1. Towards an Axiomatic Theory of Geoinformatics
Gilberto Câmara
National Institute for Space Research (INPE), Brazil
Institute for Geoinformatics, University of Münster, Germany

2. Scientists and Engineers
Photo 51(Franklin, 1952)
Scientists build in order to study
Engineers study in order to build

3. What Geoinformatics is about
Computational representations of geographical space

4. Current GIS is map-based
Big data does not fit in the “map as set of layers” model

5. What set of concepts drove GIS-20?
Map methaphor (cartographical user interfaces)
Proximal spaces (regionalized data analysis)
Object-oriented modelling and spatial reasoning
Object-relational databases (vectors and images)

6. Geoinformatics: part of e-science
Biology Bioinformatics
Medicine Medical Informatics
Genetics Genomics
Geosciences Geinformatics
GI Engineering: “Geoinformatics: “interdisciplinary field that studies representation, analysis and modelling of geographical data.”

7. Geoinformatics enables crucial links between nature and society
Nature: Physical equations
describe processes
Society: Decisions on how to
use Earth´s resources

8. mobile devices
social network
Mobile devices, crowdsourcing, massive Earth observation sets: new technologies, new challenges
sensors everywhere
ubiquitous imagery

9. Records of interaction on human societies
Semantics of big data
Records of interaction on human societies
primary aim: communication

10. Observations of nature
Semantics of big data
Observations of nature
primary aim: description

11. Measurements of nature-society interaction
Semantics of big data
Measurements of nature-society interaction
primary aim: sustainability

12. Core concepts of spatial information (Kuhn, IJGIS, 2012)
network
neighborhood
location
Core concepts of spatial information (Kuhn, IJGIS, 2012)
field
object
Na TerraLib partimos do principio que existem core concepts compartilhados na area de geoinformatica e uma idéia interessante que vem crescendo em aceitacao na comunidade\
event

13. Core concepts as abstract data types
Agent
Network
Event
Coverage Set
Time Series
Trajectory
(x)
Geometry
Field
Object

14. Earth Observation data is now free…and big
Image source: NASA
Sentinels: 3 Tb/day

15. “Remote sensing images describe landscape dynamics”
What’s in an image?
image: LAF/INPE
“Remote sensing images describe landscape dynamics”
(Câmara et al., COSIT 2001)

16. Time series analysis of land change
Forest
Pasture
Área 1
Forest
Área 2
Forest
Agriculture
Área 3
Vegetation index
time series
source: Victor Maus (INPE)

17. Big data requires new conceptual views
(x)
575 (t) x 7 (λ)
The Space-time Data Cube concept
An Australian Geoscience Data Cube

18. “Cubing” remote sensing images
Landsat images
Tile squares
 space 
 time 
& … Stack
Dice…
An Australian Geoscience Data Cube

19. Array databases: all data from a sensor put together in a single arrayX
result = analysis_function (points in space-time )

20. Data-intensive GIS is not “more maps”
Spatio-temporal data that captures change
We need new theories and methods

21. City life now guided by the hours on clock tower
Clock Towers (Prague, 1410)
City life now guided by the hours on clock tower

22. Harrison’s clocks (18th Century)
The longitude challenge was a triumph of technology over astronomy

23. The ultimate victory of time over space: GPS
GPS satellites use relativistic clocks to get precise location

24. Mental maps and mental clocks
Where is our mental central clock?

25. Temporal perception as event-ordering
The brain represents time by means of time: events are recorded as relevant personal experiences

27. The coast of Japan is an object The 2011 Tohoku tsunami was an event
Objects and events
The coast of Japan is an object
The 2011 Tohoku tsunami was an event

28. Processes and events
Flying is a process - Virgin flight VX 112 (LAX-IAD) on 26 Apr 2012 is an event

29. Aral Sea (an object) – disaster (an event)
When did the Aral Sea shrank to 10% of its original size?

30. objects exist, events occur
Mount Etna is an object
Etna’s eruption was an event

31. A view on processes and events
(Worboys & Galton)
Space
Time
Count
Objects
Events
Matter
Processes
Mass
football or game?
water or lake?

32. A pragmatic view on objects and events
Space
Time
Observable
Objects
Events
Matter
Processes
Abstract
football or game?
water or lake?

