Empirical Orthogonal Eigenfunctions Research Articles

Rapid change of erosion-deposition evolution in the Minjiang estuary, Southeast China

Extreme weather events and anthropogenic activity have severely impacted the Minjiang estuary (MJE) with evolution patterns in recent decades. Determining the estuarine alluvial evolution pattern is crucial for the sustainable development of densely populated coastal areas. Using chart data, this paper analyzed how the MJE's flushing and siltation changed over time (1950–2019). The findings indicate that, in the past 70 years, the MJE has evolved through seven stages of flushing and siltation under the condition of decreasing incoming sediment: “significant siltation - siltation - minor siltation - significant erosion - significant siltation - minor siltation - significant erosion”. Despite decreased incoming material, the estuary underwent substantial siltation from 1998 to 2005. Extreme meteorological conditions were the cause of this anomaly. The “severe flushing” phenomenon occurred in the study area from 2011 to 2019, and the comparison of pre-dam and post-dam at the estuary revealed that extreme weather can only adjust in stages and cannot change the overall state of the estuary flushing due to the reduction of incoming sediment in the basin. According to the EOF (Empirical Orthogonal/Eigen Function) study of four sections of the estuary, it is most impacted by incoming sediment from the watershed, followed by severe weather conditions. Our research is crucial for comprehending how human behavior and harsh weather might affect the development of MJE, and it offers sound advice for the long-term management of MJE.

Deciphering organic matter distribution by source-specific biomarkers in the shallow Taiwan Strait from a source-to-sink perspective

Sedimentary organic matter (OM) in coastal systems is inherently diverse, often with multiple particulate sources and transport histories. The Taiwan Strait (TS) is a typical shallow conduit region, linking the East and South China Seas. Strong ocean currents, coastal upwellings, distal large rivers, and proximal small mountainous rivers all influence the distribution of OM in the TS. We investigated the covarying patterns in the distribution of gain size classes of sand, silt, and clay; terrestrial-sourced biomarkers (n-C27+29+31 alkanes, n-C26+28+30 fatty acids (FAs), and n-C28+30+32 alkanols); marine-sourced biomarkers (phytoplankton-derived alkenones, brassicasterol, dinosterol, and zooplankton-derived cholesterol) in sea floor sediment; indicator satellite-derived primary production (Chl-a); and water-mass indicator (sea surface temperature, SST). We used an empirical orthogonal/eigen function (EOF) analysis to distinguish the influence of four hypothetical sources that entered the TS through the north, south, west, and east boundaries. Results show that input sources from the south-bound ZMCC (Zhejiang-Fujian Coastal Current) and north-bound SCSWC (South China Sea Warm Current) had the dominant influence on the OM distributions buried in the TS. Input sources via river plumes on lateral boundaries and upwellings in the TS were the secondary factors that affected the sedimentary OM distribution. Within this source-to-sink system of multiple sources and transport processes, silt and clay were the major carriers of the OM signals. Terrestrial biomarkers and primary production (Chl-a) were associated with the two major current systems and river plumes along the edge of TS. Marine biomarkers were associated with upwellings in the interior of the TS. Our finding points out that the physical systems of ocean currents, river plumes, and upwelling not only determine the distributions of biomarkers in the TS but also determine the diversity of OM in the TS.

Morphological evolution of the South Passage in the Changjiang (Yangtze River) estuary, China

Estuarine morphology of the world's major large rivers have experienced great changes due to intensive anthropogenic activities and natural forcings, especially in the Changjiang Estuary located at the end of the longest river in Asia. This study focuses on morphological changes of the South Passage (SP), which is one of the 4 channels in the branching estuary at the mouth of the Changjiang (Yangtze River), China. A multivariate analysis technique of Empirical Orthogonal/Eigen Function (EOF) method was used to examine the major modes of change in the long-term (over 26 years) water-depth data. The results show that the morphological changes at the SP could be divided into two stages: between 1987 and 1997, the SP had a single stable channel with closure of a cross-channel. Between 1997 and 2012, SP displayed southeastward elongation of a spit into the main channel, and westward shoal incision by a cross-channel. The opening of the SP developed a two-channel morphology, which stabilized and showed infilling during 2003–2012. The average deposition rate was 10 cm/yr. In the past 30 years, the most dominant morphological changes of SP included the deposition around the upstream opening of the channel. The second most important pattern of morphological change was related to the downstream elongation, retreat, and lateral migration of the spit of the Jiangyanan Shoal, which resulted in the two-channel configuration of the SP. Additionally, these morphological changes were not triggered by the decline of the distal sediment source from the upstream, but due to the input of proximal sources of the shoal at the upstream opening of the SP and spill-over sediment from the North Passage via a short-cut-channel.

