Geostatistics Without Tears

.. .. .. .. ..

Gregg Babish GIS Specialist Ecological Research Division Environmental Conservation Branch 2365 Albert Street, Room 300 Regina, Saskatchewan S4P 4K1 4K1 Edition 2006.03 July 5, 2006

Environment Environment Canada Geostatistics Without Tears  .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

 A practical Guide to Surface  In  Interpolation, Geostatistics, Variograms and Kriging Kriging

G re g g B a bi s  s h G I S  S S pe ci a  al i s  s t  E nvi r C   a n a d   a  r o n me n t  C   Re gi n S a s k a t c  n a ,  S  c hew a n

Geostatistics Without Tears

.

Geostatistics Without Tears

2

PREFACE ........................... ......................................... ............................ ............................ ............................ ............................ ............................ ........................7 ..........7 INTRODUCTION ............................ ......................................... ........................... ............................ ........................... ........................... ..........................9 ............9 INTERPOLATION INTERPOLATION TECHNIQUES........... TECHNIQUES......................... ............................ ........................... ........................... ...........................9 .............9

SPATIAL I NTERPOLATION ........................... ......................................... ............................ ............................ ........................... .........................10 ............10  Deterministic/Stochastic  Deterministic/Stochastic ........................... ......................................... ............................ ............................ ........................... ......................10 .........10 Global/Local...................................... Global/Local........................ ............................ ............................ ............................ ............................ ............................ ...............10 .10  Exact/Approximate ........................... ......................................... ............................ ........................... ........................... ............................ .................1 ...1 1 Gradual/Abrupt ........................... ......................................... ............................ ............................ ............................ ............................ .....................11 .......11 CLIMATE DATA ............................ .......................................... ............................ ............................ ............................ ........................... .........................11 ............11  ANUSPLIN....................  ANUSPLIN.................................. ............................ ............................ ............................ ............................ ............................ ......................12 ........12 Centro Internacional de Agricultura Tropical .......................... ....................................... ........................... ..................13 ....13 Climatologically Aided Interpolation ............................ .......................................... ........................... ........................... ................13 ..13  Daymet...................................  Daymet................................................ ........................... ............................ ............................ ............................ ............................14 ..............14 Gradient plus Inverse Distance Square ........................... ........................................ ........................... ............................14 ..............14 Optimal Interpolation ............................ ......................................... ........................... ............................ ............................ ..........................1 ............1 5 Parameter-evaluation Regressions on Independent Slopes Model ..........................15 ..........................15 Thin-Plate Spline ........................... ......................................... ............................ ........................... ........................... ............................ ....................16  ......16  DIGITAL ELEVATION MODELING ........................... ......................................... ............................ ............................ ............................16 ..............16 FOURIER SERIES ........................... ......................................... ............................ ............................ ............................ ........................... .........................1 ............1 8 GEOSTATISTICS ............................ .......................................... ............................ ............................ ............................ ........................... .........................18 ............18 I NVERSE DISTANCE WEIGHTED ........................... ......................................... ............................ ............................ ............................ ................1 ..1 9  NEAREST NEIGHBOUR ........................... ......................................... ............................ ............................ ............................ ............................ ................ .. 20 SPLINES ............................ .......................................... ............................ ............................ ............................ ............................ ............................ .......................20 .........20 TREND SURFACE A NALYSIS............................ .......................................... ............................ ............................ ............................ .....................2 .......2 1 TRIANGULATION ........................... ......................................... ............................ ............................ ............................ ........................... .........................21 ............21 HISTORY OF KRIGING ............................ .......................................... ........................... ........................... ............................ .........................23 ...........23 DATA ANALYSIS....................... ANALYSIS..................................... ............................ ............................ ............................ ........................... ...........................27 ..............27 SUMMARY DATA ANALYSIS ........................... ......................................... ............................ ............................ ............................27 ..............27