33. From practice to theory
Axiomatic theory of geoinformatics
Sound basis for system design
Shared conceptualization
Verifiable implementations
Image source: Deborah Estrin (UCSB)

34. The benefits of axiomatization
Euclid
(x + y)2 = x2 + 2xy + y2

35. The benefits of axiomatization
Euclid
(x + y)2 = x2 + 2xy + y2
Egenhofer
spatial topology

36. The Axiomatization of science
Newton

37. The Axiomatization of informatics
Codd

38. The Axiomatization of geoinformatics
Güting
Frank

39. What algebra is needed for spatio-temporal data?
How can this algebra be handled in an geographical databases?

40. We measure properties of the world
Observations allow us to sense external reality

41. Observations allow us to sense external reality
An observation is a measure of a value in a location in space and a position in time

42. Building blocks: Basic Types
type BASE = {Int, Real, String, Boolean}
operations: // lots of them…

43. Building blocks: Time (ISO 19108)
type TIME = {Instant, Period}
operations:
equals, before, after, begins, ends, during, contains, overlaps, meets, overlappedBy, metBy, begunBy, endedBy: TIME x TIME → Boolean

44. An observation is a measure of a value in a location in space and a position in time

45. Building blocks: Basic Types
type BASE = {Int, Real, String, Boolean}
operations: // lots of them…

46. Building blocks: Geometry (OGC)
type GEOM = {Point, LineString, Polygon, MultiPoint, MultiLineString, MultiPolygon}
operations:
equals, touches, disjoint, crosses, within, overlaps, contains, intersects: GEOM x GEOM → Bool

47. Building blocks: Time (ISO 19108)
type TIME = {Instant, Period}
operations:
equals, before, after, begins, ends, during, contains, overlaps, meets, overlappedBy, metBy, begunBy, endedBy: TIME x TIME → Boolean

48. Observation data type type Obs [T: TIME, G: GEOMETRY, B: BASE]
operations:
new: T x G x B → Obs
value: Obs → B
geom: Obs → G
time: Obs → T

49. From observations to events

50. function: Position  Value
What is a geo-sensor?
What is a geo-sensor?
Field
function: Position  Value
“Conceiving big spatiotemporal data as fields captures their nature better than the layer-oriented view”
(Câmara, Egenhofer et al., GIScience 2014)
measure (s,t) = v
s ⋲ S - set of locations in space
t ⋲ T - is the set of times.
v ⋲ V - set of values

51. Why do we need estimators?
We cannot sample every location at every moment – we need to estimate in space-time

52. Sensors: water monitoring
Brazilian Cerrado
Wells observation
50 points
50 semimonthly time series
(11 Oct 2003 – 06 March2007)
Rodrigo Manzione, Gilberto Câmara, Martin Knotters

53. Sensor data produces a time series
Are the sensor observations a time series?

54. Combined monthly sensor averages produce coverages
JUNE
JULY
MAY
AUGUST
SEPTEMBER
Manzione, Câmara, Knotters
Estimates of water table depth for an area in Brazilian Cerrado

55. Geostatistics (spatial inference of fields)
Water Availability
Index
Estimated
Surface

56. Observations: deaths in Porto Alegre

57. What is your death risk in Porto Alegre?
Santos,S.M., 1999

58. Coverage: variation of a property within a spatial extent at a time.
images: USGS
Coverage: variation of a property within a spatial extent at a time.
Two-dimensional grids whose values change.

59. individual entity that varies its location (and its extent) over time
Moving objects
individual entity that varies its location (and its extent) over time

60. Moving objects have trajectories
trajectory : represents how locations or boundaries of an object evolve over time.

61. Different Views from the Same Observation Set
Time Series
air pollution
Field
Trajectory
car location
Uma das grandes vantagens do nosso modelo é construir os data types sobre o mesmo tipo de dado. To meet different application needs!
A capacidade de criar diferentes views on the same observation set!
Extent, Position, Value
Coverage
a set of cars equipped with GPS and air pollution sensors
air pollution
within the city

62. Properties of Fields value: Position  Value
Positions at which estimations are made
Values that are estimated for each position

63. Three sets of (space,value) pairs + estimator: coverage set
LANDSAT images of the Aral Sea
images: USGS

64. Field: (time, space) pairs + estimator: trajectory
Virgin flight VX 112 (LAX-IAD) on 26 Apr 2012:
(time, space) pairs + estimator

65. Field: (time, value) pairs + estimator: time series
Buoy in Pacific ocean near the coast of Japan ( )

67. Operations on fields
Field F [P:Position, V:Value, E: Extent, G: Estimator] p1 p2 p3
pnew f1
domain(f1)= {p1,p2,p3}
value (f1, pnew) = g(f1, pnew)
Domain – defines granularity
Estimator provides value on all positions inside the extent

68. Operations on fields Three fields, same extent,
different granularities, different estimators • • • • • • • • • • f1 f2 f3
How do we express the operation
f3 = max (f1,f2)?

