A copula-based model for air pollution portfolio risk and its efficient simulation

This paper introduces a portfolio approach for quantifying pollution risk in the presence of PM\(_{2.5}\) concentration in cities. The model used is based on a copula dependence structure. For assessing model parameters, we analyze a limited data set of PM\(_{2.5}\) levels of Beijing, Tianjin, Chengde, Hengshui, and Xingtai. This process reveals a better fit for the t-copula dependence structure with generalized hyperbolic marginal distributions for the PM\(_{2.5}\) log-ratios of the cities. Furthermore, we show how to efficiently simulate risk measures clean-air-at-risk and conditional clean-air-at-risk using importance sampling and stratified importance sampling. Our numerical results show that clean-air-at-risk at 0.01 probability level reaches up to \(352\,{\mu \hbox {gm}^{-3}}\) (initial PM\(_{2.5}\) concentrations of cities are assumed to be \(100\,{\mu \hbox {gm}^{-3}}\)) for the constructed sample portfolio, and that the proposed methods are much more efficient than a naive simulation for computing the exceeding probabilities and conditional excesses.     

Effective Solar Indices for Ionospheric Modeling: A Review and a Proposal for a Real-Time Regional IRI

The first part of this paper reviews methods using effective solar indices to update a background ionospheric model focusing on those employing the Kriging method to perform the spatial interpolation. Then, it proposes a method to update the International Reference Ionosphere (IRI) model through the assimilation of data collected by a European ionosonde network. The method, called International Reference Ionosphere UPdate (IRI UP), that can potentially operate in real time, is mathematically described and validated for the period 9–25 March 2015 (a time window including the well-known St. Patrick storm occurred on 17 March), using IRI and IRI Real Time Assimilative Model (IRTAM) models as the reference. It relies on foF2 and M(3000)F2 ionospheric characteristics, recorded routinely by a network of 12 European ionosonde stations, which are used to calculate for each station effective values of IRI indices \(IG_{12}\) and \(R_{12}\) (identified as \(IG_{{12{\text{eff}}}}\) and \(R_{{12{\text{eff}}}}\)); then, starting from this discrete dataset of values, two-dimensional (2D) maps of \(IG_{{12{\text{eff}}}}\) and \(R_{{12{\text{eff}}}}\) are generated through the universal Kriging method. Five variogram models are proposed and tested statistically to select the best performer for each effective index. Then, computed maps of \(IG_{{12{\text{eff}}}}\) and \(R_{{12{\text{eff}}}}\) are used in the IRI model to synthesize updated values of foF2 and hmF2. To evaluate the ability of the proposed method to reproduce rapid local changes that are common under disturbed conditions, quality metrics are calculated for two test stations whose measurements were not assimilated in IRI UP, Fairford (51.7°N, 1.5°W) and San Vito (40.6°N, 17.8°E), for IRI, IRI UP, and IRTAM models. The proposed method turns out to be very effective under highly disturbed conditions, with significant improvements of the foF2 representation and noticeable improvements of the hmF2 one. Important improvements have been verified also for quiet and moderately disturbed conditions. A visual analysis of foF2 and hmF2 maps highlights the ability of the IRI UP method to catch small-scale changes occurring under disturbed conditions which are not seen by IRI.     

Tomography of crustal seismic attenuation in Metropolitan France: implications for seismicity analysis

In this work, we map the absorption properties of the French crust by analyzing the decay properties of coda waves. Estimation of the coda quality factor \(Q_{c}\) in five non-overlapping frequency-bands between 1 and 32 Hz is performed for more than 12,000 high-quality seismograms from about 1700 weak to moderate crustal earthquakes recorded between 1995 and 2013. Based on sensitivity analysis, \(Q_{c}\) is subsequently approximated as an integral of the intrinsic shear wave quality factor \(Q_{i}\) along the ray connecting the source to the station. After discretization of the medium on a 2-D Cartesian grid, this yields a linear inverse problem for the spatial distribution of \(Q_{i}\). The solution is approximated by redistributing \(Q_{c}\) in the pixels connecting the source to the station and averaging over all paths. This simple procedure allows to obtain frequency-dependent maps of apparent absorption that show lateral variations of \(50\%\) at length scales ranging from 50 km to 150 km, in all the frequency bands analyzed. At low frequency, the small-scale geological features of the crust are clearly delineated: the Meso-Cenozoic basins (Aquitaine, Brabant, Southeast) appear as strong absorption regions, while crystalline massifs (Armorican, Central Massif, Alps) appear as low absorption zones. At high frequency, the correlation between the surface geological features and the absorption map disappears, except for the deepest Meso-Cenozoic basins which exhibit a strong absorption signature. Based on the tomographic results, we explore the implications of lateral variations of absorption for the analysis of both instrumental and historical seismicity. The main conclusions are as follows: (1) current local magnitude \(M_{L}\) can be over(resp. under)-estimated when absorption is weaker(resp. stronger) than the nominal value assumed in the amplitude-distance relation; (2) both the forward prediction of the earthquake macroseismic intensity field and the estimation of historical earthquake seismological parameters using macroseismic intensity data are significantly improved by taking into account a realistic 2-D distribution of absorption. In the future, both \(M_{L}\) estimations and macroseismic intensity attenuation models should benefit from high-resolution models of frequency-dependent absorption such as the one produced in this study.     

