2.4. Elements of geostatistics

   Geostatistics is a collection of statistical methods which were traditionally
   used in geo-sciences. These methods describe spatial autocorrelation among sample
   data and use it in various types of spatial models. Geostatistical methods were
   recently adopted in ecology (landscape ecology) and appeared to be very useful in
   this new area.

   Geostatistics changes the entire methodology of sampling. Traditional sampling
   methods don't work with autocorrelated data and therefore, the main purpose of
   sampling plans is to avoid spatial correlations. In geostatistics there is no
   need in avoiding autocorrelations and sampling becomes less restrictive. Also,
   geostatistics changes the emphasis from estimation of averages to mapping of
   spatially-distributed populations.

   Spatial autocorrelation can be analyzed using correlograms, covariance functions
   and variograms (=semivariograms). For simplicity, here we will use correlograms
   only. Covariance functions and variograms are discussed in the [1]next lecture.

   In brief, geostatistical analysis usually has the following steps:
    1. Estimation of correlogram
    2. Estimation of parameters of the correlogram model
    3. Estimation of the surface (=map) using point kriging, or
    4. Estimation of mean values using block kriging

   Detailed description of most geostatistical methods can be found in Isaaks and
   Srivastava (1989). Here we will discuss only the most important elements of
   geostatistics.

Estimation of Correlogram

   Correlogram is a function that shows the correlation among sample points
   separated by distance h. Correlation usually decreases with distance until it
   reaches zero. Correlogram is estimated using equation:

                                    [eqcorrel.gif]

   where z[1] and z[2] are organism numbers in two samples separated by lag distance
   h, summation is performed over all pairs of samples separated by distance h; N[h]
   is the number of pairs of samples separated by distance h; M[h] and s[h] are the
   mean and the standard deviation of samples separated by distance h (each sample
   is weighted by the number of pairs of samples in which it is included).

   Notes:
    1. This is an omnidirectional correlogram; "omnidirectional" means that we don't
       care about the direction of lag h.
    2. Serious geostatistical analysis often includes estimation of directional
       correlograms (see [2]next lecture). It may happen that points are more
       closely correlated in some direction (e.g., NE-SW) than in other directions.
       If correlogram depends on direction, then the spatial pattern is called
       anisotropic. If no anisotropy detected, then it is possible to use the
       omnidirectional correlogram.
    3. Correlogram equation works only if there are no trends in population density
       in the study area. If a trend exists, then a non-ergodic correlogram should
       be used instead (see next lecture).

Estimation of parameters of correlogram model

   The correlogram can be approximated by some mathematical model. Two models are
   used most often:

   1. Exponential model:

                                     [eqexpon.gif]

   2. Spherical model:

                                     [eqspher.gif]

   where c[1] is sill, and a is range. These parameters can be found using the
   non-linear regression.

Estimation of the surface (=map) using point kriging (ordinary kriging)

   The value z'[o] at unsampled location 0 is estimated as a weighted average of
   sample values z[2] at locations i around it:

                                     [eqkrig1.gif]

   Weights depend on the degree of correlations among sample points and estimated
   point. The sum of weights is equal to 1 (this is specific to ordinary kriging):

                                     [eqkrig2.gif]

   Weights are estimated individually for each point in a regular spatial grid using
   the system of linear equations:

                                     [eqmatr.gif]

   where [eqmu.gif] is the Lagrange parameter; [eqro.gif] is the correlation between
   points i and j which is estimated from the variogram model using distance, h,
   between points i and j; 0 is the estimated point; 1,...,n are sample points.

   Using matrix notation this system can be re-written as:

                                     [eqkrig3.gif]

   The solution of this matrix equation is:

                                     [eqkrig4.gif]

   Now, weights are found, and thus, it is possible to estimate the value z'[o].
   When these values are estimated for all points in a regular grid, then we get a
   surface of population density.

   The variance of local estimation [eqkrig5.gif] is equal to: [eqkrig6.gif]

Estimation of the mean value using block kriging

   The only difference of block kriging from point kriging is that estimated point
   (0) is replaced by a block. Consequently, the matrix equation includes
   "point-to-block" correlations:

                                     [gblockr.gif]

   Point-to-block correlation [eqro2.gif] is the average correlation between sampled
   point i and all points within the block (in practice, a regular grid of points
   within the block is used, as shown in the figure).

   The variance of block estimation [eqkrig5.gif] is equal to: [eqkrig7.gif]

   Where [eqro1.gif] is the average correlation within the block (average
   correlation between all pairs of grid nodes within the block).

   Advantages of kriging:
     * It handles spatial autocorrelation
     * It is not sensitive to preferential sampling in specific areas
     * It estimates both: local population densities and block averages.
     * It can replace stratified sampling if the size of aggregations is larger than
       the inter-sample distance.

References

   Isaaks, E. H. and R. M. Srivastava. 1989. An Introduction to Applied
   Geostatistics. Oxford Univ. Press, New York, Oxford. (a very good introductory
   textbook)
   Deutsch, C. V. and A. G. Journel. 1992. GSLIB. Geostatistical software library
   and user's guide. Oxford Univ. Press, Oxford. (software code written in FORTRAN;
   I have translated a portion of this library into the C-language).

   Geostatistics in ecology (review papers):
   Rossi, R. E., D. J. Mulla, A. G. Journel and E. H. Franz. 1992. Geostatistical
   tools for modelling and interpreting ecological spatial dependence. Ecol. Monogr.
   62: 277-314.
   Liebhold, A. M., R. E. Rossi and W. P. Kemp. 1993. Geostatistics and geographic
   information systems in applied insect ecology. Annu. Rev. Entomol. 38: 303-327.

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   [6]Alexei Sharov 1/12/96

References

   1. http://alexei.nfshost.com/PopEcol/lec3/geostat.html
   2. http://alexei.nfshost.com/PopEcol/lec3/geostat.html
   3. http://alexei.nfshost.com/PopEcol/lec2/howmany.html
   4. http://alexei.nfshost.com/PopEcol/lec2/sampling.html
   5. http://alexei.nfshost.com/PopEcol/lec2/stratif.html
   6. http://alexei.nfshost.com/~sharov/alexei.html