1. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 1/45 GEOSTATISTICS INTRODUCTION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University

2. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 2/45 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 3/45 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 4/45 Deterministic vs stochastic A phenomenon which can be completely described by mathematical expressions without error and uncertainty is deterministic. In contrast, a stochastic phenomenon can not be fully described by mathematical expressions due to embedded random properties that are associated with uncertainties.

5. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 5/45 Stochastic Approach vs. Deterministic Approach

6. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 6/45 Stochastic Approach vs. Deterministic Approach P(Y>460|X=50)=? (50, 438.9794) by deterministic model.

7. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 7/45 Probability plays a key role in modeling stochastic phenomena. In reality, many practices of stochastic modeling arise from our inability of developing deterministic models for complex and complicated phenomena.

8. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 8/45 Random variables A random variable is a mapping function which assigns outcomes of a random experiment to real numbers. Occurrence of the outcome follows certain probability distribution. Therefore, a random variable is completely characterized by its probability density function (PDF).

9. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 9/45

10. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 10/45 Random processes and random Fields If random variables are time- (1-D) or space-dependent (2-D or higher), they jointly form random processes or random fields. In addition to the probability distributions, correlations between random variables are also needed to characterize random processes and random fields.

11. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 11/45

12. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 12/45 Completely stationary

13. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 13/45

14. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 14/45 Stationary up to order m Recall the definition of moment.

15. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 15/45 Second-order stationarity

16. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 16/45 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.

17. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 17/45 The Brownian motion (one-dimensional, also known as random walk) Consider a particle randomly moves on a real line. Suppose at small time intervals  the particle jumps a small distance , randomly and equally likely to the left or to the right. Let be the position of the particle on the real line at time t.

18. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 18/45 Assume the initial position of the particle is at the origin, i.e. Position of the particle at time t can be expressed as where are independent random variables, each having probability 1/2 of equating 1 and  1. ( represents the largest integer not exceeding.)

19. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 19/45 Distribution of X(t) Let the step length equal, then For fixed t, if  is small then the distribution of is approximately normal with mean 0 and variance t, i.e.,.

20. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 20/45 Graphical illustration of Distribution of X(t)

21. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 21/45 If t and h are fixed and  is sufficiently small then

22. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 22/45 Distribution of the displacement The random variable is normally distributed with mean 0 and variance h, i.e.

23. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 23/45 Variance of is dependent on t, while variance of is not. If, then, are independent random variables.

24. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 24/45 Covariance and Correlation functions of

25. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 25/45

26. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 26/45 Remarks It is custom to use capital and lower case letters respectively to represent random variables and observations. For example, X(t) and Z(x) are random variables and x(t) and z(x) are measurements. Also, X(t) and Z(x) are often used for random processes (time series) and random fields and should be made clear from the context of the manuscript.

27. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 27/45

28. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 28/45

29. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 29/45 Ensemble space The set of all realizations of a random process is called the ensemble space.

30. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 30/45 PDF of a random variable at time t t Ensemble space Figure 3. Ensemble space is a collection of all realizations.

31. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 31/45 Examples

32. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 32/45 f(t)=C 1 +C 2 t

33. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 33/45

34. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 34/45

35. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 35/45 Support of a ReV Many random variables of physical quantities are observed based on certain time and/or spatial domains. Such time/spatial domain from which observations are made is called “support” of the random variables.

36. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 36/45 A support is characterized by not only its time, area or volumetric coverage but also the geometrical shape and orientation. Even for the same physical phenomenon, measurements of different supports are considered realizations of different random variables. Change of support will change the probability distribution and spatial variation structure of the phenomenon under study and therefore define a new random variable.

37. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 37/45 Stochastic Simulation Given a random variable X, there are situations that we want to obtain a desired number of random samples, each of size n. Similarly, we may need to generate as many realizations of a random process or random field under investigation. The advances in computer technology have made it possible to generate such random samples and realizations using computers. The work of this nature is termed “simulation”, or more precisely “stochastic simulation”.

38. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 38/45

39. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 39/45 Pseudo-random number generation Pseudorandom number generation (PRNG) is the technique of generating a sequence of numbers that appears to be a random sample of random variables uniformly distributed over (0,1).

40. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 40/45 A commonly applied approach of PRNG starts with an initial seed and the following recursive algorithm (Ross, 2002) modulo m where a and m are given positive integers, and where the above means that is divided by m and the remainder is taken as the value of.

41. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 41/45 The quantity is then taken as an approximation to the value of a uniform (0,1) random variable. Such algorithm will deterministically generate a sequence of values and repeat itself again and again. Consequently, the constants a and m should be chosen to satisfy the following criteria:

42. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 42/45 For any initial seed, the resultant sequence has the “appearance” of being a sequence of independent uniform (0,1) random variables. For any initial seed, the number of random variables that can be generated before repetition begins is large. The values can be computed efficiently on a digital computer.

43. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 43/45 A guideline for selection of a and m is that m be chosen to be a large prime number that can be fitted to the computer word size. For a 32-bit word computer, m = and a = result in desired properties (Ross, 2002).

44. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 44/45 Simulating a continuous random variable probability integral transformation

45. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 45/45

46. Risk and Reliability Definition of risk and reliability in hydrology Definition of risk in risk management Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 46/45

47. 47 Another example – target cancer risk 1/23/2016 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

48. 48 Modeling MCS inorg – Log-normal 1/23/2016 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

49. 49 Cumulative distribution of the target cancer risk 1/23/2016 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU There is no need for stochastic simulation since the risk is completely dependent on only one random variable (MCS). Once the parameters of MCS are determined, the distribution of TR is completely specified.

50. An example of two-dimensional random walk simulation Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 50/45

51. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University 51/45