Semi-Periodic Sequences and Extraneous Events in Earthquake Forecasting: I. Theory and Method,Parkfield Application

We present a new method to identify semi-periodic sequences in the occurrence times of large earthquakes, which allows for the presence of multiple semi-periodic sequences and/or events not belonging to any identifiable sequence in the time series. The method, based on the analytic Fourier transform, yields estimates of the departure from periodicity of an observed sequence, and of the probability that the sequence is not due to chance. These estimates are used to make and to evaluate forecasts of future events belonging to each sequence. Numerous tests with synthetic catalogs show that the method is surprisingly capable of correctly identifying sequences, unidentifiable by eye, in complicated time series. Correct identification of a given sequence depends on the number of events it contains, on the sequence’s departure from periodicity, and, in some cases, on the choice of starting and ending times of the analyzed time window; as well as on the total number of events in the time series. Some particular data combinations may result in spectra where significant periods are obscured by large amplitudes artifacts of the transform, but artifacts can be usually recognized because they lack harmonics; thus, in most of these cases, true semi-periodic sequences may not be identified, but no false identifications will be made. A first example of an application of the method to real seismicity data is the analysis of the Parkfield event series. The analysis correctly aftcasts the September 2004 earthquake. Further applications to real data from Japan and Venezuela are shown in a companion paper.     

Short- and Long-Term Earthquake Forecasts for California and Nevada

We present estimates of future earthquake rate density (probability per unit area, time, and magnitude) on a 0.1-degree grid for a region including California and Nevada, based only on data from past earthquakes. Our long-term forecast is not explicitly time-dependent, but it can be updated at any time to incorporate information from recent earthquakes. The present version, founded on several decades worth of data, is suitable for testing without updating over a five-year period as part of the experiment conducted by the Collaboratory for Study of Earthquake Predictability  (CSEP). The short-term forecast is meant to be updated daily and tested against similar models by CSEP. The short-term forecast includes a fraction of our long-term one plus time-dependent contributions from all previous earthquakes. Those contributions decrease with time according to the Omori law: proportional to the reciprocal of the elapsed time. Both forecasts estimate rate density using a radially symmetric spatial smoothing kernel decreasing approximately as the reciprocal of the square of epicentral distance, weighted according to the magnitude of each past earthquake. We made two versions of both the long- and short-term forecasts, based on the Advanced National Seismic System  (ANSS) and Preliminary Determinations of Epicenters (PDE) catalogs, respectively. The two versions are quite consistent, but for testing purposes we prefer those based on the ANSS catalog since it covers a longer time interval, is complete to a lower magnitude threshold and has more precise locations. Both forecasts apply to shallow earthquakes only (depth 25 km or less) and assume a tapered Gutenberg-Richter magnitude distribution extending to a lower threshold of 4.0.     

A Long-term earthquake forecast for the Kuril-Kamchatka Island arc for the period 2006–2011 and a successful forecast of the MS = 8.2 middle kuril earthquake of November 15, 2006

Results are reported from continuous long-term earthquake prediction work for the Kuril-Kamchatka island arc using the patterns of seismic gaps and the seismic cycle. A five-year forecast (April 2006 to April 2011) for all portions of the Kuril-Kamchatka seismogenic zone is presented. According to this, the most likely locations of future M ≥ 7.7 earthquakes include the Petropavlovsk-Kamchatskii area where the probability of an M ≥ 7.7 earthquake causing ground motions of intensity VII to IX in the town of Petropavlovsk-Kamchatskii is 48% for 2006–2011, and the area of Onekotan I. and the Middle Kuril Islands where the probability of an M ≥ 7.7 earthquake was estimated as 26.7%. The forecast was fulfilled on November 15, 2006, when an M_s= 8.2, M_w = 8.3 earthquake occurred in the Middle Kuril Islands area. An updated long-term forecast is presented for the Kuril-Kamchatka arc for the period from November 2006 to October 2011. These forecasts provide good reasons to enhance seismic safety by strengthening buildings and structures in Kamchatka.     

