Page 130 - Modern Spatiotemporal Geostatistics
P. 130

1
                                                                       1
                                                                       1
                                                                       1
                                                                        1
                                                                        1
                                                                        1
                                                                        1
                                              th
                                              th
                                              th
                                           o
                                           o
                                           o
                                               e
                                                  BM
                                                  BM
                                              the
                                               e
                                               e
                                           o
                               Formulation
                               Formulatio
                               Formulatio
                Mathematica
                Mathematical
                Mathematica
                                         n
                                         n
                                         n
                               Formulatio
                Mathematica
                Mathematica  l l l l  Formulatio n  of f f f f  th e  BME  Method  1111 1 1 1
                                                  BM
                                                        Metho
                                                        Metho
                                                        Metho
                                                        Metho
                                                             d
                                                             d
                                                             d
                                                             d
                                                  BM
                                                     E
                                                     E
                                                     E
                                                     E
                                  physica
                                  physica
                                o
                                o
                                o
                                  physical laws..
                   BM
               5.1
                   BM
                                  physica
                   BM
                      E
                      E
                      E
             e
             e
                        analysi
                        analysi
                   BME
             e
                        analysis off
                        analysi
         Figure
               5.1.
         Figur e  5.1 . . . .  BM E  analysi s s s s  o f f f  physica l l l l  laws . . .
                                         laws
                                         laws
               5.1
         Figur
                                         laws
         Figur
               5.1
         Figur
                                                        to
                                              summation
             are
                                                                   kriging
                          in
             are
                                                                   kriging
                                                                   kriging
             are
                    again
         data
                    again
         data           are         used         again          in          terms         of          a  weighted          summation          to         derive          the          kriging
                    again
                                              summation
                                              summation
         data
         data
         data are   used  again   in  terms   of  a weighted  summation   to  derive  the   kriging
                          in
                used
                                    a
                                    a
                                  of
                                  of
                                  of
                                      weighted
                used
                used
                                    a
                                      weighted
                                      weighted
                                                                the
                            terms
                            terms
                            terms
                                                                the
                                                                the
                                                          derive
                                                          derive
                                                          derive
                          in
                                                        to
                                                        to
                 (see
                 (see
                 (see
                                       p.
                                       p.
                                   7.2,
                                       p.
                                         14$)-
                                         14$)-
                                         14$)-
                                   7.2,
                      also
                      also
                      also
                         Comment
                                   7.2,
                         Comment
                         Comment
         estimates
         estimates (see  also  Comment   7.2, p. 143)-
         estimates
         estimates
         estimates            (see  also          Comment          7.2,  p.           14$)-
             Th
             Th
                                            h
             Th
                                              th
                                              th
                                              thee
                                              th
                                                                m
                                          wit
                                          wit
                                                                m
                                                                m
                                                                m
                                          wit
             Th
                                            h
                                            h
                          p
                          p
             Thee  solutio n  ste p  (c ) ) )  i i i is concerned withh  th e  solutio n  off f f f  a system of equa-- - - -
                                                        o
                                                        o
                                                        o
                                                           a
                                                           a
                                                        o
                                                 solutio
                                                      n
                      n
                      n
                                                 solutio
                                                 solutio
                                                      n
                                                             syste
                                               e
                                               e
                                                             syste
                                                             syste
                                                             syste
                        ste
                                                 solutionn
                                                           a
                                                           a
                        stepp
                        ste
                        ste
                                                                   o
                            (c))
                                                                     equa
                                                                     equa
                            (c
                            (c
                            (c
                                                                    f
                                                                    f
                                                                    f
                                                                     equa
                                                                     equa
                                                                    f
                               s
                               s
               e
               e
               e
                               i
                                 concerne
                                 concerne
                                 concerne
                               s
                               s
                                 concerne
                 solutio
                                          wit
                                        d
                                        d
                                                                   o
                                                                   o
                                                                   o
                                        d
                 solutio
                 solutio
                 solutionn
                                        d
                                                 s
             fo
                            multipliers
                                         e
                            multipliers
                                         e
                                                   ar
                                                    e
                                                    e
                                                   ar
                                                 s
                                                 s
                                                   ar
                          e
                          e
                                     Thes
                                     Thes
                                     Thes
                                     These equations are obtainedd
                          e
                                         e
             fo
                                           equation
                                           equation
                            multipliers
                                           equation
           s
             fo
           s
           s
                                                    e
        tions for thee
                                                             b
                th
                                                                substitutin
                                                                substitutin
                                                            d
                                                            d
                                                             b
                                                             by substitutingg
                                                             b
                   Lagrang
        tion
                                                                substitutin
                 e
        tion
                 e
                th
                th
                   Lagrang
                   Lagrang
                   Lagrange multipliers..
                                                                        g
                                                      obtaine
                                                      obtaine
                                                                        g
        tion s  fo r r r r  th e  Lagrang e  multipliers . . .  Thes e  equation s  ar e  obtaine d  b y y y y  substitutin g
        tion
                                                      obtaine
                                                                  stepp
                                         into thee
                      pdf formulaa
                                         int
                                         int
                                         int
                                (Eq
                                (Eq
                                                                o
                                                                o
                                (Eq
            appropriatee
            appropriat
            appropriat
            appropriat
        th e  appropriat e  pd f f f f  formul a  (Eq.. . . .  5.6 ) ) )  int o  th e  momen t t t  equations off f f f  ste p  (a).. . . .
                                                                       (a)
                                                                       (a)
                                                                     p
                                                                       (a)
                                                                       (a)
                                     5.6
                                     5.6))
                                     5.6
                                     5.6
                                                                     p
                                                                  ste
                                                                  ste
        th
                                                                  ste
        thee
        th
        th
          e
          e
                                                                o
                                                              s
                                                              s
                                                              s
                                                              s
                          formul
                                                       equation
                                                       equation
                          formul
                          formul
                              a
                                                momentt
                                              e
                    e
                    e
                                              e
                                                momen
                                                momen
                              a
                                                momen
                                                       equation
                                           o
                      pd
                      pd
                                (Eq
                                                                o
                      pd
                                             th
                                             th
                                                       equation
                                           o
                                             th
                                           o
                                 law
                                 law
                                                                        e
                               c
                                                                        e
            e
                               c
                                 laws these are integrall
                                 law
                               c
                                             integra
                                             integra
                                                               th
                                                               th
                                             integra
                                           e
                                          ar
                                           e
                                           e
                                                    equation
                                                    equations for thee
                                                    equation
                                                    equation
                                                            fo
        Whil
                                                               th
                                                            fo
                                                            fo
                                        e
                                        e
                                                                  Lagrangee
                                                                  Lagrang
                                        e
                                      thes
                                      thes
        While in the case of algebraicc  law s s s s  thes e  ar e  integra l l l  equation s s s s  fo r r r r  th e  Lagrang e
                                      thes
                                          ar
                                          ar
        Whil
        Whil
        Whil
                                                                  Lagrang
                                                                  Lagrang
                                                                e
                                                                e
            e
                        f
              n
                th
                        f
                   cas
                        f
                        f
                th
                th
                        o
                th
                        o
                        o
                        o
                 e
                 e
              n
                 e
              n
              n
                 e
                          algebrai
                          algebrai
                          algebrai
                          algebrai
                   cas
                   cas
                   cas
              i
                      e
                      e
              i
                      e
            e
            e
                      e
              i
              i
         multipliers
         multipliers
         multipliers
                                                          m
                                                          m
         multipliers , , ,  i i i i in thee  cas e e e e  o f f f  differentia l l l  law s  the y  hav e  th e  formm  off f f f  integrodiffer - - -
                                                          m
                                            the
                                                            o
                                            the
                                            the
                            o
                                        law
                     th
                            o
                                        law
                     th
                            o
                                        law
                                        laws theyy
                                          s
                                                            o
         multipliers,,
                                                            o
                                          s
                     th
                     th
                                          s
                                                            o
                                                               integrodiffer
                                                               integrodiffer
                                                               integrodiffer
                                                               integrodiffer--
                                                hav
                                                      e
                      e
                                                hav
                                                     th
                                                     th
                      e
                      e
                                                   e
                                                   e
                                                     th
                                                   e
                                                have thee
                        cas
                                                        for
                                              y
                                                        for
                   n
                   n
                                                        for
                                                        for
                        case off
                        cas
                                                hav
                                                      e
                   n
                                              y
                        cas
                   n
                              differentia
                              differentia
                              differentiall
                              differentia
              equations
              equations
              equations
              equations
        entia
        entia
        entiall
        entia
        entia l l l  equations.. . . .
