Raster Surfaces

raster surfaces are

continuous

why are raster surfaces continuous

each grid cell has a z-value

what are some examples of terrestrial surfaces? statistical surfaces?

terrestrial

• DEM

• slope

• aspect

• precipitation

statistical 

• population density 

• crime rates 

• density of animals in area

spatial interpolation 

predicts values for cells in a raster from a limited number of sample data points 

what are some data types we can interpolate

elevation

rainfall

air quality

noise levels

how is a continuous surface calculated

from point data

what is interpolation the estimation of 

the z value (height) of a surface at unsampled data based on known surrounding z values 

what bounds the continuous surface

a mask or extend

4 more common spatial interpolation methods

1. trend

   1. local

   2. global

2. IDW

3. spline

4. Natural neighbours

is trend an approximate or exact interpolation method

approximate

trend is a _____ surface 

polynomial 

what happens with global polynomial interpolation

creates a surface that represents gradual, coarse-scale trends and is useful for capturing long-range patterns

is local variation captured in global interpolation

NO

which tool uses global polynomial interpolation 

trend 

what is a first order global polynomial

a single, flat plane (a linear function) to all data points to model

what is the goal of a polynomial

minimize error of prediction

when is global interpolation used for

large scale patterns

1st order polynomial vs 2nd order polynomial vs 3rd order polynomial

1st - linear polynomial 

2nd - one bend 

3rd - two bends

does trend pass through actual measured points

not normally - inexact interpolation

what does trend use to fit the interpolation

least squares regression

what should be true about the points above and below the trend best fit line

should be the same below and above it

when should local polynomial interpolation be used 

when you have data that exhibits short-range variation

IDW interpolation method

estimates cell values by averaging the values of sample points in neighboring cells

in IDW what is the relationship between point weight and distance

as the distance from a point increases, its weight decreases

what are TWO conditions that must be met before using IDW

1. samples are relatively evenly distributed 

2. surface characteristics do NOT change across the landscape 

what are variables of IDW

1. nearest neighbour

2. fixed radius

3. barriers

4. power

contrast nearest neighbour and fixed radius in IDW

nearest neighbour - integer value defining the MINIMUM number of points to be used for interpolation

fixed radius - distance in map units where all input points in the specified radius will be used for interpolation

what are barriers

barriers - specify location of linear features known to interrupt the surface continuity 

what do barriers NOT have

z values

examples of barriers

fences, railways, cliffs

what do barriers do to the interpolation

the selected input points for the interpolation MUST be on the SAME side of the barrier as the current processing cell

which interpolation method can use barriers

IDW

spline

kriging

what must a power be

a positive, real number

what is a common power value

is 2

what happens by defining a higher power option

more emphasis can be put on the nearest point = nearby data will have MORE influence

will a higher power result in a more or less detailed surface

more detail 

what happens in IDW with a higher power

a less smooth and more "spiky" interpolation with higher peaks and deeper valleys

what are some issues with IDW

1. min and max values must be sampled or variation will be missed

2. surface will NOT go exactly through the sampling points

spline

the bending and stretching of a surface to pass through ALL measured points

what can spline result in

a surface that captures BOTH global trends and local variation

how does spline estimate values

mathematical function that MINIMIZES the change in slope

what are two conditions of spline

1. surface must pass exactly through the data points

2. surface must have minimum curvature = minimum change in slope

what type of points is spline a good choice for 

points without abrupt changes in values 

what are the spline variables

1. regularized spline - higher weights = smoother surface

2. spline with tension - higher values = coarser surface

3. number of points - more points = smoother the surface

when to use IDW or spline?

IDW

• the variables being mapped decreases in influence with distance

spline - the variable is smooth, continuous surface that does NOT have large variability over short distance 

what does natural neighbours use twice

Thiessen polygons

where is the polygon boundary in nearest neighbours

between grid cell and nearest sampling point

which interpolation provides the best looking surface

nearest neighbours

what are Thiessen polygons

each defines an area of influence around its sample point so any location inside the polygon is closer to that point than any other

when is nearest neighbour most appropriate

sample data points are distributed with uneven density

spline

nearest neighbour

trend

IDW