Maximum entropy spectral analysis of total ozone

Total ozone data series for 1957–82 at ten locations were subjected to Maximum Entropy Spectral Analysis. Besides the annual, semi-annual, and quasi-biennial oscillations, peaks were noticed at 3.5–4, 6–7, and 10–11 years. For Arosa, Switzerland, for a longer period (1932–71), an additional peak was indicated at about 16 years.     

Space-time mapping of soil salinity using probabilistic bayesian maximum entropy

The mapping of saline soils is the first task before any reclamation effort. Reclamation is based on the knowledge of soil salinity in space and how it evolves with time. Soil salinity is traditionally determined by soil sampling and laboratory analysis. Recently, it became possible to complement these hard data with soft secondary data made available using field sensors like electrode probes. In this study, we had two data sets. The first includes measurements of field salinity (EC_a) at 413 locations and 19 time instants. The second, which is a subset of the first (13 to 20 locations), contains, in addition to EC_a, salinity determined in the laboratory (EC_2.5). Based on a procedure of cross-validation, we compared the prediction performance in the space-time domain of 3 methods: kriging using either only hard data (HK) or hard and mid interval soft data (HMIK), and Bayesian maximum entropy (BME) using probabilistic soft data. We found that BME was less biased, more accurate and giving estimates, which were better correlated with the observed values than the two kriging techniques. In addition, BME allowed one to delineate with better detail saline from non-saline areas.     

Case studies of the propagation characteristics of auroral TIDS with EISCAT CP2 data using maximum entropy cross-spectral analysis

In this paper case studies of propagation characteristics of two TIDs are presented which are induced by atmospheric gravity waves in the auroral F-region on a magnetic quiet day. By means of maximum entropy cross-spectral analysis of EISCAT CP2 data, apparent full wave-number vectors of the TIDs are obtained as a function of height. The analysis results show that the two events considered can be classified as moderately large-scale TID and medium-scale TID, respectively. One exhibits a dominant period of about 72 min, a mean horizontal phase speed of about 180 m/s (corresponding to a horizontal wavelength of about 780 km) directed south-eastwards and a vertical phase speed of 55 m/s for a height of about 300 km. The other example shows a dominant period of 44 min, a mean horizontal phase velocity of about 160 m/s (corresponding to a horizontal wavelength of about 420 km) directed southwestwards, and a vertical phase velocity of about 50 m/s at 250 km altitude.Max-Planck-Institut für Aeronomie, Germany. On leave from Department of Space Physics, Wuhan University, China     

中国的地震活动性

基于华北地震统计区近600年地震资料,采用最大熵谱分析方法,定量解析了该区及各主要地震带的地震活动时间序列特征,为多个优势周期叠加的复杂强弱交替。以最优分割法划分了华北地震区及各主要地震带的地震幕,分析了其特点。结合当前华北地震区地震平静现象等,认为目前该区地震活动趋势正处于一个复杂的十字路口,华北地区随时可能进入一个新的地震活动幕（将持续10年左右）,然后进入新一轮地震平静期（将持续几十年）;也有另一种可能,华北地区现在实际已经进入了新的地震平静期。     

Derivation of some frequency distributions using the principle of maximum entropy (POME)

The principle of maximum entropy (POME) was employed to develop a procedure for derivation of a number of frequency distributions used in hydrology. The procedure required specification of constraints and maximization of entropy, and is thus a solution of the classical optimization problem. The POME led to a unique technique for parameter estimation. For six selected river gaging stations, parameters of the gamma distribution, the log-Pearson type III distribution and extreme value type I distribution fitted to annual maximum discharges, were evaluated by this technique and compared with those obtained by using the methods of moments and maximum likelihood estimation. The concept of entropy, used as a measured of uncertainty associated with a specified distribution, facilitated this comparison.     

Revealing hidden persistence in maximum rainfall records

ABSTRACT

Clustering of extremes is critical for hydrological design and risk management and challenges the popular assumption of independence of extremes. We investigate the links between clustering of extremes and long-term persistence, else Hurst-Kolmogorov (HK) dynamics, in the parent process exploring the possibility of inferring the latter from the former. We find that (a) identifiability of persistence from maxima depends foremost on the choice of the threshold for extremes, the skewness and kurtosis of the parent process, and less on sample size; and (b) existing indices for inferring dependence from series of extremes are biased downward when applied to non-Gaussian processes. We devise a probabilistic index based on the probability of occurrence of peak-over-threshold events across multiple scales, which can reveal clustering, linking it to the persistence of the parent process. Its application shows that rainfall extremes may exhibit noteworthy departures from independence and consistency with an HK model.     