SUMMARY STATISTICS ............................ .......................................... ............................ ............................ ............................ ............................27 ..............27 MEASURES OF SHAPE ............................ .......................................... ............................ ............................ ............................ ............................ ................28 ..28 Frequency Distribution............................ Distribution.......................................... ........................... ........................... ........................... ........................28 ...........28 Cumulative Distribution ........................... ......................................... ........................... ........................... ............................ .......................28 .........28 Skewness ............................ ......................................... ........................... ............................ ............................ ........................... ........................... ...................28 .....28 Kurtosis.............................. Kurtosis............................................ ........................... ........................... ............................ ............................ ........................... ..................29 .....29  Normal Probability Plot Plot .......................... ........................................ ........................... ........................... ........................... ........................29 ...........29 MEASURES OF LOCATION ........................... ......................................... ............................ ............................ ........................... .........................31 ............31  Mean ............................ .......................................... ............................ ............................ ............................ ............................ ............................ .......................31 .........31  Median ............................ .......................................... ............................ ............................ ............................ ............................ ............................ ....................31 ......31  Mode ............................ .......................................... ............................ ............................ ............................ ............................ ............................ .......................31 .........31 Quantiles.............................. Quantiles............................................ ............................ ............................ ............................ ............................ ............................ ...............32 .32 MEASURES OF SPREAD ............................ .......................................... ............................ ............................ ............................ ............................32 ..............32  Range ........................... ......................................... ............................ ............................. ............................. ............................ ............................ ......................32 ........32 Variance.......................... Variance........................................ ............................ ............................ ............................ ............................ ............................ ....................32 ......32 Standard Deviation........................ Deviation...................................... ............................ ............................ ............................ ............................ ...................32 .....32  Interquartile Range........................ Range...................................... ............................ ........................... ........................... ............................ ....................33 ......33 Coefficient of Variation .......................... ........................................ ........................... ........................... ............................ .........................33 ...........33 CORRELATION A NALYSIS ........................... ......................................... ............................ ............................ ........................... .........................3 ............3 4 Coefficient of Correlation........................ Correlation...................................... ........................... ........................... ............................ ........................34 ..........34

Geostatistics Without Tears

3

Coefficient of Determination ............................ ......................................... ........................... ............................ ............................ ...............34 .34 Standard Error of Estimate ....................................... ..................................................... ........................... ........................... ....................35 ......35 EXPLORATORY DATA ANALYSIS...................... ANALYSIS.................................... ........................... ........................... .........................37 ...........37

CENSORED DATA............................ .......................................... ............................ ............................ ............................ ............................ .......................37 .........37 CO-LOCATED SAMPLES............................ .......................................... ............................ ............................ ............................ ............................37 ..............37 DECLUSTERING.......................... ........................................ ............................ ............................ ............................ ............................ ............................37 ..............37 EXTREME VALUES ............................ .......................................... ............................ ............................ ............................ ............................ .....................38 .......38 SKEWED DATA WITH ZEROS ........................... ......................................... ............................ ............................ ............................ .....................38 .......38 SPATIAL OUTLIERS ........................... ......................................... ............................ ............................ ............................ ............................ .....................39 .......39 STRATIFICATION (DATA SET SUBDIVISION ) ........................... ......................................... ............................ ..........................39 ............39 TIME ............................ .......................................... ............................ ............................ ............................ ............................ ............................ ............................40 ..............40 TRANSFORMATION ........................... ......................................... ............................ ............................ ............................ ............................ .....................40 .......40  Arcsine Transform ............................ ......................................... ........................... ............................ ............................ ........................... .................40 ....40  Box-Cox Transform ........................... ......................................... ........................... ........................... ............................ ............................ ................40 ..40  Lognormal Transform..................... Transform................................... ............................ ............................ ............................ ............................ ..................41 ....41  Normal Score Transform............. Transform .......................... ........................... ............................ ........................... ........................... .......................41 .........41 Scale Transform............................... Transform............................................ ........................... ............................ ............................ ........................... ..................42 .....42 Square Root Transform ............................ .......................................... ............................ ............................ ........................... ......................42 .........42 OFFSETS AND BACK -TRANSFORMS ........................... ......................................... ............................ ........................... .........................42 ............42 Offset............................. Offset........................................... ............................ ........................... ........................... ............................ ............................ .......................42 .........42  Back-transformation......................................  Back-transformation........................ ............................ ............................ ........................... ........................... ..................42 ....42 TRENDS ............................ .......................................... ............................ ............................ ............................ ............................ ............................ .......................43 .........43 APPLIED GEOSTATISTICS GEOSTATISTICS ......................... ....................................... ........................... ........................... ........................... .....................4 ........4

