An overview of Geostatistical Concepts & Examples : An overview of Geostatistical Concepts & Examples Lecture 9
Geostatistics : Geostatistics Geostatistics combines practical conceptual thoughts that facilitate the modeling of spatial variability with mathematical and statistical methods.
It is rigorous and has the ability:
analyze and integrate different types of spatial data
measure spatial autocorrelation by incorporating the statistical distribution
measure spatial relationships between the sample data
perform spatial prediction
assess uncertainty.
Geostatistics predicts the value of unsampled locations from the observed nearby samples by the defined relationships.
Geostatistics vs. Classical Statistics : Geostatistics vs. Classical Statistics Geostatistics assumes
there is spatial autocorrelation of a random function consisting of random variables spatially distributed in a 2-dimensional space
data values of a random function at different locations are spatially auto-correlated with each other.
Classical statistics assumes there is no spatial autocorrelation of a random variable, that is, data values of a random variable at different locations are independent.
Regionalized variables In geostatistics, the random variables are called regionalized variables.
the closer the locations of the data, the more similar the data values.
the similarity becomes weaker as the separation distance of data locations increases and
disappears when the distance reaches a certain value called range.
Geostatistics (Example) : Geostatistics (Example) Lets suppose we want to measure variables like rainfall and temperature
It can be possible through the meteorological stations located at specified locations.
But it is impossible to put monitoring stations everywhere.
Therefore we will establish spatial relationships between the known values of our observed locations and use these relationships too make predictions at unobserved locations.
****Geostatistics will play a role here****
Regionalized variables : Regionalized variables A variable that takes on values according to its spatial location is known as a regionalized variable.
Considering a variable z measured at location i, we can partition the total variability in z into three components:
z(i) = f(i) + s(i) + ε
where f(i) is some coarse-scale forcing or trend in the data,
s(i) is local spatial dependency, and
ε is error variance (presumed normal).
Regionalized variables : Regionalized variables
Regionalized variables : Regionalized variables Regionalized variables are variables that fall between random variables and completely deterministic variables.
Typical regionalized variables are functions describing variables that have geographic distributions
Example: elevation of ground surface).
Unlike random variables, regionalized variables exhibit spatial continuity
the change in the variable is so complex that they cannot be described by any deterministic function.
The variogram is used to describe regionalized variables
Variograms (Basic Concepts) : Variograms (Basic Concepts) Variogram: A visual exploratory tool for characterizing the spatial continuity of the variable.
Sill: the plateau that the variogram reaches;
in the variogram context it is the average squared difference between paired data values and it is approximately equal to twice the variance of the data
Range: The distance at which the variogram reaches the sill.
Nugget Effect: The vertical height of the discontinuity at the origin. It is the combination of:
(1) short-scale variations that occur at a scale smaller than the closest sample spacing; and
(2) sampling error due to the way the samples were collected, prepared, and analyzed.
Variograms (Basic Concepts) : Variograms (Basic Concepts) Kriging: The process of fitting the best linear unbiased estimate of a value at a point or of an average over a volume.
Isotropic (semi)variogram: This is when the spatial pattern is identical in all directions.
In this case, the fitting of the semivariogram model will heavily depend on the (Euclidean) distance between locations.
Anisotropic (semi)variogram: This is when the spatial pattern is strongly biased towards a specific direction.
This phenomenon is also at times referred as directional variograms because the weighting scheme depends on distance and direction.
Variograms : Variograms Nugget Range Structure Sill = nugget + structure
Variograms (Basic Concepts) : Variograms (Basic Concepts) In mathematical terms, the semi-variogram:
Where h represents a distance vector. h h h
Variograms : Variograms
Variograms (ArcGIS –Geostatistic Analysts) : Variograms (ArcGIS –Geostatistic Analysts)
Variograms : Variograms Statistical assumptions:
Stationary—mean and variance are not a function of location. Second-order stationary is required—variance is a function of the separation distance.
