1 1 Master of Science in Geospatial Master of Science in Geospatial Technologies Technologies Geostatistics Geostatistics Predictions with Predictions with Anisotropy and Simulations Anisotropy and Simulations Instituto Superior de Estatística e Gestão de Informação Universidade Nova de Lisboa Carlos Alberto Felgueiras Carlos Alberto Felgueiras [email protected][email protected]
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Master of Science in Geospatial Master of Science in Geospatial TechnologiesTechnologies
GeostatisticsGeostatisticsPredictions withPredictions with
Anisotropy and SimulationsAnisotropy and Simulations
Instituto Superior de Estatística e Gestão de InformaçãoUniversidade Nova de Lisboa
Carlos Alberto FelgueirasCarlos Alberto [email protected]@isegi.unl.pt
Master of Science in Geoespatial Technologies
ContentsStochastic Predictions with anysotropy
Introduction
Unidirectional Semivariogram – Fitting with only one model
Unidirectional Semivariogram – Fitting with nested models
Isotropy x Anisotropy
Anisotropy types – Geometric, Zonal and Combined
Modeling Anisotropic Semivariogram
Simulations and Gaussian Simulation
Problems with Geostatistic Estimators
Advantages on using geostatistics
Summary and Conclusions
Exercises
Predictions with Anisotropy and SimulationsPredictions with Anisotropy and Simulations
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Master of Science in Geoespatial Technologies
• Unidirectional Semivariograms – Fitting with only one model
Represent spatial variability of the attribute in one specific direction
Predictions with Anisotropy and SimulationsPredictions with Anisotropy and Simulations
• Introduction
Experimental Semivariogram(from samples)
[ z(ui) − z(ui + h)]22N(h)
1 ∑i = 1
N(h)γ(h) = ^
Theorical (Modeled) Semivariogram (fitted from the experimental semivariogram
using only one model)
( )
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡−⋅+=
⎟⎠⎞
⎜⎝⎛⋅+=
⎟⎟⎠
⎞⎜⎜⎝
⎛−
a10
10
eCC
aCC
h
hh
1
Expγ
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Master of Science in Geoespatial Technologies
• Unidirectional Semivariograms – Fitting with Nested Models
Predictions with Anisotropy and SimulationsPredictions with Anisotropy and Simulations
• Introduction
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪
⎨
⎧
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
>++
≤<γ=−+
≤<γ=−+
=γ
2210
21222
20
1111
10
0
a||,CCC
a||a,)(a||
3
21
a||
23CC
a||0,)(a||
3
21
a||
23CC
C,0
)(
h
hhhh
hhhh
h
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Master of Science in Geoespatial Technologies
Predictions with Anisotropy and SimulationsPredictions with Anisotropy and Simulations
• 2 semivariograms with same model function, different sills and ranges
• it can also have different nugget effects, but is not common
1010
1111
Master of Science in Geoespatial Technologies
O
Predictions with Anisotropy and SimulationsPredictions with Anisotropy and Simulations
( ) ( )
( )
( )( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛∞
⋅+−++
⎟⎟⎠
⎞⎜⎜⎝
⎛⋅−++
⎟⎟⎠
⎞⎜⎜⎝
⎛⋅−+=
2
2111011202
2
2
1
1021101
2
1
1010201
,Exp
,Exp
,Exp
aCCCC
aaCCC
aCCC
hh
hh
hhhε
γ
Where:
C01 is the nugget effect of the variogram 1 and C11 is the contribution of the variogram 1
C02 is the nugget effect of the variogram 2 and C12 is the contribution of the variogram 2
h1 is the module of the vector h in the direccion of variogram 1 ( 300 for example)
h2 is the module of the vector h in the direccion of variogram 2 (1200 for example)
O
300
1200
h1
h2
h
(0,0)
• Modeling Anisotropic Semivariogram – defining a resulting semivariogram from the two perpendicular unidirectional variograms
Master of Science in Geoespatial Technologies
Predictions with Anisotropy and SimulationsPredictions with Anisotropy and Simulations
• Modeling Anisotropic Semivariogram – Example in the laboratory
1200300
( ) ⎟⎠⎞
⎜⎝⎛⋅+=
804.961Exp880.194843.6 30
30hhγ
( ) ⎟⎠⎞
⎜⎝⎛⋅+=
548.674Exp084.190106.1 120
120hhγ
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Master of Science in Geoespatial Technologies
Predictions with Anisotropy and SimulationsPredictions with Anisotropy and Simulations
• Modeling Anisotropic Semivariogram – Example in the laboratory
( )
⎟⎠⎞
⎜⎝⎛
∞+⎟
⎠⎞
⎜⎝⎛
+⎟⎠⎞
⎜⎝⎛+=
1203012030
12030
,804.961
*533.10548.674
,804.961
*347.184
548.674,*637.5106.1
hhExphhExp
hhExpε
γ h
COMBINATION
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Master of Science in Geoespatial Technologies