Bayesian analyses of seasonal runoff forecasts

Forecasts of seasonal snowmelt runoff volume provide indispensable information for rational decision making by water project operators, irrigation district managers, and farmers in the western United States. Bayesian statistical models and communication frames have been researched in order to enhance the forecast information disseminated to the users, and to characterize forecast skill from the decision maker's point of view. Four products are presented: (i) a Bayesian Processor of Forecasts, which provides a statistical filter for calibrating the forecasts, and a procedure for estimating the posterior probability distribution of the seasonal runoff; (ii) the Bayesian Correlation Score, a new measure of forecast skill, which is related monotonically to theex ante economic value of forecasts for decision making; (iii) a statistical predictor of monthly cumulative runoffs within the snowmelt season, conditional on the total seasonal runoff forecast; and (iv) a framing of the forecast message that conveys the uncertainty associated with the forecast estimates to the users. All analyses are illustrated with numerical examples of forecasts for six gauging stations from the period 1971–1988.     

Earthquake Forecasting in Diverse Tectonic Zones of the Globe

We present a simple method for long- and short-term earthquake forecasting (estimating earthquake rate per unit area, time, and magnitude). For illustration we apply the method to the Pacific plate boundary region and the Mediterranean area surrounding Italy and Greece. Our ultimate goal is to develop forecasting and testing methods to validate or falsify common assumptions regarding earthquake potential. Our immediate purpose is to extend the forecasts we made starting in 1999 for the northwest and southwest Pacific to include somewhat smaller earthquakes and then adapt the methods to apply in other areas. The previous forecasts used the CMT earthquake catalog to forecast magnitude 5.8 and larger earthquakes. Like our previous forecasts, the new ones here are based on smoothed maps of past seismicity and assume spatial clustering. Our short-term forecasts also assume temporal clustering. An important adaptation in the new forecasts is to abandon the use of tensor focal mechanisms. This permits use of earthquake catalogs that reliably report many smaller quakes with no such mechanism estimates. The result is that we can forecast earthquakes at higher spatial resolution and down to a magnitude threshold of 4.7. The new forecasts can be tested far more quickly because smaller events are considerably more frequent. Also, our previous method used the focal mechanisms of past earthquakes to estimate the preferred directions of earthquake clustering, however the method made assumptions that generally hold in subduction zones only. The new approach escapes those assumptions. In the northwest Pacific the new method gives estimated earthquake rate density very similar to that of the previous forecast.     

Bayesian probabilities of earthquake occurrences in Greece and surrounding areas

The Bayesian extreme-value distribution of earthquake occurrences has been adopted to estimate the seismic hazard in some seismogenic zones in Greece and surrounding regions. Seismic moment, slip rate, earthquake recurrence rate and magnitude were considered as basic parameters for computing the prior estimates of the seismicity. These estimates are then updated in terms of Bayes' theorem and historical estimates of seismicity associated with each zone.High probabilities for earthquakes withM6.0 have been obtained for the northwestern part of Greece as well as for the southwestern part of the Hellenic arc.     

Probability gains of an epidemic-type aftershock sequence model in retrospective forecasting of <Emphasis Type="Italic">M</Emphasis> ≥ 5 earthquakes in Italy

A stochastic triggering (epidemic) model incorporating short-term clustering was fitted to the instrumental earthquake catalog of Italy for event with local magnitudes 2.6 and greater to optimize its ability to retrospectively forecast 33 target events of magnitude 5.0 and greater that occurred in the period 1990–2006. To obtain an unbiased evaluation of the information value of the model, forecasts of each event use parameter values obtained from data up to the end of the year preceding the target event. The results of the test are given in terms of the probability gain of the epidemic-type aftershock sequence (ETAS) model relative to a time-invariant Poisson model for each of the 33 target events. These probability gains range from 0.93 to 32000, with ten of the target events yielding a probability gain of at least 10. As the forecasting capability of the ETAS model is based on seismic activity recorded prior to the target earthquakes, the highest probability gains are associated with the occurrence of secondary mainshocks during seismic sequences. However, in nine of these cases, the largest mainshock of the sequence was marked by a probability gain larger than 50, having been preceded by previous smaller magnitude earthquakes. The overall evaluation of the performance of the epidemic model has been carried out by means of four popular statistical criteria: the relative operating characteristic diagram, the R score, the probability gain, and the log-likelihood ratio. These tests confirm the superior performance of the method with respect to a spatially varying, time-invariant Poisson model. Nevertheless, this method is characterized by a high false alarm rate, which would make its application in real circumstances problematic.     

