Interpolation & Contour Maps

Interpolation We want to draw inferences about data based on the observed data Visiting every location in a study area to measure the height, magnitude, or amount of a phenomenon is usually difficult or expensive. Instead, measure the phenomenon at strategically dispersed sample locations, and predicted values can be assigned to all other locations.

Spatial Data Perspectives Deterministic perspective: if we understand the system, we can predict the outcome with certainty Stochastic perspective: uncertainty remains in the best-designed model or experiment

Interpolation Methods Deterministic perspective: assign values to locations based on the surrounding measured values and on specified mathematical formulas that determine the smoothness of the resulting surface Stochastic perspective: assign values based on statistical models that include autocorrelation (the statistical relationship among the measured points)

Types of Interpolation Methods Deterministic perspective: Inverse Distance Weighted: points weighted by distance Spline: passes exactly through points with constraining equations Polynomial/trend analysis (nearest/natural neighbor) Stochastic perspective: Kriging: weighted by fitted semivariogram (next slides)

Natural Neighbor The colored circles, which represent the interpolating weights, are generated using the ratio of the shaded area to that of the cell area of the surrounding points. The shaded area is due to the insertion of the point to be interpolated into the Voronoi tessellation (Theissen polygons).

Inverse Distance Weighted point i known value zi location xi weight wi distance di unknown value (to be interpolated) location x The estimate is a weighted average

Trend Surface/Polynomial Flat but TILTED plane to fit data (1st order polynomial) Z = a + bx + cy Tilted but WARPED plane to fit data ( 2nd order polynomial ) Z = a + bx + cy + dx2 + exy + fy2

Trend Surfaces Simplifies the surface representation to allow visualization of general trends. Higher order polynomials can be used Robust regression methods can be used

Kriging Spatial prediction of variable Z at location x Z(x) = m(x) + γ(h) + ε Three components: structural (constant mean), random spatially correlated component residual error

Variogram Nugget Range Sill Separation Distance Semi-Variogram function Plot of the correlation of data (g) as a function of the distance between points (h) 12/10/2018

Types of Kriging • Simple Kriging: assumes mean is constant and known • Ordinary Kriging: assumes mean is constant but unknown (MapWindow) • Universal Kriging: assumes mean is varying and unknown - Modeled by a constant, linear, second or third order equation

Advantages of Kriging It handles (embraces) spatial autocorrelation Less sensitive to preferential sampling in specific areas Allows uncertainty to be estimated (Kriging error)

Example: Illinois Precipitation Formatted into a CSV with Latitude, Longitude, and Precipitation

Example: Illinois Precipitation Imported to MapWindow

Example: Illinois Precipitation Imported to MapWindow

Example: Illinois Precipitation Imported to MapWindow

Example: Illinois Precipitation Assign Projection (comes in Geographic Coordinates), use a Geographic Coordinate System (WGS 84)

Example: Illinois Precipitation Assign Projection (comes in Geographic Coordinates), use a Geographic Coordinate System (WGS 84)

Example: Illinois Precipitation Kriging requires a projected shapefile. Reproject to WGS 84 UTM 16 N

Example: Illinois Precipitation Kriging requires a projected shapefile. Reproject to WGS 84 UTM 16N

Example: Illinois Precipitation Let’s Krige! Add Layers PRECIP_STATIONS_16N.shp and IL_BOUND_16N.shp

Example: Illinois Precipitation

Example: Illinois Precipitation Enable Kriging Plugin, Interpolation Methods Menu

Example: Illinois Precipitation Select PRECIP_STATIONS_16N layer

Example: Illinois Precipitation Select PRECIP field

Example: Illinois Precipitation Semivariogram:) – Select distance classes (spatial lags) – Calculate semivariance across lags – Plot semivariance on Y axis, distance on X axis – Fit theoretical model (linear, gaussian)

Example: Illinois Precipitation Apply model (Gaussian, Range = 1,000,000, Nugget = 0, Sill = 85)

Example: Illinois Precipitation Generate surface, save as .asc

Example: Illinois Precipitation Generate contour precipitation map for the kriged surface, raster operation

Example: Illinois Precipitation Choose a location, and an interval for spacing contours

Example: Illinois Precipitation

Cease the training of impossible hedges around this life For as fast as you sow them, serendipity’s thickets will appear, And outgrow them. Lorna Goodison