Advantages Easy to understand. Disadvantages. Systematic sampling pattern Easy Samples spaced uniformly at fixed X, Y intervals Parallel lines
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1 INTERPOLATION Procedure to predict values of attributes at unsampled points Why? Can t measure all locations: Time Money Impossible (physical- legal) Changing cell size Missing/unsuitable data Past date (eg. temperature)
2 Systematic sampling pattern Easy Samples spaced uniformly at fixed X, Y intervals Parallel lines Advantages Easy to understand Disadvantages All receive same attention Difficult to stay on lines May be biases
3 Random Sampling Select point based on random number process Plot on map Visit sample Advantages Less biased (unlikely to match pattern in landscape) Disadvantages Does nothing to distribute samples in areas of high Difficult to explain, location of points may be a problem
4 Cluster Sampling Cluster centers are established (random or systematic) Samples arranged around each center Plot on map Visit sample (e.g. US Forest Service, Forest Inventory Analysis (FIA) Clusters located at random then systematic pattern of samples at that location) Advantages Reduced travel time
5 Adaptive sampling More sampling where there is more variability. Need prior knowledge of variability, e.g. two stage sampling Advantages More efficient, homogeneous areas have few samples, better representation of variable areas. Disadvantages Need prior information on variability through space
6 INTERPOLATION Many methods - All combine information about the sample coordinates with the magnitude of the measurement variable to estimate the variable of interest at the unmeasured location Methods differ in weighting and number of observations used Different methods produce different results No single method has been shown to be more accurate in every application Accuracy is judged by withheld sample points
7 INTERPOLATION Outputs typically: Raster surface Values are measured at a set of sample points Raster layer boundaries and cell dimensions established Interpolation method estimate the value for the center of each unmeasured grid cell Contour Lines Iterative process From the sample points estimate points of a value Connect these points to form a line Estimate the next value, creating another line with the restriction that lines of different values do not cross.
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10 Example Base Elevation contours Sampled locations and values
11 INTERPOLATION 1 st Method - Thiessen Polygon Assigns interpolated value equal to the value found at the nearest sample location Conceptually simplest method Only one point used (nearest) Often called nearest sample or nearest neighbor
12 INTERPOLATION Thiessen Polygon Advantage: Ease of application Accuracy depends largely on sampling density Boundaries often odd shaped as transitions between polygons are often abrupt Continuous variables often not well represented
13 Thiessen Polygon Draw lines connecting the points to their nearest neighbors. Find the bisectors of each line. Connect the bisectors of the lines and assign the resulting polygon the value of the center point Source:
14 Thiessen Polygon Start: 1) 1. Draw lines connecting the points to their nearest neighbors Find the bisectors of each line. 4 2) 3) 3. Connect the bisectors of the lines and assign the resulting polygon the value of the center point
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21 Sampled locations and values Thiessen polygons
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23 INTERPOLATION Fixed-Radius Local Averaging More complex than nearest sample Cell values estimated based on the average of nearby samples Samples used depend on search radius (any sample found inside the circle is used in average, outside ignored) Specify output raster grid Fixed-radius circle is centered over a raster cell Circle radius typically equals several raster cell widths (causes neighboring cell values to be similar) Several sample points used Some circles many contain no points Search radius important; too large may smooth the data too much
24 INTERPOLATION Fixed-Radius Local Averaging
25 INTERPOLATION Fixed-Radius Local Averaging
26 INTERPOLATION Fixed-Radius Local Averaging
27 INTERPOLATION Inverse Distance Weighted (IDW) Estimates the values at unknown points using the distance and values to nearby know points (IDW reduces the contribution of a known point to the interpolated value) Weight of each sample point is an inverse proportion to the distance. The further away the point, the less the weight in helping define the unsampled location
