## Mathematical background

The superposition of successive three-dimensional heterogeneous deformations can be expressed by a single Lagrangian equation triplet describing the superposed heterogeneous finite deformation field. This superposition is not commutative and the resulting finite deformation will differ if the order of superposition is reversed.

A deformation, which describes the transformation of an initial coordinate (x, y) to the another coordinate (x1, y1):

 (1)

is superposed by another transformation:

 (2)

deforming (x1, y1) to (x2, y2). The total finite displacement field combining Eq. (1) and (2) is given by:

 (3)

By differentiating Eq. (3) it is possible to obtain the nine components of the three-dimensional displacement gradient tensor d in Lagrangian form:

 (4)

Most of the kinematic forward modelling programs use for the functions ƒ in Eq. 1-3 a sinusoid function or Fourier series describing similar folds. Although these models do not consider layer competence contrasts that might influence the fold geometry by progressive amplification and deamplification of the layers, similar folds are mathematically simple to implement in kinematic models and a good approximation to study the geometry of natural interference structures. A simplest form of a similar fold with a vertical axial surface parallel to the xz coordinate plane with sinusoidal cross-sectional form can be mathematically described by a heterogeneous displacement:

 (5)

where a is the shear amplitude of the fold. Therefore the displacement tensor d in Lagrangian form is:

 (6)

The computer program Noddy (Jessell and Valenta, 1996) uses a more developed mathematical description of similar-type folds, where the heterogeneous displacement is defined by:

 (7)

Parameter c controls the fold cylindricity and w is the fold wavelength. The displacement tensor d in Lagrangian form is obtained by differentiating Eq. 7:

 (8)

Any spatial orientation of the modelled fold for subsequent superposition of another fold can be obtained by a displacement tensor R in Lagrangian form defining the rotation around a unit vector ν by an angle of α.

 (9)

Results can be either displayed by plotting a particular folded and refolded initial surface in three-dimensional space (structure-plot) or by visualizing fold interference patterns on arbitrary cross sections through the structure (pattern-plot).