AN ANALYSIS OF CROSS-SHORE PROFILE EVOLUTION OF A SAND AND A COMPOSITE SAND-GRAVEL BEACHES

In this paper, cross-shore morphodynamic evolution of a sand and sand-gravel composite beach, are compared and contrasted used historic surveys of beach profiles. The differential behavior of cross-shore morphodynamics at a range of time scales covering event-scale dynamics up to intra-annual scale variability are investigated and discussed. The application of equilibrium concept to both beach types is explored through Dean’s equilibrium profile and Vellinga’s beach erosion profile. Variability of cross-shore profile dynamics are quantified and discussed through bulk statistical analysis. Multi-scale morphodynamic trends are determined using Empirical Orthogonal Eigenfunction (EOF) analysis. The above analysis enables to recognise failure mechanism of each beach type during a storm.

A statistical–dynamical method for predicting estuary morphology

A statistical–dynamical model for estuary morphodynamics is presented and demonstrated with a case study on the Humber Estuary, UK. The model presented here is hybrid in nature where simplified process dynamics are combined with a data-driven approach. The modelling methodology uses an inverse technique to construct an unknown source function in the model-governing equation, using historic measurements of estuary bathymetry. Spatial and temporal variability of the source function was established using empirical orthogonal eigenfunction analysis. Predictions of estuary morphology are obtained by extrapolating the temporal coefficients of the EOFs, in order to construct a predicted source function which is then used in the hybrid morphodynamic equation to obtain the predicted estuary morphology. The model was applied to predict the morphological evolution of the Humber Estuary over a period of a decade. Predicted results were compared with measured bathymetry data. The results reveal that the model captured the decadal scale morphodynamic response of the estuary with a good qualitative accuracy, despite the simplified approach used in the model. The accuracy of model predictions can be greatly improved if high-frequency bathymetry data are available.

Characteristics of Beach Profile Change at a Sandy Coast

Sandy beach shows gradual topographic change with long-term and short-term variations corresponding to the conditions of waves and tides. Because of the irregularity in the profile change, it is difficult to investigate the characteristics of shoreline change in detail. In this research, long-term survey of the beach profiles has been carried out on three different coasts almost every week for more than ten years. The topographic data were analyzed and the characteristics of the long-term and seasonal beach profile change were discussed using the statistical methods and the empirical orthogonal eigenfunction method (EOF).

Coherent structures of the near field flow in a self-oscillating physical model of the vocal folds

Current theories of voice production depend critically upon knowledge of the near field flow which emanates from the glottis. While most modern theories predict complex, three-dimensional structures in the near field flow, few investigations have attempted to quantify such structures. Using methods of flow visualization and digital particle image velocimetry, this study measured the near field flow structures immediately downstream of a self-oscillating, physical model of the vocal folds, with a vocal tract attached. A spatio-temporal analysis of the structures was performed using the method of empirical orthogonal eigenfunctions. Some of the observed flow structures included vortex generation, vortex convection, and jet flapping. The utility of such data in the future development of more accurate, low-dimensional models of voice production is discussed.

