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2426Re: [ai-geostats] Sequential Gaussian Simulation (SGS) is not reproducing variograms (neither range nor variance)

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  • Andreas Dominik Cullmann
    Feb 26, 2006
      Dear Manuel,
      I don't know if I've got you right, but if you just wonder why the
      semivariogram of the conditionally simulated data isn't equal to the one
      of the conditioning data, this might help:
      If you use the sequential gaussian algorithm for conditional simulation,
      then the expectation E[Z_bs (s)|Z=z] of conditionally simulated values
      at location s is equal to the simple kriging estimator Z*(s), the
      variance var[Z_bs (s)|Z=z] of the cs-values is equal to the simple
      kriging variance [sigma_sk]^2 and the variogram you calculate is
      conditional to the data, too: gamma(s-t|Z=z).
      [This is the same as gamma(s-t)-0.5(c_s-c_t)'S^(-1)(c_s-c_t), where S is
      the covariance of the conditioning data and c_s (c_t) is covariance
      vector between the the data locations and location s (t). ]
      Now, the unconditional measure gamma(s-t) and the conditional
      gamma(s-t|Z=z) _shouldn't_ be the same.

      Manuel Luis Ribeiro wrote:
      > Hello,
      > I have a daily data set with 240 observations of PM10/m3 (particle
      > matter with diameter smaller tham 10um) concentrations measured in 8
      > points (for 30 days).
      > My variogram in time is quite nice (spherical, with 1 structure of 7
      > days). very good fitting.
      > My variogram in space is fitted using a standardized variable, which
      > means that each value z(s,t) in moment t in point s, is transformed in
      > z(s,t)*=[z(s,t)-m(s)]/s(s) where:
      > m(s) is the mean of the 30 days array in point s;
      > s(s) is standard error of the 30 days array in point s.
      > In this case the variogram also fits very nice, but it doesnt reach
      > the sill (my argument to explain this is because most of the
      > varibility is in time domain, not in space domain), and so i have a
      > zonal anisotropy.
      > After transforming anisotropic model in an isotropic model, i wanted
      > to create several realizations of my spatio-temporal process using
      > Sequential Gaussian Simulation (SGS). However i wanted to use my
      > original values z(s,t) (which didn't show any spatial structured
      > phenomenom), instead of z*(s,t). The result is a dramatic variogram
      > where my range and variance aren't reproduced.
      > I would like to know if anybody can help me understanding the problem
      > and how to correct it.
      > Thanks, Manel
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