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The power spectrum of a scalar field gives an indication of the spatial autocorrelation of the field, i.e. the typical size of a patch of similar values.

The power spectrum of a square or rectangular field or matrix can be determined using the Fourier transform or Fourier series.

• How can the power spectrum be determined for a country which is non-rectangular?

For example, the image below of the USA can be encoded into a matrix where all cells covering ocean are filled with NaN, while all cells covering land will have values between 25 and 375.

• What is a good strategy to deal with such large patches of NaNs?


• Can the country be mapped onto a rectangle and still provide a good measure for the original's power spectrum?


• Could spherical harmonics help, by noting the USA is a subsection of a sphere?


• Could a Wavelet transform be applied, and how?
  
  

For a given function $L^2$ function $f$ on the domain $Omega$ (your irregular country), you can define the spectral decomposition as the expansion in the basis of the set of eigenfunctions with Dirichlet boundary condition $left{(k, phi_k);Big|nabla^2 phi_k=-kphi_k, phi_kbig|_{partialOmega}=begin{cases} fbig|_{partialOmega}, &k=0 \0, &k>0end{cases};right}$. The eigenvalue equation is called the Helmholtz equation and its null boundary value problem called the Dirichlet eigenvalue problem. The set of eigenvalue function is $L^2$ norm complete, and thus forming a basis for the $L^2$ function space, and orthogonal which is guaranteed by the spectral theorem of compact self-adjoint operators. Particularly, for $k=0$, this problem is the Laplace's equation Dirichlet boundary value problem with a given function value on the boundary $fbig|_{partialOmega}$. It can be solved directly by the Green's function method through the integration of the double layer potential. So you can expand any $L^2$ function in this basis just like in the Fourier basis. Fourier series is but the same problem with $Omega$ being a rectangle.

The beauty of this method is precisely that the first $phi_0$ is essentially the average of the boundary and thus all the local minima and maxima are on the boundary, while the remaining basis $(phi_k)_{k>0}$ is only boundary geometry (shape) dependent. Incidentally, hearing the shape of a drum is an attempt at the inverse problem from the eigenvalue to the geometry of the boundary.

There is essentially the whole theory of elliptical operators and much of functional analysis behind the above solution of this problem. One has all the existence and uniqueness properties for the solution of the above eigenvalue PDE problems for the smooth boundary and boundary value. One may explore the solution properties for say fractal boundary and nowhere differentiable boundary values.