# spatial interpolation, geostatistics and sampling

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2003 All lecture materials by Austin Troy except where noted Introduction to GIS What is interpolation? Three types: Resampling of raster cell size Transforming a continuous surface from one data model to another (e.g. TIN to raster or raster to vector). Creating a surface based on a sample of values within the domain. Dense sampling networks Sparse sampling networks 2003 All lecture materials by Austin Troy except where notedTRANSCRIPT

Lecture 6:Spatial Interpolation, geostatistics and sampling

By Austin Troy

------Using GIS--Introduction to GIS

What is interpolation?

Three types:Resampling of raster cell sizeTransforming a continuous surface from one data model to another (e.g. TIN to raster or raster to vector). Creating a surface based on a sample of values within the domain. Dense sampling networksSparse sampling networks

Introduction to GIS

Sample-based interpolation

Process of creating a surface based on values at isolated sample points.Sample points are locations where we collect data on some phenomenon and record the spatial coordinatesWe use mathematical estimation to guess at what the values are in between those pointsWe can create either a raster or vector interpolated surfaceInterpolation is used because field data are expensive to collect, and cant be collected everywhere

Introduction to GIS

How does it Look

Introduction to GISLet say we have our ground water pollution samples

How does it work

Introduction to GISThis can be displayed as a 3D trend surface in 3D analyst

How does it work

Introduction to GISWe can also use interpolation methods to create contours

Also known as Isolines

Requirements of interpolation

Interpolation only works where values are spatially dependant, or spatially dependent, that is, where nearby location tend to have similar Z values.Examples of spatially dependent features: elevation, property value, crime levels, precipitationNon-dependent examples: number of drum sets per city block; cheeseburgers consumed per household. Where values across a landscape are geographically independent, interpolation does not work because value of (x,y) cannot be used to predict value of (x+1, y+1).

Introduction to GIS

Interpolation examples

Elevation:Elevation values tend to be highly spatially autocorrelated because elevation at location (x,y) is generally a function of the surrounding locationsExcept is areas where terrain is very abrupt and precipitous, such as Patagonia, or YosemiteIn this case, elevation would not be autocorrelated at local (large) scale, but still may be autocorrelated at regional (small scale)

Introduction to GIS

Interpolation examples

Elevation:

Introduction to GISSource: LUBOS MITAS AND HELENA MITASOVA, University of Illinois

Sample points

Also known as control points.These are points where you or someone else has collected data (attributes) for a spatial coordinate (point)Any number of attributes can be collected at that pointE.g.1 weather stations collect data on temperature, rainfall, wind, humidity, etc. E.g. 2 soil invertebrate samples would record abundance of numerous species at each location

Introduction to GIS

Sampling example

Imagine this elevation cross section: If each dashed line represented a sample point (in 1-D), this spacing would miss major local sources of variation, like the gorge

Introduction to GIS

Sampling example

Our interpolated surface (represented in 1-D by the blue line) would look like this

Introduction to GIS

Sampling example

If we increased the sampling rate, we would pick up that local variation

Introduction to GIS

Sampling example

Here our interpolated surface is much closer to reality at the local level, but we pay for this in the form of higher data gathering cost

Introduction to GIS

Interpolation examples

WeatherWeather tends to be modeled on a regional level (e.g. your local weather report) because, in most places, weather systems and trends happen over a very large area. Hence the need for sample point density is not so greatIn other places, local climate variability is very great, such as in the SF Bay Area where temperatures can vary 50 degrees within 10 miles due to ocean effects.

