# Random rough surfaces

The generation of stochatsticly rough surfaces is controlled in Tamaas by two abstract classes: `tamaas::SurfaceGenerator`

and `tamaas::Filter`

. The former provides access lets the user set the surface sizes and random seed, while the latter encodes the information of the spectrum of the surface. Two surface generation methods are provided:

`tamaas::SurfaceGeneratorFilter`

implements a Fourier filtering algorithm (see Hu & Tonder),`tamaas::SurfaceGeneratorRandomPhase`

implements a random phase filter.

Both of these rely on a `tamaas::Filter`

object to provided the filtering information (usually power spectrum density coefficients). Tamaas provides two such classes and allows for Python subclassing:

`tamaas::Isopowerlaw`

provides a roll-off powerlaw,`tamaas::RegularizedPowerlaw`

provides a powerlaw with a regularized rolloff.

Tamaas also provided routines for surface statistics.

## Generating a surface

Let us now see how to generate a surface. Frist create a filter object and set the surface sizes:

```
import tamaas as tm
# Create spectrum object
spectrum = tm.Isopowerlaw2D()
# Set spectrum parameters
spectrum.q0 = 4
spectrum.q1 = 4
spectrum.q2 = 32
spectrum.hurst = 0.8
```

The `spectrum`

object can be queried for information, such as the root-mean-square of heights, the various statistical moments, the spectrum bandwidth, etc. Then we create a generator object and build the random surface:

```
generator = tm.SurfaceGeneratorFilter2D([128, 128])
generator.spectrum = spectrum
generator.random_seed = 0
surface = generator.buildSurface()
```

Important

The `surface`

object is a `numpy.ndarray`

wrapped
around internal memory in the `generator`

object, so a subsequent call to
`buildSurface`

may
change its content. Note that if `generator`

goes out of scope its memory
will not be freed if there is still a live reference to the surface data.

Important

If ran in an MPI context, the constructor of
`SurfaceGeneratorFilter2D`

(and others) expects the *global*
shape of the surface. The shape can also be changed with ```
generator.shape =
[64, 64]
```

.

It is common to normalize a surface after it has been generated so that one can scale the surface to a desired stastistic, e.g. to specify the root-mean-square of slopes:

```
rms_slopes = 0.25
surface /= iso.rmsSlopes()
surface *= rms_slopes
# Compute root-mean-square of slopes in Fourier domain
print(tm.Statistics2D.computeSpectralRMSSlope(surface))
# Compute root-mean-square of slopes with finite differences
# (introduces discretization error)
print(tm.Statistics2D.computeFDRMSSlope(surface))
```

Note

The spectrum object gives the expected value of surface statistics, but the
corresponding value for a surface realization can differ (particularily if
the `SurfaceGeneratorFilter2D`

is used). One could normalize a
surface with the *actual* statistic, but this leads to **biased** quantities
when a representative sample of surfaces is used.

## Custom filter

Tamaas provides several classes that can be derived directly with Python classes, and `tamaas::Filter`

is one of them. Since it provides a single pure virtual method `computeFilter`

, it is easy to write a sub-class. Here we implement a class that takes a user-defined auto-correlation function and implements the `computeFilter`

virtual function:

```
import numpy
class AutocorrelationFilter(tm.Filter2D):
def __init__(self, autocorrelation):
tm.Filter2D.__init__(self)
self.autocorrelation = autocorrelation.copy()
def computeFilter(self, filter_coefficients):
shifted_ac = numpy.fft.ifftshift(self.autocorrelation)
# Fill in the PSD coefficients
filter_coefficients[...] = numpy.sqrt(np.fft.rfft2(shifted_ac))
# Normalize
filter_coefficients[...] *= 1 / numpy.sqrt(self.autocorrelation.size)
```

Here `filter_coefficients`

is also a `numpy.ndarray`

and is therefore easily manipulated. The creation of the surface then follows the same pattern as previously:

```
# Create spectrum object
autocorrelation = ... # set your desired autocorrelation
spectrum = AutocorrelationFilter(autocorrelation)
generator = tm.SurfaceGenerator2D()
generator.shape = autocorrelation.shape
generator.spectrum = spectrum
surface = generator.buildSurface()
```

The lifetime of the `spectrum`

object is associated to the `generator`

when `setSpectrum`

is called.

## Surface Satistics

Tamaas provides the C++ class `tamaas::Statistics`

and its wrapper `Statistics2D`

to compute statistics on surfaces, including:

power spectrum density

autocorrelation

spectrum moments

root-mean-square of heights \(\sqrt{\langle h^2 \rangle}\)

root-mean-square of slopes (computed in Fourier domain) \(\sqrt{\langle |\nabla h|^2\rangle}\)

All these quantities are computed in a discretization-independent manner: increasing the number of points in the surface should not drastically change the computed values (for a given spectrum). This allows to refine the discretization as much as possible to approximate a continuum. Note that the autocorrelation and PSD are fft-shifted. Here is a sample code plotting the PSD and autocorrelation:

```
psd = tm.Statistics2D.computePowerSpectrum(surface)
psd = numpy.fft.fftshift(psd, axes=0) # Shifting only once axis because of R2C transform
import matplotlib.pyplot as plt
from matplotlib.colors import LogNorm
plt.imshow(psd.real, norm=LogNorm())
acf = tm.Statistics2D.computeAutocorrelation(surface)
acf = numpy.fft.fftshift(acf)
plt.figure()
plt.imshow(acf)
plt.show()
```

See `examples/statistics.py`

for more usage examples of statistics.