Geisa Bugs. Trabalho final da disciplina de geoestatistica na UNL, 2007.

1. Geostatistics
Master of Science on Geospatial Technologies
Professor: Carlos A. Felgueiras
Geisa Bugs
Model elevation attribute with geostatistical procedures
This article reports the procedures, problems and results of the geostatistics proposed
exercise. The goal is to create a model with a free chosen data in order to understand how
the data is distributed and how is it spatial dependency across the study area.
1. Presentation of the spatial data to be used
The study area is composed of seven census sectors (urban area and surroundings) of the
São Gabriel municipality, located in the South Brazilian state of Rio Grande do Sul at
30° 20’9’’ latitude south and 54°19’12’’ longitude west. The whole city has an area of
5.019,646 km2 and the selected study area has an area of 618,15 km2.
For the modeling exercise was used a set of 189 elevation sample points originated from
the cartographic service of Brazilian military army’s in the scale 1:50.000. The exercise
used ArcGIS and SPRING softwares.
Figure 1: São
Gabriel location.
Figure 2: Sample points location.
2. Exploratory analysis
During the exploratory analysis graphical and visual methods and numerical techniques
are used to look for patterns in the data, look for outliers, and formulate hypothesis from
data about the symmetry for example.
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2. • Descriptive Statistic
Sample points 189
Minimum 9
Maximum 286
Mean 146,96
Standard deviation 50,739
Skewness 1,0259
Kurtosis 3,5291
st
1 Quartile 114
Median 129
rd
3 Quartile 164,5
Global variance 2560,86
Table 1: exploratory analysis.
The mean and median are measures of location and show where the center of distribution
lies. Values of the two approximately equal suggest a possible normal distribution of the
data. In the elevation data they are not much, and that may be because an erratic value is
affecting the mean value.
The standard deviation describes the variability. Skewness and kurtosis are measures of
shape and show information about symmetry. The three coefficients are also sensitive to
erratic values, since take into account the mean value.
• Histogram: a histogram plots the range of data values (x-axis) against the number
of points that have those values (y-axis).
Histogram
Transformation: None
Frequency Count : 189 Skewness : 1,0259
59 Min :9 Kurtosis : 3,5291
Max : 286 1-st Quartile : 114
Mean : 146,96 Median : 129
Std. Dev. : 50,739 3-rd Quartile : 164,5
47,2
35,4
23,6
11,8
0
0,09 0,37 0,65 0,93 1,21 1,49 1,77 2,05 2,33 2,61 2,89
-2
Data 10
Data Source:
Layer: sg_topo_pontos_cotados_Clip selection
Attribute: CODIGO
Figure 3: histogram of the elevation points.
The histogram shows that the data are almost symmetric, but there is a very low and
isolate value. The more long tail in the right indicates a bigger concentration of high values
in comparison with low values.
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3. The sequence of figures bellow helps to see where these values are located. By visual
analysis it is clear that the low values are mainly located at the north part of the study are,
the values that occur more often are mainly located in the central area and the higher
values are located mainly in the south part of the study area. By selecting the very low
value it shows that it is one value located in between the lower and the values that occur
more often. It may be considered an outlier.
Figure 4: shows the location of the values that occur more often.
Figure 5: shows the location of the higher values.
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4. Figure 6: shows the location of the lower values.
Figure 7: show the location of the lower point value.
• Normal Probability Graph: helps to see how close the distribution is to a Gaussian.
The graph bellow shows that the elevation data is almost close to a Gaussian behavior but
not actually. The points almost follow the normal line but there is an upper dispersion to
the low values, a “down belly” dispersion to the values close to the mean, and an upper
dispersion to the high values.
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5. Normal QQPlot
Transformation: None
-1
Data's Quantile 10
29,15
23,5
17,85
12,2
6,55
0,9
-2,79 -2,23 -1,67 -1,11 -0,55 0,01 0,57 1,13 1,69 2,25 2,81
Standard Normal Value
Data Source:
Layer: sg_topo_pontos_cotados_Clip selection
Attribute: CODIGO
Figure 8: the normal probability graph.
3. Experimental omnidirectional semivariogram
The semivariogram allows examining the spatial autocorrelation between the measured
sample points, besides of define the necessary parameters to do the estimations values
for not sampled areas. The principle of spatial autocorrelation tells that pairs of locations
that are close in distance should also be close in value.
In the semivariogram cloud of ARCGIS each red dot represents a pair of locations.
Figure 09: ArcGIS semivariogram cloud in 90° direction.
In order to improve the semivariogram evaluation and actually be able to see the “curve” it
was run also in SPRING.