Regeneration and utilization of nutrients during collapse of a <Emphasis Type="Italic">Mesodinium rubrum</Emphasis> red tide and its influence on phytoplankton species composition

High-biomass red tides occur frequently in some semi-enclosed bays of Hong Kong where ambient nutrients are not high enough to support such a high phytoplankton biomass. These high-biomass red tides release massive inorganic nutrients into local waters during their collapse. We hypothesized that the massive inorganic nutrients released from the collapse of red tides would fuel growth of other phytoplankton species. This could influence phytoplankton species composition. We tested the hypothesis using a red tide event caused by Mesodinium rubrum (M. rubrum) in a semi-enclosed bay, Port Shelter. The red tide patch had a cell density as high as 5.0×10⁵ cells L^?1, and high chlorophyll a (63.71 μg L^?1). Ambient inorganic nutrients (nitrate: \(\rm{NO}_3^-\), ammonium: \(\rm{NH}_4^+\), phosphate: \(\rm{PO}_4^{3-}\), silicate: \(\rm{SiO}_4^{3-}\)) were low both in the red tide patch and the non-red-tide patch (clear waters outside the red tide patch). Nutrient addition experiments were conducted by adding all the inorganic nutrients to water samples from the two patches followed by incubation for 9 days. The results showed that the addition of inorganic nutrients did not sustain high M. rubrum cell density, which collapsed after day 1, and did not drive M. rubrum in the non-red-tide patch sample to the same high-cell density in the red tide patch sample. This confirmed that nutrients were not the driving factor for the formation of this red tide event, or for its collapse. The death of M. rubrum after day 1 released high concentrations of \(\rm{NO}_3^-\), \(\rm{PO}_4^{3-}\), \(\rm{SiO}_4^{3-}\), \(\rm{NH}_4^+\), and urea. Bacterial abundance and heterotrophic activity increased, reaching the highest on day 3 or 4, and decreased as cell density of M. rubrum declined. The released nutrients stimulated growth of diatoms, such as Chaetoceros affinis var. circinalis, Thalassiothrix frauenfeldii, and Nitzschia sp., particularly with additions of \(\rm{SiO}_4^{3-}\) treatments, and other species. These results demonstrated that initiation of M. rubrum red tides in the bay was not directly driven by nutrients. However, the massive inorganic nutrients released from the collapse of the red tide could induce a second bloom in low-ambient nutrient water, influencing phytoplankton species composition.     

Stochastic modelling of relative water permeability in vegetative soils with implications on stability of bioengineered slope

