Distance

Euclidean Distance • Minimum distance from a source (Value <> NoData)• Input grid must have at least one source cell with the rest of

the grid being NoData.• Global Function• Map Algebra: Euc_Dist = EucDistance(Source_Ras)

51520 y

Point 1

10, 15

Point 2

15,20

51015 x

07.755 2222 yxD

22* yxCellsizeD Where X and Y are # of

Cells

Euclidean Distance (EucDistance)

Source Cells have values. All other cells have NoData.

Output represents the minimum distance to any source cell.

Distance is in map units.

Issues with Length Measurement• Measurements in GIS are often made on

horizontal projections of objects– length and area may be substantially lower than on

a true three-dimensional surface

Manhattan Distance

• Distance is computed between to points (cells) by moving either N-S or E-W.

51510 y

Cell 1

15, 15

Cell 2

10, 20 (row, column)

51520 x )(* yxCellsizeD

Euclidean Allocation (EucAllocation)• Calculates, for each cell, the zone of the

closest source location in Euclidean distance.

Euclidean Direction (EucDirection)

• Calculates the direction in degrees that each cell center is from the cell center of the closest source.

• The output values are based on compass directions, with 0 degrees being reserved for the source cells (90 to the east, 180 to the south, 270 to the west, and 360 to the north).

Using Points or Lines as a Source

You should define the cell size and extent of the desired distance grid, and in many cases the mask.

If not defined the function will default to the extent of the points and default cell size is the width or height (whichever is shorter)

of the extent of the feature dataset and divide by 250.

The points are intersected with the grid to create the source cells.

The source cells will have a distance of zero.

Grids are always rectangles, without a mask the entire rectangle will be

assigned values.

With extent and mask defined to the polygon.

You can select a point, polygon, or line segment and compute

the distance from selection

Distance will be computed for all cells

within the analysis area. If you want

areas “blocked out” the should be

included in a mask

Cost Distance• Cost functions determine the shortest weighted

distance (or accumulated travel cost) from each cell to the nearest cell in the set of source cells.

• The weighted distance functions apply distance in cost units (time, $$, energy), not in geographic units.

• All weighted-distance functions require a source raster and a cost surface raster.

• Outputs from weighted-distance functions include accumulated cost distance, allocation, & backlink.

Cost Distance Algorithm

• The algorithm utilizes the node/link cell representation. • In the node/link representation, each center of a cell is

considered a node and each node is connected to its adjacent nodes by links.

• Every link has an impedance associated with it. The impedance is derived from the costs associated with the cells at each end of the link (from the cost surface) and from the direction of movement.

• costpercell = cost assigned to the cell * cell resolution

Cost Distance Algorithm• When moving from a cell to one of its four directly connected

neighbors, the cost to move across the links to the neighboring node is one times cell 1, plus cell 2, divided by two:

a1 = (cost1 + cost2) / 2

where cost1 is the cost of cell 1, cost2 is the cost of cell 2, and a1 is the total cost of the link from cell 1 to cell 2.

Cost Distance Algorithm• The accumulative cost is determined by the following formula:

accum_cost = a1 + (cost2 + cost3) / 2

where cost2 is the cost of cell 2, cost3 is the cost of cell 3, and accum_cost is the accumulative cost to move into cell 3 from cell 1. a2 is the cost of moving from cell 2 to 3.

• If the movement is diagonal, the cost to travel over the link is 1.414214 (or the square root of two) times the cost of cell 1 plus the cost of cell 2, divided by two.

a1 = 1.414214 (cost1 + cost2) / 2

Cost Distance Algorithm

• Accumulated Cost Distance is computed for each additional cell until the grid is full.

Cost Distance Algorithm• Finding the shortest weighted distance is an iterative process

starting at the source(s). • Using graph theory the goal to identify the lowest cost cell and

add it to an output list. The goal of each cell is to be assigned quickly to the output cost-distance raster.

Cost Distance Algorithm• In the first iteration, the source cells are identified and assigned to zero

since there is no accumulative cost to return to themselves. • Next, all the source cell's neighbors are activated, and a cost is assigned to

the links between the source cell nodes and the neighborhood cells' nodes using the above accumulative cost formulas.

• Each of these neighborhood cells can reach a source; consequently, they can be chosen or assigned to the output accumulative cost raster. To be assigned to the output raster, a cell must have the next least-cost path to a source.

Cost Distance Algorithm• To be assigned to the output raster, a cell must have the next least-cost path

to a source. In this case 1.5. • The lowest cost cell is chosen from the active accumulative cost cell list,

and the value for that cell location is assigned to the output cost-distance raster. The list of active cells expands to include the neighbors of the chosen cell, because those cells now have a way to reach a source.

Cost Distance Algorithm• This allocation process continues. Furthermore, cells on the active list are

updated if a new, cheaper route is created by the addition of new cell locations to the output raster.

Cost Distance Algorithm• The cost distance raster identifies the accumulative cost for each cell to

return to the closest cell in the set of source cells. It does not show which source cell to return to or how to get there. The cost back link returns a raster with a value range from 0 to 8 that can be used to reconstruct the route to the source. Each value of 0 through 8 identifies the neighboring cell to move into to get back to the source.

Least-Cost Path Algorithm

• The Shortest Path function finds the shortest or least-cost path from a source or set of sources to a destination or set of destinations.

• Inputs include destination grid/feature, accumulative cost distance grid and back-link grid.

• The least-cost path travels from the destination to the source. This path is one cell wide, travels from the destination to the source, and is guaranteed to be the cheapest route relative to the cost units defined by the original cost raster that was input into the weighted-distance function.

• The path type indicates the number of paths that will be found. – For Each Cell—Finds a path for each cell in each zone; that is, each cell in

every suburb receives its own path. – For Each Zone—Finds the one least-cost path for each zone; that is, each

suburb receives only one path. – Best Single—Finds the least-cost path for all zones; that is, only the shortest

path between one suburb and one mall is computed.

Assignment 6• Goal: Design a hiking loop trail through 4 defined points (i.e

sources) in the Browns Pond area. • Create cost surface.

– Slope (flatter the better)– Roads (avoid roads, but with exceptions –”bridges”)– Human features (avoid with buffer)– Water features (create buffer, but allow crossing on existing roads)

• Create accumulative cost distance and back-link direction grids for each of the 4 source points.

• Find least-cost paths (i.e. Shortest Path) between pair of points (1-2, 2-3, 3-4, and 4-1) where one point is the source used to create the accumulative cost distance grid and the other is the destination point with the path “draining” back to the source using the back-link direction grid.

Select Function

• Extracts features from an input feature class or input feature layer and stores them in a new output feature class. The output feature class may be created with a subset of features based on a Structured Query Language (SQL) expression.

• In Lab 6 you create shapefiles for individual points based on their ID from a shapefile with multiple points.

• Query: ID == #, where # = 1, 2, 3, or 4.