Derivative in the X, Y, Z Direction (DRVXYZ)

Use the Derivative option to apply a derivative filter in the X, Y, or Z direction.

Derivative Filter dialog options

Direction

The direction of the derivative. The default is Z.

Order of differentiation

Order n of derivation. The default is to apply a 1st order derivative.

See the Application Notes below for a mathematical exemplification of the vertical derivative.

Application Notes

Derivation direction Filter operator

X

 L(u) = (i μ) n

Y

 L(v) = (i v) n

Z

 L(r) = r n

The filter operator in wavenumber domain is a function of the wavenumber domain variables μ and ν. Mathematically, the Wavenumber (Fourier) transform F(μ,v) of a 2D space domain function f(x,y) is defined as:

[1]

Where:

  • x, y are the space domain variables
  • μ, v are the wavenumber domain variables

The x derivative of (1) is:

  [2]

Comparing (1) & (2):

  • The X derivative is: - iμ F(μ,v ) (2a)
  • By analogy, the Y derivative is: - iν F(μ,v ) [3]

For potential field data, Laplace’s equation states that the second vertical derivative of the field is 0: ∇^2 φ=0

That is: 

[4]

This can be transformed to the Wavenumber domain by taking the derivative of the derivatives X(2) & Y(3) again and express it as:

[5]

Referring back to (4):

  • The filter for the Z second derivative will then be:
  • The 1st vertical derivative – the Z derivative, will be:

The horizontal derivative filter operators present a similar curve to the vertical derivative, but they are out of phase relative to the amplitude of the transform grid.

  • The horizontal derivative can be used for creating shaded images and is required for some modelling algorithms, such as Euler deconvolution.

    • Wavenumber domain variable definition

     

    k

    		 Wavenumber domain increment, used to depict a radially symmetrical variable.

    where:

    np is the number of points

    cs is the cell size

    u

    		X component in the wavenumber domain. k = 2π ( i μ+j ν )

    v

    		Y component in the wavenumber domain.

     

    r

    		Radial component in the wavenumber domain.

    also 2πk

    θ

    		 Polar component in the wavenumber domain.

    See Also: