great Question! The Ultimate Question of Life, the Universe and
Everything!" howled Loonquawl.
"Yes," said Deep Thought with the air of one, who suffers fools gladly, "but what actually is
it?"
A slow stupefied silence crept over the men as they stared at the computer and then at each
other.
"Well, you know, it's just Everything ... Everything..." offered Phouchg weakly.
"Exactly!" said Deep Thought. "So once you do know what the question actually is, you'll
know what the answer means."
From “Hitchhiker’s Guide to the Galaxy” by Douglas Adams (1979)

14. Chapter 1
Introduction
1.1 Context
Digital elevation models (DEMs) are essential components of wide ranging applications that
require a representation of natural terrain. The applications could be as diverse as
archaeology (e.g. viewshed analyses of ancient settlements; Lake and Woodman 2003) to
the modelling of zoological habitats (e.g. forest ecosystems; Mackey et al. 2000).
Naturally, an enormous quantity of research has taken place in the various aspects of
digital elevation modelling. Some recent significant works amongst many others on digital
elevation modelling include von Minusio (2002), Hutchinson and Gallant (2000) and the
proceedings of the recent ASPRS conference on “Terrain Data and Applications” (URL #1).
DEMs are an outcome of the object generalisation of continuous terrain (Weibel
and Dutton 1999) and consequently inherit uncertainty arising out of the discretisation
(von Minusio 2002). Different DEMs provide varying quantity of data on of the terrain
continuity. For example, raster DEM will provide a more complete description of terrain
heights compared to a triangulated irregular network or TIN (Kumler 1994) because of the
differences in their approach to sample the continuous terrain. Various terrain
representations reveal varying levels of abstraction which is particularly relevant when the
increasingly massive terrain datasets are stored in computers. von Minusio (2002) provides
a detailed discussion on the desirable characteristics of a reliable DEM. The choice of the
DEM to represent the continuous terrain i.e. whether TIN or raster or others is critical to
the success of the terrain analyses.
It is important to appreciate how different disciplines describe terrains, for a
broader understanding of previous research on this topic. Mathematicians have modelled
terrains primarily with an aim to decompose the shape of terrain into the basic descriptors
or elements even if it introduced over-simplification (e.g. by using the simple geometrical
shapes such as triangles) and potential loss of the structure. Such descriptions are generic
(i.e. universal to all types of surfaces), formal and robust thus crucial for applications such
as computer-aided design. The aim of these mathematical descriptions is to produce a
constrained global model of terrain. The other large group of terrain researchers from the
field of physical geography use more compound descriptors (e.g. valleys, mounds, scarps,
drainage network) with more emphasis on the preservation of the structural information of
the terrain. Although the compound descriptors are more natural, their derivation is
subjective to each individual, hence it is often difficult to derive an objective definition of
terrain features1. These researchers are more interested in the process which resulted the
surface hence the descriptors are also symbolic of the factors in the process.
1
Wolf (1993, p24) highlights the importance of exact definitions quoting Werner (1988) and Frank et al. (1986).

15. Introduction
A simple example of such a fundamental dichotomy is the description of terrain by
these two disciplines. To achieve a simple and tractable model of terrain, a typical
algebraic definition of terrain will be as follows:
Terrain is a smooth, doubly-continuous function of the form z= f(x,y), where z is the
height associated with each point (x,y). The local maxima or peak of the terrain is a
point with a zero slope and a convex curvature.
Definitions for other terrain features are defined similarly using morphometric
measures. Most physical geographers will however find these definitions very restrictive
because, a) they do not offer scope to include some common terrain features such as lakes
and overhangs that are fundamental to certain applications e.g. runoff modelling, and b)
natural terrain features cannot be localised to a point because a peak with zero local slope
doesn’t really exist in nature. In physical geography, the description of terrain surface and
terrain features is more indicative than precise. Therefore, provided the shape of the
terrain around an area could be described as a certain terrain feature type, it is the
responsibility of the geographers to locate the position of the feature on terrain based on
their opinion. Figure 1.1 shows the difference between an algebraic topology (as a
Triangulated Irregular Network or TIN) and physical geography (drainage network)
description of a terrain surface.
It follows therefore, that a combination of these two ways of terrain modelling
should provide a complete and robust approach for describing surfaces. In other words,
the terrain data model which includes both the morphological structure (in terms such as
hills and valleys) and the geometrical form (e.g. xyz coordinates) of terrain will be an
ideal digital representation of terrain. Wolf (1993) stated a more general form of this
requirement. Wolf regarded that an efficient terrain data structure will contain both the
geometrical information (e.g. coordinates, line equations) and topological information on
the geometrical data (e.g. neighbourhood relationships, adjacency relationships) of the
terrain. However, the course of research in this thesis revealed that the construction of
the topologically consistent terrain data model is non-trivial because real terrains seldom
obey the constraints required by topological rules.
It is trivial to produce terrain data models that only store the geometrical
information about a surface. We simply need to collect certain points on the surface either
on a regular lattice/grid or irregular locations. In fact many surface applications only
require geometrical information for analyses. However, storing topological information has
several significant advantages as listed below:
• If we assume certain homogeneity in surface shape (e.g. smooth and continuous),
using a topological data structure will reduce the number of points required to
construct a surface. For example, by storing only certain morphologically important
point (MIPs) (e.g. corners, inflexions) and their topological relationships we could
reconstruct the surface by using interpolation between MIPs. Thus, the quantity of
computer disk space required to store the surface will be reduced significantly.
Helman and Hesselink (1991) reported 90% compression of volumetric surface datasets
using topological data structures.
• Topological relationships are a more efficient way to gain access to a spatial database
e.g. sophisticated spatial queries such as clustering would be easily implemented by
performing trivial adjacency and proximity tests between the points.
2

16. Introduction
Terrain
Mathematician’s Model Physical Geographer’s Model
Piecewise model of terrain surface as Drainage network model of terrain surface
triangular patches. as a network of ridge and channels.
Representation simply requires the Representation requires non-trivial
storage of points and their topological derivation of consistent drainage network.
relationships.
Figure 1.1
Representation of a terrain into two different models depending on the application
• Presence of topology could provide an unified representation of the global structure of
the surface. This would be useful for analyses that require uniform and controlled
response from the entire surface e.g. morphing in computer graphics and erosion
modelling in hydrology.
• Topological data model will be useful for the visualisation of the structure of surfaces,
particularly multi-dimensional surfaces. For example, Helman and Hesselink (1991)
and Bajaj and Schikore (1996) reported that rendering of volumetric surface datasets
as skeleton-like topological data structure is faster and more comprehensible
compared to traditional volume rendering.
• Bajaj and Schikore (1996) propose that topological surface data models will be a
simple mechanism for correlating and co-registering surfaces due to the embedded
information on the structure of terrains.
3