A Bayesian approach for estimating extreme flood probabilities with upper-bounded distribution functions

Some recent research on fluvial processes suggests the idea that some hydrological variables, such as flood flows, are upper-bounded. However, most probability distributions that are currently employed in flood frequency analysis are unbounded to the right. This paper describes an exploratory study on the joint use of an upper-bounded probability distribution and non-systematic flood information, within a Bayesian framework. Accordingly, the current PMF maximum discharge appears as a reference value and a reasonable estimate of the upper-bound for maximum flows, despite the fact that PMF determination is not unequivocal and depends strongly on the available data. In the Bayesian context, the uncertainty on the PMF can be included into the analysis by considering an appropriate prior distribution for the maximum flows. In the sequence, systematic flood records, historical floods, and paleofloods can be included into a compound likelihood function which is then used to update the prior information on the upper-bound. By combining a prior distribution describing the uncertainties of PMF estimates along with various sources of flood data into a unified Bayesian approach, the expectation is to obtain improved estimates of the upper-bound. The application example was conducted with flood data from the American river basin, near the Folsom reservoir, in California, USA. The results show that it is possible to put together concepts that appear to be incompatible: the deterministic estimate of PMF, taken as a theoretical limit for floods, and the frequency analysis of maximum flows, with the inclusion of non-systematic data. As compared to conventional analysis, the combination of these two concepts within the logical context of Bayesian theory, contributes an advance towards more reliable estimates of extreme floods.     

两种短期概率预测模型在2017年九寨沟7.0级地震中的应用和比较研究

本文选用"传染型余震序列"（ETAS）模型和Reasenberg-Jones（R-J）模型,分别对九寨沟M_S7.0地震序列的模型参数稳定性、余震发生率预测和余震概率预测进行了比较研究,并利用"地震信息增益"（IGPE）、N-test和T-test检验方法对预测效果进行了评价.研究结果表明,ETAS模型和R-J模型的序列参数分别在震后t₂=2.0天和t₂=1.50天后趋于稳定,此次九寨沟M_S7.0地震序列的衰减较为正常;对未来1天的余震发生率预测和余震概率连续滑动预测表明,ETAS模型给出的余震发生率和余震概率数值均低于R-J模型预测结果;IGPE结果显示,ETAS模型在95%的置信区间上预测效果明显优于R-J模型;统计检验结果表明,在序列参数较不稳定的震后早期阶段,ETAS模型预测失效而R-J模型预测效果较好,在序列参数稳定阶段,ETAS模型预测效果较好而R-J模型预测失效.根据上述分析,在与此次九寨沟M_S7.0地震类型相同的地震的余震预测策略上,如可在序列参数不稳定的震后早期阶段使用R-J模型、在此后使用ETAS模型,或可取得较好的预测效果.     

Compositional Bayesian indicator estimation

Indicator kriging is widely used for mapping spatial binary variables and for estimating the global and local spatial distributions of variables in geosciences. For continuous random variables, indicator kriging gives an estimate of the cumulative distribution function, for a given threshold, which is then the estimate of a probability. Like any other kriging procedure, indicator kriging provides an estimation variance that, although not often used in applications, should be taken into account as it assesses the uncertainty of the estimate. An alternative approach to indicator estimation is proposed in this paper. In this alternative approach the complete probability density function of the indicator estimate is evaluated. The procedure is described in a Bayesian framework, using a multivariate Gaussian likelihood and an a priori distribution which are both combined according to Bayes theorem in order to obtain a posterior distribution for the indicator estimate. From this posterior distribution, point estimates, interval estimates and uncertainty measures can be obtained. Among the point estimates, the median of the posterior distribution is the maximum entropy estimate because there is a fifty-fifty chance of the unknown value of the estimate being larger or smaller than the median; that is, there is maximum uncertainty in the choice between two alternatives. Thus in some sense, the latter is an indicator estimator, alternative to the kriging estimator, that includes its own uncertainty. On the other hand, the mode of the posterior distribution estimator, assuming a uniform prior, is coincidental with the simple kriging estimator. Additionally, because the indicator estimate can be considered as a two-part composition which domain of definition is the simplex, the method is extended to compositional Bayesian indicator estimation. Bayesian indicator estimation and compositional Bayesian indicator estimation are illustrated with an environmental case study in which the probability of the content of a geochemical element in soil being over a particular threshold is of interest. The computer codes and its user guides are public domain and freely available.     

