Problems with Vector Overlay Analysis (esp. Polygon)Slide 2Boolean OperationsBoolean Operations with Raster LayersAlgebraic Operations w/ Raster LayersSimple Arithmetic OperationsRaster (Image) DifferenceRaster (Image) DivisionMore Complex OperationsProblems with point location dataSpatial InterpolationSpatial Interpolation: Inverse Distance Weighting (IDW)Slide 13Issues with IDWSlide 15Slide 16•Overlay analysis using the vector spatial data model is highly computationally intensive:–Complicated input layers can tax even current processors•There is a tradeoff between the complexity vs. the interpretability of results–Complex input layers with many polygons can result in many more polygon combinations… can we make sense of all those combinations?Problems with Vector Overlay Analysis (esp. Polygon)•There are often spatial mismatches between input layers–Overlay can result in spurious sliver polygons–We can “filter” out spurious slivers by querying to select all polygons with AREA less than some minimum threshold•It is difficult to choose a threshold to avoid deleting ‘real’ polygonsProblems with Vector Overlay Analysis (esp. Polygon)A•Boolean operations of OR & AND correspond to UNION & INTERSECTION, used in vector-based analysesUNIONINTERSECTIONA BORBABANDA BBoolean Operations•We can apply these concepts in the raster spatial data model, when two input layers contain true/false or 1/0 data: A B0 1 10 0 11 0 10 0 01 1 10 0 1AND=Boolean Operations with Raster Layers0 1 10 0 11 0 10 0 01 1 10 0 1OR=•The AND operation requires that the value of cells in both input layers be equal to 1 for the output to have a value of 1:•The OR operation requires that the value of a cells in either input layer be equal to 1 for the output to have a value of 1:Algebraic Operations w/ Raster Layers•We can extend this concept from Boolean logic to algebra•Map algebra:•Each cell is a number•Mathematical operations are applied to the layers, calculated on a cell-by-cell basis•The result for each cell is placed in a new layer.•Example: suitability analysis •Multiple input layers determine suitable sites:•The pixels attributes in each layer represent ‘scores’.•Layers are weighted based on their importance.•Layer scores are added on a per-pixel basis.101100110100111000+=201211110Summation101100110100111000=100100000Multiplication101100110100111000+=301322110100111000+Summation of more than two layersSimple Arithmetic OperationsNear the mall Near friend’s houseNear workGood place to live?Raster (Image) Difference•An application of taking the differences between layers is a form of change detection:•Example: you have 2 images of the same forest, 10 years apart. To measure forest growth or cuts, you can take the difference in infrared light between image dates.•More infrared light = more chlorophyll and more greenery•Question: How can the locations where a substantial changes have occurred be identified using the difference layer?517656345723541653-=-2-14115-3-32The difference between two layers:Question: Can we perform the following operation? Are there any circumstances where we cannot perform this operation? Why or why not?=Raster (Image) DivisionLinear Transformation235123421102115001+=100111000+abcMore Complex Operations•This could applied in the context of computing a statistical linear regression (e.g. output y = a*x1 + b*x2 + c*x3) using raster layers•Example: suitability analysis. •Suitable place to live example.•Near the mall – low importance (a=1)•Near work – high importance (a=4)•Near friend’s house – moderate importance (a=3)Problems with point location dataYou have point location data, but you want data for your whole site/region.Examples:–Air temperature maps created from point data.–Water pollution levels measured at points.Solution:–Estimate areas with no data.Spatial Interpolation•You have point data (temp or air pollution levels).•You want the values across your full study site.•Spatial interpolation estimates values in areas with no data.–creates a contour map by drawing isolines between the data points, or–creates a raster digital elevation model which has a value for every cellSpatial Interpolation:Inverse Distance Weighting (IDW)•One method of interpolation is inverse distance weighting:•The unknown value at a point is estimated by taking a weighted average of known values–Those known points closer to the unknown point have higher weights.–Those known points farther from the unknown point have lower weights.point iknown value zidistance diweight wiunknown value (to be interpolated) atlocation xiiiiixwzwz21iidw The estimate of the unknown value is a weighted averageSample weighting functionSpatial Interpolation:Inverse Distance Weighting (IDW)Issues with IDW•Weighted average estimates are always between the min and max known values.–If the known (sampled) points did not include the minima and maxima (e.g., mountain peaks and valleys), your data will be less extreme than reality–It is thus important to position sample points to include the extremes whenever possibleIssues with IDWThe dashed line is a hill.The x’s are sampled elevation points.The black line is the interpolated (estimated) hill elevation. With IDW, the unknown points tend towards the overall mean.Triangle Irregular Networks (TIN)Most often used for elevation surfaces.Points are known data values (of elevation). Lines are drawn between nearby points to create a set of irregular triangles.The value of the variable (e.g. elevation) moves along the lines evenly from one point to the next.The triangle between 3 lines represents a flat surface with slope and aspect.With the TIN, the feature’s value every point on the TIN is