Anterior-posterior biphonation in a finite element model of vocal fold vibration

In this paper, a finite-element model is used to simulate anterior-posterior biphonation [Neubauer et al., J. Acoust. Soc. Am. 110(6), 3179-3192 (2001)]. The anterior-posterior stiffness asymmetric factor and the anterior-posterior shape asymmetric factor describe the asymmetry properties of vocal folds. Spatiotemporal plot, spectral analysis, anterior-posterior fundamental frequency ratio, cross covariation function, and correlation length quantitatively estimate the spatial asymmetry of vocal fold oscillations. Calculation results show that the anterior-posterior stiffness asymmetry decreases the spatial coherence of vocal fold vibration. When the stiffness asymmetry reaches a certain level, the drop in spatial coherence desynchronizes the vibration modes. The anterior and posterior sides of the vocal fold oscillate with two independent fundamental frequencies (f(a) and f(p)). The complex spectral characteristics of vocal fold vibration under biphonation conditions can be explained by the linear combination of f(a) and f(p). Empirical orthogonal eigenfunctions prove the existence of higher-order anterior-posterior modes when anterior-posterior biphonation occurs. Then, it is found that the anterior-posterior shape asymmetry also decreases the spatial coherence of vocal fold vibration, and shape asymmetry is a possible reason for anterior-posterior biphonation.

Regional gravity variations in Europe from superconducting gravimeters

Recent satellite missions (CHAMP, GRACE) are now returning data on the time variation of the gravity field with harmonic coefficients computed every 4 weeks. The promise is to achieve a sub-microgal accuracy that will define continental mass variations involving large-scale hydrology. With this in mind, we examine the time varying gravity field over central Europe using a limited number of high quality ground-based superconducting gravimeter stations within the Global Geodynamics Project (GGP). Our purpose is to see whether there are coherent signals between the individual stations and to compare the regional component with that predicted from models of continental hydrology. The results are encouraging. We have found, using empirical orthogonal eigenfunctions of the gravity data that a clear annual signal is present that is consistent in phase (low amplitudes in summer) and amplitude (1–3 microgal) with that determined from a large-scale model of land water in connection with global climate modeling. More work is required to define how the gravity field is related to large-scale soil moisture and other mass variations, and we have yet to compare our results to the latest satellite-derived data. © 2004 Elsevier Ltd. All rights reserved.

Video observations of nearshore bar behaviour. Part 2: alongshore non-uniform variability

A considerable portion of the variability in nearshore sandbars is related to changes in the plan shape of quasi-rhythmic alongshore non-uniform features, such as rip channels and crescentic shapes. These changes may include changes in their alongshore length, cross-shore amplitude and alongshore position. Here, we use complex empirical orthogonal eigenfunction analysis to quantify these changes from a 3.4-year data set of almost daily time-exposure images of the double-barred coast at Noordwijk (Netherlands). The observed alongshore non-uniform features had alongshore lengths between 380 and approximately 3000 m and lifetimes in the order of months, considerably longer than the characteristic time scale of individual wave events. Transitions from one feature to another were mostly gradual, resulting from an alongshore differential growth in amplitude. Abrupt transitions, that is, the existing features disappeared entirely and were subsequently replaced by different features, were barely observed and did not always take place during high-energy wave events. The amplitude of the non-uniform features varied between 0 and 30 m on a weekly to monthly scale, unrelated to variations in the wave height. In addition, the features migrated back and forth along the shore with typical rates of O(10 m/ day) on weekly scales with the rates increasing with an increase in the alongshore component of the wave power. On the whole our observations suggest that alongshore non-uniform sandbar variability is governed by free behaviour rather than by the direct forcing of the prevailing wave conditions.

Nonlinear model reduction for dynamic analysis of cell population models

Transient cell population balance models consist of nonlinear partial differential-integro equations. An accurate discretized approximation typically requires a large number of nonlinear ordinary differential equations that are not well suited for dynamic analysis and model based controller design. In this paper, proper orthogonal decomposition (also known as the method of empirical orthogonal eigenfunctions and Karhunen Loéve expansion) is used to construct nonlinear reduced-order models from spatiotemporal data sets obtained via simulations of an accurate discretized yeast cell population model. The short-term and long-term behavior of the reduced-order models are evaluated by comparison to the full-order model. Dynamic simulation and bifurcation analysis results demonstrate that reduced-order models with a comparatively small number of differential equations yield accurate predictions over a wide range of operating conditions.

