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Geog418SptStts > Geostatistics and Kriging > Flashcards

Geostatistics and Kriging Flashcards

1
Q

Geostatistics

A
  • Applied branch of statistics that deals with spatial properties
  • Ex. Treat problems that arise when conventional statistical theory is used in estimating changes in ore grade w/in a mine
  • Deals w/ problems of spatial autocorrelation
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2
Q

Finish the sentence: data is positively correlated with a correlation that…

A

decreases as distance between data increases

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3
Q

Regionalized variable’s are?

A
  • Continuous from location to location (unlike random variables), but changes are too complex to be described by deterministic function
  • Spatially continuous and values are only known at samples, taken at specific locations
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4
Q

Regionalized Variable Def’n

A

A variable that has intermediate properties btwn a truly random variable and one that is completely deterministic

  • i.e. natural phenomena w/ geographic distribution such as elevation, population density, rainfall, etc.
  • Many earth science variables are regionalized
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5
Q

Geostats involves estimating…?

A

The form of a regionalized variable in 1, 2, or 3 dimensions

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6
Q

What is the basic statistical measure of Geostats?

A

Semivariance

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7
Q

Semivariance

A
  • Measure of degree of spatial dependence between samples at a specific point
  • Function of distance, h
  • Difference btwn attribute values as a function of their spatial separation, h, or change of a regionalized variable
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8
Q

How is semivariance estimated if spacing btwn observations is constant (change in h)

A

Semivariance (h) = sum of (zi - zi+h)^2/2n

  • zi = measurement of regionalized variable z taken at location i
  • zi-h = another measurement taken at change in h intervals away
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9
Q

Semivarince: terms inside expression

A

z, are attributes taken at intervals of size or distance, h

  • if h = 1 every point is compared to its neighbour
  • if h = 2 every point is compared to a point 2 spaces away etc. etc.
  • Then plot on semivariogram
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10
Q

Semivariance is simply half the variance of what?

A

Half of the variance of a spatial process

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11
Q

Semivariogram

A

-

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12
Q

Experimental Semivariogram

A
  • Description of how data are related/correlated w/ distance
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13
Q

Empirical Semivariogram

A
  • Smooth function defined by a model that represents the experimental semivariogram
  • Allows semivariance to be estimated at any h
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14
Q

When semivariogram = 0

A
  • h = 0
  • Same value, semivariance = 0
  • Highly related
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15
Q

As change in h increases, relatedness does what

A

Relatedness decreases, semivariance increases

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16
Q

What happens when change in h ‘critical’ is reached

A

Relatedness = 0, semivariance approximates process variance

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17
Q

When change in h is ‘small’

A

xi, xi +h is ‘similar, semivariance is small

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18
Q

Semivariogram vs. autocorrelation

A
  • Increase semivariance (increase distance), autocorrelation/relatedness decrease
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19
Q

Range

A
  • Distance at which the curve approaches the process variance (sill)
  • W/in range, closer sites are similar
  • Greater than range, point is not useful to interpolation (too far away)
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20
Q

Sill

A
  • Flat region after the range
  • At high distances the semivariance levels off
  • No spatial dependence, constant variance
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21
Q

Nugget affect

A
  • Ideally 0
  • Variable erratic over short distance
  • Variability btwn nearby points
  • Random noise due to micro-scale processes plus measurement error
  • High variance over small distance
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22
Q

Modelling a semivariogram

A
  • Trial and error process
  • Semivariance should be able to be calculated for any h
  • Should fit data as closely as possible
  • Ideally begin at origin, rise smoothly to some upper limit, continue at constant level after limit
23
Q

Parabolic semivariogram

A

Excellent continuity

- Ideal form

24
Q

Linear semivariogram

A

Moderate continuity
- No sill, never reaches critical value, may need to expand search of pairs or obtain more data out of study area to find sill, or sill may not exist

25
Q

Horizontal semivariogram

A

No spatial autocorrelation

26
Q

What are 3 important factors of semivariogram

A
  • Range
  • Sill
  • Nugget
27
Q

Spherical model

A
  • Radius of curve increases as h increases
  • Used to represent phenomena that exhibits a linear decrease in the rate of change of spatial autocorrelation as h increases
28
Q

Exponential model

A
  • Increases with distance
  • Never quite reaches a flat sill
  • Typically used for data that has long continuity distances
29
Q

Semivariance is = to?

A

Variance of the squared differences of points at distance h apart

30
Q

Regionalized variables can be regarded as what 2 parts?