大别山库区降水预报性能评估及应用对策

对降水预报进行性能评估及应用对策研究可以更好地发挥降水预报在水库调度中的决策支持作用.基于大别山库区近10 a汛期(2007—2016年5月1日—9月30日)24~168 h共7个预见期降水预报和地面降水观测资料,采用正确率、TS评分、概率统计、ROC曲线以及CTS等方法评估大别山库区降水预报性能,并以响洪甸水库为重点研究区域分析降水预报在水库调度中的应用对策.结果表明:1)大别山库区各量级的降水预报都有正预报技巧;24~72 h预见期降水预报的TS评分较高且空报率、漏报率也较低,具有较高的预报性能;但96 h及以上预见期降水预报性能明显下降,中雨以上量级空报率、漏报率较大,特别是对大暴雨及其以上量级的降水预报性能显著下降.2)大别山库区预报降水量级与实况降水量级基本符合,预报降水量级大于等于实况降水量级的概率超过75%;虽然降水预报量级上呈现出过度预报的现象,但降水过程预报对水库调度仍有较好的应用价值,应用时要考虑到降水预报量级可能存在偏差.3)转折性天气预报96 h及以上预见期CTS评分较低,但72 h以内预见期的性能明显改进,尤其是24 h预见期CTS评分也提高到了38.2%;水库调度可从长预见期的降水预报获取降水过程及其可能发生转折的信息,根据短预见期的降水预报进行调度方案调整.     

Permeability estimation conditioned to geophysical downhole log data in sandstones of the northern Galilee Basin,Queensland: Methods and application

This study uses borehole geophysical log data of sonic velocity and electrical resistivity to estimate permeability in sandstones in the northern Galilee Basin, Queensland. The prior estimates of permeability are calculated according to the deterministic log–log linear empirical correlations between electrical resistivity and measured permeability. Both negative and positive relationships are influenced by the clay content. The prior estimates of permeability are updated in a Bayesian framework for three boreholes using both the cokriging (CK) method and a normal linear regression (NLR) approach to infer the likelihood function. The results show that the mean permeability estimated from the CK-based Bayesian method is in better agreement with the measured permeability when a fairly apparent linear relationship exists between the logarithm of permeability and sonic velocity. In contrast, the NLR-based Bayesian approach gives better estimates of permeability for boreholes where no linear relationship exists between logarithm permeability and sonic velocity.     

Bayesian analysis on earthquake magnitude related to an active fault in Taiwan

It is understood that sample size could be an issue in earthquake statistical studies, causing the best estimate being too deterministic or less representative derived from limited statistics from observation. Like many Bayesian analyses and estimates, this study shows another novel application of the Bayesian approach to earthquake engineering, using prior data to help compensate the limited observation for the target problem to estimate the magnitude of the recurring Meishan earthquake in central Taiwan. With the Bayesian algorithms developed, the Bayesian analysis suggests that the next major event induced by the Meishan fault in central Taiwan should be in M_w 6.44±0.33, based on one magnitude observation of M_w 6.4 from the last event, along with the prior data including fault length of 14 km, rupture width of 15 km, rupture area of 216 km², average displacement of 0.7 m, slip rate of 6 mm/yr, and five earthquake empirical models.     

Earthquake Forecasting Using Hidden Markov Models

This paper develops a novel method, based on hidden Markov models, to forecast earthquakes and applies the method to mainshock seismic activity in southern California and western Nevada. The forecasts are of the probability of a mainshock within 1, 5, and 10 days in the entire study region or in specific subregions and are based on the observations available at the forecast time, namely the interevent times and locations of the previous mainshocks and the elapsed time since the most recent one. Hidden Markov models have been applied to many problems, including earthquake classification; this is the first application to earthquake forecasting.     