17. Introduction
While the benefits of combining topological relationships between morphological
features of terrains is clear, it is uncertain which MIPs and topological relationships should
constitute a universal surface topological data structure. In general, each terrain should
be characterised by MIPs suitable for a particular application. Many types of MIPs have
been proposed by researchers in different disciplines and referred with different names for
example landform elements (Speight 1976), critical points and lines (Pfaltz 1976),
surface-specific features (Fowler and Little 1979), symbolic surface features (Palmer
1984), surface patches (Feuchtwanger and Poiker 1987) and specific geomorphological
elements (Tang 1992). The common aim of these classifications has been to provide
sufficient resemblance to the surface relevant to a particular application.
The research presented in this thesis focuses on a DEM that models terrain as a
graph between the important morphological features. In particular, the motivation of the
research reported here was to refine and extend the surface network (Pfaltz 1976) of the
terrain. A surface network is a graph where the local peaks, passes and pits are the three
sets of vertices and the local ridges and channels, the edges (Figure 1.2). The ridges
connect the passes to the peaks and the channels connect the pits to the passes. Even with
such a general description, the surface network promises several unique and desirable
properties, as follow:
• A surface network is made up of the fundamental (Fowler and Little 1979) and minimal
set of local topographic features hence it is highly compressed (up to 90% compression
can be achieved by representing the terrain as a surface network), e.g. see Helman
and Hesselink (1991) and yet preserves critical terrain morphology.
• A surface network is an explicit expression of the terrain morphological structure
unlike TIN or raster where the morphological structure has to be represented and
stored separately.
• The graph-theoretic structure of the surface network makes it amenable to formal
graph based contraction, which has been proposed to be useful for the generalisation
of terrain structure and not merely the terrain database (Wolf 1989, Rana 1998).
Therefore, due to its explicit topographic design, surface network is a natural and
intuitive data model to represent terrains. However, several issues related to the four
main aspects of surface networks, namely their model design, automated generation,
generalisation and practical use remain unresolved, which has prevented their use as a
practical DEM. The goal of the thesis has been to fill the gaps. The remaining part of this
chapter will provide a background to the above four aspects, which will then be used to
highlight the unresolved issues and the aims of the thesis.
At this stage, a difference between the terms “data structure” and “data model”,
used in the context of terrain datasets is proposed. It is proposed that the term
(surface/terrain) data structure should merely represent a format for storing the
geometric and topological information (e.g. point heights and adjacency relationships) in a
single construction. On the other hand, the (surface/terrain) data model should be an
extended version of the surface data structure, where additional metadata information
characterising the surface (e.g. valleys, ridges i.e. characteristic properties of surface) is
also incorporated to produce a natural representation of the surface. In other words, a
surface data model is a value-added product of the surface data structure and the surface
data model explicitly represents the characteristics of the surface. Thus all surface data
models can be regarded as surface data structures but the opposite is not necessarily true.
4

18. Introduction
In the literature the term DEM has often been restricted to a raster, also called
gridded, lattice and cellular, elevation data structure. This definition seems to be a rather
limited use of the scope of the term DEM. Therefore, it is proposed that the term DEM be
used for any data structure that satisfies the following properties:
• It provides a 2.5D representation of the topographic surface.
• It provides a description that can be used to reconstruct the continuity of the
topography.
By implication the above definition will also include a TIN of a terrain as a DEM
while a collection of scattered elevation values or digitised contours will not be classified
as DEMs. Also, the term DEM here only implies the natural topography thus it excludes
vegetation and anthropogenic features which may exist on terrains. Some researchers also
use the term digital terrain model (DTM). A digital surface model (DSM) on the other hand
could consist of all desirable features including the terrain.
1.2 Unresolved Issues and Aims of the Thesis
The concept of surface network has evolved from the culmination of ideas from
mathematics, physical geography and computer science. This mixed lineage is because
terrain modelling is a part of the broader research on modelling of scalar two-dimensional
functions that are regarded as two-dimensional surfaces; specifically,
• Terrain is assumed to be a scalar function such that the scalar property z = f(x,y), and
2
• Terrain is C continuous (so there are no overhangs), defined over a domain that is
simply connected (so there are no holes), and bounded by a closed contour line.
Terrain modelling is an inter-disciplinary topic attracting interests from researchers in
physical geography (terrains); social geography (socio-economic surfaces); computer
science (digital images); metrology (metal surfaces); physics (crystallographic energy
surfaces) and many others. Koenderink and van Doorn (1998) provide an excellent review
on the various applications. The following sets of brief descriptions list the sequence of
events related to the conceptual developments of surface networks, and link the lineage
of the ideas from the various disciplines.
The primary origin of surface network research lies in the realisation of the
fundamental topographic features. Fundamental topographic features are characteristic
local topographic features that are common to all terrains and contain sufficient
information to construct the whole terrain, thus taking away the need to store each point
on the surface.
The concept of critical points of a surface (where δz/δx = δz/δy = 0), namely the
2 2 2 2 2 2 2 2 2 2
local maxima (δ z/δx > 0, δ z /δy > 0), minima (δ z/δx < 0, δ z /δy < 0), saddles also
2 2 2 2 2
called passes (δ z/δx > 0, δ z /δy < 0or vice versa) and slope lines (lines normal to
contours) also called topographic curves and gradient paths of surfaces, were proposed as
early as the mid-19th century by mathematicians De Saint-Venant (1852) (reported by
Koenderink and van Doorn 1998) and Reech(1858) (reported by Mark 1977). In physical
geography, Cayley (1859), based on contour patterns, first proposed the subdivision of a
topographic surface into a framework of summits (local peaks or maxima), immits (local
pits or minima), knots (local saddles or passes), ridge lines (slope lines from saddles that
reach up to a summit) and course lines (slope lines from saddles that go down to a minima)
5

