STOCHASTIC HYDROLOGY Fundamentals of Geostatistics I Variogram modeling 11

1. Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STOCHASTIC HYDROLOGYFundamentals of Geostatistics (I)Variogram modeling(11/28/’06) Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University Guest Professor of Kyoto University (Sept. 2006 – Feb. 2007)

2. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Regionalized phenomenon Many natural phenomena exhibit variations in time and space, for example, rainfall, temperature, elevation, hydraulic conductivity, soil moisture content, etc. When a phenomenon spreads in space and exhibits a certain spatial variation structure, it is said to be regionalized.

3. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University A regionalized variable generally possesses two characteristics: a local and random aspect which can be characterized by random variable (random feature) a spatial dependence structure characterizing correlation relationship between pairs of random variables (structural feature) A probabilistic interpretation of random function or random field (RF) is required to take into consideration both characteristics.

4. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Second-order stationarity

5. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Intrinsic stationarity The second-order stationarity assumes the existence of a covariance and a finite a priori variance, . However, there are many physical phenomena and random functions which have an infinite capacity for dispersion, i.e., which have neither an a priori variance nor a covariance. One such example is the Brownian Motion.

6. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Intrinsic stationarity A random function Z(x) is said to be intrinsically stationary if the expectation exists and does not depend on the support point x, i.e., for all vectors h the increment has a finite variance which does not depend on x. The function is called the semi-variogram, or simply the variogram.

7. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Properties of the covariance function and variogram Let Z(x) be a stationary random field with expectation ?, variance ? 2, covariance C(h) and variogram ?(h). The covariance function and variogram are related by the following equation:

8. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

9. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Other properties of the covariance function of a stationary random field include

10. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Relation between the covariance and variogram of stationary random fields.

11. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Similarly, the following properties hold for variograms:

12. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Variogram characteristics - Continuity at the origin The continuity in space of a random field Z(x) is reflected by the rate of growth of for small values of h. Theoretically , however, an experimental variogram (variogram estimated from observed data) may have a value of significantly different from 0 near the origin.

13. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Such discontinuity of the variogram near the origin is called a “nugget effect” and is possibly due to measurement errors of Z(x), or micro-scale variability of the random field under investigation. In particular, if for all h, it is called pure nugget effect.

14. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University The nugget effect

15. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Variogram characteristics – Behavior at infinity Using the property that is a conditional positive definite function, it can be shown that the variogram necessarily increases more slowly at the infinity than does , i.e.

16. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University An experimental variogram which increases at least as rapidly as for large distances h is incompatible with the intrinsic stationarity hypothesis. Such an increase in the variogram most often indicates the presence of a trend or drift, i.e., a non-stationary expectation.

17. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Variogram characteristics – the influence range The influence range is the minimum distance between two independent random variables. It is also the zone of influence of a random variable.

18. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

19. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Modeling the sample variogram Given a set of observations , the variogram can be estimated through the following procedures: define a certain number of distance classes between measurement points , calculate the number of pairs ( ) present in each distance class,

20. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University calculate the average distance in each distance class, calculate the average value of the experimental (or sample) variogram is then constructed using average distance and average value of .

21. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Calculation of the experimental variogram

22. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Basic variogram models Sample variograms usually do not satisfy the conditional negative definite condition, therefore, theoretical variogram models are needed. Some of the commonly used admissible variogram models are:

23. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

24. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

25. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University An important property of admissible variogram models is that any linear combination of admissible variogram models with positive coefficients is also an admissible model, i.e. is also an admissible variogram model and random field with such variogram model is said to have a “nested structure”.

26. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University One should also notice that, in using the nested-structure variogram model, we are not limited to combining models of the same shape.

27. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Example of a nested-structure variogram