An epidemic disease caused by a new coronavirus has spread in Northern Italy with a strong contagion rate. We implement an SEIR model to compute the infected population and the number of casualties of this epidemic. The example may ideally regard the situation in the Italian Region of Lombardy, where the epidemic started on February 24, but by no means attempts to perform a rigorous case study in view of the lack of suitable data and the uncertainty of the different parameters, namely, the variation of the degree of home isolation and social distancing as a function of time, the initial number of exposed individuals and infected people, the incubation and infectious periods, and the fatality rate. First, we perform an analysis of the results of the model by varying the parameters and initial conditions (in order for the epidemic to start, there should be at least one exposed or one infectious human). Then, we consider the Lombardy case and calibrate the model with the number of dead individuals to date (May 5, 2020) and constrain the parameters on the basis of values reported in the literature. The peak occurs at day 37 (March 31) approximately, with a reproduction ratio R 0 of 3 initially, 1.36 at day 22, and 0.8 after day 35, indicating different degrees of lockdown. The predicted death toll is approximately 15,600 casualties, with 2.7 million infected individuals at the end of the epidemic. The incubation period providing a better fit to the dead individuals is 4.25 days, and the infectious period is 4 days, with a fatality rate of 0.00144/day [values based on the reported (official) number of casualties]. The infection fatality rate (IFR) is 0.57%, and it is 2.37% if twice the reported number of casualties is assumed. However, these rates depend on the initial number of exposed individuals. If approximately nine times more individuals are exposed, there are three times more infected people at the end of the epidemic and IFR = 0.47%. If we relax these constraints and use a wider range of lower and upper bounds for the incubation and infectious periods, we observe that a higher incubation period (13 vs. 4.25 days) gives the same IFR (0.6 vs. 0.57%), but nine times more exposed individuals in the first case. Other choices of the set of parameters also provide a good fit to the data, but some of the results may not be realistic. Therefore, an accurate determination of the fatality rate and characteristics of the epidemic is subject to knowledge of the precise bounds of the parameters. Besides the specific example,

High-resolution seismic experiments, employing arrays of closely spaced, four-component ocean-bottom seismic recorders, were conducted at a site off western Svalbard and a site on the northern margin of the Storegga slide, off Norway to investigate how well seismic data can be used to determine the concentration of methane hydrate beneath the seabed. Data from P-waves and from S-waves generated by P-S conversion on reflection were inverted for P-and S-wave velocity (Vp and Vs), using 3D travel-time tomography, 2D ray-tracing inversion and 1D waveform inversion. At the NW Svalbard site, positive Vp anomalies above a sea-bottomsimulating reflector (BSR) indicate the presence of gas hydrate. A zone containing free gas up to 150-m thick, lying immediately beneath the BSR, is indicated by a large reduction in Vp without significant reduction in Vs. At the Storegga site, the lateral and vertical variation in Vp and Vs and the variation in amplitude and polarity of reflectors indicate a heterogeneous distribution of hydrate that is related to a stratigraphically mediated distribution of free gas beneath the BSR. Derivation of hydrate content from Vp and Vs was evaluated, using different models for how hydrate affects the seismic properties of the sediment host and different approaches for estimating the background velocity of the sediment host. The error in the average Vp of an interval of 20-m thickness is about 2.5%, at 95% confidence, and yields a resolution of hydrate concentration of about 3%, if hydrate forms a connected framework, or about 7%, if it is both pore-filling and framework-forming. At NW Svalbard, in a zone about 90-m thick above the BSR, a Biot-theory-based method predicts hydrate concentrations of up to 11% of pore space, and an effective-medium-based method predicts concentrations of up to 6%, if hydrate forms a connected framework, or 12%, if hydrate is both pore-filling and frameworkforming. At Storegga, hydrate concentrations of up to 10% or 20% were predicted, depending on the hydrate model, in a zone about 120-m thick above a BSR. With seismic techniques alone, we can only estimate with any confidence the average hydrate content of broad intervals containing more than one layer, not only because of the uncertainty in the layer-by-layer variation in lithology, but also because of the negative correlation in the errors of estimation of velocity between adjacent layers. In this investigation, an interval of about 20-m thickness (equivalent to between 2 and 5 layers in the model used for waveform inversion) was the smallest within which one could sensibly estimate the hydrate content. If lithological layering much thinner than 20-m thickness controls hydrate content, then hydrate concentrations within layers could significantly exceed or fall below the average values derived from seismic data.