19. Introduction
(Figure 1.2). Maxwell (1870)2 maintained Cayley’s definition of summits and immits and
added that a pass is the low point connecting two summits and a bar is the high point
connecting two immits. As can be seen in Figure 1.2, for a terrain a bar and a pass are
really the same critical point. Maxwell proposed the following relations between the
numbers of critical points on terrain, which were later proved by Morse (1925) using
differential topology:
summits = passes + 1; immits = bars + 1
∴ summits + immits – 2 = (bars + passes) / 2 OR summits + immits – passes = 2
Maxwell also described the partition of the topographic surface into Hills (areas of
terrain where all slope lines end at the same summit) and Dales (areas of terrain where all
slope lines end at the same immit) based on the fundamental topographic features.
The earliest graph-theoretic representation of the topological relationships
between the critical points of a terrain is the Reeb Graph (Reeb 1946, reported by
Takahashi et al. 1995). Reeb graph represents the splitting and merging of equi-height
contours (i.e. the cross-section) of a surface as a graph. The vertices of the graph are the
peaks, pits and passes because the contours close at the pits and the peaks, and split at
the passes. Consequently, the edges of the Reeb Graph turn out to be the ridges and
channels.
In a significant related development in mathematics, Morse (1925) derived the
relationship between the numbers of critical points of sufficiently smooth functions, which
is known as the Critical Point Theory or Morse Theory. More specifically, Morse derived
that for a two-dimensional function f with the following properties:
2
• f is sufficiently smooth i.e. f ∈ C . Thus, it is possible to calculate the curvature at
each point on the function hence cases like overhangs and lakes do not exist,
• For all points b on the boundary f(b) < f(i) where i is an interior point.
• All critical points of f are non-degenerate i.e., the Hessian matrix H(p) of the 2nd
derivatives, of critical points has a nonzero determinant (i.e. singular or regular). The
Hessian matrix for a point p(x,y) is defined as:
H(p) = δ2f/δfxδfy
• The following relationships exist between the critical points of f where P0, P1 and P2
denote the sets of pits, passes and peaks of f respectively. The last equation has been
referred as the Mountaineer’s Equation (Griffiths 1981, reported by Takahashi et al.
1995) and the Euler Formula (Takahashi et al. 1995).
|P0| ≥ 1; |P0| - |P1| ≥ 1; |P0| - |P1| + |P2|= 2
2
The anxiety with which Maxwell presented his paper is quite amusing. His note to the editor of the journal
reads “An exact knowledge of the first elements of physical geography, however, is so important, and loose
notions on the subject are so prevalent, that I have no hesitation in sending you what you, I hope, will have no
scruple in rejecting if you think it superfluous after what has been done by Professor Cayley.”
6

20. Introduction
x (1000)
z2(4000)
z1(5000) y1(2000)
x (1000)
y2(3000)
(a)
x (1000) z3(5000)
Pass
Peak
Pit
3000 z1
y1
(b) 2000
1000
z2
x 2000
2000
y2 2000
z3
Figure 1.2
(a) A contour representation of a terrain, and (b) the corresponding surface network.
Numbers in the parentheses of features in (a) are their respective elevations and the number
on the links of the graph in (b) are weights associated with edges.
7

21. Introduction
The function f that satisfies the above properties is called a Morse function. The
generic nature and wide applicability of Morse Theory led to the expansion in interest in
the critical points of surfaces amongst various disciplines.
Warntz (1966) revived the interest of geographers and social science researchers
into critical points and lines when he applied the “Hills and Dales” concept for socio-
economic surfaces, and called the data model as the Warntz network (A term apparently
first used by Mark 1977).
A data structure identical to Reeb graph is the contour tree (Morse 1968, 1969). A
contour tree represents the adjacency relations of contour loops. The tree like
hierarchical structure develops because each contour loop can enclose many other contour
loops but it can itself be enclosed by only one contour loop. As is evident, the contour tree
is the same as the Reeb graph. Interestingly, Kweon and Kanade (1994) proposed a similar
idea called the Topographic Change Tree. Similar to the Reeb graph, the vertices of such a
contour tree and topographic change tree are the peaks, pits and passes.
Pfaltz (1976) proposed the graph representation of the Warntz network and called
it the surface network (Mark 1977 refers to surface network as Pfaltz’s graph). While the
topology of Pfaltz’s Graph was based on a Warntz network, Pfaltz added the constraint
that the surface will have to be a Morse function. Since Pfaltz was in the computer science
field, his work attracted the attention of researchers in three-dimensional surfaces such as
in medical imaging, crystallography (Johnson et al. 1999, Shinagawa et al. 1991) and
computer vision (Koenderink and van Doorn 1979). Pfaltz also proposed a homomorphic
contraction of a surface network graph to reduce the number of redundant and
insignificant vertices. Along the similar lines, Mark (1977) proposed a pruning of the
contour tree to remove the nodes (representing contour loops), which do not form the
critical points, i.e., the vertices of the contour tree, and called the resultant structure,
the surface tree. This reduces the contour tree to the purely topological state of a Pfaltz’s
graph. The Reeb graph, Pfaltz’s graph and Surface tree have fundamental similarities and
are inter-convertible (Takahashi et al. 1995).
Nackman (1984) proposed a new construction for the graphs of critical points,
called the Critical Point Configuration Graph (CPCG). He proposed rules for CPCG to be a
surface network under more general conditions than those in the Pfaltz’s graph.
Wolf (1984) extended Pfaltz’s graph by introducing more topological constraints in
order it to be a consistent representation of the terrain. He proposed assigning weights to
the critical points and lines to indicate their importance in the surface and thus he
proposed the name weighted surface network (WSN) for Pfatlz’s graph. Wolf demonstrated
new weights-based criteria and methods for the contraction of the surface networks.
Later, Wolf suggested that to visualise a WSN for cartographic purposes and be useful for
spatial analyses, the vertices could be assigned metric coordinates (Wolf 1990). The
resultant representation is termed metric surface network (MSN).
Feuchtwanger and Poiker (1987) proposed a topological model for terrains, which
was a combination of ideas from the Interlocking Ridge and Channel Network (Werner
1988), Hills and Dales, Contour Tree, Surface Tree, and Pfaltz’s Graph. Sadly, although
interesting, the idea did not advance beyond the Entity-Relationship Model of the data
structure.
Recent works (c. 1999-2004) have mostly focussed on the automated extraction of
surface networks from raster and TIN DEMs, which will be discussed in a later section.
8

