Spatial Analysis

Digital Elevation Model (DEM) DEM is the representation of continuous elevation values over a topographic surface by a regular array of points with 3 coordinates (x,y for location on the surface and z-value for altitude).

DEM Derivatives Slope Hillshade Aspect A number of surfaces can be derived from a DEM: slope, aspect, hillshade, contour lines, viewsheds. Aspect DEM Analysis: http://www.youtube.com/watch?v=ukk2ciG2tDY

Slope and aspect Slope and aspect are calculated at each point in the grid, by comparing the point’s elevation to that of its neighbors Slope is the incline or steepness of a surface (measured in degrees 0 – 90, or as a percentage of a rise divided by a run) Aspect is the compass direction that a topographic slope faces usually measured in degrees from north

Draping Hillshading creates a 3-dimensional effect that provides a sense of visual relief, derived using a hypothetical illumination of a surface according to a specified azimuth and altitude for the sun Draping – a rendering of a 2 dimensional image onto a 3 dimensional surface Upper image: DEM draped over hillshade Lower image: Slope draped over hillshade

Buffering Creates a new object consisting of areas within a user-defined distance of an existing object, for example: To determine areas impacted by a proposed highway To determine the service area of a proposed hospital Can be done for both a raster and a vector

Buffering Point Polyline Polygon

Point-in-polygon transformation Determine whether a point lies inside or outside a polygon generalization: assign many points to containing polygons used to assign crimes to police precincts, voters to voting districts, accidents to reporting counties

Point-In-Polygon

Map Algebra A language that allows to transform a raster map or combine two or more raster maps by applying mathematical operations and analytical functions Local: cell-by-cell operations Focal: operations performed on a user-defined neighborhood of the focus cell Zonal: process all cells within a user-defined regions (zones) Global: the cell values for the output grid can be dependent upon all the cells in the input grid Map algebra: Creates new output Uses combinations of mathematical, logical, and Boolean operations (e.g. +, AND, >=, mean, tan)

Map Algebra Example: Sum What if we add a constant number e.g. 3 to the 1st grid? How would you do that? The neighborhoods can vary in shape and size, here a 3x3 square neighborhood is used.

Map Algebra Example: Sum Sample zones: red is urban, yellow is rural Global sum: what about the other cells?

Spatial interpolation (Tobler’s First Law of Geogaphy) The process of using points with known values to estimate values at other points. These points with known values are called known points, control points, sample points, or observations. Spatial interpolation is a process of intelligent guesswork, in which the investigator (and the GIS) attempt to make a reasonable estimate of the value of a continuous field at places where the field has not actually been measured. The one principle that underlies all spatial interpolation is the Tobler Law.

Spatial interpolation Distance Decay

Importance of the Density of Sample Points Imagine this elevation cross section: If each dashed line represented a sample point, this spacing would miss the major local sources of variation, like the gorge

Importance of the Density of Sample Points If you increase the sampling rate (take samples closer together), the local variation will be more accurately captured

Importance of the Density of Sample Points

Kriging Kriging is a spatial interpolation technique that assumes that the spatial variation of an attribute may consist of three components: a spatially correlated component, representing the variation of the regionalized variable; a ‘drift’ or structure, representing a trend; and a random error term. Developed by Georges Matheron to evaluate new GOLD mines with a limited number of borholes.

Density estimation Spatial interpolation is used to fill the gaps in a field Density estimation creates a field from discrete objects. The field’s value at any point is an estimate of the density of discrete objects at that point E.g. estimating a map of population density (a field) from a map of individual people (discrete objects)

Kernel Density Surfaces Each discrete object is replaced by a mathematical function known as a kernel. Kernels are summed to obtain a composite surface of density. The smoothness of the resulting field depends on the width of the kernel. When the kernel width is too small (in the left case it is ~80km when search area equals 20,000km2) the surface is too rugged and bumpy, and each point generates its own peak. The kernel width in the right case is ~178.5km (when search area equals 100,000km2) and produces much more plausible smooth density surface. Summary: The density is a function of a distance used to calculate it. Data shown: Ozone measurement stations in CA (NOT the measurements of ozone themselves) ArcGIS Spatial Analyst was used to generate the surfaces. Search radius: 20K km2 Search radius: 100K km2