Seismic waves and correlation autoregressive processes

Stochastic models appropriate for seismic records of earthquakes and underground nuclear explosions are considered and a selective review of the existing models is presented. Special models of stationary processes, periodically correlated processes, and uniformly modulated stationary processes as well as a new class called correlation autoregressive processes are studied. Relevant properties of correlation autoregressive processes are presented. It is shown that the successful models presently used are all subclasses of the correlation autoregressive processes. Shortcomings of the existing nonstationary models and merits of the new class for seismic wave modeling are discussed and directions for the further research and development are suggested. It is hoped that the model and the ideas introduced in this article will provide a stimulation for further examination of correlation autoregressive processes and will promote statistical modeling of seismic records.     

Trend estimation and univariate forecast of the sunspot numbers: Development and comparison of ARMA, ARIMA and Autoregressive Neural Network models

In the present study, a prominent 11-year cycle, supported by the pattern of the autocorrelation function and measures of Euclidean distances, in the mean annual sunspot number time series has been observed by considering the sunspot series for the duration of 1749 to 2007. The trend in the yearly sunspot series, which is found to be non-normally distributed, is examined through the Mann-Kendall non-parametric test. A statistically significant increasing trend is observed in the sunspot series in annual duration. The results indicate that the performance of the autoregressive neural network-based model is much better than the autoregressive moving average and autoregressive integrated moving average-based models for the univariate forecast of the yearly mean sunspot numbers.     

Variation in the value of mineral resources. Part I. The USA

The value of mineral resources produced in the USA during the 93 years, 1880–1972, deflated to 1967=100, totals 921 billions of dollars; this yields a value of $303,289 per mi² for the conterminous 48 states. Alaska has aggregated some 4160 millions of dollars during the same period, an average yield of some 7107 deflated dollars per mi². If we assume Alaska will achieve the average level of the “lower 48,”the potential value of its mineral resources is 178 billions of deflated 1967 dollars. Fuels account for about two-thirds of this value followed by about 20 percent for each of the aggregates, nonmetals and metals. The deflated dollar value of these four aggregate figures, fuels, nonmetals, metals, and total, during the period 1880–1972 are four time series and the economic processes which produced these series may be modelled through Box-Jenkins procedures. The value of fuels has steadily increased through the period, except for the depression years of the 1930s; this series may be represented as a multiplicative seasonal ARIMA(1,1,0)(1,0,1) model with periods at 4 and 3 yr for the autoregressive (AR) and moving average (MA) terms. Forecasts for 1973 to 1980, using the model, show the value of fuels produced to be about 20 billions of dollars per annum. The value of nonmetals also increased throughout the period except for a somewhat larger drop during the depression years of the 1930s. A multiplicative seasonal ARMA(1,0,0)(1,0,1) model with period at 7 yr for both the AR (autoregressive) and MA (moving average) terms appears to best reflect this series; the forecasts with this model fluctuate around their present annual value of some 6 billions of dollars. The value of metals behaves less consistently; it was much more strongly affected by the Great Depression and its subsequent growth is slower and less consistent than those of the fuels and nonmetals. It is appropriately represented by a multiplicative seasonal ARMA(1,0,0)(0,0,2) model with moving average (MA) periods at 6 and 11 yr respectively. Forecasts with this model show a decline in value for years beyond 1972; the large residual “error” \((\hat \sigma _e = 0.0925)\) , which is about twice as large as the equivalent errors for the value of fuels and nonmetals ( \(\hat \sigma _e = 0.0408\) and 0.0532, respectively), makes this forecast less firm. The total value of mineral resources is composed of all three series and, because fuels account for two-thirds of the total value, the two series closely resemble each other. The total value is not a simple aggregate of the three series; it is appropriately fitted by a multiplicative seasonal IMA(0,1,0)(0,0,2) model with periods at 7 and 11 yr (and error \(\hat \sigma _e = 0.0423\) ). Forecasts using this model imply that the total value of mineral resources produced will be over 30 billions of dollars per annum through 1980.     

