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GEOESTADÍSTICA PARA EXPLORACIONES

ProEXPLO 2013

Instituto de Ingenieros de Minas del Perú

Lima, Perú

17 y 18 de Mayo, 2013

Mario E. Rossi, MSc. Geoestadística, Ing. de Minas.

GeoSystems International, Inc.

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ESTIMACIÓN LINEAR

� Estimates are often made as weighted linear estimators. A common approach is to estimate the values as deviations from a mean or trend surface. The estimate reverts to the mean some distance away from the data, see the far right edge. The deviations from the mean surface are estimated at unsampled locations with some method of interpolation. The most common interpolation scheme is a weighted linear estimate:

� Where the * denotes an estimate, u0 denotes the unsampled location being estimated, z(·) denotes the variables value, m(·) denotes the mean or trend value and i=1,...,n is the index of data values.

� The weights λi,i=1,...,n are often constrained to be positive and to sum to one, but these constraints are not necessary and may be suboptimal.

[ ]*0 0

1

( ) ( ) ( ) ( )n

i i ii

z m z mλ=

− = −∑u u u ui

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ESTIMACIÓN LINEAR (Cont.)

� Nearest neighbor or polygonal weighting would amount to setting the weight to the closest data equal to 1.0 and all other weights to 0.0.

� Inverse distance is an estimation scheme that provides the weights according to the Euclidean distance.

� Kriging is a technique that calculate weights that are optimal according to specific criteria.

� Although estimation schemes sometimes provide a measure of how good the estimates are, there is no measure of uncertainty or risk in the estimates. Probabilistic estimation or conditional simulation is needed for this.

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ESTIMACIÓN LINEAR (Cont.)

� Certain estimator properties are intuitively desirable:� The closer the sample to the location being estimated, the

greater the weight it should receive.

� Two samples close together (clustered) should receive less weight (data redundancy).

� The weights should change from one location to the next.

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INVERS0 DE LA DISTANCIA

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COMENTARIOS

� A mayor ω, mayor ponderación reciben las muestras más cercanas.

� Si ω=0, todas las muestras reciben la misma ponderación (1/n), esto es, la estimación se reduce a un promedio aritmético.

� Si ω�∞, la muestra más cercana recibe toda la ponderación. Este es el método del “vecino más cercano”.

� Una variante del método del Inverso de la Distancia se puede implementar para que las ponderaciones tengan en cuenta la anisotropía de las muestras.

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ALGUNOS PROBLEMAS DE IMPLEMENTACIÓN

� La elección de ω es arbitraria!

� La distancia cartesiana (|x – x’|) no está relacionada con la variable que se estudia.

� El método no incorpora el concepto de redundancia de la información.

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KRIGING

� General name given to a group of minimum error variance algorithms.

� Kriging allows the estimation of conditional expectations through linear estimates (same as ID methods!).

� Provides a set of weights under the least-squares criterion.

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KRIGING (Cont)

� The main controls on the kriging weights are:� Closeness of the data to the location being estimated;� Redundancy between the data; and,� The variogram.

� There are many different types of kriging.

� The most important implicit assumptions are stationarity (partially resolved with the use of local search neighborhoods and ordinary kriging), and ergodicity (bit more of a problem).

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KRIGING ORDINARIO

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UNBIASEDNESS CONDITION

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UNBIASEDNESS CONDITION (Cont.)

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MINIMIZING THE ERROR VARIANCE

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MINIMIZING THE ERROR VARIANCE (Cont.)

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KRIGING ORDINARIO (Cont.)

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KRIGING ORDINARIO (Cont.)

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PROPERTIES OF KRIGING

� Existence and uniqueness of the solution is ensured if the matrix [C(xα,xβ)]is positive definite.

� The variance attached to kriging is smaller than any other.

� The kriging estimator is (globally) unbiased. So are all others for which the sum of weights equal 1.

� Kriging is an exact interpolator: for all x = xα, then λ α = 1, and z*(x) = z(xα).

� Kriging takes into account:� The geometry of the block being estimated through γ(V,V); the bigger V, the

smaller σ2K.

� The structural distance information, γ(x- xα); � The configuration of the data, γ(xα - xβ); � Other spatial correlation features, captured with the right variogram model γ(h).

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PROPERTIES OF KRIGING (Cont)

� Kriging estimates are additive; if V = ∑iVi, then Z*K,V = ∑iZ*

K,Vi. Thus, a sum of kriging estimates is a kriging estimate.

� Two blocks of the same size and same data configuration have the same kriging weights.

� Kriging is just a different name for the well-known Normal Equations.

� Smoothing is always present. All weighted-average estimators have this characteristic.

