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  • Basics in Geostatistics 2Geostatistical interpolation/estimation:

    Kriging methods

    Hans Wackernagel

    MINES ParisTech

    NERSC April 2013

    http://hans.wackernagel.free.fr

    http://hans.wackernagel.free.fr

  • Basic concepts

    Geostatistics

    Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 2 NERSC April 2013 2 / 35

  • Concepts

    Geostatistical model

    The experimental variogram is used to analyzethe spatial structure of the data from aregionalized variable z(x).It is fitted with a nested variogram model, thusproviding the structure function of a randomfunction.The regionalized variable (reality) is viewed asone realization of the random function Z(x),which is a collection of random variables.

    Kriging: a linear regression method for estimatingpoint values (or spatial averages) at any locationof a region.

    Conditional simulation: simulation of an ensemble ofrealizations of a random function,conditional upon data for non-linear estimation.

  • Concepts

    Geostatistical model

    The experimental variogram is used to analyzethe spatial structure of the data from aregionalized variable z(x).It is fitted with a nested variogram model, thusproviding the structure function of a randomfunction.The regionalized variable (reality) is viewed asone realization of the random function Z(x),which is a collection of random variables.

    Kriging: a linear regression method for estimatingpoint values (or spatial averages) at any locationof a region.

    Conditional simulation: simulation of an ensemble ofrealizations of a random function,conditional upon data for non-linear estimation.

  • Concepts

    Geostatistical model

    The experimental variogram is used to analyzethe spatial structure of the data from aregionalized variable z(x).It is fitted with a nested variogram model, thusproviding the structure function of a randomfunction.The regionalized variable (reality) is viewed asone realization of the random function Z(x),which is a collection of random variables.

    Kriging: a linear regression method for estimatingpoint values (or spatial averages) at any locationof a region.

    Conditional simulation: simulation of an ensemble ofrealizations of a random function,conditional upon data for non-linear estimation.

  • Stationarity

    For the top series:

    stationary mean m and covariance function C(h)

    For the bottom series:

    mean and variance are not stationary,

    actually the realization of a non-stationary processwithout drift.

    Both types of series can be characterized with a variogram.

  • Kriging of the mean

    Kriging of the mean of a random function

    Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 2 NERSC April 2013 5 / 35

  • Spatially Correlated Data

    Sample locations x in a spatial domain:

    With spatial correlation we need to consider that:

    each sample point plays a different role in estimating themean of the spatial domain,

    distances to neighboring points play a role.

    How should samples thus be weighted in an optimal way?

  • Estimation of the Mean Value

    Using the arithmetic mean:

    M? =1n

    n=1

    Z(x)

    all samples get the same weight:1n

    We rather need an estimator:

    M? =n

    =1

    w Z(x)

    with weights w reflecting the spatial correlation.

  • Stationary random function

    We assume translation-invariance of the mean:

    x D : E[Z(x)

    ]= m

    and of the covariance:

    x,x D with xx = h : cov(Z(x),Z(x)) = C(xx) = C(h)

  • Unbiased estimator

    The estimation error in our statistical model:

    M? estimated value

    m true value

    should be zero on average:

    E[M? m

    ]= 0

    No bias: the estimator M? does not on average yield avalue that is different from m.

  • No biasBias is avoided using weights of unit sum:

    n=1

    w = 1

    Consider:

    E[M? m

    ]= E

    [ n=1

    w Z(x)m]

    =n

    =1

    w E[Z(x)

    ]

    m

    m

    = mn

    =1

    w 1

    m = 0

  • Variance of the estimation error

    The variance of the estimation error is:

    2E = var(M? m) = E

    [(M? m)2

    ] E

    [M? m

    ]2

    0

    = E[M?2 2mM? +m2

    ]=

    n=1

    n=1

    ww E[Z(x)Z(x)

    ]

    2mn

    =1

    w E[Z(x)

    ]

    m

    +m2

    2E =n

    =1

    n=1

    ww C(x x)

  • Minimal estimation variance

    Aim: weights w that produce a minimal estimation variance:

    minimize 2E subject ton

    =1

    w = 1

    The objective function has n+1 parameters:

    (w1, . . . ,wn, ) = var(M? m)

    criterion

    2( n=1

    w 1)

    condition

    where is a Lagrange multiplier.

    Setting partial derivatives to zero, we obtain n+1 equations:

    : (w1, . . . ,wn, )w

    = 0,(w1, . . . ,wn, )

    = 0

  • Kriging equations

    The method of Lagrange yields the equation system forthe optimal weights wKM of the estimation of the mean:

    n=1

    wKM C(x x) KM = 0 for = 1, . . . ,n

    n=1

    wKM = 1

    The variance at the minimum:

    2KM = KM

    is equal to the Lagrange multiplier.