Receiver function estimated by maximum entropy deconvolution

Maximum entropy deconvolution is presented to estimate receiver function, with the maximum entropy as the rule to determine auto-correlation and cross-correlation functions. The Toeplitz equation and Levinson algorithm are used to calculate the iterative formula of error-predicting filter, and receiver function is then estimated. During extrapolation, reflective coefficient is always less than I, which keeps maximum entropy deconvolution stable. The maximum entropy of the data outside window increases the resolution of receiver function. Both synthetic and real seismograms show that maximum entropy deconvolution is an effective method to measure receiver function in time-domain.     

Maximum entropy spectral analysis of volcanic tremor using data from Etna (Sicily) and Merapi (central Java)

The maximum entropy spectral analysis (MESA) method is applied to synthetic and observed tremor time series using autoregressive processes and recordings from the volcanoes Etna (Sicily) and Merapi (central Java). The MESA analysis can be used to estimate power spectra with sharp peaks from short data records. If the tremor source process can be modelled by an autoregressive process, the MESA method is well-suited for determining the coefficients of the underlying difference equations. As in the standard periodogram method of power spectrum estimation, a mesagram estimate using record segmentation and MESA spectrum averaging reduces the variance of the spectral estimator. In combination with periodogram estimates, mesagram estimates confirm that the tremor source may be modelled as an ensemble of randomly excited resonators. Used together, these estimates provide a valuable method for short-term monitoring of volcanic activity. In addition, they can be applied to the determination of new source parameters such as resonator frequencies, damping coefficients, excitation probabilities, correlation of exciting forces, and resonator coupling and in the pattern recognition of source types.     

On the use of maximum entropy/autoregressive spectrum in harmonic analysis of time series

The maximum entropy (ME) spectrum, or its equivalent form of the autoregressive (AR) spectrum, has been used as a tool for harmonic analysis of time series in geophysics. This paper critically examines its usage in estimating the amplitude and the exponential decay rate of a harmonic function. The argument is based upon Prony's relation, which relates a complex-conjugate pair of poles for the AR model of the time series on one hand, to the complex frequency of one harmonic component in the time series on the other. It is found that: (i) the ME spectrum can be used as an estimator for the decay rate in a way similar to the Fourier spectral analysis; (ii) the ME spectrum contains no information whatsoever about the amplitude, contrary to what has been claimed and practiced in geophysical applications.     

The Bayesian maximum entropy method for lognormal variables

The Bayesian maximum entropy (BME) method can be used to predict the value of a spatial random field at an unsampled location given precise (hard) and imprecise (soft) data. It has mainly been used when the data are non-skewed. When the data are skewed, the method has been used by transforming the data (usually through the logarithmic transform) in order to remove the skew. The BME method is applied for the transformed variable, and the resulting posterior distribution transformed back to give a prediction of the primary variable. In this paper, we show how the implementation of the BME method that avoids the use of a transform, by including the logarithmic statistical moments in the general knowledge base, gives more appropriate results, as expected from the maximum entropy principle. We use a simple illustration to show this approach giving more intuitive results, and use simulations to compare the approaches in terms of the prediction errors. The simulations show that the BME method with the logarithmic moments in the general knowledge base reduces the errors, and we conclude that this approach is more suitable to incorporate soft data in a spatial analysis for lognormal data.     

Revisiting prior distributions,Part II: Implications of the physical prior in maximum entropy analysis

The well-known “Maximum Entropy Formalism” offers a powerful framework for deriving probability density functions given a relevant knowledge base and an adequate prior. The majority of results based on this approach have been derived assuming a flat uninformative prior, but this assumption is to a large extent arbitrary (any one-to-one transformation of the random variable will change the flat uninformative prior into some non-constant function). In a companion paper we introduced the notion of a natural reference point for dimensional physical variables, and used this notion to derive a class of physical priors that are form-invariant to changes in the system of dimensional units. The present paper studies effects of these priors on the probability density functions derived using the maximum entropy formalism. Analysis of real data shows that when the maximum entropy formalism uses the physical prior it yields significantly better results than when it is based on the commonly used flat uninformative prior. This improvement reflects the significance of the incorporating additional information (contained in physical priors), which is ignored when flat priors are used in the standard form of the maximum entropy formalism. A potentially serious limitation of the maximum entropy formalism is the assumption that sample moments are available. This is not the case in many macroscopic real-world problems, where the knowledge base available is a finite sample rather than population moments. As a result, the maximum entropy formalism generates a family of “nested models” parameterized by the unknown values of the population parameters. In this work we combine this formalism with a model selection scheme based on Akaike’s information criterion to derive the maximum entropy model that is most consistent with the available sample. This combination establishes a general inference framework of wide applicability in scientific/engineering problems.     

