Review of Random Process Theory CWR 6536 Stochastic Subsurface Hydrology

Random Process A random process may be thought of as a collection or ensemble of random variables which change through time, any realization of which might be observed on any trial of an experiment Example: Daily rainfall. Random process is the ensemble of daily rainfall profiles for each year. Each year is one realization or trial of the experiment. Daily rainfall is the r.v.

Random Field A random field may be thought of as a collection or ensemble of random variables which vary over space, any realization of which may be observed on any trial of an experiment Example: Saturated hydraulic conductivity. Random field is the ensemble of aquifers with the same geologic origin. Each aquifer is one realization or trial of the experiment,hydraulic conductivity is the r.v.

Random Processes vs Random Fields Random processes arise mostly in the analysis of spatially lumped systems (e.g. reservoir analysis) Random fields arise mostly in the analysis of spatially distributed systems (e.g. groundwater flow)

Consider a 1-D random field At a fixed depth, Ksat is a random variable which takes on different values for different realizations ( i.e. location in the field)

The r.v. K sat is characterized by univariate cdf univariate pdf moments >However must also consider another possibility. Does knowledge of K sat (z 1 ) tell you anything about K sat (z 2 )?

Relationship between K sat (z 1 ) and K sat (z 2 ) characterized by: Joint (2nd order) cdf Joint (2nd order) pdf >However to completely characterize system must of joint pdf/cdf of infinite order for the infinite number of depths in the profile. Impossible to obtain in real life so settle for knowledge of the first and second moments of the pdf/cdf

Autocovariance/Autocorrelation This function describes the degree of similarity expected between measured values of K sat at z 1 and z 2

Variogram/Correlogram This function describes the expected magnitude of the difference between K sat at z 1 and z 2. It is the variance of the increment [K sat (z 1 ) - K sat (z 2 )] Differencing the random variables gives us a handle on the relationship w/o requiring knowledge of the means of the r.v.s

Madogram/Rodogram This function also describes the expected magnitude of the difference between K sat at z 1 and z 2. Use of powers less than 2 reduce the influence of extreme values of K sat on the function Two commonly used values are  =1 (madogram) and  =1/2 (rodogram).

What is the covariance between two independent random variables? What is the value of the variogram between two independent random variables?

In general covariance and variogram functions depend on both locations z 1 and z 2.. When this is the case must have many realizations of the pair of random variables at these locations to infer these functions from field data. Examples???? Sometimes these functions depend only on the distance between z 1 and z 2, not their actual locations. When this is the case multiple pairs of data with the same separation at different locations may be used to infer functions from one realization of field data. Examples????

Stationarity A process K(Z) is strictly stationary if all its ensemble pdfs, cdfs, and moments are unaffected by a shift in origin stationarity of the ensemble mean implies stationarity of the ensemble variance implies Note that the mean can be stationary even if variance is not and vice-versa

Second Order Stationarity A process K(z) is second order stationary if its mean is independent of location and its covariance function and variogram depend only on the distance separating two points in the random field Stationarity of the covariance implies stationarity of the variogram but not vice versa Second order stationarity allows statistical inference of the first and second moments from one realization of the random field….when would this be a good assumption?

What is the relationship between the covariance and variogram functions?

What do typical covariance and variogram functions look like?

The Power Spectrum The power auto-spectrum is the Fourier transform of the auto-covariance function: For a stationary process the auto-spectrum represents the distribution of variance over frequency, and therefore must be positive and real. The spectrum divided by the variance is analogous to a pdf; hence is called the spectral density function.

Examples of Covariance- Spectrum Pairs

Cross-covariances, cross-variograms and cross-spectra The cross-covariance between two random fields X and Y is: The cross-spectrum is: The cross-variogram is:

Concept of isotropy A stationary random field is isotropic if all its ensemble pdfs, cdfs, and moments are unaffected by direction (or a rotation in axes) This implies that all autocovariance and autocorrelation coefficients are the same in all directions.

Concept of Ergodicity Averages over space (or time) equal averages over the ensemble i.e. we require that

Ergodicity implies that all variability that might occur over the ensemble occurs in each realization For a process to be ergodic it must be stationary, but not vice versa. Must generally assume both stationarity and ergodicity to infer ensemble statistics from a single realization. Often there is no way to rigorously validate these hypotheses When deriving ensemble moments of dependent random fields from physical equations stationarity of resulting process depends on physics. May be stationary or non-stationary

The Intrinsic Hypothesis A random field Z(x) is said to be intrinsic (or incrementally stationary) if The first order difference is stationary in the mean For all vectors, h, the increment [z(x+h)-z(x)] has a finite variance which does not vary with x Thus second order stationarity implies the intrinsic hypothesis but not vice-versa