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Popular  Methods  in the  Light  of  Modern  Geostatistics  257

        In  light  of the  expression




                                               2
        and  letting t  =  t',  Equation  12.47  yields X (t)  =  K x(0)  +  const,  x t 2H ',
        where H  =  v +  1/2.  In other  words, for an appropriate  range of /f-values the
        generalized  RF-i/ X(t}  can be considered as an FRF with H  =  v +  1/2.  The
        determination  of the H  and v  values depends on the situation.  Assume, e.g.,
        that X(t)  is a  Wiener  (Brownian)  RF. This  process is a generalized temporal
        RF with v =  0, which  has a zero mean and  covariance




        where a  is a constant  and r  =  \t — t'\.  In light  of  Equation  12.47,  Equation
        12.50  implies  that  K X(T)  =  —ar,  bo(t)  =  at,  and &o(*')  =  ott';  hence,
        H  =  1/2.  Equation  12.49  is satisfied  for H  =  1/2  and,  thus,  the Wiener
        (Brownian)  process X(t)  is an FRF-1/2, i.e.,  it is valid that X(ct)  =  ^/cX(t}.
        Furthermore,  a Gaussian  RF with v =  0 and K X(T)  = —ar 2H  (0 < H  <  1) is
        an  PRP-H,  also  called a fractional  Brownian  RF.  D
            The  greater  part  of  the  above  analysis  can  be  extended to  multi-FRF
        models.  The  latter include random fields that can be characterized by Equation
        12.44.  In this  case, the  fractal  exponent  is a function  of  the  space/time  point
        p,  i.e.,  the  exponent  must  be written  as H(p).  The  class  of  S/TRF-i///Lt is
        considerably  wider  than  these of  FRF.  The  former,  e.g., can  be  characterize
        by  generalized covariances that  have forms  more complicated  than  Equations
        12.46  and  12.48.

        The   class  of  wavelet RF

        As  we saw above, the  £>-spaces  of  test  functions  q  in  generalized S/TRF  are
        selected on the  basis of the physics of the situation  and the goals of the analysis.
        In  several  physical  applications,  the  test  functions  that  are  selected  possess
        properties  that  can  be  used  to  study  certain  important  characteristics of  the
        random  field  of  interest  (e.g., singularities  caused  by physical laws, multiscale
        features,  and sharp  variations).
            We start  by using the  fundamental generalized S/TRF  concept  to  define
        the  wavelet  random  field  (WRF)  model  in  the  space/time  domain.  For this
        purpose, we select a test  function  such as




        where ^(p, u, a)  is called the mother  space/time  wavelet, and the coefficient
        A(a)  is chosen so that  it  magnifies or  reduces the  sensitivity  of  the  field  to
        different  scale  parameters; Oj  and s,  (i  =  1, ..., n)  are the  spatial  scale and
        translation  parameters,  respectively,  whereas  the  at  and  Ut  are the  temporal
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