                              n
            Whe
                              n
                                                                      ,
                                                                      ,
                              n
                                                                      ,
                                 e
                                 e
                              n
                                                                      ,
                  physica
                                                   equation
                  physica
                                                               available
                                                   equation
            Whe
                  physica
            Whe
                                                   equation
            Whe
                                          differentia
                  physica
                                   for
                                                   equationss
                                 e
                                                               available
                                          differentia
                                          differentia
                                                               available
                                   for
                                                               available
                                   for
                                     m
                                                           ar
                                     m
                                                           ar
                n
                                                           ar
                        l
                n
                        l
                                                           ar
                        l
                        l
                                th
                                                                       an
            Whenn  physical laws in thee  for m  o f f f f  differentia l l l  equation s s s  are available, an-- - - -
                                   formm
                            s
                            s
                            s
                                th
                            s
                                th
                                                             e
                                                             e
                                                             e
                              i
                                                             e
                              i
                          law
                              i
                          law
                              i
                          law
                          law
                n
                                                                       an
                                                                       an
                                th
                                       o
                                                                       an
                                       o
                                       of differentiall
                                       o
                                 o
                                               differentia
                                               differentiall
                                             e
                                               differentia
                                                               n
                                 o
                                               differentia
                                                               n
                                                        a
                                                        as shown inn
                                                           show
        othe r r r  approac h  (Fig.. . . .  5.1)) ) ) )  i is too  discretize  th e  differentia l l l  a s s s s  show n  i n  Chapte r r r
                               s
                               s
                                                        a
                                                        a
                                                           show
                               i
                    h
                               i
                               i
              approac
                                                           show
                    h
              approac
                               s
                      (Fig
                                 t
                      (Fig
                      (Fig
                                                               n
                                 o
                                 t
                                 t
                                 t
        othe
        othe
                               s
        otherr
              approac
                      (Fig
        othe
              approachh
                                             e
                           5.1
                                                                 n
                                   discretize
                                   discretize
                                            th
                                            thee
                                                                 n
                                            th
                           5.1
                                                                   Chapte
                                                                   Chapterr
                                   discretize
                                   discretize
                           5.1
                           5.1
                                                                   Chapte
                                                                   Chapte
                                                                 i
                                                                 i
                                                                 i
                                            th
                                        n
                          thuss
                          thu
                          thu
                                        n
                          thu
               3.24
                               obtainin
                                                  (difference
                                       a
                               obtainin
                                                  (difference
                               obtainin
               3.24
                                                  (difference
                                                  (difference
                                                                  n
                               obtainingg
         33  (Eq . . .  3.24,, , , ,  p . . .  80) , , ,  thu s  obtainin g  ann  algebraicc c c c  (difference)) ) ) )  equationn  o f f f  th e
                                                                    o
           (Eq
                                                                    off
                                      g
                                      g
                                                                    o
                                                                  n
         3
                                                                    o
                                                                  n
               3.24
         3
         3
               3.24
                                                                         e
                                                                         e
                    p
                                                                       th
                             s
                             s
                                                                       th
                                          algebrai
           (Eq
                                                             equatio
                                                                       thee
                                                                       th
                                                             equatio
                    p
                                                             equatio
                                                             equatio
                                          algebrai
                    p..
                    p
           (Eq..
                                        n
                                       a
                                       a
                      80)
                                       a
                      80)
                                          algebrai
                                          algebrai
           (Eq
                      80),,
                      80)
                   n
                                                                      use--
                   n
                                                           s
                                                           s
                n
                                       Thi
           m
                                                           s
                n
           m
                     Equatio
                                                           s
                     Equatio
                                   77)
                                   77)
                                   77)
                     Equatio
                     Equationn
                                   77)..
