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1. Wavelet‐Based Wavenumber Spectral Estimate of Eddy Kinetic Energy: Idealized Quasi‐Geostrophic Flow

   https://doi.org/10.1029/2022MS003399  

   Uchida, Takaya; Jamet, Quentin; Poje, Andrew C.; Wienders, Nico; Dewar, William K.; Deremble, Bruno (March 2023, Journal of Advances in Modeling Earth Systems)

   Abstract A wavelet‐based method is re‐introduced in an oceanographic and spectral context to estimate wavenumber spectrum and spectral flux of kinetic energy and enstrophy. We apply this to a numerical simulation of idealized, doubly periodic quasi‐geostrophic flows, that is, the flow is constrained by the Coriolis force and vertical stratification. The double periodicity allows for a straightforward Fourier analysis as the baseline method. Our wavelet spectra agree well with the canonical Fourier approach but with the additional strengths of negating the necessity for the data to be periodic and being able to extract local anisotropies in the flow. Caution is warranted, however, when computing higher‐order quantities, such as spectral flux. 

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2. Adaptive Online Estimation of Piecewise Polynomial Trends

   Baby, Dheeraj; Wang, Yu-Xiang (December 2020, NeurIPS 2020)

   null (Ed.)

   We consider the framework of non-stationary stochastic optimization (Besbes et al., 2015) with squared error losses and noisy gradient feedback where the dynamic regret of an online learner against a time varying comparator sequence is studied. Motivated from the theory of non-parametric regression, we introduce a new variational constraint that enforces the comparator sequence to belong to a discrete k^{th} order Total Variation ball of radius C_n. This variational constraint models comparators that have piece-wise polynomial structure which has many relevant practical applications (Tibshirani, 2014). By establishing connections to the theory of wavelet based non-parametric regression, we design a polynomial time algorithm that achieves the nearly optimal dynamic regret of ~O(n^{1/(2k+3)} C_n^{2/(2k+3)}). The proposed policy is adaptive to the unknown radius C_n. Further, we show that the same policy is minimax optimal for several other non-parametric families of interest. 

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3. Learning multi-scale local conditional probability models of images

   Kadkhodaie, Zahra; Guth, Florentin; Mallat, Stéphane; Simoncelli, Eero P (March 2023, ICLR 2023)

   Deep neural networks can learn powerful prior probability models for images, as evidenced by the high-quality generations obtained with recent score-based diffusion methods. But the means by which these networks capture complex global statistical structure, apparently without suffering from the curse of dimensionality, remain a mystery. To study this, we incorporate diffusion methods into a multi-scale decomposition, reducing dimensionality by assuming a stationary local Markov model for wavelet coefficients conditioned on coarser-scale coefficients. We instantiate this model using convolutional neural networks (CNNs) with local receptive fields, which enforce both the stationarity and Markov properties. Global structures are captured using a CNN with receptive fields covering the entire (but small) low-pass image. We test this model on a dataset of face images, which are highly non-stationary and contain large-scale geometric structures. Remarkably, denoising, super-resolution, and image synthesis results all demonstrate that these structures can be captured with significantly smaller conditioning neighborhoods than required by a Markov model implemented in the pixel domain. Our results show that score estimation for large complex images can be reduced to low-dimensional Markov conditional models across scales, alleviating the curse of dimensionality. 

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4. Multiband Wavelet Age Modeling for a ∼293 m (∼600 kyr) Sediment Core From Chew Bahir Basin, Southern Ethiopian Rift

   https://doi.org/10.3389/feart.2021.594047  

   Duesing, Walter; Berner, Nadine; Deino, Alan L.; Foerster, Verena; Kraemer, K. Hauke; Marwan, Norbert; Trauth, Martin H. (March 2021, Frontiers in Earth Science)

   null (Ed.)

   The use of cyclostratigraphy to reconstruct the timing of deposition of lacustrine deposits requires sophisticated tuning techniques that can accommodate continuous long-term changes in sedimentation rates. However, most tuning methods use stationary filters that are unable to take into account such long-term variations in accumulation rates. To overcome this problem we present herein a new multiband wavelet age modeling (MUBAWA) technique that is particularly suitable for such situations and demonstrate its use on a 293 m composite core from the Chew Bahir basin, southern Ethiopian rift. In contrast to traditional tuning methods, which use a single, defined bandpass filter, the new method uses an adaptive bandpass filter that adapts to changes in continuous spatial frequency evolution paths in a wavelet power spectrum, within which the wavelength varies considerably along the length of the core due to continuous changes in long-term sedimentation rates. We first applied the MUBAWA technique to a synthetic data set before then using it to establish an age model for the approximately 293 m long composite core from the Chew Bahir basin. For this we used the 2nd principal component of color reflectance values from the sediment, which showed distinct cycles with wavelengths of 10–15 and of ∼40 m that were probably a result of the influence of orbital cycles. We used six independent 40Ar/39Ar ages from volcanic ash layers within the core to determine an approximate spatial frequency range for the orbital signal. Our results demonstrate that the new wavelet-based age modeling technique can significantly increase the accuracy of tuned age models. 

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5. Spectral diagonal ensemble Kalman filters

   https://doi.org/10.5194/npg-22-485-2015  

   Kasanický, I.; Mandel, J.; Vejmelka, M. (January 2015, Nonlinear Processes in Geophysics)

   A new type of ensemble Kalman filter is developed, which is based on replacing the sample covariance in the analysis step by its diagonal in a spectral basis. It is proved that this technique improves the approximation of the covariance when the covariance itself is diagonal in the spectral basis, as is the case, e.g., for a second-order stationary random field and the Fourier basis. The method is extended by wavelets to the case when the state variables are random fields which are not spatially homogeneous. Efficient implementations by the fast Fourier transform (FFT) and discrete wavelet transform (DWT) are presented for several types of observations, including high-dimensional data given on a part of the domain, such as radar and satellite images. Computational experiments confirm that the method performs well on the Lorenz 96 problem and the shallow water equations with very small ensembles and over multiple analysis cycles. 

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