Applications of neighborhood functions (spatial filters) Removing odd values Smooth the data Edge detection Edge sharpening Spatial variability
How to represent the real world in 3D? Data points are used to generate a continuous surface. In the below example, a color coded surface is generated from sample values
How to represent the real world in 3D? Two ways to generate real world surfaces from point data (sample values) – Vector – raster Whatever the method, what kind of data are available to represent the world?
How to represent the real world in 3D? Ways of spatial sampling
Samples could represent any quantity (value) Elevation Climate data – Temperature – Precipitation – Wind – CO2 flux Others – Ice thickness – Spatial samples (of some quantity) in a city – Gold concentrations – LiDAR data points
Elevation Data Collected by several methods – Topographic survey (very accurate) – LiDAR data (pretty accurate) – Satellite radar (surprisingly accurate) – GPS survey (much less accurate) Elevations (z-values) recorded at points
Surface Representation Regardless of vector or raster: Point elevations Triangular Irregular Networks (TINs) Contour lines Digital Elevation Models (DEMs)
Vector representation (of surfaces) Triangular Irregular Network (TIN) TIN can be used to – Generate contour lines – Slope – Aspect
Triangular Irregular Network Way of representing surfaces (vector) Elevation points connected by lines to form triangles Size of triangles may vary Each face created by a triangle is called a facet
Triangular Irregular Network The triangulation is based on the Delaunay triangulation A Delaunay triangulation is a triangulation such that no sample (point) out of all samples is inside the circumcircle of any triangle. Delaunay triangulations maximize the minimum angle of all the angles of the triangles in the triangulation; they tend to avoid skinny triangles.
Delaunay triangulation Delaunay triangles: all satisfy the condition Delaunay NOT satisfied
More about TINs No interpolation required, all elevation values are based on direct measurements Visualized using hillshade for a 3D effect
More about TINs Hillshade is one of the most common ways of displaying/visualizing TINs. Commonly Sun is shining from northwest (315deg) from 45deg above horizon. Each facet will be assigned with a color based on its orientation Products – Contour lines – Slope and aspect can be derived from
Raster Representation (of surfaces) The most commonly used term for raster representation is Digital Elevation Model (DEM) Any digital model for any other variable could be generated For DEM, each cell has an elevation (z-value) To generate DEM from sample points, interpolation is used to fill in between surveyed elevations – several methods to choose from
Comes from the word “inter” meaning between and “pole” which represent two sample points. So, you want to find a value between two points. Extrapolation is finding a value for the outside of the two points Interpolation
Linear interpolation Assume that the value for an unknown location between two known points can be estimated based on a linear assumption
Polynomial Interpolation Assume that the value for an unknown location between two known points can be estimated based on a non-linear assumption
Inverse Distance Weighted (IDW) Spatial Autocorrelation – Near objects are more similar than far objects IDW weights point values based on distance
Inverse Distance Weighted (IDW) Estimating an unknown value for a pixel (p) by weighting the sample values based on their distance to (p) i=8 in this example j In the above equation, n is the power. It is usually equals to 2, i.e., n=2. But you can pick n=1, n=1.5, etc.
IDW – Choosing the Power Power setting influences interpolation results Lower power results in smoother surfaces Higher power results in rugged surface (it become more like ….?)
Kriging Statistical regression method, whose process consists of two main components – Spatial autocorrelation (semivariance) – Some weighting scheme Advanced interpolation function, can adapt to trends in elevation data
Kriging and Semivariogram Semivarigram is a graph describing the semivariance (or simply variance) between pairs of samples at different distances (lags) The idea comes from intuition: – Things that are spatially close are more correlated than those are far way (similar to IDW)
Generating Semivariogram To generate a semivariogram, semivariance between pairs of points (for various distances/lags) are to be calculated
Kriging and Semivariogram The first step in the kriging algorithm is to compute an average semivariogram for the entire dataset. This is done by going through each single point in the dataset and calculate semivariogram. Then the semivariogram are averaged. The second step is to calculate the weights associated with each point
Evaluation of the generated surface Independent samples must be preserved for accuracy assessment of the predicted (generated) surface. These points are called check points. In other words, if you have 100 samples in the area, you’d use 90 to create the surface and 10 of them to evaluate how accurately the surface represents the actual world
Detail on Watershed Analysis Determine flow direction grid (DEM derived property). Determine flow accumulation grid (DEM derived property). Specify a "stream" threshold on the flow accumulation grid. This operation will identify all the cells in the flow accumulation grid that are greater than the provided threshold. A new grid is formed from those cells ("stream" grid). This grid will be an indication of the drainage network. Higher thresholds will result in less dense network and less internal subwatersheds, while lower thresholds will result in dense network and more internal subwatersheds. Stream grid is converted into stream segments, where each head segment and segment between the junctions has a unique identifier. Subwatersheds (in grid format) are defined for each of the stream links in the stream link grid. Subwatershed and stream grids are vectorized to produce subwatershed and stream polygon and polyline themes respectively. Additional vector processing might be needed to clean-up the data and insure correct connectivity and directionality.