Introduction to GIS

Interpolation examples

WeatherWeather is also extremely variable over time, so samples must be continually taken. This is why weather stations are usually permanent

Introduction to GISSource: LUBOS MITAS AND HELENA MITASOVA, University of IllinoisExample: precipitation varying over a season

Interpolation examples

Introduction to GISGroundwater contamination:The needed density of points will depend on the geology and the type of terrainAreas where geology allows for free groundwater flows across large areas will have less local variation and need less dense points, while areas with geologic features that inhibit or redirect flow (e.g. karst topography) will need denser points

Where interpolation does not work

Introduction to GISCannot use interpolation where values are not spatially autocorrelatedSay looking at household incomein an income-segregated city, you could take a small sample of households for income and probably interpolateHowever, in a highly income-integrated city, where a given block has rich and poor, this would not work

Sampling Approaches

Introduction to GISOften a regular gridded sampling strategy is appropriate and can eliminate sampling biasesSometimes, though, it can introduce biases if the grid pattern correlates in frequency with something in the landscape, such as trees in a plantation or irrigation linesRandom sampling can avoid this but introduces other problems including difficulty in finding sample points and uneven distribution of points, leading to geographic gaps. This depends partially on the size of the support, or sampling unit

Sampling Approaches

Introduction to GISAn intermediate approach is the stratified random sampleCreate geographic or non-geographic subpopulations, from each of which random sample is takenProportional or equal probability SRS: enforce a certain sampling rate, hj= nh/Nh for each stratum h and obs j. Simple SRS: enforce a certain sample size nhDisproportionate SRS: where hj varies such that certain strata are oversampled and certain undersampled.

Sampling Approaches

Introduction to GISDSRS is advantageous when subpopulation variances are unequal, which is frequently the case when stratum sizes are considerably different. In DSRS we sample those strata with higher variance at higher rate. We may also use this when we have an underrepresented subpopulation that will have too few observations to model if sampled with SSRS. Proportional samples are self-weighting because the rates are the same for each stratumThe other two have unequal sampling probabilities (unless a simple SRS has equal Nh) and may require weighting

Sampling Approaches

Introduction to GISWhen the stratifying unit is geographical (e.g. county, soil polygon, forest stand), this is called a cluster sample. In a one stage cluster sample (OSCS) a series of geographic units are sampled and all observations within are sampled: obviously this does not work for interpolationMore relevant is a two stage cluster sample (TSCS) in which we take a sample of cluster units and then a subsample of the population of each cluster unit. In this type of sample, variance has two components, that between clusters and that between observations

Sampling

Introduction to GISThe number of samples we want within each zone depends on the statistical certainty with which we want to generate our surfaceDo we want to be 95% certain that a given pixel is classified right, or 90% or 80%?Our desired confidence level will determine the number of samples we need per strataThis is a tradeoff between cost and statistical certaintyThink of other examples where you could stratify.

Sampling

Introduction to GISA common problem with sampling points for interpolation is what is not being sampled?Very frequently people leave out sample points that are hard to get to or hard to collect data atThis creates sampling biases and regions whose interpolated values are essentially meaninglessSo spacing of sample points from interpolation should be based on some meaningful factorif they are dense in a region in sparse in a region, it should be because the values are variable in the first area and homogeneous in the other

Sampling

Introduction to GISExample: lets say want to make an average precipitation layer and we find that in our study zone precipitation is highly spatially variable within 10 miles of the ocean Wed a coastline layer to help us sample.Wed have high density of sampling points within 10 miles of the ocean a much lower density in the inland zones

Sampling

Introduction to GISSay we were looking at an inland area, far from any ocean, and we decided that precipitation varied with elevation. How would we set up our sampling design?In this case, flat areas would need fewer sample points, while areas of rough topography would need moreIn our sampling design we would set up zones, or strata, corresponding to different elevation zones and we would make sure that we get a certain minimum number of samples within each of those zonesThis ensures we get a representative sample across, in this case, elevation;

Sampling

Introduction to GISThe number of zones we use will determine how representative our sample is; if zones are big and broad, we do not ensure that all elevation ranges are represented

Sampling and Scale dependency

Sampling strategy for interpolation depends on the scale at which you are working and the scale dependency of the phenomenon you are studyingIn many cases interpolation will work to pick up regional trends but lose the local variation in the process The density of sample points must be chosen to reflect

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