Recent research has established that the dominant P-wave attenuation mechanism in reservoir rocks at seismic frequencies is because of wave-induced fluid flow ͑mesoscopic loss͒. The P-wave induces a fluid-pressure difference at mesoscopic-scale inhomogeneities ͑larger than the pore size but smaller than the wavelength, typically tens of cen-timeters͒ and generates fluid flow and slow ͑diffusion͒ Biot waves ͑continuity of pore pressure is achieved by energy conversion to slow P-waves, which diffuse away from the interfaces͒. In this context, we consider a periodically stratified medium and investigate the amount of attenuation ͑and velocity dispersion͒ caused by different types of heterogeneities in the rock properties, namely, porosity, grain and frame moduli, permeability, and fluid properties. The most effective loss mechanisms result from porosity variations and partial saturation, where one of the fluids is very stiff and the other is very compliant, such as, a highly permeable sandstone at shallow depths, saturated with small amounts of gas ͑around 10% saturation͒ and water. Grain-and frame-moduli variations are the next cause of attenuation. The relaxation peak moves towards low frequencies as the ͑background͒ permeability decreases and the viscosity and thickness of the layers increase. The analysis indicates in which cases the seismic band is in the relaxed regime, and therefore, when the Gassmann equation can yield a good approximation to the wave velocity.

Porous media are anisotropic due to bedding, compaction, and the presence of aligned microcracks and fractures. Here, it is assumed that the skeleton (and not the solid itself) is anisotropic. The rheological model also includes anisotropic tortuosity and permeability. The poroelastic equations are based on a transversely isotropic extension of Biot’s theory, and the problem is of plane strain type, i.e., two dimensional, describing qP−qS propagation. In the high-frequency case, the (two) viscodynamic operators are approximated by Zener relaxation functions that allow a closed differential formulation of Biot’s equation of motion. A plane-wave analysis derives expressions for the slowness, attenuation, and energy velocity vectors, and quality factor for homogeneous viscoelastic waves. The slow wave shows an anomalous polarization behavior. In particular, when the medium is strongly anisotropic the polarization is quasishear and the wave presents cuspidal triangles. Anisotropic tortuosity affects mainly the slow wavefront, and anisotropic permeability produces strong anisotropic attenuation of the three modes. The diffusive characteristics of the slow mode are predicted by the plane-wave analysis. As in the single-phase case, it is confirmed that the phase velocity is the projection of the energy velocity vector onto the propagation direction. Moreover, some fundamental energy relations, valid for a single-phase medium, are generalized to two-phase media. Transient propagation is solved with a direct grid method and a time-splitting integration algorithm, allowing the solution of the stiff part of the differential equations in closed analytical form. The snapshots show that the three waves are propagative when the fluid is ideal (zero viscosity). It is confirmed that, when the fluid is viscous, the slow wave becomes diffusive and appears as a static mode at the source location. The modeling confirms the triplication (cusps) of the slow wave and the polarization behavior predicted by the plane analysis.

S U M M A R YThe presence of gas hydrate in oceanic sediments is mostly identified by bottom-simulating reflectors (BSRs), reflection events with reversed polarity following the trend of the seafloor. Attempts to quantify the amount of gas hydrate present in oceanic sediments have been based mainly on the presence or absence of a BSR and its relative amplitude. Recent studies have shown that a BSR is not a necessary criterion for the presence of gas hydrates, but rather its presence depends on the type of sediments and the in situ conditions. The influence of hydrate on the physical properties of sediments overlying the BSR is determined by the elastic properties of their constituents and on sediment microstructure. In this context several approaches have been developed to predict the physical properties of sediments, and thereby quantify the amount of gas/gas hydrate present from observed deviations of these properties from those predicted for sediments without gas hydrate.We tested four models: the empirical weighted equation (WE); the three-phase effectivemedium theory (TPEM); the three-phase Biot theory (TPB) and the differential effectivemedium theory (DEM). We compared these models for a range of variables (porosity and clay content) using standard values for physical parameters. The comparison shows that all the models predict sediment properties comparable to field values except for the WE model at lower porosities and the TPB model at higher porosities. The models differ in the variation of velocity with porosity and clay content. The variation of velocity with hydrate saturation is also different, although the range is similar. We have used these models to predict velocities for field data sets from sediment sections with and without gas hydrates. The first is from the Mallik 2L-38 well, Mackenzie Delta, Canada, and the second is from Ocean Drilling Program (ODP) Leg 164 on Blake Ridge. Both data sets have V p and V s information along with the composition and porosity of the matrix. Models are considered successful if predictions from both V p and V s match hydrate saturations inferred from other data. Three of the models predict consistent hydrate saturations of 60-80 per cent from both V p and V s from log and vertical seismic profiling data for the Mallik 2L-38 well data set, but the TPEM model predicts 20 per cent higher saturations, as does the DEM model with a clay-water starting medium. For the clay-rich sediments of Blake Ridge, the DEM, TPEM and WE models predict 10-20 per cent hydrate saturation from V p data, comparable to that inferred from resistivity data. The hydrate saturation predicted by the TPB model from V p is higher. Using V s data, the DEM and TPEM models predict very low or zero hydrate saturation while the TPB and WE models predict hydrate saturation very much higher than those predicted from V p data. Low hydrate saturations are observed to have little effect on V s . The hydrate phase appears to be connected within the sediment microstructure even at low saturations.