Adding or removing new sites to assess changes in the kriging standard error

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08-27-2019 07:31 PM
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New Contributor

Hi,

Sorry, I was wondering whether somebody might be able to help provide a brief summary of how to calculate potential improvements in the kriging prediction error by adding or removing sample sites. I would like to use Geostatistical Analyst to optimize my monitoring network. The first step I would like to do is calculate the average standard kriging error for a range of different sized networks so I can create something called a Network Density Graph (attached). From this graph I plan on selecting the optimum number of wells and then using the Densify Sample Network tool to locate my site.

Kind regards,

Simon

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Esri Regular Contributor

Hi Simon,

There are probably multiple ways to do this, but if I were going to do it myself, I would do the following:

  • Make sure the Output Type of your kriging model is set to Standard Error of Prediction.
  • Export the geostatistical layer to a raster using GA Layer To Rasters (or GA Layer To Grid in ArcMap), and use Get Raster Properties tool to find the average value of the standard error raster.  This will be the first value in your network density graph.
  • Using your kriging model, run Densify Sampling Network, and choose to only create a single new point. This will create a new point at the location of the largest standard error.  You can use the "Input weight raster" parameter to define your study area so that the new point will not be created outside of it (give a weight of 0 or NoData to all cells outside of the study area).
  • Merge the output of Densify Sampling Network and the original data that was used to create the kriging model into a new dataset, and map the "Value" field into the field that you used to interpolate (this value and the StdErr field come by interpolating the value of the new location).  This creates a new dataset with all of the original measurements and one new value.  
  • Use Create Geostatsitical Layer to create a new geostatistical layer for the merged dataset containing the newly created point.  Provide the original kriging model as the model source.  This will create a new geostatistical layer with the new dataset that uses the same parameters as the original kriging model.
  • Export the new geostatistical layer to a standard error raster.  Calculate the new average standard error with Get Raster Properties, and write the average to your network density graph.
  • Repeat the last 5 steps (starting with Densify Sampling Network) as many times as you need in order to get the average standard error beneath whatever threshold you need.  Make sure in the Create Geostatsitical Layer step to use the original model (without any of the invented points) for every iteration; do not use the model from the previous iteration.

The workflow is a bit long, but it can be automated.  It boils down to sequentially creating a new point at the location of the largest standard error within the study area, merging its predicted value into the original dataset, then recalculating the new average standard error, then repeating the process until the average standard error is smaller than some value you specify.

Please let me know if any of this is unclear or if you have any other questions.

-Eric

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Esri Regular Contributor

Also, if you want to experiment with the location of the new points rather than have them selected at the location of the largest standard error, just create the new point (or multiple new points) anywhere you want in place of running Densify Sampling Network.  You should then use GA Layer To Points to predict the value of the new points.  Merge these values into the original data before recalculating the new standard errors.

You alluded to wanting to do something like this in this post:

Optimising monitoring networks using Kriging 

Similarly, if you want to add, say, 5 new points at a time between calculating the average standard error, just specify 5 new points in Densify Sampling Network, and follow the same workflow.

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New Contributor

Hey Eric,

 Thanks for the reply again.  One question - why do you need to have predicted values for the new points? Sorry, sounds like a silly question but I thought the kriging standard error is independent of the measured value.  I'm noticing that with some of these kriging methods the value makes a difference.

Thanks again,

Simon 

 

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Esri Regular Contributor

If you employ any kind of transformation (Normal Score, logarithmic, etc), then the standard errors and the predictions will be dependent, and you will get different standard errors for different values you assign to the new point.  There is no perfect methodology for assigning a value to the new points, but interpolating the value from the original measured points is what is done in practice (this is what Densify Sampling Network does automatically).  Using GA Layer To Points just ensures that the value will be justifiable no matter the kriging model.

The issue about predictions and standard errors being independent can definitely be confusing, as you'll often see statements along the lines of "the predicted values of ordinary kriging are independent of the standard errors," with no qualifying statements or hints that there is more to the story.  While the statement is technically true, it is very easy to misunderstand due to terminology.  You might read that statement and assume that you can use Ordinary Kriging with a logarithmic transformation, and the standard errors will be independent of the predictions, but they won't be.  The problem is that the actual name for ordinary kriging with a log transformation is "Ordinary Lognormal Kriging", not just "Ordinary Kriging."  So the previous statement about predictions and standard errors being independent for ordinary kriging models was not meant to apply to ordinary lognormal kriging models.

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New Contributor

Hi Eric,

 

 That response was very useful - thanks for your explanation.  

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