When analyzing the SPRING semivariogram results it was clear to see that the data
presents trend because it was not found a stabilized behavior. In a usual behavior as the
distance between the point pairs increases, the semivariogram values also increase till
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6. stabilizes and reach a landing. In the elevation data the semivariograms values increased
continually, not showing a defined land, indicating the presence of trend in the data.
Figure 10: SPRING semivariogram 90° direction.
4. Taking off the data trend
In this step it was used a program to take off the data trend and the semivariogram
creation was tested again. The results now are very satisfactory. The sill value is
approximately the global variance value for the sample without trend (174); the range is
sensible smaller comparing to the previous results; and the nugget effect is really close to
zero.
The sill, range and nugget effect are the important parameters a semivariogram shows. Sill
represents variability in the absence of spatial dependency; in a typical behavior of an
adjusted semivariogram, the sill value is approximately like the global variance. Range
represents separation between point pairs at which the sill is reached, in other words, the
distance at which there is no evidence of spatial dependency.
Figure 11: semivariogram and numeric result for the points without trend.
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7. 5. Theoretical semivariograms
Modeled semivariograms are mathematical models representing the experimental
semivariograms. The goal is to find the best fit for the variogram (lowest fitting error). By
comparing the spherical and gaussian models, the range, sill, and contribution are very
simmilar but when looking to the akaike effect, the gaussian model suggested a little better
fitting.
Figure 12: spherical modeled semivariogram.
Figure 13: gaussian modeled semivariogram.
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8. 6. Kriging
Let’s go back to ArcGIS after had the parameters set in SPRING. By observing the surface
plot the data can be assumed as anisotropic as shows different behaviors for different
directions; the data values along certain direction are more continuous than along others.
So ArcGIS offers to calculate it automatically.
The first step is to define which interpolation method will be used. In this case it was used
the ordinary kriging stochastic interpolator. Stochastic interpolators use weighted averages
and probability models to make predictions to the points there is no samples values.
Ordinary kriging assumes that the mean is a simple constant, what means no trend on the
data. So similarly to what was done to take the trend on the data before, in ArcGIS it was
selected a second order polynomial trend removal. The input parameters were the ones
figured out from the modeled semivariogram for the point without trend made in SPRING:
number of lags: 9, nugget effect: 43, sill value: 131, model: spherical, range: 2.665.
Figure 14: ordinary kriging with anisotropy.
Figure 15: ordinary kriging with isotropy.
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9. 7. Cross validation
In the cross validation procedure one datum is removed and the rest of the data are used
to predict the removed datum.
The cross validation graphs bellow shows that the result is very satisfactory, almost all the
points are following the line, and the root mean squared is more or less 16 and it
represents only a 5% error when comparing with the range of the data. When comparing
the anisotropy and isotropy the RMS is a little slower for the isotropy.
Figure 16: cross validation - anisotropy.
Figure 17: cross validation - isotropy.
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10. Figure 18: cross validation comparison: on the left anisotropy and on the right isotropy.
8. Graph result
The graph result shows the ordinary kriging map for the elevation data from the seven
census areas of Sao Gabriel municipality. The kriging works with the hypotesis od
minimazing the error variance and creates a smooth model that can filter some details of
the original surface. The difference between isotropy and anisotropy is quite small but still
it is better the anisotropy result because is considering this difference in directions.
Figure 19: anisotropy. Figure 20: isotropy.
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11. 9. Conclusions
Different softwares offer different possibilities for data prediction using geostatistics
analysis. It is realy important to know the theory a priori in order to acept or not the
automatic parameters given by the softwares, and also to input the ones that fits better to
the expected results.
Important points to keep in mind is to observe if there is trend in the data. If there is trend
is very usefull to take the trend off in order to improve the semivariogram creation and the
result parameters used to input in the modeled semivariogram and in the interpolation
procedure. Also important to also observe if it is a isotropic or anisotropic behavior. If there
is a anisotropy is needed to combine the two information (directions) in one model.
Finally, the interactive work of generation reliable variograms really request a previous
knowledge of geostatistical concepts. If there is no previous knowledge is quite difficult to
find good results.
10. References
Isaaks, E. H. and Srivastava, R. M., 1989, An introduction to applied geostatistics, new York: Oxford
University Press.
ESRI training and education, 2007, Introduction to ArcGIS 9 Geostatistical Analyst. Avaiable online
in: http://training.esri.com/acb2000/showdetl.cfm?DID=6&Product_ID=808 (last acessed 11
November 2007).
GPS Global, 2007, Artigos geostatística. Avaiable online in:
www.gpsglobal.com.br/Artigos/Geoestat.html (last acessed 11 November 2007).
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