interpolation kriging

Download Interpolation Kriging

Post on 14-Apr-2018

219 views

Category:

Documents

0 download

Embed Size (px)

TRANSCRIPT

  • 7/28/2019 Interpolation Kriging

    1/38

    Kriging

    Using Geostatistical Analyst, ESRI

    Chang, Kang-tsung, 2006, Introduction toGeographic Information Systems,

  • 7/28/2019 Interpolation Kriging

    2/38

    So Far

    We dealt with deterministic models that

    can not provide estimates foraccuracy/certainty in predictions.

    Kriging is a stochastic model that provides

    estimates for accuracy/certainty in

    predictions.

  • 7/28/2019 Interpolation Kriging

    3/38

    Kriging

    Spatial variation consists of 3 components

    Random spatially correlated component

    A drift or structure/trend

    Random error term

  • 7/28/2019 Interpolation Kriging

    4/38

    How to measure the spatially

    correlated component?

    Uses Semivariogram to measure spatially

    correlated component, also called spatial

    autocorrelation

  • 7/28/2019 Interpolation Kriging

    5/38

    Creating a Semivariogram?

    Where:

    = semivariance between point Xi and Xj

    h = distance separating Xi and Xj

    Z = attribute value (height, ore quality, etc)

    First what is semi-variance between two point separated by distance h?

  • 7/28/2019 Interpolation Kriging

    6/38

    Semivariance between one point

    and all other points

    If I calculate all possible

    semivariance for point A

    (red dot), I get 11 points

    A

  • 7/28/2019 Interpolation Kriging

    7/38

    Repeat for Semivariance

    calculations for every possible pair

    Much more semivariance

    computations if we repeat

    for all possible pairs

  • 7/28/2019 Interpolation Kriging

    8/38

    If all pairs are considered, we get a

    Semivariogram cloud

    Semivariogram allows us to investigate spatial dependence

    If there is spatial dependence points that are closer together will have smallsemi-variance and vice versa

    Because all pairs are plotted on the semivariogram, the semivariogram

    becomes difficult to manage/interpret

    Distance

  • 7/28/2019 Interpolation Kriging

    9/38

    Binning is the solution

    A process that is used to average semivariance data by distance and

    direction

    First group pairs of sample points into lag classes. If lag size is 2000

    meters, lag classes are: (

  • 7/28/2019 Interpolation Kriging

    10/38

    Continue

    Third: For each group of pairs with similar lag and direction (in thesame grid cell), we compute the average semivariance between

    sample points separated by lag h

    Where: = average semi-variance between sample points separated by lag h

    N = is the number of pairs of sample points sorted by direction in the bin

  • 7/28/2019 Interpolation Kriging

    11/38

    Semivariogram after binning

    On a semivariogram, if spatial dependence exist, semivariance

    is expected to increase with distance

    Same lag, different directions

  • 7/28/2019 Interpolation Kriging

    12/38

    Anisotropy

    Anisotropy is a term describing the existence of directional differences in spatialdependence

    From semivariogram data, directional dependence can be extracted if it exists

    low

    Plume

    Pollutant concentration high

    Wind direction

  • 7/28/2019 Interpolation Kriging

    13/38

    Continue

    Fourth: To use the semivariogram as an interpolator, the semivariogramdata must be fitted with a mathematical function/model

  • 7/28/2019 Interpolation Kriging

    14/38

    Semivariogram

    Measures the variability of data with

    respect to spatial distribution Looks at variance between pairs of data

    points over a range of separation scales Forms the basis of every geostatistical

    study

  • 7/28/2019 Interpolation Kriging

    15/38

    Semiveriogram conditions:

    Stationarity

    The entire dataset can be described with one statistical model

    Under the condition of stationarity, you can see the distance ofcorrelation in your data

    h

    (h)

    Correlated at any distance

    Correlated at a max distance

  • 7/28/2019 Interpolation Kriging

    16/38

    Semiveriogram types

    Many functions to choose from; Geostatistical analyst provides 11 models

    Examples include: Gaussian, Linear, Spherical, Circular, and Exponential

    Popular models are the spherical and exponential

    Spatial dependencelevels after a certain

    distance

    Spatial dependenceDecreases exponentially

  • 7/28/2019 Interpolation Kriging

    17/38

    Nugget (C0

    ):

    Semivariance

    at distance 0: Represents microscale

    variations or measurement errors

    Range (a) : Distance at which semivariance

    levels off: Represents the spatially correlated portion

    Sill (C0

    + C1

    ):

    Semivariance

    at which leveling takes place

    N.B.: Every data set will have unique model features (model parameters)

    C0

    C1

    C0 &C1

    (h)a

    Semivariogram: Model Parameters

  • 7/28/2019 Interpolation Kriging

    18/38

    Fifth: Extract model parameters (C0, C1, and a) for the dataset under

    consideration from the semivariogram. This will allow us to calculate amodeled semivariance between any 2 points knowing the distance hseparating the two points.