Isotropy—no directional trends occur in the data (as contrasted with anisotropy).
However, you can compute directional variograms in order to assess directional trends in the data.
Use of trend surface analysis to remove global trends in the data (to transform a non-stationary variable [mean varies across space] to a stationary one).
Lag distances – typically we group the distance intervals into classes so that we can have enough sample points within any one distance class (typically 30 is suggested as the minimum number).
Small-scale (high resolution) variation (at the resolution implied by the original sampling scheme) may not be detectable as a result.
Variograms : Variograms The technique can provide:
a quantification of the scale of variability exhibited by natural patterns of resource distributions and
an identification of the spatial scale at which the sampled variable exhibits maximum variance.
At larger lag distances harmonic effects can be noted, in which the variogram peaks or dips at lag distances that are multiples of the natural scale.
Given the noise present in natural environmental data sets, it is unlikely that you will be able clearly to identify multiple scales.
One approach might be to fit a semivariogram model to the data, and
to examine the residuals for the presence of multiple patterns of scale.
Variograms : Variograms
Variograms : Variograms
Variogram models : Variogram models
Kriging : Kriging Kriging is a spatial interpolation technique based on semi-variograms.
Unlike every other spatial interpolation technique, kriging provides a map that shows you the uncertainty associated with the prediction.
Kriging : Kriging ? Sample data z(uα) at uα Cell u to be estimated Neighborhood used
to estimate cell u
Kriging : Kriging Kriging produces the best linear unbiased estimate of an attribute at an unmeasured site, once the variogram has been modeled.
Ordinary kriging: used when there is no drift in the data.
Universal kriging accounts for drift (in ArcGIS drift is modeled by a constant, linear, second or third order equation).
Punctual kriging: produces values for non-sampled points.
Block kriging: produces values for areas instead of points. Estimates for blocks have lower variance because several point values are averaged to get the estimated value for one block. This averaging smoothes the small scale fluctuations of the function [Z(x)] over the area of the block.
Co-kriging: uses 2 or more variables that are correlated between themselves in the estimation of values for one of them (e.g: soil bulk density and soil water content).
Geostatistics : Geostatistics Geostatistical analysis is highly useful for accounting for the small population problem and to solve the spatial prediction (will accurately predict better local estimates) and analysis
The main basis of geostatistical analysis is the regionalized variable theory.
A geostatistical analysis must be properly implemented following a solid knowledge of mathematical and statistical methods.
References & Examples of application : References & Examples of application Goovaerts, P. 1997. Geostatistics for Natural Resources Evaluation. Oxford University Press.
Wang, G., T. Oyana, M. Zhang, S. Adu-Prah, S. Zeng, H. Lin, and J. Se. 2009. Mapping and spatial uncertainty analysis of forest vegetation carbon by combining national forest inventory data and satellite images. Forest Ecology and Management 258(7):1275-1283.
Wang, G., G.Z. Gertner, H. Howard, and A.B. Anderson. 2008. Optimal spatial resolution for collection of ground data and multi-sensor image mapping of a soil erosion cover factor. Journal of Environmental management 88:1088-1098.
Wang, G., G.Z. Gertner, and A.B. Anderson. 2007. Sampling and mapping a soil erosion relevant cover factor by integrating stratification, model updating and cokriging with images. Environmental Management. 39(1):84-97.
Oyana, T.J., (2004). Statistical comparisons of positional accuracies of geocoded databases for use in medical research. In Egenhofer M, Freksa C, and Miller H. (eds.): In Proceedings of the Third International Geographic Information Science, GIScience 2004, October 20–23, 2004. Regents of the University of California: pp.309–313.
Robertson, G.P. (1987). Geostatistics in ecology: interpolating with known variance. Ecology, 68(3):744–748.
Yarus, J.M. and Chambers, R.L. (2006). Practical geostatistics—An armchair overview for petroleum reservoir engineers. Distinguished Author Series, JPT, Society of Petroleum Engineers