22. Introduction
1.2.1 Model
The basic construction of the surface network model has not changed since the proposals
by Pfaltz (1976) and Wolf (1988, 1990). Thus their description of surface networks has
been used throughout the following section. The original descriptions have been extended
in places where it was felt necessary for a better comprehension.
1.2.1.1 Concept
The surface network represents a terrain, assumed to be a two-dimensional Morse
function, as a weighted, directed, tripartite graph W = (P0, P1, P2; E), where P0, P1, P2
are three vertex sets representing the sets of all pits, passes and peaks, respectively,
while E is the set of all edges (Figure 1.2). In addition a consistent weighted surface
network (WSN) satisfies all the following rules (after Pfaltz 1976, Wolf 1988):
Rule 1: W is planar (Wolf 1988).
This means that an intersection of edges i.e. intersection of ridges and channels is not
allowed. There can only be one type of slope line passing through a point except at the
critical points.
Rule 2: The subgraphs [P0,P1] and [P1,P2] are connected (Pfaltz 1976).
This means that channels connect pits and passes, and ridges connect peaks and passes.
This property imparts a global and unified structure to surface network, which makes it
amenable for spatial analyses such as hydrological modelling.
Rule 3: |P0| - |P1| + |P2| = 2 (Pfaltz 1976).
It states that the number of pits minus the number of pass points plus the number of peaks
must always be two, as also required for the terrain to be a Morse function.
Rule 4: For all v ∈ P1, id(v) = od(v) = 2 (Pfaltz 1976); id and od represent in-degree and
out-degree respectively.
This rule requires that exactly two incoming edges (channels) and exactly two outgoing
edges (ridges) should be incident at a pass thus excluding the existence of degenerate
passes.
Rule 5: If val(u, vi) = val(vi, w) = 1 then there must exist vj ≠ vi such that
(u,vj), (vj,w) ∈ E (Pfaltz 1976); val represents valency, and u ∈ P0 , w ∈ P2 , 1 ≤ i , j ≥ |P1|.
This condition requires that if there is a path from pit u via pass vi to peak w, which
consists only of edges with valency one, then there exists another path from pit u to peak
w via a distinct saddle vj.
Rule 6a: If (u,v) is an edge of a circuit in the bipartite graph [P0,P1] then
val(v,w) ≠ 2 for all w ∈ P2 (Pfaltz 1976), and
Rule 6b: If (v,w) is an edge of a circuit in the bipartite graph [P1,P2] then
val(u,v) ≠ 2 for all u ∈ P0 (Pfaltz 1976).
This property asserts that a graph configuration as shown in Figure 1.3 is not allowed,
because it implies a violation of Rule 1. Another way of stating this rule is that val(u, v) +
val(v, w) ≤ 3 for all u ∈ P0 , v ∈ P1, w ∈ P2 (Pfaltz 1976).
9

23. Introduction
z x
y
Figure 1.3
Inconsistent topology (from Wolf 1990).
Rule 7: wt(ei) > 0 for all ei ∈ E (Wolf 1988);
wt represents weight and 1 ≤ i ≥ |E|
This means that all the edge weights must be greater than zero. For instance, if h(u), h(v)
and h(w) represents the elevations of a pit, pass and peak, respectively, then the weight
of a channel is h(v) – h(u) and the weight of a ridge is h(w) – h(v).
Rule 8: For all u ∈ P0, vi, vj ∈ P1, w ∈ P2 and (u,vi), (u,vj), (vi,w),(vj,w) ∈ E holds
w(u,vi) + w(vi,w) = w(u,vj) + w(vj,w) (Wolf 1988).
This means that for all paths from pit u to peak w the weight is the same, no matter which
saddle point is passed.
Rule 9a: If val(u,v) = 2 with ei1 = (u,v) and ei2 = (u,v) then w(ei1) = w(ei2) (Wolf 1988).
Rule 9b: If val(v,w) = 2 with ei1 = (v,w) and ei2 = (v,w) then w(ei1) = w(ei2) (Wolf 1988).
This means that all channels from a pit to a pass have the same difference in altitude; the
same holds for ridges, too.
The proof of these rules is available in Pfaltz (1976) and Wolf (1988) and it will not
be dealt with here.
1.2.1.2 Limitations of the WSN Model and Topological Rules
Although the weighted surface network model provides a natural and sophisticated
representation of the terrain, it has been considered merely as an interesting proposal by
geomorphologists. Pfaltz (1976) himself understood the several limitations and commented
that “it is unknown whether these properties are sufficient to guarantee the realizability
of G” (surface network). It is therefore no surprise that WSN is not mentioned amongst
most GI science textbooks as a DEM. It is proposed here that the surface network model
suffers from the following three main drawbacks:
(i) Non-representation of all terrains and terrain features.
The fundamental shortcoming of the surface network model is the assumption that natural
terrain is C2 continuous everywhere so that features such as overhangs (e.g. glaciated
terrains, dunes, and plateaus), holes (e.g. karstic terrains), break in slope (e.g. alluvial
fans, scarps) are absent. This requirement is a severe limitation as these features are
abundant in nature.
Commonly available DEMs are often full of sensor and interpolation noise (von
Minusio 2002), and don’t represent the relief below water level (isolated islands or large
flat areas inside terrain). Therefore, it is not possible to realise the surface networks for
all types of terrains, especially those which do not have the entire set of critical points
and lines required for the surface networks.
10

24. Introduction
It is clearly an oversimplification to assume that natural terrains are Morse
functions. The rules presented in section 1.2.1.1 e.g. Rules 3 and 4 simply ignore the
chaotic behaviour of weathering processes, which leave the terrain in a state of constant
inequilibrium3. However, it would also be unrealistic to suppose that it is impossible to
derive surface networks. These exceptional terrain patches could be regarded as well
behaved terrains or functions. Different kinds of limitations exist in representing well-
behaved terrains.
A common concern amongst geomorphologists regarding WSN is that it does not
represent many important hydrological features e.g. junctions and bifurcations because
the ridges and channels could only meet at the critical points. As a solution, Wolf (1990)
suggested that the channel junctions and ridge bifurcations could be represented as
infinitesimally closely located pit-pass and pass-peak pairs (Figures 1.4, 1.5) and termed
the new WSN model as MSN. While the proposal from Wolf (1990) correctly adjusts the
graph topology at the location of channel junctions and ridge bifurcations, the connections
of the edges incident at the artificial peaks/pits/passes is arbitrary. For example, it
assumes that free passes and peaks will be available, which connect the new artificial
peaks/pits/passes to the graph (Figure 1.4b).
Similarly, the gullies (small channels) on hill faces connecting to the main channel,
features common to any mountainous terrain, are not included. These gullies, called the
inner leaves of the channel network in the interlocking ridge and channel network model
(Figure 1.6) by Werner (1988), are a prominent terrain feature and relevant in hydrological
modelling for catchment analyses. Again, the problem here is that these gullies start from
a point on the hill face (called source nodes; Werner 1988), which is not a critical point.
(ii) Scaling
Terrain features are organised in a hierarchy, expressed as a variation in their spatial
extents (Fisher et al. 2004). For example, a gully on a slope face has small spatial extent
compared to the channel it drains into (Figure 1.6). The position of the feature in a
hierarchical arrangement can be regarded as the scale of the feature. Fisher et al. (2004)
demonstrated that a location on the terrain could be a part of different feature types at
different scales. Therefore, a terrain is inherently composed of multi-scaled features. It
therefore implies that a terrain would have multiple surface networks representing the
feature scales of a terrain. The existing surface network model (or for that matter Morse
Theory) does not address how such individual surface networks could be unified into a
single surface network model of a terrain.
(iii) Uncertainty
Surface network is an approximation of terrains based entirely on a minimal set of line and
point topographic features. As mentioned earlier, the surface network would inevitably
fail to capture all the variations present in terrain, which could lead to a considerable
heights/shape related uncertainty in the DEM. In general, the uncertainty will depend on
the deviation of the terrain from an ideal Morse function and would vary spatially across
the terrain. At present, there are no proposals for determinig the uncertainty associated
with a surface network.
3
Pfaltz (1976) suggests that points in in-equilibrium e.g. degenerate points could be decomposed into non-
degenerate points but does not provide any proposals.
11