5

VARIOGRAMS................................. VARIOGRAMS................... ............................ ............................ ............................ ............................ ............................ .....................4 .......4

5

CALCULATING THE VARIOGRAM ............................ .......................................... ............................ ............................ ............................4 ..............4 6 First Lag Distance .......................... ....................................... ........................... ............................ ............................ ........................... ...................4 ......4 6  Second Lag Distance ........................... ......................................... ............................ ............................ ............................ ...........................4 .............4 6  Third Lag Distance.............. Distance ............................ ............................ ............................ ............................ ............................ ............................ ...............4 .4 7  Fourth Lag Distance................................. Distance............................................... ............................ ............................ ............................ ......................4 ........4 7  Fifth Lag Distance .......................... ....................................... ........................... ............................ ............................ ........................... ...................4 ......4 8 Sixth Lag Distance....................... Distance.................................... ........................... ............................ ............................ ........................... ......................4 .........4 8 GRAPHING THE SEMIVARIOGRAM .......................... ........................................ ............................ ............................ ............................ .............. 50 THE SEMIVARIOGRAM ........................... ......................................... ............................ ............................ ............................ ............................ ................5 ..5 1 Coordinate System ............................ .......................................... ........................... ........................... ............................ ........................... .................5 ....5 1  Lag ............................ .......................................... ............................ ............................ ............................ ............................ ............................ ..........................5 ............5 1  Range ........................... ......................................... ............................ ............................. ............................. ............................ ............................ ......................5 ........5 3 Sill....................................... Sill..................................................... ............................ ............................ ........................... ........................... ............................ .................5 ...5 3  Nugget...............................  Nugget............................................. ............................ ............................ ............................ ............................ ............................ ..................5 ....5 4 Variance Cloud....................................... Cloud.................................................... ........................... ............................ ............................ .........................5 ...........5 6  Scattergram ............................ .......................................... ............................ ............................ ............................ ............................ ..........................5 ............5 6  SEMIVARIOGRAM MODELS ........................... ......................................... ............................ ............................ ............................ .......................5 .........5 7  Linear.........................  Linear...................................... ........................... ............................ ............................ ............................ ............................ ..........................5 ............5 7  Spherical............. Spherical ........................... ............................ ............................ ............................ ............................ ............................ ............................ ..................5 ....5 8  Exponential............  Exponential .......................... ............................ ............................ ............................ ............................ ............................ ............................ ...............5 .5 8 Gaussian ........................... ......................................... ............................ ............................ ............................ ............................ ............................ ..................5 ....5 8  Hole Effect .......................... ........................................ ............................ ........................... ........................... ............................ ............................ .................5 ...5 9 A NISOTROPIC VARIOGRAMS ........................... ......................................... ............................ ............................ ............................ ..................... ....... 60  Nugget Variance ............................ ......................................... ........................... ............................ ........................... ........................... ..................... ....... 60  Anisotropic Variogram Surface Surface .......................... ....................................... ........................... ........................... ........................... .............. 60 Geometric Anisotropy.......................... Anisotropy....................................... ........................... ............................ ............................ ........................... ..............6  .6  1  Zonal Anisotropy .......................... ........................................ ............................ ............................ ............................ ............................ ....................6  ......6  2 ALTERNATE VARIOGRAMS AND AUTOCORRELATION MEASURES..................................6 3

Geostatistics Without Tears

4

Correlogram ........................... ......................................... ............................ ............................ ............................ ............................ ..........................6  ............6  3 Covariance........................... Covariance......................................... ............................ ............................ ............................ ............................ ............................ ...............6  .6  3 Cross-K....................................... Cross-K......................... ............................ ............................ ............................ ............................ ............................ ......................6  ........6  3  Drift .................. ................................ ............................ ............................ ........................... ........................... ............................ ............................ .....................6  .......6  4 Fractal .......................... ........................................ ............................ ............................ ............................ ............................ ............................ ......................6  ........6  4 Geary’s C................................. C............................................... ............................ ............................ ............................ ............................ .........................6  ...........6  5  Inverted Covariance ........................... ......................................... ........................... ........................... ............................ ............................ ...............6  .6  5  Madogram ..................... ................................... ............................ ............................. ............................. ............................ ............................ ....................6  ......6  5  Moran’s I ............................ .......................................... ............................ ............................ ............................ ............................ ............................ ................6  ..6  6   Relative Variograms ........................... ........................................ ........................... ........................... ........................... ........................... ................6  ...6  7   Ripley’s K ................. ............................... ............................ ........................... ........................... ............................ ............................ ...........................6  .............6  8 KRIGING....................................... KRIGING......................... ............................ ............................ ............................ ............................ ............................ .........................6 ...........6

9

RIGING ALGORITHM ........................... THE K RIGING ......................................... ............................ ............................ ............................ .......................6 .........6 9 RIGING VARIATIONS ........................... K RIGING ......................................... ............................ ............................ ............................ ............................ ................ .. 70 Ordinary Kriging ........................... ......................................... ........................... ........................... ............................ ............................ .................... ...... 70 Simple Kriging ........................... ........................................ ........................... ............................ ............................ ........................... ........................ ........... 70 Universal Kriging ........................... ......................................... ............................ ............................ ............................ ............................ .................. .... 70 Kriging with External Drift...................... Drift................................... ........................... ............................ ............................ ........................7  ..........7  1 PUNCTUAL AND BLOCK K RIGING ........................................ ............................ ............................ ............................7 ..............7 1 RIGING .......................... Punctual Kriging ........................... ......................................... ............................ ............................ ........................... ........................... ....................7  ......7  1  Block Kriging............................. Kriging........................................... ............................ ........................... ........................... ............................ ........................7  ..........7  2 RIGING SEARCH NEIGHBORHOOD........................... K RIGING ......................................... ............................ ........................... .........................7 ............7 2 CROSS-VALIDATION ............................ .......................................... ............................ ............................ ............................ ........................... ..................7 .....7 3 K RIGING .......................................... ............................ ............................ ............................ ............................ ................7 ..7 4 RIGING PREDICTORS............................ Cokriging ........................... ......................................... ........................... ........................... ............................ ............................ ........................... ..................7  .....7  4  Disjunctive Kriging........................ Kriging..................................... ........................... ........................... ........................... ........................... .....................7  ........7  4  Indicator Kriging....................................... Kriging.................................................... ........................... ............................ ............................ ......................7  ........7  4  Lognormal Kriging ........................... ......................................... ............................ ............................ ........................... ........................... .................7  ...7  5  Median Indicator Kriging............................. Kriging.......................................... ........................... ............................ ............................ ...................7  .....7  6   Multiple Indicator Kriging Kriging ....................... ..................................... ............................ ............................ ........................... ......................7  .........7  6   Isofactorial Disjunctive Disjunctive Kriging .......................... ....................................... ........................... ........................... ..........................7  .............7  6  Probability Kriging.............................. Kriging........................................... ........................... ........................... ........................... ........................... ...............7  ..7  7   Regression Kriging ........................... ......................................... ............................ ........................... ........................... ............................ .................7  ...7  7   Residual Indicator Kriging.................................... Kriging................................................. ........................... ........................... ........................7  ...........7  7  RIGING R IGING AND EMOTELY E MOTELY ENSED MAGES K  R  S I .......................... ........................................ ............................ ............................7 ..............7 7

