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GEOSTATS: non-stationarity: to be or not to be

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  • Donald Myers
    First it is important to clarify what you mean when you refer to stationarity. In geostatistics there are two forms of stationarity that are used or referred
    Message 1 of 1 , May 28, 1998
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      First it is important to clarify what you mean when you refer to
      stationarity. In geostatistics there are two forms of stationarity that are
      used or referred to: second order and intrinsic. Both pertain to the random
      function and NOT to data. Second order stationarity requires that (1) the
      mean be constant (with respect to position) and (2) that the covariance
      function be a function of the separation vector rather than of the two
      positions. Covariance functions must be positive definite and also bounded.

      Intrinsic stationarity (sometimes referred to as the "intrinsic hypothesis)
      requires that the first order increments have mean zero and that the
      variance of the first order increment be a function of the separation
      vector (rather than of the two positions). Variograms must be negative
      conditionally definite and do not have to be bounded. Every covariance
      corresponds to a variogram but not every variogram corresponds to a
      covariance (e.g., a power model variogram). Note that second order
      stationarity implies that the variance is a constant whereas intrinsic
      stationarity does not require or imply that the variance exists and in
      particular that it is finite.

      The problem is that in essentially all applications encountered in
      geostatistics there are no replications, i.e., the data is considered as a
      (non-random) sample from one realization of the random function.
      Consequently one can not test for stationarity, i.e., statistical test.
      Instead one has to rely on various empirical checks to see whether one or
      the other of the two forms of stationarity is reasonable.

      One of the simplest is to make a bi-plot of the data values vs the position
      coordinates (one coordinate at a time). Analytically this means to fit a
      trend surface and ask whether the coefficients are non-zero.

      Note that even though a variogram does not have to be bounded it can not
      grow at exponential rate, in fact it must grow at less than a quadratic
      rate. The usual sample/empirical variogram does not estimate the
      variogram, i.e. 0.5Var{Z(x+h)-Z(x)}. Rather it estimates 0.5E{
      [Z(x+h)-Z(x)]^2}. If E{Z(x+h) - Z(x)} = 0 then the two are the same. If
      this latter expected value is linear (or higher order) in one or more of
      the position coordinates then the sample variogram will exhibit more than
      quadratic growth. This is evidence of non-stationarity.

      The optimal way to estimate the non-constant mean is by kriging but of
      course this requires knowing the variogram/covariance (which comes first
      the chicken or the egg). In practice one must attempt to estimate/model the
      non-constant mean (least squares, median polish, etc), then compute
      residuals and use the residuals to estimate/model the variogram/covariance.
      There are some problems with using least squares, there will be some bias
      in the sample variogram.

      There is no substitute for exploratory analysis of the data set both before
      and after variogram estimation/modeling.

      You might want to look at a paper of mine in Math Geology concerning
      stationarity, circa 1988-89, a paper by Journel and Rossi in Math Geology
      entitled "when do we need a trend model?" and a paper by two others showing
      explicitly the relationship between a trend surface and universal kriging
      with a pure nugget model.

      Finally note that since the kriging equations (simple, ordinary, universal
      and point or block) do not directly depend on the data values, only the
      data locations. Hence any valid variogram will produce results but
      obviously if the variogram is a poor fit or the stationarity conditions are
      badly violated then the results will bear little resemblence to the data.
      It is also likely that the results will be of any use.

      Donald Myers

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