Review of Random Process Theory

1
Review of Random Process Theory

• CWR 6536
• Stochastic Subsurface Hydrology

2
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.

3
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.

4
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)

5
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)

6
The r.v. Ksat is characterized by

• univariate cdf
• univariate pdf
• moments
• However must also consider another possibility.
  Does knowledge of Ksat(z1) tell you anything
  about Ksat(z2)?

7
Relationship between Ksat(z1) and Ksat(z2)
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

8
Autocovariance/Autocorrelation

• This function describes the degree of similarity
  expected between measured values of Ksat at z1
  and z2

9
Variogram/Correlogram

• This function describes the expected magnitude of
  the difference between Ksat at z1 and z2. It is
  the variance of the increment Ksat(z1) - Ksat
  (z2)
• Differencing the random variables gives us a
  handle on the relationship w/o requiring
  knowledge of the means of the r.v.s

10
Madogram/Rodogram

• This function also describes the expected
  magnitude of the difference between Ksat at z1
  and z2. Use of powers less than 2 reduce the
  influence of extreme values of Ksat on the
  function
• Two commonly used values are w1 (madogram) and
  w1/2 (rodogram).

11

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

12

• In general covariance and variogram functions
  depend on both locations z1 and z2.. 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 z1 and z2, 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????

13
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

14
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?

15
What is the relationship between the covariance
and variogram functions?
16
What do typical covariance and variogram
functions look like?
17
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.

18
Examples of Covariance-Spectrum Pairs
19
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

20
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.

21
Concept of Ergodicity

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

22

• 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

23
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(xh)-z(x) has a finite variance which does
  not vary with x
• Thus second order stationarity implies the
  intrinsic hypothesis but not vice-versa