Predictions with Anisotropy and SimulationsPredictions with Anisotropy and Simulations
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• Kriging prediction
Summary
Master of Science in Geoespatial Technologies
Predictions with Anisotropy and SimulationsPredictions with Anisotropy and Simulations
• Kriging prediction – isotropic x anisotropic modeling
(a) (b)
Anisotropy angles 170
and 1070
Examples of evaluation of the means values by kriging considering (a) isotropic and (b) anisotropic spatial variations 1515
Master of Science in Geoespatial Technologies
687.3
909.9
2.97
23.0
Predictions with Anisotropy and SimulationsPredictions with Anisotropy and Simulations
• Kriging prediction – estimate means and variance of the estimates
Maps of kriging means and kriging variances1616
Master of Science in Geoespatial Technologies
Predictions with Anisotropy and SimulationsPredictions with Anisotropy and Simulations
• Simulations – allows to get realizations from a stochastic model representing a Random Variable or a Random Field.
• Gaussian Simulation - Using the hypotheses that the mean and the variance (or standard deviation) evaluated by kriging are parameters of gaussian distributions one get (at each location for example) the following distribution equation (and graph):
( ) ( )[ ]2
21
21 σµ
πσ−−
=z
ezf
z
f(z)
If the distribution is normalized µ=0 and σ=1
( )π2
22zezf−
=
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Master of Science in Geoespatial Technologies
z
• N realizations of each RV Z are obtaining repeating n times the steps:1. Generating a random number
between 0 and 1 (cp - cumulative probability value)
2. Mapping the cp to the z value using the Gaussian cdf defined by the given µz and σz parameters.
• Problem: How can I prove (or verify) the hypothesis that the distribution in each estimated location follows a Gaussian (Normal) distribution?
cpk
zk
Predictions with Anisotropy and SimulationsPredictions with Anisotropy and Simulations
• Simulations – the process of getting realizations of the Gaussian distributionUses the cumulative distribution function (cdf) and a random number generator.
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Master of Science in Geoespatial Technologies
Predictions with Anisotropy and SimulationsPredictions with Anisotropy and Simulations
• Problems with geostochastic procedures
The main drawback of using geostatistic approaches is the need of work on variogram generations and fittings. This work is interactive and require from the user knowledge of the main concepts related to basics of the geostatistics in order to obtain reliable variograms.
The kriging approach is an estimator based on weighted mean evaluations and is uses the hypothesis of minimizing the error variance. Because of these the kriging estimates create smooth models that can filter some details of the original surfaces.
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Master of Science in Geoespatial Technologies
Predictions with Anisotropy and SimulationsPredictions with Anisotropy and Simulations
• Advantages on using geostochastic procedures
• Spatial continuity is modeled by the variogram
• Range define automatically the region of influence and number of neighbors
• Cluster problems are avoided
• It can work with isotropic and anisotropic phenomena
• Allows prediction of the Kriging variance
• Allows simulating ( get realizations from) random variables with normal distributions.
2020
Master of Science in Geoespatial Technologies
Summary and ConclusionsSummary and Conclusions
Summary and Conclusions
• Geostatistic estimators can be used to model spatial data.
• Geostatistics estimators make use of variograms that model the variation (or continuity) of the attribute in space.
• Geostatistics advantages are more highlighted when the sample set is not dense
• Current GISs allow users work with these tools mainly in Spatial Analysis Modules.
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Master of Science in Geoespatial Technologies
Predictions with Anisotropy and SimulationsPredictions with Anisotropy and Simulations
Exercises
• Run the Lab4 that is available in the geostatistics course area of ISEGI online.
• Find out if the variation of your attribute is isotropic or anisotropic. Model the anisotropy if it exists.
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Master of Science in Geoespatial Technologies
Predictions with Deterministic ProceduresPredictions with Deterministic Procedures