1. Digital Terrain Mapping and Analysis
Dr. A.K.M. Saiful Islam
Institute of Water and Flood Management (IWFM)
Bangladesh University of Engineering and Technology (BUET)

2. Lecture Topic This lecture will focus on Geo-referencing
Coordinate Systems
Map Projections
Coordinate Transform
Map & Time Distance, Scale and accuracy
Interpolation techniques
Digital Terrain Mapping and Analysis
DEM
DTM

3. Coordinate Systems
Geospatial data should be geographically referenced ( called georeferenced or geocoded) in a common coordinate system.
Plane Orthogonal Coordinates One of the most convenient way of locating points is to use plane orthogonal coordinates with x (horizontal) and y (vertical) axis.
Polar Coordinates A polar coordinate system with the angle (q ) measured from the polar axis (x axis) and distance (r) from the pole is used in some cases.
3D Orthogonal Coordinates Three dimensional (3D) orthogonal coordinates are also used to locate points with the plane coordinates (x, y) and height or depth (z).

4. Plane Orthogonal Cartesian Coordinates

5. Polar coordinates

6. 3D Coordinate System

7. The Shape of the Earth
The shape of the Earth can be represented by an ellipsoid of rotation (or called a spheroid) with the lengths of the major semi-axis (a) and the minor semi-axis (b).
Two type of coordinate system use: (i) Geodetic, (ii) Geocentric coordinates

8. Geodetic and Geocentric Latitude
Geocentric Latitude – The acute angle measured perpendicular to the equatorial plane and a line joining the center of the earth and a point on the surface of the reference ellipsoid.
Geodetic Latitude – The acute angle between the equator and a line drawn perpendicular to the tangent of the reference ellipsoid. Map coordinates are given as longitude and geodetic latitude.
[Source : ]

9. Map Projection
A map projection is a process of transforming location on the curved surface of the Earth with the geodetic coordinates (j , l) to planar map coordinates (x, y).
More than 400 difference map projections have been proposed. The map projections are classified by the following parameters.
projection plane: perspective, conical, cylindrical
aspect: normal, transverse, oblique
property: conformality, equivalence, equidistance
size: inside, tangent, secant

10. Projection property
Conformality is the characteristic of true shape, wherein a projection preserves the shape of any small geographical area. This is accomplished by exact transformation of angles around points.
The property of conformality is important in maps which are used for analyzing, guiding, or recording motion, as in navigation.
Equivalence is the characteristic of equal area. Preservation of equivalence involves an inexact transformation of angles around points and thus, is mutually exclusive with conformality except along one or two selected lines.
The property of equivalence is important in maps which are used for comparing density and distribution data, as in populations.
Equidistance is the characteristic of true distance measuring. The scale of distance is constant over the entire map.
Equidistance is important in maps which are used for analyzing velocity, e.g. ocean currents. Typically, reference lines such as the equator or a meridian are chosen to have equidistance and are termed standard parallels or standard meridians.
[ Source: ]

11. Perspective Projection
Perspective projections are classified based on the projection center or viewpoint.

12. Conical Projection
Conical projections are classified by the aspect as well as the cone size

13. Conic projection
Conic (tangent)
Conic (secant)

14. Cylindrical Projections
Cylindrical projections are classified as in case of conical projections. One of the most popular cylindrical projections is the Universal Transverse Mercator (UTM) with a transverse axis, secant cylinder and conformality (equal angle).

15. UTM Projection
Universal Transverse Mercator (UTM) with a transverse axis, secant cylinder and conformality (equal angle).
UTM is commonly used for topographic maps of the world, devided into 60 zones with a width of 6 degree longitude.

16. Coordinate Transformation
Coordinate transformation is to transform a coordinate system (x, y) to another coordinate system (u, v). The transformation is needed in the following cases:
to transform different map projections of many GIS data sources to an unified map projection in a GIS database,
to adjust errors which occur at map digitization due to shrinkage or distortion of the map measured, and
to produce geo-coded image by so called geometric correction of remote sensing imagery with geometric errors and   distortions.

17. Reference for Coordinate Transformation
Coordinate transformation is executed by a selected transformation model (or mathematical equation), with a set of reference points (or control points), that are selected as tic masks at the corner points, reseau or ground control points.

