Lab: Spatial Interpolation

The main idea of spatial interpolation is described in  Tobler's First Law of Geography  as: "Everything is related to everything else, but near things are more related than distant things". Translating 'related' into 'similar', we understand that spatial autocorrelation is a prerequisite for interpolation.

Spatial continuity is a frequent assumption for spatial interpolation, with either values for specific locations of interest (e.g. my current position) or continuous surface representations over areas of interest / study areas being estimated. 

Exercises in this lab are specified for the ArcGIS Online platform and/or ArcGIS Pro. You will find a broad range of options and parameters controlling the results of spatial interpolation – try to explore these and compare the sometimes very different results you obtain. Using alternative tools outside the ESRI domain is encouraged and should not pose any major obstacles as long as services with sample data can be accessed or replaced with local data sets.

All of Europe will serve as the stage for the following exercises, thus most of you will be on somewhat familiar terrain, 'geographically speaking'.

A sample data set from the ' Global Summary Of the Day ' (GSOD) service will be used as our demo data set throughout the following exercises (for your exercises, get your own!). From stations across the continent, we can work with precipitation totals and average daily temperatures (in °C) for May 1st, 2006. Explore this data set by clicking on points and read values in the popup.

Preparing for further analyses, start a New Map in ArcGIS Online, choose e.g. 'Light Gray Canvas' as a base map and add the layer 'World Countries (Generalized)' from the ArcGIS Living Atlas. As a next step, select a country of your choice and extract its outline from this data set.

One approach for the latter step is to 'Perform Analysis' on the 'World Countries (Generalized)' layer, 'Find Locations' / 'Find Existing Locations', then add an expression where Country IS Austria (or the country of your choice) and give it a layer name like 'AustriaGeneralized'. Subsequently, we will use the chosen country as a study area for interpolation.

For interpolation exercises, we will access the above mentioned GSOD data. Add our sample data set from layer GSOD20060501 to your current map, to be used in most subsequent maps as well.

ps, other GSOD and related data are available via    https://www.ncei.noaa.gov/weather-climate-links#gsod  

The ESRI Academy is offering several options for software training and related background knowledge related to spatial interpolation. Recommendations for useful courses are:

•  Introduction to Surface Modeling Using ArcGIS  
•  Exploring Spatial Patterns in Your Data Using ArcGIS  
•  Performing Spatial Interpolation Using ArcGIS   

(In particular the latter course is focussed on the Geostatistical Analyst Extension and significantly more advanced than the content of this unit).

Whether estimating values at a single target location or over a study area from our sample data, we distinguish several fundamental categories of interpolation methods:

• Local vs Global: relying on sample points in the vicinity of the target location, or use all available sample points (=global).
• Exact vs Approximative: an interpolated surface exactly reproduces values at sample locations, or is 'just getting close'.
• Interpolation vs Extrapolation: estimate values in between, or beyond the scope (e.g. bounding box or convex hull) of sample points.
• Linear vs Non-linear: type of function used for interpolation.
• ...

These and other basic concepts are discussed in more detail in the lecture component of this class. 

In most cases, we will achieve best results with a local interpolation technique, and this requires the choice of a suitable 'neighbourhood', a decision which and how many nearby sample points will be used to estimate a value at a target location.

A basic question is estimating a value at any location of interest. Staying with the previously set up map in ArcGIS Online, search (box in the top right corner) for a place or city you are interested in, here 'Attnang' is used. This is a small town in Austria not represented by a station in our GSOD data set.

Before continuing with discussing and performing spatial interpolation in a stricter sense, let us briefly consider the case of using only one closest sample point (the one with Near Rank = 0) for 'interpolating' a study area from a set of sample points.

With the creation of 'Thiessen polygons' (via ArcGIS Pro) data from all sample points have been transposed into their respective Thiessen ('nearest neighbourhood') polygons. By filtering for e.g. 'PREC is not 0' a map like the one on the right is created - click for popups from sample points as well as Thiessen polygons to verify the zones which experienced any amount of precipitation on this day.

As no average or other metric calculation takes place when assigning a station value toits surroundings, this 'interpolation' method is an adequate approach for a point-to-area transformation of nominal variables.

