Formal descriptions are used in this thesis to present a number of characteristics of the proposed theory. Most of these formalisms are presented in this chapter and in Chapter 4. The formalism presented here is described in terms of definitions and axioms in first- order logic, where free variables are implicitly universally quantified with maximal scope. These definitions and axioms are indexed by D and A, respectively.
A complete description of the logical framework developed to represent and reason about geographic events and processes will be given in Chapter 4. However, this chapter introduces some elements of the logical language ℜused within that framework. These elements are particularly relevant to the understanding of the proposed approach to mod- elling geographic features, and to the comprehension of the motivations behind the pro- posed method of representing spatio-temporal data. Namely, the elements introduced in this chapter are time instants and intervals; spatial regions and their coverages; and geo-
graphic features and their types.
3.3.1
Space
The Region Connection Calculus (RCC) [69] is employed as the theory of space. An overview on the RCC relations mentioned in this thesis is given in Section 2.2. Spatial
regions are used here to represent portions of the earth’s surface under some specified projection, and explicit variables riare used to denote spatial regions. These variables can be quantified over in the usual way (e.g.,∀r[φ(r)] or ∃r[φ(r)]).
The logical languageℜalso includes a number of functions to exchange information between variables and to perform spatial operations between regions. The complete set of functions will be presented in Chapter 4. However, the following auxiliary functions are used in definitions presented in this chapter and therefore are introduced now.
• union : (Vr × Vr) → Vrwhich returns a spatial region that corresponds to the spatial union of a pair of spatial regions.
• distance : (Vr × Vr) → R, which returns a non-negative number representing the 2-dimensional Cartesian minimum distance between two regions in projected units.
• concave-hull : (Vr × Vr) → Vr, which returns a concave region that encloses the two specified regions. The concave hull of a set of geometries represents a possibly concave geometry that encloses all geometries within the set. One can think of the concave hull as the geometry obtained by ‘vacuum sealing’ a set of geometries. Many different algorithms for calculating concave hulls are currently available, and they normally work based on the value of a parameter. Roughly, this parameter corresponds to the target percent of area of convex hull the algorithm solution will try to approach before giving up or exiting. Different algorithms often compute different results (even when equal values are assigned to corresponding parameters). The concave hull algorithm used to implement the system prototype is described in Chapter 5.
3.3.2
Spatial Region Coverages
A spatial region can be described in terms of characteristics of the portion of the earth’s surface it represents. The logical language ℜ includes a special type of element to de- note a certain semantic description which can be associated with spatial regions. These descriptions are called here ‘spatial region coverages’, and are denoted by explicit vari- ables ci. The meaning of ‘coverage’ employed here is not restricted to land coverages. It can also denote qualities which can be measured (by sensors or by human observation), such as ‘urbanised’, ‘arid’, ‘temperature > 10 ◦C’, ‘water covered’, or ‘heavily popu- lated’. The way coverages can be associated with spatial regions will be further clarified throughout this chapter.
3.3.3
Time
It is assumed a total linear reflexive ordering on time, and explicit variables ti and ii are used to denote time instants and proper intervals, respectively. A time interval i is con- sidered a proper interval if the time instant which represents its beginning precedes the time instant denoting its end (i.e., b(i) ≺ e(i)). These temporal variables can also be
quantified over in the usual way.
The following functions are used to exchange information between these temporal variables:
• b(i), which returns an instant t corresponding to the beginning of an interval i; • e(i), which returns an instant t corresponding to the end of an interval i.
Time instant variables can be compared by equality (t1 = t2) and by ordering (t1 ≺ t2
and t1 t2) operators. Allen temporal relations [1, 2] are employed between time in-
tervals. These relations are described in Section 2.3. The relations In(i1, i2) and In(t1, i2)
are also defined, meaning that a time interval i1(or time instant t1) is inside a proper time
interval i2. These relations are defined as shown below, in Definitions D3.1 and D3.2.
D 3.1 In(i1, i2) ≡de f (Starts(i1, i2) ∨ During(i1, i2) ∨
Finishes(i1, i2) ∨ Equals(i1, i2))
D 3.2 In(t, i) ≡de f b(i) t e(i)
The logical languageℜalso includes other functions to exchange information between variables and to perform spatial operations between temporal variables. These functions will be presented in Chapter 4.
3.3.4
Geographic Features
Geographic features will be regarded as a particular kind of endurant entity, and therefore
they are able to undergo change over time. Special attention is paid to changes affecting their spatial extensions. Of particular interest are geographic features whose spatial ex- tension at a given time instant t can be modelled as the maximal well-connected region of some particular coverage at t. The expression ‘well-connected region’ is used here in agreement with the discussion and definitions given in Cohn et al. [24]. Examples of
geographic features are forests (which can be regarded as the maximal extension of a cer- tain type of vegetation), deserts (which can be defined based on the level of precipitation) and sea (represented as the maximal extension of water body over a specified level of salinity). Section 3.6 further discusses the fundamentals underlying the representation of geographic features.
In the logical language ℜ, variables fi and ui are used to denote, respectively, indi- vidual geographic features (e.g., Amazon rainforest, Atlantic ocean) and feature types (e.g., sea, forest). This language also includes functions to exchange information between features and other types of variables. These functions will be presented in Chapter 4. However, the following function is introduced now as it is mentioned in logical defini- tions presented in this chapter.
• ext( f ,t), which returns the spatial region corresponding to the spatial extension of
a feature f at time instant denoted by t.
Relevant predicates relating to the representation of geographic features will be pre- sented in Section 3.6.