Spatial interpolation GEOG 4103 spring 06 April 3rd

Real world phenomena represented as: DISCRETE: homogeneous or spatially averaged units, e.g. subwatersheds, counties, polygons VECTOR FIELDS: grid cells or meshes RASTER

What are statistical surfaces? Features that contain Z values distributed throughout area defined by (x,y) coordinate pairs Examples? any measurable phenomena that varies across space (temperature, elevation, precipitation, etc) called “field” like (continuous) data

What is a field? a conceptual model of geographic variation at every point in the frame (x,y) there exists a single value of a variable Z e.g. a field of temperature e.g. a field of land surface elevation Field data are continuous

Representation of field phenomena A) CELLS B) REGULARLY SPACED POINTS C) IRREGULARLY D) CONTOURS E) POLYGONS F) TINs- Triangulated Irregular Network

Spatial sampling e.g. elevation Regular lattice restricted to X,Y locations Irregular lattice not restricted based on knowledge about how smooth/rugged the surface is

Interpolation procedure of estimating the value at unsampled locations from existing observations

Two types of interpolation based on the mathematical function used Linear interpolation Non-linear interpolation

Linear interpolation ? 100’ 150’ 110 120 130 140 Surface changes in a linear fashion can use simple mathematical functions

Non-linear interpolation Tobler's Law of Geography: points close together in space are more likely to have similar values than points farther apart Distance-weighted interpolation

E.g. Inverse Distance Weighted interpolation method Software measures distance from each neighboring point you can select the number of points to include in the search then distance is calculated according to a formula B • C • 100 40 ? • A 20 200 • D

Global vs. local interpolation global interpolators determine a single function which is mapped across the whole region e.g. trend surfaces local interpolators apply an algorithm repeatedly to a small portion of the total set of points e.g IDW

Point-based interpolation used for data which can be collected at point locations e.g. weather station readings, spot heights, soil measurements etc.

Green Lakes Valley Snow Survey Figure 5. Location of snow depth measurements in the Green Lakes Valley from 1997 to 2000. We also added zero depths on about a 50 meter grid for non snow covered area that was too steep to sample.

Estimated snow depth for 1999 using various interpolation techniques

Estimating error

A special case of statistical surfaces: DEM / DTM Digital elevation models = a way of representing surfaces. Quantitative model of a topographic surface in digital form. continuous surfaces.

Two methods of representing a topography inside a computer Raster surfaces: e.g. DEM (Digital Elevation Model) Vector surfaces: TIN’s (Triangulated Irregular Network)

Problems with interpolation Accuracy Visual representation Edge effects (lack of data at the margins)