Modeling short duration extreme precipitation patterns using copula and generalized maximum pseudo-likelihood estimation with censoring

The paper aims to develop researches on the spatial variability of heavy rainfall events estimation using spatial copula analysis. To demonstrate the methodology, short time resolution rainfall time series from Stuttgart region are analyzed. They are constituted by rainfall observations on continuous 30 min time scale recorded over a network composed by 17 raingages for the period July 1989–July 2004. The analysis is performed aggregating the observations from 30 min up to 24 h. Two parametric bivariate extreme copula models, the Husler–Reiss model and the Gumbel model are investigated. Both involve a single parameter to be estimated. Thus, model fitting is operated for every pair of stations for a giving time resolution. A rainfall threshold value representing a fixed rainfall quantile is adopted for model inference. Generalized maximum pseudo-likelihood estimation is adopted with censoring by analogy with methods of univariate estimation combining historical and paleoflood information with systematic data. Only pairs of observations greater than the threshold are assumed as systematic data. Using the estimated copula parameter, a synthetic copula field is randomly generated and helps evaluating model adequacy which is achieved using Kolmogorov Smirnov distance test. In order to assess dependence or independence in the upper tail, the extremal coefficient which characterises the tail of the joint bivariate distribution is adopted. Hence, the extremal coefficient is reported as a function of the interdistance between stations. If it is less than 1.7, stations are interpreted as dependent in the extremes. The analysis of the fitted extremal coefficients with respect to stations inter distance highlights two regimes with different dependence structures: a short spatial extent regime linked to short duration intervals (from 30 min to 6 h) with an extent of about 8 km and a large spatial extent regime related to longer rainfall intervals (from 12 h to 24 h) with an extent of 34 to 38 km.     

GRACE和SLR观测的地球动力学扁率最大熵谱及小波相关分析

卫星重力测量技术的实现为测定地球动力学扁率提供了新的方式和途径,GRACE卫星是目前最新的重力测量卫星,据其恢复的低阶重力场较以往精度得到大大提高,然而其观测地球动力学扁率（二阶项）却与卫星激光测距（SLR）结果相差较大.本文采用最大熵谱和小波分析方法对GRACE和SLR观测的地球动力学扁率时间序列信号进行定量比较分析,结果表明：GRACE观测的地球动力学扁率年际周期变化振幅仅为SLR观测结果的25%,并且目前GRACE观测的地球动力学扁率数据中含有系统输入信息和相位差,但前者较后者包含有较强的短周期（2～6月）信息.造成这种差异的主要原因可能来自于GRACE与SLR全球观测数据时空分布不同.     

Bayesian maximum entropy approach and its applications: a review

The present paper reviews the conceptual framework and development of the Bayesian Maximum Entropy (BME) approach. BME has been considered as a significant breakthrough and contribution to applied stochastics by introducing an improved, knowledge-based modeling framework for spatial and spatiotemporal information. In this work, one objective is the overview of distinct BME features. By offering a foundation free of restrictive assumptions that limit comparable techniques, an ability to integrate a variety of prior knowledge bases, and rigorous accounting for both exact and uncertain data, the BME approach was coined as introducing modern spatiotemporal geostatistics. A second objective is to illustrate BME applications and adoption within numerous different scientific disciplines. We summarize examples and real-world studies that encompass the perspective of science of the total environment, including atmosphere, lithosphere, hydrosphere, and ecosphere, while also noting applications that extend beyond these fields. The broad-ranging application track suggests BME as an established, valuable tool for predictive spatial and space–time analysis and mapping. This review concludes with the present status of BME, and tentative paths for future methodological research, enhancements, and extensions.     

Lake-level records support a mid-Holocene maximum precipitation in northern China

Lake level and its inferred East Asian summer monsoon (EASM) evolution in northern boundary of EASM during the Holocene are highly debated.Here,we present a 15-...     

Time series analysis using spectral techniques: Oscillatory currents

The use of time domain and frequency domain techniques in the analysis of geomorphological time series is discussed and the techniques of spectral analysis are introduced. An Excel worksheet is used to analyse wave-current data from Slapton Sands in Devon and the advantages and limitations of the technique are listed.