Optimization of sample configurations for variogram estimation

Optimization of sample configurations for variogram estimation*

Alessandro Samuel-Rosa(1), Gerard Heuvelink(2),Gustavo Vasques(3), Lúcia Anjos(1)

* Presented at Pedometrics 2015, 14 – 18 September 2015, Córdoba, Spain(1) Universidade Federal Rural do Rio de Janeiro, Seropédica, Brazil.(2) ISRIC – World Soil Information, Wageningen, the Netherlands.(3) Embrapa Soils, Rio de Janeiro, Brazil. Student

Presentation

Optimization of sample configurations for variogram estimation

Alessandro Samuel-Rosa, Gerard Heuvelink, Gustavos Vasques, Lúcia Anjos

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The problem at hand

● The variogram is a key tool in modern soil mapping

● How to design a sample to identify the form of the variogram model and estimate its parameters?

Z (s)=m(s)+ϵ(s)

NuggetPartial sill

Range

SmoothnessExponential

SphericalGaussian

Matérn

CircularAnisotropy



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What are the existing solutions?

● “Simple” designs– Regular squared grids are still commonly used

● “Less complex” designs (1980s)– Number of point-pairs per lag (Warrick and Myers, 1987)

– Commonly produces a large cluster of points

● “More complex” designs (>1990s)– Minimize the uncertainty of estimated parameters

– Require the form of the variogram to be known



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eSample configurations optimized aiming at the minimization of the uncertainty of the estimated variogram parameters

Image kindly provided by Murray Lark

Known (or assumed) spherical model!



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We came up with an idea!

● Maximize the quantity of information carried by every sample point– Maximum likelihood estimators (MLE)

● Accurately estimate the variogram intercept– The nugget variance is in the spotlight

– Concentrate on the short separation distances

● “Ideal” sample configuration– Multiple small clusters spread out across the area



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A reasonable solution would be to...

● Define an objective function aiming at the distribution of the number of unique Points Per Lag

where li* is the wanted distribution and li is the observed

distribution of points per lag, and w is a vector of weights, with i = 1, 2, …, n, n being the number of lags.

● Goal: to have each point contributing to every lag

PPL=∑i=1

nwi(li

∗−li)



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Defining the lags

● Exponential spacings● Circumradius of the

bounding box of the spatial domain

● Sequential halving● Seven lags



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Testing

● How well do we estimate the variogram parameters?– Unconditional sequential simulation of isotropic

Gaussian random fields● Nugget/sill (0.1, 0.5, 0.9) and range/extent (0.01, 0.1, 1.0)

– Random, systematic, point-pairs per lag, points per lag● Three samples with three sizes (50, 100, 200)

– Estimate the variogram parameters using REML (geoR)

NuggetPartial sill

RangeExponential



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Random fields and sample configurations

n = 200



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Preliminary results – Nugget

● Best: points per lag and point-pairs per lag– Regardless of the sample size

● More accurate with moderate to long range– More samples for short range

● Random and systematic: too few points in the first lags– Accuracy of systematic sample increased with size

Nugget (n = 50)

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Preliminary results – Partial sill

● Somewhat difficult– More than three replicates to get consistent conclusions

– Tuning of the optimizer

● Accuracy increased with sample size– For all sample configurations

– Moderate range (50 units)

– Systematic sampling

Partial sill (n = 100)

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Preliminary results – Range

● Short to moderate (5 and 50 units)– Higher accuracy as expected

● More accurate as the sample size increases– Regardless of the sample configuration

● Small sample size, moderate to low nugget– Points per lag had the highest accuracy

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Final considerations

● Accurate estimation of the nugget variance– Distribution of points or point-pairs per lag

● Remember that these are “suboptimal”● Directional constraints are not included

● Points per lag and small samples● Systematic sampling is efficient with large samples● More than three samples for consistent conclusions



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More info

● This presentation is available at

– slideshare.net/alessandrosamuelrosa● Source code:

– github.com/samuel-rosa/spsann● R-package (on CRAN):

– spsann (optimization of sample configurations using SPatial Simulated ANNealing)



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Now we want to hear you!

● Critiques● Questions● Comments● Suggestions

Contact: alessandrosamuelrosa at gmail.com

Optimization of sample configurations for variogram estimation

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