Analysis of the regional spectral properties in northwestern South China Sea based on an empirical orthogonal function

This study presents an analysis of the spectral characteristics of remote sensing reflectance(Rrs)in northwestern South China Sea based on the in situ optical and water quality data for August 2018.Rrswas initially divided into four classes,classes A to D,using the max-classification algorithm,and the spectral properties of whole Rrs were characterized using the empirical orthogonal function(EOF)analysis.Subsequently,the dominant factors in each EOF mode were determined.The results indicated that more than 95%of the variances of Rrs are partly driven by the back-scattering characteristics of the suspended matter.The initial two EOF modes were well correlated with the total suspended matter and back-scattering coefficient.Furthermore,the first EOF modes of the four classes of Rrs(A-D Rrs-EOF1)significantly contributed to the total variances of each Rrs class.In addition,the correlation coefficients between the amplitude factors of class A-D Rrs-EOF1 and the variances of the relevant water quality and optical parameters were better than those of the unclassified ones.The spectral shape of class ARrs-EOF1 was governed by the absorption characteristic of chlorophyll a and colored dissolved organic matter(CDOM).The spectral shape of class B Rrs-EOF1 was governed by the absorption characteristic of CDOM since it exhibited a high correlation with the absorption coefficient of CDOM(ag(λ)),whereas the spectral shape of class C Rrs-EOF1 was governed by the back-scattering characteristics but not affected by the suspended matter.The spectral shape of class D Rrs-EOF1 exhibited a relatively good correlation with all the water quality parameters,which played a significant role in deciding its spectral shape.

Robust digital image watermarking against shearing based on Haar orthogonal function system

In this paper, a robust digital watermarking method against shearing based on Haar orthogonal function system was introduced. The proposed method adopted the complete generalized orthogonal properties of Haar ortbogonal function system to achieve the piece-based orthogonal transform on the image. The significant middle frequency coefficients in the transformation matrix are picked up, based on characteristics of the image visual system and the Haar orthogonal transform. The watermark is adoptively weighed to the middle frequency matrix. The method improves the validity of watermarking and shows excellent advantage against shearing attack. Experimental results show that the Haar orthogonal function system based watermark approach can provide an excellent protection under geometric attacks.

A NEW ONE-STEP BAND-LIMITED EXTRAPOLATION PROCEDURE USING EMPIRICAL ORTHOGONAL FUNCTIONS

A one-step band-limited extrapolation procedure is systematically developed under an a priori assumption of bandwidth. The rationale of the proposed scheme is to expand the known signal segment based on a band-limited basis function set and then to generate a set of Empirical Orthogonal Functions (EOF’s) adaptively from the sample values of the band-limited function set. Simulation results indicate that, in addi- tion to the attractive adaptive feature, this scheme also appears to guarantee a smooth result for inexact data, thus suggesting the robustness of the proposed procedure.

A Multivariate Empirical Orthogonal Function-Based Scheme for the Balanced Initial Ensemble Generation of an Ensemble Kalman Filter

The initial ensemble perturbations for an ensemble data assimilation system are expected to reasonably sample model uncertainty at the time of analysis to further reduce analysis uncertainty. Therefore, the careful choice of an initial ensemble perturbation method that dynamically cycles ensemble perturbations is required for the optimal performance of the system. Based on the multivariate empirical orthogonal function (MEOF) method, a new ensemble initialization scheme is developed to generate balanced initial perturbations for the ensemble Kalman filter (EnKF) data assimilation, with a reasonable consideration of the physical relationships between different model variables. The scheme is applied in assimilation experiments with a global spectral atmospheric model and with real observations. The proposed perturbation method is compared to the commonly used method of spatially-correlated random perturbations. The comparisons show that the model uncertainties prior to the first analysis time, which are forecasted from the balanced ensemble initial fields, maintain a much more reasonable spread and a more accurate forecast error covariance than those from the randomly perturbed initial fields. The analysis results are further improved by the balanced ensemble initialization scheme due to more accurate background information. Also, a 20-day continuous assimilation experiment shows that the ensemble spreads for each model variable are still retained in reasonable ranges without considering additional perturbations or inflations during the assimilation cycles, while the ensemble spreads from the randomly perturbed initialization scheme decrease and collapse rapidly.