Vegetation is known to influence the hydrological state variables, suction \( \left( \psi \right) \) and volumetric water content (\( \theta_{w} \)) of soil. In addition, vegetation induces heterogeneity in the soil porous structure and consequently the relative permeability (\( k_{r} \)) of water under unsaturated conditions. The indirect method of utilising the soil water characteristic curve (SWCC) is commonly adopted for the determination of \( k_{r} \). In such cases, it is essential to address the stochastic behaviour of SWCC, in order to conduct a robust analysis on the \( k_{r} \) of vegetative cover. The main aim of this study is to address the uncertainties associated with \( k_{r} \), using probabilistic analysis, for vegetative covers (i.e., grass and tree species) with bare cover as control treatment. We propose two approaches to accomplish the aforesaid objective. The univariate suction approach predicts the probability distribution functions of \( {\text{k}}_{\text{r}} \), on the basis of identified best probability distribution of suction. The bivariate suction and water content approach deals with the bivariate modelling of the water content and suction (SWCC), in order to capture the randomness in the permeability curves, due to presence of vegetation. For this purpose, the dependence structure of \( \psi \) and \( \theta_{w} \) is established via copula theory, and the \( k_{r} \) curves are predicted with respect to varying levels of \( \psi - \theta_{w} \) correlation. The results showed that the \( k_{r} \) of vegetative covers is substantially lower than that in bare covers. The reduction in \( k_{r} \) with drying is more in tree cover than grassed cover, since tree roots induce higher levels of suction. Moreover, the air entry value of the soil depends on the magnitude of \( \psi - \theta_{w} \) correlation, which in turn, is influenced by the type of vegetation in the soil. \( k_{r} \) is found to be highly uncertain in the desaturation zone of the relative permeability curve. The stochastic behaviour of \( k_{r} \) is found to be most significant in tree covers. Finally, a simplified case study is also presented in order to demonstrate the impact of the uncertainty in \( k_{r} \), on the stability of vegetates slopes. With an increment in the parameter \( \alpha \), factor of safety (FS) is found to decrease. The trend of FS is reverse of this with parameter \( n \). Overall FS is found to vary around 4–5%, for both bare and vegetative slopes.     

Contrasted turbulence intensities in the Indonesian Throughflow: a challenge for parameterizing energy dissipation rate

Microstructure measurements were performed along two sections through the Halmahera Sea and the Ombai Strait and at a station in the deep Banda Sea. Contrasting dissipation rates (??) and vertical eddy diffusivities (K _z ) were obtained with depth-averaged ranges of \(\sim [9 \times 10^{-10}-10^{-5}]\) W kg^??1 and of \(\sim [1 \times 10^{-5}-2 \times 10^{-3}]\) m² s^??1, respectively. Similarly, turbulence intensity, \(I={\epsilon }/(\nu N^{2})\) with ν the kinematic viscosity and N the buoyancy frequency, was found to vary seven orders of magnitude with values up to \(10^{7}\). These large ranges of variations were correlated with the internal tide energy level, which highlights the contrast between regions close and far from internal tide generations. Finescale parameterizations of ?? induced by the breaking of weakly nonlinear internal waves were only relevant in regions located far from any generation area (“far field”), at the deep Banda Sea station. Closer to generation areas, at the “intermediate field” station of the Halmahera Sea, a modified formulation of MacKinnon and Gregg (2005) was validated for moderately turbulent regimes with 100 < I < 1000. Near generation areas marked by strong turbulent regimes such as “near field” stations within strait and passages, ?? is most adequately inferred from horizontal velocities provided that part of the inertial subrange is resolved, according to Kolmogorov scaling.     

Gravity Anomalies of Arbitrary 3D Polyhedral Bodies with Horizontal and Vertical Mass Contrasts

During the last 15 years, more attention has been paid to derive analytic formulae for the gravitational potential and field of polyhedral mass bodies with complicated polynomial density contrasts, because such formulae can be more suitable to approximate the true mass density variations of the earth (e.g., sedimentary basins and bedrock topography) than methods that use finer volume discretization and constant density contrasts. In this study, we derive analytic formulae for gravity anomalies of arbitrary polyhedral bodies with complicated polynomial density contrasts in 3D space. The anomalous mass density is allowed to vary in both horizontal and vertical directions in a polynomial form of \(\lambda =ax^m+by^n+cz^t\), where m, n, t are nonnegative integers and a, b, c are coefficients of mass density. First, the singular volume integrals of the gravity anomalies are transformed to regular or weakly singular surface integrals over each polygon of the polyhedral body. Then, in terms of the derived singularity-free analytic formulae of these surface integrals, singularity-free analytic formulae for gravity anomalies of arbitrary polyhedral bodies with horizontal and vertical polynomial density contrasts are obtained. For an arbitrary polyhedron, we successfully derived analytic formulae of the gravity potential and the gravity field in the case of \(m\le 1\), \(n\le 1\), \(t\le 1\), and an analytic formula of the gravity potential in the case of \(m=n=t=2\). For a rectangular prism, we derive an analytic formula of the gravity potential for \(m\le 3\), \(n\le 3\) and \(t\le 3\) and closed forms of the gravity field are presented for \(m\le 1\), \(n\le 1\) and \(t\le 4\). Besides generalizing previously published closed-form solutions for cases of constant and linear mass density contrasts to higher polynomial order, to our best knowledge, this is the first time that closed-form solutions are presented for the gravitational potential of a general polyhedral body with quadratic density contrast in all spatial directions and for the vertical gravitational field of a prismatic body with quartic density contrast along the vertical direction. To verify our new analytic formulae, a prismatic model with depth-dependent polynomial density contrast and a polyhedral body in the form of a triangular prism with constant contrast are tested. Excellent agreements between results of published analytic formulae and our results are achieved. Our new analytic formulae are useful tools to compute gravity anomalies of complicated mass density contrasts in the earth, when the observation sites are close to the surface or within mass bodies.     

Spatio-temporal analysis of particulate matter extremes in Seoul: use of multiscale approach

This paper considers a problem of analyzing temporal and spatial structure of particulate matter (PM) data with emphasizing high-level \(\text {PM}_{10}\). The proposed method is based on a combination of a generalized extreme value (GEV) distribution and a multiscale concept from scaling property theory used in hydrology. In this study, we use hourly \(\text {PM}_{10}\) data observed for 5 years on 25 stations located in Seoul metropolitan area, Korea. For our analysis, we calculate monthly maximum values for various duration times and area coverages at each station, and show that their distribution follows a GEV distribution. In addition, we identify that the GEV parameters of \(\text {PM}_{10}\) maxima hold a new scaling property, termed ‘piecewise linear scaling property’ for certain duration times. By using this property, we construct a 12-month return level map of hourly \(\text {PM}_{10}\) data at any arbitrary d-hour duration. Furthermore, we extend our study to understand spatio-temporal multiscale structure of \(\text {PM}_{10}\) extremes over different temporal and spatial scales.     

New magnitude scaling relations for earthquake early warning in the Alborz region,Iran

Rapid magnitude estimation relations for earthquake early warning systems in the Alborz region have been developed based on the initial first seconds of the P-wave arrival. For this purpose, a total of 717 accelerograms recorded by the Building and Housing Research Center in the Alborz region with the magnitude (M_w) range of 4.8–6.5 in the period between 1995 and 2013 were employed. Average ground motion period (\( \tau_{\text{c}} \)) and peak displacement (\( P_{\text{d}} \)) in different time windows from the P-wave arrival were calculated, and their relation with magnitude was examined. Four earthquakes that were excluded from the analysis process were used to validate the results, and the estimated magnitudes were found to be in good agreement with the observed ones. The results show that using the proposed relations for the Alborz region, earthquake magnitude could be estimated with acceptable accuracy even after 1 s of the P-wave arrival.     

Stochastic models of severe weather watches and warnings: transition probabilities

To alert the public to the possibility of tornado (T), hail (H), or convective wind (C), the National Weather Service (NWS) issues watches (V) and warnings (W). There are severe thunderstorm watches (SV), tornado watches (TV), and particularly dangerous situation watches (PV); and there are severe thunderstorm warnings (SW), and tornado warnings (TW). Two stochastic models are formulated that quantify uncertainty in severe weather alarms for the purpose of making decisions: a one-stage model for deciders who respond to warnings, and a two-stage model for deciders who respond to watches and warnings. The models identify all possible sequences of watches, warnings, and events, and characterize the associated uncertainties in terms of transition probabilities. The modeling approach is demonstrated on data from the NWS Norman, Oklahoma, warning area, years 2000–2007. The major findings are these. (i) Irrespective of its official designation, every warning type {SW, TW} predicts with a significant probability every event type {T, H, C}. (ii) An ordered intersection of SW and TW, defined as reinforced warning (RW), provides additional predictive information and outperforms SW and TW. (iii) A watch rarely leads directly to an event, and most frequently is false. But a watch that precedes a warning does matter. The watch type \(\{SV\), TV, \(PV\}\) is a predictor of the warning type \(\{SW\), RW, \(TW\}\) and of the warning performance: It sharpens the false alarm rate of the warning and the predictive probability of an event, and it increases the average lead time of the warning.     

