  The representation of subgrid-scale orography is a challenge in the physical
parameterization of orographic gravity-wave sources in weather forecasting. A
significant hurdle is encoding as much physical information with as simple a
representation as possible. Other issues include scale awareness, i.e., the
orographic representation has to change according to the grid cell size and
usability on unstructured geodesic grids with non-quadrilateral grid cells.
This work introduces a novel spectral analysis method approximating a
scale-aware spectrum of subgrid-scale orography on unstructured geodesic grids.
The dimension of the physical orographic data is reduced by more than two
orders of magnitude in its spectral representation. Simultaneously, the power
of the approximated spectrum is close to the physical value. The method is
based on well-known least-squares spectral analyses. However, it is robust to
the choice of the free parameters, and tuning the algorithm is generally
unnecessary. Numerical experiments involving an idealized setup show that this
novel spectral analysis performs significantly better than a straightforward
least-squares spectral analysis in representing the physical energy of a
spectrum. Studies involving real-world topographic data are conducted, and
reasonable error scores within $\pm 10\%$ error relative to the maximum
physical quantity of interest are achieved across different grid sizes and
background wind speeds. The deterministic behavior of the method is
investigated along with its principal capabilities and potential biases, and it
is shown that the error scores can be iteratively improved if an optimization
target is known. Discussions on the method's limitations and broader
applicability conclude this work.