Low-dimensional approximation and control of periodic solutions in spatially extended systems

Nonlinear model reduction is combined with numerical continuation and linear state-space control techniques to design regulators for periodic solutions in a spatially extended system. We address issues of construction and systematic evaluation of low-dimensional dynamic models using Galerkin projections on empirical orthogonal eigenfunctions (also known as proper orthogonal decomposition modes or Karhunen-Lo\`eve modes). The reduced order dynamical systems are used first to compute the open-loop bifurcation diagrams and then to design feedback controllers stabilizing unstable limit cycles. We outline the steps for discrete-time controller design and computational linear stability analysis of the resulting hybrid (continuous-discrete) closed-loop systems.

Interpretation of biomechanical simulations of normal and chaotic vocal fold oscillations with empirical eigenfunctions.

Empirical orthogonal eigenfunctions are extracted from biomechanical simulations of normal and chaotic vocal fold oscillations. For normal phonation, two dominant empirical eigenfunctions capture the vibration patterns of the folds and exhibit a 1:1 entrainment. The eigenfunctions show some correspondence to theoretical low-order normal modes of a simplified, three-dimensional elastic continuum, and to the normal modes of a linearized two-mass model. The eigenfunctions also facilitate a physical interpretation of energy transfer mechanisms in vocal fold dynamics. Subharmonic regimes and chaotic oscillations are observed during simulations of a lax cover, in which case at least three empirical eigenfunctions are necessary to capture the resulting vocal fold oscillations. These chaotic oscillations might be understood in terms of a desynchronization of a few of the low-order modes, and may be related to mechanisms of creaky voice or vocal fry. Furthermore, some of the empirical eigenfunctions captured during complex oscillations correspond to higher-order normal modes described in earlier theoretical work. The empirical eigenfunctions may also be useful in the design of lower-order models (valid over the range for which the empirical eigenfunctions remain more or less constant), and may help facilitate bifurcation analyses of the biomechanical simulation.

Ekman portion of surface currents, as measured by radar in different areas

Surface currents, as measured by the HF-radar CODAR (COstal raDAR), are investigated with respect to their dependence on wind. Time series of wind are available from single weather stations, while CODAR yields current velocities on a grid with a resolution of some 3 km. As part of various research programs experiments have been carried out by the University of Hamburg (Germany) in different areas. Two of the areas under consideration are located in the Baltic Sea, two in the North Sea, and one covering the northern part of the Dead Sea. Time series of about two weeks with 2-hourly sampling are available for some 50 grid points of each area. Vector-correlation techniques are used to determine the linear relation of surface currents on wind velocity and also windstress. In both cases, significant correlation of about the same order has been found. 35% to 60% of the variance in surface currents may be explained by linear forcing from the vectors of wind velocity or windstress. Current-to-wind ratios range from 0.015 to 0.025, and a veering to the right of currents against wind has been observed. The decomposition of horizontal two-dimensional current fields into empirical orthogonal eigenfunctions (EOF) yields higher amounts of variance in the 1. EOF as may be explained by linear windforcing. Two possible mechanisms for wind-driven currents are discussed, Ekman circulation and Stokes drift. Assuming the vertical eddy viscosity to be independent on wind velocity and water depth, it may be estimated. Values of about 5×10−4 m2s−1 are found in the Baltic and 20×10−4 m2s−1 in the North Sea, which are in reasonable agreement with those used in numerical models.

Internal wave climate near the coast of nortwest Africa during JOINT-I

Abstract Current meter measurements from six stations off the coast of northwest Africa are analysed. Decomposition of the data using empirical orthogonal eigenfunctions shows that in depths shallower than 800 m a significant part of the internal wave energy resides in first mode waves that propagate shoreward. Examination of the energy flux and energy density of these waves indicates that: (1) the origin of the internal wave energy appears to be the deep ocean; (2) the energy density of the interval waves increases shoreward in the deep coastal waters subject to a constant energy flux requirement; (3) the energy density in shallow coastal waters is nearly saturated, i.e. the maximum amplitudes are limited by a finite depth and thus the local energy levels are determined by the local Vasala frequency and depth; (4) the energy flux is largely dissipated rather than reflected back out to the deep sea.