Quantitative Analysis of Seismicity Before Large Taiwanese Earthquakes Using the G-R Law

Seismicity has been identified as an example of a natural, nonlinear system for which the distribution of frequency and event size follow a power law called the “Gutenberg–Richter (G-R) law.” The parameters of the G-R law, namely b- and a-values, have been widely used in many studies about seismic hazards, earthquake forecasting models, and other related topics. However, the plausibility of the power law model and applicability of parameters were mainly verified by statistical error σ of the b-value, the effectiveness of which is still doubtful. In this research, we used a newly defined p value developed by Clausetet al. (Power-Law Distributions in Empirical Data, SIAM Rev. 51, 661–703, 2009) instead of the statistical error σ of the b-value and verified its effectiveness as a plausibility index of the power-law model. Furthermore, we also verified the effectiveness of K–S statistics as a goodness-of-fit test in estimating the crucial parameter \(M_{\text{c}}\) of the power-law model.     

Effects of temporally correlated infiltration on water flow in an unsaturated–saturated system

Effects of temporally correlated infiltration on water flow in an unsaturated–saturated system were investigated. Both white noise and exponentially correlated infiltration processes were considered. The moment equations of the pressure head (ψ) were solved numerically to obtain the variance and autocorrelation functions of ψ at 14 observation points. Monte Carlo simulations were conducted to verify the numerical results and to estimate the power spectrum of ψ (S _ψψ ). It was found that as the water flows through the system, the variance of the ψ (\( \sigma_{\psi }^{2} \)) were damped by the system: the deeper in the system, the smaller the \( \sigma_{\psi }^{2} \), and the larger the correlation timescale of the infiltration process (λ _I ), the larger the \( \sigma_{\psi }^{2} \). The unsaturated–saturated system gradually filters out the short-term fluctuations of ψ and the damping effect is most significant in the upper part of the system. The fluctuations of ψ is non-stationary at early time and becomes stationary as time progresses: the larger the value of λ _I , the longer the non-stationary period. The correlation timescale of the ψ (λ _ψ ) increases with depth and approaches a constant value at depth: the larger the value of λ _I , the larger the value of λ _ψ . The results of the estimated S _ψψ is consistent with those of the variance and autocorrelation function.     

Conditional risk based on multivariate hazard scenarios

We present a novel methodology to compute conditional risk measures when the conditioning event depends on a number of random variables. Specifically, given a random vector \((\mathbf {X},Y)\), we consider risk measures that express the risk of Y given that \(\mathbf {X}\) assumes values in an extreme multidimensional region. In particular, the considered risky regions are related to the AND, OR, Kendall and Survival Kendall hazard scenarios that are commonly used in environmental literature. Several closed formulas are considered (especially in the AND and OR scenarios). An application to spatial risk analysis involving real data is discussed.     

Conditional heavy-tail behavior with applications to precipitation and river flow extremes

This article deals with the right-tail behavior of a response distribution \(F_Y\) conditional on a regressor vector \({\mathbf {X}}={\mathbf {x}}\) restricted to the heavy-tailed case of Pareto-type conditional distributions \(F_Y(y|\ {\mathbf {x}})=P(Y\le y|\ {\mathbf {X}}={\mathbf {x}})\), with heaviness of the right tail characterized by the conditional extreme value index \(\gamma ({\mathbf {x}})>0\). We particularly focus on testing the hypothesis \({\mathscr {H}}_{0,tail}:\ \gamma ({\mathbf {x}})=\gamma _0\) of constant tail behavior for some \(\gamma _0>0\) and all possible \({\mathbf {x}}\). When considering \({\mathbf {x}}\) as a time index, the term trend analysis is commonly used. In the recent past several such trend analyses in extreme value data have been published, mostly focusing on time-varying modeling of location or scale parameters of the response distribution. In many such environmental studies a simple test against trend based on Kendall’s tau statistic is applied. This test is powerful when the center of the conditional distribution \(F_Y(y|{\mathbf {x}})\) changes monotonically in \({\mathbf {x}}\), for instance, in a simple location model \(\mu ({\mathbf {x}})=\mu _0+x\cdot \mu _1\), \({\mathbf {x}}=(1,x)'\), but the test is rather insensitive against monotonic tail behavior, say, \(\gamma ({\mathbf {x}})=\eta _0+x\cdot \eta _1\). This has to be considered, since for many environmental applications the main interest is on the tail rather than the center of a distribution. Our work is motivated by this problem and it is our goal to demonstrate the opportunities and the limits of detecting and estimating non-constant conditional heavy-tail behavior with regard to applications from hydrology. We present and compare four different procedures by simulations and illustrate our findings on real data from hydrology: weekly maxima of hourly precipitation from France and monthly maximal river flows from Germany.     

