Spatial interpolation techniques

By:-Manisha

Definition:“Spatial interpolation is the procedure of estimating the values of properties at unsampled sites within an area covered by existing observations.”

What is Spatial Interpolation??

classification

• Global methods:

– single mathematical function applied to all points

– tends to produces smooth surfaces

• Local methods:

– single mathematical function applied repeatedly to subsets of the

total observed points

– link regional surfaces into composite surface

• Exact methods:

– honour all data points such that the resulting surface passes

exactly through all data points

– appropriate for use with accurate data

• Approximate methods:

– do not honour all data points

– more appropriate when there is high degree of uncertainty about

data points

Interpolation methods

–Thiessen polygons

–Triangulated Irregular Networks (TINs)

–Spatial moving average

–Trend Surfaces

–Kriging

–Inverse distance weighting (IDW)

• Thiessen (Voronoi) polygons:

– assume values of unsampled locations are equal to

the value of the nearest sampled point

• Vector-based method

– regularly spaced points produces a regular mesh

– irregularly spaced points produces an network of

irregular polygons

Thiessen Polygons

Thiessen polygon construction

Example Thiessen polygon

Source surface with sample points Thiessen polygons with sample points

•Another vector-based method often used to create

digital terrain models (DTMs)

– adjacent data points connected by lines (vertices) to create a

network of irregular triangles

calculate real 3D distance between data points along vertices using

trigonometry

calculate interpolated value along facets between three vertices

TINs

EXAMPLE- TIN

• Vector and raster method:– most common GIS method

– calculates new value of each location based on range of values associated with neighbouring points

– Neighbourhood determined by a filter

Spatial moving average

Example SMA (circular filter)

Source surface with sample points 21x21 circular filter SMA 41x41 circular filter SMA

Trend surfaces

• Uses a polynomial regression to fit a least-squares surface

to the data points

– normally allows user control over the order of the polynomial

used to fit the surface

– as the order of the polynomial is increased, the surface being

fitted becomes progressively more complex

higher order polynomial will not always generate the most accurate

surface, it dependent upon the data

the lower the RMS error, the more closely the interpolated surface

represents the input points

Week 17 GEOG2750 – Earth Observation and GIS of the Physical Environment 14

data point

interpolated point

Fitting a single polynomial trend surface

Spatial Analysis Lecture #8 (Exploring Continuous Data)

Example trend surfaces

Goodness of fit

(R2) = 45.42 %

Goodness of fit

(R2) = 92.72 %

Goodness of fit

(R2) = 82.11 %

Linear Quadratic Cubic

Source surface with

sample points

Inverse Distance Weighting (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

Example: IDW

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

Example: Kriging