Junyi Guo
One of the common mathematical topics of Earth and biological sciences is the treatment of data of global coverage. In many instances, data of global coverage of a quantity, e.g., the atmospheric pressure in Earth science or the density of population of a species in bioscience, are values of the quantity as a function defined on the Earth’s surface. Therefore, the mathematics behind is the representation of functions defined on a sphere. The most primitive representation of the function is to use its values over an array of points, e.g., over the intersection points of parallels of equal latitude intervals and meridians of equal longitude intervals, which are referred to as grid values. However, there should be a need of additional information in the interpretation of the data in the form of grid values: How large area the data are representative? If the data represent the averages of the quantity in some way within an area of, say, 50 km around the grid points, the data are said to have a resolution of 100 km (half wavelength). It is evident that data of higher resolution (with smaller wavelength) can be converted to data of lower resolution (with larger wavelength), but the reverse cannot be done. In this talk, we convert the data into a series and then back, so that lower resolution data can be obtained by truncating the series of the higher resolution data. Two kinds of series on a sphere will be discussed. One is the Cartesian product of a cosine series over the colatitude and a Fourier series over the longitude, and the other is the spherical harmonic series. Each kind of series has its characteristic with advantage and disadvantage for a specific application. Take for example a grid of 1∘×1∘, i.e., with equal intervals of 1∘ in both the latitude and longitude. We have 180×360 grid data (taking 180 grid points over the latitude). We can define a cosine-Fourier series with the same number of coefficients, and the grid data can be exactly recovered from the series. However, if we use the spherical harmonics, we can only expand to degree and order 179, which has 180×180 coefficients, meaning that the 180×360 grid data cannot be necessarily exactly recovered from the series. The resolution of the cosine-Fourier series is expressed as an interval in latitude and an interval in longitude; In terms of distance over the sphere, the longitudinal resolution (the length of the parallel in 1∘ longitude interval) at latitude ±60∘ is half of that over the equator (the length of the equator in 1∘ longitude interval). However, the resolution of the spherical harmonic series is homogeneous and isotropic all over the sphere in terms of distance. The choice of one over the other should be done based on the characteristic of the application.