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KRIGING SIMPLE

SIMPLE KRIGING (1)

� Consider a linear estimator:

� where Y(ui) are the residual data (data values minus the mean) and Y*(u) is the estimate (add the mean back in)� The error variance is defined as

)u(Y)u(Y i

n

1ii

* ⋅⋅⋅⋅λλλλ==== ∑====

}2* )]u(Y)u(Y{[E −−−−A2-2ab+b2

})]({[ 2* uYE )}()({2 * uYuYE ⋅⋅⋅⋅⋅⋅⋅⋅−−−− })]({[ 2uYE++++

)}()({1 1

jij

n

i

n

ji uYuYE ⋅⋅⋅⋅∑∑

==== ====λλλλλλλλ )}()({2

1∑

====⋅⋅⋅⋅⋅⋅⋅⋅−−−−

n

iii uYuYEλλλλ )0(C++++

),(1 1

jij

n

i

n

ji uuCλλλλλλλλ∑∑

==== ====∑

====⋅⋅⋅⋅−−−−

n

iii uuC

1

),(2 λλλλ )0(C++++

SIMPLE KRIGING (2)

� Optimal weights λi,i=1,…,n may be determined by taking partial derivatives of the error variance w.r.t. the weights

and setting them to zero

� This system of n equations with n unknown weights is the simple kriging (SK) system

∑====

====⋅⋅⋅⋅−−−−λλλλ⋅⋅⋅⋅====λλλλ∂∂∂∂

∂∂∂∂ n

1jijij

i

n,...,1i,)u,u(C2)u,u(C2][

∑====

========λλλλn

1jijij n,...,1i,)u,u(C)u,u(C

SIMPLE KRIGING (3)

� There are three equations to determine the three weights:

� In matrix notation: (Recall that C(h}) = C(0) - γ( h))

)3,0(C)3,3(C)2,3(C)1,3(C

)2,0(C)3,2(C)2,2(C)1,2(C

)1,0(C)3,1(C)2,1(C)1,1(C

321

321

321

====⋅⋅⋅⋅λλλλ++++⋅⋅⋅⋅λλλλ++++⋅⋅⋅⋅λλλλ====⋅⋅⋅⋅λλλλ++++⋅⋅⋅⋅λλλλ++++⋅⋅⋅⋅λλλλ

====⋅⋅⋅⋅λλλλ++++⋅⋅⋅⋅λλλλ++++⋅⋅⋅⋅λλλλ

====

λλλλλλλλλλλλ

)3,0(C

)2,0(C

)1,0(C

)3,3(C)2,3(C)1,3(C

)3,2(C)2,2(C)1,2(C

)3,1(C)2,1(C)1,1(C

3

2

1

γ 1,2

γ 2,3

γ 0,3

γ 0,2γ 0,1

γ 1,3

SIMPLE KRIGING (4)

Simple kriging with a zero nugget effect and an isotropic spherical variogram with three different ranges:

1 2 3

range = 10 0.781 0.012 0.0655 0.648 -0.027 0.0011 0.000 0.000 0.000

λλλλ λλλλ λλλλ

Changing the Range

Distance

γγγγ

range = 1

range = 5 range = 10

SIMPLE KRIGING (5)

Simple kriging with an isotropic spherical variogram with a range of 10 distance units and three different nugget effects:

1 2 3

nugget = 0% 0.781 0.012 0.06525% 0.468 0.203 0.06475% 0.172 0.130 0.053

100% 0.000 0.000 0.000

λλλλ λλλλ λλλλ

Distance

γγγγ

100%75%

nugget = 25%

Changing the Nugget Effect

SIMPLE KRIGING (6)

Simple kriging with a spherical variogram with a nugget of 25%, a principal range of 10 distance units and different “minor” ranges:

1 2 3

anisotropy 1:1 0.468 0.203 0.0642:1 0.395 0.087 0.1415:1 0.152 -0.055 0.232

20:1 0.000 0.000 0.239

λλλλ λλλλ λλλλ

Changing the Anisotropy

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KRIGING UNIVERSAL

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KRIGING UNIVERSAL (Cont.)

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KRIGING UNIVERSAL (Cont.)

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KRIGING UNIVERSAL (Cont.)

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OK and UK SYSTEMS

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Suavizamiento de Kriging

Smoothing Effect of Kriging

� Kriging is locally accurate and smooth, appropriate for visualizing trends, inappropriate for process evaluation where extreme values are important

� The “variance” of the kriged estimates is too small:

� σ2 is complete variance� σ2

SK(u) is zero at the data locations � no smoothing� σ2

SK(u) is variance σ2 far away from data locations � total smoothing

� spatial variations of σ2SK(u) depend on the variogram and data

spacing

� Missing variance is the kriging variance σ2SK(u)

( ){ }* 2 2SKVar Y σ σ= −u

Smoothing Effect of Kriging

� The idea of simulation is to correct the variance and get the right variogram

where R(u) corrects for the missing variance.

� Simulation reproduces histogram, honors spatial variability (variogram),� appropriate for process evaluation

� Allows an assessment of uncertainty with alternative realizations

� We will see simulation later - recognize that the smoothing effect of kriging can be quantified

Kriging with Anisotropic Variogram

� Sph30(h) for isotropic and Sph50,10(h) in bottom two

� Anisotropy is used in calculation of kriging weights

� Kriged estimates are still smooth

� Anisotropy is not as pronounced

Some Comments on Kriging

� “Kriged values should never be posted on a map.” A. G. Journel, 1984� the joint variability of kriged estimates is incorrect� calculated as a point-wise best estimator

� Kriging honors data with a discontinuity (yes, kriging is an exact interpolator, but …)

� Kriging weights do not depend on data values:� trends are extrapolated with negative weights� kriging variance does not depend on data values