25. Introduction
(1300)
y1
(1800)
(1100)
z1
(2100)
(a)
y2
z3
(1500) z2 y3
(1900)
(1700) (1400)
y1
x1 y4 x2 y5
(1800)
z1
(2100)
(b)
y2
z3
(1500)
z2 y3
(1900)
(1700) (1400)
Figure 1.4
(a) Two channel junctions (in circles) in a surface network and (b) their
decomposition into an infinitesimally closely located pair of pits and
passes (modified after Wolf 1990).
12

26. Introduction
z1 z1
(1600)
y3
z2
y1 y2
(1300) (1400) y1 y2
(a) (b)
x1 x1
(1000)
Figure 1.5
(a) A ridge bifurcation (in circle) in a surface network and (b) their decomposition into an
infinitesimally closely located pair of pass and peak (modified after Wolf 1990).
Figure 1.6
A mountain with gullies on slope faces (Source: Anonymous).
13

27. Introduction
1.2.2 Generation
The process of accurate automated extraction of a surface network from the digital
elevation dataset (e.g. raster, TIN, contour etc.) lies between the surface network model
and its use in practise. The methods of surface networks extraction have ranged from the
simple ones like manual digitisation (Wolf 1988) and triangulation (Takahashi et al. 1995)
methods to the complex surface fitting (Pfaltz 1976, Wood 1998). The different methods
were chosen depending on the researchers’ belief in the best way of extracting the critical
points and lines. There have been many suggestions for detecting the critical points and
lines of a surface. This thesis focuses on the afore-mentioned four works in some detail as
they represent the culmination of the most widely implemented ideas, used specifically
for surface networks.
1.2.2.1 Methodologies
In short, the generation of a surface network involves two steps - (i) extraction of the
critical points and (ii) connecting them with the critical lines. The three main categories
of methods, including a manual method, for surface network extraction are as follow:
(i) Manual Extraction
Wolf (1988) reported a successful manual generation of a topologically consistent surface
network. He picked the critical points from a contour map using a digitiser and established
the topological relationships i.e., the ridges and channels, by visual inspection.
(ii) Triangulation
Takahashi et al. (1995) proposed a modified version of the eight-neighbour method
detection of the critical points (Peucker and Douglas 1975) for grid surfaces. The eight-
neighbour method compares the height of a point, p(i,j), with its eight neighbours in a 3x3
square surrounding p (Figure 1.7) and classifies the point as a critical point based on the
criteria in Table 1.1.
Takahashi et al. (1995) showed that the eight-neighbour method based detection
depends on the value of the threshold and this ambiguity could cause the loss of the
Mountaineer’s Equation constraint i.e., pits – passes + peaks ≠ 2. They suggested that to
satisfy the Mountaineer’s Equation, the contour changes should be determined according
to the neighbour heights and not according to the threshold. They suggested the use of the
Delaunay triangulation (Guibas and Stolfi 1985) to triangulate the 3 x 3 square, centered at
p, and determine only the adjacent points (amongst the 8 surrounding neighbours) of p
(Figure 1.8). The point is then classified according to the criteria given in Table 1.2.
However, in the case of degenerate passes (Figure 1.9a) there will be more than 4
sign changes as three or more equi-height contours are merged. Takahashi et al. (1995)
derived that any degenerate pass can be decomposed into non-degenerate ones, m, where
m = (Nc – 2)/2 (Figure 1.9d). By solving this equation, we can find out that the number of
sign changes, Nc, at a degenerate pass will be equal to 2 + 2m (m = 1,2,...).
The algorithm to decompose a degenerate pass by Takahashi et al. (1995) is
unique and noteworthy. The steps are as follow:
(a) Generate a counter-clockwise (CCW) list of the adjacent neighbours of this pass, which
here is {p1, p2, p3, p4, p5, p6, p7} (Figure 1.9a).
(b) Divide this list into an upper sequence, which has the higher neighbours, i.e, {p1},{p3,
p4} and {p6}, and a lower sequence, which has the lower neighbours, i.e., {p2},{p5} and
14

28. Introduction
i –1 , j -1 i –1 , j i –1 , j +1
i , j -1 p (i,j) i , j +1
i +1 , j -1 i +1 , j i +1 , j +1
Figure 1.7
Point p(i,j) in a grid (data view) and its 8 adjacent neighbours
(after Takahashi et al. 1995).
.
Peak |∆+| > Tpeak |∆ -| = 0 Nc = 0
Pit |∆-| > Tpit |∆ +| = 0 Nc = 0
Pass |∆+| + |∆-| > Tpass Nc = 4
|∆+| The sum of all positive height differences between the point and its 8
neighbours.
|∆-| The sum of all negative height differences between the point and its 8
neighbours.
Nc The number of sign changes i.e., Σ |∆+| + |∆-| associated with the point.
Tpeak Threshold height for a point to be a peak.
Tpit Threshold height for a point to be a pit
Tpass Threshold height for a point to be a pass.
Table 1.1
Criteria for classification of critical points in the eight-neighbour method
(after Takahashi et al. 1995).
15

29. Introduction
p
Figure 1.8
Point p in a grid (analytical view) and its 7 adjacent neighbours (hollow circles)
(after Takahashi et al. 1995).
Peak |∆+| > 0 |∆-| = 0 Nc = 0
Pit |∆-| > 0 |∆+| = 0 Nc = 0
Pass |∆+| + |∆-| > 0 Nc = 4
|∆+| The sum of all positive height differences between the point and its 8
neighbours.
|∆-| The sum of all negative height differences between the point and its 8
neighbours.
Nc The number of sign changes i.e., Σ |∆ +| + |∆ -| associated with the point.
Table 1.2
Criteria for the classification of non-degenerate critical points based on Delaunay
triangulation (after Takahashi et al. 1995).
16

30. Introduction
p2 p2
86 p1 86 p1
p3 106 p3 106
110 p 110 p
103 100 89 100 89
p4 93
p7 93 104
p7
104
p5 p5
p6 p6
(a) (b)
p2 p2
86 p1 86
p3 106 p3
110 p 110 p
100 89 100
p7 93
93 104 104
p5 p5
p6 p6
(c) (d)
Figure 1.9
Decomposition of a degenerate pass (modified from Takahashi et al. 1995). Figure
shows the neighbours and their heights. Higher neighbours are placed inside a
grey region. (a) The original neighbour list, (b) the reduced neighbour list, (c) the
list in the first turn of the loop in the algorithm, and (d) the final set of
neighbours which will define the pass.
17

