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	Specifies the direction of the derivative to be applied: X, Y, or Z. The default direction is typically Z.
Order of differentiation	Sets the order n of the derivative. The default is a 1st‑order derivative. See the Application Notes below for a mathematical illustration of the vertical derivative.

Application Notes

Filter Operators by Direction

The following table summarizes the mathematical operators for the derivative filters in the wavenumber domain.

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 the wavenumber domain is expressed in terms of the wavenumber domain variables μ and ν.

Wavenumber (Fourier) Transform Definition The two-dimensional Fourier transform converts a spatial function f(x, y) into its wavenumber domain representation F(μ, ν). This new function describes the amplitude and phase of each wavenumber component that makes up the original spatial function.

Mathematically, the Wavenumber (Fourier) transform F(μ,v) of a 2D space domain function f(x,y) is defined as:

Where:

• x, y:  Spatial domain variables
• μ, v: Wavenumber domain variables
• i: The imaginary unit (√-1)

X‑ and Y- Derivatives in the Wavenumber Domain

The x derivative of Eq. (1) is:

 

Comparing (1) & (2), the Fourier transform of the X derivative is:

By analogy, the Fourier transform of the Y derivative is:

Vertical Derivatives and Laplace’s Equation

The vertical derivative is derived using Laplace's equation. For potential fields (like gravity or magnetism) in a source-free region, the field ϕ satisfies Laplace's equation, which states that the divergence of its gradient is zero.

Laplace's Equation (Spatial Domain):

In Cartesian coordinates, this is expressed as: 

    

This can be transformed to the Wavenumber domain to express the vertical second derivative in terms of the horizontal second derivatives:

    

Referring back to (4):

The second vertical derivative filter (in the wavenumber domain) is:

Therefore, the operator for the first-order vertical derivative in the wavenumber domain is:

Interpretation of Derivative Filters

• The horizontal derivatives have similar magnitude response curves to the vertical derivative. However, they are out of phase relative to the amplitude of the transform grid.

• Horizontal derivatives are commonly used for creating shaded-relief images and are required for specific modelling algorithms such as Euler deconvolution.

Wavenumber Domain Variable Definitions

The following variables are used in the wavenumber domain:
k	Wavenumber increment, used to depict a radially symmetrical variable.	Where: np: number of points cs: cell size
μ	X-component of the wavenumber.
v	Y-component of the wavenumber.
r	Radial component of the wavenumber.
θ	Angular (polar) component of the wavenumber.