An Introduction to Geostatistics

Transcription

1 An Introdction to Geostatistics András Bárdossy Universität Stttgart Institt für Wasser- nd Umweltsystemmodellierng Lehrsthl für Hydrologie nd Geohydrologie Prof. Dr. rer. nat. Dr.-Ing. András Bárdossy Pfaffenwaldring 61, Stttgart, Detschland

2 2 Introdction This is not: An introdction to a software package Nor a set of recipes how to It is an introdction to Why we se geostatistics What is important what is not What is good and what is not

3 3 Books An Introdction to Applied Geostatistics (Isaaks and Srivastava) Mining Geostatistics (Jornel and Hijbregts) Geostatistics for natral resorces evalation (Goovaerts)

4 4 Software SgeMS (Stanford Geostatistical Modelling Software) GEOEAS (Geostatistical Environmental Assessment Software) GSLIB (Geostatistical Software Library) ArcGIS Geostatistical Analyst

5 5 God does not play dice. (Albert Einstein) Qantm mechanics Draw of lottery nmbers? Or case we do not even know the exact circmstances of the processes.

6 6 The problem Discrete observations (points and blocks) Unknown in between Generating processes Physical, chemical, biological Circmstances, inpts nknown Uncertainty assmptions related to ncertainty

7

8 8 The problem We do not know reality Bt: We can se methods of statistics Deriving certain measres from observations Applying a stochastic analoge We assme that what is observed is like the realization of a stochastic process Mixtre of strctre and randomness

9 Intrinsic hypothesis 1.The expected vale of the random fnction Z() is constant all over the domain D 2.The variance of the increment corresponding to two different locations depends only on the vector separating them

10 Intrinsic hypothesis These conditions can be formlated as: EZ m for all D And for the increments 1 Var Z h Z 2 E Z h Z h

11

12

13

14

15

16

17

18 The variogram 1. (0)=0 2. (h) > 0, for all vectors h 3. (h) (-h), for all vectors h 4. variance of the increments is spposed to increase with the length of the vector h 5. limit in the continity of the parameter, vector separating two points exceeds a certain limit the variance of the increment will not increase any more 6. The variogram is often discontinos near the origin. For any h>0 we have (h)>c 0 >0, ngget effect.

19 Experimental variogram Variogram can be estimated with the help of the following formla h j i j i Z Z h N h 2 * 2 1 h Angle h j i j i, Allowing a certain difference in both the angle and the length of vector

20 Experimental variogram mm m

21 Estimation variogram Cl

22

23 Unbiasedness E Z * Point kriging n Z ii Z m for all D E n * Z EZ m ii i i i i n ii i 1

24 Z Z Var i n i i n j j i j n i i * 2 2 ) ( ) ( ) ( Estimation variance sing the variogram Minimize estimation variance:

25 i n j j i j 1 i=1,,n n j j i j K 1 2 ) ( 1 1 n j j Kriging eqations sing variogram

26 26 This is an analogy Is it better than others? (NN,ID) Rational: Distingishes between variables (variograms) Good properties Testing: Data configrations are reflected Estimates sing cross validation: Leave some ot and estimate them sing the rest Compare estimated and observed Uncertainty sing confidence intervals (CI) Leave some ot and estimate them sing the rest Where is the observed in the estimated CI

27 27 Precipitation cross validation Normed sqared error for nsed stations for each method and different time aggregations:

28 28 Is good also tre? Interpolation is a good estimator Is the obtained map the trth? NO NO NO NO NO. It is the best estimate bt Impossible as it has different properties as observations Smooth lower variance and variogram Conseqence Problems for risk assessment

29 29 Spport Observations correspond to a certain area or volme These may differ Measrement devices Sampling techniqes Distribtion of vales change when spport changes!!

30

31

32

33

34

35

36 36 Different spports: Different marginal distribtions Increase of spport decrease of variance Different variongrams Do not mix them for: Variogram calclation Yo can mix for interpolation Block Kriging

37 37 Alternate realities Interpolation is not a possible reality Generate realities: Same variogram Same observations Matching other statistics Conseqence Simlation error is higher than interpolation error Variability is realistic (?) ths good for nonlinear cases (Risk assessment)

38 38 Simlation methods Using different assmptions LR Cholewsky decomposition Trning bands Fast Forier Transform Seqential Simlations Simlated Annealing Uncertainty validation is necessary!

39 39 Goals Interpolation Simlation

40 Thank yo for yor attention!