A comparison of two spectral approaches for computing the Earth response to surface loads

When predicting the deformation of the Earth under surface loads, most models follow the same methodology, consisting of producing a unit response that is then con-volved with the appropriate surface forcing. These models take into account the whole Earth, and are generally spherical, computing a unit response in terms of its spherical harmonic representation through the use of load Love numbers. From these Love numbers, the spatial pattern of the bedrock response to any particular scenario can be obtained. Two different methods are discussed here. The first, which is related to the convolution in the classical sense, appears to be very sensitive to the total number of degrees used when summing these Love numbers in the harmonic series in order to obtain the corresponding Green's function. We will see from the spectral properties of these Love numbers how to compute these series correctly and how consequently to eliminate in practice the sensitivity to the number of degrees (Gibbs Phenomena). The second method relies on a preliminary harmonic decomposition of the load, which reduces the convolution to a simple product within Fourier space. The convergence properties of the resulting Fourier series make this approach less sensitive to any harmonic cut-off. However, this method can be more or less computationally expensive depending on the loading characteristics. This paper describes these two methods, how to eliminate Gibbs phenomena in the Green's function method, and shows how the load characteristics as well as the available computational resources can be determining factors in selecting one approach.