Power laws and inverse motion modelling: application to turbulence measurements from satellite images

Abstract

A B S T R A C TIn the context of tackling the ill-posed inverse problem of motion estimation from image sequences, we propose to introduce prior knowledge on flow regularity given by turbulence statistical models. Prior regularity is formalised using turbulence power laws describing statistically self-similar structure of motion increments across scales. The motion estimation method minimises the error of an image observation model while constraining second-order structure function to behave as a power law within a prescribed range. Thanks to a Bayesian modelling framework, the motion estimation method is able to jointly infer the most likely power law directly from image data. The method is assessed on velocity fields of 2-D or quasi-2-D flows. Estimation accuracy is first evaluated on a synthetic image sequence of homogeneous and isotropic 2-D turbulence. Results obtained with the approach based on physics of fluids outperform state-of-the-art. Then, the method analyses atmospheric turbulence using a real meteorological image sequence. Selecting the most likely power law model enables the recovery of physical quantities, which are of major interest for turbulence atmospheric characterisation. In particular, from meteorological images we are able to estimate energy and enstrophy fluxes of turbulent cascades, which are in agreement with previous in situ measurements.

Large-scale anisotropy in stably stratified rotating flows

“…This result is interesting in the light of observations of a spectrum with similar scaling and with positive flux at intermediate scales in the atmosphere [52][53][54], as it may provide another mechanism to generate such a power law (see [23,[55][56][57][58][59] for other explanations).…”

Evidence for Bolgiano-Obukhov scaling in rotating stratified turbulence using high-resolution direct numerical simulations

“…The corresponding overall effective energy dissipation rate would be ∼ U 3 0 /L 0 ≈ 1.4 × 10 −8 m 2 s −3 ; this latter value corresponds to the enhanced dissipation measured in the southern ocean [52]. As a comparison, measurements in the atmosphere indicate ≈ 10 −6 m 2 s −3 at intermediate altitude and at scales between 3 and 600 km [53]. With a rotation frequency of Ω = 10 −4 s −1 , our choice of parameters leads to a Brunt-Väisälä frequency of N ≈ 10 −3 s −1 , and F r ≈ 0.024, corresponding to the parameters of the run described above.…”

Divergence-Free Wavelets and High Order Regularization

“…For indication, also an advanced implementation of the Horn and Schunck [10] techniques is of the same order accuracy as state-of-the-art PIV method. Figure 7 shows the plot of a time-sequence of RMSE on the vorticity ω = curl(u) obtained by the proposed methods, compared to the results of [7,10,26].…”

“…Results on scalar imagery (Figure 6(b)) show that the combination of a divergence-free wavelet basis and continuous operator regularization is necessary, in order to obtain results comparable to those of the stateof-the-art. Let us note that the regularization approach proposed in [7] is accurate here since it takes advantage of an additional physical constraint (turbulence power laws parameters, whose estimation increases sig-nificantly the computational cost of the method). This type of regularization is perfectly suited to homogeneous isotropic turbulent flows as the one of this test sequence.…”

“…Indeed, optical flow standards are in general designed from almost rigid motions consideration and do not include physical considerations on the observed phenomenon occurring eventually at different spatial scales. Several attempts have been carried out in order to introduce some physical constraints, such as incompressibility [7,26] or second order regularization functional preserving blobs of curl and divergence [9].…”