18. Major Transformation Helmert Transformation Affine Transformation
scale, rotation and shift
Affine Transformation
skew, scale of x and y,and shift
Pseudo Affine Transformation
bi-linear distortion
Quadratic Transformation
parabolic distortion
Perspective Projection
rectification of aerial photo
Cubic Transformation
cubic and distortion)

19. Distance
Distance is one of the important elements in measuring spatial objects in GIS. Several different concepts of distance are defined as follows.
Euclidean Distance Euclidean distance D is the defined as the distance measured along a straight line from point (x1, y1 ) to point (x2, y2 ) in Cartesian coordinate system . D2 = ( x1 - x2 )2 + ( y1- y2 )2
Manhattan Distance Manhattan distance D is defined as the rectilinear rout measured along parallels to X and Y axes
D = | x1 - x2| + | y1-y2|

20. Distances (Contd..)
Great Circle Distance Great circle distance D is defined as distance along the great circle of the spherical Earth surface from a point (1, 1; latitude and longitude) to another point (2, 2) where R is the radius of the Earth (R = km) on the assumption that the Earth is a sphere.
Mahalanobis Distance
Mahalanobis distance D is a normalized distance in the normal distribution from the center (X0) to a point (X) in case of n dimensional normal distribution. Mahalanobis distance is used in the maximum likelihood method for the classification of multi-spectral satellite images. where S: variance-covariance matrix

21. Distances (Contd..) Time Distance
Time distance is defined as the time required to move from point B to point A by using specific transportation means.

22. Scale, Accuracy and Resolution
Scale of map refers to the ratio of distance on a map over the corresponding distance on the ground.
The scale is represented as 1: M or 1/M, where M is called the scale denominator.
The larger the scale, the more the detail described by the map and with higher accuracy.
Accuracy is generally represented by standard deviation of errors, that is difference between measurements and the true value.

23. Relationship between scale, accuracy and resolution

24. Principle of Interpolation
Interpolation is the procedure of estimating the value of properties at unsampled points or areas using a limited number of sampled observations.

25. Interpolation Techniques
1. Pointwise interpolation
1(a) Thiessen polygon
1(b) Weighted Average
2. Interpolation by curve fitting
2.1 Exact interpolation
2. 1(a). Nearest neighbor
2. 1.(b) Linear interpolation
2. 1(c) Cubic interpolation
2.2 Approximate interpolation
2.2(a) Moving Average
2.2(b) B-spline
2.2(c) Curve Fitting by Least Square Method
3. Interpolation by surface fitting
3.1 Regular grid
3.1(a) Bilinear Interpolation
3.1(b) Bicubic Interpolation
3.2 Random points
3.2(a) TIN

26. 1. Pointwise Interpolation
Pointwise interpolation is used in case the sampled points are not densely located with a limited influence or continuity in surrounding observations, for example climate observations such as rainfall and temperature, or ground water level measurements at wells.

27. 1(a) Thiessen Polygons
Thiessen polygons can be generated using distance operator which creates the polygon boundaries as the intersections of radial expansions from the observation points.
This method is also known as Voronoi tessellation.

28. 1(b) Weighted Average
A window of circular shape with the radius of dmax is drawn at a point to be interpolated, so as to involve six to eight surrounding observed points.

29. 2. Interpolation by Curve Fitting
the principle of curve fitting respectively to interpolate the value at an unsampled point using surrounding sampled points.

30. 2. Curve Fitting
Curve fitting is an important type of interpolation in many applications of. Curve fitting is divided into two categories.
2.1 exact interpolation : a fitted curve passes through all given points.
2.2 approximate interpolation : a fitted curve does not always pass through all given points.

31. 2.1 Exact interpolation There are three methods:
2.1(a) nearest neighbor : the same value as that of the observation is given within the proximal distance

32. 2.1 Exact interpolation
2.1(b) linear interpolation: a piecewise linear function is applied between two adjacent points.

33. 2.1 Exact interpolation
2.1(c) cubic interpolation : a third order polynomial is applied between two adjacent points under the condition that the first and second order differentials should be continuous.

34. 2.2. Approximate Interpolation
There are three methods;
2.2(a) Moving Average: a window with a range of -d to +d is set to average the observation within the region

35. 2.2 Approximate Interpolation
2.2(b) B-Spline: a cubic curve is determined by using four adjacent observations

36. 2.2 Approximate Interpolation
2.2(c) Curve Fitting by Least Square Method.
Least square method (sometimes called regression model) is a statistical approach to estimate an expected value or function with the highest probability from the observations with random errors. The highest probability is replaced by minimizing the sum of square of residuals in the least square method.
Equation
Slope
intercept

37. 3. Interpolation by Surface Fitting
the principle of surface fitting respectively to interpolate the value at an unsampled point using surrounding sampled points.

38. 3. Surface Fitting
Surface fitting is widely used for interpolation of points on continuous surfaces such as digital elevation model (DEM), geoid, climate model (rainfall, temperature, pressure etc.) and so on.
Surface fitting is classified into two categories:
3.1 surface fitting for regular grid and
3.2 surface fitting for random points.
3.1 Surface Fitting for Regular Grid Following two methods are commonly used.
3.1(a) Bilinear Interpolation
3.1(b) Bicubic Interpolation

39. 3.1 Surface Fitting for Regular Grid
3.1(a) Bilinear Interpolation
Bilinear function is used to interpolate z using the following formula with respect to normalized coordinates (u, v) of the original coordinates (x, y)

40. 3.1Surface Fitting for Regular Grid
3.1(b) Bicubic Interpolation
Third order polynomial is used to fit a continuous surface using 4 x 4 = 16 adjacent points.

41. 3.2 Surface Fitting for random Points
3.2. (a) Triangular network called as Triangulated Irregular Network (TIN) is applied

42. Compare Interpolation methods
Thiessen polygons are Used for service area analysis of public facilities such as hospitals. Originally proposed to estimate aerial averages precipitation in 1985.
Inverse Distance Weighted can be a good way to take a first look at an interpolated surface. However, there is no assessment of prediction errors. Accuracy depends on the selection of a power value and the neighborhood search strategy. A smaller (6) actually produce better estimations than a larger number (12).
Thin-plate Splines (applies to surface) are recommended for smooth, continuous surfaces such as elevation and water table. Also used for interpolating mean rainfall surface and land demand surface.

43. Kriging Kriging is a geostatistical method for spatial interpolation.
It can assess the quality of prediction with estimated prediction errors.
It uses statistical models that allow a variety of map outputs including predictions, prediction standard errors, probability, etc.
Semivariogram can be fitted as:
Ordinary Kriging models:
Spherical, Circular, Exponential,
Gaussian and Linear.
Universal Kriging models:
Linear with Linear drift, and
Linear with Quadratic drift

44. Semivariogram Semivariance: Y(h) = ½ [(Z(xi) - Z(xj)]2
The semivariogram functions quantifies the assumption that things nearby tend to be more similar than things that are farther apart. Semivariogram measures the strength of statistical correlation as a function of distance.
Semivariance:
Y(h) = ½ [(Z(xi) - Z(xj)]2
Covarience = Sill – Y(h)

45. Data Structure for Continuous Surface Model
In GIS, continuous surface such as terrain surface, meteorological observation (rain fall, temperature, pressure etc.) population density and so on should be modeled
Grid at regular intervals
Bi-linear surface with four points or bi-cubic surface with sixteen points is commonly used
Random points
Triangulated irregular network (TIN) is commonly used. Interpolation by weighted polynomials is also used.
Contour lines
Interpolation based on proportional distance between adjacent contours is used. TIN is also used.
Profile
Profiles are observed perpendicular to an alignment or a curve such as high ways. In case the alignment is a straight line, grid points will be interpolated. In case the alignment is a curve, TIN will be generated.

46. Different Types of DEM

47. Remote Sensing and GIS in Water Management © Dr
Remote Sensing and GIS in Water Management © Dr. Saiful Islam, IWFM, BUET
DEM
A DEM (digital elevation model) is digital representation of topographic surface with the elevation or ground height above any geodetic datum. Followings are widely used DEM in GIS:

48. DTM
A DTM (digital terrain model) is digital representation of terrain features including elevation, slope, aspect, drainage and other terrain attributes.
Usually a DTM is derived from a DEM or elevation data.
several terrain features including the following DTMs.
Slope and Aspect
Drainage network
Catchment area
Shading
Shadow
Slope stability

49. Examples of DTM

50. 1. Slope and Aspect (i) Slope
The steepest slope (s) and the direction from the east () can be computed from 3 x 3 matrix.

