Double Fourier sphere method

The Double Fourier Sphere (DFS) method is a simple technique that transforms a function defined on the surface of the sphere to a function defined on a rectangular domain while preserving periodicity in both the longitude and latitude directions.

Introduction

First, a function $f(x,y,z)$ on the sphere is written as $f(\lambda ,\theta )$ using spherical coordinates, i.e.,

$f(\lambda ,\theta )=f(\cos \lambda \sin \theta ,\sin \lambda \sin \theta ,\cos \theta ),(\lambda ,\theta )\in [-\pi ,\pi ]\times [0,\pi ].$

The function $f(\lambda ,\theta )$ is $2\pi$-periodic in $\lambda$, but not periodic in $\theta$. The periodicity in the latitude direction has been lost. To recover it, the function is "doubled up” and a related function on $[-\pi ,\pi ]\times [-\pi ,\pi ]$ is defined as

${\tilde {f}}(\lambda ,\theta )={\begin{cases}g(\lambda +\pi ,\theta ),&(\lambda ,\theta )\in [-\pi ,0]\times [0,\pi ],\\h(\lambda ,\theta ),&(\lambda ,\theta )\in [0,\pi ]\times [0,\pi ],\\g(\lambda ,-\theta ),&(\lambda ,\theta )\in [0,\pi ]\times [-\pi ,0],\\h(\lambda +\pi ,-\theta ),&(\lambda ,\theta )\in [-\pi ,0]\times [-\pi ,0],\\\end{cases}}$

where $g(\lambda ,\theta )=f(\lambda -\pi ,\theta )$ and $h(\lambda ,\theta )=f(\lambda ,\theta )$ for $(\lambda ,\theta )\in [0,\pi ]\times [0,\pi ]$. The new function ${\tilde {f}}$ is $2\pi$-periodic in $\lambda$ and $\theta$, and is constant along the lines $\theta =0$ and $\theta =\pm \pi$, corresponding to the poles.

The function ${\tilde {f}}$ can be expanded into a double Fourier series

${\tilde {f}}\approx \sum _{j=-n}^{n}\sum _{k=-n}^{n}a_{jk}e^{ij\theta }e^{ik\lambda }$