GeostatisticsGeostatistics

Mike Goodchild

Spatial interpolationSpatial interpolation

A field– variable is interval/ratio– z = f(x,y)– sampled at a set of points

How to estimate/guess the value of the field at other points?

Characteristics of interpolated surfacesCharacteristics of interpolated surfaces

Representation– raster, isolines, TIN

Form– rugged or smooth– exact or approximate– continuity

• 0-order• 1-order• 2-order

Uncertainty– variance estimators?

Linear interpolationLinear interpolation

Along a line– geocoding with address ranges

x2,y2

address2

x1,y1

address1

x,y

address

In a triangleIn a triangle

20

30

40

In a rectangleIn a rectangle

Bilinear interpolation

20

30

30

40

(24)

(34)

(29)

Characteristics of linear interpolationCharacteristics of linear interpolation

Exact 0-order continuity Contours are straight

– but not parallel in bilinear case

IDWIDW

Advantages– quick, universal, theory-free

Disadvantages– theory-free– directional effects

• non-spatial

– characteristics of a weighted average• when all weights are non-negative

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

1 4 7

10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97

100

 

Characteristics of IDW surfacesCharacteristics of IDW surfaces

Pass through each data point (exact)– if negative power distance function– 1/0b =

0-, 1-, 2-order continuous– except at data points

Underestimate peaks– volcanoes – unless peak is observation point

Extrapolate to the global mean Noisy extrapolations

KrigingKriging

Geostatistics as theoretical framework Estimation of parameters from data Use of estimated model to control

interpolation Many versions

– not a simple black box– highlights– demonstration

The variogramThe variogram

Relationship between variance and distance Formalization of Tobler's First Law Estimated from data

– how well can a given data set estimate variogram?– distribution of sample points is critical

• at peaks and pits• samples the range of possible distances• uniform spacing not desirable• but often out of the user's control

EstimationEstimation

Data points zi(xi) Interpolate at x

– stochastic process– multiple realizations

• variance obtained from variogram

A set of weights i unique to x– chosen such that the estimate is

• unbiased• minimum variance

Kriging prediction

Results of KrigingResults of Kriging

A mean surface A variance surface

– minimum at observation points Mean surface is smoother than any

realization– is not a possible realization

• a mean map is not a possible map

– compare a univariate process– average rainfall versus rainfall from a single storm– conditional simulation

Kriging standard error

Kriging variantsKriging variants

Co-Kriging– interpolation process guided by another

variable (field)– hard and soft data– observations of interpolated data are hard– guiding variable is soft

55

70

83

68

z = f (elevation)

Co-KrigingCo-Kriging

Linear relationship f Point observations are hard

– accurate, sparse Elevation observations are soft

– inaccurate (errors in measurement or prediction)

– dense

Co-Kriging prediction

Co-Kriging standard error

Indicator KrigingIndicator Kriging

Binary field– c {0,1}

Obtained by thresholding an interval/ratio field– c=1 if z>t else c=0– estimate variogram from observations of c– z is hidden

The multivariate case– sequential assignment

Indicator KrigingIndicator Kriging

Assign Class 1, notClass 1 Among notClass 1, assign Class 2,

notClass 2 Continue to Class n-1

– notClass n-1 = Class n

Universal KrigingUniversal Kriging

Simple Kriging is all second order– trend results from random walk

Stochastic process plus trend– trend is first order– remove trend before analysis– restore trend after analysis

Advantages and disadvantagesAdvantages and disadvantages

Theoretically based Not a black box Statistical

– variance estimates Sensitivity to sample design