31. Introduction
{p7}. Reduce the neighbours list by selecting the highest neighbour from each upper
sequence and the lowest neighbour from the lower sequence. For example, in the current
example, the original neighbours list is reduced to {p2, p3, p5, p6, p7, p1} (Figure 1.9.b) by
removing p4, because p3 is higher in the sequence { p3, p4}. Note that if the list has more
one neighbour then the reduced begins with a lower neighbour to ensure that the four
alternating upper and lower neighbours at the pass are selected correctly. Also it can be
seen from the reduced list that there are 6 sign changes therefore, the number of
denegerate passes m is 2.
(c) Put all the elements of the reduced list except the first two i.e., {p5, p6, p7, p1}, in a
trailing list to further reduce the neighbours list.
(d) Select the last four elements i.e., {p5, p6, p7, p1}, of the trailing list as representative
neighbours. Remove the last two elements, which are {p7, p1} in this case, of the
representative neighbours, from the trailing list.
(e) Repeat steps (c) – (d) until the trailing list is reduced to a lower and a upper neighbour
of the pass, which here are {p7, p1}. The final neighbours list of the decomposed pass has
the first two elements of the trailing list and the two elements remained after step (v)
thus here the final neigbhours of p are {p2, p3, p5, p6}.
Other degenerate points such as flat regions could either be iteratively smoothed
or perturbed to introduce a slight inclination in the terrain. The methodology to connect
the points is intuitive and simple. It is based on the assumption that a ridge line is the line
of steepest ascent from a pass while a channel is the line of steepest descent. Therefore,
the ridge (channel) line is traced by moving to the highest (lowest) neighbour and
repeating the tracing until a peak (pit) or the boundary is reached.
(iii) Polynomial surface fitting
Recall from the last chapter that according to the Morse Theory, a point is a critical point
of the surface if the local slope at the point is zero. However, not all points that have zero
slopes are critical points. To classify the locally flat areas into a peak or a pit or a pass, we
have to know the local curvature using the second derivative of the height function at the
candidate point. The local curvature could also be used to detect whether the candidate
point is a ridge or channel. The second derivative can be used to classify the critical points
and lines in two ways. First, the easier method is to compare the curvature along the
three orthogonal components (Table 1.3) (Wood 1996). The components x and y are not
necessarily parallel to the axes of the DEM, but are in the direction of maximum and
minimum profile convexity. Secondly, the eigenvalues and eigenvectors of the Hessian
matrix (see section 1.2) can indicate the gradient flow at the critical point (Figure 1.10). A
critical point is a peak if the two real parts (R1, R2) of the eigenvalues of the Hessian
matrix are positive, indicating a gradient flow away from the critical point. A critical point
is a pit if the two real parts of the eigenvalues of the Hessian matrix are negative
indicating a gradient flow towards the critical point. In the case of the pass, the two real
parts of the eigenvalues are of different signs. In addition, at a pass the eigenvector along
the positive eigenvalue indicates the ridge line while the eigenvector along the negative
eigenvalue marks the channel direction.
To calculate the derivatives, the local surface around a critical point can be
interpolated as a polynomial of desired smoothness. For example, it could be modelled as
a biquadratic function (Evans 1980, Wood 1996) or a bicubic function (Bajaj and Schikore
1996). Evidently, the complex polynomials will provide a significantly generalised surface
approximation and will take longer time to be solved. A complex polynomial will also
18

32. Introduction
require bigger kernels or filters and these lead to wider unclassified along the borders. For
instance, the surface around a DEM grid cell can be represented as the following
continuous quadratic function, made up of the sum of six terms (Wood 1998):
z = ax2 + by2 + cxy + dx + ey + f
Various methods have been used to solve the surface polynomials for the
coefficients such as simple combinations of neighbouring cells (Evans 1980, Zvenburgen
and Thorne 1987) and matrix algebra (Wood 1996, Kidner et al. 1999). The properties of
the continuous surface fitted on the discrete DEM values can now be derived analytically
from the continuous function. For example, Evans (1980) defines steepest slope and aspect
as follow:
slope = arctan (d2 + e2)1/2
aspect = arctan (e/d)
Second order derivatives such as longitudinal and cross-sectional curvature can
also be derived from the quadratic function (Wood 1998) (Table 1.3).
A potential uncertainty with these surface measures is that they represent the
value of the measure at a point at the centre of the quadratic function (Wood 1998). This
could lead to an incorrect feature classification if the centre of the critical point or line is
offset considerably from the centre of the area of interest. Wood suggested that testing
the quadratic patch for a type of conic section i.e. whether elliptic or parabolic or
hyperbolic or planar could unambiguously determine the feature type and surface flow
direction. Incidentally, the first three conic section types represent the critical points and
lines, namely pits and peaks (elliptic), channels and ridges (parabolic) and passes
(hyperbolic).
Wood (1998) used a raster DEM and classified cells by passing a square filter
window also called kernel across the DEM. The filter window is at least 3 cells by 3 cells
wide and could increase to the number of cells along the shorter side of the DEM. The
possibility of increasing the filter window means that features of varying scales could be
extracted. The procedure for connecting the critical points is more developed than the
previous one because the information about the ridge and channel axes is also available
(Wood 1998, Wood and Rana 2000). The steps are as follow:
Identify the passes,
Move upwards in the direction of any ridge axes that fall within the area of interest a
new grid is reached,
Recursively repeat (b) until no higher cell is found,
Repeat steps (a) – (c) but moving downwards along a channel axes.
1.2.2.2 Limitations
(i) Scale dependency
Scale dependency refers to the subjectivity in measurements arising out of not including
the multi-scaled nature of most spatial datasets e.g. terrain features in DEM, population
density patterns in demographic maps and many others. Many feature extraction methods
are scale-dependent because they only explore a fixed space around a point to classify the
point into a feature type. Therefore terrain features that could fit within the search
space are extracted. Takahashi et al. (1995) and Wood (1998)’s surface network
generation methods suffer from scale dependency in different ways.
19

33. Introduction
Feature Derivative Expression Description
2 2
δ z δ z Point that lies on a local convexity in all
Peak >0, 2 >0
δx 2
δy directions (all neighbours lower).
δ 2z δ 2z Point that lies on a local convexity that
Ridge >0, 2 =0 is orthogonal to a line with no
δx 2 δy convexity/concavity.
δ 2z δ 2z Point that lies on a local convexity that
Pass >0, 2 <0
δx 2 δy is orthogonal to a local concavity.
δ 2z δ 2z Points that do not lie on any surface
Plane =0, 2 =0
δx 2 δy concavity or convexity.
δ 2z δ 2z Point that lies in a local concavity that
Channel <0, 2 =0 is orthogonal to a line with no
δx 2 δy concavity/convexity.
δ 2z δ 2z Point that lies in a local concavity in all
Pit <0, 2 <0
δx 2 δy directions (all neighbours higher).
Table 1.3
Morphometric Features described by second derivatives (after Wood 1996).
Peak: R1, R2 > 0 Pit: R1, R2 > 0 Pass: R1 < 0, R2 > 0
Figure 1.10
Critical points of the surface and the configuration of their eigenvalues and
eigenvectors. R1 and R2 are the real parts of the eigenvalues.
20