Proportional integral derivative controller design using Legendre orthogonal functions

The Legendre orthogonal functions are employed to design the family of PID controllers for a variety of plants. In the proposed method, the PID controller and the plant model are represented with their corresponding Legendre series. Matching the first three terms of the Legendre series of the loop gain with the desired one gives the PID controller parameters. The closed loop system stability conditions in terms of the Legendre basis function pole(λ) for a wide range of systems including the first order, second order, double integrator, first order plus dead time, and first order unstable plants are obtained. For first order and double integrator plants, the closed loop system stability is preserved for all values of λ and for the other plants, an appropriate range in terms of λ is obtained. The optimum value of λ to attain a minimum integral square error performance index in the presence of the control signal constraints is achieved. The numerical simulations demonstrate the benefits of the Legendre based PID controller.

NON-LINEAR DYNAMIC MODEL RETRIEVAL OF SUBTROPICAL HIGH BASED ON EMPIRICAL ORTHOGONAL FUNCTION AND GENETIC ALGORITHM

Aiming at the difficulty of accurately constructing the dynamic model of subtropical high, based on the potential height field time series over 500 hPa layer of T106 numerical forecast products, by using EOF（empirical orthogonal function） temporal-spatial separation technique, the disassembled EOF time coefficients series were regarded as dynamical model variables, and dynamic system retrieval idea as well as genetic algorithm were introduced to make dynamical model parameters optimization search, then, a reasonable non-linear dynamic model of EOF time-coefficients was established. By dynamic model integral and EOF temporal-spatial components assembly, a mid-/long-term forecast of subtropical high was carried out. The experimental results show that the forecast results of dynamic model are superior to that of general numerical model forecast results. A new modeling idea and forecast technique is presented for diagnosing and forecasting such complicated weathers as subtropical high.

Nonlinear Oscillation Analysis by an Orthogonal Function Method

In this paper an orthogonal function method is presented based on the idea to suppose perioche sohuion with the method of harmonie balance The displaeement is expressed in the form of trigonometric fumctions a group of simplified digenequationsare obtained by the use of orthogonarity of trigonometric fumetions and linear mondes The method overcomes the diffieulty of a drifi term existing in systems with quadratic nonlinearities .The ealeulation examples show that the method has thd advantages of high caleulation preeision high convergenee speed and littld ealeulation work

Model Order Reduction Methods for Discrete Systems via Discrete Pulse Orthogonal Functions

This paper explores model order reduction(MOR)methods for discrete linear and discrete bilinear systems via discrete pulse orthogonal functions(DPOFs).Firstly,the discrete linear systems and the discrete bilinear systems are expanded in the space spanned by DPOFs,and two recurrence formulas for the expansion coefficients of the system’s state variables are obtained.Then,a modified Arnoldi process is applied to both recurrence formulas to construct the orthogonal projection matrices,by which the reduced-order systems are obtained.Theoretical analysis shows that the output variables of the reducedorder systems can match a certain number of the expansion coefficients of the original system’s output variables.Finally,two numerical examples illustrate the feasibility and effectiveness of the proposed methods.

Modeling the global ionospheric total electron content with empirical orthogonal function analysis

In the present work we model the global ionospheric total electron content (TEC) with the analysis of empirical orthogonal functions (EOF). The obtained statistical eigen modes, which makeup the modeled TEC, consist of two factors: the eigen vectors mapping TEC patterns at latitude and longitude (or local time LT), and the corresponding coefficients displaying the TEC variations in different time scales, i.e., the solar cycle, the yearly (annual and semiannual) and the diurnal universal time variations. It is found that the EOF analysis can separate the TEC variations into chief processes and the first two modes illustrate the most of the ionospheric climate properties. The first mode contains both the semiannual component which shows the semiannual ionospheric anomaly and the annual component which shows the annual or non-seasonal ionospheric anomaly. The second mode contains mainly the annual component and shows the normal seasonal ionospheric variation at most latitudes and local time sectors. The annual component in the second mode also manifests seasonal anomaly of the ionosphere at higher mid-latitudes around noontime. It is concluded that the EOF analysis, as a statistical eigen mode method, is resultful in analyzing the ionospheric climatology hence can be used to construct the empirical model for the ionospheric climatology.

