On the correlation functions associated with polynomials of the diffusion operator

Advancing Short‐Term Forecasts of Ice Conditions in the Beaufort Sea

“…Robustness of the effect was additionally tested against the standard NCODA run with the second‐order autoregressive (SOAR) correlation model (abbreviated by s' ). Since the SOAR model cannot be represented by the diffusion equation, the decorrelation scale was adjusted to provide the best approximation of the SOAR function by the Gaussian with the background decorrelation scale r = 6(8/π) 1/2 ~9.2 km (Yaremchuk & Smith, ). Improvement in the SOAR case was approximately twice as small ( S ( fg|s' ) = 4.6%) as compared to the run with flow‐dependent Gaussian correlations.…”

On the diffusion equation and its application to isotropic and anisotropic correlation modelling in variational assimilation

“…Hristopulos (2003), Hristopulos and Elogne (2007) and Yaremchuk and Smith (2011) have studied extensively the special case M = 1 and P = 2 on R d for which the parameter settings κ 1 = ρ L 2 and κ 2 = L 4 with ρ < 0 and satisfying ρ 2 < 4 yield a family of positive-definite, oscillatory functions such as those illustrated in Figure 3 on S 2 . With all of these approaches, however, the advantages of increasing the flexibility in the correlation model have to be carefully measured against the increase in computational cost that results from the need to solve additional or more complicated large linear systems, and the difficulty of having to estimate additional parameters.…”

Multi‐scale correlation functions associated with polynomials of the diffusion operator

“…(3) into the integral of the inverse Fourier transform and integrating over the solid angle in the wavenumber space (e.g. Yaremchuk and Smith, 2011). The integral (Eq.…”

“…Among the reasons for that limited applicability is poor conditioning of the BEC operators generated by high-degree polynomials and the necessity to link polynomial coefficients with the shape of the BEC spectrum. In the recent studies of Hristopulos andElogne (2007, 2009) and Yaremchuk and Smith (2011), correlation functions associated with an arbitrary quadratic polynomial of the homogeneous diffusion operator were obtained and relationships between the polynomial coefficients and the magnitude/length scale of the corresponding spectral peak have been provided.…”

“…In this note the result of Yaremchuk and Smith (2011) is extended for the case of an arbitrary polynomial, generating a multiple-peak BEC spectrum. Besides, it is shown that the action of the BEC operator can be reduced to a sequence of inversions of the quadratic functions of the diffusion operator, thereby relaxing the above-mentioned conditioning problem.…”