Using Ancillary Information to Improve Hypocenter Estimation: Bayesian Single Event Location (BSEL)

We have developed and tested an algorithm, Bayesian Single Event Location (BSEL), for estimating the location of a seismic event. The main driver for our research is the inadequate representation of ancillary information in the hypocenter estimation procedure. The added benefit is that we have also addressed instability issues often encountered with historical NLR solvers (e.g., non-convergence or seismically infeasible results). BSEL differs from established nonlinear regression techniques by using a Bayesian prior probability density function (prior PDF) to incorporate ancillary physical basis constraints about event location. P-wave arrival times from seismic events are used in the development. Depth, a focus of this paper, may be modeled with a prior PDF (potentially skewed) that captures physical basis bounds from surface wave observations. This PDF is constructed from a Rayleigh wave depth excitation eigenfunction that is based on the observed minimum period from a spectrogram analysis and estimated near-source elastic parameters. For example, if the surface wave is an Rg phase, it potentially provides a strong constraint for depth, which has important implications for remote monitoring of nuclear explosions. The proposed Bayesian algorithm is illustrated with events that demonstrate its congruity with established hypocenter estimation methods and its application potential. The BSEL method is applied to three events: 1) A shallow Mw 4 earthquake that occurred near Bardwell, KY on June 6, 2003, 2) the Mw 5.6 earthquake of July 26, 2005 that occurred near Dillon, MT, and 3) a deep Mw 5.7 earthquake that occurred off the coast of Japan on April 22, 1980. A strong Rg was observed from the Bardwell, KY earthquake that places very strong constraints on depth and origin time. No Rg was observed for the Dillon, MT earthquake, but we used the minimum observed period of a Rayleigh wave (7 seconds) to reduce the depth and origin time uncertainty. Because the Japan event was deep, there is no observed surface wave energy. We utilize the prior generated from the Dillon, MT event to show that even in the case when a prior is inappropriately applied, high quality data will overcome its influence and result in a reasonable hypocenter estimate.     

Spatial prediction of categorical variables: the Bayesian maximum entropy approach

 Being a non-linear method based on a rigorous formalism and an efficient processing of various information sources, the Bayesian maximum entropy (BME) approach has proven to be a very powerful method in the context of continuous spatial random fields, providing much more satisfactory estimates than those obtained from traditional linear geostatistics (i.e., the various kriging techniques). This paper aims at presenting an extension of the BME formalism in the context of categorical spatial random fields. In the first part of the paper, the indicator kriging and cokriging methods are briefly presented and discussed. A special emphasis is put on their inherent limitations, both from the theoretical and practical point of view. The second part aims at presenting the theoretical developments of the BME approach for the case of categorical variables. The three-stage procedure is explained and the formulations for obtaining prior joint distributions and computing posterior conditional distributions are given for various typical cases. The last part of the paper consists in a simulation study for assessing the performance of BME over the traditional indicator (co)kriging techniques. The results of these simulations highlight the theoretical limitations of the indicator approach (negative probability estimates, probability distributions that do not sum up to one, etc.) as well as the much better performance of the BME approach. Estimates are very close to the theoretical conditional probabilities, that can be computed according to the stated simulation hypotheses.     

Verification of SPE probability forecasts at the Space Environment Prediction Center (SEPC)

In space weather forecasting, forecast verification is necessary so that the forecast quality can be assessed. This paper provides an example of how to choose and devise verification methods and techniques according to different space weather forecast products. Solar proton events (SPEs) are hazardous space weather events, and forecasting them is one of the major tasks of the Space Environment Prediction Center (SEPC) at the National Space Science Center of the Chinese Academy of Sciences. Through analyzing SPE occurrence characteristics, SPE forecast properties, and verification requirements at SEPC, verification methods for SPE probability forecasts are identified, and verification results obtained. Overall, SPE probability forecasts at SEPC exhibit good accuracy, reliability, and discrimination. Compared with climatology and persistence forecasts, the SPE forecasts are more accurate. However, the forecasts for SPE onset days are substantially underestimated and need to be considerably improved.     