ROTATED COMPLEX EMPIRICAL ORTHOGONAL FUNCTION(RCEOF)ANALYSIS:THEORY AND EXAMPLES—PART I

In this paper,a new diagnostic method,the rotated complex empirical orthogonal function (RCEOF)analysis is developed.The general principle and the mathematical foundation of RCEOF are discussed.

ROTATED COMPLEX EMPIRICAL ORTHOGONAL FUNCTION(RCEOF)ANALYSIS—PART Ⅱ:THEORY AND EXAMPLES

The numerical solving and the program designing of the rotated complex empirical orthogonal function(RCEOF)are discussed.Some examples of RCEOF are also presented.

An equivalent linearization method for nonlinear Van der Pol oscillator subjected to random vibration using orthogonal functions

This paper deals with the idea of the orthogonal functions in the equivalent linearization of the nonlinear systems. Block Pulse （BP） function gives effective tools to approximate complex problems. The aim of this work is on using properties of the BP function as an orthogonal function in process of linearization. The BP functions have been used to propose an equivalent linearization method in the time domain to determine the unknown linearization coefficients. The accuracy of the proposed method compared with the other equivalent linearization approaches, including the regulation linearization and the dual criterion linearization methods. This study exploited the nonlinear Van der Pol oscillator system under stationary random excitation to demonstrate the feasibility of the proposed method. The validity of the analytical method is verified by applying different values of nonlinearity and intensity of excitation. Besides, by comparing the mean-square responses and frequency response functions of the linearized systems for a wide range of nonlinearity depicted the present method is in agreement with other methods.

Empirical orthogonal function analysis of short time series of sectional data in the tropical western Pacific

In this paper,some short time series of pnserved data pm sectopm 18°20′N in the tropical western Pacificwere reorganized to give mixed depth-time series,and processed by means of means of empirical orthogonal fonction analysis. It is indicated that the original form of element distribution could be obtained by linear combination of several main canonical distribution functions, and the intrinsic structure of element distribution on a certain section and its variation propertiescould be reveled by canonical distribution function and profiles in corresponding periods.

THE ORTHOGONAL 4-VALUED FUNCTION SERIES

ased on the 4-valued logic function in orthogonal area, a new series──the orthogonal 4-valued function series──is proposed. Having good performances, the new series is superior to Walsh function and block pulse function in the computation of truncation errors.

ORTHOGONAL MATRIX POLYNOMIALS WITH RESPECT TO LINEAR MATRIX MOMENT FUNCTION ALS:THEORY AND APPLICATIONS

In this paper orthogonal matrix polynomials with respect to a right matrix moment functional an introduced. Basic results, important examples and applications to the approximation of matrix integrals are studied. Error bounds for the proposed matrix quadrature rules are given.

Spatio-temporal variability of surface chlorophyll a in the Yellow Sea and the East China Sea based on reconstructions of satellite data of 2001-2020

Chlorophyll-a(Chl-a)concentration is a primary indicator for marine environmental monitoring.The spatio-temporal variations of sea surface Chl-a concentration in the Yellow Sea(YS)and the East China Sea(ECS)in 2001-2020 were investigated by reconstructing the MODIS Level 3 products with the data interpolation empirical orthogonal function(DINEOF)method.The reconstructed results by interpolating the combined MODIS daily+8-day datasets were found better than those merely by interpolating daily or 8-day data.Chl-a concentration in the YS and the ECS reached its maximum in spring,with blooms occurring,decreased in summer and autumn,and increased in late autumn and early winter.By performing empirical orthogonal function(EOF)decomposition of the reconstructed data fields and correlation analysis with several potential environmental factors,we found that the sea surface temperature(SST)plays a significant role in the seasonal variation of Chl a,especially during spring and summer.The increase of SST in spring and the upper-layer nutrients mixed up during the last winter might favor the occurrence of spring blooms.The high sea surface temperature(SST)throughout the summer would strengthen the vertical stratification and prevent nutrients supply from deep water,resulting in low surface Chl-a concentrations.The sea surface Chl-a concentration in the YS was found decreased significantly from 2012 to 2020,which was possibly related to the Pacific Decadal Oscillation(PDO).

