Aliasing and noise in core-surface flow inversions

SUMMARY The potential pitfall that core flow inversions may be aliased as a result of model underparametrization is considered. Synthetic tests involving randomly generated flows with differing energy spectra have been used to explore this problem. If underparametrizing neglects terms comparable to the largest of those already modelled, only a poor representation (not an average in space or time) of the actual flow that generated the data is obtained. It is found that the key aspects of the flow that determine whether an underparametrized inversion of the field produced by that flow will be successful are its temporal and spatial energy spectra: when the spectra fall off as (degree)−2, or faster, underparametrized inversions yield accurate results, but if the decrease is slower, the spatial and/or temporal aliasing is severe enough to corrupt the solution at all degrees. Damping does not appear 10 remedy this difficulty but in fact obscures it by forcing the flow to converge upon a single, but possibly still aliased, solution. Guided by this analysis, the ‘ufm1’ radial field model of Bloxham & Jackson (1992) was inverted for core-surface flows between 1970 and 1990. Parametrizing the flow as steady in time leads to solutions that are highly sensitive to the model truncation level. It appears that temporal (but not spatial) underparametrization and the aliasing this induces is to blame. Relaxing the steady-motion constraint produces estimates displaying convergence in the temporal domain and greatly reduces sensitivity to spatial truncation level. The core flow shows significant temporal variation, even over an interval as short as 20 years. The resulting energy spectra, however, are still too flat to be those of the true flow without incurring severe spatial aliasing that is not observed. It appears that noise in the high-degree secular variation (SV) is responsible. Damping is effective in removing most of this noise, but only because aliasing is no longer a factor and noise is restricted to that part of the SV signal which makes only a small contribution to the flow solution. Finally, tests with synthetic fields, generated by known flows and perturbed with noise, reveal that the level of real noise present in the high-degree SV may be as much as 102 times larger than the ufm1 SV convariance estimates indicate.