34. Introduction
(i.i) Scale dependency of Takahashi et al. (1995) Method
Although Takahashi et al. (1995) realised that “it is possible that small undulations hide
large undulations in the case of steep mountain regions” i.e. terrain features exist at
multiple-scales, they used wavelet filtering to eliminate such “small undulations”.
Therefore, their surface network extraction method ignored the scale dependency of
features. In addition, since the triangulation-based detection uses only the eight
surrounding neighbours for the classification of the critical points, it has a fixed scale of
observation. In a later work, Takahashi (1996) suggested referring to the scale-space
theory (Witkin 1983, Lindeberg 1994) before the extraction of the surface network.
However, it is uncertain how the current method of triangulation can be extended to
detect larger features.
(i.ii) Scale dependency of Wood (1998) Method
The Wood (1998) method allows a multi-scale extraction of the surface network but since
only the cell at the centre of the filter windows is classified, the number of cells which
could be classified reduces as the filter window grows in size (Figure 1.11a). In addition,
the extraction methodology doesn’t distinguish between a feature which can be identified
with both small and big filter windows (Figure 1.11b).
9 by 9 filter window
7 by 7 filter window
5 by 5 filter window
3 by 3 filter window
(a)
(b)
Figure 1.11
Scale dependency of the Wood (1998) method. (a) The top-left corner of a raster and
the first cells that could be classified based on different filter window sizes
and (b) two peaks with similar local shape but with different extents.
(ii) Delineation of topological links
(ii.i) Broken Surface Networks
In most methods for the generation of surface networks, including Takahashi et al. (1995)
and Wood (1998) methods, the surface network is built incrementally by tracing the ridges
and channels from the passes. It is assumed that by tracing the steepest (shallowest)
gradient (Takahashi et al. 1995) or the ridge (channel) axes (Wood 1998) starting from a
21

35. Introduction
pass will faithfully lead to either a peak (pit) or to the edge of the DEM (external pit or
peak). However, as discussed in this section, DEMs of natural topography are seldom
sufficiently smooth enough for the successful delineation of the ridges (channels). As a
result, ridges and channels don’t necessarily terminate at peaks and pits respectively.
(ii.ii) Junctions, bifurcations are not extracted
Both the Takahashi et al. (1995) and Wood (1998) methods don’t locate junctions and
bifurcations in the way suggested by Wolf (1990).
1.2.3 Simplification
Pfaltz (1976) noted that despite the significant abstraction achieved by surface networks,
they may carry too much information by storing minor peaks, passes, and pits. He
proposed a simplification of the surface network graph with the aim of extracting “those
points of equilibria which correspond to the macrostructure of the surface, and suppress
points which are part of a local microstructure” (Pfaltz 1976). Pfaltz proposed an iterative
simplification of the surface network by homomorphic contractions such that the resultant
surface network is isomorphic to the parent surface network graph. Wolf (1988)
categorised the two possible types of homomorphic contractions, which always result into
a topologically consistent surface network4. They are as follow:
(i) (yo – zo)-contraction (Wolf 1988)
Let,
W = Surface Network,
yo = Pass with Peaks R(yo) = {zo,z} and the difference in height along an adjacent ridge
h(yo, zo) ≤ h(yi, zo); i = 1,2, …., n-1 where n = degree of the peak zo.
Set of adjacent passes to zo L(zo) = {yo, y1, y2,…, yn-1}.
Then, the (yo, zo)-contracted graph W’ is the graph with the following properties:
Vertex set V(W’) = V’ = V – {yo, zo},
Edge set E(W’) = E’ = E + {(y1, z’), (y2, z’),….,(yn-1, z’)}, and
Edge elevation drops:
h(yi, z’) = h(yi, zo) – h(yo, zo) + h(yo, z); i = 1,2,…, n-1.
h(e’) = h(e) for all other edges e’ ∈ E(W’)
This transformation, which contracts the subgraph [yo,zo] and converts the original
surface network onto a condensed one is called (yo, zo)-contraction (Figure 1.12a). The
contraction removes the peak zo and its highest adjacent pass yo together with the entire
critical lines incident with at least one of these critical lines. However, this elimination
causes the loss of two properties of surface networks, which are (a) the condensed
subgraph [P’1,P’2] is no longer connected (violation of Rule 1) and (b) od(yi) = 1; i = 1,2,
…., n-1 (violation of Rule 3). The topological consistency is restored by connecting the
free passes yi to z i.e., the edge set of W’ contains the old edge set E(W) and the new
links (yi,z’). The most important part of the contraction is the choice of yo, which ensures
that the elevation differences along the new links are always greater than zero. This idea
originated from Mark (1977), where he proposed methods for the generalisation of surface
trees. Positive elevation differences are essential for the realisation of a topographic
surface for instance a situation where a higher pass connects to a lower peak is unnatural.
(ii) (xo– yo)-contraction
4
See Pfaltz (1976) and Wolf (1988) for the proof.
22

36. Introduction
A (xo – yo)-contraction can be similarly defined for the contraction of the subgraph [xo,yo]
(Figure 1.12b) except that the pass yo that is removed is the lowest pass connected to the
pit xo.
A surface network can be condensed by repeated (yo–zo)-contraction and (xo– yo)-
contraction until a desired level of simplicity or an elementary surface network (ESN) is
achieved. An elementary surface network is the most basic topologically consistent form of
surface network with a single pass with exactly two ridges and channels incident on the
pass. For example, the surface networks produced after generalisation shown in Figure
1.12 are elementary. The surrounding peak and pit is immutable as a contraction of these
will lead to a collapse of the surface network graph.
A typical surface network contraction sequence is as follow:
Step 1: For all internal x ∈ P0 ; z ∈ P2 {
Calculate w j ( x ), w k ( z ); j = 1..id ( x ), k = 1..od ( z ) ;
Add w j ( x ), w k ( z ) to contraction sequence list R .
}
Step 2: Sort R in ascending order.
Step 3: If R[ 0 ] ∈ P0 {
Do [ x 0 − y 0 ] − contraction on R[ 0 ] .
} else {
Do [ y 0 − z 0 ] − contraction on R[0] .
}
Step 4: If W is NOT elementary OR NOT generalised enough {
Go to Step 1
}
Else {
Exit
}
w are the weights associated with pits and peaks.
1.2.3.1 Selection criteria for simplification
Mark (1977) and Wolf (1988) proposed that any types of weights, used to select critical
points for contraction should be based on the elevation or in general on the value of the
mapped property of the critical point because this would ensure a topologically consistent
surface network after generalisation. They suggested the following importance measures:
(i) Height of the Peak and Pit.
w(xi) = |h(xi)|
w(zk) = |h(zk)|
(xi) is a pit, (zk) is a peak, h denotes height and w denotes weight.
Height of the critical point is perhaps the simplest and most obvious weight that could be
assigned (Mark 1977).
23