                                       Thi
                                       Thi
                                       Thi
                                                        h
                                                        h
                                                             particularl
             give
                                                             particularl
                                (p
                                (p
                                                                    y
                                          s
                                                                    y
             givenn
                                                                    y
                                (p..
                                                                      use
                                                                      use
                                                                      use
                                            approach
                                            approach, whichh
                                          s
                                (p
                                                             particularl
                                                             particularl
                  i
                                          s
        for
        for
                              5
                                                     whic
                           n
        for m  give n  i i i inn  Equatio n  3. 5  (p . . .  77) . . .  Thiss  approach , , , ,  whic h  i i i i is particularlyy  use - - -
                           n
        for
                              5
        formm
                                                     whic
                             3.
                   n
                             3.
                                            approach
                                            approach
             give
                                                     whic
                             3.
             give
                             3.55
                                                               alread
                                                                      bee
                                                               alread
                   involvin
                                                               alread
                                                                      bee
                   involvin
                   involvingg
                                                                      bee
                                                                      beenn
                                                                        n
                                                                        n
              case
              case
              case
              case
                   involvin
                                                             s
                                                             s
                                                             s
                                                  equations
                                                  equations
                                                  equations,,
        full
        fu
        fu
            i
            i
            i
                                   partia
                                   partiall
                                                  equations
                                         differentia
                                         differentia
                                         differentiall
                         g
                         g
                                         differentia
                           comple
                                   partia
                                   partia
                           comple
                           comple
                           comple
                                 x
                                                           ha
                                                           has alreadyy
                                 x
                                                           ha
                                                           ha
                                 x
            n
        fu l l l  i inn  casess s s s  involvin g  complexx  partia l l l  differentia l l l  equations , , ,  ha s  alread y y y  bee n
            n
            n
        fu
                                                                   physica
                                                                   physica
                                         "Genera
                                                                   physica
                                         "General knowledgee
                          3
                          3
                                                                  f
                                                                  f
                                                                  f
                                         "Genera
                                         "Genera
                 n
                 n
                 n
                                                                   physica
               d
                                 section
               d
                              th
               d
                            n
                            n
                                 section,,
                                 section
                                 section
                              th
                 i
                              th
                                e
                                e
                            i
                            n
                            i
                 i
                            i
                                                                  f
                                                               s
                                                 knowledg
                                                               s
                                                               s
        discusse
                                                          n
        discusse
        discusse
                                                          n
                                                          n
                   Chapterr
        discussedd  i i inn  Chapte r r r  3  i n  th e  section , , ,  "Genera l l l l  knowledg e  i i i i in termss  of physicall l l l
                   Chapte
                   Chapte
                   Chapte
                                                        e
                                                        e
                                                 knowledg
                                                 knowledg
                                                                 o
                                                                 o
                                                            term
                                                            term
                          3 in thee
                          3
                                                                 o
                                                                 o
        discusse
                                                            term
                                                            term
                                                          n
                                                     81..
                                                     81
                                                     81
                                                   p
                                   als
                                   als
              beginnin
              beginnin
              beginning onn
                                                     81
                               se
                     g
                               se
                               se
                               see alsoo
                                 e
              beginnin
                                 e
                     g
                                 e
                     g
                                   als
                        n
                        n
                                              3.10
                                              3.10
         laws,""
         laws,
         laws, " " "  beginnin g  o n  p.. . . .  76 ; ; ;  se e  als o  Exampl e  3.10 , , ,  p.. . . .  81 . . .
         laws,
                          p
                          p
                          p
                          p
                                            e
                                              3.10
                                            e
                                            e
         laws,
                            76
                       o
                            76
                            76;;
                                                   p
                                                   p
                       o
                                                   p
                                      Exampl
                                      Exampl
                                      Exampl
                                      Example 3.10,,
                                     o
                            76
                       o
                                     o
   125   126   127   128   129   130   131   132   133   134   135