GEOSTATISTICAL GEOSTATISTICAL CONDITIONAL SIMULATION ........................... ......................................... ..................7 ....7

9

CONDITIONAL COSIMULATION ............................ .......................................... ............................ ............................ ............................ ................ .. 80 GAUSSIAN SIMULATION........................... ......................................... ............................ ............................ ............................ ............................ .............. 80 PLURIGAUSSIAN .......................................... ............................ ........................... ........................... ................... ..... 80 LURIGAUSSIAN SIMULATION ............................ PROBABILITY FIELD SIMULATION.......................... ........................................ ............................ ............................ ............................ .............. 80 SEQUENTIAL GAUSSIAN SIMULATION........................... ......................................... ............................ ............................ .....................8 .......8 1 SEQUENTIAL I NDICATOR SIMULATION.......................... ........................................ ............................ ............................ .....................8 .......8 1 TRUNCATED GAUSSIAN SIMULATION ........................... ......................................... ............................ ............................ .....................8 .......8 2 TRUNCATED PLURIGAUSSIAN SIMULATION ............................ ......................................... ........................... ..........................8 ............8 2 TURNING BANDS SIMULATION ............................ .......................................... ............................ ............................ ............................ ................8 ..8 2 APPLYING GEOSTATISTICS GEOSTATISTICS .......................... ........................................ ........................... ........................... ........................... .................8 ....8

3

DATA COLLECTION........................... ......................................... ............................ ............................ ............................ ............................ .....................8 .......8 3 DATA, SUMMARY STATISTICS AND HISTOGRAM VISUALIZATION..................................8 4 SEMIVARIOGRAM VISUALIZATION............................ .......................................... ............................ ........................... .........................8 ............8 4 EVALUATION OF MODEL ............................ .......................................... ............................ ............................ ........................... .........................8 ............8 4 CONCLUSION........................................ CONCLUSION.......................... ............................ ............................ ............................ ............................ ........................... ...............8 ..8

7

Geostatistics Without Tears

5

REFERENCES REFERENCES ........................... ........................................ ........................... ............................ ............................ ........................... ........................... ................8 ..8

9

Geostatistics Without Tears

6

GEOSTATISTICS WITHOUT TEARS  A Practical Guide Guide to Surface Interpolation, Interpolation, Geostatistics, Geostatistics, Variograms and Kriging  by Gregg Babish GIS Supervisor / Data Management Environment Canada Illustrations by Mark Gilchrist Environment Canada

PREFACE Geostatistics (also known as kriging) was developed for the mining industry during the 1960s and 1970s to estimate changes in ore grade. The principles of geostatistics are now applied to many applications that require statistically based interpolation techniques. Geostatistics provides a data value estimate for locations that cannot be sampled directly by examining data taken at locations that can be sampled. If a model of spatial correlation can be established for the sampled data then this model is used to interpolate data values for those locations that cannot be sampled directly. The effectiveness of geostatistics depends upon how well the observation points represent the phenomenon under study as well as the appropriate selection of the model parameters. Matheron introduced the term geostatistics in 1962 with this definition: “Geostatistics is the application of the formalism of random functions to the reconnaissance and estimation of natural phenomena” (Journel and Huijbregts, 1978). Oliver and Webster (1991) said, “. . . there is to be no geostatistics without tears”. The intent of this paper is to provide the reader with an introduction and understanding of surface interpolation techniques and geostatistics (kriging). Hopefully no tears will be shed.

Geostatistics Without Tears

7

Geostatistics Without Tears

8