A semiempirical mathematical model of the secondary pollution of water bodies by soluble iron and manganese forms

A semiempirical mathematical model of iron and manganese migration from bottom sediments into the water mass of water bodies has been proposed based on some basic regularities in the geochemistry of those elements. The entry of dissolved forms of iron and manganese under aeration conditions is assumed negligible. When dissolved-oxygen concentration is <0.5 mg/L, the elements start releasing from bottom sediments, their release rate reaching its maximum under anoxic conditions. The fluxes of dissolved iron and manganese (Me) from bottom sediments into the water mass (J _Me) are governed by the gradients of their concentrations in diffusion water sublayer adjacent to sediment surface and having an average thickness of h = 0.025 cm: \({J_{Me}} = - {D_{Me}}\frac{{{C_{Me\left( {ss} \right)}} - {C_{Me\left( w \right)}}}}{h}\) (D _Me ≈ 1 × 10^–9 m²/s is molecular diffusion coefficient of component Me in solution; C _Me(ss) and C _Me(w) ≈ 0 are Me concentrations on sediment surface, i.e., on the bottom boundary of the diffusion water sublayer, and in the water mass, i.e., on the upper boundary of the diffusion water sublayer). The value of depends on water saturation with dissolved oxygen (\({\eta _{{O_2}}}\)) in accordance with the empiric relationship \({C_{Me\left( {ss} \right)}} = \frac{{C_{_{Me\left( {ss} \right)}}^{\max }}}{{1 + k{\eta _{{O_2}}}}}\) (k is a constant factor equal to 300 for iron and 100 for manganese; C _Me(ss) ^max is the maximal concentration of Me on the bottom boundary of the diffusion water sublayer with C _Fe(ss) ^max ≈ 200 μM (11 mg/L), and C _Mn(ss) ^max ≈ 100 μM (5.5 mg/L).     

Stochastic finite-fault simulation of ground motion from the August 11, 2012, <Emphasis Type="Italic">M</Emphasis><Subscript><Emphasis Type="Italic">w</Emphasis></Subscript> 6.4 Ahar earthquake,northwestern Iran

In this study, the 11 August 2012 M _w 6.4 Ahar earthquake is investigated using the ground motion simulation based on the stochastic finite-fault model. The earthquake occurred in northwestern Iran and causing extensive damage in the city of Ahar and surrounding areas. A network consisting of 58 acceleration stations recorded the earthquake within 8–217 km of the epicenter. Strong ground motion records from six significant well-recorded stations close to the epicenter have been simulated. These stations are installed in areas which experienced significant structural damage and humanity loss during the earthquake. The simulation is carried out using the dynamic corner frequency model of rupture propagation by extended fault simulation program (EXSIM). For this purpose, the propagation features of shear-wave including \( {Q}_s \) value, kappa value \( {k}_0 \), and soil amplification coefficients at each site are required. The kappa values are obtained from the slope of smoothed amplitude of Fourier spectra of acceleration at higher frequencies. The determined kappa values for vertical and horizontal components are 0.02 and 0.05 s, respectively. Furthermore, an anelastic attenuation parameter is derived from energy decay of a seismic wave by using continuous wavelet transform (CWT) for each station. The average frequency-dependent relation estimated for the region is \( Q=\left(122\pm 38\right){f}^{\left(1.40\pm 0.16\right)}. \) Moreover, the horizontal to vertical spectral ratio \( H/V \) is applied to estimate the site effects at stations. Spectral analysis of the data indicates that the best match between the observed and simulated spectra occurs for an average stress drop of 70 bars. Finally, the simulated and observed results are compared with pseudo acceleration spectra and peak ground motions. The comparison of time series spectra shows good agreement between the observed and the simulated waveforms at frequencies of engineering interest.     