37. Introduction
(a) (b)
y1 y1
xo z xo z
x zo x zo
yo yo
y1
y1
z
xo z
x zo
x
Figure 1.12
(a) (yo–zo)-contraction and (b) (xo–yo)-contraction of a surface network.
(ii) The maximum of the elevation differences between a peak or pit and all its adjacent
passes.
w(xi) = max {h(yj) – h(xi)}
w(zk) = max {h(zk) – h(yj)}
(xi, yj) ∈ E, (zk, yj) ∈ E
This measure ranks peaks and pits on the basis of the ridge and channel with the maximum
drop in elevation, linked to them.
(iii) The minimum of the elevation differences between a peak or pit and all its adjacent
passes.
w(xi) = min {h(yj) – h(xi)}
w(zk) = min {h(zk) – h(yj)}
(xi, yj) ∈ E, (zk, yj) ∈ E
This measure ranks peaks and pits, ranked on the basis of the ridge and channel with the
smallest drop in elevation, linked to them.
(iv) The sum of the elevation differences between a peak or pit and all its adjacent passes.
w(xi) = Σ{h(yj) – h(xi)}
w(zk) = Σ{h(zk) – h(yj)}
(xi, yj) ∈ E, (zk, yj) ∈ E
24

38. Introduction
This measure selects pits and peaks with a low number of crossings. However as can be
seen this measure could be misleading because it will be biased by the heights of the
points.
(v) The sum of the elevation differences between a peak or pit and all its adjacent passes
normalised by the degree of the peak or pit.
Σ {h ( y j ) − h ( x i )}
w(xi) =
n( x i )
Σ {h( z k ) − h ( y j )}
w(zk) =
n( z k )
(xi, yj) ∈ E, (zk, yj) ∈ E, n denotes the degree of the critical point.
The idea behind this measure is the same as in measure (iv) but this one normalises the
elevation differences. However, this is an unnecessarily involved way to find crossings. The
degree of the peak or pit is perhaps more suited. Still, the normalised sum could prove to
be useful for some other purpose.
Wolf (1988) proposed that the geometrical errors observed while creating
simplified contour maps using line simplification methods could be avoided by making the
contour maps directly from the generalised surface networks. This was particularly useful
for cartographic purposes.
1.2.3.2 Limitations
Despite the simplicity and robustness of surface network contraction methods proposed by
Pfaltz (1976) and Wolf (1988), they have three main limitations which restrict their utility
in practical terrain generalisation methods:
(i) Limitations of weight measures
According to Weibel and Dutton (1999), the first step in the generalisation of spatial
datasets is the cartometric evaluation of the dataset which involves an assessment of the
dataset to select the portions suitable for generalisation. For a surface network
generalisation above, cartometric evaluation involved the assignment of weights (based on
elevation differences) and using it to rank (by using the selection criteria) the peaks and
pits for contraction. Mark (1977) and Wolf (1988) have argued that all weights and
selection criteria must be based on elevation. However, it is simple to prove that elevation
and elevation differences provide little information about the importance of a point
(Franklin 2000). For example, two peaks could have ridges with equal elevation
differences but of different extent. Pfaltz (1976) first raised the potential arbitrariness in
assigning weights and selecting points for contraction. More crucially, the existing weight
measures do not take the morphometric measures such as slope etc. into account. In
addition, it is unclear how critical points with equal weights should be ordered for
contraction.
(ii) Sequential contraction
Pfaltz (1976), Mark (1977) and Wolf (1988) proposed an iterative and sequential (based on
rank) generalisation of a surface network until a surface with a desired simplicity has been
reached. However, it is sometimes desirable to influence the generalisation sequence for
the sake of structural integrity (Pfaltz 1976, Wolf 1988) or, when the sequence could be
anomalous (e.g. two points with equal weights). Currently, there are no proposals to
achieve an arbitrary generalisation sequence.
25

39. Introduction
(iii) Purely topological nature of generalisation
The existing generalisation method for surface networks is only able to achieve terrain
simplification at a topological level i.e. while there is a terrain corresponding to the
original surface network, simplifications in surface networks do not have a morphological
expression. For example, if after a generalisation three new ridge edges are created,
there are no corresponding ridges produced directly in the digital elevation model. This is
perhaps the most critical limitation of existing generalisation methods and prevents it
from being used for practical terrain generalisation methods. Wolf (1988) did not consider
the construction of the generalised morphology based on changes in the topological links
and merely triangulated the critical points left after generalisation.
1.3 Structure of the Thesis
Surface networks have received intensive research inputs from researchers in mainly
computer science (vision, graphics), geographic information science (terrain modellers)
and, to a limited extent, by social scientists. The research of surface networks can be
broadly divided into three main areas namely automated generation, generalisation, and
application. The research that was undertaken during the doctoral research has also been
accordingly divided into these three main topics. These chapters are self-contained
descriptions on these areas and include a conclusion either during the descriptions or at
the end of the chapters. This has been done to maintain a consistency of thoughts. These
chapter conclusions are brought together in the final thesis conclusions.
Chapter 2 presents novel techniques for an automated generation of surface
networks. Two main key proposals in this chapter are regarding a new data structure to
store surface networks and the storage of ridge junctions and channel bifurcations. The
new data structure to store surface networks ensures that both geometrical and
topological information about terrain is preserved.
Wolf (1984) and most GIS literature use the term generalisation as synonymous to
the simplification of surface networks. However, it is proposed that generalisation should
be considered as both simplification and refinement. Simplification of a data structure
involves removal of redundant and/or undesired details while on the contrary, refinement
introduces details into a coarse data structure. The literal use of the term generalisation
merely means development of a hypothesis/principle/conclusion by approximation of
many observations thus logically it is also valid for refinement. Chapter 3 identifies several
issues related to vertex-importance based simplification of surface networks. It presents
new weights measures for characterising the structure of surface networks and proposals
for the refinement of surface networks.
Chapter 4 includes a demonstration of the ideas and techniques developed in the
previous sections. Three types of common terrain analyses namely viewshed analysis,
terrain generalisation and visualisation of landscape evolution are presented where surface
networks help in improving the computation time and quality of the terrain analyses.
Chapter 5 contains a summary of the research and presents the directions for
future research.
26

40. Most of the fundamental ideas of science are essentially simple, and may,
as a rule, be expressed in a language comprehensible to everyone.
From “The Evolution of Physics” by Albert Einstein &
Leopold Infeld (1967)