The equations are as follows: ρ a ( s ) = s 2 ∫ 0 ∞ T ( λ ) J 1 ( λ s ) d s ( 4 )Where ρ a is the apparent resistivity and T ( λ ) is a resistivity transform function. Using a Hankel Transform, we get: T ( λ ) = ∫ 0 ∞ ρ a ( s ) J 1 ( λ s ) [ 1 s ] d s ( 7 )New variables are then defined by x = l n ( s ) a n d y = l n ( 1 λ ) ( 8 )Substituting (8) in (7), we get T ( y ) = ∫ − ∞ ∞ ρ a ( x ) J 1 ( 1 e y − x ) d x ( 9 )Please note that any missing equations were for another resistivity equation.

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The equations are as follows:      ρ    a    (  s  )  =      s    2        ∫          0        ∞    T  (  λ  )      J    1    (  λ  s  )  d  s    (  4  )Where       ρ    a   is the apparent resistivity and   T  (  λ  ) is a resistivity transform function. Using a Hankel Transform, we get:  T  (  λ  )  =      ∫          0        ∞        ρ    a    (  s  )      J    1    (  λ  s  )      [          1      s        ]    d  s    (  7  )New variables are then defined by  x  =  l  n  (  s  )    a  n  d    y  =  l  n      (          1      λ        )      (  8  )Substituting (8) in (7), we get  T  (  y  )  =      ∫          −      ∞        ∞        ρ    a    (  x  )      J    1        (          1              e                  y          −          x                      )    d  x    (  9  )Please note that any missing equations were for another resistivity equation.

Elliana Shelton · Accepted Answer

Starting with  T  (  λ  )  =      ∫          s      =      0              s      =      ∞            ρ    a    (  s  )      J    1    (  λ  s  )      [          1      s        ]    d  s    (  7  )and using  x  =  ln  ⁡  (  s  )    ⇒        e    x    =  s        and      d  x  =            1      s        d  s  ln  ⁡  (  0  )  →  −  ∞        and      ln  ⁡  (  ∞  )  →  ∞you now have  T  (  λ  )  =      ∫          x      =      −      ∞              x      =      ∞            ρ    a    (  x  )      J    1    (  λ      e    x    )  d  xThen using  y  =  ln  ⁡      (                  1        λ              )      ⇒        e    y    =            1      λ          ⇒        λ    =            1              e        y            you now have                                  T        (        y        )                            =                  ∫                      −            ∞                    ∞                          ρ          a                (        x        )                  J          1                          (                                    1                              e                y                                                          e            x                    )                d        x                                                        =                  ∫                      −            ∞                    ∞                          ρ          a                (        x        )                  J          1                          (                                    1                              e                                  y                  −                  x                                                              )                d        x                (        9        )

The equations are as follows: ρ a </msub> ( s ) =

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