Spectral and self-affine analysis

Spatial statistics ask how much a surface varies. Spectral analysis asks at which lateral scales that variation occurs.

Two-dimensional Fourier transform

For a sampled height field \(z(x,y)\), the discrete Fourier transform maps position to spatial frequency:

\[ Z(q_x,q_y)=\sum_{m=0}^{N_y-1}\sum_{n=0}^{N_x-1} z_{m,n}\exp\left[-2\pi i\left(\frac{nk}{N_x}+\frac{m\ell}{N_y}\right)\right]. \]

\(q_x\) and \(q_y\) are spatial frequencies in m\(^{-1}\) when pixel spacing is in metres. A wavelength is \(\lambda=1/q\) under the cycles-per-length convention. The power spectral density is proportional to \(|Z|^2\) with normalization that depends on the discrete convention; comparisons require the same convention.

Radial PSD

For an approximately isotropic surface, bins of equal \(q=\sqrt{q_x^2+q_y^2}\) are averaged to obtain a one-dimensional radial PSD.

A roll-off separates a long-wavelength plateau from the fitted self-affine band. Only the declared frequency interval contributes to the slope.

Hurst exponent and fractal dimension

For a two-dimensional self-affine height field, a common convention is

\[ \operatorname{PSD}(q)\propto q^{-\beta},\qquad \beta=2H+2, \qquad D=3-H. \]

Here \(H\) is dimensionless and, under the model, typically lies between 0 and 1; \(D\) is the corresponding surface fractal dimension between 2 and 3. These relations apply only when a stable power-law band exists. Reporting a slope from a visually selected narrow band is not a universal material property.

Correlation length

The correlation length \(\xi\) (m) is a characteristic lateral scale at which height correlations decay or the PSD changes regime. Its exact estimator and threshold convention matter. It is not automatically the grain diameter or a tip radius.

Sampling and bandwidth limits

• Field width \(L\) limits the smallest resolved spatial frequency to order \(1/L\).
• Pixel spacing \(\Delta x\) limits the highest frequency to the Nyquist value \(1/(2\Delta x)\).
• Leveling suppresses low-frequency content and can move the apparent roll-off.
• Line artifacts create directional spectral power that radial averaging can hide.
• Tip convolution suppresses high-frequency structure; the measured PSD is not the unconvolved surface PSD.
• Windowing trades leakage for spectral resolution and must be reported.

SPM-Kit implementation

spmkit.core.analysis.spectral.radial_psd computes the radial representation; fractal_dimension fits a declared log-log region and returns PSD slope, Hurst exponent, fractal dimension and R²; correlation_length estimates the lateral scale. The CLI path is:

spmkit psd scan.nid --channel Z-Axis

The command applies plane leveling before the spectral calculation. In Fathom, use Espectral (spectral).

Evidence and limitation

The spectral functions have software and synthetic tests (LEVEL 1/path-specific numerical recovery). They have not been promoted by the current external Gwyddion campaigns, which cover Sa, Sq and Sz only. Finite-size, window and tip effects remain interpretation limits even when a synthetic slope is recovered.

Roughness · Resonance