51. Slope calculation

52. Slope calculation
Slope is defined by a plane tangent to a topographic surface, as modelled by the DEM at a point (Burrough, 1986).
Slope is classified as a vector; as such it has a quantity (gradient) and a direction (aspect).
Slope gradient is defined as the maximum rate of change in altitude (tan )

53. Example: Slope from elevation data

54. (ii) Aspect
The aspect that is, the slope faced to azimuth is 180° opposite to the direction of q

55. Figure 1. Slope components, note that slope gradient can be express in percent or in degrees

56. Aspect calculation
Aspect identifies the steepest downslope direction from each cell to its neighbors. It can be thought of as slope direction or the compass direction a hill faces.
It is measured clockwise in degrees from 0 (due north) to 360, (again due north, coming full circle). The value of each cell in an aspect dataset indicates the direction the cell's slope faces. Flat areas having no downslope direction are given a value of -1.

57. Example: aspect from the elevation data

58. 2. Drainage Network and Watershed
The lowest point out of the eight neighbors is compared with the height of the central point to determine the flow direction.

59. Surface Specific points
+ is assigned if the height of the central point is higher than the one of the eight neighbors and - if lower.
A peak can be detected if all the eight neighbors are lower.
A pit or sink is formed if all the eight neighbors are higher
A pass can be extracted if the + and - alternate around the central point with at least two complete cycle.

60. Remote Sensing and GIS in Water Management © Dr
Remote Sensing and GIS in Water Management © Dr. Saiful Islam, IWFM, BUET
4. Shade and 5.Shadow
Shade is defined as reduced reflection depending on the angle between the terrain surface and the incident light such as the sun.
Shadow is projected areas that the incident light cannot reach because of visual hindrance of objects on terrain relief

61. Hill Shading
The effect of hill shading on the assumption of an ideally diffused reflecting surface (called Lambertian surface) can be computed as follows:
Relative shading = cos  = |nxsx + nysy+ nzsz |≤ 1.0
where  : angle between incident light vector s and surface normal n

62. Altitude
The altitude is the slope or angle of the illumination source above the horizon. The units are in degrees, from 0 (on the horizon) to 90 degrees (overhead). The default is 45 degrees.

63. Azimuth
The azimuth is the angular direction of the sun, measured from north in clockwise degrees from 0 to 360. An azimuth of 90 is east. The default is 315 (NW).

64. Hillshading from elevation data
The hillshade below has an azimuth of 315 and an altitude of 45 degrees.

65. Examples: A slope and hillshade maps of Glacier National Park

66. Using hill shading for display
By placing an elevation raster on top of a created hillshade, then making the elevation raster transparent, you can create realistic images of the landscape.
Hillshade + elevation

67. Generation of Contour Lines
Contour lines are one of the terrain features which represent the relief of the terrain with the same height. There are two types of contour lines in visualizing GIS data:
Vector Line Drawing In case when the terrain points are given in grid, the simplest method is to divide the square cell into two triangles mechanically.
Raster Image Contour image with painted contour terraces, belts or lines instead of vector lines will be generated in raster form.

68. Interpolation of Elevation from Contours
Digital elevation model (DEM) is very often generated by measuring terrain points along contour lines using a digitizer. DEM with contour points should be provided with an algorithm interpolate elevation at arbitrary points. There are several interpolation methods as follows.
Profile Method A profile passing through the point to be interpolated will be generated and linear or spline curve applied.
Proportional Distance Method According to distance to two adjacent contour lines, the elevation is interpolated proportionally with respect to the distance ratio.
Window Method A circular window is set up around a point to be interpolated and adjacent terrain points are used to interpolate the value using second order or third order polynomials.
TIN Method TINs are generated using terrain points along contour lines.

69. Interpolation Methods

70. Examples: A Digital Elevation Model and associated contour map of Glacier Nat'l Park

71. Automated Generation of DEM
Automated generation of DEM is achieved by photogrammetric methods based on stereo aerial photography and satellite stereo imagery.
Parallax is defined as difference between left and right photographs or image coordinates. The higher the elevation is, the bigger the parallax is. If the parallax is constant, equal elevation or contour lines will be produced.

72. Triangulated Irregular Network (TIN)
Triangulated irregular network or TIN is a DEM with a network of triangles at randomly located terrain points.
Contouring of TIN is based on the following procedure:
step 1: find the intersect of contour and a side.
step 2: assign the "reference point" with the symbol r to the vertex above the contour height and the "sub-point" with the symbol s to the vertex below the contour height.
step 3: shift over to the transversing to find the third vertex in the triangle by checking whether it is a reference point (r) or sub-point (s).

73. Example: TIN Creation

74. Thank you