A
  • Residuals
  • Drift
  • Drift <> 0 and semi-v will not flatten
  • If drift exists, it should be removed by TSA
31
Q

Irregularly spaced data

A
  • Irregular data leads to distance btwn points that is not constant
  • Necessary to partition distances btwn points into classes (lags)
32
Q

Lag distances

A
  • Should be equal to or slightly less than the average nearest Neighbour Distance
  • Number of Lags is number of intervals of lag distances that will be examined
  • Number of lags x the lag distance should be less than 1/2 the largest distance in the dataset
33
Q

Anisotropic Data

A
  • Directional
  • Stationary but not isotropic data and the semi-v will differ depending on the orientation of the analysis
  • If semi-v changes w/ direction then data is non-stationary
  • Calc semivariograms for multi directions to see if data is anisotropic and on which direction the primary axis lies
34
Q

Kriging def’n

A
  • Interpolation technique that uses regionalized variable theory to incorporate information about the stochastic aspects of spatial variation when estimating interpolation weights
35
Q

Optimal properties of Kriging

A
  • Exact estimator
  • Predicts sample points w/ 0 error
  • Provides a measure of uncertainty of the interpolated surface
36
Q

Kriging

A
  • Generalized Linear Regression (GLR) technique
  • Requires knowledge of date (from semivariogram)
  • Does not assume independence of observations
  • Does not assume randomness of observations, assumes autocorrelation
37
Q

BLUE

A
  • Best b/c aims at minimizing variance of errors
  • Linear b/c its estimates are weighted linear combinations of the available data
  • Unbiased b/c it tries to have the mean residual or error = 0
  • Estimator
38
Q

Ordinary Kriging

A
  • Simplest form
  • Dimensionless points to estimate other dimensionless points (elevation, precip)
  • Unknown value is calculated using weighted average of known values
  • Estimator is unbiased
  • Mean error = 0 for large samples b/c of weighting
39
Q

What is the crucial variable in Ordinary kriging eneqn

A
  • Weight, lambda
  • Weights change depending on location of unknown point
  • Sum of lambda = 1 ensures error variance = minimum
40
Q

Assumptions of Ordinary kriging

A
  • Partial realization of random function
  • Stationary regionalized variable (mean, spatial semivariance do not depend on the variable of interest)
  • Estimation of value at unknown location is based on values at known location (weighted average)
  • No trend or directional influence
41
Q

Remote Sensing and Ordinary Kriging

A
  • Used to fill gaps in cover of pixel information (i.e. where clouds cover)
  • Used to filter out noise
42
Q

Stochastic Processes

A
  • Drift/Trend is avg of a regionalized variable w/in a neighbourhood
  • Relatively slow varying, non-stationary part of the surface
  • Residuals are difference btwn actual measurements and the drift
  • Subtract drift and regionalized variable becomes stationary
43
Q

Universal Kriging

A
  • Generalization of Kriging procedure
  • Doesn’t require stationary variable assumption
  • Non-stationary has 2 components, Drift/Trend and Residual
  • Linear estimator not biased in presence of a trend
44
Q

Kriging performs in 1 step what would otherwise require 3

A
  1. estimate and remove trend from non-stationary variables
  2. Use Ordinary Kriging on non-stationary residuals to obtain estimated residuals at unsampled points
    - Combined estimated residuals w/ trend to obtain estimate of actual surface
    - Basically pull out trend, model residuals, put trend back in
45
Q

What does effectiveness of Kriging rely on?

A
  • Correct specification of parameters that describe semivariogram
  • Drift model
46
Q

Where does charging yield estimates of likely error?

A

Standard errors or error variances at every interpolation point

47
Q

Since kriging is robust…

A
  • Even with naive parameter selection the method will do no worse than conventional grid estimation
48
Q

Kriging: Smoothing

A
  • Smooths according to the proportion of total sample variance accounted for by random ‘noise’
  • Noisier the data, the less samples represent their immediate vicinity and the more they are smoothed
49
Q

Kriging: De-clustering

A
  • Weight assigned to a sample is lowered to the degree that its information is duplicated by nearby, highly correlated samples
  • Helps mitigate the impact of over-sampled ‘hot-spots’
50
Q

Kriging: Anisotropy

A
  • When samples are more highly correlated in a particular direction, the weights will be greater for samples in that direction
51
Q

Kriging: Precision

A
  • Most precise estimates possible will be computed for available data given a representative semivariogram
52
Q

What is price that must be paid for optimality in estimation and kriging

A
  • Price is computation complexity
  • Many simultaneous eqns must be solved for every interpolation point in kriging
  • Computer run-times will be longer using kriging over conventional interpolations
53
Q

What extensive prior study must be made for kriging?

A
  • Process stationary or not
  • Form of semivariogram
  • Set neighbourhood size/orientation
  • Select proper order of the trend if it exists
54
Q

What is the goal for validation of kriging?

A
  • Validation withholds a portion of data from semivariogram and kriging
  • Goal is to get ratio btwn the RMS Predicted Error from cross-validation/validation process and the Estimation Error of the surface to = 1

Geog418SptStts (6 decks)

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