Aliasing and noise in core-surface flow inversions

Abstract

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)-', 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 to 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 'ufml' 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 lo2 times larger than the ufml SV convariance estimates indicate.

Core flow inversion tested with numerical dynamo models

“…Furthermore, the spectral maximum of the actual flow field is near the truncation limit. Celaya & Wahr (1996) found in tests with synthetic data that ‘underparametrized’ (truncated) models are corrupted at all degrees by severe aliasing when the spectrum of the flow decays less rapidly than l − 2 . This should contribute to the problem that we face in our truncated inversions.…”

“…A necessary requirement would be that the core flow is indeed nearly geostrophic or toroidal. While we performed our inversion assuming perfect ‘data’, errors in the observational data must degrade the ability to retrieve flow structures, especially at smaller scale (Hulot et al 1992; Celaya & Wahr 1996).…”

Quasi-geostrophic flows responsible for the secular variation of the Earth's magnetic field

“…Critical to their study are the assumptions made on the statistical properties of the flow, which require that the spectrum of the flow decays with n −2 . Celaya & Wahr (1996) considered the problem of spatial but also temporal underparametrization of the flow in frozen‐flux core–surface flow inversions. They tested synthetic flows with different energy spectra and different time variations, to finally conclude that only if the spectra fall off as n −2 or faster, and the steady‐motion constraint is relaxed, can underparametrization effects (aliasing) be neglected.…”

On magnetic estimation of Earth's core angular momentum variation

“…However, this is not the case with the TG models TG27−1.0 ∗ and TG27+1.0, indicating a need of some larger SV misfit for those models to be consistent with the modeling error. Unless smoothed even more effectively, e.g., with p = 3, no TG model would be available that is simultaneously moderate fitting and not affected by the modeling error (see also Celaya and Wahr [] and Holme and Olsen [] on this issue). In view of the tendency discovered here, nevertheless, we still do not find any disadvantages of Type (III) over Type (IV) in agreeing with the observed LOD trend.…”