    Extract model parameters

    C0

    C1

    C0 &C1

    (h)a

  • 7/28/2019 Interpolation Kriging

    19/38

    Continue

    Sixth: Use the model parameters to extract semivariance values for anypoint by solving a set of simultaneous equations

    Let us consider the following Problem:

    Knowing three points: 1, 2, 3, we want to use krigging to estimate point 0

  • 7/28/2019 Interpolation Kriging

    20/38

    How?

    Where:

    Problem: Knowing three points: 1, 2, 3, we want to estimate

    semivariance values and solve equations to determine value at

    point 0

    (hij) is the semivariance between known point I & J

    (hi0) is the semivariance between known point I & point to be estimated

    Lagrange multiplier to ensure minimum estimation of errorMore like a fudge factor

    (hij) is the semivariance between known point I & J

    (hi0) is the semivariance between known point I & point to be estimated

    Lagrange multiplier to ensure minimum estimation of errorMore like a fudge factor

    (hij) is the semivariance between known point I & J

    (hi0) is the semivariance between known point I & point to be estimated

    Lagrange multiplier to ensure minimum estimation of errorMore like a fudge factor

  • 7/28/2019 Interpolation Kriging

    21/38

    Thus

    Derive semivariance between known pairs of known

    points (1,2,3) and each of the known point and unknown

    point knowing model parameters

    Derive weights solving by solving the simultaneous

    equations

    Plug in weights in the equation below to calculate thevalue of the unknown point (Z0)

    Z0

    = Z1

    W1

    + Z2

    W2

    + Z3

    W3

  • 7/28/2019 Interpolation Kriging

    22/38

    What if we had > 3 points

    (1) Apply Matrix algebra to solve simultaneous Equations

  • 7/28/2019 Interpolation Kriging

    23/38

    = Estimated value

    = Known value at point x

    = Weight associated with point x

    S = number of sample points used in estimation

    Where:

    (2) Estimator equation becomes

    What if we had > 3 points

  • 7/28/2019 Interpolation Kriging

    24/38

    How is this different from Inverse

    Distance MethodKrigging IDW

    Calculation of weightsinvolve

    (1) Variance between point to beestimated and Known points

    (2) Variance between Known points

    (1) Variance between pointto be estimated and

    Known points

    Measuring reliability A measure for the reliability of the

    estimate

    None

  • 7/28/2019 Interpolation Kriging

    25/38

    Kriging Types in ArcGIS

    Ordinary kriging

    Simple kriging Universal kriging

    Indicator kriging Probability kriging

    Disjunctive kriging Cokriging

  • 7/28/2019 Interpolation Kriging

    26/38

    From before

    Spatial variation consists of 3 components

    Random spatially correlated component

    A trend component

    Random non correlated component(error term)

  • 7/28/2019 Interpolation Kriging

    27/38

    Components of Kriging

    The value of z depends on: (1) trend component, (2) random autocorrelatedcomponent, and (3) random non-correlated component (for simplicity notrepresented in figures)

    Z(s) = (s) + (s)Where:

    Z = Value at point s

    = Trend component value at point s (first order or second order polynomial)

    = Random, autocorrolated component

  • 7/28/2019 Interpolation Kriging

    28/38

    Ordinary Kriging

    Assumes there is no trend

    Assumes m(s) is unknown and constant

    Focuses on the spatially correlated component

  • 7/28/2019 Interpolation Kriging

    29/38

    Simple Kriging

    Assumes (s) , the mean of data set is known and is constant

    Assumes there is no trend component

    In the majority of cases this is unrealistic assumption

  • 7/28/2019 Interpolation Kriging

    30/38

    Indicator Kriging

    (s) is constant, and unknown

    Values are binary (1 or 0)

    Example, a point is forest or non forest

  • 7/28/2019 Interpolation Kriging

    31/38

    Universal Kriging

    Assumes z values change because of a drift (tre

View more