THE ASYMPTOTIC FORMULA OF THE ORTHOGONAL RATIONAL FUNCTION ON THE UNIT CIRCLE

In this paper we obtained the asymptotic formula of the orthogonal rational function on the unit circle with respect to the weight function μ(z) with preasigned poles, which are in the exterior of the unit disk.

ORTHOGONAL MATRIX POLYNOMIALS WITH RESPECT TO A CONJUGATE BILINEAR MATRIX MOMENT FUNCTIONAL: BASIC THEORY

In this paper basic results for a theory of orthogonal matrix polynomials with respect to a conjugate bilinear matrix moment functional are proposed. Properties of orthogonal matrix polynomial sequences including a three term matrix relationship are given. Positive definite conjugate bilinear matrix moment functionals are introduced and a characterization of positive definiteness in terms of a block Haenkel moment matrix is established. For each positive definite conjugate bilinear matrix moment functional an associated matrix inner product is defined.

Spatiotemporal variations of parameters of internal solitary waves in the northern South China Sea

The dynamic parameters for internal solitary waves(ISWs)derived from the extended Korteweg-de Vries(eKdV)equation play an important role in the understanding and prediction of ISWs.The spatiotemporal variations of the dynamic parameters of the ISWs in the northern South China Sea(SCS)were studied based on the reanalysis of long-term temperature and salinity datasets.The results for spectrum analysis show that there are definite geographical differences for the periodic variation of the parameters:in shallow water,all parameters vary with a wave period of one year,while in deep water wave components of the parameters at other frequencies exist.Using wavelet analysis,the wavelet power spectral densities in deep water exhibited an inter-annual variation pattern.For example,the wave component of the dispersion coefficient with a wave period of about half a year reached its power peak once every two years.Based on previous work,this inter-annual variation pattern was deduced to be caused by dynamic processes.In further work on the regulatory mechanisms,empirical orthogonal function(EOF)decomposition was performed.It was found that the modes of the dispersion coefficient have different geographical distributions,explaining the reason why the wave components in different frequencies appeared in different locations.The numerical simulation results confirm that the variations in the parameters of the ISWs derived from the eKdV equation could affect the waveforms significantly because of changes in the polarity of the ISWs.Therefore,the periodic variations of the dynamic parameters are related to the geographical location because of dynamic processes operating.

Spatial and Temporal Differences of Climate Suitability of Ice and Snow Sports in Major Ski Tourism Destinations in China

Ice and snow tourism in China has grown significantly since the country successfully hosted the Beijing Winter Olympics.Climatic conditions profoundly impact the development of ice and snow tourism;however,most studies have focused on constructing different climate suitability indicators for ice and snow tourism to evaluate individual regions,lacking horizontal comparative studies across multiple regions.This study aims to enrich the connotation of climate suitability for ice and snow sports,establish an evaluation model based on snowfall amount,temperature,and wind speed,and use daily meteorological data from 1991 to 2021 to horizontally compare the climate suitability for ice and snow sports in major ski tourism destinations in China.This study boasts four major findings:1)the average ice and snow sports climate index of each region decreases over time,and the overall suitability of the climate for ice and snow sports is reducing;2)northern Xinjiang exhibits the most evident regional differentiation from‘very suitable’to‘generally suitable’;3)the spatial zoning of climate suitability for ice and snow sports exhibits heterogeneity,as northern Xinjiang is divided into two‘suitable and above’zones with rotating empirical orthogonal function(REOF).Correspondingly,the four provinces of Hebei,Heilongjiang,Jilin,and Liaoning are divided into three‘generally suitable and above’zones;4)snowfall amount is the main factor affecting the climate suitability of ice and snow sports in the major ski tourist destinations in China.