  • Special case: no spatial correlation

    When the covariance model is only nugget-effect:

    C(x x) ={2 if x = x0 if x 6= x

    the kriging of the mean system simplifies to:wKM

    2 = KM for = 1, . . . ,nn

    =1

    wKM = 1

    The solution weights are all equal: wKM =1n

    M? = 1n

    n=1

    Z(x) with variance KM = 2KM =1n

    2

  • Special case: no spatial correlation

    When the covariance model is only nugget-effect:

    C(x x) ={2 if x = x0 if x 6= x

    the kriging of the mean system simplifies to:wKM

    2 = KM for = 1, . . . ,nn

    =1

    wKM = 1

    The solution weights are all equal: wKM =1n

    M? = 1n

    n=1

    Z(x) with variance KM = 2KM =1n

    2

  • Ordinary Kriging

    Ordinary Kriging

    Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 2 NERSC April 2013 15 / 35

  • Estimation at an unsampled location

    Sample locations x (blue dots) in a spatial domain D:

    x0

    Aim: estimate Z? at an unsampled location x0.

  • Ordinary kriging

    The estimate Z? is a weighted average of data values Z(x):

    Z?(x0) =n

    =1

    w Z(x) withn

    =1

    w = 1

    The weights wOK of the Best Linear Unbiased Estimator (BLUE)are solution of the system:

    n=1

    wOK (xx) + OK = (xx0) for = 1, . . . ,n

    n=1

    wOK = 1

    Minimal variance: 2OK = OK +n

    =1

    wOK (xx0)

  • Ordinary kriging

    The estimate Z? is a weighted average of data values Z(x):

    Z?(x0) =n

    =1

    w Z(x) withn

    =1

    w = 1

    The weights wOK of the Best Linear Unbiased Estimator (BLUE)are solution of the system:

    n=1

    wOK (xx) + OK = (xx0) for = 1, . . . ,n

    n=1

    wOK = 1

    Minimal variance: 2OK = OK +n

    =1

    wOK (xx0)

  • Kriging weights

    The behavior of kriging weights

    Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 2 NERSC April 2013 18 / 35

  • Geometric anisotropySpherical variogram, weights sum up to 100%

    Isotropic model:

    L

    25%

    25%

    25%

    25%

    Anisotropic (ranges horizontal 1.5L, vertical .75L):

    L

    17.6%

    32.4%

    17.6%

    32.4%

  • Geometric anisotropySpherical variogram, weights sum up to 100%

    Isotropic model:

    L

    25%

    25%

    25%

    25%

    Anisotropic (ranges horizontal 1.5L, vertical .75L):

    L

    17.6%

    32.4%

    17.6%

    32.4%

  • The screen effectSpherical variogram with range 2L

    Left sample is at 1L and right at 2L from target 2OK = 1.14

    A B

    34.4%65.6%

    Introducing an extra sample 2OK = 0.87

    A

    49.1%

    C

    48.2%

    B

    2.7%

    Adding the sample C screens off the sample B.

  • The screen effectSpherical variogram with range 2L

    Left sample is at 1L and right at 2L from target 2OK = 1.14

    A B

    34.4%65.6%

    Introducing an extra sample 2OK = 0.87

    A

    49.1%

    C

    48.2%

    B

    2.7%

    Adding the sample C screens off the sample B.

  • Case-study

    Filtering noisy images by cokriging

    Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 2 NERSC April 2013 21 / 35

  • Application: metallography

    Trace elements are usually masked by instrumental noise.

    Data: 512512 image of phosphorus (P) trace elements.Images for chrome (Cr) and (Ni) are less noisy.

    Geostatistical filtering is used to remove the noise.

  • Structural analysis

    Image of phosphorus Nested variogram(h) = 384 nug(h) + 75 exp(h) + 13 |h|

  • Filtering the nugget-effect

    Raw image of phosphorus Filtered image

  • Multivariate data

    P Cr Ni

  • Multivariate structural analysisDirect and cross variograms

    Matrix variogram model: G(h) = B0 nug(h) + B1 exp(h) + B1 |h|

  • Filtering the nugget-effectPhosphorus

    Filtered by kriging Filtered by cokriging

  • Case study

    Space-time filteringSguret & Huchon, JGR 1990

    Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 2 NERSC April 2013 28 / 35

  • Earth magnetism

    Magnetic anomalies are essential to study earth history.

    Magnetism is influenced by several external factors like:

    solar wind explaining daily fluctuations(period: 24 hours)rotation of the moon around the earth(period: 28 days)solar pe

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