28 INTERPOLATION Inverse Distance Weighted (IDW) Zi is value of known point Dij is distance to known point Zj is the unknown point n is a user selected exponent
29 INTERPOLATION Inverse Distance Weighted (IDW)
30 INTERPOLATION Inverse Distance Weighted (IDW) Factors affecting interpolated surface: Size of exponent, n affects the shape of the surface larger n means the closer points are more influential A larger number of sample points results in a smoother surface
31 INTERPOLATION Inverse Distance Weighted (IDW)
32 INTERPOLATION Inverse Distance Weighted (IDW)
33 INTERPOLATION Trend Surface Interpolation Fitting a statistical model, a trend surface, through the measured points. (typically polynomial) Where Z is the value at any point x Where a i s are coefficients estimated in a regression model
34 INTERPOLATION Trend Surface Interpolation
35 INTERPOLATION Splines Name derived from the drafting tool, a flexible ruler, that helps create smooth curves through several points Spline functions are use to interpolate along a smooth curve. Force a smooth line to pass through a desired set of points Constructed from a set of joined polynomial functions
36 INTERPOLATION : Splines
37 INTERPOLATION Kriging Similar to Inverse Distance Weighting (IDW) Kriging uses the minimum variance method to calculate the weights rather than applying an arbitrary or less precise weighting scheme
38 Interpolation Kriging Method relies on spatial autocorrelation Higher autocorrelations, points near each other are alike.
39 INTERPOLATION Kriging A statistical based estimator of spatial variables Components: Spatial trend Autocorrelation Random variation Creates a mathematical model which is used to estimate values across the surface
40 Kriging - Lag distance Z i is a variable at a sample point h i is the distance between sample points Every set of pairs Z i,z j defines a distance h ij, and is different by the amount Z i Z j. The distance h ij is the lag distance between point i and j. There is a subset of points in a sample set that are a given lag distance apart
41 Kriging - Lag distance
42 INTERPOLATION Kriging Semi-variance Where Z i is the measured variable at one point Z j is another at h distance away n is the number of pairs that are approximately h distance apart Semi-variance may be calculated for any h When nearby points are similar (Z i -Z j ) is small so the semivariance is small. High spatial autocorrelation means points near each other have similar Z values
43 INTERPOLATION Kriging When calculating the semi-variance of a particular h often a tolerance is used Plot the semi-variance of a range of lag distances This is a variogram
44 INTERPOLATION Kriging When calculating the semi-variance of a particular h often a tolerance is used Plot the semi-variance of a range of lag distances This is a variogram
45 Idealized Variogram
46 INTERPOLATION (cont.) Kriging A set of sample points are used to estimate the shape of the variogram Variogram model is made (A line is fit through the set of semi-variance points) The Variogram model is then used to interpolate the entire surface
47 Variogram
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49 INTERPOLATION (cont.) Exact/Non Exact methods Exact predicted values equal observed Theissen IDW Spline Non Exact-predicted values might not equal observed Fixed-Radius Trend surface Kriging
50 Class Vote: Which method works best for this example? Systematic Random Original Surface: Cluster Adaptive
51 Class Vote: Which method works best for this example? Original Surface: Thiessen Polygons Fixed-radius Local Averaging IDW: squared, 12 nearest points Trend Surface Spline Kriging
52 Interpolation in ArcGIS: Spatial Analyst
53 Interpolation in ArcGIS: Geostatistical Analyst
54
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56 Validation
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58 Interpolation in ArcGIS: arcscripts.esri.com
59 What is the Core Area?
60 Core Area Identification Commonly used when we have observations on a set of objects, want to identify regions of high density Crime, wildlife, pollutant detection Derive regions (territories) or density fields (rasters) from set of sampling points.
61 Mean Circle
62 Concave Hull Convex Hull
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65 Kernal Mapping
66 Smooth density function One centered on each observation point
67 Sum these density functions
68 Sum the total, for a smooth density curve (or surface)
69 How much area should each sample cover (called bandwidth)
70 Varying bandwidths a)medium b)low h c)high h d through f are 90% density regions
71 Time-Geographic Density Estimation (TDGE)
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