Surfaces

(as introduction to Neighborhood Operations)

Objectives of Lecture:

  1. Reminders: Project team formation (P0); Discussion 2;
  2. Structure of Surfaces
  3. Introduce operations on surfaces (use of neighborhoods)

A surface is an abstract concept that presumes a single value that varies continuously across a region.
Topologically, a surface is a distorted plane located in a third dimension.
It is NOT 'two-and-a-half dimensional'; it is a two-dimensional plane in 3D.

Types of surfaces

(also called `fields' in more proper mathematical circles):

Scalar surface has single magnitude at each point.
Scalar value is a single number (eg. elevation)

Vector surface has magnitude and direction at each point.
Vector (confusing) value has a quantity (strength) and a direction (eg. slope); this usage is at the basis of "vector" hardware using graphics CRTs, then applied to any line-based representation system...

A vector field is one example of a multicomponent surface (whose single value might take two numbers to represent).

Examples of Surfaces (and DEM sources)

Visualization site (US Corps of Engineers, Champaign IL) relocated at North Carolina!.
one of the papers generated by Lubos Mitas and Helena Mitasova from the GRASS development group (Elsevier Press) [scroll down]; specific paper on terrain processing.

GRASS applications site (rediscovered! at Baylor)
Montana GIS site with different relief representations and DEMs (this one at 1 km spacing); precipitation; and more...

Washington DEM data from UW Map Library WAGDA site; and a neat poster presentation of Puget Sound.
Example of really detailed DEM development for floodplain modeling

Properties of Surfaces:

Types of continuity:

piecewise, once differentible, twice differentiable (restrictions on neighboring values).

Slope

is the composite of a gradient (often called slope) and a direction (aspect).

Computing slope requires neighboring values. There are three different neighborhood choices: orthogonal neighbors, eight neighbors (diagonals included) and slope for the area with points at corners.

Computing slope requires a rule: dominance (largest slope amongst neighbors) or a 'best fit' (least squares contributory rule). Note that the dominance result may not be as large as the value from the best fit (when the steepest slope is between the directions sampled).

Equation 7-1: slope gradient and aspect in a grid

Given a matrix Z with grid spacing S,

the least square fit plane at zi,j can be written as:

z = a + bx + cy    where:
a = (zi-1,j-1 +zi-1,j +zi-1,j+1 +zi,j-1 +zi,j +zi,j+1 +zi+1,j-1 +zi+1,j +zi+1,j+1)/9
b = (zi-1,j+1 + 2 zi,j+1 +zi+1,j+1) - (zi-1,j-1 + 2 zi,j-1 +zi+1,j-1) / 8S
c = (zi+1,j-1 +2zi+1,j +zi+1,j+1) - (zi-1,j-1 + 2 zi-1,j +zi-1,j+1) / 8S
Slope gradient tangent = square root (b2 + c2)
Aspect angle = arctan (c / b)
     Note: gradient can be represented as
     tangent, sine, or angle (degrees, grads, radians).

Convergence/ divergence

(local behavior on surface) leads to a topology of surfaces:
(Concepts due to Arthur Cayley 1856, but terms due to William Warntz 1960s)
Peaks, Pits connected by Ridges and Courses; creating Hills and Dales (watersheds)
This surface structure specifies a set of relationships...

terms vary, concepts remain (saddle for pass) | watershed = basin | "Hill" and "Dale" (Yorkshire) | ...

Downslope flow example |

Computing Properties of Surfaces

Notice that the 'properties' of continuity, slope and convergence do not occur at a location without reference to their neighbors.
Estimating these properties, and most other properties of surfaces, must be carried out using neighboring values.

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Version of 29 October 2003