Analysis of macroseismic fields using statistical data depth functions: considerations leading to attenuation probabilistic modelling

Modelling seismic attenuation is one of the most critical points in the hazard assessment process. In this article we consider the spatial distribution of the effects caused by an earthquake as expressed by the values of the macroseismic intensity recorded at various locations surrounding the epicentre. Considering the ordinal nature of the intensity, a way to show its decay with distance is to draw curves—isoseismal lines—on maps, which bound points of intensity not smaller than a fixed value. These lines usually take the form of closed and nested curves around the epicentre, with highly different shapes because of the effects of ground conditions and of complexities in rupture propagation. Forecasting seismic attenuation of future earthquakes requires stochastic modelling of the decay on the basis of a common spatial pattern. The aim of this study is to consider a statistical methodology that identifies a general shape, if it exists, for isoseismal lines of a set of macroseismic fields. Data depth is a general nonparametric method for analysis of probability distributions and datasets. It has arisen as a statistical method to order points of a multivariate space, e.g., Euclidean space \({\mathbb {R}}^{p}\), \(p \ge 1\), according to the centrality with respect to a distribution or a given data cloud. Recently, this method has been extended to the ordering of functions and trajectories. In our case, for a fixed intensity decay \(\varDelta I\), we build a set of convex hulls that enclose the sites of felt intensity \(I_s \ge I_0 -\varDelta I\), one for each macroseismic field of a set of earthquakes that are considered as similar from the attenuation point of view. By applying data depth functions to this functional dataset, it is possible to identify the most central curve, i.e., the attenuation pattern, and to consider other properties like variability, outlyingness, and possible clustering of such curves. Results are shown for earthquakes that occurred on the Central Po Plain in May 2012, and on the eastern flank of Mt. Etna since 1865.     

Statistical Significance of Minimum of the Order Parameter Fluctuations of Seismicity Before Major Earthquakes in Japan

In a previous publication, the seismicity of Japan from 1 January 1984 to 11 March 2011 (the time of the \(M9\) Tohoku earthquake occurrence) has been analyzed in a time domain called natural time \(\chi.\) The order parameter of seismicity in this time domain is the variance of \(\chi\) weighted for normalized energy of each earthquake. It was found that the fluctuations of the order parameter of seismicity exhibit 15 distinct minima—deeper than a certain threshold—1 to around 3 months before the occurrence of large earthquakes that occurred in Japan during 1984–2011. Six (out of 15) of these minima were followed by all the shallow earthquakes of magnitude 7.6 or larger during the whole period studied. Here, we show that the probability to achieve the latter result by chance is of the order of \(10^{-5}\). This conclusion is strengthened by employing also the receiver operating characteristics technique.     

Covariance functions motivated by spatial random field models with local interactions

Random fields based on energy functionals with local interactions possess flexible covariance functions, lead to computationally efficient algorithms for spatial data processing, and have important applications in Bayesian field theory. In this paper we address the calculation of covariance functions for a family of isotropic local-interaction random fields in two dimensions. We derive explicit expressions for non-differentiable Spartan covariance functions in \({\mathbb{R}}^2\) that are based on the modified Bessel function of the second kind. We also derive a family of infinitely differentiable, Bessel-Lommel covariance functions that exhibit a hole effect and are valid in \({\mathbb{R}}^{d}\), where d > 2. Finally, we define a generalized spectrum of correlation scales that can be applied to both differentiable and non-differentiable random fields in contrast with the smoothness microscale.     

Wave boundary layers in rotating stratified fluid and near-inertial oscillations

Theory of wave boundary layers (WBLs) developed by Reznik (J Mar Res 71: 253–288, 2013, J Fluid Mech 747: 605–634, 2014, J Fluid Mech 833: 512–537, 2017) is extended to a rotating stratified fluid. In this case, the WBLs arise in the field of near-inertial oscillations (NIOs) driven by a tangential wind stress of finite duration. Near-surface Ekman layer is specified in the most general form; tangential stresses are zero at the lower boundary of Ekman layer and viscosity is neglected below the boundary. After the wind ceases, the Ekman pumping at the boundary becomes a linear superposition of inertial oscillations with coefficients dependent on the horizontal coordinates. The solution under the Ekman layer is obtained in the form of expansions in the vertical wave modes. We separate from the solution a part representing NIO and demonstrate development of a WBL near the Ekman layer boundary. With increasing time t, the WBL width decays inversely proportional to \( \sqrt{t} \) and gradients of fields in the WBL grow proportionally to \( \sqrt{t} \); the most part of NIO is concentrated in the WBL. Structure of the WBL depends strongly on its horizontal scale L determined by scale of the wind stress. The shorter the NIO is, the thinner and sharper the WBL is; the short-wave NIO with L smaller than the baroclinic Rossby scale L_R does not penetrate deep into the ocean. On the contrary, for L?≥?L_R, the WBL has a smoother vertical structure; a significant long-wave NIO signal is able to reach the oceanic bottom. An asymptotic theory of the WBL in rotating stratified fluid is suggested.