69. Operations on fields Three fields, same extent,
different granularities, different estimators • • • • • • • • • • f1 f2 f3

70. Map Algebra Operations
Monday, September 17, 2018
Map Algebra Operations
Set-based algebra for operations on maps (“raster layers”)
TOMLIN, D., Geographic Information System and Cartographic Modeling, 1990.
Tomlin (1983, 1990) defined and organized operations on raster data model as local, focal and zonal according to the spatial scope of the operations.
Geographic Information System and Cartographic Modeling, Englewood Cliffs: Prentice Hall, 1990.
Menton et al (1991) add global operations.
Local
Focal
Zonal
Global

71. Map Algebra

72. Generalised map algebra
Protection areas – prot_areas ( poligons )
Roads – road_map ( lines)
Deforestation % - def_map ( cell space)

73. Events are categories (Frank, Galton)
source: USGS
Events are categories (Frank, Galton)
identity : id · a = a
composition : ∀a, ∀b, ∃c, c = a.b
associativity : a · (b · c) = (a · b) · c

74. Coverage set (hourly precipitation grid) 
When did the large flood occur in Angra? When precipitation was > 10mm/hour for 5 hours
Coverage set (hourly precipitation grid) 
Event (precipitation > 10 mm/hour for 5 hrs)
INPE has a project called PROARCO responsible for monitoring fire spots in South America, In this application, each fire is presented by its location and the time instant when it was detected.
This picture is an example of the detected fires in a Brazilian state called Pará in august 2003.
The second demand is related to a project responsible for monitoring the Amazon deforestation called PRODES. In this application, each deforested area is represented by a polygon that limits the deforested region and the time when it was detected.

75. The event data type
An event is an individual episode with a beginning and end, which define its character as a whole.
An event does not exist by itself. Its occurrence is defined as a particular condition of one spatiotemporal type.
INPE has a project called PROARCO responsible for monitoring fire spots in South America, In this application, each fire is presented by its location and the time instant when it was detected.
This picture is an example of the detected fires in a Brazilian state called Pará in august 2003.
The second demand is related to a project responsible for monitoring the Amazon deforestation called PRODES. In this application, each deforested area is represented by a polygon that limits the deforested region and the time when it was detected.

76. The event data type Type Event [T1: TIME, T2: TIME] uses ST
operations:
new: {ST x (T1, T2) → Event
compose: Event x Event → Event
intersect: Event x Event → Event
INPE has a project called PROARCO responsible for monitoring fire spots in South America, In this application, each fire is presented by its location and the time instant when it was detected.
This picture is an example of the detected fires in a Brazilian state called Pará in august 2003.
The second demand is related to a project responsible for monitoring the Amazon deforestation called PRODES. In this application, each deforested area is represented by a polygon that limits the deforested region and the time when it was detected.

77. Event composition Forest loss > 20% time Event 1 Loss > 50%
Exploração intensiva
time
Event 1
Loss > 50%
Perda >50% do dossel
Event 2
Loss > 90%
Perda >90% do dossel
Event 3
Event 4
Clear cut
Corte raso
Floresta
Floresta

78. When did the large flood occur in Angra?

79. When did the large flood occur in Angra?
Coverage prec = getData (weather forecast)
flood = new Event()
from t0 = prec.begin(); t0 <= prec.end(); t.next()
if getRain (prec, t0, t0 + 24) > 100
strong = new Event (prec, t0, t0 + 24)
flood.compose (strong)
INPE has a project called PROARCO responsible for monitoring fire spots in South America, In this application, each fire is presented by its location and the time instant when it was detected.
This picture is an example of the detected fires in a Brazilian state called Pará in august 2003.
The second demand is related to a project responsible for monitoring the Amazon deforestation called PRODES. In this application, each deforested area is represented by a polygon that limits the deforested region and the time when it was detected.

80. When did the Aral Sea shrank to 10% of its original size?
getWaterArea (Coverage cov, Time t)
area = 0
forall g inside cov.boundary()
if cov.value (g,t) == "water”
area = area + g
return area }

81. When did the Aral Sea shrank to 10% of its original size?
aralSea = new Coverage (images)
findDisaster (aralSea) {
t0 = aralSea.begin()
areaOrig = getWaterArea (aralSea,t0)
for t = aralSea.begin(); t <= aralSea.end(); t.next()
if getWaterArea (aralSea,t) < 0.1* areaOrig
disaster = new Event (aralSea, t, t.aralSea.end())
break
return disaster }

82. Algebras for spatio-temporal data are a powerful way of representing change