Interpolating Irregularly Spaced Observations for Filtering Turbulent Complex Systems

• Authors: John Harlim
• Venue: SIAM Journal on Scientific Computing
• Year: 2011

We present a numerically fast reduced filtering strategy, the Fourier domain Kalman filter with appropriate interpolations to account for irregularly spaced observations of complex turbulent signals. The design of such a reduced filter involves: (i) interpolating irregularly spaced observations to the model regularly spaced grid points, (ii) understanding under which situation the small scale oscillatory artifact from such interpolation won't degrade the filtered solutions, (iii) understanding when the interpolated covariance structure can be approximated by its diagonal terms when observations are corrupted by independent Gaussian noise, and (iv) applying a scalar Kalman filter formula on each Fourier component independently with an approximate diagonal interpolated covariance matrix. From the practical point of view, there is an emerging need to understand the effect of (i) toward the filtered solutions, for example, in utilizing the data produced from various interpolation techniques that merge multiple satellite measurements of atmospheric and ocean dynamical quantities. To understand point (iii) above and to see how many of nondiagonal terms are effectively ignored in (iv), we compute a ratio $\Lambda$ between the largest nondiagonal components and the smallest diagonal components of the interpolated covariance matrix. We find that for piecewise linear interpolation, this ratio, $\Lambda$, is always smaller than that of the alternative interpolation schemes such as trigonometric and cubic spline for any irregularly spaced observation networks. When observations are not so sparse with small noise, we find that the small scale oscillatory artifact in (ii) above is negligible when piecewise linear interpolation is used whereas for the other schemes such as the nearest neighbor, trigonometric, and cubic spline interpolation, the oscillatory artifact degrades the filtered solutions significantly. Finally, we also find that the reduced filtering strategy with piecewise linear interpolation produces more accurate filtered solutions than conventional approaches when observations are extremely irregularly spaced (such that the ratio $\Lambda$ is not so small) and very sparse.