Conditional Probabilities for Large Events Estimated by Small Earthquake Rate

We examined forecasting quiescence and activation models to obtain the conditional probability that a large earthquake will occur in a specific time period on different scales in Taiwan. The basic idea of the quiescence and activation models is to use earthquakes that have magnitudes larger than the completeness magnitude to compute the expected properties of large earthquakes. We calculated the probability time series for the whole Taiwan region and for three subareas of Taiwan—the western, eastern, and northeastern Taiwan regions—using 40 years of data from the Central Weather Bureau catalog. In the probability time series for the eastern and northeastern Taiwan regions, a high probability value is usually yielded in cluster events such as events with foreshocks and events that all occur in a short time period. In addition to the time series, we produced probability maps by calculating the conditional probability for every grid point at the time just before a large earthquake. The probability maps show that high probability values are yielded around the epicenter before a large earthquake. The receiver operating characteristic (ROC) curves of the probability maps demonstrate that the probability maps are not random forecasts, but also suggest that lowering the magnitude of a forecasted large earthquake may not improve the forecast method itself. From both the probability time series and probability maps, it can be observed that the probability obtained from the quiescence model increases before a large earthquake and the probability obtained from the activation model increases as the large earthquakes occur. The results lead us to conclude that the quiescence model has better forecast potential than the activation model.     

Prediction of response spectra via real-time earthquake measurements

The development and implementation of an earthquake early warning system (EEWS), both in regional or on-site configurations can help to mitigate the losses due to the occurrence of moderate-to-large earthquakes in densely populated and/or industrialized areas. The capability of an EEWS to provide real-time estimates of source parameters (location and magnitude) can be used to take some countermeasures during the earthquake occurrence and before the arriving of the most destructive waves at the site of interest. However, some critical issues are peculiar of EEWS and need further investigation: (1) the uncertainties on earthquake magnitude and location estimates based on the measurements of some observed quantities in the very early portion of the recorded signals; (2) the selection of the most appropriate parameter to be used to predict the ground motion amplitude both in near- and far-source ranges; (3) the use of the estimates provided by the EEWS for structural engineering and risk mitigation applications.In the present study, the issues above are discussed using the Campania–Lucania region (Southern Apennines) in Italy, as test-site area. In this region a prototype system for earthquake early warning, and more generally for seismic alert management, is under development. The system is based on a dense, wide dynamic accelerometric network deployed in the area where the moderate-to-large earthquake causative fault systems are located.The uncertainty analysis is performed through a real-time probabilistic seismic hazard analysis by using two different approaches. The first is the Bayesian approach that implicitly integrate both the time evolving estimate of earthquake parameters, the probability density functions and the variability of ground motion propagation providing the most complete information. The second is a classical point estimate approach which does not account for the probability density function of the magnitude and only uses the average of the estimates performed at each seismic station.Both the approaches are applied to two main towns located in the area of interest, Napoli and Avellino, for which a missed and false alarm analysis is presented by means of a scenario earthquake: an M 7.0 seismic event located at the centre of the seismic network.Concerning the ground motion prediction, attention is focused on the response spectra as the most appropriate function to characterize the ground motion for earthquake engineering applications of EEWS.     

基于面平均雨量误差修正的实时洪水预报修正方法

空间集总式水文模型的洪水预报精度会受到面平均雨量估计误差的严重影响.点雨量监测值的误差类型、误差大小以及流域的雨量站点密度和站点的空间分布都会影响到面平均雨量的计算.为提高实时洪水预报精度,本文提出了一种基于降雨系统响应曲线洪水预报误差修正方法.通过此方法估计降雨输入项的误差,从而提高洪水预报精度.此方法将水文模型做为输入和输出之间的响应系统,用实测流量和计算流量之间的差值做为信息,通过降雨系统响应曲线,使用最小二乘估计原理,对面平均雨量进行修正,再用修正后的面平均雨量重新计算出流过程.将此修正方法结合新安江模型使用理想案例进行检验,并应用于王家坝流域的16场历史洪水以及此流域不同雨量站密度的情况下,结果证明均有明显修正效果,且在雨量站密度较低时修正效果更加明显.该方法是一种结构简单且不增加模型参数和复杂度的实时洪水修正的新方法.