Simulation of one-dimensional correlated fields using a matrix-factorization moving average approach

The simulation of one-dimensional stationary correlated fields is of increasing importance in the earth sciences. A new method for repeated generation of independent realizations, which are long and dense relative to the correlation scale of the underlying stochastic process, is examined here. This method is conceptually simple and easy to apply. It consists of a matrix-factorization technique for derivation of moving average coefficients which are used as weights in the construction of successive observations from linear combinations of random normal deviates. The matrix-factorization procedure is fast and need be performed only once for a given correlation function and density of observations. This technique can be used to generate evenly spaced observations in time or a single space dimension for any prescribed correlation function and marginal distribution which is Gaussian with arbitrary mean and variance. Tests of ensemble properties of generation procedures have been developed and results for this method compared with those for a popular generation technique. For correlation functions and generation conditions examined, the matrix-factorization moving average approach more accurately produces ensemble characteristics of the prescribed underlying process. For repeated generation of 2001 observations spaced evenly over realizations with length equal to 100 times the correlation scale, the moving average approach requires only about one fifth the CPU time used by the Shinozuka and Jan method to obtain similar accuracy.     

Rainfall-Runoff Modeling using Clustering and Regression Analysis for the River Brahmaputra Basin

In this research, k-means, agglomerative hierarchical clustering and regression analysis have been applied in hydrological real time series in the form of patterns and models, which gives the fruitful results of data analysis, pattern discovery and forecasting of hydrological runoff of the catchment. The present study compares with the actual field data, predicted value and validation of statistical yields obtained from cluster analysis, regression analysis with ARIMA model. The seasonal autoregressive integrated moving average (SARIMA) and autoregressive integrated moving average (ARIMA) models is investigated for monthly runoff forecasting. The different parameters have been analyzed for the validation of results with casual effects. The comparison of model results obtained by K-means & AHC have very close similarities. Result of models is compared with casual effects in the same scenario and it is found that the developed model is more suitable for the runoff forecasting. The average value of R² determined is 0.92 for eight ARIMA models. This shows more accuracy of developed ARIMA model under these processes. The developed rainfall runoff models are highly useful for water resources planning and development.     

Estimation of semivariograms by the maximum entropy method

A wide variety of semivariograms may be represented in terms of a first- or second-order autoregressive (AR) process, and the nugget effect may be included by use of a moving average (MA) process. The weighting parameters for these models have a simple functional dependence on the value of the sill and the semivariance at the first and second lag. These may be estimated either graphically from the semivariogram or directly from the computed values. Improved spectral estimates of geophysical data have been obtained by the use of the maximum entropy method, and the necessary equations were adapted here for the estimation of the weighting parameters of the AR and the MA processes. Comparison among the semivariograms obtained for the ideal case, the observed case, and the estimated case for artificial series show excellent correspondence between the ideal and estimated while the observed semivariogram may show marked divergence.     

Log-uniform distribution of gold deposits along major Archean fault zones

The location of potentially unrecognized gold deposits in the close vicinity of the Cadillac–Larder Lake Fault Zone in the Archean Abitibi Subprovince (Canada) is predicted by applying a stochastic approach to the distribution of known gold deposits. The methodology uses the distances between neighboring orogenic gold deposits along the fault trace. The cumulative distribution of the curvilinear inter-distances along the fault zone, are adequately represented by a log-uniform model. The average inter-distance is 1.95 km, and an upper inter-distance of 5.6 km is observed. The same distribution pattern appears along the Destor–Porcupine Fault Zone (Abitibi). This log-uniform distribution shows that the spatial distribution of gold deposits is regionally controlled by the major crustal shear zone. Lithologies and structures only seem to exert a local influence at the deposit scale. The log-uniform spacing could be interpreted as the result of the crustal failure locations induced by hydraulic overpressure along mechanically independent segments on the main fault.     

Cumulative semivariogram models of regionalized variables

The cumulative semivariogram approach is proposed for modeling regionalized variables in the geological sciences. This semivariogram is defined as the successive summation of half-squared differences which are ranked according to the ascending order of distances extracted from all possible pairs of sample locations within a region. This procedure is useful especially when sampling points are irregularly distributed within the study area. Cumulative semivariograms possess all of the objective properties of classical semivariograms. Classical semivariogram models are evaluated on the basis of the cumulative semivariogram methodology. Model parameter estimation procedures are simplified with the use of arithmetic, semilogarithmic, or double-logarithmic papers. Plots of cumulative semivariogram values vs. corresponding distances may scatter along a straight line on one of these papers, which facilitates model identification as well as parameter estimation. Straight lines are fitted to the cumulative semivariogram scatter diagram by classical linear regression analysis. Finally, applications of the methodology are presented for some groundwater data recorded in the sedimentary basins of the Kingdom of Saudi Arabia.     