Create a similar map yourself by adding the layer 'GSODpThiessen', apply the filter, and then go one step beyond by creating a map showing different amounts of precipitation on this particular day !

Of course a precipitation or temperature map of any study area usually will not stop at the nearest-neighbour Thiessen 'interpolation' we explored right now, but we return to our original objective of creating a continuous surface or field representation across a chosen study area by area interpolation.

Every grid point or raster cell in a chosen study area now serves as target location, and a surface over the the study area is completed with a moving window process. 

For each point / cell a neighbourhood definition is applied (e.g. by using the closest 15 sample points, and then an estimate for the target point/cell is calculated through a weighted average of the sample points' temperature values. Weights are set as inverted distances (the closer to the target location, the higher the weight):

This graphic and formula demonstrate the simplest case of Inverse Distance Weighting (= 'IDW'). In a more general case the d _i  are elevated to a power, controlling the degree of smoothing across the surface. Higher power values increase the weight of nearby points, lower power values give more emphasis to distant points within the defined neighbourhood.

In this exercise, we create a daily average temperature map for Austria, for May 1st 2006. Start a New Map, choose the Light Grey Canvas basemap, and add the layers World Countries (Generalized) and our daily weather data set GSOD20060501.

For the next step in this exercise, we need to launch ArcGIS Pro. Sign In from the File menu, choose the Light Grey Canvas base map and Add the worldwide country outlines, your chosen country as well as the GSOD data set. 

Next, open the ArcToolbox, go to Spatial Analyst Tools and select the Interpolation toolset. Open the IDW dialog and explore the fields to be completed. Follow the example at the right when completing the dialog, and make sure that at the bottom of the dialog you open 'Environments ...' and under Raster Analysis you specify the country mask.

After inspecting the initial results, return to the IDW dialog, and experiment with different neighbourhood (search radius) and power settings. Always make sure that the country mask is active to avoid long processing times. Find colour schemes and cartographic classification settings you can justify. 

You will have observed that here we generate a raster data set with a 5000m resolution (you can of course change it - think about why to choose any particular value - ?) ; whereas in the previous section we had created polygonal zones with ArcGIS Online. In this raster case we can change the classification anytime, while with polygonal results the interpolation process would have to be run again for different class breaks.

As opposed to IDW as a 'local' interpolation method, sometimes it is of interest to consider all sample stations within a study area, even at equal weighting.

One example for such an approach is Trend Surface analysis, which is essentially a twodimensional case of regression analysis as we know it from statistics. An ordinary least squares fit for a polynomial surface is made, which, by the way, is also an example for an approximative interpolator.

The example at the right shows a temperature trend surface (third order polynomial) across central parts or Europe. Not very surprisingly, there is a north > south warming trend, while the cooler higher elevations of the Alps are not highlighted and likely will show up as significant residuals.  

You can reproduce this and similar results via the ArcGIS Pro  Trend  tool. 

For anyone aiming to get a bit more serious about solving interpolation problems in the ESRI ArcGIS environment, exploring the  Geostatistical Analyst toolbox  is a strong recommendation. 

This extension offers a core set of interpolation techniques around the geostatistical Kriging approaches, but its exploratory data analysis, visualisation and calibration functions are applicable for all other interpolation methods.

Areas for enhancing results include:

• considering directional characteristics of sampling or phenomena
• quality control and outlier detection
• inclusion of external controls (like e.g. elevation effects through co-kriging)
• ...

Consider these tasks as examples and suggestions, and try to create your own problem specifications:

• Create a temperature map for one European country in ArcGIS Online using the data sets introduced in this lab, carefully choosing presentation details like classification and symbols. Discuss your decisions and choices in the map description before you share the map.
• In ArcGIS Pro, use the GSOD (or any other) data set and create two IDW interpolation results with different power value settings. Compare the two results (including quantitative statistics) and discuss the different outcomes.

Briefly discuss how you suggest to approach quality assessment of interpolation results. You e.g. create an IDW output based on 12 vs 18 closest sample points - how can you judge one to be the preferable option?