Monthly karstic spring flow forecasting using a sequential gaussian simulation technique

Analyzing groundwater hydrologic equations related to karstic aquifers and spring hydrograph simulation have become the focus of many researches. Having double or triple porosity structure, mixed flow nature, and varying conduit permeability have made these formations become complex heterogenic systems with great temporal and spatial hydrodynamic variability. In this paper, a conditional sequential gaussian simulation (SGS) is used to simulate monthly flow data of five karstic springs with different hydrogeological properties, located in Zagros Mountain Chain, in western Iran. To evaluate the performance of the SGS algorithm, the results are compared with those of an autoregressive integrated moving average (ARIMA) model. The results demonstrate the efficiency of the SGS model in simulation of monthly flows compared to the ARIMA model. They also show the suitability of this model for handling uncertainty associated with karstic spring flows through generation of several equally probable stochastic realizations.     

Stochastic modeling of gold mineralization in the Champion lode system of Kolar gold fields (India)

Stochastic modeling of gold mineralization in the Champion lode of Kolar gold fields was carried out using assay data taken from developmental headings. After dividing the lode into 71 horizontal and 18 vertical strata, autoregressive (AR), moving average (MA), and autoregressive and moving average (ARMA) models were developed and applied. The model selection with the acf and pacf for the various strata showed that in most of the cases, ARMA modeling of first-order would forecast gold headings with a reasonable degree of confidence. This was substantiated by comparing the coefficients of variation. From a parsimony point of view, AR (1) model may also be considered valid. The best overall models are: ARMA (1, 1), ; AR (1), , where a_t is N (0, _a ² ), x is in logarithms of in-dwt, and t is in block units of 100 ft. The applications of these models to a specific stratum are given. These models would also be helpful to describe the characteristics of the gold mineralization process of this lode.     

应用CARMA模型评价矿井疏水对泉流量的影响

受控自回归滑动平均模型是用来描述随机现象中输入与输出关系的一类线性动态模型。本文以大柳塔井田双沟泉域为例,应用该模型分别对双沟泉域在天然条件和矿井疏水条件下的泉流量进行预测,并据此评价矿井疏水对泉流量的影响。     

Assessment of prediction performances of stochastic models: Monthly groundwater level prediction in Southern Italy

Stochastic modelling of hydrological time series with insufficient length and data gaps is a serious challenge since these problems significantly affect the reliability of statistical models predicting and forecasting skills. In this paper, we proposed a method for searching the seasonal autoregressive integrated moving average(SARIMA) model parameters to predict the behavior of groundwater time series affected by the issues mentioned. Based on the analysis of statistical indices, 8 stations among 44 available within the Campania region(Italy) have been selected as the highest quality measurements. Different SARIMA models, with different autoregressive, moving average and differentiation orders had been used.By reviewing the criteria used to determine the consistency and goodness-of-fit of the model, it is revealed that the model with specific combination of parameters, SARIMA(0,1,3)(0,1,2) _(12), has a high R~2 value,larger than 92%, for each of the 8 selected stations. The same model has also good performances for what concern the forecasting skills, with an average NSE of about 96%. Therefore, this study has the potential to provide a new horizon for the simulation and reconstruction of groundwater time series within the investigated area.     

ARMA order model detection using minimum of Kurtosis: application on seismic data

The objective of this study is to find the order and coefficients of non-low-phase causal filters for ARMA (auto regressive moving average) filter model, using the Kurtosis minimization criterion. This method is based on the Kurtosis calculation of the treated sample at the input level and its identification at the output of the ARMA model. For this purpose, the order and coefficients of the AR (auto regressive) part are identified using the Yule-Walker algorithm at order two and then extended to order four. To obtain the MA (moving average) part, the AR components are calculated at first from the ARMA filter by deconvolution. Then, spectrally equivalent and minimum phase (SEMP) MA filter is identified using the Durbin algorithm at second and fourth order. Finally, the correct filter is found when the Kurtosis value of the output ARMA filter reconstituted is the closest to the Kurtosis of introduced signal. The proposed method is then tested on simulated processes and applied to real seismic data to perform blind deconvolution and obtain the reflectivity coefficients of subsoil studied.     

The outer scale of solar-wind turbulence from GALILEO coronal-sounding data

Radio sounding experiments on of the solar plasma were carried out by the GALILEO spacecraft using S-band (2295 MHz) signals in 1995–1996 a period of minimum solar activity. Equatorial regions at heliocentric distances of 7–80 solar radii were studied. The frequency of the received signal was detected by three ground stations. By carrying out continuous observations of unprecedented duration and processing the data using spectral and correlation methods, we have obtained reliable information on large-scale inhomogeneities of the solar-wind density for the first time. The outer turbulence scale increases with heliocentric distance, the dependence being close to linear. We estimate the outer turbulence scale and analyze its dependence on distance from the Sun and local plasma parameters for a model in which the outer scale is formed due to competition between the linear amplification of Alfven waves in the irregular, moving solar-wind plasma and the nonlinear transfer of turbulent energy to higher frequencies. A comparison of predictions for various specific cases of this model with the observational data suggests that the main nonlinear processes responsible for the formation of the inertial range of the spectrum on the investigated scales are three-wave decay processes involving Alfven and magnetoacoustic waves.     

Intermittent reservoir daily-inflow prediction using lumped and distributed data multi-linear regression models

In this study, multi-linear regression (MLR) approach is used to construct intermittent reservoir daily inflow forecasting system. To illustrate the applicability and effect of using lumped and distributed input data in MLR approach, Koyna river watershed in Maharashtra, India is chosen as a case study. The results are also compared with autoregressive integrated moving average (ARIMA) models. MLR attempts to model the relationship between two or more independent variables over a dependent variable by fitting a linear regression equation. The main aim of the present study is to see the consequences of development and applicability of simple models, when sufficient data length is available. Out of 47 years of daily historical rainfall and reservoir inflow data, 33 years of data is used for building the model and 14 years of data is used for validating the model. Based on the observed daily rainfall and reservoir inflow, various types of time-series, cause-effect and combined models are developed using lumped and distributed input data. Model performance was evaluated using various performance criteria and it was found that as in the present case, of well correlated input data, both lumped and distributed MLR models perform equally well. For the present case study considered, both MLR and ARIMA models performed equally sound due to availability of large dataset.     

Structure of earthquake occurrence in space,time and magnitude

Distributions of time between consecutive earthquakes verify an approximately universal scaling law for stationary seismicity. The shape of these distributions is shown to arise as a mixture of one distribution for short‐distance events and an exponential distribution for far‐off events, the distinction from short and long distances being relative to the size of the region studied. The distributions of consecutive distances show a double power law decay and verify an approximate scaling law which guarantees the simultaneous fulfillment of the scaling laws for time. The interplay between space and time can be seen as well by looking at the distribution of distances for a fixed time separation. These results suggest that seismicity can be understood as a series of intertwined independent continuous‐time random walks, with power law‐distributed waiting times and Lévy‐flight jumps. However, a simple model based on these ideas does not capture the invariance of seismicity under renormalization.     

Features of the geological history of the Iultin Massif determined by stochastic modeling

Petrographic data from the Iultin Massif (Chuckotka) were analyzed for crystallization and postmagmatic transformation using a Markov stochastic model. The goal was to ascertain whether this model can be used to divide the massif into distinctive portions. A comparison was made between the stochastic approach and conventional geological methods for understanding the historical development of granitic massifs.     

随机序列轮长与轮次的统计规律

平稳随机序列的轮次,服从二阶线性齐次差分方程。其中,独立序列的差分方程以状态发生的平均概率为二重特征根,并能用先验条件确定差分方程通解中的常数。非独立序列的差分方程有二个相异的特征根,需根据样本资料来确定差分方程通解中的常数。经用干旱序列及实测水文序列检验,证实了方法的适用性。     

Assessing macroeconomic recovery after a natural hazard based on ARIMA—a case study of the 2008 Wenchuan earthquake in China

In this research the autoregressive integrated moving average model (ARIMA) was adopted to estimate GDP growth for assessing macroeconomic recovery from a natural hazard. ARIMA is a powerful tool for predicting future macroeconomic situations based on macroeconomic data and trends before a disaster. This research examined the economic recovery of severely affected counties 6 years after the Wenchuan Earthquake by comparing the simulated regional GDP values from 2008 to 2014 with actual regional GDP data following the disaster. A macroeconomic recovery ratio model was built to assess economic resilience of the affected counties. Although the GDP of the 10 most severely affected counties grew rapidly after the earthquake, in only one county—Pengzhou City—it recovered to the pre-disaster level under a no hazard scenario. The gaps in economic recovery between the severely affected counties were significant. Areas with competitive industries and locational advantage exhibited the highest rates of recovery. Therefore, industrial structure and relative location of the affected areas to traffic nodes and economic centers should be considered in pre-disaster assistance decisions and development plans in the future.     

粘性土壤溶质运移新模型的应用

将Moldrup等提出的模拟水流和溶质运移的运动平均斜率模型和运动浓度斜率模型联合,模拟了重粘土在入渗和蒸发条件下的水盐动态并与特征-有限差分数值模型的计算结果进行了比较。在模拟过程中新模型显示出更加高效的优点,这对粘性土壤地区盐渍化趋势的长期预报具有重要的实际意义。